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# SSC: Semantic Scan Context for Large-Scale Place Recognition
Lin Li1, Xin Kong1, Xiangrui Zhao1, Tianxin Huang1 and Yong Liu1,∗ 1Lin Li,
Xin Kong, Xiangrui Zhao, Tianxin Huang and Yong Liu are with the Institute of
Cyber-Systems and Control, Zhejiang University, Hangzhou 310027, P. R. China.
(*Yong Liu is the corresponding author, email: [email protected]).
###### Abstract
Place recognition gives a SLAM system the ability to correct cumulative
errors. Unlike images that contain rich texture features, point clouds are
almost pure geometric information which makes place recognition based on point
clouds challenging. Existing works usually encode low-level features such as
coordinate, normal, reflection intensity, etc., as local or global descriptors
to represent scenes. Besides, they often ignore the translation between point
clouds when matching descriptors. Different from most existing methods, we
explore the use of high-level features, namely semantics, to improve the
descriptor’s representation ability. Also, when matching descriptors, we try
to correct the translation between point clouds to improve accuracy.
Concretely, we propose a novel global descriptor, Semantic Scan Context, which
explores semantic information to represent scenes more effectively. We also
present a two-step global semantic ICP to obtain the 3D pose ($x$, $y$, $yaw$)
used to align the point cloud to improve matching performance. Our experiments
on the KITTI dataset show that our approach outperforms the state-of-the-art
methods with a large margin. Our code is available at:
https://github.com/lilin-hitcrt/SSC.
## I INTRODUCTION
Simultaneous Localization and Mapping (SLAM) has rapidly developed in recent
decades as critical technologies for autonomous vehicles and robots. Place
recognition represents the ability of robots to recognize previously visited
places, which can build global constraints for the SLAM system to eliminate
the odometry’s cumulative errors and establish a globally consistent map[1].
Place recognition is usually conducted by using images or point clouds. Since
point cloud data is rarely affected by environmental factors such as
illumination and seasonal changes, LiDAR-based methods have received
widespread attention in recent years.
Most existing works on LiDAR-based place recognition are achieved by encoding
the point cloud into global or local descriptors and then matching the
descriptors. They usually use low-level features such as coordinates[2, 3, 4,
5, 6], normal[7], reflection intensity[8, 9, 10, 7], etc. In recent years,
with the development of point cloud deep learning, many LiDAR-based object
detection[11] and semantic segmentation[12, 13] methods have been proposed,
making it possible to obtain semantic information from point clouds. However,
there are still only a few LiDAR-based works trying to use semantic
information[14, 15, 7].
Figure 1: An example of place recognition using semantic scan context. It is a
partial map of the KITTI sequence 08, where the frames 720 and 1500 form a
reverse loop. The lower part of the figure is the semantic scan context
corresponding to the two frames. Since the directions of them are opposite,
the descriptors are quite different, while the aligned one shown in Fig. 2 is
easy to distinguish. Figure 2: The pipeline of our approach. It mainly
consists of two parts: two-step global semantic ICP and Semantic Scan Context.
First, we conduct semantic segmentation on the raw point cloud. Then we use
semantic information to retain representative objects and project them onto
the x-y plane. The two-step global semantic ICP is performed on the projected
cloud to get the 3D pose ($\Delta x,\Delta y,\theta)$. Finally, we use the 3D
pose to align the original clouds and generate global descriptors (Semantic
Scan Context). The similarity score is obtained by matching SSC.
For place recognition, when a robot passes through a place visited before, it
does not mean that the two poses are the same. Instead, the robot may walk
through the original area from any direction, and there may be a small amount
of translation from the original position. Many existing works consider the
robot’s orientation, namely rotation, and realize the invariance of
rotation[14, 3, 4, 10]. They may think that the small translation will not
strongly impact the recognition result and therefore ignore it. However, we
find that simply ignoring the translation for the scan context-based methods
will greatly reduce the similarity of the positive samples, making them
difficult to identify.
In this paper, we propose a novel global descriptor named Semantic Scan
Context (SSC), which explores semantic information to enhance the expressive
power of descriptors. We also propose a two-step global semantic ICP that can
produce reliable results regardless of the pose initialization, to obtain the
3D pose $(x,y,yaw)$ of the point cloud. The pose is then used to align the
point clouds to reduce the influence of rotation and translation on the
similarity of the descriptors. Furthermore, it can also provide good initial
values for 6D ICP algorithms to refine the global pose further. Fig. 1 is a
demonstration of our results. The main contribution is summarized as follows:
* •
We propose a novel global descriptor for LiDAR-based place recognition, which
exploits semantic information to encode the 3D scenes effectively.
* •
We propose a two-step global semantic ICP, which doesn’t require any initial
values, to obtain the 3D pose $(x,y,yaw)$ of the point clouds.
* •
We align point clouds with the obtained 3D poses to eliminate the influence of
rotation and translation error on the similarity of the descriptors, which can
also further benefit the SLAM system as good initial poses.
* •
Exhaustive experiments on the KITTI odometry dataset show that our approach
achieves state-of-the-art performance both in place recognition and pose
estimation.
## II RELATED WORK
According to the features used, we can divide the place recognition methods
into three categories: geometry-based, semi-semantic-based, semantic-based.
Geometry-based methods: Spin image [2] establishes a local coordinate system
for each point, then projects the point into the 2D space and counts the
number of points in different areas in the 2D space to form a spin image. ESF
[16] proposes a shape descriptor that combines angle, point-distance, and area
to boost the recognition rate. M2DP [5] projects the point cloud into multiple
2D planes and generates a density signature for each plane’s points. The left
and right singular vectors of those signatures are used as the global
descriptors. Scan context [3, 4] converts the point cloud to polar coordinates
and then divides it into blocks along the azimuth and radial directions.
Lastly, it encodes the z coordinate of the highest point in each block as a 2D
global descriptor. LocNet [6] divides a point cloud into rings, generates a
distance histogram for each ring, and stitches all histograms to form a global
descriptor. Then a siamese network is used to score the similarity between the
descriptors. LiDAR Iris [17] extracts a binary signature image for each point
cloud then uses the Hamming distance of two corresponding binary signature
images as the similarity. Seed [18] segments the point cloud into different
objects and encodes the topological information of the segmented objects into
the global descriptor. The above methods have achieved good results by
encoding low-level geometric structures into descriptors. It can be expected
that integrating more advanced features can further enhance the discriminative
power of descriptors.
Semi-semantic-based methods: Some methods use non-geometric information to
construct descriptors, such as reflection intensity or learning-based features
extracted by neural networks. Such features are related to the object type but
do not clearly indicate the semantic category, so we classify these methods as
semi-semantic based. ISHOT [9] and ISC [10] exploit the intensity information
of the point cloud for place recognition. SegMatch [19] and SegMap [20]
cluster a point cloud into segments. Then they extract features for each
segment and use the kNN algorithm to identify corresponds. PointNetVLAD [21]
combines PointNet [22] and NetVLAD [23] to extract global descriptors from the
3D point clouds end-to-end. $L^{3}$-Net [24] selects key points from the given
point cloud then uses a PointNet to learn local descriptors for each key
point. OREOS [25] projects the 3D point cloud into a 2D range image and
proposes a convolutional neural network to extract the global descriptor. DH3D
[26] designs a siamese network to learn 3D local features from the raw 3D
point clouds, then use an attention mechanism to aggregate these local
features as the global descriptor. LPD-Net [27] proposes the adaptive local
feature extraction module and the graph-based neighborhood aggregation module
to extract local features of the point cloud; then, as the PointNetVLAD, they
use the NetVLAD to generate the global descriptor. MinkLoc3D [28] uses a
sparse voxelized point cloud representation and sparse 3D convolutions to
compute a discriminative 3D point cloud descriptor. SeqSphereVLAD [29]
projects the point cloud onto a spherical view, extracts features on it and
sequences those features to form a descriptor. SpoxelNet [30] voxelized the
point cloud in spherical coordinates and defines the occupancy of each voxel
in ternary values. Then they use a neural network to extract the global
descriptor. The above methods combine more advanced features with geometric
features. However, most of them use neural networks to extract abstract
features, which are more complicated and not well interpretable.
Semantic-based methods: SGPR [14] represents the scene as a semantic graph
then uses a graph similarity network to score the similarity of the graphs.
GOSMatch [15] proposes a new global descriptor that is generated from the
spatial relationship between semantics. It also proposes a coarse-to-fine
strategy to efficiently search loop closures and gives an accurate 6-DOF
initial pose estimation. The two methods represent the scene as a graph and
abstract the object as a node in the graph, which will cause the loss of
features such as the size of each object. OverlapNet [7] designed a deep
neural network that uses different types of information, such as intensity,
normal, and semantics generated from LiDAR scans, to provide overlap and
relative yaw angle estimates between paired 3D scans. However, it is too slow
in preprocessing due to the need to calculating the normal and inferring the
complex network backbone. To use the semantic information more effectively, we
propose our Semantic Scan Context approach.
## III METHODOLOGY
In this section, we present our semantic scan context approach. Different from
other scan context-based methods that use incomplete semantic information and
ignore small translations between point clouds, we explore to exploit full
semantic information and emphasize that the small translation between point
cloud pairs has a significant influence on the accuracy of recognition.
As shown in Fig. 2, our method consists of two main parts: two-step global
semantic ICP and Semantic Scan Context. The two-step global semantic ICP is
divided into Fast Yaw Angle Calculate and Fast Semantic ICP. First, we define
a point cloud frame as $P=\\{p_{1},p_{2},\cdots,p_{n}\\}$, with each point
$p_{i}=[x_{i},y_{i},z_{i},\eta_{i}]$, $\eta_{i}$ represent the semantic label
of $p_{i}$. Given a pair of point clouds $(P_{1},P_{2})$, we first use our
Fast Yaw Angle Calculate method to get the relative yaw angle $\theta$ between
them. Then we use the Fast Semantic ICP to calculate their relative
translation $(\Delta x,\Delta y)$ in the x-y plane. Through the above two
steps, we get the relative poses $(\Delta x,\Delta y,\theta)$ of the two
frames of point clouds in 3D pose space. In order to eliminate the influence
of rotation (e.g., reverse loop closures) and small translation on
recognition, we use the obtained relative pose to align point cloud $P_{2}$.
We mark the aligned point cloud as $P_{a}$. Finally, we use our global
descriptor – the Semantic Scan Context to describe $(P_{1},P_{a})$ as
$(S_{1},S_{2})$. The similarity score is obtained by comparing $S_{1}$ and
$S_{2}$.
(a) Yaw Aligned
(b) Translation Aligned
Figure 3: An illustration of the two-step global semantic ICP.
### III-A Global Semantic ICP
It is known that the general ICP algorithm based on local iterative
optimization is susceptible to local minimums [31]. For place recognition, we
usually cannot get a valid initial value, which leads to the failure of the
general ICP algorithm. To solve this, we propose the two-step global semantic
ICP algorithm consisting of Fast Yaw Angle Calculate and Fast Semantic ICP.
Benefited from the use of semantic information, our algorithm does not require
any initial values to get satisfactory results.
Fast Yaw Angle Calculate. For scan context based methods, columns of their
descriptor represent the yaw angle. The pure rotation of the LiDAR in the
horizontal plane will cause the column shift of their descriptor. Scan context
and Intensity Scan Context get the similarity score and the yaw angle at the
same time. Specifically, they calculate similarity (or distance) with all
possible column-shifted descriptors and find the maximum similarity (or
minimum distance). However, there are two main disadvantages. Firstly, it’s
inefficient to compare the whole 2D descriptors by shifting. Secondly, they
still try to get the maximum score for point clouds from different places (not
loop closure). This obviously makes it more prone to false positives. To draw
the above issues, we propose the semantic-based fast yaw angle calculate
method.
Given a point cloud pair $(P_{1},P_{2})$, we select representative objects
such as buildings, tree trunks, and traffic signs based on semantic
information. Then we convert the filtered clouds to polar coordinate in the
x-y plane:
$\displaystyle p_{i}$ $\displaystyle=[r_{i},\varphi_{i},x_{i},y_{i},\eta_{i}]$
(1) $\displaystyle r_{i}$ $\displaystyle=\sqrt{x_{i}^{2}+y_{i}^{2}}$
$\displaystyle\varphi_{i}$ $\displaystyle=arctan(\frac{y_{i}}{x_{i}})$
where $p_{i}$ is the $i^{th}$ point in each converted cloud, $r_{i}$ and
$\varphi_{i}$ represent polar diameter and polar angle, respectively. Each
converted cloud is then segmented to $N_{a}$ sectors by yaw angle. We only
keep the point with the smallest polar diameter in each sector. Finally, we
get two clouds $P_{I1}$ and $P_{I2}$, with $N_{a}$ elements. We sort the
points in $P_{I1}$ and $P_{I2}$ according to the azimuth angle and save their
corresponding polar diameters as vectors $R_{1}$ and $R_{2}$. Similar to the
scan context, the shift of the column vector is related to the yaw angle:
$\displaystyle shift$
$\displaystyle=\mathop{argmin}\limits_{i,i\in[0,N_{a}]}\varPsi(R_{1},R_{2}^{i})$
(2) $\displaystyle\theta$ $\displaystyle=360-\frac{360\times shift}{N_{a}}$
where $R_{2}^{i}$ is $R_{2}$ shifted by $i^{th}$ element and $\varPsi$ is
defined as:
$\varPsi(R_{1},R_{2}^{i})=\left\lVert R_{1}-R_{2}^{i}\right\rVert_{1}$ (3)
Compared with Scan Context and Intensity Scan Context, our method only needs
to compare one-dimensional vectors; therefore, it is more efficient. Moreover,
our method does not obtain the angle via maximizing the score, which is
helpful to identify non-loop-closure point-cloud pairs. Fig. 3 shows the
result of Fast Yaw Angle Calculate.
Figure 4: An example of generating SSC. $\rho$ and $\theta$ represent the
polar diameter and polar angle, respectively. A sector corresponds to a
descriptor column, while a ring corresponds to a row of the descriptor.
Fast Semantic ICP. Though most works ignore translation between point clouds,
ignoring the translation causes considerable declines in our experiments. In
fact, for methods based on scan context, translation will affect both the row
and column of the descriptor. We can’t get the best result just by the column-
shifted descriptor. Therefore, we propose a fast semantic ICP algorithm to
correct the translation between point clouds.
To find the relative translation, we firstly rotate $P_{I2}$ to the same
direction as $P_{I1}$, and the rotated point cloud is $P_{Ia}$, which is
defined as:
$\displaystyle x_{ai}$ $\displaystyle=x_{i}cos(\theta)-y_{i}sin(\theta)$ (4)
$\displaystyle y_{ai}$ $\displaystyle=x_{i}sin(\theta)+y_{i}cos(\theta)$
where $(x_{i},y_{i})$ and $(x_{ai},y_{ai})$ represent the $i^{th}$ point in
$P_{I2}$ and $P_{Ia}$ respectively. Our ICP problem can be defined as:
$\displaystyle(\Delta x,\Delta y)$
$\displaystyle=\mathop{argmin}\limits_{\Delta x,\Delta
y}L=\mathop{argmin}\limits_{\Delta x,\Delta
y}\sum_{i=1}^{N_{a}}\Gamma(\eta_{ai},\eta_{ri})$ (5) $\displaystyle\times$
$\displaystyle\frac{(x_{ai}+\Delta x-x_{ri})^{2}+(y_{ai}+\Delta
y-y_{ri})^{2}}{2}$
where $(x_{ri},y_{ri})$ represents the corresponding point of
$(x_{ai},y_{ai})$, which is the point closest to $(x_{ai},y_{ai})$ in
$P_{I1}$, $\eta_{ai}$ and $\eta_{ri}$ are semantic labels of the points. If
$\eta_{ai}$ is equal to $\eta_{ri}$, then the output of
$\Gamma(\eta_{ai},\eta_{ri})$ is $1$; otherwise, $0$. As our point clouds are
ordered, we can search for the corresponding points near the position where
the yaw angle is consistent with the target point. Specifically, our search
interval for the $i^{th}$ target point is:
$[i+shift-\frac{N_{l}}{2},i+shift+\frac{N_{l}}{2}]$ (6)
where $N_{l}$ is the length of search interval and $shift$ is defined in Eq.
2. After a certain number of iterations, we can get the relative translation
between the input point clouds, shown in Fig. 3.
(a) 00
(b) 02
(c) 05
(d) 06
(e) 07
(f) 08
Figure 5: Precision-Recall curves on KITTI dataset. TABLE I: $F_{1}$ max
scores and Extended Precision on KITTI dataset
Methods | 00 | 02 | 05 | 06 | 07 | 08 | Mean
---|---|---|---|---|---|---|---
SC[3] | 0.750/0.609 | 0.782/0.632 | 0.895/0.797 | 0.968/0.924 | 0.662/0.554 | 0.607/0.569 | 0.777/0.681
ISC[10] | 0.657/0.627 | 0.705/0.613 | 0.771/0.727 | 0.842/0.816 | 0.636/0.638 | 0.408/0.543 | 0.670/0.661
M2DP[5] | 0.708/0.616 | 0.717/0.603 | 0.602/0.611 | 0.787/0.681 | 0.560/0.586 | 0.073/0.500 | 0.575/0.600
LI[17] | 0.668/0.626 | 0.762/0.666 | 0.768/0.747 | 0.913/0.791 | 0.629/0.651 | 0.478/0.562 | 0.703/0.674
PV[21] | 0.779/0.641 | 0.727/0.691 | 0.541/0.536 | 0.852/0.767 | 0.631/0.591 | 0.037/0.500 | 0.595/0.621
ON[7] | 0.869/0.555 | 0.827/0.639 | 0.924/0.796 | 0.930/0.744 | 0.818/0.586 | 0.374/0.500 | 0.790/0.637
SGPR[14] | 0.820/0.500 | 0.751/0.500 | 0.751/0.531 | 0.655/0.500 | 0.868/0.721 | 0.750/0.520 | 0.766/0.545
Ours-RN | 0.939/0.826 | 0.890/0.745 | 0.941/0.900 | 0.986/0.973 | 0.870/0.773 | 0.881/0.732 | 0.918/0.825
Ours-SK | 0.951/0.849 | 0.891/0.748 | 0.951/0.903 | 0.985/0.969 | 0.875/0.805 | 0.940/0.932 | 0.932/0.868
* •
$F_{1}$ max scores and Extended Precision: $F_{1}$ max scores / Extended
Precision. The best scores are marked in bold and the second best scores are
underlined.
### III-B Semantic Scan Context
Scan Context and Intensity Scan Context uses the points’ height and reflection
intensity as features, respectively. Their methods essentially take advantage
of the different characteristics of different objects in the scene. However,
height and reflection intensity is only low-level features of the object which
are not representative enough. We explore to use the high-level semantic
features to represent scenes and thus propose the Semantic Scan Context
descriptor.
Descriptor definition. Given a point cloud $P$, we first convert it to the
polar coordinate system as we did in Section III-A. Then, like scan context,
we divide the point cloud into $N_{s}\times N_{r}$ blocks along the azimuthal
and radial directions. Each block is represented by:
$\displaystyle B_{ij}=\\{\eta_{k}|\frac{(i-1)\cdot R_{max}}{N_{r}}\leq
r_{k}<\frac{i\cdot R_{max}}{N_{r}},$ (7) $\displaystyle\frac{(j-1)\cdot
2\pi}{N_{s}}-\pi\leq\varphi_{k}<\frac{j\cdot 2\pi}{N_{s}}-\pi\\}$
where $R_{max}$ is the the maximum effective measurement distance of LiDAR,
$i\in[1,N_{r}]$ and $j\in[1,N_{s}]$. Our descriptor can be defined by:
$S(i,j)=f(B_{ij})=\mathop{argmax}\limits_{\eta\in B_{ij}}E(\eta)$ (8)
$f$ is an encoding function to encode features of $B_{ij}$. Note that if
$B_{ij}=\varnothing,~{}f(B_{ij})=0$. We manually set the priority of different
semantics in function $E$ to show their representativeness. We believe objects
that appear less frequently in the scene are more representative (e.g.,
traffic signs are more representative than roads).
Similarity Scoring. Given aligned clouds $P_{1}$ and $P_{a}$, we can get their
descriptors $S_{1}$ and $S_{2}$ by Eq. 8. Then the similarity score between
them can be calculated by:
$\displaystyle score=\frac{\sum\limits_{1\leq i\leq N_{r}}\sum\limits_{1\leq
j\leq N_{s}}I(S_{1}(i,j)=S_{2}(i,j))}{\sum\limits_{1\leq i\leq
N_{r}}\sum\limits_{1\leq j\leq N_{s}}I(S_{1}(i,j)\neq 0~{}or~{}S_{2}(i,j)\neq
0)}$ (9)
where $I$ is the indicator function, defined by:
$I(x)=\left\\{\begin{aligned} &1&&x~{}is~{}true\\\
&0&&x~{}is~{}false\end{aligned}\right.$ (10)
Fig. 4 shows Semantic Scan Context creation.
(a) Average $F_{1}$ max scores
(b) Average $EP$
Figure 6: Average $F_{1}$ max score and Average Extended Precision
corresponding to different $\alpha$.
## IV EXPERIMENTS
### IV-A Experiment Setup
We conduct experiments on the KITTI odometry dataset[32] collected by a
64-ring LiDAR, which contains 11 training sequences (00-10) with ground truth
poses. We choose sequences with loop-closure (00,02,05,06,07,08) for
evaluation and note that sequence 08 has reverse loops while others are in the
same direction. Similar to SGPR[14], we regard the point cloud pair with a
relative distance less (greater) than 3m (20m) as a positive (negative)
sample. Since there are too many negative samples, we only select a part of
the negative samples for evaluation. Specifically, if there are $N_{p}$
positive samples in a sequence, we will randomly select $\alpha\cdot N_{p}$
negative samples. We can adjust the proportion of negative samples by changing
the coefficient $\alpha$.
The ground-truth semantic labels are from the SemanticKITTI dataset[33]. We
also test our method with the semantic segmentation algorithm (RangeNet++
[34]) to prove that our method can be applied to noisy predictions in real
situations. In our experiments, we set
$N_{a}=360,~{}N_{l}=20,~{}N_{s}=360,~{}N_{r}=50$. All experiments are done on
the same system with an Intel i7-9750H @3.00GHz CPU with 16 GB RAM.
### IV-B Place Recognition Performance
As mentioned in Section IV-A, we use both ground-truth semantic labels (Ours-
SK) and predicted semantic labels (Ours-RN) for testing. We compare our
approach with the state-of-the-art methods, including Scan Context[3] (SC),
Intensity Scan Context[10] (ISC), M2DP[5], LiDAR Iris[17] (LI),
PointNetVLAD[21] (PV), OverlapNet[7] (ON), and SGPR[14]. For SGPR, we use
their pre-trained models trained with the 1-fold strategy. As we cannot
reproduce the results of OverlapNet, we use the pre-trained model provided by
the author. The model is trained on sequences 03-10, so sequences 05, 06, 07,
08 are included in the training set.
Fixed $\alpha$. In this experiment, we set $\alpha$ to 100, which means the
number of negative samples is $100N_{p}$. Fig. 5 shows the precision-recall
curve of each method. Additionally, we also use the maximum $F_{1}$ score and
Extended Precision[35] (EP) shown in Tab. I to analyze the performance. The
$F_{1}$ score is defined as:
$F_{1}=2\times\frac{P\times R}{P+R}$ (11)
where $P$ and $R$ represent the Precision and Recall, respectively; $F_{1}$ is
the harmonic mean of $P$ and $R$. It treats $P$ and $R$ as equally important
and measures the overall performance of classification. The Extended Precision
is defined as:
$EP=\frac{1}{2}(P_{R0}+R_{P100})$ (12)
where $P_{R0}$ is the precision at minimum recall, and $R_{P100}$ is the max
recall at $100\%$ precision. $EP$ is specifically designed metrics for place
recognition algorithms.
As shown in Fig. 5 and Tab. I, Ours-SK surpasses other methods in all
indicators of all sequences with a large margin. Especially in sequence 08,
which has only reverse loops, the performance of other methods drops
significantly while our method still performs well. This indicates that our
method is robust to view angle changes. OverlapNet performs well on most
sequences except 08. We guess this is because it uses the normal of the point
cloud, which will change as the point cloud rotates. Therefore, this method
cannot robustly handle reverse loops. SGPR works well on indicator the $F_{1}$
max score but poorly on the Extended Precision. We find that it gives some
negative samples a huge score, which causes the recall to be almost zero when
the accuracy reaches $100\%$. The result of Ours-RN is slightly worse than
Ours-SK as expected. As the difference is not obvious, it means that our
approach can adapt to semantic segmentation algorithms for actual systems.
Change $\alpha$. In this experiment, we change the value of $\alpha$ to
analyze the influence of the number of negative samples on those algorithms.
Fig. 6 shows the Average $F_{1}$ max score and Average Extended Precision
corresponding to different $\alpha$. It clearly shows that our method performs
better than others no matter how much $\alpha$ is taken. As $\alpha$
increases, the performance of all methods gradually decreases, but our method
is less affected, showing that our method can effectively identify negative
samples. For place recognition, negative samples are generally far more than
positive samples, which is one key reason why our method leads in metrics far
ahead. Moreover, identifying negative samples is significant as false
positives will bring fatal crashes to the SLAM system.
TABLE II: Yaw error on KITTI dataset sequences | SC (deg) | ISC (deg) | ON (deg) | Ours-SK (deg)
---|---|---|---|---
00 | 11.526 | 0.829 | 2.595 | 0.891
02 | 11.301 | 1.343 | 4.911 | 1.142
05 | 18.394 | 0.904 | 3.329 | 0.653
06 | 4.074 | 0.534 | 1.124 | 0.759
07 | 21.862 | 0.684 | 2.233 | 0.512
08 | 49.170 | 3.856 | 68.622 | 1.878
Average | 19.388 | 1.358 | 13.802 | 0.973
### IV-C Pose Accuracy
As described in Section III-A, our approach can estimate the 3D relative pose
$(\Delta x,\Delta y,\theta)$, while most other methods cannot estimate pose or
can only estimate 1D pose (yaw). We compare our method with Scan Context,
Intensity Scan Context, and Overlap. The ground-truth pose is calculated by:
$\displaystyle T$ $\displaystyle=T_{1}^{-1}T_{2}$ (13) $\displaystyle(\Delta
x,\Delta y,\theta)$
$\displaystyle=(T(1,3),T(2,3),arctan(\frac{T(2,1)}{T(1,1)}))$
where $T_{1}\in SE(3)$ and $T_{2}\in SE(3)$ represent the pose of $P^{1}$ and
$P^{2}$, respectively. Since the pitch and roll angles are hardly changed in
autonomous vehicles, we ignore them.
Tab. II shows the relative yaw error on the KITTI dataset. We can see that our
method outperforms other methods in terms of the average relative yaw error.
Especially in the challenging sequence 08, affected by the reverse loop, most
methods perform poorly, while our method can still accurately estimate the yaw
angle. This again shows that our method can handle the reverse loop well. As
mentioned in Section IV-B, OverlapNet performs poorly due to its inability to
handle reverse loops.
Fig. 7 shows the relative translation error of our approach on the KITTI
dataset. As shown, our method can estimate accurate relative translation,
which is currently not possible with other methods to our knowledge. Thus, our
Fast Yaw Angle Calculate and Fast Semantic ICP approaches can give accurate 3D
pose estimation. This can provide a good initial value for the ICP algorithm
to obtain a 6D pose or directly serve as a global constraint in the SLAM
system.
Figure 7: Translation error. TABLE III: Contribution of individual components Yaw | ICP | Semantic | $F_{1}/EP$ | Decrease
---|---|---|---|---
| $\surd$ | $\surd$ | 0.896/0.820 | 3.6%/4.8%
$\surd$ | | $\surd$ | 0.757/0.685 | 17.5%/18.3%
$\surd$ | $\surd$ | | 0.775/0.762 | 15.7%/10.6%
$\surd$ | $\surd$ | $\surd$ | 0.932/0.868 | 0.0%/0.0%
### IV-D Ablation Study
We design an ablation study to investigate the contribution of each component.
Specifically, we remove or replace a module at a time and then calculate the
$F_{1}$ max scores and Extended Precision. To show the contribution of our
Fast Yaw Angle Calculate method, we replace this module with the method used
in scan context – shift the column of descriptors and calculate the maximum
similarity score while obtaining the yaw angle. Similarly, we replace the
semantic label in the descriptor by maximum $z$ to see semantic contribution.
To evaluate the contribution of our Fast Semantic ICP approach, we directly
set $\Delta x$ and $\Delta y$ to 0. As shown in Tab. III, after removing Yaw,
ICP, and Semantic, the average $F_{1}$ max score decrease by $3.6\%$,
$17.5\%$, $15.7\%$, and the average Extended Precision decrease by $4.8\%$,
$18.3\%$, $10.6\%$. Therefore, the following conclusions can be drawn:
* •
Compared with other methods, our approach can get a more accurate yaw angle
and translation.
* •
As we emphasized, the small translation has a significant impact on scan
context-based methods. Simply ignoring the translation will greatly weaken the
performance.
* •
High-level features, like semantics, can bring considerable improvements in
the scene description.
TABLE IV: Average time cost on KITTI 08
Methods | Size | Description | Retrieval | ICP | Total
---|---|---|---|---|---
SC | $20\times 60$ | 4.825 | 0.158 | - | 4.983
ISC | $20\times 90$ | 3.094 | 0.800 | - | 3.894
Ours | $50\times 360$ | 2.563 | 0.066 | 2.126 | 4.755
* •
The unit of time in the table is milliseconds.
### IV-E Efficiency
To evaluate the efficiency, we set $\alpha$ to $1$ and compare the average
time cost of our method with Scan Context and Intensity Scan Context on
sequence 08. As shown in Tab. IV, the total time cost of our approach is
acceptable. As we use the obtained 3D pose to align the point clouds in
advance, we don’t need to shift the column of descriptors during the matching
stage, so our retrieval speed is extremely fast. Our two-step global semantic
ICP only takes 2.126 milliseconds on average. This algorithm is fast due to
the following reasons. Firstly, since we only keep $N_{a}$ (360 taken in our
experiments) points, the computational cost is greatly reduced compared to the
original point cloud (about 120,000 points). Secondly, We divide the algorithm
into two steps, first calculate the yaw angle, and then iteratively calculate
$\Delta x$ and $\Delta y$, which simplifies the algorithm and speeds up the
calculation. Thirdly, when calculating $\Delta x$ and $\Delta y$, we use the
yaw angle to align the input clouds in advance. Therefore we don’t need to
traverse the entire point cloud when looking for the corresponding points.
Instead, we can find them near the corresponding positions, which greatly
reduces the number of searches.
## V CONCLUSION
In this paper, we propose a novel semantic-based global descriptor for place
recognition. We propose a two-step global semantic ICP to obtain the 3D pose
$(x,y,yaw)$ of the point cloud pair, aligning the point clouds to improve the
descriptor matching accuracy. In addition, it can provide good initial values
for point cloud registration. We achieve leading performance on the KITTI
odometry dataset compared to the state-of-the-art methods.
Our method also has some limitations. Like most place recognition methods, our
method does not consider pitch angle and roll angle. Therefore, our method may
fail in some extreme scenarios.
In the future work, we will try to solve the above problems and further
explore the application of semantic information in LiDAR-based SLAM systems.
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|
\ulabel{5.3,-1}{S};
% \stoch{8.75,1.5}{$(\psi_H{\comp}f)|_S$}{4};
\stoch{8.75,1.5}{$c$}{4};
\rlabel{11,1.5}{H\sprime};
\rlabel{11,-1}{S};
% \showgrid
% \pbox{
% \stringdiagram{
% \ulabel{0.2,0}{H};
% \wire(-4,0) -- (2,0);
% \wire[2.25] (-3.2,0) -- (-3.2,2.25) -- (9,2.25);
% \blackdot{-3.2,0};
% \llabel{-4,0}{\Theta};
% \stoch{2,0}{$f$}{4};
% \stoch{-1.5,0}{$\psi_H$}{3};
% \wire(2,1) -- (3.5,1);
% \wire(2,-1) -- (11.5,-1);
% \rlabel{3.8,1}{H\sprime};
% \blackdot{3.5,1};
% \wire[2] (5.5,-1)--(5.5,1)--(9,1);
% \wire (9,1.5) -- (11.5,1.5);
% \blackdot{5.5,-1};
% \ulabel{5,-1}{S};
% \stoch{8.75,1.5}{$(\psi_H{\comp}f)|_S$}{4};
% \rlabel{11.5,1.5}{H\sprime};
% \rlabel{11.5,-1}{S};
%% \showgrid
% }}{.}
\end{equation}
This is an instance of definition 11.4 in <cit.>.
The conditional $c$ is now a kernel of type $\Theta\times S \to P(H')$, depending on both a data point $s\in S$ and a parameter value $\theta\in\Theta$.
The next idea is that if our parametrised distributions are suitably chosen, then for every $s$ and every $\theta$ the conditional over $H'$ will be a member of the family parametrised by $\Theta'$. This amounts to saying that there exists a deterministic function $\gamma\colon \Theta\times S \to \Theta'$ such that
\wire (0,1)--(4,1);
\wire[0.5] (0,3)--(0.5,3)--(1,2)--(2,2);
\stoch{2,1.5}{$c$}{2}
\llabel{0,3}{ S};
\llabel{0,1}{ \Theta};
\rlabel{4,1}{ H\sprime};
%\wire (-2.5,1)--(6.5,1);
%\wire[0.5] (-2.5,3)--(-2,3)--(-1,2)--(0,2);
%\llabel{-2.5,3}{ S};
%\llabel{-2.5,1}{ \Theta};
%\rlabel{6.5,1}{ H\sprime};
\stringdiagram{
\wire (0,1)--(9,1);
\wire[0.5] (0,3)--(0.5,3)--(1,2)--(2,2);
\sqbox{2,1.5}{$\gamma$}{2};
\llabel{0,3}{ S};
\llabel{0,1}{ \Theta};
\ulabel{3.6,1}{ \Theta\sprime};
\stoch{6.2,1.5}{$\Psi_{H\sprime}$}{2};
\rlabel{9,1}{H\sprime};
The equation then becomes
\begin{equation}
\label{deterministic-heterogeneous-filtering}
\stringdiagram{
\ulabel{0.2,0}{H};
\wire(-3.5,0) -- (2,0);
\llabel{-3.5,0}{\Theta};
\stoch{2,0}{$f$}{4};
\stoch{-1.5,0}{$\psi_H$}{3};
\wire(2,1) -- (4.5,1);
\wire(2,-1) -- (4.5,-1);
\rlabel{4.5,1}{H\sprime};
\rlabel{4.5,-1}{S};
\,=
\pbox{
\stringdiagram{
\ulabel{0.2,0}{H};
\wire(-4,0) -- (2,0);
\wire[2.25] (-3.2,0) -- (-3.2,2.25) -- (4.5,2.25);
\wire[0.5] (4.5,2.25) -- (5,2.25) -- (6,1) -- (8.5,1);
\blackdot{-3.2,0};
\llabel{-4,0}{\Theta};
\stoch{2,0}{$f$}{4};
\stoch{-1.5,0}{$\psi_H$}{3};
\wire(2,1) -- (3.5,1);
\wire(2,-1) -- (13,-1);
\rlabel{3.8,1}{H\sprime};
\blackdot{3.5,1};
\wire (8.5,1.5) -- (13,1.5);
\wire[2.5] (4.5,-1)--(5.5,2.25)--(8.5,2.25);
\blackdot{4.5,-1};
\ulabel{5.3,-1}{S};
\sqbox{7.75,1.5}{$\gamma$}{4};
\stoch{11,1.5}{$\psi_{H\sprime}$}{4};
\rlabel{13,1.5}{H\sprime};
\rlabel{13,-1}{S};
\ulabel{9.25,1.5}{\Theta\sprime}
% \showgrid
\end{equation}
The point is that once the appropriate families of distributions have been chosen such that this equation holds, only the function $\gamma$ needs to be implemented.
I am again considering only talking about deterministic machines. If we don't do that, the following footnote would be a paragraph at this point, and we'd use stochastic updates after that.
Later, in taking a natural science perspective, we will want to consider the case where $\gamma$ is not a function $\Theta\times S \to \Theta'$ but instead a Markov kernel $\Theta\times S \to P(\Theta')$.
That is, $\gamma$ may behave stochastically, choosing a random element of $\Theta'$ that may depend on $s\in S$ and $\theta\in \Theta$.
In this case <ref> must be changed to
\begin{equation}
\label{stochastic-heterogeneous-filtering}
\stringdiagram {
\wire (0,0) -- (11.5,0);
\ulabel{0,0}{\Theta};
\ulabel{11.5,0}{\Theta\sprime};
\blackdot{0.7,0};
\wire[3] (0.7,0) -- (0.7,3) -- (5,3);
\wire (6,2) -- (11.5,2);
\ulabel{11.5,2}{S};
\stoch{2.8,3}{$\psi_H\!$}{3};
\stoch{5.75,3}{$f$}{4};
\ulabel{4.2,3}{H};
\stoch{9.5,0.25}{$\gamma$}{3};
\blackdot{7.5,2};
\ulabel{8,2}{S};
\wire[1.5] (7.5,2) -- (7.5,0.5) -- (9.5,0.5);
\wire[2] (5,4) -- (11.5,4);
\ulabel{11.5,4}{H};
% \showgrid
\, = \!
\pbox{
\stringdiagram {
\wire (0,0) -- (15,0);
\ulabel{0,0}{\Theta};
\ulabel{15,0}{\Theta\sprime};
\blackdot{0.7,0};
\wire[3] (0.7,0) -- (0.7,3) -- (5,3);
\wire (6,2) -- (15,2);
\ulabel{15,2}{S};
\stoch{2.8,3}{$\psi_H\!$}{3};
\stoch{5.75,3}{$f$}{4};
\ulabel{4.2,3}{H};
\wire[2] (5,4) -- (8,4);
\ulabel{7.4,4}{H\sprime};
\blackdot{8,4};
\stoch{9,0.25}{$\gamma$}{3};
\blackdot{7.3,2};
\ulabel{7.8,2}{S};
\wire[1.5] (7.3,2) -- (7.3,0.5) -- (9,0.5);
\wire[4] (10.5,0) -- (10.5,4) -- (15,4);
\blackdot{10.5,0};
\ulabel{11.3,0}{\Theta\sprime};
\stoch{13.2,4}{$\psi_{H\sprime}$}{3};
\ulabel{15,4}{H\sprime};
% \showgrid
\end{equation}
This is because if $\gamma$ is stochastic then there may be several different elements of $\Theta'$ that can result from giving passing values $\theta\in\Theta$ and $s\in S$ to $\gamma$.
<Ref> demands that each of these map to a valid conditional distribution over $H'$.
In a typical Bayesian filtering setup, we would demand that $H=H'$, $\Theta=\Theta'$ and $\psi_{H} = \psi_{H'}$.
That is, instead of having one family of distributions over $H$ and another over $H'$ we instead just have one family of distributions and update its parameters in response to data.
This allows the updating process to be iterated over a sequence of inputs.
However, before we consider that case, let us first consider a slightly different kind of iteration.
Suppose that in addition to the kernel
\wire (0,-1)--(-4,-1);
\wire[0.5] (0,-3)--(-0.5,-3)--(-1,-2)--(-2,-2);
\stoch{-2,-1.5}{$f$}{2}
\rlabel{0,-3}{ S};
\rlabel{0,-1}{ H\sprime};
\llabel{-4,-1}{ H};
we also have a kernel
\wire (0,-1)--(-4,-1);
\wire[0.5] (0,-3)--(-0.5,-3)--(-1,-2)--(-2,-2);
\stoch{-2,-1.5}{$f'$}{2}
\rlabel{0,-3}{ T};
\rlabel{0,-1}{ \,\,H''};
\llabel{-4,-1}{ H'};
which we can interpret as the hidden system undergoing a second transition, changing stochastically to a new state in $H''$ and producing a second observable from a sample space $T$, which might in general be different from $S$.
Correspondingly, we can define a new parameter space $\Theta''$ and a new family of distributions
6,1 H”;
and a new update function
\wire (0,1)--(4,1);
\wire[0.5] (0,3)--(0.5,3)--(1,2)--(2,2);
\sqbox{2,1.5}{$\gamma'$}{2}
\llabel{0,3}{ T};
\llabel{0,1}{ \Theta'};
\rlabel{4,1}{ \,\,\,\Theta''};
obeying the analog of <ref>.or <ref> if we include stochastic updates.
We can see this either as two separate updates (one in response to an input in $S$ and another in response to an input in $T$), or we can see it as a single update in response to an input $(s,t)\in S\times T$, that is, an input consisting of sequence consisting of $s$ followed by $t$.
This gives us two different ways to write <ref>, and we might hope that these would be equivalent.
Indeed this is the case, and it has the interesting consequence that Bayesian filtering updates form a category.
Although this fact is somewhat technical and we do not use it extensively below, we state it as a proposition:
There is a category $\mathsf{Filt}$ of Bayesian filtering updates, in which
$(i)$ the objects are tuples $\left\langle \Theta, H, \!\!\stringdiagram{
\wire (0,1)--(4,1);
\stoch{2,1}{$\psi_H$}{2}
\llabel{0,1}{\Theta};
\rlabel{4,1}{H};
consisting of two measurable spaces and a Markov kernel; $(ii)$ a morphism $\left\langle \Theta, H, \!\!\stringdiagram{
\wire (0,1)--(4,1);
\stoch{2,1}{$\psi_H$}{2}
\llabel{0,1}{\Theta};
\rlabel{4,1}{H};
}\right\rangle \to \left\langle \Theta', H', \!\!\stringdiagram{
\wire (0,1)--(4,1);
\stoch{2,1}{$\psi_{H\sprime}$}{2}
\llabel{0,1}{\Theta'};
\rlabel{4,1}{H'};
}\right\rangle$ is an equivalence class of tuples $\left\langle S, \!\stringdiagram{
\wire (0,-1)--(-4,-1);
\wire[0.5] (0,-3)--(-0.5,-3)--(-1,-2)--(-2,-2);
\stoch{-2,-1.5}{$f$}{2}
\rlabel{0,-3}{ S};
\rlabel{0,-1}{ H\sprime};
\llabel{-4,-1}{ H};
}, \!\stringdiagram{
\wire (0,1)--(4,1);
\wire[0.5] (0,3)--(0.5,3)--(1,2)--(2,2);
\sqbox{2,1.5}{$\gamma$}{2}
\llabel{0,3}{ S};
\llabel{0,1}{ \Theta};
\rlabel{4,1}{ \Theta'};
} \right\rangle$
that consist of a measurable space, a stochastic kernel and a deterministic kernel such that <ref> holds; and $(iii)$
given morphisms
$\left\langle S, \!\stringdiagram{
\wire (0,-1)--(-4,-1);
\wire[0.5] (0,-3)--(-0.5,-3)--(-1,-2)--(-2,-2);
\stoch{-2,-1.5}{$f$}{2}
\rlabel{0,-3}{ S};
\rlabel{0,-1}{ H\sprime};
\llabel{-4,-1}{ H};
}, \!\stringdiagram{
\wire (0,1)--(4,1);
\wire[0.5] (0,3)--(0.5,3)--(1,2)--(2,2);
\sqbox{2,1.5}{$\gamma$}{2}
\llabel{0,3}{ S};
\llabel{0,1}{ \Theta};
\rlabel{4,1}{ \Theta'};
} \right\rangle$
$\left\langle T, \!\stringdiagram{
\wire (0,-1)--(-4,-1);
\wire[0.5] (0,-3)--(-0.5,-3)--(-1,-2)--(-2,-2);
\stoch{-2,-1.5}{$f$}{2}
\rlabel{0,-3}{ S};
\rlabel{0,-1}{ H\sprime};
\llabel{-4,-1}{ H'};
}, \!\stringdiagram{
\wire (0,1)--(4,1);
\wire[0.5] (0,3)--(0.5,3)--(1,2)--(2,2);
\sqbox{2,1.5}{$\gamma'$}{2}
\llabel{0,3}{ T};
\llabel{0,1}{ \Theta'};
\rlabel{4,1}{ \Theta''};
} \right\rangle$,
their composition is given by the equivalence class containing the morphism
$\left\langle S\times T,
\!\stringdiagram{
\wire (0,-1)--(-7.5,-1);
\wire[0.5] (0,-2.5)--(-0.5,-2.5)--(-1,-2)--(-2,-2);
\stoch{-2,-1.5}{$f'$}{2}
\wire[0.5] (0,-4)--(-3,-4)--(-4,-2)--(-5.5,-2);
\stoch{-5.5,-1.5}{$f$}{2}
\rlabel{0,-2.5}{ S};
\rlabel{0,-4.2}{ T};
\rlabel{0,-0.7}{ \,\,\,H''};
\llabel{-7.5,-1}{ H};
\wire (0,1)--(7.5,1);
\wire[0.5] (0,2.5)--(0.5,2.5)--(1,2)--(2,2);
\sqbox{2,1.5}{$\gamma$}{2}
\wire[0.5] (0,4)--(3,4)--(4,2)--(5.5,2);
\sqbox{5.5,1.5}{$\gamma$}{2}
\llabel{0,2.5}{ T};
\llabel{0,4.2}{ S};
\llabel{0,0.7}{ \Theta};
\rlabel{7.5,1}{ \,\,\,\Theta''};
\right\rangle$.
The appropriate notion of equivalence is defined in the supplementary information.
Alternatively, $\mathsf{Filt}$ may be viewed as a bicategory, with 2-cells given by a generalisation of sufficient statistics.
The proof and further details are given in the supplementary information.
It is interesting to note that the category $\mathsf{Filt}$ has some similarity to the categories of lenses or optics that have previously been used to model Bayesian updates two of Toby's papers, as well as agents in a game-theoretic or control-theoretic context e.g. Bayesian open games, backprop as functor, towards foundations.
However, it is different in that $\mathsf{Filt}$ lacks the bidirectional quality of lenses and optics.
Its morphisms consist of pairs of morphisms of the underlying category, but both point in the same direction instead of one of them being reversed.
We conjecture that a more lens-like bidirectional structure would be needed if we were to consider Bayesian smoothing rather than Bayesian filtering.
It will be interesting in future work to better understand the relationship between lens-like categories and the concept of interpretation developed in the present work.
Some steps have been taken in this direction recently cyber kittens, towards foundations, Translating Extensive Form Games to Open Games with Agency.
These works share a common theme with the present work, in that they identify a relationship between agency and parametrisation.
§.§ The category $\mathsf{Filt}$
Here we restate in more detail the definition of the category $\mathsf{Filt}$, in both its category form and its bicategory form.
As in the main paper, we assume we are working in a Markov category $\mathscr{C}$.
We use the term “space” or “measurable space” for objects of $\mathscr{C}$ and “Markov kernel” for morphisms of $\mathscr{C}$.
We start by defining a parametrised family of distributions.
These will be the objects of $\mathsf{Filt}$.
A parametrised family of distributions (or family) $X$ is a choice of measurable spaces $\Theta_{X}$ and $H_{X}$, together with a Markov kernel
\wire (-0.3,1)--(4.5,1);
\stoch{2,1}{$\psi_X$}{2}
\llabel{-0.3,1}{\Theta_X\,\,\,\,};
\rlabel{4.5,1}{\,\,\,\,H_X};
We then define a Bayesian filtering update between two parametrised families.
In the category version of $\mathsf{Filt}$ the morphisms of $\mathsf{Filt}$ will be equivalence classes of filtering updates.
In the bicategory version of $\mathsf{Filt}$, filtering updates will be the 1-cells.
Given parametrised families $X$ and $Y$, a Bayesian filtering update $u$ from $X$ to $Y$ is a choice of measurable space $S_u$, a Markov kernel
\wire (0,-1)--(-4,-1);
\wire[0.5] (0,-3)--(-0.5,-3)--(-1,-2)--(-2,-2);
\stoch{-2,-1.5}{$f$}{2}
\rlabel{0,-3}{ S};
\rlabel{0,-1}{ H\sprime};
\llabel{-4,-1}{ H};
temporary, just so I can copy and paste:
There is a category $\mathsf{Filt}$ of Bayesian filtering updates, in which
$(i)$ the objects are tuples $\left\langle \Theta, H, \!\!\stringdiagram{
\wire (0,1)--(4,1);
\stoch{2,1}{$\psi_H$}{2}
\llabel{0,1}{\Theta};
\rlabel{4,1}{H};
consisting of two measurable spaces and a Markov kernel; $(ii)$ a morphism $\left\langle \Theta, H, \!\!\stringdiagram{
\wire (0,1)--(4,1);
\stoch{2,1}{$\psi_H$}{2}
\llabel{0,1}{\Theta};
\rlabel{4,1}{H};
}\right\rangle \to \left\langle \Theta', H', \!\!\stringdiagram{
\wire (0,1)--(4,1);
\stoch{2,1}{$\psi_{H\sprime}$}{2}
\llabel{0,1}{\Theta'};
\rlabel{4,1}{H'};
}\right\rangle$ is an equivalence class of tuples $\left\langle S, \!\stringdiagram{
\wire (0,-1)--(-4,-1);
\wire[0.5] (0,-3)--(-0.5,-3)--(-1,-2)--(-2,-2);
\stoch{-2,-1.5}{$f$}{2}
\rlabel{0,-3}{ S};
\rlabel{0,-1}{ H\sprime};
\llabel{-4,-1}{ H};
}, \!\stringdiagram{
\wire (0,1)--(4,1);
\wire[0.5] (0,3)--(0.5,3)--(1,2)--(2,2);
\sqbox{2,1.5}{$\gamma$}{2}
\llabel{0,3}{ S};
\llabel{0,1}{ \Theta};
\rlabel{4,1}{ \Theta'};
} \right\rangle$
that consist of a measurable space, a stochastic kernel and a deterministic kernel such that <ref> holds; and $(iii)$
given morphisms
$\left\langle S, \!\stringdiagram{
\wire (0,-1)--(-4,-1);
\wire[0.5] (0,-3)--(-0.5,-3)--(-1,-2)--(-2,-2);
\stoch{-2,-1.5}{$f$}{2}
\rlabel{0,-3}{ S};
\rlabel{0,-1}{ H\sprime};
\llabel{-4,-1}{ H};
}, \!\stringdiagram{
\wire (0,1)--(4,1);
\wire[0.5] (0,3)--(0.5,3)--(1,2)--(2,2);
\sqbox{2,1.5}{$\gamma$}{2}
\llabel{0,3}{ S};
\llabel{0,1}{ \Theta};
\rlabel{4,1}{ \Theta'};
} \right\rangle$
$\left\langle T, \!\stringdiagram{
\wire (0,-1)--(-4,-1);
\wire[0.5] (0,-3)--(-0.5,-3)--(-1,-2)--(-2,-2);
\stoch{-2,-1.5}{$f$}{2}
\rlabel{0,-3}{ S};
\rlabel{0,-1}{ H\sprime};
\llabel{-4,-1}{ H'};
}, \!\stringdiagram{
\wire (0,1)--(4,1);
\wire[0.5] (0,3)--(0.5,3)--(1,2)--(2,2);
\sqbox{2,1.5}{$\gamma'$}{2}
\llabel{0,3}{ T};
\llabel{0,1}{ \Theta'};
\rlabel{4,1}{ \Theta''};
} \right\rangle$,
their composition is given by the equivalence class containing the morphism
$\left\langle S\times T,
\!\stringdiagram{
\wire (0,-1)--(-7.5,-1);
\wire[0.5] (0,-2.5)--(-0.5,-2.5)--(-1,-2)--(-2,-2);
\stoch{-2,-1.5}{$f'$}{2}
\wire[0.5] (0,-4)--(-3,-4)--(-4,-2)--(-5.5,-2);
\stoch{-5.5,-1.5}{$f$}{2}
\rlabel{0,-2.5}{ S};
\rlabel{0,-4.2}{ T};
\rlabel{0,-0.7}{ \,\,\,H''};
\llabel{-7.5,-1}{ H};
\wire (0,1)--(7.5,1);
\wire[0.5] (0,2.5)--(0.5,2.5)--(1,2)--(2,2);
\sqbox{2,1.5}{$\gamma$}{2}
\wire[0.5] (0,4)--(3,4)--(4,2)--(5.5,2);
\sqbox{5.5,1.5}{$\gamma$}{2}
\llabel{0,2.5}{ T};
\llabel{0,4.2}{ S};
\llabel{0,0.7}{ \Theta};
\rlabel{7.5,1}{ \,\,\,\Theta''};
\right\rangle$.
old version of this section. Probably just as good in the end, and the same length, but doesn't introduce $\psi_H$ as a Markov kernel
§.§ Interpretations and reasoners
some or all of this subsection could be moved to intro?
We now describe our central concept: an interpretation of a machine.
Here we will present only two kinds of interpretations, Bayesian interpretations and Bayesian filtering interpretations, but we expect these to fit naturally into a much broader family of concepts.
The idea is that an interpretation is a function that maps the physical state of a machine to some other mathematical object, which can be thought of as a set of possible beliefs about some external world.
In the case of Bayesian interpretations and Bayesian filtering interpretations, this object is the set of probability measures over some hidden variable.
An interpretation will also generally be required to satisfy some kind of consistency requirement.
In the case of Bayesian and Bayesian filtering interpretations, this consistency requirement is given by Bayes' theorem, as we explain below.
As we will show, not every machine can be consistently interpreted in every possible way; the existence of a non-trivial consistent interpretation can put strong constraints on the possible dynamics that a machine can have.
Given a machine $\gamma$ and a consistent interpretation, we will refer to the two together as a reasoner.
The idea is that a machine by itself is merely a (possibly stochastic) dynamical process, but if a consistent interpretation exists then it is at least consistent to ascribe a meaning to its states.
Specifically, we can think of each state as corresponding to a probability distribution representing a subjective state of knowledge, and these distributions will update correctly as the machine receives new inputs.
As possible extensions of this idea, one can also imagine weaker kinds of consistency requirement, which might correspond to Jeffrey updating Jeffrey; see also Jacobs, or to approximate Bayes of some form, perhaps based on some form of the free energy principle.
One could also imagine applying these ideas to machines that have outputs as well as inputs, which might allow concepts such as goals, plans and actions to be formalised via the notion of interpretation, in addition to Bayesian beliefs.
However, we leave the development of these notions to future work, and focus here only on interpretations in terms of exact Bayesian inference and filtering.
It should be noted that, for a given machine, the question of whether it can be consistently interpreted in a particular way is in principle an empirical one, since it depends on the machine's update kernel, which can in principle be measured.
However, in general a given machine might have multiple non-equivalent consistent interpretations, and one cannot distinguish between these empirically by looking only at the system's internal dynamics.
[We leave open the possibility that they could be distinguished by looking at some broader context, e.g. by discovering that a device's designer intended a particular interpretation, or that evolution selected for a particular interpretation.]
Consequently the relationship between interpretations and the empirical, physical world is rather subtle, and one should keep in mind that our notion of “reasoner” unavoidably involves an element of choice in which interpretation to adopt.
I like the following paragraph but think it's a distraction
In string diagram notation, the update kernel can be written as $\!\!\!\stringdiagram{
\wire (0,1)--(4,1);
\wire[0.5] (0,3)--(0.5,3)--(1,2)--(2,2);
\stoch{2,1.5}{$\gamma$}{2}
\llabel{0,3}{ S};
\llabel{0,1}{ Y};
\rlabel{4,1}{ Y};
The fact that it is able to take sequences of inputs follows from its type: two copies of $\gamma$ can be composed in a particular way to form $\!\!\!\stringdiagram{
\wire (0,1)--(7.5,1);
\wire[0.5] (0,2.5)--(0.5,2.5)--(1,2)--(2,2);
\stoch{2,1.5}{$\gamma$}{2}
\wire[0.5] (0,4)--(3,4)--(4,2)--(5.5,2);
\stoch{5.5,1.5}{$\gamma$}{2}
\llabel{0,2.5}{ S};
\llabel{0,4.2}{ S};
\llabel{0,0.7}{ Y};
\rlabel{7.5,1}{ Y};
[-1cm]note to self: this is related to the Para construction, e.g. from “towards foundations of categorical cybernetics”, also a similar trick is used in “backprop as functor.” Not sure whether to cite these here as this is obviously way simpler.
This can be read as a kernel that takes a state $y\in Y$ and a sequence of two inputs in $S$, and returns another state in $Y$ stochastically, as a result of feeding the two inputs to the machine in sequence.
This generalises to sequences of any length.
(Some care must be taken over associativity but this can be done in a standard way.)
\begin{equation}
\label{jacobs-equation-b}
\stringdiagram {
\wire (0,0) -- (9,0);
\ulabel{0,0}{Y};
\ulabel{9,0}{H};
\ulabel{3.5,0}{H};
\blackdot{4,0};
\wire[2] (4,0) -- (4,2) -- (9,2);
\ulabel{9,2}{S};
\stoch{2,0}{$\psi$}{3};
\stoch{6.5,2}{$\phi$}{3};
% \showgrid
\, = \!
\pbox{
\stringdiagram {
\wire (0,0) -- (14,0);
\ulabel{0,0}{Y};
\ulabel{14,0}{H};
\blackdot{1,0};
\wire[2] (1,0) -- (1,2) -- (14,2);
\ulabel{14,2}{S};
\stoch{3,2}{$\psi$}{3};
\ulabel{4.25,2}{H};
\stoch{5.5,2}{$\phi$}{3};
\sqbox{9,0}{$\gamma$}{3};
\blackdot{7,2};
\ulabel{7.5,2}{S};
\wire[1.5] (7,2) -- (7,0.5) -- (9,0.5);
\ulabel{10.3,0}{Y};
\stoch{11.5,0}{$\psi$}{3};
% \showgrid
\end{equation}
§ TECHNICAL PRELIMINARIES
I bet I can move most of this section to the SI once I've got the rest written.
In order to fill in some of the details sketched above, we introduce some concepts and notation that has been developed in the context of category theory
Fritz, Jacobs, ….
We present here only an informal sketch of the ideas. An introduction to category theory and the string diagram notation can be found in a short but accessible form in Baez and Stay or in a longer text-book format in Seven Sketches.
For a comprehensive technical exposition of their application to probability theory via so-called Markov categories, see Fritz, maybe also the representable Markov cats paper.
The framework has been used previously in a context related to Bayesian updates and the free energy principle Toby.
The central concepts are that of measurable space and of Markov kernel.
A measurable space is a set equipped with a $\sigma$-algebra.
The basic idea is that a measurable space is the kind of thing on which one can define a probability measure, but unlike a probability space it does not have any measure defined on it by default.
Often, but not always, we will take our measurable spaces to be finite sets, in which case we always take their $\sigma$-algebras to be their power sets.
In the supplementary information we give a formal definition of Markov kernels , along with the category-theoretic calculus we use to reason about them, aimed at readers without a category theory background.
However the basic idea is that for measurable spaces $X$ and $Y$, a Markov kernel $\kappa$ with source $X$ and target $Y$ is a function that maps elements of $X$ to probability measures over $Y$ (along with a technical requirement that this function be measurable in a certain sense).
Following Toby we write such a kernel as $\kappa\colon X\todot Y$, using the symbol $\todot$ to distinguish Markov kernels from ordinary functions.
Sometimes we will think of a Markov kernel as being an ordinary deterministic function that takes an element of $X$ and returns a probability distribution over $Y$.
However, we will also sometimes think of it as a stochastic function, which takes an element $x\in X$ and returns a randomly chosen element of $Y$, sampled according to a probability measure that might depend on $x$.
(These two interpretations can be distinguished formally using the machinery of representable Markov cats paper, but we will not make use of that here.)
We write $\kappa(y\mg x)$ to denote the probability that the kernel $\kappa$ assigns to the outcome $y\in Y$ when given the input $x\in X$.
We use a thick vertical line to indicate a close relationship to conditional probability while also emphasising that the concept is different:
given a kernel $\kappa\colon X\todot Y$ the quantities $\kappa(y\mg x)$ are always defined, regardless of whether any probability distribution has been defined over $X$, and regardless of whether $x$ has a nonzero probability according to such a distribution.
(In fact, with a little care this same notation can be used even in the general measure-theoretic case, where $X$ and $Y$ are not necessarily finite Fritz, section 2.8.
In this case $x$ and $y$ are interpreted as abstract indices rather than elements of $X$ and $Y$.)
two short paragraphs that mention composition and joint distributions, referring to the SI for more details
I'm entering some introductory stuff about string diagrams here, just to try and see if this will work. Nothing in this document yet is final - I'm still unsure of the form the paper will take.
my conclusion: this might have to go in the supplementary material - it could be made a bit shorter but I think most of the points here have to be said in one form or another for it to make sense. Or we just give references and don't explain it.
A measurable space is a set equipped with a $\sigma$-algebra, or formally a tuple $(X,\Sigma)$, where $X$ is a set and $\Sigma\subseteq \mathcal{P}(X)$ is a set of subsets of $X$ that obeys the axioms of a $\sigma$-algebra.
Often, but not always, our measurable spaces will be finite sets, in which case we will always assume $\Sigma=\mathcal{P}(X)$.
Given two measurable spaces $X$ and $Y$, a Markov kernel, which we write $f\colon X\todot Y$, is a measurable function $f\colon X\to P(Y)$, where $P(Y)$ is the space of probability distributions over $Y$, which is itself a measurable space.
We first introduce the framework in the case of finite probability, and then outline how it generalises to a more general measure-theoretic context that includes continuous probability.
We will make use of the more general version later on.
For now, let $X, Y, Z\dots$ represent finite sets. Given a finite set $X$, we write $P(X)$ for the set of all probability distributions over $X$.
In this finite case, $P(X)$ may be thought of as a $(|X|-1)$-dimensional simplex, given by the set of vectors in $\mathbb{R}^{|X|}$ whose components are nonnegative and sum to 1.
Given finite sets $X$ and $Y$, a Markov kernel $f\colon X\todot Y$ is defined as a function $f\colon X\to P(Y)$.
The kernel $f$ can be thought of as a set of probability distributions over $Y$, parametrised by elements of $X$.
Alternatively, it may be thought of as a stochastic function, that is, something that takes an input in the underlying set of $X$ and then produces an output in $Y$, chosen randomly according to a distribution that depends on $X$.
(A formal distinction between these interpretations can be made using the machinery of representable Markov cats, but we will not make use of that here.)
In the finite case $f$ may also be thought of as a stochastic matrix (i.e. one whose entries are nonnegative and whose columns sum to 1), representing a linear map that maps distributions over $X$ to distributions over $Y$.
Following Toby we use the symbol $\todot$ to distinguish Markov kernels from ordinary functions.
We write $f(y\mg x)$ to denote the probability that the kernel $f$ assigns to the outcome $y\in Y$ when given the input $x\in X$.
We use a thick vertical line to indicate a close relationship to conditional probability while also emphasising that the concept is different:
given a kernel $f\colon X\todot Y$ the quantities $f(y\mg x)$ are always defined, regardless of whether any probability distribution has been defined over $X$, and regardless of whether $x$ has a nonzero probability according to such a distribution.
Other common notations include $|$ or $;$ in place of $\mg$.
Given Markov kernels $f\colon X\todot Y$ and $g\colon Y\todot Z$, we can compose them to form a new kernel of type $X\todot Z$, which we write $f\comp g$, given by
\begin{equation}
(f\comp g)(z\mg x) = \sum_{y\in Y} f(x\mg y)\,g(y\mg z),
\end{equation}
which is a discrete version of the Chapman-Kolmogorov equation.
There are two important special cases of Markov kernels that have special notation.
The first is that for every finite set $X$ there is a kernel $\id_X\colon X\to X$ corresponding to the identity matrix, which simply returns
The second important special case is a Markov kernel $p\colon \one \to X$, where $\one = \{\star\}$ is a set with a single element.
(The identity of the element does not matter.)
In this case the kernel $p$ can be defined by choosing a single probability distribution over $X$, which is the kernel's output when given its only possible input.
In fact we think of distributions over $X$ and kernels of type $\one\to X$ as the same thing, and we write $p(x)$ rather than $p(x\mg \star)$.
(Note that the symbol $p$ here refers to the name of the kernel, rather than being a generic symbol for probability.)
We also use a notation called string diagrams for Markov kernels. String diagrams can be applied to a much wider range of mathematical topics than probability
(Baez+Stay is a good overview of some other applications).
Here we present them only in the context of Markov kernels, as used by Cho+Jacobs, Fritz.
We choose to orient our diagrams horizontally and generally write kernels with curved edges.
In general, in our context, every string diagram represents a Markov kernel, which might be built up out of other Markov kernels.
We denote a kernel $f\colon X\todot Y$ by the diagram $\!\!\!\stringdiagram{
\wire (0,1)--(4,1);
\stoch{2,1}{$f$}{2}
\llabel{0,1}{X};
\rlabel{4,1}{Y};
Each wire represents a finite set. Composition via the Chapman-Kolmogorov equation is written as
\begin{equation}
\stringdiagram {
\wire (0,0) -- (4,0);
\stoch{2,0}{$f\comp g$}{3};
\ulabel{0,0}{A};
\ulabel{4,0}{C};
\,=\,
\pbox{\stringdiagram {
\wire (0,0) -- (7,0);
\stoch{2,0}{$f$}{3};
\ulabel{0,0}{A};
\ulabel{3.5,0}{B};
\stoch{5,0}{$g$}{3};
\ulabel{7,0}{C};
\end{equation}
Here each side of the equation represents a Markov kernel and we are asserting that the two kernels are equal.
For this to make sense, the types have to match, meaning that the “dangling wires” on each side of the two diagrams must match.
As a special case, we write a one-element set as no wire, so that a probability distribution over $X$ (that is, a kernel $p\colon \one\to X$) is written simply as $\!\stringdiagram{
\wire (2,1)--(4,1);
\stoch{2,1}{$p$}{2}
\rlabel{4,1}{X};
String diagrams come into their own when we consider joint distributions.
For example, we might want to consider a function $h\colon A\times B\to P(C\times D)$, which we write as a Markov kernel $h\colon A\otimes B\todot C\otimes D$.
Here $A\otimes B$ and $B\otimes C$ represent the spaces of probability distributions over $A\times B$ and $C\times D$.
We write such spaces of joint distributions as pairs of parallel wires, so that the kernel
Two central concepts are measurable spaces and Markov kernels. A measurable space is a set equipped with a $\sigma$-algebra. This is a similar concept to a measure space or a probability space, but without a specified measure defined on it. Given a measurable space $X$, we write $P(X)$ for the set of all probability measures over $X$. In fact $P(X)$ itself can be made into a measurable space. We will often consider the case where a measurable space is a finite set, in which case we will always assume its associated $\sigma$-algebra is its power set.
Given measurable spaces $X$ and $Y$, a Markov kernel $f\colon X\todot Y$ can be defined as a measurable function $f\colon X\to P(Y)$. That is, $f$ can be thought of as a set of probability distributions over $Y$, parametrised by elements of $X$. Alternatively, it may be thought of as a stochastic function, that is, something that takes an input in the underlying set of $X$ and then produces an output in $Y$, chosen randomly according to a distribution that depends on $X$. Following Toby we use the symbol $\todot$ to distinguish Markov kernels from ordinary functions.
equations for the DE machine version
I am thinking there should be an introduction to Jacobs' version of conjugate priors before this. It would basically note that the update function already looks kind of like a machine, but it would note that there's a couple of issues: first we need the update function to be deterministic otherwise it doesn't work (for the reasons we discussed), and second Jacobs' equation says nothing about what the machine should do in the case of subjectively impossible inputs. The text would explain roughly why that's a problem. So we motivate the definitions below by saying they exist to solve those issues.
Given a measurable space $S$, an $S$-machine is a tuple $(M,\mu)$, consisting of a measurable space $M$, together with a Markov kernel of type $\mu\colon M\otimes S\todot M$. The set $M$ is called the state space of the machine, and $\mu$ is called its update kernel.
We can write the update kernel in string diagrams as $\!\!\!\stringdiagram{
\wire (0,1)--(4,1);
\wire[0.5] (0,3)--(0.5,3)--(1,2)--(2,2);
\stoch{2,1.5}{$\mu$}{2}
\llabel{0,3}{ S};
\llabel{0,1}{ M};
\rlabel{4,1}{ M};
Often we will refer to a machine $(M,\mu)$ by its update kernel $\mu$, leaving the state space implicit.
The intuition is that a machine can start in a state $m\in M$ and receive an input $s\in S$, and as a result change its state to some other $m'\in M$.
This next state may be chosen stochastically, and in general will depend on both the previous state and the input.
The process of receiving inputs can be iterated.
For example, the following diagram represents a kernel of type $M\otimes(S\otimes S\otimes S)\todot M$ formed by composing three copies of $\mu$.
It can be thought of as receiving a string of three inputs and outputting the resulting state.
can restructure to remove this diagram if we need to save space
\begin{equation*}
\stringdiagram{
\wire (0,1)--(10,1);
\wire[0.5] (0,3)--(0.5,3)--(1,2)--(2,2);
\stoch{2,1.5}{$\mu$}{3}
\wire[1] (0,4) -- (2,4) --(4,2)--(5,2);
\stoch{5,1.5}{$\mu$}{3}
\wire[1] (0,5) -- (4,5) --(7,2)--(8,2);
\stoch{8,1.5}{$\mu$}{3}
\llabel{0,3}{S};
\llabel{0,4}{S};
\llabel{0,5}{S};
\llabel{0,1}{M};
\rlabel{10,1}{ M};
\dlabel{3.5,1}{ M};
\dlabel{6.5,1}{ M};
\end{equation*}
It is worth noting that the notion of a machine plays two roles in our work: sometimes a machine will represent a physical system, whose state space and input space are interpreted physically, with its update function modelling its true physical dynamics.
However, machines will also sometimes play a more abstract role in our definitions, and not every machine we define will necessarily exist as a physical system.
Given an $S$-machine $(M,\mu)$ and a $T$-machine $(N,\nu)$, a map of machines $f\colon \mu\to\nu$ consists of two deterministic kernels, $f_\textnormal{state}\colon M\to N$ and $f_\textnormal{input}\colon S\to T$, such that
\begin{equation}
\stringdiagram{
\wire (-4,1)--(4,1);
\wire (-4,3)--(0,3);
\wire[0.5] (0,3)--(0.5,3)--(1,2)--(2,2);
\stoch{2,1.5}{$\nu$}{3}
\sqbox{-1.6,0.8}{$f_\textnormal{state}$}{3}
\sqbox{-1.6,3.3}{$f_\textnormal{input}$}{3}
\llabel{-4,3}{S};
\llabel{-4,1}{M};
\dlabel{0.4,1}{N};
\ulabel{0.4,3}{T};
\rlabel{4,1}{N};
\pbox{\stringdiagram{
\wire (0,1)--(7.25,1);
\wire[0.5] (0,3)--(0.5,3)--(1,2)--(2,2);
\stoch{2,1.5}{$\mu$}{3}
\sqbox{5,1.5}{$f_\textnormal{state}$}{3}
\llabel{0,3}{S};
\llabel{0,1}{M};
\dlabel{3.25,1}{M};
\rlabel{7.25,1}{N};
\end{equation}
If both of the functions $f_\text{state}$ and $f_\text{input}$ are surjective we say $f$ is a surjective map of machines.
Similarly, if both of the underlying functions $f_\text{state}$ and $f_\text{input}$ are injective we say $f$ is an injective map of machines.
if needed, we can define “isomorphism,” “isomorphic,” and “identity-on-$S$ map” here as well.
To get an intuition for the concept it is helpful to consider the surjective case first.
In this case, the functions $f_\text{input}$ and $f_\text{state}$ can be seen as coarse-graining the input space and the state space, in such a way that the dynamics are preserved.
That is, coarse-graining the state and input followed by an update of $\nu$ must be the same as first performing an update of $\mu$ and then coarse-graining the state space.
This is a similar concept to so-called lumpability of Markov processes.
The definition also demands that $\nu$ itself be a machine, which is an extra condition not implied by the lumpability condition itself.
(See the paper Martin recommended for an explanation of this point, in relation to Markov processes rather than machines.)
[-2cm]to discuss with Martin: what's the one-sentence way to correctly explain what this definition amounts to, in terms of that paper?
This intuition is probably helpful but can be removed to save space (except for the last sentence of this paragraph)
We can also consider the case of an injective map of machines $f\colon \mu\to \nu$.
In this case we can think of $\mu$ as a kind of “sub-machine” of $\nu$, in that the states in $M$ correspond to a subset of the states in $N$, and similarly $S$ corresponds to a subset of $T$.
The condition in <ref> ensures that these sets embed into $\nu$ in such a way that the dynamics are preserved, meaning in this case that if $\nu$ starts in a state corresponding to a state in $M$, and if it only receives inputs corresponding to inputs in $S$, then it will behave exactly like the machine $\mu$.
In general a map of machines need not be surjective or injective, meaning that some states in $M$ and $S$ may be coarse-grained, while some states in $T$ and $N$ might have no states that map to them.
Martin, feel free to ignore anything below this point - you can write the into however you like, it doesn't have to join up to this.
Although ultimately our goal is to understand agents in a much broader sense, we restrict ourselves here to the following very simple scenario, which is a generalisation of the one considered in Biehl and Kanai.
We consider a physical system that receives a stream of inputs.
The system is modelled as a Markov process that takes inputsis there a standard term for “Markov process that takes inputs”?, meaning that at each time step its next state is determined stochastically as a function of both its previous state and the input received.
The question we ask is, given such a system, can that system be consistently interpreted as performing exact Bayesian inference about some hidden variable?, and if it can, how many such interpretations are possible?
The latter question might at first sound surprising, but we will show in a proposition that if one non-trivial such interpretation exists then so do an infinite set of alternatives.
We will show that the requirement of consistency puts fairly strong constraints on the physical systems that can be interpreted as performing inference.
It should be noted however, that the condition of being able to perform exact inference on arbitrarily long sequences of inputs is quite a restrictive one, and one could certainly consider weaker forms of interpretation.
We focus on exact inference as a kind of case study, in order to elucidate the relationship of mutual constraint between the dynamical properties of a physical system and its interpretations as an agent performing inference.
By an “interpretation,” in this particular restricted case, we mean a function that maps the system's internal state to a space of probability distributions, which we think of as representing the agent's subjective beliefs about some hidden variable $H$.
The hidden variable may or may not in fact correspond to something that exists in the physical world:
we do not demand that the agent's beliefs be true, but only that they be consistent.
I'm removing the original content of the template, but it can be found at the bottom of the .tex file, after the end document command.
After quite a bit of work I was able to fit Simon's equation on a single line — see below. Hopefully future equations will be less work, but at least currently I have to manually position each element of the diagram, so it takes a while to enter them and get them to look nice.
Inline string diagrams are possible but it's not really feasible to convey much information. For example, we can say $\!\!\!\stringdiagram{
\wire (0,1)--(4,1);
\stoch{2,1}{$\iota$}{2}
\llabel{0,1}{M};
\rlabel{4,1}{S};
} = \!\!\!\stringdiagram{
\wire (0,1)--(6,1);
\stoch{1.75,1}{$\psi$}{2}
\stoch{4.25,1}{$\phi$}{2}
\llabel{0,1}{M};
\rlabel{6,1}{S};
but there isn't really even enough space to fit an $H$ label on the intermediate wire, without increasing the height of the equation above the height of a normal line. So probably we won't want to use inline diagrams in this paper. (It works better if you only have dots and wires.)
We require the interpretation map $M\xklto{\psi} H$ and the agent's model $H\xklto{\phi} S$ to obey the following consistency equation:
\begin{equation}
\stringdiagram {
\wire (0,0) -- (11.5,0);
\ulabel{0,0}{M};
\ulabel{11.5,0}{M};
\blackdot{1,0};
\wire[2] (1,0) -- (1,2) -- (11.5,2);
\ulabel{11.5,2}{S};
\stoch{3,2}{$\psi$}{3};
\stoch{6,2}{$\phi$}{3};
\blackdot{4.5,2};
\ddlabel{4.5,2}{H};
\stoch{9.5,0.25}{$\mu$}{3};
\blackdot{7.5,2};
\ulabel{8,2}{S};
\wire[1.5] (7.5,2) -- (7.5,0.5) -- (9.5,0.5);
\wire[2] (4.5,2) -- (4.5,4) -- (11.5,4);
\ulabel{11.5,4}{H};
% \showgrid
\, = \!
\pbox{
\stringdiagram {
\wire (0,0) -- (15,0);
\ulabel{0,0}{M};
\ulabel{15,0}{M};
\blackdot{1,0};
\wire[2] (1,0) -- (1,2) -- (15,2);
\ulabel{15,2}{S};
\stoch{3,2}{$\psi$}{3};
\ulabel{4.25,2}{H};
\stoch{5.5,2}{$\phi$}{3};
\stoch{9,0.25}{$\mu$}{3};
\blackdot{7,2};
\ulabel{7.5,2}{S};
\wire[1.5] (7,2) -- (7,0.5) -- (9,0.5);
\wire[4] (10.5,0) -- (10.5,4) -- (15,4);
\blackdot{10.5,0};
\ulabel{11.3,0}{M};
\stoch{13.2,4}{$\psi$}{3};
\ulabel{15,4}{H};
% \showgrid
\end{equation}
This says that …
The following is copied-and-pasted from elsewhere - the notation will need to be changed (the rest to match this, or this to match the rest).
We build on the formal definition of conjugate priors given by Jacobs, which we sketch here, using the language of string diagrams.
In this graphical notation, we write a Markov kernel $f\colon X\todot Y$ as
\begin{equation}
\pbox{\stringdiagram{
\wire(0,0) -- (4,0);
\stoch{2,0}{$f$}{3};
\ulabel{0,0}{X};
\ulabel{4,0}{Y};
\end{equation}
This should be thought of as a function that maps elements $x\in X$ to probability distributions over $Y$.
We can use Markov kernels to model either physical processes (in which case we interpret $X$ as the space of possible inputs and $Y$ as the space of possible outputs) or statistical models, in which case we regard $X$ as the set of possible parameter values and $Y$ as the sample space.
The Markov category framework provides various ways to compose Markov kernels together, either in series (which can be thought of as feeding the output of one process into the input of another), or in parallel, which can be thought of in terms of two processes operating on separate variables independently.
Jacobs' definition of a conjugate prior may be stated within this framework.
Let $q\colon\Theta\todot X$ be a statistical model with parameter space $\Theta$ and sample space $X$.
We say that $q$ admits a conjugate prior if there exists $(i)$ another statistical model $p\colon\Psi\to\Theta$, whose sample space is $\Theta$, and a deterministic function $h\colon \Psi\times X\to \Psi$, such thatThis equation can be made vertically smaller by laying it out in the style of <ref>. Then it will only be 2 units high, not 3.
\begin{equation}
\label{conjugate-prior-def}
\stringdiagram{
\wire(0,0) -- (3.5,0);
\stoch{2,0}{$p$}{3};
\ulabel{0,0}{\Psi};
\rlabel{3.8,0}{\Theta};
\blackdot{3.5,0};
\wire[2.5] (3.5,0) -- (3.5,2.5) -- (7.5,2.5);
\ulabel{7.5,2.5}{X};
\stoch{5.75,2.5}{$q$}{3};
\wire[2.5] (3.5,0) -- (3.5,-2.5) -- (7.5,-2.5);
\ulabel{7.5,-2.5}{\Theta};
\pbox{
\stringdiagram{
\wire(-3,0) -- (3.5,0);
\stoch{-0.3,0}{$p$}{3};
\ulabel{0.9,0}{\Theta};
\stoch{2.1,0}{$q$}{3};
\ulabel{-3,0}{\Psi};
% \uulabel{-2,0.1}{\Psi};
\blackdot{-2,0};
\wire[2.5] (-2,0) -- (-2,-2.8) -- (5.75,-2.8);
\rlabel{3.8,0}{X};
\blackdot{3.5,0};
\wire[2.5] (3.5,0) -- (3.5,2.5) -- (10,2.5);
\ulabel{10,2.5}{X};
\wire[2.2] (3.5,0) -- (3.5,-2.2) -- (5.75,-2.2);
\wire(5.75,-2.5) -- (10,-2.5);
\sqbox{5.75,-2.5}{$h$}{3};
\ulabel{7,-2.5}{\Psi};
\stoch{8.2,-2.5}{$p$}{3};
\ulabel{10,-2.5}{\Theta};
\end{equation}
The parameters $\psi\in\Psi$ of $p$ are called the hyperparameters.
The function $h$ takes the value of the hyperparameters along with a sample from $X$ and returns the updated value of the hyperparameters.
The equation expressed in string diagrams ensures that the behaviour of these functions is consistent with Bayesian inference.
The model of Biehl and Kanai may be expressed in this framework by interpreting Equation <ref> differently.
We regard $\Psi$ as the agent's physical state and $h$ as the function that updates it according to its observations $X$.
We interpret $q$ as the agent's model of $X$, and $p$ as the interpretation map, sending its internal state to a distribution over model parameters.
Only $\Psi$, $h$ and the observations $X$ are part of the physical system.
The other Markov kernels constitute the interpretation map, with $\Theta$ being a hidden variable in the agent's assumed model.
Seen this way the equation above can be seen as the defining equation for an interpretation map, for an agent that performs exact inference.
§ OLD INTRO
A question of general interest that also plays a central role in the Free Energy Principle (FEP) is which systems are performing Bayesian inference.
More precisely, what are the necessary and sufficient conditions under which it is justified to say that a system performs Bayesian inference.
This question is of general interest by itself because Bayesian inference is in some sense the optimal way to reason about uncertain events cox which makes systems (including biological ones) performing Bayesian inference special bayes in biology.
This question is also of interest as part of the broader question of which systems are intelligent since Bayesian inference plays an important role in current theories of intelligence and rationality.
The Free Energy Principle itself may be seen as a theory of intelligence that relies on Bayesian inference but it is not the only one <cit.>.
We now give some informal motivation for our proposal to answer this question. Readers mainly interested in formal development may skip ahead to refer.
§.§ Bayesian inference
Briefly, Bayesian inference is a prescription for how to update probabilistic beliefs about hypothetical causes of observations in response to such observations.
The set of hypothetical causes is represented by a family of distributions or a (statistical) model which we will informally write as $\phi(s \mg h)$.
Here $h \in H$ are hypotheses or parameters in parameter space $H$ and $s \in S$ are observations in observation space $S$.
Note that the parameterized model is often called the likelihood function and written in the style of a conditional probability $p(s|h)$ and sometimes as $p(s;h)$.
In order to keep the exposition in this introduction accessible we avoid measure theoretic notions and consider probabilistic beliefs as probability mass functions in the discrete case and probability density functions in the continuous case.
When there is no need to distinguish them we will refer to both as probability distributions.
Bayesian inference then takes a probability distribution (the prior belief) over $H$ and an observation $s \in S$ and transforms them into a new probability distribution (the posterior belief) over $H$.
This is a new attempt. Make Bayesian inference about updating probability distributions instead of measures. Then the things that are updated by the well known form of Bayes rule are not necessarily parameterizations. The other way would be to use probability measures and Bayes rule for probability measures, but only really technical people would get that...
Importantly, this transformation (and therefore Bayesian inference) is only well defined if we specify the model $\phi(s \mg h)$ whose parameters the belief ranges over.
A complicating issue is that even if a model is defined, Bayesian inference may not specify a posterior belief for every prior belief.
We will discuss this issue in the technical part refer but in this introduction we only consider the case where every prior is transformed to a unique posterior for every observation.
Next we want to consider what it could mean that a system performs Bayesian inference.
To some degree this depends on what we consider systems to be.
So we discuss this first.
§.§ Systems
For the purpose of this paper a system is a triple $\langle Y,S,\gamma\rangle$ consisting of a state space $Y$, an input space $S$, and transformation $\gamma$ that takes a state and an input and transforms them into a new state.
In this introduction we will assume that $\gamma$ is deterministic but the formal treatment works for stochastic systems as well.
Many examples of such systems are known to exist (in the mathematical sense) because they are formally defined, i.e. state space, input space, and transformation are all given in some formal language (e.g. ZFC set theory).
I am not sure that “ZFC set theory” is really a possible formal language in which all of these things can be unambiguously defined, I am pretty sure in some way it is possible.
For example, let $Y=\{0,1,2\}$, $S=\{0,1\}$, and $\gamma(y,s) \coloneqq y+s \mod 3$.
We assume that there are also parts of the physical world that are systems of this kind.
These are what we refer to as physical systems.
For example the state space may be the possible configurations of a set of degrees of freedom of some physical matter and the input space may be the set of possible configurations of a disjoint set of degrees of freedom.
[There may be other ways in which parts of the physical world forms systems of the kind we study here.]
All degrees of freedom evolve under the laws of physics with the time evolution of the state space degrees of freedom possibly depending on the input degrees of freedom.
The input dependent time evolution of the state space degrees of freedom then induces and thereby defines the transformation of such a physical system.
An important fact about systems in general is that even though two systems may have different state spaces and input spaces (which also means their transformations are different) they can still be isomorphic.
Intuitively this just means that they “behave” in the same way.
This will be explained below refer.
A simple example of a system that is isomorphic to the example above is $Y=\{0,-1,-2\}$, $S=\{0,-1\}$, and $\gamma(y,s) \coloneqq -y-s \mod 3$.
Note that among the formally defined systems there are some that were constructed specifically to model physical systems.
Note also that for some physical systems an isomorphic formal system may, by accident or due to somebodies skill, have already been defined.
With a better idea of what we mean by systems we now discuss which ones should count as performing Bayesian inference.
§.§ Systems that should count as performing Bayesian inference
As mentioned before belief updating according to Bayesian inference is only defined with respect to a model.
This means that the question of whether a system performs Bayesian inference is formulated more precisely as whether there exists a model with respect to which a system is performing Bayesian inference.
One thing we require from the model is that its observation space is equal to the system's input space.
The question is then when a system performs Bayesian inference with respect to a model of its inputs.
Note that it is straightforward to (formally) construct a system that performs Bayesian inference with respect to a model of its inputs.
At the very least it should be justified to say that such a system performs Bayesian inference.
To construct such a system we start by defining a model $\phi(s\mg h)$ together with its parameter space $H$ and observation space $S$.
The input space of the constructed system is defined as the observation space of the model.
We can then construct the space of all probability distributions (beliefs) over parameter space $H$.
[To make sure all posteriors are well defined we only allow beliefs that assign positive probability to all parameters.]
This forms the states space $Y$ of the constructed system.
Any pair $(y,s)$ of state and input is now also a pair of a probability distribution over parameters of a model (a possible prior) and an observation such that we can use Bayes rule to obtain a new probability distribution $\gamma(y,s)$, the posterior.
This completes the construction of a system that performs Bayesian inference.
For example we can choose a model for binary observations $S \coloneqq \{0,1\}$ (or equivalently $\{\text{heads},\text{tails}\}$) the family of probability distributions $\phi(s \mg h)$
\begin{align}
\phi(s \mg h) \coloneqq h^{\delta_0(s)} (1-h)^{\delta_1(s)}
\end{align}
where $h \in H$ and $H \coloneqq [0,1]$.
This is equivalent to considering $H$ as a set of hypothetical biases that could be causing the outcomes of a coin flip.
For the state space $Y$ we can choose the set of positive probability density functions over $H$.
We can then define the transformation $\gamma(y,s)$ via Bayes rule:
\begin{align}
\gamma(y,s)(h) \coloneqq \frac{\phi(s \mg h)y(h)}{\int \phi(s \mg \bar{h})y(\bar{h}) d\bar{h}}.
\end{align}
The result of $\gamma(y,s)$ is again a positive probability density function over $H$ and therefore an element of $Y$ for all $(y,s) \in Y\times S$.$\gamma$ isn't a Markov kernel in this case, for that we would need a distribution over $Y$ on the r.h.s. not just an element of $Y$. This could be done using a Dirac delta, but it would be nice to avoid it. Maybe it needs a short note.
However, apart from systems directly constructed to perfrom Bayesian inference there are three other classes of systems that we also consider to be performing Bayesian inference.
* Systems that implement Bayesian inference in a (hyper-) parameter space.
It is well known that certain statistical models have associated parameterized conjugate priors.
In this case Bayesian inference can be, and in practice often is, implemented completely in some parameter space.
This parameter space need not be isomorphic to any space of probability distributions and is often a low dimensional real space $\mathbb{R}^n$.
Maybe point to, or quickly describe an example with conjugate prior here.
Let us explain a bit more how parameterized conjugate priors allow Bayesian inference in parameter space.
The first thing to note is that for some models there are subsets of probability distributions (beliefs) over the model parameters that are closed under Bayesian updates.
This just means that if we start with a prior measure from such a subset then the posterior measure for any observation will always be in this subset again.
Often these subsets of probability distributions can be conveniently parameterized.
A set $C$ of probability distributions is parameterized by a set $D$ (not necessarily containing probability distributions and often some $\mathbb{R}^n$) if there is a bijective function $f:D \to C$.
Note that parameters of beliefs are sometimes called hyperparameters to distinguish them from the model parameters.
Let's assume we have a model that has such a closed subset $C$ of probability distributions over its parameters and is parameterized by some set $D$ of hyperparameters.
Let $f(d) \in C$ denote the probability distribution associated to hyperparameter $d \in D$ and write $f(d)(h)$ for the probability assigned to parameter/hypothesis $h \in H$ by $f(d)$.
If we then pick an arbitrary observation $s \in S$ and hyperparameter $d \in D$ we can take the belief $f(d)$ it parameterized and use Bayes rule to update it with respect to the observation $s$ to get a posterior belief $p \in C$ which is also in $C$ since $C$ is closed:
\begin{align}
\label{eq:hyperbayes}
p(h)=\frac{\phi(s \mg h)f(d)(h)}{\int \phi(s \mg \bar{h})f(d)(\bar{h}) d\bar{h}}.
\end{align}
Since $f$ is bijective we can then use its inverse to find the hyperparameter $d' \coloneqq f^{-1}(p) \in D$ of the posterior.
We can therefore define a function $\alpha(d,s)$ mapping the hyperparameter of any prior belief $f(d) \in C$ and observations $s \in S$ to the hyperparameter $d'$ of the posterior belief $f(d')=p$.
\begin{align}
\alpha(d,s) \coloneqq f^{-1}\left(\frac{\phi(s \mg h)f(d)(h)}{\int \phi(s \mg \bar{h})f(d)(\bar{h}) d\bar{h}}\right).
\end{align}
This is what we mean by implementing Bayesian inference in a (hyper)parameter space.
Note that $\alpha(d,s)$ is then also the transformation of a system whose state space is $D$ and input space is $S$.
Systems of this kind should also be considered as systems performing Bayesian inference.
* Systems where only a part performs Bayesian inference.
For example if one component of the system's state space are parameters whose dynamics implement Bayesian inference with respect to the input but another component of the system does something else. Example?
* Systems that are isomorphic to a system that performs Bayesian inference.
Note that this means that if some formal system performs Bayesian inference and is isomorphic to some physical system then we can also say that the physical system performs Bayesian inference.
The necessary and sufficient condition that we propose to identify systems that perform Bayesian inference is satsified by these three classes of systems.I guess ideally we would also know that it is satisfied by only those classes, but I don't think we can or necessarily need to. Next we give the main idea behind that condition.
§.§ Condition for systems to perform Bayesian inference
We now introduce the main idea behind the proposed formal condition (see refer) under which a system performs Bayesian inference with respect to given model.
This also leads to a condition under which a system performs Bayesian inference in general.
The condition is in form of a consistency requirement or equation that has to be satisfied between the system's transformation and Bayesian belief updating with respect to the model.
Recall that in the case of parameterized conjugate priors there was a bijective function $f$ that mapped hyperparameters to beliefs about the model parameters $H$.
In that case we constructed the transformation of the system that implemented Bayesian inference by using Bayes rule <ref>.
In the case of a given system $\langle Y,S,\gamma\rangle$ we require the existence of a (not necessarily bijective) function, which we call the interpretation map $g$, from system states $y \in Y$ to beliefs $g(y)$ about the model parameters $h \in H$ that is “consistent with Bayes rule”.
This means, the belief $g(\gamma(y,s))$ associated to the new state $\gamma(y,s)$ must be the posterior belief obtained by considering belief $g(y)$ associated to old state $y$ as the prior belief, input $s$ as the observation, $\phi(s \mg h)$ as the model and applying Bayes rule.
More formally, a (deterministic) system $\langle Y,S,\gamma\rangle$, performs Bayesian inference with respect to model $\phi(s \mg h)$ if and only if there exists a function $g$ such that for all $y \in Y,s \in S$:
\begin{align}
g(\gamma(y,s))(h) = \frac{\phi(s \mg h) g(y)(h)}{\int \phi(s \mg \bar{h}) g(y)(\bar{h}) d\bar{h}}
\end{align}
The dynamics of hyperparameters of conjugate priors obey this condition with the interpretation function being the bijective function $f$.
Systems where a only part of the system state is involved in Bayesian inference also satisfy this condition since the interpretation function can just ignore the rest of the system state.
Note that the condition is also satisfied when the state space consists of probability distributions directly and the dynamics of the system are identical to Bayesian inference.
In that case a possible interpretation function is just the identity.
Finally, if two systems are isomorphic and the first of them obeys the condition, the second does so as well.
In this case we can construct the interpretation map of the second system by composing the bijective function that exists between isomorphic systems with the interpretation function of the first system.
To be continued...
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# UnScenE: Toward Unsupervised Scenario Extraction for Automated Driving
Systems from Urban Naturalistic Road Traffic Data
Nico Weber1,2, Christoph Thiem1, and Ulrich Konigorski2 1 Opel Automobile
GmbH, Stellantis NV, 65423 Rüsselsheim am Main, Germany
<EMAIL_ADDRESS>Control Systems and Mechatronics
Laboratory, TU Darmstadt, 64283 Darmstadt, Germany
###### Abstract
Scenario-based testing is a promising approach to solve the challenge of
proving the safe behavior of vehicles equipped with automated driving systems
(ADS). Since an infinite number of concrete scenarios can theoretically occur
in real-world road traffic, the extraction of relevant scenarios that are
sensitive regarding the safety-related behavior of ADS-equipped vehicles is a
key aspect for the successful verification and validation of these systems.
Therefore, this paper provides a method for extracting multimodal urban
traffic scenarios from naturalistic road traffic data in an unsupervised
manner for minimizing the amount of (potentially biased) prior expert
knowledge needed. Rather than an (expensive) rule-based assignment by
extracting concrete scenarios into predefined functional scenarios, the
presented method deploys an unsupervised machine learning pipeline. It
includes principal feature analysis, feature extraction with so-called
scenario grids, dimensionality reduction by principal component analysis,
scenario clustering as well as cluster validation. The approach allows
exploring the unknown natures of the data and interpreting them as scenarios
that experts could not have anticipated. The method is demonstrated and
evaluated for naturalistic road traffic data at urban intersections from the
inD and the Silicon Valley dataset. The findings encourage the use of this
type of data as well as unsupervised machine learning approaches as important
pillar for a systematic construction of a relevant scenario database with
sufficient coverage for testing ADS.
## I INTRODUCTION
The maturity of technical implementations of automated driving systems (ADS)
[1] and resulting fields of application are continuously increasing. While
ADS-equipped vehicles promise to contribute significantly to a safer, more
efficient and more comfortable future mobility [2], the greatest challenge for
a market launch of such systems arises from the need for prior proof of safety
of the intended functionality [3] for future operation in real-world road
traffic [4], [5]. Regarding to an urban operational design domain (ODD),
possible increases in traffic safety are particularly relevant, as nearly 70%
of accidents involving personal injury in Germany do occur in urban areas [6]
(data collection period: 2016-2020).
Since existing safety validation approaches would require billions of test
kilometers under representative conditions before market launch [5], new
methods for the release of ADS-equipped vehicles are currently under
development. One of these methods is the scenario-based development and test
approach, as proposed by project PEGASUS [7]. Following the paradigm of this
approach, the majority of conventional driving kilometers is not challenging
enough and thus a reduction of the test effort is to be expected when testing
exclusively relevant scenarios [8]. In recent years, the focus of research
into safety validation approaches for ADS has primarily been set on highway
applications (e.g., a Highway Chauffeur) [7]. With the approval of the first
Level 3 ADS for traffic jam situations within highway environments by German
Federal Motor Transport Authority (KBA) by end of 2021 [9], series-production
ADS-equipped vehicles are expected to be introduced on public roads soon.
However, both the objective determination what relevant means with respect to
scenarios for testing ADS and a commonly accepted methodology for the
construction of a representative scenario database of sufficient coverage are
subject of ongoing research [2], [10]. The basic challenge here, independent
of the ODD, is the enormously high number of traffic situations that the ADS
has to cope with, referred to as open-context problem. The goal of a scenario
extraction method can be summarized as mapping of this infinite-dimensional
open context to a finite and manageable set of scenarios [11] that reflects
the nature of the traffic dynamics of interest in a sufficiently valid manner
for subsequent testing. Real-world data represents a valuable data source for
the construction of a comprehensive scenario database [2], [12]. While the
construction of such a database solely by expert knowledge already seems
extremely challenging for the highway environments, even when applying
sophisticated statistical approaches [13], it appears to be virtually
impossible for urban ODDs. This can be traced back to significant changes
regarding the PEGASUS six-layer model for structuring scenarios [14] for the
first four layers when transitioning to urban ODDs. The main reason for this
is a substantially increasing variability in terms of both traffic spaces and
traffic dynamics. With regard to traffic dynamics, the less rule-based
behavior and multimodal interaction of various road user (RU) types is to be
considered as crucial aspect.
Existing publications concerning data-driven scenario extraction can be
divided in rule-based approaches, machine learning based approaches or a
combination of both (e.g., [15], [16], and [17]). Rule-based approaches have
the advantage of not relying on large amounts of annotated (labeled) data.
However, the complexity of the rules to be implemented and thus the required
effort increases with the complexity of the reality of interest [18] to be
captured. On the one hand, this leads to potentially undetected known, unsafe
scenarios. On the other hand, it is impossible to explore previously unknown,
unsafe scenarios, which, however, are of great importance for a reliable
safety argumentation. Supervised machine learning approaches, in turn, require
a sufficient amount of labeled data. As these labels are partly unknown in
advance, discovering them must be an intrinsic part of a comprehensive
scenario extraction method before respective supervised approaches can be
deployed.
Therefore, this paper proposes a method for extracting multimodal urban
traffic scenarios from naturalistic road traffic data in an unsupervised
manner, minimizing the amount of prior expert knowledge required, as shown in
Fig. 1. Based on knowledge gained from first rule-based investigations [19],
we follow a generic clustering procedure [20] for exploring the unknown
natures of the data, which are interpretable as scenarios [21]. The method
utilizes principal feature analysis [22] for feature selection and proposes
the use of scenario grids for a scenario representation compatible with the
application of static data based clustering algorithms. We are emphasizing a
scenario representation compatible with established approaches in the field of
computer-vision, like convolutional neural networks, for a future deployment
of the method within a semi-supervised machine learning pipeline. For the
evaluation of the method, hierarchical agglomerative clustering (HAC) is
applied after reducing the dimensionality by principal component analysis. The
method is evaluated using the inD dataset [23] and the Silicon Valley
Intersections dataset [24]. The evaluation of the clustering results by
comparison with underlying trajectories and a rule-based baseline approach
shows promising results. It appears that unsupervised machine learning methods
can make a valuable contribution to the construction of a relevant scenario
database with sufficient coverage for testing ADS.
The main contributions of this paper are as follows: provision of an overview
of existig time series clustering methods and data-driven scenario extraction
approaches (Sec. II), description of a method for Unsupervised Scenario
Extraction (UnScenE) from naturalistic road traffic data (Sec. III), exemplary
implementation and evaluation of the scenario extraction method (Sec. IV) and,
finally, an elaboration on the results showing future research directions
(Sec. V).
## II Background
### II-A Requirements Specification
The specification of requirements for the scenario extraction method should be
accompanied by a prioritization of them in order to develop a common
understanding and to achieve a focus on the most important aspects of the
development task at hand. We utilize the MoSCoW prioritization, where each
requirement is marked as must (M), should (S), could (C) or won t (W)
according to its importance. Note, that requirements marked as W are
potentially as important as the ones marked with M, but can be left for a
future release of the implementation [25]. The following requirements for the
scenario extraction method were identified:
* •
The scenario extraction method must allow the processing of naturalistic road
traffic data by means of (multivariate) time series of different lengths (M1).
* •
The scenario extraction method must allow a clustering of multimodal urban
traffic scenarios of varying number of road users (M2).
* •
The scenario extraction method must be applicable to different urban traffic
spaces with as little adaptation effort as possible (M3).
* •
The scenario extraction method must include a feature representation that is
compatible with supervised machine learning approaches (M4).
* •
The scenario extraction method must allow a slicing of time series data in
meaningful sub-sequences (M5).
* •
The scenario extraction method should have a feature representation that
includes multiple road user dynamics simultaneously at the scenario level
(S1).
* •
The scenario extraction method should minimize the amount of prior expert
knowledge required for the application (S2).
* •
The scenario extraction method is intended to enable scenario clustering
without including road network information (C1).
* •
The scenario extraction method is intended to enable scenario clustering based
on different data sources in future releases (W1).
### II-B Clustering of Time Series Data
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The major strategies for clustering time series data can be divided into raw-
data based, feature-based, and model-based approaches [26]. Raw-data based
(direct) approaches typically compare different time series using established
clustering algorithms by replacing the distance or similarity measure for
static data with an appropriate one for time series, e.g., dynamic time
warping [20].
Figure 1: UnScenE method for extracting multimodal urban traffic scenarios.
Model-based approaches assume that the time series under investigation are
based on an underlying model or that they are determined by a combination of
probability distributions and are evaluated by means of a similarity measure
between fitted models [27]. Both, model-based and feature-based approaches can
be categorized as indirect approaches, as they first convert raw time series
data into a feature vector of lower dimension and perform the clustering on
the model parameters or the feature vector, respectively, rather than on the
raw time series data themselves [26]. Feature-based (representation-based
[28]) approaches can be further divided into strategies deploying manual or
automated feature extraction (Representation Learning [29]).
Despite the enormous number of existing approaches within the major strategies
described, we are not aware of any implementation that meets all of our
requirements specified. This is mainly caused by three characteristics within
the nature of our problem at hand: First, the scenario extraction method must
be able to deal with time series of different lengths, as different road users
have different velocities while passing through the observation area (cf. Req.
M1). Second, the method has to cope with multivariate time series, since a
relevant scenario is composed of at least two road users potentially involving
multiple time series per road user for a meaningful scenario representation
(cf. Req. M2). Third, the extraction method has to cope with a varying amount
of time series to be considered, because semantically similar scenarios can
include different numbers of road users (cf. Req. M2 and S1). Therefore, we
deployed a novel approach for scenario representation that is tailored to the
problem at hand allowing the use of established clustering algorithms for
static data.
### II-C Data-Driven Scenario Extraction
Most of the approaches dealing with data-driven scenario extraction are
developed for highway use cases (e.g., [4], [16], [30], [31], and [32]).
Elpsas et al. [17] propose a method, where rule-based detections are used to
train fully convolutional networks for extracting lane change and cut-in
scenarios. Kerber et al. [33] define a custom scenario distance measure,
considering the occupancy and relative distances of vehicles in eight slots
surrounding the ego-vehicle, for subsequent HAC. The authors mention that the
method is not suited for urban scenario extraction without modifications.
Montanari et al. [34] encourage the approach of extracting scenarios from real
driving data in an unsupervised manner. They describe a method for slicing bus
communication data of test vehicles, extract features from these slices and
cluster them into different highway scenarios using HAC.
As the traffic dynamics within urban ODDs is characterized by multimodal
interaction of various road user types, the scenario extraction method
presented in this paper shall be particularly able to extract scenarios based
on naturalistic road traffic data, e.g., recorded by unmanned aerial vehicles
(drones). This type of data collection allows an uninfluenced and objective
view of a scene [8] and thus enables capturing the traffic dynamics that an
ADS-equipped vehicle is exposed to in real-world road traffic [23].
Publications dealing with an urban ODD mostly focus on scenario extraction
based on other data sources, e.g., accident data or data of field tests with
measurement vehicles, and particularly analyze vehicle-to-vehicle interaction
scenarios (e.g., [29], [35], and [36]).
Regarding valuable work in terms of scenario extraction based on naturalistic
road traffic data, King et al. [37] propose an approach for deriving logical
vehicle-to-vehicle interaction scenarios for an unsignalized intersection.
Finally, Ries et al. [38] introduce a raw-data based clustering method for
grouping real driving sequences into semantically similar sequences. The
proposed method is the only one we are aware of that leverages clustering to
extract urban traffic scenarios with different road user types and numbers
including potentially differing sequence lengths. While this leads to
fulfilment of several of our targeted must and should requirements (cf. Req.
M1, M2, and S2), others of them are not (fully) addressed. In detail, the
authors state that the method has to be adapted for an inter-traffic-space
clustering and describe possible solutions to this challenge (cf. Req. M3).
Furthermore, we are targeting a feature representation that is compatible with
supervised machine learning approaches (cf. Req. M4) and include a slicing of
the time series data, since one ego-vehicle can encounter multiple concrete
scenarios during its lifetime (cf. Req. M5) [37]. Furthermore, instead of a
sequential and pairwise similarity estimation between trajectories belonging
to a concrete scenario, we want to compare the dynamics of multiple road users
simultaneously (cf. Req. S1). Summarizing, to the best of our knowledge, there
is no commonly accepted method for extracting relevant multimodal urban
traffic scenarios for testing ADS that meets all the requirements specified.
## III Scenario Extraction Method
The UnScenE method for extracting multimodal urban traffic scenarios developed
with respect to the specified requirements (cf. Sec. II) is shown in Fig. 1.
It follows the generic clustering procedure according to Xu and Wunsch [20]
and contains novel instantiations within the different steps to solve the task
at hand. While the amount of data decreases with the passing of the different
steps, the knowledge of the inherent patterns regarding the reality of
interest (i.e., relevant scenarios concerning specified urban traffic spaces)
increases. Since the method in the first stage of development must be able to
process naturalistic road traffic data particularly, exemplary explanations of
the application refer to this type of data source.
### III-A Spatiotemporal Filtering
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Naturalistic road traffic data contain partly irrelevant parts for subsequent
scenario extraction (e.g., vehicles driving straight through the observation
area without other RUs present). Given a non-empty set of input patterns in
form of multivariate time series data
$\textbf{{X}}=\\{\textbf{{x}}_{1},...,\textbf{{x}}_{j},...,\textbf{{x}}_{N}\\}$,
where
$\textbf{{x}}_{j}=\\{x_{j\text{1}},x_{j\text{2}},..,x_{jd}\\}~{}\epsilon~{}\mathbb{R}^{d}$,
with a $d$-dimensional feature space, the goal of this process step is to
determine the relevant subset of samples $j$ out of $N$, where $j<N$. Hence,
this process step aims at reducing noise in the data by removing the
irrelevant data proportions. Within the proposed method, this is achieved by a
search algorithm based on the post-encroachment time (PET) [39] of all
possible combinations of an ego-vehicle (ego) trajectory and those of
encountered road users (challengers) while passing through the observation
area (ego-lifetime). As an ego can encounter multiple concrete scenarios
within its lifetime (cf. Req. M5), it has to be accounted for a slicing of the
corresponding trajectories in such cases, additionally.
In detail, this step is performed by a two-layered procedure for each car
included in the dataset, as our goal is to extract relevant test scenarios for
ADS-equipped vehicles. In the first layer, a distance matrix is generated for
each combination of ego and challenger, containing the Euclidean distances of
the involved trajectories for all common time steps within the observation
area. By user-adaptive parametrization of the threshold for the distance
between trajectories $d_{\text{traj}}$, it is possible to define how close the
involved trajectories have to come to each other, so that an interaction
between the involved road users is conceivable. The value of $d_{\text{traj}}$
can be interpreted as radius of a moving circle surrounding the ego, whose
area must overlap the challenger trajectory. Within the second layer, the
minimum value of the distance matrix values that satisfy this criterion is
used to calculate the PET of the trajectories involved. By setting a PET
threshold $t_{\text{PET}}$, the temporal evolvement of the scenario is taken
into account. If the two layers of the algorithm are passed through several
times during the lifetime of an ego-vehicle, the resulting ego-challenger
combinations can be used to slice the respective time series.
It should be noted that, at the current state of implementation, this
algorithm is accompanied by two main assumptions: First, an intersection area
(rather than an intersection point) between trajectories involved in a
scenario is considered sufficient to maintain it as potentially relevant.
Second, the PET between an ego-vehicle and the nearest challenger is crucial
for the following classification of the concrete scenario to be kept. In case
of an empty set after spatiotemporal filtering, the extraction method is
aborted and, e.g., new data containing relevant scenarios must be recorded for
successful application. For further discussion, the entirety of the remaining
dataset is referred to as intersecting trajectories.
### III-B Principal Feature Analysis (Feature Selection)
While the spatiotemporal filtering involves a reduction in the number of
samples, i.e., reduces the $N$, this process step aims at reducing the number
of columns, i.e., reducing the $d$. Thus, the task can be described as
choosing $n$ distinguishing features out of $d$, whereas $n<d$. An elegant
selection of salient features can greatly decrease the storage requirements,
simplify the subsequent design process, and facilitate the understanding of
the data [20], [21].
With respect to the problem at hand, an approach has to be chosen that is able
to determine the importance of the features of the intersecting trajectories
in order to subsequently select a meaningful subset of them. While there are
various methods to reduce the dimensionality of a feature set, e.g., principal
component analysis (PCA), these approaches are characterized by resulting in a
lower-dimensional representation, where the features are not physically
interpretable anymore [20], [22]. While this is unproblematic for other tasks,
these approaches are not effective in the context of subsequent manual feature
extraction, where physical interpretability is of great importance.
Hence, the proposed method deploys feature selection by principal feature
analysis (PFA) [22], which makes it possible to exploit the structure of the
principal components of a feature set to find a subset of the original feature
vector. The method includes five steps, with the first step calculating the
covariance matrix of the standardized intersecting trajectories dataset
$\textbf{{X}}_{\text{std}}$. This is followed by the calculation of the
principal components and the eigenvalues of the covariance matrix C. By
choosing the explained variance ratio $var_{\text{PFA}}~{}\epsilon~{}[0,1]$,
$s$ columns of the matrix A, representing the orthonormal eigenvectors of C,
are kept, constructing the subspace matrix $\textbf{{A}}_{s}$. The
parametrization of $var_{\text{PFA}}$ decides how much of the variability of
the data is desired to be retained. In the third step, the rows
$\textbf{{v}}_{i}~{}\epsilon~{}\mathbb{R}^{s}$ of $\textbf{{A}}_{s}$ are
clustered using $k$-means algorithm. Since each vector $\textbf{{v}}_{i}$
represents the projection of the $i$'th feature of $\textbf{{X}}_{\text{std}}$
in the lower-dimensional space, the $s$ elements of $\textbf{{v}}_{i}$
correspond to the weights of the $i$'th feature on each axis of the subspace
$s$. As the amount of mutual information increases with the similarity of the
absolute values of these vectors, the clusters derived by $k$-means can be
used to choose one feature of a subset of highly correlated features,
respectively [22]. According to [22], these features represent each group
optimally by means of high spread in the lower dimension, reconstruction and
insensitivity to noise. It is of note, that the number of clusters should be
chosen slightly higher than $s$.
Finally, using this method it is possible to systematically reduce the
intersecting trajectories dataset regarding its dimensionality. This paves the
way for a well-founded extraction of features for the construction of the
scenario grids, as described in the following process step.
### III-C Feature Extraction
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With the spatiotemporal filtered and dimensionally reduced intersecting
trajectories dataset available, the following process step consists of a
feature extraction suitable for the subsequent clustering of multimodal urban
traffic scenarios. Ideally, such a feature representation should be of use in
distinguishing patterns belonging to different clusters, immune to noise, and
easy to obtain and interpret [20]. Considering these generic requirements as
well as the task-specific requirements (cf. Sec. II), we propose a two-step
process to finally construct a matrix feasible for serving as input for the
subsequent process step of clustering.
We utilize the principle of a grid-based representation, within which a
certain spatial area around the ego is discretized and defined as potentially
relevant in terms of the ego behavior. The basic principle has already been
proposed for use within the robotic domain [40] and various modules of an ADS,
such as decision making or motion planning [41]. Furthermore, Gruner et al.
[42] propose the use of such grid-based representations for scenario
classification based on object-list data and confirm the usability of this
type of representation for training artificial neural networks. In contrast to
[42], we propose the construction of such a discrete, multi-channel grid
structure per scenario and not per scene. The reasons for this are on the one
hand the offline character of the scenario extraction use case and the
knowledge of the evolvement of an entire scenario based on naturalistic road
traffic data. On the other hand, our approach aims at maximizing the degree of
automation with respect to scenario labeling through unsupervised machine
learning approaches, while [42] implements a rule-based approach for a semi-
automated scene labeling. In addition, the previous spatiotemporal filtering
already accounts for the temporal evolvement of a scenario and scenes are
assigned to a respective concrete scenario, even if the label of the
corresponding scenario is still unknown. Finally, our investigations showed
that the scenario-level representation requires significantly less
computational effort.
#### III-C1 Key Frame Calculation
This step comprises the determination of the point in time or frame,
respectively, of the corresponding concrete scenario (key frame), which is
used for the subsequent construction of the different grid channels defining a
scenario tensor. On the one hand, the determination of this key frame should
be applicable as generically as possible to all relevant ego-challenger
combinations identified. On the other hand, the snapshot of the scenario
created on the basis of the key frame should capture the distinguishing
characteristics of the specific scenario category as accurately as possible.
Our investigations have shown that a computation based on the maximum of the
yaw rate of the ego $\dot{\varphi}_{\text{e,max}}$ or challenger
$\dot{\varphi}_{\text{c,max}}$ within the specific concrete scenario entails
the best compromise on robustness and computational effort. In detail, first
the maximum yaw rates of the RU involved in a concrete scenario are calculated
for all ego-challenger combinations within the intersecting scenarios. Then,
the maximum of the respective yaw rate value set is calculated. In case of
$\dot{\varphi}_{\text{e,max}}\geq\dot{\varphi}_{\text{c,max}}$, the frame
associated with $\dot{\varphi}_{\text{e,max}}$ is used to construct the
scenario tensor. Since the construction of the scenario tensor is always done
with respect to the ego-vehicle state, in case of
$\dot{\varphi}_{\text{e,max}}<\dot{\varphi}_{\text{c,max}}$, a shift of time
to the associated challenger frame has to take place. In Fig. 2, the resulting
key frames are implicitly shown by the exemplary occupancy grid channels for
two concrete scenarios.
Figure 2: Exemplary concrete scenarios with respective occupancy grid channel
at Bendplatz traffic space (left: ego-to-vehicle, right: ego-to-pedestrian).
#### III-C2 Scenario Tensor Construction
The key frame for each concrete scenario defines which point in time or frame,
respectively, is to be used to construct the corresponding scenario tensor
$\Phi=\\{\textbf{{G}}_{1},...,\textbf{{G}}_{j},...,\textbf{{G}}_{l}\\}$, where
$\textbf{{G}}_{i}$ is the $i$'th grid channel matrix of size
$(a_{\text{gr}}\cdot r_{\text{gr,1}})\times(a_{\text{gr}}\cdot
r_{\text{gr,2}})$. All feature values within the $l$ grid channels are
calculated with respect to the ego coordinate system at the key frame. Thus,
the scenario tensor consists of a discrete, multi-channel grid structure,
representing the scenarios spatiotemporal characteristics from a bird's-eye
view. A feasible number of grid channels $l_{\text{gr}}$ is determined by the
result of previous PFA, whith $l_{\text{gr}}$ less or equal to the dimension
of the reduced intersecting trajectories dataset. Both the region of interest
$a_{\text{gr}}$ of the rectangular grids and the resolution vector
$\textbf{{r}}_{\text{gr}}=({r_{\text{gr,1}},r_{\text{gr,2}}})$, containing the
longitudinal and lateral grid resolution, respectively, can be adapted to the
specific application. While $a_{\text{gr}}$ determines how far the ego looks
into space and time, $\textbf{{r}}_{\text{gr}}$ determines how fine the grid
resolves the spatiotemporal evolvement of the respective scenario.
In the lower part of Fig. 2, the occupancy grid $\textbf{{G}}_{1}$, containing
both the trajectories of the ego and the nearest challenger as well as grid
cells occupied by other surrounding RUs for the corresponding key frame, is
visualized for an exemplary ego-to-vehicle and ego-to-pedestrian scenario. In
agreement with [42], this scenario representation is basically applicable to
different ODDs, adaptive by means of the region of interest and resolution.
Furthermore, other grid forms are conceivable. It should be noted that this
type of scenario representation addresses the fulfillment of essential
requirements stated (e.g., Req. M1, M2, M3, M4, and S1).
To obtain a representation suitable for clustering, each scenario tensor is
flattened column-wise into a scenario grid matrix of size $(a_{\text{gr}}\cdot
r_{\text{gr,1}})\times(a_{\text{gr}}\cdot r_{\text{gr,2}}\cdot l)$. After
standardization as well as dimensionality reduction by applying PCA, the
entirety of resulting scenario grids can finally be forwarded for the
following process step as cluster input matrix $\textbf{{M}}_{\text{c}}$.
### III-D Clustering
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The following process step of the UnScenE method consists of the actual
application of a clustering procedure. The structure of the cluster input
matrix $\textbf{{M}}_{\text{c}}$ basically allows the application of any
clustering algorithm based on static data. Since the scenario extraction
method is in general applicable to different data sources and traffic spaces,
the user can and must select the most appropriate clustering approach for the
dataset at hand.
The evaluation in the scope of this paper shows the most promising clustering
results using hierarchical agglomerative clustering (cf. Sec. IV). This is in
accordance with literature, in which HAC is described as suitable for use
cases with many clusters and many samples [21]. Furthermore, this is in line
with other publications in the context of real-data based scenario extraction,
both based on similarly structured data sources [33] as well as for other
types of data sources [34].
### III-E Cluster Validation and Result Interpretation
Clustering is a subjective process in nature due to the absence of a ground
truth [20], [37], since the main goal of the clustering and the subsequent
result interpretation itself is to find these before unknown labels as good as
possible within the given data. Clustering results heavily depend on the
choice of the clustering approach, and even for the same algorithm, the
selection of related parameters [20]. Furthermore, clustering results are a
matter of view and the definition of similarity highly depends on the problem.
In literature, three different testing criteria categories, namely external,
internal and relative indices, are defined to be able to estimate the quality
of clustering results [20]. In accordance with present work in the field of
unsupervised scenario extraction (e.g., [33], [34], and [38]), one branch of
our evaluation approach can be assigned to the external indices category,
where external information is used as standard to validate the clustering
results. For this purpose, we use the map information available within the
datasets in the form of images of the corresponding traffic spaces. In detail,
we compare the trajectories involved in different concrete scenarios belonging
to a respective cluster by visual validation. In addition, we compare the
clustering results for the Bendplatz traffic space with the results of a rule-
based baseline approach, since for the latter the elaborate rule-based
implementation was intentionally accepted to get a better reference to the
presented extraction method. Finally, we compare different clustering
structures in order to obtain a reference for deciding which one may best
reveal the characteristics of the data. Since cluster analysis is not a one-
shot process [20] the method includes feedback loops depending on the
validation results for trials and repetitions with an adjusted parametrization
at different process steps. In case that the validation criteria are met, the
extracted relevant scenarios can be used for subsequent applications, e.g.,
scenario-based testing of an ADS.
## IV Evaluation
In this chapter, we evaluate the UnScenE method for extracting relevant urban
traffic scenarios using the inD dataset [23] as well as the Silicon Valley
Intersections dataset (sv dataset) [24]. Some process steps are evaluated for
the entire datasets to get an impression of the generic character of the
method. Other process steps are demonstrated for exemplary traffic spaces in
order to illustrate specific advantages and drawbacks of the method in depth.
### IV-A Datasets and Parametrization
There are various reasons for choosing the inD dataset for the main part of
the evaluation of the scenario extraction method. The main advantages of the
inD dataset compared to other similar datasets are its size,
representativeness and accuracy [23]. The dataset consists of more than 11,500
naturalistic road user trajectories including cars, trucks and busses as well
as about 5,000 pedestrian and bicyclist trajectories recorded at four
unsignalized intersections in Aachen, Germany. In particular, the high
proportion of vulnerable road users (VRU) makes this dataset an interesting
challenger to the method (cf. Tab. ind_overview), as it can be assumed that it
is especially difficult to detect recurring patterns within the less rule-
based behavior of VRU. In addition, we use the sv dataset [24] to evaluate
individual process steps. This dataset includes naturalistic road traffic data
for seven traffic spaces in the Silicon Valley area. Due to the diversity of
the traffic spaces as well as changed boundary conditions (especially layer
1-4 of the six-layer model) compared to the inD dataset, more comprehensive
conclusions regarding the feasibility of the extraction method can be drawn.
As shown in Tab. I, Bendplatz traffic space and Frankenburg traffic space of
the inD dataset both have a high total number of trajectories and a high
percentage of VRUs. This makes these two traffic spaces particularly
interesting for in-depth studies, since a higher share of multimodal
interaction can be assumed.
TABLE I: InD Dataset Overview Traffic Space | Share in total rec. RU | VRU/Vehicles
---|---|---
Heckstraße | 0.09 | 0.07
Aseag | 0.17 | 0.12
Bendplatz | 0.28 | 0.5
Frankenburg | 0.46 | 1.6
For the subsequent studies, the following parametrization was performed:
$t_{\text{PET}}=$3\text{\,}\mathrm{s}\text{/}$$,
$d_{\text{traj}}=$1\text{\,}\mathrm{m}\text{/}$$, $var_{\text{PFA}}=0.95$,
$a_{\text{gr}}=$30\text{\,}\mathrm{m}\text{/}$$,
$r_{\text{gr,1}}=r_{\text{gr,2}}=$1\text{\,}\text{px}\mathrm{/}\mathrm{m}$$,
and $var_{\text{PCA}}=0.99$. Even if the numerical values have been chosen on
the basis of expert judgement and, if possible, literature references [42],
[43], they should not be given too much weight, since the focus is on
demonstrating the applicability of the method.
### IV-B Spatiotemporal Filtering and Feature Selection
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While the application of the search algorithm for spatio-temporal filtering is
trivial, interesting conclusions can already be drawn on the basis of the
proportion of the remaining relevant intersecting trajectories dataset. Since
the datasets provide a classification regarding the respective RU types (cars,
trucks, pedestrians, and bicyclists), high-level scenario categories relevant
from the viewpoint of proving the safety of an ADS can be defined as ego-to-
vehicle (e-to-v) scenarios, ego-to-pedestrian (e-to-p) scenarios, and ego-to-
bicyclist (e-to-b) scenarios. For the example of the Bendplatz traffic space,
it is found that on average about 4% of pedestrians, 14% of vehicles, and 15%
of bicyclists remain within the dataset for the subsequent process steps.
These numerical values may serve as an indication of the combination of
external risk to which the various RU types are exposed and internal RU type
specific risk tolerance. In addition, this gives a hint on the required amount
of recordings, since even with the conservative parametrization performed,
most of the original amount of data is classified as not relevant for the task
at hand.
Fig. 3 shows exemplary results of the principal feature analysis in terms of
the cumulative explained variance $var_{\text{PFA}}$ over the number of
features for different traffic spaces and RU types. Note, that even if this is
not a continuous function, the continuous representation of the graphs
supports the understanding of the respective results. Fig. 3 shows the curves
for the object type car for the traffic spaces with the minimum and maximum
number of features required for the defined cumulative explained variance for
the corresponding dataset. It is noticeable that the extrema of the sv dataset
wrap around those of the inD dataset. This corresponds to the intuitive
expectations, since the traffic space sv_san_jose (lat.: 37.3627, long.:
-121.8759) is located in direct proximity to a highway entrance and only
vehicles are present. It is also plausible that traffic space sv_santa_clara
(lat.: 37.3252, long.: -121.9490) requires more features for capturing the
same cumulative explained variance, since it includes a complex parking lot
with two 4-way stops and six normal stop entries.
Figure 3: Cumulative explained variance over number of selected features for
exemplary traffic spaces and road user types.
While pedestrians in general need more features for the same value of
cumulative explained variance, the analysis of the inD dataset shows that the
influence of the traffic space is marginal for this RU type, which is why only
one curve is shown here. On average, the 11 features considered (xCenter,
yCenter, heading, xVelocity, yVelocity, xAcceleration, yAcceleration,
lonVelocity, latVelocity, lonAcceleration, latAcceleration) [23] can be
reduced to about the half, with pedestrians defining the lower threshold.
While this seems intuitively logical for the inD dataset, since some features
differ only by the reference coordinate system, this process step can be an
even more valuable tool for objective feature selection for differently
structured datasets.
### IV-C Feature Extraction
The evaluation of the following process steps is conducted exemplarily for the
Bendplatz traffic space of the inD dataset (cf. Fig. 2). The PFA results in
seven features for $var_{\text{PFA}}=0.95$, with the pedestrian class forming
the feature superset of all RU classes. On this basis, the features xCenter,
yCenter, heading, xVelocity, yVelocity, xAcceleration, and yAcceleration are
used for the construction of the scenario tensor. Since the information of the
features xCenter, yCenter, and heading can be combined in the occupancy grid
channel, this results in a total of five grid channels consisting of the
occupancy grid and velocity and acceleration grids in both spatial directions
with regard to the ego state at the respective key frame. While Fig. 2 shows
the deviation of the feature representation for the case of semantically
dissimilar scenarios, Fig. 4 shows the similarity of the feature
representation for semantically similar scenarios using the example of the
x-velocity grids for two e-to-v scenarios (same scale). The underlying
scenarios were chosen randomly from the group of left turning vehicles,
respectively, out of the cluster visualized in Fig. fig_traj_v2v.
Figure 4: x-velocity grids for semantically similar ego-to-vehicle scenarios
randomly chosen from one cluster at Bendplatz (cf. Fig. fig_traj_v2v).
### IV-D Clustering
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The reasons for choosing HAC as clustering method are described in Sec. III.
HAC follows a bottom-up approach, starting with N clusters, each of which
includes exactly one sample. A series of merge operations follows based on a
linkage criterion until a threshold is reached or all samples are forced to
the same cluster [20], [33]. The results of HAC can be visualized by
dendrograms, which are characterized by an inverted tree-like structure [43].
While the root node of a dendrogram represents the entire dataset, each leaf
can be regarded as one sample. Therefore, the intermediate nodes describe the
degree of similarity between samples and the height of each branch represents
the distance between a pair of samples or clusters, or a sample and a
respective cluster [20].
Figure 5: Dendrogram depicting the clustering structure of ego-to-vehicle
scenarios at Bendplatz (color-coded for distance threshold value of 180).
Fig. fig_dendro_v2v shows the dendrogram for the e-to-v scenarios at
Bendplatz, created based on all available recordings. Our experiments have
shown the best results for this scenario category with ward linkage criterion.
On the one hand, this shows that the different process steps of the method are
consistent up to this point and that the developed feature representation is
compatible with static data based clustering methods. On the other hand,
however, clustering algorithms can always produce a partitioning given a
dataset, regardless of whether or not there exists a particular structure in
the data [20]. Thus, it is only the subsequent cluster validation and result
interpretation that provides the actual gain in knowledge and therefore
confidence in the clustering results.
### IV-E Cluster Validation and Result Interpretation
175(20.5,268) ©2022 IEEE
Figure 6: Semantically similar ego-to-vehicle scenarios.
Our approach to validate the cluster results involves three branches (cf. Sec.
II). The first branch consists of a comparison of different clustering methods
and their respective parametrizations. Thereby, we present the
parametrizations that produced the best results in the scope of this paper.
The second branch covers a visual validation approach, as described by Fig.
fig_traj_v2v and Fig. fig_traj_v2p.
In detail, this approach entails the comparison of RU trajectories in
different concrete scenarios of one specific cluster, randomly chosen for a
defined distance threshold. The trajectories shown in Fig. fig_traj_v2v belong
to all concrete e-to-v scenarios within the green, red, and purple clusters of
the left side of the dendrogram, all of which combine into a single cluster as
the distance threshold value increases up to 240. This result is promising
since only semantically similar scenarios are included within the cluster,
which can all be assigned to a left-turning ego-vehicle in the presence of
oncoming traffic (cf. Fig. fig_traj_v2v). In addition, this example
illustrates the direction independence of the feature representation, as
semantically similar scenarios are clustered regardless of their specific
position within the traffic space.
Figure 7: Semantically similar ego-to-pedestrian scenarios.
While 369 concrete e-to-v scenarios were available for the clustering at the
Bendplatz traffic space, a significantly lower number of 33 e-to-p scenarios
remained as cluster input due to their lower probability of occurrence. In
analogy to the procedure for the e-to-v scenarios, Fig. fig_traj_v2p shows
such e-to-p scenarios, all of which originate from a specific cluster. The
impression that the extraction method is able to cluster semantically similar
scenarios is strengthened. This impression also applies to the case of
multimodal interaction in Fig. fig_traj_v2p, including less rule-based
pedestrian behavior.
To get a broader impression about the performance of the scenario extraction
method, the third branch of the validation procedure includes the attempt to
get a statement about the accuracy of the overall clustering. For this
purpose, we compare the results of the clustering process with respect to the
e-to-v scenarios for the Bendplatz traffic space with a rule-based baseline
approach particularly designed for this traffic space [19]. The methodology
for determining overall accuracy is based on [43] and has been extended for
this use case. In detail, this involves assigning to each sample one of the
nine associated scenario labels originating from the rule-based approach [19].
Subsequently, the most frequently occurring labels per cluster are summed up
over all clusters and are divided by the total number of samples. Fig.
fig_clust_acc shows the results of this procedure for a varying distance
threshold. As the distance threshold increases, the number of clusters
decreases, which is accompanied by a decreasing accuracy. With respect to the
dendrogram, increasing the distance threshold corresponds to cutting the tree-
like cluster structure at a higher point. As can be seen from Fig. 8, there is
a large drop in overall acurracy from 0.75 to 0.56 in the range of the
distance threshold between 180 and 190. This corresponds analogously to a
transition from 21 to 16 clusters. With a distance threshold of 180 and a
cluster number of 21, it can be observed that many clusters have a very high
accuracy, while a few larger clusters show low accuracy. In detail, 18 of the
21 clusters have an accuracy above 0.9, while one cluster with an accuracy of
0.32 is the lower outlier. Finding the reasons for this result is subject of
future work.
Figure 8: Overall accuracy of ego-to-vehicle scenario clustering compared to
rule-based baseline approach for Bendplatz traffic space.
It should be noted that the labels of the rule-based approach used for
determining the accuracy do not represent the ground truth either. Thus, false
positive as well as false negative ratings of the accuracy can occur.
Nevertheless, this approach offers a building block for the determination of a
sufficient number of scenario clusters with respect to a reasonable balance
between scenario coverage and effort during validation and testing of an ADS.
## V Conclusion and Outlook
175(20.5,268) ©2022 IEEE
This paper proposes a method for extracting multimodal urban traffic scenarios
from naturalistic road traffic data in an unsupervised manner. The method
comprises five process steps, namely spatiotemporal filtering, principal
feature analysis, feature extraction, clustering and cluster validation. The
results of the principal feature analysis show a dependence of the cumulative
explained variance within the data on both traffic space and road user type.
In addition, the required features for the subsequent steps can be reduced by
a factor of about two within the exemplary evaluation. For feature extraction,
a discrete, multi-channel grid structure for scenario modeling is proposed,
resulting in a scenario tensor with various grids defined by the previously
selected features. This feature representation particularly adresses the
requirements M3, M4, M5, S1, S2, and C1 (cf. Sec. II). Based on the developed
feature representation it is possible to apply clustering methods which rely
on static data. To evaluate the method, hierarchical agglomerative clustering
is applied to an urban traffic space and the corresponding results are
discussed. Thereby, both the results of visual validation for selected
multimodal urban traffic scenarios and the comparison with a rule-based based
baseline approach are promising. These results confirm in particular the
fulfillment of the requirements M1 as well as M2 (cf. Sec. II).
Based on the results, future research should push for a broader evaluation of
the method by investigating more traffic spaces as well as more clustering
approaches. Furthermore, answering the question for the few, large outliers in
the cluster-individual overall accuracy represents a relevant aspect. In
addition, addressing requirement W1 by incorporating other data sources is of
interest. Moreover, the incorporation of the method into the scenario-based
simulation platform presented in [19] represents a future use case, with the
extracted scenarios representing one main input channel for the adaptive
replay-to-sim approach (ARTS), in addition to an agent-based simulation and
the ADS-under-test. Finally, the method will be integrated into a semi-
supervised machine learning pipeline to achieve the medium-term goal of a
robust and generic scenario classifier. The presented method contributes to
the quantitative real-data based extraction of relevant traffic scenarios.
This method can be seen as building block toward a systematic, data-driven
construction of a relevant scenario database with sufficient coverage in a
fully automated manner for validating the safe behavior of ADS.
## ACKNOWLEDGMENT
We want to thank Stefan Berger and Ulrich Eberle (Opel Automobile GmbH,
Stellantis NV) for the productive discussions and the feedback on our
approaches as well as for peer review of this publication prior to its
submission.
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|
# Transfer Learning in Conversational Analysis through Reusing Preprocessing
Data as Supervisors
Joshua Yee Kim
University of Sydney
<EMAIL_ADDRESS>
&Tongliang Liu
University of Sydney
<EMAIL_ADDRESS>
Kalina Yacef
University of Sydney
<EMAIL_ADDRESS>
###### Abstract
Conversational analysis systems are trained using noisy human labels and often
require heavy preprocessing during multi-modal feature extraction. Using noisy
labels in single-task learning increases the risk of over-fitting. Auxiliary
tasks could improve the performance of the primary task learning during the
same training – this approach sits in the intersection of transfer learning
and multi-task learning (MTL). In this paper, we explore how the preprocessed
data used for feature engineering can be re-used as auxiliary tasks, thereby
promoting the productive use of data. Our main contributions are: (1) the
identification of sixteen beneficially auxiliary tasks, (2) studying the
method of distributing learning capacity between the primary and auxiliary
tasks, and (3) studying the relative supervision hierarchy between the primary
and auxiliary tasks. Extensive experiments on IEMOCAP and SEMAINE data
validate the improvements over single-task approaches, and suggest that it may
generalize across multiple primary tasks.
## 1 Introduction
The sharp increase in uses of video-conferencing creates both a need and an
opportunity to better understand these conversations (Kim et al., 2019a). In
post-event applications, analyzing conversations can give feedback to improve
communication skills (Hoque et al., 2013; Naim et al., 2015). In real-time
applications, such systems can be useful in legal trials, public speaking,
e-health services, and more (Poria et al., 2019; Tanveer et al., 2015).
Analyzing conversations requires both human expertise and a lot of time.
However, to build automated analysis systems, analysts often require a
training set annotated by humans (Poria et al., 2019). The annotation process
is costly, thereby limiting the amount of labeled data. Moreover, third-party
annotations on emotions are often noisy. Deep networks coupled with limited
noisy labeled data increases the chance of overfitting (James et al., 2013;
Zhang et al., 2016). Could data be used more productively?
From the perspective of feature engineering to analyze video-conferences,
analysts often employ pre-built libraries (Baltrušaitis et al., 2016;
Vokaturi, 2019) to extract multimodal features as inputs to training. This
preprocessing phase is often computationally heavy, and the resulting features
are only used as inputs. In this paper, we investigate how the preprocessed
data can be re-used as auxiliary tasks which provide inductive bias through
multiple noisy supervision (Caruana, 1997; Lipton et al., 2015; Ghosn and
Bengio, 1997) and consequently, promoting a more productive use of data.
Specifically, our main contributions are (1) the identification of
beneficially auxiliary tasks, (2) studying the method of distributing learning
capacity between the primary and auxiliary tasks, and (3) studying the
relative supervision hierarchy between the primary and auxiliary tasks. We
demonstrate the value of our approach through predicting emotions on two
publicly available datasets, IEMOCAP (Busso et al., 2008) and SEMAINE (McKeown
et al., 2011).
## 2 Related Works and Hypotheses
Multitask learning has a long history in machine learning (Caruana, 1997). In
this paper, we focus on transfer learning within MTL, a less commonly
discussed subfield within MTL (Mordan et al., 2018). We are concerned with the
performance on one (primary) task – the sole motivation of adding auxiliary
tasks is to improve the primary task performance.
In recent years, this approach has been gaining attention in computer vision
(Yoo et al., 2018; Fariha, 2016; Yang et al., 2018; Mordan et al., 2018;
Sadoughi and Busso, 2018), speech recognition (Krishna et al., 2018; Chen and
Mak, 2015; Tao and Busso, 2020; Bell et al., 2016; Chen et al., 2014), and
natural language processing (NLP) (Arora et al., 2019; Yousif et al., 2018;
Zalmout and Habash, 2019; Yang et al., 2019; Du et al., 2017). The drawback of
adding multiple tasks increases the risk of negative transfer (Torrey and
Shavlik, 2010; Lee et al., 2016, 2018; Liu et al., 2019; Simm et al., 2014),
which leads to many design considerations. Three of such considerations are,
identifying (a) what tasks are beneficial, (b) how much of the model
parameters to share between the primary and auxiliary tasks, and (c) whether
we should prioritize primary supervision by giving it a higher hierarchy than
the auxiliary supervision.
In contrast with previous MTL works, our approach (a) identifies sixteen
beneficially auxiliary targets, (b) dedicates a primary-specific branch within
the network, and (c) investigates the efficacy and generalization of
prioritizing primary supervision across eight primary tasks.
Since our input representation is fully text-based, we dive deeper into MTL
model architecture designs in NLP, Søgaard and Goldberg (2016) found that
lower-level tasks like part-of-speech tagging, are better kept at the lower
layers, enabling the higher-level tasks like Combinatory Categorical Grammar
tagging to use these lower-level representations. In our approach, our model
hierarchy is not based on the difficulty of the tasks, but more simply, we
prioritize the primary task. Regarding identifying auxiliary supervisors in
NLP, existing works have included tagging the input text (Zalmout and Habash,
2019; Yang et al., 2019; Søgaard and Goldberg, 2016). Text classification with
auxiliary supervisors have included research article classification (Du et
al., 2017; Yousif et al., 2018), and tweet classification (Arora et al.,
2019).
Multimodal analysis of conversations has been gaining attention in deep
learning research (Poria et al., 2019). The methods in the recent three years
have been intelligently fusing numeric vectors from the text, audio, and video
modalities before feeding it to downstream layers. This approach is seen in
MFN (Zadeh et al., 2018a), MARN (Zadeh et al., 2018b), CMN (Hazarika et al.,
2018b), ICON (Hazarika et al., 2018a), DialogueRNN (Majumder et al., 2019),
and M3ER (Mittal et al., 2020). Our approach is different in two ways. (1) Our
audio and video information is encoded within text before feeding only the
text as input. Having only text as input has the benefits of interpretability,
and the ability to present the conversational analysis on paper (Kim et al.,
2019b), similar to how the linguistics community performs manual
conversational analysis using the Jefferson transcription system (Jefferson,
2004), where the transcripts are marked up with symbols indicating how the
speech was articulated. (2) Instead of using the audio and video information
as only inputs, we demonstrate how to use multimodal information in both input
and as auxiliary supervisors.
Hypothesis H1: The introduced set of auxiliary supervision features improves
primary task performance. We introduce and motivate the full set of sixteen
auxiliary supervisions, all based on existing literature: these are grouped
into four families, each with four auxiliary targets. The four families are
(1) facial action units, (2) prosody, (3) historical labels, and (4) future
labels:
(1) Facial action units, from the facial action coding system identifies
universal facial expressions of emotions (Ekman, 1997). Particularly, AU 05
(upper lid raiser), 17 (chin raiser), 20 (lip stretcher), 25 (lips part) have
been shown to be useful in detecting depression (Yang et al., 2016a; Kim et
al., 2019b) and rapport-building (Kim et al., 2021).
(2) Prosody, the tone of voice – happiness, sadness, anger, and fear – can
project warmth and attitudes (Hall et al., 2009), and has been used as inputs
in emotions detection (Garcia-Garcia et al., 2017).
(3 and 4) Using features at different historical time-points is a common
practice in statistical learning, especially in time-series modelling (Christ
et al., 2018). Lastly, predicting future labels as auxiliary tasks can help in
learning (Caruana et al., 1996; Cooper et al., 2005; Trinh et al., 2018; Zhu
et al., 2020; Shen et al., 2020). We propose using historical and future (up
to four talkturns ago or later) target labels as auxiliary targets.
Figure 1: Reusing components (dotted lines) of the feature engineering process
as auxiliary targets (in blue). The MONAH framework (Kim et al., 2021) is
introduced in section 4.2
Given that we are extracting actions and prosody families as inputs, we
propose to explore whether they can be reused as supervisors (see Fig. 1). Our
hypothesis H1 is that re-using them as auxiliary supervision improves primary
task performance. This is related to using hints in the existing MTL
literature, where the auxiliary tasks promote the learning of the feature
(Cheng et al., 2015; Yu and Jiang, 2016).
Hypothesis H2: When the primary branch is given maximum learning capacity, it
would not be outperformed by models with primary branch having less than the
maximum learning capacity. Deeper models with higher learning capacity produce
better results (Huang et al., 2019; Nakkiran et al., 2019; Menegola et al.,
2017; Blumberg et al., 2018; Romero et al., 2014). Also, since the auxiliary
branch is shared with the primary supervision, the auxiliary capacity should
be limited to improve primary task performance (Wu et al., 2020) because
limiting the auxiliary capacity will force the branch to learn common
knowledge (instead of auxiliary specific knowledge) across the auxiliary tasks
(Arpit et al., 2017). Therefore, given a fixed learning capacity budget, our
hypothesis H2 implies that we should allocate the maximum learning capacity to
the primary branch because we care only about the primary task performance.
Hypothesis H3: Auxiliary supervision at the lower hierarchy yields better
primary task performance as compared to flat-MTL. Having the auxiliary tasks
at the same supervisory level as the primary task is inherently sub-optimal
because we care only about the performance of the primary task (Mordan et al.,
2018). Information at the lower hierarchy learns basic structures that are
easy to transfer, whilst upper hierarchy learns more semantic information that
is less transferable (Zeiler and Fergus, 2014). Therefore, we propose that the
auxiliary supervision to be at the lower hierarchy than the primary
supervision.
## 3 Model Architecture
### 3.1 Flat-MTL Hierarchical Attention Model
We start with an introduction of the Hierarchical Attention Model (HAN) (Yang
et al., 2016b). We chose HAN because of its easy interpretability as it only
uses single-head attention layers. There are four parts to the HAN model, (1)
text input, (2) word encoder, (3) talkturn encoder, and (4) the predictor. In
our application, we perform our predictions at the talkturn-level for both
IEMOCAP and SEMAINE. For notation, let $s_{i}$ represent the i-th talkturn and
$w_{it}$ represent the t-th word in the i-th talkturn. Each single talkturn
can contain up to T words, and each input talkturn can contain up to L past
talkturns to give content context (see section 4.2).
Given a talkturn of words, we first convert the words into vectors through an
embedding matrix $W_{e}$, and the word selection one-hot vector, $w_{it}$. The
word encoder comprises of bidirectional GRUs (Bahdanau et al., 2014) and a
single head attention to aggregate word embeddings into talkturn embeddings.
Given the vectors $x_{it}$, the bidirectional GRU reads the words from left to
right as well as from right to left (as indicated by the direction of the GRU
arrows) and concatenates the two hidden states together to form $h_{it}$. We
then aggregate the hidden states into one talkturn embedding through the
attention mechanism. $u_{it}$ is the hidden state from feeding $h_{it}$ into a
one-layer perceptron (with weights $W_{w}$ and biases $b_{w}$). The attention
weight ($\alpha_{it}$) given to $u_{it}$ is the softmax normalized weight of
the similarity between itself ($u_{it}$) and $u_{w}$, which are all randomly
initialized and learnt jointly.
Figure 2: Forward pass of the Flat-MTL HAN architecture. Auxiliary tasks
(yellow) are added at the same level of the primary task (orange). Figure 3:
Forward pass of the HAN-ROCK architecture. There is a primary branch (orange)
where auxiliary supervision (yellow) can not influence. The fusion module
(blue) aggregates the talkturn embeddings from all tasks into one.
$\displaystyle x_{it}=W_{e}w_{it},t\in[1,T].$
$\displaystyle\overrightarrow{h}_{it}=\overrightarrow{GRU}(x_{it}),t\in[1,T].$
$\displaystyle\overleftarrow{h}_{it}=\overleftarrow{GRU}(x_{it}),t\in[T,1].$
$\displaystyle{h}_{it}=(\overrightarrow{h}_{it},\overleftarrow{h}_{it}).$
$\displaystyle u_{it}=relu(W_{w}h_{it}+b_{w}).$ $\displaystyle
s_{i}=\Sigma_{t}\alpha_{it}u_{it}.$
$\displaystyle\alpha_{it}=\frac{exp(u_{it}^{\top}u_{w})}{\Sigma_{t}exp(u_{it}^{\top}u_{w})}.$
With the current and past talkturn embeddings (content context, to discuss in
section 4.2), the talkturn encoder aggregates them into a single talkturn
representation ($v$) in a similar fashion, as shown below.
$\displaystyle\overrightarrow{h}_{i}=\overrightarrow{GRU}(s_{i}),i\in[1,L].$
$\displaystyle\overleftarrow{h}_{i}=\overleftarrow{GRU}(s_{i}),i\in[L,1].$
$\displaystyle{h}_{i}=(\overrightarrow{h}_{i},\overleftarrow{h}_{i}).$
$\displaystyle u_{i}=relu(W_{s}h_{i}+b_{s}).$
$\displaystyle\alpha_{i}=\frac{exp(u_{i}^{\top}u_{s})}{\Sigma_{i}exp(u_{i}^{\top}u_{s})}.$
$\displaystyle v=\Sigma_{i}\alpha_{i}u_{i}.$
The simplest way of adding the sixteen auxiliary task predictors would be to
append them to where the primary task predictor is, as illustrated in Fig. 2.
That way, all predictors use the same representation $v$. We refer to this
architecture as flat-MTL, but we are unable to test H2 and H3 using this
architecture. Therefore, we introduce HAN-ROCK next.
### 3.2 HAN-ROCK
We adapted111Implementation will be available at GitHub; Please see attached
supplementary material during the review phase. the ROCK architecture (Mordan
et al., 2018) which was built for Convolutional Neural Networks (LeCun et al.,
1995) found in ResNet-SSD (He et al., 2016; Liu et al., 2016) to suit GRUs
(Bahdanau et al., 2014) found in HAN (Yang et al., 2016b) (see Fig. 3). To
study H3, we bring the auxiliary task predictors forward (see Fig. 3), so that
the back-propagation from the primary supervision is able to temper the back-
propagation from the auxiliary supervision but not vice-versa. This also sets
us up to study H2. Each of the auxiliary tasks has its own talkturn encoder
but shares one word encoder in the auxiliary branch (to keep the network
small). Subscript $a$ indicates whether the word encoder is for the primary or
auxiliary branch:
$\displaystyle x_{it}=W_{e}w_{it},t\in[1,T]$
$\displaystyle\overrightarrow{h}_{ait}=\overrightarrow{GRU_{a}}(x_{it}),t\in[1,T],a\in\\{pri,aux\\}$
$\displaystyle\overleftarrow{h}_{ait}=\overleftarrow{GRU_{a}}(x_{it}),t\in[T,1],a\in\\{pri,aux\\}$
$\displaystyle{h}_{ait}=(\overrightarrow{h}_{ait},\overleftarrow{h}_{ait})$
$\displaystyle u_{ait}=relu(W_{aw}h_{ait}+b_{aw})$
$\displaystyle\alpha_{ait}=\frac{exp(u_{ait}^{\top}u_{aw})}{\Sigma_{t}exp(u_{ait}^{\top}u_{aw})}$
$\displaystyle s_{ai}=\Sigma_{t}\alpha_{ait}u_{ait}$
Figure 4: Example of a MONAH transcript.
Each task has its own talkturn encoder. Subscript $b$ indicates which of the
seventeen tasks – the primary talkturn task or one of the sixteen auxiliary
tasks – is the talkturn encoder is dedicated to:
$\displaystyle\overrightarrow{h}_{abi}=\overrightarrow{GRU_{ab}}(s_{ai}),i\in[1,L],a\in\\{pri,aux\\},$
$\displaystyle b\in\\{pri,aux1,aux2,...,aux16\\}$
$\displaystyle\overleftarrow{h}_{abi}=\overleftarrow{GRU_{ab}}(s_{ai}),i\in[L,1],a\in\\{pri,aux\\},$
$\displaystyle b\in\\{pri,aux1,aux2,...,aux16\\}$
$\displaystyle{h}_{abi}=(\overrightarrow{h}_{abi},\overleftarrow{h}_{abi})$
$\displaystyle u_{abi}=relu(W_{b}h_{abi}+b_{b})$
$\displaystyle\alpha_{abi}=\frac{exp(u_{abi}^{\top}u_{b})}{\Sigma_{i}exp(u_{abi}^{\top}u_{b})}$
$\displaystyle v_{ab}=\Sigma_{i}\alpha_{abi}u_{abi}$
The seventeen talkturn embeddings ($v_{ab}$) goes through a concatenation,
then the single head attention, aggregating talkturn embeddings across
seventeen tasks into one talkturn embedding for the primary task predictor.
Subscript $c$ pertains to the fusion module.
$\displaystyle\text{concatenation: }v_{c}=(v_{ab}),a\in\\{pri,aux\\},$
$\displaystyle b\in\\{pri,aux1,aux2,...,aux16\\}$
$\displaystyle\text{attention:
}\alpha_{c}=\frac{exp(v_{c}^{\top}u_{c})}{\Sigma_{c}exp(v_{c}^{\top}u_{c})}$
$\displaystyle\text{overall primary talkturn vector:
}v=\Sigma_{c}\alpha_{c}v_{c}$
## 4 Experiments
### 4.1 Data and Primary Tasks
We validate our approach using two datasets with a total of eight primary
tasks: the IEMOCAP (Busso et al., 2008) and the SEMAINE (McKeown et al., 2011)
datasets. Both datasets are used in multimodal emotions detection research
(Poria et al., 2019). We divided the datasets into train, development, and
test sets in an approximate 60/20/20 ratio such that the sets do not share any
speaker (Appendix A.1 details the splits).
The target labels of the eight primary tasks are all at the talkturn-level.
The four primary tasks of IEMOCAP consists of the four-class emotions
classification (angry, happy, neutral, sad), and three regression problems –
valence (1-negative, 5-positive), activation (1-calm, 5-excited), and
dominance (1-weak, 5-strong). The four-class emotions classification target is
common for IEMOCAP (Latif et al., 2020; Xia and Liu, 2015; Li et al., 2019;
Hazarika et al., 2018b; Mittal et al., 2020). For SEMAINE, there are four
regression problems – activation, intensity, power, valence. We use two
standard evaluation metrics, mean absolute error (MAE), and 4-class weighted
mean classification accuracy, MA(4).
### 4.2 Input
Multimodal feature extraction is computed using the MONAH framework (Kim et
al., 2021). This framework uses a variety of pre-trained models to extract
nine multimodal features, associated with the prosody of the speech and the
actions of the speaker, and weaves them into a multimodal text narrative. We
refer the reader to Kim et al. (2021) for the details and efficacy of the
MONAH framework. The benefit of the created narrative is that it describes
what is said together with how it is said for each talkturn, giving richer
nonverbal context to the talkturn (see Fig. 4 for an example). Being fully
text-based means that the analysis product can be printed out on paper,
without the need for speakers nor monitors to replay the conversation on a
computer.
In addition to nonverbal context, we concatenated a variable number of
preceding talkturns to the current talkturn as content context. Content
context has been proven to be useful in CMN (Hazarika et al., 2018b) and ICON
(Hazarika et al., 2018a), DialogueRNN (Majumder et al., 2019). The content-
context size is tuned as a hyperparameter. The resulting multimodal text
narrative, consisting of both nonverbal and context context, is used as the
sole input to the model.
### 4.3 Auxiliary Targets
We first clarify the method of extraction for the auxiliary families. The
OpenFace algorithm (Baltrušaitis et al., 2016) is used to extract the four
continuous facial action units (AU) – AU 05, 17, 20, 25. The Vokaturi
algorithm (Vokaturi, 2019) is used to extract the four continuous dimensions
in the tone of voice – happiness, sadness, anger, and fear. As for historical
and future features, we simply look up the target label for the past four
talkturns and future four talkturns. Any label that is not available (for
example, the label four talkturns ago is not available for the third talkturn)
is substituted with the next nearest non-missing label.
All auxiliary targets that reused the input features (actions and prosody) are
converted into a percentile rank that has the range [0,1] using the values
from the train partition. This is a subtle but note-worthy transformation.
When reusing an input as an auxiliary target, it would be trivial if the input
can easily predict the target. For example, given the following MONAH
transcript as input, “The woman sadly and slowly said no." It would be trivial
to use a binary (quantized) auxiliary target of “was the tone sad?" because we
would only be training the model to look for the word “sadly". However, if the
auxiliary target is a percentile rank (less quantized) of the sadness in tone,
then the presence of the word “sadly" increases the predicted rank, but the
model could still use the rest of the nonverbal cues (“slowly") and what is
being said (“no") to predict the degree of sadness. That way, representations
learnt for the auxiliary tasks uses more of the input.
Percentile rank also has the convenient property of having the range [0,1]. We
scaled the percentile ranks so that they all have the same range as the
primary task (see appendix A.2 for transformation details). This ensures that
if we assigned equal loss weights to all tasks, the contribution of every task
is of the same order of magnitude (Gong et al., 2019; Hassani and Haley, 2019;
Sener and Koltun, 2018).
Table 1: H1 Results. *: the model performance has a statistically significant difference with the baseline model. a: action, p: prosody, h: historical labels, f: future labels. | IEMOCAP | SEMAINE
---|---|---
Aux. Target | Classif.
MA(4) | Val.
MAE | Act.
MAE | Dom.
MAE | Act.
MAE | Int.
MAE | Pow.
MAE | Val.
MAE
None (Baseline) | 0.625 | 0.527 | 0.518 | 0.667 | 0.194 | 0.238 | 0.170 | 0.177
ap | 0.715* | 0.538 | 0.507 | 0.600* | 0.184* | 0.218* | 0.167* | 0.178
aphf | 0.706* | 0.497* | 0.504 | 0.587* | 0.187 | 0.231 | 0.165* | 0.169
Table 2: H2 Results. *: the model performance has a statistically significant difference with the baseline model (P = 256). ^: Assigning 1 GRU to the auxiliary task talkturn encoder yields a statistically significant difference with assigning 0 GRU. | IEMOCAP | SEMAINE
---|---|---
Pri. GRU | Classif.
MA(4) | Val.
MAE | Act.
MAE | Dom.
MAE | Act.
MAE | Int.
MAE | Pow.
MAE | Val.
MAE
256 (Baseline) | 0.715^ | 0.538 | 0.507 | 0.587^ | 0.187 | 0.231 | 0.165^ | 0.169^
192 | 0.736 | 0.509 | 0.518 | 0.604 | 0.187 | 0.228 | 0.165 | 0.184*
128 | 0.711 | 0.537 | 0.512 | 0.597 | 0.189 | 0.216 | 0.176* | 0.196*
64 | 0.687 | 0.540 | 0.507 | 0.593 | 0.191 | 0.234 | 0.167 | 0.192*
1 | 0.656 | 0.554 | 0.509 | 0.599 | 0.190 | 0.229 | 0.168* | 0.191*
### 4.4 Models, training, and hyperparameters tuning
The overall loss is calculated as the weighted average across all seventeen
tasks: (1) we picked a random weight for the primary task from the range
[0.50, 0.99]; this ensures that the primary task has the majority weight. (2)
For the remaining weights (1 - primary weight), we allocated them to the
sixteen auxiliary tasks by: (a) random, (b) linearly-normalized mutual
information, or (c) softmax-normalized mutual information. (a) is self-
explanatory. As for (b) and (c), mutual information has been shown that it is
the best predictor – compared to entropy and conditional entropy – of whether
the auxiliary task would be helpful (Bjerva, 2017). We computed the mutual
information (vector $m$) of each auxiliary variable with the primary target
variable (Kraskov et al., 2004; Ross, 2014) using scikit-learn (Pedregosa et
al., 2011). Then, we linearly-normalized or softmax-normalized $m$ to sum up
to 1. Finally, we multiplied the normalized $m$ with the remaining weights
from (2); this ensures that the primary weight and the sixteen auxiliary
weight sum up to one. (a), (b), and (c) have ten trials each during hyper-
parameters tuning.
Two variants of the HAN architectures are used (Fig 2 and 3). For hypotheses
testing, we bootstrapped confidence intervals (appendix A.4).
## 5 Results and Discussion
The key takeaways are: (H1) The introduced set of auxiliary supervision
improves primary task performance significantly in six of the eight primary
tasks. (H2) Maximum learning capacity should be given to the primary branch as
a default. (H3) HAN-ROCK is unlikely (in one of the eight tasks) to degrade
primary task performance significantly, and sometimes significantly improves
it (in four of the eight tasks).
(H1): To test H1 (whether the introduced set of auxiliary supervision improves
primary task performance), we first train the model with all sixteen auxiliary
targets (from families: actions, prosody, historical, and future). Then, to
differentiate the effect from the historical and future supervision, we set
the loss weights from historical and future targets to be zero; effectively,
there is only supervision from eight auxiliary targets (actions and prosody).
Lastly, for the baseline model (no auxiliary supervision), we set the loss
weights from all sixteen auxiliary targets to zero.
Table 3: H3 Results. *: the model performance has a significant difference with the baseline. | IEMOCAP | SEMAINE
---|---|---
Hierarchy | Classif.
MA(4) | Val.
MAE | Act.
MAE | Dom.
MAE | Act.
MAE | Int.
MAE | Pow.
MAE | Val.
MAE
Flat (Baseline) | 0.699 | 0.520 | 0.526 | 0.606 | 0.183 | 0.230 | 0.164 | 0.185
HAN-ROCK | 0.715 | 0.538 | 0.507* | 0.600* | 0.184 | 0.218* | 0.167* | 0.178*
Table 4: Class-wise classification F1 score on IEMOCAP. Baseline (challenger) refers to HAN-Rock architecture under the three hypotheses. *: the challenger performance has a statistically significant difference with the baseline model. Distribution | H1 | H2 | H3 | SoTA
---|---|---|---|---
Label | Count | Baseline
Aux target: None | Challenger
Aux target: aphf | Baseline
Pri GRU: 256 | Challenger
Pri GRU: 1 | Baseline
Hier-archy: Flat | Challenger
Hier-archy: HAN ROCK | M3-ER
Sad | 1,084 | 0.573 | 0.689* | 0.699 | 0.591* | 0.674 | 0.704* | 0.775
Anger | 1,103 | 0.531 | 0.683 | 0.752 | 0.672* | 0.657 | 0.720* | 0.862
Happy | 1,636 | 0.772 | 0.784 | 0.776 | 0.754* | 0.804 | 0.806 | 0.862
Neutral | 1,708 | 0.664 | 0.636 | 0.688 | 0.627* | 0.645 | 0.631 | 0.745
Given auxiliary supervision, the model significantly outperforms the baseline
of not having auxiliary supervision in six out of the eight primary tasks
(Table 1). Comparing the baseline model with the model with two auxiliary
target families, they significantly outperformed the baseline model in five
out of eight primary tasks. The addition of two auxiliary target families
(historical and future labels) sometimes significantly improved primary task
performance (valence in IEMOCAP), but it also sometimes significantly made it
worse (activation and intensity in SEMAINE). This shows that the value of
auxiliary tasks, and the associated risk of negative transfer, depends on the
auxiliary task.
(H2): To test H2 (whether maximum learning capacity should be given to the
primary branch), we let P represent the number of GRU assigned to the primary
talkturn encoder, and A represent the number of GRU assigned to each of the
sixteen auxiliary talkturn encoder. We constrained P \+ A to be equal to 257.
During our experiments, we set P to 1, 64, 128, 192, and 256. We set 256 as
the baseline model because it is the maximum learning capacity we can give to
the primary branch while giving 1 GRU ($=257-256$) to each of the sixteen
auxiliary talkturn encoders.
In all primary tasks, the baseline model of assigning 256 GRUs to the primary
branch is not significantly outperformed by models that assigned 1, 64, 128,
192 GRUs (Table 2). Generally, the performance decreased as the number of GRUs
assigned to the primary talkturn encoder decreased from 256 to 1. We observed
significantly worse performance in two out of eight tasks – in power and
valence in SEMAINE. Also, assigning 256 GRU to the primary talkturn encoders
and 1 to each of the sixteen auxiliary talkturn encoders yields the smallest
model222As opposed to assigning 1 GRU to the primary talkturn encoder and 256
GRU to each of the sixteen auxiliary encoder., and thus trains the fastest.
Therefore, we recommend that the maximum capacity be given to the primary
branch as a default.
That said, the presence of an auxiliary branch is still important. The
baseline of H1 (no auxiliary supervision, Table 1) can be approximated333Same
model architecture except that the loss weights of all auxiliary tasks are
zero as P=$256+16\times 1$, A=0 . We compared the former to the baseline in
Table 2, and found that four out of eight primary tasks have significant
improvements by changing the number of talkturn encoders assigned to each
auxiliary task from zero to one.
(H3): To test H3 (whether auxiliary supervision should be given a lower
hierarchy), we compare the results from the flat-MTL HAN architecture
(baseline) against the HAN-ROCK architecture (Table 3). Placing auxiliary
supervision at the lower hierarchy significantly improves primary task
performance in four out of eight tasks. In only one out of eight tasks (power
in SEMAINE), auxiliary supervision significantly degrades primary task
performance. Further improvements are possible through the fusion module with
future research.
### 5.1 Class-wise Performance and SoTA
Generally, we found that all hypotheses effects are stronger in lower resource
labels (sad and anger, Table 4). We also present the performance of M3ER
(Mittal et al., 2020), a previous state-of-the-art (SoTA) approach. We do not
expect the performance of our text-only input to match the SoTA approach,
which is confirmed in Table 4. By fusing numerical vectors from the three
modalities prevalent in SoTA approaches (Zadeh et al., 2018a, b; Hazarika et
al., 2018b, a; Majumder et al., 2019; Mittal et al., 2020), the inputs are of
a much higher granularity as compared to our approach of describing the
multimodal cues using discrete words. Although the text-based input is likely
to constrain model performance, the multimodal transcription could be helpful
for a human to analyze the conversation, we could also overlay the model
perspective on the multimodal transcription to augment human analysis (see
Appendix A.5).
## 6 Conclusion
We proposed to re-use feature engineering pre-processing data as auxiliary
tasks to improve performance and transfer learning. Three hypotheses were
tested. The experimental results confirm H1 – Introducing our set of sixteen
auxiliary supervisors resulted in better performance in most primary tasks.
For H2, maximum learning capacity should be given to the primary branch.
Lastly, for H3, placing the auxiliary supervision in a lower hierarchy is
unlikely to hurt performance significantly, and it sometimes significantly
improves performance. This is encouraging news for multi-modal conversational
analysis systems as we have demonstrated how pre-processed data can be used
twice to improve performance, once as inputs, and again as auxiliary tasks.
The first limitation of our paper is that the solutions are evaluated on eight
tasks in the conversational analysis domain, and it is not clear if these
would generalize outside of this domain. The second limitation is that we have
evaluated on HAN, but not on other network architectures.
A challenge to be addressed is the apriori selection of the auxiliary targets.
Future research could investigate targets selection, including how to use a
much larger range of auxiliary targets, how to decide the optimum number of
auxiliary targets, and whether it is possible to perform these automatically.
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## Appendix A Appendix
### A.1 Dataset partitions
We detail the dataset partition in Table 5 for aid reproducibility.
Table 5: Dataset partitions IEMOCAP | SEMAINE
---|---
Partition | Session | Count | Partition | Session | Count
Train | Ses01F | 861 | Train | 2008.12.05.16.03.15 | 424
Train | Ses01M | 958 | Train | 2009.01.30.12.00.35 | 404
Train | Ses02F | 889 | Train | 2009.02.12.10.49.45 | 686
Train | Ses02M | 922 | Train | 2009.05.15.15.04.29 | 436
Train | Ses03F | 958 | Train | 2009.05.29.14.30.05 | 760
Train | Ses03M | 1178 | Train | 2009.05.22.15.17.45 | 668
Dev | Ses04F | 1105 | Train | 2009.05.25.11.23.09 | 928
Dev | Ses04M | 998 | Train | 2009.05.26.10.19.53 | 790
Test | Ses05F | 1128 | Train | 2009.06.05.10.14.28 | 732
Test | Ses05M | 1042 | Train | 2009.06.15.12.13.06 | 958
| | | Train | 2009.06.19.14.01.24 | 586
| | | Train | 2009.10.27.16.17.38 | 440
| | | Dev | 2008.12.19.11.03.11 | 188
| | | Dev | 2009.01.06.14.53.49 | 472
| | | Dev | 2009.05.12.15.02.01 | 448
| | | Dev | 2009.05.08.11.28.48 | 752
| | | Dev | 2009.06.26.14.38.17 | 404
| | | Test | 2008.12.14.14.47.07 | 372
| | | Test | 2009.01.06.12.41.42 | 264
| | | Test | 2009.01.28.15.35.20 | 364
| | | Test | 2009.06.26.14.38.17 | 440
| | | Test | 2009.06.26.14.09.45 | 448
### A.2 Scaling the auxiliary targets
We detail the operations in scaling the percentile scores that range [0,1] to
various primary tasks. For IEMOCAP Primary Tasks that are regression problems,
we multiply the percentile score by 4 and add 1 to obtain the range [1,5]. For
IEMOCAP classification task, we leave the auxiliary targets in the range of
[0,1]. As for SEMAINE tasks, which are all regression problems, we multiply
the percentile score by 2 and minus 1 to obtain the range [-1,1].
### A.3 Hyperparameters tuning process
Glove word embeddings (300-dimensions) are used to represent the words
(Pennington et al., 2014). Hyper-parameters tuning is crucial because
different combinations of primary and auxiliary tasks require different sets
of hyperparameters. For hyperparameters tuning, we used random search
(Bergstra and Bengio, 2012) with thirty trials. We tuned the learning rate,
batch size, L2 regularization, the number of GRUs assigned to the primary and
auxiliary branches, the auxiliary weights assignment, the content-context
size, and lastly the GRU dropout and recurrent dropout (as detailed in Table
6).
Training is done on a RTX2070 or a V100 GPU, for up to 350 epochs. Early
stopping is possible via the median-stopping rule (Golovin et al., 2017) after
the fifth epoch and after every two epochs (i.e., at epoch number 5, 7, 9,
$...$, 349). Table 7 details the hyperparameters of models that performed the
best on the development set.
Table 6: Range of Hyperparameters tuned. U: Uniform sampling, LU: Uniform sampling on the logarithmic scale. Name | Min. | Max | Sampling
---|---|---|---
Learning rate | $2^{-10}$ | $2^{-5}$ | LU
Batch-Size | 32 | 256 | U
Pri GRU | 1, 64, 128, 192, 256 | U
Aux GRU | 257 - (minus) Pri GRU
Loss-weights Assignment | Random, Linear-Normalized, Softmax-Normalized | U
L2 Regularization | 0.0 | 0.50 | LU
Content-Size | 1 | 30 | U
GRU dropout | 0.01 | 0.50 | U
Recurrent dropout | 0.01 | 0.50 | U
Table 7: Best Hyperparameters settings for development set. R: Random, L: Linear-Normalized, S: Softmax-Normalized. | IEMOCAP | SEMAINE
---|---|---
| Classif.
MA(4) | Val.
MAE | Act.
MAE | Dom.
MAE | Act.
MAE | Int.
MAE | Pow.
MAE | Val.
MAE
Dev. set | 0.714 | 0.482 | 0.492 | 0.570 | 0.118 | 0.164 | 0.135 | 0.138
Test set | 0.747 | 0.497 | 0.499 | 0.587 | 0.184 | 0.215 | 0.164 | 0.169
Learning rate | 6.21 e-03 | 2.21 e-02 | 1.44 e-02 | 3.13 e-02 | 2.43 e-02 | 1.58 e-02 | 3.04 e-03 | 2.94 e-02
Batch size | 43 | 43 | 62 | 77 | 41 | 43 | 33 | 62
Pri GRU | 256 | 256 | 256 | 256 | 256 | 256 | 192 | 256
Aux GRU | 1 | 1 | 1 | 1 | 1 | 1 | 65 | 1
L2 regularization | 2.25 e-05 | 0 | 1.43 e-05 | 2.41 e-04 | 0 | 0 | 0 | 0
Content-Size | 18 | 4 | 3 | 3 | 17 | 21 | 19 | 14
GRU dropout | 0.27 | 0.49 | 0.33 | 0.30 | 0.06 | 0.17 | 0.05 | 0.07
Recurrent dropout | 0.04 | 0.03 | 0.02 | 0.32 | 0.25 | 0.28 | 0.08 | 0.26
Epoch No. | 232 | 161 | 74 | 14 | 48 | 116 | 110 | 110
Aux. weights assignment | S | R | L | L | L | R | S | R
Main loss | 0.840 | 0.560 | 0.810 | 0.610 | 0.870 | 0.940 | 0.950 | 0.920
AU05 loss | 0.009 | 0.013 | 0.014 | 0.000 | 0.000 | 0.002 | 0.002 | 0.008
AU17 loss | 0.009 | 0.027 | 0.009 | 0.003 | 0.002 | 0.004 | 0.002 | 0.003
AU20 loss | 0.009 | 0.028 | 0.0 | 0.001 | 0.000 | 0.007 | 0.002 | 0.006
AU25 loss | 0.009 | 0.031 | 0.006 | 0.002 | 0.015 | 0.008 | 0.002 | 0.009
Happy tone loss | 0.010 | 0.023 | 0.038 | 0.006 | 0.021 | 0.013 | 0.002 | 0.003
Sad tone loss | 0.010 | 0.028 | 0.053 | 0.008 | 0.037 | 0.010 | 0.002 | 0.003
Angry tone loss | 0.010 | 0.016 | 0.034 | 0.006 | 0.055 | 0.011 | 0.002 | 0.000
Fear tone loss | 0.010 | 0.043 | 0.036 | 0.004 | 0.000 | 0.004 | 0.002 | 0.003
$Y_{t-1}$ loss | 0.010 | 0.030 | 0.0 | 0.040 | 0.000 | 0.000 | 0.005 | 0.006
$Y_{t-2}$ loss | 0.011 | 0.022 | 0.0 | 0.049 | 0.000 | 0.000 | 0.004 | 0.004
$Y_{t-3}$ loss | 0.010 | 0.006 | 0.0 | 0.046 | 0.000 | 0.000 | 0.003 | 0.006
$Y_{t-4}$ loss | 0.010 | 0.040 | 0.0 | 0.044 | 0.000 | 0.000 | 0.003 | 0.007
$Y_{t+1}$ loss | 0.010 | 0.042 | 0.0 | 0.042 | 0.000 | 0.000 | 0.005 | 0.005
$Y_{t+2}$ loss | 0.011 | 0.0 | 0.0 | 0.052 | 0.000 | 0.000 | 0.004 | 0.006
$Y_{t+3}$ loss | 0.010 | 0.050 | 0.0 | 0.043 | 0.000 | 0.000 | 0.003 | 0.008
$Y_{t+4}$ loss | 0.010 | 0.038 | 0.0 | 0.044 | 0.000 | 0.000 | 0.003 | 0.003
### A.4 Details of computing the bootstrap confidence interval
Baseline models for each hypothesis are detailed in section 2. All non-
baseline models are referred to as challenger models. We created 1000
bootstrap samples of the test set performance by (1) resampling the
development set performance, then (2) selecting the set of hyperparameters
that resulted in the best development set performance, and (3) looking up the
test set performance given the set of best-performing hyperparameters for the
development set. To judge whether the challenger outperforms the baseline, we
computed the 95 percent confidence interval by (1) performing element-wise
subtraction between the resampled test set performance of the baseline against
the challenger, (2) removing the top and bottom 2.5 percent from the
differences, and (3) observing whether the remaining 95 percent confidence
interval includes zero. If it does not include zero, then the difference is
statistically significant.
### A.5 Visualization from HAN-ROCK
We demonstrate how the HAN-ROCK model could be used to support humans analyze
conversations using only text-based inputs. We visualized the attention
weights from two models, (1) MTL refers to the classification model with
auxiliary supervisors, whilst (2) STL refers to the same model architecture,
but its auxiliary supervisors’ loss weights are set to zero. In principle, the
MTL model should exhibit attention weights that are less likely to overfit
because the weights are tempered by auxiliary supervisors. We observe that
across the two models, both use the historical talkturns more so than the
current talkturn; secondly, both assign high attention to the second word of
the talkturns, which is interesting because the second word is where the
multimodal annotations are inserted.
Figure 5: Conversation analysis example. Both models predicted the class label
(anger) correctly. The left-most column denotes the talkturn context – $i$
refers to the current talkturn where the target emotion class is predicted.
The square boxes indicate the level of attention (N: None, L: Low, M: Medium,
H: High) assigned to the talkturn. Within the talkturn, we also enlarge and
darken the font color ti visualize higher attention weights.
As explained in Section 3.2, there are three levels of attention over the
internal representations, word ($\alpha_{it}$), talkturn ($\alpha_{abi}$), and
task ($\alpha_{c}$). To compute the overall word and talkturn attention, we
compute the weighted average of $\alpha_{it}$ and $\alpha_{abi}$ using the
$\alpha_{c}$ (task attention) as weights. Once we have the overall word and
talkturn attention, we standardize the weights by computing the z-score.
Depending on the z-score, we bucket the attention in none (z $<$ 0), low (0
$<$ z $<$ 1), medium (1 $<$ z $<$ 2), or high (2 $<$ z). We plan to validate
the efficacy of the attention weights with human users in future research.
|
# Revealing Galaxy Candidates out to $z\sim 16$ with JWST Observations of the
Lensing Cluster SMACS0723
Hakim Atek1, Marko Shuntov1, Lukas J. Furtak2, Johan Richard3, Jean-Paul
Kneib4, Guillaume Mahler5, Adi Zitrin2, H.J. McCracken1, Stéphane Charlot1,
Jacopo Chevallard6, Iryna Chemerynska1
1Institut d’Astrophysique de Paris, CNRS, Sorbonne Université, 98bis Boulevard
Arago, 75014, Paris, France
2Physics Department, Ben-Gurion University of the Negev, P.O. Box 653, Be’er-
Sheva 84105, Israel
3Univ Lyon, Univ Lyon1, Ens de Lyon, CNRS, Centre de Recherche Astrophysique
de Lyon UMR5574, F-69230, Saint-Genis Laval, France
4Laboratoire d’Astrophysique, Ecole Polytechnique Fédérale de Lausanne,
Observatoire de Sauverny, CH-1290 Versoix, Switzerland
5Institute for Computational Cosmology, Durham University, South Road, Durham
DH1 3LE, UK
6Department of Physics, University of Oxford, Denys Wilkinson Building, Keble
Road, Oxford OX1 3RH, UK
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
One of the main goals of the JWST is to study the first galaxies in the
Universe. We present a systematic photometric analysis of very distant
galaxies in the first JWST deep field towards the massive lensing cluster
SMACS0723. As a result, we report the discovery of two galaxy candidates at
$z\sim 16$, only $250$ million years after the Big Bang. We also identify two
candidates at $z\sim 12$ and 6 candidates at $z\sim 9-11$. Our search extended
out to $z\lesssim 21$ by combining color information across seven NIRCam and
NIRISS filters. By modelling the Spectral Energy Distributions (SEDs) with
EAZY and BEAGLE, we test the robustness of the photometric redshift estimates.
While their intrinsic (un-lensed) luminosity is typical of the characteristic
luminosity L∗ at $z>10$, our high-redshift galaxies typically show small sizes
and their morphologies are consistent with disks in some cases. The highest-
redshift candidates have extremely blue UV-continuum slopes $-3<\beta<-2.4$,
young ages $\sim 10-100$ Myr, and stellar masses around
$\log(M_{\star}/\mathrm{M}_{\odot})=8.8$ inferred from their SED modeling,
which indicate a rapid build-up of their stellar mass. Our search clearly
demonstrates the capabilities of JWST to uncover robust photometric candidates
up to very high redshifts, and peer into the formation epoch of the first
galaxies.
###### keywords:
galaxies: high-redshift – dark ages, reionization, first stars – galaxies:
dwarfs – galaxies: evolution – gravitational lensing: strong – cosmology:
observations
††pubyear: 2022††pagerange: Revealing Galaxy Candidates out to $z\sim 16$ with
JWST Observations of the Lensing Cluster SMACS0723–LABEL:lastpage
## 1 Introduction
Understanding the emergence of the first structures that hosted star formation
has been one of the most important goals in modern astrophysics. One of the
main routes to address this question is to identify the first generation of
galaxies that we believe formed at $z\gtrsim 15$. Great progress has been made
in the last decade to uncover the most distant galaxies out to $z\sim 11$,
essentially thanks to the Hubble Space Telescope (HST) and 8-10 m class
ground-based telescopes. The record-holder for the farthest galaxy known with
a spectroscopic confirmation is GN-z11 (Oesch et al., 2016; Jiang et al.,
2021), at a redshift of $z=10.96$. Remarkably, GN-z11 is a relatively bright
and massive galaxy ($M_{\mathrm{UV}}=-22.1$ AB), which challenges current
galaxy formation models. The existence of a galaxy with a billion solar masses
only $\sim 400$ Myr after the Big Bang indicates that galaxy formation was
well underway very early in the history of the Universe.
At slightly lower redshifts $z\sim 7-9$, large samples of nearly 2,000
galaxies have been photometrically identified, thanks mostly to the infrared
sensitivity of the Wide Field Camera Three (WFC3) aboard HST (e.g. Finkelstein
et al., 2015; Bouwens et al., 2021) or ground-based observations (e.g.,
Kauffmann et al., 2022). These large samples resulted in strong constraints on
the shape of the ultra-violet (UV) luminosity function of galaxies at those
epochs and its evolution with redshift. Moreover, Spitzer Space Telescope
observations allowed to probe the rest-frame optical emission of these high-
redshift galaxies, which enabled us to constrain their stellar mass (e.g. Song
et al., 2016; Bhatawdekar et al., 2019; Kikuchihara et al., 2020; Stefanon et
al., 2021), albeit with important uncertainties due to source confusion and
nebular emission contamination (Grazian et al., 2015; Furtak et al., 2021).
More recently, using deep Spitzer and ground-based observations, Harikane et
al. (2022) reported the detection of two galaxy candidates at $z\sim 12-13$.
Atacama Large Millimeter/submillimeter Array (ALMA) follow-up observations
have revealed the tentative ($2\sigma$) detection of an [O iii]$\lambda
88\mu\mathrm{m}$ emission line, putting HD1 at a redshift of $z=13.27$. Again,
these two candidates do not align with theoretical models, which predict a
significantly lower density of such bright galaxies at $z>10$. While wide
extra-galactic surveys have unveiled these surprisingly large numbers of
luminous galaxies at $z\sim 9-11$, deeper observations at longer wavelengths
are needed to, (i) push the redshift frontier to uncover farther away galaxies
and (ii) constrain the number density of faint galaxies at early times.
The advent of the JWST marks a new era in the detection and, more importantly,
the characterization of early star-forming galaxies. The most important
improvement is the significantly higher spatial resolution of JWST compared to
Spitzer, which enables accurate photometric measurements in the near and mid-
infrared range corresponding to the rest-frame optical for galaxies at $z>6$.
This is crucial for deriving physical properties of galaxies through spectral
energy distribution (SED) fitting, since it allows us to account for a larger
dynamic range in the age of the underlying stellar populations. Also, for the
first time, we will be able to obtain rest-frame optical spectra for these
early-star forming galaxies, giving us access to several optical emission
line, which are the gold standards for measuring galaxy properties such as
star formation rate (SFR), gas-phase metallicity, dust content, etc.
The first JWST observations consist of the Early Release Observations (ERO),
which were obtained to showcase the observatory capabilities, and the Early
Release Science (ERS) programs, aimed at testing the variety of instruments
observing modes and to help the community understand and exploit JWST data.
One of the primary targets of the ERO observations is the galaxy cluster SMACS
J0723.3-7327 (SMACS0723 hereafter; Ebeling et al., 2001; Repp & Ebeling, 2018)
which was recently observed with HST as part of the Reionization Lensing
Cluster Survey program (RELICS; Coe et al., 2019; Salmon et al., 2020). One of
the accepted ERS programs (1324, PI: Treu) also targets a lensing cluster,
Abell 2744 (A2744), which is one of the Hubble Frontier Fields (HFF; Lotz et
al., 2017) clusters. These observations follow in the steps of numerous HST
observing campaigns and efforts to use massive galaxy clusters as
gravitational telescopes. The magnification provided by the strong
gravitational lensing (SL) enables us to reach intrinsically fainter and
lower-mass galaxies and thus to constrain the very faint and low-mass ends of
the luminosity and stellar mass function (e.g. Atek et al., 2014a; Atek et
al., 2018; Livermore et al., 2017; Bouwens et al., 2017; Ishigaki et al.,
2018; Bhatawdekar et al., 2019; Kikuchihara et al., 2020; Furtak et al.,
2021).
In this work, we use the JWST ERO data of SMACS0723 to identify and
characterize the most distant galaxies in the Universe. We use imaging data in
7 broad-band filters to select $z\gtrsim 9$ galaxies based on the Lyman break
technique coupled with SED fitting to estimate their photometric redshifts and
physical properties. This paper is structured as follows. In section 2, we
present the observational data. In section 3, we describe how the galaxy
sample was selected. The SL model used in this work and our SED-fitting
analysis are shown in sections 4 and 5 respectively. Finally, we compare our
estimate with a compilation of literature results and discuss the implications
on galaxy formation models in section 6 before presenting our summary and
conclusions in section 7. Throughout the paper, magnitudes are in the AB
system (Oke & Gunn, 1983) and we adopt a cosmology with H${{}_{0}}=70$ km s-1
Mpc-1, $\Omega_{\Lambda}=0.7$, and $\Omega_{m}=0.3$.
## 2 Observations
Table 1: Limiting AB magnitudes (5$\sigma$) of the JWST/NIRCam and JWST/NIRISS imaging data used in this work. The depths were computed using random 0.3″circular apertures. SW filter | F090W | F115W | F150W | F200W
---|---|---|---|---
Depth | 28.0 | 27.8 | 28.8 | 28.9
LW filter | F277W | F356W | F444W |
Depth | 29.2 | 29.3 | 29.1 |
Figure 1: JWST/NIRCam and JWST/NIRISS through-put curves of the filter set of
the JWST ERO observations of SMACS0723 used in this work.
The first JWST science observations, targeting the lensing cluster SMACS0723,
were obtained as part of the Early Release Observations (ERO) program (ID
2736; PI: Pontoppidan), and have been released on Wednesday, July 13th. They
consist of deep multi-wavelength imaging with NIRCam (Near-Infrared Camera)
and MIRI (Mid-Infrared Instrument), NIRSpec (Near-Infrared Spectrograpgh)
multi-object Spectroscopy, and NIRISS (Near-Infrared Imager and Siltless
Spectrograpgh) wide-field slitless spectroscopy. We retrieved the data from
the Mikulski Archive for Space Telescopes (MAST). Both the short wavelength
(SW) and the long wavelength (LW) channels of NIRCam were used simultaneously
to obtain uniform spectral coverage from $\lambda\sim 1$ µm to $\lambda\sim 5$
µm. Tab. 1 summarizes the set of filters used in this work, and Fig. 1 shows
their transmission curves. Note that we complement the 6 available NIRCam
filters with F115W-band imaging data from the Near-Infrared Imager and
Slitless Spectrograph (NIRISS; Doyon et al., 2012). These however have a
smaller field-of-view and only cover the core of the cluster.
For each of the filters, we reduced the raw uncal files using the most recent
JWST calwebb pipeline v1.6.2 and the calibration context CRDS_CTX =
jwst_0970.pmap. In particular, the latest reference files contain updates from
in-flight calibration, whereas ground calibrations were used in the early
reductions of the ERO data. First, we applied the Detector1 stage to all
files. Second, we ran the images through the second reduction stage of the
pipeline Image2, which performs flat-field correction, WCS registration, and
photometric calibration. Due to significant levels of background residuals in
the calibrated images, and following recommendations from the ERS CEERS
program, we independently ran background subtraction on individual cal files.
In addition, the first NIRCam calibration observations have shown that in most
cases the detectors/filters were more sensitive than predicted (Rigby et al.,
2022). Therefore, we updated the photometric zeropoints of the cal files with
recent measurements of Boyer et al. (2022) and G.
Brammer111https://github.com/gbrammer/grizli/pull/107. Finally, cal files were
processed through the Image3 stage of the pipeline to create the final
drizzled mosaic in each filter. All images were registered to GAIA DR2
astrometry and the relative registration was refined using TweakReg. We
adopted a pixel scale of 0.03″ for both SW and LW channels.
In order to increase the detection efficiency in the extraction procedure
described in section 3.1, we created a deep LW image by stacking the three LW
filters using their respective weight maps. We also create a stacked SW image
of the non-detection filters blue-ward of the break.
One common limitation for the detection of faint sources behind massive
cluster fields is the intra-cluster light (ICL). This emission in the central
region of the cluster is emitted by tidally stripped stars and bright cluster
members (e.g. Atek et al., 2014a; Montes, 2022). In order to reduce the impact
of this diffuse light, we apply a median filter to the detection image. We
adopted a filter size of $\sim 2$″ $\times$ 2″, which significantly reduces
the diffuse ICL and BCG light while minimizing residual artifacts. Note that
the photometry measurement itself is still performed in the original image of
each filter.
Finally, we also use spectroscopic observations of SMACS0723 obtained with the
Near-Infrared Spectrograph (NIRSpec; Jakobsen et al., 2022) as part of the
same ERO program. We obtained level 3 data products from MAST which include 2D
and 1D spectra of selected targets in the cluster field.
## 3 Sample
### 3.1 Photometric Catalogs
Figure 2: Magnitude number counts from the SE++ extraction in the seven NIRCam
and NIRISS filters. Also over-plotted are the HST WFC3/IR F140W number counts
from RELICS (Coe et al., 2019; Salmon et al., 2020). The plot illustrates the
improvement in depth and the increase of the number of faint sources detected
with JWST.
We carry out source detection and photometry measurements using
SourceXtractor++ v0.17 (Bertin et al., 2020; Kümmel et al., 2020).
SourceXtractor++ (SE++) is the successor of the classic and widely used
SExtractor2, rewritten in C++ in order to optimize computational efficiency
and parallelization across multiple cores. The novelty and advantage of the
code is its support for two-level detection thresholding, flexible multi-
object and multi-frame model fitting as well as allowing for custom definition
of models. Additionally, it can operate on multiple images based on their WCS
information, bypassing the need to re-sample images on the same pixel scale.
SE++ has been extensively tested in its performance on photometry and
morphological parameter recovery from model fitting in the Euclid Morphology
Challenge, where it achieved the highest scores among all the tested codes
(Euclid Collaboration et al., 2022b, a).
For the detection of sources, we use the ICL-subtracted LW stack (cf. section
3.1) and its corresponding weight image as a weight map. The background map is
computed on a grid with cell size of 128 pixels and smoothed with a box-filter
of 5 pixels per side. Sources are detected via two-level thresholding which
ensures the detection of faint and low surface brightness sources while
minimizing false positives around bright sources. First, we detect all sources
with at least 6 contiguous pixels and an integrated SNR threshold of 1.75. A
second, higher threshold of 2.55 and 18 contiguous pixels is applied, whose
main impact is to clear false positives around bright sources. Finally, a
minimum area of 15 pixels is applied to clean spurious detections near bright
sources and random noise peaks. Detected sources that are close enough such
that they have connected pixels above the thresholds are grouped together and
later fitted simultaneously.
Measurements are then carried out on all 6 NIRCam bands simultaneously.
Additionally, we perform measurements on the NIRISS F115W-band separately
because of its smaller size and shallower depth (cf. Tab. 1), but using the
same detection image. Errors in the photometry are estimated from an RMS map,
obtained as the inverse of the square root of the weight image. The weight
image contains the contribution of the sky, dark, flat and read noise to each
pixel. As such, it considers the Poisson noise of the background, but not of
the sources. Regardless, given the fact that of interest for this paper are
the faintest sources, which are typically dominated by read and background
noise, this approach is sufficient. The photometry for each source is obtained
using AUTO apertures, as well as from model-fitting. The models and the priors
on their parameters can be user-defined, and in our case we use a Sérsic model
(Sérsic, 1963) with the radius, Sérsic index, angle, and axis ratio as free
parameters. We find that this model performs well for the faint and small
sources that we are interested in. The models are convolved with the point-
spread-function (PSF) obtained using WebbPSF v1.0.0. The PSF models are
generated on a pixel scale of $0.03$ ″/pix, on a stamp with a field-of-view of
$\sim 10$″ (301 pixels).
For the purpose of this paper, we use the model magnitudes from the
simultaneous fit of the NIRCam bands and the AUTO magnitudes for the separate
NIRISS measurement. The resulting magnitude number counts are shown in Fig. 2
for all bands used in this work. In order to check the photometric calibration
of these early JWST observations, we conducted a comparison with well-
calibrated HST images of the same cluster from RELICS. We measured fluxes in
the HST/WFC3 F140W and in the JWST/NIRCam F150W images, using the same
apertures and the HST mosaic as a detection image. The results are discussed
in appendix A and show a good agreement with a small residual offset. Fig. 2
also shows consistent number counts and magnitude depths across the JWST
filters. For comparison, we also show the number counts from the RELICS
F140W-band.
### 3.2 Dropout selection
Figure 3: Color-color selection windows using the NIRCam broadband filters for
high-redshift dropouts. Left: for $9<z<11$ Middle: for $11<z<15$; and Right:
for $16<z<21$. The solid lines, from blue to turquoise, represent the color
tracks of star-forming galaxy templates at high redshift generated with BEAGLE
(cf. text for details). The dashed lines, from yellow to red, are quiescent
galaxies, illustrating low-redshift galaxy contaminants, generated with GRASIL
(Silva et al., 1998). For each star-forming template we applied three values
of dust attenuation following the SMC dust law, $A_{V}$=[0,0.25,0.5], which
are illustrated with the brighter blue or red colors in increasing order to
illustrate the effect of dust attenuation on the color tracks. For the
quiescent galaxies, we apply attenuation values in the range $A_{V}$=[1,2,3].
The purple circles represent the colors of a library of M-class stars and
brown dwarfs (Chabrier et al., 2000; Allard et al., 2001), which also
represent potential contaminants. The high-redshift dropouts selected in this
work are represented as green diamonds.
Figure 4: Examples of NIRSpec spectra using the G395M grism for 2 dropout
galaxies at $z\sim 7-9$, displaying very strong H$\beta$ and [O iii] emission
lines.
We proceed to identify high-redshift galaxies using the Lyman-break, or
dropout, selection technique (e.g. Steidel et al., 1996; Atek et al., 2015;
Kawamata et al., 2015; Bouwens et al., 2015). In order to determine the best
color-color selection criteria, we ran a set of simulations using different
galaxy templates. We generated starburst galaxy templates using BEAGLE
(Chevallard & Charlot, 2016, cf. section 5) at increasing redshifts from $z=6$
to $z=28$, including the intergalactic medium (IGM) attenuation from Inoue et
al. (2014). In addition, we explore a range of dust attenuation from $A_{V}=0$
to $A_{V}=0.5$ using the SMC extinction law (Pei, 1992), which has been found
to well match the dust attenuation in high-redshift galaxies (e.g. Capak et
al., 2015; Reddy et al., 2015, 2018a). We then computed synthetic photometry
in the NIRCam broad-band filters and plotted the color tracks of these mock
galaxies in Fig. 3. In order to mitigate contamination from low-redshift
interlopers that mimic the IGM absorption by their intrinsically red colors,
we also explore the position of low-redshift quiescent galaxies, generated
with GRASIL (Silva et al., 1998) and available in the framework of the SWIRES
template library (Polletta et al., 2007), on this color-color diagram. These
again include several values of dust attenuation between $A_{V}=1$ and
$A_{V}=3$. Finally, we include the last significant source of contamination,
cold red stars and brown dwarfs, which can also mimic the Lyman break. We used
the stellar templates from Chabrier et al. (2000) and Allard et al. (2001).
With this information in hand, we define the best color selection window that
maximizes identification of the high-redshift sources while minimizing
contamination for each redshift range. The resulting selection criteria are
illustrated in Fig. 3.
According to these simulations, we select galaxy candidates in the redshift
ranges $z\sim 9-11$ and $z\sim 11-15$ respectively based on the following
criteria:
$\begin{array}[]{l}M_{115}-M_{150}>0.5\\\
M_{115}-M_{150}>1.0+1.4(M_{150}-M_{200})\\\ M_{150}-M_{200}<0.5\end{array}$
and
$\begin{array}[]{l}M_{150}-M_{200}>0.5\\\
M_{150}-M_{200}>1.6+1.8(M_{200}-M_{277})\\\ M_{200}-M_{277}<0.2\end{array}$
In addition to these color criteria, galaxy candidates must also satisfy a
detection significance above 5$\sigma$ in the deep IR stacked image, and
4$\sigma$ in the individual detection filters. We also reject any candidate
that shows a significant detection (at 2$\sigma$ level) in the stacked image
that combines all filters blue-wards of the Lyman break. If a source is not
detected in the dropout filter, we adopt the 1$\sigma$ lower limit measured in
the same aperture to calculate the Lyman break.
For $z\sim 16-21$ candidates, we adopt the following criteria:
$\begin{array}[]{l}M_{200}-M_{277}>0.5\\\
M_{200}-M_{277}>1.2+2.0(M_{277}-M_{356})\\\ M_{277}-M_{356}<0.4\end{array}$
Similarly to the $11<z<15$ selection, all candidates must satisfy the same
detection and non-detection criteria detailed above. All candidates are
visually inspected to check for spurious detection, artifacts, or bad PSF
residuals. We also carefully check the candidates against other potential
sources of contamination. For instance, very strong emission lines in
intermediate-redshift ($z\sim 3-5$) galaxies can contaminate the broadband
flux and create a photometric dropout (e.g. Atek et al., 2011; de Barros et
al., 2014). In particular, strong [O iii]$\lambda\lambda$4959,5007 + H$\beta$
lines are expected in high-redshift galaxies, (e.g. De Barros et al., 2019)
which can lead to red colors. To address this issue, we perform SED fitting
that takes nebular emission lines into account (cf. section 5). Note also that
most of the intermediate-redshift contaminants should be easily detected in
the bluer filters, given the depth of the F090W and F150W filters and the
observed hypothetical excess in the red filters. Finally, we discard all
sources that have point-source morphologies in order to sieve out any stars
that might scatter into our selection window. The colors of our selected
galaxies are shown in Fig. 3 as green diamonds. Note that there are a few
objects selected as high-redshift galaxies in the $z\sim 11-15$ bin which do
not completely satisfy all of the color-criteria. These objects were selected
from their photometric redshift estimates with EAZY (cf. section 5) because
they have unambiguous high-redshift solutions and do not present morphologies
consistent with stars.
In the present study, we have also analyzed the NIRSpec observations to
investigate the spectroscopic properties of the high-redshift galaxies
targeted by this program. Two examples at $z_{\mathrm{spec}}=7.66$ and
$z_{\mathrm{spec}}=8.5$ are shown in Fig. 4. These are the first rest-frame
optical spectra observed for high-redshift ($z>6$) galaxies (cf. also Schaerer
et al., 2022). They show strong H$\beta$ and [Oiii] emission lines. The rest-
frame equivalent width (EW) of these lines reaches up to $\sim 400$Å, which is
still a lower limit because the continuum is hardly detected. This means that
the flux contribution of emission lines to the broadband flux can reach up to
$\sim 30$ % in some galaxies.
Another consequence of the strong lensing is that some background sources will
have multiple images. Using the most recent available parametric lensing model
(cf. section 4), we have not identified any possible multi-image systems. The
magnification factors of the candidates range from $\mu\sim 1$ to $\sim 4$.
## 4 Lensing Model
Figure 5: The cumulative surface area behind SMACS0723 at $z=9$ as a function
of gravitational magnification $\mu$ (cf. section 4), expressed in magnitudes.
The solid line shows the area corresponding to the NIRCam field of view,
whereas the dashed line is for the NIRISS pointing. The un-lensed total survey
area is 7.23 arcmin2, and 2.1 arcmin2 for the NIRCam and NIRISS observations
respectively. For reference, the observed area is $\sim 10.8$ arcmin2 and
$\sim 4.9$ arcmin2 for NIRCam and NIRISS, respectively.
In order to compute the gravitational magnifications of our objects, we use
the most recent strong lensing mass model by Mahler et al. (2022). The model
was built using the lenstool (Kneib et al., 1996; Jullo et al., 2007; Jullo &
Kneib, 2009) software to optimize a parametric SL model for the mass
distribution of the cluster. Following the approach used by Richard et al.
(2014) for the HFF clusters (Lotz et al., 2017) and Fox et al. (2022) for the
RELICS clusters, this model uses a combination of double Pseudo Isothermal
Elliptical potentials (dPIE; Elíasdóttir et al., 2007) for the total mass
distribution at cluster- and galaxy-scales. The parameters of the galaxy-scale
dPIE potentials are matched to the identification of cluster members on the
red sequence from the RELICS HST catalogs (cf. Coe et al. 2019 for details),
fixing the center and elliptical parameters of the potential to its isophotes.
The mass parameters of the dPIE profile (velocity dispersion $\sigma$, core
radius rcore, cut radius rcut) are scaled according to the luminosity L∗ of a
reference galaxy using a constant mass-to-light ratio.
The SL constraints used in Mahler et al. (2022) are based on the released
JWST/NIRCam images (cf. section 2), and make use of spectroscopic redshifts
measured with public data from the Multi Unit Spectroscopic Explorer (MUSE;
Bacon et al., 2010) on ESO’s Very Large Telescope (VLT) by Golubchik et al.
(2022) and with JWST/NIRSpec data in Mahler et al. (2022). In summary, 48
secured images of 16 separate sources were identified, including 5
spectroscopic redshifts. The Mahler et al. (2022) SL model reproduces all
systems with a residual lens plane RMS of $\sim 0.7\arcsec$ between the
predicted and observed locations of all images.
The cumulated surface area of magnification according to the model is shown in
Fig. 5. We make use of this model to compute the magnification factors and
associated errors on each high redshift galaxy candidates. All magnifications
assume $z=15$ as there is only very little variation of magnification with
redshift at $z>7$. lenstool allows us to compute a statistical error on the
magnification by sampling the posterior probability distribution of each
parameter of the model.
## 5 Spectral Energy Distribution Fitting
We continue our selection of high-redshift galaxy candidates by computing
photometric redshifts by fitting galaxy spectral energy distributions (SEDs)
to the multi-wavelength photometry.
The photometric redshifts are obtained with EAZY (Brammer et al., 2008), a
Python-based SED fitting code. EAZY fits a non-negative linear combination of
a set of basis templates to the observed flux densities for each galaxy. The
fit is performed on the flux densities obtained from the model-fitting in SE++
(cf. section 3.1), converted to units of $\mu$Jy, and corrected for Milky Way
extinction internally in the code. Sources with valid flux density
measurements available in at least three bands are considered for fitting. We
use the template set "tweak_fsps_QSF_1_v3.param" that is derived from the
Flexible Stellar Population Synthesis models (FSPS) (Conroy et al., 2009;
Conroy & Gunn, 2010). The allowed redshift range is set to $0.01<z<22$, with a
flat prior on the luminosity. Sources of interest are then selected by
visually inspecting all sources with a best-fit $z_{\mathrm{phot}}>9$. In a
second selection step, candidates are required to show a reasonably good SED
fit.
In addition, we simultaneously infer physical parameters and photometric
redshifts of the selected high-redshift candidates by fitting galaxy SEDs to
the JWST photometry with the BayEsian Analysis of GaLaxy sEds tool (BEAGLE;
Chevallard & Charlot, 2016). This is done to refine our selection and obtain
accurate uncertainty estimates on the physical parameters of the galaxies.
Because the inclusion of nebular emission has been found to be crucial in
fitting broad-band photometry of galaxies (Schaerer & de Barros, 2009, 2010;
Atek et al., 2011; Atek et al., 2014b; McLure et al., 2011; Smit et al., 2014;
Duncan et al., 2014; Reddy et al., 2018b), BEAGLE uses galaxy templates
computed by Gutkin et al. (2016) which combine the stellar population
templates by Bruzual & Charlot (2003) with the photoionization code CLOUDY
(Ferland et al., 2013). It then accounts for IGM attenuation using the Inoue
et al. (2014) models and applies a dust attenuation law (cf. section 5.1 for
details) to the galaxy emission. The BEAGLE tool is fully Bayesian, i.e.
performs a Monte-Carlo Markov Chain (MCMC) analysis of the SED, which makes it
ideally suited to determine both photometric redshifts and several physical
parameters at the same time while robustly quantifying and combining their
uncertainties.
### 5.1 BEAGLE set-up
For our SED-fit with BEAGLE, we assume a constant star-formation history
(SFH). While this is a simplification, it is a common assumption made in high-
redshift galaxy analyses (e.g. Eyles et al., 2007; Stark et al., 2009;
González et al., 2011; Grazian et al., 2015; Kikuchihara et al., 2020) and the
results have been shown to not strongly differ from more flexible analytic SFH
assumptions such as e.g. a delayed exponential SFH (Furtak et al., 2021). In
order to avoid over-fitting due to the relatively low number of photometric
bands available for this work, we limit the fit-parameters to four:
* •
Photometric redshift $z_{\mathrm{phot}}$ both fixed to the EAZY result first
and then with a uniform prior in the limits of $z_{\mathrm{phot}}\in[0,25]$
for confirming the robustness of the photometric redshift estimate.
* •
Stellar mass $M_{\star}$ with a log-uniform prior in the limits of
$\log(M_{\star}/\mathrm{M}_{\odot})\in[6,10]$.
* •
Maximum stellar age $t_{\mathrm{age}}$ with a log-uniform prior in the limits
of $\log(t_{\mathrm{age}}/\mathrm{yr})\in[7,t_{\mathrm{universe}}]$ where
$t_{\mathrm{universe}}$ is the age of the Universe at the redshift of the
galaxy.
* •
Effective V-band dust attenuation optical depth $\hat{\tau}_{V}$ with a
uniform prior in the limits of $\hat{\tau}_{V}\in[0,0.5]$.
We furthermore fix the stellar metallicity to a constant value of
$Z=0.1\,\mathrm{Z}_{\odot}$ since it has been shown that the broad-band
photometry of high-redshift dropout galaxies is not sensitive to the
metallicity in the very low metallicity regime expected for high-redshift
galaxies (Furtak et al., 2021). Finally, we adopt an SMC dust extinction law
(Pei, 1992) which has been shown to match the observations of high-redshift
galaxies best (Capak et al., 2015; Reddy et al., 2015, 2018a), in particular
in the low-metallicity range that we are probing (Shivaei et al., 2020).
The SED fit is run on all six broad-band filters without taking the
gravitational magnification into account yet, in order to avoid including the
uncertainties of magnification in the SED fit. Instead, we correct the stellar
mass for magnification after the fit following our approach in Furtak et al.
(2021). Note though that the magnifications of our objects are not very high,
$\mu\sim 1-4$.
### 5.2 Results
Table 2: Complete list of our high-redshift candidates identified behind SMACS0723 with their derived parameters. ID | RA | Dec | $z_{\mathrm{phot}}$ | $M_{\mathrm{UV}}$ | $\beta$ | $\log(M_{\star}/\mathrm{M}_{\odot})$ | $\log(t_{\mathrm{age}}/\mathrm{yr})$ | $n$ | $r_{\rm e}$ [kpc]
---|---|---|---|---|---|---|---|---|---
$z\sim 9-11$ candidates
SMACS_z10a | 7:23:26.252 | -73:26:56.940 | $9.78^{+0.02}_{-0.02}$ | $-18.77\pm 0.20$ | $-1.72\pm$0.04 | $9.11^{+0.07}_{-0.07}$ | $8.68^{+0.2}_{-0.2}$ | $<1$ | $0.22$
SMACS_z10b | 7:23:22.709 | -73:26:06.183 | $8.88^{+0.02}_{-0.02}$ | $-20.78\pm 0.13$ | $-1.36\pm$0.19 | $10.20^{+0.03}_{-0.03}$ | $8.74^{+0.2}_{-0.2}$ | $<1$ | $0.65$
SMACS_z10c | 7:23:20.169 | -73:26:04.233 | $9.77^{+0.02}_{-0.02}$ | $-20.19\pm 0.15$ | $-2.14\pm$0.12 | $9.53^{+0.02}_{-0.02}$ | $8.68^{+0.2}_{-0.2}$ | $<1$ | $0.41$
SMACS_z10d | 7:22:46.696 | -73:28:40.898 | $9.31^{+0.06}_{-0.08}$ | $-19.76\pm 0.18$ | $-2.22\pm$0.17 | $7.77^{+0.14}_{-0.11}$ | $7.29^{+0.18}_{-0.12}$ | $<1$ | $0.55$
SMACS_z10e | 7:22:45.304 | -73:29:30.557 | $10.89^{+0.16}_{-0.14}$ | $-18.91\pm 0.26$ | $-2.03\pm$0.19 | $8.51^{+0.22}_{-0.16}$ | $7.49^{+0.26}_{-0.20}$ | $<1$ | $0.33$
SMACS_z11a | 7:22:39.505 | -73:29:40.224 | $11.05^{+0.09}_{-0.08}$ | $-18.55\pm 0.38$ | $-1.97\pm$0.38 | $8.77^{+0.17}_{-0.24}$ | $8.20^{+0.21}_{-0.32}$ | $<1$ | $0.39$
$z\sim 11-15$ candidates
SMACS_z12a | 7:22:47.380 | -73:30:01.785 | $12.20^{+0.21}_{-0.12}$ | $-19.75\pm 0.23$ | $-2.69\pm$0.16 | $8.14^{+0.21}_{-0.17}$ | $7.61^{+0.26}_{-0.20}$ | $<1$ | $0.47$
SMACS_z12b | 7:22:52.261 | -73:27:55.497 | $12.26^{+0.17}_{-0.16}$ | $-20.01\pm 0.17$ | $-2.82\pm$0.21 | $7.91^{+0.26}_{-0.17}$ | $7.56^{+0.30}_{-0.23}$ | $4.0\pm 1.0$ | $1.99$
$z>15$ candidates
SMACS_z16a | 7:23:26.393 | -73:28:04.561 | $15.92^{+0.17}_{-0.15}$ | $-20.59\pm 0.15$ | $-2.63\pm$0.13 | $8.79^{+0.32}_{-0.33}$ | $7.65^{+0.36}_{-0.39}$ | $1.1\pm 0.2$ | $0.40$
SMACS_z16b | 7:22:39.439 | -73:30:08.185 | $15.32^{+0.16}_{-0.13}$ | $-20.96\pm 0.14$ | $-2.40\pm$0.34 | $8.80^{+0.44}_{-0.25}$ | $7.46^{+0.52}_{-0.35}$ | $2.8\pm 0.6$ | $1.17$
Figure 6: Cutout images of SMACS_z16a in the JWST bands centered on the
source. The left-most column shows the LW-stack detection image (cf. section
2) and the source partition map. The following columns show the seven NIRCam
and NIRISS science images in the top rows, best-fit model image in the middle
rows and the residual in the bottom rows. The science and model images are
scaled using a linear-stretch normalization, with minimum value equal to zero
and maximum value 10 times the $2\sigma$-clipped standard deviation. The
residuals are scaled linearly between $\pm 7$ times the $2\sigma$-clipped
standard deviation.
Figure 7: Best-fit solution for the SED and photometric redshift of
SMACS_z16a. Upper row: Best-fit SED using the BEAGLE code. Left: Best-fit SED
(black solid curve) with the observed photometric data (blue points) and
expected model photometric points (black points) and associated uncertainties
(pink areas). Right: Triangle plot of the posterior probability distribution
of the four fitted galaxy parameters: redshift, stellar mass, stellar age and
attenuation. Bottom row: Best-fit SED using the EAZY code. Left panel: The
best-fit SED over-plotted over the observed flux densities (in dark squares).
Model flux densities are shown in blue circles. The Lyman-break of the SED of
this galaxy is estimated at $z=15.88$ and the redshift probability
distribution function is shown in the right panel. Both codes agree on a high-
redshift solution with a relatively narrow posterior distribution and which
does not show a secondary peak at lower redshift. Figure 8: Same as Fig. 6,
this time for the $z\sim 12$ candidate galaxy SMACS_z12a.
Figure 9: Same as Fig. 7, here showing the $z\sim 12$ galaxy candidate
SMACS_z12a. Figure 10: Example of a galaxy candidate, SMACS_z10c, at $z\simeq
9.76\pm 0.01$ showing an excess in the F444W-band. The best-fit SED (same
legend as Fig. 7) shows the presence of a Balmer break around 4.5 µm.
The final sample of high-redshift candidates consists of a total of 6 galaxies
in the redshift range $9<z<11$, two galaxies at $11<z<15$, and two galaxies at
$z>15$. All sources are new identifications, since Salmon et al. (2020) only
reported galaxies below $z\sim 8$ in this field. The complete results for our
full sample are reported in Tab. 2.
Our highest redshift galaxy, SMACS_z16a, is consistently identified by the
dropout selection and the two SED fitting procedures, as shown in Fig. 7. The
photometric redshifts derived by both codes (cf. section 5) are in excellent
agreement: $z=15.92^{+0.17}_{-0.15}$ and $z=15.89\pm 0.43$ for BEAGLE and EAZY
respectively. This places the galaxy just $\sim 250$ Myr after the Big Bang.
The galaxy exhibits an extremely blue UV continuum slope of $\beta=-2.63\pm
0.13$ which is consistent with a young and low-attenuation galaxy (cf. Fig.
7). We note in general that no template in EAZY is able to match the very blue
UV continuum, whereas BEAGLE allows for more flexibility in this range of
galaxy templates. To obtain a more accurate estimate of the physical
properties of our galaxies, we ran a second round of SED fitting with BEAGLE
using a prior in redshift from the PDF of EAZY (cf. section 5.1). We derive a
stellar mass of $\log(M_{\star}/\mathrm{M}_{\odot})=8.79^{+0.32}_{-0.33}$ and
a stellar age of $\log(t_{\mathrm{age}}/\mathrm{yr})=7.65^{+0.36}_{-0.39}$ for
SMACS_z16a. This galaxy is also very compact, as can be seen in Fig. 6, with
an estimated size of $\sim 0.4$ kpc based on the model fitting of SE++ (cf.
section 3.1) and after correcting for the lensing distortion.
In the next lower redshift range, one of the two most robust candidates,
SMACS_z12a, is presented in Fig. 8. Again, the source is well detected in all
four filters and shows a clear continuum break. As before, there is a good
agreement between the two photometric redshifts $z=12.20^{+0.21}_{-0.12}$
(BEAGLE) and $z=12.11\pm 0.19$ (EAZY). The PDF from EAZY is broader than the
BEAGLE one, because of the challenge in matching the blue colors
($\beta\sim-2.60\pm 0.32$) of this object (Fig. 9). This source is also among
the highest-redshift candidates identified photometrically in the literature
(e.g. Harikane et al., 2022; Naidu et al., 2022; Castellano et al., 2022),
around 360 Myr after the Big Bang. The derived parameters are very similar to
SMACS_z16a, with a stellar mass of
$\log(M_{\star}/\mathrm{M}_{\odot})=8.14^{+0.21}_{-0.17}$, and an age of
$\log(t_{\mathrm{age}}/\mathrm{yr})=7.61^{+0.21}_{-0.20}$.
In the lowest redshift range, $9<z<11$, some sources around $z\sim 10$ show
indication of a Balmer break as a clear excess in the F444W-band. An example
is shown in Fig. 10, where flux excess is observed relative to a flat
continuum in the bluer bands. This could indicate that an evolved stellar
population was already in place at those early epochs, which is confirmed by
the best-fit SED with a stellar age around $\sim 400$ Myr. An example for our
$9<z<11$ objects is shown in appendix B. Of course, a stronger contribution
from rest-frame optical emission lines is also possible, although this
alternative does not yield the best-fit solution.
## 6 Discussion
Figure 11: Absolute UV magnitude $M_{\mathrm{UV}}$ as a function of redshift
for the high-redshift candidates in this work, along with a compilation of
known galaxies at $z>8$. The candidates in this work are shown as orange
filled hexagons, while empty hexagon marks the less secure candidate. The
empty purple pentagons show the recently identified candidates from JWST in
the GLASS parallel field by Naidu et al. (2022). The empty blue circles show
photometric-redshift galaxies, while the solid circles represent
spectroscopically confirmed galaxies compiled from Bouwens et al. (2022).
While observing through a lensing cluster offers invaluable flux
amplification, it also results in a smaller survey area in the source plane.
The field-of-view in the NIRCam image plane is about $9.7$ arcmin2, which
translates into $7.23$ arcmin2 in the source plane after de-lensing. Although
the area reduction is still reasonable (because one of the NIRCam modules is
less affected by magnification), wide-area surveys are more prone to uncover
rare and bright high-redshift galaxies (e.g. Roberts-Borsani et al., 2016;
Oesch et al., 2016; Bowler et al., 2020; Harikane et al., 2022), while
lensing-assisted surveys are sensitive to the fainter population. While some
of our sources are relatively bright, they all have luminosities around, or
below, the characteristic luminosity $M^{\star}$ at $z\sim 9-10$ (Oesch et
al., 2018; Bouwens et al., 2021). The highest redshift candidate in our
sample, SMACS_z16a, has an intrinsic UV magnitude of
$M_{\mathrm{UV}}=-20.59\pm 0.15$, which is fainter than previous very high
redshift candidates Gn_z11 and HD1, and roughly similar to the recently
discovered candidate GLz13 (Naidu et al., 2022). Moreover, the yield of the
galaxy hunt presented in this paper is surprisingly high, with the discovery
of 2 candidates at $z\sim 16$, 2 candidates at $z\sim 12$, and 6 candidates at
$z=10-11$. In Fig. 11, we put these discoveries in the context of known high-
redshift galaxies and compare our candidates to the published samples of
galaxies at $z>8$. Not only do these results demonstrate the exceptional
capabilities of JWST, but they also suggest a higher number density than
predicted by an extrapolation of the $z\sim 9-10$ luminosity function (e.g.
Bouwens et al., 2021) and theoretical predictions (e.g., Ocvirk et al., 2020;
Dayal et al., 2022). Some early theoretical interpretations suggest that these
observations could be compatible with a total absence of dust in these
galaxies, as their very blue UV slopes might also indicate (Ziparo et al.,
2022). The evolution of the UV luminosity function observed through lensing
clusters will be the scope of a future paper.
The inferred age of SMACS_z16a indicates a recent formation history, which
points to a rapid stellar mass assembly in this galaxy. In the context of
early galaxy formation, one would expect the first galaxies to more resemble
SMACS_z16a in terms of physical parameters, than the bright and uncommon
galaxies discussed earlier. As we are closing in on the formation epoch of the
first galaxies, it will become increasingly challenging to find such evolved
and relatively massive objects. Perhaps upcoming and future deep surveys
through gravitational lenses will offer the best route to unveil these
smallest structures. This first JWST cycle already includes several ERS,
General Observer (GO), and Guaranteed Time (GTO) programs that target lensing
clusters using different observing modes.
Another interesting finding is the small sizes of these galaxies. For
instance, SMACS_z16a has an effective radius of $r_{\rm e}=0.4$ kpc, which is
consistent with recent size measurement of high-redshift galaxies (Kawamata et
al., 2018; Adams et al., 2022). More importantly, the Sérsic index, derived
from the SE++ modeling and source-plane reconstruction, hints to a disk-like
morphology. We measured $n=1.1\pm 0.2$ for SMACS_z16a, which corresponds to a
disk-like profile, although there is a real possibility that we are detecting
only the brightest star-forming region in the galaxy. Such a morphology was
also observed for GLz13 (Naidu et al., 2022), as well as in several other
high-redshift candidates reported in Adams et al. (2022). The existence of
well settled disks at early epochs would provide strong constraints on
theoretical models of galaxy formation.
While the nature of these ERO observations implies potential sources of
uncertainties, we performed photometric quality verifications, and also
combined multiple selection methods to build our sample of high-redshift
sources. The dropout selection and the photometric redshifts from two
independent codes significantly increase the robustness of the candidates. In
the near future, deeper imaging in the short wavelength range will bring
additional confirmation to these candidates, in addition to follow-up
spectroscopy, e.g with NIRSpec, which will provide definitive redshifts.
## 7 Summary
In this paper, we presented our search for very high-redshift galaxies in the
first JWST observations of the lensing cluster SMACS 0723. We report the
discovery of the highest redshift candidate known to date at $z\sim 16$. We
also discover two candidates at $z\sim 12$, and 11 candidates at $z\sim
10-11$.
We have used 6 NIRCam broadband filters, and one NIRISS filter, between $\sim
0.8$ µm and 5 µm to identify Lyman break galaxies in the redshift ranges
$9<z<11$, $11<z<15$, and $16<z<21$. We combined the dropout selection with
SED-fitting to obtain accurate photometric redshifts using two different
codes, BEAGLE and EAZY. We also simultaneously derive the physical properties
of these galaxies. Our final sample is based on all candidates that satisfy
the three selection methods and whose photometric redshifts from the two SED
fits are in agreement. To determine the intrinsic luminosity of galaxies and
account for potential multiple images, we use the one of the latest lensing
models of SMACS 0723 produced from the same JWST data set in a companion paper
(Mahler et al., 2022). The measured amplification factors range from 1 to
$\sim 4$, which gives us a lensing boost up to 1.5 magnitudes in the best
case.
With an intrinsic UV luminosity of $M_{\mathrm{UV}}=-20.59\pm 0.15$, we find
that SMACS_z16a is fainter than typical $z>10$ galaxies in the literature and
close to the characteristic luminosity $M^{\star}$ at $z\sim 10$. The best-fit
SED of SMACS_z16a is compatible with a young age of
$\log(t_{\mathrm{age}}/\mathrm{yr})=7.65^{+0.36}_{-0.39}$ and a stellar mass
of $\log(M_{\star}/\mathrm{M}_{\odot})=8.79^{+0.32}_{-0.33}$. The bulk of its
stellar mass build-up took place in the last 20 Myr. This galaxy, like some
other candidates, shows a very blue UV continuum-slope of $\beta=-2.63\pm
0.13$ consistent with young ages. In general, the galaxy candidates, from
$\sim 10$ to $z\sim 16$, have ages in the range
$\log(t_{\mathrm{age}}/\mathrm{yr})\sim=7.3-8.7$, and stellar masses of
$\log(M_{\star}/\mathrm{M}_{\odot})\sim 7.8-10.2$. Our source extraction and
photometry procedure with SE++ also provide morphological measurements, since
it relies on model-fitting. The measured sizes were reconstructed back to the
lens plane using lenstool to determine the intrinsic effective radius and the
Sersić index of the sources. Our measurements indicate very compact sources,
with $r_{\rm e}$ below 1 kpc for most of the candidates. Some of these sources
also have Sérsic indices compatible with disk-like a morphology. If true, it
is surprising, for a galaxy like SMACS_z16a, to evolve so quickly in the span
of less than $\sim 250$ Myr.
The discovery of one, possibly two, galaxy candidates only $\sim 250$ Myr
after the Big Bang, together with eight new candidates at $9<z<13$, announces
great discoveries awaiting in the deep extra-galactic fields that will be
observed by JWST. The number density, the physical properties and the
morphologies of these galaxies will be extremely valuable in constraining
hydrodynamical simulations of galaxy formation in a cosmological context.
## Acknowledgements
We would like to thank Gabriel Brammer for the grizli reduction notebooks. HA
acknowledges support from CNES. LF and AZ acknowledge support by Grant No.
2020750 from the United States-Israel Binational Science Foundation (BSF) and
Grant No. 2109066 from the United States National Science Foundation (NSF). AZ
acknowledges support by the Ministry of Science & Technology, Israel.
This work is based on observations obtained with the NASA/ESA/CSA JWST and the
NASA/ESA Hubble Space Telescope (HST), retrieved from the Mikulski Archive for
Space Telescopes (MAST) at the Space Telescope Science Institute (STScI).
STScI is operated by the Association of Universities for Research in
Astronomy, Inc. under NASA contract NAS 5-26555. This work has made use of the
CANDIDE Cluster at the Institut d’Astrophysique de Paris (IAP), made possible
by grants from the PNCG and the region of Île de France through the program
DIM-ACAV+.
The JWST Early Release Observations (ERO) and associated materials were
developed, executed, and compiled by the ERO production team: Hannah Braun,
Claire Blome, Matthew Brown, Margaret Carruthers, Dan Coe, Joseph De-
Pasquale, Nestor Espinoza, Macarena Garcia Marin, Karl Gordon, Alaina Henry,
Leah Hustak, Andi James, Ann Jenkins, Anton Koekemoer, Stephanie LaMassa,
David Law, Alexandra Lockwood, Amaya Moro-Martin, Susan Mullally, Alyssa
Pagan, Dani Player, Klaus Pontoppidan, Charles Prof- fitt, Christine Pulliam,
Leah Ramsay, Swara Ravindranath, Neill Reid, Massimo Robberto, Elena Sabbi,
Leonardo Ubeda. The EROs were also made possible by the foundational efforts
and support from the JWST instruments, STScI planning and scheduling, and Data
Management teams.
## Data Availability
The data underlying this article are publicly available on the Mikulski
Archive for Space Telescopes222https://archive.stsci.edu/ (MAST), under
program ID 2736.
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## Appendix A Photometry validation
Figure 12: Magnitude number counts in the HST WFC3/IR F140W and JWST NIRCam
F150W bands. The magnitudes are obtained for objects detected in the shallower
F140W-band and measured simultaneously in F140W and F150W. The number counts
show a good agreement and no photometric offset is present. Figure 13:
Magnitude comparison between HST-WFC3/IR F140W and JWST-NIRCam F150W bands.
The blue line marks the running median offset and the blue envelope the
$\sigma$-clipped standard deviation. The magnitudes of the two bands show a
relatively tight dispersion and an offset of about 0.24 mag, that can be
explained by the difference between the filter curves and pivot wavelengths.
We perform photometry validation by comparing our photometry with the well
known and calibrated HST WFC3/IR images from the RELICS survey. We run SE++ on
a stacked detection image and do photometric measurements simultaneously on
the WFC3/IR F140W and NIRCam F150W bands.
Fig. 12 shows the AUTO magnitude number counts in NIRCam F150W and WFC3/IR
F140W as measured with SE++. The number counts are in excellent agreement,
which is not surprising given the same detection image and with minimal or no
offsets. This confirms the accurate zero-point calibration of our JWST data
reduction, explained in section 2. The comparison of magnitudes is shown in
Fig 2, where on the $y-$axis we show the magnitude difference between the
F140W and F150W bands. There is a relatively tight scatter and a median offset
of about $0.36$ towards fainter F140W magnitudes. Most of this offset can be
attributed to the difference between the F140W and F150W filter curves and
pivot wavelengths. Using Pysynphot, we performed synthetic photometry in the
two filters, assuming a variety of galaxy templates from BC95 (Bruzual A. &
Charlot, 1993) and KC96 (Kinney et al., 1996) libraries, and a redshift range
of $z=0-5$. We found a median offset of $\sim 0.3$ magnitudes.
## Appendix B Example of a $z\sim 10$ candidate
Figure 14: Same as Fig. 6, this time for the $z\sim 12$ candidate galaxy
SMACS_z11a.
Figure 15: Same as Fig. 7, here showing the galaxy candidate SMACS_z11a.
In addition to the $z\sim 16$ and $z\sim 12$ examples shown in the main text
(cf. section 5.2), we display here an example of a $z\sim 10$ galaxy, the
candidate SMACS_z11a, in Figs 14 and 15.
## Appendix C Full candidate list
Figure 16: Photometric data and best-fit SED results for candidate SMACS_z10a.
Figure 17: Photometric data and best-fit SED results for candidate SMACS_z10b.
Figure 18: Photometric data and best-fit SED results for candidate SMACS_z10c.
Figure 19: Photometric data and best-fit SED results for candidate SMACS_z10d.
Figure 20: Photometric data and best-fit SED results for candidate SMACS_z10e.
Figure 21: Photometric data and best-fit SED results for candidate SMACS_z12b.
Figure 22: Photometric data and best-fit SED results for candidate SMACS_z16b.
|
# EasyTrack: Efficient and Compact One-stream 3D Point Clouds Tracker
Baojie Fan, Wuyang Zhou, Kai Wang, Shijun Zhou,
Fengyu Xu, Jiandong Tian, Senior Member, IEEE Baojie Fan, Wuyang Zhou, Kai
Wang and Fengyu Xu are from the College of Automation $\&$ College of
Artificial Intelligence, Nanjing University of Posts and Telecommunications.
Nanjing 210023, China, e-mail<EMAIL_ADDRESS>Jiandong Tian and Shijun Zhou
are with the State Key Laboratory of Robotics, Shenyang Institute of
Automation, Chinese Academy of Sciences, Shenyang, China and Institutes for
Robotics and Intelligent Manufacturing, Chinese Academy of Sciences, Shenyang,
China. Corresponding author: Jiandong Tian<EMAIL_ADDRESS>
###### Abstract
Most of 3D single object trackers (SOT) in point clouds follow the two-stream
multi-stage 3D Siamese or motion tracking paradigms, which process the
template and search area point clouds with two parallel branches, built on
supervised point cloud backbones. In this work, beyond typical 3D Siamese or
motion tracking, we propose a neat and compact one-stream transformer 3D SOT
paradigm from the novel perspective, termed as EasyTrack, which consists of
three special designs: 1) A 3D point clouds tracking feature pre-training
module is developed, utilizing a transformer with masks to learn patterns of
point-wise spatial relationships within three-dimensional data. 2) A unified
3D tracking feature learning and fusion network is proposed to simultaneously
learns target-aware 3D features, and extensively captures mutual correlation
through the flexible self-attention mechanism. 3) A efficient target location
network in the dense bird’s eye view (BEV) feature space is constructed for
target classification and regression. Moreover, we develop an enhanced version
named EasyTrack++, which designs the center points interaction (CPI) strategy
to reduce the ambiguous targets caused by the noise point cloud background
information. The proposed EasyTrack and EasyTrack++ set a new state-of-the-art
performance (18%, 40% and 3% success gains) in KITTI, NuScenes, and Waymo
while runing at 52.6fps with few parameters (1.3M). The code will be available
at https://github.com/KnightApple427/Easytrack.
###### Index Terms:
3D single object tracking, Lidar, point clouds pre-training, one-stream
compact tracking framework, transformer
## 1 Introduction
Single object tracking (SOT) is a classic task in computer vision. In the past
few years, camera-based 2D SOT has achieved rapid developments[1, 2, 3, 4, 5,
6, 7, 8, 9]. However, in some practical applications such as autonomous
driving, mobile robotics, and unmanned aerial vehicles[10, 11, 12, 13], 3D SOT
has attracted more and more attention to provide the pose and position of the
tracked target in the 3D space. Early 3D SOT methods mainly focus on RGB-D
tracking[14, 15]. With the development of LiDAR sensors, many trackers perform
3D SOT in the point clouds scanned by LiDAR sensors, since they can preserve
accurate geometry information[16, 17] of 3D objects and are not sensitive to
illumination changes.
Given the target’s state parameters $(x,y,z,w,l,h,\theta)$ in the first frame
of a point clouds tracklet, the purpose of 3D SOT in point clouds aims to
predict the states of the target in the coming frames. Due to the sparsity and
irregularity of the point clouds, 3D SOT in point clouds is still a
challenging task. Efficient 3D feature learning and accurate target location
are crucial to develop an effective and robust point clouds tracker.
Figure 1: Comparison of different 3D point clouds tracking framework. (a) The
framework of typical Siamese trackers. (b) The framework of the proposed
EasyTrack.
Most existing 3D SOT trackers in point clouds follow the typical 3D Siamese
multi-stage tracking paradigm, as shown in Fig. 1(a). Among it, two parallel
branches of 3D backbone with sharing parameters are designed to extract the
point cloudy features of the template and search area, respectively. Then a
heavy 3D feature fusion module is necessary to fuse 3D point cloudy features
from template and searching, and transfer the target’s information into the
search area. Finally, various target localization networks are applied to
regress the target’s location. Due to the fact that point clouds are usually
sparse, textureless, unorder and incomplete[12], many trackers introduce such
as shape completion networks [18], segmentation networks [19], or graph
networks [20] to regularize the 3D feature learning process or strengthen the
3D feature representation. Following this Siamese paradigm, P2B[21] is the
first end-to-end trained 3D Siamese tracker. It adopts PointNet++[22] as the
3D feature backbone to process point clouds. Then, the target-specific 3D
feature embedding network based on cosine similarity is designed to fuse
template and search area features. The deep Hough voting strategy is applied
for classification and regression. Different from P2B, STNet[23], SMAT[24]
design a transformer-based backbone to capture the long-range context in point
clouds. V2B[18], PTTR and PTTR++ [25] are proposed to match template and
search area features in a global manner by MLP layers or the transformer
network. C2FT[26], PTTR[25] aims to improve the voting-based localization
networks in a coarse-to-fine manner. V2B[18], GPT[20] introduce auxiliary
shape completion or graph network to improve the tracking performance.
Although these Siamese trackers present the favorable tracking performance,
they are still some problems due to the inherent properties of the multi-stage
Siamese paradigm. Firstly, the separated and parallel feature learning
branches of the template and search area points lead to the limited
discriminative ability, especially in some non-rigid categories like
Pedestrian. Because there is no mutual interaction or communication between
the template and search area in the 3D feature learning process. Secondly, the
point cloudy feature fusing operations are indispensable among these 3D
Siamese trackers since the search point features are unaware of the tracked
target. It is difficult to perform the effective feature matching in extremely
incomplete point clouds and the heavy feature fusing operation usually brings
high computational costs. Thirdly, we believe that in point cloud tracking
tasks, data may be subject to interference from environmental factors (such as
lighting changes, noise, occlusions), leading to challenges for the network to
effectively learn point-pair relationship patterns of the targets.
To address the above problems, we propose an extremely neat one-stream
framework for 3D SOT in point clouds from a novel perspective, abbreviated as
EasyTrack, as shown in Fig. 1(b). Different from the 3D Siamese tracking
framework, EasyTrack develops a target-aware unified one-stream network to
extract target-specified search area point features, based on the proposed
masked point clouds self-supervised tracking feature Learning module. And
then, the target-aware search features are directly fed into the BEV-based
localization network for classification and regression without any auxiliary
networks. The detailed network structure is shown in Fig. 3. We crop and
sample a fixed number of the original template and search area points as
input. Then a target-aware point cloud feature learning network is proposed to
extract point features and establish interaction between template and search
area points at the same time. Among it, a local embedding module based on ball
query is designed to aggregate local features for generating more global
feature representations. Then we concatenate the initial template and search
point features and apply self-perception transformer blocks to acquire point-
wise spatial relationships and facilitate regional interactions, thereby
providing target-aware search point features. In this way, the heavy feature
fusion network is no longer needed. We directly utilize a target location
network to obtain the target’s location in the search area. Specifically,
since the 3D point cloud is too sparse to accurately regression, we encode the
point cloud into a dense BEV feature space via a 2D convolution with residuals
and output the prediction parameters using decoupled positioning heads.
Although EasyTrack has already demonstrates superior tracking performance with
such a simple architecture, considering the influence of disturbances in the
dataset, we propose an enhanced version, abbreviated as Easytrack++. As
mentioned above, we feed the whole template points together with the search
points into the target-aware feature learning network for global interactions.
However, the template typically contains the target in the central region and
many background points that may cause interaction noise and ambiguous feature
description. To solve this problem, we develop a center points interaction
(CPI) strategy. As shown in Fig 6, we utilize the points in the center area of
the template to conduct secondary interaction with the search points to embed
more clear target information into the search area. It further improves the
tracking performance with acceptable computational costs.
Our main contributions can be summarized as follows:
* •
We propose a novel and neat one-stream paradigm EasyTrack for 3D SOT in point
clouds without any auxiliary networks or tricks. It runs at a speed of 52.6fps
and only has 1.30M parameters.
* •
We develop a new masked point clouds pre-training technique for 3D SOT, and
demonstrate its excellent performance on one-stream 3D SOT framework with
detailed ablations.
* •
A unified 3D tracking feature learning and interaction module is specially
designed for 3D SOT in point clouds. It generates target-aware point features
through a single-branch backbone.
* •
We further propose EasyTrack++ on top of EasyTrack. Among it, a center points
interaction strategy is applied to reduce the noise caused by background
points in the global interaction stage.
The rest of this paper is organized as follows. In section II, we introduce
the related work. Section III describes the methods of the proposed EasyTrack
and EasyTrack++. In section IV, we validate the superior performance of our
trackers in the KITTI, nuScenes and Waymo Open datasets through extensive
experiments. Finally, we conclude in section V.
## 2 Related Work
### 2.1 3D Single Object Tracking in Point Clouds
The pioneering work SC3D[27] first introduces the Siamese paradigm into 3D SOT
in point clouds. It uses the Kalman filter or exhaustive search to generate
candidate shapes in the search area. Then the candidate shape with the largest
cosine similarity is selected as the tracking result. P2B[21] adopts
PointNet++[22] as the backbone and embeds the target clues into the search
points based on cosine similarity. Proposals are obtained through a region
proposal network (RPN) based on deep Hough voting[28]. BAT[29] proposes an
additional feature representation called BoxCloud and designs a reliable
feature fusion module based on BoxCloud. MLVSNet[30] makes full use of the
multi-layer features of PointNet++ to conduct multi-layer Hough voting to
obtain more proposals. PointSiamRCNN[31] proposes a voxel-based tracker with
attention mechanism. It utilizes the multi-scale RoI pooling to form a two-
stage pipeline. C2FT[26] aims to improve the voting-based regression stage in
a coarse-to-fine manner. A local feature refinement module and a global
feature refinement module work collaboratively to refine the proposals.
V2B[18] designs a point cloud completion network to force the learned features
to include more shape information. A center-based head is adopted to locate
the target directly from BEV features. LTTR[32] first projects the point
clouds into the bird’s eye view and then uses the transformer network for
feature fusion. PTT[33] inserts two transformer blocks in the deep Hough
voting network to focus on deeper object clues in the tracking process.
PTTR[25] applies a transformer network including the self attention and cross
attention mechanisms. Among them, the global self attention operation captures
the long-range dependency and the cross attention is used to fuse two sets of
point features. GPT[20] introduces the graph neural network into point cloud
object tracking and designs a feature fusion module based on the graph neural
network. DMT[34] removes the usage of complicated 3D detectors and leverages
temporal motion cues to track. Similar to DMT, $M^{2}$-Track[19] proposes a
two-stage motion-centric paradigm. It first locates the target in consecutive
frames through motion transformation and then refines the prediction structure
through motion-assisted shape completion. STNet[18] uses the self attention
mechanism to capture the non-local information of the point clouds and the
decoder uses the cross attention mechanism to upsample discriminative point
features. An iterative correlation network based on cross attention is applied
to associate the template with the potential targets in the search area.
SMAT[24] converts point clouds into Pillars and compresses them into two-
dimensional BEV features. An encoder based on the attention mechanism realizes
the global similarity calculation between the template and search branch on
multi-scale features. CMT[35] proposes a context-matching-guided transformer
to effectively match the target template features with the search area. Many
above 3D tracking methods inherit the Siamese paradigm based on appearance
matching with the parallel classification and regression branches, based on
the supervised point clouds backbones such as PointNet++[22].
### 2.2 2D Object Tracking
Most of the existing 3D point cloud tracking algorithms refer to the idea of
2D Siamese trackers. SiamFC[1] is the pioneer of these trackers. It uses the
convolutional neural network to design a two-branch feature extraction network
for the template and search frame, respectively. Then, it obtains response
maps through cross-correlation operations. SiamRPN[2] introduces the regional
proposal (RPN) network in Faster RCNN[36] to generate proposals. SiamRPN++[3]
commits to using deeper and more powerful modern networks to extract features
such as ResNet[37]. SiamFC++[4] and SiamBAN[5] are anchor-free trackers that
directly regress the target’s location without any prior information.
TransT[6] proposes a feature fusion network based on the transformer network,
which effectively fuses the features of the template and search area through
the attention mechanism. STARK[7] uses the transformer network to capture
spatio-temporal information in video sequences. The encoder models the global
spatio-temporal feature dependency between the target and the search area, and
the decoder learns a query embedding to predict the spatial location of the
target. SwinTrack[8] designs a pure transformer tracking framework and applies
the transformer network in both feature extraction and feature fusion stages.
Mixformer[38] and OSTrack[9] follow a one-stream 2D tracking framework that
unifies the feature extraction and feature fusion processes by using the
transformer network.
### 2.3 Transformer in point clouds
The transformer network [39] has been widely adopted in various point cloud
for 3D vision tasks. In the point cloud segmentation and classification tasks,
PT[40] designs a transformer block based on the vector self attention
mechanism. Combining it with the set abstraction layer in PointNet++[22], PT
constructs a complete 3D point cloud understanding network. PCT[41] proposes a
novel offset-attention mechanism for point cloud processing. PTv2[42] develops
an effective group vector attention that enables efficient information
exchange within and among attention groups. In the 3D object detection task,
Pointformer[43] designs a local transformer module to model the interaction
between points in the local area and a global transformer to learn context-
aware representation at the scene level. VoTr[44] devotes to finding a
suitable transformer network for voxel representation. SST[45] proposes an
single-stride sparse transformer. With its local attention mechanism and
capability of handling sparse data, it overcome receptive field shrinkage in
the single-stride setting. In the 3D multi object tracking task,
TransFusion[46] proposes a two-layer transformer decoder. It first generates
initial bounding box predictions in the point clouds, and then attentively
fuse object queries with image features.
Moreover, the application of transformer-based pre-training strategies has
found extensive utility in the domain of point cloud tasks[47].
PointContrast[48] evaluates the transferability of advanced representations in
3D point clouds across diverse scenarios, demonstrating that unsupervised pre-
training effectively enhances performance on various downstream tasks and
datasets. Point-BERT[49] has introduced the concept of Masked Point Modeling
(MPM) as a pre-training task for point cloud transformers, thereby extending
the BERT paradigm into the realm of 3D point clouds. Point-MAE[50] has devised
an elegant solution for self-supervised learning on point clouds, employing an
asymmetric design and token manipulation through shift masking. It aims to
extract high-level latent features from unmasked point patches and reconstruct
the masked point patches. Point-PEFT[51] has devised a model that adjusts the
pre-training strategy with minimal learnable parameters. They freeze a
significant portion of the model’s parameters and only fine-tune the newly
introduced PEFT module on downstream tasks. Inspired above improvement of pre-
training methods on multiple point cloud tasks, we try to pre-train the point
cloud feature backbone of 3D SOT trackers, and extend the application of MAE
pre-training on point cloud 3D target tracking..
Figure 2: The 3D tracking pre-training strategy. We perform weight transfer on
the transformer blocks in the target-aware network.
## 3 METHOD
In this section, we first give an overview of one-stream 3D single point cloud
object tracking framework. Then, the detailed introduction of the proposed
EasyTrack and EasyTrack++ is presented. It mainly consists of four parts: the
3D point clouds tracking pre-training module, the unified target-aware point
clouds feature learning and interaction, target localization, and the center
points interaction strategy in EasyTrack++.
### 3.1 Overview
We propose a novel framework for 3D single object tracking, based on the
developed pre-training 3D point clouds tracking feature backbone. The network
structure is shown in Fig.3. Taking template points
$P^{t}\in\mathbb{R}^{N_{1}\times 3}$ and search points
$P^{s}\in\mathbb{R}^{N_{2}\times 3}$ as input, the tracking process can be
formulated as:
$(F^{t},F^{s})=\Phi(P^{t},P^{s})$ (1) $(x,y,z,\theta)=\Psi(F^{s})$ (2)
where $\Phi(\cdot)$ is the target-aware point cloud feature learning network,
$F^{t}\in\mathbb{R}^{N_{1}\times d}$ is the feature of template points and
$F^{s}\in\mathbb{R}^{N_{2}\times d}$ is the feature of search points.
$\Psi(\cdot)$ is the target localization network and $(x,y,z,\theta)$ is the
predicted parameters. Specifically, $(x,y,z)$ are the 3D coordinates of the
target’s center and $\theta$ is the heading angle in the X-Y plane. It is
worth noting that the size information of the target $(w,l,h)$ is given in the
template bounding box, and keeps unchanged across all frames in the point
cloud scenes. Thus we only predict $(x,y,z,\theta)$ in each frame.
Figure 3: The network structure of EasyTrack. It is mainly composed of two
parts: (1) Joint feature extraction and fusion network for template and search
area points feature learning and fusing. (2) Target location network for
classification and regression in the BEV feature space. Figure 4: The detailed
structure of the proposed local embedding network. Ball Query and MLP layers
are applied to capture the local features of point clouds.
### 3.2 3D tracking pre-training
Inspired by the success of BERT [52], we extend our pre-training strategy to
the field of 3D single target tracking. We utilize fine-tuned Point-MAE to
accommodate 3D SOT tasks, and the primary goal of pre-training is to capture
the relationships between 3D data points. We observe that the objects we need
to track in datasets such as KITTI have the following characteristics. (i)
Compared to the dozens or even hundreds of objects with completely different
shapes and structures that point cloud analysis tasks need to deal with,
tracking objects are often various types of Car, Cyclist or Motorcyclist,
Pedestrian, etc., which contain fewer categories and are more geometrically
similar. (ii) Data sets used for point cloud analysis tasks, such as ShapeNet,
often contain only a single CAD model with precise geometry and size
information. The data set nuScenes and others used by 3D SOT collected various
complex situations in real road scenes, and factors such as disturbance and
weather would cause different degrees of target occlusion. (iii) Since point
cloud data mainly comes from LiDAR sensors, the data collection process may be
limited by environmental conditions, time, location, and other factors,
resulting in the diversity and richness of data samples affected. This limits
the ability of the tracking model to learn the relationships between the
target points.
Based on the above observations, we emphasize that the tracking model learns
the correlation between points in 3D data in advance through pre-training. For
the first time, we introduce a new single-stream paradigm based on pre-
training. Specifically, for the pre-trained weights to fit a tracking target
with fewer categories and simpler geometry, we reduced the dimensions of the
attention heads and hidden layers in the encoder and the overall depth. Of
course, this operation inevitably reduces the encoder’s ability to learn, so
we increase the number of attention heads in the decoder to provide more
parallelism. This balances the encoder-decoder difference of the pre-trained
network, making the coding part more inclined to a new kind of point-pair
relational learning network.
The pre-training process is depicted in Fig. 2. Specifically, the developed
approach pre-trains a target-aware 3D tracking feature learning network based
on Point-MAE[50]. This pre-training process requires the use of sampling, KNN
algorithms, packet layers, and other methods in PointNet to obtain point-
position embedding and mask tokens and input tokens, and then learn the
underlying features through an encoder-decoder network of transformers.
Representing the complete mask labeling as $F_{m}\in\mathbb{R}^{mn\times C}$,
for a given input point cloud $P\in\mathbb{R}^{N\times 3}$, the developed pre-
training procedure can be described as follows:
$P_{e}=\text{{FPS}}(P,n),F_{v}=MLP(\text{KNN}(P,k))$ (3)
$T_{e}=\text{{Encoder}}(F_{v},P_{e})$ (4)
$H_{m}=\text{{Decoder}}(\text{concat}(T_{e},T_{m}))$ (5)
Where $Encoder$represents transformer-based point relationship learning
network, $T_{e}$ and $T_{m}$ correspond to encoding tokens and mask labels,
respectively. $P_{e}$indicates the location code. $H_{m}$ denotes the learned
mask tokens, and $F_{v}$ signifies the local features for each point. $n$
refers to the number of points that need to be sampled, and $k$ stands for K
nearest neighbor algorithm to select k samples around the center of mass point
as its neighbors.
A Multi-layer perceptron (MLP) layer is utilized as our prediction head to
reconstruct masked point patches in the coordinate space. Extracting the
output, $H_{m}$, from the decoder, the prediction head projects it onto a
vector. Subsequently, a reshaping operation is performed to yield the
predicted masked point patch, denoted as $P_{pre}$:
$P_{pre}=\text{{MLP}}(H_{m})$ (6)
EasyTrack leverages the pre-trained Point-MAE. There are some fundamental
differences between the two approaches: (i) The joint feature extraction and
fusion network is employed for both feature extraction and information fusion
in template and search area, while Point-MAE employs self attention for
feature extraction. (ii) The learning tasks are distinct, and correspondingly,
the inputs and heads differ. We utilize template and search area as inputs,
employing a voxel-to-BEV target center localization network to generate
bounding boxes, whereas Point-MAE is designed for point cloud analysis. (iii)
We further introduce center-point interaction strategies and an efficient
decoder design.
### 3.3 Target-aware 3D feature learning and interaction
Local embedding. Before feeding the point clouds into the transformer layers,
we design a local embedding network to obtain unique local features for each
point. It merges the position embedding and input embedding networks in the
typical transformer network. It composes of Ball Query[22] and multi-layer
perception (MLP) layers, aiming at local shape feature encoding for sparse and
incomplete point clouds. The detailed network structure is shown in Fig 4.
For each point $p_{i}$ in $P^{t}$, we take it as the center, $r$ as the
radius, and obtain $K$ points $P_{i}^{Q}\in\mathbb{R}^{K\times 3}$. Since we
take original point clouds as input, we treat the 3D coordinates as the point
features and apply three MLP layers to embed them into a 64-dimensional
feature space $F_{i}^{Q}\in\mathbb{R}^{K\times 64}$. Last, we perform max
pooling operation among $K$ points to obtain the local feature
$f_{i}^{lt}\in\mathbb{R}^{64}$ for $p^{i}$. The local feature embedding
process can be formulated as:
$P_{i}^{Q}=\left\\{s_{j}|\left\|s_{j}-p_{i}\right\|\leq r\right\\},j=1,...,K$
(7) $f_{i}^{lt}=max(MLP(P_{i}^{Q}))$ (8)
Figure 5: Illustration of the target-aware 3D feature learning. (a) The
detailed structure of the transformer layer. (b) The attention maps generated
by our target-aware point cloud feature learning network in the Car and
Pedestrian category in the KITTI dataset.
We unify the input embedding and the position embedding of the typical
transformer network into a local embedding module. The embedded local features
for the template $F^{lt}=\left\\{f_{i}^{lt}\right\\}_{i=1}^{N_{1}}$ provide
initial features for each point and unique local information since each point
has unique coordinates.
Target-aware feature learning. Different from previous Siamese trackers, we
develop the target-aware feature extraction and interaction method by a
unified single-branch block and avoid the time-consuming feature fusing
network, which is constructed based on the transformer network and self
attention mechanism. After local feature embedding, the template local
features $F^{lt}\in\mathbb{R}^{N_{1}\times 64}$ and search local features
$F^{ls}\in\mathbb{R}^{N_{2}\times 64}$ are concatenated as
$F^{l}=[F^{lt};F^{ls}]$. As shown in Fig. 3, we feed $F^{l}$ into three
stacked transformer blocks and obtain
$F^{g}\in\mathbb{R}^{{(N_{1}+N_{2})}\times d}$. Then we split it to obtain
target-aware search point features $F^{s}\in\mathbb{R}^{N_{2}\times d}$. The
detailed structure of the transformer network is shown in Fig. 5(a).
In each transformer block, we first use layer normalization operation to
normalize the initial point features. Then, linear layers are utilized to map
the inputs into query (Q), key (K), and value (V). Unlike the typical
transformer structure, we abandon the special position encoding module since
the coordinate-based local feature embedding network already provides unique
position information for every point. The self attention mechanism is the core
operation of our transformer network. It can be formulated as:
$attn=Softmax(\frac{QK^{T}}{\sqrt{d}}),$ (9)
$out=Softmax(\frac{QK^{T}}{\sqrt{d}})\cdot V$ (10)
where $attn$ is the attention map and $out$ is the output of the self
attention mechanism. Especially, we concatenate the template and search point
features as the input embeddings $F^{l}=[F^{lt};F^{ls}]$, thus the Q, K, V can
be represented as $[Q^{t};Q^{s}]$, $[K^{t};K^{s}]$, and $[V^{t};V^{s}]$. Eq. 9
and Eq. 10 can be expanded as:
$\displaystyle attn$
$\displaystyle=Softmax(\frac{[Q^{t};Q^{s}][K^{t};K^{s}]^{T}}{\sqrt{d}})$
$\displaystyle=[\lambda^{tt},\lambda^{ts};\lambda^{st},\lambda^{ss}],$ (11)
$\displaystyle out$
$\displaystyle=[\lambda^{tt},\lambda^{ts};\lambda^{st},\lambda^{ss}]\cdot[V^{t};V^{s}]$
$\displaystyle=[\lambda^{tt}V^{t}+\lambda^{ts}V^{s};\lambda^{st}V^{t}+\lambda^{ss}V^{s}]$
(12)
where $\lambda^{tt}$ and $\lambda^{ss}$ are attention weights, $\lambda^{ts}$
and $\lambda^{st}$ indicate the dual-interaction between template and search
area points. Eq.11 and Eq.12 explain why the proposed single-branch backbone
can learn target-aware point features for the search area. As shown in Fig.
5(b), we visualize the attention maps generated by the above transformer
network. We can find that the unified target-aware feature learning and
interaction network can focus on the target well and distinguish the similar
disturbances. It is worth noting that we apply the multi-head attention
mechanism to enhance the representative ability of the network. Besides, the
residual connection is considered and an MLP layer is designed to extract much
deeper features.
### 3.4 Efficient BEV-based Target Localization
After feature learning, we obtain target-aware search point features
$F^{s}\in\mathbb{R}^{N_{2}\times d}$. An efficient target localization is
proposed to regress the target’s location in the bird’s eye view (BEV) feature
space instead of directly regressing in sparse point clouds. We first divide
the irregular point clouds into equal voxels [53] according to their 3D
coordinates. By performing max pooling along the z-axis on the voxelized
feature maps, we obtain dense BEV feature maps where low responses in the
voxelized feature maps can be suppressed. However, voxelized features are
four-dimensional tensors and temporal fusion will superimpose features, making
the encoder computationally intensive. A pragmatic approach involves the
utilization of 3D convolutions applied to aggregate voxel features. We perform
max pooling operation along the $z$ axis to project voxel features into the
BEV feature space. However, this method is notably slow and inefficient.
To address this issue, we propose an efficient encoder module (EEM) that
reduces dimensionality with space-to-channel operations. Specifically, the
module converts the 4D voxel tensor $V\in R^{H\times W\times Z\times C}$ into
a 3D BEV tensor $V\in R^{H\times W\times(ZC)}$ via ”torch. reshape” operation,
thus avoiding the need for memory-intensive 3D convolution. Given a target-
aware search point features $F^{s}$ as input, we employ several 2D
convolutions to reduce the channel dimension and incorporate residual
connections to capture information from different layers. The module can be
represented as follows:
$F^{v}=Reshape(Voxel(F^{s}))$ (13) $F^{r}=\alpha(F^{v}+\beta F^{v})$ (14)
where $F^{r}\in\mathbb{R}^{H\times W\times d_{1}}$ represent the encoded
features of the BEV feature. $\beta$ means two 3*3 2D convolutions and a ReLU
layer. $\alpha$ represents two ReLU layers and a 3*3 2D convolution.
Figure 6: The network structure of EasyTrack++. It is built on top of
EasyTrack. Thus, the target-aware feature learning and target localization
networks are the same as Fig. 3. The center points interaction strategy crop
the center points and then make secondary interaction with the search area
points through the transformer network discussed in Fig. 5. Figure 7:
Illustration of the heads of the proposed EasyTrack. It consists of four parts
to realize accurate classification and regression collaboratively.
Inspired by [54], we design four heads from a decoupling perspective to
achieve accurate tracking, as shown in Fig. 6. A heatmap
$M\in\mathbb{R}^{H\times W\times 1}$ is predicted to find which pixel in the
BEV space in the target’s center is located. The offset map
$O\in\mathbb{R}^{H\times W\times 2}$ helps find a more accurate center and
recover the discretization error. Orientation $\Theta\in\mathbb{R}^{H\times
W\times 1}$ determines the heading angle of the target. We also predict the
height of the target’s center $Z\in\mathbb{R}^{H\times W\times 1}$ to locate
the target in the $z$-axis. After post processing, we can decode the target
location $(x,y,z,w,l,h,\theta)$ in the search area.
The total loss function of our method can be formulated as:
$L=\lambda_{1}L_{cls}+\lambda_{2}L_{off}+\lambda_{3}L_{\theta}+\lambda_{4}L_{Z}$
(15)
Among them, $L_{cls}$ is a modified focal loss for heatmap $M$, $L_{off}$ is a
$L_{1}$ loss for offset map $O$, $L_{\theta}$ and $L_{Z}$ are also $L_{1}$
losses for Orientation $\Theta$ and height $Z$ respectively.
$\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4}$ are weights for different
parts.
### 3.5 Center points interaction strategy
Although EasyTrack has demonstrated superior performance and achieved a
balance between tracking accuracy and speed, we still make improvements to it
and propose an enhanced version named EasyTrack++. In 3D single object
tracking task, the template usually contains information around the target.
This character brings heavy interaction noise when matching template and
search area. To address this problem, SimTrack[55] propose a foveal window
strategy to provide more diversified template information in 2D SOT. Here, we
develop a center points interaction strategy that specially designed for 3D
SOT in point clouds.
As shown in Fig. 8(a), the template contains points belonging to pedestrians
located in the center and background points distributed around them. We
utilize a simple but effective cropping strategy to crop the center points to
make secondary interactions with the search points to emphasize target
information in the search area. Specifically, after target-aware feature
learning, we obtain template features $F^{t}\in\mathbb{R}^{N_{1}\times d}$ and
search area features $F^{s}\in\mathbb{R}^{N_{2}\times d}$. We first use the
simplified ball query to define a central area around the geometric center $o$
of the template $P^{t}\in\mathbb{R}^{N_{1}\times 3}$. Then we collect a fixed
number of points $P^{c}\in\mathbb{R}^{N_{3}\times 3}$ inside it and aggregate
their features in $F^{t}$ as $F^{c}\in\mathbb{R}^{N_{3}\times d}$, where
$N_{3}$ is the fixed number. If the number of points inside the central area
is more than $N_{3}$, we use random sampling to sample $N_{3}$ points.
Otherwise, we repeat the existing points. After that, we concatenate features
of center points and search area and make secondary interactions through the
transformer block mentioned in Sec 3.3. The whole center points interaction
strategy can be formulated as:
$(P^{c},F^{c})=BallQuery(P^{t},F^{t},o)$ (16)
$F^{f}=Transformer(concat(F^{c},F^{s}))$ (17)
where $F^{f}$ is the final feature for target localization. We visualize the
attention maps generated by the proposed EasyTrack and EasyTrack++ in Fig. 8.
As shown in Fig. 8(c), in EasyTrack, some background points around the target
have high response that maybe confuse the target localization network and lead
to inaccurate classification and regression. However, as shown in Fig. 8(d),
the response of target is much more distinct than other backgrounds. This
further reflects that the center points interaction strategy enhances the
discrimination of feature extraction networks.
Figure 8: The visualization comparison of attention maps generated by the
global interactions in EasyTrack and EasyTrack++. We select one tracking
scenes in the Pedestrian category in the KITTI dataset. (a) the template
points. (b) the search area points. (c) the attention map generated by
EasyTrack. (d) the attention map generated by EasyTrack++. TABLE I: Tracking
results compared to other trackers in the KITTI dataset. Success / Precision
are used for evaluation. Bold denotes the best result.
Metric | Success | Precision
---|---|---
Category | Car | Pedestrian | Van | Cyclist | Mean | Car | Pedestrian | Van | Cyclist | Mean [t]
Frame Number | 6424 | 6088 | 1248 | 308 | 14068 | 6424 | 6088 | 1248 | 308 | 14068 [b]
SC3D[27] | 41.3 | 18.2 | 40.4 | 41.5 | 31.2 | 57.9 | 37.8 | 47.0 | 70.4 | 48.5 [t]
P2B[21] | 56.2 | 28.7 | 40.8 | 32.1 | 42.4 | 72.8 | 49.6 | 48.4 | 44.7 | 60.0
MLVSNet[30] | 56.0 | 34.1 | 52.0 | 34.3 | 45.7 | 74.0 | 61.1 | 61.4 | 44.5 | 66.7
LTTR[32] | 65.0 | 33.2 | 35.8 | 66.2 | 48.7 | 77.1 | 56.8 | 45.6 | 89.9 | 65.8
BAT[29] | 60.5 | 42.1 | 52.4 | 33.7 | 51.2 | 77.7 | 70.1 | 67.0 | 45.4 | 72.8
PTT[33] | 67.8 | 44.9 | 43.6 | 37.2 | 55.1 | 81.8 | 72.0 | 52.5 | 47.3 | 74.2
C2FT[26] | 67.0 | 48.6 | 53.4 | 38.0 | 57.2 | 80.4 | 75.6 | 66.1 | 48.7 | 76.4
V2B[18] | 70.5 | 48.3 | 50.1 | 40.8 | 58.4 | 81.3 | 73.5 | 58.0 | 49.7 | 75.2
PTTR[25] | 65.2 | 50.9 | 52.5 | 65.1 | 57.9 | 77.4 | 81.6 | 61.8 | 90.5 | 78.1
CMT[35] | 70.5 | 49.1 | 54.1 | 55.1 | 59.4 | 81.9 | 75.5 | 64.1 | 82.4 | 77.6
STNet[23] | 72.1 | 49.9 | 58.0 | 73.5 | 61.3 | 84.0 | 77.2 | 70.6 | 93.7 | 80.1
$M^{2}$-Track[19] | 65.5 | 61.5 | 53.8 | 73.2 | 62.9 | 80.8 | 88.2 | 70.7 | 93.5 | 83.4
STTrack[56] | 66.5 | 60.4 | 50.5 | 75.3 | 62.6 | 79.9 | 89.4 | 63.6 | 93.9 | 82.9
PCET[57] | 68.7 | 56.9 | 57.9 | 75.6 | 62.7 | 80.1 | 85.1 | 66.1 | 93.7 | 81.3
PTIT[58] | 68.6 | 56.7 | 53.8 | 74.2 | 62.6 | 81.2 | 86.3 | 70.7 | 92.5 | 82.7
CXTrack[59] | 69.1 | 67.0 | 60.0 | 74.2 | 67.5 | 81.6 | 91.5 | 71.8 | 94.3 | 85.3
CorpNet[60] | 73.6 | 74.3 | 58.7 | 55.6 | 64.5 | 84.1 | 94.2 | 66.5 | 82.4 | 82.0
SyncTrack[61] | 73.3 | 54.7 | 60.3 | 73.1 | 64.1 | 85.0 | 80.5 | 70.0 | 93.8 | 81.9
MBPTrack[62] | 73.4 | 68.6 | 61.3 | 76.7 | 70.3 | 84.8 | 93.9 | 72.7 | 94.3 | 87.9
M3SOT[63] | 75.9 | 66.6 | 59.4 | 70.3 | 70.3 | 87.4 | 92.5 | 74.7 | 93.4 | 88.6 [b]
EasyTrack(ours) | 87.9 | 88.2 | 65.3 | 85.3 | 86.0 | 90.1 | 95.6 | 78.8 | 93.8 | 91.6 [t]
EasyTrack++(ours) | 88.7 | 91.2 | 68.6 | 87.4 | 88.0 | 92.6 | 96.3 | 80.0 | 95.4 | 93.2 [b]
TABLE II: Tracking results compared to other trackers in the nuScenes dataset.
Success / Precision are used for evaluation. Bold denotes the best result.
Category | Car | Pedestrian | Truck | Bicycle | Mean | Car | Pedestrian | Truck | Bicycle | Mean [t]
---|---|---|---|---|---|---|---|---|---|---
Frame Number | 15578 | 8019 | 3710 | 501 | 27808 | 15578 | 8019 | 3710 | 501 | 27808 [b]
SC3D[27] | 25.0 | 14.2 | 25.7 | 17.0 | 21.8 | 27.1 | 16.2 | 21.9 | 18.2 | 23.1 [t]
P2B[21] | 27.0 | 15.9 | 21.5 | 20.0 | 22.9 | 29.2 | 22.0 | 16.2 | 26.4 | 25.3
BAT[29] | 22.5 | 17.3 | 19.3 | 17.0 | 20.5 | 24.1 | 24.5 | 15.8 | 18.8 | 23.0
V2B[18] | 31.3 | 17.3 | 21.7 | 22.2 | 25.8 | 35.1 | 23.4 | 16.7 | 19.1 | 29.0
STNet[23] | 32.2 | 20.4 | 27.6 | 18.5 | 26.5 | 33.4 | 32.9 | 20.8 | 26.8 | 31.5
CXTrack[59] | 29.6 | 20.4 | 27.6 | 18.5 | 26.5 | 33.4 | 32.9 | 20.8 | 26.8 | 31.5
CorpNet[60] | 35.0 | 21.3 | 39.7 | 26.9 | 31.8 | 38.4 | 33.6 | 36.3 | 43.5 | 36.8
SyncTrack[61] | 36.7 | 19.1 | 39.4 | 23.8 | 31.8 | 38.1 | 27.8 | 38.6 | 30.4 | 35.1
M3SOT[63] | 34.2 | 24.6 | 29.6 | 18.8 | 30.5 | 38.6 | 37.8 | 25.5 | 27.9 | 36.4 [b]
EasyTrack(ours) | 76.1 | 60.5 | 71.8 | 44.8 | 70.5 | 80.5 | 86.9 | 70.0 | 73.2 | 80.8 [t]
EasyTrack++(ours) | 76.8 | 61.4 | 72.1 | 45.1 | 71.2 | 80.9 | 87.2 | 70.2 | 74.5 | 81.2 [b]
TABLE III: Tracking results compared to other trackers in the WOD. Success /
Precision are used for evaluation. Bold denotes the best result.
Category | Vehicle(185731) | Pedestrian(241752) | Mean [t]
---|---|---|---
Easy | Medium | Hard | Mean | Easy | Medium | Hard | Mean | [b]
P2B[21] | 57.1/65.4 | 52.0/60.7 | 47.9/58.5 | 52.6/61.7 | 18.1/30.8 | 17.8/30.0 | 17.7/29.3 | 17.9/30.1 | 33.0/43.8 [t]
BAT[29] | 61.0/68.3 | 53.3/60.9 | 48.9/57.8 | 54.7/62.7 | 19.3/32.6 | 17.8/29.8 | 17.2/28.3 | 18.2/30.3 | 34.1/44.4
V2B[18] | 64.5/71.5 | 55.1/63.2 | 52.0/62.0 | 57.6/65.9 | 27.9/43.9 | 22.5/36.2 | 20.1/33.1 | 23.7/37.9 | 38.4/50.1
TAT[64] | 66.0/72.6 | 56.6/64.2 | 52.9/62.5 | 58.9/66.7 | 32.1/49.5 | 25.6/40.3 | 21.8/35.9 | 26.7/42.2 | 40.7/52.8
STNet[23] | 65.9/72.7 | 57.5/66.0 | 54.6/64.7 | 59.7/68.0 | 29.2/45.3 | 24.7/38.2 | 22.2/35.8 | 25.5/39.9 | 40.4/52.1
CXTrack[59] | 63.9/71.1 | 54.2/62.7 | 52.1/63.7 | 57.1/66.1 | 35.4/55.3 | 29.7/47.9 | 26.3/44.4 | 30.7/49.4 | 42.2/56.7
$M^{2}$-Track[19] | 68.1/75.3 | 58.6/66.6 | 55.4/64.9 | 61.1/69.3 | 35.5/54.2 | 30.7/48.4 | 29.3/45.9 | 32.0/49.7 | 44.6/58.2
MBPTrack[62] | 68.5/77.1 | 58.4/68.1 | 57.6/69.7 | 61.9/71.9 | 37.5/57.0 | 33.0/51.9 | 30.0/48.8 | 33.7/52.7 | 46.0/61.0 [b]
EasyTrack++(Ours) | 70.0/77.8 | 59.1/69.1 | 58.3/70.5 | 62.8/72.7 | 38.1/58.5 | 35.2/52.2 | 31.5/49.0 | 35.1/53.5 | 47.1/61.8
TABLE IV: The results for the actual tracked frames in the nuScenes dataset.
Category | Car | Pedestrian | Truck | Bicycle | Mean
---|---|---|---|---|---
Frame Number | 146871 | 75066 | 35448 | 4698 | 262123
V2B[18] | 34.5/36.1 | 18.9/23.6 | 32.9/29.3 | 23.2/30.5 | 29.6/31.5
Easytrack++ | 74.5/78.8 | 58.5/84.7 | 70.1/68.3 | 38.5/72.1 | 68.7/78.9
## 4 Experiments
In this section, we conduct extensive experiments to show the favorable
performance of EasyTrack and EasyTrack++. First, we introduce our experiment
settings including datasets and evaluation metrics. Then we give comprehensive
comparisons with other state-of-the-art trackers to validate the superiority
of our tracker. Last, adequate ablation studies validate the effectiveness of
each part in the EasyTrack and EasyTrack++.
### 4.1 Experiment settings
Datasets. We evaluate the performance of EasyTrack and EasyTrack++ on the
challenging KITTI[65], nuScenes[66] and Waymo Open Dataset (WOD)[67]. They
mainly focus on autonomous driving scenes. KITTI has 21 training scenes and we
follow [27] to split the training set since the testing set has no
annotations. Specifically, we train in scenes 0-16, validate in scenes 17-18,
and test in scenes 19-20. For the nuScenes dataset, we follow [18] to split it
into 750 training sequences and 150 validation sequences. We train our method
on the training set and test it on its validation set. Notably, we use the
official toolkit to insert ground truth for unannotated frames since only key
frames have annotations in nuScenes. For WOD, we followed LiDAR SOT[68] to
evaluate our approach on 1,121 tracks divided into easy, medium, and difficult
subsets based on point cloud sparsity. The datasets use for our pre-training
include ShapeNet,ModelNet40,ScanObjectNN. The ShapeNet[69] dataset utilized
for pre-training comprises approximately 51,300 meticulously curated 3D
models, encompassing 55 prevalent object categories. It was meticulously
compiled by collecting CAD models from open-source 3D repositories available
online. CAD models, which are digital three-dimensional representations within
computer-aided design software, exhibit highly precise geometric information.
ModelNet40 is a commonly used dataset for 3D object recognition that contains
40 different categories of 3D CAD models. The models were divided into 12,311
training samples and 1,269 test samples. The object category in ModelNet40
covers common objects such as chairs, tables, monitors, airplanes, and so on.
Unlike the first two, ScanObjectNN is a challenging real-world dataset
consisting of approximately 15,000 objects from 15 categories that provide
more light variation, noise interference, and target context information to
simulate complex and variable real-world scenarios.
Evaluation Metrics. We follow previous methods to report the Success and
Precision metrics. Among them, the Success metric is defined by the 3D IOU
between the predicted 3D bounding box and the ground truth. The Precision
metric is defined by the Area Under Curve (AUC) for the distance between
centers of the predicted and annotated bounding boxes from 0 to 2m.
Model Details. In the proposed EasyTrack and EasyTrack++, we take $N_{1}=512$
template points and $N_{2}=1024$ search area points as input. The radius
$r=0.3m$ and $K=32$ neighbor local points are considered in the local feature
embedding network. In the target-aware feature learning process, we use the
multi-head self attention in the transformer block. We stack three transformer
blocks with four heads. The feature dimension $d$ is set to 64. In the center
points interaction stage, we collect $N_{3}=128$ points to form $P^{c}$ and we
set use one layer of transformer network for secondary interaction. In the
target localization network, the voxel size is set to $(0.3m,0.3m,0.3m)$. The
size of the BEV feature map is $24\times 38\times 128$. Four 2D convolutions
and two 2D deconvolutions are designed to process the BEV feature in
combination with the residual connection.
Training. In the training stage, we merge point clouds in the $(t-1)$-th
bounding box and the first bounding box as the template. The point clouds
inside the enlarged $t$-th bounding box are sampled as the search area along
with random shift. We set
$\lambda_{1}=1.0,\lambda_{2}=1.0,\lambda_{3}=1.0,\lambda_{4}=2.0$ in the loss
function. Our model is end-to-end trained for 20 epochs. We use the Adam
optimizer, and the initial learning rate is set to 0.001. It is divided by 5
every 6 epochs.
Testing. In the testing stage, we give the target in the first frame of a
tracklet and track it across all frames. We merge point clouds in the
$(t-1)$-th predicted result and the given bounding box in the first frame as
the template. The $(t-1)$-th predicted result is enlarged by $2m$ in the
$t$-th frame and we sample 1024 points inside it as the search area.
3D Pre-training. Empirical evidence reveals that pre-trained models on 3D
analysis task datasets (e.g. ShapeNet dataset) can significantly enhance the
precision of 3D tracking tasks. The ShapeNet dataset comprises approximately
50,000 3D CAD models spanning 14 major categories and 55 subcategories. To be
more specific, our proposed target-aware feature learning network serves as
the backbone, and we pre-trained it on ShapeNet for 300 epochs using Point-
MAE. During training on the KITTI and nuScenes datasets, we initialized
EasyTrack with pre-trained weights from the backbone network of the 3D
analysis framework. The remaining layers were initialized randomly.
Figure 9: Robustness analysis in the Car category in the KITTI dataset.
### 4.2 Comparison with other trackers
Results on KITTI. We compare EasyTrack and EasyTrack++ with other state-of-
the-art 3D SOT methods in Table I. Following previous works, we report the
results in four categories including Car, Pedestrian, Van, and Cyclist. The
mean results are calculated based on all frames. It can be observed that the
proposed EasyTrack outperforms other trackers across all categories, with an
average success and precision score of 86.0/91.6. Besides, thanks to the
center points interaction strategy, the enhanced version EasyTrack++ further
improves the tracking performance. The mean Success and Precision metrics are
the highest among all trackers (88.0/93.2). Compared to EasyTrack, EasyTrack++
shows an improvement of 2.0 in mean Success and 1.6 in Precision metrics
compared to EasyTrack.
Results on nuScenes. We show the tracking results on nuScenes dataset compared
to other state-of-the-art trackers in Table II. It is important to note that
for unannotated frames, the ground truth is obtained through interpolation, as
mentioned earlier. We compare with those trackers that follow the same
settings. We report the results in four categories including Car, Pedestrian,
Truck, and Bicycle. Note that the results are computed on the key frames of
the validation set. EasyTrack outperforms other trackers in all categories.
EasyTrack++ further improves the tracking performance and achieves the best
mean Success and precision (71.2/81.2). Compared to tracking in the KITTI
dataset, it is more difficult to track the target in nuScenes due to the low-
quality interpolated ground truth. In previous work, tracking results were
reported only for keyframes (Car-15578), as shown in Table II. However, in
Table IV, we provide results for all actual tracked frames (Car-145871), and
EasyTrack++ exhibits a significant advantage over V2B.
Figure 10: Visualization of advantage cases compared to STNet[23]. We
visualize one tracking scene for each category in the KITTI dataset. Figure
11: Visualization of advantage cases compared to STNet[23]. We visualize one
tracking scene for each category in the nuScenes dataset.
Results on Waymo. To assess the generalization capabilities of our proposed
EasyTrack++, we conducted an evaluation of the KITTI pre-trained model on the
WOD, as outlined in the prior work [23]. It is worth noting that the object
categories between KITTI and WOD align as follows: Car $\rightarrow$ Vehicles
and Pedestrian $\rightarrow$ Pedestrian. The experimental results, presented
in Table III, Comparison with other methods reveal that EasyTrack++ exhibits
superior tracking performance across various challenging occlusion scenarios,
achieving the highest average success rate and precision (47.1/61.8). In
summary, our proposed methodology not only demonstrates accurate target
tracking across diverse object sizes (vehicle or pedestrian) but also exhibits
strong generalization capabilities when applied to previously unobserved
scenes.
TABLE V: Computation costs compared to other Siamese trackers. Method | Param(M) | FLOPs(G) | FPS
---|---|---|---
P2B[21] | 1.3 | 4.7 | 35.7 [t]
BAT[29] | 1.5 | 3.1 | 51.8
PTT[33] | 4.9 | 6.2 | 41.2
V2B[18] | 1.3 | 5.6 | 20.4
PTTR[25] | 2.1 | 2.7 | 35.9
STNet[23] | 2.0 | 3.1 | 27.5
$M^{2}$-Track[19] | 2.2 | 2.5 | 57.0
EasyTrack | 1.0 | 2.3 | 53.2
EasyTrack++ | 1.3 | 2.6 | 52.6 [b]
Computational cost. We propose a simple baseline for 3D SOT without the heavy
feature fusion module that is necessary for typical Siamese trackers. The
computational costs of different trackers are reported in Table V. For a fair
comparison, we use a single NVIDIA 3090 GPU to test all the trackers. Table V
reflects that EasyTrack has fewer parameters and FLOPs than other trackers.
The proposed object localization module effectively reduces the model’s
parameter training volume with the compromising accuracy. It runs at a real-
time speed (53.2FPS) and is only slow than $M^{2}$-Track but much more
accurate as shown in Table I. Compared with typical Siamese trackers,
EasyTrack is a lightweight tracker and achieves a good balance between running
speed and tracking accuracy. At the same time, EasyTrack++ further improves
tracking performance with the acceptable computation cost. It achieves the
best tracking performance as illustrated in Tab I and still outperforms most
3D SOT trackers in the speed metric and runs in a 52.6FPS.
Figure 12: (a) The impact of different pre-training epochs on tracking
performance in the KITTI dataset. (b) The effect of pre-training datasets. (c)
The convergence speed of the model in the KITTI dataset. $*$ indicates no pre-
training employed.
Robustness in Sparse Scenes. It is crucial for a 3D tracker to perform robust
tracking in sparse point clouds for practical applications. We divide the
tracing scenes into four intervals according to the number of points. We
report the Success metric in the Car category in the KITTI dataset in Fig. 9.
As the number of points increases, the performance of trackers increases
steadily. EasyTrack presents better performance than V2B[18] and STNet[23] in
three intervals. It reflects that EasyTrack has good robustness in sparse
scenes that have incomplete target information. The proposed EasyTrack++
gradually improves the performance of EasyTrack since the center points
interaction strategy has more informative points to guide the secondary
interaction in relatively dense scenes. Fig. 12(b) intuitively reflects that
EasyTrack and EasyTrack++ have good robustness in sparse scenes. This is
critical when facing occlusion conditions or tracking distant objects.
Visualization comparisons. To demonstrate the superiority of the proposed
EasyTrack more intuitively, we visualize the tracking process compared to the
competitive tracker STNet[23] in Fig. 10 and Fig. 11. In the Car category in
the KITTI dataset, as shown in the first row of Fig. 10, the point clouds of
the target are quite sparse and incomplete. EasyTrack tracks well while STNet
deviates largely. It reflects the good robustness of EasyTrack in sparse
scenes. In a scene with more disturbs in the second row, EasyTrack holds the
target pedestrian tightly while STNet gradually loses the target and locates a
similar target. This can be attributed to the superiority of our target-aware
feature learning process. The simple correlation operation in STNet leads to
the unreliable feature matching among incomplete point clouds. Compared to
KITTI dataset that uses a 64-line LIDAR, nuScenes dataset uses a 32-line
LiDAR, thus the points are usually more sparse. As shown in Fig. 11, EasyTrack
also performs better than STNet in the sparse tracking scenes.
### 4.3 Ablation study
For a more thorough assessment of the individual efficacy of the proposed
method, it is imperative to note that, unless otherwise specified, ablation
experiments were conducted without the utilization of the pre-training
strategy.
Model components. Our ablation study examined the impact of 3D SOT pre-
training (Pre), target-aware feature learning (TA), and target localization
(TL) during the training process. The results are shown in Table VI. It was
observed that the pre-training strategy significantly enhances tracking
performance ($\uparrow$21.2/$\uparrow$9.4). This finding substantiates the
superiority of the pre-training strategy specifically tailored for 3D SOT
tasks, emphasizing the complementary value of the CAD model-based ShapeNet
dataset used in pre-training for point cloud data. Even without employing the
pre-training strategy, our tracker still demonstrates competitive results,
with a mean success and precision of 64.5/83.5. Furthermore, our proposed
target-aware feature learning and object localization methods exhibited
varying degrees of performance improvement. Subsequently, with the
implementation of the proposed target-aware feature learning, the mean
performance is enhanced by 1.0/0.7. Finally, by leveraging the proposed target
localization, we achieve a further performance boost of 1.2/0.5. This
demonstrates its ability to accurately localize objects in the fusion space
based on BEV features.
TABLE VI: Ablation studies on different model components on KITTI[65] dataset.
Pre | TA | TL | Car | Pedestrian | Van | Cyclist | Mean
---|---|---|---|---|---|---|---
✗ | ✗ | ✗ | 71.5/82.0 | 53.8/82.3 | 56.9/65.9 | 72.7/93.8 | 62.6/81.0
✗ | ✓ | ✓ | 72.8/84.0 | 56.5/85.8 | 58.6/67.5 | 73.1/93.7 | 64.5/83.5
✓ | ✗ | ✗ | 85.2/88.5 | 87.1/95.0 | 60.1/76.5 | 83.2/93.7 | 83.8/90.4
✓ | ✓ | ✗ | 86.5/89.6 | 87.9/95.2 | 60.8/77.5 | 84.3/93.7 | 84.8/91.1
✓ | ✓ | ✓ | 87.9/90.1 | 88.2/95.6 | 65.3/78.8 | 85.3/93.8 | 86.0/91.6
TABLE VII: Results of different target localization networks.
Method | Success | Precision | FPS
---|---|---|---
Coordinate Space[28] | 70.3 | 81.9 | 53.4
BEV Space (w/o EEM) | 73.1 | 84.5 | 50.1
BEV Space (w/ EEM) | 73.4 | 84.4 | 52.6
TABLE VIII: Integration with Siamese-based network.
Method | Car | Pedestrian | Van | Cyclist | Mean
---|---|---|---|---|---
STNet | 72.1/84.0 | 49.9/77.2 | 58.0/70.6 | 73.5/93.7 | 61.3/80.1 [t]
STNet* | 81.1/87.5 | 71.2/84.2 | 68.7/78.4 | 80.8/93.9 | 75.7/85.4
Improvement | $\uparrow$9.0/$\uparrow$3.5 | $\uparrow$21.3/$\uparrow$7.0 | $\uparrow$10.7/$\uparrow$7.8 | $\uparrow$7.3/$\uparrow$0.2 | $\uparrow$14.4/$\uparrow$5.3
PTTR | 65.2/77.4 | 50.9/81.6 | 52.5/61.8 | 65.1/90.5 | 58.4/77.8 [t]
PTTR* | 74.1/87.2 | 72.3/88.2 | 58.8/70.5 | 73.2/93.4 | 72.0/86.3
Improvement | $\uparrow$8.9/$\uparrow$10.2 | $\uparrow$21.4/$\uparrow$6.6 | $\uparrow$6.3/$\uparrow$8.7 | $\uparrow$8.1/$\uparrow$2.9 | $\uparrow$13.6/$\uparrow$8.5
*Integrated the pre-training strategy. | |
TABLE IX: Comparison with different feature fusing modules.
Method | Success | Precision | FPS
---|---|---|---
P2B-xorr[21] | 71.5 | 82.4 | 39.4 [t]
MLVSNet-xorr[33] | 70.6 | 81.4 | 41.7
V2B-xorr[18] | 71.8 | 82.7 | 29.2
STNet-xorr[25] | 72.0 | 83.2 | 35.7
Easytrack++† | 72.4 | 83.5 | 49.8
Easytrack++ | 73.4 | 84.4 | 52.6[b]
† represents the adoption of a mask strategy.
3D pre-training. To further investigate the effectiveness of the pre-training
strategy, we conducted a series of pre-training epochs ranging from 0-600 to
explore the impact of different pre-training durations on tracking
performance. The results for different epochs within the Car category on the
KITTI dataset are presented in Fig. 12(a). The optimal performance was
observed at epoch=300, with success/precision metrics reaching 87.9/90.1.
Furthermore, to assess the generalization of the pre-training strategy, we
integrated it into the PTTR framework. The Tab VIII illustrates a significant
performance improvement when pre-training is incorporated into the PTTR
network ($\uparrow$13.6/$\uparrow$8.5). Fig. 12(c) presents a visualization of
the influence of pre-training on convergence speed. It is evident that when
utilizing pre-training, convergence is achieved as early as the 15th epoch.
This advancement of approximately 20 epochs, compared to standard training,
leads to a substantial reduction in training duration.
Layers of the transformer. As shown in the Fig. 4, We stack three layers of
the self-perception transformer in the target-aware feature learning network
in the EasyTrack++. The parameter $n$ of the self-perception transformer
layers is a key parameter. If it is set too small, the ability of feature
extraction and interaction of the network equilibrium point is insufficient..
If it is set too large, it will lead to overfitting and slow running speed. We
report the results of setting different $n$ in the Car category in the KITTI
dataset in Fig. 13. We can find that as the parameter $n$ gradually increases,
the tracking performance first increases and then decreases. The best
performance is obtained when $n=3$. The Success/Precision metrics are
73.4/84.4.
Joint or split point feature learning. The core design of our tracker is a
target-aware point cloud feature learning network. It effectively embeds the
target’s information into the search area and greatly accelerates the running
speed. To verify the superiority of our design, we also design a typical
Siamese tracker with the transform backbone in Fig. 5(a). Different from
EasyTrack++, we do not concatenate the template and search area before feature
learning. Different feature fusing modules are equipped for a full comparison.
As shown in Tab IX, we select four kinds of feature fusing networks including
P2B-xorr[21], MLVSNet-xorr[30], V2B-xorr[18], and STNet-xorr[23]. We can find
that EasyTrack++ achieves the best accuracy and fastest running speed. This
demonstrates that the proposed target-aware feature learning network can
improve the accuracy and running speed at the same time.
Localization in the BEV or 3D feature space. As discussed in Sec. 15, we
regress and classify in a relatively dense BEV feature map instead of the 3D
feature space. When facing sparse point clouds scenes, it is hard to generate
high-quality proposals in the 3D space, so these methods may not be able to
track the object effectively. To validate the superiority of our design, we
preserve the feature learning network and replace the target localization
network with the widely adopted VoteNet[28]-based strategy. However, due to
the sparsity of point clouds, the voting stage suffers from heavy outliers and
generates unreliable proposals. The results in the Car category in the KITTI
dataset are shown in Tab VII. We achieve better performance when regressing
and classifying in the BEV feature space, 3.1/2.5 higher in the
Success/Precision metrics. It further demonstrates the effectiveness of
localization in the BEV feature space.
Ways to generate center points. In the proposed EasyTrack++, we design a
center points interaction strategy to make secondary interaction to provide
more detailed target information. The quality of center points has a
significant impact on the tracking performance. In fact, we consider two ways
to define the center points including ball query and K-nearest-neighbor (KNN)
algorithms. These are two commonly used algorithms to capture local
information in the point clouds scene. Meanwhile, the number of center points
is also a key parameter. We conduct extensive ablation experiments in the Car
category in the KITTI dataset. The results are illustrated in Tab XI. We can
find that ball query shows better performance compared to KNN algorithm with
the same number of points. And as the number of points increases, the tracking
performance first increases and then stabilizes. Finally, we select ball query
to crop center points and the number of points is set to 128. since it
achieves the best tracking accuracy.
TABLE X: Results of different template generating schemes. We report the Success/Precision metrics in the Car category in the KITTI dataset. Scheme | P2B | BAT | PTT | V2B | PTTR | STNet | EasyTrack | EasyTrack++
---|---|---|---|---|---|---|---|---
First GT | 46.7/59.7 | 51.8/65.5 | 62.9/76.5 | 67.8/79.3 | 55.0/65.6 | 70.8/82.4 | 70.9/82.6 | 71.3/82.9 [t]
Previous result | 53.1/68.9 | 59.2/75.6 | 64.9/77.5 | 70.0/81.3 | 65.0/77.1 | 66.0/76.6 | 69.6/80.1 | 69.8/80.3
First GT & Previous result | 56.2/72.8 | 60.5/77.7 | 67.8/81.8 | 70.5/81.3 | 65.2/77.4 | 72.1/84.0 | 72.8/84.0 | 73.4/84.4
All previous results | 51.4/66.8 | 55.8/71.4 | 59.8/74.5 | 69.8/81.2 | 63.1/74.8 | 73.3/85.4 | 70.1/81.4 | 70.5/81.8 [b]
TABLE XI: Results of different ways to generate the center points to make secondary interaction in the proposed EasyTrack++. Method | Number | Success | Precision
---|---|---|---
KNN | 32 | 72.4 | 83.6 [t]
64 | 72.9 | 83.9
128 | 73.2 | 84.2
256 | 73.2 | 84.1 [b]
Ball Query | 32 | 72.6 | 83.7 [t]
64 | 73.1 | 83.9
128 | 73.4 | 84.4
256 | 73.2 | 84.1 [b]
Different template generating schemes. The quality of the template has a great
impact on the performance of the tracker since the target is given in the
template. Incomplete template provides ambiguous target information. To find
the best scheme, we consider four schemes to generate the template including
using the first ground truth, using the previous result, using all previous
results, and merging the first ground truth and previous result. The results
of different schemes compared to other trackers in the Car category are shown
in Tab X. EasyTrack outperforms other trackers in most cases. And EasyTrack++
further improves the tracking performance in all schemes. However, we can find
that when only using the previous result to generate template, V2B [18]
achieves the best performance. It is because V2B uses an auxiliary shape
completion network to capture more shape information. This makes the network
structure more complex. When using all previous results, STNet[23] achieves
the best performance but at a very low speed. Considering the balance between
speed and accuracy, all the trackers use the first ground truth and previous
result to generate the template and our methods achieve the best performance
in this scheme.
Figure 13: Results of setting different $n$ values in the target-aware feature
learning network.
## 5 Conclusions
In this paper, we propose a new baseline for 3D single object tracking in
point clouds abbreviated as EasyTrack, which can simplify the two-stream
multi-stage 3D Siamese trackers, and unify the process of 3D feature
extraction and target information integration, based on the masked point cloud
tracking feature pre-trained learning. After that, we design a localization
network in the BEV feature space for accurate regression and classification.
In addition, we further design an enhanced version named EasyTrack++, which
develops a center points interaction (CPI) strategy to reduce the ambiguous
target description caused by the background information in the noisy point
clouds. Extensive experiments in the KITTI, nuScenes and Waymo datasets
demonstrate that EasyTrack achieves state-of-the-art performance and runs at a
real-time speed. Due to the rich texture information provided by RGB data for
incomplete point clouds, future plans include complementing point clouds with
2D images and proposing a multi-modal one-stream 3D tracker.
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| Baojie Fan is an professor in the Department of Automation college at
Nanjing University of Posts and Telecommunications. He received the Ph.D.
degree in pattern recognition and intelligent system from State Key Laboratory
of Robotics, Shenyang Institute Automation, Chinese Academy of Sciences. His
major research interest includes robot vision system, 2D/3D object tracking,
and pattern recognition. He has published more than 20 top conferences and
journals, such as IEEE TIP, TMM, TCSVT, PR, RAL, ICRA, ECCV, etc.
---|---
| Wuyang Zhou is currently pursuing the Ph.D. degree with the Department of
Automation college at Nanjing University of Posts and Telecommunications. His
research focuses on point clouds analysis, 3D object tracking.
---|---
| Kai Wang is currently pursuing the Ph.D. degree with the Department of
Automation college at Nanjing University of Posts and Telecommunications. His
research focuses on 3D object detection and tracking, multi-model object
tracking. he has published multiple top conferences or journal papers, such as
ICRA, RAL, etc.
---|---
| Shijun Zhou is currently pursuing the Ph.D. degree with the State Key
Laboratory of Robotics, Shenyang Institute of Automation, University of
Chinese Academy of Sciences. Her research focuses on computer vision methods
in scattering media.
---|---
| Fengyu Xu received the M.S. degree from Hefei University of Technology,
Hefei, China, in 2005 and the Ph.D. degree from Southeast University, Nanjing,
China, in 2009. From May 2016 to April 2017, he was a Visiting Scientist with
the Department of Mechanical Engi neering, Michigan State University, East
Lansing, MI, USA. He is currently a Full Professor, and the Associate Dean of
the College of Automation and College of Artificial Intelligence, Nanjing
Univer-sity of Posts and Telecommunications. His current research interests
include inspection robot, soft robotics, machine vision and intelligent
manufacturing.
---|---
| Jiandong Tian (Senior Member, IEEE) received the B.S. degree from the
Department of Automation, Heilongjiang University, China, in 2005, and the
Ph.D. degree from the Shenyang Institute of Automation, Chinese Academy of
Sciences, in 2011. He is currently a Professor with the Shenyang Institute of
Automation, Chinese Academy of Sciences. He has published more than 50 top
journal and conference papers in the field of robot vision, image processing,
and pattern recognition. He has also served as reviewer for several top
journals and conferences such as T-PAMI, TIP, JMLR, TKDE, IJCV, TRO, CVPR,
ICCV, ICRA, RSS and ECCV.
---|---
|
# Array Programming with NumPy
Charles R. Harris Independent Researcher, Logan, Utah, USA K. Jarrod Millman
Brain Imaging Center, University of California, Berkeley, Berkeley, CA, USA
Division of Biostatistics, University of California, Berkeley, Berkeley, CA,
USA Berkeley Institute for Data Science, University of California, Berkeley,
Berkeley, CA, USA<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Stéfan J. van der Walt Applied Mathematics,
Stellenbosch University, Stellenbosch, South Africa Brain Imaging Center,
University of California, Berkeley, Berkeley, CA, USA Berkeley Institute for
Data Science, University of California, Berkeley, Berkeley, CA, USA
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Ralf
Gommers Quansight LLC, Austin, TX, USA<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Pauli Virtanen Department of
Physics and Nanoscience Center, University of Jyväskylä, Jyväskylä, Finland
David Cournapeau Mercari JP, Tokyo, Japan Eric Wieser Department of
Engineering, University of Cambridge, Cambridge, UK Julian Taylor
Independent Researcher, Karlsruhe, Germany Sebastian Berg Berkeley Institute
for Data Science, University of California, Berkeley, Berkeley, CA, USA
Nathaniel J. Smith Independent Researcher, Berkeley, CA, USA Robert Kern
Enthought, Inc., Austin, TX, USA Matti Picus Berkeley Institute for Data
Science, University of California, Berkeley, Berkeley, CA, USA Stephan Hoyer
Google Research, Mountain View, CA, USA Marten H. van Kerkwijk Department of
Astronomy & Astrophysics, University of Toronto, Toronto, ON, Canada Matthew
Brett Brain Imaging Center, University of California, Berkeley, Berkeley, CA,
USA School of Psychology, University of Birmingham, Edgbaston, Birmigham, UK
Allan Haldane Department of Physics, Temple University, Philadelphia, PA, USA
Jaime Fernández del Río Google, Zurich, Switzerland Mark Wiebe Department
of Physics and Astronomy, The University of British Columbia, Vancouver, BC,
Canada Amazon, Seattle, Washington, USA Pearu Peterson Quansight LLC,
Austin, TX, USA Independent Researcher, Saue, Estonia Department of
Mechanics and Applied Mathematics, Institute of Cybernetics at Tallinn
Technical University, Tallinn, Estonia Pierre Gérard-Marchant Department of
Biological and Agricultural Engineering, University of Georgia, Athens, GA
France-IX Services, Paris, France Kevin Sheppard Department of Economics,
University of Oxford, Oxford, UK Tyler Reddy CCS-7, Los Alamos National
Laboratory, Los Alamos, NM, USA Warren Weckesser Berkeley Institute for Data
Science, University of California, Berkeley, Berkeley, CA, USA Hameer Abbasi
Quansight LLC, Austin, TX, USA Christoph Gohlke Laboratory for Fluorescence
Dynamics, Biomedical Engineering Department, University of California, Irvine,
Irvine, CA, USA Travis E. Oliphant Quansight LLC, Austin, TX, USA
###### Abstract
Array programming provides a powerful, compact, expressive syntax for
accessing, manipulating, and operating on data in vectors, matrices, and
higher-dimensional arrays [1]. NumPy is the primary array programming library
for the Python language [2, 3, 4, 5]. It plays an essential role in research
analysis pipelines in fields as diverse as physics, chemistry, astronomy,
geoscience, biology, psychology, material science, engineering, finance, and
economics. For example, in astronomy, NumPy was an important part of the
software stack used in the discovery of gravitational waves [6] and the first
imaging of a black hole [7]. Here we show how a few fundamental array concepts
lead to a simple and powerful programming paradigm for organizing, exploring,
and analyzing scientific data. NumPy is the foundation upon which the entire
scientific Python universe is constructed. It is so pervasive that several
projects, targeting audiences with specialized needs, have developed their own
NumPy-like interfaces and array objects. Because of its central position in
the ecosystem, NumPy increasingly plays the role of an interoperability layer
between these new array computation libraries.
Figure 1: The NumPy array incorporates several fundamental array concepts. a,
The NumPy array data structure and its associated metadata fields. b, Indexing
an array with slices and steps. These operations return a _view_ of the
original data. c, Indexing an array with masks, scalar coordinates, or other
arrays, so that it returns a copy of the original data. In the bottom example,
an array is indexed with other arrays; this broadcasts the indexing arguments
before performing the lookup. d, Vectorization efficiently applies operations
to groups of elements. e, Broadcasting in the multiplication of two-
dimensional arrays. f, Reduction operations act along one or more axes. In
this example, an array is summed along select axes to produce a vector, or
along two axes consecutively to produce a scalar. g, Example NumPy code,
illustrating some of these concepts.
Two Python array packages existed before NumPy. The Numeric package began in
the mid-1990s and provided an array object and array-aware functions in
Python, written in C, and linking to standard fast implementations of linear
algebra [8, 9]. One of its earliest uses was to steer C++ applications for
inertial confinement fusion research at Lawrence Livermore National Laboratory
[10]. To handle large astronomical images coming from the Hubble Space
Telescope, a reimplementation of Numeric, called Numarray, added support for
structured arrays, flexible indexing, memory mapping, byte-order variants,
more efficient memory use, flexible IEEE error handling capabilities, and
better type casting rules [11]. While Numarray was highly compatible with
Numeric, the two packages had enough differences that it divided the
community, until 2005, when NumPy emerged as a “best of both worlds”
unification [12]—combining Numarray’s features with Numeric’s performance on
small arrays and its rich C _Application Programming Interface_ (API).
Now, fifteen years later, NumPy underpins almost every Python library that
does scientific or numerical computation including SciPy [13], Matplotlib
[14], pandas [15], scikit-learn [16], and scikit-image [17]. It is a
community-developed, open-source library, which provides a multidimensional
Python array object along with array-aware functions that operate on it.
Because of its inherent simplicity, the NumPy array is the _de facto_ exchange
format for array data in Python.
NumPy operates on in-memory arrays using the CPU. To utilize modern,
specialized storage and hardware, there has been a recent proliferation of
Python array packages. Unlike with the Numarray and Numeric divide, it is now
much harder for these new libraries to fracture the user community—given how
much work already builds on top of NumPy. However, to provide the ecosystem
with access to new and exploratory technologies, NumPy is transitioning into a
central coordinating mechanism that specifies a well-defined array programming
API and dispatches it, as appropriate, to specialized array implementations.
## NumPy arrays
The NumPy array is a data structure that efficiently stores and accesses
multidimensional arrays [18], also known as tensors, and enables a wide
variety of scientific computation. It consists of a pointer to memory, along
with metadata used to interpret the data stored there, notably data type,
shape, and strides (Fig. 1a).
The _data type_ describes the nature of elements stored in an array. An array
has a single data type, and each array element occupies the same number of
bytes in memory. Examples of data types include real and complex numbers (of
lower and higher precision), strings, timestamps, and pointers to Python
objects.
The _shape_ of an array determines the number of elements along each axis, and
the number of axes is the array’s dimensionality. For example, a vector of
numbers can be stored as a one-dimensional array of shape $N$, while color
videos are four-dimensional arrays of shape $(T,M,N,3)$.
_Strides_ are necessary to interpret computer memory, which stores elements
linearly, as multidimensional arrays. It describes the number of bytes to move
forward in memory to jump from row to row, column to column, and so forth.
Consider, for example, a 2-D array of floating-point numbers with shape
$(4,3)$, where each element occupies 8 bytes in memory. To move between
consecutive columns, we need to jump forward 8 bytes in memory, and to access
the next row $3\times 8=24$ bytes. The strides of that array are therefore
$(24,8)$. NumPy can store arrays in either C or Fortran memory order,
iterating first over either rows or columns. This allows external libraries
written in those languages to access NumPy array data in memory directly.
Users interact with NumPy arrays using indexing (to access subarrays or
individual elements), operators (e.g., $+$, $-$, $\times$ for vectorized
operations and $@$ for matrix multiplication), as well as array-aware
functions; together, these provide an easily readable, expressive, high-level
API for array programming, while NumPy deals with the underlying mechanics of
making operations fast.
_Indexing_ an array returns single elements, subarrays, or elements that
satisfy a specific condition (Fig. 1b). Arrays can even be indexed using other
arrays (Fig. 1c). Wherever possible, indexing that retrieves a subarray
returns a view on the original array, such that data is shared between the two
arrays. This provides a powerful way to operate on subsets of array data while
limiting memory usage.
To complement the array syntax, NumPy includes functions that perform
_vectorized_ calculations on arrays, including arithmetic, statistics, and
trigonometry (Fig. 1d). Vectorization—operating on whole arrays rather than
their individual elements—is essential to array programming. This means that
operations that would take many tens of lines to express in languages such as
C can often be implemented as a single, clear Python expression. This results
in concise code and frees users to focus on the details of their analysis,
while NumPy handles looping over array elements near-optimally, taking into
consideration, for example, strides, to best utilize the computer’s fast cache
memory.
When performing a vectorized operation (such as addition) on two arrays with
the same shape, it is clear what should happen. Through _broadcasting_ , NumPy
allows the dimensions to differ, while still producing results that appeal to
intuition. A trivial example is the addition of a scalar value to an array,
but broadcasting also generalizes to more complex examples such as scaling
each column of an array or generating a grid of coordinates. In broadcasting,
one or both arrays are virtually duplicated (that is, without copying any data
in memory), so that the shapes of the operands match (Fig. 1d). Broadcasting
is also applied when an array is indexed using arrays of indices (Fig. 1c).
Other array-aware functions, such as sum, mean, and maximum, perform element-
by-element _reductions_ , aggregating results across one, multiple, or all
axes of a single array. For example, summing an $n$-dimensional array over $d$
axes results in a $(n-d)$-dimensional array (Fig. 1f).
NumPy also includes array-aware functions for creating, reshaping,
concatenating, and padding arrays; searching, sorting, and counting data; and
reading and writing files. It provides extensive support for generating
pseudorandom numbers, includes an assortment of probability distributions, and
performs accelerated linear algebra, utilizing one of several backends such as
OpenBLAS [19, 20] or Intel MKL optimized for the CPUs at hand.
Altogether, the combination of a simple in-memory array representation, a
syntax that closely mimics mathematics, and a variety of array-aware utility
functions forms a productive and powerfully expressive array programming
language.
Figure 2: NumPy is the base of the scientific Python ecosystem. Essential
libraries and projects that depend on NumPy’s API gain access to new array
implementations that support NumPy’s array protocols (Fig. 3).
## Scientific Python ecosystem
Python is an open-source, general-purpose, interpreted programming language
well-suited to standard programming tasks such as cleaning data, interacting
with web resources, and parsing text. Adding fast array operations and linear
algebra allows scientists to do all their work within a single language—and
one that has the advantage of being famously easy to learn and teach, as
witnessed by its adoption as a primary learning language in many universities.
Even though NumPy is not part of Python’s standard library, it benefits from a
good relationship with the Python developers. Over the years, the Python
language has added new features and special syntax so that NumPy would have a
more succinct and easier to read array notation. Since it is not part of the
standard library, NumPy is able to dictate its own release policies and
development patterns.
SciPy and Matplotlib are tightly coupled with NumPy—in terms of history,
development, and use. SciPy provides fundamental algorithms for scientific
computing, including mathematical, scientific, and engineering routines.
Matplotlib generates publication-ready figures and visualizations. The
combination of NumPy, SciPy, and Matplotlib, together with an advanced
interactive environment like IPython [21], or Jupyter [22], provides a solid
foundation for array programming in Python. The scientific Python ecosystem
(Fig. 2) builds on top of this foundation to provide several, widely used
_technique specific_ libraries [16, 17, 23], that in turn underlay numerous
_domain specific_ projects [24, 25, 26, 27, 28, 29]. NumPy, at the base of the
ecosystem of array-aware libraries, sets documentation standards, provides
array testing infrastructure, and adds build support for Fortran and other
compilers.
Many research groups have designed large, complex scientific libraries, which
add _application specific_ functionality to the ecosystem. For example, the
eht-imaging library [30] developed by the Event Horizon Telescope
collaboration for radio interferometry imaging, analysis, and simulation,
relies on many lower-level components of the scientific Python ecosystem.
NumPy arrays are used to store and manipulate numerical data at every step in
the processing chain: from raw data through calibration and image
reconstruction. SciPy supplies tools for general image processing tasks such
as filtering and image alignment, while scikit-image, an image processing
library that extends SciPy, provides higher-level functionality such as edge
filters and Hough transforms. The scipy.optimize module performs mathematical
optimization. NetworkX [23], a package for complex network analysis, is used
to verify image comparison consistency. Astropy [24, 25] handles standard
astronomical file formats and computes time/coordinate transformations.
Matplotlib is used to visualize data and to generate the final image of the
black hole.
The interactive environment created by the array programming foundation along
with the surrounding ecosystem of tools—inside of IPython or Jupyter—is
ideally suited to exploratory data analysis. Users fluidly inspect,
manipulate, and visualize their data, and rapidly iterate to refine
programming statements. These statements are then stitched together into
imperative or functional programs, or notebooks containing both computation
and narrative. Scientific computing beyond exploratory work is often done in a
text editor or an integrated development environment (IDEs) such as Spyder.
This rich and productive environment has made Python popular for scientific
research.
To complement this facility for exploratory work and rapid prototyping, NumPy
has developed a culture of employing time-tested software engineering
practices to improve collaboration and reduce error [31]. This culture is not
only adopted by leaders in the project but also enthusiastically taught to
newcomers. The NumPy team was early in adopting distributed revision control
and code review to improve collaboration on code, and continuous testing that
runs an extensive battery of automated tests for every proposed change to
NumPy. The project also has comprehensive, high-quality documentation,
integrated with the source code [32, 33, 34].
This culture of using best practices for producing reliable scientific
software has been adopted by the ecosystem of libraries that build on NumPy.
For example, in a recent award given by the Royal Astronomical Society to
Astropy, they state:
> _The Astropy Project has provided hundreds of junior scientists with
> experience in professional-standard software development practices including
> use of version control, unit testing, code review and issue tracking
> procedures. This is a vital skill set for modern researchers that is often
> missing from formal university education in physics or astronomy._
Community members explicitly work to address this lack of formal education
through courses and workshops [35, 36, 37].
The recent rapid growth of data science, machine learning, and artificial
intelligence has further and dramatically boosted the usage of scientific
Python. Examples of its significant application, such as the eht-imaging
library, now exist in almost every discipline in the natural and social
sciences. These tools have become _the primary_ software environment in many
fields. NumPy and its ecosystem are commonly taught in university courses,
boot camps, and summer schools, and are at the focus of community conferences
and workshops worldwide.
NumPy and its API have become truly ubiquitous.
## Array proliferation and interoperability
Figure 3: NumPy’s API and array protocols expose new arrays to the ecosystem.
In this example, NumPy’s mean function is called on a Dask array. The call
succeeds by dispatching to the appropriate library implementation (i.e., Dask
in this case) and results in a new Dask array. Compare this code to the
example code in Fig. 1g.
NumPy provides in-memory, multidimensional, homogeneously typed (i.e., single
pointer and strided) arrays on CPUs. It runs on machines ranging from embedded
devices to the world’s largest supercomputers, with performance approaching
that of compiled languages. For most its existence, NumPy addressed the vast
majority of array computation use cases.
However, scientific data sets now routinely exceed the memory capacity of a
single machine and may be stored on multiple machines or in the cloud. In
addition, the recent need to accelerate deep learning and artificial
intelligence applications has led to the emergence of specialized accelerator
hardware, including graphics processing units (GPUs), tensor processing units
(TPUs), and field-programmable gate arrays (FPGAs). Due to its in-memory data
model, NumPy is currently unable to utilize such storage and specialized
hardware directly. However, both distributed data and the parallel execution
of GPUs, TPUs, and FPGAs map well to the _paradigm_ of array programming: a
gap, therefore, existed between available modern hardware architectures and
the tools necessary to leverage their computational power.
The community’s efforts to fill this gap led to a proliferation of new array
implementations. For example, each deep learning framework created its own
arrays; PyTorch [38], Tensorflow [39], Apache MXNet [40], and JAX arrays all
have the capability to run on CPUs and GPUs, in a distributed fashion,
utilizing lazy evaluation to allow for additional performance optimizations.
SciPy and PyData/Sparse both provide sparse arrays—which typically contain few
non-zero values and store only those in memory for efficiency. In addition,
there are projects that build on top of NumPy arrays as a data container and
extend its capabilities. Distributed arrays are made possible that way by
Dask, and labeled arrays—referring to dimensions of an array by name rather
than by index for clarity, compare x[:, 1] vs. x.loc[:, ’time’]—by xarray
[41].
Such libraries often mimic the NumPy API, because it lowers the barrier to
entry for newcomers and provides the wider community with a stable array
programming interface. This, in turn, prevents disruptive schisms like the
divergence of Numeric and Numarray. But exploring new ways of working with
arrays is experimental by nature and, in fact, several promising
libraries—such as Theano and Caffe—have already ceased development. And each
time that a user decides to try a new technology, they must change import
statements and ensure that the new library implements all the parts of the
NumPy API they currently use.
Ideally, operating on specialized arrays using NumPy functions or semantics
would simply work, so that users could write code once, and would then benefit
from switching between NumPy arrays, GPU arrays, distributed arrays, and so
forth, as appropriate. To support array operations between external array
objects, NumPy therefore added the capability to act as a central coordination
mechanism with a well-specified API (Fig. 2).
To facilitate this _interoperability_ , NumPy provides “protocols” (or
contracts of operation), that allow for specialized arrays to be passed to
NumPy functions (Fig. 3). NumPy, in turn, dispatches operations to the
originating library, as required. Over four hundred of the most popular NumPy
functions are supported. The protocols are implemented by widely used
libraries such as Dask, CuPy, xarray, and PyData/Sparse. Thanks to these
developments, users can now, for example, scale their computation from a
single machine to distributed systems using Dask. The protocols also compose
well, allowing users to redeploy NumPy code at scale on distributed, multi-GPU
systems via, for instance, CuPy arrays embedded in Dask arrays. Using NumPy’s
high-level API, users can leverage highly parallel code execution on multiple
systems with millions of cores, all with minimal code changes [42].
These array protocols are now a key feature of NumPy, and are expected to only
increase in importance. As with the rest of NumPy, we iteratively refine and
add protocol designs to improve utility and simplify adoption.
## Discussion
NumPy combines the expressive power of _array programming_ , the performance
of C, and the readability, usability, and versatility of Python in a mature,
well-tested, well-documented, and community-developed library. Libraries in
the scientific Python ecosystem provide fast implementations of most important
algorithms. Where extreme optimization is warranted, compiled languages such
as Cython [43], Numba [44], and Pythran [45], that extend Python and
transparently accelerate bottlenecks, can be used. Because of NumPy’s simple
memory model, it is easy to write low-level, hand-optimized code, usually in C
or Fortran, to manipulate NumPy arrays and pass them back to Python.
Furthermore, using array protocols, it is possible to utilize the full
spectrum of specialized hardware acceleration with minimal changes to existing
code.
NumPy was initially developed by students, faculty, and researchers to provide
an advanced, open-source array programming library for Python, which was free
to use and unencumbered by license servers, dongles, and the like. There was a
sense of building something consequential together, for the benefit of many
others. Participating in such an endeavor, within a welcoming community of
like-minded individuals, held a powerful attraction for many early
contributors.
These user-developers frequently had to write code from scratch to solve their
own or their colleagues’ problems—often in low-level languages that precede
Python, like Fortran [46] and C. To them, the advantages of an interactive,
high-level array library were evident. The design of this new tool was
informed by other powerful interactive programming languages for scientific
computing such as Basis [47], Yorick [48], R [49], and APL [50], as well as
commercial languages and environments like IDL and MATLAB.
What began as an attempt to add an array object to Python became the
foundation of a vibrant ecosystem of tools. Now, a large amount of scientific
work depends on NumPy being correct, fast, and stable. It is no longer a small
community project, but is core scientific infrastructure.
The developer culture has matured: while initial development was highly
informal, NumPy now has a roadmap and a process for proposing and discussing
large changes. The project has formal governance structures and is fiscally
sponsored by NumFOCUS, a nonprofit that promotes open practices in research,
data, and scientific computing. Over the past few years, the project attracted
its first funded development, sponsored by the Moore and Sloan Foundations,
and received an award as part of the Chan Zuckerberg Initiative’s Essentials
of Open Source Software program. With this funding, the project was (and is)
able to have sustained focus over multiple months to implement substantial new
features and improvements. That said, it still depends heavily on
contributions made by graduate students and researchers in their free time.
NumPy is no longer _just_ the foundational array library underlying the
scientific Python ecosystem, but has also become the standard API for tensor
computation and a central coordinating mechanism between array types and
technologies in Python. Work continues to expand on and improve these
interoperability features.
Over the next decade, we will face several challenges. New devices will be
developed, and existing specialized hardware will evolve, to meet diminishing
returns on Moore’s law. There will be more, and a wider variety of, data
science practitioners, a significant proportion of whom will be using NumPy.
The scale of scientific data gathering will continue to expand, with the
adoption of devices and instruments such as light sheet microscopes and the
Large Synoptic Survey Telescope (LSST) [51]. New generation languages,
interpreters, and compilers, such as Rust [52], Julia [53], and LLVM [54],
will invent and determine the viability of new concepts and data structures.
Through various mechanisms described in this paper, NumPy is poised to embrace
such a changing landscape, and to continue playing a leading role in
interactive scientific computation. To do so will require sustained funding
from government, academia, and industry. But, importantly, it will also need a
new generation of graduate students and other developers to engage, to build a
NumPy that meets the needs of the next decade of data science.
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## Methods
We use Git for version control and GitHub as the public hosting service for
our official _upstream_ repository (https://github.com/numpy/numpy). We each
work in our own copy (or fork) of the project and use the upstream repository
as our integration point. To get new code into the upstream repository, we use
GitHub’s pull request (PR) mechanism. This allows us to review code before
integrating it as well as to run a large number of tests on the modified code
to ensure that the changes do not break expected behavior.
We also use GitHub’s issue tracking system to collect and triage problems and
proposed improvements.
### Library organization
Broadly, the NumPy library consists of the following parts: the NumPy array
data structure ndarray; the so-called _universal functions_ ; a set of library
functions for manipulating arrays and doing scientific computation;
infrastructure libraries for unit tests and Python package building; and the
program f2py for wrapping Fortran code in Python [55]. The ndarray and the
universal functions are generally considered the core of the library. In the
following, we give a brief summary of these components of the library.
#### _Core._
The ndarray data structure and the universal functions make up the core of
NumPy.
The ndarray is the data structure at the heart of NumPy. The data structure
stores regularly strided homogeneous data types inside a contiguous block
memory, allowing for the efficient representation of $n$-dimensional data.
More details about the data structure are given in “The NumPy array: a
structure for efficient numerical computation” [18].
The _universal functions_ , or more concisely, _ufuncs_ , are functions
written in C that implement efficient looping over NumPy arrays. An important
feature of ufuncs is the built-in implementation of _broadcasting_. For
example, the function arctan2(x, y) is a ufunc that accepts two values and
computes $\tan^{-1}(y/x)$. When arrays are passed in as the arguments, the
ufunc will take care of looping over the dimensions of the inputs in such a
way that if, say, x is a 1-D array with length 3, and y is a 2-D array with
shape $2\times 1$, the output will be an array with shape $2\times 3$ (Fig.
1c). The ufunc machinery takes care of calling the function with all the
appropriate combinations of input array elements to complete the output array.
The elementary arithmetic operations of addition, multiplication, etc., are
implemented as ufuncs, so that broadcasting also applies to expressions such
as x + y * z.
#### _Computing libraries._
NumPy provides a large library of functions for array manipulation and
scientific computing, including functions for: creating, reshaping,
concatenating, and padding arrays; searching, sorting and counting data in
arrays; computing elementary statistics, such as the mean, median, variance,
and standard deviation; file I/O; and more.
A suite of functions for computing the _fast Fourier transform (FFT)_ and its
inverse is provided.
NumPy’s linear algebra library includes functions for: solving linear systems
of equations; computing various functions of a matrix, including the
determinant, the norm, the inverse, and the pseudo-inverse; computing the
Cholesky, eigenvalue, and singular value decompositions of a matrix; and more.
The random number generator library in NumPy provides alternative _bit stream
generators_ that provide the core function of generating random integers. A
higher-level generator class that implements an assortment of probability
distributions is provided. It includes the beta, gamma and Weibull
distributions, the univariate and multivariate normal distributions, and more.
#### _Infrastructure libraries._
NumPy provides utilities for writing tests and for building Python packages.
The testing subpackage provides functions such as assert_allclose(actual,
desired) that may be used in test suites for code that uses NumPy arrays.
NumPy provides the subpackage distutils which includes functions and classes
to facilitate configuration, installation, and packaging of libraries
depending on NumPy. These can be used, for example, when publishing to the
PyPI website.
#### _F2PY._
The program f2py is a tool for building NumPy-aware Python wrappers of Fortran
functions. NumPy itself does not use any Fortran code; F2PY is part of NumPy
for historical reasons.
### Governance
NumPy adopted an official Governance Document on October 5, 2015 [56]. Project
decisions are usually made by consensus of interested contributors. This means
that, for most decisions, everyone is entrusted with veto power. A Steering
Council, currently composed of 12 members, facilitates this process and
oversees daily development of the project by contributing code and reviewing
contributions from the community.
NumPy’s official Code of Conduct was approved on September 1, 2018 [57]. In
brief, we strive to: _be open_ ; _be empathetic, welcoming, friendly, and
patient_ ; _be collaborative_ ; _be inquisitive_ ; and _be careful in the
words that we choose_. The Code of Conduct also specifies how breaches can be
reported and outlines the process for responding to such reports.
### Funding
In 2017, NumPy received its first large grants totaling 1.3M USD from the
Gordon & Betty Moore and the Alfred P. Sloan foundations. Stéfan van der Walt
is the PI and manages four programmers working on the project. These two
grants focus on addressing the technical debt accrued over the years and on
setting in place standards and architecture to encourage more sustainable
development.
NumPy received a third grant for 195K USD from the Chan Zuckerberg Initiative
at the end of 2019 with Ralf Gommers as the PI. This grant focuses on better
serving NumPy’s large number of beginning to intermediate level users and on
growing the community of NumPy contributors. It will also provide support to
OpenBLAS, on which NumPy depends for accelerated linear algebra.
Finally, since May 2019 the project receives a small amount annually from
Tidelift, which is used to fund things like documentation and website
improvements.
### Developers
NumPy is currently maintained by a group of 23 contributors with commit rights
to the NumPy code base. Out of these, 17 maintainers were active in 2019, 4 of
whom were paid to work on the project full-time. Additionally, there are a few
long term developers who contributed and maintain specific parts of NumPy, but
are not officially maintainers.
Over the course of its history, NumPy has attracted PRs by 823 contributors.
However, its development relies heavily on a small number of active
maintainers, who share more than half of the contributions among themselves.
At a release cycle of about every half year, the five recent releases in the
years 2018 and 2019 have averaged about 450 PRs each,111 Note that before mid
2011, NumPy development did not happen on github.com. All data provided here
is based on the development which happened through GitHub PRs. In some cases
contributions by maintainers may not be categorized as such. with each release
attracting more than a hundred new contributors. Figure 4 shows the number of
PRs merged into the NumPy master branch. Although the number of PRs being
merged fluctuates, the plot indicates an increased number of contributions
over the past years.
Figure 4: Number of pull requests merged into the NumPy master branch for each
quarter since 2012. The total number of PRs is indicated with the lower blue
area showing the portion contributed by current or previous maintainers.
### Community calls
The massive number of scientific Python packages that built on NumPy meant
that it had an unusually high need for stability. So to guide our development
we formalized the feature proposal process, and constructed a development
roadmap with extensive input and feedback from the community.
Weekly community calls alternate between triage and higher level discussion.
The calls not only involve developers from the community, but provide a venue
for vendors and other external groups to provide input. For example, after
Intel produced a forked version of NumPy, one of their developers joined a
call to discuss community concerns.
### NumPy enhancement proposals
Given the complexity of the codebase and the massive number of projects
depending on it, large changes require careful planning and substantial work.
NumPy Enhancement Proposals (NEPs) are modeled after Python Enhancement
Proposals (PEPs) for “proposing major new features, for collecting community
input on an issue, and for documenting the design decisions that have gone
into Python”222https://numpy.org/neps/nep-0000.html. Since then there have
been 19 proposed NEPS—6 have been implemented, 4 have been accepted and are
being implemented, 4 are under consideration, 3 have been deferred or
superseded, and 2 have been rejected or withdrawn.
### Central role
NumPy plays a central role in building and standardizing much of the
scientific Python community infrastructure. NumPy’s docstring standard is now
widely adopted. We are also now using the NEP system as a way to help
coordinate the larger scientific Python community. For example, in NEP 29, we
recommend, along with leaders from various other projects, that all projects
across the Scientific Python ecosystem adopt a common “time window-based”
policy for support of Python and NumPy versions. This standard will simplify
downstream project and release planning.
### Wheels build system
A Python _wheel_ [58] is a standard file format for distributing Python
libraries. In addition to Python code, a wheel may include compiled C
extensions and other binary data. This is important, because many libraries,
including NumPy, require a C compiler and other build tools to build the
software from the source code, making it difficult for many users to install
the software on their own. The introduction of wheels to the Python packaging
system has made it much easier for users to install precompiled libraries.
A GitHub repository containing scripts to build NumPy wheels has been
configured so that a simple commit to the repository triggers an automated
build system that creates NumPy wheels for several computer platforms,
including Windows, Mac OSX and Linux. The wheels are uploaded to a public
server and made available for anyone to use. This system makes it easy for
users to install precompiled versions of NumPy on these platforms.
The technology that is used to build the wheels evolves continually. At the
time this paper is being written, a key component is the multibuild suite of
tools developed by Matthew Brett and other developers [59]. Currently, scripts
using multibuild are written for the continuous integration platforms Travis-
CI (for Linux and Mac OSX) and Appveyor (for Windows).
### Recent technical improvements
With the recent infusion of funding and a clear process for coordinating with
the developer community, we have been able to tackle a number of important
large scale changes. We highlight two of those below, as well as changes made
to our testing infrastructure to support hardware platforms used in large
scale computing.
### Array function protocol
A vast number of projects are built on NumPy; these projects are consumers of
the NumPy API. Over the last several years, a growing number of projects are
providers of a _NumPy-like API_ and array objects targeting audiences with
specialized needs beyond NumPy’s capabilities. For example, the NumPy API is
implemented by several popular tensor computation libraries including
CuPy333https://cupy.chainer.org/,
JAX444https://jax.readthedocs.io/en/latest/jax.numpy.html, and Apache
MXNet555https://numpy.mxnet.io/.
PyTorch666https://pytorch.org/tutorials/beginner/blitz/tensor_tutorial.html
and Tensorflow777https://www.tensorflow.org/tutorials/customization/basics
provide tensor APIs with NumPy-inspired semantics. It is also implemented in
packages that support sparse arrays such as scipy.sparse and PyData/Sparse.
Another notable example is Dask, a library for parallel computing in Python.
Dask adopts the NumPy API and therefore presents a familiar interface to
existing NumPy users, while adding powerful abilities to parallelize and
distribute tasks.
The multitude of specialized projects creates the difficulty that consumers of
these NumPy-like APIs write code specific to a single project and do not
support all of the above array providers. This is a burden for users relying
on the specialized array-like, since a tool they need may not work for them.
It also creates challenges for end-users who need to transition from NumPy to
a more specialized array. The growing multitude of specialized projects with
NumPy-like APIs threatened to again fracture the scientific Python community.
To address these issues NumPy has the goal of providing the fundamental API
for _interoperability_ between the various NumPy-like APIs. An earlier step in
this direction was the implementation of the __array_ufunc__ protocol in NumPy
1.13, which enabled interoperability for most mathematical functions [60]. In
2019 this was expanded more generally with the inclusion of the
__array_function__ protocol into NumPy 1.17. These two protocols allow
providers of array objects to be interoperable with the NumPy API: their
arrays work correctly with almost all NumPy functions [61]. For the users
relying on specialized array projects it means that even though much code is
written specifically for NumPy arrays and uses the NumPy API as import numpy
as np, it can nevertheless work for them. For example, here is how a CuPy GPU
array can be passed through NumPy for processing, with all operations being
dispatched back to CuPy:
⬇
import numpy as np
import cupy as cp
x_gpu = cp.array([1, 2, 3])
y = np.sum(x_gpu) # Returns a GPU array
Similarly, user defined functions composed using NumPy can now be applied to,
e.g., multi-node distributed Dask arrays:
⬇
import numpy as np
import dask.array as da
def f(x):
"""Function␣using␣NumPy␣API␣calls"""
y = np.tensordot(x, x.T)
return np.mean(np.log(y + 1))
x_local = np.random.random([10000, 10000]) # random local array
x_distr = da.random.random([10000, 10000]) # random distributed array
f(x_local) # returns a NumPy array
f(x_distr) # works, returns a Dask array
### Random number generation
The NumPy random module provides pseudorandom numbers from a wide range of
distributions. In legacy versions of NumPy, simulated random values are
produced by a RandomState object that: handles seeding and state
initialization; wraps the core pseudorandom number generator based on a
Mersenne Twister implementation888to be precise, the standard 32-bit version
of MT19937; interfaces with the underlying code that transforms random bits
into variates from other distributions; and supplies a singleton instance
exposed in the root of the random module.
The RandomState object makes a compatibility guarantee so that a fixed seed
and sequence of function calls produce the same set of values. This guarantee
has slowed progress since improving the underlying code requires extending the
API with additional keyword arguments. This guarantee continues to apply to
RandomState.
NumPy 1.17 introduced a new API for generating random numbers that use a more
flexible structure that can be extended by libraries or end-users. The new API
is built using components that separate the steps required to generate random
variates. Pseudorandom bits are generated by a bit generator. These bits are
then transformed into variates from complex distributions by a generator.
Finally, seeding is handled by an object that produces sequences of high-
quality initial values.
Bit generators are simple classes that manage the state of an underlying
pseudorandom number generator. NumPy ships with four bit generators. The
default bit generator is a 64-bit implementation of the Permuted Congruential
Generator [62] (PCG64). The three other bit generators are a 64-bit version of
the Philox generator [63] (Philox), Chris Doty-Humphrey’s Small Fast Chaotic
generator [64] (SFC64), and the 32-bit Mersenne Twister [65] (MT19937) which
has been used in older versions of NumPy.999The randomgen project supplies a
wide range of alternative bit generators such as a cryptographic counter-based
generators (AESCtr) and generators that expose hardware random number
generators (RDRAND) [66]. Bit generators provide functions, exposed both in
Python and C, for generating random integer and floating point numbers.
The Generator consumes one of the bit generators and produces variates from
complicated distributions. Many improved methods for generating random
variates from common distributions were implemented, including the Ziggurat
method for normal, exponential and gamma variates [67], and Lemire’s method
for bounded random integer generation [68]. The Generator is more similar to
the legacy RandomState, and its API is substantially the same. The key
differences all relate to state management, which has been delegated to the
bit generator. The Generator does not make the same stream guarantee as the
RandomState object, and so variates may differ across versions as improved
generation algorithms are introduced.101010Despite the removal of the
compatibility guarantee, simple reproducibility across versions is encouraged,
and minor changes that do not produce meaningful performance gains or fix
underlying bug are not generally adopted.
Finally, a SeedSequence is used to initialize a bit generator. The seed
sequence can be initialized with no arguments, in which case it reads entropy
from a system-dependent provider, or with a user-provided seed. The seed
sequence then transforms the initial set of entropy into a sequence of high-
quality pseudorandom integers, which can be used to initialize multiple bit
generators deterministically. The key feature of a seed sequence is that it
can be used to spawn child SeedSequences to initialize multiple distinct bit
generators. This capability allows a seed sequence to facilitate large
distributed applications where the number of workers required is not known.
The sequences generated from the same initial entropy and spawns are fully
deterministic to ensure reproducibility.
The three components are combined to construct a complete random number
generator.
⬇
from numpy.random import (
Generator,
PCG64,
SeedSequence,
)
seq = SeedSequence(1030424547444117993331016959)
pcg = PCG64(seq)
gen = Generator(pcg)
This approach retains access to the seed sequence which can then be used to
spawn additional generators.
⬇
children = seq.spawn(2)
gen_0 = Generator(PCG64(children[0]))
gen_1 = Generator(PCG64(children[1]))
While this approach retains complete flexibility, the method
np.random.default_rng can be used to instantiate a Generator when
reproducibility is not needed.
The final goal of the new API is to improve extensibility. RandomState is a
monolithic object that obscures all of the underlying state and functions. The
component architecture is one part of the extensibility improvements. The
underlying functions (written in C) which transform the output of a bit
generator to other distributions are available for use in CFFI. This allows
the same code to be run in both NumPy and dependent that can consume CFFI,
e.g., Numba. Both the bit generators and the low-level functions can also be
used in C or Cython code.111111As of 1.18.0, this scenario requires access to
the NumPy source. Alternative approaches that avoid this extra step are being
explored.
### Testing on multiple architectures
At the time of writing the two fastest supercomputers in the world, Summit and
Sierra, both have IBM POWER9 architectures [69]. In late 2018, Astra, the
first ARM-based supercomputer to enter the TOP500 list, went into production
[70]. Furthermore, over 100 billion ARM processors have been produced as of
2017 [71], making it the most widely used instruction set architecture in the
world.
Clearly there are motivations for a large scientific computing software
library to support POWER and ARM architectures. We’ve extended our continuous
integration (CI) testing to include ppc64le (POWER8 on Travis CI) and ARMv8
(on Shippable service). We also test with the s390x architecture (IBM Z CPUs
on Travis CI) so that we can probe the behavior of our library on a big-endian
machine. This satisfies one of the major components of improved CI testing
laid out in a version of our roadmap [72]—specifically, “CI for more exotic
platforms.”
PEP 599 [73] lays out a plan for new Python binary wheel distribution support,
manylinux2014, that adds support for a number of architectures supported by
the CentOS Alternative Architecture Special Interest Group, including ARMv8,
ppc64le, as well as s390x. We are thus well-positioned for a future where
provision of binaries on these architectures will be expected for a library at
the base of the ecosystem.
## Acknowledgments
We thank Ross Barnowski, Paul Dubois, Michael Eickenberg, and Perry
Greenfield, who suggested text and provided helpful feedback on the
manuscript.
We also thank the many members of the community who provided feedback,
submitted bug reports, made improvements to the documentation, code, or
website, promoted NumPy’s use in their scientific fields, and built the vast
ecosystem of tools and libraries around NumPy. We also gratefully acknowledge
the Numeric and Numarray developers on whose work we built.
Jim Hugunin wrote Numeric in 1995, while a graduate student at MIT. Hugunin
based his package on previous work by Jim Fulton, then working at the US
Geological Survey, with input from many others. After he graduated, Paul
Dubois at the Lawrence Livermore National Laboratory became the maintainer.
Many people contributed to the project including T.E.O. (a co-author of this
paper), David Ascher, Tim Peters, and Konrad Hinsen.
In 1998 the Space Telescope Science Institute started using Python and in 2000
began developing a new array package called Numarray, written almost entirely
by Jay Todd Miller, starting from a prototype developed by Perry Greenfield.
Other contributors included Richard L. White, J. C. Hsu, Jochen Krupper, and
Phil Hodge. The Numeric/Numarray split divided the community, yet ultimately
pushed progress much further and faster than would otherwise have been
possible.
Shortly after Numarray development started, T.E.O. took over maintenance of
Numeric. In 2005, he led the effort and did most of the work to unify Numeric
and Numarray, and produce the first version of NumPy.
Eric Jones co-founded (along with T.E.O. and P.P.) the SciPy community, gave
early feedback on array implementations, and provided funding and travel
support to several community members. Numerous people contributed to the
creation and growth of the larger SciPy ecosystem, which gives NumPy much of
its value. Others injected new energy and ideas by creating experimental array
packages.
K.J.M. and S.J.v.d.W. were funded in part by the Gordon and Betty Moore
Foundation through Grant GBMF3834 and by the Alfred P. Sloan Foundation
through Grant 2013-10-27 to the University of California, Berkeley.
S.J.v.d.W., S.B., M.P., and W.W. were funded in part by the Gordon and Betty
Moore Foundation through Grant GBMF5447 and by the Alfred P. Sloan Foundation
through Grant G-2017-9960 to the University of California, Berkeley.
## Author Contributions Statement
K.J.M. and S.J.v.d.W. composed the manuscript with input from others. S.B.,
R.G., K.S., W.W., M.B., and T.J.R. contributed text. All authors have
contributed significant code, documentation, and/or expertise to the NumPy
project. All authors reviewed the manuscript.
## Competing Interests
The authors declare no competing interests.
|
# Motivic homotopy theory of the classifying stack of finite groups of Lie
type
Can Yaylali Mail<EMAIL_ADDRESS>
Department of Mathematics, TU Darmstadt, Schlossgartenstraße 7, 64289
Darmstadt, Germany
ORCID: 0000-0003-0150-3622
###### Abstract
Let $G$ be a reductive group over $\mathbb{F}_{p}$ with associated finite
group of Lie type $G^{F}$. Let $T$ be a maximal torus contained inside a Borel
$B$ of $G$. We relate the (rational) Tate motives of $\textup{B}G^{F}$ with
the $T$-equivariant Tate motives of the flag variety $G/B$.
On the way, we show that for a reductive group $G$ over a field $k$, with
maximal Torus $T$ and absolute Weyl group $W$, acting on a smooth finite type
$k$-scheme $X$, we have an isomorphism $A^{n}_{G}(X,m)_{\mathbb{Q}}\cong
A^{n}_{T}(X,m)_{\mathbb{Q}}^{W}$ extending the classical result of Edidin-
Graham to higher equivariant Chow groups in the non-split case.
We also extend our main result to reductive group schemes over a regular base
that admit maximal tori. Further, we apply our methods to more general
quotient stacks. In this way, we are able to compute the motive of the stack
of $G$-zips introduced by Pink-Wedhorn-Ziegler for reductive groups over
fields of positive characteristic.
###### Contents
1. 1 Introduction
2. 2 Rational equivariant motivic homotopy theory
3. 3 Equivariant motives under reductive groups
1. 3.1 Preliminaries on reductive group schemes
2. 3.2 Torsors under finite groups
3. 3.3 The relation between the motives of $\left[X/T\right]$ and $\left[X/G\right]$
4. 4 Torsors under tori
1. 4.1 The motive of $T$-torsors
2. 4.2 Motivic cohomology of $T$-torsors
5. 5 Motivic cohomology of quotients up to isogeny
6. 6 Generalizations
7. A Existence of maximal tori over Prüfer domains
## 1 Introduction
Let $G$ be a reductive group over $\mathbb{F}_{q}$, a finite field of
characteristic $p>0$, and $\varphi\colon G\rightarrow G$ the $q$-Frobenius.
Then $G$ acts on itself via $\varphi$-conjugation, i.e. $(g,h)\mapsto
gh\varphi(g)^{-1}$. The stabilizer of the neutral element is denoted by
$G^{F}$. If $\overline{\mathbb{F}}_{q}$ denotes an algebraic closure of
$\mathbb{F}_{q}$, then $G^{F}(\overline{\mathbb{F}}_{q})=G(\mathbb{F}_{q})$ is
a finite group of Lie-type. The representation of finite groups of Lie type
over fields of characteristic $0$ was studied by Deligne and Lusztig (cf.
[DL76]). In their article they construct representations of
$G(\mathbb{F}_{q})$ by an action on the $\ell$-adic cohomology of certain
varieties, for $\ell\neq p$. Roughly, the varieties in question are
constructed by intersection of Bruhat strata and graph of Frobenius.
Let us fix a Borel $B$ of $G$.111Any reductive group over a finite field is
quasi-split and thus admits a Borel. The Bruhat strata of $\left[G/B\right]$
are induced by the Bruhat decomposition of $G$ via pullback along
$G/B\rightarrow G/B\times^{G}G/B\cong\left[B\backslash G/B\right]$. In this
article, we want to analyze the cohomological connection between
$\textup{B}G^{F}$ and $\left[B\backslash G/B\right]$, i.e. study how their
motivic categories are related.
The derived category of $\ell$-adic sheaves
$D(\textup{B}G^{F},\mathbb{Q}_{\ell})$, for $\ell\neq p$, encodes information
about the action of $G^{F}$ on $\ell$-adic cohomology. One can show that
$\textup{B}G^{F}\cong\left[G/_{\varphi}G\right]$, where $G$ acts on itself via
$\varphi$-conjugation. The restriction of the $\varphi$-conjugation to $B$
yields an adjunction
${D(\left[G/_{\varphi}G\right],\mathbb{Q}_{\ell})}$${D(\left[G/_{\varphi}B\right],\mathbb{Q}_{\ell}).}$
On the other hand there is an adjunction
${D(\left[G/_{\varphi}B\right],\mathbb{Q}_{\ell})}$${D_{B}(G/B,\mathbb{Q}_{\ell}),}$
which is induced via the graph of $\varphi$ (here the right hand side denotes
the derived category of $B$-equivariant $\mathbb{Q}_{\ell}$-modules on the
flag variety). Thus, it seems natural that the study of these two adjunctions
should lead to information about the geometric representation theory of
$G^{F}$ and connection to the classical theory of Deligne-Lusztig.
Instead of rewriting the theory of Deligne-Lusztig in the derived setting, we
want to understand the adjunctions above in the motivic setting with rational
coefficients. The idea is that first after $\ell$-adic realization, we get the
classical situation back and further this could lead to information about the
$\mathbb{Q}$-representations of $G^{F}$, as we are naturally working with
rational coefficients.
### Motives and connection to representation theory
Motives were famously envisioned by Grothendieck to capture similar behavior
of cohomology theories in an abelian category. The construction of such a
category is not an easy task and has been studied for many years. The main
approach is to define a derived category of motives with the hope to find a
$t$-structure on it, so that the heart of this $t$-structure defines the
abelian category of motives. To capture functorial behavior on cohomology
theories one demands a full six functor formalism for the derived category of
motives. There are several versions of the derived category of motives which
agree under certain assumptions. One version was constructed by Cisinki and
Déglise in the case of rational coefficients, which we denote by
$\operatorname{DM}$ (cf. [CD19]). They show that the assignment
$X\mapsto\operatorname{DM}(X)$ from smooth $k$-schemes indeed admits a six
functor formalism $(\otimes\dashv\underline{\operatorname{Hom}},f^{*}\dashv
f_{*},f_{!}\dashv f^{!})$ and agrees with the classical construction of Morel.
In particular, they show that motivic cohomology, i.e.
$\operatorname{Hom}_{\operatorname{DM}(X)}(1_{X},1_{X}(n)[m])$ agrees with
Bloch’s higher Chow groups $A^{n}(X,2n-m)_{\mathbb{Q}}$. With the help of the
$6$-functor formalism, we can define the motive of an $k$-scheme $\pi\colon
X\rightarrow\operatorname{Spec}(k)$ resp. the global sections via
$M_{k}(X)\coloneqq\pi_{!}\pi^{!}1_{Y}\quad\textup{resp.}\quad
R\Gamma_{k}(X,\mathbb{Q})\coloneqq\pi_{*}\pi^{*}1_{k}$
computing motivic cohomology resp. cohomology. The existence of a
$t$-structure is a more delicate problem and in general not known. Levine
shows that for a particular class of schemes $X$, e.g. finite fields or affine
spaces over finite fields, a $t$-structure exists on the full triangulated
subcategory of Tate motives
$\operatorname{DTM}(X)\subseteq\operatorname{DM}(X)$ generated by the
$1_{X}(n)$ for $n\in\mathbb{Z}$ (cf. [Lev93]). Further, using weight
structures one can see that $\operatorname{DTM}(\mathbb{F}_{p})$ is equivalent
to the bounded derived category of $\mathbb{Q}$-vector spaces. We also have
realization functors
$\textup{real}_{\ell}\colon\operatorname{DTM}(X)\rightarrow
D_{\operatorname{\acute{e}t}}(X,\mathbb{Q}_{\ell})$ for $\ell\neq p$ that is
conservative and $t$-exact for the perverse $t$-structure on
$D_{\operatorname{\acute{e}t}}(X,\mathbb{Q}_{\ell})$. Let us remark that if
for a morphism $f\colon X\rightarrow Y$ we have
$f_{*}1_{X}\in\operatorname{DTM}(Y)$, then this automatically induces an
adjunction of $\operatorname{DTM}(X)$ and $\operatorname{DTM}(Y)$. In
particular, the adjunction $f^{*}\dashv f_{*}$ restricts to Tate motives. We
call morphisms with such a property Tate.
Let us now explain the relation between Tate motives and geometric
representation theory. For simplicity, let us assume that $G/\mathbb{F}_{q}$
is a split reductive group. Let $T$ be a split maximal torus inside a Borel
$B$ of $G$. In [SW18], Soergel and Wendt show that for schemes stratified by
affine spaces, such as the flag variety $G/B$, one can define a subcategory of
$\operatorname{DM}$ called stratified Tate motives that admits a
$t$-structure. This $t$-structure is glued from the $t$-structure on the
strata. For $G/B$ with the Bruhat stratification, we will denote the category
of stratified Tate motives with $\operatorname{DTM}_{(B)}(G/B)$. Soergel and
Wendt show that $\operatorname{DTM}_{(B)}(G/B)$ is equivalent to the bounded
derived category $D^{b}({\mathcal{O}}^{\mathbb{Z},ev})$ of graded
${\mathcal{O}}\coloneqq H^{*}(G/B)$-modules concentrated in even degrees. To
connect this to representations, we have to go further and endow motivic
cohomology with group actions.
For this, we need to define equivariant motives. To make sense of the
following construction, we need to work in the setting of $\infty$-categories.
The idea is to define $\operatorname{DM}({\mathcal{X}})$ for an Artin stack
${\mathcal{X}}$ via gluing along the atlas. As we essentially have to glue the
derived category this only makes sense in the $\infty$-categorical framework.
Then this gluing can be defined via right Kan-extension from schemes to Artin
stacks (cf. [RS20]). As one would expect motivic cohomology of a quotient
stack $[X/G]$, where $X$ is a smooth $k$-scheme and $G$ is a linear algebraic
group, yields the equivariant Chow groups $A^{n}_{G}(X,2n-m)$ of Edidin and
Graham (cf. [RS20]). In this way, one can also extend the stratified Tate
motives to Artin stacks. In the case of the flag variety, Soergel, Virk and
Wendt show that $\operatorname{DTM}(\left[B\backslash G/B\right])$ is
equivalent to the bounded derived category of bi-Soergel-modules (cf.
[SVW18]). Further, they show that applying $K_{0}$ yields an isomorphism to
the Iwahori-Hecke algebra and Verdier duality yields the Kazhdan-Lusztig
involution. In particular, the stratified Tate motives of $\left[B\backslash
G/B\right]$ with the weight structure and $6$-functor formalism carries
information about the $\ell$-adic geometric representations of $G$.
### Connection between finite groups of Lie type and their associated flag
variety
As we have seen above the geometric representation theory of the flag variety
is linked to stratified Tate motives. This particular connection uses that the
flag variety is stratified by affine spaces and that Tate motives behave
nicely under this stratification. We expect that the geometric representation
theory of $G^{F}$ is linked to Tate motives on $\textup{B}G^{F}$, the
classifying space of $G^{F}$. In this way, we want to relate representation
theory of $G^{F}$ with the representation theory of the associated flag
variety.
We will do this by linking the Tate motives of $\textup{B}G^{F}$ to Tate
motives on $\left[B\backslash G/B\right]$ in the following way. The stack
$\textup{B}G^{F}$ is equivalent to $\left[G/_{\varphi}G\right]$, where $G$
acts on itself via $\varphi$-conjugation. Let us fix a maximal torus
$T\subseteq B$. Let $T$ act on $G$ also via $\varphi$-conjugation. We can
embed $T$ into $T\times T$ via $t\mapsto(t,\varphi(t))$. If we now let
$T\times T$ act on $G$ via $(t,t^{\prime},g)\mapsto tgt^{\prime-1}$, we get a
zigzag of Artin stacks
(1)
${\left[G/_{\varphi}G\right]}$${\left[G/_{\varphi}T\right]}$${\left[G/T\times
T\right].}$$\scriptstyle{a}$$\scriptstyle{b}$
It is a fact that $\operatorname{DM}(\left[G/T\times
T\right])\simeq\operatorname{DM}(\left[B\backslash G/B\right])$. Thus, on the
level of motivic categories, this zigzag yields adjunctions between
$\operatorname{DM}(\textup{B}G^{F})$ and $\operatorname{DM}(\left[B\backslash
G/B\right])$. Now we can formulate the leading question of this article:
($\ast$) Do these adjunctions preserve Tate motives?
The answer to this question is positive and yields a first point to access
motivic representation theory of $G^{F}$ via the motivic geometric
representation theory of $G$.
### Equivariant motivic homotopy theory of reductive groups
From now on let $\tilde{B}$ be a an excellent noetherian scheme of dimension
$\leq 2$ and $S$ a regular connected $\tilde{B}$-scheme of finite type. We
will first work with reductive $S$-group schemes $H$ that admit split maximal
tori $Z$. This is not a classical notion and any such reductive group scheme
may not be split or even quasi-split (in sense of [SGA3] or [Con14]). But in
this case, one has that the Weyl-group scheme $N_{H}(Z)/Z$ is represented by a
constant group scheme associated to a finite constant group $W_{H}$, which we
will call the Weyl group of $Z$ in $H$ (cf. Section 3.1).
As $S$ is regular noetherian, any maximal torus splits after passage to a
Galois cover and we deduce the answer to our leading question ($\ast$ ‣ 1)
from the split case.
From now on let $G$ be a reductive $S$-group scheme that admits a maximal
torus $T$. Our main question is about the behavior of Tate motives under the
induced maps on $\operatorname{DM}$ corresponding to the zigzag (1). We will
work in a more general setting and look at $a$ and $b$ separately.
#### Equivariant motives and passage to Tori
The morphism $a$ resembles the motivic version of a more classical problem on
classical problem on Chow groups. Let $X$ be an $S$-scheme locally of finite
type with $G$-action, what is the relation between $A^{\bullet}_{G}(X)$ and
$A^{\bullet}_{T}(X)$? In [EG98] Edidin and Graham answer this question for
rational Chow groups in the case where $S=\operatorname{Spec}(k)$ is the
spectrum of a field and $T$ is split, i.e.
$A^{\bullet}_{G}(X)_{\mathbb{Q}}\cong A^{\bullet}_{T}(X)^{W}_{\mathbb{Q}}$,
where $W$ denotes the Weyl group of $T$ in $G$.
This isomorphism is just a shadow of an equivalence that can be seen
motivically.
###### Theorem 1 (3.14).
Let $G$ be a reductive $S$-group scheme with split maximal torus $T$ and Weyl
group $W$. Assume $G$ acts on an $S$-scheme $X$ locally of finite type. Then
the natural map
$\left[X/T\right]\rightarrow\left[X/G\right]$
is Tate.
Further, we have
$R\Gamma_{S}(\left[X/G\right],\mathbb{Q})\simeq
R\Gamma_{S}(\left[X/T\right],\mathbb{Q})^{W}.$
The idea of the proof of Theorem 1 is to factorize
$\left[X/T\right]\rightarrow\left[X/G\right]$ into
$\left[X/T\right]\rightarrow\left[X/N_{G}(T)\right]\rightarrow\left[X/G\right].$
Then the first map of the factorization is naturally a $W$-torsor and the
second map a $G/N_{G}(T)$-bundle. For torsors under finite groups étale
descent relates motives via $W$-invariants. For $G/N_{G}(T)$-bundles it
suffices to see that $R\Gamma_{S}(G/N_{G}(T),\mathbb{Q})$ is trivial. We will
prove this by reducing the triviality to the equivalence of the map
$K_{0}(S)_{\mathbb{Q}}\rightarrow K_{0}(G/B)^{W}_{\mathbb{Q}}$, which after
comparison with rational Chow theory is classical.
Applying Theorem 1 to motivic cohomology in the case, where
$S=\operatorname{Spec}(k)$, we can extend the classical result to higher Chow
groups even in the non-split case, generalizing the analogous result from
[Kri13] from split reductive groups to non-split groups. The idea is that any
maximal torus splits, after passage to a finite Galois extension. Then we
reduce to the split case, using that motives of torsors under finite groups
are related by taking invariants.
###### Corollary 2 (3.15).
Let $k$ be a field an let $G$ be a reductive $k$-group scheme with maximal
torus $T$ and absolute Weyl group $W$. Assume $G$ acts on a smooth finite type
$k$-scheme $X$. Then for all $n,m\in\mathbb{Z}$, we have
$A^{n}_{G}(X,m)_{\mathbb{Q}}\cong A^{n}_{T}(X,m)_{\mathbb{Q}}^{W}.$
The same argumentation as before works also over general base and arbitrary
reductive group schemes, to show that
$\left[X/T\right]\rightarrow\left[X/G\right]$ is Tate. This is because over a
connected noetherian normal base any torus splits after passage to a finite
Galois extension.
###### Corollary 3 (3.16).
Let $G$ be a reductive $S$-group scheme that admits a maximal torus $T$.
Assume $G$ acts on an $S$-scheme $X$ locally of finite type. Then the natural
map
$\left[X/T\right]\rightarrow\left[X/G\right]$
is Tate.
#### Motives of $T$-torsors
We will now analyze morphism $b$. For this let $\varphi\colon G\rightarrow G$
be an isogeny that fixes $T$. Consider the embedding $T\hookrightarrow T\times
T$ given by the graph of $\varphi$. The quotient $T\times T/T$ under this
embedding is isomorphic to $T$. In particular, this isomorphism gives the map
$b\colon\left[G/_{\varphi}T\right]\rightarrow\left[G/T\times T\right]$
the structure of a $T$-torsor. So, to understand the motivic behavior of $b$
it is enough to understand motives of $T$-torsors.
Thus, let $X\rightarrow Y$ be a morphism of Artin stacks that is a $T$-torsor.
Classically, Chow groups in this setting can be computed rather easily. For
each character $\chi\in T\rightarrow\mathbb{G}_{\textup{m}}$ we get a
$1$-dimensional representation $\kappa(\chi)$ of $T$. This yields a line
bundle $L_{\chi}\coloneqq X\times^{T}\kappa(\chi)$ on $Y$. Multiplication with
the first Chern class of $L_{\chi}$ yields an action of the character group
$\hat{T}$ of $T$ on $A^{\bullet}(Y)$. If $T$ is split, the morphism $b$ yields
(2) $A^{\bullet}_{T}(G)\cong A^{\bullet}_{T\times
T}(G)/\hat{T}A^{\bullet}_{T\times T}(G)$
(cf. [Tot14]).
Again, this is just a shadow of computations for oriented cohomology theories.
The idea is the following. If $T$ is split, we have
$T\cong\mathbb{G}_{\textup{m}}^{r}$ for some $r\in\mathbb{N}$. By applying
successive $\mathbb{G}_{\textup{m}}$-quotients, we can write
$X\rightarrow X_{1}\rightarrow X_{2}\rightarrow\dots\rightarrow X_{r}\cong Y,$
where $X_{i}\coloneqq\left[X/\mathbb{G}_{\textup{m}}^{i}\right]$. Each of the
maps $X_{i-1}\rightarrow X_{i}$ is a $\mathbb{G}_{\textup{m}}$-torsor. So we
may reduce to the case, where $T=\mathbb{G}_{\textup{m}}$. In this case, we
can follow [HPL21] and assign the line bundle ${\mathcal{L}}\coloneqq
X\times^{\mathbb{G}_{\textup{m}}}\mathbb{A}^{1}$ over $Y$. Multiplication with
the first Chern class of ${\mathcal{L}}$ yields a fibre sequence
$M_{k}(X)\rightarrow M_{k}(Y)\rightarrow M_{k}(Y)(1)[2].$
Applying this construction in the split case to motivic cohomology yields a
generalization of (2).
###### Corollary 4 (4.6).
Assume that $S=\operatorname{Spec}(k)$ and $T$ is a split torus. Let
$X\rightarrow Y$ be a $T$-torsor of smooth Artin stacks over $S$. Then
$A^{\bullet}(X)_{\mathbb{Q}}\cong
A^{\bullet}(Y)_{\mathbb{Q}}/\hat{T}A^{\bullet}(Y)_{\mathbb{Q}}.$
If $X$ and $Y$ are represented by quotients of qcqs smooth schemes by
diagonalizable group schemes, for example for $b$ as above, we can actually
replace the Chow ring with equivariant $K_{0}$. More generally, the analogous
statement holds for any oriented cohomology theory that is $m$-connective (cf.
Remark 4.7).
Further, combining the above with the fact that any torus over $S$ splits
after passage to a finite Galois extension, we also see the following.
###### Proposition 5 (4.3).
Let $f\colon X\rightarrow Y$ be a $T$-torsor of smooth Artin stacks over $S$.
Then $f$ is a Tate map.
#### Applications to quotients up to conjugation by isogeny
We will summarize our results above to see that the maps relating
$\textup{B}G^{F}$ and $\left[G/T\times T\right]$ are Tate. But we haven’t
particularly used the fact that we are interested in Frobenius-conjugation but
rather conjugation up to isogeny. Thus, we can work in a more general setting,
that we describe in the following.
Let $P$ resp. $Q$ be parabolics inside $G$ with Levi-components $L$ resp. $M$.
Let $\varphi\colon L\rightarrow M$ be an isogeny. Then $L$ acts on $G$ via
$(l,g)\mapsto lg\varphi(l)^{-1}$. Let $T$ be a maximal torus of $G$ contained
in $L$. Fix a $g_{0}\in G(S)$ such that $g_{0}\varphi(T)g_{0}^{-1}=T$ and
denote by $\widetilde{\varphi}$ the composition of $\varphi$ and
$g_{0}$-conjugation. We can embed $T$ into $T\times T$ via
$t\mapsto(t,\widetilde{\varphi}(t))$.
###### Theorem 6 (5.1).
In the setting above, we have the following zigzag of Tate maps
${\left[G/_{\varphi}L\right]}$${\left[G/_{\varphi}T\right]}$${\left[G/T\times
T\right].}$$\scriptstyle{a}$$\scriptstyle{b}$
Further, if $S=\operatorname{Spec}(k)$ and $T$ is split, we can compute the
Chow ring of $\left[G/_{\varphi}L\right]$ as
$A^{\bullet}(\left[G/_{\varphi}L\right])_{\mathbb{Q}}\cong\left(A^{\bullet}_{T}(G/T)_{\mathbb{Q}}/\hat{T}A^{\bullet}_{T}(G/T)_{\mathbb{Q}}\right)^{W_{L}},$
where $W_{L}$ denotes the Weyl group of $T$ in $L$.
Using results about equivariant $K$-theory by Uma and Krishna and our results
on the cohomology theory of $T$-torsors, we can extend the situation above to
equivariant $K$-theory and generalize a result by Brokemper.
###### Corollary 7 (5.4).
In the setting above assume that $S=\operatorname{Spec}(k)$, $G$ is split with
respect to $T$ and that the derived group of $G$ is simply connected. Then we
have
$K_{0}(\left[G/_{\varphi}L\right])_{\mathbb{Q}}\cong
R(T)_{\mathbb{Q}}^{W_{L}}/(f-\tilde{\varphi}f\mid f\in
R(T)_{\mathbb{Q}}^{W}),$
where $W_{L}$ denotes the Weyl group of $T$ in $L$ and $W$ as before denotes
the Weyl group of $T$ in $G$.
###### Example 8.
Let us give two interesting examples, where Theorem 6 can be used.
1. 1.
Let $k=\mathbb{F}_{q}$ be a finite field with $q$-elements and assume
$S=\operatorname{Spec}(k)$. If we set $L=G$ and $\varphi$ the $q$-Frobenius,
then we precisely get the situation of the beginning back. In particular, we
see that there is an adjunction between $\operatorname{DTM}(\textup{B}G^{F})$
and $\operatorname{DTM}(\left[B\backslash G/B\right])$.
2. 2.
Let $k$ be a finite field of characteristic $p>0$ and assume
$S=\operatorname{Spec}(k)$. Another interesting example is the stack of
$G$-zips of Pink-Wedhorn-Ziegler (cf. Example 5.6). In particular, we can
recover the computations of Brokemper [Bro18] for Chow groups (up to some
computations of loc.cit.).
This answers our main question ($\ast$ ‣ 1) positively.
###### Remark 9.
If we work with reductive $S$-group schemes that admit split maximal tori,
then we will see in our main text, that we can drop the connectedness
hypothesis on the results above.
#### Structure of this article
We start this article by recalling properties of the $\infty$-category of
motives and how to extend this to arbitrary locally of finite type Artin
stacks. After defining the necessary notions for this article, we quickly
recollect some computational aspects.
Afterwards, we start to focus on motives on schemes with group action. First,
we explain how to achieve a group action on motives and how torsors under
finite groups of Artin stacks have a particular behavior. Then we concentrate
on the case $T\subseteq G$, a split maximal torus inside a reductive group.
Namely, we show that the relation between $T$-equivariant Chow groups and
$G$-equivariant Chow groups extend to the motivic case.
Next, we show that $T$-torsors of Artin stacks are Tate and explicitly compute
motivic cohomology implying the classical case of Chow groups.
In the end, we focus on reductive groups with conjugation up to isogeny. We
use our results from before to get the desired adjunction of Tate motives with
the $T$-equivariant flag variety. We end the paper with ideas for
generalization, that we want to address in the future.
#### Setup
1. $\bullet$
Throughout, we fix a noetherian excellent scheme $\tilde{B}$ of dimension at
most $2$ and a regular scheme $S$ of finite type over $\tilde{B}$.
2. $\bullet$
An Artin stack is an algebraic stack in the sense of [Sta22]. When we write
“stack”, we always mean “Artin stack”. Every Artin stack (and hence scheme)
will be locally of finite type over $S$ and any morphism of Artin stacks will
be an $S$-morphism.
3. $\bullet$
Throughout, we will work in the setting of $\infty$-categories and freely use
the language of $\infty$-categories. In particular, presheaves will be always
considered as presheaves with values in $\infty$-groupoids.
4. $\bullet$
Throughout $\operatorname{DM}$ denotes the Beilinson motives with coefficients
in $\mathbb{Q}$.
5. $\bullet$
We will work with reductive group schemes over $S$ (cf. [SGA3]), i.e.
$S$-affine smooth group schemes such that the geometric fibers are connected
reductive groups.
6. $\bullet$
We say that a reductive $S$-group scheme $G$ is quasi-split if it so in the
sense of [SGA3, Exp. 24 §3.9].
7. $\bullet$
Let $X$ be an $S$-scheme and $G$ an $S$-group scheme acting on $X$. Then we
denote by $\left[X/G\right]$ the associated quotient stack in the étale
topology.
8. $\bullet$
For a $S$-group scheme $G$, we denote its (étale) classifying stack by
$\textup{B}G\coloneqq\left[S/G\right]$, where $G$ acts trivially on $S$.
### Acknowledgement
I would like to thank Torsten Wedhorn for multiple discussions and comments on
the earlier version, as well as his communication of Appendix A and the idea
to use reductive group schemes with split maximal tori, generalizing the
earlier versions. Further, I would like to thank Paul Ziegler, who
communicated this project and shared his thoughts with me. Finally, I would
like to thank Rizacan Ciloguli, Arnaud Éteve, Tim Holzschuh, Marc Hoyois,
Adeel Khan, Timo Richarz, Jakob Scholbach, Fabio Tanania, Thibaud van den Hove
for fruitful discussions and feedback.
This project was funded by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) - project number 524431573, the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) TRR 326 Geometry and
Arithmetic of Uniformized Structures, project number 444845124 and by the
LOEWE grant ‘Uniformized Structures in Algebra and Geometry’.
## 2 Rational equivariant motivic homotopy theory
In this section, we want to recall some properties of the category of
(rational) motives and how to extend this to Artin stacks locally of finite
type. We expect that most readers are familiar with the notion of motives and
the $6$-functor formalism and refer to [RS20, Syn. 2.1.1] for an overview of
the properties of the $6$-functor formalism.
Nevertheless, to prevent confusion, let us quickly recall some common notation
and remarks.
###### Remark 2.1 ([RS20], [CD19]).
In the following any scheme and any morphism will be considered in the
category of finite type $S$-schemes, $\textup{Sch}_{S}^{\textup{ft}}$.
1. (i)
For any $S$-scheme $X$, $\operatorname{DM}(X)$ is a stable, presentable,
closed symmetric monoidal $\infty$-category. The $\otimes$-unit will be
denoted by $1_{X}$. It has all limits and colimits.
2. (ii)
The assignment $X\mapsto\operatorname{DM}(X)$ can be upgraded to a presheaf of
symmetric monoidal $\infty$-categories
$\operatorname{DM}^{*}\colon\textup{Sch}_{S}^{\textup{ft}}\rightarrow\textup{Cat}_{\infty}^{\otimes},\
X\mapsto\operatorname{DM}(X),\ f\mapsto f^{*}.$
For any morphism of schemes $f\colon X\rightarrow Y$, there is an adjunction
${f^{*}\colon\operatorname{DM}(Y)}$${\operatorname{DM}(X)\colon f_{*}.}$
3. (iii)
If $f$ is smooth, then $f^{*}$ has a left adjoint, denoted $f_{\sharp}$.
4. (iv)
The assignment $X\mapsto\operatorname{DM}(X)$ can be upgraded to a presheaf of
$\infty$-categories
$\operatorname{DM}^{!}:(\textup{Sch}_{S}^{\textup{ft}})^{\operatorname{op}}\rightarrow\textup{Cat}_{\infty},\
X\mapsto\operatorname{DM}(X),f\mapsto f^{!}.$
For each $f$, there is an adjunction
$f_{!}:\operatorname{DM}(X)\rightleftarrows\operatorname{DM}(Y):f^{!}.$
For any factorization $f=p\circ j$ with $j$ an open immersion and $p$ a proper
map, there is a natural equivalence $f_{!}\cong p_{*}j_{\sharp}$.
5. (v)
Both functors $\operatorname{DM}^{*}$ and $\operatorname{DM}^{!}$ are sheaves
for the $h$-topology (cf. [RS20, Thm. 2.1.13])
6. (vi)
For the projection $p:\mathbb{G}_{\textup{m},S}\times_{S}X\rightarrow X$, and
any $M\in\operatorname{DM}(X)$, the map induced by the counit
$p_{\sharp}p^{*}M[-1]\rightarrow M[-1]$ in $\operatorname{DM}(X)$ is a split
monomorphism. The complementary summand is denoted by $M(1)$. The functor
$M\mapsto M(1)$ is an equivalence with inverse denoted by $M\mapsto M(-1)$.
For any integer $n$ the $n$-fold composition is denoted by $M\mapsto M(n)$ and
in the future, we will abbreviate $\langle n\rangle\coloneqq(n)[2n]$.
Let $\underline{X}$ be a prestack222This notion is from [RS20]., i.e. presheaf
of anima on the category of rings. There are several approaches to the
$\infty$-category $\operatorname{DM}(\underline{X})$. If $\underline{X}$ is an
Artin stack over a field $k$ Hoskins-Lehalleur use a construction similar to
equivariant Chow groups. One resolves $\underline{X}$ by open substacks
$(\underline{X}_{i})$ such that on each $\underline{X}_{i}$ there is a vector
bundle $V_{i}$ together with an open $U_{i}$ with a free $G$-action such that
the codimension of $V_{i}\setminus U_{i}$ tends towards infinity (cf.
[HPL21]). This idea was further generalized to arbitrary cohomology theories
by Khan-Ravi by a construction they call lisse extension (cf. [KR21]). This
construction was already used for the motive of classifying stacks
$\textup{B}G$ by Morel-Voevodsky (cf. [MV99, §4]). Totaro then gave an
explicit computation of the motive of $\textup{B}\mathbb{G}_{\textup{m}}$ over
a field (cf. [Tot16] and Example 2.8).
Alternatively to the construction of Hoskins-Lehalleur resp. Khan-Ravi,
Richarz-Scholbach give a construction via certain left and right Kan extension
(cf. [RS20]). Their approach is based on gluing the motivic structure on
Beilinsion motives to arbitrary prestacks. Indeed, $\operatorname{DM}(-)$
satisfies $h$-descent, so it is rather formal to extend the six functor
formalism to Artin stacks locally of finite type, we will use this approach.
One should note that this was also discussed in [Kha19], to extend the
$6$-functor formalism to higher Artin stacks.
For computations of the underlying motives it seems to be better to work with
the definition of Hoskins-Lehalleur resp. Morel-Voevodsky. Let
$f\colon{\mathfrak{X}}\rightarrow\operatorname{Spec}(k)$ be a smooth Artin
stack and let $M({\mathfrak{X}})$ denote the $k$-linear motive of
${\mathfrak{X}}$ defined in [HPL21]. This defines an object in
$\operatorname{DM}^{*}(\operatorname{Spec}(k))$. Let $1_{k}$ denote the unit
in $\operatorname{DM}^{*}(\operatorname{Spec}(k))$. We will see in Corollary
2.7 that if $f$ is smooth, we have $M({\mathfrak{X}})\simeq
f_{\sharp}f^{*}1_{k}$. In particular, if we use the approach of [RS20] and
define a motive of a prestack as the $\sharp$-push/$*$-pull of the unit, we
see that the notion of motives on Artin stacks as in loc.cit. agrees with the
classical ones.
To define $\operatorname{DM}$ for prestacks, we will follow [RS20]. For the
following definition let us fix a regular cardinal $\kappa$. Let
$\textup{Aff}_{S}^{\textup{ft}}$ denote the (Nerve of the) category of affine
schemes of finite type over $S$. We let $\textup{Aff}_{S}^{\kappa}$ denote the
$\kappa$-pro-completion of $\textup{Aff}_{S}^{\textup{ft}}$. Further,
$\textup{DGCat}_{\textup{cont}}$ denotes the $\infty$-category of presentable
stable $\mathbb{Q}$-linear dg-$\infty$-categories with colimit preserving
functors.
###### Definition 2.2 ([RS20]).
Let $y\colon\textup{Aff}_{S}^{\kappa}\hookrightarrow
P(\textup{Aff}_{S}^{\kappa})$ be the Yoneda embedding. We define the functor
$\operatorname{DM}_{S}\colon
P(\textup{Aff}_{S})^{\operatorname{op}}\rightarrow\textup{DGCat}_{\textup{cont}}$
via left Kan extension along the inclusion
$\textup{Aff}_{S}^{\textup{ft}}\subseteq\textup{Aff}_{S}^{\kappa}$ and right
Kan extension along $\textup{Aff}_{S}^{\kappa}P(\textup{Aff}_{S}^{\kappa})$,
where all the functors are given via $!$-pullback. For a prestack
$\underline{X}\in P(\textup{Aff}_{S}^{\kappa})$, we define the
$\infty$-category of $S$-linear motives (with rational coefficients) of
$\underline{X}$ as $\operatorname{DM}_{S}(\underline{X})$.
As noted in [RS20], for applications we can take $\kappa$ to be large enough,
so that all affine schemes of interest are inside $\textup{Aff}_{S}^{\kappa}$.
Thus, for once and for all we may fix a regular cardinal $\kappa$ and drop it
from the notation.
Khan showed in [Kha19, Thm. A.5] that this method of extending the theory of
motives to (derived) Artin stacks, does not loose the $6$-functor formalism.
One way to see this, is that we can use the DESCENT program in [LZ12], since
the Beilinsion motives satisfy étale descent, in our context. As mentioned in
[Kha19], this is equivalent to the construction of [RS20].
###### Theorem 2.3.
Let $\widetilde{\operatorname{DM}}$ be the restriction of $\operatorname{DM}$
to Artin stacks locally of finite type over $S$. Then
$\widetilde{\operatorname{DM}}$ is compatible with the $6$-functor formalism
in the sense of [RS20, Syn. 2.1.1].
###### Proof.
The proof is the same as [Kha19, Thm A.5]333As mentioned in op.cit., the
method of extending the $6$-functor formalism works with any motivic category
that satisfies étale descent. and follows from the results of [RS20]. ∎
###### Definition 2.4 ([RS20]).
Let ${\mathfrak{X}}$ be an Artin $S$-stack locally of finite type with
structure morphism $f\colon{\mathfrak{X}}\rightarrow S$. Then we define the
(rational) $S$-linear motive of ${\mathfrak{X}}$ as
$M_{S}({\mathfrak{X}})\coloneqq f_{!}f^{!}1_{S}$. If
$S=\operatorname{Spec}(A)$ is affine, we write $M_{A}({\mathfrak{X}})$.
We further define the global sections of ${\mathfrak{X}}$ over $S$ to be
$R\Gamma_{S}({\mathfrak{X}},\mathbb{Q})\coloneqq f_{*}f^{*}1_{S}$.
###### Remark 2.5.
Let $f\colon{\mathfrak{X}}\rightarrow S$ be an Artin stack locally of finite
type over $S$. If $f$ is smooth, then relative purity implies that
$f_{\sharp}f^{*}\simeq f_{!}f^{!}$ and in particular, we see with
$\operatorname{Hom}_{\operatorname{DM}(S)}(M_{S}({\mathfrak{X}}),1(n)[m])\simeq\operatorname{Hom}_{\operatorname{DM}(S)}(1(-n)[-m],R\Gamma_{S}({\mathfrak{X}},\mathbb{Q}))$
that $M_{S}({\mathfrak{X}})$ computes motivic homology and
$R\Gamma_{S}({\mathfrak{X}},\mathbb{Q})$ motivic cohomology.
###### Notation 2.6.
Let $G$ be an $S$-group scheme locally of finite type acting on an $S$-scheme
$X$ locally of finite type, via a morphism $a$. For the quotient stack
$\left[X/G\right]$, we can define a simplicial object in $S$-schemes locally
of finite type, via its Bar-resolution
${\dots}$${G\times_{S}X}$${X}$$\scriptstyle{p}$$\scriptstyle{a}$
We denote the corresponding simplicial functor with
$\textup{Bar}^{\bullet}(X,G)$.
There is also an alternative way to define motives of algebraic stacks via the
Bar-resolution. For each $n\geq 0$ let $\mathbb{Q}(\textup{Bar}^{n}(X,G))$ be
the constant étale sheaf with coefficients in $\mathbb{Q}$ associated to
$\textup{Bar}^{n}(X,G)$. This yields a simplicial object in étale sheaves of
$S$-schemes locally of finite type with rational coefficients. The complex
associated to this simplicial object induces a motive
$M_{S}(\textup{Bar}^{\bullet}(X,G))$ in $\operatorname{DM}(S)$. For
$S=\operatorname{Spec}(k)$ the spectrum of a field, Hoskins-Lehalleur explain
in [HPL21] that this definition is equivalent to their definition of a motive
of an Artin stack. The naturally arising question is if
$M_{S}(\textup{Bar}^{\bullet}(X,G))$ is equivalent to
$M_{S}(\left[X/G\right])$ as defined in Definition 2.4. If $\left[X/G\right]$
is representable by a smooth scheme, then the answer is positive and follows
by cohomological descent for $\operatorname{DM}$ with respect to the
$h$-topology (cf. [CD19, Thm. 14.3.4, Prop. 5.2.10]). Thus, the answer stays
positive for smooth Artin stacks, as $\operatorname{DM}$ satisfies $h$-descent
by gluing (cf. [RS20, Thm. 2.2.16]).
###### Corollary 2.7.
Let $k$ be a field and $S=\operatorname{Spec}(k)$. Let $X$ be a smooth
$S$-scheme and $G$ be a smooth $S$-group scheme acting on $X$ with structure
map $f\colon\left[X/G\right]\rightarrow S$. Then $M_{S}(\left[X/G\right])$ is
equivalent to $M_{S}(\textup{Bar}^{\bullet}(X,G))$.
###### Proof.
This follows from [HPL21, Prop. A.7] and the discussion above. ∎
We can use Corollary 2.7 to compute the motive of $B\mathbb{G}_{\textup{m}}$
as in [Tot16].
###### Example 2.8.
Let $k$ be a field. Further, let $\mathbb{G}_{\textup{m},k}$ act trivially on
$\operatorname{Spec}(k)$. Then
$M_{k}(\textup{B}\mathbb{G}_{\textup{m},k})\simeq\mathop{\textup{colim}}_{i\in\mathbb{N}}M_{k}(\mathbb{P}^{i}_{k})\simeq\bigoplus_{i\geq
0}1_{k}\langle i\rangle.$
###### Remark 2.9.
In the following we want to understand the Gysin sequence for algebraic
stacks. Let us quickly recall it in the scheme case.
Let $i\colon Z\hookrightarrow X$ be a closed immersion of $S$-schemes of pure
codimension $n$ with open complement $U$. Let us assume that $Z$ and $X$ are
smooth over $S$. In particular, we see that $i$ is equivalently a regular
closed immersion of codimension $n$. Then there exists a fibre sequence of the
form
$M_{S}(U)\rightarrow M_{S}(X)\rightarrow M_{S}(Z)\langle n\rangle$
(cf. [CD19, 11.3.4]).
We are going to replace $S$ by a smooth Artin stack ${\mathfrak{Y}}$ over $S$
and $X$ by a smooth Artin stack ${\mathfrak{X}}$ over ${\mathfrak{Y}}$. For
this let us also recall the notion of a (regular) closed immersion of a
certain codimension for Artin stacks.
Let $\iota\colon{\mathfrak{Z}}\hookrightarrow{\mathfrak{X}}$ be a closed
immersion of locally noetherian Artin stacks. Let $X\rightarrow{\mathfrak{X}}$
be a smooth atlas. Then $\iota$ is representable and we define the codimension
of ${\mathfrak{Z}}$ as the codimension of
${\mathfrak{Z}}\times_{{\mathfrak{X}}}X$ in $X$ (cf. [Oss15, §6]). We can also
define the notion of a regular immersion in that way (cf. [Sta22, 06FM]) and
the notion of its codimension. In particular, a closed immersion of smooth
Artin $S$-stacks ${\mathfrak{Z}}\hookrightarrow{\mathfrak{X}}$ is
automatically regularly immersed and the codimension of the regular immersion
agrees with the codimension as a closed substack.
###### Lemma 2.10 (The Gysin sequence).
Let $f\colon{\mathfrak{X}}\rightarrow{\mathfrak{Y}}$ be a smooth schematic
morphism of smooth Artin $S$-stacks. Further let
$i\colon{\mathfrak{Z}}\hookrightarrow{\mathfrak{X}}$ be a closed immersion of
(pure) codimension $n$ such that ${\mathfrak{Z}}$ is smooth over
${\mathfrak{Y}}$ with open complement
$j\colon{\mathfrak{U}}\rightarrow{\mathfrak{X}}$. Further, let us denote
$f_{0}\coloneqq f\circ j$ and $\bar{f}\coloneqq f\circ i$. Then there exists
the following fibre sequence
$f_{0!}f_{0}^{!}1_{{\mathfrak{Y}}}\rightarrow
f_{!}f^{!}1_{{\mathfrak{Y}}}\rightarrow\bar{f}_{!}\bar{f}^{!}1_{{\mathfrak{Y}}}\langle
n\rangle.$
###### Proof.
Let $Y\rightarrow{\mathfrak{Y}}$ be a smooth atlas. Let us define $X\coloneqq
Y\times_{{\mathfrak{Y}}}{\mathfrak{X}}$ and let $\check{C}(Y)_{\bullet}$ resp.
$\check{C}(X)_{\bullet}$ denote the corresponding Čech nerves. By construction
$\check{C}(X)_{\bullet}$ is obtained by $\check{C}(Y)_{\bullet}\times_{Y}X$.
So, by functoriality we get maps
${f_{\bullet!}\colon\operatorname{DM}(\check{C}(Y)_{\bullet})}$${\operatorname{DM}(\check{C}(X)_{\bullet})\colon
f_{\bullet}^{!}}$
that induce the maps $f_{!}$ and $f^{!}$ after passing to the limit. By
construction we have a pullback diagram
${\check{C}(X)_{\bullet}}$${\check{C}(Y)_{\bullet}}$${{\mathfrak{X}}}$${{\mathfrak{Y}}.}$$\scriptstyle{f_{\bullet}}$$\scriptstyle{j_{X,\bullet}}$$\scriptstyle{j_{Y,\bullet}}$
In particular, by smoothness of the atlas and the exchange equivalence, we
have
$j^{*}_{Y,\bullet}f_{!}f^{!}1_{Y}\simeq
f_{\bullet!}f_{\bullet}^{!}1_{\check{C}(Y)_{\bullet}}.$
Thus, by smoothness we can use that
$\operatorname{DM}^{*}({\mathfrak{Y}})\simeq\operatorname{DM}^{!}({\mathfrak{Y}})$
and descent to see that
$\mathop{\textup{colim}}_{\Delta}f_{\bullet!}f_{\bullet}^{!}1_{\check{C}(Y)_{\bullet}}\simeq
f_{!}f^{!}1_{Y}.$
Analogously, we can write
$\mathop{\textup{colim}}_{\Delta}f_{0\bullet!}f_{0\bullet}^{!}1_{\check{C}(Y)_{\bullet}}\simeq
f_{0!}f^{!}_{0}1_{Y},\quad\mathop{\textup{colim}}_{\Delta}\bar{f}_{\bullet!}\bar{f}_{\bullet}^{!}1_{\check{C}(Y)_{\bullet}}\simeq\bar{f}_{!}\bar{f}^{!}1_{Y}.$
Therefore, we may assume that ${\mathfrak{Y}}$ is representable by a scheme
and by representability of $f$ also ${\mathfrak{X}},{\mathfrak{U}}$ and
${\mathfrak{Z}}$ are representable by schemes. Hence, the result now follows
from the classical Gysin sequence (cf. [CD19, 11.3.4]). ∎
Lastly, let us define Tate motives. As we mentioned in the introduction, the
existence of a motivic $t$-structure is still an open problem. For a field $k$
Levine proved that under certain vanishing assumptions on motivic cohomology
in $\operatorname{DM}(k)$, the so called Beilinson-Soulé vanishing conjecture,
such a $t$-structure exists on the full stable subcategory generated by Tate
twists $1_{k}(n)$ (cf. [Lev93]). The Beilinson-Soulé vanishing conjecture
holds for example for finite fields.
###### Definition 2.11 ([RS20]).
Let $X$ be an Artin $S$-stack locally of finite type. We define the category
of Tate motives $\operatorname{DTM}(X)$ to be the full stable subcategory of
$\operatorname{DM}(X)$ generated by $1_{X}(n)$, for $n\in\mathbb{Z}$. An
element $M\in\operatorname{DM}(X)$ is Tate, if $M\in\operatorname{DTM}(X)$.
A map $f\colon X\rightarrow Y$ of Artin stacks locally of finite type over $S$
is called Tate if $f_{*}1_{X}$ is Tate.
Levine further shows, that existence of a weight structure on Tate motives for
a field imply that the heart of $\operatorname{DTM}(\mathbb{F}_{p})$ under the
motivic $t$-structure is equivalent to the category of graded finite
dimensional $\mathbb{Q}$-vector spaces
$(\mathbb{Q}\textup{-VS}^{\mathbb{Z}})$. In particular, using a result of
Wildeshaus, we see for example that
$\operatorname{DTM}(\mathbb{F}_{p})\simeq{\mathcal{D}}^{b}(\mathbb{Q}\textup{-VS}^{\mathbb{Z}})$,
where ${\mathcal{D}}^{b}$ denotes the bounded derived category (cf. [Wil09]).
Let us give a particularly interesting example of a Tate map that will be used
later on.
###### Example 2.12.
Let $G$ be a split reductive $S$-group scheme and $B\subseteq G$ a Borel. Then
we claim that the structure map of the flag variety $G/B\rightarrow S$ is
Tate.
Indeed, the Bruhat decomposition of $G/B$ yields a stratification by affine
spaces indexed by the Weyl group. The length of each Weyl element yields a
partial order on the associated Schubert varieties. With this order, one can
show using standard arguments that
$R\Gamma_{S}(G/B,\mathbb{Q})\simeq\bigoplus_{w\in W}1_{S}\langle
l(w)-n\rangle$, where $n$ denotes the relative dimension of $G/B$ over $S$
(cf. [Yay23] or [Bac19] for more details on analogous problems).
###### Remark 2.13 ([RS20]).
If $f\colon X\rightarrow Y$ is a smooth morphism of Artin $S$-stacks locally
of finite type and $X$ is smooth over $S$, then $D_{Y}(f_{*}1_{X})\simeq
f_{!}D_{X}(1_{X})\simeq f_{!}1_{X}\langle\Omega_{X/S}\rangle$ and
$D_{Y}(f_{!}1_{X})\simeq f_{*}1_{X}\langle\Omega_{X/S}\rangle$. Thus,
$f_{*}1_{X}$ is Tate if and only if $f_{!}1_{X}$ is Tate.
In the literature one also considers the stable cocomplete $\infty$-category
generated by Tate twists (cf. [RS20]). This is usually also referred to as
“Tate motives” but we will differentiate these from our definition. An example
of such motives is given by the motive of
$\textup{B}\mathbb{G}_{\textup{m},k}\rightarrow\operatorname{Spec}(k)$, the
classifying stack of $\mathbb{G}_{\textup{m},k}$ over a field $k$. Indeed, in
Example 2.8 we have seen
$M_{k}(\textup{B}\mathbb{G}_{\textup{m}})\simeq\bigoplus_{n\in\mathbb{Z}}1_{k}\langle
n\rangle$ and thus this lies in the ind-completion of the category of Tate
motives.
###### Definition 2.14.
Let $X$ be an Artin $S$-stack locally of finite type. We will call an
$M\in\operatorname{DM}(X)$ completed Tate if it is already in the full stable
cocomplete subcategory generated by $1_{X}(n)$.
## 3 Equivariant motives under reductive groups
Let us give an outlook of the next subsections in the split case. The non-
split case will then be a corollary.
Assume that $S$ is connected and let $G$ be a reductive $S$-group scheme and
$T$ a split maximal torus in $G$ with Weyl group scheme $W_{S}$. We will see
in Section 3.1 that $W_{S}$ is a constant group scheme associated to the
finite group $W\coloneqq W_{S}(S)$, which we call Weyl group of $T$ in $G$.
Now let $X$ be an $S$-scheme locally of finite type with $G$ action. In this
section, we want to show that the natural map
$f\colon\left[X/T\right]\rightarrow\left[X/G\right]$ is Tate, i.e.
$f_{*}1_{\left[X/T\right]}$ is a Tate motive in
$\operatorname{DM}(\left[X/G\right])$.
The key idea is to use the factorization
$\left[X/T\right]\xrightarrow{g}\left[X/N\right]\xrightarrow{h}\left[X/G\right]$,
where $N$ is the normalizer of $T$ in $G$. We note that per definition the map
$g\colon\left[X/T\right]\rightarrow\left[X/N\right]$ is a $W$-torsor. As
$W_{S}$ is finite étale, we see that after passage to an étale cover,
$\left[X/T\right]$ is isomorphic to the disjoint union of $\left[X/N\right]$
indexed by $W$. Thus, automatically $g$ is Tate.
The map $h\colon\left[X/N\right]\rightarrow\left[X/G\right]$ is a
$G/N$-torsor. Up to taking $W$-invariant, we can identify $G/N$ with $G/B$. On
$G/B$, we have stratification by Schubert cells, which are affine spaces. In
this way, we can decompose $p_{*}1_{G/B}$ as a direct sum of twists and shifts
indexed by $W$, where $p\colon G/B\rightarrow S$ is the structure map. This
will enable us to reduce the question if $h$ is Tate to ordinary $K$-theory.
More precisely, it will be enough to show that
$K_{0}(S)_{\mathbb{Q}}\rightarrow K(G/T)^{W}_{\mathbb{Q}}$ is an isomorphism,
which after reduction to the case of Chow groups is classically known.
Before coming to our main result of this section, we will first introduce
group actions on motives. In the end, we will see that our computations before
showed that, we have $R\Gamma_{S}(\left[X/G\right],\mathbb{Q})\cong
R\Gamma_{S}(\left[X/T\right],\mathbb{Q})^{W}$, analogous to the case of Chow
groups.
###### Remark 3.1.
A key argument in this section, is that torsors under finite étale group
schemes have related motives by taking invariants of the action. This is a
major obstruction for the generalization to integral coefficients, as we
expect that this is only satisfied if we have étale descent (c.f. [CD19, Thm.
3.3.32]). Nevertheless, using the theory of étale motives it should not be to
difficult to have analogous results after inverting only the residue
characteristics of our base scheme.
### 3.1 Preliminaries on reductive group schemes
Let $G$ be a reductive $S$-group scheme. In the following article, we want to
work with reductive group schemes that admit maximal tori. Even though this is
automatically satisfied if $S$ is the spectrum of a field, this is not true
for general base schemes (cf. [Con15]). We will first proof our results under
the assumption that the maximal torus of $G$ is split. This is not equivalent
to splitness of $G$ or even quasi-splitness.
In general one can find two main notions of quasi-split in the literature. In
SGA 3 a quasi-split reductive group scheme admits a Borel pair as well as a
global section of the associated Dynkin diagram (cf. [SGA3, Exp. XXIV]).
Another definition is given by Conrad (cf. [Con14]), where he only enforces
existence of a Borel subgroup. If $S$ is affine, this automatically implies
the existence of a maximal torus contained in $B$.
As mentioned in the introduction, for applications it is enough to consider
only the existence of a maximal torus. If the maximal torus is split, we can
also talk about the associated Weyl group, as the next lemma shows.
We want to thank Torsten Wedhorn for explaining this and elaborating that it
is enough to consider reductive group schemes with split maximal tori instead
of split reductive group schemes.
###### Lemma 3.2.
Assume that $S$ is connected. Let $T$ be a split maximal torus inside $G$ with
normalizer $N_{G}(T)$, then $W_{S}\coloneqq N_{G}(T)/T$ is represented by a
constant finite group scheme.
###### Proof.
Let us first recall some well known fact (cf. [SGA3, Exp. XXII §3]).
The quotient $W_{S}$ is represented by a finite étale $S$-scheme and that
there is an open immersion $\iota\colon
W_{S}\hookrightarrow\underline{\operatorname{Aut}}_{S}(T)$. As $T$ is split,
$\underline{\operatorname{Aut}}_{S}(T)$ is the constant group associated to
the automorphism group $\operatorname{Aut}(X^{\bullet}(T))$. In particular,
$\underline{\operatorname{Aut}}_{S}(T)\cong\coprod_{\operatorname{Aut}(X^{\bullet}(T))}S$.
Note that $\underline{\operatorname{Aut}}_{S}(T)$ is separated and thus
$\iota$ is finite. As any finite monomorphism is a closed immersion (cf.
[Sta22, 03BB]), the morphism $\iota$ is an open and closed immersion. By
assumption $S$ is connected and therefore the image of $\iota$ has to be a
disjoint union of connected components, i.e. $W_{S}\cong\coprod_{H}S$, for
some subset $H\subseteq\operatorname{Aut}(X^{\bullet}(T))$. Finally, since
$W_{S}\rightarrow S$ is finite, the set $H$ has to be finite. ∎
###### Definition 3.3.
Assume that $S$ is connected and that $G$ admits a split maximal torus. Then
we call the finite group $W\coloneqq(N_{G}(T)/T)(S)$ the Weyl group of
$G$444If $G$ is split reductive, then this agrees with the classical notion..
Let us give some examples of reductive groups that admit maximal tori.
###### Example 3.4.
Let us give examples of schemes $S$ and $G$ as above.
1. $\bullet$
If $S=\operatorname{Spec}(k)$ is the spectrum of a field, then any reductive
group admits a maximal torus (cf. [SGA3, Exp. XIV Thm 1.1]).
2. $\bullet$
If $S$ is any connected regular $\tilde{B}$-scheme of finite type, then any
quasi-split reductive $S$-group scheme $G$ suffices.
3. $\bullet$
Assume, that $A$ is a Dedekind domain and $S=\operatorname{Spec}(A)$. Then $G$
admits a maximal torus if its generic fibre quasi-split (cf. Appendix A).
4. $\bullet$
Brian Conrad constructed examples of reductive groups over
$\operatorname{Spec}(\mathbb{Z})$ that do not admit maximal tori (cf.
[Con15]). All of these examples concern non-classical reductive groups and the
author is not aware of any other counterexamples.
### 3.2 Torsors under finite groups
To warm up, we first show that torsors under finite groups are Tate. In
particular, it will be clear that the canonical map
$\left[X/T\right]\rightarrow\left[X/N\right]$, considered in the beginning of
Section 3, is Tate. We will not need any connectedness hypothesis on $S$ for
the results of this section.
###### Lemma 3.5.
Let $\Gamma$ be a finite étale $S$-group scheme. Let $f\colon X\rightarrow Y$
be a $\Gamma$-torsor of Artin $S$-stacks locally of finite type. Then $f$ is
Tate, i.e. $f_{*}1_{X}$ is a Tate motive in $\operatorname{DM}(Y)$.
###### Proof.
Let $n$ denote the degree of $\Gamma$ over $S$. Then we claim that the natural
map $\coprod_{i=1}^{n}1_{Y}\rightarrow f_{*}1_{X}$ induced via the unit of
$f_{*}f^{*}$ is an equivalence.
Indeed, by $h$-descent we may assume that $f$ is given by the trivial
$\Gamma$-torsor $\Gamma\times_{S}Y\rightarrow Y$ and $Y$ is represented by a
scheme. In particular, $f$ is a finite étale cover of degree $n$. After
passage to an étale cover, we may assume that
$\Gamma\times_{S}Y\cong\coprod_{i=1}^{n}Y$ implying the claim. ∎
Next, let us analyze the structure of the motives with respect to the base.
For this, we need to understand group actions on motives and taking fixed
points under these actions.
An action of a group $\Gamma$ on a motive $M$ in $\operatorname{DM}(X)$ is map
$\Gamma\rightarrow\operatorname{Aut}_{\operatorname{DM}(S)}(M)$ that is a
group homomorphism on $\pi_{0}$. Or equivalently, it is a map
$\Sigma_{\Gamma}\rightarrow\operatorname{DM}(X)$ (here $\Sigma_{\Gamma}$
denotes the deloop of the group $\Gamma$ seen as a discrete category - usually
this is denoted with $\textup{B}\Gamma$ but to avoid confusion, we changed the
notation).
###### Definition 3.6.
Let $M$ be a motive in $\operatorname{DM}(X)$ with an action by a finite group
$\Gamma$. Then we define the homotopy $\Gamma$-fixed points of $M$, denoted by
$M^{h\Gamma}$, as the limit of the action map
$\Sigma_{\Gamma}\rightarrow\operatorname{DM}(X)$.
###### Definition and Remark 3.7.
Let $X$ be an Artin $S$-stack locally of finite type and with an action by a
finite group $\Gamma$. For each $\gamma$, we have an action map
$a_{\gamma}\colon X\rightarrow X$. By construction of $\operatorname{DM}(X)$
this defines a map $\gamma.\colon 1_{X}\rightarrow 1_{X}$ by lax-monoidality
of the $*$-pushforward. This endows any $M\in\operatorname{DM}(X)$ with an
action via $\gamma.$. We define the $\Gamma$-fixed points of $M$, denoted by
$M^{\Gamma}$ as the image555Note that (the homotopy category of)
$\operatorname{DM}(X)$ is pseudo-abelian and hence for any idempotent operator
we can define its image. of the map
$p=\frac{1}{\\#\Gamma}\sum_{\gamma\in\Gamma}\gamma.$
The canonical map $M^{\Gamma}\rightarrow M$ defines an equivalence
$M^{\Gamma}\xrightarrow{\sim}M^{h\Gamma}$ (cf. [CD19, 3.3.21]).
Let $X\rightarrow Y$ be a map of $\Gamma$-torsor of Artin stacks locally of
finite type over $S$ (here we see $\Gamma$ as a constant group scheme on $S$).
Then the $\Gamma$-torsor $Y$-automorphisms of $X$ is isomorphic to $\Gamma$
and thus we get a $\Gamma$-action on $f_{*}f^{*}M$ for any
$E\in\operatorname{DM}(Y)$ (via $*$-pushforward of a $\Gamma$-torsor
$Y$-automorphism). Note that $f_{*}f^{*}E$ can be used to compute the motivic
cohomology of $X$ with coefficients in $E$ for smooth $f$ as
$\operatorname{Hom}_{\operatorname{DM}(Y)}(1_{Y},f_{*}f^{*}E)=\operatorname{Hom}_{\operatorname{DM}(Y)}(f^{*}1_{Y},f^{*}E)=\operatorname{Hom}_{\operatorname{DM}(Y)}(M_{Y}(X),E).$
In particular, we see that666Kernels in triangulated categories are
monomorphisms and thus on the level of homotopy groups induce short exact
sequences, dualy cokernels induce epimorphisms. Thus, applying the homotopy
invariance functor to the Hom-spectrum
$\underline{\operatorname{Hom}}_{\operatorname{DM}(Y)}(1_{Y},f_{*}f^{*}E)$, we
see that $\pi_{0}$ precisely gives us the invariants of the induced group
action on $\pi_{0}$.
$\operatorname{Hom}_{\operatorname{DM}(Y)}(1_{Y},(f_{*}f^{*}E)^{\Gamma})=\operatorname{Hom}_{\operatorname{DM}(Y)}(M_{Y}(X),E)^{\Gamma}.$
Note that as $\Gamma$ is finite, the $\Gamma$-torsor $f$ is étale and proper,
hence $f_{*}f^{*}\simeq f_{!}f^{!}$.
Let $k$ be a field and $X$ a scheme locally of finite type over
$\operatorname{Spec}(k)$. Further, let $K/k$ be a finite Galois extension and
let us denote the base change of $X$ to $\operatorname{Spec}(K)$ by $X_{K}$.
The Chow groups of $X$ and $X_{K}$ are related by the fixed points under the
Galois group, i.e.
$\textup{CH}_{n}(X)=\textup{CH}_{n}(X_{K})^{\operatorname{Gal}(K/k)}$. As one
expects, this also holds motivically. This is due to Ayoub and Cisinki-
Déglise. As noted in Remark 3.1, we do not expect this to hold, when we do not
impose étale descent. Thus, we do not expect the next lemma to hold with
integral coefficients.
###### Lemma 3.8.
Let $\Gamma$ be a finite group. Let $f\colon X\rightarrow Y$ be a
$\Gamma$-torsor of Artin $S$-stacks locally of finite type. Then the unit
factors as $\operatorname{id}\rightarrow(f_{*}f^{*})^{\Gamma}\rightarrow
f_{*}f^{*}$ and the map $\operatorname{id}\rightarrow(f_{*}f^{*})^{\Gamma}$ is
an equivalence.
###### Proof.
The factorization of the unit follows from the description of
$(f_{*}f^{*})^{\Gamma}$ as a limit. We claim that
$\operatorname{id}\rightarrow(f_{*}f^{*})^{\Gamma}$ is an equivalence. It
suffices to check this after base change to a smooth atlas of $Y$. In
particular, we may assume that $Y$ is represented by a scheme. Since
$\operatorname{DM}_{\mathbb{Q}}$ satisfies $h$-descent, we may assume777Using
[CD19, Prop. 3.3.31], we see that
$M_{Y}(X)^{\Gamma}\simeq\varphi_{*}\varphi^{*}1_{Y}$, where
$\varphi\colon({\mathscr{X}},\Gamma)\rightarrow Y$ is the induced morphism of
the diagram $\Sigma_{\Gamma}\rightarrow\textup{Sch}_{S}$ that maps the single
point of the category $\Sigma_{\Gamma}$ to $X$ and the morphisms to the
actions. Then we can base change using [CD19, Prop. 3.1.17]. that $f$ is a
trivial $\Gamma$-torsor, where it follows from [Ayo07, Prop. 2.1.166]. ∎
###### Example 3.9.
Let us consider the $W$-torsor
$f\colon\left[X/T\right]\rightarrow\left[X/N\right]$ in the beginning of
Section 3. As $W$ is finite, Lemma 3.8 implies that
$f_{*}f^{*}1_{\left[X/T\right]}^{W}\simeq
f_{!}f^{!}1_{\left[X/T\right]}^{W}\simeq 1_{\left[X/N\right]}$. After
$\sharp$\- resp. $*$-pushforward to the base $S$ along the structure map
$\left[X/N\right]\rightarrow S$, we see that
$M_{S}(\left[X/N\right])\simeq M_{S}(\left[X/T\right])^{W}\textup{ resp.
}R\Gamma(S,\left[X/N\right])\simeq R\Gamma(S,\left[X/T\right])^{W}.$
Now assume that $S=\operatorname{Spec}(k)$ is the spectrum of a field.
Applying the latter equivalence to motivic cohomology yields
$A^{n}_{N}(X,m)\cong A^{n}_{T}(X,m)^{W}$
for the equivariant intersection theory of $X$.
### 3.3 The relation between the motives of $\left[X/T\right]$ and
$\left[X/G\right]$
We have seen, that the map $\left[X/T\right]\rightarrow\left[X/N\right]$ is
Tate and how to use the action of the Weyl group to compute the motive of
$\left[X/N\right]$ with respect to the motive of $\left[X/T\right]$. Now our
goal is to show that the map $\left[X/N\right]\rightarrow\left[X/G\right]$ is
Tate and how to compute the $!$-push/pull of this map. This will be achieved
by analyzing the motive of $G/N$. Using that $G/T\rightarrow G/N$ is again a
$W$-torsor, we will reduce to the case of the flag variety $G/B$. For this, we
will need that equivariant motives do not see the action of split unipotent
subgroups.
Now let us recall the definition of a split unipotent subgroup. These are
extensions of vector bundles, e.g. a Borel $B$ containing a maximal split
torus $T$ is an extension of $T$ by a split unipotent subgroup.
###### Definition 3.10.
An algebraic $S$-group scheme $U$ is called split-unipotent if there exists a
normal series, i.e. a filtration $U=U_{n}\supseteq
U_{n-1}\supseteq\dots\supseteq U_{0}=0$ such that $U_{i}$ is normal in
$U_{i+1}$, with each successive quotient is isomorphic a vector bundle
$\mathbb{V}({\mathcal{E}})$, where ${\mathcal{E}}$ is a finite locally free
${\mathcal{O}}_{S}$-module.
###### Example 3.11.
The $S$-subgroup scheme of unipotent matricies $\mathbb{U}_{n,S}$ in
$\operatorname{GL}_{n,S}$ is split unipotent. More generally, let $G$ be a
reductive $S$-group scheme and $P$ be a parabolic in $G$, then the unipotent
radical $R_{u}(P)$ of $P$ is split unipotent (cf. [SGA3, Exp. XXVI Prop.
2.1]).
###### Lemma 3.12.
Let $G$ be a linear algebraic $S$-group scheme. Consider a split exact
sequence of $S$-group schemes
$1\rightarrow U\rightarrow G\rightarrow H\rightarrow 1$
where $U$ is split unipotent. Chose a splitting $\pi\colon H\hookrightarrow
G$. Let $X$ be an $S$-scheme locally of finite type with an $F$-action. Then
the $!$-pullback induces an equivalence
$\pi^{!}\colon\operatorname{DM}(X/G)\xrightarrow{\sim}\operatorname{DM}(X/H).$
###### Proof.
This is analogous to the proof of [RS20, Prop. 2.2.11] but for completion, we
give a proof.
The morphism $\pi^{!}$ induces a morphism
$\pi_{n}^{!}\colon\operatorname{DM}(\textup{Bar}^{n}(X,G))\rightarrow\operatorname{DM}(\textup{Bar}^{n}(X,H))$.
Using [BN19, Lem. B.6], it suffices to show that
$\pi^{!}\colon\operatorname{DM}(G)\rightarrow\operatorname{DM}(H)$ is fully
faithful. By assumption $\pi$ is a $U$-torsor and using étale descent, we may
assume that $\pi$ is given by the trivial $U$-torsor, i.e. $\pi\colon U\times
H\rightarrow H$ is given by the projection. Replacing $H$ by $S$, it suffices
to show that
$\pi^{!}\colon\operatorname{DM}(S)\rightarrow\operatorname{DM}(U)$ is fully
faithful. As $U$ is split unipotent, it has a filtration by subgroup $U_{i}$
with successive quotients isomorphic to a vector bundle. Using the same
argumentation as above, we may assume that $U$ is a vector bundle, in which
case the assertion is clear by homotopy invariance. ∎
Let $G$ be a split reductive $S$-group scheme and let $B$ be a Borel contain
$T$ inside $G$. Let $X$ be an $S$-scheme locally of finite type with
$B$-action. The above lemma shows that
$\operatorname{DM}(\left[X/T\right])\simeq\operatorname{DM}(\left[X/B\right])$.
We also have that $G/T\rightarrow G/N$ is a $W$-torsor. The results on torsors
under finite groups combined with the above lemma then yields that
$R\Gamma_{S}(G/N,\mathbb{Q})\simeq R\Gamma_{S}(G/B,\mathbb{Q})^{W}$. The next
result will use this fact.
###### Proposition 3.13.
Let $G$ be a reductive $S$-group scheme with maximal torus $T$. Let $N$ denote
the normalizer of $T$ in $G$. Further, let $X$ be an $S$-scheme locally of
finite type with $G$-action and
$f\colon\left[X/N\right]\rightarrow\left[X/G\right]$ the canonical map. Then
the unit $1_{\left[X/G\right]}\rightarrow f_{*}f^{*}1_{\left[X/G\right]}$ is
an equivalence. In particular, $f_{*}1_{\left[X/N\right]}$ is a Tate motive in
$\operatorname{DM}(\left[X/G\right])$.
Let us summarize the idea of the proof. We will show that the unit
$1_{\left[X/G\right]}\rightarrow f_{*}f^{*}1_{\left[X/G\right]}$ is an
equivalence. To see this we will use that we have the following pullback
diagram
${G/N\times_{S}X}$${X}$${\left[X/N\right]}$${\left[X/G\right]}$$\scriptstyle{\operatorname{pr}}$
(cf. [Sta22, 04Y4]). In particular, after étale-descent, we may assume that
$f$ is given by the projection $G/N\rightarrow S$. Again, by étale descent, we
may assume that $G$ is split with respect to $T$. Let $B$ be a Borel of $G$
containing $T$. Using our calculations about torsors under finite groups, it
is enough to show that the induced map $1_{S}\rightarrow
R\Gamma_{S}(G/T,\mathbb{Q})^{W}$ is an equivalence. But as
$R\Gamma_{S}(G/T,\mathbb{Q})^{W}\simeq R\Gamma_{S}(G/B,\mathbb{Q})^{W}$ will
reduce this question to a classical question about Chow rings, at least when
$S$ is the spectrum of a field, namely if the pullback map
$\textup{CH}_{n}(S,m)_{\mathbb{Q}}\rightarrow\textup{CH}_{n}(G/B,m)_{\mathbb{Q}}^{W}$
is an isomorphism for all $n,m\in\mathbb{Z}$. But as we can identify
$\textup{CH}_{n}(G/B,m)_{\mathbb{Q}}^{W}$ with
$\textup{CH}_{n}(G/G,m)_{\mathbb{Q}}$, this is clear (cf. [Kri13, Cor. 8.7]).
In the case when $S$ is not the spectrum of a field, we have to use $K$-theory
and as the flag variety admits an integral model, we can reduce to
$S=\operatorname{Spec}(\mathbb{Z})$. Up to $K_{1}$ the rational $K$-theory of
$\mathbb{Z}$ and $\mathbb{Q}$ are isomorphic. Using that the flag variety
admits a stratification by affine spaces, we may assume the
$S=\operatorname{Spec}(\mathbb{Q})$, as the only part we have to worry about,
$K_{1}(\mathbb{Z})_{\mathbb{Q}}$, vanishes.
###### Proof of Proposition 3.13.
Throughout this proof, we will denote for readability the structure map of an
Artin stack ${\mathcal{X}}$ to $S$ with $p_{{\mathcal{X}}}$.
To show that the unit
$1_{\left[X/G\right]}\rightarrow f_{*}f^{*}1_{\left[X/G\right]}$
is an equivalence, we may assume by étale descent that the map $f$ is given by
the structure map $G/N\rightarrow S$. Again, by étale descent, we may assume
that $G$ is split with respect to $T$. Then it is enough to show that the unit
$1_{S}\rightarrow R\Gamma_{S}(G/N,\mathbb{Q})$ is an equivalence, since the
pullback of this equivalence along $p_{X}$ yields the desired equivalence.
Note that the natural map $g\colon G/T\rightarrow G/N$ is naturally a
$W$-torsor. Hence, by Lemma 3.8, we see that $1_{G/N}\simeq(g_{*}1_{G/T})^{W}$
and thus $R\Gamma_{S}(G/N,\mathbb{Q})\simeq p_{G/N*}(g_{*}1_{G/T})^{W}$. Since
$p_{G/N*}$ is a right adjoint it commutes with limits. Thus, we have
$p_{G/N*}(g_{*}1_{G/T})^{W}\simeq(p_{G/N*}g_{*}1_{G/T})^{W}\simeq(p_{G/T*}1_{G/T})^{W}\simeq
R\Gamma_{S}(G/T,\mathbb{Q})^{W}$
By Lemma 3.12, we see that $R\Gamma_{S}(G/T,\mathbb{Q})\simeq
R\Gamma_{S}(G/B,\mathbb{Q})$. By Example 2.12, we have that
$R\Gamma_{S}(G/B,\mathbb{Q})\simeq\bigoplus_{w\in W}1_{S}\langle
l(w)-n\rangle,$
where $n$ is the relative dimension of the flag variety $G/B$. In particular,
$R\Gamma_{S}(G/B,\mathbb{Q})$ and thus $R\Gamma_{S}(G/T,\mathbb{Q})$ is Tate.
As the $W$-invariants are defined as an image of a map, we see that
$R\Gamma_{S}(G/T,\mathbb{Q})^{W}$ is also Tate. The $\infty$-category of Tate
motives over $S$ is the stable subcategory of $\operatorname{DM}(S)$ generated
by $1(r)$, for $r\in\mathbb{Z}$. Therefore, the natural map $1_{S}\rightarrow
R\Gamma_{S}(G/T,\mathbb{Q})^{W}$ is an equivalence if and only if the induced
map
(3.1)
$\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(1_{S}(r),1_{S})\rightarrow\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(1_{S}(r),R\Gamma_{S}(G/T,\mathbb{Q})^{W})$
is an equivalence for all $r\in\mathbb{Z}$ (here
$\underline{\operatorname{Hom}}$ denotes the inner-$\operatorname{Hom}$). If
$r>0$, then the $m$-th homotopy group of the left hand side is isomorphic to
$K_{-2r+m}(S)^{(-r)}$ and thus vanishes as the negative Adams eigenspaces
vanish per definition (cf. [Sou85]). By Remark 3.7 and the computation of
$p_{G/B*}1_{G/B}$, we see that
$\pi_{m}\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(1_{S}(r),R\Gamma_{S}(G/T,\mathbb{Q})^{W})\cong(\bigoplus_{w\in
W}K_{-2r+m}(S)^{(l(w)-n-r)})^{W},$
which also vanishes for $r>0$ as $l(w)-n\leq 0$. Therefore, we may assume from
now on that $r\leq 0$. Before continuing with the proof, let us first assume
that $S=\operatorname{Spec}(k)$ is the spectrum of a field $k$. Then by the
above, we are reduced to show that the restriction map
$A^{r}(S,m-2r)\rightarrow A^{r}(G/B,m-2r)^{W}$
is an isomorphism (by homotopy invariance and identification of motivic
cohomology with higher Chow-theory). But as $A^{r}(G/B,m-2r)^{W}\cong
A^{r}(G/G,m-2r)=A^{r}(S,m-2r)$ (cf. [Kri13, Cor. 8.7]), the restriction above
yields the identity on $A^{r}(S,m-2r)$ which is trivially an isomorphism.
Hence, we are done in the case of $S=\operatorname{Spec}(k)$.
Next, let us show how to reduce to the case of $S=\operatorname{Spec}(k)$. As
$r\leq 0$, we may use commutativity of $\underline{\operatorname{Hom}}$ with
limits, to see that (3.1) is an equivalence if and only if
(3.2) $\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(1_{S}\langle
r\rangle,1_{S})\rightarrow\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(1_{S}\langle
r\rangle,R\Gamma_{S}(G/T,\mathbb{Q})^{W})$
is an equivalence. Further, since $S$ is finite dimensional (cf. [GW10, Prop.
14.109]), it is a fact that for every $n\in\mathbb{Z}$, the groups
$K_{n}(S)^{(i)}$ vanish for all but finitely many $i\in\mathbb{Z}$ (cf.
[Sou85, §2]). Therefore, the morphism (3.2) is an equivalence for all $r\leq
0$ if and only if
$\bigoplus_{r\in\mathbb{Z}}\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(1_{S}\langle
r\rangle,1_{S})\rightarrow\bigoplus_{r\in\mathbb{Z}}\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(1_{S}\langle
r\rangle,R\Gamma_{S}(G/T,\mathbb{Q})^{W})$
is an isomorphism. Equivalently, we can write this morphism as
$\bigoplus_{r\in\mathbb{Z}}\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(1_{S},1_{S}\langle
r\rangle)\rightarrow\bigoplus_{r\in\mathbb{Z}}\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(M_{S}(G/T),1_{S}\langle
r\rangle)^{W}$
The motive $M_{S}(G/T)$ is a direct sum of shifts and twists of the unit of
$S$ and therefore compact (cf. [CD19, Thm. 11.1.13]). By compactness of
$1_{S}$ and $M_{S}(G/T)$, the above morphism is an equivalence if and only if
the morphism
$\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(1_{S},\bigoplus_{r\in\mathbb{Z}}1_{S}\langle
r\rangle)\rightarrow\underline{\operatorname{Hom}}_{\operatorname{DM}(S)}(M_{S}(G/T),\bigoplus_{r\in\mathbb{Z}}1_{S}\langle
r\rangle)^{W}$
is an equivalence. By construction of the rational K-theory spectrum in
$\operatorname{DM}(S)$, denoted by $\textup{KGL}_{S,\mathbb{Q}}$, we see that
$\bigoplus_{r\in\mathbb{Z}}1_{S}\langle
r\rangle\simeq\textup{KGL}_{S,\mathbb{Q}}$ (cf. [CD19, Lem. 14.1.4]). Thus, as
$G/T$ is representable by a scheme (cf. [SGA3, Exp. IX Thm. 5.1]), the
$n$-homotopy group of the right hand side is isomorphic to
$K_{n}(G/T)_{\mathbb{Q}}^{W}$, the rational higher $K$-theory of $G/T$.
Further, the properties of the $K$-theory spectrum yield that the induced
morphism $K_{n}(S)_{\mathbb{Q}}\rightarrow K_{n}(G/T)_{\mathbb{Q}}$ is given
by the pullback along $G/T\rightarrow S$ (cf. [CD19, §13.1]). Taking
$W$-invariants yields the map
(3.3) $K_{n}(S)_{\mathbb{Q}}\rightarrow K_{n}(G/T)_{\mathbb{Q}}^{W}$
on the $n$-th homotopy groups. We have to show that this map is an isomorphism
for all $n\geq 0$. In the following we will freely identify
$K_{n}(G/T)_{\mathbb{Q}}$ with $K_{n}(G/B)_{\mathbb{Q}}$ and $K$-theory with
$G$-theory as we are only working with smooth schemes. As $G$ is split
reductive, there exists a model
$(G_{\mathbb{Z}},B_{\mathbb{Z}},T_{\mathbb{Z}})$ over
$\operatorname{Spec}(\mathbb{Z})$ such that $G/B$ is the pullback of the
associated flag variety $G_{\mathbb{Z}}/B_{\mathbb{Z}}$. As $G/B$ (resp.
$G_{\mathbb{Z}}/B_{\mathbb{Z}}$) is stratified by affine spaces, we have an
equivalence of $K$-theory spectra $K(G/B)_{\mathbb{Q}}\simeq
K(G_{\mathbb{Z}}/B_{\mathbb{Z}})_{\mathbb{Q}}\otimes_{K(\mathbb{Z})_{\mathbb{Q}}}K(S)_{\mathbb{Q}}$
(cf. [Jos01, Cor. 4.3]), here $K(-)_{\mathbb{Q}}$ denotes the underlying
$E_{\infty}$-ring spectrum in $\operatorname{DM}(\mathbb{Z})$, in particular
$K(\mathbb{Z})_{\mathbb{Q}}=\textup{KGL}_{\mathbb{Z},\mathbb{Q}}$ and
$K(S)_{\mathbb{Q}}$ (resp. $K(G/B)_{\mathbb{Q}}$) is just the $*$-pushforward
of $\textup{KGL}_{S,\mathbb{Q}}$ (resp. $\textup{KGL}_{G/B,\mathbb{Q}}$) to
$\operatorname{DM}(\mathbb{Z})$. We will show that the induced morphism
$K(S)_{\mathbb{Q}}\rightarrow K(G/B)_{\mathbb{Q}}^{W}$
is an equivalence if and only if
$K(\mathbb{Z})_{\mathbb{Q}}\rightarrow
K(G_{\mathbb{Z}}/B_{\mathbb{Z}})_{\mathbb{Q}}^{W}.$
This follows from the following equivalence,
$(K(G_{\mathbb{Z}}/B_{\mathbb{Z}})_{\mathbb{Q}}\otimes_{K(\mathbb{Z})_{\mathbb{Q}}}K(S)_{\mathbb{Q}})^{W}\simeq
K(G_{\mathbb{Z}}/B_{\mathbb{Z}})_{\mathbb{Q}}^{W}\otimes_{K(\mathbb{Z})_{\mathbb{Q}}}K(S)_{\mathbb{Q}},$
which we will show holds true. This is shown similarly to the analogous facts
on representations of finite groups, but we will give an argument.
For a scheme $X$ let $\textup{DK}(X)$ of $X$ denote the $\infty$-category of
$KGL_{X,\mathbb{Q}}$-modules in $\operatorname{DM}(X)$. This can also be
extended to stacks by étale descent.888Note that the étale localized
$K$-theory spectrum of stacks does not agree with the genuine rational
$K$-theory spectrum. As we will only need the existence of a six functor
formalism, we will just ignore the technicalities. By abuse of notation will
denote the tensor unit of $DK(X)$ with $1_{X}$ and just for the following will
even drop the $X$ in the notation, as it is clear in which categories the
units are. Also, we will again denote the inner-$\operatorname{Hom}$ by
$\underline{\operatorname{Hom}}$. Let us fix the following diagram with
pullback squares
${G_{\mathbb{Z}}/B_{\mathbb{Z}}}$${\mathbb{Z}}$${S}$${\left[W\backslash
G_{\mathbb{Z}}/B_{\mathbb{Z}}\right]}$${\left[\mathbb{Z}/W\right]}$${\left[S/W\right]}$${\mathbb{Z}}$${S,}$$\scriptstyle{g}$$\scriptstyle{b}$$\scriptstyle{f}$$\scriptstyle{a^{\prime}}$$\scriptstyle{a}$
here $W$ acts on $\mathbb{Z}$ resp. $S$ trivially and $a,b$ are the natural
projection. Note that by Lemma 3.8, we have that
${g_{*}g^{*}}^{W}\simeq\operatorname{id}$. Using this, smooth base change,
projection formula999Note that $f$ is proper, as $W$ is a finite constant
group scheme. and adjunctions, we get the following chain of equivalences
$\displaystyle(K(G_{\mathbb{Z}}/B_{\mathbb{Z}})_{\mathbb{Q}}\otimes_{K(\mathbb{Z})_{\mathbb{Q}}}K(S)_{\mathbb{Q}})^{W}$
$\displaystyle\simeq\underline{\operatorname{Hom}}_{\textup{DK}(\left[W/\mathbb{Z}\right])}(1,g_{*}g^{*}(b_{*}1\otimes
a^{\prime}_{*}1))^{W}$
$\displaystyle\simeq\underline{\operatorname{Hom}}_{\textup{DK}(\left[W/\mathbb{Z}\right])}(1,b_{*}1\otimes
a^{\prime}_{*}1)$
$\displaystyle\simeq\underline{\operatorname{Hom}}_{\textup{DK}(\mathbb{Z})}(1,f_{*}(b_{*}1\otimes
a^{\prime}_{*}1))$ $\displaystyle\simeq
K(G_{\mathbb{Z}}/B_{\mathbb{Z}})_{\mathbb{Q}}^{W}\otimes_{K(\mathbb{Z})_{\mathbb{Q}}}K(S)_{\mathbb{Q}}.$
Note that we also have to use that
$K(G_{\mathbb{Z}}/B_{\mathbb{Z}})_{\mathbb{Q}}$ is a direct sum of the unit,
to see that the $\underline{\operatorname{Hom}}$ agrees with the tensor
product of spectra (see below for an explicit description).
Thus, as the structure map $G/B\rightarrow S$ is induced via pullback, we see
that $K(S)_{\mathbb{Q}}\rightarrow K(G/B)^{W}_{\mathbb{Q}}$ is induced via
$K(\mathbb{Z})_{\mathbb{Q}}\otimes_{K(\mathbb{Z})_{\mathbb{Q}}}K(S)_{\mathbb{Q}}\xrightarrow{\alpha^{W}\otimes\operatorname{id}_{S}}K(G_{\mathbb{Z}}/B_{\mathbb{Z}})^{W}_{\mathbb{Q}}\otimes_{K(\mathbb{Z})_{\mathbb{Q}}}K(S)_{\mathbb{Q}},$
where $\alpha\colon
G_{\mathbb{Z}}/B_{\mathbb{Z}}\rightarrow\operatorname{Spec}(\mathbb{Z})$
denotes the structure map. In particular, we may assume that
$S=\operatorname{Spec}(\mathbb{Z})$.
As $G/B$ admits an affine cell decomposition, we see that
$K(G/B)_{\mathbb{Q}}\simeq\bigoplus_{w\in
W}K(\mathbb{Z})_{\mathbb{Q}},\textup{ and
}K(G_{\mathbb{Q}}/B_{\mathbb{Q}})_{\mathbb{Q}}\simeq\bigoplus_{w\in
W}K(\mathbb{Q})_{\mathbb{Q}},$
where $G_{\mathbb{Q}}/B_{\mathbb{Q}}$ denotes the generic fiber of $G/B$ (cf.
[Yay23]). It is well known that for any $n\neq 1$ there is an isomorphism
$K_{n}(\mathbb{Z})_{\mathbb{Q}}\cong K_{n}(\mathbb{Q})_{\mathbb{Q}}$ and
$K_{1}(\mathbb{Z})_{\mathbb{Q}}=0$. Thus, it is clear that
$K_{1}(\mathbb{Z})_{\mathbb{Q}}\rightarrow K_{1}(G/B)_{\mathbb{Q}}^{W}$ is an
isomorphism. For the other $K$-groups, we can use the identification of
$K_{n}(\mathbb{Z})_{\mathbb{Q}}$ with $K_{n}(\mathbb{Q})_{\mathbb{Q}}$ and may
assume that $S=\operatorname{Spec}(\mathbb{Q})$. This concludes the proof via
the argument in the Chow group case. ∎
Combining Proposition 3.13 and the results of Section 3.2 now yields the main
result of this section.
###### Theorem 3.14.
Let $G$ be a reductive $S$-group scheme that admits a split maximal torus $T$
with Weyl group $W$. Assume $G$ acts on an $S$-scheme $X$ locally of finite
type. Then the natural map
$\left[X/T\right]\rightarrow\left[X/G\right]$
is Tate.
Further, if $S$ is connected, we have
$R\Gamma_{S}(\left[X/G\right],\mathbb{Q})\simeq
R\Gamma_{S}(\left[X/T\right],\mathbb{Q})^{W}.$
###### Proof.
Let $N$ be the normalizer of $T$ in $G$. We can factor the map in the theorem
as
$\left[X/T\right]\xrightarrow{f}\left[X/N\right]\xrightarrow{g}\left[X/G\right]$.
The first part of the theorem follows immediately with Lemma 3.5 and
Proposition 3.13.
For the proof of the second claim, we apply Lemma 3.8 and Proposition 3.13 and
get
$\displaystyle(p_{\left[X/T\right]*}1_{\left[X/T\right]})^{W}\simeq(p_{\left[X/G\right]*}g_{*}f_{*}1_{\left[X/T\right]})^{W}$
$\displaystyle\simeq
p_{\left[X/G\right]*}g_{*}(f_{*}1_{\left[X/T\right]})^{W}$
$\displaystyle\simeq p_{\left[X/G\right]}g_{*}1_{\left[X/N\right]}\simeq
p_{\left[X/G\right]*}1_{\left[X/G\right]}.$
∎
Let $S=\operatorname{Spec}(k)$ be the spectrum of a field and $G$ a split
reductive group over $k$ with split maximal torus $T$ and Weyl group $W$.
Edidin and Graham have shown in [EG98] that for any smooth scheme $X$ with
$G$-action, we have $A^{\bullet}_{G}(X)\cong A^{\bullet}_{T}(X)^{W}$. Later on
Krishna has shown in [Kri13, Cor. 8.7] that this result can be generalized to
higher Chow groups, at least when $X$ is quasi-projective. Our Theorem 3.14 is
the motivic analog of both of these results. In fact, we will see in the next
corollary, that applying Theorem 3.14 to motivic cohomology recovers both
Edidin-Graham’s and Krishna’s result. Moreover, as any reductive group over a
field splits after passage to a finite Galois extension, we can use the
results of Section 3.2 to extend this comparison of (higher) equivariant Chow
groups to arbitrary reductive groups over fields. We also note that Levine
showed that Bloch’s higher Chow groups also satisfy a localization sequence,
even in the non quasi-projective case. The following corollary summarizes this
discussion.
###### Corollary 3.15.
Assume that $S=\operatorname{Spec}(k)$ is the spectrum of a field. Let $G$ be
a reductive $S$-group scheme with maximal torus $T$. Let $W$ denote the
absolute Weyl101010Let $\bar{k}$ be an algebraic closure of $k$, then
${\mathcal{W}}\coloneqq N_{G}(T)/T$ is a finite étale group scheme of $S$ and
we set the absolute Weyl group to be $W\coloneqq{\mathcal{W}}(\bar{k})$ group
of $T$ in $G$. Assume $G$ acts on a smooth finite type $S$-scheme $X$. Then we
have
$R\Gamma_{S}(\left[X/G\right],\mathbb{Q})\simeq
R\Gamma_{S}(\left[X/T\right],\mathbb{Q})^{W}.$
In particular, applying this result to motivic cohomology yields for all
$n,m\in\mathbb{Z}$
$A^{n}_{G}(X,m)_{\mathbb{Q}}\cong A^{n}_{T}(X,m)_{\mathbb{Q}}^{W}.$
###### Proof.
Any reductive group over $k$ becomes split after passing to a finite Galois
extension $K/k$ (cf. [SGA3, Exp. XXII Cor. 2.4]). Thus, let $K$ be such an
extension, so that $T_{K}$ is a split maximal torus. Then we have the
following diagram with pullback squares
${\left[X_{K}/T_{K}\right]}$${\left[X_{K}/G_{K}\right]}$${\operatorname{Spec}(K)}$${\left[X/T\right]}$${\left[X/G\right]}$${\operatorname{Spec}(k).}$$\scriptstyle{f}$$\scriptstyle{p_{T_{K}}}$$\scriptstyle{g}$$\scriptstyle{p_{G_{K}}}$$\scriptstyle{p_{K}}$$\scriptstyle{p_{T}}$$\scriptstyle{p_{G}}$
As $K/k$ is a finite Galois extension, the morphisms $p_{K}$ and thus also $f$
are $\Gamma\coloneqq\operatorname{Gal}(K/k)$-torsors. Then Lemma 3.8 yields
the equivalence
$p_{G*}1_{\left[X/G\right]}\simeq(p_{K*}p_{K}^{*}p_{G*}1_{\left[X/G\right]})^{\Gamma}.$
As the diagram above has cartesian squares, we can use smooth base change, to
see that $p_{K}^{*}p_{G*}\simeq p_{G_{K}*}g^{*}$. But by Theorem 3.14, we have
$p_{G_{K}*}g^{*}1_{\left[X/G\right]}\simeq
R\Gamma_{K}(\left[X_{K}/G_{K}\right],\mathbb{Q})\simeq
R\Gamma_{K}(\left[X_{K}/T_{K}\right],\mathbb{Q})^{W}.$
Commutativity of the above diagram yields
$p_{K*}R\Gamma_{K}(\left[X_{K}/T_{K}\right],\mathbb{Q})\simeq
p_{T*}f_{*}1_{\left[X_{K}/T_{K}\right]}.$ Thus, we have
$p_{G*}1_{\left[X/G\right]}\simeq((p_{T*}f_{*}1_{\left[X_{K}/T_{K}\right]})^{W})^{\Gamma}.$
As limits commute with limits, we may write the right hand side as
$(p_{T*}(f_{*}f^{*}1_{\left[X/T\right]})^{\Gamma})^{W}$ and again by Lemma
3.8, we have $(f_{*}f^{*}1_{\left[X/T\right]})^{\Gamma}\simeq
1_{\left[X/T\right]}$ concluding the proof.
The result about motivic cohomology follows from Remark 3.7 by using [RS20,
Thm. 2.2.10] which proves that motivic cohomology for smooth quotient stacks
is computed by the higher equivariant Chow groups of Edidin-Graham.111111In
loc.cit. they assume some properties on the groups and on $X$, as Edidin-
Graham need these assumptions to compare higher Chow theory of stacks and
equivariant higher Chow theory [EG98, Prop. 13 (b)]. The assumptions in
loc.cit. are needed as Bloch only shows the existence of a long exact sequence
for higher Chow groups in the case where $X$ is quasi-projective. This result
was extended by Levine to all finite type $k$-schemes (cf. [Lev01]). Thus, the
comparison of Edidin-Graham and hence also of Richarz-Scholbach go through in
the case of the corollary. ∎
We can apply the same reasoning as above to see in the next corollary that the
map $\left[X/T\right]\rightarrow\left[X/G\right]$ is Tate even if $T$ is not
split. The main point is that groups of multiplicative type over locally
noetherian normal schemes are isotrivial, i.e. become diagonalizable after
passing to a finite étale cover (cf. [SGA3, Exp. X Thm. 5.16]). In particular,
after passing to a Galois cover of our base, we can reduce to Theorem 3.14
using the results of Section 3.2.
###### Corollary 3.16.
Let $S$ be a connected scheme. Let $G$ be a reductive $S$-group scheme that
admits a maximal torus $T$. Assume $G$ acts on an $S$-scheme $X$ locally of
finite type. Then the natural map
$\left[X/T\right]\rightarrow\left[X/G\right]$
is Tate.
###### Proof.
As any étale cover of a connected scheme can be refined by a Galois cover (cf.
[Sta22, 03SF]), there exists a Galois cover $\tilde{S}\rightarrow S$ such that
$\tilde{T}\coloneqq T\times_{S}\tilde{S}$ splits (cf. [SGA3, Exp. X Thm.
5.16]). Let us denote the base change of $X$ under $\tilde{S}\rightarrow S$
with $\tilde{X}$. We get an induced action of $\tilde{G}$ on $\tilde{X}$. In
particular, we get a pullback square diagram
${\left[\tilde{X}/\tilde{T}\right]}$${\left[\tilde{X}/\tilde{G}\right]}$${\left[X/T\right]}$${\left[X/G\right].}$$\scriptstyle{\tilde{f}}$$\scriptstyle{h}$$\scriptstyle{g}$$\scriptstyle{f}$
Let $\Gamma$ denote the finite étale $S$-automorphism group of $\tilde{S}$.
Then $g$ and $h$ are $\Gamma$-torsors. Thus, we can compute
$f_{*}1_{\left[X/T\right]}\simeq(f_{*}h_{*}1_{\left[\tilde{X}/\tilde{T}\right]})^{\Gamma}\simeq(g_{*}\tilde{f}_{*}1_{\left[\tilde{X}/\tilde{T}\right]})^{\Gamma}.$
The morphism $\tilde{f}$ is Tate by Theorem 3.14 and $g$ is also Tate by Lemma
3.5. Further, taking invariants under finite group actions is given by taking
the image of the fixed point operator (cf. Remark 3.7), i.e. by extensions,
and thus preserves Tate motives. In particular, $f_{*}1_{\left[X/T\right]}$ is
a Tate motive. ∎
## 4 Torsors under tori
Let us fix a torus $T$ over $S$ (cf. [SGA3, Exp. IX]).
In this subsection, we want to understand the motivic homotopy theory of
torsors under $T$. More precisely, let us consider the following situation.
Let $X\rightarrow Y$ be a $T$-torsors of Artin $S$-stacks locally of finite
type. We want to understand the relation between $M_{S}(X)$ and $M_{S}(Y)$.
Let us recall the classical case of Chow theory. For this paragraph let us
assume that $X$ and $Y$ are smooth Artin stacks over $\operatorname{Spec}(k)$,
where $k$ is a field. Then the group $\hat{T}$ of characters of $T$ acts on
$A^{\bullet}(Y)$ in the following way. Let $\chi\in\hat{T}$ be a character and
consider its associated $1$-dimensional representation $\kappa(\chi)$. The
quotient $L_{\chi}\coloneqq X\times^{T}\kappa(\chi)$ is representable by a
line bundle over $Y$. Multiplication with the first Chern class of $L_{\chi}$
yields an action of $\hat{T}$ on $A^{\bullet}(Y)$. If $T$ is split, then for
Chow rings it is known that
$A^{\bullet}(X)\cong A^{\bullet}(Y)/\hat{T}A^{\bullet}(Y).$
Our goal is to extend this result to motivic homotopy theory, so that it
generalizes the result about Chow theory and yields a similar statement for
$K$-theory. Even though we work with Beilinson motives, we will remark in the
end how to extend this to integral $K$-theory under some assumptions on $X$
and $Y$.
### 4.1 The motive of $T$-torsors
Let us denote the character group of $T$ with $\hat{T}$. Let $X\rightarrow Y$
be a $T$-torsor of Artin stacks locally of finite type over $S$ and
$\chi\in\hat{T}$ a character. Let $\mathbb{G}_{\textup{m},S}$ act via left
multiplication in $\mathbb{A}^{1}_{S}$. Then $T$ acts via $\chi$ on
$\mathbb{A}^{1}_{S}$ and thus, $X\times^{T}\mathbb{A}^{1}_{S}\rightarrow
X/T\cong Y$ yields a line bundle over $Y$. The action of the first Chern class
of $X\times^{T}\mathbb{A}^{1}_{S}$ on the motivic cohomology will be described
by a Gysin sequence (cf. Proposition 4.2).
###### Notation 4.1.
Assume that $T$ is split. In the following we want to split up the $T$-torsor
$f\colon X\rightarrow Y$ into a sequence of
$\mathbb{G}_{\textup{m},S}$-torsors. Note that by splitness
$T\cong\mathbb{G}_{\textup{m},S}^{r}$ for some $r\in\mathbb{N}$. Fixing a
numbering of the $\mathbb{G}_{\textup{m},S}$-components of $T$, we can embed
for any $1\leq k\leq r$ the product $\mathbb{G}_{\textup{m},S}^{k}$ into $T$
by $\operatorname{id}_{\mathbb{G}_{\textup{m},S}^{k}}\times 1^{r-k}$. Then
$\mathbb{G}_{\textup{m},S}^{k}$ acts on $X$ via this embedding. We get a
sequence
$X\rightarrow X/\mathbb{G}_{\textup{m},S}\rightarrow
X/\mathbb{G}_{\textup{m},S}^{2}\rightarrow\dots\rightarrow X/T\cong Y$
of $\mathbb{G}_{\textup{m},S}$-torsors. We denote the induced maps
$X/\mathbb{G}_{\textup{m},S}^{i}\rightarrow Y$ with $f_{i}$.
###### Proposition 4.2.
Assume that $T\cong\mathbb{G}_{\textup{m},S}^{r}$ for some $r\in\mathbb{N}$.
Let $X\rightarrow Y$ be a $T$-torsor of smooth Artin stacks over $S$. Then,
there exists a filtration
$M_{Y}(X)=M_{0}\rightarrow M_{1}\rightarrow\dots\rightarrow M_{r}=1_{Y}$
in $\operatorname{DM}(S)$, where $M_{i}\coloneqq
M_{Y}(X/\mathbb{G}_{\textup{m},S}^{i})$ such that the cofiber of
$M_{Y}(X_{i-1})\rightarrow M_{Y}(X_{i})$ is given by $M_{Y}(X_{i})\langle
1\rangle$ and the map $M_{Y}(X_{i})\rightarrow M_{Y}(X_{i})\langle 1\rangle$
is induced by multiplication with
$c_{1}(X_{i-1}\times^{\mathbb{G}_{\textup{m}}}\mathbb{A}^{1}_{S})$.
###### Proof.
This follows by successively using [HPL21, Prop. 2.32]. But for completion let
us give a proof by recalling the argument.
The morphism $X_{i-1}\rightarrow X_{i}$ is a $\mathbb{G}_{m}$-torsor. In
particular, the scheme ${\mathcal{L}}_{i}\coloneqq
X_{i-1}\times_{S}^{\mathbb{G}_{\textup{m},S}}\mathbb{A}^{1}_{S}$ is a line
bundle over $X_{i}$. Let $s\colon X_{i}\hookrightarrow V_{i}$ denote the zero
section. Certainly, the complement of the closed immersion $s$ is isomorphic
to $X_{i-1}$. Then the Gysin sequence of Lemma 2.10 yields a fibre sequence
$M_{Y}(X_{i-1})\rightarrow M_{Y}(V_{i})\simeq
M_{Y}(X_{i})\xrightarrow{\varphi}M_{Y}(X_{i})\langle 1\rangle.$
By construction of the Gysin sequence $\varphi$ is given by multiplication
with $c_{1}({\mathcal{L}}_{i})$. This concludes the proof. ∎
###### Corollary 4.3.
Assume $S$ is connected and let $f\colon X\rightarrow Y$ be a $T$-torsor of
smooth Artin stacks over $S$. Then $f$ is a Tate map.
If $T$ is split, we may drop the connectedness hypothesis on $S$.
###### Proof.
If $T$ is split, then Proposition 4.2 implies that $M_{Y}(X)$ is a successive
extension of $1_{Y}$. Thus, the result follows from Remark 2.13.
In the non-split case, this is completely analogous to the proof of Corollary
3.16. ∎
### 4.2 Motivic cohomology of $T$-torsors
Now we will assume that $T$ is split and fix an $r\in\mathbb{N}$ such that
$T\cong\mathbb{G}_{\textup{m},S}^{r}$.
For any Artin stack $X$ locally of finite type over $S$, we will denote its
motivic cohomology with
$H^{p}(X,\mathbb{Q}(n))\coloneqq\operatorname{Hom}_{\operatorname{DM}(S)}(M_{S}(X),\mathbb{Q}(n)[p]).$
If $X$ is representable by a smooth $S$-scheme, then we have
$H^{p}(X,\mathbb{Q}(n))\cong K_{2n-p}(X)^{(n)}$. If $S=\operatorname{Spec}(k)$
is the spectrum of a field and $X$ is smooth and over $S$, we have
$H^{p}(X,\mathbb{Q}(n))\cong A^{n}(X,2n-p)$. Note that for a smooth Artin
$S$-stack $X$ the motivic cohomology vanishes automatically in certain degrees
by descent and vanishing of negative $K$-theory for regular schemes, i.e. we
have that $H^{p}(X,\mathbb{Q}(n))\cong 0$ for $p>2n$.
###### Notation 4.4.
Consider a $\chi\in\hat{T}$. The associated line bundle $L_{\chi}\coloneqq
X\times_{S}^{T}\mathbb{V}({\mathcal{E}}_{\chi})$ yields a map
$H^{p+2}(Y,\mathbb{Q}(n+1))\rightarrow H^{p}(Y,\mathbb{Q}(n))$, by
multiplication with the Chern class of $L_{\chi}$. We denote the image of this
map with $c_{1}(L_{\chi})H^{p}(Y,\mathbb{Q}(n))$.
###### Proposition 4.5.
In the setting of Proposition 4.2 let us further fix an $n\in\mathbb{Z}$.
Then, we have
$H^{2n}(X,\mathbb{Q}(n))\cong
H^{2n}(Y,\mathbb{Q}(n))/\hat{T}H^{2n}(Y,\mathbb{Q}(n)).$
###### Proof.
First let us note that it is enough121212Any character is generated by
primitive characters and the corresponding $1$-dimensional representation is
given by the associated tensor product. to show that
$H^{2n}(X,\mathbb{Q}(n))=H^{2n}(Y,\mathbb{Q}(n))/\langle
c_{1}(X\times_{S}^{T}\mathbb{V}({\mathcal{E}}_{\chi_{i}}))H^{2n}(Y,\mathbb{Q}(n))\rangle,$
where $\chi_{i}$ is a primitive character in $\hat{T}$. Let
$X=X_{0}\rightarrow X_{1}\rightarrow\dots\rightarrow X_{r}=Y$ be the sequence
of Proposition 4.2. For each $0\leq i\leq r$ this yields a long exact sequence
on motivic cohomology
$\displaystyle\dots\rightarrow H^{2n+2}(X_{i},\mathbb{Q}(n+1))$
$\displaystyle\rightarrow H^{2n}(X_{i},\mathbb{Q}(n))$
$\displaystyle\rightarrow H^{2n}(X_{i-1},\mathbb{Q}(n))\rightarrow
H^{2n+3}(X_{i},\mathbb{Q}(n+1))\rightarrow\dots.$
We have $H^{2n+3}(X_{i},\mathbb{Q}(n+1))=0$ and thus get an exact sequence of
the form
$H^{2n+2}(X_{i},\mathbb{Q}(n+1))\xrightarrow{a}H^{2n}(X_{i},\mathbb{Q}(n))\xrightarrow{b}H^{2n}(X_{i-1},\mathbb{Q}(n))\rightarrow
0.$
The map $b$ is the usual pullback on motivic cohomology. The map $a$ is
induced by multiplication with the Chern class of the line bundle
${\mathcal{L}}_{i}=X_{i-1}\times^{\mathbb{G}_{\textup{m}}}_{S}V({\mathcal{E}}_{\chi_{i}})$.
As $X_{r}=Y$, we have $H^{2n}(X_{r-1},\mathbb{Q}(n))\cong
H^{2n}(Y,\mathbb{Q}(n))/c_{1}({\mathcal{L}}_{r})H^{2n}(Y,\mathbb{Q}(n))$.
Hence, inductively we see that
$H^{2n}(X,\mathbb{Q}(n))\cong H^{2n}(Y,\mathbb{Q}(n))/\langle
c_{1}({\mathcal{L}}_{i})H^{2n}(Y,\mathbb{Q}(n))\rangle_{1\leq i\leq r}.$
We are left to show that
$c_{1}({\mathcal{L}}_{i})H^{2n}(Y,\mathbb{Q}(n))=c_{1}(X\times_{S}^{T}V({\mathcal{E}}_{i}))H^{2n}(X,\mathbb{Q}(n))$.
For this let us start with $i=r$. Then by construction
$X\times_{S}^{T}V({\mathcal{E}}_{\chi_{r}})\cong
X_{r-1}\times_{S}^{\mathbb{G}_{\textup{m}}}V({\mathcal{E}}_{\chi_{r}})$.
Inductively, we may replace $Y$ by $X_{i}$, where the claim again follows by
construction.
∎
###### Corollary 4.6.
Let $S=\operatorname{Spec}(k)$ be the spectrum of a field and Let
$X\rightarrow Y$ be a $T$-torsor of smooth Artin stacks of over $S$. Then
$A^{\bullet}(X)_{\mathbb{Q}}\cong
A^{\bullet}(Y)_{\mathbb{Q}}/\hat{T}A^{\bullet}(Y)_{\mathbb{Q}}.$
###### Proof.
This follows immediately from Proposition 4.5. ∎
###### Remark 4.7.
Proposition 4.2 and Proposition 4.5 can be extended to other cohomology
theories in the following way. Let us fix a $T$-torsor $X\rightarrow Y$ of
smooth Artin $S$-stacks.
1. (1)
(Rational étale localized cohomology theories) Let us fix a $T$-torsor
$X\rightarrow Y$ of smooth Artin $S$-stacks. Let
$M\in\textup{SH}(S)_{\mathbb{Q},\operatorname{\acute{e}t}}$ be an oriented
$E_{\infty}$-ring spectrum and let us denote its pullback to any smooth Artin
$S$-stack $Z$ with $M_{Z}$. The orientation of $M$ yields a Chern class map
$c_{1}\colon\operatorname{Pic}(Z)\rightarrow
H^{2}(Z,M(1))\coloneqq\operatorname{Hom}_{\textup{SH}(Z)}(1_{Z},M_{Z}(1)[2]).$
In the same fashion as before, for any character $\chi\in\hat{T}$, we can
define $c_{1}(L_{\chi})H^{n}(Y,M)$.
Assume there exists a $p\in\mathbb{Z}$ such that $M$ satisfies
$H^{m}(Z,M)\coloneqq\operatorname{Hom}_{\textup{SH}(Z)}(1_{Z},M_{Z}[m])=0$
for all $m>p$. Then
$H^{p}(X,M)\cong H^{p}(Y,M)/\langle
c_{1}(X\times_{S}^{T}\mathbb{V}({\mathcal{E}}_{\chi_{i}}))H^{p}(Y,M)\rangle.$
If we take for example $M=\textup{KGL}_{\mathbb{Q},S}$, the rational
$K$-theory spectrum in $\textup{SH}(S)$, then we get
$K_{0}(X)_{\mathbb{Q}}^{\operatorname{\acute{e}t}}\cong
K_{0}(Y)_{\mathbb{Q}}^{\operatorname{\acute{e}t}}/\langle
c_{1}(X\times_{S}^{T}\mathbb{V}({\mathcal{E}}_{\chi_{i}}))K_{0}(Y)_{\mathbb{Q}}^{\operatorname{\acute{e}t}}\rangle,$
where $K_{0}(-)_{\mathbb{Q}}^{\operatorname{\acute{e}t}}$ denotes the étale
localized rational $K$-theory.
2. (2)
(Integral $K$-theory) The extension to integral $K$-theory is more subtle, as
we have to restrict ourselves to certain algebraic stacks131313These are
called scalloped stacks (cf. [KR21]). to make sense of the stable homotopy
category. For simplicity, we may assume that $X=\left[X^{\prime}/H\right]$ and
$Y=\left[Y^{\prime}/F\right]$ are represented by quotients of quasi-projective
schemes by diagonalizable group schemes. Then there is a well defined notion
of a stable homotopy category $\textup{SH}(X)$ resp. $\textup{SH}(Y)$ together
with a functorial $E_{\infty}$-ring object KGL that represents equivariant
$K$-theory (cf. [KR21]). Further, Bott-periodicity yields an orientation on
KGL. In particular, using that $\textup{KH}(X)$ and $\textup{KH}(Y)$ are
connective141414This holds Nisnevich locally by loc.cit. and by descent for
KH. (cf. [HK19, Thm. 5.7]), we see that Corollary 4.6 holds for integral
$K_{0}$ in this case, i.e.
$K_{0}(X)\cong K_{0}(Y)/\hat{T}K_{0}(Y).$
This result can be glued to the class of so called scalloped stacks (cf.
[KR21] for the notion of scalloped stacks and the construction of SH).
## 5 Motivic cohomology of quotients up to isogeny
We are now ready to answer the main question of this article ($\ast$ ‣ 1). Let
us assume that $S$ is connected and let us fix a reductive $S$-group scheme
$G$ and assume that it admits a maximal torus $T$.
For completeness, we will work in a more general setting than in the beginning
of the introduction.
Let $P$ resp. $Q$ be parabolics inside $G$ with Levi-components $L$ resp. $M$.
Assume that $T$ is contained in $L$. Let $\varphi\colon L\rightarrow M$ be an
isogeny. Then $L$ acts on $G$ via $(l,g)\mapsto lg\varphi(l)^{-1}$. We are
interested in the quotient of this action, which we denote by
$\left[G/_{\varphi}L\right]$ or rather its motive. To do so, we follow the
idea of Brokemper in the proof of [Bro16, Prop. 1.2].
As $\varphi$ is an isogeny the image of $T$ is again a maximal torus. In
particular, up to conjugation by an element $g_{0}\in G(S)$, we may identify
$T$ with $\varphi(T)$. The $g_{0}$-conjugation of $G$ induces an isomorphism
$G\rightarrow G$ that is $L$-equivariant, where $L$ acts on the right hand
side via $(l,g)\mapsto lgg_{0}^{-1}\varphi(l)^{-1}g_{0}$. In particular, after
replacing $M$ resp. $Q$ by their $g_{0}^{-1}$-conjugation, we may assume that
$\varphi(T)=T$. Then we have the following embedding $T\hookrightarrow T\times
T$, via $t\mapsto(t,\varphi(t))$. The quotient under this embedding is
$T\times T/T\cong T$. Thus, the naturally induced morphism
$\left[G/_{\varphi}T\right]\rightarrow\left[G/T\times T\right]$, where
$T\times T$ acts on $G$ via $(t,t^{\prime},g)\mapsto tgt^{\prime-1}$, is a
$T$-torsor. This leaves us with the following picture
${\left[G/_{\varphi}L\right]}$${\left[G/_{\varphi}T\right]}$${\left[G/T\times
T\right].}$$\scriptstyle{a}$$\scriptstyle{b}$
For morphism $a$ we can apply Corollary 3.16 and see that it is Tate. The
morphism $b$ is by the above a $T$-torsor. Therefore, we can use Corollary 4.3
to see that $b$ is also Tate.
If $T$ is split, we can say more. In the split case, we have
$R\Gamma_{S}(\left[G/_{\varphi}L\right],\mathbb{Q})\simeq
R\Gamma_{S}(\left[G/_{\varphi}T\right],\mathbb{Q})^{W_{L}},$ where $W_{L}$
denotes the Weyl group of $T$ in $L$ (cf. Theorem 3.14). Further, we can
compute the motive resp. the motivic cohomology of
$\left[G/_{\varphi}T\right]$ via the motive resp. the motivic cohomology of
$\left[G/T\times T\right]$ using Proposition 4.2 and Proposition 4.5.
If $G$ admits a Borel $B$ containing $T$, then by invariance under extensions
of unipotent groups, we can identify $\operatorname{DM}(\left[G/T\times
T\right])$ with $\operatorname{DM}(\left[T\backslash G/B\right])$ (cf. Lemma
3.12). Therefore, with all of the above we see that the $T$-equivariant
motivic cohomology resp. motive of the flag variety $G/B$ yields results about
the motivic cohomology resp. the motive of $\left[G_{\varphi}/L\right]$. But
the author has shown151515Even though in the cited article, we assume that $S$
is affine, all the arguments go through in $\operatorname{DM}$ without this
assumption. in [Yay23] that the motive of $\left[T\backslash G/B\right]$ is
computed by $M_{S}(G/B)\otimes M_{S}(BT)$.
If further $S=\operatorname{Spec}(k)$, we have seen that
$M_{S}(\textup{B}T)\cong\bigotimes_{i=1}^{r}M_{S}(\textup{B}\mathbb{G}_{\textup{m}})=\bigotimes_{i=1}^{r}\bigoplus_{j\geq
0}1_{k}\langle i\rangle,$
which is completed Tate (cf. Example 2.8). As the motive of the flag variety
$G/B$ is also Tate (cf. Example 2.12), we see that $M_{S}(T\backslash G/B)$ is
completed Tate. Summarizing all of the above yields the following theorem.
###### Theorem 5.1.
We have the following diagram of Tate maps
${\left[G/_{\varphi}L\right]}$${\left[G/_{\varphi}T\right]}$${\left[G/T\times
T\right].}$$\scriptstyle{a}$$\scriptstyle{b}$
If $T$ is split, we may drop the connectedness hypothesis on $S$.
Further, if $S=\operatorname{Spec}(k)$ and $T$ is split, the motives
$R\Gamma_{S}(\left[G/_{\varphi}L\right],\mathbb{Q})$ and
$M_{S}(\left[G/_{\varphi}L\right])$ are completed Tate motives in
$\operatorname{DM}(S)$. And, we can compute the Chow ring of
$\left[G/_{\varphi}L\right]$ as
$A^{\bullet}(\left[G/_{\varphi}L\right])_{\mathbb{Q}}\cong\left(A^{\bullet}_{T}(G/T)_{\mathbb{Q}}/\hat{T}A^{\bullet}_{T}(G/T)_{\mathbb{Q}}\right)^{W_{L}}$
###### Proof.
The first assertion is the discussion above (for the non-connected case use
Theorem 3.14). Together with the discussion above, the second resp. third
assertion follows again from the discussion above and Remark 2.13 resp.
Corollary 4.6. ∎
###### Remark 5.2.
The last isomorphism in Theorem 5.1 is also valid in the case, where $S$ is
not the spectrum of a field, after replacing the Chow groups with the $(2n,n)$
motivic cohomology group for $n\in\mathbb{Z}$ as in Proposition 4.5.
###### Remark 5.3.
If $G$ is a split reductive group with split maximal torus $T$, Brokemper has
shown that one can give a more explicit computation of the Chow ring of
$\left[G_{\varphi}/L\right]$ using the computations of Brion [Bri97] (cf.
[Bro16, Prop. 1.2]). To be more precise, we can write the last isomorphism of
Theorem 5.1 as
$A^{\bullet}(\left[G/_{\varphi}L\right])_{\mathbb{Q}}\cong
S^{W_{L}}/(f-\varphi f\mid f\in S_{+}^{W_{G}}),$
where $S=\operatorname{Sym}_{\mathbb{Q}}(\hat{T})\cong
A^{\bullet}_{T}(\ast)_{\mathbb{Q}}$, $S_{+}$ are the elements of positive
degree and $W_{G}$ is the Weyl group of $T$ in $G$. A more detailed
computation can be found in the proof of [Bro16, Prop. 1.2].
Remark 4.7 enables us to get an analogous result of Remark 5.3 for rational
$K_{0}$.
###### Proposition 5.4.
In the setting above, let us assume that $S=\operatorname{Spec}(k)$ is the
spectrum of a field, the derived group of $G$ is simply connected and $T$ is a
split maximal torus. Then we have
$K_{0}(\left[G/_{\varphi}L\right])_{\mathbb{Q}}\cong
R(T)_{\mathbb{Q}}^{W_{L}}/(f-\tilde{\varphi}f\mid f\in
R(T)_{\mathbb{Q}}^{W_{G}}).$
###### Proof.
For $K_{0}$, we could not produce an analogue of Corollary 3.15 using Theorem
3.14 and thus have to use a result of Krishna on equivariant $G$-theory (cf.
[Kri14, Lem. 9.2]). For completion we recall the main argument.
First, we may replace $Q$ and $M$ by $\prescript{g_{0}}{}{Q}$ and
$\prescript{g_{0}}{}{M}$ and assume that $\varphi(T)=T$. In particular,
$\tilde{\varphi}$-conjugation of $G$ by $T$ is just $\varphi$-conjugation.
Now we embed $T$ into $T\times_{S}T$ by $t\mapsto(t,\varphi(t))$. Let
$T\times_{S}T$ act on $G$ by $(t,t^{\prime}).g\coloneqq tgt^{\prime-1}$. This
yields a morphism
$\left[G/_{\varphi}T\right]\rightarrow\left[G/T\times_{S}T\right]$. This is
$T\cong T\times_{S}T/T$-torsor. Thus, by Remark 4.7, we have
$\displaystyle K_{0}(\left[G/_{\varphi}T\right])_{\mathbb{Q}}$
$\displaystyle\cong
K_{0}(\left[G/T\times_{S}T\right])_{\mathbb{Q}}/\hat{T}K_{0}(\left[G/T\times_{S}T\right])_{\mathbb{Q}}.$
By homotopy invariance, we have
$K_{0}(\left[G/T\times_{S}T\right])_{\mathbb{Q}}\cong
K_{0}^{T}(G/B)_{\mathbb{Q}}$. Therefore, we are reduced to classical
statements about $T$-equivariant $K$-theory of flag varieties (cf. [Uma13])
and get
$K_{0}(\left[G/_{\varphi}T\right])_{\mathbb{Q}}\cong
R(T)_{\mathbb{Q}}/(f-\tilde{\varphi}f\mid f\in R(T)_{\mathbb{Q}}^{W_{G}}).$
It follows161616Even though Krishna works with $G=\operatorname{GL}_{n}$ his
proof of [Kri14, Lem. 9.2] goes through in our case. from [Kri14, Lem. 9.2]
that
$K_{0}(\left[G/_{\varphi}L\right])_{\mathbb{Q}}\cong
K_{0}(\left[G/_{\varphi}T\right])_{\mathbb{Q}}^{W_{L}}.$
Thus, it suffices to show that
$IR(T)_{\mathbb{Q}}^{W_{L}}=IR(T)_{\mathbb{Q}}\cap R(T)^{W_{L}}_{\mathbb{Q}}$,
where $I=(f-\tilde{\varphi}f\mid f\in R(T)_{\mathbb{Q}}^{W_{G}})$, but this
follows from faithfully flatness of $R(T)^{W_{L}}\hookrightarrow R(T)$ (cf.
[Ste75, Thm. 1.2] \- see also the proof of [Bro16, Prop. 1.2] resp. [Bro18,
Prop. 2.3.2] for a detailed argument in the Chow group case). ∎
We want to give two motivating examples of quotients
$\left[G/_{\varphi}L\right]$ as above, that appear naturally and apply Theorem
5.1 to see that their motives are Tate. The classifying stack of finite groups
of Lie-type and the stack of $G$-zips. Both are examples in characteristic
$p>0$. In the last section, we want to use Theorem 5.1 to give an idea how we
want to approach geometric representation theory of finite groups of Lie-type
$G^{F}$ by relating it motivically to geometric representation of the
Langlands dual of $G$ (see below for the notation).
###### Example 5.5.
Let us assume that $S=\operatorname{Spec}(\mathbb{F}_{q})$ be a finite field
of characteristic $p>0$. We set $\varphi\colon G\rightarrow G$ to be the
$q$-Frobenius. This is an isogeny and thus, we can apply Theorem 5.1 in this
setting. Further, the stack $\left[G/_{\varphi}G\right]$ is isomorphic to
$\textup{B}G^{F}$, where $G^{F}$ is the stabilizer group scheme of the neutral
element (cf. [Bro16, Lem. 2.1]). It is well known that
$G^{F}(\overline{\mathbb{F}}_{q})\cong G(\mathbb{F}_{q})$, where
$\overline{\mathbb{F}}_{q}$ denotes an algebraic closure of $\mathbb{F}_{q}$.
Thus, we see that the motive of the classifying stack of a finite group of
Lie-type is Tate. Further, we are able to relate Tate motives of
$\left[T\backslash G/B\right]$, where $B$ is a Borel containing $T$ with Tate
motives of $\textup{B}G^{F}$ via the diagram in Theorem 5.1. This answers our
main question ($\ast$ ‣ 1) of the article.
One of Brokemper’s applications of his computations is the computation of the
Chow ring of $G$-zips. In a similar fashion we will apply the above results
and show that the motive of the stack of $G$-zips over a field is completed
Tate.
###### Example 5.6.
Let $k$ be a field of characteristic $p>0$ and let $S=\operatorname{Spec}(k)$.
Let $G,P,Q$ be as above. Let us denote the unipotent radical of $P$ resp. $Q$
with $R_{u}(P)$ resp. $R_{u}(Q)$. Further, let $\varphi\colon
P/R_{u}(P)\rightarrow Q/R_{u}(Q)$ be an isogeny. The datum
${\mathcal{Z}}\coloneqq(G,P,Q,\varphi)$ is called algebraic zip-datum. To
every algebraic zip-datum like above, we can associate the group
$E_{{\mathcal{Z}}}\coloneqq\\{(p,q)\in P\times
Q\mid\varphi(\bar{p})=\bar{q}\\}.$
The group $E_{{\mathcal{Z}}}$ acts on $G$ via conjugation $(p,q).g\coloneqq
pgq^{-1}$. The quotient stack
$G\textup{-Zip}\coloneqq\left[G/E_{{\mathcal{Z}}}\right]$ is called the stack
of G-zips. There are also alternative constructions using a Tannakian
formalism on the stack of $F$-zips (cf. [PWZ15]). In op.cit. there is also an
explicit description of the points of $G\textup{-Zip}$. Let $L\subseteq P$ be
a Levi-component of $P$. Then as seen in the proof of [Bro18, Thm. 2.4.4]
there is a split exact sequence
$1\rightarrow R_{u}(P)\times R_{u}(Q)\rightarrow E_{{\mathcal{Z}}}\rightarrow
L\rightarrow 1,$
where the splitting is induced by $L\hookrightarrow E_{{\mathcal{Z}}}$,
$l\mapsto(l,\varphi(l))$. Therefore, by homotopy invariance, we have
$M_{S}(G\textup{-Zip})\simeq M_{S}(\left[G/_{\varphi}L\right])$ which is
completed Tate by Theorem 5.1 and the discussion before the theorem.
## 6 Generalizations
In this section, we want to give an overview on the integral version of
Theorem 3.14 and Theorem 5.1. We want to mention three questions that came up
naturally during the work on this article that we want to address in the
future.
1. (1)
Under what assumptions can we transport all of these results to motives
defined via Spitzweck’s motivic cohomology ring spectrum?
2. (2)
Is it enough to invert the residue characteristics of the base for all of our
results?
3. (3)
Can the results of this article be extended to other cohomology theories?
Let us go a bit more into detail. So, let $S=\operatorname{Spec}(k)$ be the
spectrum of a field and $X$ be an $S$-scheme locally of finite type. further,
let $G$ be a split reductive group over $k$ with split maximal torus $T$ and
associated Weyl group $W$. Assume $G$ acts on $X$.
Question (1) is rather straightforward. For Chow groups one can see that if
$A^{\bullet}_{G}(X)\cong A^{\bullet}_{T}(X)^{W}$ holds integrally if any only
if $G$ is special. But if we assume that $G$ is special, then any $G$-torsor
is trivial Zariski-locally. If we define integral motives
$\operatorname{DM}_{\mathbb{Z}}$ on Prestacks via right Kan extension of
Spitzweck motives (cf. [Spi18]), we see that $\operatorname{DM}_{\mathbb{Z}}$
satisfies Nisnevich and in particular Zariski descent. Thus, up to
technicalities, we expect that all the arguments after Section 3.3 go through.
Crucially, Section 3.2 has to be worked out in this context, as both Ayoub and
Cisinski-Déglise use rational coefficients. This is not surprising, as one can
see that the particular motivic behavior of torsors under finite groups should
yield étale descent. We hope that in the case of special group there is a work
around.
Question (2) is addressed in a similar fashion, instead of Spitzweck motives,
we can use étale motive (cf. [CD16]). As these still satisfy étale descent
again all the arguments after Section 3.3 should go through. We expect that
inverting the residue characteristics should be enough to recover the
statements of Section 3.2 in this case. But still one needs to prove the
necessary results, which we expect to be rather straightforward.
Question (3) needs more careful treatment. The results about $T$-torsors of
Section 4 can be extended to other oriented cohomology theories, as we have
seen. Section 3.3 is more difficult as it boils down to vanishing results on
cohomology theories. The key part is Proposition 3.13. We expect that this
proposition still holds for modules over the étale $K$-theory ring spectrum
but we did not check this thoroughly. If one wants to work with genuine
$K$-theory, then this statement can not be proven via étale descent and thus
needs a more careful treatment. For other cohomology theories one can possibly
give a precise vanishing assumption to extend the results of this article.
For completion let us note, that the computation of the global sections in
Theorem 3.14 will not work for arbitrary cohomology theories. If one takes for
example Chow-Witt cohomology and $G=\operatorname{SL}_{2}$, then the Chow-Witt
ring of $\textup{B}\mathbb{G}_{\textup{m}}$ is trivial, while the Chow-Witt
ring of $\textup{B}G$ is not (cf. [Lev22]).
## Appendix A Existence of maximal tori over Prüfer domains
The following is essentially from the authors notes of a seminar talk from
Torsten Wedhorn and we do not claim originality (cf. [Wed23]). I want to thank
him for allowing me to use his ideas.
For our main result (Theorem 5.1), we need to work with reductive group
schemes that admit maximal tori. As we mentioned in Section 6 not every
reductive group scheme admits such a maximal tori. In this short appendix, we
want to prove that a reductive group scheme over sufficiently nice base, e.g.
Dedekind domains, with quasi-split generic fibre admits a maximal torus. In
fact, to prove this, we only need to assumptions on the base. First, we will
need that the base is affine and secondly the stalks of the base are valuation
rings. The following idea was communicated to us by Torsten Wedhorn.
Rings with the property that the stalks are valuation rings were studied
before and we will give them a name according to literature.
###### Definition A.1 ([FS01]).
An integral domain $A$ is called Prüfer domain, if for any prime
${\mathfrak{p}}\subseteq A$, the localization $A_{{\mathfrak{p}}}$ is a
valuation ring.
###### Example A.2.
Any Dedekind domain is a Prüfer domain conversely, any noetherian Prüfer
domain is in fact a Dedekind domain (cf. [Sta22, 00II]). So, Prüfer domains
can be seen as the non-noetherian analog of Dedekind domains.
Another example of Prüfer domains, are so called Bézout domains, i.e. integral
domains in which every finitely generated ideal is principal. In particular,
the Picard group of a Bézout domain is trivial. This includes all PID’s and
valuation rings, but also rings such as the algebraic integers
$\overline{\mathbb{Z}}$ (cf. [DB72, pp. 243-245] for more examples).
In the following let us fix a Prüfer domain $A$ and set
$S\coloneqq\operatorname{Spec}(A)$.
###### Lemma A.3.
Let us denote $K\coloneqq\operatorname{Quot}(A)$. Further, let $X\rightarrow
S$ be a proper morphism of finite presentation. Then the map $X(A)\rightarrow
X(K)$ is bijective.
###### Proof.
Let $\sigma\in X(K)$. By the valuative criterion for properness, we can find
for any $s\in S$ a unique lift $\sigma_{s}\in X({\mathcal{O}}_{S,s})$ of
$\sigma$. As $X$ is of finite presentation, we can find an open neighborhood
$U_{s}$ of $s$ and a lift $\sigma_{U_{s}}\in X(U_{s})$ of $\sigma_{s}$. By
uniqueness of the lifts $\sigma_{s}$, the $\sigma_{U_{s}}$ glue to a unique
lift $\sigma_{A}\in X(A)$ of $\sigma$. ∎
Applying this lemma to the scheme representing Borels in a reductive $S$-group
scheme $G$, we see that if the generic fiber of $G$ is quasi-split, then $G$
is quasi-split. The existence of a maximal torus in $G$ follow now from the
fact maximal tori contained in $B$ are a $R_{u}(B)$-torsor. Writing everything
out we get the following proposition.
###### Proposition A.4.
Let $G$ be a reductive $S$-group scheme. If the generic fibre of $G$ is quasi-
split, then $G$ admits a Borel pair. Further, if $\operatorname{Pic}(A)=0$ and
the generic fibre of $G$ is split, then $G$ is split.
###### Proof.
Let $K\coloneqq\operatorname{Quot}(A)$ and $\operatorname{Bor}$ be the scheme
classifying the Borels of $G$ over $A$. As $G_{K}$ is quasi-split, there
exists an element $B_{K}\in\operatorname{Bor}(K)$. The scheme
$\operatorname{Bor}$ is proper and of finite presentation (cf. [SGA3, Exp.
XXII Cor. 5.8.3]) and thus we can apply Lemma A.3 to $\operatorname{Bor}$ and
see that $B_{K}$ lifts uniquely to a Borel $B$ of $G$. Let $U$ denote the
unipotent radical of $B$. The functor of maximal tori in $B$ is representable
by a smooth affine $S$-scheme $X$ and is further a $U$-torsor (cf. [SGA3, Exp.
XXII Cor. 5.6.13]). But $H^{1}(S,U)=0$, as $S$ is affine (cf. [SGA3, Exp. XXII
Cor. 5.9.6]), and therefore $X\rightarrow S$ admits a section.
If further $T_{K}$ is split and $\operatorname{Pic}(A)=0$, then we claim that
$G$ is split. For this let ${\bar{\eta}}$ be the geometric point corresponding
to $K$. As $S$ is integral and normal (cf. [Sta22, 00IC]), $T$ splits after
passage to a finite Galois cover of $S$ (cf. [Con15, Cor. B.3.6]). This
induces an action of $\pi_{1}^{\operatorname{\acute{e}t}}(A,{\bar{\eta}})$ on
the character group $X^{\bullet}(T)$, which is trivial if and only if $T$ is
split. As $T_{K}$ is split, the action of
$\pi_{1}^{\operatorname{\acute{e}t}}(K,{\bar{\eta}})$ on $X^{\bullet}(T_{K})$
is trivial. Since $\pi_{1}^{\operatorname{\acute{e}t}}(A,{\bar{\eta}})$ is a
quotient of $\pi_{1}^{\operatorname{\acute{e}t}}(K,{\bar{\eta}})$ (cf. [Sta22,
0BQM]), we deduce that $\pi_{1}^{\operatorname{\acute{e}t}}(A,{\bar{\eta}})$
acts trivially on $X^{\bullet}(T)$. Therefore, $T$ is split and since
$\operatorname{Pic}(A)=0$, we see that $G$ is split (cf. [SGA3, Exp. XXII
Prop. 2.2]). ∎
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# A Robust Unscented Transformation for Uncertain Moments
Hugo T. M. Kussaba<EMAIL_ADDRESS>João Y. Ishihara Leonardo R. A. X.
Menezes Department of Electrical Engineering, University of Brasília – UnB,
70910-900, Brasília, DF, Brazil
###### Abstract
This paper proposes a robust version of the unscented transform (UT) for one-
dimensional random variables. It is assumed that the moments are not exactly
known, but are known to lie in intervals. In this scenario, the moment
matching equations are reformulated as a system of polynomial equations and
inequalities, and it is proposed to use the Chebychev center of the solution
set as a robust UT. This method yields a parametrized polynomial optimization
problem, which in spite of being NP-Hard, can be relaxed by some algorithms
that are proposed in this paper.
###### keywords:
Unscented Transform , Polynomial Optimization , Lasserre’s hierarchy ,
Statistics , Filtering.
t1t1footnotetext: ©2019. This manuscript version is made available under the
CC BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/.
This article has been accepted for publication in a future issue of the
Journal of Franklin Institute, but has not been fully edited. Content may
change prior to final publication. The final version of record is available at
https://doi.org/10.1016/j.jfranklin.2019.02.018.
## 1 Introduction
In numerous problems of statistics and stochastic filtering, one is often
interested in calculating the posterior expectation of a continuous random
variable $X$ that undergoes a nonlinear transform $f$, viz.:
$\textrm{E}\left\\{f(X)\right\\}=\int_{\mathbb{R}}f(\xi)p_{X}(\xi)d\xi.$ (1)
It is not always possible to have a closed-form expression for this integral
in terms of elementary functions: if this integral does not satisfy the
hypothesis of Liouville’s theorem (see for instance, Section 12.4 of [1]),
then the antiderivative of this integral cannot be expressed in terms of
elementary functions. Thus, instead of using analytical methods to calculate
(1), in many situations numerical methods must be employed.
A common way to numerically calculate (1) is by using the Monte Carlo
integration method [2], which is a stochastic sampling technique: by taking a
sufficiently large number of samples of the random variable $X$, one can
approximate the probability density function $p_{X}$ and obtain an estimate
for (1). However, this method can be very demanding computationally, since it
frequently employs several thousands of simulations to obtain the statistics
of the final result.
Another way to calculate (1) with less computational burden than Monte Carlo
integration method is the technique of Unscented Transform (UT). Originally
proposed for the problem of extending the Kalman filter for nonlinear
dynamical systems [3], this method has also been applied in several problems
of engineering, such as in the analysis of the sensitivity of antennas [4] and
in circuit optimization [5].
Different from the Monte Carlo integration method, the UT is a deterministic
sampling technique: in place of choosing random points to approximate $p_{X}$,
points known as sigma points are deterministically selected to capture the
statistics of $p_{X}$. This is accomplished by generating a discrete
distribution $p^{\prime}_{X}$ having the same first and second (and possibly
higher) moments of $p_{X}$. The mean, covariance and higher moments of the
transformed ensemble of sigma points can then be computed as the estimate of
the nonlinear transformation $f$ of the original distribution [3], [6]. Thus,
at least the first moments of $p_{X}$ must exist and be exactly known. In
several circumstances, however, this is not valid: for instance, it may be the
case that the distribution does not even have the first moment (one example is
the Cauchy distribution [7]). In the case that the moments can be assumed to
exist, it is usual that they are not precisely known, but it is still possible
to have upper and lower bounds for the exact value of moments from statistical
experiments or by the use, for instance, of Chebychev’s inequality [2].
In this paper it is designed a technique for generating sigma points when the
exact value of the moments are not known, but upper and lower bounds for the
unknown moments are known. In this case, the moment matching equations of UT
are no longer just a system of polynomial equations but a system of polynomial
equations with polynomial inequalities. Furthermore, since this system can
have more than one solution, it is possible to choose a set of sigma points
which minimizes a given cost function by formulating the problem as a
polynomial optimization problem (POP). Although the solution to this problem
is in general computationally infeasible [8], by using Lasserre’s hierarchy of
semidefinite programming relaxations one can approximate the solution of the
original POP problem by the solution of a computationally feasible convex
optimization problem[9, 10].
The main contribution of this paper is the introduction of the concept of UT
robustness in the sense of exploiting the upper and lower bounds for moments.
A robust UT is proposed in [11] but robustness has different meaning and it is
achieved by matching precisely known high order moments. It is important to
note that Lasserre’s hierarchy of relaxations was applied in earlier
investigations of the moment matching problem [12], but its use was limited to
polynomial equations while here polynomial inequalities are also taken into
account.
This paper is organized as follow. Some preliminaries for the robust UT are
presented in Section 2. The robust UT itself is detailed in Section 3. Details
about the computation of robust sigma points is presented in Section 4. The
computation of UT transform is presented in Section 5. Finally, the conclusion
is presented in Section 7.
## 2 Unscented Transform
The rationale behind the unscented transformation is that it is easier to
calculate a moment for a discrete distribution than a continuous one. In fact,
to compute the posterior distribution of a random scalar variable $X$ with
distribution $p_{X}$ by a function $f$, one must calculate the integral (1)
while, in other hand, if $Z$ is a discrete random variable with $m+1$ atoms
$z_{i}$ and distribution $p_{Z}$, one only needs to calculate the sum
$E\left\\{f(Z)\right\\}=\sum_{i=1}^{m+1}f(z_{i})p_{Z}(z_{i}).$ (2)
Henceforth, if it is possible to choose the atoms and their weights of an
adequate $p_{Z}$ to approximate $p_{X}$, then the value $E\\{f(Z)\\}$ would be
a good approximation to $E\\{f(X)\\}$, but with (2) being easier to compute
than (1). Thus, the principle behind the UT is to approximate the continuous
distribution $p_{X}$ by the discrete distribution $p_{Z}$ by equating the
first $m$ moments of these distributions. In other words, the following
equations must be satisfied:
$\displaystyle E\\{X^{k}\\}$ $\displaystyle=E\\{Z^{k}\\}$ (3)
$\displaystyle=\sum_{i=1}^{m+1}z_{i}^{k}p_{Z}(z_{i}),\ k=1,\ldots,m,$ (4)
where is supposed the access to $E\\{X^{k}\\}$ for $k=1,\ldots,m$.
Given $E\\{X^{k}\\}$ it is always possible to find $z_{i}$ and $p_{Z}(z_{i})$,
$i=1,\ldots,m+1$ (which will be called sigma point and its weight,
respectively) satisfying (4). In fact, as it will be stated next, there is at
least one solution for the equations
$E\left\\{g_{k}(Z)\right\\}=E\left\\{g_{k}(X)\right\\},\ k=1,\ldots,m,$ (5)
where $g_{k}:\mathbb{R}\rightarrow\mathbb{R}$ are continuous functions,
$g_{k}\not\equiv g_{j}$ for $k\neq j$ and from which (4) is a particular case
with $g_{k}(x)=x^{k}$. That (5) has at a least one solution is stated next in
Theorem 1.
###### Theorem 1.
Consider that the $m$ moments $E\\{g_{k}(X)\\}$, $k=1,\ldots,m$, are given.
The system (5) in terms of variables $z_{i}$ and $p_{Z}(z_{i})$ has at least
one solution with at most $m+1$ sigma points.111The proof of the theorem is
known in the mathematical literature in the context of the Caratheodory’s
theorem. Since it is less known in the context of UT literature, the proof is
presented here for easy reference. See, e.g., [13].
###### Proof.
Since the point
$P=\left(E\left\\{g_{1}(X)\right\\},\ldots,E\left\\{g_{m}(X)\right\\}\right)$
belongs to the convex hull of the set
$G=\left\\{\left(g_{1}(x),\ldots,g_{m}(x)\right)\mid x\in\mathbb{R}\right\\}$,
then Caratheodory’s theorem [14, Theorem 1.3.6] gives that $P$ can be written
as a convex combination of at most $m+1$ points in $G$. Thus, one has that
there exist $\theta_{i}\geq 0$ and $z_{i}\in\mathbb{R}$ such that
$E\\{g_{k}(X)\\}=\sum_{i=1}^{m+1}\theta_{i}g_{k}(z_{i}),\ k=1,\ldots,m$ (6)
and
$\sum_{i=1}^{m+1}\theta_{i}=1.$ (7)
Taking $Z$ to be the discrete random variable with probability distribution
given by
$p_{Z}(k)=\begin{cases}\theta_{i},&\text{if }k=z_{i},\\\
0,&\text{otherwise,}\end{cases}$
one has that equations (6) and (7) are exactly the equations given by (5). ∎
It is important to note that Theorem 1 only states the existence of sigma
points satisfying (5), but it does not give any conclusion about uniqueness.
In fact, many solutions are possible, and the choice of an adequate set of
sigma points has been investigated thoroughly in the literature [6, 15].
###### Remark 1.
Specifically for the first moments of $X$ focused in this paper, i.e. the case
that $g_{k}(x)=x^{k}$, $k=0,\ldots,m+1$, one can also see (4) as a Gaussian
quadrature integration scheme with $p_{X}$ being the weighting function and
$z_{i}$ and $\omega_{i}:=p_{Z}(z_{i})$, $i=1,\ldots,m+1$, being respectively
the nodes and their weights in the quadrature formula. Thus, depending on the
probability density function $p_{X}$, the sigma points can be readily
calculated as the roots of an orthogonal polynomial [16, Section 4.6]. For
instance, if $p_{X}$ is a normal distribution, then the sigma points are the
roots of a Hermite polynomial.
One can note that if the only intention were to find some discrete random
variable such that $E\\{f(X)\\}=E\\{f(Z)\\}$, it is always possible to find a
$Z$ variable with two sigma points. In fact, in Theorem 1, consider $m=1$ and
$f=g_{1}$. However, this would imply the knowledge of the function $f$. In the
UT reasoning, it is sought a greater number of sigma points in order that the
approximation $E\\{f(X)\\}=E\\{f(Z)\\}$ be valid for a greater number of
functions $f$. In [17] it is shown that the approximation is good for any
function $f$ which can be well approximated by its $m$-order Taylor
representation.
On top of that, the larger is $m$, the more precise is the approximation for
$E\\{f(X)\\}$: an estimate for the error
$\int_{a}^{b}p_{X}(x)f(x)\,dx-\sum_{i=1}^{m}\omega_{i}f(z_{i})$ is given by
$\frac{f^{(2m)}(\xi)}{(2m)!}\int_{a}^{b}p_{X}(x)h_{m}^{2}(x)\,dx,$
where $\xi\in\left(a,b\right)\subset\mathbb{R}$ and $h_{m}$ is the associated
monic orthogonal polynomial of degree $m$ associated to $p_{X}$ [18, Theorem
3.6.24]. As a matter of fact, the Gauss quadrature computed integral for
$E\\{f(X)\\}$ is exact for all $f(x)$ that are polynomials of degree less or
equal than $2m-1$ [18, pp. 172-175].
Since Theorem 1 takes in account generic functions $g_{k}$, one can analyze
very general moment settings (for instance, the case of fractional moments is
worked in [19]).
Summing up, one can note that the basic assumption for the UT theory until now
is that the moments are precisely known. However, this assumption can be too
strong in practical situations where the moments are estimated from
experiments and then, their values are only known to be in some intervals. In
this work, we deal with the scenario of unknown moments and propose to
calculate a robust set of sigma-points in the sense of minimizing the worst
possible error between this robust choice of sigma-points and the sigma-points
computed by using the real values of the moments.
## 3 Robust UT
To motivate the proposed robust UT consider a normally distributed random
variable $X$ with mean $\mu$ and variance $V$. In the case where $\mu$ and $V$
are given and $m=2$, it is known that a set of sigma points are given by [6]
$\begin{array}[]{ccc}z_{1}&=&\mu-\sqrt{3V},\\\ z_{2}&=&\mu,\\\
z_{3}&=&\mu+\sqrt{3V},\end{array}$ (8)
with its weights given respectively by $\omega_{1}=\frac{1}{6}$,
$\omega_{2}=\frac{2}{3}$ and $\omega_{3}=\frac{1}{6}$.
Suppose now that the value of $V$ is not exactly known, but an upper bound
$\overline{V}$ and a lower bound $\underline{V}$ for $V$ is known, i.e.
$V\in\left[\underline{V},\overline{V}\right]$. A naive approach for choosing
the sigma points would be to use the mean value between $\overline{V}$ and
$\underline{V}$ for $V$ in (8). In this case, the sigma points $z_{1}$ and
$z_{3}$ would be given by
$\displaystyle z_{1}=$ $\displaystyle
z_{1}^{M}\coloneqq\mu-\sqrt{3(\underline{V}+\overline{V})/2},$ $\displaystyle
z_{3}=$ $\displaystyle
z_{3}^{M}\coloneqq\mu+\sqrt{3(\underline{V}+\overline{V})/2},$
and $z_{2}$, and the weights $\omega_{1}$, $\omega_{2}$ and $\omega_{3}$
unchanged.
As Figure 1 illustrates, this can be a pessimistic choice for the sigma
points, since if, for example, the real value of $V$ is in fact nearer of
$\bar{V}$, a better choice of sigma points would be nearer to the point
$\left(z_{1},z_{3}\right)=\left(\mu-\sqrt{3\overline{V}},\mu+\sqrt{3\overline{V}}\right)$.
In fact, as can be seen in Figure 1, a preferable choice for the sigma points
would be at the center of the region:
$\displaystyle z_{1}=$ $\displaystyle
z_{1}^{C}\coloneqq\mu-(\sqrt{3}/2)(\sqrt{\underline{V}}+\sqrt{\overline{V}}),$
$\displaystyle z_{3}=$ $\displaystyle
z_{3}^{C}\coloneqq\mu+(\sqrt{3}/2)(\sqrt{\underline{V}}+\sqrt{\overline{V}}).$
This choice is precisely the Chebychev center222There are two non-equivalent
definitions of Chebychev center of a bounded set with non-empty interior: the
first definition is the center of the minimal-radius ball enclosing this set,
and the second is the center of the largest incribed ball in this set [20]. In
this paper, only the first definition will be used. of the set of possibles
sigma points given that $V\in\left[\underline{V},\overline{V}\right]$. It has
the property of having the minimum worst possible error between the chosen
sigma point set and the sigma point set corresponding to the true value of
$V$.
Figure 1: Possible choices of sigma points.
Consider now a general scenario where some of the moments are not known, but
upper and lower bounds for them are. In this case, the set of possible sigma
points and their weights would be in the semialgebraic set
$S\subset\mathbb{R}^{2m+2}$ of elements
$x=(z_{1},\ldots,z_{m+1},\omega_{1},\ldots,\omega_{m+1})$, defined as solution
of the system
$S:\qquad\begin{array}[]{cccc}\omega_{j}&\geq&0,&j\in\\{1,\ldots,m\\},\\\
\sum_{i=1}^{m+1}\omega_{i}&=&1,\\\
\sum_{i=1}^{m+1}z_{i}^{k_{1}}\omega_{i}&=&E\\{X^{k_{1}}\\},&k_{1}\in\mathcal{K}_{1},\\\
\sum_{i=1}^{m+1}z_{i}^{k_{2}}\omega_{i}&\leq&u_{k_{2}},&k_{2}\in\mathcal{K}_{2},\\\
\sum_{i=1}^{m+1}z_{i}^{k_{2}}\omega_{i}&\geq&\ell_{k_{2}},&k_{2}\in\mathcal{K}_{2},\end{array}$
(9)
where $\mathcal{K}_{1}$ (resp. $\mathcal{K}_{2}$) is the set of indexes $k$
for which the $k$-moment is known (resp. unknown),
$\mathcal{K}_{1}\cup\mathcal{K}_{2}=\\{1,\ldots,m\\}$ and $u_{k_{2}}$ and
$\ell_{k_{2}}$ are respectively known upper and lower bounds for the
$k_{2}$-moment.
Although Theorem 1 guarantees a solution for (9), this solution may not be
unique. On that account, there is more than one set of sigma points and
corresponding weights, and it is natural to ask which is a good choice of
sigma points and weights. Based on the previous example, a so-called robust
choice would be the Chebychev center of the semialgebraic set defined by (9)
since this choice minimizes the worst possible error between the chosen sigma
point set and the sigma point set corresponding to the possible true values of
the moments of $X$.
Different from the previous example, however, in this generic scenario it may
be the case that an analytic formula to express the sigma points as the
function of the moments of the random variable $X$ is not available, and the
following optimization problem must be solved to find the Chebychev center:
$(z_{1}^{C},\ldots,z_{m+1}^{C},\omega_{1}^{C},\ldots,\omega_{m+1}^{C})=\underset{\hat{x}\in\mathbb{R}^{2m+2}}{\arg\min}\
\underset{x\in S}{\max}\|x-\hat{x}\|^{2},$ (10)
where $S$ is the solution set for (9) and
$x=(z_{1},\ldots,z_{m+1},\omega_{1},\ldots,\omega_{m+1})$.
It is important to note that the optimization problem in (10) is not always
guaranteed to have a solution, since the set $S$ may not be bounded (if, for
example, $\omega_{1}$ is zero, then $z_{1}$ can take any value). Nevertheless,
it still makes sense to use the Chebychev center of a large bounded subset of
$S$ to try to choose a set of sigma points minimizing the worst possible
estimation error. In fact, for any (bounded or unbounded) set $S$, one can
take an $\varepsilon>0$ and consider the Chebychev center of the semi-
algebraic set $S_{\varepsilon}\subseteq S$ defined as the set of points
$x=(z_{1},\ldots,z_{m+1},\omega_{1},\ldots,\omega_{m+1})$
satisfying
$S_{\varepsilon}:\qquad\begin{array}[]{cccc}\omega_{j}&\geq&\varepsilon,&j\in\\{1,\ldots,m\\},\\\
\sum_{i=1}^{m+1}\omega_{i}&=&1,\\\
\sum_{i=1}^{m+1}z_{i}^{k_{1}}\omega_{i}&=&E\\{X^{k_{1}}\\},&k_{1}\in\mathcal{K}_{1},\\\
\sum_{i=1}^{m+1}z_{i}^{k_{2}}\omega_{i}&\leq&u_{k_{2}},&k_{2}\in\mathcal{K}_{2},\\\
\sum_{i=1}^{m+1}z_{i}^{k_{2}}\omega_{i}&\geq&\ell_{k_{2}},&k_{2}\in\mathcal{K}_{2}.\end{array}$
(11)
As $\varepsilon$ decreases, $S_{\varepsilon}$ covers a larger part of $S$.
Fig. 2 illustrates how the set $S_{\varepsilon}$ approximates the set $S$ by
choosing a sufficiently small $\varepsilon>0$.
While the Chebychev center of $S$ may not exist, the next theorem assures the
existence of the Chebychev center of $S_{\varepsilon}$ for any
$\varepsilon>0$.
###### Theorem 2.
If $m\geq 2$ and $\varepsilon>0$, then the optimization problem
$(z_{1}^{C},\ldots,z_{m+1}^{C},\omega_{1}^{C},\ldots,\omega_{m+1}^{C})=\underset{\hat{x}\in\mathbb{R}^{2m+2}}{\arg\min}\
\underset{x\in S_{\varepsilon}}{\max}\|x-\hat{x}\|^{2},$ (12)
has a solution.
###### Proof.
It suffices to prove that the set $S_{\varepsilon}$ is bounded. The variables
$\omega_{i}$, $i=1,\ldots,m+1$ are all inside the $m+1$ dimensional simplex
and thus bounded. Since
$\sum_{i=1}^{m+1}z_{i}^{2}\omega_{i}\leq u_{2},$
and $\omega_{i}>0$, it is impossible for any variable $z_{i}$ to grow without
bound. ∎
###### Remark 2.
It is interesting to note that alternatively, instead of the compact (closed
and bounded) set $S_{\varepsilon}$ one could consider the set $\hat{S}$
defined as the set $S$ up to the first inequalities replaced by a strict
inequality (that is, $\omega_{j}\geq 0$ is replaced by $\omega_{j}>0$,
$j\in\\{1,\ldots,m\\}$). This set is indeed bounded as $S_{\varepsilon}$.
However, strict inequalities are in general not well-handled by numeric
solvers.
Figure 2: The regions $S\setminus S_{\varepsilon}$ (red) and $S_{\varepsilon}$
(blue) are plotted in various viewpoints to illustrate the approximation of
the semi-algebraic set $S$ by $S_{\varepsilon}$ with $\varepsilon=0.01$,
$m=1$, $\mathcal{K}_{1}=\emptyset$, $\mathcal{K}_{2}=\\{1,2\\}$,
$\ell_{1}=\ell_{2}=0$ and $u_{1}=u_{2}=1$. Only $\omega_{1}$, $z_{1}$ and
$z_{2}$ are exhibited in the graphic, since $\omega_{2}=1-\omega_{1}$.
The inner maximization problems of (10) and (12) are polynomial optimization
problems (POP). Such problems are ubiquitous and are encountered in several
fields [21], such as: finance [22, 23, 24], robust and nonlinear control [25,
26], signal processing [27, 28], quantum physics [29] and materials science
[30]. It is known that this problem is NP-Hard in general [8], but despite
this, it is possible to approximate a POP by convex optimization problems
which are computationally feasible using Lasserre’s hierarchy of semidefinite
programming relaxations [9, 10]. In fact, by using the GloptiPoly 3
package333Available for download at
http://homepages.laas.fr/henrion/software/gloptipoly3/. [31] or the SparsePOP
package444Available for download at
http://sourceforge.net/projects/sparsepop/. [32] for MATLAB, or ncpol2sdpa
library555Available for download at https://gitlab.com/peterwittek/ncpol2sdpa.
[33] for Python, these relaxations can be easily implemented and enables the
user to transparently construct an increasing sequence of convex LMI
relaxations whose optima are guaranteed to converge monotonically to the
global optimum of the original non-convex global optimization problem [26].
Moreover, it is possible to numerically certify the global optimality of the
problem.
A direct approach to solve (10) would be to use the mentioned Lasserre’s
hierarchy to solve the inner maximization problem for a fixed
$\hat{x}=\hat{x}_{0}$ and a local optimization algorithm to search which
$\hat{x}_{0}$ minimizes (10). This however, besides being a hard numeric
problem, would not guarantee a good approximation to the real value of the
Chebychev center. In the next section alternative ways to compute (10) with
different trade-off between precision of the result and speed of algorithm are
proposed.
## 4 Computation of robust sigma points
### 4.1 Computation of an outer box to approximate the Chebychev center
The first proposed approach to find an approximate solution to (12) is to
compute an outer-bounding box $B$ of $S_{\varepsilon}$ and approximate the
Chebychev center of $S_{\varepsilon}$ by the Chebychev center of $B$. By
Theorem 2, for $m\geq 2$ the constraint set $S_{\varepsilon}$ is bounded.
Thus, each one of the following polynomial optimization problems on
$x=(x_{1},\ldots,x_{2m})\in S_{\varepsilon}$ has solution
$\displaystyle\underline{z}_{i}$
$\displaystyle=\arg\min_{x\in\mathbb{R}^{2m+2}}x_{i}\text{ s.t. }x\in
S_{\varepsilon},\ i=1,\ldots,m+1,$ $\displaystyle\underline{\omega}_{i}$
$\displaystyle=\arg\min_{x\in\mathbb{R}^{2m+2}}x_{i+m+1}\text{ s.t. }x\in
S_{\varepsilon},\ i=1,\ldots,m+1,$ $\displaystyle\overline{z}_{i}$
$\displaystyle=\arg\max_{x\in\mathbb{R}^{2m+2}}x_{i}\text{ s.t. }x\in
S_{\varepsilon},\ i=1,\ldots,m+1,$ $\displaystyle\overline{\omega}_{i}$
$\displaystyle=\arg\max_{x\in\mathbb{R}^{2m+2}}x_{i+m+1}\text{ s.t. }x\in
S_{\varepsilon},\ i=1,\ldots,m+1,$ (13)
and can be used to construct an outer box
$\displaystyle
B=\\{(z_{1},\ldots,z_{m+1},\omega_{1},\ldots,\omega_{m+1})\in\mathbb{R}^{2m+2}:\underline{z}_{i}\leq
z_{i}\leq\overline{z}_{i},$
$\displaystyle\underline{\omega}_{i}\leq\omega_{i}\leq\overline{\omega}_{i},i=1,\ldots,m+1\\}$
such that $S_{\varepsilon}\subset B$. The next theorem gives an estimate of
the error of the outer-bounding box approximation.
###### Theorem 3.
Let $c_{B}$ be the center of Chebychev of $B$ and let $c_{S}$ be the center of
Chebychev of $S_{\varepsilon}$. If $d\coloneqq\|c_{B}-c_{S}\|$ is the defined
as the distance between the centers of Chebychev, then
$d\leq\frac{\textrm{diam}(B)}{2},$
where $\textrm{diam}(B)$ is the diameter of $B$, that is, the least upper
bound of the set of all distances between pairs of points in $B$.
###### Proof.
Suppose that $d>\textrm{diam}(B)/2$. Thus $c_{S}\not\in B$, and since $c_{S}$
is the center of Chebychev of $S_{\varepsilon}$, one has that $c_{S}$ is in
the convex hull of $S_{\varepsilon}$. But this implies that $c_{S}$ is in $B$,
and the result follows by contradiction. ∎
### 4.2 Polynomial optimization program to compute minimum enclosing ball
Another approximate solution to problem (12) can be found by using the two-
stage approach of [34]. Since the Chebychev center of $S_{\varepsilon}$ is
always inside of the outer box $B$ computed in Section 4.1, problem (10) is
equivalent to
$(z_{1}^{C},\ldots,z_{m+1}^{C},\omega_{1}^{C},\ldots,\omega_{m+1}^{C})=\underset{\hat{x}\in
B}{\arg\min}\ \underset{x\in S_{\varepsilon}}{\max}\|x-\hat{x}\|^{2}.$ (14)
In the first stage, the inner optimization problem of (14), namely,
$J(\hat{x})\coloneqq\max_{x\in S_{\varepsilon}}\|x-\hat{x}\|^{2},$
is approximated by a polynomial function $\tilde{J}_{\tau}(\hat{x})$ of degree
$2\tau$ using the semidefinite optimization program outlined in [34]. Then, in
the second stage, the outer minimization problem is replaced with the
polynomial optimization problem
$(z_{1}^{C},\ldots,z_{m+1}^{C},\omega_{1}^{C},\ldots,\omega_{m+1}^{C})=\underset{\hat{x}\in
B}{\arg\min}\ \tilde{J}_{\tau}(\hat{x}),$ (15)
which can be solved using the Lasserre’s hierarchy. As the degree $2\tau$ of
the approximation polynomial increases, the solution of problem (15) converges
to the solution of problem (14) in the sense of [34]. Alternatively, problem
(12) can be shown to be equivalent to the following polynomial optimization
problem:
$\begin{array}[]{cc}\min_{\hat{x},r}r&\text{ s.t. }\|x-\hat{x}\|^{2}\leq r,\\\
&x\in S_{\varepsilon},\,\hat{x}\in B,\end{array}$ (16)
where $\hat{x}$ is the Chebychev center of $S_{\varepsilon}$. In other words,
the Chebychev center is the center of the radius of the minimum volume ball
that encloses $S_{\varepsilon}$.
###### Remark 3.
It is interesting to note that if the gap between $\ell_{i}$ and $u_{i}$ is
not large enough, the resulting semi-definite programs from relaxing the
polynomial optimization problems (13) and (16) can be numerically unstable. In
this case, however, one may use the naive approach of computing sigma points
by using the arithmetic mean of the moments without loss, since the difference
between the real Chebychev center and the point computed by this method would
be not so great due to the small difference between the upper and lower bounds
of the moments.
## 5 Computation of UT transform
Suppose that
$x_{CB}\coloneqq(z_{1}^{C},\ldots,z_{m+1}^{C},\omega_{1}^{C},\ldots,\omega_{m+1}^{C})$
is the Chebychev center of $S$ computed by one of the methods of Section 4.
Based on (2), define the function
$UT_{f}(z_{1},\ldots,z_{m+1},w_{1},\ldots,w_{m+1})\coloneqq\sum_{i=1}^{m+1}w_{i}f(z_{i}).$
As discussed above, $UT_{f}$ is a good approximation for $E\\{f(X)\\}$ for a
sufficiently large $m$. While it is true that $x_{CB}$ approximates of the
center of Chebychev of $S$, one also wish to know if $UT_{f}(x_{CB})$ is near
to the Chebychev center of $UT_{f}(S)$ (that is, the image of the set $S$ by
the function $UT_{f}$). A class of functions $f$ such that $UT_{f}(x_{CB})$ is
near the Chebychev center of $UT_{f}(S)$ is given by the functions such that
the solution of the following optimization problem
$\displaystyle\min\,D\ \text{s.t.\ }$
$\displaystyle(1-D)\|x-y\|\leq\|UT_{f}(x)-UT_{f}(y)\|$ (17)
$\displaystyle\leq(1+D)\|x-y\|,$ $\displaystyle x,y\in S$
is sufficiently small. If these values are sufficiently small, the function
$UT_{f}$ is a low-distortion geometric embedding [35], and this implies that
$UT_{f}(x_{CB})$ is near the Chebychev center of $UT_{f}(S)$. Finally, to
compute (17), one can estimate a solution for (17) by uniformly sampling
random values $(x_{i},y_{i})$ of $S$ and computing $\max\,D_{i}$, where
$\displaystyle D_{i}\coloneqq\min\,D\ \text{s.t.\ }$
$\displaystyle(1-D)\|x_{i}-y_{i}\|\leq\|UT_{f}(x_{i})-UT_{f}(y_{i})\|$ (18)
$\displaystyle\leq(1+D)\|x_{i}-y_{i}\|.$
## 6 Numerical experiments
In this example, the computation of Chebychev center using the methods
proposed in this paper will be illustrated. Consider that one desires to
compute $2$ sigma points in a scenario where the values of the first and
second moment are not precisely known, but it is known that $E\\{X\\}\in[0,1]$
and that $E\\{X^{2}\\}\in[0,1]$. In this case, one has (9) with $m=1$,
$\mathcal{K}_{1}=\emptyset$, $\mathcal{K}_{2}=\\{1,2\\}$, and $\ell_{1}=-3$
and $u_{1}=4$, and $\ell_{2}=0$ and $u_{2}=5$.
Naive method: For fixed values of $E\\{X\\}=\mu$ and $E\\{X^{2}\\}=V$, one can
compute a point inside the set $S$ by using the canonical formula
$\begin{array}[]{c}\omega_{1}=0.5,\ \omega_{2}=0.5,\\\
z_{1}=\mu+\sqrt{V-\mu^{2}},\ z_{2}=\mu-\sqrt{V-\mu^{2}}.\end{array}$ (19)
A naive choice of sigma points would be the use of the arithmetic mean of the
lower and upper bounds for the moments in (19). Using (19) with
$\mu=\frac{1}{2}(\ell_{1}+u_{1})=0.5$ and $V=\frac{1}{2}(\ell_{2}+u_{2})=2.5$
results in the sigma points $z_{1}=2$, $z_{2}=-1$ with weights given by
$w_{1}=0.5$ and $w_{2}=0.5$.
Method 1: By using the method proposed in Section 4.1 it is possible to
compute an outer-bounding box approximation to the semi-algebraic set
$S_{\varepsilon}$. Using $\varepsilon=0.01$, the computation of an approximate
point of the Chebychev center using outer-bounding box approximation results
in the sigma points $z_{1}=0$, $z_{2}=0$ with weights respectively given by
$w_{1}=0.5$ and $w_{2}=0.5$.
Method 2: Finally, by computing a point using the method proposed in Section
4.2 with $\varepsilon=0.01$ results in the sigma points $z_{1}=-0.0001$,
$z_{2}=0.0001$ with weights respectively given by $w_{1}=0.1$ and $w_{2}=0.9$.
In both Method 1 and 2, the MATLAB toolbox Gloptipoly 3 was used to relax the
polynomial optimization problems (13) and (16). A relaxation order of $2$ was
used and the global optimality of the problems was numerically certified by
Gloptipoly3. Moreover, higher relaxation orders could not be used, since using
bigger relaxation orders results in semi-definite programs with a large number
of variables and constraints, and thus yields an unstable numeric problem.
Figure 3 illustrates the semi-algebraic variety $S_{\varepsilon}$ and the
coordinates of the sigma points calculated by the three methods. It can be
seen by Figure 3 that the point computed by using the outer-bounding box
approximation is the best approximation to the real Chebychev center of the
variety. Moreover, it is important to note that the point calculated by Method
2 is very far from the real Chebychev center of the set $S_{\varepsilon}$.
This is due to the semi-definite relaxation of (16) being far from the exact
solution. In principle, one can increase the relaxation, but this can lead to
an unstable numeric problem, which global optimality cannot be more certified.
Figure 3: Semi-algebraic variety $S_{\varepsilon}$ with $\varepsilon=0.01$,
$m=1$, $\mathcal{K}_{1}=\emptyset$, $\mathcal{K}_{2}=\\{1,2\\}$,
$\ell_{1}=-3$, $u_{1}=4$, $\ell_{2}=0$ and $u_{2}=5$. The blue point is the
coordinates of the sigma-point obtained by the naive method, the red point is
the coordinates of the sigma-point obtained by the outer-bounding box
approximation, and the green point is obtained by solving the polynomial
optimization problem (16) directly. Only $\omega_{1}$, $z_{1}$ and $z_{2}$ are
illustrated in the graphic, since $\omega_{2}=1-\omega_{1}$.
Finally, to illustrate the robustness of the computed point, $100$ random
samples from $E\\{X\\}$ and $E\\{X^{2}\\}$ are respectively draw from the
uniform distributions $U(\ell_{1},u_{1})$ and $U(\ell_{2},u_{2})$. Solving
(18) for $f(x)=\sin(x)$ with $500$ samples gives an estimate for $D$ of
$0.9956$, and the posterior expectation in (1) is approximated as
$\textrm{E}\left\\{\sin(X)\right\\}\approx\omega_{1}\sin(z_{1})+\omega_{2}\sin(z_{2}),$
where $\omega_{1}$, $\omega_{2}$, $z_{1}$ and $z_{2}$ are computed according
to the above methods. The mean error
$\left|E\\{\sin(X)\\}-\sum_{i=1}^{2}\omega_{i}\sin(z_{i})\right|$ by the naive
method is $0.0339$, while by Method 1 is $3.6932\cdot 10^{-6}$ and by Method 2
is $1.2085\cdot 10^{-4}$.
## 7 Conclusion
In this paper, it was proposed a way to devise a robust unscented transform by
the computation of the Chebychev center of the semialgebraic set defined by
the possible choices of sigma points and its weights. Although, in general,
this problem is NP-Hard, some methods are proposed in this paper to
approximate the solution of the original problem by convex optimization
problems. Further works intend to generalize the present work to higher
dimensions, as this could enable novel filter designs for multivariate
dynamical systems. Another possible extension to this work is to consider
adjustments of the proposed algorithm to work with confidence bounds for the
moments instead of absolute upper and lower bounds, since the former is more
common to be used on interval estimation techniques, such as the bootstrapping
method.
Finally, one limitation of the method is that a great number of sigma points
can lead to a numerically unstable relaxation of the polynomial optimization
problems. Nevertheless, recent advances in polynomial optimization techniques
such as novel relaxation hierarchies based on linear programming [36] could
help to scale the presented approaches to a higher number of sigma points.
## Acknowledgments
The authors would like to thanks Dario Piga and Henrique M. Menegaz for the
worthy suggestions and discussions relevant to this paper. We would also like
to thanks the Brazilian agencies CNPq and CAPES which partially supported this
work.
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|
# Content Delivery over Broadcast Erasure Channels with Distributed Random
Cache
Alireza Vahid Shih-Chun Lin I-Hsiang Wang Yi-Chun Lai Alireza Vahid is with
the Electrical Engineering Department of the University of Colorado Denver,
Denver, CO, USA. Email<EMAIL_ADDRESS>Shih-Chun Lin and Yi-Chun
Lai are with Department of Electrical and Computer Engineering, NTUST, Taipei,
Taiwan. Email<EMAIL_ADDRESS>I-Hsiang Wang is with Department of
Electrical Engineering, National Taiwan University, Taipei, Taiwan. Email:
<EMAIL_ADDRESS>
###### Abstract
We study the content delivery problem between a transmitter and two receivers
through erasure links, when each receiver has access to some random side-
information about the files requested by the other user. The random side-
information is cached at the receiver via the decentralized content placement.
The distributed nature of receiving terminals may also make the erasure state
of two links and indexes of the cached bits not perfectly known at the
transmitter. We thus investigate the capacity gain due to various levels of
availability of channel state and cache index information at the transmitter.
More precisely, we cover a wide range of settings from global delayed channel
state knowledge and a non-blind transmitter (i.e. one that knows the exact
cache index information at each receiver) all the way to no channel state
information and a blind transmitter (i.e. one that only statistically knows
cache index information at the receivers). We derive new inner and outer
bounds for the problem under various settings and provide the conditions under
which the two match and the capacity region is characterized. Surprisingly,
for some interesting cases the capacity regions are the same even with single-
user channel state or single-user cache index information at the transmitter.
## I Introduction
Available receiver-end side-information can greatly enhance content delivery
and increase the attainable data rates in wireless systems. In particular, in
various applications such as caching [1, 2], coded computing [3], private
information retrieval [4, 5], and index coding [6, 7, 8], side-information is
intentionally and strategically placed at each receiver’s cache during some
placement phase in order to lighten future communication loads. However, in a
more realistic setting, there is no centralized mechanism to populate the
local caches. On the other hand, receivers may obtain some side-information by
simply over-hearing the signals intended for other nodes over the shared
wireless medium. The transmitter(s) may or may not be aware of the exact
content of the side-information at each receiver, and if the transmitter is
not aware of the exact content, the problem is referred to as Blind Index
Coding [9]. Further, the transmitter(s) may be enhanced by receiving channel
state feedback. This paper focuses on content delivery in wireless networks
with potential channel state feedback and/or random available side-information
at the receivers.
In a packet-based communication network, instead of the classic Gaussian
channel, the network-coding-based approaches generally model each
communication hop as a packet erasure channel. In this work, as [10], we focus
on broadcast erasure channels with random receiver side-information. More
specifically, we consider a single transmitter communicating with two
receiving terminals through erasure links. Through decentralized content
placement [9][10], each receiver has randomly cached some side-information
about the message bits (file) of the other user. In [10], for both users,
their indexes of cached bits during the placement and (delayed) channel
erasure state during the delivery are globally known at the transmitter.
However, as pointed out in [11][12], in distributed networks, acquiring such
global information at transmitters is prohibitive due to the extensive date
exchange and the heterogenous capabilities for receiving terminals to feed
back. Thus we consider three scenarios about the transmitter’s knowledge of
the receiver-end side-information: (1) the blind-transmitter case where only
the statistics of the _random_ cache index information is known to the
transmitter, (2) the non-blind-transmitter case where the transmitter knows
exactly what each receiver has access to, and (3) the semi-blind-transmitter
case that knows the exact cache index information at only one of the
receivers. Meanwhile, a range of assumptions on the availability of channel
state information at the transmitter (CSIT) are also considered: (1) when both
receivers provide delayed CSI to the transmitter (DD); (2) when only one
receiver provides delayed CSI to the transmitter (DN); and (3) when receivers
do not provide any CSI to the transmitter (NN).
The blind-transmitter case under scenario NN in [9] corresponds to the setting
that during the whole content placement and delivery, both receivers can not
have capabilities to feed back through the control channel or shared wireless
medium. Then the transmitter has neither cache index information nor CSI. On
the other hand, for each receiver, if the feedback resource is available for
some period of time and then becomes unavailable or intermittent [13, 14],
then we have other scenarios. For example, though during the placement both
receivers can feed back cache indexes, during the delivery one of the receiver
may not be able to feed back the fast varying channel state of its own link.
Then we have a non-blind transmitter under scenario DN. This setting could
also arise due to new security threats that aim to disrupt the flow of control
packets in distributed systems [15, 16, 17], that is, during the delivery the
CSI feedback from one of the receivers is severely attacked.
Related Work and Literature Review: To better place our work within the
literature for cached network or index coding, we summarize some of the main
results in the literature. The classic index coding problem [18, 6, 19]
assumes the transmitter is aware of the exact side-information at each node
[7, 20], and the network does not have wireless links directly. This problem
has been a powerful tool in studying the data communication network with
receiver caching [21, 8, 22].
For index coding over wireless links, as aforementioned, two extreme cases
have been studied [9][10]. In [10], the capacity regions for two and three-
user broadcast erasure channels, when the transmitter has access to global
delayed CSI and cached index information, are characterized. This setting
assumes rather strong and stable feedback channels from all receivers. On the
other hand, blind wireless index coding is introduced in [9] where the
transmitter only knows the statistics of the CSI and cached index information
since there is no feed back at all from the two users. However, a limitation
is added in [9] such that only the user with weaker link (higher erasure
probability) has cache. Even under this limitation, the capacity region is not
fully known In conclusion, there remains a gap in the index coding literature
between these two extreme points when it comes to wireless setting, which is
the target of our paper. More comparisons with [9][10] can be found in Section
III.
Contributions: In this work, we consider the two-user erasure broadcast
channel with the erasure states of the two links being independent of each
other. We first present the capacity for the two-user broadcast erasure
channel with a non-blind transmitter under scenario NN, that is, no CSI
feedback. For scenario NN, we observe that the stronger receiver, _i.e._ the
one whose channel has a smaller erasure probability, will eventually be able
to decode messages intended for both receivers. Note that this observation
does not imply that the broadcast channel is degraded due to the cache.
Interestingly, the achievability derived from this observation indicates that
our optimal protocol works even when the transmitter does not know the exact
cache index information at the stronger receiver, _i.e._ with a “semi-blind”
transmitter. To derive the outer-bounds, besides using the aforementioned
observations, we also show an extremal entropy inequality between the two
receivers that captures the availability of receiver-end side-information.
With a blind transmitter, these outer-bounds can be automatically applied
since the transmitter has more knowledge in the non-blind case, and we show
that the outer bounds can be achieved when the channel is symmetric. When some
delayed CSI is available, we also demonstrate the capacity with a non-blind
transmitter and global CSI (scenario DD) [10] can be achieved when the
transmitter has less information. In particular, for the blind-transmitter
case under scenario DD, we provide an optimal protocol for the symmetric
channel. For the non-blind-transmitter case or semi-blind-transmitter case, we
also extend our earlier non-cached capacity result for scenario DN [12], and
show that even with cache the capacity region with only single-user CSI feed
back can match that with global CSI in [10].
TABLE I: Summary of Contributions Categorized by Transmitter’s Information State | Cache | Contributions | State | Cache | Contributions
---|---|---|---|---|---
W | S | W | S | | W | S | W | S |
✗ | ✗ | ✓ | ✓ | *$\mathcal{C}^{\mathrm{non-blind}}_{\mathrm{NN}}$ in Theorem 1 | ✓ | ✗ | ✗ | ✓ | $\mathcal{C}^{\mathrm{semi-blind}}_{\mathrm{DN}}$ in Theorem 2 Case B
✗ | ✗ | ✓ | ✗ | *$\mathcal{C}^{\mathrm{semi-blind}}_{\mathrm{NN}}$ in Corollary 1 | ✗ | ✓ | ✗ | ✗ | Remains open
✗ | ✗ | ✗ | ✓ | Inner-bounds in Theorem 4 | ✓ | ✓ | ✓ | ✓ | $\mathcal{C}^{\mathrm{non-blind}}_{\mathrm{DD}}$ previously known [10]
✗ | ✗ | ✗ | ✗ | Symmetric $\mathcal{C}^{\mathrm{blind}}_{\mathrm{NN}}$ in Theorem 3 | ✓ | ✓ | ✓ | ✗ | $\mathcal{C}^{\mathrm{semi-blind}}_{\mathrm{DD}}$ in Theorem 2 Case B
✗ | ✓ | ✓ | ✓ | *$\mathcal{C}^{\mathrm{non-blind}}_{\mathrm{DN}}$ in Theorem 2 Case C | ✓ | ✓ | ✗ | ✓ | $\mathcal{C}^{\mathrm{semi-blind}}_{\mathrm{DD}}$ in Theorem 2 Case B
✗ | ✓ | ✓ | ✗ | $\mathcal{C}^{\mathrm{semi-blind}}_{\mathrm{DN}}$ in Theorem 2 Case B | ✓ | ✓ | ✗ | ✗ | Symmetric $\mathcal{C}^{\mathrm{blind}}_{\mathrm{DD}}$ in Theorem 2 Case A
Table I summarizes our contributions, which we will further elaborate upon in
Section III. In this table, “W” stands for the weak user (the one with higher
erasure probability) and “S” stands for the strong user. Further, “State”
columns are for CSIT, and “Cache” columns are for the transmitter’s knowledge
of bit indices cached at the receiver: “✓” indicates availability and “✗”
indicates missing information. For example, “✗✗✓✗” is the no CSI case (NN)
with semi-blind Tx that knows the cache index information at the weaker
receiver. For the three cases marked with “*”, the capacity regions are fully
identified without additional limitations on system parameters.
Paper Organization: The rest of the paper is organized as follows. In Section
II, we present the problem setting and the assumptions we make in this work.
Section III presents the main contributions and provides further insights and
interpretations of the results. The proof of the main results will be
presented in the following sections. Finally, Section VIII concludes the
paper.
## II Problem Formulation
Following [10], we consider the canonical two-user broadcast erasure channel
in Figure 1 to understand how transmitters can exploit the available side-
information at the receivers to improve the capacity region. In this network,
a transmitter, $\mathsf{Tx}$, wishes to transmit two independent messages
(files), $W_{1}$ and $W_{2}$, to two receiving terminals $\mathsf{Rx}_{1}$ and
$\mathsf{Rx}_{2}$, respectively, over $n$ channel uses. Each message, $W_{i}$,
contains $m_{i}$ data packets (or bits) which we denote by
$\vec{a}=\left(a_{1},a_{2},\ldots,a_{m_{1}}\right)$ for $\mathsf{Rx}_{1}$, and
by $\vec{b}=\left(b_{1},b_{2},\ldots,b_{m_{2}}\right)$ for $\mathsf{Rx}_{2}$.
Here, we note that each packet is a collection of encoded bits, however, for
simplicity and without loss of generality, we assume each packet is in the
binary field, and we refer to them as bits. Extensions to broadcast packet
erasure channels where packets are in large finite fields are straightforward
as in [10][23].
Channel model: At time instant $t$, the messages are mapped to channel input
$X[t]\in\mathbb{F}_{2}$, and the corresponding received signals at
$\mathsf{Rx}_{1}$ and $\mathsf{Rx}_{2}$ are
$\displaystyle
Y_{1}[t]=S_{1}[t]X[t]~{}~{}\;\mbox{and}\;~{}~{}Y_{2}[t]=S_{2}[t]X[t],$ (1)
respectively, where $\left\\{S_{i}[t]\right\\}$ denotes the Bernoulli
$(1-\delta_{i})$ process that governs the erasure at $\mathsf{Rx}_{i}$, and is
independently and identically distributed (i.i.d.) over time and across users.
When $S_{i}[t]=1$, $\mathsf{Rx}_{i}$ receives $X[t]$ noiselessly; and when
$S_{i}[t]=0$, it receives an erasure. In other words, as we assume receivers
are aware of their local channel state information, each receiver can map the
received signal when $S_{i}[t]=0$ to an erasure.
CSI assumptions: We assume the receivers are aware of the channel state
information (_i.e._ global CSIR). For the transmitter, on the other hand, we
assume the following scenarios:
1. 1.
NN or No CSIT model: The transmitter knows only the erasure probabilities and
not the actual channel realizations;
2. 2.
DN model: The transmitter knows the erasure probabilities and the actual
channel realizations of one receiver with unit delay;
3. 3.
DD or delayed CSIT model: The transmitter knows the erasure probabilities and
the actual channel realizations of both receivers with unit delay.
Figure 1: Two-user broadcast erasure channels with channel state feedback and
random side-information at each receiver.
Available receiver side-information: Decentralized content placement [9][10]
is adopted, where each user independently caches a subset of the message bits
(file). In particular, we assume a random fraction $(1-\epsilon_{i})$ of the
bits intended for receiver $\mathsf{Rx}_{\bar{i}}$ is cached at
$\mathsf{Rx}_{{i}}$, $\bar{i}\overset{\triangle}{=}3-i$, and we denote this
side information with $W_{\bar{i}|i}$ as in Figure 1. This assumption on the
available side-information at each receiver could also be represented using an
erasure side channel. More precisely, we can assume available side-information
to receiver $i$ is created through
$\displaystyle
E_{i}[\ell]W_{\bar{i}}[\ell],\qquad\ell=1,2,\ldots,nR_{\bar{i}},$ (2)
where $W_{\bar{i}}[\ell]$ is the $\ell^{\mathrm{th}}$ bit of message
$W_{\bar{i}}$, while the cache index information $E_{i}[\ell]$ is an i.i.d.
Bernoulli $(1-\epsilon_{i})$ process independent of all other channel
parameters and known at receiver $i$. To concisely present our result, in our
placement model each user does not cache its own message but only the
interference. Our results can be easily extended to the case when both the own
message and interference bits are cached.
The Transmitter’s knowledge of the cache index: Following the convention which
presents the length $nR_{\bar{i}}$ sequence $E_{i}[1],\ldots
E_{i}[nR_{\bar{i}}]$ as $E^{nR_{\bar{i}}}_{i}$, we consider three scenarios
for the blindness of the cache index at the transmitter
1. 1.
Blind Transmitter: In this scenario, the transmitter’s knowledge of the
receiver side-information is limited to the values of $\epsilon_{1}$ and
$\epsilon_{2}$, while the cache index information $E^{nR_{2}}_{1}$ and
$E^{nR_{1}}_{2}$ in (2) is unknown.
2. 2.
Semi-Blind Transmitter: In this scenario, the transmitter’s knowledge of the
side-information at $\mathsf{Rx}_{i}$ is limited to the value of
$\epsilon_{i}$, while the transmitter knows $W_{i|\bar{i}}$ through
$E^{nR_{i}}_{\bar{i}}$
3. 3.
Non-Blind Transmitter: In this scenario, the transmitter knows exactly what
fraction of each message is available to the unintended receiver. In other
words, the transmitter knows $W_{2|1}$ and $W_{1|2}$ through the the cache
index information $E^{nR_{2}}_{1}$ and $E^{nR_{1}}_{2}$.
Encoding: We start with the NN model where the constraint imposed at the
encoding function $f_{t}(.)$ at time index $t$ for the blind scenario is
$\displaystyle X[t]=f_{t}\left(W_{1},W_{2},\mathsf{PI}\right),$ (3)
for the non-blind scenario is
$\displaystyle
X[t]=f_{t}\left(W_{1|2},\bar{W}_{1|2},W_{2|1},\bar{W}_{2|1},\mathsf{PI}\right),$
(4)
and
$\displaystyle
X[t]=f_{t}\left(W_{i|\bar{i}},\bar{W}_{i|\bar{i}},W_{\bar{i}},\mathsf{PI}\right),$
(5)
where $\mathsf{PI}$ represents the knowledge of statistical parameters
$\delta_{1},\delta_{2},\epsilon_{1},$ and $\epsilon_{2}$, and
$\bar{W}_{i|\bar{i}}$ is the complement of $W_{i|\bar{i}}$ with respect to
$W_{i}$, $i=1,2$. Although not ideal, this notation is adopted to highlight
the transmitter’s knowledge of the available side-information at the
receivers.
For the DD case, $S^{t-1}$ is added to the inputs of $f_{t}(.)$, while under
DN model, only of the channels is revealed to the transmitter up to time
$(t-1)$. Rather than enumerating all possibilities, we present an example, to
clarify the encoding constraints. Suppose the transmitter is knows the side-
information available to $\mathsf{Rx}_{2}$ (semi-blind), and has access to the
delayed CSI from $\mathsf{Rx}_{1}$ (DN model), then, we have
$\displaystyle
X[t]=f_{t}\left(W_{1|2},\bar{W}_{1|2},W_{2},S_{1}^{t-1},\mathsf{PI}\right),$
(6)
Decoding: Each receiver $\mathsf{Rx}_{i}$, $i=1,2$ knows its own CSI across
entire transmission block $S^{n}_{i}$, and the CSI $S^{n}_{\bar{i}}$ if the
other receiver $\bar{i}$ provides feedback. Under scenario NN it uses a
decoding function
$\varphi_{i,n}\left(Y_{i}^{n},S_{i}^{n},W_{\bar{i}|i}\right)$ to get an
estimate $\widehat{W}_{i}$ of $W_{i}$, while under scenario DD the decoding
function becomes $\varphi_{i,n}\left(Y_{i}^{n},S^{n},W_{\bar{i}|i}\right)$
where $S^{n}=(S_{1}^{n},S^{n}_{2})$. Note that under scenario DN, only the no-
feedback receiver has global $S^{n}$. An error occurs whenever
$\widehat{W}_{i}\neq W_{i}$. The average probability of error is given by
$\displaystyle\lambda_{i,n}=\mathbb{E}[P(\widehat{W}_{i}\neq W_{i})],$ (7)
where the expectation is taken with respect to the random choice of the
transmitted messages.
Capacity region: We say that a rate pair $(R_{1},R_{2})$ is achievable, if
there exists a block encoder at the transmitter, and a block decoder at each
receiver, such that $\lambda_{i,n}$ goes to zero as the block length $n$ goes
to infinity. The capacity region, $\mathcal{C}$, is the closure of the set of
the achievable rate pairs. Throughout the paper, we will distinguish the
capacity region under different assumptions. For example,
$\mathcal{C}^{\mathrm{blind}}_{\mathrm{NN}}$ is the capacity region of the
two-user broadcast erasure channels with a blind transmitter and no CSIT.
## III Main Results
In this section, we present the main contributions of this paper and provide
some insights and intuitions about the findings.
### III-A Statement of the Main Results
We start with scenarios in which we characterize the capacity region, and
then, we present cases for which we derive new inner-bounds. In Theorem 1, for
the no CSIT scenario, we establish the capacity region with a non-blind
transmitter. We will highlight the importance of side-information at the
weaker receiver and how a semi-blind transmitter may achieve the same region
in Remarks 2 and 1, respectively. Next, we present new capacity results when
(some) delayed CSI is available to the transmitter in Theorem 2. Next, for the
no CSIT assumption and a blind transmitter, in Theorem 3 we present new
conditions beyond [9] under which the capacity region is achievable, and a new
achievable region is presented in Theorem 4.
###### Theorem 1.
For the two-user broadcast erasure channel with a non-blind transmitter and no
CSIT as described in Section II, we have
$\mathcal{C}^{\mathrm{non-
blind}}_{\mathrm{NN}}\equiv\left\\{\begin{array}[]{ll}0\leq\beta^{\mathrm{no}}_{1}R_{1}+R_{2}\leq\left(1-\delta_{2}\right),&\\\
0\leq
R_{1}+\beta^{\mathrm{no}}_{2}R_{2}\leq\left(1-\delta_{1}\right).&\end{array}\right.$
(8)
where
$\displaystyle\beta^{\mathrm{no}}_{i}=\epsilon_{\bar{i}}\min\left\\{\frac{1-\delta_{\bar{i}}}{1-\delta_{i}},1\right\\}.$
(9)
The derivation of the outer-bounds has two main ingredients. First, as
detailed in upcoming Remark 1, even the channel is not degraded, the stronger
receiver can decode both messages regardless of the values of $\epsilon_{1}$
and $\epsilon_{2}$ for receiver cache. Second, as detailed in upcoming Lemma
2, we derive an extremal entropy inequality between the two receivers that
captures the availability of receiver-end side-information, including the
channel state and cache index information. The outer-bound region holds for
the non-blind setting and thus, includes the capacity region with a blind
transmitter as well. The following two remarks provide further insights.
###### Remark 1 (Simplified expressions and degradedness).
Without loss of generality, assume $\delta_{2}\geq\delta_{1}$, meaning that
receiver $1$ has a stronger channel. Then, the region of Theorem 1 can be
written as
$\left\\{\begin{array}[]{ll}0\leq\epsilon_{2}\frac{1-\delta_{2}}{1-\delta_{1}}R_{1}+R_{2}\leq\left(1-\delta_{2}\right),&\\\
0\leq R_{1}+\epsilon_{1}R_{2}\leq\left(1-\delta_{1}\right).\end{array}\right.$
(10)
Unlike the scenario with no side-information at the receivers, this assumption
does not mean the channel is degraded. However, the stronger receiver, ${\sf
Rx}_{1}$ in this case, will be able to decode both $W_{1}$ and $W_{2}$ by end
of the communication block regardless of the values of $\epsilon_{1}$ and
$\epsilon_{2}$. The reason is as follows. After decoding $W_{1}$, receiver
${\sf Rx}_{1}$ has access to the side-information of receiver ${\sf Rx}_{2}$,
_i.e._ $W_{1|2}$, and can emulate the channel of ${\sf Rx}_{2}$ as it has a
stronger channel ($\delta_{2}\geq\delta_{1}$). Finally, we note that although
the stronger receiver is able to decode both messages, this does not imply
that the stronger receiver will have a higher rate. As an example, suppose
$\delta_{1}=1/3,\delta_{2}=1/2,\epsilon_{1}=2/3,$ and $\epsilon_{2}=1/6$.
Then, from the Theorem 1, the maximum sum-rate point is:
$\displaystyle\left(R_{1},R_{2}\right)=\left(\frac{4}{11},\frac{5}{11}\right).$
(11)
###### Remark 2 (Importance of side-information at the weaker receiver).
Under the same assumption of the previous remark, $\delta_{2}\geq\delta_{1}$,
from the outer-bounds of Theorem 1, we conclude that if the weaker receiver
has no side-information, _i.e._ $\epsilon_{2}=1$, then, the capacity region is
the same as having no side-information at either receivers. In other words, as
long as the weaker receiver has no side-information, additional information at
the stronger receiver does not enlarge the region. On other other hand, any
side-information at the weaker receiver results in an outer-bound region that
is strictly larger than the capacity region with no side-information at either
receivers.
Based on these remarks, we can provide more details on the achievability
protocol. Under the no-CSIT assumption, the stronger receiver will eventually
be able to decode both messages. Thus, the first step is to deliver the
message intended for the weaker receiver. The stronger receiver will be able
to decode this message faster than the intended receiver and thus, in the
second step, we include the part of the message for the stronger user that is
available at the weaker receiver. In other words, this second step is
beneficial to both receivers. We note that to accomplish this task, the
transmitter at least needs to know the side-information of the weaker
receiver. This latter fact is further explained in the Corollary below. During
the final step, the remaining part of the message intended for the stronger
receiver is delivered.
As will be detailed in Section V, to achieve the outer-bounds, indeed the
transmitter only needs to know the side-information available to the weaker
receiver. Thus, we have
###### Corollary 1.
The capacity region with a semi-blind transmitter $\mathcal{C}^{\mathrm{semi-
blind}}_{\mathrm{NN}}$ equals to $\mathcal{C}^{\mathrm{non-
blind}}_{\mathrm{NN}}$ in Theorem 1 if the cache index information (2) of the
weaker receiver (link with larger erasure probability) is known at the
transmitter.
The following lemma from [10] establishes the outer-bounds on the capacity
region of the two-user broadcast erasure channel with a non-blind transmitter
and delayed CSIT, $\mathcal{C}^{\mathrm{non-blind}}_{\mathrm{DD}}$. We provide
several new achievability strategies to achieve these bounds when the
transmitter has less knowledge compared to what these bounds assume, and
establish interesting capacity results.
###### Lemma 1 ([10]).
For the two-user broadcast erasure channel with a non-blind transmitter and
delayed CSIT as described in Section II, we have outer-bound region
$\mathcal{C}^{\mathrm{non-
blind}}_{\mathrm{DD}}\subseteq\left\\{\begin{array}[]{ll}\vspace{1mm}0\leq
R_{i}\leq(1-\delta_{i}),&i=1,2,\\\
\beta^{\mathrm{delayed}}_{i}R_{i}+R_{\bar{i}}\leq\left(1-\delta_{\bar{i}}\right),&i=1,2.\end{array}\right.$
(12)
where
$\displaystyle\beta^{\mathrm{delayed}}_{i}=\epsilon_{\bar{i}}\frac{1-\delta_{\bar{i}}}{1-\delta_{1}\delta_{2}}.$
(13)
Now, we show that this outer-bound region is achievable under the following
scenarios.
###### Theorem 2.
For the two-user broadcast erasure channel, the capacity region is achieved
when:
Case A: with a blind transmitter and global delayed CSIT, the capacity region
$\mathcal{C}^{\mathrm{blind}}_{\mathrm{DD}}$ equals to (12) when the channel
is symmetric (_i.e._ $\delta_{1}=\delta_{2}=\delta$ and
$\epsilon_{1}=\epsilon_{2}=\epsilon$);
Case B: with the transmitter knowing full side-information from one receiver
and only the delayed CSI of the other receiver (e.g., the semi-blind-
transmitter case with $\epsilon_{1}=0$ for $\mathsf{Rx}_{1}$), the capacity
region $\mathcal{C}^{\mathrm{semi-blind}}_{\mathrm{DN}}$ (and thus
$\mathcal{C}^{\mathrm{semi-blind}}_{\mathrm{DD}}$) equals to (12)
Case C: with a non-blind transmitter having access to only the delayed CSI of
one receiver, the capacity region $\mathcal{C}^{\mathrm{non-
blind}}_{\mathrm{DN}}$ equals to (12).
Note that having full side-information at a receiver (as in Case B above)
immediately implies the transmitter is not blind with respect to that
receiver. For the blind-transmitter case, if $\epsilon_{i}=0$ then the
transmitter knows $\mathsf{Rx}_{i}$ has full side-information, and thus
$\mathcal{C}^{\mathrm{blind}}_{\mathrm{DN}}$ is also partially known from Case
B.
Without the global channel state and/or cache index information from both
receivers, our new achievability results of Theorem 2 differ significantly
from the those in [10]. In particular in [10], overheard bits and cached bits
are both known at the transmitter to create network coding opportunities
simultaneously benefit for both receivers. In our achievability, network
coding opportunities can only be opportunistically created. Rather
interestingly, in three cases identified by our Theorem, transmitter blindness
or one-sided feedback may not result in any capacity loss compared with [10].
To achieve capacity $\mathcal{C}^{\mathrm{blind}}_{\mathrm{DD}}$ in Case A, we
present an opportunistic protocol where the transmitter first sends out linear
combinations of the packets for both receivers. Then, using the feedback
signals and the fact that some of the packets for one receiver are available
at the other, the transmitter sends bits for intended for one receiver in such
as way to help one receiver remove interference and the other to obtain new
information about its bits. Depending on channel parameters, this may follow
with a multicast phase. This idea could be interpreted as an opportunistic
reverse network coding for erasure channels. Both Cases B and C focus on
cached capacity with only single-user delayed CSI. Interestingly, our four-
phase opportunistic network coding for Case C will also create blind side-
information at the “N” user for the recycled bits. Indeed, the last two phases
in Case C is modified form the achievability for Case B. Compared with [12],
the new ingredient for Case C is the non-blind cache, and we propose to
generalize [12] by carefully mixing the fresh cached bits with the recycled
un-cached bits. Specifically, the recycling in [12] is done by mixing two pre-
encoded bit sequences. On the contrary to [12] where the input sequence of
each pre-encoder is always recycled, it may contain fresh bits in our Case C
as detailed in Sec. VII.
The following two theorems focus on the no-CSIT blind-transmitter assumption.
The first one identifies conditions under which the outer-bound region of
Theorem 1 can be achieved even when the transmitter does not know what side-
information is available to each receiver. The second theorem presents an
achievable region when the stronger receiver has full side-information, but
this achievable region does not match the outer-bounds.
###### Theorem 3.
For the two-user broadcast erasure channel with no channel feedback, a blind
transmitter, and available receiver side-information as described in Section
II, the capacity $\mathcal{C}^{\mathrm{blind}}_{\mathrm{NN}}$ equals to
$\mathcal{C}^{\mathrm{non-blind}}_{\mathrm{NN}}$ in Theorem 1 when:
1. 1.
When $\delta_{2}\geq\delta_{1}$ and $\epsilon_{2}\in\\{0,1\\}$;
2. 2.
Symmetric setting: when $\delta_{1}=\delta_{2}=\delta$, and
$\epsilon_{1}=\epsilon_{2}=\epsilon$.
###### Theorem 4.
For the two-user broadcast erasure channel with no channel feedback, a blind
transmitter, as described in Section II, when $\delta_{2}\geq\delta_{1}$ and
$\epsilon_{1}=0$, we can the following region is achievable:
$\mathcal{R}^{\mathrm{in}}\equiv\left\\{\begin{array}[]{ll}0\leq\left(\epsilon_{2}+\delta_{1}(1-\epsilon_{2})\right)\frac{1-\delta_{2}}{1-\delta_{1}}R_{1}+R_{2}\leq\left(1-\delta_{2}\right),&\\\
0\leq R_{1}\leq\left(1-\delta_{1}\right).&\end{array}\right.$ (14)
We note that for $\delta_{1}=0$ (no erasure at the stronger receiver), the
inner-bound region of Theorem 4 matches the outer-bounds of Theorem 1, _i.e._
$\mathcal{R}^{\mathrm{in}}\equiv\mathcal{C}^{\mathrm{blind}}_{\mathrm{NN}}$ .
Note that the blind-transmitter case under no-CSIT was also considered in [9]
but only the weaker receiver has side-information. In that setting, outer and
inner bounds were presented that match only when erasure probabilities are
equal to zero, which is no longer an erasure setting. In contrast, in this
work, we have the capacity region $\mathcal{C}^{\mathrm{non-
blind}}_{\mathrm{NN}}$ in Theorem 1 for the non-blind-transmitter case, which
recovers the outer-bounds of [9] as a special case. The capacity region
$\mathcal{C}^{\mathrm{blind}}_{\mathrm{NN}}$ of blind index coding over no-
CSIT broadcast erasure channel remains open in general.
### III-B Illustration of the results
In this subsection, we briefly illustrate the results through a few examples
to further clarify and discuss some of the insights and intuitions provided
above.
We start with Theorem 2 where the transmitter has access to (some) delayed
CSI. Figure 2 illustrates the capacity region
$\mathcal{C}^{\mathrm{blind}}_{\mathrm{DD}}=\mathcal{C}^{\mathrm{non-
blind}}_{\mathrm{DN}}$ from Theorem 2 when $\delta_{1}=\delta_{2}=\delta=0.5$
and $\epsilon_{1}=\epsilon_{2}=\epsilon\in\\{0,0.5,1\\}$. In particular,
$\epsilon=1$ is the case in which no side-information is available and our
results recover [24]. The other extreme is $\epsilon=0$ where the entire
message of one user is available to the other and maximum individual point-to-
point rates can be achieved. Finally, $\epsilon=0.5$ is an intermediate case
and the capacity is strictly larger than that of no side-information.
Figure 2: Capacity region
$\mathcal{C}^{\mathrm{blind}}_{\mathrm{DD}}=\mathcal{C}^{\mathrm{non-
blind}}_{\mathrm{DN}}$ for $\delta_{1}=\delta_{2}=\delta=0.5$ and
$\epsilon_{1}=\epsilon_{2}=\epsilon\in\\{0,0.5,1\\}$.
We then consider the no-CSIT scenario. For the first example in this case, we
consider $\delta_{1}=\frac{1}{2}$ and $\delta_{2}=\frac{3}{4}$. The capacity
region of the broadcast erasure channel with these parameters and no side-
information at either receiver is described by all non-negative rates
satisfying:
$\displaystyle\frac{1}{2}R_{1}+R_{2}\leq\frac{1}{4}.$ (15)
Note that we no side-information, the channel is degraded. Further, as
discussed earlier, as long as the weaker receiver (${\sf Rx}_{2}$ in this
case) has no side-information, _i.e._ $\epsilon_{2}=1$, the capacity region
remains identical to the one described in (15) with no side-information at
either receivers. This region is included in Figure 3(a) and (b) as benchmark.
Note that significant caching or index coding gains are obtained for all
settings presented in Figure 3(a) and (b).
Figure 3: (a) Illustration of the capacity region $\mathcal{C}^{\mathrm{non-
blind}}_{\mathrm{NN}}$ when ${\sf Rx}_{i}$ has full side-information
($\epsilon_{i}=0$); (b) increase in the achievable rates as $\epsilon_{1}$
goes from $1$ to $0$ for $\epsilon_{2}=0.5$.
Next, we examine the region $\mathcal{C}^{\mathrm{non-blind}}_{\mathrm{NN}}$
when one receiver has full side-information. Figure 3(a) includes both these
cases and depicts how the capacity region enlarges as more side-information is
available to the receivers. We note that with full side-information at both
receivers ($\epsilon_{1}=\epsilon_{2}=0$), maximum individual rates given by
$R_{i}=(1-\delta_{i})$, $i=1,2$, are achievable simultaneously. Figure 3(b)
depicts the gradual increase in achievable rates when
$\epsilon_{2}=\frac{1}{2}$ and $\epsilon_{1}$ goes from $1$ (no side
information) to $0$ (full side-information). Note that receiver ${\sf Rx}_{1}$
has a stronger channel so the illustrated region also equals to the capacity
region $\mathcal{C}^{\mathrm{semi-blind}}_{\mathrm{NN}}$ when the transmitter
has no cache index information of ${\sf Rx}_{1}$.
Figure 4: (a) Although the stronger receiver will always be able to decode
both messages, the weaker receiver may have a higher rate. In this example for
$\mathcal{C}^{\mathrm{non-blind}}_{\mathrm{NN}}$, we have
$\delta_{1}=\frac{1}{3},\delta_{2}=\frac{1}{2},\epsilon_{1}=\frac{2}{3},$ and
$\epsilon_{2}=\frac{1}{6}$; (b) Capacity region with symmetric parameters
$\delta_{1}=\delta_{2}=\delta$ and $\epsilon_{1}=\epsilon_{2}=\epsilon$. The
maximum sum-rate point is given by $\frac{2(1-\delta)}{(1+\epsilon)}$. The
capacity region with symmetric parameters can be achieved under the blind-
transmitter scenario as well. See Theorem 3.
As the second example, we consider $\delta_{1}=\frac{1}{3}$ and
$\delta_{2}=\frac{1}{2}$. Similar to the previous case for capacity region
$\mathcal{C}^{\mathrm{non-blind}}_{\mathrm{NN}}$, receiver ${\sf Rx}_{1}$ has
a stronger channel. However, as illustrated in Figure 4(a), with
$\epsilon_{1}=\frac{2}{3}$ and $\epsilon_{2}=\frac{1}{6}$, the weaker receiver
has a higher rate as given in (11). Finally, Figure 4(b) depicts the capacity
region $\mathcal{C}^{\mathrm{non-
blind}}_{\mathrm{NN}}=\mathcal{C}^{\mathrm{blind}}_{\mathrm{NN}}$ with
symmetric channel parameters ($\delta_{1}=\delta_{2}=\delta$ and
$\epsilon_{1}=\epsilon_{2}=\epsilon$): with no side-information, the maximum
achievable sum-rate is $(1-\delta)$; and with side-information (even blind),
the maximum sum-rate is given by:
$\displaystyle\frac{2(1-\delta)}{(1+\epsilon)}.$ (16)
Figure 5: Comparing the inner-bounds of Theorem 4 for
$\mathcal{C}^{\mathrm{blind}}_{\mathrm{NN}}$ to the outer-bounds of Theorem 1
for $\delta_{2}\geq\delta_{1}$ and $\epsilon_{1}=0$. Black dashed lines define
the outer-bound region, while the solid green lines are the inner-bounds. The
shaded area is the gap between the two.
Finally, Figure 5 illustrates the outer-bounds of Theorem 1 and the inner-
bounds of Theorem 4 for $\delta_{2}\geq\delta_{1}$ and $\epsilon_{1}=0$. In
other words, in this case, the stronger receiver has full side-information. We
note that the inner-bounds of Theorem 4 for the blind-transmitter case match
the outer-bounds when $\delta_{1}=0$.
### III-C Organization of the Proofs
In the following sections, we provide the proof of our main contributions. We
prove the capacity region of Theorem 1 in Sections IV and V. We then move to
the achievability part of Theorem 2 as they are capacity-achieving and include
several interesting new ingredients, as in Section VI and VII. The proofs of
other results are deferred to the Appendix.
## IV Converse Proof of $\mathcal{C}^{\mathrm{non-blind}}_{\mathrm{NN}}$ in
Theorem 1
In this section, we derive the outer-bounds of Theorem 1. We note that as the
capacity region of the non-blind setting includes that of the blind
assumption, the derivation in this section is for the non-blind transmitter
case and the bounds apply to the blind-transmitter case as well.
The point-to-point outer-bounds, _i.e._ $R_{i}\leq(1-\delta_{i})$, are those
of erasure channels, and thus, omitted. Without loss of generality, we assume
$\delta_{2}\geq\delta_{1}$, meaning that receiver $1$ has a stronger channel.
As discussed in Remark 1, unlike the scenario with no side-information at the
receivers, this assumption does not mean the channel is degraded. In what
follows, we derive the following outer-bounds:
$\displaystyle\mathbf{B1}:\frac{\epsilon_{2}(1-\delta_{2})}{(1-\delta_{1})}R_{1}+R_{2}\leq\left(1-\delta_{2}\right),$
$\displaystyle\mathbf{B2}:R_{1}+\epsilon_{1}R_{2}\leq\left(1-\delta_{1}\right).$
(17)
Suppose rate-tupe $\left(R_{1},R_{2}\right)$ is achievable. We first derive
$\mathbf{B2}$ to get some insights.
Derivation of $\mathbf{B2}$: As discussed in Remark 1, the stronger receiver,
${\sf Rx}_{1}$ in this case, is able to decode both messages by the end of the
communication block using its available side information. Thus, we have
$\displaystyle H(W_{1}|G^{n})+H(\bar{W}_{2|1}|G^{n})\leq$ $\displaystyle
I(W_{1},\bar{W}_{2|1};Y_{1}^{n}|W_{2|1},G^{n})+n\upxi_{n}\leq
H(Y_{1}^{n}|W_{2|1},G^{n})+n\upxi_{n}\leq n(1-\delta_{1})+n\upxi_{n}.$ (18)
We also note that
$\displaystyle
nH(\bar{W}_{2|1}|G^{n})=\sum^{m_{2}}_{\ell=1}H\big{(}(1-E_{1}[\ell])W_{2}[\ell]\big{|}E_{1}[\ell]\big{)}=n\epsilon_{1}H(W_{2})=n\epsilon_{1}R_{2}.$
(19)
Thus, from (IV) and (19), we get
$\displaystyle n(R_{1}+\epsilon_{1}R_{2})\leq n(1-\delta_{1})+n\upxi_{n}.$
(20)
Dividing both sides by $n$ and let $n\rightarrow\infty$, we get the second
outer-bound in (IV).
Derivation of $\mathbf{B1}$: We enhance receiver $\mathsf{Rx}_{1}$ by
providing the entire $W_{2}$ to it, as opposed to the already available
$W_{2|1}$, and we note that this cannot reduce the rates. Moreover, motivated
from the Derivation of $\mathbf{B2}$, this enhancement should only have
limited rate increase. From the decentralized placement model (2), we define
the global channel state and cache index information as
$G^{n}:=\\{S^{n}_{1},S^{n}_{2},E^{nR_{2}}_{1},E^{nR_{1}}_{2}\\}:=\\{S^{n},E^{n}\\}$
(21)
then, using
$\displaystyle\beta_{1}^{\mathrm{no}}=\frac{\epsilon_{2}(1-\delta_{2})}{(1-\delta_{1})},$
(22)
we have
$\displaystyle n$
$\displaystyle\left(\beta_{1}^{\mathrm{no}}R_{1}+R_{2}\right)=\beta_{1}^{\mathrm{no}}H(W_{1})+H(W_{2})$
$\displaystyle\overset{(a)}{=}\beta_{1}^{\mathrm{no}}H(W_{1}|W_{2},G^{n})+H(W_{2}|W_{1|2},G^{n})$
$\displaystyle\overset{(\mathrm{Fano})}{\leq}\beta_{1}^{\mathrm{no}}I(W_{1};Y_{1}^{n}|W_{2},G^{n})+I(W_{2};Y_{2}^{n}|W_{1|2},G^{n})+n\upxi_{n}$
$\displaystyle=\beta_{1}^{\mathrm{no}}H(Y_{1}^{n}|W_{2},G^{n})-\beta_{1}^{\mathrm{no}}\underbrace{H(Y_{1}^{n}|W_{1},W_{2},G^{n})}_{=~{}0}$
$\displaystyle\quad+H(Y_{2}^{n}|W_{1|2},G^{n})-H(Y_{2}^{n}|W_{1|2},W_{2},G^{n})+n\upxi_{n}$
$\displaystyle\overset{(b)}{\leq}H(Y_{2}^{n}|W_{1|2},G^{n})+2n\upxi_{n}$
$\displaystyle\overset{(c)}{\leq}n\left(1-\delta_{2}\right)+2\upxi_{n},$ (23)
where $\upxi_{n}\rightarrow 0$ as $n\rightarrow\infty$; $(a)$ follows from the
independence of messages and captures the enhancement of receiver
$\mathsf{Rx}_{1}$; $(b)$ follows from Lemma 2 below; $(c)$ is true since the
entropy of a binary random variable is at most one and the channel to the
second receiver is only on a fraction $\left(1-\delta_{2}\right)$ of the
communication time. Dividing both sides by $n$ and let $n\rightarrow\infty$,
we get the first outer-bound in (IV).
###### Lemma 2.
For the two-user broadcast erasure channel with no channel feedback but with
non-blind side information at the receivers as described in Section II,
whether blind or not, and $\beta_{1}^{\mathrm{no}}$ given in (9), we have
$\displaystyle
H\left(Y_{2}^{n}|W_{1|2},W_{2},G^{n}\right)+n\upxi_{n}\geq\beta_{1}^{\mathrm{no}}H\left(Y_{1}^{n}|W_{2},G^{n}\right),$
(24)
where $\upxi_{n}\rightarrow 0$ as $n\rightarrow\infty$, where $G^{n}$ is the
global channel state and cache index information in (21).
###### Proof.
We first have the following fact
$\displaystyle
H\left(Y_{2}^{n}|W_{1|2},W_{2},G^{n}\right)\geq\frac{1-\delta_{2}}{1-\delta_{1}}H\left(Y_{1}^{n}|W_{1|2},W_{2},G^{n}\right),$
(25)
which is modified from our previous result [25]. For completeness, we still
present the details in Appendix A. Now, to prove this Lemma with (25), we note
that
$0=H\left(Y_{1}^{n}|W_{1},W_{2},G^{n}\right)=H\left(Y_{1}^{n}|W_{1|2},\bar{W}_{1|2},W_{2},G^{n}\right),$
(26)
where $\bar{W}_{1|2}$ is the complement of $W_{1|2}$ in $W_{1}$, and then
$\displaystyle
H\left(Y_{1}^{n}|W_{1|2},W_{2},G^{n}\right)=I(\bar{W}_{1|2};Y_{1}^{n}|W_{1|2},W_{2},G^{n})$
$\displaystyle=$ $\displaystyle
H(\bar{W}_{1|2}|W_{1|2},W_{2},G^{n})-H(\bar{W}_{1|2}|Y_{1}^{n},W_{1|2},W_{2},G^{n}).$
(27)
Since $H\left(W_{1}|Y_{1}^{n},W_{2},G^{n}\right)\leq n\upxi_{n},$ the second
term in the RHS of (27) will also be less than $n\upxi_{n}$ due to the
$W_{1}=(W_{1|2},\bar{W}_{1|2})$ and the chain rule. For the first term in the
RHS, as (19), it equals to $H(\bar{W}_{1|2}|E_{2}^{n})=\epsilon_{2}H(W_{1}).$
Then we get
$\displaystyle H$
$\displaystyle\left(Y_{1}^{n}|W_{1|2},W_{2},G^{n}\right)\geq\epsilon_{2}H\left(W_{1}\right)-n\upxi_{n}\overset{(a)}{\geq}\epsilon_{2}H\left(Y_{1}^{n}|W_{2},G^{n}\right)-n\upxi_{n},$
(28)
and $(a)$ is obtained from (26) as
$\displaystyle H\left(Y_{1}^{n}|W_{1},W_{2},G^{n}\right)=0\Rightarrow
H\left(W_{1}\right)\geq H\left(Y_{1}^{n}|W_{2},G^{n}\right).$
Finally, from (25) and (28), we obtain
$\displaystyle H$
$\displaystyle\left(Y_{2}^{n}|W_{1|2},W_{2},G^{n}\right)\overset{\eqref{Eq:withSideInfoN}}{\geq}\frac{1-\delta_{2}}{1-\delta_{1}}H\left(Y_{1}^{n}|W_{1|2},W_{2},G^{n}\right)$
$\displaystyle\overset{\eqref{Eq:RemovingSideInfoN}}{\geq}\frac{\epsilon_{2}\left(1-\delta_{2}\right)}{\left(1-\delta_{1}\right)}H\left(Y_{1}^{n}|W_{2},G^{n}\right)-n\upxi_{n}$
$\displaystyle\overset{\eqref{Eq:Beta_No}}{=}\beta_{1}^{\mathrm{no}}H\left(Y_{1}^{n}|W_{2},G^{n}\right)-n\upxi_{n}.$
(29)
This completes the proof of Lemma 2. ∎
## V Achievability Proof of $\mathcal{C}^{\mathrm{non-blind}}_{\mathrm{NN}}$
in Theorem 1
In this section, we provide an achievability protocol for the non-blind
transmitter case, and we show that the achievable regions matches the outer-
bounds of Theorem 1. Thus, we characterize the capacity region of the problem
when the transmitter is aware of the available side-information at the
receivers. For the proof, without loss of generality, we assume
$\delta_{2}\geq\delta_{1}$, meaning that ${\sf Rx}_{1}$ has a stronger channel
than ${\sf Rx}_{2}$.
Our achievability proof reveals a surprising result, only the cache index
information $E^{nR_{1}}_{2}$ of the weaker receiver 2 in (2) is needed at the
transmitter. Thus the transmitter can be semi-blind as Corollary 1 to achieve
the outer-bounds. As a warm-up, we first focus on an example where
$\epsilon_{2}=0$. In this case, receiver ${\sf Rx}_{2}$ (the weaker receiver)
has full side-information of the message for ${\sf Rx}_{1}$, _i.e._
$W_{1|2}=W_{1}$, and receiver ${\sf Rx}_{1}$ has access to $(1-\epsilon_{1})$
of the bits intended for ${\sf Rx}_{2}$. The outer-bounds of Theorem 1 in this
case become:
$\left\\{\begin{array}[]{ll}0\leq R_{2}\leq\left(1-\delta_{2}\right),&\\\
0\leq R_{1}+\epsilon_{1}R_{2}\leq\left(1-\delta_{1}\right).\end{array}\right.$
(30)
Thus, the non-trivial corner point is given by:
$\displaystyle R_{1}=(1-\delta_{1})-\epsilon_{1}(1-\delta_{2}),$
$\displaystyle R_{2}=(1-\delta_{2}).$ (31)
In this case, we set
$\displaystyle\eta=\frac{R_{1}}{R_{2}}=\frac{(1-\delta_{1})}{(1-\delta_{2})}-\epsilon_{1}>0.$
(32)
Achievability protocol for $\epsilon_{2}=0$ : Recall $\eta=m_{1}/m_{2}$, we
start with $m$ bits for ${\sf Rx}_{2}$ and $\eta m$ bits for ${\sf Rx}_{1}$.
The achievability protocol is carried a single phase with two segments
(another Phase will be added later for general $\epsilon_{2}$). The total
communication time is set to $\frac{m}{(1-\delta_{2})}.$
Segment a: This segment has a total length of
$\displaystyle
t_{a}=\frac{\epsilon_{1}m}{(1-\delta_{1})}<\frac{m}{(1-\delta_{2})},$ (33)
where the last inequality is from $R_{1}>0$ in (V). During this segment, the
transmitter sends $t_{a}$ of the random combinations of the $m$ bits intended
for ${\sf Rx}_{2}$. During this segment, stronger ${\sf Rx}_{1}$ obtains
$\displaystyle(1-\delta_{1})t_{a}=\epsilon_{1}m$ (34)
random equations of the $m$ bits for ${\sf Rx}_{2}$ and in combination with
the available side-information $W_{2|1}$ with
$\left|W_{2|1}\right|=(1-\epsilon_{1})m$, ${\sf Rx}_{1}$ has sufficient
linearly independent equations to decode $W_{2}$ when the code length is large
enough.
Segment b: This segment has a total length of
$\displaystyle
t_{b}=\frac{(1-\delta_{1})-\epsilon_{1}(1-\delta_{2})}{(1-\delta_{1})(1-\delta_{2})}m=\frac{m}{(1-\delta_{2})}-t_{a}>0.$
(35)
During this segment, the transmitter creates $t_{b}$ random linear
combinations of the $\eta m$ bits in $W_{1}$, and creates the XOR of these
combinations with $t_{b}$ random combinations of the $m$-bit $W_{2}$ for ${\sf
Rx}_{2}$. The transmitter sends the resulting XORed sequence during Segment b.
The decodability comes as follows. In segment b, ${\sf Rx}_{1}$ can remove the
interference since $W_{2}$ is known from Segment a, and gets
$t_{b}(1-\delta_{1})=\eta m$ linearly independent equations for decoding
$W_{1}$ correctly. Also in this segment, ${\sf Rx}_{2}$ can remove the
interference from $W_{1}$ using the side-information $W_{1|2}=W_{1}$, then the
total linearly independent equations it has will be
$(t_{b}+t_{a})(1-\delta_{2})=m$. Thus, by the end of the communication block,
${\sf Rx}_{2}$ can decode $W_{2}$, and ${\sf Rx}_{1}$ can decode both $W_{1}$
and $W_{2}$.
Achievable rates: Since the total communication time is
$\displaystyle\frac{m}{(1-\delta_{2})},$ (36)
we immediately conclude the achievability of rates in (V).
Note that in the toy example aforementioned, only cache index information
$E^{nR_{1}}_{2}$ for $W_{1|2}=W_{1}$ is used at the transmitter in Segment b,
but not the other $E^{nR_{2}}_{1}(W_{2|1})$. Now we present the proof for the
general case $\epsilon_{2}\neq 0$ and show that the achievability also needs a
“semi-blind” transmitter. From (10), the non-trivial corner point of the
region defined in (8) is given by:
$\displaystyle
R_{1}=\frac{(1-\delta_{1})-\epsilon_{1}(1-\delta_{2})}{1-\frac{\epsilon_{1}\epsilon_{2}(1-\delta_{2})}{(1-\delta_{1})}},$
$\displaystyle
R_{2}=\frac{(1-\epsilon_{2})(1-\delta_{2})}{1-\frac{\epsilon_{1}\epsilon_{2}(1-\delta_{2})}{(1-\delta_{1})}}.$
(37)
We define
$\displaystyle\eta\overset{\triangle}{=}\frac{R_{1}}{R_{2}}=\frac{(1-\delta_{1})-\epsilon_{1}(1-\delta_{2})}{(1-\epsilon_{2})(1-\delta_{2})}.$
(38)
We note that if
$\displaystyle(1-\epsilon_{2}+\epsilon_{1})>\frac{(1-\delta_{1})}{(1-\delta_{2})},$
(39)
then $\eta<1$.
Achievability protocol for general $\epsilon_{2}$: We start with $m$ bits for
${\sf Rx}_{2}$ and $\eta m$ bits for ${\sf Rx}_{1}$. The achievability
protocol is carried over two phases with the first phase having two segments
as those for $\epsilon_{2}=0$. As revealed by the decodability in toy example,
the idea that after the first phase, receiver ${\sf Rx}_{2}$ (the weaker
receiver) decodes its message $W_{2}$. Since the first receiver has a stronger
channel, it can recover interference $W_{2}$ in a shorter time horizon after
the first segment of Phase I. Then, during the second segment of Phase I, the
transmitter starts delivering cached $W_{1|2}$ to ${\sf Rx}_{1}$, while it
continues delivering $W_{2}$ to ${\sf Rx}_{2}$. Note that since ${\sf Rx}_{2}$
knows $W_{1|2}$ and ${\sf Rx}_{1}$ has recovered $W_{2}$ in the first segment,
the second segment benefits both receivers. Finally, during the newly-added
second phase, $\bar{W}_{1|2}$ (the non-cached part of $W_{1}$ outside
$W_{1|2}$) is delivered to ${\sf Rx}_{1}$. The whole process is summarized in
Figure 6.
Figure 6: Semi-blind two-phase protocol to achieve outer-bounds
$\mathcal{C}^{\mathrm{non-blind}}_{\mathrm{NN}}$ in Theorem 1.
Phase I: The transmitter creates
$\displaystyle\frac{m}{(1-\delta_{2})}+O\left(m^{2/3}\right)$ (40)
random linear combinations of the $m$ bits intended for ${\sf Rx}_{2}$ such
that any randomly chosen $m$ combinations are randomly independent. This can
be accomplished by a random linear codebook of which each element is generated
from i.i.d. Bernoulli $\mathrm{Ber}(1/2)$. In practice, Fountain codes can be
used.
###### Remark 3 (Expected values of concentration results).
The $O\left(m^{2/3}\right)$ is to ensure sufficient number of equations will
be received given the stochastic nature of the channel. At the end we let
$m\rightarrow\infty$, such terms do not affect the achievability of the
overall rates. Thus, for simplicity of expressions, we omit these terms and
only work with the expected value of the number of equations in what follows,
since the actual number will converge to this expection.
Segment a of Phase I: This segment has a total (on average) length of $t_{a}$
as in (33). During this segment, the transmitter sends $t_{a}$ of the random
combinations it created for ${\sf Rx}_{2}$ as described above and illustrated
in Figure 6.
Segment b of Phase I: This segment has a total (on average) length of $t_{b}$
as (35). During this segment, the transmitter creates $t_{b}$ random linear
combinations of the $(1-\epsilon_{2})\eta m$ bits in $W_{1|2}$, and XORs these
combinations with the another $t_{b}$ random combinations it created for
$W_{2}$. The transmitter sends the XORed sequence during Segment b of Phase I.
Phase II: This phase has a total (on average) length of
$\displaystyle
t_{2}=\frac{\epsilon_{2}\left((1-\delta_{1})-\epsilon_{1}(1-\delta_{2})\right)}{(1-\epsilon_{2})(1-\delta_{1})(1-\delta_{2})}m=\frac{\epsilon_{2}\eta
m}{(1-\delta_{1})}.$ (41)
During this phase, the transmitter creates $t_{2}$ random linear combinations
of the bits in $\bar{W}_{1|2}$ and sends these combinations.
The decodability for ${\sf Rx}_{1}$ comes as follows. At the end of the second
phase, ${\sf Rx}_{1}$ gathers $\epsilon_{2}\eta m$ linearly independent
equations of non-cached $\bar{W}_{1|2}$, and as
$\left|\bar{W}_{1|2}\right|=\epsilon_{2}\eta m$, ${\sf Rx}_{1}$ can decode
$\bar{W}_{1|2}$. For cached $W_{1|2}$, as in toy example, during segment b of
Phase I, ${\sf Rx}_{1}$ can remove the interference ($W_{2}$ is known in
segment a) and decode $W_{1|2}$. Thus, ${\sf Rx}_{1}$ can decode ${W}_{1|2}$
and $\bar{W}_{1|2}$, meaning that it can recover $W_{1}$. The decodability for
${\sf Rx}_{2}$ after Phase I directly follows from that in the toy example.
Achievable rates: Recall the total length $t_{a}+t_{b}$ of Phase I is
$\frac{m}{(1-\delta_{2})}$ from (35), then with (41), the total communication
time is given by:
$\displaystyle
t_{a}+t_{b}+t_{2}=\frac{(1-\delta_{1})-\epsilon_{1}\epsilon_{2}(1-\delta_{2})}{(1-\epsilon_{2})(1-\delta_{1})(1-\delta_{2})}m,$
(42)
which immediately implies the achievability of rates in (V).
## VI Achievability Proof for $\mathcal{C}^{\mathrm{blind}}_{\mathrm{DD}}$ in
Case A of Theorem 2
First, we remind the reader of the conditions in Case A of Theorem 2: global
delayed CSIT and symmetric channel where $\delta_{1}=\delta_{2}=\delta$ and
$\epsilon_{1}=\epsilon_{2}=\epsilon$. For this setup, we present an
opportunistic communication protocol that enables the transmitter to use the
available delayed CSI and the statistical knowledge of the available side-
information at the receivers. This protocol starts by sending the summation
(_i.e._ XOR in the binary field) of individual bits intended for the two
receivers, and then, based on the available channel feedback and the
statistics of receiver side-information, the transmitter is able to
efficiently create recycled bits for retransmission. In this regard, the
protocol has some similarities to the reverse Maddah-Ali-Tse scheme [26],
which was originally designed for multiple-input fading broadcast channels
[27]. As discussed in [28], the channel setting is fundamentally different:
discrete erasure channel vs. continuous Rayleigh channel, single antenna vs.
multiple-input transmitter. Together with the blind receiver side-information
in our case; thus, our protocols end up sharing little ingredients with [26].
We skip the protocol for achieving $R_{i}=\left(1-\delta\right)$, as it is a
well-established result. Instead, we provide the achievability protocol for
the maximum symmetric sum-rate point given by
$\displaystyle R_{1}=R_{2}=\frac{1-\delta^{2}}{1+\delta+\epsilon}.$ (43)
Figure 7: The proposed four-phase protocol when $\epsilon\leq\delta$: Phase I
delivers combinations of $\vec{a}$ and $\vec{b}$; Phases II and III deliver
interfering bits to unintended receivers alongside useful information to the
intended receivers; Phase IV multicasts XOR of available bits at each receiver
needed at the other.
We break the scheme based on the relationship between $\epsilon$ and $\delta$.
We note that since we focus on the homogeneous setting in this section, the
transmitter has $m$ bits for each receiver: $a_{j}$’s for receiver ${\sf
Rx}_{1}$ and $b_{j}$’s for receiver ${\sf Rx}_{2}$, $j=1,2,\ldots,m$.
Scenario 1 ($\epsilon\leq\delta$): This case assumes the side channel that
provides each receiver its side-information is stronger than the channel from
the transmitter. The transmitter first creates
$\vec{c}=\left(c_{1},c_{2},\ldots,c_{m}\right)$ according to
$\displaystyle c_{j}=a_{j}\oplus b_{j},\qquad j=1,2,\ldots,m.$ (44)
The protocol is divided into four phases as described below and depicted in
Figure 7.
Phase I: During this phase, the transmitter sends out individual bits from
$\vec{c}$ until at least one receiver obtains this bit, and then, moves on to
the next bit. Due to the statistics of the channel, this phase takes on
average
$\displaystyle t_{1}=\frac{m}{1-\delta^{2}}$ (45)
time instants.
###### Remark 4.
To keep the description of the protocol simple, we use the expected value of
different random variable (_e.g._ length of phase, number of bits received by
each user, etc). A more precise statement would use a concentration theorem
result such as the Bernstein inequality to show the omitted terms do not
affect the overall result and the achievable rates as done in [29, 30].
Moreover, when talking about the number of bits or time instants, we are
limited to integer numbers. If a ratio is not an integer number, we can use
$\lceil\cdot\rceil$, the ceiling function, and since at the end we take the
limit for $m\rightarrow\infty$, the results remain unchanged.
After Phase I is completed, receiver $\mathsf{Rx}_{1}$ obtains on average
$m/(1+\delta)$ bits from $\vec{c}$. The transmitter, using channel feedback
during Phase I, knows which bits out of $\vec{b}$ where among those received
by $\mathsf{Rx}_{1}$ as part of $\vec{c}$, denoted by $\vec{\tilde{b}}$ in
Figure 7. Furthermore, $\mathsf{Rx}_{1}$ statistically knows a fraction
$(1-\epsilon)$ of the interfering $\vec{\tilde{b}}$ from its side-information.
Thus, if $\mathsf{Rx}_{1}$ obtains an additional fraction $\epsilon$ of
$\vec{\tilde{b}}$, it can resolve interference in Phase I to get
$m/(1+\delta)$ pure bits from $\vec{a}$. A similar statement holds for
$\mathsf{Rx}_{2}$.
Phase II: The transmitter creates $\epsilon m/(1+\delta)$ linearly independent
combinations of $\vec{\tilde{b}}$, and encodes them using an erasure code of
rate $(1-\delta)$ and sends them out. This phase takes
$\displaystyle t_{2}=\frac{\epsilon m}{1-\delta^{2}},$ (46)
time instant, and upon its completion, $\mathsf{Rx}_{1}$ gets the additional
equations to remove interference during Phase I and recover $m/(1+\delta)$
bits from $\vec{a}$, while $\mathsf{Rx}_{2}$ obtains further $\epsilon
m/(1+\delta)$ equations of its intended bits.
Phase III: This phase is similar to Phase II, but the transmitter sends out
$\vec{\tilde{a}},$ those bits out of $\vec{a}$ that were received by
$\mathsf{Rx}_{2}$ as part of $\vec{c}$. This phase takes
$\displaystyle t_{3}=\frac{\epsilon m}{1-\delta^{2}},$ (47)
time instants, and upon its completion, $\mathsf{Rx}_{2}$ gets the additional
equations to remove interference during Phase I and recover $m/(1+\delta)$
bits from $\vec{b}$, while $\mathsf{Rx}_{1}$ obtains further $\epsilon
m/(1+\delta)$ equations of its intended bits.
Number of equations at each receiver: After the first three phases, each
receiver has a total of
$\displaystyle\frac{m}{1+\delta}+\frac{\epsilon m}{1+\delta}$ (48)
linearly independent equations of its bits, and thus, needs an additional
$\displaystyle\frac{\delta-\epsilon}{1+\delta}m$ (49)
new equations to complete recovery of its bits. Note that if
$\epsilon=\delta$, the protocol ends here.
Phase IV: The transmitter creates
$\displaystyle\frac{\delta-\epsilon}{1+\delta}m$ (50)
further linearly independent random combinations of the bits intended for
$\mathsf{Rx}_{i}$ but received at $\mathsf{Rx}_{\bar{i}}$ as part of $\vec{c}$
during Phase I, denoted by $\vec{\tilde{a}}$ and $\vec{\tilde{b}}$ in Figure
7. Note that at this point, each receiver has full knowledge of the
interfering bits during Phase I and retransmission of such bits will no longer
create any interference. Thus, the transmitter encodes these two sets of bits
(one for each receiver) using an erasure code of rate $\left(1-\delta\right)$
and send the XOR of these encoded bits. This phase takes
$\displaystyle t_{4}=\frac{\delta-\epsilon}{1-\delta^{2}}m$ (51)
time instants.
We note that in Phases II, III, and IV, the transmitter needs to create
linearly independent combinations of the bits. Thus, we need to guarantee the
feasibility of these operations. In Phase I, as part of $\vec{c}$, a total of
$\displaystyle\frac{\delta}{1+\delta}m$ (52)
bits intended for one receiver arrive at the unintended receiver and
effectively, during the next phases, we deliver these bits to the intended
receiver. In fact, (52) equals the summation of the number of linearly
equations needed for $\mathsf{Rx}_{1}$ during Phases III and IV, and for
$\mathsf{Rx}_{2}$ during Phases II and IV. Thus, the feasibility of creating
sufficient number of linearly independent combinations is guaranteed.
Upon completion of Phase IV, each receiver first removes the contribution of
the bits intended for the other user, and then, recovers the additional
equations needed as indicated in (49), and thus, is able to complete recovery
of its message.
Achievable rates: The total communication time is
$\displaystyle\sum_{j=1}^{4}{t_{j}}=\frac{1+\delta+\epsilon}{1-\delta^{2}}m,$
(53)
which immediately results in target rates of (43).
Scenario 2 ($\epsilon>\delta$): This scenario corresponds to the case in which
the side channel that provides each receiver with its side-information is
weaker than the channel from the transmitter. The protocol has four phases as
before with some modifications. Phase I remains identical to the previous
scenario; during Phases II and III, instead of $\epsilon m/(1+\delta)$, the
transmitter creates $\delta m/(1+\delta)$ linearly independent equations of
$\vec{\tilde{b}}$ and $\vec{\tilde{a}}$, respectively, and sends them out as
done in the previous scenario. With these modifications, after the first three
phases, $\mathsf{Rx}_{1}$, still has
$\displaystyle\frac{\epsilon-\delta}{1+\delta}m$ (54)
equations interfered by $\vec{\tilde{b}}$. Thus, for $\mathsf{Rx}_{1}$ to
successfully recover $\vec{a}$, the transmitter has two options: $(i)$ to
deliver the same number as in $\eqref{Eq:Sc2AfterPh12}$, new combinations of
$\vec{\tilde{b}}$ to $\mathsf{Rx}_{1}$, and $(ii)$ to provide
$\mathsf{Rx}_{1}$ with additional combinations, same as in
$\eqref{Eq:Sc2AfterPh12}$ , of its own bits $\vec{\tilde{a}}$. With the first
choice, receiver $\mathsf{Rx}_{1}$ fully resolves the interference and
recovers its bits; while with the second choice, it simply obtains $m$
linearly independent equations of $\vec{a}$. Interestingly, either choice is
also good for $\mathsf{Rx}_{2}$: with the first choice, $\mathsf{Rx}_{2}$
obtains $m$ linearly independent equations of $\vec{b}$; while with the first
choice, $\mathsf{Rx}_{2}$ fully resolves the interference and recovers its
bits. In other words, in this scenario, during Phase IV, no XOR operation is
needed and only bits intended for one user would enable both receivers to
decode their bits.
Based on the discussion above, during Phase IV, the transmitter creates
$\left(\epsilon-\delta\right)/\left(1+\delta\right)m$ linearly independent
combinations, same as (54), of the bits in $\vec{\tilde{a}}$ intended for
$\mathsf{Rx}_{1}$ but received as part of $\vec{c}$ at $\mathsf{Rx}_{2}$
during Phase I. Then, the transmitter encodes these equations using an erasure
code of rate $\left(1-\delta\right)$ and sends them to both receivers. Similar
to the previous scenario, we can guarantee the feasibility of creating these
linearly independent equations.
After Phase IV, as discussed above, $\mathsf{Rx}_{1}$ will have sufficient
number of equations to recover $\vec{a}$; while $\mathsf{Rx}_{2}$ first needs
to resolve the interference using the equations it obtains during this phase,
and then, recover $\vec{b}$. We note that, the transmitter could send
combinations of $\vec{b}$ instead of $\vec{a}$ during the last phase, and the
decoding strategy of the receivers would get swapped.
## VII Achievability Proofs for $\mathcal{C}^{\mathrm{semi-
blind}}_{\mathrm{DN}}$ in Case B and
$\mathcal{C}^{\mathrm{non-blind}}_{\mathrm{DN}}$ in Case C of Theorem 2
Case B : Achievability for $\mathcal{C}^{\mathrm{semi-blind}}_{\mathrm{DN}}$
when $\epsilon_{1}=0$ : Now $\epsilon_{1}=0$ and thus $W_{2|1}=W_{2}$. Recall
message $W_{2}$ is also represented by a bit vector $\vec{b}$. The encoding
process come as follows. At time index $t$, the $j$-th bit $a_{j}$ in message
$\vec{a}$ for $\mathsf{Rx}_{1}$ is repeated according to the state feedback
$S_{1}$, and after XORing a random linear combination
$(\vec{g}_{t})^{\intercal}\vec{b}$, the resulting superposition is sent. Each
entry of $\vec{g}_{t}$ is generated from $i.i.d.$ $\mathrm{Ber}(1/2)$. Each
$a_{j}$ is repeated until the corresponding state feedback is $S_{1}=1$. In
other words, prior to the superposition via XORing, $\vec{a}$ is pre-encoded
and repeated as in standard ARQ, while $\vec{b}$ is pre-encoded by a fountain-
like random linear code. the termination of this fountain code is determined
by the state feedback of $\mathsf{Rx}1$, that is whether or not all bits in
$\vec{a}$ are successfully delivered to $\mathsf{Rx}1$.
The decoding at $\mathsf{Rx}_{1}$ follows from the standard ARQ, since full
side-information $\vec{b}$ is known. By setting total transmission length as
$\frac{m_{1}}{1-\delta_{1}},$ (55)
the achievable rate is
$R_{1}=1-\delta_{1}$ (56)
For user $\mathsf{Rx}_{2}$, it has side-information $W_{1|2}$ for
$(1-\epsilon_{2})$ bits of the interference $\vec{a}$ and each reception of
the corresponding XOR transmitted will result in a linear equation of
$\vec{b}$. To see this, consider the $j$-th bit $a_{j}$ of interference
$\vec{a}$. Suppose it is repeated $L_{i}$ times until its mixture with
$\vec{b}$ is successfully arrived at $\mathsf{Rx}1$. Within this span,
$\mathsf{Rx}2$ gets
$K_{i}\triangleq\sum_{\ell=1}^{L_{j}}S_{2,j}[\ell]$ (57)
linear equations mixing $a_{j}$ and $\vec{b}$, where $S_{2,j}[\ell]$ is the
erasure state at $\mathsf{Rx}2$ for the $\ell$-th transmission of $a_{j}$.
Then $\mathsf{Rx}2$ gets $K_{i}$ pure equations of $\vec{b}$. In total, the
number of pure linear equations of message $\vec{b}$ is
$m_{1}(1-\epsilon_{2})\mathbb{E}[K_{i}]=m_{1}(1-\epsilon_{2})\frac{1-\delta_{2}}{1-\delta_{1}},$
(58)
For interference bits without side-information, by using interference
alignment in [12], the number of pure linear equations is
$m_{1}\epsilon_{2}\mathbb{E}[(K_{i}-1)^{+}]=m_{1}\epsilon_{2}\left(\frac{\delta_{1}-\delta_{2}}{1-\delta_{1}}+\frac{\delta_{2}-\delta_{1}\delta_{2}}{1-\delta_{1}\delta_{2}}\right)$
(59)
Thus the total number of pure equations from (58) and (59) is
$m_{1}(1-\delta_{2})\left(\frac{1}{1-\delta_{1}}-\frac{\epsilon_{2}}{1-\delta_{1}\delta_{2}}\right),$
which results in achievable rate
$R_{2}=(1-\delta_{2})\left(1-\epsilon_{2}\frac{1-\delta_{1}}{1-\delta_{1}\delta_{2}}\right)$
(60)
where (55) is applied. It can be checked that $(R_{1},R_{2})$ in (56)(60) is
the corner point of outer-bound region (12) when $\epsilon_{1}=0$
$\displaystyle\epsilon_{2}\frac{1-\delta_{2}}{1-\delta_{1}\delta_{2}}$
$\displaystyle R_{1}+R_{2}\leq\left(1-\delta_{2}\right)$ (61) $\displaystyle
0\leq$ $\displaystyle R_{i}\leq(1-\delta_{i}),i=1,2,$ (62)
Other corner point can be trivially achieved by time-sharing.
Figure 8: Proposed four-phase protocol for achieving
$\mathcal{C}^{\mathrm{non-blind}}_{\mathrm{DN}}$.
Case C : Achievability for $\mathcal{C}^{\mathrm{non-blind}}_{\mathrm{DN}}$ :
As in Fig. 8, we now introduce the four-phase scheme for this achievability,
of which the third and fourth phases are similar to that in Case B. We first
represent $W_{2|1}$ and $W_{1|2}$ using bit vectors $\vec{b}_{1}$ and
$\vec{a}_{2}$ respectively, then the encoding process is
Phase I : The transmitter sends bits from $\vec{a}$ which are not cached at
$\mathsf{Rx}_{2}$ and not in $\vec{a}_{2}$. The total length $t_{1}$ of Phase
I is $\epsilon_{2}m_{1}$. After Phase I, the transmitter knows length
$t_{1}\delta_{1}$ sequence $\bar{\tilde{a}}_{2}$, which is formed by bits
erased at $\mathsf{Rx}1$ in Phase I where $S_{1}[t]=0$.
Phase II : The transmitter selects $\epsilon_{1}m_{2}$ bits from $\vec{b}$
which are not cached at $\mathsf{Rx}_{1}$ and not in $\vec{b}_{1}$, and send
then random linear combinations of them. The total length $t_{2}$ of Phase II
is $\epsilon_{1}m_{2}/(1-\delta_{1}\delta_{2})$. After Phase II, the
transmitter knows length $t_{2}(1-\delta_{1})$ sequence $\bar{\tilde{b}}_{1}$,
which is formed by bits received at $\mathsf{Rx}1$ in Phase I where
$S_{1}[t]=1$.
Phase III : The transmission is similar to that in semi-blind Case B, the
differences are as follows. Now the transmitter is non-blind to $\vec{a}$ so
it pre-encodes cached $\vec{a}_{2}$ instead of the whole $\vec{a}$ using ARQ.
Also the transmitter is only sure that $(\bar{\tilde{b}}_{1},\vec{b}_{1})$ is
known at $\mathsf{Rx}_{1}$, it pre-encodes $(\bar{\tilde{b}}_{1},\vec{b}_{1})$
instead of full $\vec{b}$ using the random linear code. More specifically, the
output of the second pre-encoder at time $t$ becomes the XOR of random linear
combinations
$(\vec{g}_{t})^{\intercal}\bar{\tilde{b}}_{1}\oplus(\vec{g^{\prime}}_{t})^{\intercal}\vec{b}_{1}$,
where each entry of $\vec{g}_{t}$ or $\vec{g^{\prime}}_{t}$ is generated from
i.i.d. $\mathrm{Ber}(1/2).$ For the first pre-encoder, each bit in
$\vec{a}_{2}$ cached at user 2 is repeated according to the delayed $S_{1}$ as
described in Case B. Finally the XOR of outputs of these two pre-encoders is
sent.
Phase IV: The transmission in phase is same as that in Phase III, by replacing
the input of the first ARQ pre-encoder by the recycled $\bar{\tilde{a}}_{2}$.
Though $(1-\delta_{2})$ of sequence $\bar{\tilde{a}}_{2}$ will be known at
$\mathsf{Rx}_{2}$ in Phase IV, the transmitter is blind to these bits. On the
contrary, in Phase III the transmitter knows that the input $\vec{a}_{2}$ of
the ARQ pre-encoder is totally cached at $\mathsf{Rx}_{2}$.
Note that without receiver side-information $\epsilon_{1}=\epsilon_{2}=1$,
there will be no Phase III and our scheme reduces to the three-phase scheme in
[12].
We focus on the decodability for receiver $\mathsf{Rx}1$ first. In Phase III
and IV, since $(\bar{\tilde{b}}_{1},\vec{b}_{1})$ is already known at
$\mathsf{Rx}_{1}$, $\vec{a}_{2}$ and $\bar{\tilde{a}}_{2}$ can be recovered,
if the lengthes of Phases are respectively chosen as
$\displaystyle t_{3}(1-\delta_{1})=(1-\epsilon_{2})m_{1}$ $\displaystyle
t_{4}(1-\delta_{1})=\delta_{1}t_{1}=\delta_{1}\epsilon_{2}m_{1}$ (63)
Together with bits received in Phase I, receiver $\mathsf{Rx}1$ gets all
$m_{1}$ bits. Now, we turn to the decodability at $\mathsf{Rx}2$. Receiver
$\mathsf{Rx}2$ will first decode super-$(\bar{\tilde{b}}_{1},\vec{b}_{1})$
from its received bits during the entire three phases. With the
$\bar{\tilde{b}}_{1}$ and the received equations during Phase II, it will have
$t_{2}(1-\delta_{1}\delta_{2})=\epsilon_{1}m_{2}$ equations to decode uncached
bits in $\vec{b}$. Together with $\vec{b}_{1}$, the whole message for user 2
is decoded. To ensure successful decoding of the
super-$(\bar{\tilde{b}}_{1},\vec{b}_{1})$, we calculate the corresponding
expected number of linearly independent equations as follows. In Phase III,
every reception at $\mathsf{Rx}_{2}$ will result in a new equation since
$\vec{a}_{2}$ is cached, and we have
$(1-\epsilon_{2})m_{1}\mathbb{E}[K_{i}]$
equations after Phase III. In Phase IV, as (58) and (59), we will have
additional
$t_{1}\delta_{1}(1-\delta_{2})\mathbb{E}[K_{i}]+t_{1}\delta_{1}\delta_{2}\mathbb{E}[(K_{i}-1)^{+}]$
equations since $(1-\delta_{2})$ of $\bar{\tilde{a}}_{2}$ will be received
during Phase I. By using equations of $\bar{\tilde{b}}_{1}$ received at
$\mathsf{Rx}_{2}$ in Phase II as additional cache, we need
$\displaystyle t_{2}(1-\delta_{1})+(1-\epsilon_{1})m_{2}\leq$ $\displaystyle
t_{2}(1-\delta_{1}-\delta_{2}+\delta_{1}\delta_{2})+(1-\epsilon_{2})m_{1}\frac{1-\delta_{2}}{1-\delta_{1}}+t_{1}\delta_{1}(1-\delta_{2})\left(\frac{1}{1-\delta_{1}}-\frac{\delta_{2}}{1-\delta_{1}\delta_{2}}\right)$
(64)
for successful decoding the length $t_{2}(1-\delta_{1})+(1-\epsilon_{1})m_{2}$
super-$(\bar{\tilde{b}}_{1},\vec{b}_{1})$. Note that by collecting
$\bar{\tilde{b}}_{1}$ received in Phase II (with standard basis for
$\bar{\tilde{b}}_{1}$) and the linear equations produced in Phase III and IV,
one can form a set of linear equations of $(\bar{\tilde{b}}_{1},\vec{b}_{1})$
described by a full (column) rank matrix, when codelengths are long enough.
For $(R_{1},R_{2})$ satisfying outer-bound
$R_{1}/(1-\delta_{1})+\epsilon_{1}R_{2}/(1-\delta_{1}\delta_{2})=1$ in (12),
the total communication time must meet
$\sum^{4}_{j=1}t_{j}=\frac{m_{1}}{1-\delta_{1}}+\frac{\epsilon_{1}m_{2}}{1-\delta_{1}\delta_{2}}.$
From selected lengths of Phase I and II, $m_{1}=t_{1}/\epsilon_{2}$ and
$m_{2}=t_{2}(1-\delta_{1}\delta_{2})/\epsilon_{1}$, together with (63), this
constraint is meet since
$\sum^{4}_{j=1}t_{j}=t_{1}\left(1+\frac{(1-\epsilon_{2})/\epsilon_{2}+\delta_{1}}{1-\delta_{1}}\right)+t_{2}=\frac{t_{1}}{\epsilon_{2}(1-\delta_{1})}+t_{2}$
For the corner point $(R_{1},R_{2})$ which also satisfies outer-bound
$R_{2}/(1-\delta_{2})+\epsilon_{2}R_{1}/(1-\delta_{1}\delta_{2})=1$ in (12),
we further show that the decodability (64) will also be met. From (12),
$\frac{t_{1}}{1-\delta_{1}\delta_{2}}+\frac{1-\delta_{1}\delta_{2}}{\epsilon_{1}(1-\delta_{2})}t_{2}=\sum^{4}_{j=1}t_{j}=\frac{t_{1}}{\epsilon_{2}(1-\delta_{1})}+t_{2},$
which implies
$t_{2}\left(\frac{1-\delta_{1}\delta_{2}}{\epsilon_{1}}-(1-\delta_{2})\right)=t_{1}(1-\delta_{2})\left(\frac{1}{\epsilon_{2}(1-\delta_{1})}-\frac{1}{1-\delta_{1}\delta_{2}}\right),$
or equivalently
$t_{2}\left(\delta_{2}-\delta_{1}\delta_{2}+\frac{1-\epsilon_{1}}{\epsilon_{1}}(1-\delta_{1}\delta_{2})\right)=t_{1}(1-\delta_{2})\left(\frac{(1-\epsilon_{2})/\epsilon_{2}}{1-\delta_{1}}+\left(\frac{1}{1-\delta_{1}}-1\right)-\left(\frac{1}{1-\delta_{1}\delta_{2}}-1\right)\right).$
Then (64) is met since $m_{1}=t_{1}/\epsilon_{2}$ and
$m_{2}=t_{2}(1-\delta_{1}\delta_{2})/\epsilon_{1}$.
## VIII Conclusion
We studied the problem of communications over two-user broadcast erasure
channels with random receiver side-information. We assumed the transmitter may
not have access to global channel state information and global cache index
information for both receivers. For the non-blind-transmitter case, we
characterized the capacity region, while with a blind transmitter we showed
the outer-bounds can be achieved under certain conditions. Thus, in general
with a blind transmitter, the capacity region of the problem, also known as
blind index coding over the broadcast erasure channel, remains open.
## Appendix A Proof of (25) in Lemma 2
For time instant $t$ where $2\leq t\leq n$, we have
$\displaystyle
H\left(Y_{2}[t]|Y_{2}^{t-1},W_{1|2},W_{2},S_{2}^{t},E^{n}\right)$ (65)
$\displaystyle\quad=\left(1-\delta_{2}\right)H\left(X[t]|Y_{2}^{t-1},W_{1|2},W_{2},S_{2}[t]=1,S_{2}^{t-1},E^{n}\right)$
$\displaystyle\quad\overset{(a)}{=}\left(1-\delta_{2}\right)H\left(X[t]|Y_{2}^{t-1},W_{1|2},W_{2},S_{2}^{t},E^{n}\right)$
$\displaystyle\quad\overset{(b)}{\geq}\left(1-\delta_{2}\right)H\left(X[t]|Y_{1}^{t-1},W_{1|2},W_{2},S_{1}^{t},E^{n}\right)$
$\displaystyle\quad\overset{(c)}{=}\frac{1-\delta_{2}}{1-\delta_{1}}H\left(Y_{1}[t]|Y_{1}^{t-1},W_{1|2},W_{2},S_{1}^{t},E^{n}\right),$
(66)
where $(a)$ holds since $X[t]$ is independent of the channel realization at
time instant $t$; $(b)$ follows from the following arguments. Consider a
virtual channel state $\tilde{S}[t]$ is an i.i.d. Bernoulli
$(\delta_{2}-\delta_{1})/\delta_{2}$ process independent of $S_{2}[t]$ and
transmitted signal, then, we have
$\displaystyle H\left(X[t]|Y_{2}^{t-1},W_{1|2},W_{2},S_{2}^{t},E^{n}\right)$
$\displaystyle=H\left(X[t]|Y_{2}^{t-1},W_{1|2},W_{2},S_{2}^{t},\tilde{S}^{t-1},E^{n}\right)$
$\displaystyle\geq
H\left(X[t]|\\{(1-S_{2}[\ell])\tilde{S}[\ell]X[\ell]\\}_{\ell=1}^{\ell=t-1},Y_{2}^{t-1},W_{1|2},W_{2},S_{2}^{t},\tilde{S}^{t-1},E^{n}\right)$
$\displaystyle=H\left(X[t]|Y_{1}^{t-1},W_{1|2},W_{2},S_{1}^{t},E^{n}\right),$
(67)
where the third equality holds since receiving both virtual
$(1-S_{2}[\ell])\tilde{S}[\ell]X[\ell]$ and $Y_{2}[\ell]=S_{2}[\ell]X[\ell]$
is statistically the same as receiving $S_{1}[\ell]X[\ell]=Y_{1}[\ell]$ (note
if $S_{2}[\ell]=0$ there is $(\delta_{2}-\delta_{1})/\delta_{2}$ probability
$(1-S_{2}[\ell])\tilde{S}[\ell]=1$), and $X[t]$ is independent of the channel
states and virtual $\tilde{S}^{t-1}$; $(c)$ comes from the fact that
$\Pr\left(S_{1}[t]=0\right)=\delta_{1}$. Next, taking the summation over $t$
from $1$ to $n$, and using the fact that the transmit signal at time instant
$t$ is independent of future channel realizations, we get
$H\left(Y_{2}^{n}|W_{1|2},W_{2},S^{n}_{2},E^{n}\right)\geq\frac{1-\delta_{2}}{1-\delta_{1}}H\left(Y_{1}^{n}|W_{1|2},W_{2},S^{n}_{1},E^{n}\right)$
and also (25).
## Appendix B Proof of $\mathcal{C}^{\mathrm{blind}}_{\mathrm{NN}}$ in
Theorem 3: Opportunistic Transmission
Case 1 : $\delta_{2}\geq\delta_{1}$ and $\epsilon_{2}\in\\{0,1\\}$: First, we
note that as stated in Remark 2, when the weaker receiver has no side-
information, _i.e._ $\epsilon_{2}=1$, then, the capacity region is the same as
having no side-information at either receivers. The case with $\epsilon_{2}=0$
is already given in Section V
Case 2 : $\delta_{1}=\delta_{2}=\delta$, and
$\epsilon_{1}=\epsilon_{2}=\epsilon$ (symmetric setting):
Now we focus on the blind-transmitter assumption where the transmitter no
longer has the luxury of knowing $W_{1|2}$ to send bits in such a way to
benefit both receivers (as was done in Segment b of Phase I in the previous
section V). We note that if both $\epsilon_{1}$ and $\epsilon_{2}$ are equal
to zero, the problem becomes trivial as each receiver has full side-
information of the other user’s message and $R_{i}=(1-\delta_{i})$ is
achievable for $i=1,2$. For case $\epsilon_{1}=\epsilon_{2}=\epsilon>0$, if
each receiver obtains a total of $(1+\epsilon)m$ linearly independent
observation of both $W_{1}$ and $W_{2}$, then, it can recover both messages.
It turns out that this protocol is capacity-achieving. The non-trivial corner
point in this case is given by:
$\displaystyle R_{1}=R_{2}=\frac{(1-\delta)}{(1+\epsilon)}.$ (68)
The protocol is straightforward. The transmitter starts with $m$ bits for each
receiver and creates $(1+\epsilon)m$ linearly independent combinations of the
total $2m$ bits for the two receivers. Then, the transmitter encodes these
combinations using an erasure code of rate $(1-\delta)$ and communicates the
encoded message. Each receiver by the end of the communication block will have
$2m$ linearly independent observations of the $2m$ unknown variables and can
decode both messages. This immediately implies the achievability of the corner
point described in (68).
## Appendix C Proof of Theorem 4
In this case, receiver ${\sf Rx}_{1}$ (the stronger receiver) has full side-
information of the message for ${\sf Rx}_{2}$, _i.e._ $W_{2|1}=W_{2}$, and
receiver ${\sf Rx}_{2}$ has access to $(1-\epsilon_{2})$ of the bits intended
for ${\sf Rx}_{1}$. The outer-bounds of Theorem 1 in this case become:
$\left\\{\begin{array}[]{ll}0\leq\epsilon_{2}\frac{1-\delta_{2}}{1-\delta_{1}}R_{1}+R_{2}\leq\left(1-\delta_{2}\right),&\\\
0\leq R_{1}\leq\left(1-\delta_{1}\right).\end{array}\right.$ (69)
Thus, the non-trivial corner point is given by:
$\displaystyle R_{1}=(1-\delta_{1}),$ $\displaystyle
R_{2}=(1-\epsilon_{2})(1-\delta_{2}).$ (70)
In this case. we cannot achieve the corner point given in (C). To achieve the
region described in Theorem 4, we need to prove the achievability of the
following corner point:
$\displaystyle R_{1}=(1-\delta_{1}),$ $\displaystyle
R_{2}=(1-\epsilon_{2})(1-\delta_{1})(1-\delta_{2}).$ (71)
Achievability protocol: We start with $m$ bits for ${\sf Rx}_{2}$ and $\eta m$
bits for ${\sf Rx}_{1}$, where
$\displaystyle\eta=\frac{R_{1}}{R_{2}}=\frac{1}{(1-\epsilon_{2})(1-\delta_{2})}>1.$
(72)
The achievability protocol is carried over two phases. During the first phase,
the transmitter creates $\eta m$ random linear combinations of the $m$ bits
for ${\sf Rx}_{2}$ such that each subset of $m$ combinations are linearly
independent. The transmitter then sends out the XOR of these combinations with
the uncoded $\eta m$ bits of ${\sf Rx}_{1}$. Thus, this phase has a length of
$\displaystyle t_{1}=\eta m.$ (73)
During this phase, ${\sf Rx}_{1}$ obtains $(1-\delta_{1})\eta m$ of its bits
as it has access to $W_{2}$ as side-information and can cancel out the
interference. Moreover, ${\sf Rx}_{2}$ obtains $(1-\delta_{2})\eta m$ XORed
combinations, and since ${\sf Rx}_{2}$ statistically knows $(1-\epsilon_{2})$
of the bits for ${\sf Rx}_{1}$, we conclude that ${\sf Rx}_{2}$ gathers
$\displaystyle(1-\epsilon_{2})(1-\delta_{2})\eta m=m$ (74)
linearly independent combinations of its $m$ bits and is able to decode its
message $W_{2}$.
The second phase has a total length of
$\displaystyle t_{2}=\frac{\delta_{1}}{(1-\delta_{1})}\eta m.$ (75)
During this second phase, the transmitter creates $t_{2}$ random linear
combinations of the $\eta m$ bits intended for ${\sf Rx}_{1}$ and sends them
out. At the end of this phase, the first receiver gathers additional
$(1-\delta_{1})\eta m$ random equations of its intended bits and combined with
the $(1-\delta_{1})\eta m$ bits it already knows from the first phase, ${\sf
Rx}_{1}$ is able to decode $W_{1}$.
Achievable rates: The total communication time is
$\displaystyle t_{1}+t_{2}=\frac{\eta m}{(1-\delta_{1})}.$ (76)
This immediately implies the achievability of the rates given in (C).
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|
# Matrix-pairing states in the alkaline Fe-selenide superconductors: exotic
Josephson junctions
Emilian M. Nica Department of Physics Box 871504 Arizona State University
Tempe, Arizona 85287-1504<EMAIL_ADDRESS>Qimiao Si Department of Physics and
Astronomy, Rice University, 6100 Main St, Houston, TX, 77005, USA Rice Center
for Quantum Materials, Rice University, 6100 Main St, Houston 77005 TX, USA
Onur Erten Department of Physics Box 871504 Arizona State University Tempe,
Arizona 85287-1504
###### Abstract
True to their unconventional nature, multi-band alkaline Fe-selenides and,
more recently, the heavy-fermion CeCu2Si2 have shown experimental signatures
of fully-gapped but sign-changing superconductivity. A two-orbital pairing
state, called $s\tau_{3}$, with non-trivial matrix structure, was proposed as
a candidate able to reconcile the seemingly contradictory properties of these
superconductors. Motivated by the non-trivial orbital structure of the
proposed $s\tau_{3}$ state, which has opposite signs for the pairing functions
of the two orbitals, we study prototypical Josephson junctions where at least
one of the leads is in a superconducting state of this kind. An analysis of
these junctions in the limit of two degenerate orbitals (bands) and with a
simple form of junction hybridization reveals several remarkable properties.
One is the emergence of gapless, purely electron- and hole-like bound states
for $s\tau_{3}-N-s\tau_{3}$ junctions with arbitrary global phase difference
between the leads, and likewise for $s\tau_{3}-N-I$ junctions. The other is
the absence of static Josephson currents when both leads are superconducting.
In both of these signatures, $s\tau_{3}$ junctions are dramatically different
from more conventional Josephson junctions. We also find that the gapless
bound states are protected by an orbital-exchange symmetry, although the
protection is not topological. Junctions which break this symmetry, such as
$s\tau_{3}-N-s$, have gapped Andreev bound states. In general, the Josephson
effect also re-emerges once the degeneracy of the two orbitals is lifted. We
support these conclusions via analytical and numerical results for the bound
states, together with microscopic calculations of the Josephson current. Our
results indicate that junctions involving $s\tau_{3}$ pairing in alkaline Fe-
selenides will generically have bound states with a small gap together with a
greatly suppressed Josephson current.
## I Introduction
Unconventional superconductors (SC’s) have led to some of the most important
questions in condensed matter physics. One such puzzle came to light not too
long ago in a branch of the Fe-based family, the alkaline Fe-selenides Lee
(2017). On one hand these SC’s are fully-gapped, as evidenced by ARPES studies
Mou _et al._ (2011); X.-P. Wang and T. Qian and P. Richard and P. Zhang and
J. Dong and H.-D. Wang and C.-H. Dong and M.-H. Fang and H. Ding (2011); Xu
_et al._ (2012); Wang _et al._ (2012), while on the other hand, they exhibit
an in-gap spin-resonance in inelastic neutron scattering experiments Park _et
al._ (2011); Friemel _et al._ (2012). The first of these features points
toward an $s$-wave pairing state, while the second implies a pairing state
which changes sign under a $\pi/2$ rotations Eschrig (2006); Stockert _et
al._ (2011); Maier _et al._ (2011); Dai (2015); Si _et al._ (2016), such as
$d$-wave pairing. These seemingly mutually-exclusive, traits were shown to be
reconcilable in a pairing state which transforms as a sign-changing $B_{1g}$
representation of the point group, which nonetheless leads to a fully-gapped
SC state. We called this state $s\tau_{3}$ Nica _et al._ (2017) since it
consists of a $s_{x^{2}y^{2}}$ form factor multiplied by a $\tau_{3}$ Pauli
matrix in a $d_{xz},d_{yz}$ two-orbital space appropriate to the alkaline Fe-
selendides. Underlying the remarkable properties of this pairing candidate is
the non-trivial $\tau_{3}$ matrix structure in orbital space, which ensures
that it transforms as $B_{1g}$. In the tight-binding appropriate to the
alkaline Fe-selenides this pairing state can also be thought of as an
effective $d+d$ intra- and inter-_band_ pairing Nica and Si (2021).
$s\tau_{3}$ pairing was stabilized in a realistic five-orbital model of the
alkaline Fe-selenides Nica _et al._ (2017). Remarkably, a similar
experimental landscape has also recently emerged in the venerable heavy-
fermion CeCu2Si2, the first-discovered unconventional SC Steglich _et al._
(1979). Believed to be a typical $d$-wave for almost it’s entire history,
CeCu2Si2 was recently found to exhibit a small gap for temperatures well below
$T_{c}$ from specific heat Kittaka _et al._ (2014) and London penetration
depth measurements Pang _et al._ (2018); Yamashita _et al._ (2017). In-gap
spin resonances in the inelastic neutron spectrum Stockert _et al._ (2011) as
well as other indicators single out CeCu2Si2 as another example of gapped but
sign-changing superconductivity. A pairing candidate, which incorporates the
orbital and spin-structure of this compound into a non-trivial matrix pairing
was also recently proposed by two of us Nica and Si (2021). More generally,
the $s\tau_{3}$ pairing state represents one of the most dramatic forms of the
overall notion of orbital-selective superconducting pairing introduced in the
context of 111 iron pnictides Yu _et al._ (2014), which has also been
subsequently discussed in other iron pnictides Yin _et al._ (2014); Ong _et
al._ (2016) and the nematic FeSe Sprau _et al._ (2017); Hu _et al._ (2018);
Yu _et al._ (2018).
Figure 1: Summary of our results for Josephson junctions along $x$ where at
least one of the Left (L) and Right (R) leads is in a $s\tau_{3}$ pairing
state. In that case, the pairing functions of the two orbitals have opposite
signs and thus have non-trivial structure on orbital space. We show the most
important results in the limit where the two orbitals are degenerate and
couple identically to a single orbital in the Center (C) metallic part. (a)
$s\tau_{3}-N-s\tau_{3}$ junctions where the L and R leads are identical up to
a global phase difference. For arbitrary global phase differences between L
and R leads, the bound states are either electron- or hole-like and become
gapless for a set of conserved momenta $k_{y}$ . In this limit, the static
Josephson current is zero, since the contributions of the two orbital sectors
effectively cancel. For almost degenerate bands, as expected for systems with
$s\tau_{3}$ pairing, the bound states acquire a small gap via Andreev
scattering, although exceptions can occur for junctions along $z$. A weak
static Josephson current, determined by band-splitting near the FS is also
expected. $s\tau_{3}-N-I$ junctions, where the R lead is in an insulating
state, has similar bound states. (b) $s\tau_{3}-N-s$ junction, where the R
lead consists of a single orbital in a trivial $s$-wave pairing state. In the
same degenerate limit, there are gapped Andreev bound states, but a vanishing
Josephson current. For small band splitting, a weak Josephson current is
expected.
In Ref. Nica and Si (2021), we also pointed out that the $s\tau_{3}$ pairing
state and a similarly-constructed microscopic candidate in CeCu2Si2 are both
equivalent to $d+d$ intra- and inter-band pairing in the band basis. This
equivalence holds when periodic boundary conditions are imposed along all of
the axes. Moreover, we discussed the similarities between $d+d$ and 3He-B,
where a similar $p+p$ pairing can be realized in spin space. In view of the
topologically-protected edge states of 3He-B, a natural question is whether
$d+d$ pairing can also have non-trivial edge states. This consideration opens
up a new direction in the exploration of both the general $d+d$ pairing and
actual microscopic states such as $s\tau_{3}$. For definiteness, in this work
we focus on Josephson junctions as these show features which are specific to
this type of matrix-pairing and therefore can facilitate its identification by
experiments. We reserve a more detailed discussion of the edge state spectrum
for future work.
Although some of the bulk properties of pairing candidates with non-trivial
matrix structure have already been mentioned, much remains to be explored as
far as characteristic experimental signatures are concerned. One promising
route involves Josephson junctions, which in principle can take advantage of
the inherent phase difference between the two orbital sectors in $s\tau_{3}$
pairing. Motivated by this observation, we study prototypical S-N-S and S-N-I
junctions where at least one of the leads is in a $s\tau_{3}$ pairing state.
Although our study is specifically geared toward junctions with $s\tau_{3}$
pairing states and thus target the alkaline Fe-selenides, we expect that some
of the features discussed here can also manifest for similar pairing
candidates with non-trivial matrix structure as proposed for CeCu2Si2 Nica and
Si (2021).
To isolate the most striking features of this pairing state, we describe the
superconducting leads via effective two-orbital models. Furthermore, in order
to capture the effects of the intrinsic phase difference between the pairing
functions of the two orbitals, we consider junctions where the two orbitals
couple to a single orbital in the metallic part. As we show, the bound states
in these rather unusual setups differ sharply from those in more conventional
junctions, which ignore cross-coupling between the various orbital. More
precisely, we examine setups where the Left (L) lead, Center (C) metallic, and
Right (R) leads are arranged in $s\tau_{3}-N-s\tau_{3}$, $s\tau_{3}-N-I$
(insulator), and $s\tau-N-s$ (trivial single-channel s-wave SC) junctions. We
assume that the coupling to the junction does not destabilize $s\tau_{3}$
pairing. Our salient results are as summarized below.
### I.1 Summary of main results
Our most striking results occur when the two orbitals of the $s\tau_{3}$ SC
leads are degenerate. In this limit, both intra- and inter-orbital
hybridization terms which lead to band splitting are absent, and therefore the
two orbitals correspond to two degenerate bands. The precise definitions of
the intra- and inter-orbital hybridizations are discussed in Sec. II. In
addition, the two orbitals couple identically to a single orbital of the C
part. Due to the effects of the non-trivial orbital structure of $s\tau_{3}$
pairing and to the equal coupling to a single C orbital, the usual Andreev
scattering processes are effectively “frustrated”, leading to gapless bound
states which are either electron- or hole-like for $s\tau_{3}-N-s\tau_{3}$ for
arbitrary global phase differences between L and R leads. This also holds for
$s\tau_{3}-N-I$ junctions. The bound states are non-trivial since they decay
into the SC leads. We also find that the gapless bound states are also
protected by an orbital-exchange symmetry which is present for these junctions
in the limit considered here, although the protection is not topological in
origin. We find that breaking the orbital-exchange symmetry leads to a
restoration of Andreev scattering which mixes electron- and hole-like states,
which, in turn, leads to a gap. An extreme example of this symmetry breaking
is provided by $s\tau_{3}-N-s$ junctions, which exhibit Andreev bound states.
A second important feature is that the bound state spectra for the
$s\tau_{3}-N-s\tau_{3}$ and $s\tau_{3}-N-s$ junctions are invariant to changes
in the _global_ phase difference across the junction. This is also in striking
contrast to the typical single-channel Josephson junction, where the Andreev
bound state spectrum changes with phase, leading to a static Josephson effect.
Our unusual findings are confirmed via microscopic calculations of the
Josephson currents in the tunneling limit. We show in this limit that the
contributions from the two sectors of $s\tau_{3}$ pairing, which have opposite
intrinsic phases, cancel.
We also consider more realistic junctions where the SC leads include intra-
and inter-orbital hybridization. We focus on cases where the band splitting in
the vicinity of the Fermi surface (FS) is much smaller than the pairing
amplitudes. We show that intra-orbital hybridization terms break the orbital-
exchange symmetry, leading to bound states more akin to the usual, gapped
Andreev bound states. The same terms also make the bound state spectrum
sensitive to a global phase difference across the junction. These findings are
further confirmed by microscopic calculation of the Josephson current in the
tunneling limit. As a counter-example to the gapped bound state spectrum, we
consider setups where the junction is along the $z$-direction, while the
intra- and inter-orbital hybridization are predominantly in-plane. We show
that gapless states can still be found for in-plane momenta along the
diagonals, where the orbital-exchange symmetry is preserved. In all cases, we
show that the analytical solutions are consistent with numerical results. Fig.
1 provides a summary of our main results.
As discussed in Refs. Nica _et al._ (2017); Nica and Si (2021), $s\tau_{3}$
pairing is expected to provide a superconducting state whose single-particle
excitations are gapped everywhere on the FS. The simplest case to consider is
well established for the alkaline Fe-selenides. In this case, the splitting of
the relevant bands near the FS, which is essentially determined by intra- and
inter-orbital hybridization terms, is small when compared to the pairing
amplitudes. This is likely the case for a similar proposal in CeCu2Si2 Nica
and Si (2021). In light of our preceding discussion, the band-splitting in
alkaline Fe-selenides and CeCu2Si2 necessarily leads to gapped bound states
and to a finite Josephson current for arbitrary global phase differences
between the two leads. However, we expect that these effects are small.
Therefore, $s\tau_{3}-N-s\tau_{3}$ and $s\tau_{3}-N-I$ junctions will
typically exhibit bound states with gaps which are much smaller than the bulk
gap, although exceptions with gapless states are also possible for arbitrary
global phase differences. Likewise, $s\tau_{3}-N-s\tau_{3}$ and
$s\tau_{3}-N-s$ junctions will exhibit static Josephson currents which are
determined by band-splitting near the FS, and are therefore small when
compared to similar junctions with orbitally-trivial pairing states. While
these predictions do not provide unequivocal probes for $s\tau_{3}$ pairing,
their combined signatures can provide significant supporting evidence.
The remainder of the article is divided into the following sections. In Sec.
II, we introduce the microscopic models for the junctions. In Sec. III, we
discuss both analytical and numerical solutions for $s\tau_{3}-N-s\tau_{3}$
junctions, and we briefly compare these to junctions with orbitally-trivial
pairing. $s\tau_{3}-N-I$ junctions have similar bound states, and are briefly
discussed at the end of the section. Section IV is devoted to $s\tau_{3}-N-s$
junctions. In the final Sec. V, we summarize our findings and discuss possible
experimental realizations of the junctions. The appendices contain detailed
discussions of some of the points presented in the main text. In Appendix A,
we present the analytical bound state solutions for $s\tau_{3}-N-s\tau_{3}$
junctions in the important limit of degenerate orbitals. Appendix B briefly
touches on the effect of unequal coupling of the two orbitals to the metallic
part, while Appendix C discusses junctions with orbitally-trivial pairings.
Appendices D and E present the analytical solutions for $s\tau_{3}-N-I$ and
$s\tau_{3}-N-s$ junctions, respectively, both in the limit of degenerate
orbitals. The final Appendix F, presents our results for the Josephson current
for $s\tau_{3}-N-s\tau_{3}$ junctions, as well as for junctions with
orbitally-trivial pairing.
## II Models
The main distinction of the setups considered here are the superconducting
leads which are in a $s\tau_{3}$ pairing state, which was originally proposed
for an effective $d_{xz},d_{yz}$ two-orbital model for the alkaline Fe-
selenides Raghu _et al._ (2008); Si and Abrahams (2008); Daghofer _et al._
(2010). The _spin-singlet_ pairing state consists of a
$s_{x^{2}y^{2}}(k_{x},k_{y})$ form factor multiplied by a $\tau_{3}$ Pauli
matrix in orbital space. Due to the non-trivial $\tau_{3}$ orbital structure,
$s\tau_{3}$ pairing transforms as a $B_{1g}$ irreducible representation of the
tetragonal $D_{4h}$ point-group, and thus it changes sign under a $\pi/2$
rotation about $z$. In the simplest case, with the pairing amplitude being
larger than the band-splitting near the FS, which is due to intra- and inter-
orbital hybridization terms in the normal-state Hamiltonian, the Bogoliubov-de
Gennes (BdG) spectrum of this state is _fully gapped_. In the more general
cases, it is always gapped on the FS. In Ref. Nica _et al._ (2017),
$s\tau_{3}$ pairing was stabilized in a realistic five-orbital model for the
alkaline Fe-selenides.
When considering Josephson junctions, the gapped nature of the $s\tau_{3}$
state implies that the relative phases of the pairing of the two
$d_{xz},d_{yz}$ orbital sectors are preserved by the junction. Furthermore,
the common $s_{x^{2}y^{2}}$ form factor has nodes along the $k_{x}=\pm\pi/2$
and $k_{y}=\pm\pi/2$ axes. For FS’s away from these axes, we can ignore the
momentum-dependence of the pairing, which does not introduce any qualitatively
new effects.
### II.1 Junctions along $x$
For all of the three types of junctions along $x$, we adopt the following
model
$\displaystyle H=$ $\displaystyle
H_{\text{L}}+H_{\text{L-C}}+H_{\text{C}}+H_{\text{C-R}}+H_{\text{R}}.$ (1)
$H_{\text{L}}$, $H_{\text{R}}$, and $H_{\text{C}}$ are the bulk Hamiltonians
for the left, right leads and the center metallic parts, respectively.
$H_{\text{L-C}}$ and $H_{\text{C-R}}$ contain the terms at the lead-center
part interfaces, and by convention, are defined on the last and first sites of
the L and R leads, respectively. The L lead which includes the pairing terms
extends along $x[a]\leq(L_{x}-l)/2$, where $L_{x}$ is the number of sites for
the entire L-C-R system , while $l$ is the number of sites of the metallic C
part, along the $x$-direction, respectively. The length is defined in units of
the lattice spacing $a$. The most general form of the bulk L lead Hamiltonian
for $s\tau_{3}$ pairing is
$\displaystyle H_{\text{L}}=$ $\displaystyle H_{\text{L, TB}}+H_{\text{L,
Pair}},~{}\text{for}~{}x[a]<\left(\frac{L_{x}-l}{2}-1\right).$ (2)
$\displaystyle H_{\text{L, TB}}=$
$\displaystyle\sum_{\alpha\textbf{r}\sigma}\bigg{[}-\left(t_{x\alpha}c^{{\dagger}}_{\mathbf{r},\alpha\sigma}c_{\mathbf{r}+\hat{\mathbf{x}},\alpha\sigma}+t_{y\alpha}c^{{\dagger}}_{\mathbf{r},\alpha\sigma}c_{\mathbf{r}+\hat{\mathbf{y}},\alpha\sigma}+\text{H.c.}\right)-\mu
c^{{\dagger}}_{\mathbf{r},\alpha\sigma}c_{\mathbf{r},\alpha\sigma}$
$\displaystyle+$
$\displaystyle\sum_{\beta\neq\alpha}t_{4}\left(c^{{\dagger}}_{\mathbf{r},\alpha\sigma}c_{\mathbf{r}+\hat{\mathbf{x}}+\hat{\mathbf{y}},\beta\sigma}-c^{{\dagger}}_{\mathbf{r},\alpha\sigma}c_{\mathbf{r}+\hat{\mathbf{x}}-\hat{\mathbf{y}},\beta\sigma}+\text{H.c.}\right)\bigg{]}$
(3)
$\displaystyle H_{\text{L,
Pair}}=\sum_{\alpha\mathbf{r}}\Delta_{\alpha,\text{L}}\left(c^{{\dagger}}_{\mathbf{r},\alpha\uparrow}c^{{\dagger}}_{\mathbf{r},\alpha\downarrow}-c^{{\dagger}}_{\mathbf{r},\alpha\downarrow}c^{{\dagger}}_{\mathbf{r},\alpha\uparrow}\right)+\text{H.c.}$
(4)
The indices $\alpha\in\\{1,2\\}$ stand for the $d_{xz}$ and $d_{yz}$ orbitals,
respectively, while the spin is represented by
$\sigma\in\\{\uparrow,\downarrow\\}$. $t_{x\alpha}$ are the nearest-neighbor
(NN) hopping coefficients for orbital $\alpha$, while $t_{4}$ is a NN orbital
hybridization, with all coefficients taken to be real. These are based on a
simplified two-orbital model introduced in Ref. Raghu _et al._ (2008). Note
the absence of terms proportional to a $\tau_{2}$ matrix in the tight-binding
part, since these break time-reversal symmetry. The pairing satisfies
$\Delta_{1}=-\Delta_{2}=\Delta$. As mentioned previously, we neglect the
spatial dependence of the pairing which is not of immediate importance to the
effects considered here.
For periodic boundary conditions (BC’s), the BdG Hamiltonian for the L lead
and for single spin sector reads
$\displaystyle
H_{\text{L}}(\mathbf{k})=\begin{pmatrix}\xi_{0}+\xi_{3}&\xi_{1}&\Delta&0\\\
\xi_{1}&\xi_{0}-\xi_{3}&0&-\Delta\\\
\Delta^{*}&0&-(\xi_{0}+\xi_{3})&-\xi_{1}\\\
0&-\Delta^{*}&-\xi_{1}&-(\xi_{0}-\xi_{3})\end{pmatrix}$ (5)
where
$\displaystyle\xi_{0}(\mathbf{k})=$
$\displaystyle(t_{1}+t_{2})\left[\cos(k_{x}a)+\cos(k_{y}a)\right]-\mu$ (6)
$\displaystyle\xi_{3}(\mathbf{k})=$
$\displaystyle(t_{1}-t_{2})\left[\cos(k_{x}a)-\cos(k_{y}a)\right]$ (7)
$\displaystyle\xi_{1}(\mathbf{k})=$
$\displaystyle-4t_{4}\sin(k_{x}a)\sin(k_{y}a).$ (8)
$D_{4h}$ symmetry restricts
$\displaystyle t_{x1}=$ $\displaystyle t_{y2}=t_{1}$ (9) $\displaystyle
t_{x2}=$ $\displaystyle t_{y1}=t_{2}.$ (10)
Three terms $\xi_{0}\tau_{0},\xi_{1}\tau_{1}$, and $\xi_{3}\tau_{3}$ determine
the normal state, corresponding to a common dispersion, inter- and intra-
orbital hybridization terms, respectively. Note that terms
$\propto\cos(k_{x}a)\cos(k_{y}a)$, which preserve the lattice symmetry can
also be added to the orbital-_diagonal_ $\xi_{0}$ terms Raghu _et al._
(2008). While these can lead to a change in shape of the FS, they preserve the
orbital structure of $H_{\text{L}}$, i.e. do not induce any additional band
splitting, which is essential to the results of this work. Therefore, we
ignore these additional contributions. The important degenerate-orbital limit,
which we mention in the following, occurs when both inter- and intra-orbital
hybridization terms are set to zero with $\xi_{1}=\xi_{3}=0$. For junctions
along $x$ which break the translation symmetry along this direction, this
corresponds to $t_{1}-t_{2}=t_{4}=0.$
The positive eigenvalues are
$\displaystyle
E_{1,2}=\sqrt{\xi^{2}_{0}+\xi^{2}_{3}+\xi^{2}_{1}+|\Delta|^{2}\pm
2\sqrt{\xi^{2}_{0}\left(\xi^{2}_{1}+\xi^{2}_{3}\right)+\xi^{2}_{1}|\Delta|^{2}}}.$
(11)
As mentioned previously, we consider cases where the band splittings near
either of the two FS’s are smaller than the pairing amplitude
$\displaystyle\sqrt{\xi^{2}_{1}(\mathbf{k})+\xi^{2}_{3}(\mathbf{k})}\big{|}_{\mathbf{k}\in\text{FS}}<|\Delta|^{2}.$
(12)
which ensures that the leads are gapped in the bulk.
The Hamiltonian at the L-C interface is
$\displaystyle H_{\text{L-C}}=$ $\displaystyle\sum_{\alpha
y}\bigg{\\{}\sum_{\sigma}\left[-\left(V_{\alpha}c^{{\dagger}}_{\mathbf{r},\alpha\sigma}c_{\mathbf{r}+\hat{\mathbf{x}},\sigma}+t_{y\alpha}c^{{\dagger}}_{\mathbf{r},\alpha\sigma}c_{\mathbf{r}+\hat{\mathbf{y}},\alpha\sigma}+\text{H.c.}\right)-\mu
c^{{\dagger}}_{\mathbf{r},\alpha\sigma}c_{\mathbf{r},\alpha\sigma}\right]+\left[\Delta_{\alpha}\left(c^{{\dagger}}_{\mathbf{r},\alpha\uparrow}c^{{\dagger}}_{\mathbf{r},\alpha\downarrow}-c^{{\dagger}}_{\mathbf{r},\alpha\downarrow}c^{{\dagger}}_{\mathbf{r},\alpha\uparrow}\right)+\text{H.c.}\right]\bigg{\\}}$
$\displaystyle,~{}$ $\displaystyle\text{for}~{}x[a]=\frac{L_{x}-l}{2}$ (13)
As mentioned earlier, the $k$-dependence of the $s(k)$ factor in the
$s\tau_{3}$ pairing is unimportant for our purpose, and this is reflected in
the form of the pairing term of Eq. 13. Crucially, the terms proportional to
$V_{\alpha}$ denote the hybridization of the two orbitals in the L lead to _a
single orbital_ in the C part along $x$. Since the point-group symmetry is
necessarily broken at the lead-center interfaces, this symmetry does not _a
priori_ restrict the values of the $V_{\alpha}$’s. In the following, we allow
these couplings to take arbitrary values and we comment on the effects of the
$d_{xz},d_{yz}$ nature of the orbitals in the leads in Sec. V. The remaining
terms are identical to those of $H_{\text{L}}$ while the summation is over the
$y$ coordinate along the junction. We typically choose $V_{1}=V_{2}=t_{1}=1$,
unless stated otherwise. As already mentioned, we neglect the spatial
dependence of the pairing.
The Hamiltonian for the bulk of the central (C) part reads
$\displaystyle H_{\text{C}}=$
$\displaystyle\sum_{\alpha\mathbf{r}\sigma}\left[-t_{1}\left(c^{{\dagger}}_{\mathbf{r},\sigma}c_{\mathbf{r}+\hat{\mathbf{x}},\sigma}+c^{{\dagger}}_{\mathbf{r},\sigma}c_{\mathbf{r}+\hat{\mathbf{y}},\sigma}+\text{H.c.}\right)-\mu
c^{{\dagger}}_{\mathbf{r},\sigma}c_{\mathbf{r},\sigma}\right],~{}\text{for}~{}\frac{L_{x}-l}{2}+1\leq
x[a]\leq\frac{L_{x}+l+2}{2}-1.$ (14)
As mentioned previously, the C part involves a _single channel_ without any
pairing. We consider for simplicity nearest-neighbor hopping in this sector,
taken to be equal to $t_{1}$.
In the case of $s\tau_{3}-N-s\tau_{3}$ pairing we choose the Hamiltonians at
the C-R junction $H_{\text{C-R}}$ for $x[a]=(L_{x}+l+2)/2$ and for the bulk of
the R lead $H_{\text{R}}$ for $x[a]\geq(L_{x}+l+2)/2+1$ to have identical form
and coefficients to $H_{\text{L-C}}$ and $H_{\text{L}}$, respectively, with
the exception of a possible _global non-zero phase_ $\phi$ for the pairing
terms:
$\displaystyle\Delta_{\text{L}}=$ $\displaystyle\Delta$ (15)
$\displaystyle\Delta_{\text{R}}=$ $\displaystyle\Delta e^{i\phi}.$ (16)
For $s\tau_{3}-N-s$, $H_{\text{C-R}}$ and $H_{\text{R}}$ reflect the presence
of a single channel in the R lead as
$\displaystyle H_{\text{C-R}}=$ $\displaystyle\sum_{\alpha
y}\bigg{\\{}\sum_{\sigma}\bigg{[}-\left(Vc^{{\dagger}}_{\mathbf{r},\sigma}c_{\mathbf{r}-\hat{\mathbf{x}},\sigma}+t_{y}c^{{\dagger}}_{\mathbf{r},\sigma}c_{\mathbf{r}+\hat{\mathbf{y}},\sigma}+\text{H.c.}\right)$
$\displaystyle-$ $\displaystyle\mu
c^{{\dagger}}_{\mathbf{r},\alpha\sigma}c_{\mathbf{r},\alpha\sigma}\bigg{]}+\left[\Delta
e^{i\phi}\left(c^{{\dagger}}_{\mathbf{r},\uparrow}c^{{\dagger}}_{\mathbf{r},\downarrow}-c^{{\dagger}}_{\mathbf{r},\downarrow}c^{{\dagger}}_{\mathbf{r},\uparrow}\right)+\text{H.c.}\right]\bigg{\\}}$
$\displaystyle,~{}\text{for}~{}x[a]=\frac{L_{J}+l+2}{2}$ (17)
$\displaystyle H_{\text{R, TB}}=$
$\displaystyle-\sum_{\textbf{r}\sigma}\bigg{[}-t\left(c^{{\dagger}}_{\mathbf{r},\sigma}c_{\mathbf{r}-\hat{\mathbf{x}},\sigma}+c^{{\dagger}}_{\mathbf{r},\alpha\sigma}c_{\mathbf{r}+\hat{\mathbf{y}},\sigma}+\text{H.c.}\right)$
$\displaystyle-$ $\displaystyle\mu
c^{{\dagger}}_{\mathbf{r},\sigma}c_{\mathbf{r},\sigma}\bigg{]}$ (18)
$\displaystyle H_{\text{R, Pair}}=\sum_{\mathbf{r}}\Delta
e^{i\phi}\left(c^{{\dagger}}_{\mathbf{r},\uparrow}c^{{\dagger}}_{\mathbf{r},\downarrow}-c^{{\dagger}}_{\mathbf{r},\downarrow}c^{{\dagger}}_{\mathbf{r},\uparrow}\right)+\text{H.c.}.$
(19)
For simplicity, we always consider $V=t=1$.
### II.2 Junctions along $z$
For $s\tau_{3}-N-s\tau_{3}$ junctions along $z$ we define the complete
Hamiltonian via
$\displaystyle\tilde{H}=$
$\displaystyle\tilde{H}_{\text{L}}+\tilde{H}_{\text{L-C}}+\tilde{H}_{\text{C}}+\tilde{H}_{\text{C-R}}+\tilde{H}_{\text{R}}.$
(20)
In addition to the terms already present in the pure two-dimensional case, we
consider additional NN hopping along $z$ for the L/R leads as
$\displaystyle\tilde{H}_{\text{L, TB}}=$ $\displaystyle H_{\text{L,
TB}}-\sum_{\alpha\textbf{r}\sigma}t_{z}\left(c^{{\dagger}}_{\mathbf{r},\alpha\sigma}c_{\mathbf{r}+\hat{\mathbf{z}},\alpha\sigma}+\text{H.c.}\right),$
(21)
where $H_{\text{L, TB}}$ was defined in Eq. 3, and the sum over lattice sites
is now along all of the three axes. We take the lengths of the L, C, and R
parts along $z$ to be the same as for the junctions along $x$. For simplicity,
we consider only orbital-diagonal, NN hopping along the $z$-direction.
Additional terms are of course possible, but they do not qualitatively change
our conclusions. For clarity, we ignore the momentum dependence of the two-
dimensional pairing, as for junctions along $x$. Similar expressions hold for
the bulk of the R lead, with the exception of a global phase difference
$\phi$. Likewise, the C part amounts to
$\displaystyle\tilde{H}_{\text{C}}=$ $\displaystyle
H_{\text{C}}-\sum_{\alpha\textbf{r}}t_{z}\left(c^{{\dagger}}_{\mathbf{r},\sigma}c_{\mathbf{r}+\hat{\mathbf{z}},\sigma}+\text{H.c.}\right).$
(22)
We use a NN hopping in the C part which is equal in amplitude to that of the
L/R leads $t_{z}=0.2t_{1}$, unless stated otherwise. The reduced value of
$t_{z}$ reflects the tetragonal anisotropy of our target systems.
Finally, the Hamiltonians for the L-C and C-R interfaces are obtained via a
straightforward generalization to hopping along $z$
$\displaystyle H_{\text{L-C}}=$
$\displaystyle-\sum_{\alpha,xy}\bigg{\\{}\sum_{\sigma}\bigg{[}\left(-V_{\alpha
z}c^{{\dagger}}_{\mathbf{r},\alpha\sigma}c_{\mathbf{r}+\hat{\mathbf{z}},\sigma}-t_{x\alpha}c^{{\dagger}}_{\mathbf{r},\alpha\sigma}c_{\mathbf{r}+\hat{\mathbf{x}},\alpha\sigma}-t_{y\alpha}c^{{\dagger}}_{\mathbf{r},\alpha\sigma}c_{\mathbf{r}+\hat{\mathbf{y}},\alpha\sigma}+\text{H.c.}\right)-\mu
c^{{\dagger}}_{\mathbf{r},\alpha\sigma}c_{\mathbf{r},\alpha\sigma}$
$\displaystyle+$
$\displaystyle\sum_{\beta\neq\alpha}t_{4}\left(c^{{\dagger}}_{\mathbf{r},\alpha\sigma}c_{\mathbf{r}+\hat{\mathbf{x}}+\hat{\mathbf{y}},\beta\sigma}-c^{{\dagger}}_{\mathbf{r},\alpha\sigma}c_{\mathbf{r}+\hat{\mathbf{x}}-\hat{\mathbf{y}},\beta\sigma}+\text{H.c.}\right)\bigg{]}+\left[\Delta_{\alpha}\left(c^{{\dagger}}_{\mathbf{r},\alpha\uparrow}c^{{\dagger}}_{\mathbf{r},\alpha\downarrow}-c^{{\dagger}}_{\mathbf{r},\alpha\downarrow}c^{{\dagger}}_{\mathbf{r},\alpha\uparrow}\right)+\text{H.c.}\right]\Bigg{\\}}$
(23)
$\tilde{H}_{\text{C-R}}$ can be obtained via a similar generalization of
$H_{\text{C-R}}$.
Unless stated otherwise, we choose units where the nearest-neighbor hopping
and hybridization at the interfaces are $t_{1}=V_{1}=1$ and $\Delta=0.4$.
### II.3 Orbital-exchange symmetry
We consider the $s\tau_{3}-N-s\tau_{3}$ and $s\tau_{3}-N-I$ setups in the
important limit where the two orbitals have identical tight-binding
coefficients, zero intra-orbital hybridization ($\xi_{3}=0$) and inter-orbital
hybridization ($\xi_{1}=0$), and couple identically to the single orbital of
the C part. This limit corresponds to
$\displaystyle t_{1}=$ $\displaystyle t_{2}$ (24) $\displaystyle t_{4}=$
$\displaystyle 0$ (25) $\displaystyle V_{1}=$ $\displaystyle V_{2}.$ (26)
The two orbitals in the L (or R if appropriate) lead correspond to degenerate
bands in the normal state. We define a _local_ transformation which acts on
the L (and R) spinors
$\Psi^{T}=(c_{\mathbf{r},1\uparrow},c_{\mathbf{r},2\uparrow},c^{{\dagger}}_{\mathbf{r},1\downarrow},c^{{\dagger}}_{\mathbf{r},2\downarrow})$
as
$\displaystyle\Psi\rightarrow\hat{R}\Psi$ (27)
where
$\displaystyle\hat{R}=\begin{pmatrix}0&i&0&0\\\ i&0&0&0\\\ 0&0&0&-i\\\
0&0&-i&0\end{pmatrix}.$ (28)
This corresponds to an exchange of the two orbitals followed by a gauge
transformation which changes the sign of the pairing $\Delta$. Similarly, in
the C part the transformation acting on the single-orbital spinor
$\psi^{T}=(c_{\mathbf{r},\uparrow},c^{{\dagger}}_{\mathbf{r},\downarrow})$
$\displaystyle\psi\rightarrow\hat{S}\psi$ (29)
where
$\displaystyle\hat{S}=\begin{pmatrix}i&0\\\ 0&-i\\\ \end{pmatrix}.$ (30)
This operation, corresponding to a gauge transformation for the C part, is
required to compensate for the factors of $i$ in $H_{\text{L-C}}$ and
$H_{\text{C-R}}$ if appropriate. Both $\hat{R}$ and $\hat{S}$ are anti-
hermitian operators which obey
$\displaystyle\hat{R}^{{\dagger}}=$ $\displaystyle-\hat{R}$ (31)
$\displaystyle\hat{R}^{2}=$ $\displaystyle-\hat{1}$ (32)
and similarly for $\hat{S}$.
It is straightforward to define an orbital-exchange operator for the entire
lattice models, corresponding to each of the junctions. For
$s\tau_{3}-N-s\tau_{3}$ junctions it reads
$\displaystyle\hat{Q}=\begin{cases}\hat{R},~{}\forall~{}x\in\text{L, R}\\\
\hat{S},~{}\forall~{}x\in\text{C}\end{cases}$ (33)
The $s\tau_{3}-N-s\tau_{3}$ and $s\tau_{3}-N-I$ junction Hamiltonians in the
degenerate-orbital limit considered here are invariant under $\hat{Q}$. This
orbital exchange symmetry plays an important role in classifying the electron-
like and hole-like solutions encountered in these cases.
## III $s\tau_{3}-N-s\tau_{3}$ and $s\tau_{3}-N-I$ junctions
We treat $s\tau_{3}-N-s\tau_{3}$ junctions in some detail in the following and
only briefly discuss $s\tau_{3}-N-I$ junctions at the end of the section and
in Appendix C, since the latter exhibit similar bound states.
### III.1 Degenerate orbitals
We consider a $s\tau_{3}-N-s\tau_{3}$ junction as defined previously, in the
limit of degenerate orbitals ( Eqs. 24-26 ). We find the bound states in the
continuum limit by linearizing the BdG equations in the vicinity of points
$(\alpha K_{Fx},K_{Fy})$ on the FS Sauls (2018), where $\alpha=\pm 1$. The
detailed solution is presented in Appendix A. Here, we summarize some of the
most important results. In contrast to the typical single-channel junction,
the presence of two orbitals in either leads, which couple to a single orbital
in the C part, imposes additional BC’s. Indeed, in the limit considered here,
only the linear combination $c_{1}+c_{2}$ of the two orbitals couples to the C
part. The corresponding BdG coefficients must vary continuously at the L-C or
C-R interfaces, much like in a single-channel junction. By contrast, the anti-
symmetric combination does not couple to the C part and thus the corresponding
BdG coefficients obey open BC’s at the interface. As shown in Appendix A, the
two channels, corresponding to symmetric and anti-symmetric linear
combinations, do not decouple in the bulk of the leads, due to the non-trivial
matrix structure of $s\tau_{3}$ pairing. Consequently, the continuity and open
BC’s cannot be satisfied simultaneously for solutions which involve both
electron- and hole-like solutions in the C part. The solutions are either
electron- or hole-like in the C part, in contrast to the typical Andreev bound
states which mix the two.
The electron-like bound states with eigenvalues
$\displaystyle\frac{\epsilon}{|\Delta|}=$
$\displaystyle\pm\cos\left(\frac{\epsilon
l}{2v_{Fx}}+\frac{K_{Fx}l}{2}+\frac{(a-m)\pi}{2}\right).$ (34)
have the form
$\displaystyle\Psi^{L/R}_{K_{Fy};\text{Electron}}=$
$\displaystyle\left(u^{L/R}_{1e},u^{L/R}_{2e},v^{L/R}_{1e},v^{L/R}_{2e}\right)^{T}$
(35)
$\displaystyle u^{L/R}_{1e}=$
$\displaystyle\frac{1}{\sqrt{2}}|A||\Delta|e^{i\theta^{0}_{A}}\sin\left[K_{Fx}x\mp\frac{\epsilon
l}{2v_{Fx}}+\frac{(a+m)\pi}{2}\right]$ (36) $\displaystyle u^{L/R}_{2e}=$
$\displaystyle u^{L/R}_{1}$ (37) $\displaystyle v^{L/R}_{1e}=$
$\displaystyle\frac{1}{\sqrt{2}}|A||\Delta|e^{i\theta^{0}_{A}}\sin\left[K_{Fx}\left(x\pm\frac{l}{2}\right)+\gamma_{L/R}\pi\right]$
(38) $\displaystyle v^{L/R}_{2e}=$ $\displaystyle-v^{L/R}_{1}$ (39)
$\displaystyle\gamma_{L}=$ $\displaystyle a$ (40) $\displaystyle\gamma_{R}=$
$\displaystyle m$ (41)
in the L/R leads and
$\displaystyle\Psi^{C}_{K_{Fy};\text{Electron}}=$
$\displaystyle\left(u^{C}_{e},v^{C}_{e}\right)^{T}$ (42)
$\displaystyle u^{C}_{e}=$ $\displaystyle
2|A||\Delta|e^{i\theta^{0}_{A}}\sin\left[K_{Fx}x+\frac{\epsilon
x}{v_{Fx}}+\frac{(a+m)\pi}{2}\right]$ (43) $\displaystyle v^{C}_{e}=$
$\displaystyle 0.$ (44)
in the C part. $|A|$ is a normalization constant, $\theta^{0}_{A}$ is an
arbitrary phase, and $a,m$ are arbitrary integers. Crucially, the C $v$ BdG
coefficient is identically zero.
Similarly, the hole-like solutions with eigenvalues
$\displaystyle\frac{\epsilon}{|\Delta|}=$
$\displaystyle\pm\cos\left(\frac{\epsilon
l}{2v_{Fx}}-\frac{K_{Fx}l}{2}+\frac{(n-b)\pi}{2}\right).$ (45)
are of the form
$\displaystyle\Psi^{L/R}_{K_{Fy};\text{Hole}}=$
$\displaystyle\left(u^{L/R}_{1h},u^{L/R}_{2h},v^{L/R}_{1h},v^{L/R}_{2h}\right)^{T}$
(46)
$\displaystyle u^{L/R}_{1h}=$
$\displaystyle\frac{1}{\sqrt{2}}|B|e^{i\theta^{B}_{0}}\sin\left[K_{Fx}\left(x\pm\frac{l}{2}\right)+\gamma_{L/R}\pi\right]$
(47) $\displaystyle u^{L/R}_{2h}=$ $\displaystyle-u^{L}_{1h}$ (48)
$\displaystyle v^{L/R}_{1h}=$
$\displaystyle\frac{1}{\sqrt{2}}|B|e^{i\theta^{B}_{0}}\sin\left[K_{Fx}x\pm\frac{\epsilon
l}{2v_{Fx}}+\frac{\pi}{2}+\frac{(n+b)\pi}{2}\right]$ (49) $\displaystyle
v^{L/R}_{2h}=$ $\displaystyle v^{L}_{1h}$ (50) $\displaystyle\gamma_{L}=$
$\displaystyle b$ (51) $\displaystyle\gamma_{R}=$ $\displaystyle n$ (52)
for the L/R leads and
$\displaystyle\Psi^{C}_{K_{Fy};\text{Hole}}=$
$\displaystyle\left(u^{C}_{h},v^{C}_{h}\right)^{T}$ (53)
$\displaystyle u^{C}_{h}=$ $\displaystyle 0$ (54) $\displaystyle v^{C}_{h}=$
$\displaystyle\sin\left[K_{Fx}x-\frac{\epsilon
x}{v_{Fx}}+\frac{(n+b)\pi}{2}\right]$ (55)
in the C part. As before, $|B|$ is a normalization constant, $\theta^{0}_{B}$
is an arbitrary phase, while $b,n$ are integers. In contrast to the electron-
like solutions, $u^{C}=0$.
These detailed analytical solutions present a number of important features.
Firstly, the solutions are either electron- or hole-like since the
corresponding BdG coefficient $v$ or $u$ vanishes in the C part. This is in
clear contrast to a single-channel junction, where electron-like and hole-like
solutions are mixed via Andreev scattering. The presence of both open and
continuity BC’s at L-C and C-R interfaces, as well as the non-trivial
structure of the pairing, prevents the existence of solutions which mix
electron and hole states.
Secondly, electron- and hole-like solutions become gapless when
$\displaystyle K_{Fx}l=$ $\displaystyle(m-a)\pi$ (56) $\displaystyle K_{Fx}l=$
$\displaystyle(n-b)\pi,$ (57)
respectively, as indicated by Eqs. 34 and 45. This happens when $K_{Fx}l$ is
either 0 or a multiple of $\pi$. Recall that the solutions are labeled by the
momenta $(K_{Fx},K_{Fy})$ which vary continuously along the FS. Consequently,
the gapless conditions can be realized in multiple instances along the FS if
the latter and/or junction length $l$ are sufficiently large. Linearizing Eqs.
34 and 45 about these points indicates that the electron- and hole-like states
are counter-propagating, as a function of the conserved momentum $K_{Fy}$.
Thirdly, the gaplessness of the electron- and hole-like states is protected by
the orbital-exchange symmetry defined in Sec. II.3. Indeed, it is
straightforward to check that
$\displaystyle\hat{R}\Psi^{L/R}_{K_{Fy};\text{Electron}}=$ $\displaystyle
i\Psi^{L/R}_{K_{Fy};\text{Electron}}$ (58)
$\displaystyle\hat{R}\Psi^{L/R}_{K_{Fy};\text{Hole}}=$ $\displaystyle
i\Psi^{L/R}_{K_{Fy};\text{Hole}},$ (59)
together with
$\displaystyle\Psi^{L/R,{\dagger}}_{K_{Fy};\text{Electron}}\hat{R}=$
$\displaystyle i\Psi^{L/R,{\dagger}}_{K_{Fy};\text{Electron}}.$ (60)
The last relation follows from the anti-hermitian nature of $\hat{R}$. Similar
relations hold for $\hat{S}$ acting on the C spinors. Together, these imply
that any operator $\hat{O}$ added to the Hamiltonian of the junction, which
commutes with $\hat{R}$ and $\hat{S}$, will not mix electron- and hole-like
states via
$\displaystyle\braket{\Psi^{{\dagger}}_{\text{Electron}}}{\hat{O}\hat{R}}{\Psi_{\text{Hole}}}=\braket{\Psi^{{\dagger}}_{\text{Electron}}}{\hat{R}\hat{O}}{\Psi_{\text{Hole}}}$
(61)
which implies that
$\displaystyle(-i)\braket{\Psi^{{\dagger}}_{\text{Electron}}}{\hat{O}}{\Psi_{\text{Hole}}}=$
$\displaystyle(+i)\braket{\Psi^{{\dagger}}_{\text{Electron}}}{\hat{O}}{\Psi_{\text{Hole}}}$
(62)
and subsequently that
$\displaystyle\braket{\Psi^{{\dagger}}_{\text{Electron}}}{\hat{O}}{\Psi_{\text{Hole}}}=0.$
(63)
Note that $\hat{O}$ is not necessarily local at the lattice level. A similar
reasoning can be applied to $\hat{S}$. Therefore, the orbital-exchange
symmetry protects the gapless electron- and hole-like states, in analogy to
the time-reversal operator for spin-polarized edge states in quantum spin Hall
systems Kane and Mele (2005), although we stress that the gapless states are
not due to any topological property of the bulk in our cases. In practice, the
orbital-exchange symmetry is preserved when no intra-orbital hybridization
terms (such as $\xi_{3}$ defined in Eq. 5) are present, although inter-orbital
hybridization terms (such as $\xi_{1}$ in Eq. 5) preserve the orbital-exchange
symmetry.
Fourthly, the bound state spectrum is independent of the global relative phase
$\phi$ between L and R leads, as indicated by Eqs. 34 and 45. This is in clear
contrast to the single-channel junctions, where the change in the Andreev
bound state spectrum with relative phase is proportional to the static
Josephson current Sauls (2018). In the $s\tau_{3}-N-s\tau_{3}$ case, a bound
state spectrum which is insensitive to the relative phase indicates the
absence of a static Josephson current. This is confirmed by microscopic
calculations of the Josephson current in the tunneling approximation,
discussed in detail in Appendix F, in the degenerate orbital limit considered
here. As shown there, the Josephson currents due to each orbital sector cancel
due essentially to the $\pi$ phase difference between the two components of
$s\tau_{3}$ pairing.
Next, we proceed to confirm the analytical solutions via numerical solutions
of the lattice model introduced in Sec. II, in the degenerate-orbital limit.
The results were obtained for a lattice of 100 sites along $x$ with a C part
of 5 sites extending along $49\leq x\leq 53$, in units of the lattice constant
$a$.
In Fig. 2, we show the bound state spectrum as a function of the conserved
momentum $k_{y}$, for varying chemical potential $\mu$ and global relative
phase difference between the two superconducting leads $\phi$. For $\mu=-3.0$
in panel (a), we illustrate that the spectrum is independent of $\phi$, in
accordance with the eigenvalues in Eqs. 34 and 45. This implies the absence of
static Josephson current, as confirmed in Appendix F by a microscopic
calculation of the Josephson current in the tunneling limit. Also note that
the bound states become gapless and cross twice near $k_{y}a\approx 0.3\pi$.
This is consistent with the condition for gapless states as determined by the
analytical solution (Eqs. 56, 57). Indeed, for increasing chemical potential
$\mu=-2.0$, as shown in panel (b), an additional crossing occurs near
$k_{y}=0$. This is due to an $K_{Fx}$ (near $K_{Fy}=0$) which increases by
$\pi$ as the FS expands. The crossings in (a) likewise shift to higher
$k_{y}$. A similar picture presents itself with increasing chemical potential
$\mu=-0.25$ in panel (c). Together, these results are consistent with the
analytical solution.
We also consider the nature of the bound states. In Fig. 3, we show a close-up
of panel (a) of Fig. 2 for $\phi=0$. One pair of hole- and electron-like
states in the vicinity of a crossing are marked by a square and triangle,
respectively. To elucidate the nature of these states, in Fig. 4 we illustrate
the real parts of the BdG coefficients of the hole-like state marked in Fig. 3
by a square. Note that the imaginary pars are trivially zero and are not
shown. The $u$ BdG coefficients shown in panel (a) are consistent with the
analytical solution in Eq. 46. Indeed, $u^{C}$ vanishes in the C part for
$49\leq x[a]\leq 53$, as expected for a hole-like solution. Furthermore,
$u^{L/R}_{1}$ and $u^{L/R}_{2}$ both vanish at the L-C and C-R interfaces,
respectively, and their signs are consistent with Eq. 46 for $n-b$ an odd
multiple of $\pi$. In panel (b), the $v^{C}$ is finite and continuous across
the junction. We conclude that the numerical solution is consistent with the
hole-like solution indicated by the analytical results.
A similar picture emerges for the electron-like state marked with a triangle
in Fig. 3. In Fig. 5, we show the real parts of the BdG coefficients of this
state across the junction, and compare these with the analytical solution in
Eq. 35. $u^{C}$ is now finite, while $v^{C}$ vanishes in the C region, as
expected. Furthermore, the signs of $u^{L/R}_{1}$ and $u^{L/R}_{2}$ are
consistent with a solution with $m-a$ an odd multiple of $\pi$. Together with
the BdG coefficient for the hole-like state, these numerical solutions confirm
the analytical results.
We briefly discuss the case for a $s\tau_{3}-N-s\tau_{3}$ junction along $z$
for the model introduced in Sec. II.2. The analytical solution is similar to
the junction along $x$ with the exception that both $k_{x},k_{y}$ are
conserved quantities. We therefore linearize about the pair of points
$(K_{Fx},K_{Fy},\alpha K_{Fz})$ and obtain the bound-state spectrum as in Eqs.
34 and 45 which now extend along the $z$-direction. The eigenvalues are
obtained via the replacement $K_{Fx}\rightarrow K_{Fz}.$
We comment on the effects of unequal coupling of the two orbitals in either
lead to the C part. In this case, both symmetric and anti-symmetric linear
combinations of the two orbitals (Appendix A), couple to the C part, in
contrast to the solutions presented here. As shown by numerical results
presented in Appendix B, unequal couplings induce a gap, as well as a spectrum
which depends on the global phase difference $\phi$. The first can be
understood via a breaking of the orbital-exchange symmetry, while the second
can be confirmed via a direct calculation of the Josephson current, as
discussed in Appendix F.
Finally, we note that the surprising behavior for $s\tau_{3}$ junctions
coupled to a single orbital in the C part differs dramatically from that of
junctions consisting of leads in a pairing state which has trivial orbital
structure. We consider junctions with such a state, which we call $s\tau_{0}$,
where the two orbitals have identical pairing functions, including the signs.
For degenerate orbitals, we can apply the same analysis as in the
$s\tau_{3}-N-s\tau_{3}$ case (Appendix A) and we again find that the symmetric
linear combination of the two orbitals in either leads couples across the
junction. However, the anti-symmetric linear combination is entirely decoupled
along the entire junction, and in contrast to $s\tau_{3}$ junctions, imposes
no additional constraints on the bound-state solutions. Thus $s\tau_{0}$
junctions for degenerate orbitals behave essentially like conventional single-
channel junctions. This is illustrated in Appendix C.
Figure 2: Bound state spectrum of a $s\tau_{3}-N-s\tau_{3}$ junction along
$x$, as defined in Sec. II.1, with degenerate orbitals in the L and R leads
which also couple identically to a single orbital in the N part. (a)
Eigenvalues for a fixed chemical potential $\mu=-3.0$ and varying global
relative phase $\phi$ between the R and L superconducting leads, as functions
of the conserved momentum $k_{y}$. Note that the spectrum is insensitive to
$\phi$, in accordance to the analytical solution in Eqs. 34 and 45. Also note
the presence of gapless states near $k_{y}a\approx 0.3\pi$. (b) Same as (a)
for fixed $\mu=-2.0$ and $\phi=0.0$. Although not shown, the spectrum is also
insensitive to $\phi$. There is an additional crossing of gapless states near
$k_{y}a=0.0$. This is consistent with the condition for gapless states
obtained from the analytical solutions in Eqs. 56-57. With decreasing $\mu$
the FS increases, allowing $K_{Fx}l$ to reach values which are multiples of
$\pi$ near $k_{y}=0$. (c) Same as (b) for $\mu=-0.25$ and $\phi=0$. Note that
the number of gapless crossings increases with the size of the FS, as
predicted by the analytical solution. Figure 3: Close-up of panel (a) of Fig.
2 (a). The square and triangle identify the hole- and electron-like states
illustrated in Fig. 4 and 5, respectively. Figure 4: Real parts of the BdG
coefficients for a hole-like solution as a function of $x$ across the
junction, for the momentum indicated by the square in Fig. 3. These are
obtained from the numerical calculations for $\mu=-3.0,\phi=0$. The imaginary
parts vanish identically and are not shown. (a) The $u$ coefficients are
consistent with Eq. 46, as $u^{C}=0$ (blue squares) in the C part extending
from $49\leq x[a]\leq 53$, as expected for a pure hole-like state. Similarly,
$u^{L/R}_{1}$ and $u^{L/R}_{2}$ vanish at the L-C and C-R interfaces,
respectively, in accordance to the analytical calculations. The opposite signs
of these coefficients within the L and R leads and across the junction are
also consistent with the analytical solution with $n-b$ an odd number. (b) The
$v$ coefficients are finite throughout, including the C part, and are
continuous across the junction. These are also consistent with the analytical
solution in Eq. 46. Figure 5: Real part of the BdG coefficients for an
electron-like solution as a function of $x$ across the junction, for the
momentum indicated by the triangle in Fig. 3. As in Fig. 4, these are obtained
from the numerical calculations for $\mu=-3.0,\phi=0$, and the imaginary parts
vanish identically. (a) $u^{C}$ is finite, as expected for a pure electron-
like state, and in agreement with the analytical solution in Eq. 35. The
remaining $u$ coefficients are also consistent with the latter. (b) $v^{C}=0$
again indicates a pure electron-like state. $v^{L/R}_{1}$ and $v^{L/R}_{2}$
vanish at the L-C and C-R interfaces, and their signs are consistent with
$m-a$ and odd, as indicated by Eq. 35.
### III.2 Non-degenerate orbitals
So far, we have focused on the special case of degenerate orbitals for a
$s\tau_{3}-N-s\tau_{3}$ junction along $x$. We now consider more realistic
cases, where the L and R leads include terms which lift the degeneracy between
the two orbitals. As discussed in Sec. II.1, we consider two types of
symmetry-allowed terms which can lift the orbital degeneracy: (i) intra-
orbital hybridization terms corresponding to $\xi_{3}\tau_{3}$ in the non-
pairing part of the bulk lead Hamiltonians (Eq. 5), and (ii) inter-orbital
hybridization terms corresponding $\xi_{1}\tau_{1}$ in the same model. We
examine the effects of each of these terms on the bound state spectrum
separately.
We first consider intra-orbital hybridization terms only by fixing $t_{1}=1$
and allowing $\delta t=t_{1}-t_{2}$ to vary (Eq. 7). All other parameters are
identical to those in the top panel of Fig. 2. In Fig. 6 (a) we show the bound
state spectrum for the orbital-degenerate case with $\delta t=0$ together with
a case with significant intra-orbital hybridization corresponding to $\delta
t=0.8$. The gapless states for the degenerate orbital case become gapped with
the inclusion of intra-orbital hybridization terms. These break the orbital-
exchange symmetry and thus mix the electron- and hole-like states of the
orbital-degenerate case, in accordance with Sec. III.1. In panel (b), we show
the spectrum for $\delta t=0.8$ as a function of the global phase difference
$\phi$. The eigenvalues change with $\phi$ and we recover gapless states at
$\pi$ phase difference, as is the case for the typical single-channel
junction. The evolution of the bound state spectrum with $\phi$ is also
confirmed via a calculation of the Josephson current in Appendix F.
We now consider the effect of the inter-orbital hybridization terms only,
corresponding to $\xi_{1}$ in Eq. 5. All other parameters are the same in
porevious cases. As shown in Fig. 7 (a) the bound state spectrum remains
gapless under the inclusion of inter-orbital hybridization, in contrast to
intra-orbital hybridization terms. This can also be understood via the
orbital-exchange symmetry, since the inter-orbital hybridization terms
preserve the former. Similarly, the spectrum is invariant under changes in the
global phase difference $\phi$, as indicated in Fig. 7 (b). It is then clear
that, due to intra-orbital hybridization terms, a general
$s\tau_{3}-N-s\tau_{3}$ junction along $x$ exhibits the usual features of a
single-channel junction.
As a final example, we consider the effects of intra- and inter-orbital
hybridization on the junction along $z$. The spectrum in this case can be
labeled by both $k_{x},k_{y}$ conserved momenta. For general values of the
latter, the spectrum is gapped, in analogy to the junction along $x$. However,
along the diagonals of the two-dimensional Brillouin zone $|k_{x}|=|k_{y}|$,
the intra-orbital hybridization terms vanish, and the bound state states again
become gapless. This is illustrated in Fig. 8, and can be understood via the
orbital-exchange symmetry, which is recovered along the diagonals. Thus, in
contrast to the junction along $x$, the junction along $z$ can exhibit gapless
edge states for a rather general model of $s\tau_{3}$ pairing.
Figure 6: The effects of intra-orbital hybridization terms corresponding to
$\xi_{3}$ in the bulk of the L/R leads (Eq. 5) on the bound state spectrum for
a $s\tau_{3}-N-s\tau_{3}$ junction along $x$. We introduce these terms by
fixing $t_{1}=1$ and allowing $\delta t=t_{1}-t_{2}$ to vary (Eq. 7). The
inter-orbital hybridization corresponding to $\xi_{1}$ in Eq. 5 is set to
zero. All other parameters are the same as in Fig. 2 (a) and $\phi=0$. The two
orbitals couple identically to the C part. (a) A gap develops as intra-orbital
hybridization terms are introduced. For clarity, we show the case for
degenerate orbitals $\delta t=0$ (green rhombi), and a case with extreme
intra-orbital hybridization for $\delta t=0.8$ (red squares). Note that a
finite gap opens for any $\delta t\neq 0$. Intra-orbital hybridization terms
break the orbital-exchange symmetry discussed in Sec. II.3, allowing the
electron- and hole-like states in Eqs. 35 and 46 to mix via Andreev
scattering. (b) Effect of a global phase difference $\phi$ across the junction
on the bound state spectrum with finite intra-orbital hybridization $\delta
t=0.8$. In contrast to the degenerate case shown in Fig. 2 (a), the inclusion
of finite intra-orbital hybridization allows the spectrum to change with
$\phi$. Note that the states become gapless at $\phi=\pi$, much like in a
conventional single-channel junction. The sensitivity of the spectrum with
$\phi$ is also confirmed via a calculation of the Josephson current in
Appendix F. Figure 7: The effects of an inter-orbital hybridization term,
corresponding to $\xi_{1}$ (Eq. 8) in the bulk L and R leads, on the bound
state spectrum for a $s\tau_{3}-N-s\tau_{3}$ junction along $x$. The intra-
orbital hybridization terms corresponding to $\xi_{3}$ are set to zero. The
remaining parameters are the same as in Fig. 6. (a) The bound states remain
gapless as $t_{4}$ (Eq. 8) increases, in contrast to the intra-orbital
hybridization case in Fig. 6. The inter-orbital hybridization preserves the
orbital-exchange symmetry defined in Sec. II.3 and thus does not mix the
electron- and hole-like states, which remain adiabatically connected with the
solutions of the orbital-degenerate junction in Eqs. 35 and 46. (b) The
spectrum is insensitive to a global phase difference $\phi$, in contrast to
leads with intra-orbital hybridization. These results are also confirmed via a
calculation of the Josephson current (Appendix F). Figure 8: Bound state
spectrum for a $s\tau_{3}-N-s\tau_{3}$ junction along $z$, calculated from the
model in Sec. II.2, with finite intra- and inter-orbital hybridization. We use
$\delta t=0.4$ and $4t_{4}=0.4$, $\mu=-3.0$ with a NN $t_{z}=0.2$ along $z$.
Note that both $k_{x}$ and $k_{y}$ are conserved in this case. In contrast to
the junction along $x$, gapless states are possible here even when intra-
orbital hybridization is included. The latter vanishes for $|k_{x}|=|k_{y}|$.
For these momenta, we recover the orbital-exchange symmetry defined in Sec.
II.3. (a) bound state eigenvalues as a function of $k_{y}$ for fixed
$k_{x}=0$. The intra-orbital hybridization terms are finite for these momenta,
and the states are gapped, as when the junction is along $x$ (Fig. 6. (b) Same
as (a) for $k_{x}=0.4k_{y}$. The gaps shrink but remain finite. (c) Same as
(a) and (b) along the diagonal of the 2D Brillouin zone for $k_{x}=k_{y}$. The
gap closes as the model recovers the orbital-exchange symmetry, and the
spectrum is similar to that of the junction along $x$ with inter-orbital
hybridization only (Fig. 7).
The bound-state spectra of $s\tau_{3}-N-I$ junctions are very similar to those
of $s\tau_{3}-N-s\tau_{3}$ junctions. In the limit of degenerate orbitals, the
bound states of the former are also electron- or hole like, and they also
become gapless for a set of conserved momenta. The analytical solutions in
this limit are discussed in Appendix C. The evolution with either intra- or
inter-orbital hybridization is also very similar to that in
$s\tau_{3}-N-s\tau_{3}$ junctions and, as such, will not be discussed here in
detail.
## IV $s\tau_{3}-N-s$ junctions
We consider a $s\tau_{3}-N-s$ junction, where the L lead is in a $s\tau_{3}$
pairing state, while the R lead is in a single-channel $s$-wave state. As in
the case of $s\tau_{3}-N-s\tau_{3}$ junctions, both orbitals on the L side
couple identically to a single orbital in the C metallic part, which in turn
couples to a single orbital in the trivially-paired R lead. The model is
described in Sec. II.1.
We first solve the system in the limit of degenerate orbitals in the L leads,
which couple identically to the single C orbital. The analytical solution is
presented in detail in Appendix E. Here, we summarize some of the most
important results. In contrast to the $s\tau_{3}-N-s\tau_{3}$ case, only the
L-C interface is subject to both open and continuity BC’s. Furthermore, as
explained in Sec. II.3, this model does not preserve the orbital-exchange
symmetry, due to the presence of a single channel in the R lead, even when the
two orbitals are degenerate. Thus, in contrast to the $s\tau_{3}-N-s\tau_{3}$
case, the $s\tau_{3}-N-s$ junction allows for Andreev scattering mixing
electron and hole-like states, and leading to bound states which are generally
gapped. However, in the degenerate orbital limit, the bound states states of
the $s\tau_{3}-N-s$ are invariant under a change in the global relative phase
$\phi$.
The analytical results are confirmed by the numerical solutions. In Fig. 9
(a), we show the bound state spectrum for a $s\tau_{3}-N-s$ junction, whith
parameters similar to those of Fig. 2, for two values of $\mu$. We see the
presence of a gap in both instances, a gap which also occurs for any value of
$\mu$. An analysis of the eigenstates, not shown here for brevity, likewise
indicates that the C spinors involve a mixture of electron- and hole-like
states. Remarkably, the spectrum is invariant under a change in global phase
difference $\phi$ in the degenerate-orbital limit, as shown in Fig. 9 (b). The
spectrum does change with $\phi$ once intra-orbital hybridization terms are
introduced, as in the case of $s\tau_{3}-N-s\tau_{3}$ junctions. The bound-
state variation with $\phi$ is also confirmed via a calculation of the
Josephson current in Appendix F.
Figure 9: Bound state spectrum for a $s\tau_{3}-N-s$ junction along $x$. In
all cases, the two orbitals in the L lead couple identically to the single C
orbital. All other parameters, unless explicitly stated, are the same as in
Fig. 2. (a) Evolution of the spectrum for degenerate orbitals in the L lead
with chemical potential $\mu$. Note that the states are gapped for both values
shown here, as well as for any $\mu$. This stands in contrast to the
$s\tau_{3}-N-s\tau_{3}$ junction, as shown in Fig. 2. The difference can be
attributed to the orbital-exchange symmetry, which is broken for the
$s\tau_{3}-N-s$, thus allowing Andreev scattering which mixes electron- and
hole-like states. (b) The spectrum for the degenerate-orbital case of panel
(a) with $\mu=-3.0$ is invariant under a change in global relative phase
$\phi$. This is due to the absence of intra-orbital hybridization terms, as is
also the case with the $s\tau_{3}-N-s\tau_{3}$ junctions in Fig. 2 (a). (c)
The spectrum does change with $\phi$ once intra-orbital hybridization terms
are introduced for the L lead, in a manner analogous to Fig. 6. This is also
confirmed by the calculation of the Josephson current in Appendix F.
## V Summary and Discussion
We now summarize our main results and subsequently discuss possible
experimental realizations.
### V.1 Summary of results
We studied Josephson junctions where the left lead is in an unconventional
$s\tau_{3}$ gapped pairing state which involves two orbitals with opposite-
sign pairing. Similar pairing states were advanced as promising candidates in
alkaline Fe- selenides Nica _et al._ (2017), where $d_{xz},d_{yz}$ orbitals
provided the two-dimensional manifold for $s\tau_{3}$, as well as in the
heavy-fermion superconductor CeCu2Si2 Nica and Si (2021). We considered
junctions where both orbitals of the left lead couple to a single orbital in
an intermediate metallic part. Such arrangements, although unusual, allow for
a richer phenomenology when compared to the more typical junctions without
cross-coupling. We considered three different types of junctions, referred to
as $s\tau_{3}-N-s\tau_{3}$, $s\tau_{3}-N-I$, and $s\tau_{3}-N-s$, where the
right lead is in a two-orbital $s\tau_{3}$ pairing, insulating, and trivial
single-orbital $s$-wave pairing states, respectively. We discussed both two-
dimensional arrangements with junctions along the $x$-axis, as well as
junctions along $z$.
We studied the three types of junctions in the important limit where the two
orbitals of the $s\tau_{3}$ pairing state, in the left and, when appropriate,
right leads, are degenerate and couple identically to a single orbital of the
central part. Our most striking results can be grouped under two headings. One
is the emergence of purely electron and hole-like bound states, which become
gapless and degenerate for a set of conserved momenta. The other is that the
bound state spectrum is invariant under a change in the global phase
difference between left and right superconducting leads. In both aspects, the
junctions differ sharply from the typical single- or multi-channel junctions
without cross-coupling. The first of these two effects occurs for
$s\tau_{3}-N-s\tau_{3}$ and $s\tau_{3}-N-I$ junctions along $x$. Here, the
bound states differ from to the typical single-channel junction, where Andreev
scattering mixes purely electron- and hole-like states. The absence of Andreev
scattering in these cases, which leads to gapless electron- and hole-like
states, is due to the combined effects of non-trivial orbital structure of
$s\tau_{3}$ pairing and coupling to a single C orbital. We also find that
these gapless bound states are protected by an orbital-exchange symmetry.
However, we stress that this protection is not due to any topological property
of the junction, unlike the well-known gapless states which occur in a single-
channel $\pi$ junction. The second property, the invariance of the bound state
spectrum with global phase difference, manifests in $s\tau_{3}-N-s\tau_{3}$
and $s\tau_{3}-N-s$ junctions along $x$. The invariance of the bound state
spectrum points toward a vanishing Josephson current Sauls (2018). We
confirmed that this is the case by calculating the Josephson current directly
in the tunneling limit. We stress that $s\tau_{3}-N-s\tau_{3}$ junctions in
the limit of degenerate orbitals differ dramatically from
$s\tau_{0}-N-s\tau_{0}$ junctions in the same limit. The latter involve
orbitally-trivial pairing states which are identical to both orbitals,
including the sign. In clear contrast to the $s\tau_{3}-N-s\tau_{3}$
junctions, the $s\tau_{0}-N-s\tau_{0}$ bound state spectrum is essentially
that of a typical single-channel junction.
We also considered deviations from the degenerate-orbital limit by introducing
intra- and inter-orbital hybridization terms for the normal part of the leads.
These terms are allowed by tetragonal symmetry for $d_{xz},d_{yz}$ orbitals in
the context of alkaline Fe-selenides Raghu _et al._ (2008). We found that
intra-orbital hybridization terms break the orbital-exchange symmetry and lead
to Andreev scattering which mixes electron and hole-like states, gapping the
bound state spectrum of $s\tau_{3}-N-s\tau_{3}$ and $s\tau_{3}-N-I$ junctions
along $x$. By contrast, inter-orbital hybridization terms do not break the
orbital-exchange symmetry and preserve the gapless electron- and hole-like
states in the cases mentioned above. The bound states thus behave as in the
intra-orbital hybridization case when both the former and inter-orbital
hybridization are present. Important exceptions can occur as exemplified by
$s\tau_{3}-N-s\tau_{3}$ and $s\tau_{3}-N-I$ junctions along $z$, with dominant
NN hopping along $z$. In this case, the bound states become gapless at the
zeroes of the intra-orbital hybridization terms, along the diagonal of the
two-dimensional Brillouin zone in our case, since the junction recovers
orbital-exchange symmetry at these points.
We also considered the effects of lifting the orbital degeneracy in the bulk
of the leads when a finite global phase difference across the junction is
present. Here again we find that the bound state spectrum becomes sensitive to
the phase when intra-orbital hybridization terms are in effect for both
$s\tau_{3}-N-s\tau_{3}$ and $s\tau_{3}-N-s$ junctions. This is also confirmed
by calculations of the Josephson current. In contrast, under the addition of
inter-orbital hybridization terms only, the bound state spectrum remains
invariant, also in accordance with a vanishing Josephson current.
The two crucial aspects behind the striking behavior of all of the junctions
considered here involve two-orbital matrix-pairing $s\tau_{3}$ in the leads,
together with identical coupling of both of these orbitals to a single orbital
in the central metallic parts. These setups ensure that the two orbital
sectors of $s\tau_{3}$ pairing are entangled in a non-trivial way near the
interface with the metallic part, even when the two sectors are decoupled far
into the bulk of the leads. This is to be contrasted with cases where the
pairing has trivial matrix structure, as exemplified by $s\tau_{0}$ pairing.
In those cases, the trivial pairing matrix structure means that two orbitals
which are decoupled in the bulk of the leads remain so near the interfaces
with the metallic part. Likewise, $s\tau_{3}$ junctions with central metallic
regions which likewise involve two orbitals, which couple in a ”one-to-one”
manner with those of $s\tau_{3}$ would fail to fully capture the effects of
the non-trivial matrix structure of $s\tau_{3}$. As exemplified by
$s\tau_{3}-N-I$ junctions, the interplay between non-trivial interfaces and
non-trivial matrix-pairing suggests that exotic edge states could also be
engineered for materials which likely involve $s\tau_{3}$ pairing, such as the
alkaline Fe-selenides.
### V.2 Experimental signatures
Having given a summary of our main results, we now discuss their potential
experimental signatures. We stress that, in the limit of degenerate orbitals,
junctions involving $s\tau_{3}$ pairing are dramatically different from the
typical single-channel junction as well as from junctions involving pairing
states with trivial orbital structure, such as the $s\tau_{0}$ states.
However, any realizations of the $s\tau_{3}$ junctions proposed here will
involve deviations from the ideal case. We argue that such deviations can be
made small, in the sense discussed below, allowing a partial observation of
the striking properties of the ideal case.
We consider three likely deviations from the ideal case in the form of: (i)
unequal coupling of the two orbitals to the single-orbital of the metallic
part, (ii) lifting of the degeneracy of the two orbitals in the leads via
symmetry-allowed intra- and inter-orbital hybridization, and (iii) weak
disorder in the junction. For (i), unequal coupling to the C part gaps the
electron- and hole-like states and induces a nonzero Josephson effect, even
when the orbitals of the leads are degenerate. A single-orbital intermediate
region amounts to an effectively single-band system which remains non-
superconducting for temperatures above the critical temperature of
$s\tau_{3}$, which can be estimated from the transition temperature in
alkaline Fe-selenides Lee (2017). The condition of equal coupling to the
center part is unlikely to occur for junctions along the in-plane axes of the
tetragonal Fe-selenides i.e. along $x$ or $y$, since $d_{xz}$ and $d_{yz}$
orbitals typically cannot couple identically to any other in-plane orbital.
However, junctions along $z$ provide better candidates in this context, as the
the lobes of $d_{xz}$ and $d_{yz}$ are likely to have comparable overlap with
such orbitals as $p_{z}$, $d_{z^{2}}$, and $d_{xy}$ along the $z$ axis.
A second deviation from the ideal case occurs due to the presence of both
intra- and inter-orbital hybridization terms in the leads for junctions
involving alkaline Fe-selenide leads. As we have shown, such terms will lead
to gapping of the electron- and hole-like states and will furthermore ensure
that a static Josephson effect is present. Note however, that, as proposed in
Refs. Nica _et al._ (2017); Nica and Si (2021), and as reiterated in our
text, $s\tau_{3}$ pairing induces a full gap in the bulk of the leads when the
pairing amplitude exceeds the band splitting near the FS, which is governed by
the intra- and inter-orbital hybridization terms. We expect that this limits
the effects of band splitting in the case of the junctions, such that the
induced gap for the bound states will be small compared to the bulk gap in the
leads. Similarly, the Josephson current, which depends on the strength of the
intra-orbital hybridization terms, will likely be finite but strongly
suppressed, as compared to a similar setup for pairing states with trivial
orbital structure. However, it should be borne in mind that exceptions to this
behavior can occur for junctions along $z$. The intra-orbital hybridization
terms can vanish by symmetry along certain directions in the two-dimensional
Brillouin zone. Here, the junction recovers the orbital-exchange symmetry and
the gapless bound states. We do expect a finite Josephson current in these
cases, due to the contribution of bulk states away from these points.
The third source of deviation from the ideal case is the presence of disorder
in the junction. The gapless bound states in the ideal degenerate cases are
not due to any non-trivial topology, although they are protected by the
orbital-exchange symmetry. Therefore, these states are not robust against
disorder.
Due to these inherent limitations, the junctions discussed here cannot
singlehandedly indicate $s\tau_{3}$ pairing, but they can nonetheless provide
strong supporting evidence.
## VI Acknowledgements
OE acknowledge support from National Science Foundation Awards No. DMR
1904716. EMN is supported by ASU startup grant. Work at Rice has been
supported by the U.S. Department of Energy, Office of Science, Basic Energy
Sciences, under Award No. DE-SC0018197 and the Robert A. Welch Foundation
under Grant No. C-1411 (Q.S.). One of us (Q.S.) acknowledges the hospitality
of the Aspen Center for Physics, which is supported by NSF grant No.
PHY-1607611.
## Appendix A Analytical solutions of the $s\tau_{3}-N-s\tau_{3}$ junction
with degenerate orbitals
We consider a $s\tau_{3}-N-s\tau_{3}$ junction with degenerate orbitals/bands
in the normal states corresponding to $t_{1}=t_{2},t_{4}=0$ in Eqs. 9, 8
together with identical coupling to the single channel of the C part
corresponding to $V_{1}=V_{2}$ in Eq. 13. In this limit, only the symmetric
linear combination of the two orbitals at either L-C and C-R interfaces
couples to the single orbital of the C part.
We apply a local unitary transformation to the L lead
$\displaystyle\begin{pmatrix}c_{\mathbf{r},I,\uparrow}\\\
c_{\mathbf{r},II,\uparrow}\\\ c^{{\dagger}}_{\mathbf{r},I,\downarrow}\\\
c^{{\dagger}}_{\mathbf{r},II,\downarrow}\end{pmatrix}=\frac{1}{\sqrt{2}}\begin{pmatrix}1&1&0&0\\\
1&-1&0&0\\\ 0&0&1&1\\\
0&0&1&-1\end{pmatrix}\begin{pmatrix}c_{\mathbf{r},1,\uparrow}\\\
c_{\mathbf{r},2,\uparrow}\\\ c^{{\dagger}}_{\mathbf{r},1,\downarrow}\\\
c^{{\dagger}}_{\mathbf{r},2,\downarrow}\end{pmatrix}.$ (64)
Under this transformation, only channel $I$, corresponding to the symmetric
linear combination of the operators for the two orbitals, couples to the
single orbital of the C part. Furthermore, the transformation leaves the
orbital-diagonal tight-binding part of the Hamiltonian invariant
(corresponding to $\xi_{0}$ in Eq. 5) while it transforms the $s\tau_{3}$
pairing as
$\displaystyle H_{\text{L, Pair}}=$
$\displaystyle\sum_{\mathbf{r}}\Delta\left(c^{{\dagger}}_{\mathbf{r},1\uparrow}c^{{\dagger}}_{\mathbf{r},1\downarrow}-c^{{\dagger}}_{\mathbf{r},2\uparrow}c^{{\dagger}}_{\mathbf{r},2\downarrow}\right)+\text{H.c.}$
$\displaystyle\rightarrow\sum_{\mathbf{r}}\Delta\left(c^{{\dagger}}_{\mathbf{r},I\uparrow}c^{{\dagger}}_{\mathbf{r},II\downarrow}-c^{{\dagger}}_{\mathbf{r},II\uparrow}c^{{\dagger}}_{\mathbf{r},I\downarrow}\right)\text{+H.c.},$
(65)
where for simplicity we only consider one spin sector. The model thus reduces
to solving for channel $I$ as in a typical, single-channel Josephson junction,
while channel $II$ must obey open BC’s at the L-C interface.
We proceed to find solutions of the model in the transformed basis which lie
below the bulk gap, corresponding to solutions which decay into the L lead. We
first apply a Fourier transform along $y$ and linearize the BdG equations in
the vicinity of two points $(\alpha K_{Fx}+q_{x},K_{Fy}+q_{y})$ on the FS,
with $q_{x},q_{y}$ small, and $\alpha=\pm 1$. This is in analogy to the
typical single-channel solution Sauls (2018). After identifying
$q_{x}\rightarrow-i\partial_{x}$ in the continuum limit, we separate the BdG
ansatz of the L lead into fast- and slowly-varying parts as
$\displaystyle\begin{pmatrix}\tilde{u}_{I,k_{y};L}\\\
\tilde{u}_{II,k_{y};L}\\\ \tilde{v}_{I,k_{y};L}\\\
\tilde{v}_{II,k_{y};L}\end{pmatrix}=$ $\displaystyle
e^{iK_{Fy}y}e^{iK_{Fx}x}e^{\kappa\left(x+\frac{l}{2}\right)}\begin{pmatrix}u_{\alpha=1,I;L}\\\
u_{\alpha=1,II;L}\\\ v_{\alpha=1,I;L}\\\
v_{\alpha=1,II;L}\end{pmatrix}+e^{iK_{Fy}y}e^{-iK_{F}x}e^{\kappa\left(x+\frac{l}{2}\right)}\begin{pmatrix}u_{\alpha=\bar{1},I;L}\\\
u_{\alpha=\bar{1},II;L}\\\ v_{\alpha=\bar{1},I;L}\\\
v_{\alpha=\bar{1},II;L}\end{pmatrix}.$ (66)
Note that the C part is between $-l/2\leq x\leq l/2$. Since we are looking for
solutions which decay into the L lead, we take $\kappa$ to be real and
positive. Also note that we consider an ansatz which is a linear superposition
of solutions with opposite momenta along $x$. This is due to the BdG
coefficients of channel $II$ which must obey open BC at the L-C junction,
while the momentum along $y$ is conserved. We ignore corrections due to
$q_{y}a$.
The slowly-varying parts obey the BdG equation
$\displaystyle\begin{pmatrix}\alpha\left[-iv_{Fx}\kappa\right]-\epsilon&0&0&\Delta\\\
0&\alpha\left[-iv_{Fx}\kappa\right]-\epsilon&\Delta&0\\\
0&\Delta^{*}&-\alpha\left[-iv_{Fx}\kappa\right]-\epsilon&0\\\
\Delta^{*}&0&0&-\alpha\left[-iv_{Fx}\kappa\right]-\epsilon\end{pmatrix}\begin{pmatrix}u_{\alpha,I}\\\
u_{\alpha,II}\\\ v_{\alpha,I}\\\ v_{\alpha,II}\end{pmatrix}=$ $\displaystyle
0,$ (67)
where we introduced the Fermi velocities along $x$ via
$\displaystyle v_{Fx}=$ $\displaystyle 2t_{1}\sin(K_{Fx}a).$ (68)
Note that the first and fourth, and second and third rows respectively
decouple in the bulk Hamiltonian and can be solved independently. We find that
the solutions in the bulk of the L lead are
$\begin{pmatrix}u_{\alpha,I}\\\ u_{\alpha,II}\\\ v_{\alpha,I}\\\
v_{\alpha,II}\end{pmatrix}=\begin{pmatrix}A_{\alpha}\left(\epsilon-i\alpha\Lambda\right)\\\
B_{\alpha}\left(\epsilon-i\alpha\Lambda\right)\\\ B_{\alpha}\Delta^{*}\\\
A_{\alpha}\Delta^{*}\end{pmatrix}$ (69)
where $A_{\alpha},B_{\alpha}$ are coefficients to be determined from the BC’s
and
$\displaystyle\kappa=$ $\displaystyle\frac{\Lambda}{v_{Fx}}$ (70)
$\displaystyle\Lambda=$
$\displaystyle\sqrt{\Delta^{2}-\epsilon^{2}},~{}\text{for}~{}\epsilon<|\Delta|.$
(71)
The general solution in the C part, extending from $-l/2\leq x\leq l/2$ can be
similarly determined to be of the form
$\displaystyle\begin{pmatrix}\tilde{u}_{k_{y};C}\\\ \tilde{v}_{k_{y};C}\\\
\end{pmatrix}=$ $\displaystyle
e^{iK_{F}x}\begin{pmatrix}E_{1}e^{i\frac{\epsilon x}{v_{Fx}}}\\\
G_{1}e^{-i\frac{\epsilon
x}{v_{Fx}}}\end{pmatrix}+e^{-iK_{F}x}\begin{pmatrix}E_{\bar{1}}e^{-i\frac{\epsilon
x}{v_{Fx}}}\\\ G_{\bar{1}}e^{i\frac{\epsilon x}{v_{Fx}}}\end{pmatrix}.$ (72)
The solutions in the R lead, which decay away from the C-R interface for
$x>l/2$ are
$\displaystyle\begin{pmatrix}\tilde{u}_{I,k_{y};R}\\\
\tilde{u}_{II,k_{y};R}\\\ \tilde{v}_{I,k_{y};R}\\\
\tilde{v}_{II,k_{y};R}\end{pmatrix}=$ $\displaystyle
e^{iK_{F}x}e^{-\kappa\left(x-\frac{l}{2}\right)}\begin{pmatrix}M_{1}(\epsilon+i\Lambda)\\\
N_{1}(\epsilon+i\Lambda)\\\ N_{1}|\Delta|e^{-i\phi}\\\
M_{1}|\Delta|e^{-i\phi}\end{pmatrix}+e^{-iK_{F}x}e^{-\kappa\left(x-\frac{l}{2}\right)}\begin{pmatrix}M_{\bar{1}}(\epsilon-i\Lambda)\\\
N_{\bar{1}}(\epsilon-i\Lambda)\\\ N_{\bar{1}}|\Delta|e^{-i\phi}\\\
M_{\bar{1}}|\Delta|e^{-i\phi}\end{pmatrix}.$ (73)
Note that we have introduced a global phase $\phi$ in the pairing of the R
lead.
For our purposes, it proves convenient to parameterize all of the coefficients
in terms of an overall phase and a relative phase as in
$\displaystyle A_{\alpha}=$
$\displaystyle|A|e^{i\theta^{0}_{A}}e^{i\alpha\theta_{A}},$ (74)
and similarly for all $\alpha$-dependent quantities. We also parameterize the
factors which enter in the general solutions for the leads as
$\displaystyle\epsilon\pm i\alpha\Lambda=$ $\displaystyle|\Delta|e^{\pm
i\alpha\theta}$ (75) $\displaystyle\theta=$
$\displaystyle\arg\left(\epsilon+i\Lambda\right).$ (76)
We now consider the BC’s. As discussed previously, channel $II$ in the L and R
leads does not couple to the C part, and thus must obey open BC’s at the L-C
and C-R interfaces, respectively:
$\displaystyle\tilde{u}_{II,k_{y};L}\left(x=\frac{-l}{2}\right)=$
$\displaystyle 0$ (77)
$\displaystyle\tilde{v}_{II,k_{y};L}\left(x=\frac{-l}{2}\right)=$
$\displaystyle 0$ (78)
$\displaystyle\tilde{u}_{II,k_{y};R}\left(x=\frac{l}{2}\right)=$
$\displaystyle 0$ (79)
$\displaystyle\tilde{v}_{II,k_{y};R}\left(x=\frac{l}{2}\right)=$
$\displaystyle 0.$ (80)
By contrast, channel $I$ couples across the junction and therefore satisfies
the continuity conditions
$\displaystyle\tilde{u}_{I,k_{y};L}\left(x=\frac{-l}{2}\right)=$
$\displaystyle\tilde{u}_{k_{y};C}\left(x=\frac{-l}{2}\right)$ (81)
$\displaystyle\tilde{v}_{I,k_{y};L}\left(x=\frac{-l}{2}\right)=$
$\displaystyle\tilde{v}_{k_{y};C}\left(x=\frac{-l}{2}\right)$ (82)
$\displaystyle\tilde{u}_{I,k_{y};R}\left(x=\frac{l}{2}\right)=$
$\displaystyle\tilde{u}_{k_{y};C}\left(x=\frac{l}{2}\right)$ (83)
$\displaystyle\tilde{v}_{I,k_{y};R}\left(x=\frac{l}{2}\right)=$
$\displaystyle\tilde{v}_{k_{y};C}\left(x=\frac{l}{2}\right)$ (84)
Expressed in term of the $\alpha$\- dependent quantities these amount to
$\displaystyle-\frac{K_{Fx}l}{2}+\theta_{B}-\theta=$
$\displaystyle\frac{\pi}{2}+b\pi$ (85)
$\displaystyle-\frac{K_{Fx}l}{2}+\theta_{A}=$
$\displaystyle\frac{\pi}{2}+a\pi$ (86)
$\displaystyle\frac{K_{Fx}l}{2}+\theta_{N}+\theta=$
$\displaystyle\frac{\pi}{2}+n\pi$ (87)
$\displaystyle\frac{K_{Fx}l}{2}+\theta_{M}=$
$\displaystyle\frac{\pi}{2}+m\pi,$ (88)
for the open BC’s. $a,b,m$, and $n$ are arbitrary integers. The continuity
conditions imply that
$\displaystyle\theta_{A}-\theta+\frac{\epsilon l}{v_{Fx}}=$
$\displaystyle\theta_{M}+\theta$ (89) $\displaystyle\theta_{B}-\frac{\epsilon
l}{v_{Fx}}=$ $\displaystyle\theta_{N}$ (90)
$\displaystyle|A|=$ $\displaystyle|M|$ (91) $\displaystyle|B|=$
$\displaystyle|N|$ (92)
$\displaystyle\theta^{0}_{A}=$ $\displaystyle\theta^{0}_{M}$ (93)
$\displaystyle\theta^{0}_{B}=$ $\displaystyle\theta^{0}_{M}-\phi.$ (94)
Note that the _overall phase_ $\theta^{0}_{B}$ can incorporate the global
phase difference of the pairing $\phi$. This is in contrast to the typical
single-channel case and it leads to an insensitivity of the bound state
spectrum w.r.t. to $\phi$.
Note that there are five unknowns consisting of the five relative phase
$\theta_{A},\theta_{B},\theta_{M},\theta_{N}$ together with the eigenenergies
$\epsilon$. However, due to the open BC’s, there are six equations. Therefore,
non-trivial solutions cannot be found for this system of equations. We
consider instead solutions where either $A_{\alpha},M_{\alpha}$ or
$B_{\alpha},N_{\alpha}$ are trivially zero. In either of these cases, the
system involving the relative phase and $\epsilon$ reduces to three equations
with three unknowns. Importantly, these solutions are found to be either hole-
or electron-like, as either the $u$ or $v$ BdG coefficients vanish in the C
part.
The electron-like solutions obtained in this manner are
$\displaystyle\begin{pmatrix}\tilde{u}_{I,k_{y};L-e}\\\
\tilde{u}_{II,k_{y};L-e}\\\ \tilde{v}_{I,k_{y};L-e}\\\
\tilde{v}_{II,k_{y};L-e}\end{pmatrix}=$ $\displaystyle
2|A||\Delta|e^{i\theta^{0}_{A}}\begin{pmatrix}\cos\left[K_{Fx}x-\frac{\epsilon
l}{2v_{Fx}}+\frac{(a+m+1)\pi}{2}\right]\\\ 0\\\ 0\\\
\cos\left[K_{Fx}\left(x+\frac{l}{2}\right)+\frac{\pi}{2}+a\pi\right]\end{pmatrix}.$
(95)
$\displaystyle\begin{pmatrix}\tilde{u}_{k_{y};C-e}\\\ \tilde{v}_{k_{y};C-e}\\\
\end{pmatrix}=$ $\displaystyle
2|A||\Delta|e^{i\theta^{0}_{A}}\begin{pmatrix}\cos\left[K_{Fx}x+\frac{\epsilon
x}{v_{Fx}}+\frac{(a+m+1)\pi}{2}\right]\\\ 0\end{pmatrix}.$ (96)
$\displaystyle\begin{pmatrix}\tilde{u}_{I,k_{y};R-e}\\\
\tilde{u}_{II,k_{y};R-e}\\\ \tilde{v}_{I,k_{y};R-e}\\\
\tilde{v}_{II,k_{y};R-e}\end{pmatrix}=$ $\displaystyle
2|A||\Delta|e^{i\theta^{0}_{A}}\begin{pmatrix}\cos\left[K_{Fx}x+\frac{\epsilon
l}{2v_{Fx}}+\frac{(a+m+1)\pi}{2}\right]\\\ 0\\\ 0\\\
\cos\left[K_{Fx}\left(x-\frac{l}{2}\right)+\frac{\pi}{2}+m\pi\right]\end{pmatrix}.$
(97)
Using
$\displaystyle\arctan(\theta)=\frac{\Lambda}{\epsilon}$ (98)
we determine the eigenvalues
$\displaystyle\frac{\epsilon}{|\Delta|}=$
$\displaystyle\pm\cos\left(\frac{\epsilon
l}{2v_{Fx}}+\frac{K_{Fx}l}{2}+\frac{(a-m)\pi}{2}\right).$ (99)
Similarly, the hole-like solutions are
$\displaystyle\begin{pmatrix}\tilde{u}_{I,k_{y};L-h}\\\
\tilde{u}_{II,k_{y};L-h}\\\ \tilde{v}_{I,k_{y};L-h}\\\
\tilde{v}_{II,k_{y};L-h}\end{pmatrix}=$ $\displaystyle
2|B|e^{i\theta^{B}_{0}}\begin{pmatrix}0\\\
\cos\left[K_{Fx}\left(x+\frac{l}{2}\right)+\frac{\pi}{2}+b\pi\right]\\\
\cos\left[K_{Fx}x+\frac{\epsilon
l}{2v_{Fx}}+\frac{\pi}{2}+\frac{(n+b+1)\pi}{2}\right]\\\ 0\end{pmatrix}.$
(100)
$\displaystyle\begin{pmatrix}\tilde{u}_{k_{y};C-h}\\\ \tilde{v}_{k_{y};C-h}\\\
\end{pmatrix}=$ $\displaystyle 2|B|e^{i\theta^{B}_{0}}\begin{pmatrix}0\\\
\cos\left[K_{Fx}x-\frac{\epsilon
x}{v_{Fx}}+\frac{(n+b+1)\pi}{2}\right]\end{pmatrix}.$ (101)
$\displaystyle\begin{pmatrix}\tilde{u}_{I,k_{y};R-h}\\\
\tilde{u}_{II,k_{y};R-h}\\\ \tilde{v}_{I,k_{y};R-h}\\\
\tilde{v}_{II,k_{y};R-h}\end{pmatrix}=$ $\displaystyle
2|B|e^{i(\theta^{B}_{0}+\phi)}\begin{pmatrix}0\\\
\cos\left[K_{Fx}\left(x-\frac{l}{2}\right)+\frac{\pi}{2}+n\pi\right]\\\
\cos\left[K_{Fx}x-\frac{\epsilon l}{2v_{Fx}}+\frac{(n+b+1)\pi}{2}\right]\\\
0\end{pmatrix}.$ (102)
with eigenvalues
$\displaystyle\frac{\epsilon}{|\Delta|}=$
$\displaystyle\pm\cos\left(\frac{\epsilon
l}{2v_{Fx}}-\frac{K_{Fx}l}{2}+\frac{(n-b)\pi}{2}\right).$ (103)
The solutions in the original basis, obtained via the inverse transformation
of Eq. 64 can be obtained from the solutions shown above.
## Appendix B $s\tau_{3}-N-s\tau_{3}$ junctions with unequal coupling to the
C part
In Fig. 10 (a), we show the bound state spectrum for a $s\tau_{3}-N-s\tau_{3}$
junction along $x$ as a function of $\delta V=V_{1}-V_{2}$, the difference
between the coupling constants of the two degenerate orbitals in either lead
to the single orbital of the C part. The results show that, in clear contrast
to the case of equal-coupling $\delta V=0$, a finite $\delta V$ breaks the
orbital-exchange symmetry and gaps the purely electron- and hole-like states.
In panel (b) we show the spectrum as a function of $\phi$, for fixed $\delta
V=0.7$. Again in contrast to the $\delta V=0$ cases, the spectrum becomes
dependent on $\phi$, a feature which is also confirmed via a calculation of
the Josephson current in Appendix F.
Figure 10: Bound state spectrum for a $s\tau_{3}-N-s\tau_{3}$ junction along
$x$, where the couplings to the single orbital of the C part, $V_{1}$ and
$V_{2}$ (Eq. 13), are unequal. We consider degenerate orbitals and compare to
the results for $V_{1}=V_{2}$ in Fig. 2 (a). In contrast to the case with
equal $V_{1}=V_{2}$, the bound states acquire a gap with increasing $\delta
V=V_{1}-V_{2}$. This can be understood via a broken orbital-exchange symmetry
which mixes the purely electron- and hole-like states via Andreev scattering.
(b) The spectrum changes as a function of global phase difference across the
junction $\phi$. The results are for $\delta V=0.3$. This is in contrast to
the case with $\delta V=0$, which is shown in Fig. 2.
## Appendix C $s\tau_{0}-N-s\tau_{0}$ junctions
In this section, we consider $s\tau_{0}-N-s\tau_{0}$ junctions along $x$,
where the leads are in $s\tau_{0}$ pairing states. In these cases, the pairing
functions for both orbitals are identical. We consider degenerate orbitals
which couple to a single orbital in the C part identically. We show that, in
clear contrast to $s\tau_{3}-N-s\tau_{3}$ junctions, $s\tau_{0}-N-s\tau_{0}$
junctions behave essentially like a typical single-channel junction. In Fig.
11 we illustrate the evolution of the bound state spectrum as a function of
global phase difference $\phi$.
Figure 11: Bound state spectrum for a $s\tau_{0}-N-s\tau_{0}$ junction along
$x$, where the two orbitals have identical pairing functions, as a function of
global phase $\phi$. In contrast to $s\tau_{3}$ pairing, $s\tau_{0}$ is an
orbitally-trivial pairing state. The overall setup of the junction is the same
as in the $s\tau_{3}$ cases, with each of the two degenerate orbitals in
either L and R leads coupling identically to a single orbital in the C part.
All of the parameters of the model are the same as in Fig. 2 (a). As indicated
by the results, the bound state spectrum for a $s\tau_{0}-N-s\tau_{0}$
junction differs dramatically from a $s\tau_{3}-N-s\tau_{3}$ junction in two
main aspects: (i) The spectrum for $\phi=0$ is gapped, and (ii) for varying
$\phi$ the spectrum evolves much like a typical single-channel junction,
becoming gapless at $\phi=\pi$. In $s\tau_{0}$ junctions, the symmetric linear
combination of the two orbitals couples across the junction as in the single-
orbital case, while the remaining antisymmetric linear combination decouples
entirely.
## Appendix D Analytical solutions of the $s\tau_{3}-N-I$ junction with
degenerate orbitals
The solution in the continuum limit is very similar to that of the
$s\tau_{3}-N-s\tau_{3}$ case. Instead of continuity BC’s at the C-R interface,
the C spinors obey open BC’s. Using the ansatze and the conventions of
Appendix A, these conditions amount to
$\displaystyle\frac{K_{Fx}l}{2}+\frac{\epsilon l}{2v_{Fx}}+\theta_{E}=$
$\displaystyle\frac{\pi}{2}+p\pi$ (104)
$\displaystyle\frac{K_{Fx}l}{2}-\frac{\epsilon l}{2v_{Fx}}+\theta_{G}=$
$\displaystyle\frac{\pi}{2}+r\pi,$ (105)
where $p,r$ are integers. As for the $s\tau_{3}-N-s\tau_{3}$ junctions, we
obtain electron- and hole-like solutions of the form
$\displaystyle\begin{pmatrix}u^{L}_{1e}&u^{L}_{2e}&v^{L}_{1e}&v^{L}_{2e}\end{pmatrix}^{T}$
$\displaystyle=$ $\displaystyle
2|A|e^{i\theta^{A}_{0}}|\Delta|\begin{pmatrix}\sin\left[K_{Fx}\left(x-\frac{l}{2}\right)-\frac{\epsilon
l}{v_{Fx}}+p\pi\right]\\\
\sin\left[K_{Fx}\left(x-\frac{l}{2}\right)-\frac{\epsilon
l}{v_{Fx}}+p\pi\right]\\\
\sin\left[K_{Fx}\left(x+\frac{l}{2}\right)+a\pi\right]\\\
-\sin\left[K_{Fx}\left(x+\frac{l}{2}\right)+a\pi\right]\end{pmatrix}$ (106)
$\displaystyle\begin{pmatrix}u^{C}_{e}v^{C}_{e}\end{pmatrix}^{T}$
$\displaystyle=$ $\displaystyle
2|A|e^{i\theta_{0}}|\Delta|\begin{pmatrix}\sin\left[K_{Fx}\left(x-\frac{l}{2}\right)+\frac{\epsilon}{v_{Fx}}\left(x-\frac{l}{2}\right)+p\pi\right]\\\
0\end{pmatrix}$ (107)
with eigenvalues
$\displaystyle\frac{\epsilon}{|\Delta|}=$
$\displaystyle\pm\cos\left(\frac{\epsilon l}{v_{Fx}}+K_{Fx}l-(p-a)\pi\right)$
(108)
for the electron-like states. The hole-like solutions are
$\displaystyle\begin{pmatrix}u^{L}_{1h}&u^{L}_{2h}&v^{L}_{1h}&v^{L}_{2h}\end{pmatrix}^{T}$
$\displaystyle=$ $\displaystyle
2|B|e^{i\theta^{B}_{0}}|\Delta|\begin{pmatrix}\sin\left[K_{Fx}\left(x+\frac{l}{2}\right)+b\pi\right]\\\
-\sin\left[K_{Fx}\left(x+\frac{l}{2}\right)+b\pi\right]\\\
\sin\left[K_{Fx}\left(x-\frac{l}{2}\right)+\frac{\epsilon
l}{v_{Fx}}+r\pi\right]\\\
\sin\left[K_{Fx}\left(x-\frac{l}{2}\right)+\frac{\epsilon
l}{v_{Fx}}+r\pi\right]\end{pmatrix}$ (109)
$\displaystyle\begin{pmatrix}u^{C}_{h}v^{C}_{h}\end{pmatrix}^{T}$
$\displaystyle=$ $\displaystyle
2|B|e^{i\theta^{0}_{B}}|\Delta|\begin{pmatrix}0\\\
\sin\left[K_{Fx}\left(x-\frac{l}{2}\right)-\frac{\epsilon}{v_{Fx}}\left(x-\frac{l}{2}\right)+r\pi\right]\end{pmatrix}$
(110)
with eigenvalues
$\displaystyle\frac{\epsilon}{|\Delta|}=$
$\displaystyle\pm\cos\left(\frac{\epsilon l}{v_{Fx}}-K_{Fx}l+(r-b)\pi\right).$
(111)
As before, $|A|,|B|$ are normalization constants, $\theta^{0}_{A/B}$ are
arbitrary phases, while $a,b,p,r$ are arbitrary integers.
## Appendix E Analytical solutions of the $s\tau_{3}-N-s$ junction with
degenerate orbitals
Consider a junction of the $s\tau_{3}-N-s$ along $x$, where the R lead is in a
single-channel, $s$-wave pairing state. The model for this junction was
introduced in Sec. II.1. Here, we tackle this analytically for the case where
the two channels of the L leads, which are in a $s\tau_{3}$ pairing state, are
degenerate, and couple identically to the orbital in the C part.
We proceed along the same lines as the $s\tau_{3}-N-s\tau_{3}$ case in the
continuum limit. As in that case, only the symmetric linear combination of the
two channels of the L lead couples to the C part. We therefore consider the
same ansatz for the BdG coefficients in the L lead which also obey the same
open and continuity BC at the L-C interface as before (Eqs. 81-82). The main
distinction is due to the presence of a single channel in the R lead with
general solutions in the bulk given by
$\displaystyle\begin{pmatrix}\tilde{u}_{k_{y};R}\\\
\tilde{v}_{k_{y};R}\end{pmatrix}=$ $\displaystyle
e^{iK_{F}x}e^{-\kappa\left(x-\frac{l}{2}\right)}\begin{pmatrix}M_{1}(\epsilon+i\Lambda)\\\
M_{1}|\Delta|e^{-i\phi}\end{pmatrix}+e^{-iK_{F}x}e^{-\kappa\left(x-\frac{l}{2}\right)}\begin{pmatrix}M_{\bar{1}}(\epsilon-i\Lambda)\\\
M_{\bar{1}}|\Delta|e^{-i\phi}\\\ \end{pmatrix}.$ (112)
Furthermore, these BdG coefficients obey a single continuity BC at the C-R
interface. Adopting the same conventions as in the $s\tau_{3}-N-s\tau_{3}$
case, we summarize the boundary conditions as
$\displaystyle-\frac{K_{Fx}l}{2}+\theta_{B}-\theta=$
$\displaystyle\frac{\pi}{2}+b\pi$ (113)
$\displaystyle-\frac{K_{Fx}l}{2}+\theta_{A}=$
$\displaystyle\frac{\pi}{2}+a\pi$ (114)
$\displaystyle\theta_{A}-\theta+\frac{\epsilon l}{v_{Fx}}=$
$\displaystyle\theta_{M}+\theta$ (115) $\displaystyle\theta_{B}-\frac{\epsilon
l}{v_{Fx}}=$ $\displaystyle\theta_{M}$ (116)
for the relative phases and
$\displaystyle|A|=$ $\displaystyle|M|$ (117) $\displaystyle|B|=$
$\displaystyle|N|$ (118)
$\displaystyle\theta^{0}_{A}=$ $\displaystyle\theta^{M}_{0}$ (119)
$\displaystyle\theta^{0}_{B}=$ $\displaystyle\theta^{0}_{M}-\phi.$ (120)
for the amplitudes and global phases of the BdG coefficients. $a,b$ are
arbitrary integers.
In contrast to the case of the $s\tau_{3}-N-s\tau_{3}$ junction, the absence
of additional open BC for the R lead ensures that the equations for the
relative phases and $\epsilon$ have a unique solution. After straightforward
algebra, we determine the eigenvalues from
$\displaystyle\frac{\epsilon}{|\Delta|}=$
$\displaystyle\pm\cos\left(\frac{2\epsilon
l}{3v_{Fx}}+\frac{(a-b)\pi}{3}\right).$ (121)
The corresponding states are
$\displaystyle\begin{pmatrix}\tilde{u}_{1,k_{y},L}\\\
\tilde{u}_{2,k_{y},L,e}\\\ \tilde{v}_{1,k_{y},L,e}\\\
\tilde{v}_{2,k_{y},L,e}\end{pmatrix}=$ $\displaystyle
2|A||\Delta|e^{i\theta^{A}_{0}}\begin{pmatrix}\sin\left[K_{Fx}\left(x+\frac{l}{2}\right)-\frac{2\epsilon
l}{3v_{Fx}}+\frac{(2a+b)\pi}{3}\right]+e^{-i\phi}\sin\left[K_{Fx}\left(x+\frac{l}{2}\right)+b\pi\right]\\\
\\\ \sin\left[K_{Fx}\left(x+\frac{l}{2}\right)-\frac{2\epsilon
l}{3v_{Fx}}+\frac{(2a+b)\pi}{3}\right]-e^{-i\phi}\sin\left[K_{Fx}\left(x+\frac{l}{2}\right)+b\pi\right]\\\
\\\ e^{-i\phi}\sin\left[K_{Fx}\left(x+\frac{l}{2}\right)+\frac{2\epsilon
l}{3v_{Fx}}+\frac{(a+2b)\pi}{3}\right]+\sin\left[K_{Fx}\left(x+\frac{l}{2}\right)+a\pi\right]\\\
\\\ e^{-i\phi}\sin\left[K_{Fx}\left(x+\frac{l}{2}\right)+\frac{2\epsilon
l}{3v_{Fx}}+\frac{(a+2b)\pi}{3}\right]-\sin\left[K_{Fx}\left(x+\frac{l}{2}\right)+a\pi\right]\end{pmatrix}.$
(122)
$\displaystyle\begin{pmatrix}\tilde{u}_{k_{y},C}\\\ \tilde{v}_{k_{y},C}\\\
\end{pmatrix}=$ $\displaystyle
2|A||\Delta|e^{i\theta_{0}}\begin{pmatrix}\sin\left[K_{Fx}\left(x+\frac{l}{2}\right)+\frac{\epsilon
x}{v_{Fx}}-\frac{\epsilon l}{6v_{Fx}}+\frac{(2a+b)\pi}{3}\right]\\\ \\\
e^{-i\phi}\sin\left[K_{Fx}\left(x+\frac{l}{2}\right)-\frac{\epsilon
x}{v_{Fx}}+\frac{\epsilon
l}{6v_{Fx}}+\frac{(a+2b)\pi}{3}\right]\end{pmatrix}.$ (123)
$\displaystyle\begin{pmatrix}\tilde{u}_{k_{y},R}\\\ \tilde{v}_{k_{y},R}\\\
\end{pmatrix}=$ $\displaystyle
2|A||\Delta|e^{i\theta_{0}}\begin{pmatrix}\sin\left[K_{Fx}\left(x+\frac{l}{2}\right)+\frac{\epsilon
l}{3v_{Fx}}+\frac{(2a+b)\pi}{3}\right]\\\ \\\
e^{-i\phi}\sin\left[K_{Fx}\left(x+\frac{l}{2}\right)-\frac{\epsilon
l}{3v_{Fx}}+\frac{(a+2b)\pi}{3}\right]\end{pmatrix}.$ (124)
Note that the C spinor is a mixture of electron- and hole-like solutions, as
in the case with typical Andreev bound states.
## Appendix F Josephson current in the tunneling limit
In Sec. III.1 and IV, we showed that in the limit of degenerate orbitals which
couple identically to a single orbital in the C part, $s\tau_{3}-N-s\tau_{3}$
and $s\tau_{3}-N-s$ junctions exhibit bound state spectra which are
insensitive to changes in the relative global phase $\phi$. In view of the
relation between the Josephson current and the derivative of the ground-state
energy with $\phi$, which is typically determined by the Andreev bound state
spectrum, our results suggest the absence of a static Josephson effect.
In order to confirm these findings, we consider simplified models of the
$s\tau_{3}-N-s\tau_{3}$ and $s\tau_{3}-N-s$ junctions and determine the
Josephson current in the tunneling limit. We follow the standard approach in
Ref. Sauls (2018). More precisely, we introduce a Hamiltonian which includes
the leads but which also involves direct tunneling between the latter:
$\displaystyle H=H_{\text{R}}+H_{\text{L}}+H_{\text{T}}.$ (125)
For $s\tau_{3}-N-s\tau_{3}$ junctions, $H_{\text{L/R}}$ are those of Eq. 5,
while for $s\tau_{3}-N-s$ junctions, $H_{\text{R}}$ is a single-channel
$s$-wave bulk Hamiltonian, determined by the dispersion $\xi_{0}$ and $\Delta$
which also enter the expression for the two-orbital $H_{\text{L}}$. The two
leads are connected via a tunneling Hamiltonian $H_{\text{T}}$. For the
$s\tau_{3}-N-s\tau_{3}$ junctions, this takes the form
$\displaystyle H_{\text{T},s\tau_{3}-N-s\tau_{3}}=$
$\displaystyle\sum_{\mathbf{k}\mathbf{p}\sigma}\bigg{(}T^{11}_{\mathbf{k}\mathbf{p}}c^{{\dagger}}_{\text{R},1\mathbf{k}\sigma}c_{\text{L},1\mathbf{p}\sigma}+T^{21}_{\mathbf{k}\mathbf{p}}c^{{\dagger}}_{\text{R},2\mathbf{k}\sigma}c_{\text{L},1\mathbf{p}\sigma}$
$\displaystyle+$ $\displaystyle
T^{12}_{\mathbf{k}\mathbf{p}}c^{{\dagger}}_{\text{R},1\mathbf{k}\sigma}c_{\text{L},2\mathbf{p}\sigma}+T^{22}_{\mathbf{k}\mathbf{p}}c^{{\dagger}}_{\text{L},2\mathbf{k}\sigma}c_{\text{L},2\mathbf{p}\sigma}+\text{H.c.}\bigg{)},$
(126)
where $T^{ij}_{\mathbf{k}\mathbf{p}}$ are tunneling matrix elements, and where
we introduced R and L indices for the two leads in addition to the $1,2$
indices for the two orbitals. For the $s\tau_{3}-N-s$ junction $H_{\text{T}}$
has the simpler form
$\displaystyle H_{\text{T},s\tau_{3}-N-s}=$
$\displaystyle\sum_{\mathbf{k}\mathbf{p}\sigma}\bigg{(}T^{(1)}_{\mathbf{k}\mathbf{p}}c^{{\dagger}}_{\text{R},\mathbf{k}\sigma}c_{\text{L},1\mathbf{p}\sigma}+T^{(2)}_{\mathbf{k}\mathbf{p}}c^{{\dagger}}_{\text{R},\mathbf{k}\sigma}c_{\text{L},2\mathbf{p}\sigma}$
$\displaystyle+$ $\displaystyle\text{H.c.}\bigg{)},$ (127)
where the tunneling matrix elements are labeled by a single orbital index. We
shall neglect the spin indices for simplicity.
The total currents out of the L lead read
$\displaystyle\dot{I}_{\text{Tot},s\tau_{3}-N-s\tau_{3}}=$ $\displaystyle-
ie\left(\dot{N}_{\text{L},1}+\dot{N}_{\text{L},2}\right)$ (128)
$\displaystyle=$ $\displaystyle-
ie\sum_{\textbf{kp}}\sum_{ij}\left(T^{ij}_{kp}c^{{\dagger}}_{\text{R},i\mathbf{k}}c_{\text{L},j\mathbf{p}}-\text{H.c.}\right)$
(129)
$\displaystyle\dot{I}_{\text{Tot},s\tau_{3}-N-s}=$ $\displaystyle-
ie\left(\dot{N}_{\text{L},1}+\dot{N}_{\text{L},2}\right)$ (130)
$\displaystyle=$ $\displaystyle-
ie\sum_{\textbf{kp}}\sum_{i}\left(T^{(i)}_{kp}c^{{\dagger}}_{\text{R},\mathbf{k}}c_{\text{L},j\mathbf{p}}-\text{H.c.}\right),$
(131)
where $N_{\text{L},i}$ is the total charge associated with either orbital in
the L lead.
As shown in Ref. Mahan (2000), the Josephson current to leading order in the
tunneling matrix elements is determined from
$\displaystyle I_{\text{J}}(t)=$ $\displaystyle
2e\text{Im}\left[e^{-2ieVt/\hbar}\theta(eU)\right],$ (132)
where $eU$ is a potential drop across the junction.
### F.1 $s\tau_{4}-N-s\tau_{3}$ and $s\tau_{3}-N-s$ junction
For these cases the quantity $\theta$ is obtained via analytical continuation
from
$\displaystyle\theta_{s\tau_{3}-N-s\tau_{3}}(eU)=$
$\displaystyle\lim_{i\omega\rightarrow
eU+i\eta}\left[2\sum_{ij}\sum_{mn}\sum_{\mathbf{kp}}T^{ij}_{\mathbf{kp}}T^{mn}_{\mathbf{-k,-p}}\frac{1}{\beta}\sum_{i\nu}F^{{\dagger}}_{\text{R},im}(\mathbf{k},i\nu)F_{\text{L},jn}(\mathbf{p},i\nu-i\omega)\right]$
(133)
$\displaystyle\theta_{s\tau_{3}-N-s}(eU)=$
$\displaystyle\lim_{i\omega\rightarrow
eU+i\eta}\left[2\sum_{ij}\sum_{\mathbf{kp}}T^{(i)}_{\mathbf{kp}}T^{(j)}_{\mathbf{-k,-p}}\frac{1}{\beta}\sum_{i\nu}F^{{\dagger}}_{\text{R}}(\mathbf{k},\nu)F_{\text{L},ij}(\mathbf{p},i\nu-i\omega)\right],$
(134)
where the anomalous Green’s functions are determined from the corresponding L
and R lead Hamiltonians as
$\displaystyle F_{\text{L/R},11}(\mathbf{k},i\nu)=$
$\displaystyle\frac{-\Delta_{\text{L/R}}\left[(i\nu)^{2}-\left(\xi_{0}-\xi_{3}\right)^{2}-|\Delta_{\text{L/R}}|^{2}+\xi^{2}_{1}\right]}{\Gamma(\mathbf{k},i\nu)}$
(135)
$\displaystyle F_{\text{L/R},12}(\mathbf{k},i\nu)=$
$\displaystyle\frac{-2\xi_{1}\Delta_{\text{L/R}}\left(i\nu+\xi_{3}\right)}{\Gamma(\mathbf{k},i\nu)}$
(136)
$\displaystyle F_{\text{L/R},21}(\mathbf{k},i\nu)=$
$\displaystyle\frac{2\xi_{1}\Delta_{\text{L/R}}\left(i\nu-\xi_{3}\right)}{\Gamma(\mathbf{k},i\nu)}$
(137)
$\displaystyle F_{\text{L/R},22}(\mathbf{k},i\nu)=$
$\displaystyle\frac{\Delta_{\text{L/R}}\left[(i\nu)^{2}-\left(\xi_{0}+\xi_{3}\right)^{2}-|\Delta_{\text{L/R}}|^{2}+\xi^{2}_{1}\right]}{\Gamma(\mathbf{k},i\nu)}$
(138)
for $s\tau_{3}$ leads, where
$\displaystyle\Gamma(\mathbf{k},i\nu)=$ $\displaystyle\left(i\nu-
E_{1}\right)\left(i\nu+E_{1}\right)\left(i\nu-
E_{2}\right)\left(i\nu+E_{2}\right)$ (139)
and
$\displaystyle
E_{1,2}=\sqrt{\xi^{2}_{0}+\xi^{2}_{3}+\xi^{2}_{1}+|\Delta|^{2}\pm
2\sqrt{\xi^{2}_{0}\left(\xi^{2}_{1}+\xi^{2}_{3}\right)+\xi^{2}_{1}|\Delta|^{2}}}$
(140)
are the BdG bands corresponding to the eigenvalues of the Hamiltonian in Eq.
5. Note that all $\xi$ and $E$ terms are functions of $\mathbf{k}$. Since we
consider $|\Delta_{\text{L}}|=|\Delta_{\text{R}}|$, $\Gamma$’s are independent
of the lead index.
For the single-channel $s$-wave lead, we have the standard expression
$\displaystyle F_{\text{R}}(\mathbf{k},\nu)=$
$\displaystyle\frac{-\Delta_{\text{R}}}{\left[i\nu-\sqrt{\xi^{2}_{0}+|\Delta|^{2}}\right]\left[i\nu+\sqrt{\xi^{2}_{0}+|\Delta|^{2}}\right]}$
(141)
These expressions simplify considerably if we consider the case where all of
the tunneling coefficients are identical for all channels as in
$T^{ij}_{\mathbf{kp}}=T_{\mathbf{kp}}$ and
$T^{(i)}_{\mathbf{kp}}=T_{\mathbf{kp}}$. These correspond to junctions and the
bound state spectra discussed in sections III and IV. The expressions for the
$\theta$ functions are
$\displaystyle\theta_{s\tau_{3}-N-s\tau_{3}}(eU)=$
$\displaystyle\lim_{i\omega\rightarrow
eU+i\eta}2\sum_{\mathbf{kp}}T_{\mathbf{kp}}T_{\mathbf{-k,-p}}\frac{1}{\beta}\sum_{i\nu}\left[\sum_{im}F^{{\dagger}}_{\text{R},im}(\mathbf{k},i\nu)\right]\left[\sum_{jn}F_{\text{L},jn}(\mathbf{p},i\nu-i\omega)\right]$
(142)
$\displaystyle\theta_{s\tau_{3}-N-s}(eU)=$
$\displaystyle\lim_{i\omega\rightarrow
eU+i\eta}2\sum_{\mathbf{kp}}T_{\mathbf{kp}}T_{\mathbf{-k,-p}}\frac{1}{\beta}\sum_{i\nu}F^{{\dagger}}_{\text{R}}(\mathbf{k},\nu)\left[\sum_{ij}F_{\text{L},ij}(\mathbf{p},i\nu-i\omega)\right],$
(143)
The sums in the square brakets amount to
$\displaystyle\left[\sum_{ij}F_{\text{L/R},ij}(\mathbf{k},i\nu)\right]=$
$\displaystyle\frac{-4\Delta_{\text{L/R}}\xi_{3}\left(\xi_{0}+\xi_{1}\right)}{\Gamma(\mathbf{k},i\nu)}$
(144)
From this expression, it is clear that, in the limit of zero intra-orbital
hybridization $\xi_{3}=0$, the _total_ Josephson currents in the tunneling
limit for either $s\tau_{3}-N-s\tau_{3}$ or $s\tau_{3}-N-s$ junctions vanish
identically. This holds even for a finite inter-orbital hybridization
$\xi_{1}$ on the L (and R lead if appropriate). The immediate reason for this
remarkable result is that the L lead continues to preserve the orbital
exchange symmetry defined in Sec. II.3 _in the bulk_ which ensures that
$\displaystyle F_{\text{L},11}(\mathbf{k},i\nu)=$ $\displaystyle-
F_{\text{L},22}(\mathbf{k},i\nu)$ (145) $\displaystyle
F_{\text{L},12}(\mathbf{k},i\nu)=$ $\displaystyle-
F_{\text{L},21}(\mathbf{k},i\nu)$ (146)
as long as the intra-orbital hybridization terms $\xi_{3}$ vanish, even though
the complete Hamiltonian does not preserve the symmetry for the
$s\tau_{3}-N-s$ junction. In the limit considered here, where all of the
orbitals couple identically across the junction, this symmetry in the bulk of
the L lead ensures that the total current out of the L lead vanishes.
Josephson currents which vanish in the limit of zero intra-orbital
hybridization terms are completely consistent with the results of sections III
and IV where the bound state spectra were shown to be insensitive to the
relative phase precisely when the same intra-orbital hybridization terms were
ignored.
### F.2 $s\tau_{0}-N-s\tau_{0}$ and $s\tau_{0}-N-s$ junction
The expression for the current and the associated $\theta$ functions in Eqs.
142 is the same as for the $s\tau_{3}$ cases. However, the anomalous Green’s
functions for $s\tau_{0}$ are
$\displaystyle F_{\text{L/R},11}(\mathbf{k},i\nu)=$
$\displaystyle\frac{\Delta\left[-(i\omega)^{2}+\left(\xi_{0}-\xi_{3}\right)^{2}_{2}+\xi^{2}_{1}+|\Delta|^{2}\right]}{\chi(\textbf{k},i\nu)}$
(147)
$\displaystyle F_{\text{L/R},12}(\mathbf{k},i\nu)=$
$\displaystyle\frac{2\Delta\xi_{1}\xi_{0}}{\chi(\textbf{k},i\nu)}$ (148)
$\displaystyle F_{\text{L/R},21}(\mathbf{k},i\nu)=$
$\displaystyle\frac{2\Delta\xi_{1}\xi_{0}}{\chi(\textbf{k},i\nu)}$ (149)
$\displaystyle F_{\text{L/R},22}(\mathbf{k},i\nu)=$
$\displaystyle\frac{\Delta\left[-(i\omega)^{2}+\epsilon^{2}_{1}+\xi^{2}_{1}+|\Delta|^{2}\right]}{\chi(\textbf{k},i\nu)}$
(150)
where
$\displaystyle\chi(\textbf{k},i\nu)=\left(i\nu-
E_{1}\right)\left(i\nu+E_{1}\right)\left(i\nu-E_{2}\right)\left(i\nu-
E_{2}\right)$ (151)
and
$\displaystyle E_{1/2}=$
$\displaystyle\sqrt{\left(\xi_{0}\pm\sqrt{\xi^{2}_{1}+\xi^{2}_{3}}\right)^{2}+|\Delta|^{2}}$
(152)
is the bulk BdG spectrum for $s\tau_{0}$ pairing.
We calculate
$\displaystyle\left[\sum_{ij}F_{\text{L/R},ij}(\mathbf{k},i\nu)\right]=$
$\displaystyle\frac{-2\Delta\left\\{(i\omega)^{2}-\left[(\xi_{0}+\xi_{1})^{2}+\xi^{2}_{3}+|\Delta|^{2}\right]\right\\}}{\chi(\textbf{k},i\nu)}$
(153)
This is clearly distinct from the expression for $s\tau_{3}$ pairing in Eq.
144. Indeed, in the $\xi_{1}=\xi_{3}$ limit corresponding to degenerate
orbitals this reduces to
$\displaystyle\left[\sum_{ij}F_{\text{L/R},ij}(\mathbf{k},i\nu)\right]=$
$\displaystyle\frac{-2\Delta}{\left[(i\omega)^{2}-\xi^{2}_{0}-|\Delta|^{2}\right]}$
(154)
which amounts to twice the result for a single-channel junction, due to the
presence of the two, decoupled orbitals. This is consistent with the
dependence of the bound state spectrum on $\phi$ (Appendix C).
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|
# Gradient Descent Temporal Difference-difference Learning
Rong J.B. Zhu
Institute of Science and Technology for Brain-inspired Intelligence
Fudan University, Shanghai 200433, People’s Republic of China
<EMAIL_ADDRESS>
&James M. Murray
Institute of Neuroscience
University of Oregon, Eugene, OR 97403, USA
<EMAIL_ADDRESS>
###### Abstract
Off-policy algorithms, in which a behavior policy differs from the target
policy and is used to gain experience for learning, have proven to be of great
practical value in reinforcement learning. However, even for simple convex
problems such as linear value function approximation, these algorithms are not
guaranteed to be stable. To address this, alternative algorithms that are
provably convergent in such cases have been introduced, the most well known
being gradient descent temporal difference (GTD) learning. This algorithm and
others like it, however, tend to converge much more slowly than conventional
temporal difference learning. In this paper we propose gradient descent
temporal difference-difference (Gradient-DD) learning in order to improve
GTD2, a GTD algorithm [1], by introducing second-order differences in
successive parameter updates. We investigate this algorithm in the framework
of linear value function approximation, theoretically proving its convergence
by applying the theory of stochastic approximation. Studying the model
empirically on the random walk task, the Boyan-chain task, and the Baird’s
off-policy counterexample, we find substantial improvement over GTD2 and, in
several cases, better performance even than conventional TD learning.
## 1 Introduction
Off-policy algorithms for value function learning enable an agent to use a
behavior policy that differs from the target policy in order to gain
experience for learning. However, because off-policy methods learn a value
function for a target policy given data due to a different behavior policy,
they are slower to converge than on-policy methods and may even diverge when
applied to problems involving function approximation [2, 3].
Two general approaches have been investigated to address the challenge of
developing stable and effective off-policy temporal-difference algorithms. One
approach is to use importance sampling methods to warp the update distribution
so that the expected value is corrected [4, 5]. This approach is useful for
the convergence guarantee, but it does not address stability issues. The
second main approach to addressing the challenge of off-policy learning is to
develop true gradient descent-based methods that are guaranteed to be stable
regardless of the update distribution. [6, 1] proposed the first off-policy
gradient-descent-based temporal difference algorithms (GTD and GTD2,
respectively). These algorithms are guaranteed to be stable, with
computational complexity scaling linearly with the size of the function
approximator. Empirically, however, their convergence is much slower than
conventional temporal difference (TD) learning, limiting their practical
utility [7, 8]. Building on this work, extensions to the GTD family of
algorithms (see [9] for a review) have allowed for incorporating eligibility
traces [10, 11], non-linear function approximation such as with a neural
network [12], and reformulation of the optimization as a saddle point problem
[13, 14]. However, due to their slow convergence, none of these stable off-
policy methods are commonly used in practice.
In this work, we introduce a new gradient descent algorithm for temporal
difference learning with linear value function approximation. This algorithm,
which we call gradient descent temporal difference-difference (Gradient-DD)
learning, is an acceleration technique that employs second-order differences
in successive parameter updates. The basic idea of Gradient-DD is to modify
the error objective function by additionally considering the prediction error
obtained in the last time step, then to derive a gradient-descent algorithm
based on this modified objective function. In addition to exploiting the
Bellman equation to get the solution, this modified error objective function
avoids drastic changes in the value function estimate by encouraging local
search around the current estimate. Algorithmically, the Gradient-DD approach
only adds an additional term to the update rule of the GTD2 method, and the
extra computational cost is negligible. We prove its convergence by applying
the theory of stochastic approximation. This result is supported by numerical
experiments, which also show that Gradient-DD obtains better convergence in
many cases than conventional TD learning.
### 1.1 Related Work
In related approaches to ours, some previous studies have attempted to improve
Gradient-TD algorithms by adding regularization terms to the objective
function. These approaches have used have used $l_{1}$ regularization on
weights to learn sparse representations of value functions [Liu:12], or
$l_{2}$ regularization on weights [Ghiassian:20]. Our work is different from
these approaches in two ways. First, whereas these previous studies
investigated a variant of TD learning with gradient corrections, we take the
GTD2 algorithm as our starting point. Second, unlike these previous
approaches, our approach modifies the error objective function by using a
distance constraint rather than a penalty on weights. The distance constraint
works by restricting the search to some region around the evaluation obtained
in the most recent time step. With this modification, our method provides a
learning rule that contains second-order differences in successive parameter
updates.
Our approach is similar to trust region policy optimization [Schulman:15] or
relative entropy policy search [Peters:10], which penalize large changes being
learned in policy learning. In these methods, constrained optimization is used
to update the policy by considering the constraint on some measure between the
new policy and the old policy. Here, however, our aim is to find the optimal
value function, and the regularization term uses the previous value function
estimate to avoid drastic changes in the updating process.
Our approach bears similarity to the natural gradient approach widely used in
reinforcement learning [Amari:98, Bh:09, Degris:12, DT:14, Thomas:16], which
also features a constrained optimization form. However, Gradient-DD is
distinct from the natural gradient. The essential difference is that, unlike
the natural gradient, Gradient-DD is a trust region method, which defines the
trust region according to the difference between the current value and the
value obtained from the previous step. From the computational cost viewpoint,
unlike natural TD [DT:14], which needs to update an estimate of the metric
tensor, the computational cost of Gradient-DD is essentially the same as that
of GTD2.
## 2 Gradient descent method for off-policy temporal difference learning
### 2.1 Problem definition and background
In this section, we formalize the problem of learning the value function for a
given policy under the Markov decision process (MDP) framework. In this
framework, the agent interacts with the environment over a sequence of
discrete time steps, $t=1,2,\ldots$. At each time step the agent observes a
state $s_{t}\in\mathcal{S}$ and selects an action $a_{t}\in\mathcal{A}$. In
response, the environment emits a reward $r_{t}\in\mathbb{R}$ and transitions
the agent to its next state $s_{t+1}\in\mathcal{S}$. The state and action sets
are finite. State transitions are stochastic and dependent on the immediately
preceding state and action. Rewards are stochastic and dependent on the
preceding state and action, as well as on the next state. The process
generating the agent’s actions is termed the behavior policy. In off-policy
learning, this behavior policy is in general different from the target policy
$\pi:\mathcal{S}\rightarrow\mathcal{A}$. The objective is to learn an
approximation to the state-value function under the target policy in a
particular environment:
$V(s)=\text{E}_{\pi}\left[\sum\nolimits_{t=1}^{\infty}\gamma^{t-1}r_{t}|s_{1}=s\right],$
(1)
where $\gamma\in[0,1)$ is the discount rate.
In problems for which the state space is large, it is practical to approximate
the value function. In this paper we consider linear function approximation,
where states are mapped to feature vectors with fewer components than the
number of states. Specifically, for each state $s\in\mathcal{S}$ there is a
corresponding feature vector $\mathbf{x}(s)\in\mathbb{R}^{p}$, with
$p\leq|\mathcal{S}|$, such that the approximate value function is given by
$V_{\mathbf{w}}(s):=\mathbf{w}^{\top}\mathbf{x}(s).$ (2)
The goal is then to learn the parameters $\mathbf{w}$ such that
$V_{\mathbf{w}}(s)\approx V(s)$.
### 2.2 Gradient temporal difference learning
A major breakthrough for the study of the convergence properties of MDP
systems came with the introduction of the GTD and GTD2 learning algorithms
[SSM09, SMP09]. We begin by briefly recapitulating the GTD algorithms, which
we will then extend in the following sections. To begin, we introduce the
Bellman operator $\mathbf{B}$ such that the true value function
$\mathbf{V}\in\mathbb{R}^{|\mathcal{S}|}$ satisfies the Bellman equation:
$\mathbf{V}=\mathbf{R}+\gamma\mathbf{P}\mathbf{V}=:\mathbf{B}\mathbf{V},$
where $\mathbf{R}$ is the reward vector with components
$\text{E}(r_{n}|s_{n}=s)$, and $\mathbf{P}$ is a matrix of the state
transition probabilities under the behavior policy. In temporal difference
methods, an appropriate objective function should minimize the difference
between the approximate value function and the solution to the Bellman
equation.
Having defined the Bellman operator, we next introduce the projection operator
$\mathbf{\Pi}$, which takes any value function $\mathbf{V}$ and projects it to
the nearest value function within the space of approximate value functions of
the form Eqn. (2). Letting $\mathbf{X}$ be the matrix whose rows are
$\mathbf{x}(s)$, the approximate value function can be expressed as
$\mathbf{V}_{\mathbf{w}}=\mathbf{X}\mathbf{w}$. The projection operator is
then given by
$\mathbf{\Pi}=\mathbf{X}(\mathbf{X}^{\top}\mathbf{D}\mathbf{X})^{-1}\mathbf{X}^{\top}\mathbf{D},$
where the matrix $\mathbf{D}$ is diagonal, with each diagonal element $d_{s}$
corresponding to the probability of visiting state $s$.
The natural measure of how closely the approximation $\mathbf{V}_{\mathbf{w}}$
satisfies the Bellman equation is the mean-squared Bellman error:
$\text{MSBE}(\mathbf{w})=\|\mathbf{V}_{\mathbf{w}}-\mathbf{B}\mathbf{V}_{\mathbf{w}}\|_{\mathbf{D}}^{2},$
(3)
where the norm is weighted by $\mathbf{D}$, such that
$\|\mathbf{V}\|^{2}_{\mathbf{D}}=\mathbf{V}^{\top}\mathbf{D}\mathbf{V}$.
However, because the Bellman operator follows the underlying state dynamics of
the Markov chain, irrespective of the structure of the linear function
approximator, $\mathbf{B}\mathbf{V}_{\mathbf{w}}$ will typically not be
representable as $\mathbf{V}_{\mathbf{w}}$ for any $\mathbf{w}$. An
alternative objective function, therefore, is the mean squared projected
Bellman error (MSPBE), which we define as
$\displaystyle
J(\mathbf{w})=\|\mathbf{V}_{\mathbf{w}}-\mathbf{\Pi}\mathbf{B}\mathbf{V}_{\mathbf{w}}\|_{\mathbf{D}}^{2}.$
(4)
Following [SMP09], our objective is to minimize this error measure. As usual
in stochastic gradient descent, the weights at each time step are then updated
by $\Delta\mathbf{w}=-\alpha\nabla J(\mathbf{w})$, where $\alpha>0$, and
$\displaystyle-\frac{1}{2}\nabla J(\mathbf{w})$
$\displaystyle=-\text{E}[(\gamma\mathbf{x}_{n+1}-\mathbf{x}_{n})\mathbf{x}_{n}^{\top}][\text{E}(\mathbf{x}_{n}\mathbf{x}_{n}^{\top})]^{-1}\text{E}(\delta_{n}\mathbf{x}_{n}).$
(5)
For notational simplicity, we have denoted the feature vector associated with
$s_{n}$ as $\mathbf{x}_{n}=\mathbf{x}({s_{n}})$. We have also introduced the
temporal difference error
$\delta_{n}=r_{n}+(\gamma\mathbf{x}_{n+1}-\mathbf{x}_{n})^{\top}\mathbf{w}_{n}$.
Let $\boldsymbol{\eta}_{n}$ denote the estimate of
$[\text{E}(\mathbf{x}_{n}\mathbf{x}_{n}^{\top})]^{-1}\text{E}(\delta_{n}\mathbf{x}_{n})$
at the time step $n$. Because the factors in Eqn. (5) can be directly sampled,
the resulting updates in each step are
$\displaystyle\delta_{n}=$ $\displaystyle
r_{n}+(\gamma\mathbf{x}_{n+1}-\mathbf{x}_{n})^{\top}\mathbf{w}_{n}$
$\displaystyle\boldsymbol{\eta}_{n+1}=$
$\displaystyle\boldsymbol{\eta}_{n}+\beta_{n}(\delta_{n}-\mathbf{x}_{n}^{\top}\boldsymbol{\eta}_{n})\mathbf{x}_{n}$
$\displaystyle\mathbf{w}_{n+1}=$
$\displaystyle\mathbf{w}_{n}-\alpha_{n}(\gamma\mathbf{x}_{n+1}-\mathbf{x}_{n})(\mathbf{x}_{n}^{\top}\boldsymbol{\eta}_{n}).$
(6)
These updates define the GTD2 learning algorithm, which we will build upon in
the following section.
## 3 Gradient descent temporal difference-difference learning
In this section we modify the objective function by additionally considering
the difference between $\mathbf{V}_{\mathbf{w}}$ and
$\mathbf{V}_{\mathbf{w}_{n-1}}$, which denotes the value function estimate at
step $n-1$ of the optimization. We propose a new objective
$J_{\text{GDD}}(\mathbf{w}|\mathbf{w}_{n-1})$, where the notation
“$\mathbf{w}|\mathbf{w}_{n-1}$" in the parentheses means that the objective is
defined given $\mathbf{V}_{\mathbf{w}_{n-1}}$ of the previous time step $n-1$.
Specifically, we modify Eqn. (4) as follows:
$J_{\text{GDD}}(\mathbf{w}|\mathbf{w}_{n-1})=J(\mathbf{w})+\kappa\|\mathbf{V}_{\mathbf{w}}-\mathbf{V}_{\mathbf{w}_{n-1}}\|_{\mathbf{D}}^{2},$
(7)
where $\kappa\geq 0$ is a parameter of the regularization. We show in Section
A.1 of the appendix that minimizing Eqn. (7) is equivalent to the following
optimization
$\arg\min\limits_{\mathbf{w}}J(\mathbf{w})\text{ s.t.
}\|\mathbf{V}_{\mathbf{w}}-\mathbf{V}_{\mathbf{w}_{n-1}}\|_{\mathbf{D}}^{2}\leq\mu$
(8)
where $\mu>0$ is a parameter which becomes large when $\kappa$ is small, so
that the MSPBE objective is recovered as $\mu\to\infty$, equivalent to
$\kappa\to 0$ in Eqn. (7).
Figure 1: Schematic diagram of Gradient-DD learning with
$\mathbf{w}\in\mathbb{R}^{2}$. Rather than updating $\mathbf{w}$ directly
along the gradient of the MSPBE (black arrow), the update rule selects
$\mathbf{w}_{n}$ (red arrow) that minimizes the MSPBE while satisfying the
constraint
$\|\mathbf{V}_{\mathbf{w}}-\mathbf{V}_{\mathbf{w}_{n-1}}\|_{\mathbf{D}}^{2}\leq\mu$
(shaded ellipse).
Rather than simply minimizing the optimal prediction from the projected
Bellman equation, the agent makes use of the most recent update to look for
the solution, choosing a $\mathbf{w}$ that minimizes the MSPBE while following
the constraint that the estimated value function should not change too
greatly, as illustrated in Fig. 1. In effect, the regularization term
encourages searching around the estimate at previous time step, especially
when the state space is large.
Eqn. (8) shows that the regularized objective is a trust region approach,
which seeks a direction that attains the best improvement possible subject to
the distance constraint. The trust region is defined by the value distance
rather than the weight distance, meaning that Gradient-DD also makes use of
the natural gradient of the objective around $\mathbf{w}_{n-1}$ rather than
around $\mathbf{w}_{n}$ (see Section A.2 of the appendix for details). In this
sense, our approach can be explained as a trust region method that makes use
of natural gradient information to prevent the estimated value function from
changing too drastically.
For comparison with related approaches using natural gradients, in Fig. 8 of
the appendix we compare the empirical performance of our algorithm with
natural GTD2 and natural TDC [DT:14] using the random walk task introduced
below in Section 5.
With these considerations in mind, the negative gradient of
$J_{\text{GDD}}(\mathbf{w}|\mathbf{w}_{n-1})$ is
$\displaystyle-\frac{1}{2}\nabla J_{\text{GDD}}(\mathbf{w}|\mathbf{w}_{n-1})$
$\displaystyle=$
$\displaystyle-\text{E}[(\gamma\mathbf{x}_{n+1}-\mathbf{x}_{n})\mathbf{x}_{n}^{\top}][\text{E}(\mathbf{x}_{n}\mathbf{x}_{n}^{\top})]^{-1}\text{E}(\delta_{n}\mathbf{x}_{n})-\kappa\text{E}[(\mathbf{x}_{n}^{\top}\mathbf{w}_{n}-\mathbf{x}_{n}^{\top}\mathbf{w}_{n-1})\mathbf{x}_{n}].$
(9)
Because the terms in Eqn. (3) can be directly sampled, the stochastic gradient
descent updates are given by
$\displaystyle\delta_{n}=$ $\displaystyle
r_{n}+(\gamma\mathbf{x}_{n+1}-\mathbf{x}_{n})^{\top}\mathbf{w}_{n}$
$\displaystyle\boldsymbol{\eta}_{n+1}=$
$\displaystyle\boldsymbol{\eta}_{n}+\alpha_{n}(\delta_{n}-\mathbf{x}_{n}^{\top}\boldsymbol{\eta}_{n})\mathbf{x}_{n}$
$\displaystyle\mathbf{w}_{n+1}=$
$\displaystyle\mathbf{w}_{n}-\kappa\alpha_{n}(\mathbf{x}_{n}^{\top}\mathbf{w}_{n}-\mathbf{x}_{n}^{\top}\mathbf{w}_{n-1})\mathbf{x}_{n}-\alpha_{n}(\gamma\mathbf{x}_{n+1}-\mathbf{x}_{n})(\mathbf{x}_{n}^{\top}\boldsymbol{\eta}_{n}).$
(10)
These update equations define the Gradient-DD method, in which the GTD2 update
equations (2.2) are generalized by including a second-order update term in the
third update equation, where this term originates from the squared bias term
in the objective (7). Since Gradient-DD is not sensitive to the step size of
updating $\boldsymbol{\eta}$ (see Fig. 7 in the appendix), the updates of
Gradient-DD only have a single shared step size $\alpha_{n}$ rather than two
step sizes $\alpha_{n},\beta_{n}$ as GTD2 and TDC used. It is worth noting
that the computational cost of our algorithm is essentially the same as that
of GTD2. In the following sections, we shall analytically and numerically
investigate the convergence and performance of Gradient-DD learning.
## 4 Convergence Analysis
In this section we establish the convergence of Gradient-DD learning. Denote
$\mathbf{G}_{n}=\left[\begin{array}[]{cc}\mathbf{x}_{n}\mathbf{x}_{n}^{\top}&\mathbf{x}_{n}(\mathbf{x}_{n}-\gamma\mathbf{x}_{n+1})^{\top}\\\
-(\mathbf{x}_{n}-\gamma\mathbf{x}_{n+1})\mathbf{x}_{n}^{\top}&\mathbf{0}\end{array}\right],\text{
and }\mathbf{H}_{n}=\left[\begin{array}[]{cc}\mathbf{0}&\mathbf{0}\\\
\mathbf{0}&\mathbf{x}_{n}\mathbf{x}_{n}^{\top}\end{array}\right]$. We rewrite
the update rules in Eqn. (3) as a single iteration in a combined parameter
vector with $2n$ components,
$\boldsymbol{\rho}_{n}=(\boldsymbol{\eta}_{n}^{\top},\mathbf{w}_{n}^{\top})^{\top}$,
and a new reward-related vector with $2n$ components,
$\mathbf{g}_{n+1}=(r_{n}\mathbf{x}_{n}^{\top},\mathbf{0}^{\top})^{\top}$, as
follows:
$\displaystyle\boldsymbol{\rho}_{n+1}=$
$\displaystyle\boldsymbol{\rho}_{n}-\kappa\alpha_{n}\mathbf{H}_{n}(\boldsymbol{\rho}_{n}-\boldsymbol{\rho}_{n-1})+\alpha_{n}(\mathbf{G}_{n}\boldsymbol{\rho}_{n}+\mathbf{g}_{n+1}),$
(11)
###### Theorem 1.
Consider the update rules (11) with step-size sequences $\alpha_{n}$. Let the
TD fixed point be $\mathbf{w}^{*}$, such that
$\mathbf{V}_{\mathbf{w}^{*}}=\mathbf{\Pi}\mathbf{B}\mathbf{V}_{\mathbf{w}^{*}}$.
Suppose that (A0) $\alpha_{n}\in(0,1)$,
$\sum\nolimits_{n=1}^{\infty}\alpha_{n}=\infty$,
$\sum\nolimits_{n=1}^{\infty}\alpha_{n}^{2}<\infty$, (A1)
$(\mathbf{x}_{n},r_{n},\mathbf{x}_{n+1})$ is an i.i.d. sequence with uniformly
bounded second moments, (A2)
$\text{E}[(\mathbf{x}_{n}-\gamma\mathbf{x}_{n+1})\mathbf{x}_{n}^{\top}]$ and
$\text{E}(\mathbf{x}_{n}\mathbf{x}_{n}^{\top})$ are non-singular, (A3)
$\sup_{n}\|\boldsymbol{\rho}_{n+1}-\boldsymbol{\rho}_{n}\|$ is bounded in
probability, (A4) $\kappa\in[0,\infty)$. Then as $n\rightarrow\infty$,
$\mathbf{w}_{n}\rightarrow\mathbf{w}^{*}$ with probability 1.
Due to the second-order difference term in Eqn. (11), the analysis framework
in [BM:00] does not directly apply to the Gradient-DD algorithm when (A0)
holdes, i.e., step size is tapered. Likewise, the two-timescale convergence
analysis [Bh:09] is also not directly applicable. Defining
$\mathbf{u}_{n+1}=\boldsymbol{\rho}_{n+1}-\boldsymbol{\rho}_{n}$, we rewrite
the iterative process in Eqn. (11) into two parallel processes which are given
by
$\displaystyle\boldsymbol{\rho}_{n+1}$
$\displaystyle=\boldsymbol{\rho}_{n}-\kappa\alpha_{n}\mathbf{H}_{n}\mathbf{u}_{n}+\alpha_{n}(\mathbf{G}_{n}\boldsymbol{\rho}_{n}+\mathbf{g}_{n+1}),$
(12) $\displaystyle\mathbf{u}_{n+1}$
$\displaystyle=-\kappa\alpha_{n}\mathbf{H}_{n}\mathbf{u}_{n}+\alpha_{n}(\mathbf{G}_{n}\boldsymbol{\rho}_{n}+\mathbf{g}_{n+1}).$
(13)
We analyze the parallel processes Eqns. (12) & Eqn. (13) instead of directly
analyzing Eqn. (11). Our proofs have three steps. First we show
$\sup_{n}\|\boldsymbol{\rho}_{n}\|$ is bounded by applying the stability of
the stochastic approximation [BM:00] into the recursion Eqn. (12). Second,
based on this result, we shall show that $\mathbf{u}_{n}$ goes to 0 in
probability by analyzing the recursion Eqn. (13). At last, along with the
result that $\mathbf{u}_{n}$ goes to 0 in probability, by applying the
convergence of the stochastic approximation [BM:00] into the recursion Eqn.
(12), we show that $\boldsymbol{\boldsymbol{\rho}}_{n}$ goes to the TD fixed
point which is given by the solution of
$\mathbf{G}\boldsymbol{\rho}+\mathbf{g}=0$. The details are provided in
Section A.3 of the Appendix.
Theorem 1 shows that Gradient-DD maintains convergence as GTD2 under some mild
conditions. The assumptions (A0), (A1), and (A2) are standard conditions in
the convergence analysis of Gradient TD learning algorithms [SSM09, SMP09,
Maei:11]. The assumption (A3) is weak since it means only that the incremental
update in each step is bounded in probability. The assumption (A4) requires
that $\kappa$ is a constant, meaning $\kappa=O(1)$. Given this assumption, the
contribution of the term $\kappa\mathbf{H}_{n}\mathbf{u}_{n}$ is controlled by
$\alpha_{n}$ as $n\rightarrow\infty$.
## 5 Empirical Study
In this section, we assess the practical utility of the Gradient-DD method in
numerical experiments. To validate performance of Gradient-DD learning, we
compare Gradient-DD learning with GTD2 learning, TDC learning (TD with
gradient correction [SMP09]), TDRC learning (TDC with regularized correction
[Ghiassian:20]) and TD learning in both tabular representation and linear
representation. We conducted three tasks: a simple random walk task, the
Boyan-chain task, and Baird’s off-policy counterexample. In each task, we
evaluate the performance of a learning algorithm by empirical root mean-
squared (RMS) error:
$\sqrt{\sum\nolimits_{s\in\mathcal{S}}d_{s}(V_{\mathbf{w}_{n}}(s)-V(s))^{2}}$.
The reason we choose the empirical RMS error rather than root projected mean-
squared error or other measures as [Gh:18, Ghiassian:20] used is because it is
a direct measure of concern in practical performance. We set the $\kappa=1$.
We also investigate the sensitivity to $\kappa$ in the Appendix, where we show
that $\kappa=1$ is a good and natural choice in our empirical studies,
although tuning $\kappa$ may be necessary for some tasks in practice.
### 5.1 Random walk task
As a first test of Gradient-DD learning, we conducted a simple random walk
task [Sutton:2018]. The random walk task has a linear arrangement of $m$
states plus an absorbing terminal state at each end. Thus there are $m+2$
sequential states, $S_{0},S_{1},\cdots,S_{m},S_{m+1}$, where $m=$ 10, 20, or
40. Every walk begins in the center state. At each step, the walk moves to a
neighboring state, either to the right or to the left with equal probability.
If either edge state ($S_{0}$ or $S_{m+1}$) is entered, the walk terminates. A
walk’s outcome is defined to be $r=0$ at $S_{0}$ and $r=1$ at $S_{m+1}$. Our
aim is to learn the value of each state $V(s)$, where the true values are
$(1,\cdots,m)/(m+1)$. In all cases the approximate value function is
initialized to the intermediate value $V_{0}(s)=0.5$. We first consider
tabular representation of the value function. We also consider linear
approximation representation and obtain the similar results, which are
reported in Fig. 10 of the appendix. We consider that the learning rate
$\alpha_{n}$ is tapered according to the schedule
$\alpha_{n}=\alpha(10^{3}+1)/(10^{3}+n)$. We tune
$\alpha\in\\{10^{-12/4},10^{-11/4},\cdots,10^{-1/4},1\\}$. We also consider
the constant step sizes and obtain the similar results, which are reported in
Fig. 9 of the appendix. For GTD2 and TDC, we set $\beta_{n}=\zeta\alpha_{n}$
with $\zeta\in\\{1/64,1/16,1/4,1,4\\}$.
We first compare the methods by plotting the empirical RMS error from
averaging the final 100 steps as a function of step size $\alpha$ in Fig. 2,
where 20,000 episodes are used. We also plot the average error of all episodes
during training and report these results in Fig. 5 of the Appendix. From these
figures, we can make several observations. (1) Gradient-DD clearly performs
better than the GTD2 and TDC methods. This advantage is consistent in various
settings, and gets bigger as the state space becomes large. (2) Gradient-DD
performs similarly to TDRC and conventional TD learning, with a similar
dependence on $\alpha$, although Gradient-DD exhibits greater sensitivity to
the value of $\alpha$ in the log domain than these other algorithms. In
summary, Gradient-DD exhibits clear advantages over the GTD2 algorithm, and
its performance is also as good as TDRC and conventional TD learning.
Figure 2: The random walk task with tabular representation and tapering step
size $\alpha_{n}=\alpha(10^{3}+1)/(10^{3}+n)$. Upper: Performance as a
function of $\alpha$; Lower: performance over episodes. In each row, state
space size 10 (left), 20 (middle), or 40 (right). The curves are averaged over
50 runs, with error bars denoting the standard error of the mean, though most
are vanishingly small.
Next we look closely at the performance during training in Fig. 2. For each
method, we tuned $\alpha\in\\{10^{-12/4},\cdots,10^{-1/4},1\\}$ by minimizing
the average error of the last 100 episodes. We also compare the performance
when $\alpha$ is tuned by minimizing the average error of the last 100
episodes and report in Fig. 5 of the appendix. From these results, we draw
several observations. (1) For all conditions tested, Gradient-DD converges
much more rapidly than GTD2 and TDC. The results also indicate that Gradient-
DD even converges as fast as TDRC and conventional TD learning. (2) The
advantage of Gradient-DD learning over GTD2 grows as the state space increases
in size. (3) Gradient-DD has consistent and good performance under both the
constant step size setting and under the tapered step size setting. In
summary, the Gradient-DD learning curves in this task show improvements over
other gradient-based methods and performance that matches conventional TD
learning.
Like TDRC, the updates of Gradient-DD only have a single shared step size
$\alpha_{n}$, i.e., $\beta_{n}=\alpha_{n}$, rather than two independent step
sizes $\alpha_{n}$ and $\beta_{n}$ as in the GTD2 and TDC algorithms. A
possible concern is that the performance gains in our second-order algorithm
could just as easily be obtained with existing methods by adopting this two-
timescale approach, where the value function weights are updated with a
smaller step size than the second set of weights. Hence, in addition to
investigating the effects of the learning rate, size of the state space, and
magnitude of the regularization parameter, we also investigate the effect of
using distinct values for the two learning rates, $\alpha_{n}$ and
$\beta_{n}$, although we set $\beta_{n}=\zeta\alpha_{n}$ with $\zeta=1$. To do
this, we set $\beta_{n}=\zeta\alpha_{n}$ for GDD, with
$\zeta\in\\{1/64,1/16,1/4,1,4\\}$, and report the results in Fig. 7 of the
appendix. The results show that comparably good performance of Gradient-DD is
obtained under these various $\zeta$, providing evidence that the second-order
difference term in our approach provides an improvement beyond what can be
obtained with previous gradient-based methods using the two time scale
approach.
### 5.2 Boyan-chain task
We next investigate Gradient-DD learning on the Boyan-chain problem, which is
a standard task for testing linear value-function approximation [Boyan02]. In
this task we allow for $4p-3$ states, each of which is represented by a
$p$-dimensional feature vector, with $p=20,50,$ or $100$. The $p$-dimensional
representation for every fourth state from the start is $[1,0,\cdots,0]$ for
state $s_{1}$, $[0,1,0,\cdots,0]$ for $s_{5}$, $\cdots$, and
$[0,0,\cdots,0,1]$ for the terminal state $s_{4p-3}$. The representations for
the remaining states are obtained by linearly interpolating between these. The
optimal coefficients of the feature vector are $(-4(p-1),-4(p-2),\cdots,0)/5$.
In each state, except for the last one before the end, there are two possible
actions: move forward one step or move forward two steps, where each action
occurs with probability 0.5. Both actions lead to reward -0.3. The last state
before the end just has one action of moving forward to the terminal with
reward -0.2. We tune
$\alpha\in\\{10^{-2},10^{-1.5},10^{-1},10^{-3/4},10^{-1/2},10^{-1/4},10^{-1/8},1,10^{1/8},10^{1/4}\\}$
for each method by minimizing the average error of the final 100 episodes. The
step size is tapered according to the schedule $\alpha_{n}=\alpha(2\times
10^{3}+1)/(2\times 10^{3}+n)$. For GTD2 and TDC, we set
$\beta_{n}=\zeta\alpha_{n}$ with $\zeta\in\\{1/64,1/16,1/4,1,4\\}$. In this
task, we set $\gamma=1$.
We report the performance as a function of $\alpha$ and the performance over
episodes in Fig. 3, where we tune $\alpha$ by the performance based on the
average error of the last 100 episodes. We also compare the performance based
on the average error of all episodes during training and report the results in
Fig. 11 of the appendix. These figures lead to conclusions similar to those
already drawn in the random walk task. (1) Gradient-DD has much faster
convergence than GTD2 and TDC, and generally converges to better values. (2)
Gradient-DD is competitive with TDRC and conventional TD learning despite
being somewhat slower at the beginning episodes. (3) The improvement over GTD2
or TDC grows as the state space becomes larger. (4) Comparing the performance
with constant step size versus that with tapered step size, the Gradient-DD
method performs better with tapered step size than it does with constant step
size.
Figure 3: The Boyan Chain task with linear approximation and tapering step
size $\alpha_{n}=\alpha(2\times 10^{3}+1)/(2\times 10^{3}+n)$. Upper:
Performance as a function of $\alpha$; Lower: performance over episodes. In
each row, the feature size is 20 (left), 50 (middle), or 100 (right). The
curves are averaged over 50 runs, with error bars denoting the standard error
of the mean, though most are vanishingly small across runs.
### 5.3 Baird’s off-policy counterexample
We also verify the performance of Gradient-DD on Baird’s off-policy
counterexample [Baird:95, Sutton:2018], illustrated schematically in Fig. 4,
for which TD learning famously diverges. We show results from Baird’s
counterexample with $N=7,20$ states. The reward is always zero, and the agent
learns a linear value approximation with $N+1$ weights $w_{1},\cdots,w_{N+1}$:
the estimated value of the $j$-th state is $2w_{j}+w_{N+1}$ for $j\leq N-1$
and that of the $N$-th state is $w_{N}+2w_{N+1}$. In the task, the importance
sampling ratio for the dashed action is $(N-1)/N$, while the ratio for the
solid action is $N$. Thus, comparing different state sizes illustrates the
impact of importance sampling ratios in these algorithms. The initial weight
values are $(1,\cdots,1,10,1)^{\top}$. Constant $\alpha$ is used in this task.
We set $\gamma=0.99$. For TDC and GTD2, thus we tune
$\zeta\in\\{4^{-2},4^{-1},1,4^{2},4^{3}\\}$. Meanwhile we tune $\alpha$ for
TDC in a wider region. For Gradient-DD, we tune $\kappa\in\\{4^{-1},1,4\\}$.
We tune $\alpha$ separately for each algorithm by minimizing the average error
from the final 100 episodes.
Fig. 4 demonstrates that Gradient-DD works better on this counterexample than
GTD2, TDC, and TDRC. It is worthwhile to observe that when the state size is
20, TDRC become unstable, meaning serious unbalance of importance sampling
ratios may cause TDRC unstable. We also note that, because the linear
approximation leaves a residual error in the value estimation due to the
projection error, the RMS errors of GTD2, TDC, and TDRC in this task do not go
to zero. In contrast to other algorithms, the errors from our Gradient-DD
converge to zero.
(a) Illustration of the extended Baird’s off-policy counterexample. The
“solid" action usually goes to the $N$-th state, and the “dashed" action
usually goes to one of the other $N-1$ states, each with equal probability.
(b) The performance of various algorithms.
Figure 4: Baird’s off-policy counterexample. Upper in (b): Performance as a
function of $\alpha$; Lower in (b): performance over episodes. From left to
right in (b): 7-state and 20-state.
## 6 Conclusion and discussion
In this work, we have proposed Gradient-DD learning, a new gradient descent-
based TD learning algorithm. The algorithm is based on a modification of the
projected Bellman error objective function for value function approximation by
introducing a second-order difference term. The algorithm significantly
improves upon existing methods for gradient-based TD learning, obtaining
better convergence performance than conventional linear TD learning.
Since GTD learning was originally proposed, the Gradient-TD family of
algorithms has been extended to incorporate eligibility traces and learning
optimal policies [Maei:10, GS:14], as well as for application to neural
networks [Maei:11]. Additionally, many variants of the vanilla Gradient-TD
methods have been proposed, including HTD [Hackman12] and Proximal Gradient-TD
[Liu16]. Because Gradient-DD just modifies the objective error of GTD2 by
considering an additional squared-bias term, it may be extended and combined
with these other methods, potentially broadening its utility for more
complicated tasks.
One potential limitation of our method is that it introduces an additional
hyperparameter relative to similar gradient-based algorithms, which increases
the computational requirements for hyperparameter optimization. This is
somewhat mitigated by our finding that the algorithm’s performance is not
particularly sensitive to values of $\kappa$, and that $\kappa\sim 1$ was
found to be a good choice for the range of environments that we considered.
Another potential limitation is that we have focused on value function
prediction in the two simple cases of tabular representations and linear
approximation, which has enabled us to provide convergence guarantees, but not
yet for the case of nonlinear function approximation. An especially
interesting direction for future study will be the application of Gradient-DD
learning to tasks requiring more complex representations, including neural
network implementations. Such approaches are especially useful in cases where
state spaces are large, and indeed we have found in our results that Gradient-
DD seems to confer the greatest advantage over other methods in such cases.
Intuitively, we expect that this is because the difference between the optimal
update direction and that chosen by gradient descent becomes greater in
higher-dimensional spaces (cf. Fig. 1). This performance benefit in large
state spaces suggests that Gradient-DD may be of practical use for these more
challenging cases.
## Acknowledgments and Disclosure of Funding
This work was partially supported by the National Natural Science Foundation
of China (No.11871459) and by the Shanghai Municipal Science and Technology
Major Project (No.2018SHZDZX01). Support for this work was also provided by
the National Science Foundation NeuroNex program (DBI-1707398) and the Gatsby
Charitable Foundation.
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## Appendix A Appendix
### A.1 On the equivalence of Eqns. (7) & (8)
The Karush-Kuhn-Tucker conditions of Eqn. (8) are the following system of
equations
$\begin{cases}\frac{d}{d\mathbf{w}}J(\mathbf{w})+\kappa\frac{d}{d\mathbf{w}}(\|\mathbf{V}_{\mathbf{w}}-\mathbf{V}_{\mathbf{w}_{n-1}}\|_{\mathbf{D}}^{2}-\mu)=0;\\\
\kappa(\|\mathbf{V}_{\mathbf{w}}-\mathbf{V}_{\mathbf{w}_{n-1}}\|_{\mathbf{D}}^{2}-\mu)=0;\\\
\|\mathbf{V}_{\mathbf{w}}-\mathbf{V}_{\mathbf{w}_{n-1}}\|_{\mathbf{D}}^{2}\leq\mu;\\\
\kappa\geq 0.\end{cases}$
These equations are equivalent to
$\begin{cases}\frac{d}{d\mathbf{w}}J(\mathbf{w})+\kappa\frac{d}{d\mathbf{w}}\|\mathbf{V}_{\mathbf{w}}-\mathbf{V}_{\mathbf{w}_{n-1}}\|_{\mathbf{D}}^{2}=0\text{
and }\kappa>0,\\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \text{ if
}\|\mathbf{V}_{\mathbf{w}}-\mathbf{V}_{\mathbf{w}_{n-1}}\|_{\mathbf{D}}^{2}=\mu;\\\
\frac{d}{d\mathbf{w}}J(\mathbf{w})=0\text{ and }\kappa=0,\text{ if
}\|\mathbf{V}_{\mathbf{w}}-\mathbf{V}_{\mathbf{w}_{n-1}}\|_{\mathbf{D}}^{2}<\mu.\end{cases}$
Thus, for any $\mu>0$, there exists a $\kappa\geq 0$ such that
$\frac{d}{d\mathbf{w}}J(\mathbf{w})+\mu\frac{d}{d\mathbf{w}}\|\mathbf{V}_{\mathbf{w}}-\mathbf{V}_{\mathbf{w}_{n-1}}\|_{\mathbf{D}}^{2}=0$.
### A.2 The relation to natural gradients
In this section, we shall show that Gradient-DD is related to, but distinct
from, the natural gradient. We thank a reviewer for pointing out the
connection between Gradient-DD and the natural gradient.
Following [Amari:98] or [Thomas:14], the natural gradient of $J(\mathbf{w})$
is the direction obtained by solving the following optimization:
$\lim_{\epsilon\rightarrow
0}\arg\min\limits_{\Delta}J(\mathbf{w}+\epsilon\Delta)\text{ s.t.
}\epsilon^{2}\Delta^{\top}\mathbf{X}^{\top}\mathbf{D}\mathbf{X}\Delta\leq\mu.$
(A.1)
We can note that this corresponds to the ordinary gradient in the case where
the metric tensor $\mathbf{X}^{\top}\mathbf{DX}$ is proportional to the
identity matrix.
Now we rewrite Eqn. (8) as
$\displaystyle\|\mathbf{V}_{\mathbf{w}}-\mathbf{V}_{\mathbf{w}_{n-1}}\|_{\mathbf{D}}^{2}=(\mathbf{w}-\mathbf{w}_{n-1})^{\top}\mathbf{X}^{\top}\mathbf{D}\mathbf{X}(\mathbf{w}-\mathbf{w}_{n-1}).$
Denote $\epsilon\Delta=\mathbf{w}-\mathbf{w}_{n-1}$, where $\epsilon$ is the
radius of the circle of $\mathbf{w}$ around $\mathbf{w}_{n-1}$ and $\Delta$ is
a unit vector. Thus, we have
$\displaystyle\|\mathbf{V}_{\mathbf{w}}-\mathbf{V}_{\mathbf{w}_{n-1}}\|_{\mathbf{D}}^{2}=\epsilon^{2}\Delta^{\top}\mathbf{X}^{\top}\mathbf{D}\mathbf{X}\Delta.$
For the MSPBE objective, we have
$\displaystyle
J(\mathbf{w})=J(\mathbf{w}_{n-1}+\mathbf{w}-\mathbf{w}_{n-1})=J(\mathbf{w}_{n-1}+\epsilon\Delta).$
Minimizing Eqn. (8) is equivalent to the following optimization
$\arg\min\limits_{\Delta}J(\mathbf{w}_{n-1}+\epsilon\Delta)\text{ s.t.
}\epsilon^{2}\Delta^{\top}\mathbf{X}^{\top}\mathbf{D}\mathbf{X}\Delta\leq\mu.$
(A.2)
In the limit as $\epsilon\rightarrow 0$, the above optimization is equivalent
to
$\arg\min\limits_{\Delta}\Delta^{\top}\nabla J(\mathbf{w}_{n-1})\text{ s.t.
}\epsilon^{2}\Delta^{\top}\mathbf{X}^{\top}\mathbf{D}\mathbf{X}\Delta\leq\mu.$
Thus, given the metric tensor
$\mathbf{G}=\mathbf{X}^{\top}\mathbf{D}\mathbf{X}$, $-\mathbf{G}^{-1}\nabla
J(\mathbf{w}_{n-1})$ is the direction of steepest descent, i.e. the natural
gradient, of $J(\mathbf{w}_{n-1})$. The natural gradient of $J(\mathbf{w})$,
on the other hand, is the direction of steepest descent at $\mathbf{w}$,
rather than at $\mathbf{w}_{n-1}$.
Therefore, our Gradient-DD approach makes use of the natural gradient of the
objective around $\mathbf{w}_{n-1}$ rather than around $\mathbf{w}_{n}$ in
Eqn. (A.1). This explains the distinction of the updates of our Gradient-DD
approach from the updates of directly applying the natural gradient of the
objective $\mathbf{w}$.
### A.3 Proof of Theorem 1
We introduce an ODE result on stochastic approximation in the following lemma,
then show the convergence of our GDD approach in Theorem 1 by applying this
result.
###### Lemma 1.
(Theorems 2.1 & 2.2 of [BM:00]) Consider the stochastic approximation
algorithm described by the $d$-dimensional recursion
$\mathbf{y}_{n+1}=\mathbf{y}_{n}+a_{n}[f(\mathbf{y}_{n})+\mathbf{M}_{n+1}].$
Suppose the following conditions hold: (c1) The sequence $\\{\alpha_{n}\\}$
satisfies $0<\alpha_{n}<1$, $\sum_{n=1}^{n}\alpha_{n}=\infty$,
$\sum_{n=1}^{n}\alpha_{n}^{2}<\infty$. (c2) The function $f$ is Lipschitz, and
there exists a function $f_{\infty}$ such that
$\lim_{r\rightarrow\infty}f_{r}(\mathbf{y})=f_{\infty}(\mathbf{y})$, where the
scaled function $f_{r}:\mathbb{R}^{d}\rightarrow\mathbb{R}^{d}$ is given by
$f_{r}(\mathbf{y})=f(r\mathbf{y})/r$. (c3) The sequence
$\\{\mathbf{M}_{n},\mathcal{F}_{n}\\}$, with
$\mathcal{F}_{n}=\sigma(\mathbf{y}_{i},\mathbf{M}_{i},i\leq n)$, is a
martingale difference sequence. (c4) For some $c_{0}<\infty$ and any initial
condition $y_{0}$, $\text{E}(\|\mathbf{M}_{n+1}\|^{2}|\mathcal{F}_{n})\leq
c_{0}(1+\|\mathbf{y}_{n}\|^{2})$. (c5) The ODE
$\dot{\mathbf{y}}=f_{\infty}(\mathbf{y})$ has the origin as a globally
asymptotically stable equilibrium. (c6) The ODE
$\dot{\mathbf{y}}(t)=f(\mathbf{y}(t))$ has a unique globally asymptotically
stable equilibrium $\mathbf{y}^{*}$. Then (1) under the assumptions (c1-c5),
$\sup_{n}\mathbf{y}_{n}<\infty$ in probability. (2) under the assumptions
(c1-c6), as $n\rightarrow\infty$, $\mathbf{y}_{n}$ converges to
$\mathbf{y}^{*}$ with probability 1 .
Now we investigate the stochastic gradient descent updates in Eqn. (11), which
is recalled as follows:
$\displaystyle\boldsymbol{\rho}_{n+1}$
$\displaystyle=\boldsymbol{\rho}_{n}-\kappa\alpha_{n}\mathbf{H}_{n}(\boldsymbol{\rho}_{n}-\boldsymbol{\rho}_{n-1})+\alpha_{n}(\mathbf{G}_{n}\boldsymbol{\rho}_{n}+\mathbf{g}_{n+1}).$
(A.3)
The iterative process in Eqn. (A.3) can be rewritten as
$\displaystyle(\boldsymbol{\rho}_{n+1}-\boldsymbol{\rho}_{n})$
$\displaystyle=-\kappa\alpha_{n}\mathbf{H}_{n}(\boldsymbol{\rho}_{n}-\boldsymbol{\rho}_{n-1})+\alpha_{n}(\mathbf{G}_{n}\boldsymbol{\rho}_{n}+\mathbf{g}_{n+1}).$
(A.4)
Defining
$\mathbf{u}_{n+1}=\boldsymbol{\rho}_{n+1}-\boldsymbol{\rho}_{n}.$
Eqn. (A.4) becomes
$\displaystyle\mathbf{u}_{n+1}$
$\displaystyle=-\kappa\alpha_{n}\mathbf{H}_{n}\mathbf{u}_{n}+\alpha_{n}(\mathbf{G}_{n}\boldsymbol{\rho}_{n}+\mathbf{g}_{n+1}).$
Thus, the iterative process in Eqn. (A.3) is rewritten as two parallel
processes that are given by
$\displaystyle\boldsymbol{\rho}_{n+1}$
$\displaystyle=\boldsymbol{\rho}_{n}-\kappa\alpha_{n}\mathbf{H}_{n}\mathbf{u}_{n}+\alpha_{n}(\mathbf{G}_{n}\boldsymbol{\rho}_{n}+\mathbf{g}_{n+1}),$
(A.5) $\displaystyle\mathbf{u}_{n+1}$
$\displaystyle=-\kappa\alpha_{n}\mathbf{H}_{n}\mathbf{u}_{n}+\alpha_{n}(\mathbf{G}_{n}\boldsymbol{\rho}_{n}+\mathbf{g}_{n+1}).$
(A.6)
Our proofs have three steps. First we shall show
$\sup_{n}\|\boldsymbol{\rho}_{n}\|$ is bounded by applying the ordinary
differential equation approach of the stochastic approximation (the 1st result
of Lemma 1) into the recursion Eqn. (A.5). Second, based on this result, we
shall show that $\mathbf{u}_{n}$ goes to 0 in probability by analyzing the
recursion Eqn. (A.6). At last, along with the result that $\mathbf{u}_{n}$
goes to 0 in probability, by applying the 2nd result of Lemma 1 into the
recursion Eqn. (A.5), we show that $\boldsymbol{\boldsymbol{\rho}}_{n}$ goes
to the TD fixed point, which is given by the solution of
$\mathbf{G}\boldsymbol{\rho}+\mathbf{g}=0$.
First, we shall show $\sup_{n}\|\boldsymbol{\rho}_{n}\|$ is bounded. Eqn.
(A.5) is rewritten as
$\boldsymbol{\rho}_{n+1}=\boldsymbol{\rho}_{n}+\alpha_{n}(f(\boldsymbol{\rho}_{n})+\mathbf{M}_{n+1}),$
(A.7)
where
$f(\boldsymbol{\rho}_{n})=(\mathbf{G}\boldsymbol{\rho}_{n}+\mathbf{g})-\kappa\mathbf{H}\mathbf{u}_{n}$
and
$\mathbf{M}_{n+1}=((\mathbf{G}_{n}-\mathbf{G})\boldsymbol{\rho}_{n}+\mathbf{g}_{n+1}-\mathbf{g})-\kappa(\mathbf{H}_{n}-\mathbf{H})\mathbf{u}_{n}$.
Let
$\mathcal{F}_{n}=\sigma(\mathbf{u}_{0},\boldsymbol{\rho}_{0},\mathbf{M}_{0},\mathbf{u}_{1},\boldsymbol{\rho}_{1},\mathbf{M}_{1},\cdots,\mathbf{u}_{n},\boldsymbol{\rho}_{n},\mathbf{M}_{n})$
be $\sigma$-fields generated by the quantities
$\mathbf{u}_{i},\boldsymbol{\rho}_{i},\mathbf{M}_{i}$, $i\leq n$.
Now we verify the conditions (c1-c5) of Lemma 1. Condition (c1) is satisfied
under the assumption of the step sizes. Clearly, $f(\mathbf{u})$ is Lipschitz
and $f_{\infty}(\boldsymbol{\rho})=\mathbf{G}\boldsymbol{\rho}$, meaning
Condition (c2) is satisfied. Condition (c3) is satisfied by noting that
$(M_{n},\mathcal{F}_{n})$ is a martingale difference sequence, i.e.,
$\text{E}(\mathbf{M}_{n+1}|\mathcal{F}_{n})=0$.
We next investigate $\text{E}(\|\mathbf{M}_{n+1}\|^{2}|\mathcal{F}_{n})$. From
the triangle inequality, we have that
$\displaystyle\|\mathbf{M}_{n+1}\|^{2}\leq
2\|(\mathbf{G}_{n}-\mathbf{G})\|^{2}\|\boldsymbol{\rho}_{n}\|^{2}+2\|\kappa(\mathbf{H}_{n}-\mathbf{H})\|^{2}\|\mathbf{u}_{n}\|^{2}.$
(A.8)
From Assumption A3 in Theorem 1 that $\|\mathbf{u}_{n}\|$ is bounded and the
Assumption A1 that ($\mathbf{x}_{n},r_{n},\mathbf{x}_{n+1}$) is an i.i.d.
sequence with uniformly bounded second moments, there exists some constant
$c_{0}$ such that
$\displaystyle\|\mathbf{G}_{n}-\mathbf{G}\|^{2}\leq c_{0}/2,\text{ and
}\|\kappa(\mathbf{H}_{n}-\mathbf{H})\|^{2}\|\mathbf{u}_{n}\|^{2}\leq c_{0}/2.$
Thus, Condition (c4) is satisfied.
Note that $\mathbf{G}$ is defined here with opposite sign relative to
$\mathbf{G}$ in [Maei:11]. From [SSM09] and [Maei:11], the eigenvalues of the
matrix $-\mathbf{G}$ are strictly negative under the Assumption A2. Therefore,
Condition (c5) is satisfied. Thus, applying the 1st part of Lemma 1 shows that
$\sup_{n}\|\boldsymbol{\rho}_{n}\|$ is bounded in probability.
Second, we investigate the recursion Eqn. (A.6). Let
$\mathbf{y}_{n+1}=(\mathbf{G}_{n}\boldsymbol{\rho}_{n}+\mathbf{g}_{n+1})$.
Then
$\displaystyle\mathbf{u}_{n+1}$
$\displaystyle=\alpha_{n}[-\kappa\mathbf{H}_{n}\mathbf{u}_{n}+\mathbf{y}_{n+1}]$
$\displaystyle=\alpha_{n}\mathbf{y}_{n+1}+\alpha_{n}\alpha_{n-1}(-\kappa\mathbf{H}_{n})\mathbf{y}_{n}+\alpha_{n}\alpha_{n-1}\alpha_{n-2}(-\kappa\mathbf{H}_{n})(-\kappa\mathbf{H}_{n-1})\mathbf{y}_{n-1}$
$\displaystyle\quad+\cdots+\alpha_{n}\prod_{k=0}^{n-1}\alpha_{k}(-\kappa\mathbf{H}_{k+1})\mathbf{y}_{1}+\prod_{k=0}^{n}\alpha_{k}(-\kappa\mathbf{H}_{k})\mathbf{u}_{0}.$
(A.9)
Note that $\|\mathbf{H}_{n}\|\leq 1/\kappa$ due to $\|\mathbf{x}_{n}\|\leq
1/\kappa$ and that there exists a constant $c$ such that
$\|\boldsymbol{\rho}_{n}\|\leq c$ due to the above result that
$\sup_{n}\|\boldsymbol{\rho}_{n}\|<\infty$ in probability. Without loss of
generality, we assume that $\|\mathbf{x}_{n}\|\leq 1/\kappa$. Eqn. (A.3)
implies that
$\displaystyle\|\mathbf{u}_{n+1}\|$ $\displaystyle\leq
c\left(\alpha_{n}+\alpha_{n}\alpha_{n-1}+\alpha_{n}\alpha_{n-1}\alpha_{n-2}+\cdots+\prod_{k=0}^{n}\alpha_{k}\right)+\prod_{k=0}^{n}\alpha_{k}\|\mathbf{u}_{0}\|.$
(A.10)
Under Assumption A0, $\alpha_{n}\rightarrow 0$ as $n\rightarrow 0$. Based on
this, Lemma 2 (given in the following section) tells us that
$\alpha_{n}+\alpha_{n}\alpha_{n-1}+\alpha_{n}\alpha_{n-1}\alpha_{n-2}+\cdots+\prod_{k=0}^{n}\alpha_{k}\rightarrow
0$ as $n\rightarrow 0$. Thus, Eqn. (A.10) implies that
$\mathbf{u}_{n}\rightarrow 0$ in probability.
Finally, for applying the 2nd part of Lemma 1, we just need to verify
Condition (c6). Because $\mathbf{u}_{n}$ goes to 0 with probability 1, Eqn.
(A.7) tells us that the associated ODE corresponds to
$\mathbf{G}\boldsymbol{\rho}+\mathbf{g}=0.$
Thus, Condition (c6) is satisfied. The theorem is proved.
### A.4 A Lemma
###### Lemma 2.
Denote
$\epsilon_{n}=\alpha_{n}+\alpha_{n}\alpha_{n-1}+\cdots+\alpha_{n}\alpha_{n-1}\cdots\alpha_{0}$.
If $\alpha_{n}\rightarrow 0$ as $n\rightarrow\infty$, then
$\epsilon_{n}\rightarrow 0$ as $n\rightarrow\infty$.
###### Proof.
Because $\alpha_{n}\rightarrow 0$ as $n\rightarrow\infty$, there exists
$\alpha\in(0,1)$ and some integer $N$ such that $\alpha_{n}\leq\alpha<1$ when
$n\geq N$. Define a sequence $\varepsilon_{n}$ such that
$\displaystyle\varepsilon_{n}$ $\displaystyle=1+\alpha\varepsilon_{n-1}\text{
for }n\geq N+1;$ $\displaystyle\varepsilon_{N}$ $\displaystyle=\epsilon_{N}.$
Obviously,
$\epsilon_{n}\leq\varepsilon_{n},\forall n\geq N.$ (A.11)
Now we investigate the sequence $\varepsilon_{n}$.
$\displaystyle\varepsilon_{n}$
$\displaystyle=1+\alpha\varepsilon_{n-1}=1+\alpha(1+\alpha\varepsilon_{n-2})=\cdots=1+\alpha+\cdots+\alpha^{n-N-1}+\alpha^{n-N}\varepsilon_{N}$
$\displaystyle\leq\sum\nolimits_{k=0}^{\infty}\alpha^{k}+\alpha^{n-N}\varepsilon_{N}=1/(1-\alpha)+\alpha^{n-N}\varepsilon_{N}.$
Thus, we have that
$\sup_{n\geq N}\varepsilon_{n}<\infty.$ (A.12)
From Eqns. (A.11) & (A.12), we have
$\sup_{n\geq 0}\epsilon_{n}<\infty.$ (A.13)
From the definition of $\epsilon_{n}$, we have that
$\epsilon_{n}=\alpha_{n}+\alpha_{n}\epsilon_{n-1}$. It follows that
$\alpha_{n}=\frac{\epsilon_{n}}{1+\epsilon_{n-1}}\geq\frac{\epsilon_{n}}{1+\sup_{k\geq
0}\epsilon_{k}}.$
From the assumption $\alpha_{n}\rightarrow 0$ as $n\rightarrow\infty$ and Eqn.
(A.13), we have $\epsilon_{n}\rightarrow 0$ as $n\rightarrow\infty$. ∎
### A.5 Additional empirical results
Figure 5: The random walk task with tabular representation. The setting is
similar to Fig. 2, but the performance is evaluated by the average error of
all episodes, and $\alpha$ is tuned by minimizing the average error of all
episodes. Upper: Performance as a function of $\alpha$; Lower: performance
over episodes. From left to right: state space size 10 (left), 20 (middle), or
40 (right).
Figure 6: Performance of Gradient-DD in the random walk task in the tabular
representation with $\kappa\in\\{0.25,0.5,1,2,4\\}$. From left to right: state
space size 10 (left), 20 (middle), or 40 (right). In each figure, $\alpha$ is
tuned for each algorithm by minimizing the average error of the last 100
episodes. Results are averaged over 50 runs, with error bars denoting standard
error of the mean.
Figure 7: The random walk task in the tabular representation. Performance for
various $\beta_{n}=\zeta\alpha_{n}$, with
$\zeta\in\\{4^{-3},4^{-2},4^{-1},1,4\\}$. From left to right in each row: the
size of the state space is $m=10$, $m=20$, and $m=40$. In each case $\alpha$
is tuned by by minimizing the average error of the last 100 episodes according
to the their performance of corresponding algorithms. Results are averaged
over 50 runs, with error bars denoting standard error of the mean. Figure 8:
Performance of natural TDC and natural GTD2 in the random walk task with the
tabular representation and $m=20$. Performance for various
$\beta_{n}=\zeta\alpha_{n}$, with $\zeta\in\\{4^{-3},4^{-2},4^{-1},1,4\\}$. In
each case $\alpha$ is tuned by by minimizing the average error of the last 100
episodes according to the their performance of corresponding algorithms.
“NGTD" and “NTDC" denote the natural gradient-based algorithm of GTD2 and TDC,
respectively. Results are averaged over 50 runs, with error bars denoting
standard error of the mean.
Figure 9: The random walk task with the tabular representation and constant
step size. The state size is 20. The curves are averaged over 50 runs, with
error bars denoting the standard error of the mean, though most are
vanishingly small.
Figure 10: The random walk task with linear value-function approximation,
where $\alpha_{n}=\alpha(10^{3}+1)/(10^{3}+n)$. The state size is $m=20$. Each
state is represented by a $p$-dimensional feature vector with $p=5$,
corresponding to $m$-state with $m=20$. The $p$-dimensional representation for
every fourth state from the start is $[1,0,\cdots,0]$ for state $s_{1}$,
$[0,1,0,\cdots,0]$ for $s_{5}$, $\cdots$, and $[0,0,\cdots,0,1]$ for state
$s_{4p+1}$. The representations for the intermediate states are obtained by
linearly interpolating between these. The sequential states,
$S_{1},\cdots,S_{m}$ are obtained by using the first $m$ states above. Left:
Performance as $\alpha$; Right: performance over episodes. The curves are
averaged over 50 runs.
Figure 11: The Boyan Chain task with linear approximation. The setting is
similar to Fig. 3, but we evaluate the performance the average error of all
episodes, and $\alpha$ is tuned by minimizing the average error of all
episodes. Upper: Performance as a function of $\alpha$; Lower: performance
over episodes.
Figure 12: Performance of Gradient-DD in the Boyan Chain task with
$\kappa\in\\{2^{-2},2^{-1},1,2,2^{2}\\}$ and tapered $\alpha_{n}$.
“GDD($\kappa$)" denotes the Gradient-DD with regularization parameter
$\kappa$.
|
Bayesian prediction regions and density estimation with type-2 censored data
111.
Akbar Asgharzadeha, Éric Marchandb & Ali Saadati Nika
a Department of Statistics, University of Mazandaran, P.O. Box 47146-1407,
Balbosar, IRAN
b Département de mathématiques, Université de Sherbrooke, Sherbrooke Qc,
CANADA, J1K 2R1
(e-mails<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Summary
For exponentially distributed lifetimes, we consider the prediction of future
order statistics based on having observed the first $m$ order statistics. We
focus on the previously less explored aspects of predicting: (i) an arbitrary
pair of future order statistics such as the next and last ones, as well as
(ii) the next $N$ future order statistics. We provide explicit and exact
Bayesian credible regions associated with Gamma priors, and constructed by
identifying a region with a given credibility $1-\lambda$ under the Bayesian
predictive density. For (ii), the HPD region is obtained, while a two-step
algorithm is given for (i). The predictive distributions are represented as
mixtures of bivariate Pareto distributions, as well as multivariate Pareto
distributions. For the non-informative prior density choice, we demonstrate
that a resulting Bayesian credible region has matching frequentist coverage
probability, and that the resulting predictive density possesses the
optimality properties of best invariance and minimaxity.
AMS 2020 subject classifications: 62F15, 62N01, 62N05, 62C10, 62C20.
Keywords and phrases: Bayesian predictive density; Credibility; Coverage
probability; Mixtures; Multivariate Pareto; Prediction region; Type-2
censoring.
## 1 Introduction
Predictive analysis based on censored data in life testing experiments is
fundamental and leads to interesting challenges. In this work, we are
concerned with prediction regions for future order statistics based on the
first $m$ order statistics generated by exponentially distributed data. There
has been some previous work on such problems, but we focus here on the less
explored: (a) multivariate aspects, and (b) use of Bayesian predictive
densities to generate prediction regions and their related theoretical
properties.
We consider an i.i.d. sample of size $n$ from an exponential distribution with
density $\theta e^{-\theta t}\mathbb{I}_{(0,\infty)}(t)$, but we are only able
to observe the first $m$ order statistics $X_{1:n},\ldots,X_{m:n}$, commonly
referred to as type-II censoring. Our objectives include:
1. (I)
the joint prediction of two future order statistics $X_{r:n}$ and $X_{s:n}$,
with $m<r<s\leq n$, as well as
2. (II)
the joint prediction of the next $N$ order statistics with $2\leq N\leq n-m$.
Scenario (II) seems a natural one to consider and includes the case of all
future order statistics with $N=n-m$. Scenario (I) in more relevant in
situations where the focus is for instance on the next and last order
statistics (i.e., $r=m+1,s=n$), or the last two order statistics (i.e.,
$r=n-1,s=n$). Obviously, both scenarios overlap for $N=2$. Surprisingly
perhaps, for Gamma priors which we consider, the specification of Bayesian
predictive densities and regions leads to more complex representations in the
bivariate case. The bivariate case (I) was considered recently by Bagheri et
al. (2022) and we refer to this work for further motivation and historical
aspects of the problem. Their prediction regions are non-Bayesian however and
based on a pair pivotal quantities which will not arise as a Bayesian
solution.
It seems natural to generate prediction regions for future order statistics
via a Bayesian predictive density, but such problems seem to have been
relatively unexplored. An exception is given by Dunsmore (1974), but his work
concerns a single order statistic. We proceed in doing so for Gamma prior
densities as well as the non-informative density
$\pi_{0}(\theta)=\frac{1}{\theta}\,\mathbb{I}_{(0,\infty)}(\theta)$. The
obtained expressions are quite tractable and, interestingly, bring into play
finite mixtures of bivariate Pareto distributions, and multivariate Pareto
distributions. The obtained mixtures are “non-probabilistic” in the sense that
the coefficients $a_{i}$ in $\sum_{i}a_{i}f_{i}(t)$; the $f_{i}$’s being
densities; take on both positive and negative values. This aspect is less
appreciated than “probabilistic mixtures”, but it does not hinder the
usefulness of the representation for computational purposes of moments and
cumulative distributions functions.
For a given credibility $1-\lambda$, we fully describe the highest posterior
density (HPD) prediction region for scenario (II), while we propose an
algorithm to generate exact prediction regions for the bivariate scenario (I),
based on a natural decomposition of the joint predictive density into its
marginal and conditional parts. The analysis is carried out and much
facilitated by considering the prediction of spacings between future order
statistics, which can be converted back to the prediction of the order
statistics themselves.
Many of the recent studies on predictive density estimation focus on
theoretical properties of the density itself in a decision-theoretic context.
For scenarios (I) and (II), we report on the best invariant and minimax
properties for Kullback-Leibler divergence loss of the Bayesian predictive
density associated with the usual non-informative prior density
$\pi(\theta)=\frac{1}{\theta}$, attributable to the work of Liang & Barron
(2004). Moreover, we show that the frequentist probabilty of coverage for a
prediction region generated by such a Bayesian predictive density matches its
exact credibility. Therefore, such a Bayesian density not only possesses
optimality properties in a decision-theoretic framework, but also provides a
satisfactory frequentist option that compares favourably with previous
solutions (e.g., Bagheri et al., 2022).
The organization of the manuscript is as follows. The preliminary results of
Chapter 2 cover model densities for scenarios (I) and (II), some general
aspects on Bayesian predictive densities, and multivariate Pareto
distributions. Bayesian predictive densities and regions are derived and
illustrated in Section 3. In Section 4, we demonstrate in a more general
context including ours that a Bayesian credible region with credibility
$1-\lambda$ associated with the non-informative prior density $\pi_{0}$ yields
matching frequentist coverage probability $1-\lambda$ for all $\theta>0$.
Finally in Section 5, we concisely review optimality properties of the
Bayesian predictive densities again associated with density $\pi_{0}$ and
pertaining to the best invariant and minimaxity properties.
## 2 Preliminary results
### 2.1 Model densities
For the first $m$ order statistics $X_{1:n},\ldots,X_{m:n}$ among $n$
generated from i.i.d. $Exp(\theta)$ data, it is well known (e.g., Lawless
(1971)) $X=\sum_{i=1}^{m}X_{i:n}+(n-m)X_{m:n}$ is a sufficient statistic and
Gamma distributed $\mathcal{G}(m,\theta)$. Hereafter, we therefore consider
such a summary and corresponding density, that is
$X\sim\frac{\theta^{m}}{\Gamma(m)}\,x^{m-1}\,e^{-\theta
x}\,\mathbb{I}_{(0,\infty)}(x).$ (2.1)
For the prediction of $X_{r:n}$ and $X_{s:n}$ based on $X$, it is convenient
and equivalent to consider
$Y=(Y_{1},Y_{2})^{\top}\hbox{ with
}Y_{1}\,=\,X_{r:n}-X_{m:n},\,Y_{2}\,=\,X_{s:n}-X_{r:n}.$ (2.2)
The equivalence stems from the correspondence between a prediction region $R$
for $Y$ and the inverse mapping
$\\{(y_{1}+x_{m:n},y_{1}+y_{2}+x_{m:n}):(y_{1},y_{2})\in R\\}$ as a prediction
region for $(X_{r:n},X_{s:n})$.
For the multivariate version where the objective is to predict jointly the $N$
next future order statistics, it is analogously useful to consider
$Z=(Z_{1},\ldots,Z_{N})\,,\hbox{ with }\,Z_{i}=X_{m+i:n}-X_{m+i-1:n}\hbox{ for
}i\in\\{1,\ldots,N\\},2\leq N\leq n-m\,,$ (2.3)
as the objects of prediction.
As described with the next result, both transformations (2.2) and (2.3) lead
to convenient underlying model densities for $Y$ or $Z$.
###### Lemma 2.1.
Given set-up (2.1, 2.2, and 2.3) and fixed $\theta$:
1. (a)
$Y$ and $X$ are independently distributed;
2. (b)
$Z$ and $X$ are independently distributed;
3. (c)
$Y_{1}$ and $Y_{2}$ are independently distributed with joint density on
$\mathbb{R}_{+}^{2}$ given by $q_{\theta}(y)\,=\,\theta^{2}\,q_{1}(\theta
y_{1},\theta y_{2})$ and with
$q_{1}(y)\,=\,\frac{(n-m)!}{(r-m-1)!\,(s-r-1)!\,(n-s)!}\,\big{(}1-e^{-y_{1}}\big{)}^{r-m-1}\,\big{(}1-e^{-y_{2}}\big{)}^{s-r-1}\,e^{-(n-r+1)y_{1}-(n-s+1)y_{2}}\,;$
(2.4)
4. (d)
$Z_{1},\ldots,Z_{N}$ are independently distributed with $Z_{i}\sim
Exp\big{(}(n-m-i+1)\,\theta\big{)}$.
Proof. Part (b) follows from a familiar renewal property of the exponential
distribution, and furthermore implies (a) since $Y$ is a function $h(Z)$ of
$Z$. For (c), since $(U=X_{r:n}-X_{m:n},V=X_{s:n}-X_{m:n})$ and $X_{m:n}$ are
independently distributed, the distribution of $(U,V)$ matches that of its
conditional distribution given $X_{m:n}$, and the latter joint distribution
can by seen to be equivalent, with the above-mentioned renewal property, to
that of the $(r-m)^{\hbox{th}}$ and $(s-m)^{\hbox{th}}$ order statistics from
a sample of size $n-m$ from an $Exp(\theta)$ distribution. Such a joint
density is given by
$f(u,v)\,=\,\frac{(n-m)!}{(r-m-1)!\,(s-r-1)!\,(n-s)!}\,\big{(}1-e^{-\theta
u}\big{)}^{r-m-1}\,\big{(}e^{-\theta u}-e^{-\theta
v}\big{)}^{s-r-1}\,(e^{-\theta v})^{n-s}\,\theta^{2}\,e^{-\theta(u+v)},\,$
(2.5)
for $0<u<v$. The result follows by transforming $(U,V)$ to $Y$. Finally for
(d), since $Z$ and $X_{m:n}$ are independently distributed, the distribution
of $Z$ matches that of its conditional distribution given $X_{m:n}$, and as
above the latter joint distribution matches that of the distribution of the
first $N$ order statistics spacings from a sample of size $N$ from an
$Exp(\theta)$ distribution. It is well known that such order statistics
spacings are independently distributed as $Exp\big{(}(n-m-i+1)\theta\big{)}$
(e.g., Lehmann & Casella, 1998, problem 6.18, page 71) which yields the
result. ∎
###### Remark 2.1.
The marginal densities of $Y_{1}$ and $Y_{2}$ in (2.4) can be extracted from
part (c) and are of the form
$\mathcal{B}^{-1}(c_{1},c_{2})\,(1-e^{-u})^{c_{1}-1}\,e^{-c_{2}u}\,\mathbb{I}_{(0,\infty)}(u)$,
with $\mathcal{B}(c_{1},c_{2})$ the Beta function. An alternative and readily
verified representation for the distribution of the above $Y_{1}$ and $Y_{2}$
is:
$e^{-\theta Y_{1}}\sim\hbox{Beta}(n-r+1,r-m)\,,\,e^{-\theta
Y_{2}}\sim\hbox{Beta}(n-s+1,s-r)\,\hbox{ independent}\,.$
### 2.2 Bayesian predictive densities
A general set-up for predictive density estimation relates to the following
model density:
$(X,Y)|\theta\sim
p_{\theta}(x)\,q_{\theta}(y|x)\,,x\in\mathbb{R}^{d_{1}},y\in\mathbb{R}^{d_{2}}.$
We observe $X$ according to $p_{\theta}$ and we wish to estimate the density
$q_{\theta}(\cdot|x)$. Except for $\theta$, the densities are known. The
observed $X$ provides information about $\theta$ and determines the
conditional density $q_{\theta}(\cdot|x)$ to estimate. Much, but not all
(e.g., Fourdrinier et al. 2019), previous work on properties of predictive
densities focusses on models for which $X$ and $Y$ are conditionally
independent, i.e., $q_{\theta}(\cdot|x)\equiv q_{\theta}(\cdot)$ for all $x$,
but it is natural in general to estimate the density $q_{\theta}(\cdot|x)$
when there is dependence.
Assume that we have a prior density $\pi$ for $\theta$, and a resulting
posterior density $\pi(\cdot|x)$ taken to be absolutely continuous with
respect to measure $\nu$. A natural estimator of the density
$q_{\theta}(\cdot|x)$ is the Bayes predictive density given by the conditional
density $q(\cdot|x)$ of $Y|X=x$. Integrating out $\theta$, we obtain the
estimator
$\hat{q}_{\pi}(y|x)\,=\,\int_{\Theta}q_{\theta}(y|x)\,\pi(\theta|x)\,d\nu(\theta).$
(2.6)
The above is a fully Bayesian procedure that can be used for obtaining
predictive point estimates or prediction regions for $Y$. The estimator or
density $\hat{q}_{\pi}(\cdot;X)$ will naturally depend on $X$ through a
sufficient statistics $T(X)$.
### 2.3 Multivariate Pareto densities
The predictive densities that we elicit below bring into play univariate,
bivariate, and multivariate type II Pareto distributions, as well as non-
probabilistic mixtures of such distributions. Such distributions have been
extensively studied (see for instance Section 52.4 of Kotz et al. 2000, or
Arnold 2014, and the references therein). With convenient expressions for the
moments and cumulative distribution functions (c.d.f.’s) relative to the
$f_{i}$’s, the mixture representations clearly facilitate expressions for
moments and c.d.f.’s for the full distribution.
Multivariate Pareto distributions possess Pareto marginals with parameters
$\ell,h>0$, densities and survival functions, which we denote and write as
$f_{\ell,h}(t)\,=\,\frac{\ell\,h}{\;\;\;(1+ht)^{\ell+1}}\,\mathbb{I}_{(0,\infty)}(t)\,,\hbox{
and }\bar{F}_{l,h}(t)\,=\,(1+ht)^{-\ell}\,,$ (2.7)
for $t>0$, respectively. Here is a multivariate Pareto definition; which
includes the bivariate case; and some useful properties.
###### Definition 2.1.
A random vector $Z=(Z_{1},\ldots,Z_{N})^{\top}$ has a multivariate Pareto type
II distribution, denoted $Z\sim\mathcal{P}2(m,h_{1},\ldots,h_{N})$ when it has
density
$g_{m,h_{1},\ldots,h_{N}}(z)\,=\,\frac{(m)_{N}\prod_{i=1}^{N}h_{i}}{(1+\sum_{i=1}^{N}h_{i}z_{i})^{m+N}}\,\mathbb{I}_{\mathbb{R}_{+}^{N}}(z),$
(2.8)
for parameters $h_{1},\ldots h_{N},m>0$, and
$(m)_{N}=\frac{\Gamma(m+N)}{\Gamma(m)}$.
Such distributions form a $N+1$ parameter family with scale parameters
$1/h_{i},i=1,\ldots,N$, and shape parameter $m$. As recorded with the
following lemma containing a selection of properties that are known and
readily verified, such multivariate Pareto distributions possess univariate
Pareto marginals and conditionals, subvectors that are also distributed as
multivariate Pareto with densities, and a joint survival function having a
rather simple form.
###### Lemma 2.2.
Consider $Z\sim\mathcal{P}2(m,h_{1},\ldots,h_{N})$. Let $i\neq j$. Then, we
have: (i) $Z_{i}\sim f_{m,h_{i}}$, (ii) $Z_{j}|Z_{i}=z_{i}\sim
f_{m+1,h_{j}/(1+h_{i}z_{1})}$, (iii) $(h_{1}Z_{i},\ldots,h_{N}Z_{N})$ has
density $g_{m,1,\ldots,1}$, (iv) the joint survival function is given by
$\mathbb{P}(\cap_{i=1}^{N}\\{Z_{i}\geq
z_{i}\\})\,=\,\\{(1+\sum_{i=1}^{N}h_{i}z_{i})\\}^{-m}$ for $z_{i}\geq
0,i=1,\ldots,N$, (v) the simple regressions are linear with (vi)
$\mathbb{E}(Z_{j}|Z_{i})\,=\,\frac{1+h_{i}Z_{i}}{h_{j}m}\,$, and (vi) the
correlation between $Z_{i}$ and $Z_{j}$ is given by
$\rho(Z_{i},Z_{j})=\frac{1}{m}$ for $m>2$.
Furthermore, we will require the following.
###### Lemma 2.3.
Let $Z\sim\mathcal{P}2(m,h_{1},\ldots,h_{N})$ and
$W=\sum_{i=1}^{N}h_{i}Z_{i}$. Then $W$ is Beta type-II distributed (denoted
$W\sim B2(N,m)$) with p.d.f.
$f_{W}(w)\,=\,\frac{\Gamma(N+m)}{\Gamma(N)\Gamma(m)}\frac{w^{N-1}}{(1+w)^{N+m}}$,
for $w\in(0,\infty)$.
Proof. The result is known but we provide a proof for completeness. Given the
multivariate Pareto representation
$(h_{1}Z_{1}\,\ldots,h_{N}Z_{N})=^{d}(\frac{E_{1}}{G},\ldots,\frac{E_{N}}{G})$
with $E_{1},\ldots E_{N},G$ independently distributed, and with $E_{i}\sim
Exp(1)$ and $G\sim\mathcal{G}(m,1)$, we see that $W$ is distributed as the
ratio of two independent $\mathcal{G}(N,1)$ and $\mathcal{G}(m,1)$ variables,
hence Beta type II with the given parameters. ∎
## 3 Predictive densities and regions
Based on $X$ as in (2.1), we provide in this section Bayesian predictive
densities and regions for $Y$ and $Z$ as defined in (2.2) and (2.3). We
consider Gamma $\mathcal{G}(\alpha,\beta)$ prior densities
$\pi_{\alpha,\beta}(\theta)\propto\theta^{\alpha-1}\,e^{-\beta\theta}\,\mathbb{I}_{(0,\infty)}(\theta)$,
including the usual non-informative case
$\pi_{0}(\theta)\,=\,\frac{1}{\theta}\,\mathbb{I}_{(0,\infty)}(\theta)$ for
$\alpha=\beta=0$.
### 3.1 Predictive densities
We begin with the future next $N$ order statistics.
###### Theorem 3.1.
The Bayes predictive density of $Z$ in (2.3), based on
$X\sim\mathcal{G}(m,\theta)$ and prior density $\pi_{\alpha,\beta}$ for
$\theta$, is that of $\mathcal{P}2(m+\alpha,h_{1},\ldots,h_{N})$ distribution
with $h_{i}=\frac{n-m-i+1}{x+\beta}$ for $i=1,\ldots,N$.
Proof. With
$q_{\theta}(z|x)\,=\,\frac{\theta^{N}\,(n-m)!}{(n-m-N)!}\,e^{-\theta\sum_{i=1}^{N}(n-m-i+1)z_{i}}$
and $\theta|x\sim\mathcal{G}(\alpha+m,x+\beta)$, we obtain from (2.6):
$\displaystyle\hat{q}_{\pi_{\alpha,\beta}}(z|x)\,$ $\displaystyle=$
$\displaystyle\,\frac{(n-m)!\,(\beta+x)^{\alpha+m}}{(n-m-N)!\,\Gamma(\alpha+m)\,}\int_{0}^{\infty}\theta^{N+\alpha+m-1}\,e^{-\theta\big{(}\beta+x+\sum_{i=1}^{N}(n-m-i+1)z_{i}\big{)}}\,d\theta\,$
$\displaystyle=$
$\displaystyle\,\frac{(n-m)!\,(\beta+x)^{\alpha+m}}{(n-m-N)!\,\Gamma(\alpha+m)\,}\,\Gamma(N+\alpha+m)\,\big{\\{}\beta+x+\sum_{i=1}^{N}(n-m-i+1)z_{i}\big{\\}}^{-(N+\alpha+m)},$
which is indeed a $\mathcal{P}2(m+\alpha,h_{1},\ldots,h_{N})$ density. ∎
Observe that the density has univariate Pareto marginals and multivariate
Pareto distributed subvectors in accordance to Lemma 2.2.
We now turn to the Bayesian predictive density for two future order statistics
and demonstrate a non-probabilistic mixture of bivariate Pareto densities as
given in Definition 2.1 for $N=2$.
###### Theorem 3.2.
The Bayes predictive density of $Y$ in (2.2), based on
$X\sim\mathcal{G}(m,\theta)$ and prior density $\pi_{\alpha,\beta}$ for
$\theta$, is given by
$\hat{q}_{\pi_{\alpha,\beta}}(y|x)\,=\,\sum_{i=0}^{r-m-1}\sum_{j=0}^{s-r-1}\omega_{i,j}\,g_{m+\alpha,\frac{a_{i}}{x+\beta},\frac{b_{j}}{x+\beta}}\big{(}y\big{)}\,,$
(3.9)
with $g$ a bivariate Pareto density given in (2.8) for $N=2$,
$a_{i}=n-r+i+1,b_{j}=n-s+j+1,\hbox{and
}\omega_{i,j}\,=\,\frac{(n-m)!}{(n-s)!}\,\frac{(-1)^{i+j}}{i!\,j!}\,\,\frac{1}{(r-m-i-1)!\,(s-r-j-1)!}\frac{1}{a_{i}b_{j}}.$
Proof. From (2.4) and (2.6), with $\theta|x\sim\mathcal{G}(m+\alpha,x+\beta)$,
we obtain
$\hat{q}_{\pi_{\alpha,\beta}}(y|x)\,=\,\frac{(n-m)!}{(n-s)!}\frac{\,(x+\beta)^{m+\alpha}}{\Gamma(m+\alpha)}\,\int_{0}^{\infty}\theta^{m+\alpha+1}\,\frac{(1-e^{-\theta
y_{1}})^{r-m-1}}{(r-m-1)!}\,\frac{(1-e^{-\theta
y_{2}})^{s-r-1}}{(s-r-1)!}e^{-\theta(L(y)+x+\beta)}\,d\theta,$
with $L(y)\,=\,(n-r+1)y_{1}+(n-s+1)y_{2}$. Binomial expansions and an
interchange of sum and integral yield
$\displaystyle\hat{q}_{\pi_{\alpha,\beta}}(y|x)\,$ $\displaystyle=$
$\displaystyle\,\frac{\,(x+\beta)^{m+\alpha}}{\Gamma(m+\alpha)}\,\sum_{i=0}^{r-m-1}\sum_{j=0}^{s-r-1}a_{i}b_{j}\,\omega_{i,j}\,\int_{0}^{\infty}\theta^{m+\alpha+1}\,e^{-\theta(L(y)+iy_{1}+jy_{2}+x+\beta)}\,d\theta\,$
$\displaystyle=$
$\displaystyle\,(m+\alpha)\,(m+\alpha+1)\,(x+\beta)^{m+\alpha}\,\sum_{i=0}^{r-m-1}\sum_{j=0}^{s-r-1}a_{i}b_{j}\,\omega_{i,j}\,\big{(}L(y)+iy_{1}+jy_{2}+x+\beta\big{)}^{-(m+\alpha+2)},$
which leads to (3.9). ∎
We point out the following result for the next two order statistics, which
follows immediately from either Theorem 3.1 or Theorem 3.2.
###### Corollary 3.1.
For the particular case where $r=m+1,s=m+2$, the predictive density of $Y$ as
in (2.2) is that of a $\mathcal{P}2(m+\alpha,h_{1},h_{2})$ distribution (3.9)
with $h_{1}=\frac{n-m}{x+\beta}$ and $h_{2}=\frac{n-m-1}{x+\beta}$.
###### Remark 3.2.
Interestingly, the above weights $\omega_{i,j}$ arise through a series
expansion of the Beta function. Indeed, by the the binomial expansion of
$(1-t)^{d-1}$ for $d\in\mathbb{N}_{+}$, we obtain for $c>0$
$\displaystyle\int_{0}^{1}t^{c-1}\,(1-t)^{d-1}\,dt\,$ $\displaystyle=$
$\displaystyle\,\frac{\Gamma(c)\Gamma(d)}{\Gamma(c+d)}$
$\displaystyle\Longrightarrow$
$\displaystyle\sum_{k=0}^{d-1}\gamma_{c,d,k}=1\,,$
with
$\gamma_{c,d,k}\,=\,\frac{\Gamma(c+d)}{\Gamma(c)}\,\frac{(-1)^{k}}{k!\,(d-1-k)!}\,\frac{1}{c+k}$.
Furthermore, observe that $\omega_{i,j}=\omega_{1,i}\,\omega_{2,j}$ with
$\omega_{1,i}=\frac{(n-r)!}{(n-s)!}\,\gamma_{n-r+1,r-m,i}$ and
$\omega_{2,j}=\frac{(n-s)!}{(n-r)!}\,\gamma_{n-s+1,s-r,j}$, demonstrating
alternatively that $\sum_{i,j}\omega_{i,j}=1$.
The mixture representation of the predictive density in Theorem 3.2 coupled
with the bivariate Pareto properties of Lemma 2.2 facilitate the evaluation of
the marginal and conditional distributions associated with (3.9). As shown
below, mixture representations of univariate Pareto distributions arise.
###### Corollary 3.2.
For $m<r\leq n-1$, the marginal density of $Y_{1}$ associated with the Bayes
predictive density (3.9) is given by
$\hat{q}_{\pi}(y_{1}|x)\,=\,\sum_{i=0}^{r-m-1}\gamma_{n-r+1,r-m,i}\,f_{m+\alpha,\frac{a_{i}}{x+\beta}}(y_{1})\,,$
(3.10)
where $\gamma_{c,d,k}$ is given in Remark 3.2 and $f_{l,h}$ is a univariate
Pareto density as given in (2.7). In the particular case where $r=m+1$, we
have that $\hat{q}_{\pi}(y_{1};x)=f_{m+\alpha,\frac{n-m}{x+\beta}}(y_{1})$ for
$y_{1}>0$.
Proof. We have from Theorem 3.2
$\displaystyle\hat{q}_{\pi}(y_{1}|x)\,$ $\displaystyle=$
$\displaystyle\,\int_{0}^{\infty}\hat{q}_{\pi}(y|x)\,dy_{2}$ $\displaystyle=$
$\displaystyle\,\sum_{i=0}^{r-m-1}\gamma_{n-r+1,r-m,i}\sum_{j=0}^{s-r-1}\gamma_{n-s+1,s-r,j}\;\int_{0}^{\infty}g_{m+\alpha,\frac{a_{i}}{x+\beta},\frac{b_{j}}{x+\beta}}\big{(}y\big{)}\,dy_{2},$
which leads to (3.10) since the joint density
$g_{m+\alpha,\frac{a_{i}}{x+\beta},\frac{b_{j}}{x+\beta}}$ for $Y$ has
marginal $f_{m+\alpha,\frac{a_{i}}{x+\beta}}$ for $Y_{1}$, for all $i,j$
(Lemma 2.2) and since $\sum_{j=0}^{s-r-1}\gamma_{n-s+1,s-r,j}=1$. ∎
The result in itself is not new and was obtained with the univariate analysis
carried out by Dunsmore (1974). A similar development establishes that (3.10)
holds for the marginal predictive density of $X_{n:n}-X_{m:n}$. For the
conditional distributions, we have the following.
###### Corollary 3.3.
For the Bayes predictive density (3.9):
1. (a)
The conditional density of $Y_{2}$ given $Y_{1}=y_{1}$ is given by the mixture
representation
$\hat{q}_{\pi}(y_{2}|y_{1};x)\,=\,\sum_{i=0}^{r-m-1}\sum_{j=0}^{s-r-1}\alpha_{i}(y_{1})\,\beta_{j}\,f_{m+\alpha+1,\frac{b_{j}}{x\,+\,\beta\,+\,a_{i}\,y_{1}}}(y_{2}),$
with
$\alpha_{i}(y_{1})\propto\gamma_{n-r+1,r-m,i}\,f_{m+\alpha,\frac{a_{i}}{x+\beta}}(y_{1})$
such that $\sum_{i=0}^{r-m-1}\alpha_{i}(y_{1})=1$, and
$\beta_{j}=\gamma_{n-s+1,s-r,j}\,$;
2. (b)
The conditional density of $Y_{1}$ given $Y_{2}=y_{2}$ is given by the mixture
representation
$\hat{q}_{\pi}(y_{1}|y_{2};x)\,=\,\sum_{i=0}^{r-m-1}\sum_{j=0}^{s-r-1}\xi_{j}(y_{2})\,\nu_{i}\,f_{m+\alpha+1,\frac{a_{i}}{x\,+\,\beta\,+\,\beta_{j}\,y_{2}}}(y_{1}),$
with
$\xi_{j}(y_{2})\propto\gamma_{n-s+1,s-r,j}\,f_{m+\alpha,\frac{b_{j}}{x+\beta}}(y_{2})$
such that $\sum_{j=0}^{s-r-1}\xi_{j}(y_{2})=1$, and
$\nu_{i}=\gamma_{n-r+1,r-m,i}$.
Proof. Part (a) follows directly with the properties of Lemma 2.2 upon writing
$g_{m+\alpha,\frac{a_{i}}{x+\beta},\frac{b_{j}}{x+\beta}}\big{(}y\big{)}\,=f_{m+\alpha,\frac{a_{i}}{x\,+\,\beta}}(y_{1})\,f_{m+\alpha+1,\frac{b_{j}}{x\,+\,\beta\,+\,a_{i}y_{1}}}(y_{2})$.
The same approach leads to (b). ∎
###### Remark 3.3.
With the above marginal and conditional distributions expressible as mixtures
of univariate Pareto distributions, corresponding moments are readily
available. For instance, one obtains
$\displaystyle\mu_{1}=\mathbb{E}_{\pi}(Y_{1}|x)$ $\displaystyle=$
$\displaystyle\frac{x+\beta}{m+\alpha-1}\,\sum_{i=0}^{r-m-1}\gamma_{n-r+1,r-m,i}\,\frac{1}{a_{i}}\,$
$\displaystyle\hbox{ and
}\mu_{2}(y_{1})\,=\,\mathbb{E}_{\pi}(Y_{2}|Y_{1};x)\,$ $\displaystyle=$
$\displaystyle\,\sum_{i=0}^{r-m-1}\sum_{j=0}^{s-r-1}\alpha_{i}(y_{1})\,\beta_{j}\,\frac{x+\beta+a_{i}y_{1}}{(m+\alpha)b_{j}}$
$\displaystyle=$
$\displaystyle\,y_{1}\,\sum_{i=0}^{r-m-1}\sum_{j=0}^{s-r-1}\alpha_{i}(y_{1})\,\beta_{j}\,\frac{a_{i}}{(m+\alpha)b_{j}}\,+\,\frac{x+\beta}{m+\alpha}\sum_{j=0}^{s-r-1}\,\,\frac{\beta_{j}}{b_{j}}.$
Observe that the conditional expectation $\mathbb{E}_{\pi}(Y_{2}|y_{1};x)$ is
affine linear of the form $C+Dy_{1}$.
### 3.2 HPD Bayesian prediction regions
A Bayesian prediction region with credibility $1-\lambda$ for $Z|\theta$ based
on an observed value $x$ of $X|\theta$ and for a given prior density $\pi$ is
such that
$\int_{R(x)}\hat{q}_{\pi}(z|x)\,dz\,=\,1-\lambda,$ (3.11)
with $\hat{q}_{\pi}$ given in (2.6). One such choice which minimizes volume is
the ubiquitous HPD region which is of the form
$R_{HPD}(x)\,=\,\big{\\{}z\in\mathbb{R}_{+}^{N}\,:\,\hat{q}_{\pi}(z|x)\,\geq
k\big{\\}},$ (3.12)
where $k$ is chosen so that (3.11) is satisfied. Here is an explicit form of
the HPD credible region for the future order statistics spacings
$Z_{1},\dots,Z_{N}$.
###### Theorem 3.3.
Based on $X_{i:n},i=1,\ldots,m$, the first $m$ order statistics among $n$ from
i.i.d. Exp$(\theta)$ data, setting $X$ as in (2.1) and $Z$ as in (2.3), the
HPD region of credibility $1-\lambda$ for $Z$ associated with Gamma prior
$\pi_{\alpha,\beta}$ for $\theta$ is given by
$R_{HPD}(x)\,=\,\big{\\{}z\in\mathbb{R}_{+}^{N}\,:\,\sum_{i=1}^{N}(n-m-i+1)\,z_{i}\,\leq
c_{0}(x+\beta)\big{\\}},$ (3.13)
with $c_{0}$ the quantile of order $1-\lambda$ of a $B2(N,m+\alpha)$
distribution.
Proof. It follows from Theorem 3.1 that $R_{HPD}$ defined generally in (3.12)
is of form (3.13). From Lemma 2.3, the posterior predictive distribution of
$W=\sum_{i=1}^{N}\frac{(n-m-i+1)\,z_{i}}{x+\beta}$ is $B2(N,m+\alpha)$
distributed and the result follows by setting $c_{0}$ such that
$\mathbb{P}(W\leq c_{0})=1-\lambda$. ∎
###### Remark 3.4.
With simple integral or finite sum forms for the $B2(N,m+\alpha)$ c.d.f., the
evaluation of the above quantiles is rather straightforward. For instance in
the bivariate case, we have $\mathbb{P}(W\leq
c)=1-\frac{1+c\,(m+\alpha+1)}{(1+c)^{m+\alpha+1}}$, so that $c_{0}$ is
expressible as the solution in $c>0$ of
$\frac{1+c\,(m+\alpha+1)}{(1+c)^{m+\alpha+1}}\,=\,\lambda.$
###### Remark 3.5.
HPD credible regions are in general not invariant with respect to
transformations, but they are whenever the transformation is affine linear.
Since the original order statistics $T=(X_{m+1:n},\ldots,X_{m+N;n})^{\top}$
are related to the spacings $Z=(Z_{1},\ldots,Z_{N})^{\top}$ by the affine
linear transformation $T=b+AZ$, with $b=(X_{m:n},\ldots,X_{m:n})^{\top}$ and
$A$ a lower triangular matrix with non-zero elements $a_{i,j}=1$ for $i\leq
j$, the transformation of $R_{HPD}(x)$ given by (3.13) to the order statistics
$X_{m:n},\ldots,X_{m+N;n}$ is also HPD.
### 3.3 Prediction regions: an algorithm for the bivariate case
We present here a general bivariate case solution to obtain a prediction
region with a given credibility based on Theorem 3.2’s predictive density for
$Y$ as in (2.2). There are several options available, but we opt for a rather
explicit strategy to construct a Bayesian credible region. It can be applied
for any choice of $(r,s,m,n)$, prior $\pi_{\alpha,\beta}$ such that
$m+\alpha>1$, and the targeted credibility $1-\lambda$. We refer to notation
used throughout this paper, namely $a_{i}$ and $b_{j}$ as in Theorem 3.2,
$\gamma_{c,d,k}$ as in Remark 3.2, $f_{l,h}$ and $\bar{F}_{l,h}$ as the
univariate Pareto density and survival functions in (2.7), $\beta_{j}$ and
$\alpha_{i}(y_{1})$ as in Corollary 3.3, and $\mu_{1}$ and $\mu_{2}(y_{1})$ as
in Remark 3.3.
1. Step 1. Select a prediction region $A$ for $Y_{1}$ of credibility $\sqrt{1-\lambda}$ based on the marginal density in Corollary 3.2. Such a choice would desirably be an interval with relatively high levels of the predictive density. A suitable choice is:
$A\,=\,\big{[}\mu_{1}-\Delta_{1},\mu_{1}+\Delta_{1}\big{]}\,\cap[0,\infty)\,,$
with $\Delta_{1}>0$ uniquely chosen such that
$\displaystyle\int_{\mu_{1}-\Delta_{1}}^{\mu_{1}+\Delta_{1}}\hat{q}_{\pi_{\alpha,\beta}}(y_{1}|x)\,dy_{1}\,=\,\sqrt{1-\lambda}$
$\displaystyle\Longleftrightarrow$
$\displaystyle\sum_{i=0}^{r-m-1}\gamma_{n-r+1,r-m,i}\,\Big{(}\bar{F}_{m+\alpha,\frac{a_{i}}{x+\beta}}(\mu_{1}-\Delta_{1})_{+}\,-\,\bar{F}_{m+\alpha,\frac{a_{i}}{x+\beta}}(\mu_{1}+\Delta_{1})\Big{)}\,=\,\sqrt{1-\lambda},$
with $z_{+}=\max\\{0,z\\}$. Since this last expression is strictly increasing
in $\Delta_{1}$, it is rather straightforward to approach $\Delta_{1}$
numerically.
2. Step 2. For each $y_{1}\in A$, select a prediction region $B(y_{1})$ of conditional credibility $\sqrt{1-\lambda}$ for $y_{2}$ based on the conditional density of $Y_{2}|Y_{1}=y_{1}$ given in part (a) of Corollary 3.3. The challenge here is similar to the one in Step 1 but to be repeated for all $y_{1}$. Analogously to Step 1, $B(y_{1})$ can reasonably be constructed pivoting around the mean as
$B(y_{1})\,=\,\big{[}\mu_{2}(y_{1})-\Delta_{2}(y_{1}),\mu_{2}(y_{1})+\Delta_{2}(y_{1})\big{]}\,\cap[0,\infty),$
with $\Delta_{2}(y_{1})>0$ chosen such that
$\displaystyle\\!\sum_{i=0}^{r-m-1}\sum_{j=0}^{s-r-1}\alpha_{i}(y_{1})\,\beta_{j}\,\int_{\mu_{2}(y_{1})-\Delta_{2}(y_{1})}^{\mu_{2}(y_{1})+\Delta_{2}(y_{1})}\,f_{m+\alpha+1,\frac{b_{j}}{x+\beta+a_{i}y_{1}}}(y_{2})\,dy_{2}\,=\,\sqrt{1-\lambda}$
$\displaystyle\Longleftrightarrow\\!$
$\displaystyle\sum_{i=0}^{r-m-1}\sum_{j=0}^{s-r-1}\alpha_{i}(y_{1})\,\beta_{j}\,\,\Big{(}\bar{F}_{m+\alpha+1,\frac{b_{j}}{x+\beta+a_{i}y_{1}}}(\mu_{2}(y_{1})-\Delta_{2}(y_{1}))_{+}\,-\,\bar{F}_{m+\alpha+1,\frac{b_{j}}{x+\beta+a_{i}y_{1}}}(\mu_{2}(y_{1})+\Delta_{2}(y_{1}))\Big{)}\,\,$
$\displaystyle=\sqrt{1-\lambda}.$
The resulting prediction region $R=\\{(y_{1},y_{2}):y_{1}\in A,y_{2}\in
B(y_{1})\\}$ has credibility $1-\lambda$ indeed since
$\displaystyle\mathbb{P}_{\pi}\big{(}Y_{1}\in A,Y_{2}\in B(y_{1})|x\big{)}$
$\displaystyle=$
$\displaystyle\int_{A}\hat{q}_{\pi}(y_{1}|x)\big{\\{}\int_{B(y_{1})}\hat{q}_{\pi}(y_{2}|y_{1};x)\,dy_{2}\big{\\}}\,dy_{1}$
$\displaystyle=$
$\displaystyle\int_{A}\hat{q}_{\pi}(y_{1}|x)\,(\sqrt{1-\lambda})\,dy_{1}$
$\displaystyle=$ $\displaystyle 1-\lambda.$
###### Remark 3.6.
Other regions can also be selected. For instance, one-sided choices with $A$
of the form $[0,\bar{\Delta}_{1}]$ or $[\underline{\Delta}_{1},\infty)$, and
with $B$ of the form $[0,\bar{\Delta}_{2}(y_{1})]$ or
$[\underline{\Delta}_{2}(y_{1}),\infty)$. Another alternative would be to aim
for different credibilities $1-\lambda_{1}$ and $1-\lambda_{2}$ in Steps 1 and
2, respectively, such that $(1-\lambda_{1})(1-\lambda_{2})=1-\lambda$.
### 3.4 Example
The following dataset from Murthy et al. (2004) shows $n=30$ ordered failure
times for repairable items:
0.11 | 0.30 | 0.40 | 0.45 | 0.59 | 0.63 | 0.70 | 0.71 | 0.74 | 0.77
---|---|---|---|---|---|---|---|---|---
0.94 | 1.06 | 1.17 | 1.23 | 1.23 | 1.24 | 1.43 | 1.46 | 1.49 | 1.74
1.82 | 1.86 | 1.97 | 2.23 | 2.37 | 2.46 | 2.63 | 3.46 | 4.36 | 4.73
For the purpose of illustration, suppose that only the $m=20$ first values are
observed and that we wish to predict either:
1. (i)
the next two order statistics spacings
$(Z_{1},Z_{2})\,=\,\big{(}X_{21:30}-X_{20:30},X_{22:30}-X_{21:30}\big{)}$,
or
1. (ii)
the next and last order statistics $\big{(}X_{21:30},X_{30:30}\big{)}$.
In both cases, we use the non-informative prior density
$\pi_{(}\theta)\,=\,\frac{1}{\theta}\,\mathbb{I}_{(0,\infty)}(\theta)$, and
consider credibility $1-\lambda=0.95$. As expanded upon in the next section,
the frequentist coverage of such a prediction region matches the credibility
for all $\theta>0$. The data yields
$x\,=\,\sum_{i=1}^{20}x_{i:30}\,+\,10\,x_{20:30}\,=\,35.79$.
For case (i), Theorem 3.3 applies and the HPD credible region for
$(Z_{1},Z_{2})$ is given by
$\displaystyle
R_{\textrm{HPD}}(35.79)\,=\,\bigl{\\{}(z_{1},z_{2})\in\mathbb{R}_{+}^{2}\,:\,0.2794\,z_{1}\,+\,0.2515z_{2}\,\leq
0.2606\big{\\}},$
$c_{0}\,=\,0.2606$ being the quantile of order $0.95$ of a $B2(2,20)$
distribution.
For case (ii), we illustrate the use of Section 3.3’s algorithm which is
prescribed for $(Y_{1},Y_{2})$ with $Y_{1}\,=\,X_{21:30}-X_{20:30}$ and
$Y_{2}\,=X_{30:30}-X_{21:30}$. Figure 1 (a) presents the resulting Bayesian
prediction region along with the regression function or conditional
expectation $\mathbb{E}(Y_{2}|y_{1},x=35.79)$ as a function of $y_{1}$. The
first step yields $A=[0,0.722]$, while the $B(y_{1})$ intervals are shown in
the figure. For example, $B(0.50)=\,[0,13.059]$. Observe that a left
neighbourhood of $0$ is included in $B(y_{1})$ for all $y_{1}\in A$ as the
second step credibility, equal to $(0.95)^{1/2}\approx 0.9747$, is quite
large. In contrast, the Bayesian prediction region with credibility $0.80$ is
displayed in Figure 1 (b) resulting in $A=[0,0.4258]$ and intervals $B(y_{1})$
centered at $\mathbb{E}(Y_{2}|y_{1},x=35.79)$ which exclude values close to
$0$ for all $y_{1}\in A$. Finally, the corresponding prediction regions for
$\big{(}X_{21:30},X_{30:30}\big{)}\,$ and credibilities $0.95$ and $0.80$ are
obtained as $\\{(y_{1}+x_{20:30},y_{1}+y_{2}+x_{20:30}):(y_{1},y_{2})\in R\\}$
and displayed in Figure 2.
(a) $1-\lambda=0.95$ (b) $1-\lambda=0.80$
Figure 1: Bayesian prediction regions and $\mathbb{E}(Y_{2}|y_{1},x=35.79)$
(dashed)
(a) $1-\lambda=0.95$ (b) $1-\lambda=0.80$
Figure 2: Bayesian prediction regions for $(X_{21:30},X_{30:30})$ in case
(ii)
## 4 Frequentist coverage and credibility
Through Theorem 3.2’s Bayesian predictive densities, the prediction regions
for future order statistics described in the previous section are constructed
in order to attain exact Bayesian credibility. This includes the non-
informative prior prediction densities $\hat{q}_{\pi_{0}}(\cdot;X)$ given by
Theorems 3.1 and 3.2 for $\alpha=\beta=0$. Moreover, prediction regions based
on $\hat{q}_{\pi_{0}}(\cdot;X)$ also lead to exact frequentist coverage as
expanded on below. In fact, we cast the result in a more general scale
parameter family model setting with scale parameter densities
$p_{\sigma}(x)=\frac{1}{\sigma}\,p_{1}(\frac{x}{\sigma})\hbox{ and
}q_{\sigma}(y^{\prime})=\frac{1}{\sigma^{d_{2}}}\,q_{1}(\frac{y_{1}^{\prime}}{\sigma},\ldots,\frac{y_{d_{2}}^{\prime}}{\sigma}).$
(4.14)
We will make use of the following intermediate result, a univariate version of
which was first given by L’Moudden et al. (2017) (i.e., $d_{2}=1$).
###### Lemma 4.4.
Under model (4.14) and prior density
$\pi_{0}(\sigma)\,=\,\frac{1}{\sigma}\,\mathbb{I}_{(0,\infty)}(\sigma)$, the
Bayesian predictive density $\hat{q}_{\pi_{0}}$ is given by
$\hat{q}_{\pi_{0}}(y^{\prime}|x)\,=\,\frac{1}{x^{d_{2}}}\,h(\frac{y^{\prime}}{x}),$
(4.15)
where $h$ is the frequentist density of
$R=\frac{Y^{\prime}}{X}=(R_{1},\ldots,R_{d_{2}})$, which is free of $\sigma$,
and given by
$h(r)\,=\,\int_{0}^{\infty}u^{d/2}\,q_{1}(r_{1}u,\ldots,r_{d_{2}}u)\,p(u)\,du.$
(4.16)
Furthermore, it is the case that
$\big{(}\frac{Y_{1}^{\prime}}{x},\ldots,\frac{Y^{\prime}_{d_{2}}}{x}\big{)}\big{|}x\,=^{d}\,\big{(}\frac{Y_{1}^{\prime}}{X},\ldots,\frac{Y^{\prime}_{d_{2}}}{X}\big{)}\big{|}\sigma\,\hbox{
for all }x,\sigma,$ (4.17)
i.e., the posterior predictive and frequentist distributions of $R$ match and
are furthermore independent of the observed value of $x$ and the parameter
value of $\sigma$.
Proof. Identity (4.17) follows from the first part of the lemma. A direct
evaluation yields (4.16) as the density of $R|\sigma$. Finally, the posterior
density of $\sigma$ is given by
$\pi(\sigma|x)\,=\,\frac{x}{\sigma^{2}}\,p_{1}(\frac{x}{\sigma})$, which leads
to the predictive density
$\displaystyle\hat{q}_{\pi_{0}}(y^{\prime}|x)\,$ $\displaystyle=$
$\displaystyle\,\int_{0}^{\infty}\frac{1}{\sigma^{d_{2}}}\,q_{1}(\frac{y_{1}^{\prime}}{\sigma},\ldots,\frac{y_{d_{2}}^{\prime}}{\sigma})\,\frac{x}{\sigma^{2}}\,p_{1}(\frac{x}{\sigma})\,d\sigma$
$\displaystyle=$
$\displaystyle\frac{1}{x^{2}}\int_{0}^{\infty}u^{d_{2}}q_{1}\big{(}\frac{y_{1}^{\prime}u}{x},\ldots,\frac{y_{d_{2}}^{\prime}u}{x}\big{)}\,p(u)\,du$
$\displaystyle=$
$\displaystyle\frac{1}{x^{d_{2}}}\,h(\frac{y^{\prime}}{x})\,.\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\qed$
###### Remark 4.7.
Identity (4.17) is quite general and clarifies why Bayesian analysis with
respect to the prior density $\pi_{0}$ matches pivotal based analysis that
stems from the right-hand side. In the univariate case and for the prediction
of a single future order statistic, as remarked upon by Dunsmore (1974), the
identity explains leads to his Bayesian solutions matching the pivotal based
solution of Lawless (1971).
###### Theorem 4.4.
Consider model (4.14) and a Bayesian prediction region $R(X)$ with credibility
$1-\lambda$ associated with the prior density
$\pi_{0}(\theta)=\frac{1}{\theta}\,\mathbb{I}_{(0,\infty)}(\theta)$. Then,
$R(X)$ has exact frequentist coverage probability, i.e., $\mathbb{P}(R(X)\ni
Y^{\prime}|\theta)\,=\,1-\lambda$ for all $\theta>0$.
Proof. Since $R(X)$ has credibility $1-\lambda$, we have
$\mathbb{P}\big{(}Y^{\prime}\in
R(x))|x\big{)}\,=\,\mathbb{P}\big{(}\frac{Y^{\prime}}{x}\in
R^{*}(x)\big{|}x\big{)}\,=\,1-\lambda,$
where $R^{*}(x)\,=\,\\{r\in\mathbb{R}_{+}^{d_{2}}:\frac{r}{x}\in R(x)\\}$.
Now, since the predictive distribution of $\frac{Y^{\prime}}{x}$ is free of
$x$ with p.d.f. $h$ (Lemma 4.4), it follows that $R^{*}(x)$ is free of $x$
such that $\int_{R^{*}(x)}h(z)\,dz\,=\,1-\lambda$. On the other hand, the
frequentist coverage of $R(X)$ is given by
$\mathbb{P}(Y^{\prime}\in
R(X)|\,\theta\,)\,=\,\mathbb{P}\big{(}\frac{Y^{\prime}}{X}\in
R^{*}(X)\big{|}\theta\big{)}\,=\,\int_{R^{*}(x)}h(r)\,dr\,=\,1-\lambda,$
since $h$ is also the density of $\frac{Y^{\prime}}{X}|\theta$ (Lemma 4.4). ∎
To conclude, the above (with $Y^{\prime}=Y$ or $Y^{\prime}=Z$) applies to our
set-ups as follows.
###### Corollary 4.4.
Based on $X$ as in (2.1) and the non-informative prior density
$\pi_{0}(\theta)=\frac{1}{\theta}\,\mathbb{I}_{(0,\infty)}(\theta)$, Bayesian
predictive regions for $Z$ or $Y$ with credibility $1-\lambda$, based on the
predictive densities given in Theorems 3.1 and 3.2, have matching frequentist
coverage probability $1-\lambda$ for all $\theta>0$.
## 5 Best invariance and minimaxity
As seen in Section 3, Bayesian predictive densities such as those given in
Theorem 3.2 facilitate the construction of a prediction region for future
values of $Y$ or $Z$ with a given credibility. Non-Bayesian predictive
densities are also available, such as plug-in densities and those obtained by
likelihood or pivotal-based methods. Kullback-Leibler (KL) divergence loss and
accompanying risk can be used to evaluate the frequentist performance of
density estimators $\hat{q}(\cdot;X)$. For our problem, these are given by
$L_{KL}(\theta,\hat{q}(\cdot;x))\,=\,\int
q_{\theta}(t)\,\log\big{(}\frac{q_{\theta}(t)}{\hat{q}(t;x)}\big{)}\,dt\,$
(assuming Lebesgue densities), and
$R_{KL}(\theta,\hat{q})\,=\,\mathbb{E}_{\theta}\big{\\{}L_{KL}(\theta,\hat{q}(\cdot;X))\big{\\}}.$
It is of interest to assess the efficiency of predictive densities for KL
risk, and decision-theoretic properties of invariance and minimaxity
applicable to our contexts are reviewed in this section. An early contribution
to the determination of a best invariant density is due to Murray (1977),
while a more exhaustive treatment of best invariant densities, as well as
minimaxity in predictive density estimation, appears in Liang & Barron
(2004).
A density $\hat{q}_{m}$ is minimax whenever $\hat{q}_{m}$ minimizes among all
densities the supremum frequentist risk, i.e., in our cases when
$\sup_{\theta>0}R_{KL}(\theta,\hat{q}_{m})\,=\,\inf_{\hat{q}}\sup_{\theta>0}R_{KL}(\theta,\hat{q})\,.$
For our problems, and more generally model (4.14) with $\sigma=1/\theta$,
Kullback-Leibler divergence loss, a predictive density $\hat{q}$ is invariant
under changes of scale whenever it satisfies the scale parameter family
property
$\hat{q}(y^{\prime};cx)\,=\,\frac{1}{c^{d_{2}}}\,\hat{q}(\frac{y^{\prime}}{c};x)\,,y^{\prime}\in\mathbb{R}_{+}^{d_{2}}\,,$
(5.18)
for all $c,x>0$. The class of invariant densities here includes
$\hat{q}_{\pi_{0}}$; as can be verified directly from the expressions given in
Theorems 3.1 and 3.2 with $\alpha=\beta=0$, or even by (4.15); as well as
plug-in densities $q_{\hat{\theta}}$ with $\hat{\sigma}(x)=kx$, i.e., a scale
invariant point estimator $\hat{\sigma}$ of $\theta$ satisfying
$\hat{\sigma}(cx)\,=\,c\,\hat{\theta}(x)$ for $c,x>0$; such as the maximum
likelihood choice with $c=1/m$.
The present invariance structure implies that an invariant density has
constant risk as a function of $\theta$ as long as it is finite, from which it
follows that there exists an optimal choice among invariant densities. The
risk constancy follows for vastly more general settings (e.g., Berger, 1985),
but can be also derived directly as follows:
$\displaystyle R_{KL}(c\sigma,\hat{q})\,$ $\displaystyle=$
$\displaystyle\int_{(0,\infty)}p_{c\sigma}(x)\,\int_{(0,\infty)^{d_{2}}}q_{c\sigma}(y^{\prime})\,\log\big{(}\frac{q_{c\sigma}(y^{\prime})}{\hat{q}(y;x)}\big{)}\,dy^{\prime}\,dx$
$\displaystyle=$
$\displaystyle\int_{(0,\infty)}\frac{1}{c}\,p_{\sigma}(\frac{x}{c})\,\int_{(0,\infty)^{d_{2}}}q_{\sigma}(\frac{y^{\prime}}{c})\frac{1}{c^{d_{2}}}\,\log\big{(}\,\frac{q_{\sigma}(\frac{y^{\prime}}{c})}{\hat{q}(\frac{y^{\prime}}{c};\frac{x}{c})}\big{)}\,dy^{\prime}\,dx$
$\displaystyle=$
$\displaystyle\int_{(0,\infty)}p_{\sigma}(t)\,\int_{(0,\infty)^{d_{2}}}q_{\sigma}(u)\,\log\big{(}\frac{q_{\sigma}(u)}{\hat{q}(u;t)}\big{)}\,du\,dt$
$\displaystyle=$ $\displaystyle\,R_{KL}(\sigma,\hat{q}),$
for $c,\sigma>0$, with the change of variables
$(t,u)\,=\,\frac{1}{c}(x,y^{\prime})$, and by making use of (5.18).
Under general conditions for problems that are invariant, which are met here,
a best invariant procedure exists and coincides with the generalized Bayes
estimator associated with a (right) invariant prior density (e.g., Berger,
1985, section 6.6.2). In out set-up, such a prior density is the non-
informative $\pi_{0}$, and it leads to the best invariant property of the
density $\hat{q}_{\pi_{0}}$. Furthermore, $\hat{q}_{\pi_{0}}$ is minimax (see
Liang & Barron (2004), Theorem 2 and Proposition 3). We conclude this section
by summarizing the above as applicable (with $Y^{\prime}=Y$ or $Y^{\prime}=Z$)
to the problems at hand.
###### Theorem 5.5.
Consider $X_{i:n},i=1,\ldots,m$, the first $m$ order statistics among $n$ from
i.i.d. Exp$(\theta)$ data, and $X$, $Y$, and $Z$ as in (2.1), (2.2) and (2.3).
Then the best invariant densities for estimating the densities of $Y$ and $Z$,
respectively, under KL divergence loss are $\hat{q}_{\pi_{0}}$, as given in
Theorems 3.1 and 3.2 for $\alpha=\beta=0$. Furthermore, its KL risk is
constant as a function of $\theta$ and $\hat{q}_{\pi_{0}}$ is minimax.
## 6 Concluding remarks
We have illustrated the natural usage of Bayesian credible regions for the
prediction of future order statistics under a type-2 censoring scheme with
exponentially distributed data. We have emphasized multivariate aspects and
provided explicit expressions for Bayesian predictive densities and resulting
prediction credible regions. We have also addressed optimality properties
achieved with the non-informative prior density choice, such as the matching
of Bayesian credibility and frequentist probabilty coverage, as well as the
best invariant and minimax properties of the corresponding Bayesian predictive
density.
It would be interesting to extend the analysis more broadly to other types of
probability models and censoring schemes. We have provided a Bayesian HPD
credible region (Theorem 3.3) for the next $N$ future order statistics.
However, for the general bivariate case, such a solution is lacking. We do not
know for instance if the Bayesian predictive density is unimodal which would
facilitate the determination of such a region.
## Acknowledgements
The authors are thankful to the Université de Sherbrooke for its financial
support awarded as part of its visiting researcher’s program. Éric Marchand’s
research is supported in part by the NSERC of Canada.
## References
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* Berger (1985) Berger, J. (1985). Statistical Decision Theory and Related Topics. Springer Texts in Statistics. Springer-Verlag, New York. Second edition.
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* Lawless (1971) Lawless, J.F. (1971). A prediction problem concerning samples from the exponential distributions, with application in life testing. Technometrics, 13, 725–730.
* Lehmann & Casella (1998) Lehmann, E.L. & Casella, G. (1998). Theory of Point Estimation. Second edition, Springer, New York.
* Liang & Barron (2004) Liang, F. & Barron, A. (2004). Exact minimax strategies for predictive density estimation, data compression, and model selection. IEEE Transactions on Information Theory, 62, 2708–2726.
* L’Moudden et al. (2017) L’Moudden, A., Marchand, É., Kortbi, O. & Strawderman, W.E. (2017). On predictive density estimation for Gamma models with parametric constraints. Journal of Statistical Planning and Inference, 185, 56–68.
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* Murthy et al. (2004) Murthy, D.P., Xie, M. & Jiang, R. (2004). Weibull models. John Wiley & Sons.
|
# A New Look at the YY CrB Binary System
Somayeh Soomandar Independent astrophysics researcher, Kerman, Iran Atila
Poro Astronomy Department of the Raderon Lab., BC., Burnaby, Canada
###### Abstract
This study presented a new analysis for the TESS-observed W Ursae Majoris (W
UMa) binary star YY Coronea Borealis (YY CrB). The light curve was analyzed by
the PHysics Of Eclipsing BinariEs (PHOEBE) Python version together with the
Markov chain Monte Carlo (MCMC) method. The light curve solutions required a
hot spot and $l_{3}$. New eclipse times from the TESS observations were
extracted, and the O-C curve of primary and secondary minima showed an anti-
correlated manner. In order to study the O-C curve of minima, minima times
between 1991 and 2023 were collected. This investigation reported a new linear
ephemeris and by fitting a quadratic function to the O-C curve of minima,
calculated the orbital period rate of $\mathop{P}\limits^{.}\approx
5.786\times{10^{-8}}\frac{{day}}{{year}}$. Assuming mass conservation, a mass
exchange rate of $\mathop{{M_{2}}}\limits^{.}=2.472\times{10^{-8}}$ calculated
from the more massive component to the less massive one. Then, by using the
light travel time function, the possible third body was determined in the
binary and derived the mass of the third body as $0.498{M_{\odot}}$ with a
period of $\simeq 7351.018$ days. The O-C curve analysis and the quantity of
mass indicate that the presence of a third body is unlikely. This binary is
expected to evolve into a broken-contact phase and is a good case to support
the thermal relaxation oscillation model.
binaries: eclipsing – method: photometric
## 1 Introduction
W UMa-type systems are recognised by their eclipsing light curves with almost
equal minima and a short orbital period. These stars have spectral types
ranging from A to middle K, and the convective atmosphere is the main reason
for chromosphere activity, as well as starspots, which are signs of the
existence of dynamo-generated magnetic activity.
YY CrB (HIP 77598, TIC 29287800) is a W UMa binary system discovered by
Hipparcos ESA 1997. This system has been studied in some works; first
Rucinski, et al. (2000) revealed the spectral type F8V for two
components.Vaňko et al. (2004) found the light curves to be asymmetric and
mentioned the existence of starspots on the components. Gazeas et al. (2005)
analyzed the light curve and derived the geometric and photometric parameters
and concluded that this target is a contact binary with weak magnetic
activity.
Essam et al. (2010) combined photometric and spectroscopic solutions and
calculated the fill-out factor approximately equal to 64 percent and mass
ratio of 0.241. In addition, they studied the changes in the orbital period
using the O-C diagram and concluded that the orbital period is decreasing. Yu,
Xiang, & Xiao (2015) studied the orbital period changes and implied that the
decreased period rate and concluded that the sinusoidal oscillatory can be
interrupted as magnetic activity. Essam et al. (2010) and Yu, Xiang, & Xiao
(2015) on the rate of period decrease demonstrate that the value of reduced
rate is lowering progressively, indicating that this system was going through
an orbital expansion stage of thermal relaxation oscillation (TRO) cycles.
Also, understanding the evolutionary status of this target could prove
invaluable. Using new space-based data, we have re-analyzed the light curve
solution and studied the O-C curve in detail. Moreover, we studied the
possibility of a third body in this interesting system. The structure of the
paper is as follows: Section 2 provides information on TESS observations and a
data reduction process. The light curve solution and the estimation of
absolute parameters are included in Sections 3 and 4 respectively, the orbital
period variation analysis is presented in Section 5, and finally, Section 6
contains the discussion and conclusion.
## 2 Observation and Data Reduction
YY CrB was observed by the TESS during sectors 24 and 51 (April 16, 2020-May
13, 2020, and April 22, 2022-May 18, 2022) on Cameras 1 and 3. There is two-
minute cadence data for sector 24 that are processed by the Science Processing
Operations Center (SPOC) pipeline (Jenkins et al. (2016); Jenkins (2015)).
Photometric photos were downloaded using the Lightkurve package (Lightkurve
Collaboration et al., 2018) that provides the functions to download TESS data
from the public data archive at MAST111https://mast.stsci.edu. For sector 24,
we used the Pre-search Data Conditioning flux of the Simple Aperture
Photometry (PDCSAP). There is no detrended light curve for sector 51.
Therefore, we download the TESS Full Frame Images (FFIs) from the MAST and
used Lightkurve package to extract the SAP light curve with a mask that are
defined by the pixels shown in the left panel of Figure 1. We used create
threshold mask function to produce an aperture mask using a threshold equal to
10. This function identifies the pixels in the target pixel file and shows
that a median flux that is brighter than the threshold times the standard
deviation above the overall median. The right panel of Figure 1 shows the
phased light curve that was produced.
Figure 1: Left panel: TESS target pixel file of YY CrB in sector 51. The
pixels included in the computation of the SAP as red bordered pixels. Right
panel: phased light curve during sector 51.
## 3 Light curve solution
Essam et al. (2010) calculated the optimal parameters by combining
simultaneous radial velocity and light curve solutions. We started with the
initial values of the parameters taken from the solution by the Essam et al.
(2010) study. One-day data from TESS sector 24 were utilized to light curve
solution. The observation of 2-min cadence help to better analysis of the
effect of spots on the components. Photometric analysis of the YY CrB system
was carried out using the PHOEBE 2.4.9 version, TESS filter of the code, and
the MCMC approach (Prša & Zwitter 2005, Prša et al. 2016, Conroy et al. 2020,
Poro et al. 2022).
We selected the TESS passband from the code and chose the contact binary mode
in the PHOEBE based on the light curve’s shape and solutions of previous
studies.
The initial and input parameters were as follows: The mass ratio $q=0.241$ and
the effective temperature of primary component ${T_{1}}=5819$ are (Essam et
al., 2010), the gravity darkening coefficients, ${g_{1}}={g_{2}}=0.32$ and the
albedo coefficients, ${A_{1}}={A_{2}}=0.5$ (Lucy 1967, Ruciński 1969).
The limb-darkening coefficients were employed as free parameters, and the
Castelli & Kurucz (2004) method was used to model the stellar atmosphere. The
parameters searched in MCMC include: the orbital inclination $i$, the mean
temperature of the stars $T_{1,2}$, the mass ratio $q$, the fillout factor
$f$, the bandpass luminosity of the primary star (${L_{1}}$), and the third
light in total light (${l_{3}}$). We applied 46 walkers and 1000 iterations to
each walker in MCMC. According to the asymmetry in the brightness of maxima in
the light curve of the close eclipsing binary, the solution requires the
assumption of a hot spot on the primary component (O’Connell 1951). According
to observational and theoretical light curves in this study, it has not been
possible to provide the solution without considering $l_{3}$. The theoretical
fit on the observational light curve for the YY CrB system is given in Figure
2. The corner plot that MCMC produced is displayed in Figure 3. Also, the
geometrical structure is plotted in Figure 4, which has a lower temperature at
the point of contact between companion stars due to the gravity darkening
(Prša et al. 2016). The calculated parameters together with the values
obtained by Essam et al. (2010) are listed in Table 1.
Figure 2: Light curve solution of the eclipsing binary YY CrB. Observational light curve (blank circle), synthetic light curve (solid red line), and the residuals (blue circle). Figure 3: The corner plots of the light curve solution. Figure 4: The geometrical structure of YY CrB. Table 1: The parameters of the eclipsing binary YY CrB. Parameter | This study | Essam et al. (2010)
---|---|---
$q=M_{2}/M_{1}$ | $0.2498_{\rm-0.0024}^{+0.0031}$ | $0.241\pm 0.002$
$T_{1}$ (K) | $5621_{\rm-3}^{+3}$ | 5819
$T_{2}$ (K) | $5944_{\rm-8}^{+6}$ | $6010\pm 72$
$i$ (deg) | $81.50_{\rm-0.29}^{+0.36}$ | $80.26\pm 0.05$
${\Omega_{1}=\Omega_{2}}$ | $2.295\pm 0.079$ | 2.237
$l_{1}/l_{tot}$ | $0.730_{\rm-0.001}^{+0.001}$ | $0.7508\pm 0.0154$
$l_{2}/l_{tot}$ | $0.264\pm 0.001$ | 0.2492
$l_{3}/l_{tot}$ | $0.006_{\rm-0.001}^{+0.001}$ |
$f$ | $0.363_{\rm-0.031}^{+0.025}$ | 0.64
${r_{1}}_{mean}$ | $0.522\pm 0.018$ | 0.537
${r_{2}}_{mean}$ | $0.287\pm 0.028$ | 0.282
Phase shift | $0.08\pm 0.005$ |
Spot on the star 1: | |
Colatitude $\theta$ | $99\pm 1$ | 90
Longitude $\lambda$ | $325\pm 1$ | $11.25\pm 1.638$
Angular radii $\gamma$ | $18\pm 1$ | $5.250\pm 0.573$
${T_{star}}/{T_{spot}}$ | $1.04\pm 0.02$ | 0.750
Spot on the star 2: | |
Colatitude $\theta$ | | $99.487\pm 3.919$
Longitude $\lambda$ | | 325
Angular radii $\gamma$ | | $16.300\pm 1.326$
${T_{star}}/{T_{spot}}$ | | 1.351
## 4 Absolute parameters
The absolute parameters of the binary system including ${M_{v1,2}}$,
${M_{bol1,2}}$, $L_{1,2}$, $R_{1,2}$, $M_{1,2}$, $log(g)_{1,2}$, and $a$ were
calculated. We used Gaia DR3 parallaxes and the parameters of the light curve
solution in this study. We followed the same method as done by Poro et al.
(2022). First the absolute magnitude ${M_{v}}$ of the system was calculated by
Equation (1).
${M_{v(system)}}=V_{max}-5\log(d)+5-{A_{v}}$ (1)
where the distance of the system from Gaia DR3 ($d_{pc}=90.07\pm 0.1$) was
derived and $V_{max}=8.64\pm 0.08$ comes from the
VSX222https://www.aavso.org/vsx/ database. Extinction coefficient
${A_{v}}=0.015\pm 0.002$ was obtained using the DUST-MAPS package in Python
(Green et al., 2019). Also, Equation (2) can be utilized to determine the
primary and secondary components’ absolute magnitude.
${M_{v1,2}}-{M_{vtot}}=-2.5\log(\frac{{{l_{1,2}}}}{{{l_{tot}}}})$ (2)
The bolometric magnitude ${M_{bol}}$ of each component of the binary was
obtained by Equation (3),
${M_{bol}}={M_{v}}+BC$ (3)
where the effective temperature of the stars is employed to obtain the
bolometric correction for the primary and secondary components retrieved
$BC_{1}=-0.111$ and $BC_{2}=-0.052$ respectively (Flower, 1996). The
bolometric correction is presented as polynomial fits in Equation (4).
$BC=a+b(\log{T_{eff}})+c{(\log{T_{eff}})^{2}}+d{(\log{T_{eff}})^{3}}+e{(\log{T_{eff}})^{4}}$
(4)
Then, the luminosity of two components is determined from Pogson’s relation
(Pogson, 1856),
${M_{bol}}-{M_{bol\odot}}=-2.5\log(\frac{L}{{{L_{\odot}}}})$ (5)
where ${M_{bol\odot}}$is taken as $4.73^{mag}$ from Torres (2010). The radius
of primary and secondary components is calculated by the equation (6),
$R={(\frac{L}{{4\pi\sigma{T^{4}}}})^{1/2}}$ (6)
where the $\sigma$ is the Stephen-Boltzmann constant and $T$ is the
temperatures of each components.
Additionally, with considering ${r_{mean1,2}}$ and $a=\frac{R}{{{r_{mean}}}}$,
we calculated the separation $a$ in average ${a_{1}}$ and ${a_{2}}$. The
resulting parameters and the values obtained by the Essam et al. (2010) study
are listed in Table LABEL:tab2.
Table 2: The absolute parameters of YY CrB. Absolute parameters | This study | Essam et al. (2010)
---|---|---
${M_{bol1}}(mag)$ | $4.083\pm 0.074$ | 3.939
${M_{bol2}}(mag)$ | $5.246\pm 0.072$ | 5.173
${L_{1}}({L_{\odot}})$ | $1.832\pm 0.121$ | 2.580
${L_{2}}({L_{\odot}})$ | $0.628\pm 0.041$ | 0.668
${R_{1}}({R_{\odot}})$ | $1.430\pm 0.049$ | 1.427
${R_{2}}({R_{\odot}})$ | $0.749\pm 0.026$ | 0.757
$a({R_{\odot}})$ | $2.674\pm 0.080$ | 2.64
$M_{1}({M_{\odot}})$ | $1.448\pm 0.131$ | $1.467$
$M_{2}({M_{\odot}})$ | $0.362\pm 0.037$ | $0.357$
$log(g)_{1}(cgs)$ | $4.288\pm 0.008$ | 4.295
$log(g)_{2}(cgs)$ | $4.248\pm 0.012$ | 4.232
## 5 The Orbital Period Changes
To calculate the eclipse times of minima, we used the same method as done by
Soomandar & Abedi (2020). First, split the detrended light curves for
individual eclipses and fit a Lorentzian function to each eclipse by using the
least-squares method. We used the Scipy.curve-fit package in Python to fit the
Lorentzian function to the individual eclipses. We used the
np.sqrt(np.diag(cov)) function to calculate the standard deviation errors on
the parameters. TESS observations yielded a total of 220 primary and secondary
minima, as displayed in the table 3. The new observational eclipse times were
calculated. Then, we performed an analysis of observed (O) minus calculated
(C) eclipse times (Sterken, 2005). We calculated the O-C curve using the
following linear ephemeris (Kreiner 2004, Yu, Xiang, & Xiao 2015):
$Min.I=2452500.1757+0.3765545\times E$ (7)
The O-C curve of primary and secondary minima for sectors 24 and 51 are
plotted in Figure 5. The anti-correlated manner between primary and secondary
minima is obvious which is a confirmation of the presence of spots on the
contact binary components (Tran et al. (2013); Balaji et al. (2015)). We
averaged primary and secondary minima to eliminate the anti-correlated impact
when analyzing orbital period changes (Balaji et al., 2015). The calculated
values are shown in Table 3. 109 YY CrB observational minima times were
recorded in the literature over a 31-year period. The appendix contains a list
of the data gathered with uncertainty. Observational minima times converted to
BJD-TDB333https://astroutils.astronomy.osu.edu/time/hjd2bjd.html. Figure 6’s
left panel depicts the O-C curve of the minima. We presented a new ephemeris
for this target as Equation (8) by fitting a linear function on the O-C curve
of primary.
$Min.I=2458955.8598(\pm 1.1e-4)+0.3765581(\pm 1.1e-7)\times E$ (8)
Figure 5: Left panel: primary and secondary O-C curve of minima for sector 24.
Right panel: primary and secondary minima for sector 51 (primary O-C curve in
black circles and secondary O-C curve in red circles.).
The O-C curve of minima is calculated with the new ephemeris and the resulting
curve shows the same shape as the left panel of Figure 6. we fitted a
quadratic function to the O-C curve in order to investigate the variations in
the orbital period:
${T_{mid}}(E)={T_{0}}+PE+\frac{1}{2}\frac{{dP}}{{dt}}{E^{2}}$ (9)
where mid-eclipse times ${T_{mid}}$ are described by ${T_{0}}$ is the
reference mid-eclipse time, P is the orbital period and, E is the epoch of
eclipses (Patra et al., 2017). And the quadratic fit showed a drop in period,
which corresponded to the quadratic plot in Figure 6’s left panel. We
determined the rate of period decline using the model’s quadratic coefficient
as Equation (10)
$\mathop{P}\limits^{.}=\frac{{2\times-1.124\times{{10}^{-11}}}}{{0.3765545}}=-5.786\times{10^{-8}}\pm
9.965\times{10^{-9}}\frac{{day}}{{year}}$ (10)
Considering ${M_{1}}=1.448{M_{\odot}}$ for the primary and
${M_{2}}=0.362{M_{\odot}}$ for the secondary one calculated in this study and
using Equation (11) and mass conservation, the rate of mass exchange between
primary and secondary components was estimated.
$\frac{{\mathop{P}\limits^{.}}}{P}=-3\frac{{{{\mathop{M}\limits^{.}}_{2}}({M_{1}}-{M_{2}})}}{{{M_{1}}{M_{2}}}}\Rightarrow\mathop{{M_{2}}}\limits^{.}=+2.472\times{10^{-8}}\pm
0.190\times{10^{-8}}{M_{\odot}}y{r^{-1}}$ (11)
The positive sign indicates the direction of mass transfer from the more
massive to the less massive component. The cyclic changes are shown by the
residuals of the quadratic fit. As a result, we investigated the Light Travel
Time Effect (LTTE) as a possible cause of the O-C curve variations. The
following periodogram analysis was performed with the Period 04 software (Lenz
& Breger, 2005) for the residuals of a quadratic fit. The peak of frequencies
in the periodogram analysis of residuals shows a period of 7351.018 days.
Then, we used the least square method to fit the Light Travel Time (LTT)
formula on the O-C curve (Irwin, 1952):
${(O-C)_{LTT}}=A\times\left({\frac{{1-{e^{2}}}}{{1+e\cos\upsilon}}\sin(\upsilon+\omega)+e\sin(\omega)}\right)$
(12)
where $A=\frac{{{a_{12}}\sin i}}{c}$, and ${a_{12}}$ is the semi-major axis of
the relative orbit of the eclipsing system around the center of mass (in Au
unit), $i$ is the inclination of the third-body orbit, $e$ is the eccentricity
of the supposed third body, $\omega$ is the longitude of periastron passage in
the plane of the orbit and, $\upsilon$ is the true anomaly. To fit the LTT
function to the residuals of the O-C curve, we have to convert the epoch to
the true anomaly and Kepler’s formula provides the link between the eccentric
anomaly and the observed eclipse time:
${E_{3}}-e\sin E{}_{3}=\frac{{2\pi}}{{{P_{3}}}}(t-{T_{0}})$ (13)
The equation (13) was calculated using Newton-Raphson’s method for every
eclipse time of minima and considering equation (14) the epochs converted to
the true anomaly.
$\begin{array}[]{l}\tan\frac{\nu}{2}={(\frac{{1+e}}{{1-e}})^{1/2}}\tan\frac{{{E_{3}}}}{2}\\\
t={T_{0}}+epoch\times{P_{binary}}\end{array}$ (14)
where ${P_{b}inary}$, t, ${P_{3}}$, ${E_{3}}$, and ${T_{0}}$ are the period of
binary, the time of observed minima, the period of the third body, the
eccentric anomaly and, the time of periastron passage respectively. By
assuming a coplanar orbit ($i=90$), we determined the lower limit for the mass
of the third body. The calculated parameters of the third body are listed in
Table 4 and the related curve is plotted in the right panel of Figure 6.
Figure 6: Left Panel: The O-C data points of the minima (black circle) and, the polynomial fit (red line). Right Panel: LTTE on the residuals of the polynomial fit; LTTE (black line), the O-C data points after subtracted polynomial fit (red circles), and the residuals of the LTT effect fit (blue circles). Table 3: Extracted the times of minima from TESS observations. All times of minima have been reduced to 2450000. Additionally, the minimum error is equal to 0.0001. Min. | Epoch | O-C | Min. | Epoch | O-C | Min. | Epoch | O-C | Min. | Epoch | O-C
---|---|---|---|---|---|---|---|---|---|---|---
8955.8597 | 17144 | 0.0337 | 8966.5927 | 17172.5 | 0.0348 | 8978.8326 | 17205 | 0.0368 | 9702.0170 | 19125.5 | 0.0482
8956.0504 | 17144.5 | 0.0360 | 8966.7802 | 17173 | 0.0340 | 8979.0187 | 17205.5 | 0.0345 | 9702.2047 | 19126 | 0.0351
8956.2360 | 17145 | 0.0336 | 8966.9693 | 17173.5 | 0.0348 | 8979.2092 | 17206 | 0.0368 | 9702.2047 | 19126 | 0.0476
8956.6126 | 17146 | 0.0334 | 8967.1567 | 17174 | 0.0340 | 8979.3952 | 17206.5 | 0.0344 | 9702.3936 | 19126.5 | 0.0482
8956.8034 | 17146.5 | 0.0359 | 8967.3459 | 17174.5 | 0.0349 | 8979.5859 | 17207 | 0.0369 | 9702.5813 | 19127 | 0.0476
8956.9892 | 17147 | 0.0334 | 8967.5334 | 17175 | 0.0341 | 8979.7718 | 17207.5 | 0.0345 | 9702.7702 | 19127.5 | 0.0482
8957.1799 | 17147.5 | 0.0359 | 8967.7225 | 17175.5 | 0.0349 | 8979.9626 | 17208 | 0.0370 | 9702.9578 | 19128 | 0.0475
8957.3657 | 17151 | 0.0002 | 8967.9099 | 17176 | 0.0340 | 8980.1483 | 17208.5 | 0.0344 | 9703.1468 | 19128.5 | 0.0482
8957.3657 | 17148 | 0.0334 | 8968.0990 | 17176.5 | 0.0349 | 8980.3392 | 17209 | 0.0371 | 9703.3343 | 19129 | 0.0476
8957.5565 | 17148.5 | 0.0359 | 8969.4165 | 17180 | 0.0344 | 8980.5249 | 17209.5 | 0.0344 | 9703.5233 | 19129.5 | 0.0482
8957.7421 | 17149 | 0.0333 | 8969.6050 | 17180.5 | 0.0348 | 8980.9014 | 17210.5 | 0.0345 | 9703.7111 | 19130 | 0.0477
8957.9331 | 17149.5 | 0.0359 | 8969.7931 | 17181 | 0.0345 | 8981.2781 | 17211.5 | 0.0346 | 9703.8998 | 19130.5 | 0.0482
8958.1187 | 17150 | 0.0333 | 8969.9817 | 17181.5 | 0.0348 | 8981.4691 | 17212 | 0.0373 | 9704.0876 | 19131 | 0.0478
8958.3096 | 17150.5 | 0.0359 | 8970.1697 | 17182 | 0.0346 | 9692.9793 | 19101.5 | 0.0478 | 9704.2763 | 19131.5 | 0.0481
8958.4952 | 17151 | 0.0332 | 8970.3582 | 17182.5 | 0.0348 | 9693.1681 | 19102 | 0.0482 | 9704.4642 | 19132 | 0.0478
8958.6861 | 17151.5 | 0.0358 | 8970.5464 | 17183 | 0.0347 | 9693.3556 | 19102.5 | 0.0475 | 9704.6531 | 19132.5 | 0.0484
8958.8717 | 17152 | 0.0332 | 8970.7348 | 17183.5 | 0.0348 | 9693.5445 | 19103 | 0.0481 | 9705.9703 | 19136 | 0.0476
8959.0624 | 17152.5 | 0.0356 | 8970.9231 | 17184 | 0.0348 | 9693.7323 | 19103.5 | 0.0476 | 9706.1590 | 19136.5 | 0.048
8959.2482 | 17153 | 0.0332 | 8971.1114 | 17184.5 | 0.0349 | 9693.9211 | 19104 | 0.0482 | 9706.3469 | 19137 | 0.0478
8959.4390 | 17153.5 | 0.0356 | 8971.2997 | 17185 | 0.0349 | 9694.1092 | 19104.5 | 0.0479 | 9706.5354 | 19137.5 | 0.0479
8959.6249 | 17154 | 0.0333 | 8971.4880 | 17185.5 | 0.0349 | 9694.2977 | 19105 | 0.0482 | 9706.7234 | 19138 | 0.0477
8959.8155 | 17154.5 | 0.0355 | 8971.6763 | 17186 | 0.0349 | 9694.4855 | 19105.5 | 0.0478 | 9706.9121 | 19138.5 | 0.0481
8960.0014 | 17155 | 0.0332 | 8971.8645 | 17186.5 | 0.0348 | 9695.2398 | 19107.5 | 0.0489 | 9707.1000 | 19139 | 0.0477
8960.1919 | 17155.5 | 0.0354 | 8972.0531 | 17187 | 0.0351 | 9695.4272 | 19108 | 0.0481 | 9707.2889 | 19139.5 | 0.0483
8960.3778 | 17156 | 0.0331 | 8972.2411 | 17187.5 | 0.0348 | 9695.6154 | 19108.5 | 0.048 | 9707.4766 | 19140 | 0.0477
8960.5685 | 17156.5 | 0.0354 | 8972.4297 | 17188 | 0.0352 | 9695.8037 | 19109 | 0.0480 | 9708.2306 | 19142 | 0.0486
8960.7544 | 17157 | 0.0331 | 8972.6176 | 17188.5 | 0.0348 | 9695.9922 | 19109.5 | 0.0483 | 9709.5480 | 19145.5 | 0.0481
8960.9450 | 17157.5 | 0.0355 | 8972.8063 | 17189 | 0.0353 | 9696.1804 | 19110 | 0.0482 | 9710.4893 | 19148 | 0.048
8961.1310 | 17158 | 0.0331 | 8972.9941 | 17189.5 | 0.0348 | 9696.3688 | 19110.5 | 0.0483 | 9710.6776 | 19148.5 | 0.0480
8961.3215 | 17158.5 | 0.0354 | 8973.1830 | 17190 | 0.0354 | 9696.5567 | 19111 | 0.0479 | 9710.8660 | 19149 | 0.0481
8961.5077 | 17159 | 0.0333 | 8973.3707 | 17190.5 | 0.0348 | 9697.3101 | 19113 | 0.0482 | 9711.0543 | 19149.5 | 0.0481
8961.6980 | 17159.5 | 0.0354 | 8973.5597 | 17191 | 0.0355 | 9697.4983 | 19113.5 | 0.0481 | 9711.2425 | 19150 | 0.0481
8961.8842 | 17160 | 0.0332 | 8973.7471 | 17191.5 | 0.0347 | 9697.6865 | 19114 | 0.0480 | 9711.4307 | 19150.5 | 0.0481
8962.0746 | 17160.5 | 0.0354 | 8973.9364 | 17192 | 0.0357 | 9697.8749 | 19114.5 | 0.0482 | 9711.6190 | 19151 | 0.048
8962.2608 | 17161 | 0.0331 | 8974.1236 | 17192.5 | 0.0346 | 9697.8749 | 19114.5 | 0.0482 | 9711.8073 | 19151.5 | 0.0481
8962.4511 | 17161.5 | 0.0353 | 8974.3130 | 17193 | 0.0375 | 9698.0631 | 19115 | 0.0481 | 9711.9956 | 19152 | 0.0481
8962.6374 | 17162 | 0.0333 | 8974.5001 | 17193.5 | 0.0345 | 9698.2516 | 19115.5 | 0.0483 | 9712.1838 | 19152.5 | 0.048
8962.8276 | 17162.5 | 0.0353 | 8975.0664 | 17195 | 0.0360 | 9698.4395 | 19116 | 0.0479 | 9712.3722 | 19153 | 0.0482
8963.0140 | 17163 | 0.0334 | 8975.2533 | 17195.5 | 0.0347 | 9698.6281 | 19116.5 | 0.0482 | 9712.5602 | 19153.5 | 0.0479
8963.2041 | 17163.5 | 0.0352 | 8975.4430 | 17196 | 0.0360 | 9698.8160 | 19117 | 0.0478 | 9712.7487 | 19154 | 0.0481
8963.3905 | 17164 | 0.0333 | 8975.6298 | 17196.5 | 0.0346 | 9699.0047 | 19117.5 | 0.0483 | 9712.9368 | 19154.5 | 0.0479
8963.5807 | 17164.5 | 0.0351 | 8975.8196 | 17197 | 0.0361 | 9699.1926 | 19118 | 0.0479 | 9713.1252 | 19155 | 0.0481
8963.7673 | 17165 | 0.0335 | 8976.0063 | 17197.5 | 0.0346 | 9699.3812 | 19118.5 | 0.0483 | 9713.5019 | 19156 | 0.0481
8963.9570 | 17165.5 | 0.035 | 8976.1962 | 17198 | 0.0362 | 9699.5691 | 19119 | 0.0479 | 9713.6898 | 19156.5 | 0.0478
8964.1438 | 17166 | 0.0335 | 8976.3828 | 17198.5 | 0.0346 | 9699.7578 | 19119.5 | 0.0482 | 9713.8784 | 19157 | 0.0481
8964.3335 | 17166.5 | 0.035 | 8976.5729 | 17199 | 0.0363 | 9699.9456 | 19120 | 0.0479 | 9714.0664 | 19157.5 | 0.0478
8964.5204 | 17167 | 0.0336 | 8976.7594 | 17199.5 | 0.0345 | 9700.1344 | 19120.5 | 0.0483 | 9714.2550 | 19158 | 0.0482
8964.7101 | 17167.5 | 0.0349 | 8976.9495 | 17200 | 0.0364 | 9700.3221 | 19121 | 0.0478 | 9714.4429 | 19158.5 | 0.0477
8964.8970 | 17168 | 0.0336 | 8977.1360 | 17200.5 | 0.0345 | 9700.3222 | 19122 | 0.004 | 9714.6315 | 19159 | 0.0481
8965.0865 | 17168.5 | 0.0349 | 8977.3260 | 17201 | 0.0009 | 9700.5108 | 19121.5 | 0.0482 | 9714.8195 | 19159.5 | 0.0478
8965.2736 | 17169 | 0.0337 | 8977.5125 | 17201.5 | 0.0345 | 9700.6986 | 19122 | 0.0477 | 9715.0082 | 19160 | 0.0482
8965.4631 | 17169.5 | 0.0349 | 8977.7028 | 17202 | 0.0364 | 9700.8874 | 19122.5 | 0.0482 | 9715.1961 | 19160.5 | 0.0478
8965.6503 | 17170 | 0.0338 | 8977.8891 | 17202.5 | 0.0345 | 9701.0752 | 19123 | 0.0478 | 9715.1961 | 19160.5 | 0.0478
8965.8397 | 17170.5 | 0.0349 | 8978.0794 | 17203 | 0.0366 | 9701.2639 | 19123.5 | 0.0482 | 9715.3847 | 19161 | 0.0482
8966.0269 | 17171 | 0.0338 | 8978.2656 | 17203.5 | 0.0345 | 9701.4517 | 19124 | 0.0478 | 9715.5727 | 19161.5 | 0.0479
8966.2162 | 17171.5 | 0.0349 | 8978.4560 | 17204 | 0.0367 | 9701.6405 | 19124.5 | 0.0481 | 9715.7612 | 19162 | 0.0482
8966.4035 | 17172 | 0.0339 | 8978.6422 | 17204.5 | 0.0345 | 9701.8282 | 19125 | 0.0476 | | |
Table 4: Parameters of the third body. Parameters of third body | | Value
---|---|---
Eccentricity (e) | | $0.689\pm 0.005$
The longitude of periastron passage ($\omega$) | | $57.6\pm 1.8$
Period (days) | | 7351.018
Amplitude (minutes) | | $17.28\pm 1.44$
The time of periastron passage ($T_{0}$) | | 2460201
Projected semi-major axis$\times\sin i$ | | $2.11\pm 0.17$
Mass Function $(MassFunction({f_{m}}))$ | | $0.023\pm 0.006$
${M_{3}}\sin i(i=90,{M_{\odot}})$ | | 0.498
$\sum{{{(O-C)}^{2}}}$ | | 0.001
Assuming the third body is a main-sequence star, this corresponds to the M1V
spectral type with a brightness of $0.041{L_{\odot}}$
444http://www.pas.rochester.edu/~emamajek/EEM_dwarf_UBVIJHK_colors_Teff.txt ,
or $0.016$ of the total luminosity which doesn’t agree with the value $l_{3}$
determined in section 3.
We are not certain that $l_{3}$ produced from the TESS light curve analysis
represents a valid observation of flux from a third body in the system,
despite the computation of the effect of the third body, especially as there
is no exact radial-velocity curve. We estimate that this figure is most likely
due to systematic errors in background flux level readings in TESS photos
and/or an underestimation of photometric aperture contamination by other stars
in the image. So, we investigated the Applegate’s effect as a plausible
explanation for the cyclical fluctuations in the O-C curve. We calculated the
observed relative change of the orbital period throughout one cycle of the
binary using the previously obtained modulation period and the O-C amplitude
computed from the orbit of the third body simulated in the previous section.
$\frac{{\Delta P}}{P}=2\pi\frac{{(O-C)}}{{{P_{\bmod}}}}=1.025\times{10^{-5}}$
(Applegate, 1992). The value of $\frac{{\Delta P}}{P}$suggests that the
Applegate effect can explain the cyclic changes in the O-C curve of minima.
This system shows unequal maxima that is known as the O’Connell effect
(O’Connell, 1951) due to the presence of the hot spots (Wilsey & Beaky, 2009).
So, this difference implies that the hemisphere of a component emits a
different amount of radiation than the other hemisphere. These types of
systems have active chromospheres because of the existence of large spots
(Knote et al., 2022). Starspots can alter the depth of minima and have an
obvious effect on the eclipse light curve (Han, Muirhead, & Swift, 2019).
To investigate the effect of the spots on the light curve over both sectors of
TESS observations, we calculated the difference between the two depths of the
primary and secondary minima. We considered the relative fluxes in phases 0
and 0.5 for every complete individual light curve. The curves that resulted
were displayed in Figure 7. YY CrB has the values of the DepthI - DepthII as
large as about $10\%$ of the variable light amplitude. And this is possible
because of the migration and evolution of spots with time on the surface of
two components that cause cyclic magnetic activity.
Figure 7: Right panel: the difference between the depths of Primary and
Secondary minima over the 24th sector of TESS observations. Left panel: the
difference between the depths of Primary and Secondary minima over the 51st
sector of TESS observations.
## 6 Discussion and conclusion
Based on the estimated mass ratio, fill-out factor and inclination angle, YY
CrB is an over-cantact binary with an increased orbital period. Essam et al.
(2010) calculated the decreased period rate
$1.194\times{10^{-6}}\frac{{day}}{{year}}$. Yu, Xiang, & Xiao (2015)
considered all of the minima time published until 2013 and calculated a
secular period decrease with a rate of
$6.727\times{10^{-7}}\frac{{day}}{{year}}$. In this study, the decreasing
value of period rate $5.786\times{10^{-8}}\frac{{day}}{{year}}$ indicates that
the rate of period changes has been decreased. And, when mass conservation is
taken into account, the mass transfer from the Roche-lobe-filling primary
component to the secondary component is
${\mathop{M}\limits^{.}_{2}}=2.472\times{10^{-8}}{M_{\odot}}y{r^{-1}}$. When
the stated results in Yu, Xiang, & Xiao (2015) are compared to the value of
mass transfer in this study, it is clear that mass transfer has been decreased
and the distance between two components is growing while the value of fill-out
factor is decreasing (compare the values of fill-out factors in Table 1). And
this target may evolve to shallow-contact binary via the thermal relaxation
oscillation (TRO) model (Flannery (1976); Robertson & Eggleton (1977)) and
ultimately reach a broken-contact phase ( Lucy (1976)). The mass ratio of the
components, which is related to the mass transfer, is the crucial parameter in
the evolution of the close binary stars. Table 5 contains a list of contact
binaries with low mass ratios $<0.25$. In order to explain the evolutionary
status of the YY CrB system, we provide the mass-luminosity ($M-L$) diagram
displayed in Figure 8. The Zero-Age Main Sequence (ZAMS) and the Terminal‐Age
Main Sequence (TAMS) are plotted along with the selected contact binaries with
low mass ratios. It is obvious that the more massive primary components are
around the ZAMS line, meaning they are not evolved or little evolved. Also,
the less massive secondary components have evolved away from the main sequence
stars and over-luminosity comparing the stars with the same mass in the main
sequence. In addition, the orbital angular momentum of YY CrB has a value of
$51.585\pm 0.067$. The $logJ_{0}-logM$ diagram shows the position of the
system (Figure 9), and this diagram shows that YY CrB is in a contact binary
systems region.
Table 5: Absolute parameters for low mass ratio contact binaries. System | $q$ | ${M_{1}}({M_{\odot}})$ | ${M_{2}}({M_{\odot}})$ | ${R_{1}}({R_{\odot}})$ | ${R_{2}}({R_{\odot}})$ | ${L_{1}}({L_{\odot}})$ | ${L_{2}}({L_{\odot}})$ | Reference
---|---|---|---|---|---|---|---|---
V429 Cam | 0.206 | 1.36(12) | 0.28(3) | 1.55(3) | 0.78(2) | 3.56(9) | 0.85(2) | Li et al. (2021)
V830 Cep | 0.23 | 0.84(5) | 0.19(1) | 0.91(1) | 0.47(1) | 0.98(1) | 0.29(1) | Li et al. (2021)
FP Boo | 0.096 | 1.614(52) | 0.154(21) | 2.310(25) | 0.774(8) | 11.193(99) | 0.920(13) | Gazeas et al. (2006)
DN Boo | 0.103 | 1.428(39) | 0.148(6) | 1.710(67) | 0.670(110) | 3.750(280) | 0.560(170) | Şenavcı et al. (2008)
FG Hya | 0.112 | 1.444(25) | 0.161(7) | 1.405(9) | 0.591(8) | 2.158(86) | 0.412(17) | Qian & Yang (2005)
CK Boo | 0.0108 | 1.442(14) | 0.154(2) | 1.453(3) | 0.577(10) | 2.74(1) | 0.47(2) | Kalci & Derman (2005)
GR Vir | 0.122 | 1.37(16) | 0.17(6) | 1.42(7) | 0.61(4) | 2.87(28) | 0.48(6) | Qian & Yang (2004)
CSS J234807.2$+$193717 | 0.176 | 1.19(4) | 0.21(3) | 1.36 (2) | 0.66 (1) | 1.45(24) | 0.42(5) | Christopoulou et al. (2022)
J170307 | 0.092 | 1.134(253) | 0.105(24) | 1.204(120) | 0.436(48) | 1.874(572) | 0.271(72) | Liu et al. (2023)
J1641000 | 0.095 | 1.402(287) | 0.133(28) | 1.580(144) | 0.577(58) | 3.912(1.109) | 0.512(142) | Liu et al. (2023)
J223837 | 0.093 | 1.541(306) | 0.144(30) | 1.784(159) | 0.646(64) | 5.463(1.534) | 0.704(192) | Liu et al. (2023)
CSS J222607.8$+$062107 | 0.221 | 1.49(3) | 0.33(13) | 1.51(5) | 0.81(2) | 3.35(65) | 1.15(24) | Sun et al. (2020)
CSS J012559.7$+$203404 | 0.231 | 1.38(3) | 0.32(12) | 1.42(4) | 0.77(2) | 2.46(56) | 0.81(18) | Sun et al. (2020)
CSS J153855.6$+$042903 | 0.187 | 1.44(5) | 0.27(12) | 1.37(4) | 0.66(2) | 2.94(96) | 0.30(10) | Sun et al. (2020)
CSS J141923.2$-$013522 | 0.168 | 1.31(5) | 0.22(11) | 1.23(4) | 0.57(2) | 1.97(68) | 0.33(11) | Sun et al. (2020)
CSS J130111.2$-$132012 | 0.108 | 1.38(3) | 0.15(12) | 1.49(5) | 0.61(2) | 2.49(57) | 0.40(9) | Sun et al. (2020)
CSS J165813.7$+$390911 | 0.183 | 1.09(3) | 0.20(9) | 1.05(3) | 0.49(1) | 0.92(24) | 0.24(6) | Sun et al. (2020)
V870 Ara | 0.082 | 1.546(54) | 0.127(37) | 1.64(6) | 0.63(5) | 2.64(17) | 0.42(6) | Poro et al. (2021)
TYC 6995-813-1 | 0.111 | 1.23(1) | 0.135(1) | 1.46(1) | 0.60(1) | 2.293(4) | 0.58(2) | Wadhwa et al. (2021)
NSVS 13602901 | 0.171 | 1.19(2) | 0.203(10) | 1.69(1) | 0.79(1) | 2.05(4) | 0.58(2) | Wadhwa et al. (2021)
NSVS 5029961 | 0.151 | 1.872(468) | 0.284(72) | 1.573(119) | 0.680(53) | 3.403(14) | 0.610(26) | Zheng et al. (2021)
CSS J022914.4$+$044340 | 0.201 | 1.44(25) | 0.29(5) | 1.26(8) | 0.65(4) | 1.718(191) | 0.416(50) | Liu & Li (2021)
HV Aqr | 0.15 | 1.240(28) | 0.186(17) | 1.456(12) | 0.601(5) | 3.326(213) | 0.638(44) | Gazeas et al. (2021)
ZZ PsA | 0.078 | 1.213(8) | 0.095(1) | 1.422(4) | 0.559(4) | 2.20(4) | 0.63(4) | Wadhwa et al. (2021)
NSVS 1926064 | 0.160 | 1.558(38) | 0.249(6) | 1.605(13) | 0.755(42) | 3.91(28) | 0.641(33) | Kjurkchieva et al. (2020)
Figure 8: $M-L$ diagram of selected contact binaries with low mass ratio. The
primary and secondary components of YY CrB are plotted in red and blue colors,
respectively. Figure 9: The location of YY CrB on the $logJ_{0}-logM$ diagram.
The quadratic line is based on a study by Eker et al. (2006).
According to the study Yu, Xiang, & Xiao (2015) and the periodic changes in
the residuals of the quadratic fit on the O-C curve, the potential of
excitability of the third body was investigated. The light-time function
fitting revealed the existence of a body with the value of $0.498{M_{\odot}}$.
This number equates to 0.016 of total brightness, which differs from the value
of $l_{3}$ determined in section 3. We explored the Applegate effect as a
possible explanation for the fluctuation in the O-C curve because there is no
exact radial-velocity curve.
YY CrB is a contact binary with a mass ratio less than $0.3$, so considering
Hut’s criteria (Hut, 1980) to investigate the stability is necessary. We used
Equation (15) to calculate the ratio of the spin angular momentum to the
orbital angular momentum (Yang & Qian, 2015).
$\frac{{{J_{s}}}}{{{J_{o}}}}=\frac{{q+1}}{q}[{({k_{1}}{r_{1}})^{2}}+{({k_{2}}{r_{2}})^{2}}q]$
(15)
where $r_{1}$ and $r_{2}$ are the relative radii for the primary and secondary
components and ${k^{2}}_{1,2}=0.06$ (Li & Zhang, 2006), are the dimensionless
gyration radii. The calculated value of $\frac{{{J_{s}}}}{{{J_{o}}}}=0.087$,
which is less than the threshold value therefore this system is stable. This
target shows a period increase which is attributed to mass transfer. According
to the existing data and the analyses done in this study, the existence of a
third body is unlikely for this system, and detailed spectroscopic and
photometric observations over a longer length of time are required for the
definitive determination.
## Acknowledgements
This manuscript has made use of data from the TESS mission. Funding for the
TESS mission is provided by the NASA Science Mission Directorate. This
research has made use of the SIMBAD and VIZIER databases, operated at CDS,
Strasbourg, France. The time of minima data from the Variable Star Observers
League in Japan (VSOLJ) websites proved invaluable to the assessment of
potential period changes experienced by this variable star. The authors would
like to thank Marco Brentel for his help. We are grateful to Ehsan Paki from
the BSN project (https://bsnp.info/) for providing Figure 4 of this
manuscript, which also shows the color-temperature scale.
## ORCID iDs
Somayeh Soomandar: 0000-0002-9520-9573
Atila Poro: 0000-0002-0196-9732
## Appendix A Available Minima Times
The appendix table displays the minima times along with their error in the
first column, the epochs in the third column, the O-C values in the fourth
column, and the references in the final column.
Table 1: Available mid-eclipse times of YY CrB system. ons. All minimum have been reduced to 2450000 Min.($BJD_{TDB}$) | Epoch | O-C | Reference | Min.($BJD_{TDB}$) | Epoch | O-C | Reference
---|---|---|---|---|---|---|---
955.8695(12) | -4101 | -0.0562 | Pribulla et al. (2001) | 4308.3899(2) | 4802 | -0.0005 | Parimucha et al. (2009)
955.8718(6) | -4101 | -0.0539 | Rucinski, et al. (2000) | 4564.2595 | 5481.5 | 0.0003 | Nagai (2009)
1318.4993 | -3138 | -0.04847 | Essam et al. (2010) | 4604.36215(1) | 5588 | -0.0001 | Yilmaz et al. (2009)
1318.5001 | -3138 | -0.0475 | Essam et al. (2010) | 4605.4917(2) | 5591 | -0.0002 | Yilmaz et al. (2009)
1361.4275(3) | -3024 | -0.0475 | Keskin et al. (2000) | 4628.4657(2) | 5652 | 0.0039 | Parimucha et al. (2009)
1361.4275(2) | -3024 | -0.0474 | Keskin et al. (2000) | 4632.4144(3) | 5662.5 | -0.0012 | Parimucha et al. (2009)
1368.3965(4) | -3005.5 | -0.0447 | Keskin et al. (2000) | 4648.4191(30) | 5706 | 0.0000 | Parimucha et al. (2009)
1368.3966(4) | -3005.5 | -0.0446 | Keskin et al. (2000) | 4648.4201(10) | 5705 | 0.0009 | Hubscher et al. (2009)
1370.4659(6) | -3000 | -0.0463 | Keskin et al. (2000) | 4688.3352(3) | 5811 | 0.0013 | Yilmaz et al. (2009)
1372.3494(3) | -2995 | -0.0457 | Keskin et al. (2000) | 4931.4009(1) | 6456.5 | 0.0011 | Hubscher et al. (2011)
1668.3359 | -2209 | -0.0309 | Essam et al. (2010) | 4931.5883(1) | 6457 | 0.0001 | Hubscher et al. (2011)
1669.4602 | -2206 | -0.0363 | Essam et al. (2010) | 4958.5136(20) | 6528.5 | 0.0018 | Hubscher et al. (2011)
1670.3976 | -2203.5 | -0.0403 | Essam et al. (2010) | 4983.7401(3) | 6595.5 | -0.0009 | Diethelm (2009)
1670.3984 | -2203.5 | -0.0395 | Essam et al. (2010) | 5017.44086(3) | 6685 | -0.0017 | Parimucha et al. (2009)
1674.3548 | -2193 | -0.0369 | Karska & Maciejewski (2003) | 5213.6297(3) | 7206 | 0.0022 | Parimucha et al. (2011)
1692.4299 | -2144 | -0.0365 | Essam et al. (2010) | 5219.6532(1) | 7222 | 0.0009 | Parimucha et al. (2011)
1692.4319 | -2145 | -0.0345 | Essam et al. (2010) | 5261.8279(1) | 7334 | 0.0014 | Dvorak (2011)
1975.6050 | -1392 | -0.0304 | Pribulla, et al. (2003) | 5264.4630(3) | 7341 | 0.0006 | Parimucha et al. (2011)
1975.6061(1) | -1392 | -0.0293 | Pribulla et al. (2001) | 5311.3444(2) | 7465.5 | 0.001 | Parimucha et al. (2011)
1975.6064(1) | -1393 | -0.0290 | Pribulla et al. (2001) | 5311.5317(2) | 7466 | 0.000 | Parimucha et al. (2011)
1975.6108(7) | -1392 | -0.0246 | Pribulla & Vanko (2002) | 5351.4463(1) | 7572 | -0.0002 | Hubscher, et al. (2012)
2029.4398(7) | -1250 | -0.0426 | Pribulla & Vanko (2002) | 5354.4587(2) | 7580 | -0.0002 | Parimucha et al. (2011)
2031.5168(7) | -1244.5 | -0.0369 | Pribulla & Vanko (2002)) | 5420.3573(3) | 7755 | 0.0137 | Parimucha et al. (2011))
2045.4589 | -1207.5 | -0.0273 | Essam et al. (2010) | 5652.8828(30) | 8372.5 | 0.0045 | Diethelm (2011)
2060.3320 | -1167 | -0.0281 | Essam et al. (2010) | 5665.4931(3) | 8406 | 0.0088 | Parimucha et al. (2013)
2060.3352 | -1167 | -0.0249 | Essam et al. (2010) | 5705.4093(23) | 8512 | 0.0016 | Hubscher, et al. (2012)
2400.1804(2) | -265.5 | -0.0202 | Pribulla et al. (2002) | 6011.5475(2) | 9325 | 0.0051 | Parimucha et al. (2013)
2400.3660 | -264 | -0.0227 | Karska & Maciejewski (2003) | 55987.6371(2) | 9261.5 | 0.0018 | Parimucha et al. (2013)
2469.4699(4) | -81.5 | -0.017 | Demircan et al. (2003) | 5987.6371(2) | 9261.5 | 0.0018 | Parimucha et al. (2013)
2472.2898(2) | -74 | -0.0209 | Petropoulou et al. (2015) | 5992.5319(2) | 9274.5 | 0.0014 | Parimucha et al. (2013)
2473.4197(2) | -71 | -0.0206 | Petropoulou et al. (2015) | 6005.5237(2) | 9309 | 0.0021 | Parimucha et al. (2013)
2473.4247(4) | -71 | -0.0156 | Demircan et al. (2003) | 6005.5237(2) | 9309 | 0.0021 | Parimucha et al. (2013)
2500.1757 | 0 | 0.000 | Kreiner (2004) | 6149.3679(2) | 9691 | 0.0026 | Parimucha et al. (2013)
2719.3200 | 582 | -0.0104 | Nagai (2004) | 2456199.26169(2) | 9823.5 | 0.0028 | Parimucha et al. (2013)
2764.5082(23) | 702 | -0.0088 | Hubscher et al. (2005) | 6742.4439(14) | 11266 | 0.00513 | Hubscher & Lehmann (2015)
2786.3500(2) | 761 | -0.0071 | Ak & Filiz (2003) | 6749.4115(5) | 11284.5 | 0.0066 | Hubscher & Lehmann (2015)
2793.5038(2) | 779 | -0.0079 | Ak & Filiz (2003) | 6011.5483(2) | 9325 | 0.0018 | Parimucha et al. (2013)
7074.9466(1) | 12149 | 0.010 | Nelson (2016) | 6754.4970(35) | 11298 | 0.0086 | Hubscher & Lehmann (2015)
2793.5042(21) | 779 | -0.0075 | Hubscher et al. (2005) | 7074.9466(1) | 12149 | 0.010 | Nelson (2016)
2814.4011(1) | 834.5 | -0.0093 | Selam et al. (2003) | 7084.1734 | 12173.5 | 0.0114 | Nagai (2016)
3151.0470 | 1728.5 | -0.0032 | Nagai (2005) | 7123.5238(28) | 12278 | -0.011 | Hubscher (2017))
3458.8835(2) | 2546 | 0.000 | Dvorak (2006) | 6749.6002(8) | 11285 | 0.0068 | Hubscher & Lehmann (2015)
3466.41483(1) | 2566 | 0.000 | Pribulla et al. (2005) | 7513.0723 | 13312.5 | 0.0147 | Nagai (2017)
4201.4509(16) | 4518 | 0.0020 | Hubscher (2007) | 7489.5378(18) | 13250 | 0.015 | Hubscher (2017)
4201.6350(17) | 4518.5 | -0.0022 | Hubscher (2007) | 9024.3943 | 17326 | 0.0353 | Nagai (2021)
4245.5055(9) | 4635 | -0.0003 | Brát, et al. (2007) | 9037.3884 | 17360.5 | 0.0382 | Nagai (2021)
4224.4161(2) | 4579 | -0.0028 | Parimucha et al. (2009) | 9269.5368(10) | 17977 | 0.0408 | Paschke (2021)
4300.4828(1) | 4781 | 0.0000 | Parimucha et al. (2009) | 9328.4664(7) | 18133.5 | 0.0398 | Lienhard (2022)
4500.6209(3) | 5312.5 | -0.0006 | Parimucha et al. (2009) | 9605.99382(1) | 18870.5 | 0.0464 | Nelson & Alton (2022)
4504.5751(8) | 5323 | -0.0002 | Parimucha et al. (2009) | 10064.4578(30) | 20088 | 0.0553 | Paschke (2023)
14513.6131(6) | 5347 | 0.0005 | Parimucha et al. (2009) | | | |
14560.1297 | 5470.5 | 0.0126 | Nagai (2009) | | | |
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|
# Effects of strangeness on the chiral pseudo-critical line
Mahammad Sabir Ali<EMAIL_ADDRESS>School of Physical Sciences, National
Institute of Science Education and Research, An OCC of Homi Bhabha National
Institute, Jatni-752050, India Deeptak Biswas<EMAIL_ADDRESS>School of
Physical Sciences, National Institute of Science Education and Research, An
OCC of Homi Bhabha National Institute, Jatni-752050, India Amaresh Jaiswal
<EMAIL_ADDRESS>School of Physical Sciences, National Institute of
Science Education and Research, An OCC of Homi Bhabha National Institute,
Jatni-752050, India Institute of Theoretical Physics, Jagiellonian
University, ul. St. Łojasiewicza 11, 30-348 Krakow, Poland Hiranmaya Mishra
<EMAIL_ADDRESS>School of Physical Sciences, National Institute of
Science Education and Research, An OCC of Homi Bhabha National Institute,
Jatni-752050, India
###### Abstract
Within a 2+1 flavor Nambu–Jona-Lasinio model, we have calculated the curvature
coefficients and checked them against available lattice QCD estimations. With
the observation that the flavor mixing due to the ‘t Hooft determinant term
significantly affects the $\kappa_{2}^{S}$, we explore the effect of $\mu_{S}$
on the $T-\mu_{B}$ cross-over lines. With the novel determination of negative
$\kappa_{2}^{B}$ at large $\mu_{S}$, we advocate the importance of studying
the same in lattice QCD.
## I Introduction
The phase diagram of the strongly interacting matter necessitates the
determination of the chiral transition line in the high-density and high-
temperature regions. The chiral symmetry is broken in the low-density
(temperature) phase of quantum chromodynamics (QCD), which gets restored as
the temperature and/or density increases. At vanishing baryon density, the
restoration of the chiral symmetry is determined to be a cross-over with a
pseudo-critical temperature $T_{pc}=156.5\pm 1.5\text{MeV}$ [1]. On the other
hand, the transition is expected to be a first order at high density, which is
connected to the cross-over line through a critical endpoint (CEP). Although
the determination of the cross-over line for small values of the baryon
chemical potential ($\mu_{B}$) is quite settled with the recent advancements
of lattice QCD (LQCD) [2, 1] calculations, the extension of the line at finite
$\mu_{B}$ suffers from the infamous sign problem which leads to the
oscillatory behaviour of the Monte Carlo sampling method.
For small chemical potential ($\mu_{X}$), the pseudo-critical line can be
Taylor expanded at the lowest order in $\mu_{X}^{2}$, where one defines the
line with the following ansatz, [2, 3, 1],
$\frac{T_{pc}(\mu_{X})}{T_{pc}(0)}=1-\kappa_{2}^{X}\left(\frac{\mu_{X}}{T_{pc}(0)}\right)^{2}-\kappa_{4}^{X}\left(\frac{\mu_{X}}{T_{pc}(0)}\right)^{4}~{}.$
(1)
Here, $\mu_{X}$ corresponds to chemical potential associated with various
charges like baryon charge $B$, electric charge $Q$, and strangeness $S$. Such
parametrization allows for the comparison of results from different models and
lattice QCD calculations within the same baseline. The curvature coefficients
$\kappa_{2}$ and $\kappa_{4}$ have been examined by the Taylor expansion
method on the lattice [4, 5, 1]. Another standard approach relies on
performing the calculations at imaginary chemical potential, followed by
analytic continuation to the real plane [2, 6, 7]. The above-mentioned results
are in good agreement with each other within the respective variances. Similar
studies have been performed within the perturbative QCD [8] as well as in the
ideal and mean-field hadron resonance gas (HRG) model [9, 10], quark-meson
model [11, 12, 13, 14, 15, 16]. Moreover, the Nambu–Jona-Lasinio (NJL) model
has also been employed in this context[17] considering 2 flavors of light
quark.
The effective models, considering the symmetries of the QCD Lagrangian, enable
one to probe the matter at extreme conditions like high temperature and(or)
density, even in the presence of a magnetic field, to understand the phases of
the QCD matter, and provide a bulk description [18, 19]. The NJL model relies
on chiral symmetry and provides a qualitative description of the QCD matter
considering the pseudo-scalar mesons [20, 21]. Despite the analytical
simplicity and the dependence on the parameter sets of such an effective
model, the estimations made with NJL are quite robust [22, 23]. It acts as a
suitable alternative for benchmark estimation at high-density and low-
temperature regions [24], as there is no restriction over the applicability of
this model at finite density.
Over the past few decades, LQCD and NJL have complemented each other while
broadening our understanding of strong interaction in various scenarios. For
example, magnetic catalysis (MC) was first shown within a NJL framework [25,
26]. Two decades later, lattice QCD not only looked at the MC feature [27, 28,
29, 30, 31], but also observed inverse magnetic catalysis (IMC) around the
cross-over temperature [31]. This results in better versions of NJL models
with nonlocal interactions [32] and external agent dependent interaction
strength [33, 34]. Further, in an NJL-like model, the anomalous breaking of
$U(1)_{A}$ symmetry is addressed by explicitly adding the ’t Hooft determinant
interaction (characterized by coupling $G_{d}$), which also represents the
flavor mixing. Recently, Ref. [35, 36] have explored the effect of $G_{d}$ on
isospin-sensitive observables in 2-flavor NJL model and constrained $G_{d}$
using the same from LQCD. In the context of the 3-flavor NJL, $G_{d}$ is the
most ill-constrained parameter with a large allowed range while reproducing
acceptable values of physical observables [22, 23].
In this letter, for the first time in the $2+1$ flavor case with isospin
symmetry, the effect of the $G_{d}$ has been explored by incorporating a
finite strangeness chemical potential. This provides an opportunity to study
the effect of a large $\mu_{S}$ on the pseudo-critical line and provide novel
estimations. Although the large variation of $\mu_{S}$ ($0-200$) MeV is beyond
the scope of the freeze-out lines in heavy-ion collisions owing to strangeness
neutrality, the present investigation is of particular interest for extending
the NJL model at very high density. We have organized the letter as follows,
in Sec. II we describe the model formalism for a $2+1$ flavor NJL model with
the isospin symmetry. We have presented our results in Sec. III and summarized
our findings in Sec. IV.
## II Formalism
The $2+1$-flavor NJL model Lagrangian is given by
${\cal
L}_{\text{NJL}}=\bar{\psi}\left(i\gamma_{\mu}\partial^{\mu}-\hat{m}\right)\psi+{\cal
L}_{\text{S}}+{\cal L}_{\text{D}},$ (2)
where the four- and six-point interaction terms are given by
$\begin{split}{\cal
L}_{\text{S}}&=G_{s}\sum_{a=0}^{8}\left[\left(\bar{\psi}\lambda_{a}\psi\right)^{2}+\left(\bar{\psi}\,i\gamma_{5}\lambda_{a}\psi\right)^{2}\right],\\\
{\cal
L}_{\text{D}}&=G_{d}\left[\det\bar{\psi}_{i}(1-\gamma_{5})\psi_{j}+\det\bar{\psi}_{i}(1+\gamma_{5})\psi_{j}\right].\end{split}$
(3)
Here, $\psi^{\text{T}}=(u,d,s)$ is the quark triplet in flavor space with up,
down and strange quark; and $\hat{m}=\text{diag}(m_{u},m_{d},m_{s})$ is the
current quark mass matrix. In the interaction, the $\lambda$’s are the Gell-
Mann matrices, and in ${\cal L}_{\text{D}}$, the determinant is taken in the
flavor space. ${\cal L}_{\text{S}}$ represents the four-quark interaction,
with the coupling strength $G_{s}$, which is symmetric under $U(3)\times U(3)$
symmetry. On the other hand, ${\cal L}_{\text{D}}$, with coupling strength
$G_{d}$, describes the six-quark interactions, known as the ’t Hooft
determinant. ${\cal L}_{\text{D}}$ is included to break the $U(1)_{A}$
symmetry explicitly as $U(1)_{A}$ is anomalous in quantum theory.
To obtain the free energy, it is standard to introduce auxiliary fields using
Hubbard-Stratonovich transformation [37] to make the Lagrangian quadratic in
fermion fields. Within mean-field approximation, we can have nonzero vacuum
expectation values of these auxiliary fields. In the absence of any other
external agents (like a magnetic field, isospin chemical potential, etc.),
symmetry only allows $\bar{\psi}\psi$ channel to acquire nonzero vacuum
expectation values. And the meanfield Lagrangian becomes
${\cal
L}_{\text{MFA}}=\bar{\psi}\left(i\gamma_{\mu}\partial^{\mu}-\hat{M}\right)\psi-2G_{s}\sum_{i}\sigma_{i}^{2}+4G_{d}\prod_{i}\sigma_{f},$
(4)
where $\hat{M}$ is the constituent mass matrix, and the constituent masses are
given by [22]
$M_{i}=m_{i}-4G_{s}\sigma_{i}+2G_{d}\epsilon_{ijk}\sigma_{j}\sigma_{k},$ (5)
with $\sigma_{i}=\langle\bar{\psi}_{i}\psi_{i}\rangle$ being the condensate
that works as the order parameter of chiral symmetry breaking. As it is
evident from the above equation, $G_{d}$ mixes different flavors.
It is straightforward to integrate out the fermion degrees of freedom from Eq.
(4) to obtain the free energy. To introduce temperature (T) and chemical
potentials ($\mu_{f}$), it is customary to perform the following
transformations [38]
$p_{0}\rightarrow ip_{4}-\mu_{f},\qquad p_{4}=(2n+1)\pi T.$ (6)
With the above transformation, the integration over $p_{0}$ gets replaced by
the sum over Mastubara frequencies, $n$. Moreover, the free energy is given by
[39, 40]
$\Omega=\Omega_{\text{MF}}+\Omega_{\text{Vac}}+\Omega_{\text{Th}},$ (7)
where
$\displaystyle\Omega_{\text{MF}}$ $\displaystyle=$ $\displaystyle
2G_{s}\sum_{i}\sigma_{i}^{2}+4G_{d}\prod_{i}\sigma_{i},$ (8)
$\displaystyle\Omega_{\text{Vac}}$ $\displaystyle=$
$\displaystyle-2N_{c}\sum_{i}\int^{\Lambda}\frac{d^{3}p}{(2\pi)^{3}}\,\varepsilon_{i}(p),$
(9) $\displaystyle\Omega_{\text{Th}}$ $\displaystyle=$
$\displaystyle-2N_{c}T\sum_{i}\int\frac{d^{3}p}{(2\pi)^{3}}\left[\ln{\left(1+e^{-(\varepsilon_{i}(p)-\mu_{i})/T}\right)}\right.$
(10)
$\displaystyle+\left.\ln{\left(1+e^{-(\varepsilon_{i}(p)+\mu_{i})/T}\right)}\right].$
With, $N_{c}=3$ is the number of colors,
$\varepsilon_{i}(p)=\sqrt{\vec{p}^{2}+M_{i}^{2}}$ is the energy of the $i$-th
flavor quark, and $\Lambda$ is the three-momentum cutoff.
To obtain the ground state, one can minimize the free energy, defined in Eqs.
(7), by solving the following gap equations simultaneously
$\displaystyle\frac{\partial\Omega}{\partial\sigma_{u}}=\frac{\partial\Omega}{\partial\sigma_{d}}=\frac{\partial\Omega}{\partial\sigma_{s}}=0.$
(11)
In this study, we have considered the isospin symmetric case; in other words,
the electric charge and associated chemical potential ($\mu_{Q}$) are ignored,
which implies that $\sigma_{u}=\sigma_{d}=\sigma_{l}$. The quark chemical
potential can be written in terms of baryon and strangeness chemical potential
$\begin{split}\mu_{u}&=\mu_{d}=\frac{1}{3}\,\mu_{B},\\\
\mu_{s}&=\frac{1}{3}\,\mu_{B}-\mu_{S}.\end{split}$ (12)
Finally, for a fixed $\mu_{B}$ and $\mu_{S}$, we define the pseudo-critical
temperature ($T_{pc}$) as the inflection temperature where the curvature of
$\sigma_{l}$ changes sign [15]. In the context of LQCD, $T_{pc}$ is generally
determined from the maximum of the chiral susceptibility [15].
| $\Lambda$ (MeV) | $G_{s}\Lambda^{2}$ | $G_{d}\Lambda^{5}$ | $m_{l}$(MeV) | $m_{s}$(MeV)
---|---|---|---|---|---
set I | 631.4 | 1.835 | 9.29 | 5.5 | 135.7
set II | 602.3 | 1.835 | 12.36 | 5.5 | 140.7
Table 1: Parameter sets of NJL model used in the present work. Set I and set
II are from Hatsuda and Kunihiro[22] and Rehberg, Klevansky, and Hufner[23]
respectively
Let us note that there are five parameters in this three-flavor NJL model,
namely the current quark mass for the strange and light quarks ($m_{s}$ and
$m_{l}$), two coupling $G_{s}$ and $G_{d}$ and the three-momentum cutoff
$\Lambda$. After choosing the $m_{l}=5.5$ MeV, consistent with chiral
perturbation theory [41], the remaining four parameters are fixed by fitting
the pion decay constant and the masses of pion, kaon, and $\eta^{\prime}$ [22,
23] to their empirical values. With the parametrization of set I, the mass of
$\eta$ meson is underestimated by $11\%$ while for set II, the same is
underestimated by $6\%$. As may be seen from Tab. 1, the dimensionless
coupling $G_{d}\Lambda^{5}$ defer by $30\%$ between the two sets. The value of
$G_{d}$ even differs by about $70\%$. We intend to prescribe here a means of
constraining $G_{d}$. The parameter sets of the present study are given in
Table. 1. With this set of parameters, the mass of the $\eta$ meson is
underestimated by $6\%$ [17] and the constituent masses in the vacuum turn out
to be $M_{l}=368$ MeV and $M_{s}=549$ MeV. Among the widely used parameter
sets [23, 22] (see Tab. 1), $G_{d}\Lambda^{5}$ or $G_{d}$ differ most and in
this work, we intend to provide a prescribe a way to constrain it more
precisely.
## III Results
For a given value $\mu_{B}$ and $\mu_{S}$, the pseudo-critical temperature
($T_{pc}$) is defined to be the inflection point of light quark condensate
(the order parameter of chiral symmetry breaking) as a function of
temperature. Before proceeding to investigate the effect of finite $\mu_{S}$
on $T-\mu_{B}$ line, it is essential to check the model estimation against the
available lattice QCD results of the curvature coefficients ($\kappa_{2,4}$).
Considering $\mu_{Q}=0$, we have first investigated the $T-\mu_{B}$
($\mu_{S}=0$) and $T-\mu_{S}$ ($\mu_{B}=0$) plane, and find the $\kappa_{2,4}$
by parameterizing the respective pseudo-critical lines with the ansatz of Eq.
(1) for the range $\mu_{B,S}/T_{pc}(0)\leq 1.0$ with $T_{pc}(0)=171.1$ and
$173.4$ MeV for parameter set I and II, respectively.
We have tabulated our estimations for the curvature coefficients $\kappa_{2}$
and $\kappa_{4}$ in Tab. 2 and Tab. 3, respectively. The lattice results are
taken from the HotQCD collaboration [1] and WB collaboration [2] There is
excellent agreement with LQCD estimations for $\kappa_{2}$ in both the
$T-\mu_{B}$ and $T-\mu_{S}$ line. We want to emphasize that the values of
$\kappa_{2}^{B}$ are similar for both the parameter sets, which infers that
the large difference in $G_{d}$ between two parameter sets does not influence
the $T-\mu_{B}$ phase line. On the contrary, for the $\mu_{B}=0$ plane,
$\kappa_{2}^{S}$ is distinctly different for the two parameter sets. Although
these $\kappa_{2}^{S}$ values match with the lattice estimations within the
variances, the estimation with parameter set II has a better agreement with
the mean value. The difference in the $\kappa_{2}^{S}$ is attributed to
$G_{d}$, which brings the influence of strange quark to the light quarks as
pointed out in Eq. 5. This motivates us to examine the effect of $G_{d}$ on
$\kappa_{2}^{B}$ by exploring the $T-\mu_{B}$ line at various values of
$\mu_{S}$.
| $\kappa_{2}^{B}$ ($\mu_{S}=0$) | $\kappa_{2}^{S}$($\mu_{B}=0$) | $\kappa_{2}^{B,n_{S}=0}$
---|---|---|---
NJL, set I | 0.01627 | 0.01345 | 0.01478
NJL, set II | 0.01619 | 0.01719 | 0.01350
Lattice QCD | 0.016(6) [1] | 0.017(5) [1] | 0.012(4) [1]
| | | 0.0153(18) [7]
Table 2: Estimations of $\kappa_{2}$ for the two parameter sets. The Lattice
QCD results are taken from Ref. [1, 7] (a) Figure 1: Phase line in
$T-\mu_{B}$ plane for different $\mu_{S}$, evaluated with parameter set I. For
representative purposes, axes are scaled with respective $T_{pc}(\mu_{B}=0)$.
The continuous lines represent our estimations for various values of
$\mu_{S}$. The data and band are from the HotQCD [1]. The dashed line
corresponds to the strangeness neutral case.
To appreciate the effects arising from the strange sector, we have restricted
this study to smaller values of baryon chemical potential (up to
$\mu_{B}/T_{c}(0)\leq 1.0$) and varied the $\mu_{S}$ from 0 to 200 MeV. We
have restricted the $\mu_{S}$ within half the kaon mass to exclude the
possibility of kaon condensation [42]. Due to the difference in the magnitude
of the $T_{pc}$ between NJL and Lattice studies, we have scaled the results
with their respective $T_{pc}(\mu_{B}=0)$ as shown in Fig. 1. As may be
observed, for finite values of $\mu_{S}$, the $T_{pc}$ initially increases
with $\mu_{B}$ and then decreases. For smaller values of $\mu_{B}$, a finite
$\mu_{S}$ decreases the thermal weight in the strange sector ($\mu_{S}$ comes
with a negative sign in the strange thermal distribution, see Eq. (12)) and
therefore leads to a higher value of $T_{pc}/T_{pc}(0)$ for the same $\mu_{B}$
as shown in Fig. 1. As $\mu_{B}$ increases further, this rise in $T_{pc}$ gets
saturated and eventually starts decreasing.
For the first time, such a prominent increase in the pseudo-critical
temperature ($T_{pc}$) along the $T-\mu_{B}$ line is observed, which arises
due to a finite strangeness chemical potential. This trend was not observed in
earlier studies within LQCD [1, 43] and HRG [10], as most of them were
performed along the $\mu_{S}=0$ line or along the freeze-out line, where the
strangeness neutrality sets up the limit of $\mu_{S}\leq\mu_{B}/3$ [43].
(a) Figure 2: $\kappa_{2}^{B}$ as a function of $\mu_{S}$ in MeV. The red
(dashed) and green (dashed dot) lines are the central values for the set I and
II, respectively. The blue and cyan bands are associated with $\pm 10\%$
change in $G_{d}$ and $\pm 5\%$ change in $G_{s}$ respectively for parameter
set I.
To quantify the increase in the $T_{pc}$ with $\mu_{B}$ for a given value of
$\mu_{S}$, we have used the ansatz of Eq. (1) to extract the curvature
coefficients. We have presented the variation of $\kappa_{2}^{B}$ with
$\mu_{S}$ in Fig. 2 for both the parameter sets. The curvature coefficient
$\kappa_{2}^{B}$ starts from a positive value for $\mu_{S}=0$ and decreases as
we increase the strangeness chemical potential. We wish to emphasize that with
$\mu_{S}$, $\kappa_{2}^{B}$ decreases from its positive value at $\mu_{S}=0$
and eventually becomes negative at some $\mu_{S}=\mu_{S}^{c}$. This negative
sign of $\kappa_{2}^{B}$ is one of the novel results of the present
investigation. This was earlier not observed in the context of pseudo-critical
line [1, 7]. One important observation is that the $\mu_{S}^{\text{c}}$ are
distinctively different for the two parameter sets. $G_{s}\Lambda^{2}$ being
the same for both the sets, this difference in $\mu_{S}^{c}$ is essentially
due to the variance in $G_{d}$. A large $G_{d}$ provides a stronger influence
of the strange quark sector on the light quarks, resulting in a faster
decrease in $\kappa_{2}^{B}$.
At this juncture, it is instructive to check $\kappa_{2}^{B}$ along the
strangeness neutrality line ($n_{S}=0$). A finite $\mu_{B}$ requires the
strangeness chemical potential $\mu_{S}\neq 0$ to achieve zero net
strangeness. This corresponds to $\mu_{S}=\mu_{B}/3$ as we are considering
$\mu_{Q}=0$ and there is no vector interaction in the present model. We have
found $\kappa_{2}^{B,n_{S}=0}<\kappa_{2}^{B,\mu_{S}=0}$ as listed in Table.2.
The decrease of $\kappa_{2}^{B}$ for the strangeness neutral case is
commensurate with the lattice estimations [1, 43] and in accordance with our
findings of the reduction of $\kappa_{2}^{B}$ with $\mu_{S}$. We would like to
comment here that while the behaviour of the $\kappa_{2}^{B}$ ($\mu_{S}\neq
0$) is similar to the lattice QCD calculations. The lattice estimations of
$n_{S}=0$ correspond to values of $\mu_{S}$ which are not large enough to
constrain the flavor mixing determinant coupling. This necessitates LQCD
simulations at a larger value of $\mu_{S}$.
It would be interesting to check the robustness of this negative
$\kappa_{2}^{B}$ on the parametrization of the NJL model itself. For this
purpose, we have varied $G_{s}$, $G_{d}$ by $5\%$ and $10\%$, respectively and
examined the effect on $\kappa_{2}^{B}$ variation as shown in Fig. 2. As
discussed earlier, a larger value for $G_{d}$ increases the coupling between
the light and strange sector resulting in a faster decrease of
$\kappa_{2}^{B}$. It is needless to say that $\kappa_{2}^{B}$ becomes
independent of $\mu_{S}$ at $G_{d}=0$ as the strange and light quark sector
decouple which is evident in the Lagrangian of the NJL model. On the contrary,
the variation of $G_{s}$ has a weaker effect on the features mentioned above.
| $\kappa_{4}^{B}$ ($\mu_{S}=0$) | $\kappa_{4}^{S}$ ($\mu_{B}=0$) | $\kappa_{4}^{B,n_{S}=0}$
---|---|---|---
NJL, set I | 0.00006 | 0.001477 | 0.000081
NJL, set II | 0.00005 | 0.001892 | 0.000742
Lattice QCD | 0.001(7) [1] | 0.004(6) [1] | 0.000(4) [1]
| | | 0.00032(67) [7]
Table 3: Values of $\kappa_{4}$ for the pseudo-critical line for different
cases. (a) Figure 3: $\kappa_{4}^{B}$ as a function of $\mu_{S}$ in MeV.
Color code is same as Fig.2.
Within LQCD, the numerical value of $\kappa_{4}^{B}$ is consistent with zero
[1, 7], as for the small value of $\mu_{B,S}/T$, the $4^{th}$ order
coefficients of $\mu_{X}/T$ expansion is prone to have a weaker effect on the
$T-\mu_{X}$ line. In the present study, we have found $\kappa_{4}$ to have
good agreement for the case (${\mu_{B}\neq 0,\ \mu_{S}=0}$) and (${\mu_{B}=0,\
\mu_{S}\neq 0}$) as shown in Table 3. It would be essential to investigate the
same for the $T-\mu_{B}$ line at various $\mu_{S}$. For larger values of
$\mu_{S}$, we have found the $\kappa_{4}$ to be finite (as in Fig.3), even
with the different parameter sets, as mentioned earlier. These findings
suggest that even within the small $\mu_{B}$ range, a non-zero $\kappa_{4}$ is
possible by switching on a finite strangeness-chemical potential $\mu_{S}$,
which is relevant in the context of lattice simulations.
## IV Summary and conclusion
In this letter, we explore the chiral phase boundary of the QCD matter within
a $2+1$ flavor Nambu–Jona-Lasinio model with special emphasis on the effect of
strangeness on the curvature coefficients $\kappa_{2}^{B}$ and
$\kappa_{4}^{B}$. To our knowledge, this is the first such exploration within
a $2+1$ NJL model. We have considered the isospin symmetric case and
$\mu_{Q}=0$. To have better control over the lowest-order coefficients
($\kappa_{2}^{X}$), we have limited the study within the range $\mu_{X}/T\leq
1$. As benchmark, we have first estimated the $\kappa_{2,4}^{X}$ for three
separate cases, 1) $T-\mu_{B}$ plane ($\mu_{S}=0$) i.e $\kappa_{2,4}^{B}$ 2)
$T-\mu_{S}$ plane ($\mu_{B}=0$) i.e $\kappa_{2,4}^{S}$ and 3) along the
strangeness neutrality line $\kappa_{2,4}^{B,n_{S}=0}$. We have used two
standard sets of parametrization of $2+1$ NJL model that differ significantly
regarding the flavor mixing determinant interaction. Although we have an
excellent agreement of $\kappa_{2}^{B}$ with the available LQCD finding for
both the parameter sets, we observed that $\kappa_{2}^{S}$ has a strong
dependence on the flavor mixing and $U(1)_{A}$ breaking ’t Hooft interaction.
Between the two parameter sets used, the set II with a higher value of $G_{d}$
reproduces lattice estimation of $\kappa_{2}^{S}$ better. To explore the
effects of flavor mixing through $G_{d}$, it is instructive to study the
$\mu_{S}$ dependence of the $T-\mu_{B}$ lines, which have been quantified by
estimating $\kappa_{2}^{B}$ as a function of $\mu_{S}$.
It is observed that within the model $\kappa_{2}^{B}$ decreases with $\mu_{S}$
and becomes negative for sufficiently large values of $\mu_{S}$. Further, it
is observed that the value of $\mu_{S}$ where $\kappa_{2}^{B}$ vanishes are
different for two parameter sets. This difference is attributed to the fact
that a larger value of $G_{d}$ strengthens the strange contribution to the
light sector, resulting in a faster decrease. This novel observation provides
a direct measure of the flavor mixing through ‘t Hooft interaction. We expect
that outcomes from LQCD investigations for $\kappa_{2}^{B}$ at large enough
$\mu_{S}$ will assist in better constraining the ‘t Hooft coupling $G_{d}$,
thereby enhancing our understanding of effective models like NJL and QCD.
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|
###### Abstract
The Cosmic Ray Extremely Distributed Observatory (CREDO) is a newly formed,
global collaboration dedicated to observing and studying cosmic rays (CR) and
cosmic ray ensembles (CRE): groups of a minimum of two CR with a common
primary interaction vertex or the same parent particle. The CREDO program
embraces testing known CR and CRE scenarios, and preparing to observe
unexpected physics, it is also suitable for multi-messenger and multi-mission
applications. Perfectly matched to CREDO capabilities, CRE could be formed
both within classical models (e.g. as products of photon-photon interactions),
and exotic scenarios (e.g. as results of decay of Super Heavy Dark Matter
particles). Their fronts might be significantly extended in space and time,
and they might include cosmic rays of energies spanning the whole cosmic ray
energy spectrum, with a footprint composed of at least two extensive air
showers with correlated arrival directions and arrival times. Since CRE are
mostly expected to be spread over large areas and, because of the expected
wide energy range of the contributing particles, CRE detection might only be
feasible when using available cosmic ray infrastructure collectively, i.e. as
a globally extended network of detectors. Thus, with this review article, the
CREDO Collaboration invites the astroparticle physics community to actively
join or to contribute to the research dedicated to CRE, and in particular to
share any cosmic ray data useful for the specific CRE detection strategies.
###### keywords:
cosmic rays; cosmic ray ensembles; extensive air showers; large scale cosmic
ray correlations; CREDO Collaboration
xx 1 5 Received: date; Accepted: date; Published: date Cosmic Ray Extremely
Distributed Observatory Piotr Homola1*, Dmitriy Beznosko2, Gopal Bhatta3,
Łukasz Bibrzycki4, Michalina Borczyńska5, Łukasz Bratek6, Nikolai Budnev7,
Dariusz Burakowski8, David E. Alvarez-Castillo1,9, Kevin Almeida Cheminant1,
Aleksander Ćwikła10, Punsiri Dam-o11, Niraj Dhital12, Alan R. Duffy13, Piotr
Głownia6, Krzysztof Gorzkiewicz1, Dariusz Góra1, Alok C. Gupta14, Zuzana
Hlávková15, Martin Homola15, Joanna Jałocha6, Robert Kamiński1, Michał
Karbowiak16, Marcin Kasztelan17, Renata Kierepko1, Marek Knap8, Péter
Kovács18, Szymon Kuliński5, Bartosz Łozowski19, Marek Magryś20, Mikhail V.
Medvedev21,22, Justyna Mędrala23, Jerzy W. Mietelski1, Justyna Miszczyk1,
Alona Mozgova24, Antonio Napolitano25, Vahab Nazari1,9, Y. Jack Ng26, Michał
Niedźwiecki10, Cristina Oancea27,28, Bogusław Ogan8, Gabriela Opiła23,
Krzysztof Oziomek20, Maciej Pawlik20,23, Marcin Piekarczyk4, Bożena
Poncyljusz5, Jerzy Pryga29, Matías Rosas30, Krzysztof Rzecki23, Jilberto
Zamora-Saa31, Katarzyna Smelcerz10, Karel Smolek32, Weronika Stanek23,
Jarosław Stasielak1, Sławomir Stuglik1, Jolanta Sulma33, Oleksandr Sushchov1,
Manana Svanidze34, Kyle Tam35, Arman Tursunov36, José M. Vaquero37, Tadeusz
Wibig${}^{16}\orcidY{}$, Krzysztof W. Woźniak1 Piotr Homola, Dmitriy Beznosko,
Gopal Bhatta, Łukasz Bibrzycki, Michalina Borczyńska, Łukasz Bratek, Nikolai
Budnev, Dariusz Burakowski, David E. Alvarez-Castillo, Kevin Almeida
Cheminant, Aleksander Ćwikła, Punsiri Dam-o, Niraj Dhital, Alan R. Duffy,
Piotr Głownia, Krzysztof Gorzkiewicz, Dariusz Góra, Alok C. Gupta, Zuzana
Hlávková, Martin Homola, Joanna Jałocha, Robert Kamiński, Michał Karbowiak,
Marcin Kasztelan, Renata Kierepko, Marek Knap, Péter Kovács, Szymon Kuliński,
Bartosz Łozowski, Marek Magryś, Mikhail V. Medvedev, Justyna Mędrala, Jerzy W.
Mietelski, Justyna Miszczyk, Alona Mozgova, Antonio Napolitano, Vahab Nazari,
Y. Jack Ng, Michał Niedźwiecki, Cristina Oancea, Bogusław Ogan, Gabriela
Opiła, Krzysztof Oziomek, Maciej Pawlik, Marcin Piekarczyk, Bożena Poncyliusz,
Jerzy Pryga, Matías Rosas, Krzysztof Rzecki, Jilberto Zamora-Saa, Katarzyna
Smelcerz, Karel Smolek, Weronika Stanek, Jarosław Stasielak, Sławomir Stuglik,
Jolanta Sulma, Oleksandr Sushchov, Manana Svanidze, Kyle Tam, Arman Tursunov,
José M. Vaquero, Tadeusz Wibig, Krzysztof W. Woźniak Correspondence:
<EMAIL_ADDRESS>Tel.: +48-12-662-8341
## Introduction
While state-of-the-art cosmic ray research to date has been focused on the
detection and analysis of cosmic particles observed through individual
detectors or arrays, the correlated observations of cosmic rays on the global
scale remain insufficiently explored, yet no less promising. This
collaborative perspective could provide a deeper insight into the physical
processes within energy ranges rarely considered, including the highest
energies known. Here we discuss a general approach to research dedicated to
detecting and studying the astroparticle physics phenomena called Cosmic Ray
Ensembles (CRE) defined as groups of a minimum of two correlated, be it
spatially or temporally, cosmic rays with a common primary interaction vertex
or the same parent particle. Such particles, constituents of CRE, are
messengers of the primary physical processes – probes of the physics that
happened even billions of years ago, at energies even millions of times higher
than energies to which we can accelerate particles using man-made
infrastructure.
Figure 1: Cosmic-Ray Ensembles: a novelty in cosmic-ray research and in multi-
messenger astroparticle physics.
Armed with the particle physics background telling us that cosmic ray
particles are expected to interact with radiation and matter on their way
through the Cosmos and give birth to CRE, we ask not whether CRE exist, but
under which circumstances and with which conditions they can reach the Earth
and be detected with the available or possible infrastructure. As illustrated
in Fig. 1, the signatures of CRE might be spread over very large surfaces
which might make them hardly detectable by existing detector systems operating
individually. On the other hand, if these detector systems operate under a
planetary (and beyond) network, as proposed by The Cosmic-Ray Extremely
Distributed Observatory (CREDO) Homola and et al. (2018) (CREDO Collab.); Góra
and et al. (2018) (CREDO Collab.); Homola and et al. (2020) (CREDO Collab.),
the chances for detection of CRE will naturally increase. The components of
CRE might have energies that practically span the whole cosmic-ray energy
spectrum. Thus, all the cosmic ray detectors working in this range, beginning
from smartphones (e.g. DECO Vandenbroucke and et al (2016), CRAYFIS Whiteson
et al. (2016), CREDO Detector cre ; Bibrzycki and et al. (2020) (CREDO
Collab.); Niedzwiecki and et al. (2019) (CREDO Collab.), Cosmic Ray App cos )
and pocket size open-hardware scintillator detectors (e.g. Cosmic Watch Axani
et al. (2018); Schaub et al. (2018) or CosmicPi cos ), through numerous larger
educational detectors and arrays (e.g. HiSPARC Colle et al. (2007); Fokkema
(2012), QuarkNet Bardeen et al. (2018), Showers of Knowledge Bychkov and
Guskov (2012), CZELTA Smolek and et al. (2008)) to the professional
infrastructure that receives or will receive cosmic rays as a signal or as a
background Pierre Auger Observatory Pierre Auger Collaboration (2015),
Telescope Array Kawai and et al. (2008), JEM-EUSO Adams and et al. (2015a)
(JEM-EUSO Collab.); Adams and et al. (2015b) (JEM-EUSO Collab.), HAWC DeYoung
(2012), MAGIC Ferenc (2005), H.E.S.S. Vasileiadis and et al. (2005), VERITAS
Krennrich and et al. (2004); Park and et al. (2015), IceCube Aartsen and et
al. (2017) (IceCube Collab.), Baikal-GVD D and et al. (2016) (Baikal-GVD
Collab.), ANTARES Telescope Ageron and et al. (2011) (ANTARES Collab),
European Southern Observatory Franza and Wilson (1982), other astronomical
observatories, underground observatories, accelerator experiments in the off-
beam mode) could contribute to a common effort towards a hunt for CRE.
Therefore it is not only desirable, but also feasible to put the CRE research
into a routine implementation. So far, the experimental searches for cosmic-
ray correlations have been realized on different scales with arrays of
detectors located at schools and universities. However, all those efforts were
dedicated to very specific scenarios, concerning mostly fragmentation of
nuclei in background electromagnetic radiation fields Gerasimova and Zatsepin
(1960), which limits the number of particles in the group (ensemble) to just a
few. Some of these projects include e.g. CHICOS Carlson et al. (2005) in the
U.S., ALTA Pospıšil et al. (2009) in Canada, GELATICA (Svanidze et al., 2011)
in Georgia, EEE Abbrescia and et al (2016) in Italy, LAAS Ochi and et al
(2003); Matsumoto and et al (2019) in Japan, and the aforementioned CZELTA in
Czech Republic. Time correlation of registered showers was studied at
distances from 100 m to 7000 km, and in some cases evidence for unexpected
coincidences have been found. However, these were without any convincing
follow-up studies and data taking campaigns, which is hard without global
coordination. Only very recently the idea of looking for large scale
correlations in a general and global way took shape with the CREDO
Collaboration, formalized in September 2019 Collaboration . CREDO is meant to
be a multi-technique (different detector types) and doubly open (for both data
upload and offering access) infrastructure enabling global research programs
concerning radiation (both cosmic and terrestrial), with a number of multi-
messenger, multi-mission and transdisciplinary opportunities. With this review
article we invite the community to both benefit from the openness of CREDO and
to contribute to its program with the research dedicated to a general search
for ensembles of cosmic rays, especially photons of different energies. Since
the concept of CREDO assumes the openness for independent data streams, it is
expected that the projects mentioned above, as well as the other cosmic ray
infrastructure including private, widely spread detectors such as smartphones,
will have a direct interest in connecting to the global CREDO system, thus
reinforcing its scientific attractiveness, and chances of individual research
programs.
As this article is meant to serve as a review of the current status and
perspectives of CREDO and the research related to CRE, its structure follows
the scientific and logical roadmap to observing and studying cosmic ray
ensembles. We begin with a general description of the field in Section 1
“Foundations of the CREDO methodology” and 2 “CREDO within the cosmic ray
landscape”, and theoretical modeling of CRE sources in Section 3 “UHECR
sources and cosmic ray ensembles”. Then we present and discuss example CRE
scenarios with an emphasis on the simulations of propagation of the CRE
components through the Cosmos and within the Earth’s atmosphere (Section 4
“CRE simulations”), describe the status of the observational efforts (Sections
5 “CREDO detectors: cloud of clouds”, 6 “Data management and analysis”, and 7
“Building the scale: public engagement as a scientific need”), and conclude
the article with the outlook, summary and conclusion (Sections 8 and 9).
## 1 Foundations of the CREDO methodology
The CREDO experiment by its very idea of making discoveries and approaching
truth in various research areas, touches something one can call the borderline
of the current state of our knowledge. As explained below, there are reasons
to anticipate that the experiment has the potential to falsify some of the
adopted models. In the context of CREDO one can even ask questions about
science itself and its methodology.
As an example one can consider exotic QED processes which are potentially
within the observational scope of CREDO. Namely, pair creation or photon
splitting in strong magnetic fields. These predictions of QED can now be
tested in the context of cosmic rays physics under extreme astrophysical
conditions met in pulsar magnetospheres (with typical magnetic field strengths
of $10^{12}$ Gauss, or even $10^{14}$ Gauss in magnetars). In such conditions
there are several observational signatures of the two processes Usov (2002);
Harding and et al (1996). What is more, the photon splitting phenomenon (in
exotic scenarios) fits very well in the context of cosmic ray ensembles (CRE)
that are potentially within the observational scope of CREDO (as long as the
opening angle of the secondary photons would not be too large). Therefore, the
CREDO experiment opens up new opportunities to test the well-established QED
theory as well as consider more exotic scenarios. The experiment is
potentially capable of changing our view of the basis of science, specifically
by providing a new type of means to falsify generally accepted theories in an
energy range that has not been yet available for to date observatories. In
this way one arrives at the strictly philosophical question to be addressed
and that naturally arises in the context of CREDO – the issue of parsimony of
the scientific method or Occam’s razor.
It has been accepted that the scientific method of explaining and
understanding facts should follow the principle of not multiplying entities
without necessity. Explaining all new phenomena should be based on what one
has already had, namely, based on the theories and models that successfully
worked so far, and only if this attempt is found to have failed one may
consider rejecting these theories and models. Occasionally, the parsimony
principle, when understood in a fundamentalistic manner, would stand in the
way of knowledge. This kind of epistemological attitude might lead to
supplementing old models and theories with new “epicycles”, rather than
encouraging one to humbly admit that some of the assumptions being adopted so
far need to be rejected as inconsistent with reality. A solution to this state
of things is to presume that the parsimony principle, although an invaluable
component of the scientific method, is not something the scientific method
entirely stands on. Sometimes one has to bypass the strict rule (which in
practice is done by the majority of the scientific community). Accordingly, if
the CREDO experiment (or some other experiment) comes up with an anomalous
result, not falling within the framework of accepted models and theories,
first the possibility of an error should be considered, be it in the
measurement or interpretation side. However, one should also not be afraid of
pushing the limits and going beyond the realm of modern knowledge by extending
or redefining the adopted concepts as well as the language with which we
describe reality.
The CREDO initiative inspires one to deliberate on falsifiability, another
important element of the scientific method. It is often said that scientific
statements should be falsifiable – refutable by contradicting evidence. This
is not definitely true. For example, general statements about existence are
most often unfalsifiable, nevertheless they are easy to be verified
empirically, which makes the statements quite scientific. Even though science
is not entirely determined by falsifiability, this attribute is very
advantageous and often characterizes statements formulated by science, either
as models or scientific theories.
Classical electrodynamics provides a good example of a theory that is
falsifiable. For the purpose of illustration imagine that CREDO (or another
experiment of this kind) comes up with some evidence for spontaneous photon
splitting (to be in touch with the experimental scope of CREDO pointed out
earlier) – a phenomenon there is no place for in the linear Maxwell
electrodynamics. There are effective terms that could be incorporated to the
electromagnetic Lagrangian to mimic at the classical level such effects. To be
more specific, some QED vacuum polarization effects such as photon-photon
scattering or photon splitting in external fields can be effectively described
by classical theory (the Euler-Heisenberg Lagrangian is used in this context),
however, these nonlinear phenomena are merely interaction effects arising upon
quantization of linear Maxwell fields in accordance with methods of quantum
field theory, therefore already explainable within the current paradigm. The
real change in the electromagnetic theory required to incorporate new
phenomena such as spontaneous photon splitting that could occur in free space
and observed through cosmic ray ensemble evolution, would be to replace the
Maxwell Lagrangian by an alternative one resulting in nonlinear equations
before quantization, or maybe even to alter the quantization method as such.
Is that to mean that Maxwell’s electrodynamics would just be falsified by an
observation of a spontaneous photon splitting?
Stated clearly, it is not that simple – scientific theories are deeply
founded, routinely withstanding repeated attempts at falsification. In the
first place, one should try to explain a particular observational result
within the current theory. Only if such attempts turn out absolutely
unsuccessful and a growing number of anomalies are observed at the same time,
one would rather conclude that the theory has been falsified – though the old
theory will probably still survive as a good approximation (as this was the
case with Newtonian and relativistic mechanics). Thus, falsifying a theory is
not an easy task, at least not possible based merely on a single observation
or measurement. This being said, experiments such as CREDO appear even more
valuable, since both for the verification or falsification to be feasible, one
needs as many channels as possible through which the universe is looked at.
## 2 CREDO within the cosmic ray landscape
Although CRE detectable on or around the Earth can be initiated by particles
spanning a wide range of energies, and since it is not a priori evident
whether the chances of observing a CRE increase with the energy of the primary
particle, one should stay open-minded about possible focus concerning the
energy regimes of investigation. Here, for clarity, we chose to focus on the
cosmic rays of the highest energies known, $E$ > 1018 eV, hereafter referred
to as ultra-high energy cosmic rays (UHECR), capable of initiating CRE
composed of billions of component particles which can propagate unaffected
large astrophysical distances, as in case of GeV-TeV photons (see e.g. Risse
and Homola (2007)), of which an observable fraction may reach the human
technosphere.
The surprisingly constant, power-law character of the energy spectrum of
cosmic radiation observed by more than ten orders of magnitude with an almost
constant exponent of about (-3) could be, in principle, continued on without
any obstacles. At least until the mid-1960s there was no reason to expect any
definite end. Although the sources and mechanisms of acceleration of single
elementary particles to the energy of a dozen joules both then and now are
unknown (the search is still ongoing). Review of objects in the sky that would
potentially come into question, their spatial sizes ($L$) and magnetic fields
($B$) suggests that their capabilities ($E_{max}<ZeBL$) do not reach far
beyond the limit of 1020 eV. It should be remembered that any acceleration
mechanism by its nature would have to be of a statistical character and
therefore, when analysing the average or typical parameters of primary cosmic
ray particles, it is difficult to exclude the occurrence of large and
extremely large fluctuations Tsallis et al. (2003).
The situation changed substantially after the discovery of the cosmic
microwave background (CMB) radiation. If we assume that the primary particles
of cosmic radiation are protons, as Greisen Greisen (1966), Zatsepin and
Kuzmin Zatsepin and Kuzmin (1966) noticed immediately, a sudden end of the
spectrum must be caused by collisions of them with CMB photons and resonant
production of the $\Delta^{+}$ particle. $\Delta^{+}$ decays instantly again
into a nucleon, and its energy is on average about 20% less than before the
collision. This process occurs as long as the nucleons have enough energy,
which happens at about 5-6$\times 10^{19}$ eV. However, particles of cosmic
radiation do not have to be protons. Observations suggest (e.g. Wibig and
Wolfendale (2005)), that heavier nuclei with a higher charge ($Ze$), thus
easier to be accelerated, may entirely dominate the highest energy cosmic
radiation flux. For nuclei with energies of about 1020 eV, when energies are
calculated separately for each nucleon, the GZK mechanism does not work, but
this does not mean that the Universe for them remains transparent. In the
centre of mass system, when colliding with photons of intergalactic radiation
(infrared mostly) they excite to a giant dipole resonance, and then fragment,
most preferably emitting few neutrons. Although fragments retain the same
energy per nucleon, as a whole they have correspondingly reduced their total
energy.
Searching for the end of the cosmic radiation spectrum, according to what has
been just said, could lead to solving the problem of mass composition of
cosmic radiation of the highest energy, and thus bring us closer to
identifying its sources (and acceleration mechanisms). However, it
unfortunately happens that both the GZK cut-off and the photodisintegration of
the nuclei start to work effectively in a similar range of energy, just where
today’s observations of cosmic radiation end because of the scarce cosmic ray
flux. In addition to the statistics, another circumstance is that two giant
experiments in operation today which statistically dominate the measurements
of giant showers in the highest energy range: the Pierre Auger Observatory in
Argentina and the Telescope Array in Utah, US, while claiming a general
agreement of the energy spectra within experimental uncertainties up to 10
EeV, they admit the need for non-linearity to bring the spectra in agreement
at the higher energies, and within the range of common declinations. However,
the sources of this non-linearity have not been identified, yet Deligny
(2020). It turns out, then, that the current status of the UHECR observations
does not allow definite conclusions about the exact location and nature of the
observed cut-off of the energy spectrum, implying the uncertainty about the
cosmic ray composition at the highest energies known.
Either way, the results of the leading UHECR observatories and their
conclusions from widely quoted publications (Abbasi and et al. (2008) (HiRes
Collaboration); Abbasi and et al (2014); Abraham and et al. (2008) (Pierre
Auger Collaboration); Valino (2016)) indicate that the spectrum of cosmic
radiation is significantly suppressed at the highest energies, although it is
not known at which energy, and whether this energy depends e.g. on the kind of
the sources. Such a picture is widely accepted and despite minor scratches,
small inconsistencies and doubts it seems that we understand it. There
shouldn’t be many particles above the cut-off and in fact this is the case.
However, among the aforementioned minor doubts, there are still recorded in
the last century in several (or actually almost all) large and significant
giant air shower experiments, the cases initiated by particles with estimated
energies exceeding $10^{20}$ eV.
The first, historical, Volcano Ranch event was recorded by Linsley in 1962
with the energy of $10^{20}$ eV, Linsley (1963). In the 1980s the Haverah Park
experiment reported a significant increase in the number of showers with
energies exceeding $\sim 10^{20}$ eV Cunningham et al. (1980); Watson (1991);
Lawrence et al. (1991) . In the Yakutsk array, the record energy was rated at
$1.5\times 10^{20}$ eV Afanasiev and et al (1995). The Japanese AGASA
experiment published the event from December 3, 1993 which had an energy of
$2\times 10^{20}$ eV Teshima and el al (2000). However, a world record was set
in the USA in the Fly’s Eye experiment: $3.2\times 10^{20}$ eV Bird and et al
(1995). These cases have not been discussed recently in literature, but it
seems that they require some explanation. The first, more straightforward
explanation is that the experimenters might have been misled about the energy
reconstruction of their record event by the imperfectness of the tools
available at that time, with particular emphasis on numerical tools.
Thankfully, the Monte-Carlo methods developed in the 21st century are
certainly more precise and they allow today making more adequate corrections
than decades ago, in particular in experimental procedures like localization
of shower axis, or estimation of shower energy, whether by collecting
fluorescent light or determining the charged particle density distribution on
the ground. And this explanation could be enough, provided one does not
discuss in detail the statistics of the “above 1020 eV” cosmic ray detections
of 20th century experiments in contemporary analyses. But, as already
mentioned, the two great experiments, the Pierre Auger Observatory and the
Telescope Array, also reported around 20 UHECR cases of energies above 1020 eV
observed already with the newest tools and methods (see e.g. Ref. Aab and et
al. (2020) (Pierre Auger Collaboration) where 15 events with energies above
1020 eV are mentioned collectively). To list just a few such events in detail
we mention the Pierre Auger Observatory measurements which contain an event
with energy $1.4\times 10^{20}$ eV Abreu and et al. (2010) (Pierre Auger
Collaboration) – or $1.3\times 10^{20}$ eV Aab and et al. (2015) (Pierre Auger
Collaboration), and the Telescope Array announcement concerning an event of
similar energy $1.39\times 10^{20}$ eV Abbasi and et al (2014). It is of
course clear that if the cosmic ray energy spectrum breaks around 5-6$\times
10^{19}$ eV, this is most likely a (gradual) suppression rather than an abrupt
cut-off, so there must be a few events exceeding $10^{20}$ eV. However, the
current statistics of these events does not allow telling whether they are
compatible with the spectrum cut-off or not.
It is a great achievement that today we can tell that the UHECR spectrum has
been quite well measured – see for example two recent Pierre Auger Observatory
papers Aab and et al. (2020) (Pierre Auger Collaboration, Pierre Auger
Collaboration), and that a lot of effort and resources are being currently
invested into explaining the nature of the observed spectrum suppression (GZK-
like versus acceleration limit). Nevertheless the current results are still
inconclusive, and it is therefore advisable to continue research in the
highest energy regime of cosmic rays, and to try alternative methods whenever
possible. In other words, the status of the dispute in the UHECR area
encourages a closer look at the field and being ready for a major revision or
breakthrough in the understanding of physics at the highest energies known.
The CREDO initiative with its objectives dedicated to going beyond studying
individual cosmic rays and taking under investigation also UHECR products –
cosmic ray ensembles, may provide a precious complementary approach to UHECR
studies.
## 3 UHECR sources and cosmic ray ensembles
The Standard Model (SM) of particle physics predicts that if cosmic rays (CR)
are primarily composed by protons, there should exist a limit on the maximum
energy of the CR coming from distances farther than $\sim 100$ Mpc. This bound
is called the GZK limit and was presented in Refs. Greisen (1966); Zatsepin
and Kuzmin (1966); Stecker (1968). Despite this, UHECR with energies above the
GZK limit have been reported, by experimentalist, from directions where there
are no nearby111The current experimental data suggests an extragalactic origin
for UHECRs with energies above the GZK cutoff Aab et al. . sources Zotov et
al. (2017); Verzi et al. (2017). Therefore, it seems that there is a missing
piece in our understanding of the sourcing, nature, and/or propagation of the
CR.
The cosmic ray background has been simulated by means of numerical propagation
codes Kalashev et al. (2001); Aloisio and et al (2012), and the results have
shown that it is very unlikely that CR with energy greater than GZK limit to
be photons Aab and et al. (2017) (Pierre Auger Collaboration).
In general, one can distinguish two qualitatively different approaches in
unveiling the physics of UHECRs: theoretical models assuming interactions of
exotic super-heavy matter (including extra dimensions, Lorentz invariance
violation, cosmic strings, existence of new particles etc Weiler (1982, 1984);
Aloisio and et al (2015); Tyler et al. (2001); Domokos and Kovesi-Domokos
(1999); Coleman and Glashow (1997); Bhattacharjee and Sigl (2000); Bietenholz
(2011); Scully and Stecker (2009); Gorham et al. (2012); Rubtsov et al.
(2014); Tasson (2014); Rubtsov et al. (2017); Mohanmurthy et al. (2016);
Klinkhamer and Schreck (2008); Alcantara et al. (2019); Aloisio and Tortorici
(2008); Supanitsky and Medina-Tanco (2019)) and acceleration scenarios
describing processes, in which the particles are accelerated by a particular
astrophysical object (shocks in relativistic plasma jets, unipolar induction
mechanisms, second-order Fermi acceleration, etc.). Acceleration scenarios
rely on the existence of powerful astrophysical sources with available energy
that is sufficient for the energy transfer from these objects to cosmic ray
particles.
In the age of multi-wavelength and multi messenger astronomy, transient
astronomical objects are of the great interest for UHECRs emission. There are
several classes of astronomical objects which can be prime targets for UHECR
observations e.g. gamma-ray bursts (GRBs); supernovae (SN); fast radio bursts
(FRBs); various classes of active galactic nuclei (AGN) such as Seyfert
galaxies, radio galaxies and blazars; and possible neutrino emitting blazars.
In the recent past there are evidences that these objects emit or can emit
UHECRs. A 5 millisecond bright fast radio burst of extragalactic origin was
detected (Lorimer et al., 2007). It is claimed that the radio galaxies have
emitted UHECRs (Nagar and Matulich, 2008). There is an evidence of hadronic
$\gamma-$ray emission from supernova remnants (Moskalenko et al., 2008).
Blazars are one of the most prominent sources of UHERCs emission. There are
several papers which estimated / predicted neutrino and UHERCs emission from
blazars (e.g., Murase et al., 2012; Rodrigues et al., 2018, and references
therein). About 2 decades ago, it was predicted that there were some very
bright high energy peaked blazars which would be neutrino loud(Neronov and
Semikoz, 2002). And recently, the IceCube Collaboration found the evidence of
neutrino emission from the blazar TXS 0506$+$056 (Collaboration, 2018) which
opened a new window to search for such another blazars too.
### 3.1 Supermassive black holes as UHECR sources
Among the most powerful astrophysical sources, one can highlight supermassive
black holes (SMBHs) located in the centers of most galaxies due to their
compactness and enormous energy available for the extraction. Below we shall
briefly show the capability of SMBHs to power the UHECRs in a given model. It
appears that up to $29\%$ of the total energy ($M_{\rm BH}c^{2}$) of the black
hole is rotational energy and available for extraction (Bekenstein, 1973).
Nowadays, all tests of general relativity indicate that astrophysical black
holes can be fully characterized by their masses and spins, while other
properties of black holes are hidden inside the event horizon and unavailable
to the external observer. For a typical SMBH of $10^{9}$ solar masses the
extractable energy is of the order of $10^{74}$eV, turning SMBHs into the
Universe’s largest and the most compact energy reservoirs. Therefore, it is
important to search for the processes, which tap these enormous energy sources
in the most efficient way.
Attempts to tap the energy of black holes started in 1969 by Roger Penrose
Penrose (1969), followed by many authors (see, e.g. references Blandford and
Znajek (1977); Wagh et al. (1985); Parthasarathy et al. (1986); Dadhich et al.
(2018) and references therein), who used the existence of negative energy
states of particles with respect to a stationary observer at infinity. If a
particle decays into two fragments near a black hole with one of the fragments
attaining negative energy, the other fragment (respecting energy conservation)
may escape from the black hole with greater energy than those of the primary
particle. The remarkable property of the rotating black hole is the existence
of a special region outside the event horizon called the ergosphere, where the
energy of a particle relative to infinity can be negative. However, the
negative energy states may also appear due to purely electromagnetic
interactions between the SMBH and surrounding matter without the need for
ergosphere (Tursunov and Dadhich, 2019; Tursunov et al., 2020).
Black holes, as any astrophysical object, are immersed into magnetic fields.
Rotation of the black hole in an external magnetic field leads to the twisting
of magnetic field lines due to the frame-dragging effect. Similar to a
classical Faraday unipolar inductor, a black hole rotating in a magnetic field
attains non-zero induced electric charge (Wald, 1974). This charge has been
shown to be non-negligible for any astrophysical black hole and is potentially
observationally measurable (Zajacek et al., 2018). Since the induced charge of
the black hole is coupled and proportional to the black hole spin, its
discharge is equivalent to the slowing down of the black hole and decreasing
the black hole’s rotational energy. To demonstrate the process of black hole
energy extraction and resulting acceleration of particles to ultra-high
energy, one can consider the ionization or decay of a neutral particle into
charged fragments in the vicinity of rotating SMBH immersed into an external
magnetic field. If the black hole possesses induced electric charge, the
energy of charged fragments after decay of the neutral particle obtains a
strong Coulombic contribution in addition to the gravitational negative energy
in the ergosphere. Since the induced charge of SMBHs is more likely positive
(Zajacek et al., 2018), these are the protons, which would escape from the
black hole with tremendous energy (Tursunov et al., 2020). This is the ultra-
efficient regime of the so-called magnetic Penrose process that can serve as a
possible mechanism for acceleration of UHECRs when applied to SMBH candidates.
The energy of an escaping UHE particle depends on its charge to mass ratio,
strength of external magnetic field and the mass of the black hole. For
protons accelerated in this mechanism, the maximum energy is predicted to be
$\centering E_{\rm p}=1.3\times 10^{21}{\rm
eV}~{}\frac{Z~{}m_{p}}{m}~{}\frac{B}{10^{4}{\rm
G}}~{}\frac{M}{10^{10}M_{\odot}}.\@add@centering$ (1)
In Figure 2 we demonstrate the acceleration of UHE protons resulting from the
hydrogen ionization in the SMBH vicinity. Similar results are obtained for the
neutron beta decay ($n\rightarrow p^{+}+e^{-}+\bar{\nu}_{\rm e}$). Here we
provide verifiable constraints on the SMBH mass and magnetic field as UHECR
source. As an example we indicate the few famous nearby SMBH candidates, which
are capable of producing UHE protons of certain energies. It is interesting to
note that the black hole located at the centre of our Galaxy can accelerate
particles up to the energies corresponding to the knee of the cosmic ray
spectrum. On the right side of Figure 2 we demonstrate the results of
numerical simulation of the magnetic Penrose process (for details, see
Tursunov et al., 2020; Stuchlík et al., 2020; Tursunov and Dadhich, 2019).
Remarkably, the mechanism operates in viable physical conditions for typical
SMBHs, without the need for a large acceleration zone or exotic assumptions
for black holes.
Figure 2: Left: constraints on the SMBH mass and magnetic field for UHE
protons and chosen nearby sources (at the distance $<20$ Mpc from the Solar
system) fitting UHECRs of certain energy. Data is taken from (Baczko and et
al, 2016; Eckart et al., 2012; Doeleman et al., 2012; Kino et al., 2015; Daly,
2019; Piotrovich et al., 2020), tSMBH source corresponds to a typical SMBH of
mass $10^{9}M_{\odot}$ and magnetic field $10^{4\pm 0.5}$G. Right: numerical
simulation of a decay (ionization) of a freely falling neutral particle (grey
thick) into two charged fragments in the vicinity of a rotating black hole
immersed in an external magnetic field. Positively charged fragment (blue) is
accelerated by the black hole and escapes to infinity along the symmetry axis.
Negatively charged fragment (red) ultimately falls into the black hole,
extracting its rotational energy (see, details in Tursunov et al., 2020).
### 3.2 Axion-like particles as UHECRs
A different approach to explain the observation of UHECR is avoiding the GZK
cutoff via particles beyond the SM. In such a case UHECR must be composed by
new particles that must fulfill the following conditions: (i) be stable enough
to reach the Earth from cosmological distances; (ii) interact very weakly with
the Cosmic Microwave Background and the extra galactic magnetic fields, in
order they do not lose very much energy; (iii) have a significant flux at
their origin; and (iv) interact strong enough in the near galaxy, with the Sun
or Earth magnetic field.
One of the most popular candidates are axions, which were first proposed by
Peccei and Quinn, by introducing an additional global axial symmetry to the
standard model Lagrangian Peccei and Quinn (1977a, b) allowing to solve the
strong CP problem dynamically. Additionally, axions were proposed as
candidates to avoid GZK cutoff in reference Mirizzi et al. (2008); Csaki et
al. (2003). However, it has been shown that it is extremely unlikely that the
axion production, together with their conversion to photons by the galactic
magnetic field, accounts for UHECR within the present exclusion limits
Gorbunov et al. (2001).
A different scenario comes from the consideration of particles with similar
features of axions, called Axion-Like Particles (ALPs). The general Lagrangian
model involving ALPs read as
$\mathscr{L}_{\textsc{alp}}=\frac{1}{2}\partial_{\mu}A\
\partial^{\mu}A-\frac{1}{2}m_{A}^{2}A^{2}-\frac{g_{\textsc{alp}}}{4}aF_{\mu\nu}\tilde{F}^{\mu\nu},$
(2)
here $A$ is the ALP field, $m_{A}$ is the ALP mass, $g_{\textsc{alp}}$ is a
model-dependent coupling between ALPs and photons, $F_{\mu\nu}$ and
$\tilde{F}_{\mu\nu}$ are the electromagnetic field strength and its dual,
respectively.
The Lagrangian in Eq. (2) provides a decay channel for ALPs to photons
($A\to\gamma\gamma$), which plays a fundamental role in experimental searches.
To leading order, the decay width for the above mentioned decay is given by
Beringer and et al (2012); Cadamuro and Redondo (2012)
$\Gamma_{A}\left(A\to\gamma\gamma\right)=\frac{g_{\textsc{alp}}^{2}m_{A}^{3}}{64\pi}.$
(3)
An important feature of ALPs is that they can be converted into photons (and
vice-versa) by means of the Primakoff effect Halprin et al. (1966); this
effect occurs when a strong external magnetic field exist in the region where
the ALPs are propagating. The Primakoff effect can induce an ALP-photon
oscillation, similar to the neutrinos flavor oscillation Sikivie (1983). This
ALP-Photon oscillation changes the polarization of photons traveling in
external magnetic fields, providing an additional mechanism in order to detect
these pseudoscalar particles Maiani et al. (1986). The ALP-Photon oscillation
could also produce an apparent dimming of distant sources as well, affecting
the luminosity-redshift relation of Ia supernovae, the dispersion of quasar
spectra, and the spectrum of the CMB Mirizzi et al. (2008).
On the other hand, due to the weak interaction between ALPs and the CMB, this
can travel across the Universe essentially without decaying. The decay length
of a particle is given by
$\lambda_{A}=\frac{E_{A}}{\Gamma_{A}m_{A}},$ (4)
where $E_{A}$ is the energy of ALP. If we require that the decay length has to
be at least of the order of magnitude of the observed universe $R_{U}$ as
considered in Ref. Gorbunov et al. (2001), one finds the following condition
$R_{U}\lesssim\lambda_{A}\equiv\frac{E_{A}}{\Gamma_{A}m_{A}}\quad\Rightarrow\quad\Gamma_{A}\lesssim\frac{E_{A}}{R_{U}m_{A}}.$
(5)
The Eq. (5) allows one to establish a restriction on the ALPs coupling, which
allow it to reach the Earth from distances beyond the $R_{\textsc{gzk}}$
radius
$g_{\textsc{alp}}\lesssim\left(\frac{64\pi
E_{a}}{R_{U}m_{a}^{4}}\right)^{1/2}.$ (6)
This scenario has been studied in Refs. Gorbunov et al. (2001); Csaki et al.
(2003); Fairbairn et al. (2011), constraining their parameter space according
to current experimental data.
### 3.3 Dark Matter as a source of UHECR
Two long-standing problems of contemporary astrophysics can be formulated as:
(i) What is the nature of Dark Matter? and (ii) How to explain the existence
of cosmic rays with energies greater than $10^{20}$ eV? A plausible hypothesis
is that these two mysteries of science can be explained with just one
scenario. This is in agreement with “Occam’s razor”, which favors the least
possible set of solutions to a collection of seemingly independent problems.
According to this scenario, Super Heavy Dark Matter (SHDM) may decay or be
destroyed via annihilation (see e.g., (Chung et al., 1999)). This implies a
production of super massive – with rest energies of $E\geq 10^{23}$ eV –
particles in the early Universe, during the inflation phase. Such particles
could annihilate or decay presently, leading to the production of jets
containing copious amounts of less massive secondaries, possibly or even
mainly photons. The energies of these secondaries could easily be of the order
of $10^{20}$ eV, the value that seems to be out of reach in the acceleration
processes in the potential sources.
The key prediction of such scenarios in the SHDM group is that the UHECR flux
observed on Earth should be dominated by photons (see e.g., (Rubtsov and et
al, 2006)). On the other hand, the highest energy events observed by the
leading collaborations: Pierre Auger Observatory and Telescope Array, are not
considered photon candidates if the present state of the art air shower
reconstruction procedures are applied. In fact there are no photon candidates
within the energy range where SHDM model should give a photon flux, i.e., for
energies above above $10^{18}$ eV, and this non-observation result leads to
very stringent upper limits on photon fluxes (Petrera, 2019; Aab and et al.
(2019), Pierre Auger Collaboration; Abu-Zayyad and et al. (2013), Telescope
Array Collaboration).
However, there are two main concerns about such conclusions. First, the
present state of the art analysis does not take into account mechanisms that
could potentially lead to a good mimicking of hadronic air showers with the
showers induced by photons, e.g., a very efficient splitting of the primary
energy into secondary photons/particles and underestimation of photonuclear
interaction cross-sections. Second, the present state-of-the-art analysis does
not take into account mechanisms that could lead to an efficient screening or
cascading of the very-high or ultra-high energy photons on their way to the
Earth, e.g., interactions under special cases of Lorentz invariance violation
(Klinkhamer and Schreck, 2008), so that the products of such screening or
cascading are spread over large areas and thus out of reach of the presently
operating observatories, which is interpreted as non-observation of ultra-high
energy (UHE) photons. If the first possibility occurs, this would mean that
UHE photons might have already been detected and they might be in the data but
not properly identified. If the second possibility – namely that the real
properties of cosmic rays and the relevant propagation mechanisms are not well
understood – takes place, then one has little or no chance to detect most of
the very high energy photons that travel towards us. Both possibilities
obviously question the conclusion about limitations imposed on the SHDM
scenarios by the presently accepted upper limits to photon fluxes.
Furthermore, such a conclusion can be accepted only if both concerns above are
alleviated. A detailed study of this issue is imperative in the forthcoming
years.
An inherent part of the SHDM scenario is the very existence of such DM
particles. It is not an overstatement that science has currently no clue of
what DM particles are, nor what their properties would be.
For a long time, the Weakly Interacting Massive Particles (WIMPs) were the
most favored candidate. There was a reason for that – the “WIMP Miracle”. The
number density of the particles that freeze out in the early universe in a
certain epoch is set by the balance of the two rates: the production-
annihilation rate $n_{X}\langle\sigma v\rangle$ and the Universe’s expansion
rate $H$ (the Hubble constant at that epoch). Note that $H$ characterizes the
temperature of plasma in the universe $T$, so the number density of particles
which are in thermal equilibrium can drop exponentially fast with lowering
temperature if particles are heavy, $m_{X}c^{2}\gg k_{B}T$. On the other hand,
the cross-section depends on the coupling constant (i.e., the type of
interaction) and the particle’s mass. Next, the mass density of DM in the
universe depends on both the density and mass, $\Omega_{X}\propto n_{X}m_{X}$,
where $\Omega_{X}$ is the ratio (at present) of the particle-“X” mass density
to the critical density in the universe. Furthermore, the coupling constant
describing weak interactions – the Fermi constant, $G_{F}\approx 1.1\times
10^{5}$ GeV-2 naturally introduces the new mass scale of about $100$ GeV.
Finally, if the cross-section is the weak cross-section and the particle’s
mass is about $m_{X}\sim 100$ GeV, then the abundance of “$X$” is
$\Omega_{X}\propto\langle\sigma v\rangle^{-1}\sim m^{2}_{X}/g_{X}^{4}$ and
$g_{X}\sim 0.6$, so $\Omega_{X}\sim 0.1$. Thus, “$X$” is Dark Matter. This is
the “WIMP miracle”: particle physics independently predicts particles with the
right mass density to be Dark Matter. In this scenario, $\langle\sigma
v\rangle\simeq 3\times 10^{-26}\textrm{ cm}^{3}\textrm{s}^{-1}$; since $v\leq
c$, the gross-section should be $\sigma\geq 10^{-36}\textrm{ cm}^{2}$.
Numerous ongoing direct-detection DM experiments have reached sensitivity
levels that correspond to $\sigma\sim 10^{-44}-10^{-45}\textrm{ cm}^{2}$ in
the mass range of interest – from tens of GeV to a few TeV, without a
statistically significant and reproducible detection. Thus, the WIMP miracle
is at odds with experiment.
The dismissal of the most favorable WIMP miracle in Dark Matter theory has
ignited great interest in alternative scenarios. At present, all bets
regarding the non-standard DM scenarios are on the table. There is no deficit
of the putative candidates, including the super-heavy dark sector candidates.
Here are some examples, as follows. (1) Magnetic monopoles (’t Hooft, 1974;
Polyakov, 1974) are topological defects that naturally appear in Grand Unified
Theories (GUT) and carry a magnetic charge. The natural mass scale for them is
the GUT scale, $\sim 10^{25}$ eV. However, the actual mass of a monopole is
not constrained and all mass scales above the one that can be probed in an
experiment are considered. Being topological defects, monopoles are copiously
produced in a GUT phase transition, creating a severe over-closing problem
$\Omega_{X}\gg 1$. This problem is remedied by inflation, which reduces the
monopole abundance in the amount enough to not contradict observational and
recent cosmological constraints (Medvedev and Loeb, 2017). (2) Wimpzillas
(Kolb and Long, 2017) are superheavy DM particles, which mass scale is many
orders of magnitude above the conventional WIMP scale, possibly as large as
the GUT scale. The WIMP mass cannot exceed hundreds of TeV. Otherwise, heavier
WIMPs would over-close the universe, $\Omega_{X}>1$. Therefore, Wimpzillas,
like monopoles, are not ‘thermal relics’. They emerge out of thermal
equilibrium right after inflation and their density is not determined by the
balance of the production-annihilation rate and the universe expansion rate.
(3) Planckian-scale particles (Garny et al., 2016, 2018, 2019) is a whole
class of candidates that appears in a minimal scenario of DM, where only
gravitational interactions with the standard model are assumed. There is only
one parameter in the scenario – the particle mass, which ranges from TeV up to
the GUT scale. These particles are assumed to be produced by gravitational
scattering in the thermal plasma of the Standard Model sector after inflation.
For example, the Kaluza-Klein excitation of the graviton in the string theory
is one of the realizations of this scenario. The above discussion does not
present a complete list of SHDM candidates, of course, but just a few popular
examples. There are more theoretically predicted candidates.
The search for signatures of propagation and/or decay of particle primaries
beyond the Standard Model is to be taken seriously and thoroughly. In this
regard, CREDO observatory can serve as an indirect search for one of the non-
standard Dark Matter candidates: super heavy particles with masses equal or
exceeding $10^{23}$ eV. Such particles could be produced in the early Universe
around the inflation phase and decay or annihilate presently leading to
observable products in the UHE range (Bhattacharjee, 2000). The interest in
SHDM scenario is supported by certain difficulties the electromagnetic theory
faces at explaining UHECR acceleration, which is discussed below, in Section
3.4.
### 3.4 Constraint on electromagnetic acceleration of UHECR
Electrically charged UHECR traversing a region of size $R$ filled with
magnetic field $B$ assumed to be uniform, loses its energy due to synchrotron
radiative energy loss (Medvedev, 2003) according to
$\frac{dE}{dx}=F_{rad}=-\frac{2}{3}\left(\frac{Ze}{Am_{p}c^{2}}\right)^{4}B^{2}E^{2},$
(7)
where $A$ and $Z$ are the atomic mass and charge of an accelerated nucleus,
$m_{p}$ is the proton mass. The solution to this equation is trivial:
$E=\left(E_{0}^{-1}+E_{cr}^{-1}\right)^{-1}.$ (8)
Thus, for an arbitrarily large initial energy, $E_{0}\to\infty$, the particle
energy cannot exceed the ‘critical energy’ threshold:
$E_{cr}=\frac{3}{2}\left(\frac{Am_{p}c^{2}}{Ze}\right)^{4}\left({B^{2}R}\right)^{-1}\sim
3\times 10^{16}\frac{(A/Z)^{4}}{B_{G}^{2}R_{kpc}}~{}\textrm{eV},$ (9)
where $B_{G}$ is in Gauss and $R_{kpc}$ is in kiloparsec.
Furthermore, even if one is to continuously accelerate the particle along its
curved path via an inductive electric force $F_{EM}=Ze\,E_{ind}\simeq
Ze(v/c)B\sim Ze\,B$, one has the following energy evolution:
$\frac{dE}{dx}=F_{EM}+F_{rad}=ZeB-\frac{2}{3}\left(\frac{Ze}{Am_{p}c^{2}}\right)^{4}B^{2}E^{2}.$
(10)
It has a beautiful solution (assuming the initial energy is small compared to
the final energy):
$E=\sqrt{E_{acc}E_{cr}}\tanh\sqrt{E_{acc}/E_{cr}},$ (11)
where
$E_{acc}=Ze\,B\,R\sim 9\times 10^{23}Z\,B_{G}\,R_{kpc}$ (12)
is the maximum energy the accelerated particle can reach without losses. This
solution has two obvious asymptotic scalings. If $E_{acc}\ll E_{cr}$, i.e.,
losses are small, one recovers the Hillas constraint (Hillas, 1984) $E\simeq
E_{acc}$, and in the opposite limit one has
$E_{\rm max}\simeq\sqrt{E_{acc}E_{cr}}\sim 10^{20}A^{2}Z^{-3/2}B^{-1/2}\
~{}\textrm{eV}.$ (13)
Figure 3: The $B$-$R$ “diagram of state” for UHECR acceleration (Medvedev,
2003). The long-dashed line is the original Hillas relation, Eqn. (12), for a
proton of energy $3\times 10^{20}$ eV. The short-dashed and dot-dashed lines
represent the radiative cooling constraints given by Eqns. (9) and (13), for
the same proton energy. Above the dot-dashed line, the force of radiative
friction dominates over an electromagnetic force. The solid lines represent
the boundaries of the allowed parameter regions for protons with energies
$3\times 10^{20}$, $10^{22}$, and $3\times 10^{23}$ eV, from the outermost to
the innermost “wedge”, respectively. The dotted lines are the same boundaries
for an iron nuclei with energy $3\times 10^{20}$ eV. Only those astrophysical
objects which fall inside the “wedges” are, in principle, capable of
accelerating the particles to such energies. The grey vertical lines mark two
characteristic spatial scales: the GZK attenuation distance ($\sim 20$ Mpc)
and the Hubble horizon size ($\sim 4$ Gpc). For GRB sources we took into
account that the Lorentz boost changes with radius.
These results are summarized in Figure 3. All the curves, except for two
dotted ones, correspond to protons ($A=Z=1$), and the dotted are for iron
nuclei ($A=56,\,Z=26$). The details are discussed in the figure caption. Here
we make a few important conclusions. First, there is an absolute maximum
energy of UHECR accelerated electromagnetically and subject to radiative
losses (e.g., in dipolar magnetospheres, galactic and extragalactic shocks and
such), which is given by the balance of the acceleration and losses. This puts
the upper limit on the magnetic field strength, given by the dot-dashed
horizontal line and by Eq. (13):
$B\lesssim E_{20}^{-2}A^{4}Z^{-3}\textrm{ Gauss},$ (14)
where $E_{20}=E/(10^{20}\textrm{ eV})$. Second, there is a corresponding limit
on the size of an accelerator. This is the size that corresponds to the “tip
of the wedges”, which can be obtained, for instance, from Equations (13) and
(12), namely
$R\gtrsim 6\times 10^{-2}E_{20}^{3}A^{-4}Z^{2}\textrm{ pc}.$ (15)
This constraint effectively rules out compact accelerators, such as neutron
stars, white dwarfs and such. Large objects, such as galactic halos, radio
lobes, and galaxy clusters become favorable. Third, electromagnetic
acceleration of UHECR beyond $\sim 3\times 10^{22}$ eV is hard because the
size of an accelerator becomes comparable to the GZK distance. Furthermore,
the acceleration beyond $\sim 3\times 10^{23}$ eV would require an accelerator
of the size of the observable universe, which is questionable. Fourth, our
analysis above does not take into account that the source may be moving
relativistically with a large Lorentz factor, as in a gamma-ray burst outflow,
for example. Accounting for this, relaxes the size and field strength
constraints but not very dramatically, as far as UHECR are concerned. Fifth,
our analysis excludes special arrangements such as linear accelerators. If one
can arrange particle acceleration along a straight path, e.g., strictly along
a static straight magnetic field line, then the particle would experience no
synchrotron energy loss, regardless of the field strength (still well below
the QED, ‘Schwinger field’ strength). In this case, our analysis is
inapplicable.
### 3.5 SQM objects
Recently, strange quark matter (SQM) objects (either in the form of stars or
planets) have been shown to efficiently convert mechanical energy into
hadronic energy when they oscillate Kutschera et al. (2020). This is possible
thanks to the property that the mass density at the edge of SQM objects of
$4.7\times 10^{14}\mathrm{g}/\mathrm{cm}^{3}$ is the critical density below
which SQM is unstable with respect to decay into photons, hadrons, and
leptons. Either compression or expansion of the SQM object, such as
oscillation induced deformations, releases energy. Oscillations of SQM objects
could be induced in stellar or planetary systems where tidal interactions are
ubiquitous. The excitation energy is converted into electromagnetic energy in
a short time of $1\,\mathrm{ms}$, during a few oscillations. Higher amplitude
oscillations result in faster energy release that could lead even to
fragmentation or dissolution of SQM objects. In the context of CREDO, it would
be interesting to observe periodic signatures of such events.
SQM stars and planets are very sensitive to radial oscillations. The amplitude
of oscillations is physically determined by the excitation process in close
encounters of SQM objects and another compact star or black hole. Depending on
the closest approach distance, the energy transferred to the oscillations can
reach $10^{53}\,\mathrm{erg}$ when the closest approach distance is 3 times
the star radius. By mode couplings, monopole (that is, spherically symmetric
radial) oscillations also would be excited. For violent encounters, the
induced oscillations of the radius result in excitation energy in the surface
layer of every SQM object, equally for stars of pulsar masses and planetary-
mass SQM objects. Fractional amplitudes of radial oscillations $\chi=10^{-6}$
are quite possible. The corresponding deposited energy to be radiated away for
a star of mass $1.4\mathrm{M}_{\odot}$ is estimated to be $6.6\times
10^{36}\,\mathrm{erg}$. For more violent encounters (such as inspiraling of a
tightly bound binary system), one can expect even amplitudes $\chi={10}^{-3}$
and the corresponding energy orders of magnitudes higher. A tightly bound
binary system could induce periodic distortions of a SQM companion and
associated periodic bursts of intense radiation (associated with gravitational
radiation). In the evolution of binary systems with the SQM objects, the
energy loss due to excitations of SQM stars and planets must be accounted for
and this can change significantly the predictions obtained with unexcited SQM
objects (this would be particularly relevant to binary gravitational wave
sources, its exact template calculation should account for excitations of SQM
objects).
To understand the possible radiation scenarios, one may bring to attention the
following estimations taken from Kutschera et al. (2020). For small SQM
objects called planets, the excited energy is deposited in the whole volume of
the object. The energy then scales quadratically with the fractional amplitude
of radial oscillations $\chi$ – for planets $E=7.4\times
10^{48}\cdot\frac{M}{\mathrm{M}_{\oplus}}\chi^{2}\,\mathrm{erg}$. In
particular, $E=7.4\times 10^{38}\,\mathrm{erg}$ for an Earth mass SQM planet
undergoing radial deformation of amplitude $\chi=10^{-5}$. As the mass of the
considered objects increases, the deposition zone shrinks toward the surface
and the energy law changes. In the intermediate region of masses the energy
scaling law can be obtained only numerically. For masses on the order of solar
mass, the energy deposition zone becomes a thin layer and the total energy to
be eventually radiated away can again be estimated analytically and it scales
cubically with the fractional amplitude of oscillations $\chi$ (the resulting
formula is given in Kutschera et al. (2020)). Furthermore, one also has to
take into account a more complicated mass-radius relation for relativistic SQM
stars implied by the equilibrium state numerical solution. For the
representative star model of $M=1.4\,\mathrm{M}_{\odot}$ the radius is
$R=10.3\,\mathrm{km}$ and the energy to be released behaves as $E=6.6\times
10^{54}\chi^{3}\mathrm{erg}$. For the amplitude $\chi=0.001$ this gives
$E=6.6\times 10^{45}\,\mathrm{erg}$ in a $20$m-wide deposition shell.
The process of excitation of quark matter is an irreversible one. The
excitation energy will be dissipated eventually into heat and radiation. In
the intermediate phase the excitation energy is being converted into
electromagnetic energy. The generation rate of radial oscillations of the SQM
star can be estimated by assuming that the timescale of radial oscillations is
comparable with that already known in the model without the excitation energy,
which is $T=0.37\times 10^{-3}\mathrm{s}$. The period is much larger than the
timescale for electromagnetic interactions of $10^{-16}\mathrm{s}$. Thus the
excitation energy can be assumed to be converted into electromagnetic energy
inside the SQM star instantaneously. The rate of electromagnetic energy
generation within a quarter of the oscillation period is $7.2\times
10^{49}\mathrm{erg}/\mathrm{s}$ for $\chi=10^{-3}$ (and it increases with
$\chi$, eg. $6.7\times 10^{55}\mathrm{erg}/\mathrm{s}$ for $\chi=0.1$).
Further investigations are required to find how much of the energy will be
radiated away and how fast this will proceed. A rough conservative estimation
made in Kutschera et al. (2020) shows that the luminosity for the energy
radiated away by the outermost shell of thickness on the order of the photon
mean free path would amount to $L=1.3\times 10^{34}\mathrm{erg}/\mathrm{s}$
(for $\chi=10^{-3}$) with a corresponding effective temperature of $2.0\times
10^{6}\,\mathrm{K}$. The total energy of $E=6.6\times 10^{45}\,\mathrm{erg}$
released in the star would thus sustain radiation for $1.6\times 10^{4}$
years. However, neutrino cooling will switch on in $10^{-6}\,\mathrm{s}$ and
the star will cool fast. The estimated neutrino cooling time for the volume of
the star $1.4\times 10^{3}\,\mathrm{s}$ is $3.5\times 10^{8}$ times shorter
(respectively, $1.5\times 10^{5}\,\mathrm{s}$ for the volume of the initial
deposition shell is $3.4\times 10^{6}$ times shorter) than the electromagnetic
radiation time. With the considered excitation mechanism accounted for, the
extreme, e.g. $\chi=0.1$ large oscillations, are rather excluded in nature,
since the energy of such oscillations would be less than the value of
excitation energy generated during the time of a single oscillation, therefore
oscillations would be damped during the first cycle). For lower amplitudes the
energy generation time is longer. More accurate calculations would require
taking into account the evolution of the temperature of the SQM star. The
damping effect on radial oscillations by weak quark processes, although
efficient, is not expected to determine the discussed strong interaction
phase, although it might dominate the thermalization of excited SQM.
### 3.6 Neutron star collapse to third family
Massive neutron stars may support in their cores exotic states of matter
different from hadronic, like deconfined quark matter. These stars are denoted
hybrid stars. In the case of a strong first order phase transition inside
compact stars, their mass-radius relation presents a gap of unstable
configurations between pure hadronic and hybrid stars. The latter ones will
populate the so-called “third branch”, following the hadronic neutron stars
and white dwarfs branches. The transition scenarios may correspond to a
neutron star configuration lying at the top end of the hadronic branch whose
central density is right below the critical value for deconfinement. There are
possible ways for the increase of central density of such a neutron star, thus
triggering the transition to a third branch configuration. For instance, a
fast rotating neutron star may lose energy by dipole emission resulting in a
spin down, or a neutron star in a binary system may undergo an accretion-
induced spin-up by matter from a companion. The latter case has been recently
proposed as an explanation for eccentric orbits of binary systems where the
neutron star is able to accrete matter from a circumbinary disk created after
a low-mass X-ray episode. Consequently, the neutrino burst which accompanies
the deconfinement phase transition in the neutron star interior may trigger a
pulsar kick producing the observed eccentric orbit Alvarez-Castillo and et al
(2019). A pair of compact stars of about the same mass, each one of them lying
in different branches of their mass-radius relation are usually called “twin
stars”, see Benic et al. (2015); Alvarez-Castillo and Blaschke (2017);
Blaschke et al. (2020). The aforementioned dynamical scenarios, where one of
the neutron stars collapses into its twin star, may conserve the total baryon
number resulting in a mass defect of less than a tenth of the solar mass for
state-of-the-art equations of state, which corresponds to an energy of a few
$10^{51}$ ergs Alvarez-Castillo et al. (2015).
It is therefore feasible that the possible deconfinement phase transition in
compact stars produce energetic emissions, for instance acting as an engine
for Gamma Ray Bursts, with an accompanied characteristic neutrino signal as
well as high energy cosmic rays. An analogous situation can be discussed in
the framework of neutron star mergers, where both electromagnetic and
gravitational radiation are emitted, together with cosmic rays. For the
GW170817 event , it has been estimated that the cosmic rays flux is able to
support the population of cosmic rays detected at Earth below the “ankle”
Rodrigues et al. (2019).
### 3.7 UHECR as the spacetime structure probe?
The CRE hypothesis can be considered a candidate scenario for a number of
unexplained cosmic-ray measurements. The examples include Smith et al. 1983
Smith et al. (1983) (32 TeV EAS within 5 minutes while only one such event
would have been expected) and Fegan et al. 1983 Fegan et al. (1983)
(simultaneous increase in the cosmic-ray shower rate at two recording stations
250 km apart). On one hand the mentioned measurements were single
observations, not seen by other groups, but on the other, nobody checked
further on a global scale. We are going to do so within this project – using
the CREDO meta-structure composed of the detectors operated by numerous
research groups in all the available energy ranges. This will offer a chance
to confirm or question the aforementioned historical reports, potentially
leading to the observation of New Physics effects, including probing the
spacetime structure. Although the expectations concerning the potential
observations of spacetime structure manifestations through effects accumulated
over large astrophysical distances are highly uncertain due to missing physics
formalism below the Planck scale, as stated e.g. in Ng, Y Jack (2003); Carlip
(2015), the cumulation of time delays of high energy photons travelling
astrophysical distances is thinkable, especially if one considers photons of
different energies emitted simultaneously. The cumulative time delay of higher
energy photons with respect to the lower energy ones might or might not depend
on photon energies – it is hard to tell due to the missing formalism regarding
the proper averaging of spacetime foam fluctuations Ng, Y Jack (2003). (Due to
quantum fluctuations, spacetime, when probed at very small scales, will appear
very complicated – something akin in complexity to a chaotic turbulent froth,
which physicist John Wheeler dubbed quantum foam, also known as spacetime
foam.) If the cumulative effects of spacetime foam fluctuations are
independent of photon energies, then they might be too small to be observed.
On the other hand, if they depend on energy, one cannot exclude time delays
even as large as of the order of minutes – comparable to the observations
mentioned above. Therefore if, as hoped in CREDO, we can observe CRE composed
of high energy photons of different energies, possibly spanning the whole
cosmic ray energy spectrum, i.e. from GeV to ZeV, then in any case we will
have a new “spacetime probe” at hand – to bring a new input for tuning the
existing, or even building the new spacetime models, be it delay evidence or
null results that further constrain the available theories. Thus, in any case,
we are entitled to expect that within this proposal we will contribute in a
novel way to getting closer to understanding the cumulative effects in
spacetime foam fluctuations, which would mean a contribution to the
foundations of science. While proper averaging methods of cumulative spacetime
foam fluctuations are yet to be developed (as stated in Ng, Y Jack (2003)),
and despite the uncertainty concerning energy dependence of such effects, not
exploring the new research opportunity offered by CREDO would be a
methodological mistake. The research in the proposed direction should begin
with a wide review of spacetime structure models which predict differences in
time delays between the arrival times of high energy photons of different
energies. With such a review the experimental efforts dedicated to CRE could
be prioritized accordingly so that appropriate detection and monitoring
algorithms could be developed, including the alerting mechanisms on a
subthreshold level to be analysed in accordance with the multi-messenger
astrophysics strategies. The latter direction gives promising perspectives for
new advances in astrophysics, reflected also in the new attention of
theorists, also those addressing the questions related to space time structure
(see e.g. Carlip (2019); Wang and Unruh (2019); Ng, Y Jack (2019)) and private
interest in discussions222P. Homola’s private communication with S. Carlip, Y.
J. Ng, and Q. Wang..
## 4 CRE simulations
High energy particles that propagate through the Universe unavoidably interact
with background radiation, magnetic fields, and matter, initiating cascades of
secondary particles which might be observed with available techniques – like
in artificial particle colliders but on a much larger scale. While an
interaction of a particle during its propagation through the Cosmos is
commonly approximated with extinction, one has to admit that the logically
correct question about chances for observing CRE is not if they exist, but
when and how we could observe them. This argument is illustrated with Fig. 4.
Let us consider a CRE initiated near the Earth, taking as an example the
preshower effect Erber (1966); McBreen and Lambert (1981); Homola and et al
(2005); Dhital and et al. (2018) (CREDO Collab.), i.e. an interaction of an
UHE photon and subsequent secondary electrons with the geomagnetic field above
the Earth’s atmosphere. The word “preshower” is meant to describe the result
of the initial interaction: “pre” emphasizes the location of the interaction
vertex above the atmosphere, i.e. before the extensive air showers (EAS) are
initiated; “shower” addresses the number of particles that emerge instead of
one primary UHE photon. Since the distribution of preshower particles (mostly
photons) above the atmosphere is confined to a very small region (a fraction
of cm2) the resultant extensive air shower will have properties very similar
to air showers initiated by single primary particles of the energies
corresponding to the photon which initiated this preshower. Thus the
observation of a preshower-like CRE will require nothing else but a standard
infrastructure dedicated to the detection of UHECR, and in Fig. 4 (left panel)
we name this scenario an “obvious detection” example. On the other hand, if a
CRE particle distribution is so sparse that only one particle can reach the
ideal detector system at a time, there is no technical chance to interpret
this individual particle as a component of a CRE. Thus we consider this
scenario an obvious technical limitation for CRE observation, in Fig. 4 named
“obvious extinction” (right panel). Consequently, the focus of any research
dedicated to CRE is limited to the scenarios that can result in particle
distributions less sparse than the “obvious extinction” limit, as also
illustrated in Fig. 4 with the “obvious between” CRE case (middle panel), so
far unexplored with a globally coordinated observational effort.
Figure 4: Obviously in experimental reach, although not yet probed, detection
of Cosmic Ray Ensembles, using ultra-high energy photons as example primary
particles.
As far as CRE scenarios are concerned, the ongoing simulation studies include
synchrotron radiation of high energy electrons in the presence of planetary,
stellar and galactic magnetic fields.
As shown e.g. in Ref. Risse and Homola (2007), in case of the preshower effect
occurring due to UHE photon interactions with the geomagnetic field the
resultant EAS can hardly be distinguished from those initiated by individual
particles based on standard air shower observables like the atmospheric depth
of shower maximum development or muon content in the particle distribution on
the ground – unless a large statistics of events is available. Given the
stringent UHE photon limits one does not expect many photon- or preshower-
induced events to be observed even with the largest air shower detector arrays
like the Pierre Auger Observatory or Telescope Array. Instead, one might
consider alternative observables and alternative infrastructures, more
sensitive to electromagnetic multi-primary EAS origins, e.g. those connected
with Cherenkov emission induced by air shower particles and being observed by
gamma-ray telescopes. A study in this direction was presented in Ref. Almeida
Cheminant and et al. (2020) (CREDO Collab.), where the authors analyze the
feasibility of detecting preshower-induced EAS using Monte-Carlo simulations
of nearly horizontal air showers for the example of the La Palma site of the
Cherenkov Telescope Array (CTA), as illustrated in Fig. 5.
Figure 5: An ultra-high energy photon interacting with the transverse
component of the geomagnetic field produces an $e^{+}/e^{-}$ pair ${\sim}$1000
km above sea level which emits bremsstrahlung photons. As such a process can
repeat itself for some of these photons, a collection of particles (mainly
photons and a few $e^{+}$ and $e^{-}$) reaches the top of the atmosphere.
Consequently, atmospheric air showers are produced and in the case of nearly
horizontal showers, only the muonic component reaches the Imaging Atmospheric
Cherenkov Telescopes (IACTs) on the ground, which detect the Cherenkov
emission of this component. Almeida Cheminant and et al. (2020) (CREDO
Collab.)
It was demonstrated that there is a realistic chance for identification of
preshowers induced by 40 EeV photons coming from an astrophysical point source
during 30 hours of observation. This result confirms the Imaging Atmospheric
Cherenkov Telescope (IACT) technique could be used to probe physical phenomena
not only in the TeV domain, but also in the EeV regime, with a particular
connection to the CRE-related physics. Although, as inferred from the upper
limits to UHE photons, the rate of expected preshower events for the studied
configuration of the observatory is quite low, the gamma/hadron separation
obtained by adopting the nearly-horizontal observation mode allows for strong
filters to be applied in order to identify such events with a high degree of
confidence. Searches for particles with low expected flux using IACTs have
been previously performed, as it is the case for the tau neutrino Ahnen et al.
(2018) or UHE cosmic rays, as demonstrated in Neronov et al. (2016). Moreover,
multimessenger alerts obtained from other operating observatories may allow
fast pointing of the telescopes towards sources suspected to be capable of
producing UHE photons, e.g. via interactions between UHE cosmic rays possibly
produced by AGNs and the cosmic microwave background, or during gamma-ray
bursts, which would significantly increase the chance probability to observe
UHE photons and preshower-like CRE. Such potential is well illustrated by the
correlation of the arrival direction of a 290 TeV neutrino observed by IceCube
with gamma emissions from blazar TXS 0506+056 observed by MAGIC and FERMI-LAT
IceCube Collaboration et al. (2018). Moreover, a program of observation could
be run on catalogs of high energy sources, with an observation time
significantly higher than the 30h presented in Ref. Almeida Cheminant and et
al. (2020) (CREDO Collab.).
Figure 6: An example energy spectrum of a cosmic ray ensemble generated by an
ultra-high energy photon in the vicinity of the Sun, after the interaction
with the solar magnetic field (primary photon energy: 1020 eV, the
heliocentric latitude of “the closest approach point”: 0o, and the impact
parameter: 3 R⊙). Dhital and et al. (2018) (CREDO Collab.)
To generalize the notion “preshower”, the term “super-preshower” was
introduced (see e.g. Dhital and et al. (2018) (CREDO Collab.)). Super-
preshowers (SPS) are cascades of electromagnetic particles, thus a subset of
potentially wider CRE family, originated above the Earth’s atmosphere, no
matter the initiating process and distance to the Earth. Super-preshowers can
be classified with respect to their potentially principal observable
properties: spread in space and arrival times. For an example, a cascade
initiated due to the preshower effect in the geomagnetic field (even at
altitudes as high as 10000 km a.s.l.) is expected to have the lateral spread
above the atmosphere (100 km a.s.l.) in the order of millimeters, and
negligible spread of arrival times Homola and et al (2005); Homola et al.
(2013). If the preshower effect was to occur in the vicinity of the Sun, one
would expect still negligible spread of arrival times, but the lateral spread
then might reach the size of the Earth Dhital and et al. (2018) (CREDO
Collab.) or even larger. The resultant SPS signature is then expected to be
composed of even hundreds of thousands of extensive air showers, forming a
characteristic, very thin (order of centimeters) and elongated (up to millions
of kilometers) pattern. This example shows that by analyzing the properties of
CRE/SPS one might approach attributing a non-trivial physical scenario to the
observable event category, provided the underlying uncertainties are properly
understood and quantified – as planned by CREDO. The few calculations
performed so far in this direction can still serve only as qualitative
indications concerning potential CRE/SPS observables. For instance, in
Ref.Bednarek (1999) the lateral spread of an SPS originated nearby the Sun is
simulated using a private code and assuming that the SPS is composed only of
photons of energies larger than E=1017 eV, while from the more detailed
calculation done in Ref. Dhital and et al. (2018) (CREDO Collab.), obtained
with an open source public code PRESHOWER Homola and et al (2005); Homola et
al. (2013), one learns that the energy spectrum of SPS particles might be
extended over a wide range, down to TeV and lower, as illutrated in Fig. 6.
TeV photons would certainly induce air showers that would contain particles,
mostly muons, observable on the Earth’s surface. While in large cosmic-ray
observatories these muons are treated as background, the complete CRE/SPS-
oriented research we propose here has to include a proper handling of this
“unwanted muon background” which can be processed to extract a signal induced
by ensembles of low energy air showers arriving simultaneously at the detector
– very clear, unique, and so far untested signature. We make one step further
and propose a global analysis of the data from the available detectors to
search for extremely spread CRE/SPS events, inaccessible by the largest
observatories taken individually. The experimental question which can be
addressed in this regard is: “which CRE/SPS fronts, and in which
circumstances, can be detected by a network of devices located on Earth or
around it?”. It is the question about the SPS/CRE particle density on the top
of the atmosphere (or speaking more general: on top and within the
technosphere) and the particle density, or any other observable information,
like e.g. Cherenkov light, related to the resultant air shower ensembles.
Another type of a CRE scenario, where electromagnetic particles play a role,
is the propagation of very high energy electrons through intergalactic and
galactic magnetic fields. One of the analysis in this direction currently
being carried out within the CREDO Collaboration concerns the propagation of
electrons of energies between 1017 eV and 1019 eV within the Galaxy, using
CRPropa Batista et al. (2016) – a Monte Carlo simulation of cosmic ray
propagation and the state-of-the art modeling of the galactic magnetic fields
to quantify the chances of observing a CRE on Earth. The intermediate results
indicate qualitatively that one might have a chance of observing a CRE
originating from synchrotron radiation occurring within the Galaxy or maybe
event at some other extragalactic sources. Quantification of the observational
chances is still on the way, but already the qualitative study conveys a very
important message: even “conventional” and abundant electromagnetic processes
like synchrotron radiation in galactic magnetic fields are expected to
generate CRE reaching the Earth.
At the end of this section it is worth emphasizing that the experimental
strategies dedicated to CRE do not need to rely on specific theoretical
scenarios – one might also “fish” for clearly non-random cosmic-ray global
footprints. This approach is justified by the ultra-high energy physics
uncertainties we are aware of – large enough to admit that we might not be
able to imagine all the possible physics scenarios predicting a CRE signal
that would be observable on Earth. It is therefore sensible “just” to explore
fully the potential of the infrastructure we have at hand and go fishing for
signatures we are not able to predict, but which we can distinguish from the
diffuse (random) cosmic-ray background.
## 5 CREDO detectors: cloud of clouds
As explained above, any experimental strategy oriented on observation and
investigation on large scale cosmic ray correlations in the form of widely
defined CRE require a global approach to cosmic-ray research. Since both
extensive air showers and incoherent cosmic rays might originate from a CRE,
one realizes that in fact any detectors capable of detecting cosmic ray
signals, both on the ground or on satellites, can potentially serve as
valuable components of a global data acquisition system. In other words, the
chances for CRE observation increase with any single detector or observatory
joining the collective observational effort. Within this general concept,
schematically depicted in Fig. 7 the role of CREDO can be understood as an
umbrella research program that enables a collaborative effort dedicated to CRE
with the use of existing infrastructure and expertise, and with openness for
designing and building complementary detectors or arrays – if required and
justified by specific CRE models and research plans.
Figure 7: The concept of the Cosmic Ray Extremely Distributed Observatory
(CREDO): open on two ends (data upload and access), using both professional,
dedicated cosmic ray infrastructure and off-the-shelf detection solutions,
including smartphones.
Widening the participation in the CREDO program increases its scientific
potential. The CREDO program benefits all participants by being as open as
possible through the following: removing or reducing as many non-scientific
barriers that block findability, availability and interoperability of
contributing data sets as possible, and facilitating the usage of the systems,
taking into account different needs and levels of expertise of the individual
participants and institutional stakeholders. To illustrate technological
diversity behind the general CREDO concept in this section, we briefly sketch
a landscape of detection techniques used in the state-of-the-art cosmic ray
research with particular emphasis on the current situation, needs and
interests of the CREDO Collaboration.
### 5.1 Cosmic ray detection techniques
The current CREDO collaboration with existing experiments and all the future
planned detector installations allows for the extension of the widely used EAS
(Extensive Air Shower) detection and analysis techniques. EAS detector systems
are typically designed as a grid of individual detectors that spreads over a
large area. This is due to two reasons: the disk of the EAS at the observation
level can be up to a few kilometers in diameter for ultra-high energy events,
and the frequency of such events per km2 is low so higher grid area means
higher event rate.
The most common technique that is applied as part of data analysis uses the
particle density distribution within the EAS disk. The average distribution of
the particle density in the disk varies with the distance from the shower axis
in a known way; it additionally depends on the energy of the primary – E0.
Figure 8 Beisembaev et al. (2019) illustrates the distribution of the particle
densities in simulated EAS disks at the different distance from the axis of
simulated showers. This is shown for several values of the E0 of the primary
protons. The simulations were done using CORSIKA Heck et al. (1998) software
package. Note that the particle density distribution is non-linear (approx.
inverse quadratic far from the axis). Additionally, each individual detector
within the grid provides the time of the hit, e.g. some timing information
when the signal in this detector went over a preset threshold. This
information is used to determine the EAS arrival direction.
Figure 8: Particle density distribution in simulated EAS disk vs. distance
from axis for different E0 Beisembaev et al. (2019).
However, there is an unused piece of information here. As the EAS disk passes
the detection level, the detectors can measure not only the integral number of
detected particles but their distribution over time as well, getting an
effective width of the disk Beznosko et al. (2020). This technique was first
developed and used at the Horizon-T Beisembaev et al. (2020) experiment that
holds close ties with the CREDO collaboration. This approach uses the width of
the EAS disk as measured by different detectors during the event. Figure 9
Beisembaev et al. (2019) illustrates the widths of the same simulated EAS from
Figure 8. The grey lines of the plot show approximated widths at the distances
from the axis where particle density is low for reliable determination. The
width behaves in a more linear way when compared to particle density,
specifically closer to the axis, and is weakly dependent on the primary
particle energy E0. Using all available information from arrival time ,
particle density distribution and disk width, one gains advantages such as the
ability to do some analysis on the events with axis outside of the detector
active area. This ability is due to the fact that the approximate position of
the axis position can be reasonably estimated from the disk width information.
However, such detectors and specifically electronics are more costly as fast
Flash ADC and fast PMT with matching detection medium are required Beznosko et
al. (2017).
Figure 9: Simulated EAS disk width vs. distance from axis for different E0
Beisembaev et al. (2019).
### 5.2 Overview of the different EAS detection techniques – scintillator,
water Cherenkov, CMOS/CCD, air fluorescence, radio
Numerous methods are used by existing experiments to detect charged particles
from the EAS disk. As we cannot see the particles themselves, the principle is
to convert the passage of the particle through the detector into a signal that
is easy to process, or observe the result of the particle interaction with the
detection medium.
### CMOS/CCD
The most direct method involves using a silicon-based pixelated strip. The
most commonly found detectors of this type are in everyone’s cell phones and
photo cameras – they are the CMOS/CCD photosensors (more information is
available in Fossum and Hondongwa (2014)). On passage through the pixel,
charged particles as well as gamma and x-rays affect CMOS/CCD sensors in a
similar way as light does. If the camera cap is closed (or cell phone lens is
well covered) so that the sensor is in the dark, the pixels affected, or hit,
by a particle passage will produce a response as if they were exposed to
light. If more than a single pixel is hit, a part of the particle track may be
detected as well as shown in Figure 10. This method is currently used by the
CREDO Collaboration using the cell phone cameras and a special app Bibrzycki
and et al. (2020) (CREDO Collab.); Niedzwiecki and et al. (2019) (CREDO
Collab.) extendable or portable to a browser application.
Figure 10: A sample event of a cosmic ray leaving a track on a CCD sensor
Niedzwiecki and et al. (2019) (CREDO Collab.).
### Water-based Cherenkov detector
Another detection method that is based on observing the results of particle-
matter interaction relies on the detection of the Cherenkov light – an
electromagnetic shock wave from the charged particle passing the transparent
medium (water, air, glass, plastic) faster than the speed of light inside that
medium (read more on Cherenkov radiation in Watson (2011)). The shock wave
consists of thousands of photons per 1 cm of particle path in the medium that
can be detected using fast photodetectors with internal signal amplification,
such as PMT (Photo Multiplier Tube).
The design of the water-based Cherenkov detector is very simple – an insulated
volume of water and at least one PMT that is optically connected with this
volume. A charged particle creates a cone of Cherenkov light that is detected
by a PMT. If the volume is very large and many PMTs are used, the Cherenkov
light cone can be imaged and used for measuring particle momentum and
direction: this technique is used, for example, by Super-Kamiokande detector
(more information please see Walter (2008)).
The advantages of water-based detectors are the detection speed (on the order
of one nanosecond Beznosko et al. (2017)) and relatively lower cost since the
detection medium is water (typically, ultra-pure). However, the Cherenkov
signal is relatively weak. Other disadvantages are low to no mobility and the
insulation of high voltage electric components like PMT from the water.
### Scintillator-based particle detectors
The scintillators are the materials that produce a so-called scintillation
light in response to a charged particle passage in addition to Cherenkov
light. The main advantages of scintillators over water or glass as detection
medium is that there is about 10-15 times more scintillation light photons
than from Cherenkov radiation. Also, most scintillators are solid and the
detector is easily movable in most cases and does not require a lot of
additional external support, but there are liquid and even water-based
scintillators Bignell et al. (2015). Specifically, plastic scintillators are
lightweight and inert solids that can be used for production of detectors that
are suitable for installation at schools. In such cases, PMT cannot be used
due to high voltage being a hazard, but there are silicon versions of PMT that
work at the safe voltage such as SiPM, MPPC, MRS etc. Duspayev et al. (2016)
Beznosko (2009). Thus, scintillator-based detectors are most suited for CREDO
long-term outreach and educational goals. The main disadvantages are the cost
and a slower response when compared to Cherenkov light Beznosko et al. (2017)
– on the order of 10 ns for plastic and liquid scintillators, on the order of
100 ns for inorganic.
### Air fluorescence detectors
As the EAS disk travels through the air, the charged particles that comprise
it ionize the air molecules along the entire EAS path. As the electrons return
to their ground state, the energy is released as light. This light can be
collected by typically the collection of mirrors with multiple PMTs as light
detectors (there are various designs and detectors are used). The main
advantage for this method is that with some luck when viewed at the right
angle, an actual development of EAS can be captured and accurate estimates of
the properties of the parent particle can be made. The main con is that the
observations can be done only during Moonless and clear nights. Fluorescent
detectors are often combined with scintillator or water-Cherenkov detectors.
More information can be found in Tkachev and et al (2013).
### Radio signal detectors
As the EAS disk moves in the Earth magnetic field, the positive and negative
charges within it experience Lorentz force in opposite directions and radiate
in a radio frequency range around 30-80 MHz or so Aab and et al. (2016)
(Pierre Auger Collab.). This effect is highest for the EAS closer to the
horizon and moving perpendicular to the magnetic lines of Earth. While many
experiments are actively trying to use this method, it is a supplement to all
other detection methods listed.
### 5.3 The CREDO extension proposals
The involvement of non-scientists and outreach is one of the pillars for CREDO
collaboration. There are currently proposals to design and produce portable,
most likely scintillator-based EAS detectors that could be used at education
establishments (schools, universities) as both demonstrators and part of the
education process. The designs are all being proposed with the flash ADC
capabilities to utilize this expanded set of analysis techniques that involve
using advanced EAS timing information and possibly extend the searches for new
phenomena within EAS such as ‘unusual’ events described in Beisembaev et al.
(2019) Beisembaev and et al (2019).
One of such designs is CREDO-Maze, a project that will create a global, unique
physical apparatus, which will consist of a network of local (school)
measuring stations. The concept of the CREDO-Maze array was developed based on
the 20 years old Roland Maze Project Gawin et al. (2002); Feder et al. (2005).
The technology today has developed greatly and the local shower array idea of
Linsley Linsley (1985) can now be implemented much more easily and,
critically, much more cheaply. Eventually it is planned to equip high-schools
with sets of at least four professional portable cosmic ray detectors
connected locally and forming the small school EAS array. Feasibility of EAS
detection with such a mini-network was demonstrated in Ref. Karbowiak et al.
(2020) where pocket-size (sensitive surface of $\sim$ ($25cm^{2}$) affordable
(cost $\sim$ 100 $ / piece) scintillator detectors Axani et al. (2018) were
used. The CREDO-Maze project uses technologically sophisticated measuring
equipment in extracurricular activities: detectors of charged relativistic
elementary particles will be made of small ($0.02m^{2}$) plastic
scintillators. The light pulses will be collected by optical fibers shifting
the wavelength from ultraviolet to green and then light will be converted into
electrical signals by Silicon Photomultipliers. Further electronics will be
based on high speed digital circuits and microcontrollers to connect to
higher-level servers via the Internet and WiFi links. Prototypes of individual
components of the apparatus have been largely developed independently by
several institutions.
One of the important parameters of proposed equipment is the cost. It is easy
to build expensive and complicated, 100$\%$ effective professional arrays. We
are on the way of building the prototype which cost (including scintillators,
SiPMs, trigger electronics, storage and data transmission micro-computer) is
below 200 $ (compared to 3000 € per detector for $\mu$Cosmics detector in
Petropoulos et al. (2020). Prototypes of individual elements of the apparatus
have been largely developed independently in several partner academic centres:
University of Lodz, National Centre for Nuclear Studies and Institute of
Nuclear Physics in Poland, Institute of Experimental and Applied Physics, in
Czech Republic, Swinburne University of Technology in Australia. Completion of
the whole and its technical adaptation for replication will be one of the
interesting tasks of the project. It is an interesting concept to deliver to
some of the end-users (schools) kits, which are adapted for this purpose,
assembled only in basic, skill-intensive parts. This would allow students in
their local project groups to build and assemble from them a fully operational
and efficient whole, under the supervision, of course, of the staff of the
institutes managing the local project networks. The independent construction
of the operating scientific equipment is an additional motivating element and
undoubtedly increases the involvement of young people and the general interest
of those not participating in the project. These effects were observed in
previous attempts to implement similar activities on a smaller scale.
On the other side it should be mentioned that proposed devices are designed
and implemented in such a way that, while maintaining high standards, they are
as inexpensive as possible. Technologies will be developed to ensure that the
measurement kits can be duplicated and distributed to end users as “self-
assembly kits” with different degrees of sophistication of the finished
components. As potential business projects they will be able, together with
educational material pledges and software, to provide a ready-made market
product. With positive recommendations based on our research results, the
potential market, the demand of educational institutions, seems to be quite
considerable. The creation of local structures comprising young people
involved and organised in research groups (led by teachers/educators) using
network communication and based on science centres, as, e.g., higher education
institutions, universities, is an important step in the development and
institutional activities research performing organisations, including , as
well as research funding organisations. The proposed actions open up new areas
of innovation in non-formal non-school education. Creating a model system of
social communication networks and demonstrating its effectiveness in the
proposed field being an element of STEM will allow to plan and create similar
networks realized in other areas of education. There are no contraindications
for such networks to cover various groups of young people and research
centres.
Another concept of building a large scale cosmic-ray network is based on
engaging the wide community, thanks to the attractive properties and
portability of devices such as pixel cameras. A benchmark standard here is
Minipix Timepix-EDU, a hybrid semiconductor pixel detector, with a silicone
sensor of various thicknesses (e.g. 300, 500 $\mu m$) provided by ADVACAM adv
, using the technology developed within the Medipix Collaboration at CERN med
. This detector provides specific characteristics: fast-data acquisition,
portable, lightweight, easy to use and easy to place in difficult-to-access
environments, and can be operated remotely. Minipix Timepix EDU is a
simplified and price effective version of the Minipix Timepix detector. It was
designed and created with the purpose to make science accessible to the public
(schools, scientific research centers, non-commercial institutions). The
simplified version of the software Pixet, with predefined settings, makes it
possible to operate the detector without advanced training in data acquisition
just by connecting the device to the USB port of the PC.
Figure 11: Illustration of the Minipix Timepix EDU detector dedicated for
educational purposes, www.advacam.com, accessed on 22.09.2020.
The ASIC read–out chip contains a matrix of 256 by 256 pixels (total 65,536
independent channels) and an active sensor area of 14 mm by 14 mm, where one
pixel corresponds to 55 $\mu m$, see Fig. 11. The dark noise signal enables
measurements of low–Linear Energy Transfer (LET) particles with high
precision. Given to the per–pixel calibration, an adjustable threshold is made
in each pixel. This allows a detection efficiency close to 100% for heavy
charged particles, making these detectors unique and suitable for the
detection of cosmic rays.
### Online track visualization and processing software
The software package Pixet Pro (D. Turecek, J. Jakubek, 2020, Advacam s.r.o.,
Prague, Czech Republic) is used to operate the detector and to control the
readout, data acquisition and recording. The detectors can be connected to a
standard notebook using a USB 2.0 port. The PC connectivity and cross-platform
operating system compatibility includes Windows, Linux and Mac OS.
Figure 12: An example of per-pixel energy deposited by various particles in a
mixed radiation field measured using a Minipix Timepix detector with Silicone
sensor. The detector was operated in ToT-frame mode with an acquisition time
of 10 ms. Low-energy, narrow, curly tracks are typical for electrons, high-
energy, wide, straight tracks for energetic heavy charged particles such as
protons.
An example of a mixed field frame can be seen in Figure 12. Large roundish
blobs are created by alpha particles, long strikes by cosmic muons, curving
tracks by electrons or small dots by gamma or X-rays. Moreover, rare and
exotic events can be observed: Delta electrons, recoiled nuclei, cascade of
two or more nuclear transitions, proton tracks. According to the frame
occupancy the acquisition time can be set. Other parameters such as threshold
are already predefined. Further, the data can be processed as dose rates,
absorbed dose, fluence maps, energy deposited, LET spectra Granja and et al
(2016).
A notable role in the implementation of the CRE-oriented detection strategies
is also going to be played by professional, medium size detectors giving a
high quality signal.
Since 2018, low-background, digital gamma gamma-ray spectrometer with active
shield has been operating in a ground level laboratory in the Department of
Nuclear Physical Chemistry, Institute of Nuclear Physics Polish Academy of
Sciences (IFJ PAN), Krakow, Poland. The spectrometer is equipped with Broad
Energy Germanium detector BE5030 (Canberra, USA), multi-layer passive shield
and five large, plastic scintillators (Scionix, Netherlands), playing a role
of additional active shield (veto system). Areas of scintillation detectors
range from ca. 0.14 up to 0.49 m2. Data acquisition as well as signal
processing are performed by means of digital analyzer (so called digitizer)
DT5725 (CAEN, Italy) Gorzkiewicz et al. (2019).
The main role of both passive and active shielding is reduction of the
radiation background of the germanium detector. However, thanks to digital
data acquisition it has become possible to expand the research potential of
the constructed spectrometer. Registering and storing data generated by all
spectrometer’s detectors and using manifold, off-line data exploration
techniques allowed one to initiate continuous monitoring of the cosmic-ray
muon flux.
From the CREDO Project point of view, such device may be used as a reference
detector because scintillators register several dozen of muons per second and
thanks to their non-collinear orientation (Fig. 1 in Gorzkiewicz et al.
(2019)) it is possible to detect at least 3 muons correlated both
geometrically and in time. Preliminary research showed that during single
gamma-ray spectrometry measurement, that lasts 426721 s (about 5 days),
scintillators registered 329 events of 5-fold detection coincidences of muons
(Fig. 13) that could originate from air showers.
Figure 13: Distribution of maximum time intervals of 329 events of five-fold
coincidences registered by spectrometer’s active shield during single gamma-
ray spectrometry measurement (time of measurement 426721 s).
Additional advantage of continuous monitoring of cosmic-ray muons flux is the
possibility to investigate correlation of changes in its intensity and
Earthquakes that modulate the local geomagnetic field Kovalyov and Kovalyov
(2014).
### 5.4 Inter-detector communication
An important aspect of R$\&$D concerning the CRE detection feasibility is
communication between detectors constituting an array. It requires attention
especially in deployment areas with limited access to internet and/or
electricity, where data received by an individual detector cannot be
transferred directly to the central acquisition system. Communication might
also be critically important in situations which require some preprocessing of
the data collected by a subset of the array, before sending a trigger message
to the central system. The communication issues concerning the CRE-related
applications were addressed in Ref. Smelcerz and et al. (2020) (CREDO
Collab.). They are the subject of further ongoing investigations and
engineering works, below we briefly summarize the current status of these
efforts.
We focus on a scenario, when designing the network, which assumes that the
detectors are mobile devices that can be located in hard-to-reach areas, such
as deserts or forests, as in highly urbanized areas, i.e. cities, and even
inside blocks (Fig. 14). Such assumptions force the network to have the
following features:
* •
scalability – it is easy to connect new devices to the network, the network
should be self-configuring
* •
low energy consumption during data transmission – an indispensable parameter
to ensure support for mobile devices (without access to power from the
network)
* •
universality – the network must operate both in dense urban buildings and in
desert or forest areas
* •
wireless – provides the ability to collect data without the need for expensive
infrastructure in the form of cables and the need to use human force (manual
data collection from SD cards), in other words, significantly reduces the cost
of maintaining the project
* •
as long range as possible – guarantees stable transmission without the need to
use re-transmitters or uneconomical use of too many gates
Figure 14: Symbolic diagram for the CREDO network scenario.
#### 5.4.1 Solutions available on the market in wireless networks
In order to choose the right solution for wireless transmission in the CREDO
network, the solutions already existing on the market were compared in terms
of parameters important for the CREDO project. The results are presented in
Table 1. The table completely omits Bluetooth technology due to the very small
range (10-50m) blu , which does not meet the long range criterion.
Feature | ZigBeeZig | LoRaLoR | SigFoxSig | WiFiWiF | GPRSGPR
---|---|---|---|---|---
Frequency | 868/915 MHz and 2.4 GHz | 100 MHz to 1.67 GHz | 868/915 MHz | 2.4 GHz | 900–1800 MHz
Power consumption tx | 37 mW | 100 mW | 122 mW | 835 mW | 560 mW
Range | 100 m | 5 km | 10 km | 100 m | 1–10 km
Cons | Requires infrastructure | Available Gateways on the market are only for 438 and 868MHz | Requires infrastructure similar to GSM (masts and receiving stations) | No Internet connection in non-urbanized areas; high power consumption | No access in non-urbanized areas; high power consumption
Suitable for battery devices? | Yes | Yes | Yes | No, too much power consumption | No, too much power consumption
Table 1: Comparison of wireless communication solutions available on the
market.
Based on the table above, ZigBee, LoRa and SigFox were selected for testing
and deeper analysis. It is worth noting that the ranges given in the table are
the best result, often achievable only in an open area, not in built-up areas.
It was decided that all tests will be carried out in built-up areas,
considering city centers as the most problematic environment for establishing
long-range wireless communication. No retransmitters were used in the tests.
The first tests were conducted for the ZigBee network. We managed to achieve a
range of about 30m, similar to that presented by Kuzminykh et al. (2017).
Then, tests were carried out for the sigFox network Dis (a). Due to the
necessity to use ready-made SigFox infrastructure and the still incomplete
coverage of the area with this infrastructure, further development work using
this standard was abandoned. Finally, tests were carried out on LoRa modules
Dis (a, b). We did not manage to achieve satisfactory results (100-300 m).
Experiments carried out by Centenaro et al. (2016) in built-up areas in cities
that used the actual LoRa network and its ready infrastructure also did not
reach the maximum range of 5 km.
After the tests, it became clear to us that in order to obtain good coverage
both in the city without the use of repeaters, and in open areas, where there
is no network infrastructure. We need to develop our own system. We
immediately decided to work at lower frequencies (169 MHz) so as to minimize
the attenuation of waves by obstacles Saunders and Aragón-Zavala (2007).
#### 5.4.2 Definition of CREDO Wireless Sensor Detector Network
The CREDO Wireless Sensor Detector Network (CREDO WSDN) is a fusion of the
well-known Wireless Sensor Network (WSN) currently used most frequently in
solutions for the Internet of Things (IoT).
CREDO WSDN is a wireless network, organized in a star topology, based on radio
waves at the frequency of 169MHz, used to transmit information to the Sink
collective station, from dedicated mobile cosmic ray detectors. Sink has a COM
connection to the PC station, which in turn is connected to the Internet. The
internet provides the end user (e.g. a scientist) with access to the collected
data. Additionally, additional sensors, such as humidity or temperature
sensors, can be connected to the CREDO WSDN to investigate the correlation
between events detected by the detectors and other parameters in the area.
Fig.15 presents the CREDO WSDN topology.
Figure 15: CREDO WSDN topology.
Due to the fact that the network is to be specially adapted to communication
on mobile devices (mobile detectors), in the current version of the system
communication is only one-way, i.e. the data from the detector is sent to
Sink. The radio transmitter will be integrated with the detector itself, not a
separate device as it was in the first version of the system. These measures
are primarily intended to reduce the power consumption of the detector itself,
which is to operate on battery power.
The CREDO network consists of several main elements, such as radio
communication on 169 MHz, a Sink station, a microcontroller or a power source.
The features and role of the most important elements of the entire system are
discussed below.
### Radio communication
Radio communication takes place at 169 MHz, unidirectional in a star topology.
These features ensure not only excellent coverage in built-up areas or in
buildings, but also the optimization of energy consumption – only the Sink
station must remain listening all the time. At the moment, the best result
that we have achieved is a range of 8 km in built-up areas.
### Transmitter
We plan to introduce a new version of the transmitter. This year, a new
STM32WL chip is to appear on the market – it is a microcontroller integrated
with the radio STM . It has excellent sensitivity and transmitter power. At
the same time the energy consumption is half less of the previous solution,
thanks to the use of a built-in impulse converter. Usually, such a solution
significantly worsens the sensitivity of the receiver, but this time it does
not have this effect. Additional feature will be that the same system will
count pulses caused by particle impacts.
### Receiving station (Sink)
The receiving station in current version already has very good sensitivity
Smelcerz and et al. (2020) (CREDO Collab.) however, as we plan to upgrade the
transmitters, there’s a prototype of a new receiving station as well,
currently during testing. The new version with a new LNA (Low Noise
Amplifiers) amplifier with lower noise and much lower distortion for now
increased the range by 4km compared to the previous version (from 4km to 8km).
A new version will be tested as well with several antennas to increase the
sensitivity.
### Mobile Detectors
The reason why the network was created was the need to handle the transmission
between battery-powered devices, and more specifically mobile cosmic ray
detectors. At the moment, the first prototype of the scintillation detector
has been developed, the tests are showing promising results, and the device is
already adapted to battery power.
### Microcontroller
The microcontroller acts as the brain of the entire system, makes decisions
about how to configure the network, when to transmit the collected data, and
informs Sink about the transmitter battery status. It is found both in Sink,
transmitting stations and in the detector.
#### 5.4.3 CREDO network tests in the field
During the tests on the first version, it was possible to achieve a
satisfactory range of 4 km in built-up areas Smelcerz and et al. (2020) (CREDO
Collab.). Currently, after changes to the design of the transmitter and the
collection station (Sink), the range has increased to 8 km. Also there was
significant improvement in range in Elevated Areas (mountains). In the new
version of the receiving station, we managed to establish communication on a
cloudy day, in the area where the mountain was located. An interesting
phenomenon that we were able to observe was the reflection of the wave from
the stratosphere on a cloudy day (Fig. 16) Saunders and Aragón-Zavala (2007).
Figure 16: Reflection of the radio wave. Saunders and Aragón-Zavala (2007)
#### 5.4.4 Future work and conclusion
The CREDO communication system, thanks to its scalability and universality,
can be used in many areas of IoT and can be an alternative to existing
solutions, where due to the lack of infrastructure or too high costs of their
implementation, they cannot be launched. There are still many challenges ahead
of us, we are constantly working on increasing the range. Maintaining security
in the network and possibly encrypting data is also a challenge. The
phenomenon that we want to study better is the reflection of waves from the
stratosphere. We will also be conducting tests with new mobile scintillation
detectors in the near future, which will be immediately integrated with
transmitting stations.
### 5.5 Detection efficiency
Independently of the available cosmic ray infrastructures and expertise that
is or might be contributing to the implementation of the CREDO strategies, the
novel hardware extensions of the global network of detectors require new
studies and tools providing information of detection efficiency on different
identification and reconstruction levels: a) individual particles and the
corresponding detection rates, b) extensive air showers and the corresponding
cosmic-ray fluxes, and finally c) Cosmic Ray Ensembles. The level a) is being
addressed elsewhere within this Special Issue Bibrzycki and et al. (2020)
(CREDO Collab.), considerations of level c) were initiated with Ref. Verbetsky
and Svanidze (2020), and a study dedicated to arrays of portable detectors
like CREDO-Maze described above is under preparation.
## 6 Data management and analysis
### 6.1 CREDO IT Infrastructure
From its very beginning the CREDO initiative assumed a necessity for a
scalable data acquisition and processing infrastructure. While most of the
infrastructure is supplied by volunteers, in the form of smartphones and other
detectors, there still exists a need to create a central repository of
detection events. Information about each individual detection event is
recorded first by the end device, then it is transferred to the central
repository. This repository allows for managing and sharing information among
interested parties, it serves as a base for CREDO data analysis. It also
fulfills the role of the system’s central point, providing APIs (Application
Programming Interface) and information about stored data itself. The CREDO it
ecosystem, with some of the implementation details, is depicted in Figure 17.
Figure 17: Diagram representing the architecture of storage application.
The CREDO data repository uses a dedicated API, tailored for the project’s
needs. The object structure used in the API closely models the information
about physical detection events and additionally contains metadata, e.g. what
detector was used or what was the operating time of the detector. Data storage
in CREDO project is provided with respect to the FAIR principles Wilkinson et
al. (2016), namely we provide means to Find the data, Access it, enable
Interoperability and Reuse of the data. One of the means to achieve this goal
is the API, which was implemented as a set of REST services adhering to the
OpenAPI standard ope . OpenAPI is a widely accepted service standard, which
includes community-developed guidelines for developing interfaces exposing
data and metadata. This standard focuses on data openness, technical
accessibility and discoverability, which allow for effortless integration. The
stored data is publicly accessible, and so it will be in the future, which
should enable the reuse of information and verification of experiments based
on the given data.
Central processing for CREDO data storage was implemented as an extensible
database, a storage system running in a containerized environment with aim to
enhance reliability, portability and security. The main component, responsible
for gathering and managing data, was implemented as a Django dja web
application. User devices communicate with the API exposed by the main
application, the data is subjected to basic filtering and then placed in the
storage back end. The storage is supplied through the usage of MariaDB mar
relational database cluster. Redis red , a key-value in memory object store is
used as a temporary cache, implemented in order to speed up queries and data
presentation. The external service providing S3 compatible access is used for
backups and exporting, prepared beforehand, larger chunks of data.
Due to the nature of observed physics phenomena and the community factor, data
stream characteristics are often difficult to predict. Additionally, data
analysis is done mainly in an exploratory manner, so the underlying hardware
infrastructure needs to be flexible enough to adapt to requirements of the
moment. The presented system has been deployed in a production environment (in
this case, virtual machines are hosted in the cloud provided by ACC Cyfronet
AGH) and is continuously gathering information about detection events from the
CREDO detector network. There are several performance metrics, i. e. CPU usage
and request latency, which are constantly monitored thus allowing us to
determine if the current hardware configuration delivers required performance.
In the case of inadequate resources, the cloud management system is able to
autonomously and proactively spawn additional service instances which will
allow for load distribution and handling of the data influx.
It is important to emphasize that the adopted solutions and protocols enable
two-end open access: data collected by various detectors can be transmitted to
the central system if only their format is kept compatible with the CREDO API
structure, and also the access to the data being stored centrally is being
granted to everybody upon request including a sensible motivation. The web-
based monitor of the CREDO data acquisition system including basic user and
detection statistics is available publicly api and the technical information
regarding the full data access is being provided upon approving of individual
requests.
### 6.2 The current data set
Regardless of the ongoing efforts concerning the FAIR principles, the CREDO
data set is continuously available on request for individual users if a
sensible motivation for the usage is presented. In addition, all the currently
available scripts and tools facilitating data access, selection, and further
processing are made freely available in the public CREDO Collaboration
repository cre as a project that can be imported into the PyCharm integrated
development environment Pyc .
The CREDO data set is divided into three basic categories:
1. 1.
Detections – a set of detections containing detailed information about
individual events on all devices;
2. 2.
Pings – activity logs of devices, including the information about their
connections to the database and time of work in the detecting mode;
3. 3.
Mappings – three collections containing information about users, devices and
teams.
Most of the data collected by CREDO to date comes from smartphones with the
CREDO Detector app, operating on the Android system. The data statistics since
the premiere of the app until September 1, 2020 includes:
* •
11,150 users (unique accounts) have registered
* •
15,739 devices were used for particle detection
* •
4,941,133 candidate detections registered
* •
the total operating time of the devices is over 379,629 days (over 1039 years)
The raw data files currently occupy 44 GB (detections: 39.3 GB; pings: 1.6 GB;
mappings: 3.1 GB). Assuming the single CMOS camera sensor has a diagonal in
size of $1/3^{\prime\prime}$ (on average), the total area of all the presently
registered devices is 0.56 m2. The daily average value of detections per
device calculated from the number of candidate detections registered and the
number of total operating time of the devices is about 13 candidate detections
registered daily. Currently standard data filtering and clusterization are
being performed continuously, and very early results show a perspective for
more advanced image processing and classification using machine learning or
deep learning techniques to perform more efficient classification of candidate
detections.
A single detection record is stored in JavaScript Object Notation (JSON) open
standard data format and it contains the following information:
1. 1.
user – detection user information: “team_id”, “user_id”;
2. 2.
location – geographical coordinates: “latitude”, “longitude”;
3. 3.
time – detection time information: “timestamp” (detection unix time in
milliseconds), “time_received”: reception time in the database;
4. 4.
picture – detection image information: “id” (unique detection identification),
“frame_content”: image (a fragment of the snapshot, typically containing a
margin of 30 pixels around the brightest pixel position – see below) code in
base64, “height”: resolution “vertical” dimension, “width”: resolution
“horizontal” dimension;
5. 5.
server side visibility – “visible” (tells whether a detection pass through the
server side filters, sensitive e.g. to repeatedly flashing pixels or
incompatible versions of the applications sending the data);
6. 6.
brightest pixel position – “x”, “y” (row and column number of the brightest
pixel).
It often happens that a single picture taken by a smartphone in the detection
mode contains more than one pixel that fulfill trigger conditions and can be
classified as detections. If these pixels are located sufficiently far one
from another, i.e. more than the extraction margin explained in p. 4 above,
then they are considered separate detections and each of them is assigned an
individual detection record. In this way clearly distinguishable particle hits
collected in one shot are easily identifiable as detections with the same
“timestamp”. An example set of particle track candidates collected by CREDO
Detector is presented in Fig. 18.
Figure 18: Example particle candidate tracks recorded with a smartphone using
the CREDO Detector mobile application. [source: the CREDO Collaboration
materials and measurements]
The nature of the penetrating radiation measurements made by CMOS sensors
assumes identification of pixels that are significantly brighter than the
background. The currently used algorithms allow such identification only when
the visible light does not reach the sensor, i.e. with the smartphone camera
covered tightly. Intentional or unintentional uncovering of the smartphone
camera might result in collecting images generated by visible light, sometimes
hardly distinguishable from signal excesses induced by penetrating particles
(see Fig. 19 for some examples).
Figure 19: Example artifact tracks recorded with a smartphone using the CREDO
Detector mobile application. [source: the CREDO Collaboration materials and
measurements]
One might apply a specific software filter to remove such a contamination from
the data sample being analysed. The internally available preliminary studies
in this direction show that the number of bright pixels that compose the image
might be a sufficient data quality measure. The rule of thumb currently used
within the CREDO Collaboration is based on the data collected by a number of
“trustable” devices (operated by non-anonymous team members engaged in the
application development) and it shows that one does not expect detections of
penetrating particles that generate tracks composed of more than 70 pixels
with brightness above 70, in the 0-255 scale.
Another type of contamination is the electronic noise which typically gives
images composed of individual (or very few) bright pixels only that are being
recorded very frequently compared to the expected detection frequency of the
cosmic and local radiation. These electronic artifacts can be “detected” e.g.
if the pixel brightness threshold is improperly set, and such a situation can
be identified by monitoring the detection frequency and comparing it to the
expected 100$\%$ sensor efficiency for the sensor in use. For an example, a
typical CMOS sensor in a smartphone with a surface of 0.2 cm2 can see at most
one cosmic background muon in 5 minutes (the expected integrated background
muon flux is 1/cm2/minute). Assuming that the local radioactive sources might
induce a signal at most $\sim$10 times more often, one does not expect
detection rates larger than 2/minute, and this is in fact the case when
looking at the statistics collected from the reference (“trustable”) devices.
An example of the corresponding filter being used with the CREDO Collaboration
requires that detection frequency is less than 10 detections with different
timestamps per minute.
Despite the fact that with full access to the raw data sets users are able and
welcome to apply their own filters and perform their own analyses, the CREDO
Collaboration will periodically release its official data sets and
recommendations concerning the data quality which will be based on the
appropriate and publically available studies. If it comes to scientific
results, the researchers will be expected either to refer to the official data
sets, or to describe and justify their own selections.
The examples of the already ongoing analyses concerning the CREDO data set
include image classification based on the shapes of particle track candidates
Niedzwiecki and et al. (2019) (CREDO Collab.), monitoring detections collected
by individual devices in search for temporal event clustering in 5 minute
intervals Eur and muon identification and zenith angle reconstruction based
on the lengths of the rectilinear particle tracks Karbowiak and et al. (2020)
(CREDO Collaboariton).
One of the examples of novel and promising directions of analyses related to
the CREDO data is exploitation of the techniques of cyclostationary signal
processing and its generalizations Napolitano (2012), Napolitano (2019).
Cyclostationarity is a statistical property of science data generated by the
combination/interaction of periodic and random phenomena. That is, these data
have second- or higher-order statistical functions that are periodic functions
of time. More general models can account for the presence of multiple,
possibly incommensurate, and irregular periodicities (Napolitano, 2019,
Chapters 1,2). The observed signals are not periodic, but the hidden
periodicity can be restored by estimating statistical functions from the data.
These statistical functions contain information of the generating mechanism of
the data that cannot be extracted starting from the classical stationary
modeling of the observed signals.
The extensions of cyclostationarity can be of interest if time dilation
effects have to be accounted for (Napolitano, 2012, Chapter 7). In particular,
the effects of constant relative radial speed and/or constant relative radial
acceleration between cosmic source and receiver can be suitably modeled by
exploiting the generalizations of cyclostationarity. General time-warping of
the source signal can be modeled by new and recently proposed signal models
(Napolitano, 2012, Chapter 6), (Napolitano, 2019, Chapters 12-14).
Source location and parameter estimation problems based on measurements taken
at sensors very far apart separated are very interference tolerant if the
|
# Continuous Differentiability of the Value Function of Semilinear Parabolic
Infinite Time Horizon Optimal Control Problems on $L^{2}(\Omega)$ under
Control Constraints ††thanks: The authors were supported by the ERC advanced
grant 668998 (OCLOC) under the EU’s H2020 research program.
Karl Kunisch Institute for Mathematics and Scientific Computing, University of
Graz, Heinrichstrasse 36, A-8010 Graz, Austria, and Radon Institute, Austrian
Academy of Science, Linz, Austria. ([email protected]). Buddhika
Priyasad Institute for Mathematics and Scientific Computing, University of
Graz, Heinrichstrasse 36, A-8010 Graz, Austria. (b.sembukutti-liyanage@uni-
graz.at).
###### Abstract
An abstract framework guaranteeing the local continuous differentiability of
the value function associated with optimal stabilization problems subject to
abstract semilinear parabolic equations subject to a norm constraint on the
controls is established. It guarantees that the value function satisfies the
associated Hamilton-Jacobi-Bellman equation in the classical sense. The
applicability of the developed framework is demonstrated for specific
semilinear parabolic equations.
## 1 Introduction.
Continuous differentiability of the value function with respect to the initial
datum is an important problem in optimal feedback control theory. Indeed, if
the value function is $C^{1}$ then it is the solution of a Hamilton Jacobi
Bellman (HJB) equation and its negative gradient can be used to define on
optimal state feedback law. The subject matter of this paper addresses local
continuous differentiability of the value function $\mathcal{V}$ for infinite
horizon optimal control problems subject to semilinear parabolic control
problems and norm constraints on the control. Such problems are intimately
related to stabilization problems which are often cast as infinite horizon
optimal control problems. Investigating infinite horizon problems constitutes
one of the specificities of this paper. Another one is the fact that we focus
on the differentiability of $\mathcal{V}$ on (subsets of) $L^{2}(\Omega)$.
Thus we need to consider the semilinear equations with initial data $y_{0}\in
L^{2}(\Omega)$. As a consequence the solutions of the semilinear equations
only enjoy low Sobolev-space regularity. This restricts the class of
nonlinearities, compared to those which are admissible if the states are in
$L^{\infty}((0,\infty)\times\Omega)$, which is the situation typically
addressed in the literature on optimal control [Cas] and [Tro2]. The latter
necessitates to take the initial conditions in spaces strictly smaller than
$L^{2}(\Omega)$. Here we consider $L^{2}(\Omega)$, first due to intrinsic
interest, secondly because ultimately the HJB equation should be solved
numerically, which is easier in an $L^{2}(\Omega)$ setting than in other
topologies, like $H^{1}(\Omega)$. Let us also recall that one of the
approaches to solve the HJB equation is given by the policy iteration. It
assumes that the value function is $C^{1}$.
The underlying analysis demands stability and sensitivity analysis of infinite
dimensional optimal control problems subject to nonlinear equations. For this
purpose we utilize the theory of generalized equations as established by [Don]
and [Rob]. It involves first order approximations of the state and adjoint
equations, which lead to restrictions on the class of nonlinearities which can
be admitted. We refer to the section on examples in this respect.
The current investigations are to some degree a continuation of work the first
author’s work on optimal feedback control for infinite dimensional systems. In
[BKP1, BKP2, BKP3] Taylor approximations of the value function for problems
with a concrete structure, namely, bilinear control systems, and the Navier
Stokes equations were investigated and differentiability of the value function
was obtained as a by product. In these investigations norm constraints were
not considered. Here we admit norm constraints and we focus on semilinear
equations. Let us also notice that the systems investigated in [BKP1, BKP2,
BKP3] share the property that the second derivatives with respect to the state
variable of the nonlinearity in the state equation do not depend on the state
itself anymore.
Let us also compare our work to the developments in the field of parametric
sensitivity analysis of semilinear parabolic equations under control
constraints. There are many papers focusing on stability and sensitivity
analysis of finite time horizon problems along with pointwise control
constraints, see e.g. [BM, GHH, Gri, GV, Mal, MT, Tro1, Wac], and the
literature there. First, of these papers, except for [GV], consider the case
with initial data in $L^{2}(\Omega)$. In [GV] again the third derivative of
the nonlinearity is zero. Secondly, all of them consider the finite horizon
case. Since we treat infinite horizon problems we have to guarantee
stabilizability (for small initial data) under control constraints. Then we
use a fixed point argument to obtain well-posedness of the system. Well-
posedness and stability with respect to parameters of the adjoint equation is
significantly more involved for infinite horizon problems than for finite
horizon problems. It requires techniques, differently from those used in the
finite horizon case. Another aspect is the proper characterization of the
adjoint state at $t=\infty$.
In the finite dimensional case, there is, of course a tremendous amount of
work on the treatment of the value function if it is not $C^{1}$. Fewer papers
concentrate on the case where the value function enjoys smoothness properties.
We mention [Goe] and [CF] in this respect.
In order to achieve the goal we desire, we lay out the following setup. In
Section 2, we consider an abstract parametric optimization problem with an
equality constraint and another convex constraint. Existence of an optimal
solution, of a multiplier associate to the equality constraint, and Lipschitz
stability of the component of the state variable which lies in the complement
of the kernel of the linearized constraint will be established. This result is
necessary but not sufficient for the further developments, since stability is
obtained in a norm which is too weak and since the stability estimate does not
involve the component in the kernel of the linearized constraint and the
multiplier, i.e. the adjoint states, yet. At the level of Section 2 this
remains as Assumption (H7). In Section 3 we specify the concrete optimal
stabilization problem and a set of conditions, most importantly on the
nonlinearity of the state equation, under which Assumption (H7) can be
established, for initial data $y_{0}\in L^{2}(\Omega)$. Section 3 also
contains a summary of the main results of this paper. They are stated as
theorems with a little stronger assumptions than eventually necessary, for the
saker of easing the presentation. Section 4 is dedicated to the proof of
verifying the assumptions of the general setup of Section 2 for the concrete
optimal control problem stated in Section 3. As conclusion we obtain the
Lipschitz continuity in the appropriate norms of the all variables appearing
in the optimality system with respect to the parameter of interest, which is
the initial condition $y_{0}$, in our case. Since our analysis is a local one
involving second order optimality conditions, solutions to the optimality
system are related to local solutions to the optimal control problem. As a
corollary to these results we obtain that the local value function is Fréchet
differentiable. In Section 5, we show that in the neighborhood of global
solutions the value function $\mathcal{V}$ satisfies the Hamilton-Jacobi-
Bellman (HJB) equation in the strong sense. Finally, Section 6 is devoted to
demonstrating that the developed framework is applicable for some concrete
examples, namely for linear systems, Fisher’s equations, and parabolic
equations with global Lipschitz nonlinearities. All our results require a
smallness assumption on the initial conditions $y_{0}$. Two aspects need to be
taken into consideration in this respect. First $y_{0}$ has to be sufficiently
small so that the controlled system is stable. Secondly a second order
optimality condition is needed. For this to hold a sufficient condition is
provided by smallness of the adjoint state, which in turn can be implied by
smallness of $y_{0}$. We stress that these two issues are of related, but
independent nature.
## 2 Lipschitz stability for an abstract optimization problem.
Here we present a stability result for an abstract, infinite dimensional
optimization problem which will be the building block for the results below.
This result is geared towards exploiting the specific nature of optimization
problem with differential equations as constraints. First existence of a dual
variable will result from a regular point condition. Subsequently the
Lipschitz stability result is obtained in two steps. In the first one, we rely
on the relationship between the linearized optimality conditions and an
associated linear-quadratic optimal optimization problem, with an extra convex
constraint. This approach is useful since it provides the existence of
solutions to the linearized system on the basis of variational techniques.
However it dictates a certain norms for the involved quantities. These norms
are too weak for our goal of obtaining Lipschitz continuity of the adjoint
variables in such a manner that differentiability of the cost with respect to
the initial conditions can be argued. Therefore, in a second step we exploit
the specific structure of the optimality systems, using the fact the it is
related to a parabolic optimal control problem, to obtain the Lipschitz
continuity in the stronger norms. This two step approach is also present in
some of the earlier work on stability and sensitivity analysis which was
quoted in the introduction. But due to that fact these papers considered
finite horizon problems it came as a byproduct which improved the regularity
of the adjoints. In our work it is essential to reach our goal. This is why we
decided to formalize this two step approach which was not done in earlier
work.
Concretely, we consider the optimization problem
$\begin{cases}\min\ f(x)\\\ e(x,q)=0,\ x\in C.\end{cases}$ ($P_{q}$)
with a parameter dependent equality constraint, and a general constraint
described by $x\in C$, where $C$ is a closed convex subset of a real Hilbert
space $X$. Further, $W$ is a real Hilbert space and $P$ is a normed linear
space. In the application that we have in mind, the parameter $q$ will appear
as the initial condition in the dynamical system. The following Assumption
(H1) is assumed to hold throughout.
###### Assumption H1.
$q_{0}\in P$ is a nominal reference parameter,
$x_{0}$ is a local solution ($P_{q_{0}}$),
$f:X\longrightarrow\mathbb{R}^{+}$ is twice continuously differentiable in a
neighborhood of $x_{0}$,
$e:X\times P\longrightarrow W$ is continuous, and twice continuously
differentiable w.r.t. $x$, with first and second derivative Lipschitz
continuous in a neighborhood of $(x_{0},q_{0})$.
The derivatives with respect to $x$ will be denoted by primes and the
derivatives w.r.t. $y$ and $u$ later on, are denoted by subscripts. They are
all considered in the sense of Lebesgue derivatives.
We introduce the Lagrangian ${\mathcal{L}}:X\times P\times
W^{*}\longrightarrow\mathbb{R}$ associated to ($P_{q}$) by
${\mathcal{L}}(x,q,\lambda)=f(x)+\langle\lambda,e(x,q)\rangle_{W^{*},W}.$
(2.1)
Next further relevant assumptions are introduced:
###### Assumption H2 (regular point condition).
$0\in\text{int }e^{\prime}(x_{0},q_{0})(C-x_{0}),$
where $int$ denotes the interior in the $W$ topology. This regularity
condition implies the existence of a Lagrange multiplier $\lambda_{0}\in
W^{*}$, see e.g. [MZ] such that the following first order condition holds:
$\begin{cases}\langle{\mathcal{L}}^{\prime}(x_{0},q_{0},\lambda_{0}),c-x_{0}\rangle_{X^{*},X}\geq
0,\ \forall c\in C,\\\ e(x_{0},q_{0})=0.\end{cases}$ (2.2)
It is equivalent to
$\displaystyle\begin{cases}0\in{\mathcal{L}}^{\prime}(x_{0},q_{0},\lambda_{0})+\partial\mathbf{I}_{C}(x_{0}),\quad&\text{in
}X^{*},\\\ e(x_{0},q_{0})=0,\quad&\text{in }W,\end{cases}$ (2.3)
where $\partial\mathbf{I}_{C}(x)$ denotes the subdifferential of the indicator
function of the set $C$ at $x\in X$.
Let $\displaystyle A\in{\mathcal{L}}(X,X^{*})$ denote the operator
representation of
$\displaystyle{\mathcal{L}}^{\prime\prime}(x_{0},q_{0},\lambda_{0})$, i.e.
$\langle
Ax_{1},x_{2}\rangle_{X^{*},X}={\mathcal{L}}^{\prime\prime}(x_{0},q_{0},\lambda_{0})(x_{1},x_{2})$
(2.4)
and define
$E=e^{\prime}(x_{0},q_{0})\in{\mathcal{L}}(X,W).$ (2.5)
We further require
###### Assumption H3 (positive definiteness).
$\exists\kappa>0:\ \langle Ax,x\rangle_{X^{*},X}\geq\kappa\left\lVert
x\right\rVert^{2}_{X},\ \forall x\in\text{ker }E.$
The stability result of $(x_{0},\lambda_{0})$ with respect to perturbation of
$q$ at $q_{0}$ will be based on Robinson’s strong regularity condition which
involves the following linearized form of the optimality condition,
$\displaystyle\begin{cases}0\in{\mathcal{L}}^{\prime}(x_{0},q_{0},\lambda_{0})+A(x-x_{0})+E^{*}(\lambda-\lambda_{0})+\partial\mathbf{I}_{C}(x)&\text{in
}X^{*},\\\ 0=e(x_{0},q_{0})+E(x-x_{0})&\text{in }W.\end{cases}$ (2.6)
We define a multivalued operator $\displaystyle{\mathcal{T}}:X\times
W^{*}\longrightarrow X^{*}\times W$ by
${\mathcal{T}}\begin{pmatrix}x\\\
\lambda\end{pmatrix}=\begin{pmatrix}A&E^{*}\\\
E&0\end{pmatrix}\begin{pmatrix}x\\\
\lambda\end{pmatrix}+\begin{pmatrix}f^{\prime}(x_{0})-Ax_{0}\\\
-Ex_{0}\end{pmatrix}+\begin{pmatrix}\partial\mathbf{I}_{C}(x)\\\
0\end{pmatrix},$ (2.7)
and observe that (2.6) is equivalent to
$\displaystyle 0\in{\mathcal{T}}\begin{pmatrix}x\\\ \lambda\end{pmatrix}.$
Here it is understood that ${\mathcal{T}}$ is evaluated at
$(x_{0},q_{0},\lambda_{0})\in X\times P\times W^{*}$. But ${\mathcal{T}}$ is
not yet the mapping for which we need to verify the Robinson-Dontchev strong
regularity condition in our context. It relates to the fact that we must to
treat the multiplier $\lambda$ in smaller space than $W^{*}$. Before we can
properly specify this condition some additional preparation is necessary. We
first introduce Banach spaces:
$\underline{X}\subset X,\ \underline{W^{*}}\subset W^{*},\
\underline{X^{*}}\subset X^{*},$ (2.8)
with continuous injections. A restriction of ${\mathcal{T}}$ will be defined
as multivalued operator
$\displaystyle\underline{{\mathcal{T}}}:\underline{X}\times\underline{W^{*}}\to\underline{X^{*}}\times
W$. Indeed, in applications to optimal control problems extra regularity of
multipliers can be obtained by investigating the solutions (2.3), see e.g.
Section 3. In the context of optimal stabilization problems this structural
property will become transparent in Proposition 4.1 and Proposition 4.2, see
also [BKP3, Proposition 15]. It will turn out to be essential for our
purposes. But this situation where the multiplier has extra regularity is also
of abstract interest. When studying stability in this setting this means that
the second coordinate of the domain of ${\mathcal{T}}$ needs to be changed
from $W^{*}$ to $\underline{W^{*}}$. This entails that the range space of
${\mathcal{T}}$ has to be modified appropriately, in order to obtain stability
of the $\lambda$ coordinate. For this purpose we introduce
$\underline{X^{*}}\subset X^{*}$. The reason for further restricting $X$ to
$\underline{X}$ will become evident in the proof of Proposition 4.2. It is
related to the fact that we consider infinite horizon problems.
Now we adapt the conditions on $f$ and $e$ to the choice of the spaces in
(2.8).
###### Assumption H4.
There exists a neighborhood
$\displaystyle\widetilde{U}_{1}\times\widetilde{U}_{2}\subset\underline{X}\times
P$ of $(x_{0},q_{0})$ such that
1. (i)
the restriction of $x\mapsto f^{\prime}(x)$ to $\underline{X}$ defines a
mapping from $\underline{f^{\prime}}(x)$ from
$\widetilde{U}_{1}\subset\underline{X}$ to $\underline{X^{*}}$,
2. (ii)
the restriction $e^{\prime}(x,q)^{*}\in{\mathcal{L}}(W^{*},X^{*})$ to
$\underline{W^{*}}$ defines operators
$\underline{e^{\prime}(x,q)^{*}}\in{\mathcal{L}}(\underline{W^{*}},\underline{X^{*}})$
for every $(x,q)\in\widetilde{U}_{1}\times\widetilde{U}_{2}$.
With these assumption holding we define the restricted linearized Lagrangian
$\underline{{\mathcal{L}}^{\prime}}:\widetilde{U}_{1}\times\widetilde{U}_{2}\times\underline{W^{*}}\subset\underline{X}\times
P\times\underline{W^{*}}\longrightarrow\underline{X^{*}}\quad\text{by}\quad\underline{{\mathcal{L}}^{\prime}}(x,q,\lambda)=\underline{f^{\prime}}(x)+\underline{e^{\prime}(x,q)^{*}}\lambda.$
(2.9)
Next we adapt $\partial\mathbf{I}_{C}\subset X^{*}$ to the situation of (2.8)
and define for $x\in\underline{X}$ the set valued mapping
${\underline{\partial\mathbf{I}_{C}}(x)}=\left\\{y\in\underline{X^{*}}:\langle
y,v-x\rangle_{X^{*},X}\leq 0,\ \forall v\in
C\cap\underline{X}\right\\}\subset\underline{X^{*}}.$ (2.10)
We henceforth assume that
$(x_{0},\lambda_{0})\in\underline{X}\times\underline{W^{*}}$, it will also
follow as a special case of (H7) below. The following assumption will
guarantee that the restriction $\underline{{\mathcal{T}}}$ of ${\mathcal{T}}$
is well-defined as operator from $\underline{X}\times\underline{W^{*}}$ to
$\underline{X^{*}}\times W$, and the one beyond is needed for Lipschitz
continuous dependence of local solutions to ($P_{q}$) with respect to $q$.
###### Assumption H5.
$\displaystyle\underline{{\mathcal{L}}^{\prime}}:\widetilde{U}_{1}\times\widetilde{U}_{2}\times\underline{W^{*}}\subset\underline{X}\times
P\times\underline{W^{*}}\longrightarrow\underline{X^{*}}$ is Fréchet
differentiable with respect to $x$, and
$(\underline{{\mathcal{L}}^{\prime}})^{\prime}$ is continuous at
$(x_{0},q_{0},\lambda_{0})\in\underline{X}\times P\times\underline{W^{*}}$.
###### Assumption H6.
There exists $\nu>0$ such that:
$\displaystyle\left\lVert e(x,q_{1})-e(x,q_{2})\right\rVert_{W}$
$\displaystyle\leq\nu\left\lVert q_{1}-q_{2}\right\rVert_{P},\
\forall(x,q_{1})\text{ and
}(x,q_{2})\in\widetilde{U}_{1}\times\widetilde{U}_{2},$ (2.11a)
$\displaystyle\left\lVert{\underline{e^{\prime}(x,q_{1})^{*}}-\underline{e^{\prime}(x,q_{2})^{*}}}\right\rVert_{{\mathcal{L}}(\underline{W^{*}},\underline{X^{*}})}$
$\displaystyle\leq\nu\left\lVert q_{1}-q_{2}\right\rVert_{P},\
\forall(x,q_{1})\text{ and
}(x,q_{2})\in\widetilde{U}_{1}\times\widetilde{U}_{2}.$ (2.11b)
Let us further set
$\underline{E^{*}}=\underline{e^{\prime}(x_{0},q_{0})^{*}}\in{\mathcal{L}}(\underline{W^{*}},\underline{X^{*}})\text{
and
}\underline{A}=(\underline{{\mathcal{L}}^{\prime}}(x_{0},q_{0},\lambda_{0}))^{\prime}\in{\mathcal{L}}(\underline{X},\underline{X^{*}}).$
With Assumptions (H1)-(H5) holding (2.3) can be expressed as
$\displaystyle\begin{cases}0\in\underline{{\mathcal{L}}^{\prime}}(x_{0},q_{0},\lambda_{0})+\underline{\partial\mathbf{I}_{C}}(x_{0}),\quad&\text{in
}\underline{X^{*}},\\\ e(x_{0},q_{0})=0,\quad&\text{in }W.\end{cases}$ (2.12)
Moreover (2.6) restricted to $\underline{X}\times\underline{X^{*}}$ result in:
$\displaystyle
0\in\begin{cases}\underline{{\mathcal{L}}^{\prime}}(x_{0},q_{0},\lambda_{0})+\underline{A}(x-x_{0})+\underline{E^{*}}(\lambda-\lambda_{0})+\underline{\partial\mathbf{I}_{C}}(x)&\text{
in }\underline{X^{*}},\\\ e(x_{0},q_{0})+E(x-x_{0})&\text{ in }W,\end{cases}$
(2.13)
and the multivalued operator
$\displaystyle\underline{{\mathcal{T}}}:\underline{X}\times\underline{W^{*}}\longrightarrow\underline{X^{*}}\times
W$ related to (2.7) is defined as
$\underline{{\mathcal{T}}}\begin{pmatrix}x\\\
\lambda\end{pmatrix}=\begin{pmatrix}\underline{A}&\underline{E^{*}}\\\
E&0\end{pmatrix}\begin{pmatrix}x\\\
\lambda\end{pmatrix}+\begin{pmatrix}\underline{f^{\prime}}(x_{0})-\underline{A}x_{0}\\\
-Ex_{0}\end{pmatrix}+\begin{pmatrix}\underline{\partial\mathbf{I}_{C}}(x)\\\
0\end{pmatrix}.$ (2.14)
Observe that (2.13) is equivalent to
$\displaystyle 0\in\underline{{\mathcal{T}}}\begin{pmatrix}x\\\
\lambda\end{pmatrix}.$
Existence and Lipschitz continuity of solutions in a neighborhood
$(x_{0},q_{0},\lambda_{0})$ will follow from the strong regularity assumption
which requires us to show that there exist neighborhoods
$\hat{V}\subset\underline{X^{*}}\times W$ of $0$ and
$\hat{U}=\hat{U}_{1}\times\hat{U}_{2}\subset\underline{X}\times\underline{W^{*}}$
of $(x_{0},q_{0})$ such that $\underline{{\mathcal{T}}}^{-1}$ has the
properties that $\underline{{\mathcal{T}}}^{-1}(\hat{V})\cap\hat{U}$ is
single-valued and that it is Lipschitz continuous from $\hat{V}$ to $\hat{U}$,
see [Don], (and also [Rob], [IK, Definition 2.2, p 31], in case
$\underline{X}=X,\ \underline{W^{*}}=W^{*},\ \underline{X^{*}}=X^{*}$). We
approach the strong regularity assumption in two steps. In the first one we
argue invertibility of ${\mathcal{T}}$ and Lipschitz continuity of the
variable $x$ in $X$. For this purpose we exploit the symmetry of
${\mathcal{T}}$ and consider an associated variational problem. In our
specific situation the inverse of ${\mathcal{T}}$ \- and consequently of
$\underline{{\mathcal{T}}}$ \- is single-valued and thus the restriction to
the neighborhood $\hat{U}$ is not needed. Existence and Lipschitz continuity
of $\lambda$ as well as Lipschitz continuity of $x$ in the small space
$\underline{X}\times\underline{W^{*}}$ remains an assumption in the generality
of problem ($P_{q}$). It will be verified in a second step for the optimal
stabilization problems in the following sections.
###### Assumption H7.
For $(\beta_{1},\beta_{2})\in\hat{V}\subset\underline{X^{*}}\times W$ , the
solution
$\displaystyle\left(x_{(\beta_{1},\beta_{2})},\lambda_{(\beta_{1},\beta_{2})}\right)$
to $\displaystyle{\mathcal{T}}\begin{pmatrix}x\\\
\lambda\end{pmatrix}=\begin{pmatrix}\beta_{1}\\\ \beta_{2}\end{pmatrix}$ lies
in $\underline{X}\times\underline{W^{*}}$. Moreover there exists a constant
$k>0$ such that
$\left\lVert
x_{(\beta_{1},\beta_{2})}-x_{(\hat{\beta}_{1},\hat{\beta}_{2})}\right\rVert_{\underline{X}}+\left\lVert\lambda_{(\beta_{1},\beta_{2})}-\lambda_{(\hat{\beta}_{1},\hat{\beta}_{2})}\right\rVert_{\underline{W^{*}}}\leq
k\left[\left\lVert(\beta_{1},\beta_{2})-(\hat{\beta}_{1},\hat{\beta}_{2})\right\rVert_{\underline{X^{*}}\times
W}+\left\lVert
x_{(\beta_{1},\beta_{2})}-x_{(\hat{\beta}_{1},\hat{\beta}_{2})}\right\rVert_{X}\right]$
for all
$(\beta_{1},\beta_{2})\in\hat{V},(\hat{\beta}_{1},\hat{\beta}_{2})\in\hat{V}$.
This condition is used after the existence of
$x_{\beta}=x_{(\beta_{1},\beta_{2})}$ was already established. Note that for
$(\beta_{1},\beta_{2})^{T}=0$ we have
$(x_{(0,0)},\lambda_{(0,0)})=(x_{0},\lambda_{0})$ and hence (H7) in particular
implies that $(x_{0},\lambda_{0})\in\underline{X}\times\underline{W^{*}}$. We
arrive at the announced stability result.
###### Theorem 2.1.
Assume that (H1)-(H7) hold at a local solution $x_{0}$ of ($P_{q_{0}}$). Then
there exist a neighborhood
$U=U(x_{0},\lambda_{0})\subset\underline{X}\times\underline{W^{*}}$, a
neighborhood $N=N(q_{0})\subset P$, and a constant $\mu$ such that for all
$q\in N$ there exists a unique $(x(q),\lambda(q))\in U$ satisfying
$\displaystyle
0\in\begin{cases}\underline{{\mathcal{L}}^{\prime}}(x(q),q,\lambda(q))+\underline{\partial\mathbf{I}_{C}}(x(q)),\quad&\text{
in }\underline{X^{*}},\\\ e(x(q),q),\quad&\text{ in }W,\end{cases}$ (2.15)
and
$\left\lVert(x(q_{1}),\lambda(q_{1}))-(x(q_{2}),\lambda(q_{2}))\right\rVert_{\underline{X}\times\underline{W^{*}}}\leq\mu\left\lVert
q_{1}-q_{2}\right\rVert_{P},\ \forall q_{1},q_{2}\in N.$ (2.16)
In addition there exists a nontrivial neighborhood $\widetilde{N}\subset N$ of
$q_{0}$ such that $x(q)$ is a local solution of ($P_{q}$) for
$q\in\widetilde{N}$.
For the proof we shall employ the following lemma in which
$A\in{\mathcal{L}}(X,X^{*})$ and $E\in{\mathcal{L}}(X,W)$ denote generic
operators. For the sake of completeness we also include its proof.
###### Lemma 2.1.
Let $(\tilde{a},\tilde{b})\in X^{*}\times W$, assume that
$A\in{\mathcal{L}}(X,X^{*})$ is self-adjoint and satisfies (H3), and that the
set $\displaystyle S(\tilde{b})=\\{x\in C:\ Ex=\tilde{b}\\}$ is nonempty. Then
the problem
$\begin{cases}\min_{x\in C}\tilde{J}(x)=\min_{x\in C}\frac{1}{2}\langle
Ax,x\rangle_{X^{*},X}+\langle\tilde{a},x\rangle_{X^{*},X},\\\
Ex=\tilde{b},\end{cases}$ (2.17)
admits a unique solution $x=x(\tilde{a},\tilde{b})$ satisfying
$\langle Ax+\tilde{a},v-x\rangle_{X^{*},X}\geq 0,\ \text{for all }v\in
S(\tilde{b}).$ (2.18)
If moreover the regular point condition $0\in\text{int
}E(C-x(\tilde{a},\tilde{b}))$ holds, then there exists
$\lambda=\lambda(\tilde{a},\tilde{b})\in W^{*}$ such that
$0\in\begin{cases}\begin{pmatrix}A&E^{*}\\\
E&0\end{pmatrix}\begin{pmatrix}x\\\
\lambda\end{pmatrix}+\begin{pmatrix}\tilde{a}\\\
-\tilde{b}\end{pmatrix}+\begin{pmatrix}\partial\mathbf{I}_{C}(x)\\\
0\end{pmatrix}.\end{cases}$ (2.19)
###### Proof.
Since $C$ is a closed and convex, $S(\tilde{b})$ is closed and convex. By
assumption $S(\tilde{b})$ is nonempty. Hence there exists an $x\in C$ such
that $Ex=\tilde{b}$. Note that $x$ can be uniquely decomposed as $x=w+y$, with
$y\in\text{ker}E,\,w\in\text{ker}E^{\perp}$ and $Ew=\tilde{b}$. By (H3) the
functional $\tilde{J}$ is bounded from below and coercive on $S(\tilde{b})$.
Hence there exists a minimizing sequence $\\{x_{n}\\}$ in $S(\tilde{b})$ such
that $\displaystyle\lim_{n\rightarrow\infty}\tilde{J}(x_{n})=\inf_{x\in
S(\tilde{b})}\tilde{J}(x)$. Each $x_{n}$ can be decomposed as $x_{n}=w+y_{n}$,
with $y_{n}\in\text{ker}E$. By (H3) the sequences
$\displaystyle\\{y_{n}\\}_{n=1}^{\infty}$ and hence
$\displaystyle\\{x_{n}\\}_{n=1}^{\infty}$ are bounded. Thus there exists a
subsequence $\displaystyle\\{x_{n_{k}}\\}$ with weak limit
$x=x(\tilde{a},\tilde{b})$ in $S(\tilde{b})$. Since $\tilde{J}$ weakly lower
semi-continuous, we have that
$\displaystyle\tilde{J}(x)\leq\liminf_{k\rightarrow\infty}\tilde{J}(x_{n_{k}})$
and $x$ minimizes $\tilde{J}$ over $S(\tilde{b})$. This further implies that
$\displaystyle\langle Ax+\tilde{a},v-x\rangle_{X^{*},X}\geq 0$ for all $v\in
S(\tilde{b})$. Uniqueness of $x$ follows from (H3).
The regular point condition implies the existence of a multiplier
$\lambda=\lambda(\tilde{a},\tilde{b})\in W^{*}$ such that (2.19) holds. See
e.g. [IK, Theorem 1.6] ∎
Proof of the Theorem 2.1.
1. (i)
The proof of the first assertion of the Theorem 2.1 is based on the implicit
function theorem of Dontchev for generalized equations, see [Don, Theorem 2.4,
Remark 2.5]. We introduce the mapping
$\underline{{\mathcal{F}}}:\underline{X}\times
P\times\underline{W^{*}}\longrightarrow\underline{X^{*}}\times W$ given by
$\underline{{\mathcal{F}}}(x,q,\lambda)=\begin{pmatrix}\underline{{\mathcal{L}}^{\prime}}(x,q,\lambda)\\\
e(x,q)\end{pmatrix},$
and observe that Assumption (H6) implies that for all $(x,q_{1},\lambda),$ and
$(x,q_{2},\lambda)\in\widetilde{U}_{1}\times\widetilde{U}_{2}\times\underline{W^{*}}$
$\left\lVert\underline{{\mathcal{F}}}(x,q_{1},\lambda)-\underline{{\mathcal{F}}}(x,q_{2},\lambda)\right\rVert_{\underline{X^{*}}\times
W}\leq\nu\left(1+\left\lVert\lambda\right\rVert_{\underline{W^{*}}}\right)\left\lVert
q_{1}-q_{2}\right\rVert_{P}.$ (2.20)
By (H1) and (H5), and using the integral mean value theorem it can be argued
that
$\begin{pmatrix}x\\\
\lambda\end{pmatrix}\to\begin{pmatrix}\underline{A}&\underline{E^{*}}\\\
E&0\end{pmatrix}\begin{pmatrix}x\\\
\lambda\end{pmatrix}+\begin{pmatrix}\underline{f^{\prime}}(x_{0})-\underline{A}x_{0}\\\
-Ex_{0}\end{pmatrix}$
strongly approximates $\underline{{\mathcal{F}}}$ at
$(x_{0},q_{0},\lambda_{0})$, [Don]. In the next two steps the strong
regularity condition for $\underline{{\mathcal{T}}}$ will be verified.
2. (ii)
(Existence). Let, at first, $(\beta_{1},\beta_{2})\in X^{*}\times W$ and
consider $\displaystyle{\mathcal{T}}\begin{pmatrix}x\\\
\lambda\end{pmatrix}=\begin{pmatrix}\beta_{1}\\\ \beta_{2}\end{pmatrix}$ which
is equivalent to,
$0\in\begin{pmatrix}a\\\ -b\end{pmatrix}+\begin{pmatrix}A&E^{*}\\\
E&0\end{pmatrix}\begin{pmatrix}x\\\
\lambda\end{pmatrix}+\begin{pmatrix}\partial\mathbf{I}_{C}(x)\\\
0\end{pmatrix},$ (2.21)
with $a=f^{\prime}(x_{0})-Ax_{0}-\beta_{1},\ b=Ex_{0}+\beta_{2}$, and $A,E$
defined in (2.4), (2.5). To solve (2.21) we consider,
$\begin{cases}\min_{x\in C}\ \frac{1}{2}\langle Ax,x\rangle_{X^{*},X}+\langle
a,x\rangle_{X^{*},X},\\\ Ex=b.\end{cases}$ (2.22)
This corresponds to (2.17) with $\tilde{a}=a$, $\tilde{b}=b$ and feasible set
$\displaystyle S(\beta_{2})=\\{x\in C:Ex=b\\}$. Clearly $\displaystyle
x_{0}\in S(0)=\\{x\in C:Ex=Ex_{0}\\}$. By (H2) and [IK, Theorem I.2.8], there
exists a neighborhood of the origin $\displaystyle\tilde{V}\subset X^{*}\times
W$ such that $S(\beta_{2})$ is not empty for all
$(\beta_{1},\beta_{2})\in\tilde{V}$. Thus by Lemma 2.1 there exists a unique
solution $x=x(\beta_{1},\beta_{2})$ to (2.22) for each
$(\beta_{1},\beta_{2})\in\tilde{V}$. By [IK, Theorem I.2.11, I.2.12, I.2.15],
possibly after reducing $\tilde{V}$, these solutions depend Hölder
continuously on $(\beta_{1},\beta_{2})\in\tilde{V}\subset X^{*}\times W$, with
exponent $\frac{1}{2}$. The regular point condition for the solution
$x(\beta_{1},\beta_{2})$ is
$0\in\text{int }E(C-x(\beta_{1},\beta_{2}))=\text{int }E(C-x_{0})-\beta_{2},$
which is satisfied due to (H2), possibly after again reducing $\tilde{V}$.
Hence there exists a Lagrange multiplier
$\lambda=\lambda(\beta_{1},\beta_{2})$ associated to $Ex=b$, and (2.21) admits
a unique solution $(x(\beta_{1},\beta_{2}),\lambda(\beta_{1},\beta_{2}))$
since it is the first order optimality condition for (2.17).
3. (iii)
(Uniqueness and Lipschitz continuity) Let $(\beta_{1},\beta_{2})\in\tilde{V}$
and $(\hat{\beta_{1}},\hat{\beta_{2}})\in\tilde{V}$ with corresponding
solutions $(x,\lambda)\in X\times W^{*}$ and $(\hat{x},\hat{\lambda})\in
X\times W^{*}$. This implies that
$\begin{cases}\langle a+Ax+E^{*}\lambda,c-x\rangle_{X^{*},X}\geq 0,\ \forall
c\in C,\\\ Ex=b,\text{ with }a=f^{\prime}(x_{0})-Ax_{0}-\beta_{1},\
b=Ex_{0}+\beta_{2},\end{cases}$ (2.23)
and
$\begin{cases}\langle\hat{a}+A\hat{x}+E^{*}\hat{\lambda},c-\hat{x}\rangle_{X^{*},X}\geq
0,\ \forall c\in C,\\\ E\hat{x}=\hat{b},\text{with
}\hat{a}=f^{\prime}(x_{0})-Ax_{0}-\hat{\beta_{1}},\
\hat{b}=Ex_{0}+\hat{\beta_{2}}.\end{cases}$ (2.24)
By the first equations in (2.23) and (2.24) we obtain that
$\langle a+Ax+E^{*}\lambda,\hat{x}-x\rangle_{X^{*},X}\geq 0,\
\langle\hat{a}+A\hat{x}+E^{*}\hat{\lambda},x-\hat{x}\rangle_{X^{*},X}\geq 0,\
x,\hat{x}\in C.$ (2.25)
Combining these inequalities, we have that
$\langle
a-\hat{a}+A(x-\hat{x})+E^{*}(\lambda-\hat{\lambda}),x-\hat{x}\rangle_{X^{*},X}\leq
0.$ (2.26)
The second equalities in (2.23) and (2.24) imply that
$E(x-\hat{x})=b-\hat{b}.$ (2.27)
Let us set
$\delta x=\hat{x}-x,\ \delta\lambda=\hat{\lambda}-\lambda,\ \delta
a=\hat{a}-a,\ \delta b=\hat{b}-b.$
Then $\delta\beta_{1}=-(\hat{\beta_{1}}-\beta_{1})$ and
$\delta\beta_{2}=\hat{\beta_{2}}-\beta_{2}$, and (2.26), (2.27) result in
$\langle\delta x,A\delta x\rangle_{X,X^{*}}+\langle\delta\lambda,E\delta
x\rangle_{W^{*},W}-\langle\delta\beta_{1},\delta x\rangle_{X^{*},X}\leq 0,$
(2.28)
and
$E\delta x=\delta b.$ (2.29)
By (H2) the operator $E$ is surjective. Hence by the closed range theorem
expressing $\delta x=\delta v+\delta w\in\text{ker }E+\text{range }E^{*}$
implies that $\displaystyle E\delta x=E\delta w=\delta\beta_{2}$. Again by the
closed range theorem there exists $k_{1}>0$:
$\left\lVert\delta w\right\rVert_{X}\leq
k_{1}\left\lVert\delta\beta_{2}\right\rVert_{W}.$ (2.30)
From the first equation in (2.21) we have
$Ax+E^{*}\lambda-
Ax_{0}+f^{\prime}(x_{0})-\beta_{1}\in-\partial{\mathbf{I}}_{C}(x).$
Next we restrict the perturbation parameters to satisfy
$(\beta_{1},\beta_{2})\in(\underline{X^{*}}\times W)\cap\tilde{V}$. Due to
Assumptions (H4) and (H7) we have
$(x,\lambda)\in\underline{X}\times\underline{W^{*}}$,
$\underline{A}x+{\underline{E^{*}}}\lambda-\underline{A}x_{0}+\underline{f^{\prime}}(x_{0})-\beta_{1}\in\underline{X^{*}}$
and hence
$\underline{A}x+{\underline{E^{*}}}\lambda-\underline{A}x_{0}+\underline{f^{\prime}}(x_{0})-\beta_{1}\in-\underline{\partial\mathbf{I}_{C}}(x).$
The analogous equation holds with $(x,\lambda,\beta_{1})$ replaced by
$(\hat{x},\hat{\lambda},\hat{\beta}_{1})$.
By (H3), (2.28) and Assumption (H7) we find
$\displaystyle\kappa\left\lVert\delta v\right\rVert^{2}$
$\displaystyle\leq\langle\delta v,A\delta v\rangle_{X,X^{*}}=\langle\delta
x,A\delta x\rangle_{X,X^{*}}-2\langle\delta v,A\delta
w\rangle_{X,X^{*}}-\langle\delta w,A\delta w\rangle_{X,X^{*}}$ (2.31)
$\displaystyle\leq-\langle\delta\lambda,E\delta
x\rangle_{W^{*},W}+\langle\delta\beta_{1},\delta
x\rangle_{X,X^{*}}-2\langle\delta v,A\delta w\rangle_{X,X^{*}}-\langle\delta
w,A\delta w\rangle_{X,X^{*}}$
$\displaystyle\leq\tilde{k}\left\lVert\delta\lambda\right\rVert_{\underline{W}^{*}}\left\lVert\delta\beta_{2}\right\rVert_{W}+\left\lVert\delta\beta_{1}\right\rVert_{X^{*}}\left\lVert\delta
x\right\rVert_{X}+\left\lVert A\right\rVert\left\lVert\delta
w\right\rVert_{X}\left(2\left\lVert\delta v\right\rVert_{X}+\left\lVert\delta
w\right\rVert_{X}\right)$ $\displaystyle\leq\tilde{k}k(\left\lVert\delta
w\right\rVert_{X}+\left\lVert\delta
v\right\rVert_{X}+\left\lVert\delta\beta\right\rVert_{\underline{X}^{*}\times
W})\left\lVert\delta\beta_{2}\right\rVert_{W}+(\left\lVert\delta\beta_{1}\right\rVert_{X^{*}}+2\left\lVert
A\right\rVert\left\lVert\delta w\right\rVert_{X})(\left\lVert\delta
v\right\rVert_{X}+\left\lVert\delta w\right\rVert_{X}),$
where $\tilde{k}$ denotes the embedding constant of $\underline{W}^{*}$ into
$W^{*}$. Using (2.30) and rearranging terms there exists a constant $k_{2}>0$
such that
$\left\lVert\delta v\right\rVert_{X}\leq
k_{2}\left(\left\lVert\delta\beta_{1}\right\rVert_{\underline{X}^{*}}+\left\lVert\delta\beta_{2}\right\rVert_{W}\right).$
(2.32)
Applying (2.30) again this implies the existence of $k_{3}$ such that
$\left\lVert\delta x\right\rVert_{X}\leq
k_{3}\left(\left\lVert\delta\beta_{1}\right\rVert_{\underline{X}^{*}}+\left\lVert\delta\beta_{2}\right\rVert_{W}\right)\text{
for all }(\beta_{1},\beta_{2})\in(\underline{X^{*}}\times W)\cap\tilde{V}.$
(2.33)
Another application of (H7) and (2.33) imply the existence of a constant
$k_{4}$ and a neighborhood $\hat{V}$ of the origin in $\underline{X^{*}}\times
W$ such that the desired Lipschitz stability estimate for
$(\underline{{\mathcal{T}}})^{-1}$
$\left\lVert\delta
x\right\rVert_{\underline{X}}+\left\lVert\delta\lambda\right\rVert_{\underline{W^{*}}}\leq
k_{4}\left(\left\lVert\delta\beta_{1}\right\rVert_{\underline{X^{*}}}+\left\lVert\delta\beta_{2}\right\rVert_{W}\right)\text{
for all }(\beta_{1},\beta_{2})\in\hat{V}\subset\underline{X^{*}}\times W$
(2.34)
holds.
4. (iv)
As a consequence of the previous two steps $\underline{{\mathcal{T}}}$ is
strongly regular at $(x_{0},q_{0},\lambda_{0})$. Together with step (i),
Dontchev’s theorem is applicable [Don, Theorem 2.4, Remark 2.5], and (2.15),
and (2.16) follow.
5. (v)
(Local solution to ($P_{q}$)) Now we show that there exists a neighborhood
$\tilde{N}$ of $q_{0}$ such that for $q\in\tilde{N}$ the second order
sufficient optimality condition is satisfied at $x(q)$, so that $x(q)$ is a
local solution of ($P_{q}$) by eg. [IK, Theorem 2.12, p42]. Due to (H3) and
regularity of $f,e$ we obtain
${\mathcal{L}}^{\prime\prime}(x(q),q,\lambda(q))(h,h)\geq\frac{\kappa}{2}\left\lVert
h\right\rVert^{2},\ \text{for all }h\in\text{ker }E,\ \text{if }q\in
N(q_{0}).$ (2.35)
Let us define $\displaystyle E_{q}=(e_{y}(x(q),q))$ for $q\in N(q_{0})$. By
the surjectivity of $E_{q_{0}}$ and regularity of $e$ there exists a
neighborhood $\tilde{N}\subset N(q_{0})$ such that $E_{q}$ is surjective for
all $q\in\tilde{N}$. There exist $\delta_{0},\gamma>0$ such that
${\mathcal{L}}^{\prime\prime}(x(q),q,\lambda(q))(h+z,h+z)\geq\delta_{0}\left\lVert
h+z\right\rVert^{2},\ \text{for all }h\in\text{ker }E,z\in X$ (2.36)
satisfying $\left\lVert z\right\rVert\leq\gamma\left\lVert h\right\rVert$ by
[IK, Lemma 2.13, p43]. Let us define the orthogonal projection onto ker$\
E_{q}$ given by $\displaystyle P_{\text{ker
}E_{q}}=I-E_{q}^{*}(E_{q}E_{q}^{*})^{-1}E_{q}$. We choose $\tilde{N}$ so that
$\left\lVert
E_{q}^{*}(E_{q}E_{q}^{*})^{-1}E_{q}-E_{q_{0}}^{*}(E_{q_{0}}E_{q_{0}}^{*})^{-1}E_{q_{0}}\right\rVert\leq\frac{\gamma}{1+\gamma}$
for all $q\in\tilde{N}$. For $x\in\text{ker }E_{q}$, we have $x=h+z$ for
$h\in\text{ker }E,\ z\in(\text{ker }E)^{\perp}$ and $\left\lVert
x\right\rVert^{2}=\left\lVert h\right\rVert^{2}+\left\lVert
z\right\rVert^{2}$. Thus,
$\left\lVert z\right\rVert\leq\left\lVert
E_{q}^{*}(E_{q}E_{q}^{*})^{-1}E_{q}x-E_{q_{0}}^{*}(E_{q_{0}}E_{q_{0}}^{*})^{-1}E_{q_{0}}x\right\rVert\leq\frac{\gamma}{1+\gamma}\big{(}\left\lVert
h\right\rVert+\left\lVert z\right\rVert\big{)}$
and hence $\left\lVert z\right\rVert\leq\gamma\left\lVert h\right\rVert$. From
(2.35) this implies
${\mathcal{L}}^{\prime\prime}(x(q),q,\lambda(q))\geq\delta_{0}\left\lVert
x\right\rVert^{2},\ \text{for all }x\in\text{ker }E.$
This concludes the proof. ∎
## 3 Differentiability of value function for optimal stabilization subject to
semi-linear parabolic equations.
Here we describe the optimal control problems which we shall analyze and state
the main results.
### 3.1 Notation
Let $\Omega$ be an open connected bounded subset of $\mathbb{R}^{d}$ with a
Lipschitz continuous boundary $\Gamma$. The associated space-time cylinder is
denoted by $Q=\Omega\times(0,\infty)$ and the associated lateral boundary by
$\Sigma=\Gamma\times(0,\infty)$. We define the Hilbert spaces
$Y=L^{2}(\Omega),\quad V=H^{1}_{0}(\Omega),\text{ and }\quad
U=L^{2}(0,\infty;\,{\mathcal{U}}),$
where ${\mathcal{U}}$ is a Hilbert space which will be identified with its
dual. Observe that the embedding $V\subset Y$ is dense and compact. Further
$V\subset Y\subset V^{*}$, is a Gelfand triple. Here $V^{*}$ denotes the
topological dual of $V$ with respect to the pivot space $Y$. For any
$T\in(0,\infty)$ we define the space
$W(0,T)=\bigg{\\{}y\in L^{2}(0,T;V);\ \frac{dy}{dt}\in
L^{2}(0,T;V^{*})\bigg{\\}},$
and for $T=\infty$, we write $W_{\infty}$ and $I=(0,\infty)$. We further set
$W^{0}_{\infty}=\\{y\in W_{\infty}:y(0)=0\\}$. We also set
$W(T,\infty)=\bigg{\\{}y\in L^{2}(T,\infty;V);\ \frac{dy}{dt}\in
L^{2}(T,\infty;V^{*})\bigg{\\}}.$
We shall frequently use that $W_{\infty}$ embeds continuously into
$C([0,\infty),Y)$, see e.g. [LM, Theorem 4.2] and that
$\displaystyle\lim_{t\to\infty}y(t)=0$, for $y\in W_{\infty}$, see e.g. [CK].
The set of admissible controls $U_{ad}$ is chosen to be
$U_{ad}\subset\\{u\in U:\|u(t)\|_{{\mathcal{U}}}\leq\eta,\text{ for }a.e.\
t>0\\}$ (3.1)
where $\eta$ is a positive constant. We further set
${\mathcal{U}}_{ad}=\\{v\in{\mathcal{U}}:\|v\|_{{\mathcal{U}}}\leq\eta\\}$ and
denote by $\displaystyle\mathbb{P}_{{\mathcal{U}}_{ad}}$ the projection of
${\mathcal{U}}$ on ${\mathcal{U}}_{ad}$. For this choice of admissible
controls the dynamical system can be stabilized for all sufficiently small
initial conditions in $Y$, see Corollary 4.3 and Remark 4.1.
For $\delta>0$ and $\bar{y}\in Y$, we define the open neighborhoods
$B_{Y}(\delta)=\left\\{y\in Y:\left\lVert y\right\rVert_{Y}<\delta\right\\},$
and $\quad B_{Y}(\bar{y},\delta)=\left\\{y\in Y:\left\lVert
y-\bar{y}\right\rVert_{Y}<\delta\right\\}$.
### 3.2 Problem formulation and assumptions.
We focus on the stabilization problem for an abstract semi-linear parabolic
equation formulated as infinite horizon optimal control problem under control
constraints:
$\displaystyle(\mathcal{P})\qquad\mathcal{V}(y_{0})=\min_{(y,u)\in
W_{\infty}\times U_{ad}}\ J(y,u)$ $\displaystyle=\min_{(y,u)\in
W_{\infty}\times U_{ad}}\ \frac{1}{2}\int_{0}^{\infty}\left\lVert
y(t)\right\rVert^{2}_{Y}dt+\frac{\alpha}{2}\int_{0}^{\infty}\left\lVert
u(t)\right\rVert^{2}_{Y}dt,$ (3.2a) subject to the semilinear parabolic
equation $\displaystyle y_{t}$
$\displaystyle={\mathcal{A}}y+{\mathcal{F}}(y)+Bu\quad\text{ in
}L^{2}(I;V^{*})$ (3.2b) $\displaystyle y(x,0)$ $\displaystyle=y_{0}\quad\text{
in }Y.$ (3.2c)
Throughout ${\mathcal{F}}$ is the substitution operator associated to a
mapping $\mathfrak{f}:\mathbb{R}\to\mathbb{R}$ so that
$({\mathcal{F}}y)(t)=\mathfrak{f}(y(t))$. Sufficient conditions which
guarantee the existence of solutions to (3.2b), (3.2c), as well as solutions
$(\bar{y},\bar{u})$ to ($\mathcal{P}$), for $y_{0}\in Y$ sufficiently small,
will be given below. We shall also make use of the adjoint equation associated
to an optimal state $\bar{y}$, given by
$-p_{t}-{\mathcal{A}}^{*}p-{\mathcal{F}}^{\prime}(\bar{y})^{*}p=-\bar{y}\quad\text{
in }L^{2}(I;V^{*}).$ (3.2d)
Its adjoint state $p$ which will be considered in $L^{2}(I;V)$ or in
$W_{\infty}$. The following assumption will be essential.
#### 3.2.1 Assumptions A.
* A1
The operator ${\mathcal{A}}$ with domain ${\mathcal{D}}({\mathcal{A}})\subset
Y$ and range in $Y$, generates a strongly continuous analytic semigroup
$\displaystyle e^{{\mathcal{A}}t}$ on $Y$ and can be extended to
${\mathcal{A}}\in{\mathcal{L}}(V,V^{*})$.
* A2
$B\in{\mathcal{L}}({\mathcal{U}},Y)$ and there exists a stabilizing feedback
operator $K\in{\mathcal{L}}(Y,{\mathcal{U}})$ such that the semigroup
$\displaystyle e^{({\mathcal{A}}-BK)t}$ is exponentially stable on $Y$.
* A3
The nonlinearity $\displaystyle{\mathcal{F}}:W_{\infty}\to L^{2}(I;V^{*})$ is
twice continuously Fréchet differentiable, with second Fréchet derivative
${\mathcal{F}}^{\prime\prime}$ bounded on bounded subsets of $W_{\infty}$, and
${\mathcal{F}}(0)=0$.
* A4
${\mathcal{F}}:W(0,T)\to L^{1}(0,T;{\mathcal{H}}^{*})$ is weak-to-weak
continuous for every $T>0$, for some Hilbert space ${\mathcal{H}}$ which
embeds densely in $V$.
Note that
$\displaystyle\left(L^{1}(0,T;{\mathcal{H}}^{*})\right)^{*}=L^{\infty}(0,T;{\mathcal{H}})$,
see [Emm, Theorem 7.1.23(iv), p 164]. Moreover,
$L^{\infty}(0,T;{\mathcal{H}})$ is dense in $L^{2}(0,T;V)$, see [MS, Lemma
A.1, p 2231].
* A5
${\mathcal{F}}^{\prime}(\bar{y})\in{\mathcal{L}}{(L^{2}(I;V),L^{2}(I;V^{*}))}$.
###### Remark 3.1.
The requirement that $\mathcal{F}(0)=0$ in (A2) is consistent with the fact
that we focus on the stabilization problem with $0$ as steady state for
(3.2b). Without loss of generality we further assume that
$\mathcal{F}^{\prime}(0)=0,$ (3.3)
which can always be achieved by making $\mathcal{F}^{\prime}(0)$ to be
perturbation of $\mathcal{A}$.
###### Remark 3.2.
Let us assume that (A3) holds. Then in view of the fact that ${\mathcal{F}}$
is a substitution operator we have
$[{\mathcal{F}}^{\prime}(y)v](t)=\mathfrak{f}^{\prime}(y(t))v(t)$ for $y$ and
$v$ in $W_{\infty}$, and
${\mathcal{F}}^{\prime}(y)\in{\mathcal{L}}(W_{\infty},L^{2}(I;V^{*}))$. Its
adjoint $[{\mathcal{F}}^{\prime}(y)^{*}v](t)=\mathfrak{f}^{\prime}(y(t))v(t)$,
for $v\in L^{2}(I;V)$, satisfying
${\mathcal{F}}^{\prime}(y)^{*}\in{\mathcal{L}}(L^{2}(I;V),W_{\infty}^{*})$. It
has a natural restriction to an operator
$\underline{{\mathcal{F}}^{\prime}(y)^{*}}\in{\mathcal{L}}(W_{\infty},L^{2}(I;V^{*}))$.
With (A3) holding it is differentiable and
$[\underline{{\mathcal{F}}^{\prime}(y)}^{*}]^{\prime}$ is a bilinear form on
$W_{\infty}\times W_{\infty}$ with values in $L^{2}(I;V^{*})$.
#### 3.2.2 Abstract setup.
Here we relate problem ($\mathcal{P}$) to the abstract problem ($P_{q}$),
which is used with the following spaces:
$\begin{array}[]{l}X=W_{\infty}\times U,\;W=L^{2}(I;V^{*})\times
Y,\;P=Y,\;C=U_{ad},\;X^{*}=W_{\infty}^{*}\times U,\;W^{*}=L^{2}(I;V)\times
Y,\\\\[7.3194pt] \underline{X}=W_{\infty}\times(U\cap
C(\bar{I};{\mathcal{U}})),\quad\underline{X^{*}}=L^{2}(I;V^{*})\times(U\cap
C(\bar{I};{\mathcal{U}})),\quad\underline{W^{*}}=\widetilde{W}_{\infty},\end{array}$
(3.4)
where $\widetilde{\mathcal{W}}=\\{(\varphi,\varphi(0)):\varphi\in
W_{\infty}\\}$, endowed with the norm of $W_{\infty}$. At times we identify
$\widetilde{\mathcal{W}}$ with $W_{\infty}$. We recall that the dual space of
$\displaystyle W_{\infty}=L^{2}(I;V)\cap W^{1,2}(I;V^{*})$ is $\displaystyle
W^{*}_{\infty}=L^{2}(I;V^{*})+(W^{1,2}(I;V^{*}))^{*}$, endowed with the norm
$\displaystyle\left\lVert
z\right\rVert_{W^{*}_{\infty}}=\inf_{z=z_{1}+z_{2}}\left\lVert
z_{1}\right\rVert_{L^{2}(I;V^{*})}+\left\lVert
z_{2}\right\rVert_{W^{1,2}(I;V^{*})^{*}}$, where $\displaystyle z_{1}\in
L^{2}(I;V^{*}),z_{2}\in(W^{1,2}(I;V^{*}))^{*}$.
To express ($P_{q}$) for the present case, we set $x=(y,u)\in W_{\infty}\times
U$, and the parameter $q$ becomes the initial condition $y_{0}\in Y$. Further
$f:W_{\infty}\times U\longrightarrow\mathbb{R}$ is given by
$f(y,u)=\frac{1}{2}\int_{0}^{\infty}\left\lVert
y(t)\right\rVert^{2}_{Y}dt+\frac{\alpha}{2}\int_{0}^{\infty}\left\lVert
u(t)\right\rVert^{2}_{{\mathcal{U}}}dt,$ (3.5)
and $e(x,q)=e(y,u,y_{0})$ is
$e(y,u,y_{0})=\begin{pmatrix}y_{t}-{\mathcal{A}}y-{\mathcal{F}}(y)-Bu\\\
y(0)-y_{0}\end{pmatrix}:W_{\infty}\times U\times Y\longrightarrow
L^{2}(I;V^{*})\times Y$ (3.6)
By (A3) the mapping $e$ is Fréchet differentiable with respect to
$\displaystyle x=(y,u)\in W_{\infty}\times U$ and thus for $(y,u,y_{0})\in
W_{\infty}\times U\times Y$ we have
$e^{\prime}(y,u,y_{0})(v,w)=\begin{pmatrix}v_{t}-{\mathcal{A}}v-{\mathcal{F}}^{\prime}(y)v-Bw\\\
v(0,\cdot)\end{pmatrix}:W_{\infty}\times U\longrightarrow L^{2}(I;V^{*})\times
Y.$ (3.7)
The Lagrange functional ${\mathcal{L}}:W_{\infty}\times U\times Y\times
L^{2}(I;V)\times Y\longrightarrow\mathbb{R}$ corresponding to our optimal
control problem is given by
${\mathcal{L}}(y,u,y_{0},p,p_{1})=J(y,u)+\int_{0}^{\infty}\langle
p,y_{t}-{\mathcal{A}}y-{\mathcal{F}}(y)-Bu\rangle_{V,V^{*}}dt+(p_{1},y(0)-y_{0})_{Y},$
where $(p,p_{1})\in L^{2}(I;V)\times Y$ corresponds to the abstract Lagrange
multiplier $\lambda\in W^{*}$.
In the remainder of this subsection we specify the mappings ${\mathcal{T}}$
and $\underline{{\mathcal{T}}}$ for problem ($\mathcal{P}$). This will
facilitate the proofs of the main results further below.
At first we take a closer look to the adjoint
$\displaystyle\widetilde{E}^{*}:=e^{\prime}(y,u,y_{0})^{*}\in{\mathcal{L}}(L^{2}(I;V)\times
Y,W^{*}_{\infty}\times U)$ at a generic element $(y,u,y_{0})\in
W_{\infty}\times U\times Y$. It is characterized by the property that for all
$(v,w)\in W_{\infty}\times U,\ (p,p_{1})\in L^{2}(I;V)\times Y$ we have
$\langle\widetilde{E}(v,w),(p,p_{1})\rangle_{L^{2}(I;V^{*})\times
Y,L^{2}(I;V)\times Y}=\langle
v,\widetilde{E}_{1}^{*}(p,p_{1})\rangle_{W_{\infty},W^{*}_{\infty}}+(w,\widetilde{E}_{2}^{*}(p,p_{1}))_{U}$
where
$\langle
v,\widetilde{E}_{1}^{*}(p,p_{1})\rangle_{W_{\infty},W^{*}_{\infty}}=\langle
v_{t}-{\mathcal{A}}v-{\mathcal{F}}^{\prime}(y)v,p\rangle_{L^{2}(I;V^{*}),L^{2}(I;V)}+(v(0),p_{1})_{Y},$
and
$(w,\widetilde{E}_{2}^{*}(p,p_{0}))_{U}=-(w,B^{*}p)_{U}.$
If for some $\tilde{\beta}_{1}\in L^{2}(I;V^{*})$ the pair $(p,p_{1})\in
L^{2}(I;V)\times Y$ is a solution to
$\tilde{E}_{1}^{*}(p,p_{1})=\tilde{\beta}_{1}$ then for all $v\in W_{\infty}$:
$\langle
v_{t}-{\mathcal{A}}v-{\mathcal{F}}^{\prime}(y)v,p\rangle_{L^{2}(V^{*}),L^{2}(V)}+(v(0),p_{1})_{Y}-(w,B^{*}p)_{U}=\langle\tilde{\beta}_{1},v\rangle_{L^{2}(I;V^{*}),L^{2}(I;V)}.$
(3.8)
Now we assume that ${\mathcal{F}}^{\prime}(y)$ is not only an element of
$\displaystyle{\mathcal{L}}(W_{\infty},L^{2}(I;V^{*}))$ but rather that it can
be extended to an operator
${{\mathcal{F}}^{\prime}(y)}\in{\mathcal{L}}(L^{2}(I;V),L^{2}(I;V^{*}))$. This
is guaranteed by (A5) at minimizers $\bar{y}$. Then (3.8) implies that $p\in
W_{\infty}$, and $p_{1}=p(0)$. In particular
$(p,p_{1})=(p,p(0))\in\widetilde{W}_{\infty}$, and (3.8) can equivalently be
expressed as
$\langle
v,\widetilde{E}_{1}^{*}(p,p_{1})\rangle_{L^{2}(I;V),L^{2}(I;V^{*})}=\langle
v_{t}-{\mathcal{A}}v-{\mathcal{F}}^{\prime}(y)v,p\rangle_{L^{2}(I;V^{*}),L^{2}(I;V)}=\langle
v,\widetilde{\beta}_{1}\rangle_{L^{2}(I;V),L^{2}(I;V^{*})},$ (3.9)
for all $v\in L^{2}(I;V)$, where we assumed that $\widetilde{\beta}_{1}\in
L^{2}(I;V^{*})$. Conversely, of course, if $p\in\widetilde{W}_{\infty}$, then
$\displaystyle\widetilde{E}^{*}_{1}(p,p(0))=-p_{t}-{\mathcal{A}}^{*}p-{\mathcal{F}}^{\prime}(y)^{*}p\in
L^{2}(I;V^{*})$.
From now on, let $q_{0}=\bar{y}_{0}$ denote a reference (or nominal) parameter
with associated solution $x_{0}=(\bar{y},\bar{u})$. In Proposition 4.1 we
shall argue that the regular point condition Assumption (H2) is satisfied and
that consequently there exists a Lagrange multiplier $(\bar{p},\bar{p}_{1})$
such that the pair $(x_{0},\lambda_{0})=(\bar{y},\bar{u},\bar{p},\bar{p}_{1})$
satisfies (2.3). Moreover, it will turn out that $\bar{p}\in W_{\infty}$,
$\bar{p}_{1}=p(0)$, and that $\bar{u}\in U\cap C(I;{\mathcal{U}})$. For
convenience let us present (2.3) for the present case
$\displaystyle 0\in\begin{cases}\bar{y}+E^{*}_{1}(\bar{p},\bar{p}(0)),\\\
\alpha\bar{u}-B^{*}\bar{p}+\partial\mathbf{I}_{U_{ad}}(\bar{u}),\\\
\bar{y}_{t}-{\mathcal{A}}\bar{y}-{\mathcal{F}}(\bar{y})-B\bar{u},\\\
{\bar{y}(0)}-y_{0},\end{cases}$ (3.10)
where $\displaystyle E=\begin{pmatrix}E_{1}\\\
E_{2}\end{pmatrix}=e^{\prime}(\bar{y},\bar{u},\bar{y}_{0})$. We stress that
while the Lagrange multiplier $\bar{p}\in W_{\infty}$, the operator
$E^{*}_{1}$ in (3.9) is still considered as an element of
${\mathcal{L}}(L^{2}(I;V)\times Y,W^{*}_{\infty})$.
We are now prepared to specify the multivalued operators
$\displaystyle{\mathcal{T}}:$ $\displaystyle W_{\infty}\times U\times
L^{2}(I;V)\times Y\longrightarrow W_{\infty}^{*}\times U\times
L^{2}(I;V^{*})\times Y,\quad\text{and}$ (3.11)
$\displaystyle\underline{{\mathcal{T}}}:$ $\displaystyle
W_{\infty}\times(U\cap
C(I;{\mathcal{U}}))\times\widetilde{W}_{\infty}\longrightarrow
L^{2}(I;V^{*})\times(U\cap C(I;{\mathcal{U}}))\times L^{2}(I;V^{*})\times Y$
(3.12)
corresponding to (2.7) and (2.14) by
${\mathcal{T}}\begin{pmatrix}y\\\ u\\\ p\\\
p_{1}\end{pmatrix}=\begin{pmatrix}E^{*}_{1}(p,p_{1})+y-[{\mathcal{F}}^{\prime}(\bar{y})^{*}p]^{\prime}(y-\bar{y})\\\
\alpha u-B^{*}p\\\
y_{t}-{\mathcal{A}}y-Bu-{\mathcal{F}}^{\prime}(\bar{y})(y-\bar{y})-{\mathcal{F}}(\bar{y})\\\
y(0)-{\bar{y}_{0}}\end{pmatrix}+\begin{pmatrix}0\\\
\partial\mathbf{I}_{U_{ad}}(u)\\\ 0\\\ 0\end{pmatrix},$ (3.13)
and
$\underline{{\mathcal{T}}}\begin{pmatrix}y\\\ \underline{u}\\\
\underline{p}\\\
\underline{p}(0)\end{pmatrix}=\begin{pmatrix}-\underline{p}_{t}-{\mathcal{A}}^{*}\underline{p}-\underline{{\mathcal{F}}^{\prime}(\bar{y})^{*}}\,\underline{p}+y-[\underline{{\mathcal{F}}^{\prime}(\bar{y})^{*}}\,\underline{\bar{p}}]^{\prime}(y-\bar{y})\\\
\alpha\underline{u}-B^{*}\underline{p}\\\
y_{t}-{\mathcal{A}}y-B\underline{u}-{\mathcal{F}}^{\prime}(\bar{y})(y-\bar{y})-{\mathcal{F}}(\bar{y})\\\
y(0)-{\bar{y}_{0}}\end{pmatrix}+\begin{pmatrix}0\\\
\underline{\partial\mathbf{I}_{U_{ad}}}(\underline{u})\\\ 0\\\
0\end{pmatrix}.$ (3.14)
where
$\underline{\partial\mathbf{I}_{U_{ad}}}(\underline{u})=\left\\{\widetilde{u}\in
U\cap C(I;{\mathcal{U}}):\
(\widetilde{u}(t),v-\underline{u}(t))_{\mathcal{U}}\leq 0,\ \forall t\in I,\
v\in B_{\eta}(0)\right\\},$ (3.15)
with $\displaystyle B_{\eta}(0)=\left\\{v\in{\mathcal{U}};\left\lVert
v\right\rVert_{{\mathcal{U}}}\leq\eta\right\\}$. In (3.14), we underline the
elements which are taken from different domains when compared to (3.13). The
range of the first two coordinates of $\underline{{\mathcal{T}}}$ is smaller
than that of ${\mathcal{T}}$. Accordingly we can make use of (3.9) when moving
from the first row of (3.13) to the first row of (3.14).
For convenience for the subsequent work we recall that the strong regularity
condition introduced below (2.14) requires us to find neighborhoods of $0$ and
$(\bar{y},\bar{u},\bar{p},\bar{p}(0))$ of the form $\hat{V}\subset
L^{2}(I;V^{*})\times(U\cap C(\overline{I};{\mathcal{U}}))\times
L^{2}(I;V^{*})\times Y$ and $\hat{U}\subset W_{\infty}\times(U\cap
C(\overline{I};{\mathcal{U}}))\times\widetilde{W}_{\infty}$, such that for all
$\boldsymbol{\beta}=(\beta_{1},\beta_{2},\beta_{3},\beta_{4})\in\hat{V}$ the
equation
$\underline{{\mathcal{T}}}\left(y,u,\underline{p},\underline{p}(0)\right)^{T}=\left(\beta_{1},\beta_{2},\beta_{3},\beta_{4}\right)^{T}$
(3.16)
admits a unique solution
$(y,\underline{u},\underline{p},\underline{p}(0))\in\hat{U}$ depending
Lipschitz-continuously on $\boldsymbol{\beta}$.
###### Remark 3.3.
We observe that as a consequence of (A3) and Remark 3.2 the operator
$\underline{{\mathcal{T}}}$ is continuous.
Subsequently we shall frequently refrain from the underline-notation since the
meaning should be clear from the context.
### 3.3 Main Theorems.
In this subsection, we present the main theorems of this paper. The first
theorem asserts local continuous differentiability of the value function
$\mathcal{V}$ w.r.t. $y_{0}$, with $y_{0}$ small enough. The second theorem
establishes that $\mathcal{V}$ satisfies the HJB equation in the classical
sense. The proof of the first theorem is based on Theorem 2.1. It will be
given in Section 4 below. For this purpose it will be shown that assumptions A
imply (H1)-(H7). Moreover we need to assert the underlying assumption that
problem ($\mathcal{P}$) is well-posed. This will lead to a smallness
assumption on the initial states $y_{0}$. Consequently it would suffice to
assume that (A3) and (A4) only hold locally in the neighborhood of the origin.
Concerning (A5) observe that it is not implied by (A3). It is vacuously
satisfied for $\bar{y}=0$, which is the case for $y_{0}=0$, since then
$\mathcal{F}^{\prime}(0)=0$, see (3.3).
We invoke Theorem 2.1 to assert the Lipschitz continuity of the state, the
adjoint state, and the control with respect to the initial condition $y_{0}\in
Y$ in the neighborhood of a locally optimal solution $(\bar{y},\bar{u})$
corresponding to a sufficiently small reference initial state $\bar{y}_{0}$.
This will imply the differentiability of the value function associated to
local minima. We shall refer to the value function associated to local minima
as ’local value function’.
###### Theorem 3.1.
Let the assumptions (A) hold. Then associated to each local solution
$(\bar{y}(y_{0}),\bar{u}(y_{0}))$ of ($\mathcal{P}$) there exists a
neighborhood of $U(y_{0})$ such that the local value function
${\mathcal{V}}:U(y_{0})\subset Y\to\mathbb{R}$ is continuously differentiable,
provided that $y_{0}$ is sufficiently close to the origin in $Y$.
To obtain a HJB equation we require additionally that
$t\to({\mathcal{F}}(\bar{y}))(t)$ is continuous with values in $Y$ for global
solutions $(\bar{y},\bar{u})$ to ($\mathcal{P}$), with
$y_{0}\in{\mathcal{D}}({\mathcal{A}})$. In view of the fact that for $y_{0}\in
V$ we can typically expect that the solution of semilinear parabolic equations
satisfy $y\in L^{2}(I;{\mathcal{D}}({\mathcal{A}}))\cap W^{1,2}(I;Y)\subset
C([0,\infty),V)$ this is not a restrictive assumption beyond that what is
already assumed in (A3).
###### Theorem 3.2.
Let the assumptions (A) hold, and let $(\bar{y}(y_{0}),\bar{u}(y_{0}))$ denote
a global solution of ($\mathcal{P}$), for
$y_{0}\in{\mathcal{D}}({\mathcal{A}})$ with sufficiently small norm in $Y$.
Assume that there exists $T_{y_{0}}>0$ such that ${\mathcal{F}}(\bar{y})\in
C([0,T_{y_{0}});Y)$. Then the following Hamilton-Jacobi-Bellman equation holds
at $y_{0}$:
$\mathcal{V}^{\prime}(y)({\mathcal{A}}y+{\mathcal{F}}(y))+\frac{1}{2}\left\lVert
y\right\rVert^{2}_{Y}+\frac{\alpha}{2}\left\lVert\mathbb{P}_{\mathcal{U}_{ad}}\left(-\frac{1}{\alpha}B^{*}\mathcal{V}^{\prime}(y)\right)\right\rVert^{2}_{Y}+\left\langle
B^{*}\mathcal{V}^{\prime}(y),\mathbb{P}_{\mathcal{U}_{ad}}\left(-\frac{1}{\alpha}B^{*}\mathcal{V}^{\prime}(y)\right)\right\rangle_{Y}=0.$
(3.17)
Moreover the optimal feedback law is given by
${\bar{u}}(0)=\mathbb{P}_{\mathcal{U}_{ad}}\left(-\frac{1}{\alpha}B^{*}\mathcal{V}^{\prime}({\bar{y}}(0))\right).$
(3.18)
The condition on the smallness of $y_{0}$ will be discussed in Remark 4.2
below. Roughly it involves well-posedness of the optimality system and second
order sufficient optimality at local solutions. A more detailed, respectively
stronger statement of Theorem 3.1 and Theorem 3.2, will be given in Theorem
4.1 and Theorem 5.1 below. The regularity assumptions
${\mathcal{F}}(\bar{y})\in C([0,T_{y_{0}});Y)$ of Theorem 3.2 will be
addressed in Section 6.
## 4 Proof of Theorem 3.1.
In this section we give the proof for Theorem 3.1. Many of the technical
difficulties arise from the fact that we are working with an infinite horizon
optimal control problem. In this respect we can profit from techniques which
were developed in [BKP3], which, however, do not include the case of
constraints on the norm. Throughout we assume that assumptions (A1) - (A4)
hold.
### 4.1 Well-posedness of problem ($\mathcal{P}$).
Here we prove well-posedness for ($\mathcal{P}$) with small initial data.
First, we recall two consequences of the assumption that ${\mathcal{A}}$ is
the generator of an analytic semigroup.
###### Consequence 1.
Since ${\mathcal{A}}$ generates a strongly continuous analytic semigroup on
$Y$, there exist $\rho\geq 0$ and $\theta>0$ such that
$\langle(\rho I-{\mathcal{A}})v,v\rangle_{V^{*},V}\geq\theta\left\lVert
v\right\rVert^{2}_{V}$
See [BPDM, Part II, Chaptor 1, p 115], [Paz, Theorem 4.2, p14].
###### Consequence 2.
For all $y_{0}\in Y,f\in L^{2}(0,T;V^{*})$, and $T>0$, there exists a unique
solution $y\in W(0,T)$ to
$\dot{y}={\mathcal{A}}y+f,\quad y(0)=y_{0}.$ (4.1)
Furthermore, $y$ satisfies
$\left\lVert y\right\rVert_{W(0,T)}\leq c(T)\Big{(}\left\lVert
y_{0}\right\rVert_{Y}+\left\lVert f\right\rVert_{L^{2}(0,T;V^{*})}\Big{)}$
(4.2)
for a continuous function $c$. Assuming that $y\in L^{2}(0,\infty;Y)$,
consider the equation
$\dot{y}=\underbrace{({\mathcal{A}}-\rho
I)y}_{{\mathcal{A}}_{\rho}}+\underbrace{\rho y+f}_{f_{\rho}},\quad
y(0)=y_{0},$
where $f_{\rho}\in L^{2}(I;V^{*})$. Then the operator ${\mathcal{A}}_{\rho}$
generates a strongly continuous analytic semigroup on $Y$ which is
exponentially stable, see [BPDM, p 115, Theorem II.1.2.12]. It follows that
$y\in W_{\infty}$, that there exists $M_{\rho}$ such that
$\left\lVert y\right\rVert_{W_{\infty}}\leq M_{\rho}\Big{(}\left\lVert
y_{0}\right\rVert_{Y}+\left\lVert
f_{\rho}\right\rVert_{L^{2}(I;V^{*})}\Big{)},$ (4.3)
and that $y$ is the unique solution to (4.1) in $W_{\infty}$, see [BKP3,
Section 2.2] .
###### Lemma 4.1.
There exists a constant $C>0$, such that for all $\delta<(0,1]$ and for all
$y_{1}$ and $y_{2}$ in $W_{\infty}$ with $\displaystyle\left\lVert
y_{1}\right\rVert_{W_{\infty}}\leq\delta$ and $\displaystyle\left\lVert
y_{2}\right\rVert_{W_{\infty}}\leq\delta$, it holds that
$\left\lVert{\mathcal{F}}(y_{1})-{\mathcal{F}}(y_{2})\right\rVert_{L^{2}(I;V^{*})}\leq\delta
C\left\lVert y_{1}-y_{2}\right\rVert_{W_{\infty}}.$ (4.4)
###### Proof.
Let $y_{1},y_{2}$ be as in the statement of the lemma. Using Remark 3.1 we
obtain the estimate
$\displaystyle\left\lVert{\mathcal{F}}(y_{1})-{\mathcal{F}}(y_{2})\right\rVert_{L^{2}(I,V^{*})}\leq\int_{0}^{1}\left\lVert{\mathcal{F}}^{\prime}(y_{1}+t(y_{2}-y_{1}))-{\mathcal{F}}^{\prime}(0)\right\rVert_{{\mathcal{L}}(W_{\infty},L^{2}(I,V^{*}))}\,dt\,\|y_{2}-y_{1}\|_{W_{\infty}}$
$\displaystyle\leq\int_{0}^{1}\int_{0}^{1}\left\lVert{\mathcal{F}}^{\prime\prime}(s(y_{1}+t(y_{2}-y_{1})))(ty_{2}+(1-t)y_{1})\right\rVert_{{\mathcal{L}}(W_{\infty},L^{2}(I,V^{*}))}\,dsdt\,\|y_{2}-y_{1}\|_{W_{\infty}}.$
Now the claim follows by assumption (A3). ∎
###### Lemma 4.2.
Let $\displaystyle{\mathcal{A}}_{s}$ be the generator of an exponentially
stable analytic semigroup $\displaystyle e^{{\mathcal{A}}_{s}t}$ on $Y$. Let
$C$ denote the constant from Lemma 4.1. Then there exists a constant $M_{s}$
such that for all $y_{0}\in Y$ and $f\in L^{2}(I;V^{*})$ with
$\tilde{\gamma}=\left\lVert y_{0}\right\rVert_{Y}+\left\lVert
f\right\rVert_{L^{2}(I;V^{*})}\leq\frac{1}{4CM^{2}_{s}}$
the system
$y_{t}={\mathcal{A}}_{s}y+{\mathcal{F}}(y)+f,\quad y(0)=y_{0}$ (4.5)
has a unique solution $y\in W_{\infty}$, which satisfies
$\left\lVert y\right\rVert_{W_{\infty}}\leq 2M_{s}\tilde{\gamma}.$
With Lemma 4.1 holding, this lemma can be verified in the same manner as
[BKP3, Lemma 5, p 6]. In the following corollary we shall use Lemma 4.2 with
${\mathcal{A}}_{s}={\mathcal{A}}-BK$, and the constant corresponding to
$M_{s}$ will be denoted by $M_{K}$. Further $\|{\mathcal{I}}\|$ denotes the
norm of the embedding constant of $W_{\infty}$ into $C(I;Y)$, $\|i\|$ is the
norm of the embedding $V$ into $Y$, and we recall the constant $\eta$ from
(3.1).
###### Corollary 4.3.
For all $\displaystyle y_{0}\in Y$ with
$\left\lVert
y_{0}\right\rVert_{Y}\leq\min\left\\{\frac{1}{4CM^{2}_{K}},\frac{\eta}{2M_{K}\left\lVert
K\right\rVert_{{\mathcal{L}}(Y)}\left\lVert{\mathcal{I}}\right\rVert}\right\\}$
there exists a control $u\in{U_{ad}}$ such that the system
$y_{t}={\mathcal{A}}y+{\mathcal{F}}(y)+Bu,\quad y(0)=y_{0}$ (4.6)
has a unique solution $y\in W_{\infty}$ satisfying
$\left\lVert y\right\rVert_{W_{\infty}}\leq 2M_{K}\|y_{0}\|_{Y}\quad\text{ and
}\quad\left\lVert
u\right\rVert_{U}\leq\|K\|_{{\mathcal{L}}(Y,\,{\mathcal{U}})}\|{\mathcal{I}}\|\|y\|_{W_{\infty}}\leq
2M_{K}\|y_{0}\|_{Y}\|K\|_{{\mathcal{L}}(Y,\,{\mathcal{U}})}\|{\mathcal{I}}\|.$
(4.7)
###### Proof.
By Assumption (A2), there exists $K$ such that ${\mathcal{A}}-BK$ generates an
exponentially stable analytic semigroup on $Y$. Taking $u=-Ky$, equation (4.6)
becomes
$y_{t}=({\mathcal{A}}-BK)y+{\mathcal{F}}(y),\quad y(0)=y_{0}.$ (4.8)
Then by Lemma 4.2 with $\tilde{\gamma}=\|y_{0}\|_{Y}$ there exists $M_{K}$
such that (4.8) has a solution $y\in W_{\infty}$ satisfying
$\left\lVert y\right\rVert_{W_{\infty}}\leq 2M_{K}\|y_{0}\|_{Y},$
and thus the first inequality in (4.7) holds. For every $t\in I$ we have
$\|u\|_{U}=\|Ky\|_{U}\leq\|K\|_{{\mathcal{L}}(Y,\,{\mathcal{U}})}\|y\|_{Y}\leq\|K\|_{{\mathcal{L}}(Y,\,{\mathcal{U}})}\|{\mathcal{I}}\|\|y\|_{W_{\infty}}\leq
2M_{K}\|y_{0}\|_{Y}\|K\|_{{\mathcal{L}}(Y,\,{\mathcal{U}})}\|{\mathcal{I}}\|,$
(4.9)
and thus the second inequality in (4.7) holds.We still need to assert that
$u\in U_{ad}$. This follows from the second smallness condition on
$\left\lVert y_{0}\right\rVert_{Y}$ and (4.9). ∎
###### Remark 4.1.
In the above proof stabilization was achieved by the feedback control $u=-Ky$.
For this $u$ to be admissible it is needed that ${\mathcal{U}}_{ad}$ has
nonempty interior. The upper bound $\eta$ could be allowed to be time
dependent as long as it satisfies $\displaystyle\inf_{t\geq 0}|\eta(t)|>0$.
###### Corollary 4.4.
Let $\displaystyle y_{0}\in Y$ and let $u\in U_{ad}$ be such that the system
$y_{t}={\mathcal{A}}y+{\mathcal{F}}(y)+Bu,\quad y(0)=y_{0}$ (4.10)
has a unique solution $y\in L^{2}(I;Y)$. If
$\displaystyle\gamma:=\left\lVert y_{0}\right\rVert_{Y}+\left\lVert\rho
y+Bu\right\rVert_{L^{2}(I;V^{*})}\leq\min\left\\{\frac{1}{4CM^{2}_{\rho}},\frac{\eta}{2M_{\rho}\left\lVert
K\right\rVert_{{\mathcal{L}}(Y)}\left\lVert{\mathcal{I}}\right\rVert}\right\\},$
then $y\in W_{\infty}$ and it holds that
$\left\lVert y\right\rVert_{W_{\infty}}\leq 2M_{\rho}\gamma.$
###### Proof.
Since $y\in L^{2}(I;Y)$, we can apply Lemma 4.2 to the equivalent system
$y_{t}=({\mathcal{A}}-\rho I)y+{\mathcal{F}}(y)+\tilde{f},$
where $\tilde{f}=\rho y+Bu$. This proves the assertion. ∎
###### Lemma 4.5.
There exists $\delta_{1}>0$ such that for all $\displaystyle y_{0}\in
B_{Y}(\delta_{1})$, problem ($\mathcal{P}$) possesses a solution
$(\bar{y},\bar{u})\in W_{\infty}\times U_{ad}$. Moreover, there exists a
constant $M>0$ independent of $y_{0}$ such that
$\max\big{\\{}\left\lVert\bar{y}\right\rVert_{W_{\infty}},\left\lVert\bar{u}\right\rVert_{U}\big{\\}}\leq
M\left\lVert y_{0}\right\rVert_{Y}$ (4.11)
###### Proof.
The proof of this lemma follows with analogous argumentation as provided in
[BKP3, Lemma 8]. Let us choose,
$\delta_{1}\leq\min\left\\{\frac{1}{4CM^{2}_{K}},\frac{\eta}{2M_{K}\left\lVert
K\right\rVert_{{\mathcal{L}}(Y)}\left\lVert{\mathcal{I}}\right\rVert}\right\\}$,
where $C$ as in Lemma 4.1 and $M_{K}$ denotes the constant from the Corollary
4.3. We obtain that for each $y_{0}\in B_{Y}(\delta_{1})$, there exists a
control $u\in U_{ad}$ with associated state $y$ satisfying
$\max\big{\\{}\left\lVert u\right\rVert_{U},\left\lVert
y\right\rVert_{W_{\infty}}\big{\\}}\leq\tilde{M}\left\lVert
y_{0}\right\rVert_{Y},$ (4.12)
where $\displaystyle\tilde{M}=2M_{K}\text{ max }\big{(}1,\left\lVert
i\right\rVert\left\lVert
K\right\rVert_{{\mathcal{L}}(Y,\,{\mathcal{U}})}\big{)}$. We can thus consider
a minimizing sequence $\displaystyle(y_{n},u_{n})_{n\in\mathbb{N}}\in
W_{\infty}\times U_{ad}$ with $\displaystyle
J(y_{n},u_{n})\leq\frac{1}{2}M^{2}\left\lVert
y_{0}\right\rVert^{2}_{Y}(1+\alpha)$. For all $n\in\mathbb{N}$ that
$\left\lVert y_{n}\right\rVert_{L^{2}(I;Y)}\leq\tilde{M}\left\lVert
y_{0}\right\rVert_{Y}\sqrt{1+\alpha}\quad\text{and}\quad\left\lVert
u_{n}\right\rVert_{L^{2}(I;\,{\mathcal{U}})}\leq\tilde{M}\left\lVert
y_{0}\right\rVert_{Y}\sqrt{\frac{1+\alpha}{\alpha}}.$ (4.13)
We set $\eta(\alpha,\tilde{M})=\Big{[}1+\tilde{M}\|i\|\
\sqrt{(1+\alpha)}\Big{(}\rho+\frac{\left\lVert
B\right\rVert_{{\mathcal{L}}({\mathcal{U}},Y)}}{\sqrt{\alpha}}\Big{)}\Big{]}$.
Then we have $\left\lVert y_{0}\right\rVert+\left\lVert\rho
y_{n}+Bu_{n}\right\rVert_{L^{2}(I;V^{*})}\leq\eta(\alpha,\tilde{M})\left\lVert
y_{0}\right\rVert_{Y}$. After further reduction of $\delta_{1}$, we obtain
with $M_{\rho}$ from Corollary 4.4:
$\left\lVert y_{0}\right\rVert+\left\lVert\rho
y_{n}+Bu_{n}\right\rVert_{L^{2}(I;V^{*})}\leq\frac{1}{4CM^{2}_{\rho}}.$
It follows from this corollary that the sequence
$\\{y_{n}\\}_{n\in\mathbb{N}}$ is bounded in $W_{\infty}$ with
$\sup_{n\in\mathbb{N}}\left\lVert y_{n}\right\rVert_{W_{\infty}}\leq
2M_{\rho}(1+\eta(\alpha,\tilde{M}))\left\lVert y_{0}\right\rVert_{Y}.$ (4.14)
Extracting if necessary a subsequence, there exists
$\displaystyle(\bar{y},\bar{u})\in W_{\infty}\times U$ such that
$\displaystyle({y}_{n},{u}_{n})\rightharpoonup(\bar{y},\bar{u})\in
W_{\infty}\times U$, and $(\bar{y},\bar{u})$ satisfies (4.12).
Let us prove that $(\bar{y},\bar{u})$ is feasible and optimal. Since $U_{ad}$
is weakly sequentially closed and $u_{n}\in U_{ad}$, we find $\bar{u}\in
U_{ad}$. For each fixed $T>0$ and arbitrary $z\in
L^{\infty}(0;T;{\mathcal{H}})\subset L^{2}(0,T;V)$, see (A4), we have for all
$\displaystyle n\in\mathbb{N}$ that
$\int_{0}^{T}\langle\dot{y}_{n}(t),z(t)\rangle_{V^{*},V}dt=\int_{0}^{T}\langle{\mathcal{A}}y_{n}(t)+{\mathcal{F}}(y_{n}(t))+Bu_{n}(t),z(t)\rangle_{V^{*},V}dt.$
(4.15)
Since $\dot{y}_{n}\rightharpoonup\dot{y}$ in $L^{2}(0,T;V^{*})$, we can pass
to the limit in the l.h.s. of the above equality. Moreover, since
${\mathcal{A}}y_{n}\rightharpoonup{\mathcal{A}}y$ in $L^{2}(0,T;V^{*})$,
$\int_{0}^{T}\langle{\mathcal{A}}y_{n}(t),z(t)\rangle_{V^{*},V}dt\xrightarrow[n\rightarrow\infty]{}\int_{0}^{T}\langle{\mathcal{A}}\bar{y}(t),z(t)\rangle_{V^{*},V}dt.$
Analogously, we obtain that
$\int_{0}^{T}\langle
Bu_{n}(t),z(t)\rangle_{V^{*},V}dt\xrightarrow[n\rightarrow\infty]{}\int_{0}^{T}\langle
B\bar{u}(t),z(t)\rangle_{V^{*},V}dt.$
If moreover $z\in L^{\infty}(0,T;{\mathcal{H}})\subset L^{2}(0,T;V)$, we use
(A4) to assert
$\int_{0}^{T}\langle{\mathcal{F}}(y_{n}(t))-{\mathcal{F}}(\bar{y}(t)),z(t)\rangle_{V^{*},V}dt=\int_{0}^{T}\langle{\mathcal{F}}(y_{n}(t))-{\mathcal{F}}(\bar{y}(t)),z(t)\rangle_{{\mathcal{H}}^{*},{\mathcal{H}}}dt\xrightarrow[n\rightarrow\infty]{}0.$
Thus we have for all $z\in L^{\infty}(0,T;{\mathcal{H}})$
$\int_{0}^{T}\langle\dot{y}(t)-{\mathcal{A}}y(t)-Bu(t),z(t)\rangle_{V^{*},V}dt=\int_{0}^{T}\langle{\mathcal{F}}(y(t)),z(t)\rangle_{V^{*},V}dt.$
(4.16)
Since $\displaystyle\dot{y}-{\mathcal{A}}y-Bu\in L^{2}(0,T;V^{*})$ and
$L^{\infty}(0,T;{\mathcal{H}})$ is dense in $L^{2}(0,T;V)$ we conclude that
(4.16) holds for all $z\in L^{2}(0,T;V)$ and $T>0$. This yields
$e(\bar{y},\bar{u})=(0,0)$, and thus $(\bar{y},\bar{u})$ is feasible. By weak
lower semicontinuity of norms it follows that $\displaystyle
J(\bar{y},\bar{u})\leq\liminf_{n\rightarrow\infty}J(\bar{y}_{n},\bar{u}_{n})$,
which proves the optimality of $(\bar{y},\bar{u})$, and (4.11) follows from
(4.13). ∎
For the derivation of the optimality system for ($\mathcal{P}$), we need the
following lemma which is taken from [BKP1, Lemma 2.5].
###### Lemma 4.6.
Let $\displaystyle G\in{\mathcal{L}}(W_{\infty},L^{2}(I;V^{*}))$ such that
$\displaystyle\left\lVert G\right\rVert<\frac{1}{M_{K}}$, where $\left\lVert
G\right\rVert$ denotes the operator norm of $G$. Then for all $\displaystyle
f\in L^{2}(I;V^{*})$ and $y_{0}\in Y$, there exists a unique solution to the
problem:
$y_{t}=({\mathcal{A}}-BK)y(t)+(Gy)(t)+f(t),\quad y=y_{0}.$
Moreover,
$\left\lVert
y\right\rVert_{W_{\infty}}\leq\frac{M_{K}}{1-{M_{K}\|G\|}}\left(\left\lVert
f\right\rVert_{L^{2}(I;V^{*})}+\left\lVert y_{0}\right\rVert_{Y}\right).$
We close this section by deriving the optimality conditions for
($\mathcal{P}$).
###### Proposition 4.1.
Let the assumptions (A1) - (A4) hold. Then there exists
$\delta_{2}\in(0,\delta_{1}]$ such that each local solution
$(\bar{y},\bar{u})$ with $y_{0}\in B_{Y}(\delta_{2})$ is a regular point, i.e.
(2.3) is satisfied, and there exists an adjoint state
$(\bar{p},\bar{p}_{1})\in L^{2}(I;V)\times Y$ satisfying
$\displaystyle\langle
v_{t}-{\mathcal{A}}v-{\mathcal{F}}^{\prime}(\bar{y})v,\bar{p}\rangle_{L^{2}(I;V^{*}),L^{2}(I;V)}+(v(0),\bar{p}_{1})_{Y}+(\bar{y},\bar{p})_{L^{2}(I;V)}$
$\displaystyle=0,\quad\text{for all }v\in W_{\infty},$ (4.17)
$\displaystyle\langle\alpha\bar{u}-B^{*}\bar{p},u-\bar{u}\rangle_{U}$
$\displaystyle\geq 0,\quad\text{for all }u\in U_{ad}.$ (4.18)
If the assumption (A5) is satisfied, then
$-\bar{p}_{t}-{\mathcal{A}}^{*}\bar{p}-{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}=-\bar{y}\quad\text{
in }L^{2}(I;V^{*}),$
and hence $\bar{p}\in W_{\infty}$ and
$\lim_{t\rightarrow\infty}\bar{p}(t)=0.$ (4.19)
Moreover, there exists $\widetilde{M}>0$, independent of $y_{0}\in
B_{Y}(\delta_{2})$, such that
$\left\lVert\bar{p}\right\rVert_{W_{\infty}}\leq\widetilde{M}\left\lVert
y_{0}\right\rVert_{Y},\text{ and }u\in C(\overline{I},{\mathcal{U}}).$ (4.20)
###### Proof.
To verify the regular point condition, we evaluate $e$ defined in (3.6) at
$(\bar{y},\bar{u},y_{0})$. To check the claim on the range of
$e^{\prime}(\bar{y},\bar{u},y_{0})$ we consider for arbitrary
$\displaystyle(r,s)\in L^{2}(I,V^{*})\times Y$ the equation
$z_{t}-{\mathcal{A}}z-{\mathcal{F}}^{\prime}(\bar{y})z-B(w-\bar{u})=r,\
z(0)=s,$ (4.21)
for unknowns $(z,w)\in W_{\infty}\times{U_{ad}}$. By taking $w=-Kz\in U$ we
obtain
$z_{t}-({\mathcal{A}}-BK)z-{\mathcal{F}}^{\prime}(\bar{y})z+B\bar{u}=r,\
z(0)=s.$
We apply Lemma 4.6 to this equation with $\displaystyle
G=-{\mathcal{F}}^{\prime}(\bar{y})$ and $f=r-B\bar{u}$. By Lemma 4.5 and (3.3)
in Remark 3.1 there exists $\delta_{2}\in(0,\delta_{1}]$ such that
$\|{\mathcal{F}}^{\prime}(\bar{y})\|_{{\mathcal{L}}(W_{\infty},L^{2}(I;V^{*}))}\leq\frac{1}{2}M_{K}$.
Consequently by Lemma 4.6 there exists $\widetilde{M}$ such that
$\displaystyle\left\lVert z\right\rVert_{W_{\infty}}$
$\displaystyle\leq\widetilde{M}\big{(}\left\lVert
r\right\rVert_{L^{2}(I;V^{*})}+\left\lVert s\right\rVert_{Y}+\left\lVert
B\right\rVert_{{\mathcal{L}}({\mathcal{U}},Y)}\left\lVert\bar{u}\right\rVert_{U}\big{)}$
$\displaystyle\leq\widetilde{M}\big{(}\left\lVert
r\right\rVert_{L^{2}(I;V^{*})}+\left\lVert s\right\rVert_{Y}+\left\lVert
B\right\rVert_{{\mathcal{L}}({\mathcal{U}},Y)}M\left\lVert
y_{0}\right\rVert_{Y}\big{)},$ (4.22)
with $M$ as in (4.11). We shall need to check whether $w=-Kz$ is feasible,
which will be the case if $w(t)\leq\eta$ for a.e. $t\in I$. Indeed we have
$\left\lVert w(t)\right\rVert_{Y}\leq\left\lVert
K\right\rVert_{{\mathcal{L}}(Y,\,{\mathcal{U}})}\left\lVert
z(t)\right\rVert_{Y}\leq\left\lVert
K\right\rVert_{{\mathcal{L}}(Y,\,{\mathcal{U}})}\left\lVert{\mathcal{I}}\right\rVert\widetilde{M}\big{(}\left\lVert
r\right\rVert_{L^{2}(I;V^{*})}+\left\lVert s\right\rVert_{Y}+\left\lVert
B\right\rVert_{{\mathcal{L}}({\mathcal{U}},Y)}M\left\lVert
y_{0}\right\rVert_{Y}\big{)}.$
Consequently, possibly after further reducing $\delta_{2}$, and choosing
$\tilde{\delta}>0$ sufficiently small we have
$\left\lVert w\right\rVert_{L^{\infty}(I;Y)}\leq\eta\text{ for all
}\displaystyle y_{0}\in B_{Y}(\delta_{2})\text{ and all }(r,s)\text{
satisfying }\left\lVert(r,s)\right\rVert_{L^{2}(I;V^{*})\times
Y}\leq\tilde{\delta}.$ (4.23)
Consequently the regular point condition is satisfied. Hence there exists a
multiplier $\displaystyle\lambda=(p,p_{1})\in L^{2}(I;V)\times Y$ satisfying,
$\langle{\mathcal{L}}_{y}(\bar{y},\bar{u},y_{0},\bar{p},\bar{p}_{1}),v\rangle_{L^{2}(I;V^{*}),L^{2}(I;V)}=0,\quad\langle{\mathcal{L}}_{u}(\bar{y},\bar{u},y_{0},\bar{p},\bar{p}_{1}),u-\bar{u}\rangle_{U}\geq
0,\ \forall u\in U_{ad}$ (4.24)
where
${\mathcal{L}}(y,u,y_{0},p,p_{1})=J(y,u)+\int_{0}^{\infty}\langle
p,y_{t}-{\mathcal{A}}y-{\mathcal{F}}(y)-Bu\rangle_{V,V^{*}}dt+\langle
p_{1},y(0)-y_{0}\rangle_{Y}.$
This implies that (4.17) holds.
Now, if we impose the additional assumption (A5), we have
${\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}\in L^{2}(I;V^{*})$. Thus
$-{\mathcal{A}}^{*}\bar{p}-{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}+\bar{y}\in
L^{2}(I;V^{*})$ and the previous identity implies that $\bar{p}\in
W_{\infty}$. Thus we derive
$-\bar{p}_{t}-{\mathcal{A}}^{*}\bar{p}-{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}=-\bar{y}\
\text{in }L^{2}(I;V^{*})\text{ and }\lim_{t\rightarrow\infty}\bar{p}(t)=0,$
and (4.17)-(4.19). Testing the first identity in (4.24) with $v\in L^{2}(I;V)$
we also have $\bar{p}_{1}=\bar{p}(0)$, which is well-defined since $\bar{p}\in
W_{\infty}\subset C(I;Y)$. The second identity in (4.24) gives (4.18). It
remains to estimate $\displaystyle\bar{p}\in W_{\infty}$.
Let $\displaystyle r\in L^{2}(I;V^{*})$ with $\displaystyle\left\lVert
r\right\rVert_{L^{2}(I;V^{*})}\leq\tilde{\delta}$ and consider
$z_{t}-{\mathcal{A}}z-{\mathcal{F}}^{\prime}(\bar{y})z-B(w-\bar{u})=-r,\
z(0)=0.$ (4.25)
Arguing as in (4.21)-(4.22) there exists a solution to (4.25) with $w=-Kz$
such that
$\left\lVert
z\right\rVert_{W_{\infty}}\leq\widetilde{M}\big{(}\tilde{\delta}+\left\lVert
B\right\rVert_{{\mathcal{L}}({\mathcal{U}},Y)}M\left\lVert
y_{0}\right\rVert_{Y}\big{)}\leq\widetilde{M}\big{(}\tilde{\delta}+\left\lVert
B\right\rVert_{{\mathcal{L}}({\mathcal{U}},Y)}M\delta_{2}\big{)}=:C_{1}.$
(4.26)
From (4.1) we have that $\displaystyle\left\lVert
w\right\rVert_{L^{\infty}(I,\,{\mathcal{U}})}\leq\eta$. Let us now observe
that
$\displaystyle\langle\bar{p},r\rangle_{L^{2}(I,V),L^{2}(I,V^{*})}$
$\displaystyle=\langle\bar{p},-z_{t}+{\mathcal{A}}z+{\mathcal{F}}^{\prime}(\bar{y})z\rangle_{L^{2}(I,V),L^{2}(I,V^{*})}+\langle\bar{p},B(w-\bar{u})\rangle_{L^{2}(I;Y)}$
$\displaystyle=\langle\bar{p}_{t}+{\mathcal{A}}^{*}\bar{p}+{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p},z\rangle+\langle
B^{*}\bar{p},w-\bar{u}\rangle_{U},$
where we have used that $z(0)=0$ and
$\displaystyle\lim_{t\rightarrow\infty}\bar{p}(t)=0$, since $\bar{p}\in
W_{\infty}$. We next estimate using (4.17), (4.19) and (4.26)
$\displaystyle\langle\bar{p},r\rangle_{L^{2}(I,V),L^{2}(I,V^{*})}$
$\displaystyle\leq\left\lVert\bar{y}\right\rVert_{L^{2}(I,V^{*})}\left\lVert
z\right\rVert_{L^{2}(I,V)}+\alpha\langle\bar{u},w-\bar{u}\rangle_{U}$
$\displaystyle\leq\left(\left\lVert\bar{y}\right\rVert_{L^{2}(I,V^{*})}+\alpha\left\lVert\bar{u}\right\rVert_{U}\right)\left(C_{1}+\eta+\left\lVert\bar{u}\right\rVert_{U}\right).$
By (4.11), this implies the existence of a constant $C_{2}$ such that
$\sup_{\left\lVert
r\right\rVert_{L^{2}(I,V^{*})}\leq\tilde{\delta}}\langle\bar{p},r\rangle_{L^{2}(I,V),L^{2}(I,V^{*})}\leq
C_{2}\left\lVert y_{0}\right\rVert_{Y}$
and thus
$\left\lVert\bar{p}\right\rVert_{L^{2}(I,V)}\leq\frac{C_{2}}{\tilde{\delta}}\left\lVert
y_{0}\right\rVert_{Y},\quad\text{for all }y_{0}\in B_{Y}(\delta_{2}).$ (4.27)
Now we estimate, again using (A5)
$\displaystyle\left\lVert\bar{p}_{t}\right\rVert_{L^{2}(I;V^{*})}$
$\displaystyle\leq\left\lVert{\mathcal{A}}^{*}\bar{p}+{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}-\bar{y}\right\rVert_{L^{2}(I;V^{*})}\leq
C_{3}\left\lVert\bar{p}\right\rVert_{L^{2}(I,V)}+C_{4}\left\lVert\bar{p}\right\rVert_{L^{2}(I,V)}+\left\lVert\bar{y}\right\rVert_{L^{2}(I;V^{*})}.$
By (4.11) and (4.27) we obtain
$\left\lVert\bar{p}_{t}\right\rVert_{L^{2}(I;V^{*})}\leq C_{5}\left\lVert
y_{0}\right\rVert_{Y}.$ Combining this estimate with (4.27) yields (4.20).
Finally, by (4.18) we find
$\displaystyle\bar{u}(t)=\mathbb{P}_{{\mathcal{U}}_{ad}}\left(\frac{1}{\alpha}B^{*}\bar{p}(t)\right)$.
Since $\bar{p}\in C(\bar{I};Y)$ and $B^{*}\in{\mathcal{L}}(Y,U)$ this implies
that $u\in C(\bar{I};{\mathcal{U}})$. ∎
### 4.2 Verification of (H1)-(H6).
In this section we specialize the previously proved abstract results in
Section 2 to the semilinear parabolic setting. We start with the following
lemma which shows that assumptions A imply (H1)-(H6).
###### Lemma 4.7.
Consider problem ($\mathcal{P}$) with assumptions (A1)-(A4) holding. Then
(H1)–(H4), (H6) are satisfied for ($\mathcal{P}$) uniformly for all $y_{0}\in
B_{Y}(\widetilde{\delta}_{2})$ for some
$\displaystyle\widetilde{\delta}_{2}\in(0,\delta_{2}]$. If moreover, (A5)
holds, then (H5) holds as well.
###### Proof.
Throughout $y_{0}\\!\in\\!B_{Y}(\delta_{2})$, $(\bar{y},\bar{u})$ denotes a
local solution to ($\mathcal{P}$), and $(p,p_{1})\in L^{2}(I;V^{*})\times Y$
the associated Lagrange multiplier.
1. (i)
Verification of (H1): The initial condition $y_{0}$ is our nominal reference
parameter $q$. Lemma 4.5 guarantees the existence of a local solution
$(\bar{y},\bar{u})\sim x_{0}$ to ($\mathcal{P}$)$\sim$ ($P_{q_{0}}$). Clearly
$f$ defined in (3.5) satisfies the required regularity assumptions. Moreover
$e$ satisfies the regularity assumptions as a consequence of (A3).
2. (ii)
Verification of (H2): Proposition 4.1 implies that $(\bar{y},\bar{u})$ is a
regular point.
3. (iii)
Verification of (H3): The second derivative of $e$ is given by
$e^{\prime\prime}(\bar{y},\bar{u},y_{0})((v_{1},w_{1}),(v_{2},w_{2}))=\begin{pmatrix}{\mathcal{F}}^{\prime\prime}(\bar{y})(v_{1},v_{2})\\\
0\end{pmatrix},\quad\forall\ v_{1},v_{2}\in W_{\infty},\ \forall
w_{1},w_{2}\in U.$ (4.28)
For the second derivative of ${\mathcal{L}}$ w.r.t. $(y,u)$, we find
${\mathcal{L}}^{\prime\prime}(\bar{y},\bar{u},y_{0},\bar{p},\bar{p}_{1})((v_{1},w_{1}),(v_{2},w_{2}))=\int_{0}^{\infty}(v_{1},v_{2})_{Y}dt+\alpha\int_{0}^{\infty}(w_{1},w_{2})_{Y}dt\\\
+\int_{0}^{\infty}\langle\bar{p},{\mathcal{F}}^{\prime\prime}(\bar{y})(v_{1},v_{2})\rangle_{V,V^{*}}dt.$
(4.29)
By (A3) for ${\mathcal{F}}^{\prime\prime}$ and Lemma 4.5 , there exists
$\widetilde{M}_{1}$ such that
$\int_{0}^{\infty}\langle\bar{p},{\mathcal{F}}^{\prime\prime}(\bar{y})(v,v)\rangle_{V,V^{*}}dt\leq\widetilde{M}_{1}\|\bar{p}\|_{L^{2}(I;V)}\left\lVert
v\right\rVert^{2}_{W_{\infty}},\quad\forall\ v\in W_{\infty},$ (4.30)
for each solution $(\bar{y},\bar{u})$ of ($\mathcal{P}$) with $y_{0}\in
B_{Y}(\delta_{2})$. Then we obtain
${\mathcal{L}}^{\prime\prime}(\bar{y},\bar{u},y_{0},\bar{p},\bar{p}_{1})((v,w),(v,w))\geq\int_{0}^{\infty}\left\lVert
v\right\rVert_{Y}^{2}dt+\alpha\int_{0}^{\infty}\left\lVert
w\right\rVert^{2}_{{\mathcal{U}}}dt-\widetilde{M}_{1}\|\bar{p}\|_{L^{2}(I;V)}\left\lVert
v\right\rVert^{2}_{W_{\infty}}.$ (4.31)
Now let $0\neq(v,w)\in\ker E\subset W_{\infty}\times U_{ad}$, where $E$ as
defined in (3.7) is evaluated at $(\bar{y},\bar{u})$. Then,
$v_{t}-{\mathcal{A}}v-{\mathcal{F}}^{\prime}(\bar{y})v-Bw=0,\quad v(0)=0.$
Next choose $\rho>0$, such that the semigroup generated by
$({\mathcal{A}}-\rho I)$ is exponentially stable. This is possible due to
(A1). We equivalently write the system in the previous equation as,
$v_{t}-({\mathcal{A}}-\rho I)v-{\mathcal{F}}^{\prime}(\bar{y})v-\rho
v-Bw=0,\quad v(0)=0.$
Now, we invoke Lemma 4.6 with ${\cal A}-BK$ replaced by ${\mathcal{A}}-\rho
I$, $\displaystyle G={\mathcal{F}}^{\prime}(\bar{y})$, and $f(t)=\rho
v(t)+Bw(t)$, and the role of the constant $M_{K}$ will now be assumed by a
parameter $M_{\rho}$. By selecting $\widetilde{\delta}_{2}\in(0,\delta_{2}]$
such that $\left\lVert\bar{y}\right\rVert_{W_{\infty}}$ sufficiently small, we
can guarantee that
$\displaystyle\left\lVert{\mathcal{F}}^{\prime}(\bar{y})\right\rVert_{{\mathcal{L}}(W_{\infty};L^{2}(I;V^{*}))}\leq{}^{1}\\!/_{2M_{\rho}}$,
see (4.11) and (3.3) in Remark 3.1. Then the following estimate holds,
$\left\lVert v\right\rVert_{W_{\infty}}\leq 2{M_{\rho}}\left\lVert
v+Bw\right\rVert_{L^{2}(I;V^{*})}.$
This implies that
$\left\lVert v\right\rVert^{2}_{W_{\infty}}\leq\widetilde{M}_{2}(\left\lVert
v\right\rVert^{2}_{L^{2}(I;Y)}+\left\lVert w\right\rVert^{2}_{L^{2}(I;Y)}).$
(4.32)
for a constant $\widetilde{M}_{2}$ depending on $M_{\rho},\|B\|$, and the
embedding of $Y$ into $V^{*}$. These preliminaries allow the following lower
bound on ${\mathcal{L}}^{\prime\prime}$:
$\displaystyle{\mathcal{L}}^{\prime\prime}(\bar{y},\bar{u},y_{0},\bar{p},\bar{p}_{1})((v,w),(v,w))$
$\displaystyle\geq\int_{0}^{\infty}\left\lVert
v\right\rVert_{Y}^{2}dt+\alpha\int_{0}^{\infty}\left\lVert
w\right\rVert^{2}_{Y}dt-\widetilde{M}_{1}\|\bar{p}\|_{L^{2}(I;V)}\left\lVert
v\right\rVert^{2}_{W_{\infty}}$ $\displaystyle\text{by ~{} \eqref{ker_vw}
~{}}\geq\int_{0}^{\infty}\left\lVert v\right\rVert_{Y}^{2}+\alpha$
$\displaystyle\int_{0}^{\infty}\left\lVert
w\right\rVert^{2}_{Y}-\widetilde{M}_{1}\widetilde{M}_{2}\|\bar{p}\|_{L^{2}(I;V)}\left[\left\lVert
v\right\rVert^{2}_{L^{2}(I;Y)}+\left\lVert
w\right\rVert^{2}_{L^{2}(I;Y)}\right]$
$\displaystyle=\left(1-\widetilde{M}_{1}\widetilde{M}_{2}\right.$
$\displaystyle\left.\|\bar{p}\|_{L^{2}(I;V)}\right)\left\lVert
v\right\rVert^{2}_{L^{2}(I;Y)}+\left(\alpha-\widetilde{M}_{1}\widetilde{M}_{2}\|\bar{p}\|_{L^{2}(I;V)}\right)\left\lVert
w\right\rVert^{2}_{L^{2}(I;Y)}$
$\displaystyle\geq\tilde{\gamma}\left[\left\lVert
v\right\rVert^{2}_{L^{2}(I;Y)}+\left\lVert
w\right\rVert^{2}_{L^{2}(I;Y)}\right]$ (4.33)
where
$\displaystyle\tilde{\gamma}=\min\left\\{1-\widetilde{M}_{1}\widetilde{M}_{2}\|\bar{p}\|_{L^{2}(I;V)},\alpha-\widetilde{M}_{1}\widetilde{M}_{2}\|\bar{p}\|_{L^{2}(I;V)}\right\\}$.
By possible further reduction of $\tilde{\delta}_{2}$ it can be guaranteed
that $\tilde{\gamma}>0$, see (4.27). Then by (4.32), we obtain,
$\displaystyle{\mathcal{L}}^{\prime\prime}(\bar{y},\bar{u},y_{0},\bar{p},\bar{p}_{1})((v,w),(v,w))$
$\displaystyle\geq\frac{\tilde{\gamma}}{2}\left[\left\lVert
v\right\rVert^{2}_{L^{2}(I;Y)}+\left\lVert
w\right\rVert^{2}_{L^{2}(I;Y)}\right]+\frac{\tilde{\gamma}}{2\widetilde{M}_{2}}\left\lVert
v\right\rVert^{2}_{W_{\infty}},$
$\displaystyle\geq\frac{\tilde{\gamma}}{2\widetilde{M}_{2}}\left\lVert
v\right\rVert^{2}_{W_{\infty}}+\frac{\tilde{\gamma}}{2}\left\lVert
w\right\rVert^{2}_{L^{2}(I;Y)}.$
By selecting
$\displaystyle\bar{\gamma}=\min\left\\{\frac{\tilde{\gamma}}{2\widetilde{M}_{2}},\frac{\tilde{\gamma}}{2}\right\\}$,
we obtain the positive definiteness of ${\mathcal{L}}^{\prime\prime}$, i.e.
${\mathcal{L}}^{\prime\prime}(\bar{y},\bar{u},y_{0},\bar{p},\bar{p}_{1})((v,w),(v,w))\geq\bar{\gamma}\left\lVert(v,w)\right\rVert^{2}_{W_{\infty}\times
U},\ y_{0}\in B_{Y}(\widetilde{\delta}_{2}),\ (v,w)\in\text{ker }E.$ (4.34)
Thus (H3) is satisfied.
4. (iv)
Verification of (H4): It can easily be checked that $f^{\prime}(y,u)$ can be
extended to an element in $\underline{X^{*}}=L^{2}(I;V^{*})\times(U\cap
C(\bar{I};{\mathcal{U}}))$ for each $(y,u)\in X=W_{\infty}\times U$. We refer
to Remark 3.2 to show that the restriction of $e^{\prime}(y,u,y_{0})^{*}$ to
$\underline{W^{*}}$ satisfies
$\underline{e^{\prime}(y,u,y_{0})^{*}}\in{\mathcal{L}}(\underline{W^{*}},\underline{X^{*}})={\mathcal{L}}(W_{\infty}\times
Y,L^{2}(I;V^{*})\times(U\cap C(\bar{I};{\mathcal{U}})))$.
5. (v)
Verification of (H6): This is trivially satisfied.
Thus we have proved that assumptions (A1)-(A4) imply (H1)-(H4), and (H6) for
all $\displaystyle y_{0}\in B_{Y}(\widetilde{\delta}_{2})$.
6. (vi)
Verification of (H5): Here we use (A5) and have
$(p,p_{1})=(p,p(0))\in\widetilde{W}_{\infty}$. Observe that
$\underline{{\mathcal{L}}^{\prime}}:W_{\infty}\times(U\cap
C(\bar{I};{\mathcal{U}}))\times Y\times W_{\infty}\to
L^{2}(I;V^{*})\times(U\cap C(\bar{I};{\mathcal{U}}))$ evaluated at
$(\bar{y},\bar{u},y_{0},p)$ is given by
$\underline{{\mathcal{L}}^{\prime}}(\bar{y},\bar{u},y_{0},\bar{p})=\begin{pmatrix}\bar{y}+\underline{e^{\prime}(\bar{y},\bar{u},y_{0})^{*}}\bar{p}\\\
\alpha\bar{u}-B^{*}\bar{p}\end{pmatrix}=\begin{pmatrix}\bar{y}-\bar{p}_{t}-{\mathcal{A}}^{*}\bar{p}-{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}\\\
\alpha\bar{u}-B^{*}\bar{p}\end{pmatrix}.$
Further for $(v.w)\in W_{\infty}\times U$ we have
$(\underline{{\mathcal{L}}^{\prime}})^{\prime}(\bar{y},\bar{u},y_{0},\bar{p})(v,w)=\begin{pmatrix}v-[{\mathcal{F}}^{\prime}(\bar{y})^{*}]^{\prime}(\bar{p},v)\\\
\alpha w\end{pmatrix}\in L^{2}(I;V^{*})\times(U\cap
C(\bar{I};{\mathcal{U}})).$
By (A3) and Remark 3.2 we have that $\underline{{\mathcal{L}}^{\prime}}$ and
$(\underline{{\mathcal{L}}^{\prime}})^{\prime}$ are continuous as mappings
from $W_{\infty}\times(U\cap C(\bar{I};{\mathcal{U}}))\times Y\times
W_{\infty}$ to $L^{2}(I;V^{*})\times(U\cap C(\bar{I};{\mathcal{U}}))$,
respectively to ${\mathcal{L}}(W_{\infty}\times(U\cap
C(\bar{I};{\mathcal{U}}));L^{2}(I;V^{*})\times(U\cap
C(\bar{I};{\mathcal{U}})))$.
This proves the lemma. ∎
###### Remark 4.2.
Let us summarize our findings so far. There exists $\tilde{\delta}_{2}$ such
that for each $y_{0}\in B_{Y}(\tilde{\delta}_{2})$ problem ($\mathcal{P}$)
posesses a solution $(\bar{y},\bar{u})\in W_{\infty}\times(U\cap
C(\bar{I};{\mathcal{U}}))$, with an adjoint
$\bar{p}\in\widetilde{W}_{\infty}$. Further (A1)-(A5) imply (H1)-(H6) for
($\mathcal{P}$) with $y_{0}\in B_{Y}(\tilde{\delta}_{2})$. As a consequence
for each $y_{0}\in B_{Y}(\tilde{\delta}_{2})$ and each associated local
solution $(\bar{y},\bar{u})$ there exists a neighborhood $\hat{V}$ of the
origin in ${\mathcal{Y}}:=L^{2}(I;V^{*})\times(U\cap
C(\bar{I};{\mathcal{U}}))\times L^{2}(I;V^{*})\times Y$ such that for each
$\boldsymbol{\beta}\in\hat{V}$ there exists a unique solution
$\displaystyle\left(y_{(\boldsymbol{\beta})},u_{(\boldsymbol{\beta})},p_{(\boldsymbol{\beta})},{p_{1}}_{(\boldsymbol{\beta})}\right)\in
W_{\infty}\times U\times L^{2}(I;V)\times Y$ to
${\mathcal{T}}(y,u,p,p_{1})=\boldsymbol{\beta}$, see step (ii) of the proof of
Theorem 2.1. – To verify the remaining assumption (H7) we need to argue that
$\left(u_{(\boldsymbol{\beta})},p_{(\boldsymbol{\beta})}\right)\in(U\cap
C(\bar{I};{\mathcal{U}}))\times W_{\infty}$ and that
$\displaystyle\boldsymbol{\beta}\mapsto\left(y_{(\boldsymbol{\beta})},u_{(\boldsymbol{\beta})},p_{(\boldsymbol{\beta})}\right)$
is Lipschitz continuous from $\hat{V}\subset\mathcal{Y}$ to
$W_{\infty}\times(U\cap C(\bar{I};{\mathcal{U}}))\times W_{\infty}$ .
###### Remark 4.3.
Here we remark on the smallness assumption on $y_{0}$ expressed by
$\delta_{2}$, respectively $\tilde{\delta}_{2}$. The condition $y_{0}\in
B_{Y}(\delta_{2})$ guarantees the well-posedness of ($\mathcal{P}$), existence
and boundedness of adjoint states as expressed in Proposition 4.1. The
additional condition $y_{0}\in B_{Y}(\tilde{\delta}_{2})$ implies that the
second order optimality condition (H3) is satisfied, for each local solution
associated to an initial condition $y_{0}\in B_{Y}(\delta_{2})$. In what
follows formulate the results for all $y_{0}\in B_{Y}(\tilde{\delta}_{2})$.
Alternatively we could narrow down the claims to neighborhoods of single local
solutions $(\bar{y},\bar{u})$ with $y_{0}\in B_{Y}(\delta_{2})$ and
additionally assuming that the second order condition is satisfies at
$(\bar{y},\bar{u})$. Concerning the second order condition itself, in some
publications, see e.g. [Gri], it is required to hold only for elements
$x=(y,u)\in\text{ker}E$ and $u=u_{1}-u_{2}$, with $u_{1},u_{2}$ in $U_{ad}$.
By a scaling argument it can easily be seen that this condition is equivalent
to the one we use.
### 4.3 Verification of (H7) and Lipschitz stability of the linearized
problem.
Throughout the remainder, we assume that (A1)-(A5) are satisfied and that
$y_{0}\in B_{Y}(\tilde{\delta}_{2})$ so that Proposition 4.1 and Lemma 4.7 are
applicable. In the following, the triple $(y,u,p)$ refers to the solution
${\mathcal{T}}(y,u,p,p_{1})=\boldsymbol{\beta}$. Throughout without loss of
generality, we also assume that $\hat{V}$ is bounded.
###### Lemma 4.8.
Let assumptions (A) hold and let $(\bar{y},\bar{u})$, and $\bar{p}$ denote a
local solution and associated adjoint state to ($\mathcal{P}$) corresponding
to an initial datum $y_{0}\in B_{Y}(\tilde{\delta}_{2})$. Then the mapping
$\boldsymbol{\beta}\mapsto p_{(\boldsymbol{\beta})}$ is continuous from
$\hat{V}$ to $W_{\infty}$.
###### Proof.
Step 1: For $\boldsymbol{\beta}\in\hat{V}$, with $\hat{V}$ as in Remark 4.2,
let
$\displaystyle(y_{(\boldsymbol{\beta})},u_{(\boldsymbol{\beta})},p_{(\boldsymbol{\beta})},{p_{1}}_{(\boldsymbol{\beta})})$
be the solution to ${\mathcal{T}}(y,u,p,p_{1})=\boldsymbol{\beta}$. As a
consequence of (A5) it is also a solution to
$\underline{{\mathcal{T}}}(y,u,p,p(0))=\boldsymbol{\beta}$ with $p\in
W_{\infty}$. Thus the first two equations of this latter equality can be
expressed as
$\displaystyle-
p_{t}-{\mathcal{A}}^{*}p-{\mathcal{F}}^{\prime}(\bar{y})^{*}p+y-[{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}]^{\prime}(y-\bar{y})$
$\displaystyle=\beta_{1}\quad\text{ in }L^{2}(I;V^{*}),$ (4.35a)
$\displaystyle\langle\alpha u-B^{*}p-\beta_{2},w-u\rangle_{U}$
$\displaystyle\geq 0\quad\text{for all }w\in U_{ad},$ (4.35b)
where we dropped the dependence of
$\displaystyle\left(y_{(\boldsymbol{\beta})},u_{(\boldsymbol{\beta})},p_{(\boldsymbol{\beta})}\right)$
on $\displaystyle\boldsymbol{\beta}$. Since $p\in\widetilde{W}_{\infty}\subset
C(\bar{I};Y)$ inequality (4.35b) implies that $u\in C(\bar{I};{\mathcal{U}})$.
Step 2: (Boundedness of
$\\{p_{(\boldsymbol{\beta})}:\boldsymbol{\beta}\in\hat{V}\\}$). Since
$\hat{V}$ is assumed to be bounded, the discussion in Remark 4.2 shows that
there exists a constant $M>0$ such that
$\left\lVert y_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}+\left\lVert
u_{(\boldsymbol{\beta})}\right\rVert_{U}\leq M\quad\text{for all
}\boldsymbol{\beta}\in\hat{V}.$
To argue the boundedness of $p_{(\boldsymbol{\beta})}$, we use similar
techniques as in the proof of Proposition 4.1. In the following
$\tilde{\delta},z$ and $C_{1}$ are taken as in the proof of that proposition.
For arbitrary $r\in R=\left\\{r\in L^{2}(I;V^{*}):\left\lVert
r\right\rVert_{L^{2}(I;V^{*})}\leq\tilde{\delta}\right\\}$, $z$ denotes the
solution to
$z_{t}-{\mathcal{A}}z-{\mathcal{F}}^{\prime}(\bar{y})z-B(w-u)=-r,\ z(0)=0,\
w=-Kz.$ (4.36)
We know that $\left\lVert z\right\rVert_{W_{\infty}}\leq C_{1}$ and
$\left\lVert w\right\rVert_{L^{\infty}(I;{\mathcal{U}})}\leq\eta$
independently of $r\in R$. Due to (4.35a), we have
$\displaystyle\langle p,r\rangle_{L^{2}(I;V),L^{2}(I;V^{*})}$
$\displaystyle=\langle
p,-z_{t}+{\mathcal{A}}z+{\mathcal{F}}^{\prime}(\bar{y})z\rangle_{L^{2}(I;V),L^{2}(I;V^{*})}+\langle
B^{*}p,w-u\rangle_{U},$ $\displaystyle=\langle
y-[{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}]^{\prime}(y-\bar{y})-\beta_{1},z\rangle_{L^{2}(I;V^{*}),L^{2}(I;V)}+\langle\alpha
u-\beta_{2},w-u\rangle_{U},$ (4.37)
where we used (4.35b) and the feasibility of $w\in U_{ad}$. Consequently
$\langle p,r\rangle_{L^{2}(I;V),L^{2}(I;V^{*})}\leq\left\lVert
z\right\rVert_{L^{2}(I;V)}\left(\left\lVert
y\right\rVert_{L^{2}(I;V^{*})}+\left\lVert\beta_{1}\right\rVert_{L^{2}(I;V^{*})}+\left\lVert[{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}]^{\prime}(y-\bar{y})\right\rVert_{L^{2}(I;V^{*})}\right)\\\
+\left(\alpha\left\lVert
u\right\rVert_{U}+\left\lVert\beta_{2}\right\rVert_{U}\right)\left\lVert
w-u\right\rVert_{U}.$
The right hand side is uniformly bounded for $\boldsymbol{\beta}$ in the
bounded set $\hat{V}$ and w.r.t. $r\in R$. Hence taking the supremum w.r.t.
$r\in R$ we verified that $\displaystyle\left\\{\left\lVert
p_{(\boldsymbol{\beta})}\right\rVert_{L^{2}(I;V^{*})}:\boldsymbol{\beta}\in\hat{V}\right\\}$
is bounded. Boundedness of $\displaystyle\left\\{\left\lVert
p_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}:\boldsymbol{\beta}\in\hat{V}\right\\}$
follows from (4.35a).
Step 3: (Continuity of $p_{(\boldsymbol{\beta})}$ in $W_{\infty}$). Let
$\\{\boldsymbol{\beta}_{n}\\}$ be a convergent sequence in $\hat{V}$ with
limit $\boldsymbol{\beta}$. Since $\displaystyle\left\\{\left\lVert
p_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}:n\in\mathbb{N}\right\\}$ is
bounded, there exists a subsequence $\\{\boldsymbol{\beta}_{n_{k}}\\}$ such
that $\displaystyle p_{(\boldsymbol{\beta}_{n_{k}})}\rightharpoonup\tilde{p}$
weakly in $W_{\infty}$ and strongly $L^{2}(I;Y)$. Here we need the compactness
of $W_{\infty}$ in $L^{2}(I;Y)$, see e.g. [Emm, Satz 8.1.12, pg 213]. Passing
to the limit in the variational form of
$-\partial_{t}p_{(\boldsymbol{\beta}_{n_{k}})}-{\mathcal{A}}^{*}p_{(\boldsymbol{\beta}_{n_{k}})}-{\mathcal{F}}^{\prime}(\bar{y})^{*}p_{(\boldsymbol{\beta}_{n_{k}})}+y-[{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}]^{\prime}\left(y_{(\boldsymbol{\beta}_{n_{k}})}-\bar{y}\right)=(\boldsymbol{\beta}_{n_{k}})_{1},$
we obtain
$-\partial_{t}\tilde{p}-{\mathcal{A}}^{*}\tilde{p}-{\mathcal{F}}^{\prime}(\bar{y})^{*}\tilde{p}+y_{(\boldsymbol{\beta})}-[{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}]^{\prime}\left(y_{(\boldsymbol{\beta})}-\bar{y}\right)=(\boldsymbol{\beta})_{1}.$
(4.38)
Since the solution to this equation is unique we have $\displaystyle
p_{(\boldsymbol{\beta}_{n})}\rightharpoonup p_{(\boldsymbol{\beta})}$ weakly
in $W_{\infty}$. To obtain strong convergence we set
$\delta\boldsymbol{\beta}=\boldsymbol{\beta}_{n}-\boldsymbol{\beta},\ \delta
p=p_{(\boldsymbol{\beta}_{n})}-p_{(\boldsymbol{\beta})},\ \delta
y=y_{(\boldsymbol{\beta}_{n})}-y_{(\boldsymbol{\beta})}$. From (4.35a) we
derive that
$-\partial_{t}(\delta p)-{\mathcal{A}}^{*}(\delta
p)-{\mathcal{F}}^{\prime}(\bar{y})^{*}(\delta
p)+(I-[{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p})]^{\prime}(\delta
y)=(\delta\boldsymbol{\beta})_{1}.$ (4.39)
Consider again (4.36) with $r\in R$. Then we obtain
$\langle\delta
p,r\rangle_{L^{2}(I;V),L^{2}(I;V^{*})}=\langle(I-[{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}]^{\prime})(\delta
y)-(\delta\boldsymbol{\beta})_{1},z\rangle_{L^{2}(I;V^{*}),L^{2}(I;V)}+\langle\delta
p,BKz-\bar{u}\rangle_{U}.$
This implies that for some $C_{2}>0$,
$\sup_{r\in R}\ \langle\delta p,r\rangle_{L^{2}(I;V),L^{2}(I;V^{*})}\leq
C_{2}\left(\left\lVert\delta y\right\rVert_{W_{\infty}}+\left\lVert\delta
p\right\rVert_{L^{2}(I;Y)}+\left\lVert(\delta\boldsymbol{\beta})_{1}\right\rVert_{L^{2}(I;V^{*})}\right).$
(4.40)
Since $\left\lVert\delta y\right\rVert_{W_{\infty}}\to 0,\ \left\lVert\delta
p\right\rVert_{L^{2}(I;Y)}\to 0,\
\left\lVert(\delta\boldsymbol{\beta})_{1}\right\rVert_{L^{2}(I;V^{*})}\to 0$
for $n\to 0$ this implies that $\left\lVert\delta
p\right\rVert_{L^{2}(I;V)}\to 0$. Together with (4.39) this implies that
$\displaystyle\lim_{n\rightarrow\infty}\left\lVert\delta
p\right\rVert_{W_{\infty}}=0$. ∎
###### Proposition 4.2.
Let assumptions (A) hold and let $(\bar{y},\bar{u})$, and $\bar{p}$ denote a
local solution and associated adjoint state state to ($\mathcal{P}$)
corresponding to an initial condition $\bar{y}_{0}\in
B_{Y}(\tilde{\delta}_{2})$. Then there exists $\varepsilon>0$ and $C>0$ such
that for all $\boldsymbol{\hat{\beta}}$ and $\boldsymbol{\beta}\in\hat{V}\cap
B_{{\mathcal{Y}}}(\varepsilon)$
$\left\lVert\hat{y}_{(\hat{\boldsymbol{\beta}})}-y_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}+\left\lVert\hat{p}_{(\hat{\boldsymbol{\beta}})}-p_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}+\left\lVert
u_{(\hat{\boldsymbol{\beta}})}-u_{(\boldsymbol{\beta})}\right\rVert_{C(\bar{I};{\mathcal{U}})}\\!\\\
\leq
C\left(\left\lVert\hat{y}_{(\hat{\boldsymbol{\beta}})}-y_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}+\left\lVert
u_{(\hat{\boldsymbol{\beta}})}-u_{(\boldsymbol{\beta})}\right\rVert_{U}+\left\lVert\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right\rVert_{{\mathcal{Y}}}\right)$
(4.41)
holds.
###### Proof.
Since $\bar{p}\in W_{\infty}$, there exists $T>0$ such that
$\frac{1}{\alpha}\left\lVert
B^{*}\bar{p}\right\rVert_{Y}\leq\frac{\eta}{2},\quad\forall t>T.$
Since $p(0)=\bar{p}$ and since by Lemma 4.8,
$\displaystyle\boldsymbol{\beta}\in{\mathcal{Y}}\mapsto
p_{(\boldsymbol{\beta})}\in W_{\infty}\subset C(\bar{I};Y)$ is continuous,
there exists $\varepsilon>0$ such that
$\frac{1}{\alpha}\left\lVert
B^{*}p_{(\boldsymbol{\beta})}(t)+\beta_{2}(t)\right\rVert_{Y}\leq\frac{\eta}{4},\quad\forall
t\geq T,\ \forall\boldsymbol{\beta}\in\hat{V}\cap
B_{{\mathcal{Y}}}(\varepsilon).$
Consequently the constraints are inactive for these parameter values, i.e. we
have
$u_{(\boldsymbol{\beta})}(t)=\frac{1}{\alpha}\left[B^{*}p_{(\boldsymbol{\beta})}(t)+\beta_{2}(t)\right],\
\left\lVert u_{(\boldsymbol{\beta})}(t)\right\rVert_{Y}\leq\eta,\quad\forall
t\geq T,\ \forall\boldsymbol{\beta}\in\hat{V}\cap
B_{{\mathcal{Y}}}(\varepsilon).$ (4.42)
Let us henceforth set
$(y,u,p)=\left(y_{(\boldsymbol{\beta})},u_{(\boldsymbol{\beta})},p_{(\boldsymbol{\beta})}\right)$,
and
$\left(\hat{y},\hat{u},\hat{p}\right)=\left(\hat{y}_{(\boldsymbol{\hat{\beta}})},\hat{u}_{(\boldsymbol{\hat{\beta}})},\hat{p}_{(\boldsymbol{\hat{\beta}})}\right)$.
We shall use that
$\left\lVert\hat{p}-p\right\rVert_{L^{2}(T,\infty;V)}=\sup_{\left\lVert
r\right\rVert_{L^{2}(T,\infty;V^{*})}\leq
1}\int_{T}^{\infty}\langle\hat{p}(t)-p(t),r(t)\rangle_{V,V^{*}}dt.$
Let $z\in W(T,\infty)$ be such that,
$z_{t}-({\mathcal{A}}-BK)z-{\mathcal{F}}^{\prime}(\bar{y})z=r,\ z(T)=0,$
From Lemma 4.6, see also the proof of Proposition 4.1, we know that there
exists a constant $C_{1}>0$ such that $\displaystyle\left\lVert
z\right\rVert_{W(T,\infty)}\leq C_{1}\left\lVert
r\right\rVert_{L^{2}(T,\infty;V^{*})}$. Then we can estimate
$\displaystyle\left\lVert\hat{p}-p\right\rVert_{L^{2}(T,\infty;V)}$
$\displaystyle=\sup_{\left\lVert r\right\rVert_{L^{2}(T,\infty;V^{*})}\leq
1}\int_{T}^{\infty}\langle\hat{p}-p,r\rangle_{V,V^{*}}dt$
$\displaystyle\leq\sup_{\left\lVert r\right\rVert\leq
1}\int_{T}^{\infty}-\langle(\hat{p}_{t}-p_{t})+{\mathcal{A}}^{*}(\hat{p}-p)+{\mathcal{F}}^{\prime}(\bar{y})^{*}(\hat{p}-p),z\rangle_{V^{*},V}dt$
$\displaystyle\hskip 199.16928pt+\sup_{\left\lVert r\right\rVert\leq
1}\int_{T}^{\infty}\langle B^{*}(\hat{p}-p),Kz\rangle_{V,V^{*}}dt.$
In the following, $C_{i}$ denote constants independent of
$\boldsymbol{\hat{\beta}}\text{ and }\boldsymbol{\beta}\in\hat{V}\cap
B_{{\mathcal{Y}}}(\varepsilon)$. From (4.35a) and (4.42) we obtain, for
$C_{2}>0$,
$\left\lVert\hat{p}-p\right\rVert_{L^{2}(T,\infty;V)}\leq
C_{2}\sup_{\left\lVert r\right\rVert\leq
1}\int_{T}^{\infty}\left[\left\lVert\hat{y}-y\right\rVert_{V^{*}}+\left\lVert[{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}]^{\prime}(\hat{y}-y)\right\rVert_{V^{*}}+\left\lVert\hat{\beta}_{1}-\beta_{1}\right\rVert_{V^{*}}\right.\\\
\left.+\left\lVert\hat{\beta}_{2}-\beta_{2}\right\rVert_{U\cap
C(I;{\mathcal{U}})}+\alpha\left\lVert B^{*}\right\rVert\left\lVert
K\right\rVert\left\lVert\hat{u}-u\right\rVert\right]\left\lVert
z\right\rVert_{V}dt.$
From (A3) recall that
$\displaystyle\left\lVert[{\mathcal{F}}^{\prime}(\bar{y})^{*}\bar{p}]^{\prime}(\hat{y}-y)\right\rVert_{L^{2}(I;V^{*})}\leq
C_{3}\left\lVert\hat{y}-y\right\rVert_{W_{\infty}}$. This gives the following
estimate for $C_{4}>0$,
$\left\lVert\hat{p}-p\right\rVert_{L^{2}(T,\infty;V)}\leq
C_{4}\left(\left\lVert\hat{y}-y\right\rVert_{W_{\infty}}+\left\lVert\hat{u}-u\right\rVert_{U}+\left\lVert\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right\rVert_{{\mathcal{Y}}}\right).$
(4.43)
By (4.35a) we have $\displaystyle\left(\hat{p}_{t}-p_{t}\right)\in
L^{2}(T,\infty;V^{*})$. Then we obtain $\hat{p}-p\in W(T,\infty)$. Then there
exists $C_{5}>0$ independent of $\boldsymbol{\hat{\beta}}\text{ and
}\boldsymbol{\beta}\in\hat{V}\cap B_{{\mathcal{Y}}}(\varepsilon)$ such that,
$\left\lVert\hat{p}-p\right\rVert_{W(T,\infty)}\leq
C_{5}\left(\left\lVert\hat{y}-y\right\rVert_{W_{\infty}}+\left\lVert\hat{u}-u\right\rVert_{U}+\left\lVert\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right\rVert_{{\mathcal{Y}}}\right).$
(4.44)
By the embedding $\displaystyle W(T,\infty)\subset C(T,\infty;Y)$, there
exists a constant $C_{6}>0$:
$\left\lVert\hat{p}-p\right\rVert_{C([T,\infty);Y)}\leq
C_{6}\left(\left\lVert\hat{y}-y\right\rVert_{W_{\infty}}+\left\lVert\hat{u}-u\right\rVert_{U}+\left\lVert\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right\rVert_{{\mathcal{Y}}}\right).$
(4.45)
Similarly, we estimate on $[0,T]$:
$\left\lVert\hat{p}-p\right\rVert_{L^{2}(0,T;V)}=\sup_{\left\lVert
r\right\rVert_{L^{2}(0,T;V^{*})}\leq
1}\int_{0}^{T}\langle\hat{p}-p,r\rangle_{V,V^{*}}dt.$ (4.46)
Choose $z$ as
$z_{t}-\left({\mathcal{A}}z+{\mathcal{F}}^{\prime}(\bar{y})z\right)=r,\
z(0)=0,$
Then there exists $\displaystyle C_{7}>0$ such that $\displaystyle\left\lVert
z\right\rVert_{W(0,T)}\leq C_{7}\left\lVert r\right\rVert_{L^{2}(0,T;V^{*})}$
by Lemma 4.6. Note that $C_{7}$ depends on $T$, but $T$ is fixed. We obtain
the following estimate,
$\left\lVert\hat{p}-p\right\rVert_{L^{2}(0,T;V)}\leq\sup_{\left\lVert
r\right\rVert\leq
1}\int_{0}^{T}-\langle(\hat{p}_{t}-p_{t})+{\mathcal{A}}^{*}(\hat{p}-p)+{\mathcal{F}}^{\prime}(\bar{y})^{*}(\hat{p}-p),z\rangle_{V^{*},V}dt\\\
+\left\lVert\hat{p}(T)-p(T)\right\rVert_{Y}\left\lVert z(T)\right\rVert_{Y}.$
Then by a similar computation to that for $t\in[T,\infty)$ case, we obtain,
$\left\lVert\hat{p}-p\right\rVert_{L^{2}(0,T;V)}\leq
C_{8}\left(\left\lVert\hat{y}-y\right\rVert_{W_{\infty}}+\left\lVert\hat{\beta}_{1}-\beta_{1}\right\rVert_{L^{2}(I;V^{*})}\right)+\left\lVert\hat{p}(T)-p(T)\right\rVert_{Y}\left\lVert
z(T)\right\rVert_{Y}.$ (4.47)
By (4.45) with $\left\lVert z(T)\right\rVert_{Y}\leq C_{9}$, we obtain
$\left\lVert\hat{p}(T)-p(T)\right\rVert_{Y}\left\lVert
z(T)\right\rVert_{Y}\leq
C_{7}C_{9}\left(\left\lVert\hat{y}-y\right\rVert_{W_{\infty}}+\left\lVert\hat{u}-u\right\rVert_{U}+\left\lVert\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right\rVert_{{\mathcal{Y}}}\right).$
Combining this estimate with (4.43) and (4.47), we obtain for some $C_{10}>0$
$\left\lVert\hat{p}_{(\hat{\boldsymbol{\beta}})}-p_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}\leq
C_{10}\left(\left\lVert\hat{y}_{(\hat{\boldsymbol{\beta}})}-y_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}+\left\lVert\hat{u}_{(\hat{\boldsymbol{\beta}})}-u_{(\boldsymbol{\beta})}\right\rVert_{U}+\left\lVert\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right\rVert_{{\mathcal{Y}}}\right).$
(4.48)
We also have
$u_{(\boldsymbol{\beta})}=\mathbb{P}_{\mathcal{U}_{ad}}\left[-\frac{1}{\alpha}\left(B^{*}p_{(\boldsymbol{\beta})}+\beta_{2}\right)\right]\in
U\cap C(\bar{I};{\mathcal{U}}),$
and thus
$\displaystyle\left\lVert\hat{u}_{(\hat{\boldsymbol{\beta}})}(t)-u_{(\boldsymbol{\beta})}(t)\right\rVert_{{\mathcal{U}}}$
$\displaystyle\leq\left\lVert\mathbb{P}_{\mathcal{U}_{ad}}\left[-\frac{1}{\alpha}\left(B^{*}\hat{p}_{(\hat{\boldsymbol{\beta}})}(t)+\hat{\beta}_{2}(t)\right)\right]-\mathbb{P}_{\mathcal{U}_{ad}}\left[-\frac{1}{\alpha}\left(B^{*}p(t)_{(\boldsymbol{\beta})}+\beta_{2}(t)\right)\right]\right\rVert_{{\mathcal{U}}}$
$\displaystyle\leq\frac{1}{\alpha}\left(\left\lVert
B^{*}\right\rVert\left\lVert\hat{p}_{(\hat{\boldsymbol{\beta}})}(t)-p(t)_{(\boldsymbol{\beta})}\right\rVert_{Y}+\left\lVert\hat{\beta}_{2}(t)-\beta_{2}(t)\right\rVert_{{\mathcal{U}}}\right).$
This yields
$\left\lVert\hat{u}_{(\hat{\boldsymbol{\beta}})}-u_{(\boldsymbol{\beta})}\right\rVert_{C(\bar{I};{\mathcal{U}})}\leq
C_{11}\left(\left\lVert\hat{p}_{(\hat{\boldsymbol{\beta}})}-p_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}+\left\lVert\hat{\beta}_{2}-\beta_{2}\right\rVert_{C(\bar{I};{\mathcal{U}})}\right),$
(4.49)
and (4.41) follows. Combining Remark 4.2 and (4.41) there exists a constant
$L$ such that
$\left\lVert\hat{y}_{(\hat{\boldsymbol{\beta}})}-y_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}+\left\lVert\hat{p}_{(\hat{\boldsymbol{\beta}})}-p_{(\boldsymbol{\beta})}\right\rVert_{W_{\infty}}+\left\lVert\hat{u}_{(\hat{\boldsymbol{\beta}})}-u_{(\boldsymbol{\beta})}\right\rVert_{U\cap
C(\bar{I};Y)}\leq
L\left\lVert\hat{\boldsymbol{\beta}}-\boldsymbol{\beta}\right\rVert_{{\mathcal{Y}}}$
(4.50)
for all $\displaystyle\boldsymbol{\hat{\beta}}$ and
$\displaystyle\boldsymbol{\beta}\in\hat{V}\cap
B_{{\mathcal{Y}}}(\varepsilon)$. Here and in the following the $p_{1}$
coordinate of the adjoint state coincides with $p(0)$. Therefore it is not
indicated. ∎
With the above proposition, the verification of (H1)–(H7) is concluded. We
also obtain the following corollary to Theorem 2.1.
###### Corollary 4.9.
Let assumptions (A) hold and let $(\bar{y},\bar{u})$ be a local solution of
($\mathcal{P}$) corresponding to an initial datum $\bar{y}_{0}\in
B_{Y}(\tilde{\delta}_{2})$. Then there exist $\delta_{3}>0$, a neighborhood
$\hat{U}=\hat{U}(\bar{y},\bar{u},p)\subset W_{\infty}\times(U\cap
C(\bar{I};{\mathcal{U}}))\times W_{\infty}$, and a constant $\mu>0$ such that
for each $y_{0}\in B_{Y}(\bar{y}_{0},\delta_{3})$ there exists a unique
$(y(y_{0}),u(y_{0}),p(y_{0}))\in\hat{U}$ satisfying the first order condition,
and
$\left\lVert\left(y(\hat{y}_{0}),u(\hat{y}_{0})),p(\hat{y}_{0}))\right)-\left(y(\tilde{y}_{0}),u(\tilde{y}_{0}),p(\tilde{y}_{0}))\right)\right\rVert_{W_{\infty}\times(U\cap
C(\bar{I};{\mathcal{U}}))\times
W_{\infty}}\leq\mu\left\lVert\hat{y}_{0}-\tilde{y}_{0}\right\rVert_{Y},$
(4.51)
for all $\hat{y}_{0},\tilde{y}_{0}\in B_{Y}(\bar{y}_{0},\delta_{3})$, and
$\left(y(y_{0}),u(y_{0})\right)$ is a local solution of ($\mathcal{P}$).
Next we obtain one of the main results of this paper, the Fréchet
differentiability of the local value function associated to ($\mathcal{P}$).
By local value function we refer to the fact that for some $y_{0}\in
B_{Y}(\tilde{\delta}_{2})$ may not be unique. But since due to the second
order optimality condition local solutions are locally unique under small
perturbations of $y_{0}$, there is a well-defined local value function. We
continue to use the notation for $\hat{U}$ and $B_{Y}(\bar{y}_{0},\delta_{3})$
of Corollary 4.9.
###### Theorem 4.1.
(Sensitivity of Cost) Let assumptions (A) hold and let $(\bar{y},\bar{u})$ be
a local solution of ($\mathcal{P}$) corresponding to an initial datum
$\bar{y}_{0}\in B_{Y}(\tilde{\delta}_{2})$. Then for each $y_{0}\in
B_{Y}(\bar{y}_{0},\delta_{3})$ the local value function $\mathcal{V}$
associated to ($\mathcal{P}$) is Fréchet differentiable with derivative given
by
$\mathcal{V}^{\prime}(y_{0})=-p(0;y_{0}).$ (4.52)
###### Proof.
Let $\bar{y}_{0}\in B_{Y}(\tilde{\delta}_{2}),y_{0}\in
B_{Y}(\bar{y}_{0},\delta_{3})$, and choose $\delta y_{0}$ sufficiently small
so that $y_{0}+\delta y_{0}\in B_{Y}(\bar{y}_{0},\delta_{3})$ as well.
Following Corollary 4.9 let $(\tilde{y}(y_{0}+s(\delta
y_{0})),\tilde{u}(y_{0}+s(\delta y_{0})),\tilde{p}(y_{0}+s(\delta
y_{0})))\in\hat{U}$ for $s\in[0,1]$ be solutions of the optimality system with
$(\tilde{y}(y_{0}+s(\delta y_{0})),\tilde{u}(y_{0}+s(\delta y_{0})))$ local
solutions to ($\mathcal{P}$). We obtain
$\mathcal{V}(y_{0}+s(\delta
y_{0}))-\mathcal{V}(y_{0})=\left(\frac{1}{2}\left\lVert\tilde{y}\right\rVert^{2}_{L^{2}(I,Y)}+\frac{\alpha}{2}\left\lVert\tilde{u}\right\rVert^{2}_{U}\right)-\left(\frac{1}{2}\left\lVert
y\right\rVert^{2}_{L^{2}(I,Y)}+\frac{\alpha}{2}\left\lVert
u\right\rVert^{2}_{U}\right)\\\ =\langle
y,\tilde{y}-y\rangle_{L^{2}(I,Y)}+\alpha\langle
u,\tilde{u}-u\rangle_{U}+\frac{1}{2}\left\lVert\tilde{y}-y\right\rVert^{2}_{L^{2}(I,Y)}+\frac{{\alpha}}{2}\left\lVert\tilde{u}-u\right\rVert^{2}_{U}.$
(4.53)
Observe the identity
$\langle y,\tilde{y}-y\rangle_{L^{2}(I,Y)}+\alpha\langle
u,\tilde{u}-u\rangle_{U}={-}(p(0),s(\delta
y_{0}))_{Y}{-}\langle(\tilde{y}_{t}-y_{t})-{\mathcal{A}}(\tilde{y}-y)-{\mathcal{F}}^{\prime}(y)(\tilde{y}-y),p\rangle+\alpha\langle
u,\tilde{u}-u\rangle_{U}\\\ ={-}(p(0),s(\delta
y_{0}))_{Y}{-}\langle{\mathcal{F}}(\tilde{y})-{\mathcal{F}}(y)-{\mathcal{F}}^{\prime}(y)(\tilde{y}-y),p\rangle_{L^{2}(I;V^{*}),L^{2}(I;V)}+\langle\alpha
u-B^{*}p,\tilde{u}-u\rangle_{U},$
where $p=p(y_{0})$. Now we have for $\displaystyle\mathcal{V}(y_{0}+s(\delta
y_{0}))-\mathcal{V}(y_{0})$,
$\mathcal{V}(y_{0}+{s}(\delta y_{0}))-\mathcal{V}(y_{0})={-}(p(0),{s}(\delta
y_{0}))_{Y}+\langle{\mathcal{F}}(y)-{\mathcal{F}}(\tilde{y})+{\mathcal{F}}^{\prime}(y)(\tilde{y}-y),p\rangle_{L^{2}(I,V^{*}),L^{2}(I,V)}\\\
+\langle\alpha
u-B^{*}p,\tilde{u}-u\rangle_{U}+\frac{1}{2}\left\lVert\tilde{y}-y\right\rVert^{2}_{L^{2}(I,Y)}+\frac{{\alpha}}{2}\left\lVert\tilde{u}-u\right\rVert^{2}_{U}.$
(4.54)
Since $p\in L^{2}(I;V)$,
$\left\lVert\tilde{y}-y\right\rVert_{W_{\infty}}=O(s)$, and by the continuous
Fréchet differentiability of ${\mathcal{F}}^{\prime}$ due to (A3) we have
$\left\lvert\langle{\mathcal{F}}(\tilde{y})-{\mathcal{F}}(y)+{\mathcal{F}}^{\prime}(y)(\tilde{y}-y),p\rangle_{L^{2}(I,V^{*}),L^{2}(I,V)}\right\rvert=o(s)$
(4.55)
Let $s_{n}\to 0$ be an arbitrary convergent sequence. By Corollary 4.9 we have
that
$\left\lVert\tilde{u}(y_{0}+s_{n}(\delta
y_{0}))-u(y_{0})\right\rVert_{U}\leq\mu s_{n}(\delta y_{0})$
for all $s_{n}$ sufficiently small. Hence there exists a subsequence, denoted
by the same notation and some $\dot{u}$ such that
$s_{n}^{-1}\left(\tilde{u}(y_{0}+s_{n}(\delta
y_{0}))-u(y_{0})\right)\rightharpoonup\dot{u}\text{ weakly in }U.$
Using (4.18), we have
$\lim_{n\rightarrow\infty}s_{n}^{-1}\langle\alpha
u-B^{*}p,\tilde{u}-u\rangle_{U}=\langle\alpha u-B^{*}p,\dot{u}\rangle_{U}\geq
0.$
Analogously
$\lim_{n\rightarrow\infty}s_{n}^{-1}\langle\alpha\tilde{u}-B^{*}p,u-\tilde{u}\rangle_{U}=\langle\alpha
u-B^{*}p,\dot{u}\rangle_{U}\leq 0.$
and hence $\langle\alpha u-B^{*}p,\dot{u}\rangle_{U}=0$. Since the sequence
$\\{s_{n}\\}$ is arbitrary, we obtain
$\langle\alpha u-B^{*}p,\tilde{u}-u\rangle_{U}=o(s).$ (4.56)
Corollary 4.9 yields,
$\left\lVert\tilde{y}(y_{0}+{s_{n}}(\delta
y_{0}))-y(y_{0})\right\rVert^{2}_{L^{2}(I;Y)}+\alpha\left\lVert\tilde{u}(y_{0}+{s_{n}}(\delta
y_{0}))-u(y_{0})\right\rVert^{2}_{L^{2}(I;Y)}=o(s_{n}).$ (4.57)
Combining (4.55), (4.56), and (4.57) we obtain
$\lim_{s\rightarrow 0^{+}}{s^{-1}}\left(\mathcal{V}(y_{0}+s(\delta
y_{0}))-\mathcal{V}(y_{0})\right)={-}(p(0),(\delta y_{0}))_{Y}.$ (4.58)
This implies the Gateaux differentiability. Since $y_{0}\to p(y_{0})$ is
continuous from $B_{Y}(\bar{y}_{0};\delta_{3})$ to $C(\bar{I},Y)$ the mapping
$y_{0}\to\mathcal{V}(y_{0})$ is Fréchet differentiable in
$B_{Y}(\bar{y}_{0};\delta_{3})$. ∎
###### Remark 4.4 (Sensitivity w.r.t. other parameters).
We have developed a technique to verify the continuous differentiability of
the local value function $\mathcal{V}$ pertaining to a semilinear parabolic
equation on infinite time horizon subject to control constraints with respect
to small initial data $y_{0}\in Y$. Thus the parameter $q$ in ($P_{q}$) is the
initial condition $y_{0}$. The reason to focus on this case is due to feedback
control. Without much additional effort the sensitivity analysis of the value
function could be carried out with respect to other parameters as for instance
additive noise on the right hand side of the state equation. The papers cited
in the introduction, see e.g. [GHH], [GV], consider such situations for the
finite horizon case.
## 5 Proof of Theorem 3.2: Derivation of the HJB Equation.
Utilizing the results established so far we now verify that the (global) value
function $\mathcal{V}$ (i.e. the value function associated to global minima)
is a solution to a Hamilton-Jacobi-Bellman equation. The initial conditions
will be chosen from the neighborhood $Y_{0}$ of the origin in $Y$ so that the
assertions of Theorem 4.1 and Corollary 4.9 are available. It will be
convenient to recall the dynamic programming principle for the infinite time
horizon problem: let $y_{0}$ be an initial condition for which a solution to
($\mathcal{P}$) exists. Then for all $\tau>0$, we have
$\mathcal{V}(y_{0})=\inf_{u\in
L^{2}(0,\tau;\mathcal{U}_{ad})}\int_{0}^{\tau}\ell(S(u,y_{0};t),u(t))dt+\mathcal{V}(S(u,y_{0};\tau)),$
(5.1)
where $\displaystyle\ell(y,u)=\frac{1}{2}\left\lVert
y\right\rVert^{2}_{Y}+\frac{\alpha}{2}\left\lVert
u\right\rVert^{2}_{{\mathcal{U}}}$, and $S(u,y_{0};t)$ denotes the solution to
(3.2b), (3.2c) on $(0,\tau]$.
For convenience we restate Theorem 3.2. Utilizing the notation that we have
already established we can now slightly ease the assumption on the regularity
of ${\mathcal{F}}(\bar{y})$.
###### Theorem 5.1.
Let assumptions (A) hold and let $(\bar{y},\bar{u})$ be a global solution of
($\mathcal{P}$) corresponding to an initial datum $\bar{y}_{0}\in
B_{Y}(\tilde{\delta}_{2})$. Let $Y_{0}$ denote the subset of initial
conditions in $B_{Y}(\bar{y}_{0},\delta_{3})$ which allow global solutions in
$\hat{U}$, and assume that for each $y_{0}\in{\mathcal{D}}({\mathcal{A}})\cap
Y_{0}$ there exists $T_{y_{0}}>0$ such that ${\mathcal{F}}(\bar{y})\in
C([0,T_{y_{0}});Y)$. Then the following Hamilton-Jacobi-Bellman equation holds
at $y_{0}$:
$\mathcal{V}^{\prime}(y)({\mathcal{A}}y+{\mathcal{F}}(y))+\frac{1}{2}\left\lVert
y\right\rVert^{2}_{Y}+\frac{\alpha}{2}\left\lVert\mathbb{P}_{\mathcal{U}_{ad}}\left(-\frac{1}{\alpha}B^{*}\mathcal{V}^{\prime}(y)\right)\right\rVert^{2}_{Y}+\left\langle
B^{*}\mathcal{V}^{\prime}(y),\mathbb{P}_{\mathcal{U}_{ad}}\left(-\frac{1}{\alpha}B^{*}\mathcal{V}^{\prime}(y)\right)\right\rangle_{Y}=0.$
(5.2)
If for the optimal trajectory $\bar{y}(t)\in
B_{Y}(\bar{y}_{0},\delta_{3})\cap{\mathcal{D}}({\mathcal{A}})$ for a.a.
$t\in(0,\infty$) and $T_{y_{0}}=\infty$, then (5.2) holds at a.a.
$t\in(0,\infty$) and
${\bar{u}}(t)=\mathbb{P}_{\mathcal{U}_{ad}}\left(-\frac{1}{\alpha}B^{*}\mathcal{V}^{\prime}({\bar{y}}(t))\right).$
(5.3)
###### Proof.
The proof is similar to that of [BKP1, Proposition 10]. For the sake of
completeness and since it also requires some changes we provide it here.
Choose and fix some $y_{0}\in{\mathcal{D}}({\mathcal{A}})\cap Y_{0}$. Then the
existence of a (globally) optimal pair $(\hat{y},\hat{u})\in W_{\infty}\times
U_{ad}$ to ($\mathcal{P}$) and of an associated adjoint state $\hat{p}\in
W_{\infty}$ with $(\hat{y},\hat{u},\hat{p})\in\hat{U}$ are guaranteed, see
Corollary 4.9. In particular we have that
$\displaystyle\hat{u}(t)=\mathbb{P}_{\mathcal{U}_{ad}}\left(\frac{1}{\alpha}B^{*}{\hat{p}}(t)\right)$,
and since $\displaystyle\hat{p}\in C([0,\infty);Y)$ we have that
$\displaystyle\hat{u}\in C([0,\infty);Y)$. Let $u_{0}$ denote the limit of
$\hat{u}$ as time $t$ tends to $0$. Since $\hat{y}\in C([0,\infty);Y)$ and
since $B_{Y}(y_{0},\delta_{3})$ is open there exists $\tau_{y_{0}}>0$ such
that $\hat{y}(t)\in B_{Y}(y_{0},\delta_{3})$, for all $t\in[0,\tau_{y_{0}})$.
Step 1: Let us first prove that
$\mathcal{V}^{\prime}(y_{0})\big{(}{\mathcal{A}}y_{0}+{\mathcal{F}}(y_{0})+Bu_{0}\big{)}+\ell(y_{0},u_{0})=0.$
(5.4)
For this purpose we invoke the dynamic programing principle: We have
$\frac{1}{\tau}\int_{0}^{\tau}\ell(\hat{y}(s),\hat{u}(s))ds+\frac{1}{\tau}\big{(}\mathcal{V}(\hat{y}(\tau))-\mathcal{V}(y_{0})\big{)}=0,$
(5.5)
where we choose $\tau\in(0,\min(T_{y_{0}},\tau_{y_{0}}))$ . By continuity of
$\hat{y}$ and $\hat{u}$ at time $0$, the first term converges to
$\displaystyle\ell(y_{0},u_{0})$ as $\tau\to 0$. To take $\tau\to 0$ in the
second term we first consider
$\frac{1}{\tau}\big{(}\hat{y}(\tau)-y_{0}\big{)}=\frac{1}{\tau}\big{(}e^{{\mathcal{A}}\tau}y_{0}-y_{0}\big{)}+\frac{1}{\tau}\int_{0}^{\tau}e^{{\mathcal{A}}(\tau-s)}\big{[}{\mathcal{F}}(\hat{y}(s))+B\hat{u}(s)\big{]}ds.$
(5.6)
Using the facts that $y_{0}\in{\mathcal{D}}({\mathcal{A}})$, that the terms in
square brackets are continuous with values in $Y$, and that ${\mathcal{A}}$
generates a strongly continuous semigroup on $Y$, we can pass to the limit in
(5.6) to obtain that
$\lim_{\tau\to
0^{+}}\frac{1}{\tau}\big{(}\hat{y}(\tau)-y_{0}\big{)}={\mathcal{A}}y_{0}+{\mathcal{F}}(y_{0})+Bu_{0}\text{
in }Y.$ (5.7)
Now we return to the second term in (5.5) which we express as
$\frac{1}{\tau}\big{(}\mathcal{V}(\hat{y}(\tau))-\mathcal{V}(y_{0})\big{)}=\int_{0}^{1}\mathcal{V}^{\prime}\big{(}y_{0}+s(\hat{y}(\tau)-y_{0})\big{)}ds\;\frac{1}{\tau}(\hat{y}(\tau)-y_{0}).$
(5.8)
Using (5.7) and since $y\to\mathcal{V}^{\prime}(y)$ is continuously
differentiable at $y_{0}$, we can pass to the limit in (5.8) to obtain
$\lim_{\tau\to
0^{+}}\frac{1}{\tau}\big{(}\mathcal{V}(\hat{y}(\tau))-\mathcal{V}(y_{0})\big{)}=\mathcal{V}^{\prime}(y_{0})\big{(}{\mathcal{A}}y_{0}+{\mathcal{F}}(y_{0})+Bu_{0}\big{)}.$
(5.9)
Now we can pass to the limit in (5.5) and obtain (5.4).
Step 2: For $u\in\mathcal{U}_{ad}$ we define $\tilde{u}\in U_{ad}$ by,
$\widetilde{u}(\tau,x)=\begin{cases}u\quad\text{for }\tau\in(0,1)\\\
0\quad\text{ for }\tau\in[1,\infty)\end{cases}$
and define $\tilde{y}=S(y_{0},\tilde{u})$ as the solution to (3.2b), (3.2c).
Then $\tilde{y}(t)\in B_{Y}(\bar{y}_{0},\delta_{3})$, for all $t$ sufficiently
small, and by (5.1) we have,
$\frac{1}{\tau}\int_{0}^{\tau}\ell(\tilde{y}(s),u(s))ds+\frac{1}{\tau}\big{(}\mathcal{V}(\tilde{y}(\tau))-\mathcal{V}(y_{0})\big{)}\geq
0,$
for all $\tau$ sufficiently small. We pass to the limit $\tau\to 0^{+}$ with
the same arguments as in Step 1 and obtain
$\mathcal{V}^{\prime}(y_{0})\big{(}{\mathcal{A}}y_{0}+{\mathcal{F}}(y_{0})+Bu\big{)}+\ell(y_{0},u)\geq
0.$ (5.10)
This inequality becomes an equality if $u=u_{0}$, and thus the quadratic
function on the left had side of (5.10) reaches its minimum $0$ at $u=u_{0}$.
This implies that
$u_{0}=\mathbb{P}_{\mathcal{U}_{ad}}\left(-\frac{1}{\alpha}B^{*}\mathcal{V}^{\prime}(y_{0})\right).$
Inserting this expression into (5.4) we obtain
$\mathcal{V}^{\prime}(y_{0})({\mathcal{A}}y_{0}+{\mathcal{F}}(y_{0}))+\frac{1}{2}\left\lVert
y_{0}\right\rVert^{2}_{Y}+\frac{\alpha}{2}\left\lVert\mathbb{P}_{\mathcal{U}_{ad}}\left(-\frac{1}{\alpha}B^{*}\mathcal{V}^{\prime}(y_{0})\right)\right\rVert^{2}_{Y}+\left\langle
B^{*}\mathcal{V}^{\prime}(y_{0}),\mathbb{P}_{\mathcal{U}_{ad}}\left(-\frac{1}{\alpha}B^{*}\mathcal{V}^{\prime}(y_{0})\right)\right\rangle_{Y}=0.$
(5.11)
Under the additional assumptions on the trajectory, (5.3) follows.
∎
## 6 Some Applications
In this section we discuss the applicability of the framework in two specific
cases. It should be noted that even for linear state equations, the
sensitivity result for the constraint infinite horizon optimal control problem
may be new.
### 6.1 Fisher’s Equation
We consider the optimal stabilization problem for the Fisher equation in an
open connected bounded domain $\Omega$ in $\mathbb{R}^{d},\
d\in\\{1,2,3,4\\}$, with Lipschitzian boundary $\Gamma=\partial\Omega$:
$\displaystyle(\mathcal{P}_{Fis})\qquad\mathcal{V}(y_{0})=\min_{\begin{matrix}(y,u)\in
W_{\infty}\times U_{ad}\end{matrix}}\ \frac{1}{2}\int_{0}^{\infty}\left\lVert
y\right\rVert^{2}_{Y}dt+\frac{\alpha}{2}\int_{0}^{\infty}\left\lVert
u\right\rVert^{2}_{{\mathcal{U}}}dt,$ (6.1a) subject to $\displaystyle y_{t}$
$\displaystyle=\Delta y+y(1-y)+Bu\quad$ $\displaystyle\text{ in
}Q=(0,\infty)\times\Omega$ (6.1b) $\displaystyle y$ $\displaystyle=0\quad$
$\displaystyle\text{ on }\Sigma=(0,\infty)\times\Gamma$ (6.1c) $\displaystyle
y(0)$ $\displaystyle=y_{0}\quad$ $\displaystyle\text{ in }\Omega.$ (6.1d)
where ${\mathcal{U}}$ and $U_{ad}$ are as in Section 3.1,
$B\in{\mathcal{L}}({{\mathcal{U}},Y})$, with $Y=L^{2}(\Omega)$ and
$V=H^{1}_{0}(\Omega)$. To further cast this problem in the framework of
Section 3, we define the operator
${\mathcal{A}}y=(\Delta+\mathbf{I})y\quad\text{and}\quad
y|_{\Gamma}=0,\quad{\mathcal{D}}({\mathcal{A}})=H^{2}(\Omega)\cap V.$
Clearly ${\mathcal{A}}$ has an extension as operator
${\mathcal{A}}\in{\mathcal{L}}(V,V^{*})$. Moreover it generates an analytic
semigroup on $Y$. Thus (A1) holds. For ${\mathcal{U}}=Y$ and $B=\mathbf{I}$,
condition (A2) is trivially satisfied. Feedback stabilization by finite
dimensional controllers was analyzed in [Tri], for example.
It can readily be checked that the nonlinearity ${\mathcal{F}}(y)=-y^{2}$ is
twice continuously differentiable as mapping ${\mathcal{F}}:W_{\infty}\to
L^{2}(I;V^{*})$. We note that the second derivative is constant, so the
required boundedness of the second derivative is automatic. For the sake of
illustration we verify the boundedness of the bilinear form of the second
derivative on $W_{\infty}\times W_{\infty}$. For this purpose, for arbitrary
$y\in W_{\infty},v_{1},v_{2}\in W_{\infty},\phi\in L^{2}(I;V)$ we estimate
$\displaystyle\int_{0}^{\infty}\langle{\mathcal{F}}^{\prime\prime}({y})(v_{1},v_{2}),\phi\rangle_{V^{*},V}dt$
$\displaystyle\leq 2\int_{0}^{\infty}\int_{\Omega}v_{1}v_{2}\phi\
dxdt\leq\int^{\infty}_{0}\left\lVert
v_{1}\right\rVert_{L^{2}(\Omega)}\left\lVert
v_{2}\right\rVert_{L^{4}(\Omega)}\left\lVert\phi\right\rVert_{L^{4}(\Omega)}\,dt,$
(6.2) $\displaystyle\leq C_{1}\left\lVert
v_{1}\right\rVert_{W_{\infty}}\int^{\infty}_{0}\left\lVert
v_{2}\right\rVert_{V}\left\lVert\phi\right\rVert_{V}\,dt\leq C_{2}\left\lVert
v_{1}\right\rVert_{W_{\infty}}\left\lVert
v_{2}\right\rVert_{L^{2}(I;V)}\left\lVert\phi\right\rVert_{L^{2}(I;V)},$
$\displaystyle\leq C_{3}\left\lVert v_{1}\right\rVert_{W_{\infty}}\left\lVert
v_{2}\right\rVert_{W_{\infty}}\left\lVert\phi\right\rVert_{L^{2}(I;V)},$
where $C_{i}$ are embedding constants, independent of $y\in W_{\infty},v\in
W_{\infty},\phi\in L^{2}(I;V)$. We use that $V$ embeds continuously into
$L^{4}(\Omega)$ in dimension up to 4. This implies that
$\left\lVert{\mathcal{F}}^{\prime\prime}({y})(v_{1},v_{2})\right\rVert_{L^{2}(I;V^{*})}\leq
C_{3}\left\lVert v_{1}\right\rVert_{W_{\infty}}\left\lVert
v_{2}\right\rVert_{W_{\infty}}$. Finally we have
${\mathcal{F}}(0)={\mathcal{F}}^{\prime}(0)=0$ and thus (A3) and (3.3) are
satisfied.
Turning to (A4) we show that ${\mathcal{F}}(y):W(0,T)\rightarrow
L^{1}(0,T;V^{*})$ is continuous for every $T>0$. We consider the sequence
$\displaystyle y_{n}\rightharpoonup\hat{y}$ in $W_{\infty}$ and let $z\in
L^{\infty}(0,T;V)$ be given. Then we estimate
$\displaystyle\int_{0}^{T}\langle{\mathcal{F}}(y_{n})-{\mathcal{F}}(\hat{y}),z\rangle_{V^{*},V}dt$
$\displaystyle=\int_{0}^{T}\langle
y_{n}^{2}-\hat{y}^{2},z\rangle_{V^{*},V}=\int_{0}^{T}\int_{\Omega}(y_{n}-\hat{y})(y_{n}+\hat{y})z\
dxdt$ $\displaystyle\leq C_{4}\int_{0}^{T}\left\lVert
y_{n}-\hat{y}\right\rVert_{Y}\left\lVert
y_{n}+\hat{y}\right\rVert_{L^{4}(\Omega)}\left\lVert
z\right\rVert_{L^{4}(\Omega)}\ dt$ $\displaystyle\leq C_{4}\left\lVert
y_{n}-\hat{y}\right\rVert_{L^{2}(0,T;Y)}\left[\left\lVert
y_{n}\right\rVert_{L^{2}(0,T;V)}+\left\lVert\hat{y}\right\rVert_{L^{2}(0,T;V)}\right]\left\lVert
z\right\rVert_{L^{\infty}(0,T;V)}.$
Since $V$ is compactly embedded in $Y$, we obtain by the Aubin Lions lemma
that $\displaystyle\left\lVert y_{n}-\hat{y}\right\rVert_{L^{2}(0,T;Y)}\to 0$
for $n\to\infty$. This implies
$\int_{0}^{T}\langle{\mathcal{F}}(y_{n})-{\mathcal{F}}(\hat{y}),z\rangle_{V^{*},V}dt\xrightarrow[n\rightarrow\infty]{}0,$
and (A4) follows. It is simple to check that
$\displaystyle{\mathcal{F}}^{\prime}(\bar{y})=2\bar{y}\in{\mathcal{L}}(L^{2}(I;V),L^{2}(I;V^{*}))$
and thus (A5) holds as well.
We turn to the assumption ${\mathcal{F}}(\bar{y})\in C([0,T_{y_{0}});Y)$, for
$y_{0}\in{\mathcal{D}}({\mathcal{A}})$ and some $T_{y_{0}}$, arising in
Theorem 3.2 for $y_{0}\in{\mathcal{D}}({\mathcal{A}})$. Utilizing the fact
that $V$ embeds continuously into $L^{4}(\Omega)$ in dimension $d\leq 4$ and
$\bar{y}\in L^{2}(I;V)$, we have ${\mathcal{F}}(\bar{y})\in L^{2}(I;Y)$. Hence
parabolic regularity theory implies that $\bar{y}\in C([0,\infty);V)$ for
$y_{0}\in V$, and ${\mathcal{F}}(\bar{y})\in C([0,\infty);Y)$ follows.
###### Remark 6.1.
The specificity of this example rests in the fact that the second derivative
is independent of the point were it is taken. Other nontrivial cases of
analogous structure are reaction diffusion systems with bilinear coupling, see
[Gri] where the finite horizon case was treated. Even the case of the Navier
Stokes equations falls in this category. Sensitivity for the infinite horizon
problems was treated by independent techniques in [BKP3].
### 6.2 Nonlinearities induced by functions with globally Lipschitz
continuous second derivative.
Consider the system ($\mathcal{P}$) with ${\mathcal{A}}$ associated to a
strongly elliptic second order operator with domain $H^{2}(\Omega)\cap
H^{1}_{0}(\Omega)$, so that (A1)-(A2) are satisfied. Let
${\mathcal{F}}:W_{\infty}\to L(I;V^{*})$ be the Nemytskii operator associated
to a mapping $\mathfrak{f}:\mathbb{R}\to\mathbb{R}$ which is assumed to be
$C^{2}(\mathbb{R})$ with first and second derivatives globally Lipschitz
continuous, and second derivative globally bounded. The regularity assumption
${\mathcal{F}}(\bar{y})\in C([0,T_{y_{0}});Y)$ for $y_{0}\in
V=H^{1}_{0}(\Omega)$ is satisfied by parabolic regularity theory. We discuss
assumption (A3)-(A5) for such an ${\mathcal{F}}$, and show that they are
satisfied for dimensions $d\in\\{1,2\\}$. For the finite horizon problem it
will turn out that $d=3$ is also admissible. By direct calculation it can be
checked that ${\mathcal{F}}$ is continuously Fréchet differentiable for
$d\in\\{1,2,3\\}$. We leave this part to the reader and immediately turn to
the second derivative.
We proceed by considering the general dimension $d$ to highlight, how the
restrictions on the dimension arise. Thus let $d\in\mathbb{N}$ with $d>1$. The
case $d=1$ can be treated with minor modifications from those in the following
steps.
#### 6.2.1 Second derivative of ${\mathcal{F}}(y)$.
For $y,h_{1},h_{2}\in W_{\infty}$ the relevant expression is given by
$\left\lVert{\mathcal{F}}^{\prime}(y+h_{2})h_{1}-{\mathcal{F}}^{\prime}(y)h_{1}-{\mathcal{F}}^{\prime\prime}(y)(h_{1},h_{2})\right\rVert_{L^{2}(I;V^{*})}\\\
\quad=\sup_{\left\lVert\varphi\right\rVert_{L^{2}(I;V)}\leq
1}\langle{\mathcal{F}}^{\prime}(y+h_{2})h_{1}-{\mathcal{F}}^{\prime}(y)h_{1}-{\mathcal{F}}^{\prime\prime}(y)(h_{1},h_{2}),\varphi\rangle_{L^{2}(I;V^{*}),L^{2}(I;V)}\\\
\quad=\sup_{\left\lVert\varphi\right\rVert_{L^{2}(I;V)}\leq
1}\int_{0}^{\infty}\int_{\Omega}(\mathfrak{f}^{\prime}(y(t,x)+h_{2}(t,x))-\mathfrak{f}^{\prime}(y(t,x))-\mathfrak{f}^{\prime\prime}(y(t,x))h_{2}(t,x))h_{1}(t,x)\varphi(t,x)dxdt\\\
\quad=\sup_{\left\lVert\varphi\right\rVert_{L^{2}(I;V)}\leq
1}\int_{0}^{\infty}\int_{\Omega}g(t,x)\ h_{2}(t,x)h_{1}(t,x)\varphi(t,x)dxdt,$
where $\displaystyle
g(t,x)=\int_{0}^{1}(\mathfrak{f}^{\prime\prime}(y(t,x)+sh_{2}(t,x))-\mathfrak{f}^{\prime\prime}(y(t,x)))ds$.
Note that $g$ is bounded on $I\times\Omega$ and $g\in W_{\infty}$. Here we use
that $\mathfrak{f}^{\prime\prime}$ is globally Lipschitz continuous and that
$h_{1}\in W_{\infty}$. Henceforth we let $r\in(1,\frac{2d}{d-2}]$ so that
$W^{1,2}(\Omega)\subset L^{r}(\Omega)$ continuously. Let $r^{\prime}$ denote
the conjugate of $r$ so that $r^{\prime}\in[\frac{2d}{d+2},\infty)$ for $d>2$
and $r^{\prime}\in(1,\infty)$ for $d=2$. We further choose $\rho>1,\sigma>2$
|
# Modelling the development of counting with memory-augmented neural networks
Zack Dulberg<EMAIL_ADDRESS>
Princeton Neuroscience Institute, Princeton, NJ Taylor Webb
<EMAIL_ADDRESS>
University of California Los Angeles, Los Angeles, CA Jonathan Cohen
<EMAIL_ADDRESS>
Princeton Neuroscience Institute, Princeton, NJ
###### Abstract
Learning to count is an important example of the broader human capacity for
systematic generalization, and the development of counting is often
characterized by an inflection point when children rapidly acquire proficiency
with the procedures that support this ability. We aimed to model this process
by training a reinforcement learning agent to select N items from a binary
vector when instructed (known as the give-$N$ task). We found that a memory-
augmented modular network architecture based on the recently proposed Emergent
Symbol Binding Network (ESBN) exhibited an inflection during learning that
resembled human development. This model was also capable of systematic
extrapolation outside the range of its training set - for example, trained
only to select between 1 and 10 items, it could succeed at selecting 11 to 15
items as long as it could make use of an arbitrary count sequence of at least
that length. The close parallels to child development and the capacity for
extrapolation suggest that our model could shed light on the emergence of
systematicity in humans.
Keywords: counting; development; give-$N$; reinforcement learning; memory-
augmented neural networks
## Introduction
Humans are capable of systematic generalization, that is, performing well
outside the range of values on which they were trained (?, ?, ?). For example,
a human could fetch 12 apples if asked, despite having only ever grabbed up to
9 apples in the past. Although this capacity falls short of perfect
systematicity (?, ?), artificial neural networks have much greater difficulty
performing well in contexts outside the convexity of their training data (?,
?, ?). Learning to count is one of the earliest systematic behaviours acquired
in human development, and is foundational with respect to further development
of abstract procedures like mathematics. Here, we present a counting model
that exhibits both a developmental trajectory similar to humans as well as
systematicity.
The development of counting in childhood has previously been summarized by the
knower-level theory (?, ?, ?, ?, ?). This theory suggests a number of distinct
stages in acquiring an understanding of the cardinal meaning of numbers. A
child begins as a pre-numeral knower, with no understanding of cardinality,
and then becomes a subset-knower, learning subsequent numbers in order (i.e.,
becomes a one-knower, then a two-knower, then a three-knower, etc). Around the
time a child becomes a five-knower, there appears to be an inductive
transition, after which the child becomes a cardinal-principle-knower (CP-
knower), and understands the cardinal meaning of numbers as high as they can
count. Some data have supported this view, though children pass through
knower-levels at different rates (?, ?), and early stages may be more
approximate than previously thought (?, ?), while other data have called into
question whether a true semantic induction underlies this apparent transition
(?, ?).
The knower-level theory has generally been based on data from the widely used
give-$N$ task (?, ?, ?, ?). In this task, a child is instructed to ’give $N$
objects’, and is typically considered an $N$-knower if they can select the
correct number of objects twice as often as they make an error (66 %
accuracy). Our goal was to train a neural network to perform the give-$N$ task
and examine its performance and developmental trajectory. We suggest the
give-$N$ task is most realistically modelled using a reinforcement learning
framework. First, the task can be successfully completed in many different
ways (i.e. by selecting objects in various orders), so a reinforcement rather
than a more specific instructive signal is appropriate. Furthermore, children
receive a substantial amount of reinforcement from adults when learning tasks
like these (”good job!”). Finally, the give-$N$ task can easily be expressed
as a decision process with discrete actions, which therefore lends itself
naturally to reinforcement learning.
We report simulations of a model that, trained to perform the give-$N$ task
using reinforcement learning, displays both an inflection point consistent
with the human developmental trajectory, as well as the capacity for
systematic extrapolation when presented with stimuli out of the range of its
training. The model extended the Emergent Symbol Binding Network (ESBN) (?,
?), in which internal representations of a control network were separated from
task inputs, interacting only through binding in a differentiable external
memory module. We developed a variant of the ESBN within a reinforcement
learning framework to address how its capacity for symbol-like behavior might
support the development of systematic counting proficiency. We also softened
the separation constraint, giving the ESBN access to both internal and input
streams (and thus the option to over-fit to training inputs), and it still
exhibited symbol-like behavior and the capacity for systematic generalization.
Baseline models did not display inflection or extrapolation, suggesting
variable binding and dot-product similarity evaluation as the relevant
architectural inductive biases that promoted systematicity and emulated human
development. This work also supports the broader idea that the development of
abstract conceptual knowledge may be supported by the learning of systematic
procedures.
## Related Work
There have been various attempts to model counting. One line of inquiry
investigated the ability of neural networks to estimate numerosity by, for
example, looking at an image of objects (?, ?, ?, ?). Another study used
counting equivariance relations to support the learning of visual
representations (?, ?). However, such perceptual tasks do not address the
capacity for systematic counting behavior. Other work has shown that recurrent
neural networks can keep track of counts in order to predict the next
character in a string derived from some grammar (?, ?, ?). Though accuracy was
limited, some of these networks could extrapolate to longer strings than those
on which they were trained. However, this was a character prediction task
rather than a counting task, making the parallel to human counting development
unclear. Finally, Bayesian models have been proposed to address the
development of systematic counting in humans (?, ?, ?). These models generally
assumed knower-levels (including CP-knower) as primitive hypotheses. The model
we propose does not include such primitives, seeking to explain systematicity
as an emergent property that arises through learning.
Some neural network modeling studies have directly addressed learning to
count. ? (?) introduced a feed-forward network with a visual attention
mechanism that was trained through a combination of reinforcement learning and
‘social scaffolding’ (demonstrations from a teacher) to perform the how-many
task (a simpler counting task that children often master before they can
perform the give-$N$ task). ? (?) developed an extension of this model that
used a recurrent network and operated over more realistic two-dimensional
inputs. Although this network learned numbers in the correct order (passing
appropriately through subset-knower levels), it did not display an inflection
point for higher numbers, and was not tested on extrapolation for numbers
outside the range on which it was trained. ? (?) expanded further on this work
by training a recurrent convolutional network to perform a variety of counting
tasks, including give-$N$. However, the give-$N$ task did not display a
familiar developmental trajectory (for example, it learned to give-1 last
rather than first), and this model was also not tested on numbers outside the
training range. A common element of these studies was the use of teacher-
guided learning; that is, networks were trained to imitate a specific way of
solving the task, rather than learning from reinforcement alone. Imitation has
in fact been shown to improve learning of multiple tasks in a rich 3D
environment, but interestingly, in that work, agents still struggled most with
learning to count (?, ?). Although imitation likely plays a role in the
development of many skills, here we investigated to what extent a model
trained from reinforcement alone could account for the relevant developmental
phenomena.
## Methods
### Environment
We represented a set of objects in the give-$N$ task as a binary vector in
which 1’s corresponded to the presence of an object at a given location. For
example, the vector [0,1,0,1,1,0] indicated that there were objects at
locations 2, 4, and 5. Vectors were of length 40 so that the space of possible
object arrangements was combinatorically large ($2^{40}$ states), in order to
prevent memorization. All models were trained on the give-$N$ task as follows:
given an initial object vector $\bm{o_{0}}$, and an instruction to give $N$
objects (represented by the one-hot vector $\bm{x_{N}}$), the model should
select $N$ unique objects one at a time from the object vector to produce a
correct response. If a location with an object was selected at time $t-1$, the
value at that location changed from 1 to 0 in $\bm{o_{t}}$. The binary object
vector can be interpreted as the output of an object segmentation model;
focusing on this intermediate level of abstraction was motivated by the desire
to understand the acquisition of counting competency separately from the
details of sensory processing and object segmentation.
### Model
(a) Counter
(b) ESBN
(c) Dot-product
(d) LSTM
(e) Transformer
Figure 1: Schematics. Temporal count sequence shown in (a), spatial
architecture of models in (b-e). The counter (a) consisted of the encoder $e$
(which translated one-hot vectors $\bm{x_{N}}$ / $\bm{x_{1}}$ into count
embeddings $\bm{z_{0}}$ / $\bm{z_{1}}$), and the successor function $s$ (which
iteratively produced $\bm{z_{t}}$ from $\bm{z_{t-1}}$ starting with
$\bm{z_{1}}$). The ESBN model (b) consisted of an LSTM controller which at
each time step received a concatenation of key $\bm{k_{r_{t}}}$ (read from
memory), object vector $\bm{o_{t}}$ and count embedding $\bm{z_{t}}$ as
inputs, wrote key $\bm{k_{w_{t}}}$ to memory, and selected action
$\bm{a_{t}}$. The count embeddings also interacted with the controller via a
key/value memory system indicated by matrices $\bm{M_{k}}$/$\bm{M_{v}}$. In
the dot product model (c), only the cosine similarity (dot symbol in brackets)
of each count embedding $\bm{z_{t}}$ with the instruction embedding
$\bm{z_{0}}$ was input to the LSTM controller along with $\bm{o_{t}}$. In the
LSTM baseline (d), $\bm{z_{t}}$ and $\bm{o_{t}}$ were input directly into the
LSTM controller. In the transformer baseline (e), the full history of count
embeddings and object vectors up until time $t$ was input into a transformer
layer.
First, we pre-trained a counter module that consisted of an encoding function
$e$ (a 128 unit linear projection), a successor function $s$ (another 128 unit
linear projection), and a decoding function $d$ (a 15 unit linear projection).
Given a 15-unit one-hot vector $\bm{x_{n}}$ representing an integer
$n\in\\{1,15\\}$, the counter was trained to produce an embedding $\bm{z}$
such that $\bm{z}=e(\bm{x_{n}})$, $\bm{x_{n}}=d(e(\bm{x_{n}}))$, and
$\bm{x_{n+1}}=d(s(e(\bm{x_{n}})))$. In this way, $s$ was a successor operation
that could iterate through a learned sequence of $\bm{z}$ embeddings, $e$ was
an encoder that could translate a one-hot instruction into one of the
embeddings in that sequence, and $d$ was a decoder that could translate
embeddings back into one-hots (in the our models, $d$ only played a role
during pre-training). These components implemented a counting capability
assumed to have been acquired by children at the time they are asked to
perform the give-$N$ task.
The temporal structure of the counter outputs was the same for all models
(Fig. 1(a)), where $\bm{z_{t}}$ represented the embedding produced by the
counter and passed to the networks at teach time step $t$. At $t=0$, the task
instruction $\bm{x_{N}}$ was passed through the encoder $e$ to generate
$\bm{z_{0}}$. At $t=1$, the one-hot representing the start of the count
sequence, $\bm{x_{1}}$, was passed through $e$ to generate $\bm{z_{1}}$. For
each subsequent time step, the embedding from the previous time step
$\bm{z_{t-1}}$ was passed through the successor function $s$ to generate the
embedding for the next number in the count list $\bm{z_{t}}=s(\bm{z_{t-1}})$
(representing the counter reciting its learned count sequence).
The ESBN model consisted of a set of components outlined in Fig. 1(b). The
controller was an LSTM (?, ?) augmented with a differentiable (external)
memory separated into keys ($\bm{M_{k}}$) and values ($\bm{M_{v}}$). The
memory was initialized with one learned key/value pair so it did not start out
empty, and an additional key/value pair ($\bm{k_{w_{t}}}$ / $\bm{z_{t}}$) was
written to memory at each subsequent time-step. Note the value here was the
$\bm{z_{t}}$ just described.
As in the original ESBN, the LSTM controller had a single layer with 512
units. The controller received a concatenation of three vectors as input at
each time step: $\bm{k_{r_{t}}}$, $\bm{z_{t}}$, and $\bm{o_{t}}$. The input
$\bm{k_{r_{t}}}$ was a key initialized to zeros at $t=0$, and retrieved from
memory at each subsequent time-step. To retrieve $\bm{k_{r_{t}}}$, the cosine
similarity was calculated between $\bm{z_{t}}$ and all previous values in
memory $\bm{M_{v_{t-1}}}$, passed through a softmax to produce a set of
weights $\bm{w_{k_{t}}}$, and used to calculate a weighted sum over
$\bm{M_{k_{t-1}}}$, the keys in memory. The second input to the controller,
$\bm{z_{t}}$, was the count embedding itself (so that the decision to use
$\bm{k_{r_{t}}}$, $\bm{z_{t}}$ or some combination thereof to solve the task
was left up to the model). The third input was the object vector $\bm{o_{t}}$.
The ESBN model had two output heads. The action head was a fully-connected
layer with input size $512$ and output size 41 (corresponding to the 40
possible object locations, plus an additional done action). A softmax
activation was applied to the output logits to produce a vector of action
probabilities. The controller also had a key output head (256 units with ReLU
nonlinearities) that produced the key $\bm{k_{w_{t}}}$ written to memory at
each time step .
We compared the ESBN model to a set of models that lacked key/value memory
modules. The dot-product model (Fig. 1(c)) was meant to elucidate the role of
similarity-based memory retrieval in the success of the ESBN model, albeit in
a manner that is specifically tailored to the counting task (unlike the
original ESBN). We computed the cosine-similarity between $\bm{z_{t}}$ and
$\bm{z_{0}}$ directly, so the controller only received $\bm{o_{t}}$ and the
scalar similarity score as inputs. In the LSTM baseline (Fig. 1(d)), the
controller simply received $\bm{o_{t}}$ and $\bm{z_{t}}$ as inputs. Finally,
in the transformer baseline, the controller was a single transformer layer (?,
?) (8 self-attention heads, 512-unit MLP, positional encoding) which at each
time step received the entire past sequence of $\bm{o_{0..t}}$ and
$\bm{z_{0..t}}$, concatenated as shown in Fig. 1(e).
### Training
#### Pre-training Counter
The counter was pre-trained on its auto-encoding and successor functions.
Given a one-hot input vector $\bm{x_{n}}$ where $n\sim$ Uniform$(1,15)$, the
counter produced an output of $d(s^{i}(e(\bm{x_{n}})))$. $s^{i}$ represented
iterating $s$, the successor function, $i$ times in a row, with $i\sim$
Uniform$(0,15-n)$. This allowed for interleaved learning of both encoding and
decoding between one-hot space and embedding space, as well as iterating
through the learned embeddings sequentially with the successor function. The
loss was computed as the mean-squared-error between the output vector and the
desired one-hot vector $\bm{x_{n+i}}$. As well, a similarity penalty on the
embeddings was added to the loss (for $i\neq 0$) as the dot product
$e(\bm{x_{n}})\cdot s^{i}(e(\bm{x_{n}}))$. Without this penalty, repeated
applications of the successor function caused embeddings to drift apart from
their corresponding one-hot encodings. The Adam optimizer (?, ?) was used to
perform weight updates on mini-batches of 1 with a learning rate of $10^{-4}$.
The weights of the counter network were frozen before being used in the
subsequent reinforcement learning task.
#### Reinforcement Learning of give-$N$ Task
During training, action $a_{t}$ was sampled at each time-step from a
categorical distribution using action probabilities produced by our models. We
used a two-step training curriculum.
In the first step, agents were trained only to select 1s and not 0s. The
object vector was the only input (all other input units were set to zero), and
the agent received a reward of 0 if it selected an object slot that contained
a 1 and a reward of -1 otherwise. Episodes ended after 20 time-steps.
In the second step, we switched to the give-$N$ task. Each training episode
started by randomly selecting an integer $N$ between 1 and $N_{max}$,
represented as the one-hot instruction vector $\bm{x_{N}}$. $N_{max}$ was
initialized to 1, and incremented by 1 once the network achieved at least
$66\%$ accuracy on give-$N_{max}$. This curriculum progressed until $N_{max}$
was fixed at 10. Having selected $N$ for a given episode, the object vector
was populated with $j$ objects, where $j$ $\sim$ Uniform ($N+10,35$). This was
done to reduce the correlation between $N$ and the number of objects in the
object vector, while keeping the space of possible object arrangements very
large.
If the agent selected an object location containing a $1$, that object was
replaced with a $0$ on the next time step, and the agent received a reward of
$0$. If the agent selected a location containing a $0$, it received a reward
of $-1$. Finally, if the agent selected done (ending the episode), it received
a reward of $+5$ if it had by that time selected exactly $N$ objects, and
otherwise a reward of $-|N-n|$ if it had selected $n\neq N$ objects.
At the end of each episode, one gradient descent step in weight space was
performed according to the REINFORCE policy gradient algorithm (?, ?) (chosen
because it was the simplest algorithm capable of learning the task). The Adam
optimizer with a learning rate of $5\times 10^{-5}$ was used for a total of
500,000 episodes (50,000 episodes on step 1 and the remaining 450,000 episodes
on give-$N$). A set of 30 randomly initialized models were trained for each
condition.
### Testing
In order to track developmental trajectories, models were tested on give-$N$
at check-points during training every 1000 episodes. At each check-point, an
accuracy score was produced by calculating the proportion of correct responses
out of 30 new object vectors (unseen during training) for each requested
$N\in\\{1,10\\}$. Actions were selected during testing based on the maximum
action probability rather than categorical sampling. A correct response was
defined as receiving no negative rewards during the complete episode (based on
our reward scheme, this meant the agent selected exactly $N$ objects, and did
not select any empty locations, before indicating done). The episode at which
training accuracy exceeded a threshold of $66\%$ was recorded for each
$N\in\\{1,10\\}$. This produced a developmental trajectory for the accuracy of
each model, and we compared how well these trajectories were fit by linear,
exponential, logarithmic or sigmoidal functions using the Bayesian information
criterion (?, ?).
(a) ESBN Model
(b) Dot-product Model
(c) LSTM Baseline
(d) Transformer Baseline
Figure 2: Accuracy on the give-$N$ task for all trained models. Two bars are
displayed for each $N$: the point during training with the highest average
accuracy across N from 1 to 10 (dark blue) and the best accuracy for each N at
any point in training (light blue, left out for extrapolation to avoid using
extrapolation performance to select when to test the model). Bars in each
subplot represent mean and standard error across an ensemble of 30 trained
models. The red line separates the training regime (left) from the
extrapolation regime (right).
Finally, best performance as well as extrapolation performance was determined
by calculating the accuracy of each model at the episode during training that
had the highest average accuracy across $N\in\\{1,10\\}$. The extrapolation
set was defined as $N=\\{11,15\\}$, a set of instructions for give-$N$ that
was never presented to the model during training, but was nevertheless tested
for accuracy at each checkpoint. Occasionally, our models exhibited unstable
behavior (i.e., initially learning the task well, but then dropping in
performance prior to the final episode). This required us to select an
appropriate testing point, intentionally not using accuracy on the
extrapolation set to determine this point in order to avoid test-set leakage
into our results. In order to evaluate more modest success, particularly for
the baseline models, we also report the best performance for each
$N\in\\{1,10\\}$ at any point during training individually.
(a) ESBN Model
(b) Dot-product Model
(c) LSTM Baseline
(d) Transformer Baseline
Figure 3: Developmental trajectories for all models. For each requested $N$ on
the x-axis, the episode at which a threshold of $66\%$ accuracy was crossed is
displayed on the y-axis (which begins after 50,000 episodes of the step-1
curriculum). Insets show zoomed-in trajectories, colours represent individual
models, and error bars represent standard deviation across models. Out of a
training ensemble of 30 models, only those that reached threshold performance
for all $N$ at some point during training were included (n=30/30 for ESBN and
dot-product models, n=25/30 for LSTM baseline, and n=3/30 for the transformer
baseline).
## Results
Overall performance and extrapolation performance are shown in Figure 2. The
ESBN model learned the task well, and achieved significant extrapolation. The
dot-product model did even better in these respects. In contrast, while the
baseline models performed well for $N$ up to around 5, performance degraded in
various ways past that. For example, unlike the ESBN and dot-product models,
the LSTM baseline struggled to perform well for higher values of $N$
simultaneously, indicated by the difference between best accuracy at any time
during training, and best accuracy when the model was doing its best on
average. The transformer baseline struggled as $N$ increased, with only 3
models ever crossing threshold on Give-10. The baseline models were also
incapable of extrapolation ($N$=11-15; Fig. 2(c) and 2(d)).
The developmental trajectory of performance on the training set is shown in
Figure 3. In order to be included in the plot, a model had to have crossed
threshold at some point during training for all $N$ displayed. The ESBN and
dot-product models achieved criterial accuracy (66%) for all values of $N$ for
30/30 models; in contrast, only 25/30 LSTM baseline and 3/30 transformer
baseline models met this criterion. Though not shown, the transformer model
also failed when we input the object vector after rather than before the
transformer layer.
All models that met criterion displayed sequential learning, with higher
values of $N$ crossing the threshold later than lower values of $N$ (this
occurred even without being enforced by curriculum training, but those results
are not shown). However, the ESBN and dot product models achieved these
thresholds much earlier in training ($\sim 100k$ episodes) compared to the
baseline models ($\sim 450k$ episodes). However, only the ESBN displayed an
inflection point, past which criterial performance for higher $N$’s were
reached after many fewer epochs of training, and sometimes almost immediately.
This was confirmed using a Bayesian information criterion to compare linear,
exponential, logarithmic and sigmoidal fits to the development trajectory of
the ESBN model. The sigmoidal fit best; and, when fit to the trajectory of
individual instances of the model to quantify the $N$ of inflection, exhibited
a mean of $4.38\pm 0.39$ (mean $\pm$ sd).
## Discussion
We showed that a model of counting based on the ESBN architecture, and trained
with reinforcement learning, exhibited a developmental trajectory
qualitatively similar to the one observed in humans learning to count, as well
as the capacity for systematic extrapolation. A model that implemented only
the retrieval operation required for the counting task (the dot-product
similarity operation) displayed good performance and extrapolation, but not a
clear inflection in its developmental trajectory. Baseline models using either
an LSTM or transformer as a controller, but without the external memory
component, displayed much slower learning overall, no inflection in the
learning trajectory, and no capacity for extrapolation. The transformer model
did particularly poorly, possibly because standard transformers are ill-suited
to processing adjacent time-steps (?, ?).
One explanation for the success of the ESBN model was identified where it was
first described (?, ?). There, the authors argued that because the information
stream accessible to the controller was isolated from the incoming data stream
by the key/value memory, the controller was free to produce and respond to
abstract representations needed to perform the task, without being shaped or
tied to individual items (tokens); these could be thought of as fulfilling the
role of symbols in traditional architectures. Here, the key associated with
(i.e. bound to) the instruction embedding $\bm{z_{0}}$ was this symbol, which
functioned as $N$ in the give-$N$ task. Since the controller’s job was to
report when the correct count was reached, it simply had to recognize when
$\bm{k_{r}}$ was close enough to this key. Once learned, it could quickly gain
the capacity to give any $N$ for as high as it could count.
Here, we relaxed the isolation of the controller from the data, giving it
access to both the key and value streams at every time-point, and allowed it
to learn which source of information was most useful. In principle, the
network could have ignored the input from its external memory (the retrieved
keys), performing the task only on the basis of the count embeddings that it
received directly as input. However, this likely would have resulted in
overfitting to the count embeddings observed in the training set, preventing
extrapolation to new count embeddings, as was observed for the LSTM and
transformer baseline models. Surprisingly, the ESBN did not display this
overfitting, suggesting that it was indeed performing the task on the basis of
information retrieved from its external memory.
We hypothesize that this occurred because the gradients associated with the
direct input of count embeddings at the beginning of an episode tended to
vanish over the course of the episode, whereas the information retrieved from
external memory was available at the time point immediately before the
relevant action (done) was taken. This suggests that the strict architectural
separation in the original ESBN model might not be necessary to achieve
systematic behavior. It is also interesting to note that both baseline models
had good performance until around $N=5$, past which performance degraded,
reflecting the possibility that increased task difficulty after this point
pressured the transition from a specific to a systematic solution in the ESBN
model. It could be that a similar mechanism is responsible for the transition
seen around $N=5$ in children.
We found that a simpler version of the model, incorporating only the dot-
product similarity operation, displayed a comparable level of task performance
and extrapolation. This suggests that, in the context of this task, this dot-
product operation was the key inductive bias that enabled the ESBN to display
systematic counting behavior. However, this simpler version of the model was
specifically designed for the counting task, since the relevant similarity
value (between the instruction and the count at each time step) was passed
directly to the controller, rather than being embedded in a more general-
purpose memory architecture. Furthermore, this version of the model did not
exhibit an inflection in its learning trajectory, suggesting that this
inflection may reflect the difficulty of learning to interact with memory. For
these reasons, this model should be viewed as an attempt to better understand
the operations of the ESBN, rather than a competing theoretical account.
One concern with the developmental trajectory displayed by the ESBN might be
that, although it exhibits an inflection, that is softer than a more discrete
transition to the CP-knower stage suggested by some developmental data (?, ?).
It is worth emphasizing that this data was collected at longitudinal intervals
of a minimum of 5-8 weeks, and so may not have adequate temporal resolution to
clearly distinguish between these two possibilities. Future work involving
more fine-grained longitudinal evaluation might test whether this transition
is truly discrete, or closer to the sigmoidal trajectory displayed by the
ESBN.
We exploited a form of curriculum learning in the present work, first pre-
training networks to iterate through the count sequence, then training them to
select objects, and finally to perform the give-$N$ task. It was possible for
networks to learn all of these tasks at the same time, but we chose a
curricular approach to mirror the fact that children are typically able to
memorize arbitrary sequences early on, and often can count to numbers much
higher than they are able to successfully employ in tasks such as give-$N$ (?,
?). Importantly, the pre-learned count sequence could be used as scaffolding
to support extrapolation to numbers outside the range of training on the
give-$N$ task.
Some have argued that the transition to being a CP-knower, as measured by the
give-$N$ task, marks a more general semantic induction of abstract number
concepts, as measured by other related tasks (such as the ability to judge
which of two numbers is greater) (?, ?). Our model, which was only trained to
perform the give-$N$ task, offers an alternative interpretation: the
developmental trajectory observed in this task might reflect the learning of a
systematic, but narrow, procedure, rather than a general understanding of
cardinality. In line with this view, there is some evidence to suggest that
children can often succeed on the give-$N$ task while failing to perform
closely related tasks (?, ?). Thus, similar to our model, children may indeed
undergo a phase in which their ability to perform related counting tasks is
not yet integrated, and is better characterized as a set of ‘blind’ procedures
specific to each task, despite being able to perform those procedures in a
systematic manner. This is in line with the ‘knowledge-in-pieces’ view of
development, whereby early concepts are not immediately integrated into a
coherent whole (?, ?).
#### Limitations and Future Work
Despite the limitation of having trained our models on only one task, it is
possible that the ESBN architecture may not only facilitate the learning of
systematic procedures in specific tasks (as demonstrated here) but in a multi-
task learning context as well. In future work, we plan to train networks on
multiple related tasks (e.g. the how-many task, unit task, and direction task
(?, ?)) and study whether the ESBN affords a similar benefit in terms of the
ability to perform these tasks systematically, and in a manner that mirrors
the human developmental trajectory. We are also interested in allowing the
controller to select the most useful stream of data to bind to its memory
depending on its goal, so that we don’t need to hard-code its use of the pre-
trained counter. Preliminary data suggest the model is capable of making this
selection.
Additionally, some instances of the baseline models failed to learn the full
task, and some of the ESBN models were unstable, learning the task well
initially but then having performance drop as training progressed. This could
have been due to the high-variance of the policy gradient estimator (the
REINFORCE algorithm). We tried to replicate the human ability to learn from a
single episode at a time, but off-policy learning methods (e.g. replay) might
be required to ensure all networks reliably learn the task. As well, more
advanced policy gradient algorithms might improve performance of our models,
and are left to future work for implementation.
Finally, our model was restricted in that it could only count as high as the
length of the count sequence it was trained to memorize, representing an
intermediate level of human development (i.e. the stage when children cannot
count higher than their memorized count sequence, despite being presumed CP-
knowers). Most adults are capable of counting to arbitrarily large numbers,
using a recursive, hierarchical algorithm; that is, cycling through digits and
keeping track of place values. We believe that introducing hierarchical
structure, context segmentation and normalization (?, ?), together with
recursive application of the counter’s ability to iterate through a fixed
sequence, might permit the capacity for unbounded counting. This offers the
promise of providing a neurally plausible mechanism for symbolic counting,
which lies at the heart of many powerful cognitive functions of which humans
are capable, including mathematical reasoning.
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## Acknowledgements
This project / publication was made possible through the support of a grant
from the John Templeton Foundation. Thank you to Steven Frankland, Simon
Segert, Randall O’Reilly, and Alexander Petrov for their helpful discussions.
|
# Time-dependent orbital-optimized coupled-cluster methods families for
fermion-mixtures dynamics
Haifeng Lang<EMAIL_ADDRESS>Department of Nuclear Engineering and
Management, Graduate School of Engineering, The University of Tokyo, 7-3-1
Hongo, Bunkyo-ku, Tokyo 113-8656, Japan Takeshi Sato
<EMAIL_ADDRESS>Department of Nuclear Engineering and Management,
Graduate School of Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-
ku, Tokyo 113-8656, Japan Photon Science Center, Graduate School of
Engineering, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-8656,
Japan Research Institute for Photon Science and Laser Technology, The
University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033 Japan
###### Abstract
Five time-dependent orbital optimized coupled-cluster (TD-ooCC) methods, of
which four can converge to the complete active space self-consistent-field
method, are presented for fermion-mixtures with arbitrary fermion kinds and
numbers. Truncation schemes maintaining the intragroup orbital rotation
invariance, as well as equations of motion of CC amplitudes and orbitals, are
derived. Present methods are compact CC-parameterization alternatives to the
time-dependent multiconfiguration self-consistent-field method for systems
consisting of arbitrarily different kinds and numbers of interacting fermions.
Theoretical analysis of applications of present methods to various chemical
systems are reported.
## I introduction
Simulating dynamics of fermions and fermion mixtures quantitatively is
essential for understanding and predicting strong-field and ultrafast
phenomenaProtopapas, Keitel, and Knight (1997); Agostini and DiMauro (2004);
Krausz and Ivanov (2009); Gallmann, Cirelli, and Keller (2013); Nisoli _et
al._ (2017) of molecular systems (Noticing that Hilbert spaces of vibrational
systems and photons are isomorphic to fermion mixtures with one occupation in
each kindMordovina _et al._ (2020); Christiansen (2004); Hansen _et al._
(2019); Christiansen (2004); Sverdrup Ofstad _et al._ (2023)). However,
solving the time-dependent Schrödinger equation (TDSE) exactly is impractical
for even moderate size systems due to the factorial growth of computational
costs. Commonly used benchmark methods to describe such dynamics are the
multiconfiguration time-dependent Hartree-Fock (MCTDHF) method,Zanghellini
_et al._ (2003); Kato and Kono (2004); Caillat _et al._ (2005); Nest,
Klamroth, and Saalfrank (2005); Kato and Kono (2008); Hochstuhl and Bonitz
(2011); Haxton, Lawler, and McCurdy (2012); Haxton and McCurdy (2014) as well
as its extensions for fermion-mixturesLode _et al._ (2020); Alon, Streltsov,
and Cederbaum (2007). In these methods, the wavefunction is expanded as a full
configuration-interaction (CI) form inside an active space, of which single
particle orbitals are time-dependent. Equations of motion (EoMs) of parameters
are determined by time-dependent variational principle (TDVP). Due to the
compactness of the ansatz, the numbers of optimized time-dependent orbitals
are usually much less than the numbers of grid points in TDSE, and thus
reducing the computational cost hugely. However, the full CI expansion makes
methods still suffer from the factorial growth of the computational cost with
respect to the number of fermions. Therefore, the methods can be only applied
to moderate size systems. The more general time-dependent multiconfiguration
self-consistent-field (TD-MCSCF) methodNguyen-Dang _et al._ (2007); Ishikawa
and Sato (2015); Anzaki, Sato, and Ishikawa (2017) further restricts
configurations in the wavefunction expansion, which bridges the gap bewteen
time-dependent Hartree-Fock and MCTDHF. One of the attractive TD-MCSCF
methods, the time-dependent complete-active-space self-consistent-field (TD-
CASSCF) method Sato and Ishikawa (2013); Li, Sato, and Ishikawa (2021),
significantly reduces the cost by assigning some occupied orbitals as cores
(which must be occupied in all configurations within the expansion), resulting
in a computational cost that scales factorially with respect to the number of
fermions in the active orbitals rather than the total number of fermions. It
is also possible to restrict configurations in the active space by the
excitation levels with respect to the reference state, for instance,
MCTDH[n]Madsen _et al._ (2020a) and multireference MCTDH[n]Madsen _et al._
(2020b) for the vibrational dynamics, and time-dependent restricted active
space self-consistent field (TD-RASSCF)Miyagi and Madsen (2013, 2014); Haxton
and McCurdy (2015) and time-dependent occupation-restricted multiple active
space (TD-ORMAS)Sato and Ishikawa (2015) for the electron dynamics, to achieve
the polynomial cost scalingHaxton and McCurdy (2015); Sato _et al._ (2016);
Sawada, Sato, and Ishikawa (2016); Omiste, Li, and Madsen (2017) with a
drawback of not being size extensiveHelgaker, Jørgensen, and Olsen (2002).
Recently, the polynomial scaling and size-extensive CC parameterizationSzabo
and Ostlund (1996); Helgaker, Jørgensen, and Olsen (2002); Kümmel (2003);
Shavitt and Bartlett (2009) in the time-dependent active space gains lots of
popularity. In the CC parameterization, the ket vector is described by an
exponential truncated excitation operator acting on the reference state, while
the bra vector is not the Hermitian conjugate of the ket vector. For this
reason, the time-dependent bivariational principleArponen (1983) and
biorthogonal orbitals are also widely used in additional to the TDVP and
orthogonal orbitals. Many methods have been developed and successively
implemented, e.g., time-dependent optimized coupled-cluster method (TD-
OCC)Sato _et al._ (2018); Pathak, Sato, and Ishikawa (2020a, b, 2021) and
orbital-adapted time-dependent coupled-clusterKvaal (2012) (OATDCC) method for
the electron dynamics, and time-dependent vibrational coupled cluster with
time-dependent modals (TDMVCC) familiesMadsen _et al._ (2020c); Højlund,
Zoccante, and Christiansen (2024); Højlund and Christiansen (2024); Sverdrup
Ofstad _et al._ (2023); Højlund _et al._ (2022); Jensen _et al._ (2023) for
the vibrational dynamics. These methods can be considered as the time-
dependent extensions of stationary OCCG. E. Scuseria and H. F. Schaefer III
(1987); Sherrill _et al._ (1998); Krylov _et al._ (1998); Köhn and Olsen
(2005) and nonorthogonal orbital optimized coupled cluster (NOCC)Pedersen,
Fernández, and Koch (2001).
Despite the fruitful results of CC parameterization of the active space
wavefunction, it is still less explored than the CI parameterization. First,
previous works mainly focus on either electron dynamics (one kind fermion) or
vibrational dynamics (fermion mixtures but only one occupation in each fermion
kind). A general theory of arbitrary occupations of fermion mixtures is still
unclear. Second, numerically stable and convergentKöhn and Olsen (2005); Myhre
(2018) (in the sense of converging to the CASSCF/TD-CASSCF) CC
parameterizations using orthogonal orbitals, for instance, the standard
CCHuber and Klamroth (2011); D. R. Nascimento and A. E. DePrince III (2016)
and Brueckner coupled cluster (BCC)Raghavachari _et al._ (1990); Hampel,
Peterson, and Werner (1992); Chiles and Dykstra (1981); Handy _et al._ (1989)
types of parameterization in the active space, are unknown. In this paper, we
advance in these directions by presenting five time-dependent orbital-
optimized coupled-cluster (TD-ooCC) methods for a system composed of
arbitrarily different kinds and numbers of interacting fermions (with particle
conservation).
This paper is organized as follows. In the theory section Sec. II, we first
analyze the convergence condition of time-dependent CC with time varying
orthogonal orbitals to the TD-CASSCF. Further, the selections of both single
CC (de)exication operators ($\hat{T}_{1}$ and $\hat{\Lambda}_{1}$, defined in
Eq. (3,4)), which determines the type of the method, and non-single CC
(de)exication operators ($\hat{T}_{0}$, $\hat{T}_{1}$, $\cdots$,
$\hat{\Lambda}_{0}$ $\hat{\Lambda}_{2}$, $\cdots$, defined in Eq. (3,4)) ,
which can reduce the computational resources potentially, in the wavefunction
ansatz of TD-ooCC methods are introduced. The equations of motion (EoMs) for
CC amplitudes and orbitals are presented. Finally, we discuss their advantages
and potential drawbacks. In the application section Sec. III, we discuss three
major applications of our methods to the chemical systems, electronic
dynamics, vibrational dynamics, and non-adiabatic dynamics. Sec. IV summarizes
our theories. The Hartree atomic units are used throughout unless otherwise
noted.
## II Theory
### II.1 Full coupled cluster expansion in the complete active space
Following the CASSCF method, general orthogonal time-dependent orbitals
$\mu^{m},\nu^{m},\cdots$ of fermion kind $m$ are classified into virtual
orbitals $\alpha^{m},\beta^{m},\cdots$, and occupied orbitals
$\bar{p}^{m},\bar{q}^{m},\cdots$, which can be further classified into frozen-
cores $i^{\prime\prime m},j^{\prime\prime m}\cdots$, dynamical-cores
$i^{\prime m},j^{\prime m}$, and active orbitals $t^{m},u^{m},\cdots$ that
contain hole orbitals $i^{m},j^{m},\cdots$, and particle orbitals
$a^{m},b^{m},\cdots$. Here, the hole orbitals and particle orbitals refer to
active orbitals that are occupied and unoccupied, respectively, in the
reference state. EoMs of orbitals can be formally expressed as
$i|\dot{\psi}_{\nu^{m}}\rangle=i|\psi_{\mu^{m}}\rangle
X^{\mu^{m}}_{\nu^{m}}\,,$ (1)
where $X^{\mu^{m}}_{\nu^{m}}=\braket{\psi_{\mu^{m}}}{\dot{\psi}_{\nu^{m}}}$ is
anti-Hermitian to enforce the orthonormality of orbitals
$\langle\psi_{\mu^{m}}(t)|\psi_{\nu^{m}}(t)\rangle=\delta^{\mu^{m}}_{\nu^{m}}$
at any time, and thus called the constraints of orbital rotations. There are
two types of constraints of orbitals: the first type includes pre-determined
constraints that are intra-group rotation constraints and frozen-core related
orbital constraintsCon ; the second type includes others that are variational
parameters which should be determined by the time-dependent variational
principle. In this article, we will use the Einstein notation except for
fermion kind index $m,n,\cdots$. For the convenience, we use the notations
$p^{m},q^{m},\cdots$ to label time-dependent occupied orbitals except for the
frozen-cores due to the fact that frozen-cores are pre-determineded by the
gauge choiceSato _et al._ (2016). Additionally, $\hat{c}_{\mu^{m}}^{\dagger}$
and $\hat{c}_{\mu^{m}}$ are defined as the creation and annihilation operators
for $\psi_{\mu^{m}}$ (as well as other orbital labels).
The most general ansatz of the CC wave function with time-dependent orbital
can be expressed as
$\ket{\Psi_{\rm R}}=e^{\hat{T}}\ket{\Phi}\,,\quad\bra{\Psi_{\rm
L}}=\bra{\Phi}\hat{\Lambda}e^{-\hat{T}}\,,$ (2)
where $\ket{\Phi}$ and $\bra{\Phi}$ are the reference states, i.e., tensor
product of Slater determinants of all fermion kinds
$\ket{\Phi}=\prod_{m}\prod_{i^{\prime\prime m}i^{\prime
m}i^{m}}\hat{c}^{\dagger}_{i^{\prime\prime m}}\hat{c}^{\dagger}_{i^{\prime
m}}\hat{c}^{\dagger}_{i^{m}}\ket{0}$, and
$\displaystyle\hat{T}$
$\displaystyle=\hat{T}_{0}+\hat{T}_{1}+\hat{T}_{2}+\hat{T}_{3}+\cdots$ (3)
$\displaystyle=\tau_{0}\hat{E}_{0}+\sum_{m}\tau_{i^{m}}^{a^{m}}\hat{E}_{i^{m}}^{a^{m}}+\sum_{m_{1}m_{2}}\tau_{i^{m_{1}}j^{m_{2}}}^{a^{m_{1}}b^{m_{2}}}\hat{E}_{i^{m_{1}}j^{m_{2}}}^{a^{m_{1}}b^{m_{2}}}$
$\displaystyle+\sum_{m_{1}m_{2}m_{3}}\tau_{i^{m_{1}}j^{m_{2}}k^{m_{3}}}^{a^{m_{1}}b^{m_{2}}c^{m_{3}}}\hat{E}_{i^{m_{1}}j^{m_{2}}k^{m_{3}}}^{a^{m_{1}}b^{m_{2}}c^{m_{3}}}+\cdots$
$\displaystyle=\tau^{\mathring{A}}_{\mathring{I}}\hat{E}^{\mathring{A}}_{\mathring{I}}=\sum_{m}\tau_{i^{m}}^{a^{m}}\hat{E}_{i^{m}}^{a^{m}}+\tau^{{A}}_{{I}}\hat{E}^{{A}}_{{I}}\,,$
$\displaystyle\hat{\Lambda}$
$\displaystyle=\hat{\Lambda}_{0}+\hat{\Lambda}_{1}+\hat{\Lambda}_{2}+\hat{\Lambda}_{3}+\cdots$
(4)
$\displaystyle=\lambda_{0}\hat{E}_{0}+\sum_{m}\lambda^{i^{m}}_{a^{m}}\hat{E}^{i^{m}}_{a^{m}}+\sum_{m_{1}m_{2}}\lambda^{i^{m_{1}}j^{m_{2}}}_{a^{m_{1}}b^{m_{2}}}\hat{E}^{i^{m_{1}}j^{m_{2}}}_{a^{m_{1}}b^{m_{2}}}$
$\displaystyle+\sum_{m_{1}m_{2}m_{3}}\lambda^{i^{m_{1}}j^{m_{2}}k^{m_{3}}}_{a^{m_{1}}b^{m_{2}}c^{m_{3}}}\hat{E}^{i^{m_{1}}j^{m_{2}}k^{m_{3}}}_{a^{m_{1}}b^{m_{2}}c^{m_{3}}}+\cdots$
$\displaystyle=\lambda_{\mathring{A}}^{\mathring{I}}\hat{E}_{\mathring{A}}^{\mathring{I}}=\sum_{m}\lambda^{i^{m}}_{a^{m}}\hat{E}^{i^{m}}_{a^{m}}+\lambda_{{A}}^{{I}}\hat{E}_{{A}}^{{I}}\,,$
Here, $\mathring{I}=ijk\cdots$ and $\mathring{A}=abc\cdots$ (can be empty)
denote hole and particle orbital strings to make the notation brevity. The
general-rank excitation (deexcitation) operators and amplitudes are
$\hat{E}^{\mathring{A}}_{\mathring{I}}$
($\hat{E}_{\mathring{A}}^{\mathring{I}}$) and
$\tau^{\mathring{A}}_{\mathring{I}}$
($\lambda_{\mathring{A}}^{\mathring{I}}$), where
${\hat{E}}^{\mu_{1}^{m_{1}}\mu_{2}^{m_{2}}\mu_{3}^{m_{3}}\cdots}_{\nu_{1}^{m_{1}}\nu_{2}^{m_{2}}\nu_{3}^{m_{3}}\cdots}=\hat{c}^{\dagger}_{\mu_{1}^{m_{1}}}\hat{c}^{\dagger}_{\mu_{2}^{m_{2}}}\hat{c}^{\dagger}_{\mu_{3}^{m_{3}}}\cdots{\hat{c}}_{\nu_{3}^{m_{3}}}{\hat{c}}_{\nu_{2}^{m_{2}}}{\hat{c}}_{\nu_{1}^{m_{1}}}$,
and $\hat{E}_{0}=\hat{I}$ for the zero-rank excitation (empty strings of
$\mathring{I}$ and $\mathring{A}$). We will also use the notation $I$ and $A$
to label non-single hole and particle orbital strings (the orbital number of
the strings is not one) in this article.
Before working on time-dependent variational principles, we stress that single
amplitudes terms $\hat{T}_{1}$ and $\hat{\Lambda}_{1}$ are included in the
ansatz due to the convenience of the theoretical analysis although they can
cause the numerical instability in practical simulations and are omitted in
previous workSato _et al._ (2018); Köhn and Olsen (2005). Additionally, zero-
rank amplitudes $\tau_{0}$ and $\lambda_{0}$ are also explicitly included in
the ansatz to recover the CASSCF wavefucntion in the full CC expansion limit.
The normalization condition makes $\lambda_{0}\equiv 1$ conserved, and
$\tau_{0}$ has no impacts on observables but essential for the
autocorrelations.
Following the previous workPedersen, Koch, and Hättig (1999); Kvaal (2012);
Helgaker, Jørgensen, and Olsen (2002); Sato _et al._ (2018), we begin with
the coupled-cluster Lagrangian $L$ and real action functional $S$,
$\displaystyle L$ $\displaystyle=\langle\Psi_{\rm
L}|(\hat{H}-i{\partial_{t}})|\Psi_{\rm R}\rangle\,,$ (5) $\displaystyle S$
$\displaystyle=\Re\int_{t_{0}}^{t_{1}}Ldt=\frac{1}{2}\int_{t_{0}}^{t_{1}}\left(L+L^{*}\right)dt\,.$
(6)
Here, the Hamiltonian $\hat{H}$ can be an arbitrary Hamiltonian with particle
conservations. EoMs of parameters are given by the stationary condition of the
action with respect to the variation of all parameters of left- and right-
state blackwavefunctions (with necessary constraints),
black
$\displaystyle{\color{black}2}\delta S$ $\displaystyle=$
$\displaystyle\delta\tau^{\mathring{A}}_{\mathring{I}}\left\\{\langle\Psi_{\rm
L}|[\hat{H}-i\hat{X},\hat{E}^{\mathring{A}}_{\mathring{I}}]|\Psi_{\rm
R}\rangle+i\dot{\lambda}_{\mathring{A}}^{\mathring{I}}\right\\}+\delta\lambda_{\mathring{A}}^{\mathring{I}}\left\\{\langle\Phi^{\mathring{A}}_{\mathring{I}}|e^{-\hat{T}}(\hat{H}-i\hat{X})|\Psi_{\rm
R}\rangle-i\dot{\tau}^{\mathring{A}}_{\mathring{I}}\right\\}+c.c$ (7)
$\displaystyle+$
$\displaystyle\sum_{m}\Delta^{\mu^{m}}_{\nu^{m}}\left\\{\langle\Psi_{\rm
L}|[\hat{H}-i(\partial^{\hskip
0.20999pt\prime}_{t}\hat{T})-i\hat{X},\hat{E}^{\mu^{m}}_{\nu^{m}}]|\Psi_{\rm
R}\rangle+i\langle\Phi|(\partial^{\hskip
0.20999pt\prime}_{t}\hat{\Lambda})e^{-\hat{T}}\hat{E}^{\mu^{m}}_{\nu^{m}}|\Psi_{\rm
R}\rangle\right.$ $\displaystyle\hskip 13.00005pt\left.-\langle\Psi_{\rm
L}|[\hat{H}-i(\partial^{\hskip
0.20999pt\prime}_{t}\hat{T})-i\hat{X},\hat{E}_{\mu^{m}}^{\nu^{m}}]|\Psi_{\rm
R}\rangle^{*}+i\langle\Phi|(\partial^{\hskip
0.20999pt\prime}_{t}\hat{\Lambda})e^{-\hat{T}}\hat{E}_{\mu^{m}}^{\nu^{m}}|\Psi_{\rm
R}\rangle^{*}\right\\}\,.$
where $\partial^{\hskip
0.20999pt\prime}_{t}\hat{T}=\dot{\tau}^{\mathring{A}}_{\mathring{I}}\hat{E}^{\mathring{A}}_{\mathring{I}}$,
$\partial^{\hskip
0.20999pt\prime}_{t}\hat{\Lambda}=\dot{\lambda}_{\mathring{A}}^{\mathring{I}}\hat{E}_{\mathring{A}}^{\mathring{I}}$,
$\hat{X}=\sum_{m}X^{\mu^{m}}_{\nu^{m}}\hat{E}^{\mu^{m}}_{\nu^{m}}$,
$\hat{\Delta}=\sum_{m}\Delta^{\mu^{m}}_{\nu^{m}}\hat{E}^{\mu^{m}}_{\nu^{m}}$,
and
$\Delta^{\mu^{m}}_{\nu^{m}}=\braket{\psi_{\mu^{m}}}{\delta{\psi}_{\nu^{m}}}$.
Similar to the matrices $X^{\mu^{m}}_{\nu^{m}}$, $\Delta^{\mu^{m}}_{\nu^{m}}$
are also anti-Hermitian to enforce the orthonormality of orbitals. Meanwhile,
$\Delta^{\mu^{m}}_{i^{\prime\prime m}}\equiv 0$ due to the fact that EoMs of
frozen-cores are not determined by variational principle. Additionally, we
only consider the ansatz that satisfies the intragroup rotation invariance
including dynamical-cores-dynamical-cores, hole-hole, particle-particle, and
virtual-virtual rotation invariance, i.e., the wavefunction is invariant up to
a phase factor with these orbital rotations and the corresponding inverse
rotations of CC amplitudes. These invariances give the redundancy of
$X^{i^{\prime m}}_{j^{\prime m}}$, $X_{i^{m}}^{j^{m}}$, $X_{a^{m}}^{b^{m}}$,
and $X_{\alpha^{m}}^{\beta^{m}}$. For simplicity, we always set all of them as
zero. The details of the wavefunction ansatz that maintain the hole-hole,
particle-particle, and virtual-virtual rotation invariance will be discussed
in the Sec. II.2.2 and B.
Variations with respect to CC amplitudes, $\delta
S/\delta\lambda_{\mathring{A}}^{\mathring{I}}=0$ and $\delta
S/\delta\tau^{\mathring{A}}_{\mathring{I}}=0$ give
$\displaystyle i\dot{\tau}^{A}_{I}$
$\displaystyle=\langle\Phi^{{A}}_{{I}}|e^{-\hat{T}}(\hat{H}-i\hat{X})|\Psi_{\rm
R}\rangle\,,$ (8) $\displaystyle i\dot{\tau}_{i^{m}}^{a^{m}}$
$\displaystyle=\bra{\Phi_{i^{m}}^{a^{m}}}e^{-\hat{T}}(\hat{H}-i\hat{X})\ket{\Psi_{\rm
R}}\,,$ (9) $\displaystyle-i\dot{\lambda}^{I}_{A}$
$\displaystyle=\langle\Psi_{\rm
L}|e^{-\hat{T}}[\hat{H}-i\hat{X},\hat{E}^{A}_{I}]|\Psi_{\rm R}\rangle\,,$ (10)
$\displaystyle-i\dot{\lambda}^{i^{m}}_{a^{m}}$ $\displaystyle=\bra{\Psi_{\rm
L}}[\hat{H}-i\hat{X},\hat{E}_{i^{m}}^{a^{m}}]\ket{\Psi_{\rm R}}\,.$ (11)
Here, we separate EoMs of CC amplitudes into non-single amplitudes part and
single amplitudes part. The variation with respect to $\Delta_{a^{m}}^{i^{m}}$
gives
$\displaystyle\bra{\Psi_{\rm
L}}[\hat{H}-i\hat{X}-i(\partial_{t}^{\prime}\hat{T}),\hat{E}^{i^{m}}_{a^{m}}]\ket{\Psi_{\rm
R}}$ (12) $\displaystyle+$ $\displaystyle
i\bra{\Phi}(\partial_{t}^{\prime}\hat{\Lambda})e^{-\hat{T}}\hat{E}^{i^{m}}_{a^{m}}|\Psi_{\rm
R}\rangle$ $\displaystyle-$ $\displaystyle\bra{\Psi_{\rm
L}}[\hat{H}-i\hat{X}-i(\partial_{t}^{\prime}\hat{T}),\hat{E}_{i^{m}}^{a^{m}}]\ket{\Psi_{\rm
R}}^{*}$ $\displaystyle+$ $\displaystyle
i\bra{\Phi}(\partial_{t}^{\prime}\hat{\Lambda})e^{-\hat{T}}\hat{E}_{i^{m}}^{a^{m}}|\Psi_{\rm
R}\rangle^{*}=0\,,$
Substituting Eq. (8,10) into Eq. (12) yields,
$\displaystyle
i(\delta_{b^{m}}^{a^{m}}D_{i^{m}}^{j^{m}}-D_{b^{m}}^{a^{m}}\delta_{i^{m}}^{j^{m}})(\dot{\tau}_{j^{m}}^{b^{m}}+X_{j^{m}}^{b^{m}})+i\frac{1}{2}(\dot{\lambda}^{i^{m}}_{a^{m}})^{*}=$
(13) $\displaystyle
B_{i^{m}}^{a^{m}}-i\frac{1}{2}\sum_{n}\left\\{\langle\Phi_{j^{n}}^{b^{n}}|e^{-\hat{T}}\hat{E}^{i^{m}}_{a^{m}}|\Psi_{\rm
R}\rangle\dot{\lambda}^{j^{n}}_{b^{n}}+A^{a^{m}j^{n}}_{i^{m}b^{n}}(X^{b^{n}}_{j^{n}})^{*}\right\\}\,,$
where
$\displaystyle A^{a^{m}j^{n}}_{i^{m}b^{n}}$ $\displaystyle=\bra{\Psi_{\rm
L}}[\hat{E}^{A}_{I},\hat{E}^{j^{n}}_{b^{n}}]\ket{\Psi_{\rm
R}}\langle\Phi^{A}_{I}|e^{-\hat{T}}\hat{E}^{i^{m}}_{a^{m}}\ket{\Psi_{\rm R}}$
$\displaystyle-\bra{\Psi_{\rm
L}}[\hat{E}^{A}_{I},\hat{E}^{i^{m}}_{a^{m}}]\ket{\Psi_{\rm
R}}\langle\Phi_{I}^{A}|e^{-\hat{T}}\hat{E}^{j^{n}}_{b^{n}}\ket{\Psi_{\rm R}},$
(14) $\displaystyle B_{i^{m}}^{a^{m}}$
$\displaystyle=-\frac{1}{2}\braket{\Psi_{\rm
L}}{[\hat{H},\hat{E}^{i^{m}}_{a^{m}}]}{\Psi_{\rm
R}}-\frac{1}{2}\braket{\Psi_{\rm
R}}{[\hat{H},\hat{E}^{i^{m}}_{a^{m}}]}{\Psi_{\rm L}}$
$\displaystyle-\frac{1}{2}\bra{\Psi_{\rm
L}}[\hat{E}^{A}_{I},\hat{H}]\ket{\Psi_{\rm
R}}\langle\Phi^{A}_{I}|e^{-\hat{T}}\hat{E}^{i^{m}}_{a^{m}}\ket{\Psi_{\rm R}}$
$\displaystyle+\frac{1}{2}\bra{\Psi_{\rm
L}}[\hat{E}^{A}_{I},\hat{E}^{i^{m}}_{a^{m}}]\ket{\Psi_{\rm
R}}\langle\Phi_{I}^{A}|e^{-\hat{T}}\hat{H}\ket{\Psi_{\rm R}}\,.$ (15)
Solving coupled equations Eq. (9,11,13), EoMs of single amplitudes and
particle-hole orbital rotations $\dot{\tau}_{i^{m}}^{a^{m}}$,
$\dot{\lambda}^{i^{m}}_{a^{m}}$, and $X_{i^{m}}^{a^{m}}$ can be obtained.
Substituting $X_{i^{m}}^{a^{m}}$ into Eq. (8,10), EoMs of non-single
amplitudes can be solved.
In the full CC expansion limit, the particle-hole orbital rotations
$X_{i^{m}}^{a^{m}}$ are redundant, and can be freely selected. For an explicit
proof, see the Appendix. B. The simplest and the most common choice is to set
them as zero. However, coupling equations Eq. (9,11,13) suggest that the
redundancy of $X_{i^{m}}^{a^{m}}$ can be transferred to
$\lambda^{i^{m}}_{a^{m}}$ or $\tau_{i^{m}}^{a^{m}}$. For instance, one can set
$\lambda_{a^{m}}^{i^{m}}\equiv 0$ as redundant parameters, and this type of
wavefunction parameterization is known as the BCCRaghavachari _et al._
(1990); Hampel, Peterson, and Werner (1992); Chiles and Dykstra (1981); Handy
_et al._ (1989). In fact, one can arbitrarily select one of
$X_{i^{m}}^{a^{m}}$, $\tau_{i^{m}}^{a^{m}}$ and ${\lambda}^{i^{m}}_{a^{m}}$ as
redundant parameter for each kind of fermion, and set them as zero. In the
Appendix. B, more rigorous arguments of transferring redundancy and
preparation of wavefunctions are presented.
### II.2 Wavefunction Ansatz of Time dependent orbital-optimized coupled-
cluster families
#### II.2.1 Selection of single amplitudes
In principle, the redundancy caused by including all $X_{i^{m}}^{a^{m}}$,
$\tau_{i^{m}}^{a^{m}}$, and $\lambda^{i^{m}}_{a^{m}}$ only occurs at the full
CC expansion limit. Nonetheless, our experience shows that including them all
is numerically instable even when only double (rank two) excitations are
considered, which suggests the compactness of CC wavefunction
parameterization. One has to exclude (at least) one of $X_{i^{m}}^{a^{m}}$,
$\tau_{i^{m}}^{a^{m}}$ and $\lambda^{i^{m}}_{a^{m}}$ in practical simulations.
The first method we will consider is TD-OCC (terminology adapted from the
referenceSato _et al._ (2018)), of which the ansatz only includes
$X_{i^{m}}^{a^{m}}$. This method fails to converge to the CASSCF limitKöhn and
Olsen (2005); Myhre (2018); Højlund, Zoccante, and Christiansen (2024), but
shows an acceptable accuracy in ultrafast electron dynamicsSato _et al._
(2018) and vibrational systemsHøjlund, Zoccante, and Christiansen (2024)
according to previous reports. Specifically, TD-OCC with double and triple
excitations (TD-OCCDT) is in nearly perfect agreement with the CASSCF results
for benchmarks of ultrafast electron dynamicsSato _et al._ (2018). We will
also present three methods with the absence of one of $X_{i^{m}}^{a^{m}}$,
$\tau_{i^{m}}^{a^{m}}$, and $\lambda^{i^{m}}_{a^{m}}$ in the ansatz, thus all
three methods can converge to the CASSCF limit. The details of four methods
are summarized as:
(a). TD-OCC: only $X_{i^{m}}^{a^{m}}$ are included in the ansatz, or
equivalently
$\tau_{i^{m}}^{a^{m}}=\lambda^{i^{m}}_{a^{m}}=\delta\tau_{i^{m}}^{a^{m}}=\delta\lambda^{i^{m}}_{a^{m}}\equiv
0$.
(b). TD-OCCT1: $\tau_{i^{m}}^{a^{m}}$ and $X_{i^{m}}^{a^{m}}$ are included in
the ansatz, or equivalently
$\lambda^{i^{m}}_{a^{m}}=\delta\lambda^{i^{m}}_{a^{m}}\equiv 0$.
(c). TD-BCC: $X_{i^{m}}^{a^{m}}$ and $\lambda^{i^{m}}_{a^{m}}$ are included in
the ansatz, or equivalently
$\tau_{i^{m}}^{a^{m}}=\delta\tau_{i^{m}}^{a^{m}}\equiv 0$.
(d). TD-OCCX0: $\tau_{i^{m}}^{a^{m}}$ and $\lambda^{i^{m}}_{a^{m}}$ are
included in the ansatz, or equivalently
$X_{i^{m}}^{a^{m}}=\Delta_{i^{m}}^{a^{m}}\equiv 0$.
Here, the latter definition of each method can be regarded as the constraints
in the sense that some parameters are pre-determined (to be zero). We stress
that the absence of particle-hole orbital rotations $X_{i^{m}}^{a^{m}}$ of TD-
OCCX0 is only justified for the dynamics without external fields. In the
presence of external fields $X_{i^{m}}^{a^{m}}$ has to be assigned different
values to ensure the gauge invariance, akin to the treatment of frozen-
coresSato _et al._ (2016). Additionally, we will consider the most general
hybrid form of wavefunction parameterization, i.e., the parameter selections
(a), (b), (c), or (d) is separately, and flexibly adopted for each fermion
kinds, which is named as TD-OCCH and will be discussed with all the details in
the Appendix.D.
#### II.2.2 Selection of non-single excitations
Inspired by TD-MCSCF, one can restrict the non-single excitations in TD-ooCC
to make the ansatz more compact. To ensure the intragroup rotation invariance
of particle and hole orbitals, one can select non-single excitations that
satisfy the property
$\displaystyle\hat{E}^{\mu^{m}}_{\nu^{m}}\ket{\Psi_{\rm
R}}=e^{\hat{T}}\hat{\Pi}e^{-\hat{T}}\hat{E}^{\mu^{m}}_{\nu^{m}}\ket{\Psi_{\rm
R}}\,,$ (16) $\displaystyle\bra{\Psi_{\rm
L}}\hat{E}^{\mu^{m}}_{\nu^{m}}=\bra{\Psi_{\rm
L}}\hat{E}^{\mu^{m}}_{\nu^{m}}e^{\hat{T}}\hat{\Pi}e^{-\hat{T}}\,,$
where
$\hat{\Pi}=\ket{\Phi_{\mathring{I}}^{\mathring{A}}}\bra{\Phi_{\mathring{I}}^{\mathring{A}}}$.
Here $\mu^{m}$ and $\nu^{m}$ should be particle or hole orbital index
simultaneously. For the details of the proof, see the Appendix. B. One of the
simplest selection scheme is the $[k]$-truncation scheme, i.e., non-single
excitation is included in the ansatz if and only if it is not larger than the
certain truncation order $k$, which has been successfully implemented in TD-
OCC.
It is also possible to assign different truncation orders for different kinds
to adapt their contribution to the dynamics. Here, we propose two possible
schemes. The first, the weighted excitation truncation scheme, is the modified
version of the truncation scheme reported in Ref. [Madsen _et al._ , 2020d].
Define the nonnegative excitation weight and the excitation number of each
kind as $w^{m}$ and $n_{\rm ext}^{m}$, respectively. The generalized
excitation level of a given excitation amplitude is
$w=\sum_{m}n_{\rm ext}^{m}w^{m}\,,$ (17)
and all non-single excitation amplitude that satisfy $w\leq k$ should be
included in the ansatz, where $k$ is the maximum generalized excitation level.
Here, different spin fermion configurations can be regarded as different
“fermion kinds” for both unrestricted and restricted spin orbitals if the each
spin component is conserved. This truncation scheme can reduce to the scheme
in Ref. [Madsen _et al._ , 2020d] in the vibrational system, in which $n_{\rm
ext}^{m}$ can only be 1 or 0. One can set the weights of important kinds as
relatively small numbers, even zero. When all $w^{m}=1$, the weighted
excitation truncation scheme reduces to the $[k]$-truncation scheme.
The second one, group excitation truncation scheme, is the following: (1) all
kinds of fermions can be divided into several groups A,B,C,$\cdots$, and their
truncation order of excitations can be assigned as $n^{A}_{A}$, $n^{B}_{B}$,
$n^{C}_{C}$,$\cdots$. Here, $n=1$ if only zero excitations (and single
excitations when included) are included in the ansatz. All possible non-single
excitations beyond the truncation order must be excluded while all others must
be included in the ansatz. The different spin components of a fermion can also
be regarded as different kinds if they are conserved; (2) for each type of
excitations across groups, truncation order of the total excitation and all
single groups should be assigned, for instance,
$(n_{AB},n_{AB}^{A},n_{AB}^{B})$ should be assigned for the excitation
acrossing groups $A$ and $B$, and
$(n_{ABC},n_{ABC}^{A},n_{ABC}^{B},n_{ABC}^{C})$ should be assigned for the
excitation acrossing groups $A$, $B$ and $C$. Iff both the total excitation
number and the excitation number of each group are not larger than the
corresponding truncation parameters, it is included in the ansatz; (3) $n$
must satisfy the inequalities: for all $X$, $n^{X}_{Y}\leq n^{X}_{Z}$ if the
group numbers of $Y$ is larger than $Z$. We will list two explicit examples on
the electron dynamics. We set the $\alpha$ spin and $\beta$ spin are two
different groups. The first example is
$n_{\alpha}^{\alpha}=n_{\beta}^{\beta}=n_{\alpha\beta}^{\alpha}=n_{\alpha\beta}^{\beta}=2$
and $n_{\alpha\beta}=3$, which means all double excitations of $\alpha$ and
$\beta$ spins are included, and all $2\alpha 1\beta$ and $2\beta 1\alpha$ type
triple excitations are included. The second is
$n_{\alpha}^{\alpha}=n_{\beta}^{\beta}=n_{\alpha\beta}^{\alpha}=n_{\alpha\beta}^{\beta}=1$,
and $n_{\alpha\beta}=2$, which means only $1\alpha 1\beta$ excitations are
included. For a slightly more complicated example, see Fig. (1).
Figure 1: Illustration of the Truncation of non-single CC amplitudes (cores
and virtual orbitals are neglected). In the example of this figure, three
kinds fermions ($m_{1}$, $m_{2}$, and $m_{3}$) are classified into two groups:
Group A containing $m_{1}$ and $m_{2}$, and Group B containing $m_{3}$. We set
$n_{A}^{A}=n_{B}^{B}=n_{AB}^{B}=2$, $n_{AB}=3$, and $n_{AB}^{A}=1$. Fig. (1a)
is the reference configuration, where dashed line is the fermion surface.
Figs. (1b,c) show two of allowed CC excitations across the group, where blue
rectangles circle the excitation kinds. Fig. (1d) shows the forbidden
excitations, where red rectangles circle the excitation kinds. In the Fig.
(1d), although the total excitation and excitation of group B satisfy the
conditions, $3=n_{AB}$ and $1<n_{AB}^{B}$, the excitation is still forbidden
due to the condition violation of the group A excitation $2>n_{AB}^{A}$.
### II.3 Equation of motions of Time dependent optimized coupled cluster
families
EoMs of single amplitudes and particle-hole orbital rotations of five methods
can be obtained by setting corresponding constraints on Eq. (9,11,13), which
will be presented in the corresponding subsubsections and the Appendix. EoMs
of other variables are the same for all five methods [Eq. (8,10) ]. EoMs of
intergroup rotations $X^{\alpha^{m}}_{p^{m}}$ and $X^{t^{m}}_{i^{\prime m}}$
are obtained from the variation with respect to $\Delta_{\alpha^{m}}^{p^{m}}$
and $\Delta_{t^{m}}^{i^{\prime m}}$,
$\displaystyle\bra{\Psi_{\rm
L}}[\hat{H}-i\hat{X},\hat{E}_{\alpha^{m}}^{p^{m}}]\ket{\Psi_{\rm
R}}+\bra{\Psi_{\rm
R}}[\hat{H}-i\hat{X},\hat{E}_{\alpha^{m}}^{p^{m}}]\ket{\Psi_{\rm L}}=0\,,$
(18) $\displaystyle\bra{\Psi_{\rm
L}}[\hat{H}-i\hat{X},\hat{E}_{t^{m}}^{i^{\prime m}}]\ket{\Psi_{\rm
R}}+\bra{\Psi_{\rm R}}[\hat{H}-i\hat{X},\hat{E}_{t^{m}}^{i^{\prime
m}}]\ket{\Psi_{\rm L}}=0\,,$
which can be further simplified as
$\displaystyle 2iX^{u^{m}}_{i^{\prime m}}$
$\displaystyle(2\delta_{u^{m}}^{t^{m}}-D_{u^{m}}^{t^{m}})=$
$\displaystyle\braket{\Psi_{\rm L}}{\hat{E}_{t^{m}}^{i^{\prime
m}}\hat{H}}{\Psi_{\rm R}}+\braket{\Psi_{\rm R}}{\hat{E}_{t^{m}}^{i^{\prime
m}}\hat{H}}{\Psi_{\rm L}}\,,$ (19) $\displaystyle 2iX^{\alpha^{m}}_{q^{m}}$
$\displaystyle D^{q^{m}}_{p^{m}}=\braket{\Psi_{\rm
L}}{\hat{E}_{\alpha^{m}}^{p^{m}}\hat{H}}{\Psi_{\rm R}}+\braket{\Psi_{\rm
R}}{\hat{E}_{\alpha^{m}}^{p^{m}}\hat{H}}{\Psi_{\rm L}}\,,$ (20)
with
$\displaystyle\rho_{q^{m}}^{p^{m}}$ $\displaystyle=\bra{\Psi_{\rm
L}}\hat{E}^{q^{m}}_{p^{m}}\ket{\Psi_{\rm R}}\,,\hskip
5.0ptD_{q^{m}}^{p^{m}}=\frac{1}{2}(\rho_{q^{m}}^{p^{m}}+(\rho^{q^{m}}_{p^{m}})^{*})$
(21)
In practical simulations, virtual orbitals are never propagated. Instead, one
can convertBeck _et al._ (2000); Anzaki, Sato, and Ishikawa (2017) the
propagation of occupied orbitals arisen from the virtual-occupied orbital
rotations to
$\displaystyle
i\braket{\psi_{\bar{\mu}^{m}}}{\psi_{\alpha^{m}}}X^{\alpha^{m}}_{p^{m}}=(D^{-1})_{q^{m}}^{p^{m}}$
(22) $\displaystyle\times$ $\displaystyle\Big{\\{}\braket{\Psi_{\rm
L}}{\hat{E}_{\bar{\mu}^{m}}^{q^{m}}\hat{H}}{\Psi_{\rm R}}-\braket{\Psi_{\rm
L}}{\hat{E}_{\bar{o}^{m}}^{q^{m}}\hat{H}}{\Psi_{\rm
R}}\braket{\psi_{\bar{\mu}^{m}}}{\psi_{\bar{o}^{m}}}$
$\displaystyle+\braket{\Psi_{\rm
R}}{\hat{E}_{\bar{\mu}^{m}}^{q^{m}}\hat{H}}{\Psi_{\rm L}}-\braket{\Psi_{\rm
R}}{\hat{E}_{\bar{o}^{m}}^{q^{m}}\hat{H}}{\Psi_{\rm
L}}\braket{\psi_{\bar{\mu}^{m}}}{\psi_{\bar{o}^{m}}}\Big{\\}}\,,$
where $\bar{\mu}^{m}$ is the primary time-independent basis to expand orbitals
that can be discrete variables representationBeck _et al._ (2000), grid
pointsLi, Sato, and Ishikawa (2021), etc.
#### II.3.1 TD-OCC
Setting $\tau_{i^{m}}^{a^{m}}={\lambda}^{i^{m}}_{a^{m}}\equiv 0$ in Eq. (13),
$i(\delta_{b^{m}}^{a^{m}}D_{i^{m}}^{j^{m}}-D_{b^{m}}^{a^{m}}\delta_{i^{m}}^{j^{m}})X_{j^{m}}^{b^{m}}+i\sum_{n}\frac{1}{2}A^{a^{m}j^{n}}_{i^{m}b^{n}}(X^{b^{n}}_{j^{n}})^{*}=B_{i^{m}}^{a^{m}}\,.$
(23)
In the case of one occupation of each kind of fermion and a single kind of
fermion, the method reduces to oTDMVCCHøjlund, Zoccante, and Christiansen
(2024) for vibrational systems and TD-OCCSato _et al._ (2018) for electrons,
respectively.
#### II.3.2 TD-OCCT1
Setting ${\lambda}^{i^{m}}_{a^{m}}\equiv 0$ in Eq. (11,13),
$i\left(\rho_{j^{m}}^{i^{m}}\delta_{b^{m}}^{a^{m}}-\rho_{a^{m}}^{b^{m}}\delta_{j^{m}}^{i^{m}}\right)X^{j^{m}}_{b^{m}}=\bra{\Psi_{\rm
L}}[\hat{H},\hat{E}_{i^{m}}^{a^{m}}]\ket{\Psi_{\rm R}}\,,$ (24) $\displaystyle
i(\delta_{b^{m}}^{a^{m}}D_{i^{m}}^{j^{m}}-D_{b^{m}}^{a^{m}}\delta_{i^{m}}^{j^{m}})\dot{\tau}_{j^{m}}^{b^{m}}=$
(25) $\displaystyle
B_{i^{m}}^{a^{m}}-i(\delta_{b^{m}}^{a^{m}}D_{i^{m}}^{j^{m}}-D_{b^{m}}^{a^{m}}\delta_{i^{m}}^{j^{m}})X_{j^{m}}^{b^{m}}-i\sum_{n}\frac{1}{2}A^{a^{m}j^{n}}_{i^{m}b^{n}}(X^{b^{n}}_{j^{n}})^{*}\,.$
Interestingly, TD-OCCT1 is mathematically equivalent to spilt OATDCC (sOATDCC,
or equivalently, sTDMVCC)Højlund and Christiansen (2024), which uses
orthogonal virtual orbitals and bi-orthogonal active orbitals without single
CC amplitudes and is reported in the vibrational systems and electron dynamics
recently. The biorthogonal orbitlas in sOATDCC have the same role as
$\hat{T}_{1}$ in the TD-OCCT1, however, they will give different
$X_{i^{m}}^{j^{m}}$ and $X_{a^{m}}^{b^{m}}$ constraints, which might give
different numerical performances. For more details, see Appendix. C. When
virtual orbitals do not exist, TD-OCCT1 and sOATDCC are equivalent to
TDMVCCMadsen _et al._ (2020c) and OATDCCKvaal (2012) (NOCC)Pedersen,
Fernández, and Koch (2001) for vibrational systems and electron dynamics,
respectively.
#### II.3.3 TD-BCC
Setting ${\tau}^{a^{m}}_{i^{m}}\equiv 0$ in Eq. (9,13),
$\displaystyle iX_{i^{m}}^{a^{m}}+\sum_{n}i$
$\displaystyle\langle\Phi_{i^{m}}^{a^{m}}|e^{-\hat{T}}\hat{E}^{j^{n}}_{b^{n}}|\Psi_{\rm
R}\rangle
X^{j^{n}}_{b^{n}}=\langle\Phi_{i^{m}}^{a^{m}}|e^{-\hat{T}}\hat{H}|\Psi_{\rm
R}\rangle\,,$ (26) $\displaystyle
i\frac{1}{2}\left\\{\sum_{n}\langle\Phi_{j^{n}}^{b^{n}}|e^{-\hat{T}}\hat{E}^{i^{m}}_{a^{m}}|\Psi_{\rm
R}\rangle\dot{\lambda}^{j^{n}}_{b^{n}}+(\dot{\lambda}^{i^{m}}_{a^{m}})^{*}\right\\}=$
$\displaystyle
B_{i^{m}}^{a^{m}}-i(\delta_{b^{n}}^{a^{m}}D_{i^{m}}^{j^{n}}-D_{b^{n}}^{a^{m}}\delta_{i^{m}}^{j^{n}})X_{j^{n}}^{b^{n}}-i\sum_{n}\frac{1}{2}A^{a^{m}j^{n}}_{i^{m}b^{n}}(X^{b^{n}}_{j^{n}})^{*}\,.$
(27)
#### II.3.4 TD-OCCX0
TD-OCCX0 can be considered as the most straightforward extension of TD-CC when
virtual orbitals are considered, due to the similarity of the parameterization
of the CC amplitudes parts and the absence of particle-hole orbital rotations.
The EoMs of the single amplitudes of TD-OCCX0 are given by Eq. (9,11). The
nonvariance of particle-hole orbital rotations makes the method not fully
variational. Therefore, the Ehrenfest theorem should be modifiedSato _et al._
(2016) even when frozen-cores are excluded,
$\displaystyle 2i\frac{d}{dt}\braket{\hat{O}}$
$\displaystyle=\braket{\Psi_{\rm L}}{[\hat{O},\hat{H}]}{\Psi_{\rm
R}}+\braket{\Psi_{\rm R}}{[\hat{O},\hat{H}]}{\Psi_{\rm L}}$ (28)
$\displaystyle+\sum_{m}O_{t^{m}}^{u^{m}}\\{\braket{\Psi_{\rm
L}}{(\hat{H}-i\hat{X})e^{\hat{T}}\bar{\hat{\Pi}}e^{-\hat{T}}\hat{E}_{t^{m}}^{u^{m}}}{\Psi_{\rm
R}}$ $\displaystyle-\braket{\Psi_{\rm
L}}{\hat{E}_{t^{m}}^{u^{m}}e^{\hat{T}}\bar{\hat{\Pi}}e^{-\hat{T}}(\hat{H}-i\hat{X})}{\Psi_{\rm
R}}$ $\displaystyle+\braket{\Psi_{\rm
R}}{(\hat{H}-i\hat{X})e^{\hat{T}}\bar{\hat{\Pi}}e^{-\hat{T}}\hat{E}_{t^{m}}^{u^{m}}}{\Psi_{\rm
L}}$ $\displaystyle-\braket{\Psi_{\rm
R}}{\hat{E}_{t^{m}}^{u^{m}}e^{\hat{T}}\bar{\hat{\Pi}}e^{-\hat{T}}(\hat{H}-i\hat{X})}{\Psi_{\rm
L}}\\}\,,$
where $\bar{\hat{\Pi}}=\hat{I}-\hat{\Pi}$. Similarly, TD-OCCH is not fully
variational either when TD-OCCX0-type parameterization is considered, and its
modified Ehrenfest theorem (as well as frozen-cores including scenario) will
be presented in the Appendix. D. As a comparison, Ehrenfest theorem holds for
other three methods since they are fully variational.
When frozen orbitals are included, the modified Ehrenfest theoremsSato _et
al._ (2016) are necessary for all five methods: two additional terms that
replace $t^{m}$, $u^{m}$ with $i^{\prime\prime m}$, $\mu^{m}$ and $\mu^{m}$,
$i^{\prime\prime m}$ in the last term of Eq. (28) should be added in case of
TD-OCCX0, and replace $t^{m}$, $u^{m}$ with $i^{\prime\prime m}$, $\mu^{m}$
and $\mu^{m}$, $i^{\prime\prime m}$ in the last term of Eq. (28) in case of
TD-BCC, TD-OCC, and TD-OCCT1. The modified Ehrenfest theorem of TD-OCCH with
frozen-cores will be reported in the Appendix. D.
### II.4 Discussion
One of the major problems of TD-CC is that the growth of CC amplitudes could
cause numerical instabilityPedersen and Kvaal (2019); Kristiansen _et al._
(2020), which can be mitigated by rotations of reference determinants that
could compensate for excitations and reduce the amplitude norms $\tau$ and
$\lambda$ in all five methods considered in this article. However, one must be
careful when using TD-OCCX0 since such a compensation might not be huge when
virtual orbital subspaces are relatively small due to the absence of particle-
hole orbital rotations.
Although the absence of particle-hole orbital rotations in TD-OCCX0 poses
inconveniences for observable evaluations and numerical stability, it does
save computing resources, especially for vibrational systems. Inverse matrix
operations required to solve $X_{i^{m}}^{a^{m}}$ in other methods are
unnecessary for TD-OCCX0. Comparable methods in this regard are TD-OCCT1, in
which $X_{i^{m}}^{a^{m}}$ [Eq. (24)] and $\tau_{i^{m}}^{a^{m}}$ [Eq. (25)] of
different modes do not couple with each other, and TD-OCCH with few fermion
kinds using BCC and OCC types parameterization. For the details, see the
Appendix. D. In both two methods, only several small-dimensional matrix
inverse operations are required, which is much cheaper than TD-OCC [Eq. (23)]
and TD-BCC [Eq. (26,II.3.3)] cases, requiring huge-dimensional matrix inverse
operations. However, when strong external fields exist, the advantage of TD-
OCCX0 and TD-OCCH can be dismissed due to the unfavorable resource cost of the
evaluations of position matrix elementsSato _et al._ (2016); Sato, Teramura,
and Ishikawa (2018), which will be discussed in Sec. III.1 and Appendix. D.
## III Applications
In this section, we will discuss three major applications of our methods that
cover most of chemical systems: electron dynamics, nuclei dynamics, and
nuclei-electron coupling dynamics.
### III.1 Electron dynamics in a strong laser field
For a many-electron system in external semi-classical electromagnetic fields,
the Hamiltonian in the second quantization can be expressed as
$\displaystyle\quad\quad\quad\quad\hat{H}(t)=h^{\mu}_{\nu}(t)\hat{E}^{\mu}_{\nu}+\frac{1}{2}g^{\mu\gamma}_{\nu\delta}\hat{E}^{\mu\gamma}_{\nu\delta},$
(29) $\displaystyle\quad\quad\quad h^{\mu}_{\nu}(t)=\int
dx\psi^{*}_{\mu}[h_{0}+V_{\rm ext}(t)]\psi_{\nu},$ (30) $\displaystyle
g^{\mu\gamma}_{\nu\delta}=\int\int
dx_{1}dx_{2}\psi^{*}_{\mu}(x_{1})\psi^{*}_{\gamma}(x_{2})r^{-1}_{12}\psi_{\nu}(x_{1})\psi_{\delta}(x_{2}),$
(31)
where $x\\!=\\!(\boldsymbol{r},\sigma)$ labels spatial-spin coordinate. Here,
we drop off the kind lable for the simplicity (although one can treat
different spin component as different kinds and use group excitation
truncation scheme to save the computational resources). $V_{\rm ext}$ is the
electron-laser interaction term. $h_{0}$ is the one-electron field free
Hamiltonian that contains the kinetic term and Coulomb’s terms induced by
fixed nuclei. Under the framework of the electric dipole approximation,
$V_{\rm ext}=\boldsymbol{E}(t)\cdot\boldsymbol{r}$ in the length gauge (LG) or
$V_{\rm ext}=-i\boldsymbol{A}(t)\cdot\boldsymbol{\nabla}_{\boldsymbol{r}}$ in
the velocity gauge (VG), with $\boldsymbol{E}(t)$ and
$\boldsymbol{A}(t)=-\int^{t}dt^{\prime}\boldsymbol{E}(t^{\prime})$ being the
electric field and vector potential, respectively. In principle, two gauges
are equivalent in the basis-set limit (basis numbers tend to the infinity),
but inequivalent in practical numerical simulations due to different
convergence speeds of basis-sets. Empirically, VG is more suitable for ultra
strong fields scenario while LG is more suitable for other casesHan and Madsen
(2010).
When methods are fully variational, the gauge invariance of variational
methods are satisfied automatically. Otherwise, the non-variance part of
orbital rotations should be adjusted to ensure the gauge invariance. For the
methods considered in this article, FC-part of $\hat{X}$ need to be selected
as
$X_{\mu}^{i^{\prime\prime}}=i\boldsymbol{E}(t)\cdot\braket{\psi_{i^{\prime\prime}}}{\boldsymbol{r}}{\psi_{\mu}}\,,$
(32)
and the particle-hole orbital rotations of TD-OCCX0 should be
$X_{i}^{a}=i\boldsymbol{E}(t)\cdot\braket{\psi_{a}}{\boldsymbol{r}}{\psi_{i}}\,.$
(33)
in the VGSato _et al._ (2016). These adjustments are also necessary in the
many-kind fermion systems, and the kind label should be explicitly included.
Specifically, Eq. (33) should be applied in all kinds with TD-OCCX0 type
parameterization in TD-OCCH method, see also Appendix. D.
### III.2 Vibrational systems
The application of methods to vibrational problems is straightforward. A
vibrational system with $m$ vibrational modes is isomorphic to a $m$-specie
fermion system with only one fermion occupation in each specie. We point out
that quantum photons, which are essential for molecules in cavities, can also
be described in the same framework.
Previous works of TDVCC familiesHøjlund, Zoccante, and Christiansen (2024);
Højlund and Christiansen (2024); Højlund _et al._ (2022); Hansen _et al._
(2019); Madsen _et al._ (2020c) focus mainly on the $[k]$-truncation scheme,
which is sufficient for ordinary systems. Nonetheless, for the systems with
double well potentials and vibronic coupling systems (see next subsection),
weighted excitation truncation scheme or group excitation truncation scheme is
necessary for the faithful simulations since the degree of freedom (DOF) with
double well potentials has more contribution than other DOFs to the dynamics.
A practical strategy of weighted excitation truncation scheme for such a
system is to set the DOF with double well potentials as 0 and other DOFs as 1.
Currently, special truncation schemes have been successfully implemented in
TDVCCMadsen _et al._ (2020d) and TDMVCCJensen _et al._ (2023); TDH , but
lacking theoretical discussion on orbital rotation invariance. The proof of
the intragroup rotations of particle and hole orbitals in the Appendix. B also
holds for the biorthogonal orbitals scenario. Therefore, we show that weighted
excitation truncation scheme can be safely used in TDVCC families without
compromising the intragroup rotations of particle and hole orbitals, which
benefits future numerical implementations. Despite existing TDVCC families and
weighted excitation truncation schemes, our work provides new ansatz, TD-
OCCT1, TD-BCC, TD-OCCX0, and TD-OCCH, and the group excitation truncation
scheme as new candidates for future implementations. Additionally, although
TD-OCCT1 and sTDMVCC are mathematically equivalent, different constraints
$X_{i^{m}}^{j^{m}}$ and $X_{a^{m}}^{b^{m}}$, as well as using orthogonal or
nonorthogonal orbitals, might give difference performances in practical
implementations. Therefore, they should be benchmarked systematically.
### III.3 Vibronic coupling systems
There are two ways to simulate vibronic coupling systems. Traditionally, the
electronic DOF is described as a $S$-level system, thus a vibronic coupling
system is isomorphic to a vibrational system. The major differences are: 1.
almost always all $S$ orbitals of the electronic DOF are active in the
wavefunction ansatz (no virtual orbitals); 2. vibrational contributions to the
dynamics associated with different electronic levels are comparableWorth and
Cederbaum (2004); Köuppel, Domcke, and Cederbaum (1984); Bao, Raymond, and
Nooijen (2024). These make TD-OCCX0 and TD-OCC unsuitable for the vibronic
coupling simulations, and one should also avoid using OCCX0 and OCC types
parameterization for the electronic DOF in TD-OCCH. Additionally, the same
weighted excitation truncation scheme as systems with the double well
potentials (proposed in the last subsection) should be used.
In recent years, treating electronic and nuclear DOF on an equal footing has
attracted increasing attention due to advancements in theorySibaev _et al._
(2020); Muolo _et al._ (2020); Sasmal and Vendrell (2020, 2022); Sasmal,
Schröder, and Vendrell (2024). The five CC ansatz and two special truncation
schemes proposed in this article can be regarded as the prototype of future
advanced CC simulations.
## IV Summary
We have successfully formulated five new orbital-optimized time-dependent
coupled-cluster methods, which are distinguished by whether single amplitudes,
$\tau_{i^{m}}^{a^{m}}$, $\lambda_{a^{m}}^{i^{m}}$ and particle-hole orbital
rotations $X_{i^{m}}^{a^{m}}$ are constraints, for arbitrary fermion mixtures.
When one of $\tau_{i^{m}}^{a^{m}}$, $\lambda_{a^{m}}^{i^{m}}$, and
$X_{i^{m}}^{a^{m}}$ is a constraint for each fermion kind $m$, the method can
converge to the CASSCF. Two advanced truncation schemes of higher-order
amplitudes which maintain the intragroup rotation invariance are introduced.
The applications to electronic dynamics, vibrational dynamics, and non-
adiabatic dynamics are also discussed. Our methods are more compact CC-
parameterization alternatives of the CI-parameterization of TD-MCSCF method,
and would shed light on the high-accuracy numerical simulations of large
systems. The numerical implementations on the electronic dynamics will be
presented in a forthcoming article.
###### Acknowledgements.
This research was supported in part by a Grant-in-Aid for Scientific Research
Grant Number JP22H05025 from the Ministry of Education, Culture, Sports,
Science and Technology (MEXT) of Japan. This research was also partially
supported by MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) Grant Number
JPMXS0118067246 and JST COI-NEXT Grant Number JPMJPF2221. This work is partly
supported by IBM-UTokyo lab.
## AUTHOR DECLARATIONS
### Conflict of Interest
The authors have no conflicts to disclose.
### Author Contributions
Haifeng Lang: Conceptualization (equal); Methodology (lead); Investigation
(lead); Writing – original draft (lead); Writing – review & editing (equal).
Takeshi Sato: Conceptualization (equal); Methodology (supporting); Funding
acquisition (lead); Writing –review & editing (equal).
## DATA AVAILABILITY
The data that supports the findings of this study are available within the
article.
## Appendix A Redundancy transfer of $X_{i^{m}}^{a^{m}}$,
$\tau_{i^{m}}^{a^{m}}$ and ${\lambda}^{i^{m}}_{a^{m}}$ in the full CC limit
In this section, we will demonstrate that the redundancy of
$X_{i^{m}}^{a^{m}}$ can be transferred to $\lambda^{i^{m}}_{a^{m}}$ or
$\tau_{i^{m}}^{a^{m}}$, and the preparation of the exact wavefunction without
$\lambda^{i^{m}}_{a^{m}}$ or $\tau_{i^{m}}^{a^{m}}$ is possible in the full CC
limit. The analysis on $\lambda^{i^{m}}_{a^{m}}$ and $\tau_{i^{m}}^{a^{m}}$
are extremely similar, for convenience, we will consider the case of all
$\lambda^{i^{m}}_{a^{m}}$ redundancy first and briefly discuss the most
general scenario.
Mathematically, the redundancy of $X_{i^{m}}^{a^{m}}$ means that the
coefficient of $\Delta_{i^{m}}^{a^{m}}$ is identically zero, i.e, Eq. (13) is
identiy regardless the choice of $X_{i^{m}}^{a^{m}}$. This can be rigorously
proved using the same method in the next section. Now we assume that the time-
dependent trajectories of $\lambda^{i^{m}}_{a^{m}}$ are arbitrarily given. Eq.
(11) becomes equations of $X_{i^{m}}^{a^{m}}$. If the solution of Eq. (11)
exists and is unique, Eq. (9,13) should hold due to the redundancy of
$X_{i^{m}}^{a^{m}}$. Further, if one treats Eq. (13) as equations of
$\dot{\tau}_{i^{m}}^{a^{m}}$ and assumes the solution exists and is unique,
the redundancy of $X_{i^{m}}^{a^{m}}$ has been successfully transferred to
$\lambda^{i^{m}}_{a^{m}}$. In general, it is very hard to prove the existence
and uniqueness of the solution even without constraint
$X_{a^{m}}^{b^{m}}=X_{i^{m}}^{j^{m}}\equiv 0$, but one can expect that the
statement is true, at least when the state is dominantly of single reference
characterMyhre (2018).
To completely exclude $\lambda^{i^{m}}_{a^{m}}$ in the time-dependent full CC
expansion, one needs to prepare the initial state with the parameterization
$\lambda^{i^{m}}_{a^{m}}(0)=0$ and use the constraint
$\dot{\lambda}^{i^{m}}_{a^{m}}\equiv 0$. Here, we provide two possible methods
to convert the parameterization of an arbitrary state $\ket{\Psi}$ into the
vanishing $\lambda^{i^{m}}_{a^{m}}$ form. The first is to prepare the initial
state as an arbitrary Slater determinant $\ket{\Phi_{0}}$, and let it evolve
from $t=0$ to $t=\pi/2$ under the Hamiltonian
$\hat{H}=-i(\ket{\Phi_{0}}\bra{\Psi}-\ket{\Psi}\bra{\Phi_{0}})$ with
$\dot{\lambda}^{i^{m}}_{a^{m}}\equiv 0$. The second is to let the state
$\ket{\Psi}$ in an arbitrary parameterization form evolve from $t=0$ to $t=1$
under the Hamiltonian $\hat{H}=0$ with
$\dot{\lambda}^{i^{m}}_{a^{m}}\equiv-\lambda^{i^{m}}_{a^{m}}(0)$.
Now we are in the position to give the most general redundancy analysis of
parameters. Assuming one of $X_{i^{m}}^{a^{m}}$, $\lambda^{i^{m}}_{a^{m}}$,
and $\tau_{i^{m}}^{a^{m}}$ is pre-determined for each kind of fermion,
equations corresponding to the variation coefficients in $\delta S$ should be
removed in coupled equations Eq. (9,11,13). If the solution of remaining
equations exists and is unique, the removing equations are identities and the
redundancy of parameters have been proved. To exclude these redundant
parameters in the time-dependent full CC expansion, one may follow the same
procedures as for the $\lambda^{i^{m}}_{a^{m}}$-redundancy scenario described
above, with the replacements with corresponding pre-determined variables
(constraints).
## Appendix B Redundancy proof of particle-particle and hole-hole orbital
rotations
We will explicitly show the redundancy of $X_{i^{m}}^{j^{m}}$ and
$X_{a^{m}}^{b^{m}}$ from $\delta S$. For the scenarios
$\mu^{m},\nu^{m}\in\\{i^{m}\\}$, and $\mu^{m},\nu^{m}\in\\{a^{m}\\}$,
$\displaystyle\langle\Psi_{\rm L}|[\hat{H}-i(\partial^{\hskip
0.20999pt\prime}_{t}\hat{T})-i\hat{X},\hat{E}^{\mu^{m}}_{\nu^{m}}]|\Psi_{\rm
R}\rangle$ $\displaystyle+$ $\displaystyle i\langle\Phi|(\partial^{\hskip
0.20999pt\prime}_{t}\hat{\Lambda})e^{-\hat{T}}\hat{E}^{\mu^{m}}_{\nu^{m}}|\Psi_{\rm
R}\rangle$ $\displaystyle=$ $\displaystyle\langle\Psi_{\rm
L}|[\hat{H}-i\hat{X},\hat{E}^{\mu^{m}}_{\nu^{m}}]|\Psi_{\rm R}\rangle$
$\displaystyle-$ $\displaystyle
i\dot{\tau}_{\mathring{I}}^{\mathring{A}}\langle\Psi_{\rm
L}|[\hat{E}_{\mathring{I}}^{\mathring{A}},\hat{E}^{\mu^{m}}_{\nu^{m}}]|\Psi_{\rm
R}\rangle+i\dot{\lambda}^{\mathring{I}}_{\mathring{A}}\langle\Phi_{\mathring{I}}^{\mathring{A}}|e^{-\hat{T}}\hat{E}^{\mu^{m}}_{\nu^{m}}|\Psi_{\rm
R}\rangle$ (34) $\displaystyle=$ $\displaystyle\braket{\Psi_{\rm
L}}{\hat{E}_{\mathring{I}}^{\mathring{A}}(\hat{H}-i\hat{X})}{\Psi_{\rm
R}}\langle\Phi_{\mathring{I}}^{\mathring{A}}|e^{-\hat{T}}\hat{E}^{\mu^{m}}_{\nu^{m}}|\Psi_{\rm
R}\rangle$ $\displaystyle-$
$\displaystyle\braket{\Phi_{\mathring{I}}^{\mathring{A}}}{e^{-\hat{T}}(\hat{H}-i\hat{X})}{\Psi_{\rm
R}}\langle\Psi_{\rm
L}|\hat{E}_{\mathring{I}}^{\mathring{A}}\hat{E}^{\mu^{m}}_{\nu^{m}}|\Psi_{\rm
R}\rangle\,,$ (35)
Here, time derivatives of all single amplitudes are included in Eq. (34).
Converting Eq. (34) to Eq. (35), one needs to replace all EoMs of CC
amplitudes with Eq. (8,9,10,11) and use Eq. (16). The equations are
straightforward for TD-OCCX0, and we will explain more on other methods. We
first consider TD-OCCT1. $\lambda_{a^{m}}^{i^{m}}$ in the ansatz should be
understood as a constraint $\lambda_{a^{m}}^{i^{m}}\equiv 0$. Therefore, Eq.
(34) holds. Noticing that the multiplier of $\dot{\tau}^{a^{m}}_{i^{m}}$ in
Eq. (34) vanishes, one can safely replace $\dot{\tau}^{a^{m}}_{i^{m}}$ with
Eq. (9). Additionally, Eq. (11) also ensures
$\dot{\lambda}_{a^{m}}^{i^{m}}\equiv 0$, and we have finished the proof of
equations for TD-OCCT1. For the discussion on TD-BCC, one just needs to swap
$\lambda_{a^{m}}^{i^{m}}$ and $\tau^{a^{m}}_{i^{m}}$ in the proof of TD-OCCT1.
For the TD-OCC method, both $\lambda_{a^{m}}^{i^{m}}$ and
$\tau^{a^{m}}_{i^{m}}$ should be regarded as constraints and their multipliers
in Eq. (34) are always zero, and thus equations hold. Additionally, equations
of TD-OCCH automatically hold when equations of four other methods hold.
Notice that
$\langle\Phi^{\mathring{B}}_{\mathring{J}}|\hat{E}_{\mathring{I}}^{\mathring{A}}$
is not zero iff $\mathring{A}$ and $\mathring{I}$ are substrings of
$\mathring{B}$ and $\mathring{J}$, respectively. For convenience, we use
${\mathring{B}}-{\mathring{A}}$ to express the absolute complementary string
of $\mathring{A}$ with respect to $\mathring{B}$. Expanding $\bra{\Psi_{\rm
L}}$ in Eq. (35) as
$\lambda_{\mathring{B}}^{\mathring{J}}\bra{\Phi^{\mathring{B}}_{\mathring{J}}}$,
Eq. (35) can be converted to
$\displaystyle\lambda_{\mathring{B}}^{\mathring{J}}$
$\displaystyle\big{(}\bra{\Phi^{\mathring{B}-\mathring{A}}_{\mathring{J}-\mathring{I}}}(\hat{H}-i\hat{X})\ket{\Psi_{\rm
R}}{\bra{\Phi^{\mathring{A}}_{\mathring{I}}}}e^{-\hat{T}}\hat{E}_{\nu^{m}}^{\mu^{m}}\ket{\Psi_{\rm
R}}$ (36)
$\displaystyle-{\bra{\Phi^{\mathring{A}}_{\mathring{I}}}}(\hat{H}-i\hat{X})\ket{\Psi_{\rm
R}}\bra{\Phi^{\mathring{B}-\mathring{A}}_{\mathring{J}-\mathring{I}}}e^{-\hat{T}}\hat{E}_{\nu^{m}}^{\mu^{m}}\ket{\Psi_{\rm
R}}\big{)}\,.$
Resuming the second term by swapping ${\mathring{B}}-{\mathring{A}}$,
${\mathring{J}}-{\mathring{I}}$ and ${\mathring{A}}$, ${\mathring{I}}$, one
can immediately obtain that Eq. (35) is zero, which completes the proof that
$X_{i^{m}}^{j^{n}}$ and $X_{a^{m}}^{b^{n}}$ are redundant.
Following the same discussion, one can also explicitly show that $\hat{X}$ is
redundant for TD-ooCC with $\hat{T}_{1}$ and $\hat{\Lambda}_{1}$ in the CASSCF
limit. In this case, it is easy to check that Eq. (34) and Eq. (35) also hold,
and applying the same resummation argument as the proof of redundancy of
$X_{i^{m}}^{j^{m}}$ and $X^{b^{m}}_{a^{m}}$, the proof of redundancy of
$X_{i^{m}}^{a^{m}}$ can be obtained.
## Appendix C OATDCC and OATDCC with split orthogonal and biorthogonal
orbitals
Before presenting the equivalence proof of sTDMVCCHøjlund and Christiansen
(2024) (sOATDCC) and TD-OCCT1, we will briefly review TDVCC families,
including TDVCCHansen _et al._ (2019), TDMVCC, restricted polar TDMVCC
(rpTDMVCC)Højlund _et al._ (2022), orthogonal TDMVCC (oTDMVCC)Højlund,
Zoccante, and Christiansen (2024), and sTDMVCCHøjlund and Christiansen (2024).
First, we will focus on the EoMs and parameterization of singlel amplitudes of
CC coefficients and orbitals. In TDVCC, all orbitals are active, and TD-OCCX0
is equivalent to TDVCC in this case. The fast growth of norms of CC amplitudes
arisen from the absence of orbital rotations makes applications of TDVCC
limited. It is the vibrational analog of TD-CC. To resolve this problem,
orbital rotations and virtual orbitals are introduced. All other four methods,
TDMVCC, rpTDMVCC, sTDMVCC and oTDMVCC have formally same (but not equivalent
due to orthogonal/biorthogonal orbitals) CC part parameterizations and EoMs,
and their major differences are the parameterization of orbitals and their
EoMs. oTDMVCC is completely identical to TD-OCCSato, Teramura, and Ishikawa
(2018) of this article, which uses orthogonal orbitals but suffers from the
huge matrix inversion and nonconvergence to CASSCF. Instead of using
orthogonal orbitals, TDMVCC, which is the vibrational analog of OATDCC, solves
the matrix inversion and convergence problems by using biorthogonal orbitals,
but the non-orthogonality causes numerical instabilities in long time
simulations. One can force active orbitals lie in the conjugate ket and bra
spaces to mitigate instabilities, which is implemented by using orthogonal
virtual orbitals and biorthogonal active orbitals in rpTDMVCC and sTDMVCC.
EoMs of active-virtual orbital rotations are determined by the variational
principle and projection in sTDMVCC and rpTDMVCC, respectively, while EoMs of
CC amplitudes and particle-hole rotations of two methods are completely
identical. In fact, the contribution of biorthogonal active orbitals is
equivalent to the one-rank excitation in TD-OCCT1, and the TDVP ensures the
equivalence of TD-OCCT1 and sTDMVCC.
For convenience, we only consider models without dynamical-core in this
Appendix. For models with dynamical-core, the treatments of dynamical-core
orbitals are almost identical to the virtual orbitals except that they should
be evolved explicitly. We will first review the standard OATDCC, which uses
birothogonal bra and ket orbitals rather than orthogonal orbitals in TD-OCC.
The highly flexible variational space choices could cause serious numerical
instabilities arisen from the different spaces spanned by ket and bra active
orbitals in practical implementationsHøjlund _et al._ (2022). Next, we will
review sOATDCC, i.e., OTADCC with virtual orbitals of ket and bra are
orthogonal in the Lagrangian and active orbitals are biorthogonal. Such a
parameterization forces the spaces spanned by active orbitals of ket and bra
are identical, which eliminates the numerical instability. The modifications
only appear in the intergroup rotation between active orbitals and other types
of orbitals. OATDCC and sOATDCC reviewed in this Appendix are describing
fermion mixtures with arbitrary occupations, which are more general than the
references results. When sOATDCC is applied to the vibrational problem, it is
the sTDMVCC and inherits the advantage of TDMVCCMadsen _et al._ (2020c);
Højlund, Zoccante, and Christiansen (2024): EoMs of different vibrational
modes are separate and converge to the CASSCF. In contrast to rpTDMVCCHøjlund
_et al._ (2022), this approach is fully variational. We further prove that
this approach is equivalent to TD-OCCT1.
We define $\ket{\phi_{\mu^{m}}}$ and $\bra{\tilde{\phi}_{\mu^{m}}}$ as
biorthogonal bases of OATDCC, and their corresponding creation (annihilation)
operators are denoted as $\hat{a}_{\mu^{m}}^{\dagger}$ ($\hat{a}_{\mu^{m}}$)
and $\tilde{\hat{a}}_{\mu^{m}}^{\dagger}$ ($\tilde{\hat{a}}_{\mu^{m}}$). They
satisfy the following relations,
$\displaystyle\braket{\tilde{\phi}_{\mu^{m}}}{\phi_{\nu^{m}}}=\delta_{\mu^{m}\nu^{m}}\,,$
(37)
$\displaystyle\\{\hat{a}_{\nu^{m}}^{\dagger},\tilde{\hat{a}}_{\mu^{m}}\\}=\braket{\tilde{\phi}_{\mu^{m}}}{\phi_{\nu^{m}}}=\delta_{\mu^{m}\nu^{m}}\,.$
(38)
In the standard OATDCC, $\hat{a}_{\mu^{m}}$ and
$\tilde{\hat{a}}_{\mu^{m}}^{\dagger}$ are never used, nonetheless, they play
important roles in the sOATDCC. We also introduce the operator
$\tilde{\hat{E}}^{\mu_{1}^{m_{1}}\mu_{2}^{m_{2}}\mu_{3}^{m_{3}}\cdots}_{\nu_{1}^{m_{1}}\nu_{2}^{m_{2}}\nu_{3}^{m_{3}}\cdots}=\hat{a}^{\dagger}_{\mu_{1}^{m_{1}}}\hat{a}^{\dagger}_{\mu_{2}^{m_{2}}}\hat{a}^{\dagger}_{\mu_{3}^{m_{3}}}\cdots\tilde{\hat{a}}_{\nu_{3}^{m_{3}}}\tilde{\hat{a}}_{\nu_{2}^{m_{2}}}\tilde{\hat{a}}_{\nu_{1}^{m_{1}}}$,
and all other quantities in OATDCC are denoted as corresponding quantities in
TD-OCC with tildes, similarly. $m_{1},m_{2},\cdots,m_{3}$ can be the same
index of fermion kind. The ket and bra reference wavefunctions of OATDCC are
tensor products Slater determinants of $\ket{\phi_{\mu^{m}}}$ and
$\bra{\tilde{\phi}_{\mu^{m}}}$ for all $m$, denoted as $\ket{{\Phi}}$ and
$\bra{\tilde{\Phi}}$, respectively. Wavefunction and their time derivatives
and variations as well as the Lagrangian in OATDCC are formally identical to
the TD-OCC correspondence with tildes replacements. Due to the loss of
orthogonality,
$(\tilde{\hat{E}}_{I}^{A})^{\dagger}\neq\tilde{\hat{E}}_{A}^{I}$,
$(\tilde{X}^{\mu^{m}}_{\nu^{m}})^{*}\neq\tilde{X}_{\mu^{m}}^{\nu^{m}}$, and
$(\tilde{\Delta}^{\mu^{m}}_{\nu^{m}})^{*}\neq\tilde{\Delta}_{\mu^{m}}^{\nu^{m}}$
in general. Also, the anti-Hermicity of $\tilde{\hat{X}}$ and
$\tilde{\hat{\Delta}}$ cannot be expected. Therefore, all variables and their
complex conjugate are independent quantities in OATDCC. The action of OATDCC
is a complex functional
$\displaystyle\tilde{S}$ $\displaystyle=$
$\displaystyle\int_{t_{0}}^{t_{1}}dt\tilde{L}\,,$ (39)
and require the action to be stationary
$\displaystyle{\color{black}}\delta\tilde{S}$ $\displaystyle=$
$\displaystyle\langle\delta\tilde{\Psi}_{\rm L}|\hat{H}|\tilde{\Psi}_{\rm
R}\rangle+\langle\tilde{\Psi}_{\rm L}|\hat{H}|\delta\tilde{\Psi}_{\rm
R}\rangle$ (40) $\displaystyle-$ $\displaystyle
i\langle\delta\tilde{\Psi}_{\rm L}|\dot{\tilde{\Psi}}_{\rm
R}\rangle+i\langle\dot{\tilde{\Psi}}_{\rm L}|\delta\tilde{\Psi}_{\rm
R}\rangle=0\,.$
The non-existence of complex conjugate variables in OATDCC action makes the
action complex analytic, and ensures the existence of the solution. The
complex even dimension of the variational manifold of OATDCC ensures
uniqueness of the solution. OTADCC converges to CASSCF when all excitation and
deexcitation operators except for the single excitation and deexcitation ones
are included. We also point out that the real action choice will lead the same
solutions due to the independence of all variables and their complex
conjugates.
Working out the variations explicitly black
$\displaystyle\delta\tilde{S}$ $\displaystyle=$
$\displaystyle\delta\tilde{\tau}^{A}_{I}\left\\{\langle\tilde{\Psi}_{\rm
L}|[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}^{A}_{I}]|\tilde{\Psi}_{\rm
R}\rangle+i\dot{\tilde{\lambda}}^{I}_{A}\right\\}+\delta\tilde{\lambda}^{I}_{A}\left\\{\langle\tilde{\Phi}^{A}_{I}|e^{-\tilde{\hat{T}}}(\hat{H}-i\tilde{\hat{X}})e^{\tilde{\hat{T}}}|\tilde{\Phi}\rangle-i\dot{\tilde{\tau}}^{A}_{I}\right\\}$
(41) $\displaystyle+$
$\displaystyle\tilde{\Delta}^{\mu^{m}}_{\nu^{m}}\left\\{\langle\tilde{\Psi}_{\rm
L}|[\hat{H}-i(\partial^{\hskip
0.20999pt\prime}_{t}\tilde{\hat{T}})-i\tilde{\hat{X}},\tilde{\hat{E}}^{\mu^{m}}_{\nu^{m}}]|\tilde{\Psi}_{\rm
R}\rangle+i\langle\tilde{\Phi}|(\partial^{\hskip
0.20999pt\prime}_{t}\tilde{\hat{\Lambda}})e^{-\tilde{\hat{T}}}\tilde{\hat{E}}^{\mu^{m}}_{\nu^{m}}e^{\tilde{\hat{T}}}|\tilde{\Phi}\rangle\right\\}\,,$
black EoMs of CC coefficients,
$\displaystyle i\dot{\tilde{\tau}}^{A}_{I}$ $\displaystyle=$
$\displaystyle\langle\tilde{\Phi}_{I}^{A}|e^{-\tilde{\hat{T}}}(\hat{H}-i\tilde{\hat{X}})e^{\tilde{\hat{T}}}|\tilde{\Phi}\rangle\,,$
(42) $\displaystyle-i\dot{\tilde{\lambda}}^{I}_{A}$ $\displaystyle=$
$\displaystyle\langle\tilde{\Psi}_{\rm
L}|[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}^{A}_{I}]|\tilde{\Psi}_{\rm
R}\rangle\,,$ (43)
and EoMs of orbital rotations,
$\displaystyle\bra{\tilde{\Psi}_{\rm
L}}[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}^{\alpha^{m}}_{p^{m}}]\ket{\tilde{\Psi}_{\rm
R}}=0\,,$ (44) $\displaystyle\bra{\tilde{\Psi}_{\rm
L}}[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}_{\alpha^{m}}^{p^{m}}]\ket{\tilde{\Psi}_{\rm
R}}=0\,,$ (45) $\displaystyle\bra{\tilde{\Psi}_{\rm
L}}[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}_{i^{m}}^{a^{m}}]\ket{\tilde{\Psi}_{\rm
R}}=0\,,$ (46) $\displaystyle\langle\tilde{\Psi}_{\rm
L}|[\hat{H}-i(\partial^{\hskip
0.20999pt\prime}_{t}\tilde{\hat{T}})-i\tilde{\hat{X}},\tilde{\hat{E}}_{a^{m}}^{i^{m}}]|\tilde{\Psi}_{\rm
R}\rangle$ $\displaystyle+$ $\displaystyle
i\langle\tilde{\Phi}|(\partial^{\hskip
0.20999pt\prime}_{t}\tilde{\hat{\Lambda}})e^{-\tilde{\hat{T}}}\tilde{\hat{E}}_{a^{m}}^{i^{m}}e^{\tilde{\hat{T}}}|\tilde{\Phi}\rangle=0\,,$
(47)
can be obtained. Similar to the TD-OCC scenario, all intragroup rotations are
redundant, thus should be pre-selected, for instance, setting all the
redundant rotations as zero.
In principle, $\\{\ket{\phi_{p^{m}}}\\}$ and
$\\{\ket{\tilde{\phi}_{p^{m}}}\\}$ can span different spaces, which can make
the propagation unstable. To resolve this problem, we adapt real action with
the constraints that virtual orbitals are orthogonal,
$\displaystyle S_{s}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\int_{t_{0}}^{t_{1}}dt\left(\tilde{L}+\tilde{L}^{*}\right),$
(48)
We still use the notations of OATDCC in sOATDCC when there is no ambiguity. In
this case, corresponding ket and bra virtual orbitals are identical,
$\ket{\phi_{\alpha^{m}}}^{*}=\bra{\tilde{\phi}_{\alpha^{m}}}$,
$\hat{a}_{\alpha^{m}}=\tilde{\hat{a}}_{\alpha^{m}}$ and
$\hat{a}_{\alpha^{m}}^{\dagger}=\tilde{\hat{a}}_{\alpha^{m}}^{\dagger}$.
Therefore, $\\{\ket{\phi_{p^{m}}}\\}$ and $\\{\ket{\tilde{\phi}_{p^{m}}}\\}$
span the same space, and the following relations hold,
$\displaystyle\ket{\tilde{\phi}_{p^{m}}}=\braket{\tilde{\phi}_{q^{m}}}{\tilde{\phi}_{p^{m}}}\ket{\phi_{q^{m}}}\,,\quad\tilde{\hat{a}}_{p^{m}}^{\dagger}=\braket{\tilde{\phi}_{q^{m}}}{\tilde{\phi}_{p^{m}}}\hat{a}_{q^{m}}^{\dagger}\,,$
(49)
$\displaystyle\bra{\phi_{p^{m}}}=\bra{\tilde{\phi}_{q^{m}}}\braket{\phi_{p^{m}}}{\phi_{q^{m}}}\,,\quad\hat{a}_{p^{m}}=\braket{\phi_{p^{m}}}{\phi_{q^{m}}}\tilde{\hat{a}}_{q^{m}}\,.$
The constraints also force
$\ket{\delta\phi_{\alpha^{m}}}\equiv\ket{\delta\tilde{\phi}_{\alpha^{m}}}$ and
$\ket{\dot{\phi}_{\alpha^{m}}}\equiv\ket{\dot{\tilde{\phi}}_{\alpha^{m}}}$.
Using above relations, one could find that
$(\tilde{X}^{\alpha^{m}}_{p^{m}})^{*}$ and $\tilde{X}_{\alpha^{m}}^{p^{m}}$,
as well as $(\tilde{\Delta}^{\alpha^{m}}_{q^{m}})^{*}$ and
$\tilde{\Delta}_{\alpha^{m}}^{q^{m}}$ are not independent,
$\displaystyle(\tilde{X}^{\alpha^{m}}_{p^{m}})^{*}=-\tilde{X}_{\alpha^{m}}^{q^{m}}\braket{\phi_{p^{m}}}{\phi_{q^{m}}}\,,$
(50)
$\displaystyle\tilde{X}_{\alpha^{m}}^{p^{m}}=-(\tilde{X}^{\alpha^{m}}_{q^{m}})^{*}\braket{\tilde{\phi}_{p^{m}}}{\tilde{\phi}_{q^{m}}}\,,$
(51)
$\displaystyle(\tilde{\Delta}^{\alpha^{m}}_{p^{m}})^{*}=-\tilde{\Delta}_{\alpha^{m}}^{q^{m}}\braket{\phi_{p^{m}}}{\phi_{q^{m}}}\,,$
(52)
$\displaystyle\tilde{\Delta}_{\alpha^{m}}^{p^{m}}=-(\tilde{\Delta}^{\alpha^{m}}_{q^{m}})^{*}\braket{\tilde{\phi}_{p^{m}}}{\tilde{\phi}_{q^{m}}}\,,$
(53)
$\displaystyle(\tilde{X}^{\alpha^{m}}_{p^{m}}\tilde{\hat{E}}^{\alpha^{m}}_{p^{m}})^{\dagger}=-\tilde{X}_{\alpha^{m}}^{p^{m}}\tilde{\hat{E}}_{\alpha^{m}}^{p^{m}}\,,$
(54)
$\displaystyle(\tilde{\Delta}^{\alpha^{m}}_{p^{m}}\tilde{\hat{E}}^{\alpha^{m}}_{p^{m}})^{\dagger}=-\tilde{\Delta}_{\alpha^{m}}^{p^{m}}\tilde{\hat{E}}_{\alpha^{m}}^{p^{m}}\,.$
(55)
Therefore, the explicit form of the variation with independent variational
variables is black
$\displaystyle 2\delta{S}_{s}$ $\displaystyle=$
$\displaystyle\delta\tilde{\tau}^{A}_{I}\left\\{\langle\tilde{\Psi}_{\rm
L}|[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}^{A}_{I}]|\tilde{\Psi}_{\rm
R}\rangle+i\dot{\tilde{\lambda}}^{I}_{A}\right\\}+\delta\tilde{\lambda}^{I}_{A}\left\\{\langle\tilde{\Phi}^{A}_{I}|e^{-\tilde{\hat{T}}}(\hat{H}-i\tilde{\hat{X}})e^{\tilde{\hat{T}}}|\tilde{\Phi}\rangle-i\dot{\tilde{\tau}}^{A}_{I}\right\\}+c.c.$
(56) $\displaystyle+$
$\displaystyle\tilde{\Delta}^{p^{m}}_{q^{m}}\left\\{\langle\tilde{\Psi}_{\rm
L}|[\hat{H}-i(\partial^{\hskip
0.20999pt\prime}_{t}\tilde{\hat{T}})-i\tilde{\hat{X}},\tilde{\hat{E}}^{p^{m}}_{q^{m}}]|\tilde{\Psi}_{\rm
R}\rangle+i\langle\tilde{\Phi}|(\partial^{\hskip
0.20999pt\prime}_{t}\tilde{\hat{\Lambda}})e^{-\tilde{\hat{T}}}\tilde{\hat{E}}^{p^{m}}_{q^{m}}e^{\tilde{\hat{T}}}|\tilde{\Phi}\rangle\right\\}\,+c.c.$
$\displaystyle+$
$\displaystyle\tilde{\Delta}^{p^{m}}_{\alpha^{m}}\left\\{\langle\tilde{\Psi}_{\rm
L}|[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}^{p^{m}}_{\alpha^{m}}]|\tilde{\Psi}_{\rm
R}\rangle-\braket{\phi_{q^{m}}}{\phi_{p^{m}}}\langle\tilde{\Psi}_{\rm
L}|[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}_{q^{m}}^{\alpha^{m}}]|\tilde{\Psi}_{\rm
R}\rangle^{*}\right\\}$ $\displaystyle+$
$\displaystyle\tilde{\Delta}_{p^{m}}^{\alpha^{m}}\left\\{\langle\tilde{\Psi}_{\rm
L}|[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}_{p^{m}}^{\alpha^{m}}]|\tilde{\Psi}_{\rm
R}\rangle-\braket{\tilde{\phi}_{p^{m}}}{\tilde{\phi}_{q^{m}}}\langle\tilde{\Psi}_{\rm
L}|[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}^{q^{m}}_{\alpha^{m}}]|\tilde{\Psi}_{\rm
R}\rangle^{*}\right\\}\,.$
Compared with Eq. (41), it is clear that the constraints do not change the
EoMs of coefficients and intergroup orbital rotations between particle and
hole orbitals. The variations with respect to $\Delta_{\alpha^{m}}^{p^{m}}$
give EoMs of intergroup rotation between active orbitals and virtual orbitals
$\displaystyle\bra{\tilde{\Psi}_{\rm
L}}[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}_{\alpha^{m}}^{p^{m}}]\ket{\tilde{\Psi}_{\rm
R}}+\bra{\tilde{\Psi}_{\rm
R}}[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}_{\alpha^{m}}^{p^{m}}]\ket{\tilde{\Psi}_{\rm
L}}=0\,,$ (57) $\displaystyle\bra{\tilde{\Psi}_{\rm
L}}[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}^{\alpha^{m}}_{p^{m}}]\ket{\tilde{\Psi}_{\rm
R}}+\bra{\tilde{\Psi}_{\rm
R}}[\hat{H}-i\tilde{\hat{X}},\tilde{\hat{E}}^{\alpha^{m}}_{p^{m}}]\ket{\tilde{\Psi}_{\rm
L}}=0\,.$ (58)
sOATDCC is equivalent to TD-OCCT1. To see this, one just needs to prove their
manifolds are identical up to normalization factors, which corresponds to a
point transformation in Lagrange MechanicsGoldstein (1980). More explicitly
speaking, for any states $\ket{\tilde{\Psi}_{\rm R}}$ and
$\bra{\tilde{\Psi}_{\rm L}}$ parameterized via the sOATDCC formalism, one can
always find states parameterized via the TD-OCCT1 formalism $\ket{\Psi_{\rm
R}}$ and $\bra{\Psi_{\rm L}}$ that satisfy $\ket{\tilde{\Psi}_{\rm
R}}=A\ket{\Psi_{\rm R}}$, and $\bra{\tilde{\Psi}_{\rm L}}=A^{-1}\bra{\Psi_{\rm
L}}$ and vice versa. Here, $A$ is a normalization factor and only gives a
trivial total derivative in Lagrange. In both transformations, one can always
choose $\ket{\psi_{\alpha}}\equiv\ket{\phi_{\alpha}}$ and they only appear in
the time derivatives of wavefunctions.
We will first consider converting TD-OCCT1 to sOATDCC. The transformation is
given by
$\displaystyle\ket{\phi_{i^{m}}}=\ket{\psi_{i^{m}}}+\tau_{i^{m}}^{a^{m}}\ket{\psi_{a^{m}}}\,,\quad\hat{a}^{\dagger}_{i^{m}}=\hat{c}_{i^{m}}^{\dagger}+\tau_{i^{m}}^{a^{m}}\hat{c}^{\dagger}_{a^{m}}\,,$
(59)
$\displaystyle\bra{\tilde{\phi}_{a^{m}}}=\bra{\psi_{a^{m}}}-\tau_{i^{m}}^{a^{m}}\bra{\psi_{i^{m}}}\,,\quad\tilde{\hat{a}}_{a^{m}}=\hat{c}_{a^{m}}-\tau_{i^{m}}^{a^{m}}\hat{c}_{i^{m}}\,,$
(60)
$\displaystyle\ket{\phi_{a^{m}}}=\ket{\psi_{a^{m}}}\,,\quad\hat{a}^{\dagger}_{a^{m}}=\hat{c}_{a^{m}}^{\dagger}\,,$
(61)
$\displaystyle\bra{\tilde{\phi}_{i^{m}}}=\bra{\psi_{i^{m}}}\,,\quad\tilde{\hat{a}}_{i^{m}}=\hat{c}_{i^{m}}\,,$
(62)
$\displaystyle\tilde{\tau}_{I}^{A}=\tau_{I}^{A}\,,\quad\tilde{\lambda}_{A}^{I}=\lambda_{A}^{I}\,,\quad\ket{\tilde{\Psi}_{\rm
R}}=\ket{\Psi_{\rm R}}\,,\quad\bra{\tilde{\Psi}_{\rm L}}=\bra{\Psi_{\rm
L}}\,.$ (63)
These identities are ensured by the invariance of particle-hole orbital
rotation in OATDCC parameterization:
$\displaystyle\hat{a}_{p^{m}}^{\dagger}\rightarrow
e^{\tilde{\hat{T}}_{1}}\hat{a}_{p^{m}}^{\dagger}e^{-\tilde{\hat{T}}_{1}}\,,\quad\tilde{\hat{a}}_{p^{m}}\rightarrow
e^{\tilde{\hat{T}}_{1}}\tilde{\hat{a}}_{p^{m}}e^{-\tilde{\hat{T}}_{1}}\,,$
(64)
$\displaystyle\bra{\tilde{\Phi}}\rightarrow\bra{\tilde{\Phi}}e^{-\tilde{\hat{T}}_{1}}=\bra{\tilde{\Phi}}\,,\quad\ket{{\Phi}}\rightarrow
e^{\tilde{\hat{T}}_{1}}\ket{{\Phi}}\,,$ $\displaystyle\bra{\tilde{\Psi}_{\rm
L}}\rightarrow\bra{\tilde{\Phi}}(1+e^{\tilde{\hat{T}}_{1}}\tilde{\hat{\Lambda}}e^{-\tilde{\hat{T}}_{1}})e^{-\tilde{\hat{T}}+\tilde{\hat{T}}_{1}}=\bra{\tilde{\Psi}_{\rm
L}}\,,$ $\displaystyle\ket{\tilde{\Psi}_{\rm R}}\rightarrow
e^{\tilde{\hat{T}}-\tilde{\hat{T}}_{1}}e^{\tilde{\hat{T}}_{1}}\ket{\tilde{\Phi}}=\ket{\tilde{\Psi}_{\rm
R}}\,.$
where
$\tilde{\hat{T}}_{1}=\tilde{\tau}_{i^{m}}^{a^{m}}\tilde{\hat{E}}_{i^{m}}^{a^{m}}$
is an arbitrary single excitation operator. Such a rotation invariance causes
the $\tilde{\hat{T}}_{1}$ redundancy in OATDCC and has been extensively
discussed in the supplementary material of the original OATDCC paperKvaal
(2012).
To convert sOATDCC wavefunctions to TD-OCCT1 correspondence needs two steps.
The first step is to orthonormalize $\ket{\phi_{a^{m}}}$ and
$\bra{\tilde{\phi}_{i^{m}}}$,
$\ket{\phi_{a^{m}}^{\prime}}=G_{b^{m}}^{a^{m}}\ket{\phi_{b^{m}}}\,,\quad\bra{\tilde{\phi}_{i^{m}}^{\prime}}=\tilde{G}_{j^{m}}^{i^{m}}\bra{\tilde{\phi}_{j^{m}}}\,,$
(65)
where $\ket{\phi_{a^{m}}^{\prime}}$ and $\bra{\tilde{\phi}_{i^{m}}^{\prime}}$
are orthonormalized vectors, $G$ and $\tilde{G}$ are transformation matrices.
We also denote quantities after the orthonormalization as original quantities
with primes. Dual vectors of $\ket{\phi_{a^{m}}^{\prime}}$ and
$\bra{\tilde{\phi}_{i^{m}}^{\prime}}$ are
$\ket{\phi_{i^{m}}^{\prime}}=(\tilde{G}^{-1})_{i^{m}}^{j^{m}}\ket{\phi_{j^{m}}}\,,\quad\bra{\tilde{\phi}_{a^{m}}^{\prime}}=({G}^{-1})_{a^{m}}^{b^{m}}\bra{\tilde{\phi}_{b^{m}}}\,.$
(66)
We also define general-rank transformations of $G$ and $\tilde{G}$ as
$G_{I}^{J}$ and $\tilde{G}_{A}^{B}$, which are the tensor products of single-
rank transformations of $G_{i^{m}}^{j^{m}}$ and $\tilde{G}_{a^{m}}^{b^{m}}$,
respectively. Wavefunctions as well as relevant quantities under such a
transformation are given by
$\displaystyle\tilde{\hat{E}}_{I}^{A}\rightarrow(\tilde{\hat{E}}_{I}^{A})^{\prime}=G^{A}_{B}\tilde{\hat{E}}_{J}^{B}\tilde{G}^{I}_{J}\,,$
(67)
$\displaystyle\tilde{\hat{E}}_{A}^{I}\rightarrow(\tilde{\hat{E}}_{A}^{I})^{\prime}=(\tilde{G}^{-1})_{I}^{J}\tilde{\hat{E}}_{B}^{J}({G}^{-1})_{A}^{B}\,,$
$\displaystyle\tilde{{\tau}}_{I}^{A}\rightarrow(\tilde{\tau}_{I}^{A})^{\prime}=(G^{-1})_{A}^{B}\tilde{\tau}_{J}^{B}(\tilde{G}^{-1})_{I}^{J}\,,$
$\displaystyle\tilde{{\lambda}}_{A}^{I}\rightarrow(\tilde{\lambda}_{A}^{I})^{\prime}=\tilde{G}^{I}_{J}\tilde{\lambda}_{B}^{J}{G}_{B}^{A}\,,$
$\displaystyle\ket{\tilde{\Psi}_{\rm R}}\rightarrow\ket{\tilde{\Psi}_{\rm
R}^{\prime}}=\prod_{m}(\det{(\tilde{G}_{j^{m}}^{i^{m}})})^{-1}\ket{\tilde{\Psi}_{\rm
R}}\,,$ $\displaystyle\bra{\tilde{\Psi}_{\rm
L}}\rightarrow\bra{\tilde{\Psi}_{\rm
L}^{\prime}}=\prod_{m}\det{(\tilde{G}_{j^{m}}^{i^{m}})}\bra{\tilde{\Psi}_{\rm
L}}\,.$
In fact, if one replaces single-rank $G$ and $\tilde{G}$ with arbitrary
invertible matrices, the above transformation invariance still holds and that
is the reason of the redundancy of $\tilde{X}_{i^{m}}^{j^{m}}$ and
$\tilde{X}_{a^{m}}^{b^{m}}$ Kvaal (2012).
The second step is to express $\ket{{\Psi}_{\rm R}}$ and $\bra{{\Psi}_{\rm
L}}$ via $\ket{\tilde{\Psi}_{\rm R}^{\prime}}$ and $\bra{\tilde{\Psi}_{\rm
L}^{\prime}}$,
$\displaystyle\ket{\psi_{a^{m}}}=\ket{\phi_{a^{m}}}\,,\quad\hat{c}_{a^{m}}^{\dagger}=(\hat{a}^{\dagger}_{a^{m}})^{\prime}\,,$
(68)
$\displaystyle\bra{\psi_{i^{m}}}=\bra{\tilde{\phi}_{i^{m}}}\,,\quad\hat{c}_{i^{m}}=(\tilde{\hat{a}}_{i^{m}})^{\prime}\,,$
(69)
$\displaystyle\tau_{I}^{A}=(\tilde{\tau}_{I}^{A})^{\prime}\,,\quad\lambda_{A}^{I}=(\tilde{\lambda}_{A}^{I})^{\prime}\,,\quad\tau_{i^{m}}^{a^{m}}=\braket{\phi_{a^{m}}^{\prime}}{\phi_{i^{m}}^{\prime}}\,,$
(70) $\displaystyle\ket{\Psi_{\rm R}}=\ket{\tilde{\Psi}_{\rm
R}^{\prime}}\,,\quad\bra{\Psi_{\rm L}}=\bra{\tilde{\Psi}_{\rm L}^{\prime}}\,.$
(71)
Now we have finished the proof of the equivalence of sOATDCC and TD-OCCT1.
Although it is unnecessary for the proof of the equivalence of two methods, we
will present the explicit form of the transformation and show that EoMs of two
methods coincide for the scenario that converting TD-OCCT1 to sOATDCC since it
will give readers a straightforward understanding. Useful transformations are
$\displaystyle\tilde{\hat{E}}_{i^{m}}^{a^{m}}=\hat{E}_{i^{m}}^{a^{m}}\,,\quad\tilde{X}_{i^{m}}^{a^{m}}=X_{i^{m}}^{a^{m}}+\dot{\tau}_{i^{m}}^{a^{m}}-\tau_{j^{m}}^{a^{m}}\tau_{i^{m}}^{b^{m}}X^{j^{m}}_{b^{m}}\,,$
(72)
$\displaystyle\tilde{\hat{E}}_{i^{m}}^{j^{m}}=\hat{E}_{i^{m}}^{j^{n}}+\tau_{j^{m}}^{a^{m}}\hat{E}_{i^{m}}^{a^{m}}\,,\quad\tilde{X}_{i^{m}}^{j^{m}}=X_{a^{m}}^{j^{m}}\tau_{i^{m}}^{a^{m}}\,,$
$\displaystyle\tilde{\hat{E}}_{a^{m}}^{b^{m}}=\hat{E}_{a^{m}}^{b^{n}}-\tau_{i^{m}}^{a^{m}}\hat{E}_{i^{m}}^{b^{m}}\,,\quad\tilde{X}_{a^{m}}^{b^{m}}=-\tau_{i^{m}}^{b^{m}}X^{i^{m}}_{a^{m}}\,,$
$\displaystyle\tilde{\hat{E}}^{i^{m}}_{a^{m}}=\hat{E}^{i^{m}}_{a^{m}}-\tau_{j^{m}}^{a^{m}}\hat{E}_{j^{m}}^{i^{m}}+\tau_{i^{m}}^{b^{m}}\hat{E}_{a^{m}}^{b^{m}}-\tau_{j^{m}}^{a^{m}}\tau_{i^{m}}^{b^{m}}\hat{E}_{j^{m}}^{b^{m}}\,,$
$\displaystyle\tilde{\hat{E}}_{\alpha^{m}}^{i^{m}}=\hat{E}_{\alpha^{m}}^{i^{m}}+\tau_{i^{m}}^{a^{m}}\hat{E}_{\alpha^{m}}^{a^{m}}\,,\quad\tilde{X}_{\alpha^{m}}^{i^{m}}=X_{\alpha^{m}}^{i^{m}}\,,\quad\tilde{X}^{i^{m}}_{a^{m}}=X^{i^{m}}_{a^{m}}\,,$
$\displaystyle\tilde{\hat{E}}_{\alpha^{m}}^{a^{m}}=\hat{E}_{\alpha^{m}}^{a^{m}}\,,\quad\tilde{X}_{\alpha^{m}}^{a^{m}}=X_{\alpha^{m}}^{a^{m}}-\tau_{i^{m}}^{a^{m}}{X}_{\alpha^{m}}^{i^{m}}\,,$
$\displaystyle\tilde{\hat{E}}^{\alpha^{m}}_{i^{m}}=\hat{E}^{\alpha^{m}}_{i^{m}}\,,\quad\tilde{X}^{\alpha^{m}}_{i^{m}}=X^{\alpha^{m}}_{i^{m}}+\tau_{i^{m}}^{a^{m}}{X}^{\alpha^{m}}_{a^{m}}\,,$
$\displaystyle\tilde{\hat{E}}^{\alpha^{m}}_{a^{m}}=\hat{E}^{\alpha^{m}}_{a^{m}}-\tau_{i^{m}}^{a^{m}}\hat{E}^{\alpha^{m}}_{i^{m}}\,,\quad\tilde{X}^{\alpha^{m}}_{a^{m}}=X^{\alpha^{m}}_{a^{m}}\,,$
$\displaystyle\tilde{\hat{X}}=\hat{X}+(\partial^{\hskip
0.20999pt\prime}_{t}{\hat{T}_{1}})\,.$
where we already set $X_{i^{m}}^{j^{m}}$ and $X_{a^{m}}^{b^{m}}$ as zero. We
stress that the constraint selection of $\tilde{X}_{i^{m}}^{j^{m}}$ and
$\tilde{X}_{a^{m}}^{b^{m}}$ in sOATDCC in this case is no longer zero, which
is not a usual choice in practical simulations. It is straightforward to
verify that EoMs of active-virtual orbital rotations and CC coefficients are
satisfied. To show the EoMs of intergroup rotations of active orbitals also
hold, one notices that their EoMs can be expressed uniformly as
$\displaystyle\langle\tilde{\Psi}_{\rm L}|[\hat{H}-i(\partial^{\hskip
0.20999pt\prime}_{t}\tilde{\hat{T}})-i\tilde{\hat{X}},\tilde{\hat{E}}_{p^{m}}^{q^{m}}]|\tilde{\Psi}_{\rm
R}\rangle$ (73) $\displaystyle+$ $\displaystyle
i\langle\tilde{\Phi}|(\partial^{\hskip
0.20999pt\prime}_{t}\tilde{\hat{\Lambda}})e^{-\tilde{\hat{T}}}\tilde{\hat{E}}_{p^{m}}^{q^{m}}e^{\tilde{\hat{T}}}|\tilde{\Phi}\rangle=0\,,$
which also holds since it is the summation (with some weights) of
$\displaystyle\langle\Psi_{\rm L}|[\hat{H}-i(\partial^{\hskip
0.20999pt\prime}_{t}\hat{T})-i\hat{X},\hat{E}^{q^{m}}_{p^{m}}]|\Psi_{\rm
R}\rangle$ (74) $\displaystyle+$ $\displaystyle i\langle\Phi|(\partial^{\hskip
0.20999pt\prime}_{t}\hat{\Lambda})e^{-\hat{T}}\hat{E}^{q^{m}}_{p^{m}}|\Psi_{\rm
R}\rangle=0\,.$
## Appendix D EoMs of TD-OCCH
In this section, we will present the EoMs of single amplitudes and particle-
hole orbital rotations of TD-OCCH. EoMs of other parameters of the method have
already been presented in the Sec. II.3. We will use notations $m_{\rm O}$,
$m_{\rm T}$, $m_{\rm B}$, and $m_{\rm X}$ to label the fermion kinds with
parameterization of TD-OCC, TD-OCCT1, TD-BCC, and TD-OCCX0. $m$ without any
subscripts represent an arbitrary fermion kind. $X_{i^{m_{\rm X}}}^{a^{m_{\rm
X}}}$ are set as Eq. (33) in VG and zero in LG. EoMs of $\lambda^{i^{m_{\rm
X}}}_{a^{m_{\rm X}}}$ and $X_{i^{m_{\rm T}}}^{a^{m_{\rm T}}}$ are decoupled
with other EoMs
$i\left(\rho_{j^{m_{\rm T}}}^{i^{m_{\rm T}}}\delta_{b^{m_{\rm T}}}^{a^{m_{\rm
T}}}-\rho_{a^{m_{\rm T}}}^{b^{m_{\rm T}}}\delta_{j^{m_{\rm T}}}^{i^{m_{\rm
T}}}\right)X_{b^{m_{\rm T}}}^{j^{m_{\rm T}}}=\bra{\Psi_{\rm
L}}[\hat{H},\hat{E}_{i^{m_{\rm T}}}^{a^{m_{\rm T}}}]\ket{\Psi_{\rm R}}\,,$
(75)
$\displaystyle-i\dot{\lambda}^{i^{m_{\rm X}}}_{a^{m_{\rm X}}}=$ (76)
$\displaystyle\bra{\Psi_{\rm L}}[\hat{H},\hat{E}_{i^{m_{\rm X}}}^{a^{m_{\rm
X}}}]\ket{\Psi_{\rm R}}-i\left(\rho_{j^{m_{\rm X}}}^{i^{m_{\rm
X}}}\delta_{b^{m_{\rm X}}}^{a^{m_{\rm X}}}-\rho_{a^{m_{\rm X}}}^{b^{m_{\rm
X}}}\delta_{j^{m_{\rm X}}}^{i^{m_{\rm X}}}\right)X_{b^{m_{\rm X}}}^{j^{m_{\rm
X}}}\,.$
After obtaining $\dot{\lambda}^{i^{m_{\rm X}}}_{a^{m_{\rm X}}}$ and
$X_{i^{m_{\rm T}}}^{a^{m_{\rm T}}}$, one can solve the following coupled
equations to get $\dot{\lambda}^{i^{m_{\rm B}}}_{a^{m_{\rm B}}}$,
$X_{i^{m_{\rm B}}}^{a^{m_{\rm B}}}$, and $X_{i^{m_{\rm O}}}^{a^{m_{\rm O}}}$.
$\displaystyle i\frac{1}{2}\left\\{\sum_{n_{\rm B}}\langle\Phi_{j^{n_{\rm
B}}}^{b^{n_{\rm B}}}|e^{-\hat{T}}\hat{E}^{i^{m_{\rm B}}}_{a^{m_{\rm
B}}}|\Psi_{\rm R}\rangle\dot{\lambda}^{j^{n_{\rm B}}}_{b^{n_{\rm
B}}}+(\dot{\lambda}^{i^{m_{\rm B}}}_{a^{m_{\rm B}}})^{*}+\sum_{n\in n_{\rm
B},n_{\rm O}}A^{a^{m_{\rm B}}j^{n}}_{i^{m_{\rm
B}}b^{n}}(X^{b^{n}}_{j^{n}})^{*}\right\\}+i(\delta_{b^{m_{\rm B}}}^{a^{m_{\rm
B}}}D_{i^{m_{\rm B}}}^{j^{m_{\rm B}}}-D_{b^{m_{\rm B}}}^{a^{m_{\rm
B}}}\delta_{i^{m_{\rm B}}}^{j^{m_{\rm B}}})X_{j^{m_{\rm B}}}^{b^{m_{\rm B}}}=$
$\displaystyle B_{i^{m_{\rm B}}}^{a^{m_{\rm B}}}-i\sum_{n\in n_{\rm T},n_{\rm
X}}\frac{1}{2}A^{a^{m_{\rm B}}j^{n}}_{i^{m_{\rm
B}}b^{n}}(X^{b^{n}}_{j^{n}})^{*}-i\frac{1}{2}\sum_{n_{\rm
X}}\langle\Phi_{j^{n_{\rm X}}}^{b^{n_{\rm X}}}|e^{-\hat{T}}\hat{E}^{i^{m_{\rm
B}}}_{a^{m_{\rm B}}}|\Psi_{\rm R}\rangle\dot{\lambda}^{j^{n_{\rm
X}}}_{b^{n_{\rm X}}}\,,$ (77)
$\displaystyle i\frac{1}{2}\left\\{\sum_{n_{\rm B}}\langle\Phi_{j^{n_{\rm
B}}}^{b^{n_{\rm B}}}|e^{-\hat{T}}\hat{E}^{i^{m_{\rm O}}}_{a^{m_{\rm
O}}}|\Psi_{\rm R}\rangle\dot{\lambda}^{j^{n_{\rm B}}}_{b^{n_{\rm
B}}}+\sum_{n\in n_{\rm B},n_{\rm O}}A^{a^{m_{\rm O}}j^{n}}_{i^{m_{\rm
O}}b^{n}}(X^{b^{n}}_{j^{n}})^{*}\right\\}+i(\delta_{b^{m_{\rm O}}}^{a^{m_{\rm
O}}}D_{i^{m_{\rm O}}}^{j^{m_{\rm O}}}-D_{b^{m_{\rm O}}}^{a^{m_{\rm
O}}}\delta_{i^{m_{\rm O}}}^{j^{m_{\rm O}}})X_{j^{m_{\rm O}}}^{b^{m_{\rm O}}}=$
$\displaystyle B_{i^{m_{\rm O}}}^{a^{m_{\rm O}}}-i\sum_{n\in n_{\rm T},n_{\rm
X}}\frac{1}{2}A^{a^{m_{\rm O}}j^{n}}_{i^{m_{\rm
O}}b^{n}}(X^{b^{n}}_{j^{n}})^{*}-i\frac{1}{2}\sum_{n_{\rm
X}}\langle\Phi_{j^{n_{\rm X}}}^{b^{n_{\rm X}}}|e^{-\hat{T}}\hat{E}^{i^{m_{\rm
O}}}_{a^{m_{\rm O}}}|\Psi_{\rm R}\rangle\dot{\lambda}^{j^{n_{\rm
X}}}_{b^{n_{\rm X}}}\,,$ (78)
$\displaystyle iX_{i^{m_{\rm B}}}^{a^{m_{\rm B}}}+\sum_{n\in n_{\rm B},n_{\rm
O}}i\langle\Phi_{i^{m_{\rm B}}}^{a^{m_{\rm
B}}}|e^{-\hat{T}}\hat{E}^{j^{n}}_{b^{n}}|\Psi_{\rm R}\rangle
X^{j^{n}}_{b^{n}}=\langle\Phi_{i^{m_{\rm B}}}^{a^{m_{\rm
B}}}|e^{-\hat{T}}\hat{H}|\Psi_{\rm R}\rangle-\sum_{n\in n_{\rm T},n_{\rm
X}}i\langle\Phi_{i^{m_{\rm B}}}^{a^{m_{\rm
B}}}|e^{-\hat{T}}\hat{E}^{j^{n}}_{b^{n}}|\Psi_{\rm R}\rangle
X^{j^{n}}_{b^{n}}\,.$ (79)
Finally, $\dot{\tau}_{i^{m_{\rm T}}}^{a^{m_{\rm T}}}$ and
$\dot{\tau}_{i^{m_{\rm X}}}^{a^{m_{\rm X}}}$ can be solved
$\displaystyle i(\delta_{b^{m_{\rm T}}}^{a^{m_{\rm T}}}D_{i^{m_{\rm
T}}}^{j^{m_{\rm T}}}-D_{b^{m_{\rm T}}}^{a^{m_{\rm T}}}\delta_{i^{m_{\rm
T}}}^{j^{m_{\rm T}}})\dot{\tau}_{j^{m_{\rm T}}}^{b^{m_{\rm T}}}=B_{i^{m_{\rm
T}}}^{a^{m_{\rm T}}}$ $\displaystyle-$ $\displaystyle i(\delta_{b^{m_{\rm
T}}}^{a^{m_{\rm T}}}D_{i^{m_{\rm T}}}^{j^{m_{\rm T}}}-D_{b^{m_{\rm
T}}}^{a^{m_{\rm T}}}\delta_{i^{m_{\rm T}}}^{j^{m_{\rm T}}})X_{j^{m_{\rm
T}}}^{b^{m_{\rm T}}}-\sum_{n}\frac{i}{2}A^{a^{m_{\rm T}}j^{n}}_{i^{m_{\rm
T}}b^{n}}(X^{b^{n}}_{j^{n}})^{*}$ $\displaystyle-$
$\displaystyle\frac{i}{2}\sum_{n\in n_{\rm B},n_{\rm
X}}\langle\Phi_{j^{n}}^{b^{n}}|e^{-\hat{T}}\hat{E}^{i^{m_{\rm T}}}_{a^{m_{\rm
T}}}|\Psi_{\rm R}\rangle\dot{\lambda}^{j^{n}}_{b^{n}}\,,$ (80)
$i\dot{\tau}_{i^{m_{\rm X}}}^{a^{m_{\rm X}}}=\bra{\Phi_{i^{m_{\rm
X}}}^{a^{m_{\rm X}}}}e^{-\hat{T}}(\hat{H}-i\hat{X})\ket{\Psi_{\rm R}}\,.$ (81)
It is straightforward to verify that TD-OCCH reduce to other four methods when
the corresponding parameterization is used in all fermion kinds. It is also
possible to use two or three types of parameterization in TD-OCCH method, and
one needs to remove unused types of parameters in above EoMs. An attractive
parameterization is the OCCT1-OCCX0 mixed one, in which all single amplitudes
and particle-hole rotations of different kinds are decoupled. One can also
assign few kinds fermions OCC and BCC type parameterization, in which only few
of coupled equations need to be solved.
Similar to the TD-OCCX0 method, the non-variance of $X_{i^{m_{\rm
X}}}^{a^{m_{\rm X}}}$ makes the Ehrenfest theoremSato _et al._ (2016)
modified even without frozen-cores. One only needs to account for the TD-
OCCX0-type parameterization fermion kinds
$\displaystyle 2i\frac{d}{dt}\braket{\hat{O}}$
$\displaystyle=\braket{\Psi_{\rm L}}{[\hat{O},\hat{H}]}{\Psi_{\rm
R}}+\braket{\Psi_{\rm R}}{[\hat{O},\hat{H}]}{\Psi_{\rm L}}$ (82)
$\displaystyle+\sum_{m\in m_{\rm X}}O_{t^{m}}^{u^{m}}\\{\braket{\Psi_{\rm
L}}{(\hat{H}-i\hat{X})e^{\hat{T}}\bar{\hat{\Pi}}e^{-\hat{T}}\hat{E}_{t^{m}}^{u^{m}}}{\Psi_{\rm
R}}$ $\displaystyle-\braket{\Psi_{\rm
L}}{\hat{E}_{t^{m}}^{u^{m}}e^{\hat{T}}\bar{\hat{\Pi}}e^{-\hat{T}}(\hat{H}-i\hat{X})}{\Psi_{\rm
R}}$ $\displaystyle+\braket{\Psi_{\rm
R}}{(\hat{H}-i\hat{X})e^{\hat{T}}\bar{\hat{\Pi}}e^{-\hat{T}}\hat{E}_{t^{m}}^{u^{m}}}{\Psi_{\rm
L}}$ $\displaystyle-\braket{\Psi_{\rm
R}}{\hat{E}_{t^{m}}^{u^{m}}e^{\hat{T}}\bar{\hat{\Pi}}e^{-\hat{T}}(\hat{H}-i\hat{X})}{\Psi_{\rm
L}}\\}\,.$
When frozen orbitals are included, two additional terms that replace $t^{m}$,
$u^{m}$ with $i^{\prime\prime m}$, $\mu^{m}$ and $\mu^{m}$, $i^{\prime\prime
m}$ in the last term of Eq. (82) should be added. Additionally, the summation
for the new terms should be for all kinds of fermions.
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|
$\mathcal{W}$, characterized by $\mathbf{n}\in\mathcal{W}_{i}\Leftrightarrow
n_{i}=n_{i+1}$. Then (8.2.1) is immediate from (8.1). Hence we may assume that
$d\geq 2$ and work in our usual framework using $d$-diagrams.
2. (ii)
We will show more precisely: The map
(8.2.2) $\displaystyle\delta\colon\operatorname{diag}^{(d)}(\mathbf{n})$
$\displaystyle\longrightarrow\operatorname{diag}^{(d)}(\mathbf{n}\hat{})$
$\displaystyle(i,v)$ $\displaystyle\longmapsto(r+1-i,n_{1}+d-1-v)$
is well-defined and bijective, and sends
(8.2.3) $B_{k}^{(d)}(\mathbf{n})\text{ to
}B_{rd+1-k}^{(d)}(\mathbf{n}\hat{}).$
3. (iii)
As both sets $\operatorname{diag}^{(d)}(\mathbf{n})$ and
$\operatorname{diag}^{(d)}(\mathbf{n}\hat{})$ have the same cardinality, it
suffices for (8.2.2) to show that $\delta$ is well-defined and satisfies
$\delta^{2}=\operatorname{id}$. Now
$\displaystyle(i,v)\in\operatorname{diag}^{(d)}(\mathbf{n})\Longleftrightarrow
n_{i}\leq v<n_{i}+d$ $\displaystyle(1\leq i<r)$
and, after a little calculation, this turns out to be equivalent with
$\mathbf{n}\hat{}_{r+1-i}\leq n_{1}+d-1-v<\mathbf{n}\hat{}_{r+1-i}+d$
for all $i$, that is, $\delta$ maps in fact to
$\operatorname{diag}^{(d)}(\mathbf{n}\hat{})$. The equation $\delta^{2}=1$ is
trivial.
4. (iv)
Recall the order relation “$\leq$” on $\operatorname{diag}^{(d)}(\mathbf{n})$
given by (4.1.1). It is obvious from the definition of $\delta$ that it
reverses this order, which gives (8.2.3).
∎
### 8.3.
Coming back to the arithmetic significance of
$\mathcal{BT}(d,k)=\mathcal{BT}({}_{a}\ell_{k})=\lambda(\Omega({}_{a}\ell_{k}))$,
where $a\in A=\mathds{F}[T]$ has degree $d$, we find the following
relationship. Let $a\in A$ of degree $d$ be given, and consider for $1\leq
k<rd$ the modular forms ${}_{a}\ell_{k}$. Then the zero sets
$\Omega({}_{a}\ell_{k})$ of ${}_{a}\ell_{k}$ and $\Omega({}_{a}\ell_{rd-k})$
of ${}_{a}\ell_{rd-k}$ are related by
(8.3.1)
$\lambda(\Omega({}_{a}\ell_{k}))\hat{}=\lambda(\Omega({}_{a}\ell_{rd-k})).$
### 8.4.
Note that the considerations of this section also apply for $r=2$. Here
$\mathcal{BT}(d,k)=\mathcal{BT}^{2}(d,k)$ is a finite set of vertices of the
Bruhat-Tits tree $\mathcal{BT}=\mathcal{BT}^{2}$. The involution
$(\,.\,)\hat{}$ is trivial, and (8.2.1) resp. (8.3.1) degenerate to
$\mathcal{W}(d,k)=\mathcal{W}(d,k)\hat{}$ resp.
$\lambda(\Omega({}_{a}\ell_{k}))=\lambda(\Omega({}_{a}\ell_{2d-k}))$. This is
in accordance with the symmetry property in Theorem 5.1 of [Gekeler2011].
## 9\. Examples and concluding remarks
Note that our Theorems 5.17, 6.10, 7.4 and 8.2 along with Proposition 3.8
assert all the ingredients of the Main Theorem 1.8, which thereby is proved.
Here we present examples for the simplicial complexes $\mathcal{W}(d,k)$. For
reasons of presentation, we restrict to the case $r=3$, where
$\mathcal{W}(d,k)$ has dimension 1, i.e., is a graph embedded into
$\mathcal{W}=\mathcal{W}^{3}$ (see 3.2 and the legend given there).
###### Example 9.1 ($r=3$, the graphs $\mathcal{W}(2,k)$, $1\leq k\leq 5$):
$\mathcal{W}=\mathcal{W}^{3}$$\mathcal{W}_{1}=\mathcal{W}(2,5)$$\mathcal{W}_{2}=\mathcal{W}(2,1)$$\sigma$$x\hat{}$$x$$=\mathcal{W}(2,2)$$=\mathcal{W}(2,3)$$=\mathcal{W}(2,4)$
Note that $\mathcal{W}(2,1)=\mathcal{W}(1)=$ the wall $\mathcal{W}_{2}$ and
$\mathcal{W}(2,2)$ have been drawn in [V] Figure 1. By symmetry, this gives
$\mathcal{W}(2,5)=\mathcal{W}(2,1)\hat{}=\mathcal{W}_{1}$ and
$\mathcal{W}(2,4)=\mathcal{W}(2,2)\hat{}$. Let $a\in A$ have degree 2, e.g.,
$a=T^{2}$. Then $f\vcentcolon={}_{a}\ell_{1}{}_{a}\ell_{2}{}_{a}\ell_{3}$ has
the property: The edges of the leftmost triangle $\sigma$ of $\mathcal{W}$
belong to $\mathcal{W}(f)$, while
$\overset{\circ}{\sigma}(\mathds{Q})\cap\mathcal{W}(f)=\varnothing$. Hence the
zero set $\mathcal{W}(f)$ is not a full subcomplex of $\mathcal{W}$ and $f$ is
not simplicial. Examples of modular forms of this shape that are not
simplicial are abundant.
###### Example 9.2 ($r=3$, the graphs $\mathcal{W}(3,k)$, $k=3$ or $4$):
$\mathcal{W}=\mathcal{W}^{3}$$=\mathcal{W}(3,3)=\mathcal{W}(3)$$=\mathcal{W}(3,4)$
As $\mathcal{W}(3,k)=\mathcal{W}(k)$ for $k=1,2$ has been given in 9.1 and
$\mathcal{W}(3,k)\hat{}=\mathcal{W}(3,9-k)$, we may restrict to $k=3$ or $4$.
###### Example 9.3 ($r=3$, the graphs $\mathcal{W}(4,k)$, $k=4,5,6$):
$\mathcal{W}=\mathcal{W}^{3}$$=\mathcal{W}(4,4)=\mathcal{W}(4)$$=\mathcal{W}(4,5)$$=\mathcal{W}(4,6)$
As $\mathcal{W}(4,k)=\mathcal{W}(k)$ for $k=1,2,3$ is in 9.1 and 9.2 and
$\mathcal{W}(4,k)\hat{}=\mathcal{W}(4,12-k)$, we restrict to $k=4,5,6$. Note
that the graph $\mathcal{W}(4,6)$ has a non-trivial cycle.
###### Remark 9.4 (about the involution $(\,.\,)\hat{}$):
A very satisfactory explanation/interpretation of Theorem 8.2 would be given
by the existence of a lift of $(\,.\,)\hat{}$ to $\Omega$, i.e., an involution
$\boldsymbol{\omega}\mapsto\boldsymbol{\omega}\boldsymbol{\hat{}}$ on $\Omega$
such that
(9.4.1)
${\Omega}$${\Omega}$${\mathcal{BT}}$${\mathcal{BT}}$$\scriptstyle{\lambda}$$\scriptstyle{(\,.\,)\boldsymbol{\hat{}}}$$\scriptstyle{\lambda}$$\scriptstyle{(\,.\,)\hat{}}$
commutes. In view of (8.1), a natural candidate for
$(\,.\,)\boldsymbol{\hat{}}$ would be the map
(9.4.2)
$\boldsymbol{\omega}=(\omega_{1},\dots,\omega_{r-1},1)\longmapsto\left(\omega_{1},\frac{\omega_{1}}{\omega_{r-1}},\dots,\frac{\omega_{1}}{\omega_{2}},1\right).$
This however fails for a number of reasons. First, and most important,
$\Omega$ is not stable under the map described in (9.4.2) (I owe this hint to
Andreas Schweizer). Second, a reasonable lift $(\,.\,)\boldsymbol{\hat{}}$ as
in (9.4.1) had to interchange the zeroes of ${}_{a}\ell_{k}$ with those of
${}_{a}\ell_{rd-k}$, which however is excluded for “weight reasons”.
Let ${}_{d}L_{k}$ be the $\mathds{Q}$-valued function on
$\mathcal{BT}(\mathds{Q})$ given by
(9.4.3)
${}_{d}L_{k}(\boldsymbol{x})\vcentcolon=\log_{q}\lVert{}_{a}\ell_{k}\rVert_{\mathbf{x}},$
where $\lVert f\rVert_{\boldsymbol{x}}$ is the spectral norm
$\sup_{\boldsymbol{\omega}\in\lambda^{-1}(\boldsymbol{x})}\lvert
f(\boldsymbol{x})\rvert$ on the affinoid $\lambda^{-1}(\boldsymbol{x})$ (see
[I], [II], [IV]). (Actually, $\lVert{}_{a}\ell_{k}\rVert_{\boldsymbol{x}}$
depends only on $d=\deg a$, see [V] Proposition 3.8.) Now, as ${}_{a}\ell_{k}$
has turned out to be simplicial, ${}_{d}L_{k}$ encodes the zero set
$\mathcal{BT}({}_{a}\ell_{k})=\mathcal{BT}(d,k)$, see Proposition 1.8 in [V].
Therefore, as a substitute for some diagram (9.4.1), we expect(?) a simple
relation between ${}_{d}L_{k}\hat{}$ and ${}_{d}L_{rd-k}$ or perhaps between
the van der Put transforms $P({}_{a}\ell_{k})\hat{}$ and
$P({}_{a}\ell_{rd-k})$ (see [V] Section 1) that implies Theorem 8.2.
### 9.5.
Similar to $\mathcal{BT}(d,k)$, which presents a coarse picture of the zero
locus $\Omega({}_{a}\ell_{k})$ of ${}_{a}\ell_{k}$ in $\Omega$, the object
$\Gamma\backslash\mathcal{BT}(d,k)$ may be seen as a coarse picture of the
quotient
${\Gamma\backslash\Omega({}_{a}\ell_{k})}$${\Gamma\backslash\Omega}$${M^{r}({}_{a}\ell_{k})}$${M^{r}.}$$\scriptstyle{\cong}$$\scriptstyle{\cong}$
Here $M^{r}$ is (the set of $C_{\infty}$-points of) the moduli scheme of
rank-$r$ Drinfeld $A$-modules. Now, despite the fact that $\Gamma$ acts
simplicially on $\mathcal{BT}$ and its subcomplex $\mathcal{BT}(d,k)$, the
quotient modulo $\Gamma$ doesn’t inherit a structure as a simplicial complex.
This comes from the (well-known but annoying) fact that, e.g. in the case of
dimension 1, endpoints of $1$-simplices $\sigma,\tau$ are possibly identified
under $\Gamma$, but $\sigma$ and $\tau$ are not. This leads to double edges
$[\sigma]$$[\tau]$
and similar phenomena in $\Gamma\backslash\mathcal{BT}$. Hence the topological
space $\Gamma\backslash\mathcal{BT}(d,k)(\mathds{R})$ has only the weaker
structure of cell complex. Nevertheless, the next topics to deal with should
be:
* •
Study the spaces $\Gamma\backslash\mathcal{BT}(d,k)(\mathds{R})$, the natural
map
$\mathcal{W}(d,k)(\mathds{R})\to\Gamma\backslash\mathcal{BT}(d,k)(\mathds{R})$,
their Betti numbers, etc.;
* •
Work out the relationship between, say, the $\ell$-adic cohomology of the
moduli scheme $M({}_{a}\ell_{k})$ and the (co-)homology of
$\Gamma\backslash\mathcal{BT}(d,k)(\mathds{R})$;
* •
Instead of the zero sets of one modular form ${}_{a}\ell_{k}$ in $\Omega$,
$\mathcal{BT}$, $\Gamma\backslash\mathcal{BT}$, study the simultaneous zero
sets of families of modular forms in $\Omega$, $\mathcal{BT}$ and their
quotients modulo $\Gamma$, or modulo congruence subgroups $\Gamma^{\prime}$ of
$\Gamma$. For the relatively simple case of the family $\\{g_{k}\mid 1\leq
k<r,k\neq j\\}$ where $j$ is fixed, see [Gekeler2019].
### 9.6.
As the reader will have noticed, $q=\\#(\mathds{F})$ has completely vanished
from the largest part of the paper. This is due to the fact that we mainly
worked with $\mathcal{A}$ and $\mathcal{W}$ and the combinatorics of its
subcomplexes. These are independent of $q$ and are entirely determined through
the root system $\Phi$ and its Weyl group $W$. The quantity $q$ occurs only in
the way how the different apartments $\mathcal{A}^{\prime}\cong\mathcal{A}$
are glued together, i.e., how many neighbors
$v^{\prime}\in\mathcal{BT}(\mathds{Z})$ there are for a given vertex
$v=\mathbf{n}$ in $\mathcal{A}$. Therefore, the simplicial complexes
$\mathcal{A}(k)$, $\mathcal{A}(d,k)$, $\mathcal{BT}(k)$, $\mathcal{BT}(d,k)$
are fundamental structures for $\Phi$, and should play a role beyond their
describing the zero sets of the Drinfeld modular forms ${}_{a}\ell_{k}$.
## References
|
# Bijective proof of a conjecture on unit interval posets
Wenjie Fang
LIGM, Univ Gustave Eiffel, CNRS, ESIEE Paris, F-77454 Marne-la-Vallée, France
###### Abstract
In a recent preprint, Matherne, Morales and Selover conjectured that two
different representations of unit interval posets are related by the famous
zeta map in $q,t$-Catalan combinatorics. This conjecture was proved recently
by Gélinas, Segovia and Thomas using induction. In this short note, we provide
a bijective proof of the same conjecture with a reformulation of the zeta map
using left-aligned colored trees, first proposed in the study of parabolic
Tamari lattices.
## 1 Introduction
The study of unit interval posets started in statistics (see [WF57]) and
psychology (see [Sco64]). However, it has since been connected to other
objects in algebraic combinatorics, such as $(3+1)$-free posets [SR03] and
positroids [CG18]. The connection to $(3+1)$-free posets is particularly
important, as these posets are at the center of the Stanley-Stembridge
conjecture [SS93], which states that the chromatic symmetric function of the
incomparability graph of a $(3+1)$-free poset has only positive coefficients
when expanded in the basis of elementary symmetric functions. Unit interval
posets thus receive some attention as it can be used to study the structure of
$(3+1)$-free posets [SR03, GPMR14].
It is known that unit interval posets are counted by Catalan numbers [WF57],
and researchers have given two different bijections to represent a unit
interval poset by Dyck paths [SR03, GPMR14]. In a recent preprint of Matherne,
Morales and Selover [MMS], it was conjectured that the two bijections are
related by the famous zeta map in $q,t$-Catalan combinatorics (see [Hag08]),
which was first given in [AKOP02]. In this short note, we settle this
conjecture using bijections (see 4.6), in contrast to a recent inductive proof
of the same conjecture by Gélinas, Segovia and Thomas [GST]. To this end, we
first introduce a bijection $\Xi_{\mathrm{poset}}$ between unit interval
posets and plane trees. We then show that the two different known bijections
from unit interval posets are conjugates of special cases of some bijections
$\Xi_{\mathrm{steep}}$ and $\Xi_{\mathrm{bounce}}$ in [CFM20] constructed for
the study of parabolic Tamari lattices. Using the link between
$\Xi_{\mathrm{steep}},\Xi_{\mathrm{bounce}}$ and the zeta map also established
in the same article, we conclude our proof.
The rest of the article is organized as follows. In Section 2 we give the
basic definitions. Then we give our bijection between unit interval posets and
plane trees in Section 3, along with the related bijections in [CFM20].
Finally in Section 4 we recall the two known bijections between unit interval
posets and Dyck paths and establish their link with the bijections in Section
3, leading to a bijective proof of our main result (4.6).
### Acknowledgment
We thank Adrien Segovia for bringing this conjecture to our attention. This
work is not supported by any funding with precise predefined goal, but it is
supported by the publicly funded laboratory LIGM of Université Gustave Eiffel.
## 2 Preliminary
For convenience, we write $[n]$ for the set $\\{1,2,\ldots,n\\}$. We consider
finite partially order sets (or _poset_ for short) of the form
$P=(P_{\mathrm{elem}},\preceq)$, where $P_{\mathrm{elem}}$ is a finite set and
$\preceq$ a partial order on $P_{\mathrm{elem}}$. Given a set $S$ of reals
$x_{1}<x_{2}<\cdots<x_{n}$, we define a partial order $\preceq_{S}$ on $[n]$
by taking $i\preceq_{S}j$ if and only if $x_{i}+1<x_{j}$. The order
$\preceq_{S}$ can also be seen as defined on intervals of unit length starting
at the $x_{i}$’s, where $i\preceq_{S}j$ if the interval starting with $x_{i}$
is on the left of that with $x_{j}$ without overlap. A poset $P$ with $n$
elements is a _unit interval order_ if there is some $S\subset\mathbb{R}$ with
$n$ elements such $P\cong([n],\preceq_{S})$. In this case, we call $S$ a
_starting set_ of $P$. We sometimes represent unit interval orders as
$([n],\preceq_{S})$ for some $S$ hereinafter.
We have the following characterization of unit interval orders.
###### Theorem 2.1 (Theorem 2.1 of [Sco64]).
A poset $P$ is a unit interval poset if and only if it is $(3+1)$-free and
$(2+2)$-free, that is, the order induced on any four elements cannot be a
chain of $3$ elements plus an incomparable elements, or two disjoint chains
each containing $2$ elements.
Unit interval orders are counted by Catalan numbers.
###### Proposition 2.2 ([WF57]).
The number of unit interval posets with $n$ elements is the $n$-th Catalan
number $\mathrm{Cat}_{n}=\frac{1}{2n+1}\binom{2n+1}{n}$.
By definition, we may represent a unit interval poset $P$ by a set of reals
$S$, though the choice of $S$ is clearly not unique. In the following, for
convenience, we use this perspective of (starting points of) intervals, which
is arguably easier to manipulate. We denote by $\mathcal{P}_{n}$ the set of
unit interval posets with $n$ elements.
There are many other families of combinatorial objects counted by Catalan
numbers, such as Dyck paths and plane trees. A _Dyck path_ is a lattice path
formed by north steps $\uparrow=(0,1)$ and east steps $\rightarrow=(1,0)$,
starting at $(0,0)$ and ending on the diagonal $y=x$ while staying weakly
above it. The _size_ of a Dyck path is the number of its north steps. We
denote by $\mathcal{D}_{n}$ the set of Dyck paths of size $n$. We define plane
trees recursively: a _plane tree_ $T$ is either a single node (a _leaf_) or an
_internal node_ $u$ linked by edges to a sequence of plane trees called _sub-
trees_. In the latter case, the node $u$ is also called the _root_ of $T$, and
the roots of sub-trees are called _children_ of $u$. We denote by
$\mathcal{T}_{n}$ the number of plane trees with $n$ non-root nodes. For a
node $u$ in a plane tree $T$, its _depth_ is the distance (number of edges)
between $u$ and the root of $T$.
## 3 Unit interval posets and plane trees
We start by a new bijection between unit interval posets and plane trees.
###### Construction 3.1.
Let $S=\\{x_{1}<\cdots<x_{n}\\}$ be a starting set of a unit interval poset
$P=([n],\preceq_{S})$. We define a plane tree $T$ as follows (see Figure 1 for
an example). We set $x_{0}=x_{1}-2$. We denote by $v_{i}$ the node of $T$
corresponding to $x_{i}$. For $i\in[n]$, the parent of $v_{i}$ is $v_{j}$ if
and only if $j$ is the largest index such that $j\preceq_{S}i$. By the
definition of $x_{0}$, the parent of each $v_{i}$ is well-defined, and all
nodes are descendants of $x_{0}$. We then order children of the same node from
left to right with _decreasing_ index. We thus conclude that $T$ is a well-
defined plane tree. We note that $T$ depends only on $\preceq_{S}$. We define
$\Lambda_{\mathrm{poset}}(P)=T$.
Figure 1: Example of $\Lambda_{\mathrm{poset}}$ from a unit interval poset
defined by a set of unit intervals to a plane tree
###### Construction 3.2.
Given a plane tree $T$ with $n$ non-root nodes, we define an order relation
$\preceq_{S}$ on $[n]$ induced by some $S\subset\mathbb{R}$. Let $m$ be the
maximal arity of $T$, and $r_{T}$ the root of $T$. Given a node $u$ in $T$, if
it is the $i$-th child of its parent _from right to left_ , then we define
$c(u)=i$. We observe that $0<c(u)<m$. For a non-root node $u$ in $T$, let
$u_{0}=r_{T},u_{1},\ldots,u_{\ell}=u$ be the nodes on the unique path from the
root $u_{0}=r_{T}$ to $u_{\ell}=u$. We then define a real number $x_{u}$
associated to $u$ using base $(m+2)$:
$x_{u}=\ell+(0.c(u_{1})c(u_{2})\cdots
c(u_{\ell}))_{(m+2)}=\ell+\sum_{i=1}^{\ell}c(u_{i})(m+2)^{-i}.$ (1)
It is clear from (1) that $x_{u}$ is strictly decreasing from left to right
for nodes of the same depth in $T$. Let $S$ be the set of $x_{u}$ for non-root
nodes $u$ in $T$. We define $\Xi_{\mathrm{poset}}(T)=([n],\preceq_{S})$.
The following lemma gives a direct connection between a plane tree and its
corresponding unit interval poset obtained from $\Xi_{\mathrm{poset}}$.
###### Lemma 3.3.
For $T\in\mathcal{T}_{n}$, we take $S$ to be the set constructed in 3.2, and
$x_{u}$ the number constructed from a non-root node $u$ of $T$. Then for non-
root nodes $u,v$, the parent of $u$ in $T$ is $v$ if and only if $x_{v}$ is
the largest element in $S$ smaller than $1+x_{u}$.
###### Proof.
Suppose that $T$ has maximal arity $m$. Let $v^{\prime}$ be the parent of $u$.
By (1), we have $1<x_{u}-x_{v^{\prime}}<1+(m+2)^{-\ell+1}$, with $\ell$ the
distance from $v^{\prime}$ to the root. Then, $x_{v}$ satisfying
$x_{v^{\prime}}<x_{v}<1+x_{u}$ means $0<x_{v}-x_{v^{\prime}}<(m+2)^{-\ell+1}$,
which means $v=v^{\prime}$ by (1), as it requires $v$ and $v^{\prime}$ to have
the same distance to the root, but in this case $|x_{v}-x_{v^{\prime}}|$
cannot be smaller than $(m+2)^{-\ell+1}$ if $v\neq v^{\prime}$. We thus have
the equivalence. ∎
We now show that $\Lambda_{\mathrm{poset}}$ and $\Xi_{\mathrm{poset}}$ are
bijections.
###### Proposition 3.4.
For all $n\geq 1$, the map $\Lambda_{\mathrm{poset}}$ is a bijection from
$\mathcal{P}_{n}$ to $\mathcal{T}_{n}$, with $\Xi_{\mathrm{poset}}$ its
inverse.
###### Proof.
It is well-known that there are $\mathrm{Cat}_{n}$ elements in
$\mathcal{T}_{n}$. By 2.2, we have $|\mathcal{P}_{n}|=|\mathcal{T}_{n}|$. We
thus only need to show that
$\Lambda_{\mathrm{poset}}\circ\Xi_{\mathrm{poset}}=\operatorname{id}$.
Given a plane tree $T\in\mathcal{T}_{n}$, let
$P=([n],\preceq_{S})=\Xi_{\mathrm{poset}}(T)$ and
$T^{\prime}=\Lambda_{\mathrm{poset}}(P)$. Let $S=\\{x_{1}<\cdots<x_{n}\\}$ be
the set of real numbers constructed in 3.2 for $\preceq_{S}$. Given $i\in[n]$,
let $u_{i}$ be the node in $T$ that gives rise to $x_{i}$ in $S$, and
$u^{\prime}_{i}$ the node in $T^{\prime}$ that represents $x_{i}$. By 3.3 and
3.1, for any $i,j\in[n]$, the node $u_{i}$ is the parent of $u_{j}$ in $T$ if
and only if $u^{\prime}_{i}$ is the parent of $u^{\prime}_{j}$ in
$T^{\prime}$. Then, $T^{\prime}$ has the same order of siblings as $T$, as in
both we order siblings with decreasing order in their corresponding reals in
$S$. We thus conclude that $T=T^{\prime}$. ∎
We now restate two previously known bijections between plane trees and Dyck
paths that will be used to prove our main result.
###### Construction 3.5.
Let $n\geq 1$ and $T\in\mathcal{T}_{n}$, we construct a Dyck path $D$ by
taking the _clockwise_ contour walk starting from the top of the root of $T$.
During the walk, when we pass an edge for the first (resp. second) time, we
append $\uparrow$ (resp. $\rightarrow$) to $D$. We define
$\Xi_{\mathrm{steep}}(T)=D$. The map $\Xi_{\mathrm{steep}}$ is a bijection
from $\mathcal{T}_{n}$ to $\mathcal{D}_{n}$ for all $n\geq 1$, and we denote
its inverse by $\Lambda_{\mathrm{steep}}$.
###### Construction 3.6.
Let $n\geq 1$ and $T\in\mathcal{T}_{n}$, we construct a Dyck path $D$ by
appointing the number of north steps at each $x$-coordinate. Let
$\alpha_{\ell}$ be the number of nodes of depth $\ell$ (thus distance $\ell$
to the root), and $\ell_{\max}$ the maximal depth of nodes in $T$. We take
$s_{\ell}=\alpha_{1}+\cdots+\alpha_{\ell}$. Given $1\leq k\leq n-1$, there is
a unique way to write $k=s_{\ell}-r$ with $0\leq r<\alpha_{\ell}$ and
$1\leq\ell\leq\ell_{\max}$. In this case, the number of north steps in $D$ on
$x=k$ is the number of children of the $(r+1)$-st node of depth $\ell$. The
number of north steps in $D$ on $x=0$ is the number of children of the root.
We define $\Xi_{\mathrm{bounce}}(T)=D$. The map $\Xi_{\mathrm{bounce}}$ is a
bijection from $\mathcal{T}_{n}$ to $\mathcal{D}_{n}$ for all $n\geq 1$ (see
Lemma 3.18 in [CFM20]), and we denote its inverse by
$\Lambda_{\mathrm{bounce}}=\Xi_{\mathrm{bounce}}^{-1}$.
###### Proposition 3.7 (Theorem IV of [CFM20]).
Let $\zeta$ be the zeta map from $\mathcal{D}_{n}$ to $\mathcal{D}_{n}$ with
$n\geq 1$. We have $\zeta=\Xi_{\mathrm{bounce}}\circ\Lambda_{\mathrm{steep}}$.
See Figure 2 for an illustration of 3.7.
Figure 2: The zeta map as composition of bijections mediated by trees. The
nodes of the same depth of the tree are grouped together.
###### Remark 1.
Albeit the same notation, the maps $\Xi_{\mathrm{steep}}$ and
$\Xi_{\mathrm{bounce}}$ are only special cases of the ones in [CFM20,
Construction 3.10 and Equation 14]. More precisely, the central objects in
[CFM20] are called _LAC trees_ , which are plane trees with non-root nodes
divided into regions algorithmically. We are only using the special case where
the region $d$ consists of nodes of depth $d$, which is also the one used in
[CFM20, Section 3.3] to prove 3.7.
###### Remark 2.
The bijection $\Xi_{\mathrm{steep}}$ is classical, except that we do the
contour walk from right to left instead of from left to right. The bijection
$\Xi_{\mathrm{bounce}}$ is quite close to the classical bijection between
plane trees and Łukasiewicz words (see [FS09, Section I.5.3]), which can be
seen as sequences $(x_{0},x_{2},\ldots,x_{n-1})$ with $x_{i}\geq-1$ such that
$\sum_{i=0}^{k}x_{i}\geq 0$ for $0\leq k<n-1$, and the sum of all $x_{i}$’s is
$-1$. However, in the classical bijection of Łukasiewicz, we deal with nodes
in a depth-first order, but in $\Xi_{\mathrm{bounce}}$ they are processed in a
breadth-first order.
## 4 Unit interval posets and the zeta map
In the following, we give the two representations of unit interval posets as
Dyck paths, and we detail how they are related to the bijections detailed in
Section 3, leading to a proof of our main result (4.6).
We start by the first representation, which was first defined implicitly using
anti-adjacency matrices of unit interval posets in [SR03], only involving the
poset structure. The following form was first given in [CG18].
###### Construction 4.1.
Given $P=([n],\preceq_{S})$ a unit interval poset with $S\subset\mathbb{R}$,
we define a Dyck path $D$ as follows. Let $S^{+}=\\{x+1\mid x\in S\\}$.
Without loss of generality, we can choose $S$ such that $S\cap
S^{+}=\varnothing$. Then suppose that $S\cup S^{+}=\\{y_{1}<\cdots<y_{2n}\\}$.
For $i\in[2n]$, the $i$-th step of $D$ is $\uparrow$ if $y_{i}\in S$, and is
$\rightarrow$ otherwise. We define $\varphi(P)=D$. See Figure 3 for an
example.
Although defined here using the set $S$, the map $\varphi$ in 4.1, with this
form given in [CG18] and proved in Lemma 5.7 therein to be equivalent to the
original definition, does not depend on the choice of $S$.
Figure 3: Example of the bijections $\varphi$ and $\psi$
###### Proposition 4.2.
We have $\varphi\circ\Xi_{\mathrm{poset}}=\Xi_{\mathrm{bounce}}$, and
$\varphi$ is a bijection from $\mathcal{P}_{n}$ to $\mathcal{D}_{n}$ for all
$n\geq 1$.
###### Proof.
Let $T\in\mathcal{T}_{n}$ and $D=\Xi_{\mathrm{bounce}}(T)$. We take
$P=\Xi_{\mathrm{poset}}(T)=([n],\preceq_{S})\in\mathcal{P}_{n}$ with
$S=\\{x_{1}<\cdots<x_{n}\\}$ to be that given in 3.2. Let
$D^{\prime}=\varphi(P)$. To show that $D^{\prime}=D$, we first notice that a
Dyck path is determined by the number of north steps on each line $x=k$ for
$0\leq k\leq n-1$. Thus, we only need to show that for each $k$, $D^{\prime}$
has the same number of north steps on $x=k$ as in $D$ given in 3.6.
For the line $x=0$, we observe that the number of elements in $S$ strictly
smaller than $x_{1}+1$ must be those with integer part equal to $1$, thus
those from the children of the root of $T$. For $1\leq k\leq n-1$, from 4.1,
we know that the number of north steps on the line $x=k$ is the number of
elements $x_{i}$ such that $x_{k-1}+1<x_{i}<x_{k}+1$, as $x_{k}+1$ corresponds
to the $k$-th east step in $D$. Let $u_{k}$ be the node in $T$ corresponding
to $x_{k}$. By 3.3, the nodes in $T$ corresponding to such $x_{i}$ are
children of $u_{k}$, meaning that the total number of north steps on $x=k$ is
the number of children of $u_{k}$. Furthermore, suppose that $u_{k}$ is of
depth $\ell$, then we can write $k=s_{\ell}-r$ with $0\leq r\leq\alpha_{\ell}$
as in 3.6. We know that $u_{k}$ is the $(r+1)$-st node of depth $\ell$ in $T$,
as in 3.1 the values corresponding to nodes of the same depth are decreasing
from left to right. We thus conclude that $D=D^{\prime}$, and
$\varphi\circ\Xi_{\mathrm{poset}}=\Xi_{\mathrm{bounce}}$. As both
$\Xi_{\mathrm{bounce}}$ (Lemma 3.18 in [CFM20] and $\Xi_{\mathrm{poset}}$
(3.4) are bijections, we conclude that $\varphi$ is also a bijection. ∎
The second presentation was first defined in [GPMR14] for $(3+1)$-free posets,
and it takes a simpler form for unit interval posets.
###### Construction 4.3.
For $n\geq 1$, let $D\in\mathcal{D}_{n}$ be a Dyck path of size $n$. Its _area
vector_ , denoted by $\operatorname{Area}(D)$, is a vector
$(a_{1},\ldots,a_{n})$ with $a_{i}$ the number of full unit squares with the
upper edge on $y=i$ between $D$ and the diagonal $y=x$. We then define a poset
$P=([n],\preceq)$ by taking $i\prec j$ for $i,j\in[n]$ such that
* •
Either $a_{i}+2\leq a_{j}$;
* •
Or $a_{i}+1=a_{j}$ and $i<j$.
We define $\psi(D)=P$, and its inverse $\psi^{-1}(P)=D$. See Figure 3 for an
example.
The following result ensures that $\psi$ is well-defined.
###### Proposition 4.4 (Special case of Theorem 5.2 in [MMS]).
The map $\psi$ is a bijection from Dyck paths to unit interval posets
preserving sizes.
###### Proposition 4.5.
We have $\psi=\Xi_{\mathrm{poset}}\circ\Lambda_{\mathrm{steep}}$.
###### Proof.
Let $P=\psi(D)$, $T=\Lambda_{\mathrm{steep}}(D)$ and
$P^{\prime}=\Xi_{\mathrm{poset}}(D)$. We write $P=([n],\preceq)$ and
$P^{\prime}=([n],\preceq_{S})$, with $S$ constructed in 3.2.
For $i\in[n]$, let $u_{i}$ be the $i$-th non-root node in $T$ that is visited
in the clockwise contour walk of $T$. Suppose that
$\operatorname{Area}(D)=(a_{1},\ldots,a_{n})$. By 3.5, the depth of $u_{i}$ is
$a_{i}+1$. We now define a partial order $\preceq_{T}$ on non-root nodes in
$T$ by taking $u_{i}\preceq_{T}u_{j}$ if and only if $i\preceq j$ in $P$. We
denote by $d(v)$ the depth of $v$ in $T$. By 4.3, two non-root nodes $v,w$ in
$T$ satisfy $v\prec_{T}w$ if and only if
* •
Either $d(v)+2\leq d(w)$;
* •
Or $d(v)+1=d(w)$, and the parent of $w$ is either $v$ or on the left of $v$.
Now, for $v$ a non-root node in $T$, we take $x_{v}$ defined in (1). Suppose
that there are two non-root nodes $v,w$ of $T$ such that $x_{v}+1<x_{w}$. Then
we have $d(v)+1\leq d(w)$. There are two possibilities: either $d(v)+1=d(w)$,
or $d(v)+2\leq d(w)$. In the first case, let $w^{\prime}$ be the parent of
$w$, we have $x_{w^{\prime}}+1<x_{w}$. By 3.3, we know that $x_{v}\leq
w_{w^{\prime}}$. From (1), we know that either $v=w^{\prime}$ or $v$ is on the
right of $w^{\prime}$, as $x_{v}$ is decreasing from left to right for nodes
in $T$ of the same depth. Combining with the second case, we conclude that
$x_{v}+1<x_{w}$ if and only if $v\prec_{T}w$. By 3.2, we have $P\cong
P^{\prime}$, which concludes the proof. ∎
###### Theorem 4.6 (Conjecture 7.1 in [MMS]).
We have $\varphi\circ\psi=\zeta$.
###### Proof.
This is a consequence of 3.7, 4.2 and 4.5. ∎
## References
* [AKOP02] G. E. Andrews, C. Krattenthaler, L. Orsina, and P. Papi. ad-nilpotent $\mathfrak{b}$-ideals in ${\rm sl}(n)$ having a fixed class of nilpotence: Combinatorics and enumeration. Trans. Amer. Math. Soc., 354(10):3835–3853, 2002.
* [CFM20] C. Ceballos, W. Fang, and H. Mühle. The Steep-Bounce Zeta Map in Parabolic Cataland. J. Combin. Theory Ser. A, 172:105210, 2020.
* [CG18] A. Chavez and F. Gotti. Dyck paths and positroids from unit interval orders. J. Combin. Theory Ser. A, 154:507–532, 2018.
* [FS09] Ph. Flajolet and R. Sedgewick. Analytic combinatorics. Cambridge University Press, Cambridge, 2009.
* [GPMR14] M. Guay-Paquet, A. H. Morales, and E. Rowland. Structure and Enumeration of $(3+1)$-Free Posets. Ann. Comb., 18(4):645–674, 2014.
* [GST] F. Gélinas, A. Segovia, and H. Thomas. Proof of a conjecture of Matherne, Morales, and Selover on encodings of unit interval orders. arXiv:2212.12171 [math.CO].
* [Hag08] J. Haglund. The $q,t$-Catalan Numbers and the Space of Diagonal Harmonics, volume 41. American Mathematical Society, Providence, RI, 2008.
* [MMS] J. P. Matherne, A. H. Morales, and J. Selover. The Newton polytope and Lorentzian property of chromatic symmetric functions. arXiv:2201.07333 [math.CO].
* [Sco64] D. Scott. Measurement structures and linear inequalities. J. Math. Psychol., 1(2):233–247, 1964.
* [SR03] M. Skandera and B Reed. Total nonnegativity and $(3+1)$-free posets. J. Combin. Theory Ser. A, 103(2):237–256, 2003.
* [SS93] Richard P Stanley and John R Stembridge. On immanants of Jacobi-Trudi matrices and permutations with restricted position. J. Combin. Theory Ser. A, 62(2):261–279, 1993.
* [WF57] R. L. Wine and J. E. Freund. On the Enumeration of Decision Patterns Involving $n$ Means. Ann. Math. Stat., 28(1):256–259, 1957.
|
# Experimental Contexts _Can_ Facilitate Robust Semantic Property Inference in
Language Models, but Inconsistently
Kanishka Misraτ Allyson Ettingerα Kyle Mahowaldτ
τThe University of Texas at Austin αAllen Institute for Artificial
Intelligence
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Recent zero-shot evaluations have highlighted important limitations in the
abilities of language models (LMs) to perform meaning extraction. However, it
is now well known that LMs can demonstrate radical improvements in the
presence of experimental contexts such as in-context examples and
instructions. How well does this translate to previously studied meaning-
sensitive tasks? We present a case-study on the extent to which experimental
contexts can improve LMs’ robustness in performing property
inheritance—predicting semantic properties of novel concepts, a task that they
have been previously shown to fail on. Upon carefully controlling the nature
of the in-context examples and the instructions, our work reveals that they
can indeed lead to non-trivial property inheritance behavior in LMs. However,
this ability is inconsistent: with a minimal reformulation of the task, some
LMs were found to pick up on shallow, non-semantic heuristics from their
inputs, suggesting that the computational principles of semantic property
inference are yet to be mastered by LMs.
Experimental Contexts _Can_ Facilitate Robust Semantic Property Inference in
Language Models, but Inconsistently
Kanishka Misraτ Allyson Ettingerα Kyle Mahowaldτ τThe University of Texas at
Austin αAllen Institute for Artificial Intelligence<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
## 1 Introduction
Carefully controlled behavioral analyses on meaning-sensitive tasks have
revealed holes in the ability of language models (LMs) to demonstrate robust
meaning extraction and use (Pandia and Ettinger, 2021; Elazar et al., 2021;
Schuster and Linzen, 2022; Misra et al., 2023; Kim and Schuster, 2023, i.a).
However, since a large subset of these investigations uses zero-shot
evaluation as the primary methodology, there are growing concerns that they do
not paint a complete picture of LMs’ abilities (Lampinen, 2022; Sinclair et
al., 2022; Sinha et al., 2023). Conclusions that LMs lack a particular ability
may be overhasty if it turns out the ability is easily accessed through in-
context learning, different question formulations, or particular instructions
(Lampinen, 2022; Wei et al., 2022).
Figure 1: We prompt LMs with in-context examples that are compatible with
both, robust property inheritance, as well as position-based heuristics. At
test time, we evaluate the model on cases where the heuristics support
desirable behavior and on cases where they do not. We use stimuli from comps
and its reformulation as a QA task (comps-qa).
Our focus111Our code can be found in this link in this paper is a particularly
challenging data set for meaning-sensitive behavior: comps (Misra et al.,
2023), a dataset of minimal pair sentences that tests the ability of LMs on
property knowledge of everyday concepts (a beaver/gorilla has a flat tail) and
their inheritance for novel concepts (a wug is a beaver/gorilla. therefore a
wug has a flat tail). Contemporary LMs failed miserably on the hardest subset
of the comps stimuli, the examples of which contain two novel concepts (wug
vs. dax), where only one of them inherits the target property (has a flat
tail):
* (1)
A wug is a beaver. A dax is a gorilla. Therefore, a wug/dax has a flat tail.
Given the success of LMs on a wide variety of complicated tasks, their utter
failure on this seemingly straightforward task remains puzzling. Here, we
systematically explore comps on five modern LMs ranging from 1.5–13B
parameters, varying (a) whether models are evaluated zero-shot or with
multiple examples and (b) whether or not instructions are present.
Unlike other minimal-pair datasets, using comps in an in-context learning
setting is non-trivial (and thus potentially informative). This is because the
task can be solved using a position-based heuristic. For example, in one
subset of comps, the target property is always attached to the first novel
concept—like in (1). Importantly, models’ failures on comps were shown to be
in part a result of models’ tendencies towards heuristic behavior: the
performance of LMs is particularly bad when the distractor (a dax is a
gorilla) is recent—i.e., autoregressive LMs show a recency bias in attributing
properties to novel concepts. In that sense, comps follows a rich body of work
in which tasks are set up in a manner that two types of generalization
mechanisms can lead to the same prediction, but only one of which is desirable
(McCoy et al., 2019, 2020; Warstadt et al., 2020b; Mueller et al., 2022; Si et
al., 2023; Mueller et al., 2023).
We find that experimental contexts, as operationalized using in-context
examples and instructions, can in fact demonstrate robust improvements in LMs’
property inheritance behavior as measured by the stimuli in comps. However,
this improvement comes with a caveat: With a minimal reformulation of comps
into a QA task, where there is a direct link between the LMs’ output space and
the features of the input that control the heuristic, LMs show a strong
preference towards the heuristic, and are therefore at chance. This
discrepancy suggests that the improvements on the original task do not
necessarily indicate that the models have successfully mastered the reasoning
ability required to perform property inheritance, which remains a key
challenge for them.
## 2 Methodology
#### Dataset
We use the most difficult subset of the comps dataset (Misra et al.,
2023)—comps-wugs-dist—for our experiments. This dataset contains 13,828
sentence pairs of the form similar to (1), constructed using 152 animal
concepts and 991 properties.
#### Stimuli re-design
We take a number of steps to minimize noise from other (likely
uninterpretable) heuristics beyond the ones we have set out to target. First,
we enforce that the concepts and properties that appear in the in-context
examples are disjoint from ones that are used in tests. To this end, we sample
50 concepts and their relevant properties and reserve it for our in-context
examples, leaving the rest to be sampled for our test set. We also enforce
this constraint for our novel concepts—i.e., all in-context examples contain
different nonce words, and the collection of nonce words for the in-context
examples and the test set is disjoint. Furthermore, we counterbalance the
nonce words in the test set in a manner that having a bias towards one of them
would lead to chance performance. We additionally also use multiple different
sets of in-context examples, to add variability and to ensure that the results
are not only due to one particular choice of in-context examples. In total, we
use 10 different in-context learning example sets, each containing 6 different
comps stimuli. For our test set, we use a constant set of 256 unique pairs
sampled from our pool of stimuli containing unused concepts and properties.
#### Heuristics
Our most important design decision is to consider two distinct sets of
stimuli—each separately making available the two types of heuristics that the
LMs could rely upon: first-correct and recent-correct, where the property is
inherited by the first and the most recent novel concept, respectively. That
is, for the same set of in-context examples, we have a version where the first
concept is correct like in (1), and one where the most recent concept is
correct:
* (2)
A wug is a gorilla. A dax is a beaver. Therefore, a wug/dax has a flat tail.
For each type of in-context stimuli, we similarly have two versions of test
stimuli: one that is consistent with the target heuristic, and one that is
not. That is, a test example that is consistent with the first-correct
heuristic will also have its first concept be the one that inherits the
property in question, while one which is inconsistent will have the most
recent concept be the inheritor of the property. Therefore, a model that shows
a preference for a given heuristic will succeed only on one test set and
succumb on the other, while a model that is robust to the heuristics will
succeed on both.
#### Reformulation into QA
The original comps stimuli test for property inheritance using declarative
statements, where models are tested for the log-probability they asign to the
property (has a flat tail) given either of the two concepts (wug vs. dax).
Here we additionally consider an alternate formulation of comps as a question
answering task (comps-qa), where we make the property explicit in the prompt
to the model and instead ask which of the two concepts possesses it:
* (3)
A wug is a beaver. A dax is a gorilla. Question: Which one of them has a flat
tail? Answer: wug/dax
Since the shallow heuristics we consider are controlled by the relative
ordering of the novel concepts, this formulation of the task directly allows
us to link the models’ output space (the novel concepts) to the heuristics
(positions).
#### Testing setup
For the original comps setting we follow Misra et al. (2023) and compare the
log-probability of the property phrase given the correct vs. the incorrect
prefix. For comps-qa however, since we have a constant prefix (same premises
and question), we evaluate the relative log-probability of the two novel
concepts, only one of which is the correct answer. Accuracy in both cases is
the proportion of cases the correct surface form was assigned relatively
higher log-probability. Since we use pairwise comparisons throughout, chance
performance is 50%.
#### Instructions
We consider four different kinds of instruction templates, with varying levels
of detail (see appendix 0.B) per formulation (comps and comps-qa). In our
experiments we report results on the instruction that gives the best average
performance for a given model.
#### LMs tested
We evaluated 5 different open-source LMs, all of which are decoder-only, and
were accessed using the huggingface hub (Wolf et al., 2020): GPT-2 XL (Radford
et al., 2019); OPT-6.7b (Zhang et al., 2022); Llama-2 (we used the 7b and the
13b versions; Touvron et al., 2023); and Mistral-7b (Jiang et al., 2023).
Details about the models can be found in the appendix.
Figure 2: Overall results from our experiments testing LMs on comps and comps-
qa using in-context examples, with and without instructions. Results are
aggregated across both heuristics: first-correct and recent-correct. Error
bars are over different sets of in-context examples. All models start off near
chance in the 0-shot case, but many improve as more examples are given. Some
(e.g., GPT-2 XL on comps-qa) show strongly heuristically driven behavior, as
evidenced by the diverging performance on items where heuristics work and
those where they do not. Figures 3 and 4 show fine-grained results.
## 3 Analyses and Results
We evaluate on comps and comps-qa, with and without instructions. In each
case, we progressively supply 0 through 6 in-context examples, allowing us to
track the dynamics of the models’ performance with an increasing amount of
demonstrations. Together with our separate types of test sets and heuristics
encoded in the in-context examples, along with five different instruction
settings (four with and one without) we run 2420 experiments per LM. We
hypothesize that LMs would be more sensitive to the positional heuristics in
comps-qa because of the clear link between their output space and the relative
position of the novel concepts—the feature that controls our target
heuristics.
Figure 2 shows accuracies of the tested LMs on our four different comps
settings as a function of the number of in-context examples provided to them,
for both: cases where the heuristics are consistent with success on the test
set, and cases where they are not. We also show an additional curve denoting
the average performance across the two types of test sets to paint an overall
picture of the models’ performance. In this figure, the extent to which a
model relies on heuristic is indicated by the gap between the dotted
($\newmoon$) and the dashed ($\blacktriangle$) lines. A model that is robust
to the heuristics will have curves of both colors rise above chance, with no
gap between the two, while one that is prone to using heuristics will have its
dotted ($\newmoon$) curve be substantially greater than its dashed
($\blacktriangle$) curve.
#### Experimental context can improve attribution of properties to concepts…
On comps, models unsurprisingly start off at chance performance on average,
corroborating the previous findings of Misra et al. (2023). However, in the
presence of in-context examples and instructions, they are able to improve
monotonically as the number of in-context examples increases. It is worth
noting Llama-2-13b does occasionally show a slight preference for heuristics
in the absence of instructions (e.g., 84% vs. 62% when prompted with 2
examples). An intermediate conclusion that we draw here is that LMs can indeed
demonstrate non-trivial property inheritance on observing a few examples that
reflect that behavior.
#### …but not the attribution of concepts to properties
While experimental context seems to aid models in attributing properties to
the right concept in context, the same does not hold on comps-qa. Similar to
comps, models start off at chance performance on average with a zero-shot set
up, however, unlike in the case of comps, LMs seem to consistently prefer the
heuristics available in the prompt, showing worse than chance performance on
cases where the test set does not follow the heuristic. This is most apparent
for GPT-2 XL, OPT6.7b, and Llama-2-7b—here the gap between the accuracy for
cases where heuristics support performance on the test set and the accuracy
for cases where they do not almost always worsens with an increase in the
number of exemplars, on average. This is especially notable for OPT6.7b, which
attains perfect performance on cases where the heuristics match up with the
test set and at the same time it is close to 0% on cases where they do not. A
notable exception to this trend is Mistral-7b, which seems to be resilient to
the spurious heuristics, showing a net-positive improvement from the zero-shot
case, especially in the presence of instructions. Nevertheless in the absence
of instructions, it too shows a slight preference for position-based
heuristics—for instance, its accuracy with 6 in-context exemplars when the
heuristics support success on the test set is 82% and on cases where the
heuristics oppose the test set is 65%.
Our results suggest that LMs are more likely to show behavior that is
compatible with the use of positional heuristics when their output space
(choice between the two novel concepts) has a clear connection with positional
artifacts in their input (relative ordering of the novel concepts). This is
consistent with our hypothesis in about 8 out of 10 cases. When this link is
not clear and models must instead predict likely properties given a novel
concept (i.e., in comps), instructions and in-context examples do seem to lead
to robust performance. It is important to note that instructions alone do not
always account for the observed improvement—LMs’ performance on zero-shot
settings are consistently still at chance in all cases, suggesting that it is
the in-context examples that critically alter models’ output distribution to
support desirable property inference behavior.
## 4 Conclusion
In this work, we investigated the extent to which in-context examples and
instructions—key components that drive impressive performance in contemporary
LMs—can overcome important limitations of LMs at tests that have poked holes
in their ability to extract conceptual meaning from text. As a case study, we
analyzed how well such experimental contexts can improve LM abilities to
perform property inheritance (Murphy, 2002; Misra et al., 2023) in
context—binding of novel concepts to existing concepts, and endowing them with
valid property inferences as a result. Our findings suggest that mastery of
this ability has yet to be robustly achieved, and that LMs in general are
still prone to using shallower patterns in their context (when available)
rather than systematically extracting conceptual meaning. At the same time,
exploring precisely what makes Mistral less susceptible to heuristics will be
useful to design more robust LMs, which we leave for future work.
## 5 Limitations
#### Single dataset
A clear limitation of this work is that it exclusively focuses on a single
dataset: comps (Misra et al., 2023). So, a question that arises here is to
what extent are our findings localized to the chosen dataset vs. meaning-
sensitive evaluations in general. This would require a further non-trivial,
non-straightforward amount of work, since: (1) different meaning sensitive
evaluations focus on different (though equally useful) operationalizations of
meaning; and more importantly (2) not all prior work in this area focuses on a
standardized and well-defined usage of heuristics that is directly
transferable to the experimental setup we have used in this work (following
McCoy et al., 2019, 2020; Warstadt et al., 2020b; Mueller et al., 2022; Si et
al., 2023).
We do hope that our work contributes to the larger-scale vision of carefully
benchmarking different types of meaning extraction abilities in LMs in a
controlled manner.
#### Lack of mechanistic insight
Our work continues the long-standing precedent of using carefully constructed
behavioral experiments to conclude about the competence of LMs (Linzen et al.,
2016; Gulordava et al., 2018; Futrell et al., 2019; Ettinger, 2020; Warstadt
et al., 2020a) However, recent works have made impressive strides in
localizing the kinds of computations that give rise to the observed behavior
in LMs (Hanna et al., 2023; Wang et al., 2023, i.a.) Therefore, it is entirely
possible that our conclusions about the precise nature of computations carried
out by LMs can be greatly strengthened when supplemented by the methods
developed in these aforementioned works.
#### Single Language
Finally, this work only focuses on property inheritance problems stated in the
English language. This does little to contribute towards diversity in NLP
research.
## 6 Acknowledgments
We thank Tom McCoy and Andrew Lampinen for providing comments on a previous
version of the draft. (KM)2 acknowledge funding from NSF Grant 2104995 awarded
to Kyle Mahowald.
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## Appendix 0.A Dataset and implementation details
Our experiments use the stimuli from comps, released with an MIT License by
Misra et al. (2023), but with a modification that involves changing of the
nonce words to obey the constraint that the in-context examples all have
different nonce word pairs. To this end, we use the following nonce-words:
* •
In-context examples: wug, dax, fep, zek, blick, toma, kiki, glorp, bova, zup,
tufa, flib (counter-balanced)
* •
Test examples: gek, wif (counter-balanced)
### 0.A.1 Methodological details
Following comps, as well as the precedent set by a number of previous minimal
pair analyses (Linzen et al., 2016; Gulordava et al., 2018; Futrell et al.,
2019; Wilcox et al., 2019; Warstadt et al., 2020a; Hu et al., 2020), we use a
forced choice task to evaluate our LM subjects. Like in comps, we compare the
log-probability of the property phrase (here, has a flat tail) given the
choice of left contexts (which indicate whether the right vs. the wrong
concept has the property). For example, we measure:
$\displaystyle\log P_{\theta}(\textit{has a flat tail}\mid$ a gek is a beaver.
a wif is a $\displaystyle\textit{ gorilla. therefore, a
{{\color[rgb]{0.23828125,0.5234375,0.77734375}\definecolor[named]{pgfstrokecolor}{rgb}{0.23828125,0.5234375,0.77734375}gek}/{\color[rgb]{0.9453125,0.76171875,0.1953125}\definecolor[named]{pgfstrokecolor}{rgb}{0.9453125,0.76171875,0.1953125}wif}}}),$
and for comps-qa, we compare the relative probabilities of the two novel
concepts given a fixed left prefix which contains a question about the
property. For example, we measure:
$\displaystyle\log
P_{\theta}(\textit{{{\color[rgb]{0.23828125,0.5234375,0.77734375}\definecolor[named]{pgfstrokecolor}{rgb}{0.23828125,0.5234375,0.77734375}gek}/{\color[rgb]{0.9453125,0.76171875,0.1953125}\definecolor[named]{pgfstrokecolor}{rgb}{0.9453125,0.76171875,0.1953125}wif}}}\mid$
a gek is a beaver. a wif is a gorilla. Question: Which one of them has a
$\displaystyle\textit{ flat tail? {Answer:}})$
In both cases above, gek is the concept that should inherit the property.
While these examples show the zero-shot case, cases with in-context examples
and instructions simply add more context to the prefix, therefore the surface
form of the output space remains the same regardless of the number of in-
context examples or the presence of instructions.
Log-probabilities for all models were accessed using minicons (Misra,
2022),222https://github.com/kanishkamisra/minicons a library that wraps around
transformers (Wolf et al., 2020) by huggingface, and is written in pytorch.
For our experiments with Llama-13B, we quantize the model to 4-bits in order
to fit it onto a single GPU. All experiments were run on a cluster with 4
NVIDIA A40 GPUs, though each individual experiment on a model was computed on
a single A40 GPU.
### 0.A.2 Model Metadata
Model | Params | Pre-training Tokens | Vocab size
---|---|---|---
GPT-2 XL | 1.5B | 8B | 50,257
OPT-6.7b | 6.7B | 180B | 50,272
Llama-2-7b | 7B | 2T | 32,000
Llama-2-13b | 13B | 2T | 32,000
Mistral-7b | 7B | ? | 32,000
Table 1: Overview of the LMs used in this work. ‘?’ indicates that the given
value was not made available in the LM’s release.
Table 1 shows the LMs used in this work, along with their total parameters,
tokens encountered during training, and vocabulary size.
## Appendix 0.B Instructions
Tables 2, 3, 4, 5 show our instruction templates.
comps version | Instruction Template
---|---
comps | Given a pair of statements that introduce novel entities as types of real world animals, write a true statement about the properties of the novel entities:
{exemplars} (omitted in zero-shot)
{test-stimulus}
comps-qa | Given a pair of statements that introduce novel entities as types of real world animals, answer the question that follows:
{exemplars} (omitted in zero-shot)
{test-stimulus}
Table 2: Instructions for comps and comps-qa with instruction type: “minimal” comps version | Instruction Template
---|---
comps | Some aliens have come to earth, and it turns out they have their own language for talking about our animals here on Earth. Your job is to help the aliens learn about our Earthling animals by giving them some information about the animals.
Let’s get started:
{exemplars} (omitted in zero-shot)
{test-stimulus}
comps-qa | Some aliens have come to earth, and it turns out they have their own language for talking about our animals here on Earth. Your job is to help the aliens learn about our Earthling animals by answering some questions about them.
Let’s get started:
{exemplars} (omitted in zero-shot)
{test-stimulus}
Table 3: Instructions for comps and comps-qa with instruction type: “aliens”
comps version | No. of shots | Instruction Template
---|---|---
comps | Zero-shot | It is important to know and reason about the properties of entities in the world. The following is a pair of premise statements that introduce novel entities as new types of real world animals. Your task is to make a conclusion about the properties of one of the entities by reasoning over the premise statements.
{test_stimulus}
Few-shot | It is important to know and reason about the properties of entities in the world. The following example(s) show a pair of premise statements that introduce novel entities as new types of real world animals, followed by another statement that attributes a property to one of the entities introduced in the premise statements.
Examples:
{examples}
Here is another pair of premise statements. Your task is to make a conclusion
about the properties of one of the entities by reasoning over the premise
statements.
{test_stimulus}
comps-qa | Zero-shot | It is important to know and reason about the properties of entities in the world. The following is a pair of premise statements that introduce novel entities as new types of real world objects. The statements are followed by a question that asks which novel entity in the premise can a specific property can be attributed to. Answer the question by reasoning over the premise statements.
{test_stimulus}
Few-shot | It is important to know and reason about the properties of entities in the world. The following example(s) show a pair of premise statements that introduce novel entities as new types of real world animals. The statements are followed by a question that asks which novel entity in the premise can a specific property can be attributed to, and the answer to the question, obtained by reasoning over the premise statements.
Examples:
{examples}
Here is another pair of premise statements. Answer the question that follows.
{test_stimulus}
Table 4: Instructions for comps and comps-qa with instruction type:
“Detailed-1”
comps version | No. of shots | Instruction Template
---|---|---
comps | Zero-shot | It is important to know and reason about the properties of entities in the world. The following is a pair of premise statements that introduce novel entities as new types of real world animals. Your task is to write a true statement about the properties of the novel entities.
{test_stimulus}
Few-shot | It is important to know and reason about the properties of entities in the world. The following example(s) show a pair of premise statements that introduce novel entities as new types of real world animals, followed by another statement that attributes a property to one of the entities introduced in the premise statements.
Examples:
{examples}
Here is another pair of premise statements. Your task is to write a true
statement about the properties of the novel entities.
{test_stimulus}
comps-qa | Zero-shot | It is important to know and reason about the properties of entities in the world. The following is a pair of premise statements that introduce novel entities as new types of real world animals. The statements are followed by a question that asks which of the introduced entities a specific property can be attributed to. Answer the question by reasoning over the premise statements.
{test_stimulus}
Few-shot | It is important to know and reason about the properties of entities in the world. The following example(s) show a pair of premise statements that introduce novel entities as new types of real world animals. The statements are followed by a question that asks which of the introduced entities a specific property can be attributed to, and the answer to the question, obtained by reasoning over the premise statements.
Examples:
{examples}
Here is another pair of premise statements. Answer the question that follows.
{test_stimulus}
Table 5: Instructions for comps and comps-qa with instruction type:
“Detailed-2”
## Appendix 0.C Fine-grained results
While Figure 2 shows results aggregated over both types of heuristics that we
have used in this work, we additionally display finer-grained, heuristics-wise
results in this section. Again, in each of these plots, the extent to which a
model relies on heuristic is indicated by the gap between the dotted
($\newmoon$) and the dashed ($\blacktriangle$) lines. This is now separately
shown for each of our heuristics. Figure 3 shows results on comps with and
without instructions for both the heuristics, and similarly Figure 4 shows
results on comps-qa with and without instructions for both the heuristics.
(a) comps
(b) comps with Instructions
Figure 3: Fine-grained results on COMPS as a function of the number of in-
context examples (with and without instructions).
(a) comps-qa
(b) comps-qa with Instructions
Figure 4: Fine-grained results on COMPS-QA as a function of the number of in-
context examples (with and without instructions).
|
# Mixing matters
Douglas Rennehan1
1Department of Physics & Astronomy, University of Victoria, BC V8P 5C2, Canada
E-mail<EMAIL_ADDRESS>(DR)
(Accepted XXX. Received YYY; in original form ZZZ)
###### Abstract
All hydrodynamical simulations of turbulent astrophysical phenomena require
sub-grid scale models to properly treat energy dissipation and metal mixing.
We present the first implementation and application of an anisotropic eddy
viscosity and metal mixing model in Lagrangian astrophysical simulations,
including a dynamic procedure for the model parameter. We compare these two
models directly to the common Smagorinsky and dynamic variant. Using the mesh-
free finite mass method as an example, we show that the anisotropic model is
best able to reproduce the proper Kolmogorov inertial range scaling in
homogeneous, isotropic turbulence. Additionally, we provide a method to
calibrate the metal mixing rate that ensures numerical convergence. In our
first application to cosmological simulations, we find that all models
strongly impact the early evolution of galaxies leading to differences in
enrichment and thermodynamic histories. The anisotropic model has the
strongest impact, with little difference between the dynamic and the constant-
coefficient variant. We also find that the metal distribution functions in the
circumgalactic gas are significantly tighter at all redshifts, with the
anisotropic model providing the tightest distributions. This is contrary to a
recent study that found metal mixing to be relatively unimportant on
cosmological scales. In all of our experiments the constant-coefficient
Smagorinsky and anisotropic models rivaled their dynamic counterparts,
suggesting that the computationally inexpensive constant-coefficient models
are viable alternatives in cosmological contexts.
###### keywords:
methods: numerical – turbulence – hydrodynamics – galaxies: evolution –
galaxies: general
††pubyear: 2021††pagerange: Mixing matters–B
## 1 Introduction
Galaxies form and evolve in tempestuous gaseous environments where
hydrodynamics, radiative cooling, and gravity synergize to produce rich
emergent phenomena on a myriad of spatial scales. The immense dynamic range of
scales involved and their interconnectedness prove to be limiting factors in
advancing our understanding of the complete picture of galaxy evolution (see
Naab & Ostriker 2017 for an excellent review). At the forefront of the issue
is hydrodynamical turbulence as it is a multi-scale, non-linear phenomenon
that occurs in almost all galactic environments — directly impacting our
theoretical understanding of galaxy evolution.
While the importance of turbulence in the interstellar medium of galaxies has
been long recognized (see Elmegreen & Scalo 2004 and Scalo & Elmegreen 2004
for reviews), only recently has the role of turbulence in halo gas come under
careful consideration. Indeed, both the circumgalactic medium (CGM) of $L^{*}$
galaxies and the intracluster medium (ICM) of groups and clusters of galaxies
show signs of turbulence playing an important role in their evolution (see,
for example, Prasad et al. 2018 and Wang et al. 2020). Observationally, there
is evidence of complex kinematic structure in the CGM of $L^{*}$-galaxies that
emerged through the revolutionary Cosmic Origins Spectrograph halo survey on
the Hubble space telescope (COS-halos; Tumlinson et al. 2013, 2017). For
instance, Werk et al. (2016) found that turbulent velocities of $50$-$75$
$\mathrm{km}\,\mathrm{s}^{-1}$ explains the broadening of absorption lines in
the CGM that is not otherwise explainable that has subsequently been confirmed
using numerical studies (Buie et al., 2020). Moving up in mass scale, the ICM
also shows evidence of turbulence through indirect observational methods such
as X-ray surface brightness and Sunyaev-Zeldovich fluctuations (Zhuravleva et
al., 2014; Pinto et al., 2015; Zhuravleva et al., 2015; Khatri & Gaspari,
2016; Zhuravleva et al., 2018).
There are two main drivers of turbulence in galactic environments: (a) global
outflows that emerge from star formation processes and supermassive black
holes (SMBHs) within galaxies (Prasad et al., 2015, 2018; Karen Yang &
Reynolds, 2016; Bourne & Sijacki, 2017; Fielding et al., 2017; Fielding et
al., 2018; Sokołowska et al., 2018; Li et al., 2020) and (b) shearing motion
driven by gas in-fall during structure formation (Dekel et al., 2009; Vazza et
al., 2010, 2012; Vazza et al., 2017; Wittor et al., 2017; Bennett & Sijacki,
2020), mergers (ZuHone et al., 2013), and ram-pressure stripping of galaxies
moving through the ICM (Ruggiero & Lima Neto, 2017; Simons et al., 2020). In
both the CGM and ICM, turbulence could provide additional pressure support
(Poole et al., 2006; Vazza et al., 2018; Lochhaas et al., 2020) that prevents
the gas from collapsing and rapidly converting into stars as well as a
physical mechanism to transport energy and metals throughout gas directly –
impacting the cooling profile, star formation cycle, and metal distribution
functions (Shen et al., 2010; Shen et al., 2012, 2013; Brook et al., 2014;
Sokołowska et al., 2018; Escala et al., 2018; Tremmel et al., 2019; Rennehan
et al., 2019; Hafen et al., 2019, 2020). Therefore, understanding the nature
of turbulence is imperative to understand the complete picture of galaxy
evolution.
In a general sense, turbulence results from a large-scale injection of kinetic
energy into a gas, followed by the eventual transformation of that kinetic
energy into thermal energy through a small-scale viscous dissipation process.
Physically, the injected kinetic energy on large scales breaks the gas into
eddies of smaller and smaller sizes until the energy thermalises at the
dissipation scale. This transfer of kinetic energy from the large scale to the
small scale is the turbulent cascade and the ratio between the energy
injection scale $L$ and the viscous dissipation scale, $\eta$, determines the
dynamic range of scales within the cascade. The dissipation scale $\eta$ is
proportional to viscosity of the gas; a more viscous fluid has a larger $\eta$
and a smaller range over which the turbulent cascade may exist (Landau &
Lifshitz, 1987). It is in the cascade where the redistribution of energy and
metals occurs as the eddies transport the fluid properties as mixing
approaches a homogeneous state.
Understanding of galaxy evolution requires treating turbulence simultaneously
with the complexity of the astrophysics — a fact that demands the use of
numerical simulations. Contemporary computational power limits both the
spatial resolution, $h$, of cosmological simulations and the volume of the
Universe, $V$, that we may simulate. To obtain a statistically sound sample of
galaxies requires a representative (large) volume, while capturing the
relevant astrophysics, such as star formation, stellar and supermassive black
hole feedback, requires a small spatial resolution. Therefore, the dynamic
range of a simulation, $L/h$ (where $L\sim V^{1/3}$), is the limiting factor
for cosmological simulations. As a result, those important processes related
to star formation and supermassive black holes usually occur on scales much
smaller than $h$ and are, therefore, unresolved in most simulations. That fact
necessitates sub-grid astrophysical models in cosmological simulations that
reduce the relevant small-scale physics into a coarse-grained representation,
allowing the physics below a scale $h$ to interact with those on scales
$\ell$, where $L>\ell>h$ (for a more in-depth overview, see Somerville & Davé
2015). While there are many successful simulations that use a variety of sub-
grid assumptions to broadly reproduce galaxy populations (e.g. Guedes et al.
2011; Hopkins et al. 2014; Vogelsberger et al. 2014; Schaye et al. 2015; Genel
et al. 2014; Davé et al. 2017; Davé et al. 2019; Tremmel et al. 2019;
Pillepich et al. 2018), one aspect that is often overlooked is the treatment
of sub-grid turbulence.
Therein lies a problem: in hydrodynamical simulations, the physical
dissipation scale $\eta$ is almost always much smaller than the resolution
scale, $\eta\ll h$ (Pope, 2000). For that reason, the kinetic energy that is
flowing in the turbulent cascade reaches some scale $H\gtrsim h$ where it may
no longer progress. If the numerical viscosity of the hydrodynamical method is
not sufficiently strong to thermalise the kinetic energy, there will be a
build-up of kinetic energy at that scale $H\gtrsim h\gg\eta$ (Garnier et al.,
2009). The kinetic energy build-up is a completely unphysical representation
of turbulence and not only impacts the energetics, but the large scale flow
properties such as the redistribution of metals.
Although not explicitly stated, many cosmological hydrodynamical simulation
studies implicitly assume that numerical dissipation is sufficient to mimic
sub-grid turbulence. However, numerical dissipation is not be sufficient to
reduce the kinetic energy build-up and may not represent turbulent flow
statistics in all cases (Sagaut, 2006). This has led to the development of
sub-grid turbulence models whose purpose is to eliminate the build-up of
kinetic energy at the resolution scale by providing a new viscous dissipation
mechanism (or transport mechanism, in the case of metals).
In simulations that use Eulerian (i.e. grid-based) hydrodynamics, there has
been extensive effort to developing models for sub-grid turbulence (see Sagaut
2006; Garnier et al. 2009 and Schmidt 2015 for extensive reference lists). For
astrophysically-relevant work, we refer the reader to Scannapieco & Brüggen
(2008), Pan et al. (2013), Federrath (2013), Schmidt et al. (2014), and
Semenov et al. (2016) (for a review, see Schmidt 2015). In this paper, we
focus on the Lagrangian hydrodynamical methods.
In Lagrangian hydrodynamics, the fluid equations are approximated via fluid
elements that move with the flow. There are three main approaches that are
most common in cosmological simulations: smoothed particle hydrodynamics (SPH)
(Gingold & Monaghan, 1977; Lucy, 1977; Hernquist & Katz, 1989; Hopkins, 2013),
the moving-mesh method (MM) (Springel, 2010), and mesh-free methods (MF)
(Lanson & Vila, 2008a, b; Gaburov & Nitadori, 2011; Hopkins, 2015). Each of
these methods track fluid elements using different discretisation techniques
that lead to different levels of numerical dissipation. In SPH, there is no
inherent numerical dissipation and it, counter-intuitively, produces a deficit
of kinetic energy near the resolution scale rather than a build-up (Bauer &
Springel, 2012; Price, 2012). The MM and MF methods use Riemann solvers to
approximate the fluid equations of motion between neighbouring fluid elements.
Riemann solvers are generally diffusive due to their approximate nature, and
the build-up of kinetic energy is present in both methods on scales up to
$\sim 10$ times the resolution scale (Bauer & Springel, 2012; Hopkins, 2015).
A solution to kinetic energy build-up is to model the action of turbulent
eddies as a viscous process that diffuses momentum (and metals) and dissipates
kinetic energy via a diffusion equation and source term in the energy
equation, respectively. Usually the assumption is that the viscosity, or
diffusivity, depends on velocity fluctuations $v_{\mathrm{eddy}}$ near the
resolution scale $h$. The resulting diffusivity is $D\propto
hv_{\mathrm{eddy}}$, where $v_{\mathrm{eddy}}$ may be a characteristic
velocity, or velocity difference, within the neighbourhood of a fluid element
(Wadsley et al., 2008; Greif et al., 2009).
One particularly important choice of $v_{\mathrm{eddy}}$ is the Smagorinsky
model (Smagorinsky, 1963), which assumes that velocity shear fluctuations
drive dissipation and mixing through $v_{\mathrm{eddy}}\sim h|S^{*}|$, where
$|S^{*}|$ is the magnitude of the trace-free shear tensor. Wadsley et al.
(2008) showed, using SPH, that a Smagorinsky-like model for thermal energy
mixing improved resolved turbulent mixing in their hydrodynamical experiments
– a problem that plagued SPH for decades (Agertz et al., 2007). The
Smagorinsky model has been successfully used to treat metal and thermal energy
mixing in SPH (Wadsley et al., 2008; Shen et al., 2010; Shen et al., 2012;
Brook et al., 2014; Williamson et al., 2016; Tremmel et al., 2017; Wadsley et
al., 2017; Su et al., 2017) and has been extended to other Lagrangian
hydrodynamical methods, such as the MFM method (Rennehan et al., 2019). In
particular, the Smagorinsky model produces a reasonable description of
turbulence in isotropic scenarios (Colbrook et al., 2017), as well as
essential for understanding metal distribution functions of stellar
populations in dwarf galaxies (Escala et al., 2018), and understanding gas
flows in the CGM (Hafen et al., 2019, 2020). While the Smagorinsky model
improves mixing in Lagrangian simulations, it is important to consider two
assumptions in the model: (a) that shear fluctuations (i.e., changes in
$|S^{*}|$) always represent turbulence and (b) that the diffusive process acts
isotropically through the magnitude of the trace-free (i.e. ignoring
compression) shear tensor over the scale $h$. Are either of these assumptions
reasonable?
To address point (a), consider two incompressible fluids of different
densities in contact and in parallel opposing motion. Will one fluid encroach
the other? Will the flow become turbulent? Such a situation results in the
Kelvin-Helmholtz instability and, given certain conditions on the density and
velocities, we know that perturbations on the contact interface will grow
exponentially and mix the fluids, generating vorticity and turbulence
(Chandrasekhar, 1961). However, if the instability condition is not met then
there remains a large $|S^{*}|$ at the interace without turbulence. If we used
the simple assumption of a diffusivity $D\propto h|S^{*}|$ in a simulation we
would diffuse rapidly at the interface even though there are no turbulent
eddies! Piomelli & Liu (1995) recognized the problem and presented a solution:
a method of dynamically calculating the model coefficient at simulation time.
That method was employed in Rennehan et al. (2019) for the first time in
Lagrangian hydrodynamics and they found that the model significantly reduced
over-diffusion in non-turbulent shear flows, such as in rotating galactic
disks and the Kelvin-Helmholtz instability.
The second point (b) concerns the isotropy in $D$ and the discounting of
compression in $|S^{*}|$. Recently Hu & Chiang (2020) showed that a better
representation of sub-grid scale turbulence is obtained by treating the full
velocity tensor $\boldsymbol{\nabla}\otimes\boldsymbol{u}$ for the diffusivity
$\boldsymbol{D}$, now a tensor. The model is the gradient model (Clark et al.,
1979) and is a major change since diffusion now depends on the directionality
encoded in $\boldsymbol{\nabla}\otimes\boldsymbol{u}$ rather than equally in
every spatial direction. The trace of
$\boldsymbol{\nabla}\otimes\boldsymbol{u}$ is automatically included and,
therefore, compression is automatically handled111The trace of
$\boldsymbol{\nabla}\otimes\boldsymbol{u}$ is
$\boldsymbol{\nabla}\cdot\boldsymbol{u}$, the divergence of the flow. – an
important point for highly-compressible turbulence in cosmological flows.
However, Hu & Chiang (2020) used a common approach from the fluid mechanics
community to post-process driven turbulence simulations in order to check if
the model would have improved the results. Our goal is to implement the model
for the mesh-free finite mass method, a method growing in popularity, and
determine its feasibility at simulation time.
We introduce an implementation of the gradient model for Lagrangian
astrophysical simulations and additionally provide methods for computing the
model coefficient at simulation time. In Section 2 we provide a derivation of
the model, as well as a derivation of the dynamic procedure in Section 2.1
that allows calculation of the model coefficient at simulation time. Section 3
describes driven turbulence validation tests of the gradient model, included
at run time, for both eddy viscosity and metal mixing. As a first application,
we describe the qualitative impact of the eddy viscosity and metal mixing
model on cosmological gas phases in Section 4. We present our conclusions and
recommendations in Section 5.
## 2 The gradient model
In finite-mass Lagrangian hydrodynamics, such as the mesh-free finite mass
(MFM) method (Lanson & Vila, 2008a, b; Gaburov & Nitadori, 2011; Hopkins,
2015), the build-up of kinetic energy at the resolution scale demands an
additional dissipation mechanism. Additionally, metals follow the fluid mass
elements throughout the simulation volume and, therefore, the exchange of
metals between fluid elements due to sub-grid scale turbulent motion does not
occur. The crux of the issue is that discretisation of the fluid field leads
to damping out of the high-frequency turbulent fluctuations that should
continue down to the physical dissipation scale. It is useful to think of the
damping action as a high-pass filter acting on the fluid equations of motion.
By applying a general filter to the momentum conservation equation, it is
possible to derive the correct level of mixing that should occur between fluid
elements due to unresolved turbulence.
In general, the filtering action over a scalar field $f_{i}(\boldsymbol{r})$
can be represented as,
$\overline{f}_{i}(\boldsymbol{r})=\int_{D}f_{i}(\boldsymbol{r}^{\prime})G(|\boldsymbol{r}-\boldsymbol{r}^{\prime}|,h)d\boldsymbol{r}^{\prime},$
(1)
where $h$ is the smoothing scale over the domain $D$. We apply this to the
momentum equation to determine the correction terms due to unresolved
turbulence,
$\frac{\partial(\rho\boldsymbol{u})}{\partial
t}+\nabla\cdot(\rho\boldsymbol{u}\otimes\boldsymbol{u}+P\boldsymbol{I})=0,$
(2)
where $\rho$ is the gas density, $P$ is the pressure, and $\boldsymbol{u}$ is
the velocity vector. If we apply equation (1) to equation (2) it follows that,
assuming the filtering operation and derivatives commute,
$\frac{\partial(\overline{\rho\boldsymbol{u}})}{\partial
t}+\nabla\cdot(\overline{\rho\boldsymbol{u}\otimes\boldsymbol{u}}+\overline{P}\boldsymbol{I})=0.$
(3)
For simplicity, we switch to density-weighted variables such that
$\widetilde{\boldsymbol{u}}\equiv\overline{\rho\boldsymbol{u}}/\overline{\rho}$
and
$\frac{\partial(\overline{\rho}\tilde{\boldsymbol{u}})}{\partial
t}+\nabla\cdot(\overline{\rho}\widetilde{\boldsymbol{u}\otimes\boldsymbol{u}}+\overline{P}\boldsymbol{I})=0.$
(4)
The term $\widetilde{\boldsymbol{u}\otimes\boldsymbol{u}}$ is unknown at
simulation time because it relies on information below the resolution scale.
To put the equation in a more manageable form, we add
$\nabla\cdot(\overline{\rho}[\widetilde{\boldsymbol{u}}\otimes\widetilde{\boldsymbol{u}}-\widetilde{\boldsymbol{u}\otimes\boldsymbol{u}}])$
and rearrange,
$\frac{\partial(\overline{\rho}\widetilde{\boldsymbol{u}})}{\partial
t}+\nabla\cdot(\overline{\rho}\widetilde{\boldsymbol{u}}\otimes\widetilde{\boldsymbol{u}}+\overline{P}\boldsymbol{I})=\nabla\cdot(\overline{\rho}[\widetilde{\boldsymbol{u}}\otimes\widetilde{\boldsymbol{u}}-\widetilde{\boldsymbol{u}\otimes\boldsymbol{u}}]).$
(5)
Therefore, we define the sub-grid scalar flux $\boldsymbol{F}$,
$\boldsymbol{F}\equiv\overline{\rho}(\widetilde{\boldsymbol{u}\otimes\boldsymbol{u}}-\widetilde{\boldsymbol{u}}\otimes\widetilde{\boldsymbol{u}}),$
(6)
and retrieve a new equation,
$\frac{\partial(\overline{\rho}\widetilde{\boldsymbol{u}})}{\partial
t}+\nabla\cdot(\overline{\rho}\widetilde{\boldsymbol{u}}\otimes\widetilde{\boldsymbol{u}}+\overline{P}\boldsymbol{I})=-\nabla\cdot\boldsymbol{F}.$
(7)
The sub-grid scale momentum flux $\boldsymbol{F}$ is unknown at simulation
time and must be modelled yet is widely ignored in cosmological simulation
studies which usually focus only on the thermal energy and metal fluxes via
the Smagorinsky model.
There are a myriad of models in the literature for $\boldsymbol{F}$ (see
Garnier et al. 2009 for extensive lists) but here we take the direct approach
of using a Taylor series approximation following Clark et al. (1979) and Hu &
Chiang (2020). We expand222See Appendix A for a full derivation.
$\widetilde{\boldsymbol{u}\otimes\boldsymbol{u}}$ via a Taylor expansion as
$\widetilde{\boldsymbol{u}\otimes\boldsymbol{u}}\approx\widetilde{\boldsymbol{u}}\otimes\widetilde{\boldsymbol{u}}+\epsilon\nabla^{2}(\widetilde{\boldsymbol{u}}\otimes\widetilde{\boldsymbol{u}})$
(8)
and, therefore, the flux becomes
$\boldsymbol{F}=\overline{\rho}(\widetilde{\boldsymbol{u}\otimes\boldsymbol{u}}-\widetilde{\boldsymbol{u}}\otimes\widetilde{\boldsymbol{u}})\approx\overline{\rho}\epsilon\nabla^{2}(\widetilde{\boldsymbol{u}}\otimes\widetilde{\boldsymbol{u}}),$
(9)
where $\epsilon$ is a constant that depends on the kernel scale $h$ as
$\epsilon\propto h^{2}$ (Monaghan, 1989, 2002, 2011). Expanding the term on
the right hand side of equation (9) and keeping only the first derivative
terms we find
$\boldsymbol{F}=2\overline{\rho}Ch^{2}(\nabla\otimes\widetilde{\boldsymbol{u}})(\nabla\otimes\widetilde{\boldsymbol{u}})^{\mathrm{T}},$
(10)
where $C$ is our model parameter.
A similar result emerges when considering the mass flux of metals in a fluid,
$\frac{\partial(\rho Z)}{\partial
t}+\boldsymbol{\nabla}\cdot(\rho\boldsymbol{u}Z)=0.$ (11)
Applying the same approach as before gives an equation for the sub-grid flux
of metals,
$\boldsymbol{F}=2\overline{\rho}C_{\mathrm{Z}}h^{2}(\nabla\otimes\widetilde{\boldsymbol{u}})\cdot\mathbf{\nabla}Z$
(12)
The method for solving equation (7) is detailed in Hopkins (2017). From their
equation (2),
$\boldsymbol{F}=\boldsymbol{K}\cdot(\nabla\otimes\boldsymbol{q}),$ (13)
where $\boldsymbol{K}$ is the tensor describing the diffusive strength, and
$\boldsymbol{q}$ is the fluid field property. Therefore, we identify,
$\boldsymbol{K}\equiv
2\overline{\rho}Ch^{2}(\nabla\otimes\widetilde{\boldsymbol{u}}),$ (14)
$\boldsymbol{q}\equiv\widetilde{\boldsymbol{u}}.$ (15)
The model in equation (10) is known to lead to numerical instability due to
locally stretching flows. Balarac et al. (2013) showed that ignoring the
positive eigenvalues of $\boldsymbol{S}$ (the stretching effect) ensures the
model is always well behaved. Therefore, we follow Balarac et al. (2013) and
only keep the contribution due to the negative eigenvalues of the shear
tensor. However, we must first decompose
$\nabla\otimes\widetilde{\boldsymbol{u}}$ into the symmetric and anti-
symmetric parts,
$\nabla\otimes\widetilde{\boldsymbol{u}}=\frac{1}{2}(\nabla\otimes\widetilde{\boldsymbol{u}}+[\nabla\otimes\widetilde{\boldsymbol{u}}]^{\mathrm{T}})+\frac{1}{2}(\nabla\otimes\widetilde{\boldsymbol{u}}-[\nabla\otimes\widetilde{\boldsymbol{u}}]^{\mathrm{T}})\equiv\boldsymbol{S}+\boldsymbol{\Omega}.$
(16)
We further decompose $\boldsymbol{S}$ into the contribution from the positive
eigenvalues and negative eigenvalues as
$\boldsymbol{S}\equiv\boldsymbol{S}_{\oplus}+\boldsymbol{S}_{\ominus}$. It
then follows that
$\boldsymbol{S}_{\ominus}\equiv\sum_{k=1}^{3}\mathrm{min}(0,\lambda^{(k)})\boldsymbol{e}^{(k)}\otimes\boldsymbol{e}^{(k)},$
(17)
where $\lambda^{(k)}$ is the $k$th eigenvalue and $\boldsymbol{e}^{(k)}$ is
the corresponding eigenvector. Therefore, the new diffusion coefficient tensor
is
$\boldsymbol{K}=2\overline{\rho}Ch^{2}\boldsymbol{S}_{\ominus},$ (18)
where we have reduced the shear contribution into $\boldsymbol{S}_{\ominus}$.
Removing the negative eigenvalues appears to not be necessary for the metal
mixing case as we found no cases of numerical instability in all of our
hydrodynamical tests and cosmological simulations. However, we find that it is
absolutely necessary in cosmological simulations for the momentum flux.
It is important to note that our choice of $h$ differs from that in Hopkins et
al. (2018). We choose $h$ as the kernel radius of compact support as we found
better results in our tests in Section 3. Our value is approximately twice
that found in the FIRE studies (Su et al., 2017; Escala et al., 2018; Hafen et
al., 2019, 2020), but produces $4$ times as much dissipation and metal mixing
since the dependence is squared in equation (18). Our choice of $h$ is the
same as Wadsley et al. (2017) who use the radius of compact support for
turbulent mixing in the GASOLINE-2 code.
### 2.1 The dynamic gradient model
We apply the same procedure in Balarac et al. (2013) combined with the
density-weighted filtering procedure in Rennehan et al. (2019). Following
their notation, we replace $\tilde{f}$ with $\overline{f}$ since fluid
properties are inherently density-weighted in the mesh-free finite mass
method. Although we focus on the velocity fluctuations in the following
procedure, we note that it applies equally to the metal field. The resolved
fluctuations in the flow are,
$\boldsymbol{\mathcal{L}}=\widehat{\boldsymbol{\overline{u}}\otimes\overline{\boldsymbol{u}}}-\widehat{\boldsymbol{\overline{u}}}\otimes\widehat{\overline{\boldsymbol{u}}},$
(19)
where $\widehat{\overline{f}}$ represents the quantity filtered on twice the
resolution scale ($\sim 2\overline{h}$, see Section 2.4 of Rennehan et al.
2019). If we use only resolved quantities (i.e. doubly-filtered) then the
gradient model in equation (10) should reproduce $\boldsymbol{\mathcal{L}}$,
$\boldsymbol{\mathcal{L}}=\widehat{\boldsymbol{\overline{u}}\otimes\overline{\boldsymbol{u}}}-\widehat{\boldsymbol{\overline{u}}}\otimes\widehat{\overline{\boldsymbol{u}}}=2C\widehat{\overline{h}}^{2}\widehat{\overline{\boldsymbol{S}}}_{\ominus}(\boldsymbol{\nabla}\otimes\widehat{\overline{\boldsymbol{u}}})^{\mathrm{T}}.$
(20)
The above equation results in a solution for the one unknown parameter $C$,
$C=\frac{\boldsymbol{\mathcal{L}}\cdot\big{[}2\widehat{\overline{h}}^{2}\widehat{\overline{\boldsymbol{S}}}_{\ominus}(\boldsymbol{\nabla}\otimes\widehat{\overline{\boldsymbol{u}}})^{\mathrm{T}}\big{]}}{||2\widehat{\overline{h}}^{2}\widehat{\overline{\boldsymbol{S}}}_{\ominus}(\boldsymbol{\nabla}\otimes\widehat{\overline{\boldsymbol{u}}})^{\mathrm{T}}||^{2}}=\frac{1}{2}\frac{\boldsymbol{\mathcal{L}}\cdot\boldsymbol{\alpha}}{||\boldsymbol{\alpha}||^{2}},$
(21)
where we have defined
$\alpha\equiv\widehat{\overline{h}}^{2}\widehat{\overline{\boldsymbol{S}}}_{\ominus}(\boldsymbol{\nabla}\otimes\widehat{\overline{\boldsymbol{u}}})$.
Using this determined value of $C$, we use equation (18) as the diffusivity
tensor in the additional flux term from equation (10). Note that
$\overline{h}$ is the resolution scale and not the radius of compact support
of the kernel. The resolution scale is approximately half of the radius of
compact support, or the mean interparticle spacing.
Applying the same procedure to the metal field yields a separate equation for
$C_{\mathrm{Z}}$. The resolved fluctuations take a similar form to equation
(19),
$\boldsymbol{\mathcal{L}}_{\mathrm{Z}}=\widehat{\boldsymbol{\overline{u}}\,\overline{Z}}-\widehat{\boldsymbol{\overline{u}}}\,\widehat{\overline{Z}}.$
(22)
Note that $\boldsymbol{\mathcal{L}}$ is now a vector rather than a rank-2
tensor. Following the procedure we outline above results in an equation for
$C_{\mathrm{Z}}$
$C_{\mathrm{Z}}=\frac{\boldsymbol{\mathcal{L}}_{\mathrm{Z}}\cdot\big{[}2\widehat{\overline{h}}^{2}(\boldsymbol{\nabla}\otimes\widehat{\overline{\boldsymbol{u}}})\cdot\boldsymbol{\nabla}\widehat{\overline{Z}}\big{]}}{||2\widehat{\overline{h}}^{2}(\boldsymbol{\nabla}\otimes\widehat{\overline{\boldsymbol{u}}})\cdot\boldsymbol{\nabla}\widehat{\overline{Z}}||^{2}}=\frac{1}{2}\frac{\boldsymbol{\mathcal{L}}_{\mathrm{Z}}\cdot\boldsymbol{\beta}}{||\boldsymbol{\beta}||^{2}},$
(23)
where we have defined
$\boldsymbol{\beta}\equiv\widehat{\overline{h}}^{2}(\boldsymbol{\nabla}\otimes\widehat{\overline{\boldsymbol{u}}})\cdot\boldsymbol{\nabla}\widehat{\overline{Z}}$.
### 2.2 Comparison to the Smagorinsky model
The gradient model differs from the widely used Smagorinsky model in a subtle
yet important way. Returning to the definition of the sub-grid flux in
equation (10), the Smagorinsky model represents $\boldsymbol{F}$ as,
$\boldsymbol{F}=2\overline{\rho}(C_{\mathrm{s}}h)^{2}||\boldsymbol{S}^{*}||\boldsymbol{S}^{*},$
(24)
where $\boldsymbol{S}^{*}$ is the trace-free symmetric shear
tensor333$\boldsymbol{S}^{*}\equiv\boldsymbol{S}-\frac{1}{3}\mathrm{tr}(\boldsymbol{S})\cdot\boldsymbol{I}$.
The diffusivity tensor is isotropic,
$\boldsymbol{K}_{\mathrm{Smag}}\equiv
2\overline{\rho}(C_{\mathrm{s}}h)^{2}||\boldsymbol{S}^{*}||\,\boldsymbol{I}.$
(25)
and
$||\boldsymbol{K}_{\mathrm{Smag}}||=2\overline{\rho}(C_{\mathrm{s}}h)^{2}||\boldsymbol{S}^{*}||$
with $C_{\mathrm{s}}\sim 0.15$. The constant nature of $C_{\mathrm{s}}$
implies that the diffusivity scales with the magnitude of the symmetric shear
or, more directly, that the strength of turbulent fluctuations is only
determined by fluctuations in the shear strength. That is a good assumption in
purely turbulent flows but fails dramatically in laminar shear flows, where
the shear is not a good indicator of the presence of turbulence.
One solution to over-diffusion in the Smagorinsky model is to dynamically
calculate the coefficient $C_{\mathrm{s}}$ at simulation time based on the
local fluid properties. We showed in Rennehan et al. (2019) that the dynamic
procedure predicts much lower values of $C_{\mathrm{s}}$ in the majority of
our simple hydrodynamical tests. However, we did not consider the impact of
altering the isotropic nature of the diffusivity.
The gradient model has
$\boldsymbol{K}_{\mathrm{Grad}}\propto\boldsymbol{S}_{\ominus}$ with
$||\boldsymbol{K}_{\mathrm{Grad}}||=2\overline{\rho}Ch^{2}||\boldsymbol{S}_{\ominus}||$.
The constant $C$ is yet to be determined but the direction differs from the
dynamic and non-dynamic Smagorinsky models. The diffusivity itself no longer
acts in each direction equally but acts in the direction of the eigenvectors
of $\boldsymbol{S}_{\ominus}$. However, in a simple incompressible, low Mach
number turbulent flows we expect that
$||\boldsymbol{K}_{\mathrm{Smag}}||\sim||\boldsymbol{K}_{\mathrm{Grad}}||$
given that the velocity derivative tensor
$\boldsymbol{\nabla}\otimes\boldsymbol{\tilde{u}}$ is approximately isotropic
in that regime.
Our application of the dynamic method (Piomelli & Liu, 1995) to the gradient
model simultaneously allows $C=C(\boldsymbol{x},t)$ and the diffusivity to be
anisotropic, $\boldsymbol{K}_{\mathrm{Grad}}\propto\boldsymbol{S}_{\ominus}$.
This should be important for any complicated astrophysical flows such as those
we investigate in the following sections.
### 2.3 Models
Table 1: Turbulence models and parameters. The dynamic model calculates the model parameters at simulation and, therefore, the values listed below are the forced upper limit. Label | Dynamic | (Max.) Parameter Value | Anisotropic
---|---|---|---
None | N/A | N/A | N/A
Smag. | $\times$ | 0.15 | $\times$
Dyn. Smag. | ✓ | 0.20 | $\times$
FIRE | $\times$ | 0.05 | $\times$
Grad. | $\times$ | 0.22 | ✓
Dyn. Grad. | ✓ | 1.00 | ✓
Table 1 contains a compact description of our model set. In all cases where
there is a sub-grid turbulence model, we treat both metals and viscosity
simultaneously. There are three categories of models: no sub-grid model
(None), the Smagorinksy model, and the gradient model. The dynamic procedure
allows us to extend the Smagorinsky and gradient models with a model parameter
that depends on spatio-temporal coordinates. Additionally, we test the only
other calibration of the Smagorinsky model in the mesh-free finite mass method
from the FIRE collaboration (Escala et al., 2018).
For the Smagorinsky model, Smag., we use the theoretical value of
$C_{\mathrm{s}}$ $\sim 0.15$ and limit $C_{\mathrm{s}}$ to $0.20$ for the
dynamic Smagorinsky model (Dyn. Smag.) to avoid numerical instability. The
FIRE calibration of the Smagorinsky model (labelled FIRE) is $C_{\mathrm{s}}$
$\approx 0.046$ and we adopt $C_{\mathrm{s}}$ $=0.05$ for simplicity. For
these three versions of the Smagorinsky model the value of $C_{\mathrm{s}}$ is
the same for both metals and eddy-viscosity.
In the new gradient model we use fixed values of $C=0.22$ and
$C_{\mathrm{Z}}=0.22$ for the baseline comparison and label these as Grad. In
our other tests, we use the dynamic procedure outlined in Section 2.1 and
label these tests as Dyn. Grad. We derive our fixed values of $C$ and
$C_{\mathrm{Z}}$ from the approximate median value predicted by the dynamic
procedure in the driven turbulence tests in Section 3.
## 3 Homogeneous Turbulence
Turbulence is ubiquitous in astrophysical flows on a myriad of scales and Mach
numbers. Therefore, in this section, we investigate the impact of the gradient
model on the velocity statistics and metal distributions in homogeneous,
isotropic, driven turbulence at Mach numbers
$\mathcal{M}\in\\{0.3,0.7,2.1\\}$.
For each $\mathcal{M}$, our control (i.e., no sub-grid turbulence model)
simulation set comprises $5$ simulations with particle counts
$N\in\\{64^{3},128^{3},256^{3},512^{3},768^{3}\\}$ within a box of side length
$L=1$ with initial pressure, density, and specific internal energy of $P=1$,
$\rho=1$, $u=1000$, respectively444We use arbitrary code units in all of our
hydrodynamical tests.. Initially, we place equal mass particles on a uniform
grid and then subsequently mix the gas via the prescription of Bauer &
Springel (2012) and Hopkins (2015). We list our turbulent driving parameters
for GIZMO in Table 2 (cf. Table 1 in Bauer & Springel 2012). For more details
of our approach, see Section 3.1 of Rennehan et al. (2019).
Table 2: Parameters for our driven turbulence experiments. Units are arbitrary code units. $\sim\mathcal{M}$ | $\sigma$ | $\Delta t$ | $t_{\mathrm{s}}$ | $k_{\mathrm{min}}$ | $k_{\mathrm{max}}$ | $\tau_{\mathrm{mix}}$
---|---|---|---|---|---|---
0.3 | 0.014 | 0.005 | 1 | 6.27 | 12.57 | 3.33
0.7 | 0.045 | 0.005 | 1 | 6.27 | 12.57 | 1.67
2.1 | 0.21 | 0.005 | 0.5 | 6.27 | 12.57 | 0.50
Our interest lies in measuring the impact of (1) the eddy-viscosity model on
the velocity power spectra of these driven turbulence volumes and (2) the
convergence of metal distribution functions in these volumes. However, these
properties rely on driven turbulence volumes that are in statistical
equilibrium. To gauge whether our simulations are in equilibrium, we define a
mixing timescale $\tau_{\mathrm{mix}}\equiv L/\langle v\rangle$, where
$\langle v\rangle$ is the expected average velocity of the particles in each
volume. However, $\langle v\rangle=v_{\mathrm{s}}\mathcal{M}$, where $v_{s}=1$
is the isothermal sound speed of the gas and, additionally, $L=1$. Therefore,
$\tau_{\mathrm{mix}}=1/\mathcal{M}$. We evolved each control simulation for
several mixing timescales ($\sim 4\tau_{\mathrm{mix}}$) to ensure that the gas
is in steady-state statistical equilibrium and we confirmed the stability of
the Mach number. The mixing timescale and steady-state Mach numbers are listed
in Table 2. We measure the velocity power spectra following the same method as
Bauer & Springel (2012), which is available in the public version of GIZMO.
It may seem out of place to study low resolutions, such as $64^{3}$ and
$128^{3}$, in this study when it is clearly possible to study resolutions up
to $768^{3}$. The ultimate goal of our work is to apply the model to
cosmological simulations which have a huge dynamic range and, therefore, low
resolutions in individual galaxies. For example, IllustrisTNG (Pillepich et
al., 2018) and the RomulusC (Tremmel et al., 2019) simulations both have
particle mass resolutions of $\sim 10^{5}$ $\mathrm{M}_{\odot}$ for each gas
particle, at their best. Consider that in an $L^{*}$-galaxy, we expect perhaps
$\lesssim 10^{10}$ $\mathrm{M}_{\odot}$ of hot gas in the circumgalactic
medium (Anderson & Bregman, 2010). At the best resolutions we have today, that
gives $\sim 10^{5}$ particles per $L^{*}$-halo or $\sim 50^{3}$ particles.
Evidently, contemporary cosmological simulations that capture both hundreds of
$\mathrm{Mpc}$ on the large-scale as well as individual galaxies are far off
from $\gtrsim 256^{3}$ per galactic halo.
### 3.1 Velocity power spectra
A standard measure to determine if eddy viscosity models improve the accuracy
of hydrodynamical simulations is whether the velocity power spectra reproduces
the theoretically predicted Kolmogorov scaling, $E(k)\sim k^{-5/3}$. That
scaling holds for incompressible, low Mach number turbulence
($\mathcal{M}\lesssim 1$) but is shallower than the apparent scaling in
supersonic turbulence, $E(k)\sim k^{-2}$ (Federrath, 2013). In physical
turbulence, the dissipation scale is demarcated by a sharp decline from the
Kolmogorov slope on the smallest scales. In simulations, the resolution scale
forces dissipation to occur on a much larger scale than what would occur in
nature as the physical dissipation scale is unresolved. If the numerical
viscosity of a hydrodynamical method cannot rapidly dissipate that energy,
there will be a build-up of kinetic energy near the resolution scale that
causes an unphysical representation of turbulence.
Eddy-viscosity models introduce additional dissipation in the gas by
accounting for the unrepresented scales in the flow, or equivalently by
minimizing the error from the missing terms in the equations of motion. The
build-up of kinetic energy is usually observed as a "bump" in the velocity
power spectra where there is artificial correlation in the velocities on small
scales. However, before we discuss the impact of eddy viscosity models on the
power spectrum we must first test the convergence of the mesh-free finite mass
(MFM) method in simulations without eddy viscosity.
Figure 1: The velocity power spectra of our simulations with no sub-grid eddy
viscosity model at three Mach numbers from top to bottom, respectively:
$\mathcal{M}=0.3$, $0.7$, and $2.1$. We compensate the power spectra by $k$
for easy comparison with Bauer & Springel 2012. The coloured lines show
resolutions $64^{3}$, $128^{3}$, $256^{3}$, $512^{3}$, and $768^{3}$ from
lightest to darkest, respectively. In each panel, we show the predicted
scaling with a dotted line of arbitrary normalisation. Only in the highest
Mach number case do we see many of the scales in the inertial range
represented and clear convergence of the hydrodynamical method.
Fig. 1 shows the velocity power spectra for our set of simulations with
particle counts $64^{3}$, $128^{3}$, $256^{3}$, $512^{3}$, and $768^{3}$
coloured by lines from lightest to darkest, respectively. The power spectra
are compensated by $k$ for easy comparison to Bauer & Springel (2012). The
panels are ordered from lowest to highest Mach number from top to bottom —
$\mathcal{M}\sim 0.3$, $0.7$ and $2.1$, respectively. In each panel, the
dotted line represents the predicted scaling but at an arbitrary
normalisation.
From the top panel of Fig. 1, it is apparent that $64^{3}$ and $128^{3}$ do
not faithfully represent a turbulent gas as they are dominated by the bump.
More precisely, the $E(k)$ scaling is much too shallow compared to Kolmogorov
turbulence for a wide range of $k$. Our $256^{3}$ simulation shows an inkling
of the inertial range scaling but is slightly too steep below $k\lesssim 20$
and dominated by the bump at $k\gtrsim 30$. As we move up in resolution the
inertial range only grows slightly. At our highest resolution, the inertial
range spans $k\sim 40$ to $k\sim 60$ and the bump dominates the small scales.
We skip discussion of the middle panel as the results are qualitatively
equivalent between $\mathcal{M}\sim 0.3$ and $0.7$.
In the bottom panel of Fig. 1, we show the compensated power spectra for
supersonic turbulence at $\mathcal{M}\sim 2.1$. It is immediately evident that
the simulations converge much more rapidly to the proper scaling than in
subsonic turbulence. Still, at $64^{3}$ resolution the power spectra is
dominated by the build-up of kinetic energy near the resolution scale, with
the inertial range only beginning to appear at $128^{3}$ resolution. At our
highest resolution, $768^{3}$, there is arguably an entire order-of-magnitude
resolved in the inertial range before the build-up of kinetic energy dominates
at the smallest scales.
Figure 2: The velocity power spectrum of our turbulence volumes with various
eddy viscosity models. Rows are Mach numbers $0.3$, $0.7$, and $2.1$ from top
to bottom, respectively. The left column has no additional increase in
dissipation whereas the right column has an order-of-magnitude boost in
dissipation on particles with $\mathcal{M}<1$. The black curve shows the
$256^{3}$ simulation with no eddy viscosity model. The other solid lines show
the Smag. and Dyn. Smag. models, and the dashed and dotted lines show the
Grad. and Dyn. Grad. models, respectively. While all of the models improve the
inertial scaling, an additional boost factor ($\gamma\sim 10$) for subsonic
particles is required to reproduce the proper scaling at all Mach numbers.
Evidently, sub-grid eddy viscosity models are required for the mesh-free
finite mass (MFM) method at any resolutions that may be used in astrophysical
environments. We show the impact of our eddy viscosity models in Fig. 2. The
left column shows the compensated power spectra, $E(k)$, as a function of
wavenumber $k$. The rows represent the same three Mach numbers
$\mathcal{M}=0.3$, $0.7$, and $2.1$ from top to bottom, respectively. The
dotted black line shows the Kolmogorov scaling at each Mach number. All of the
simulations were run at $256^{3}$ resolution since that is the resolution
where, with no eddy viscosity model, we begin to see an extended inertial
range and distinguish the kinetic energy "bump". The coloured curves show the
eddy viscosity models, shaded from lightest to darkest: Dyn. Smag (solid
salmon), Smag. (solid magenta), Dyn. Grad. (dotted purple), and Grad. (dashed
purple). To explain the right column, we must first explain the results in the
left column.
The left column of Fig. 2 shows the velocity power spectra of the turbulent
gas in our simulated volumes with the same models presented in Section 2. The
black curve shows the control experiment at $256^{3}$ resolution, i.e. the
simulation with no eddy viscosity model and only numerical dissipation as in
Fig. 1. All Mach numbers show the same trend: the eddy viscosity models have
little impact on reducing the build-up of kinetic energy at small scales.
Especially important is that the sub-sonic ($\mathcal{M}\lesssim 1$)
simulations are much less improved than the supersonic case. However, the new
gradient model variants, Grad. and Dyn. Grad., dissipate slightly more rapidly
and allow for a steeper slope closer to the Kolmogorov scaling.
We did not expect that all of the eddy viscosity implementations would fail to
reduce the kinetic energy build-up a priori. The fact that there is not enough
dissipation suggests that some physical property in the diffusion tensor was
assigned incorrectly. As we outlined in Section 2.2, the diffusion strength
for the Smagorinsky and the gradient model classes should effectively scale
with each other
($||\boldsymbol{K}_{\mathrm{Smag}}||\sim||\boldsymbol{K}_{\mathrm{Grad}}||$)
in isotropic, homogeneous turbulence. Therefore, in both classes of models,
there are only two physically-motivated quantities that control the strength
of diffusion: the length-scale $h$ and the velocity tensor
$\boldsymbol{\nabla}\otimes\boldsymbol{u}$. The velocity tensor should not be
the issue since it has been verified through the hydrodynamical tests in
Hopkins (2015) and would cause the MFM method to fail drastically if the
velocity gradients were incorrect. That leaves $h$ as the issue, suggesting
that our estimate of the scale over which the eddy viscosity interactions
propagate is underestimated555Recall that our definition of $h$ is already
twice as large as Hopkins (2017), leading to $4$ times as much dissipation and
mixing.. Therefore, we introduce a boost factor $\gamma$ to the diffusion
tensor $||\boldsymbol{K}||$ (i.e.,
$||\boldsymbol{K}||\rightarrow\gamma||\boldsymbol{K}||$) in order to get a
more reasonable scaling in the inertial range.
We determined the boost factor by running a series of driven turbulence tests
with discrete $\gamma$ from $\gamma=1$ to $\gamma=100$. We did not perform a
quantitative fit to the Kolmogorov scaling as our interest is in the
approximate offset required to improve the inertial scaling and the exact
$\gamma$ is unimportant for the statistics of the flow. We additionally found
that we only need to correct the diffusion strength in particles that are
subsonic, $\mathcal{M}<1$, with the Mach number for each particle derived from
the current velocity of the particle divided by its thermal sound speed.
The right column of Fig. 2 shows the velocity power spectra of our simulations
with a dissipation boost factor of $\gamma\sim 10$ on only the subsonic
particles in each simulation. At all Mach numbers the additional dissipation
causes the kinetic energy to convert into thermal energy much more readily,
causing the disappearance of the additional power at small scales, $k\gtrsim
40$. The gradient models Grad. and Dyn. Grad. perform better than the
Smagorinksy variants, but only slightly. That is expected in isotropic
homogeneous turbulence since the dissipation strength is effectively the same,
and confirms that our implementation of the gradient model is a successful
eddy viscosity model.
It is very important that we emphasise the results we observe in Fig. 1 are
very similar to those found in Bauer & Springel (2012) and Hopkins (2015) for
the moving-mesh method (MM; as implemented in AREPO) and the mesh-free finite
mass method (MFM; as implemented in GIZMO), respectively. The only difference
is that we show results for much higher resolutions where we begin to see an
extended inertial range. Both the MFM and MM hydrodynamical methods produce
the same kinetic energy bump that exists in grid-based methods and, therefore,
we would expect an eddy viscosity model to also solve the problem in the MM
method, although we do not test that in this work. To reiterate, it is
necessary that all hydrodynamical simulations resolve the inertial range
combined with an immediate sharp drop-off in power, or they are not
reproducing what we physically observe as turbulence below a certain scale,
where the build-up of kinetic energy begins to dominate.
### 3.2 Metal mixing
After each simulation reached $\sim 4\tau_{\mathrm{mix}}$, we treated each
steady-state volume as new initial conditions for our metal mixing study. In
each volume, we gave the densest $50\%$ of particles a metal mass fraction of
$Z=1$ while keeping the rest $Z=0$. The metals in our simulations act as
passive scalars and have no impact on the flow properties. We will test the
model with more realistic metal distributions in Section 4.
First, we must determine if our simulations converge toward a solution for the
metal distribution as we increase resolution. We ran each of the metal
enriched volumes for an additional $4\tau_{\mathrm{mix}}$ to sample a wide
variety of metal distribution states. We expect a priori that by $\sim
2\tau_{\mathrm{mix}}$ the metal-enriched particles should be scattered
approximately homogeneously since a particle with the typical velocity
$\langle v\rangle$ should have crossed the volume twice in that time. Although
that is true for all resolutions, how can we compare each resolution on equal
footing after it has reached equilibrium?
The appropriate comparison involves smoothing the spatial distribution of
metals on the same scale in all of our simulations. The main assumption is
that our simulations with particle counts $\geqslant 128^{3}$ contain more
accurate information on the scales equivalent to our $64^{3}$ simulations.
Equivalently, if we degrade the resolution of the highest resolution
simulations to the lowest resolution, we should hope to obtain a result
similar to the lowest resolution simulation. To degrade the resolution for
each simulation, we first kernel-weight the particle data onto a grid with
resolution twice as fine as minimum smoothing length in the simulation,
$\Delta x_{\mathrm{sim,i}}$. Next, we smooth the grid data on a physical scale
equivalent to our $64^{3}$ simulation using a uniform top-hat
filter666Specifically, we use the uniform_filter function from the scipy
package in Python, with periodicity enabled. with width $w=\Delta
x_{\mathrm{low}}/\Delta x_{\mathrm{sim,i}}$, where $\Delta x_{\mathrm{low}}$
is always $\Delta x_{\mathrm{low}}\equiv 1/64$ since our box has length $L=1$.
See Appendix B for a comparison of the smoothed metal distributions.
Figure 3: The standard deviation of the Gaussian fit to the metal
distributions in our driven turbulence experiments as a function of
resolution, $N_{\mathrm{x}}$. Mach numbers are $\mathcal{M}=0.3$, $0.7$, and
$2.1$ from top to bottom, respectively. The stars show the simulations with no
sub-grid metal mixing model at resolutions given by the number associated with
each point. The simulations at $64^{3}$ resolution with sub-grid metal mixing
models are given by the figure labels. The dotted line shows an exponential
fit to the no metal mixing model simulations. Clearly, the gradient model
allows us, at $64^{3}$ resolution, to reproduce the spatial redistribution of
metals due to turbulence at a level better than $4-8$ times the resolution.
Fig. 3 shows the standard deviation ($\sigma_{\mathrm{Z}}$) of the smoothed
metal distribution as a function of $N_{\mathrm{x}}=\\{64,128,256,512,768\\}$
in our simulations at $t\sim 2\tau_{\mathrm{mix}}$, for Mach numbers $0.3$,
$0.7$, and $2.1$ from top to bottom, respectively. The stars correspond to the
simulations without a sub-grid metal mixing model at the resolution given by
their labels. The remaining symbols in the legend correspond to the
simulations at $64^{3}$ with a sub-grid metal mixing model (see Table 1 for a
description). The dotted line shows an exponential decay fit to test
convergence in the simulations without metal mixing. We expect decreasing
$\sigma_{\mathrm{Z}}$ with increasing resolution since hydrodynamical mixing
is more resolved and the metal value in each grid cell approaches the mean.
At $\mathcal{M}\sim 0.3$ in the top panel of Fig. 3, $\sigma_{\mathrm{Z}}$
follows an exponentially decreasing trend with resolution in the simulations
without metal mixing, as expected. However, there is not evidence for strong
convergence at our highest resolution — although it appears the curve is
beginning to flatten. The inverted triangle shows the results for the Dyn.
Smag. model and the left-pointing triangle shows the result for the Smag.
model. Both the dynamic and standard Smagorinsky models predict a more
reasonable $\sigma_{\mathrm{Z}}$, reproducing a metal distribution closer to a
resolution of $512^{3}$. The FIRE calibration of the Smagorinsky model, marked
as $\times$, shows little improvement in $\sigma_{\mathrm{Z}}$; the result is
effectively equivalent to having no model at all. The Dyn. Grad. and Grad.
models are marked by a plus sign and diamond, respectively. The gradient model
obviously improves $\sigma_{\mathrm{Z}}$ and can, at $64^{3}$ resolution, also
reproduce a metal distribution equivalent to a resolution $512^{3}$. The
trends are equivalent for $\mathcal{M}\sim 0.7$ turbulence, so we continue to
the next panel.
The bottom panel of Fig. 3 shows how the sub-grid metal mixing models impact
supersonic turbulence at $\mathcal{M}\sim 2.1$. There is much better
convergence of the metal distribution at $t\sim 2\tau_{\mathrm{mix}}$ than in
subsonic turbulence at this time. The qualitative trend remains the same for
the $64^{3}$ simulations with sub-grid metal mixing: both the gradient models
and Smagorinsky models perform equally as well. However, the width
$\sigma_{\mathrm{Z}}$ of the distributions are much wider in $\mathcal{M}\sim
2.1$ turbulence. With the exception of the FIRE calibration, the other metal
mixing models at $64^{3}$ resolution predict metal distributions similar to
$256^{3}$.
Our turbulence tests with metals demonstrate that sub-grid metal mixing models
are necessary in the mesh-free finite mass (MFM) method if one desires more
accurate metal distributions. The method we provided for calibrating the metal
mixing models is important and, we argue, must be investigated whenever one
implements a new metal mixing model into any hydrodynamics solver.
Particularly, one must calibrate the model parameter at the resolution they
desire for their mixing model if a dynamic procedure is not applied. That is
an important point: the dynamic procedure (Dyn. Smag. and Dyn. Grad.) allows
us to approximate the calibrated model parameter for the corresponding model
(Smag. and Grad.) without carrying out the calibration. However, the true
power of the dynamic procedure is in simulations with mixtures of non-
turbulent and turbulent gas at various Mach numbers. In those complex
environments the dynamic procedure automatically adjusts the model parameter
and, as we showed in Rennehan et al. (2019) for the Dyn. Smag. model,
drastically alters the resulting metal distributions.
As expected, in pure homogeneous, isotropic turbulence all of the sub-grid
metal mixing models improved the accuracy of the metal distributions when
using a proper calibration. That is expected in the homogeneous, isotropic
case since all of the models effectively act in the same way, on average. The
power of the dynamic procedure, and the new anisotropic model is in
cosmological simulations where many complex flows interact.
### 3.3 Applicability to other hydrodynamical methods
While our interest lies in the mesh-free finite mass (MFM) method, the models
in Section 2 and experiments in Section 3 are applicable to other
hydrodynamical solvers. Specifically, our results extrapolate with minor
modification to grid-based hydrodynamics. Only slight changes to the filtering
method must be implemented, as outlined in Schmidt (2015). More care must be
taken when applying the model to smoothed particle hydrodynamics (SPH).
However, as we outline below, there is much broader applicability to the
moving-mesh (MM) method.
First we consider eddy viscosity. In SPH, it is well-established that there is
a deficit in power near the resolution scale rather than a build-up of kinetic
energy (Bauer & Springel, 2012; Price, 2012; Hopkins, 2013, 2015). That fact
suggests that SPH reproduces turbulence better than the MFM or MM methods, but
produces results at a much lower effective resolution. The lack of power,
rather than the overabundance of power, implies that an eddy viscosity model
would only further degrade the resolution of SPH results and not improve the
inertial scaling. Therefore, we do not recommend eddy viscosity models for SPH
but rather the work of Di Mascio et al. (2017), who recently provided an SPH
equivalent.
For the MM method, the build-up of kinetic energy at the resolution scale is
present (Bauer & Springel, 2012) and of equivalent magnitude to our results in
the MFM method. Therefore, we recommend investigation into eddy viscosity
models for the MM method as they could improve the inertial scaling. In terms
of implementation, all of the derivations in Section 2 apply to the MM method.
Metal mixing using the Smagorinsky model has been studied in cosmological
simulations involving SPH but widely ignored in the MM method. However, no
calibration technique has been provided by the community for SPH and the
calibrations usually follow the theoretical value for the Smagorinsky model
(e.g., Shen et al. 2010; Williamson et al. 2016) or calibrations that depend
on sub-grid astrophysics models (e.g., Wadsley et al. 2017; Escala et al.
2018). The metal mixing calibration technique in Section 3 is completely
applicable to SPH since metals are treated equivalently to the MFM method, and
they are both constant mass methods. Additionally, our calibration technique
does not depend on the uncertainties within astrophysical sub-grid models —
only pure hydrodynamics.
For the MM method, all of Section 2 is applicable for metal mixing since the
MM method relies on transport equations such as equation (2) for advecting
metals throughout the fluid (Springel, 2010). In fact, Balarac et al. (2013)
find that the gradient model improves the inertial scaling in the power in the
metal field through identical transport equations. However, convergence tests
such as those in Fig. 3 are necessary in order to determine whether they are
truly required, or if numerical dissipation is adequate.
## 4 Cosmological Simulations
Understanding the evolution of galaxies is a complex enterprise involving
highly non-linear coupled physical processes. Not only do stellar feedback and
active galactic nuclei produce powerful outflows that drive turbulence locally
in the interstellar media of galaxies, but also in the gas reservoirs
surrounding galaxies. Turbulence also appears through the Kelvin-Helmholtz
instability in ram pressure stripping of galaxies moving through a hot medium,
and through the stellar winds from stars making their way out from the galaxy
into the circumgalactic medium.
The physical processes above occur on spatial scales much smaller than
currently possible to resolve in the average contemporary cosmological
simulation. For that reason, the majority of astrophysics in cosmological
simulations are encoded into parametrised sub-grid models that use the
information on the largest scales to predict what occurs below the resolution
of the simulation. The resulting calculations usually indicate how much mass
and energy should be injected (or removed) from the large-scale gas and
stellar components. However, there is no one correct way to approximate the
astrophysics on the sub-grid scale since it highly depends on the maximum
possible resolution, hydrodynamical method, as well as other complex numerical
effects. All of the issues with numerics and missing physics usually ends up
in one or more tunable free parameters in the model.
Assuming such a sub-grid astrophysical model is developed, how do we know that
it is correct? Or, at least approximating reality? Normally, one or more
trusted astronomical observation is used to test the validity of all of the
sub-grid astrophysics that may exist. A common example would be the galaxy
stellar mass function, or the $\mathrm{M}_{\mathrm{BH}}$ \-
$\sigma_{\mathrm{*}}$ relationship that links supermassive black hole masses
to the stellar masses of their host galaxies. However, two different
hydrodynamical methods may provide different parameter values for the same
sub-grid astrophysical models. Additionally, there may be two completely
different approaches to modelling the same physical phenomenon with no clear
mapping between free parameters! Calibrating sub-grid astrophysical models is
obviously a complicated endeavour and must be built on a strong hydrodynamics
base. How can we begin to trust that our understanding of the astrophysics of
small scale approximate reality if the hydrodynamics, as we showed in Section
3, does not reproduce reality?
Our goal is in determining whether the converged and separately calibrated
sub-grid turbulence models we presented in Section 2 have any significant
impact over the sub-grid astrophysical models that we use in large-volume
cosmological simulations. As a first step, we only investigate the broad,
qualitative impact on the gas properties in gaseous halos in a single set of
sub-grid astrophysical models. We stress that we do not intend to reproduce
the full galaxy population in a calibrated and predictive sense. Additionally,
we note that more testing is required across the all of sub-grid astrophysical
models that exist in the literature, as the turbulent mixing models may
interact in unexpected ways due to the non-linearity of the problem.
### 4.1 The simulations
For our comparison, we choose to use the SIMBA galaxy formation model. SIMBA
includes robust sub-grid models of star formation, cooling, stellar feedback,
chemical enrichment, active galactic nuclei feedback, and dust evolution — all
evolved in concert with the mesh-free mass method (MFM) (Davé et al., 2016,
2019). For this study, we implemented the SIMBA models into the public version
of GIZMO as described in Davé et al. (2019) and we point the interested reader
to that study for the details of the sub-grid models. We follow the approach
of Schaye et al. (2015) and calibrate our implementation of the SIMBA model
only to the galaxy stellar mass function and the $M_{\mathrm{BH}}$-$M_{*}$
relationship at $z=0$ for the purposes of this study.
Table 3: Cosmological parameters and simulation information for our simulation set. Our parameters follow Davé et al. 2019 except begin at a higher redshift. Cosmological Parameters |
---|---
$\Omega_{\mathrm{m,0}}$ | $0.3$
$\Omega_{\mathrm{\Lambda,0}}$ | $0.7$
$\Omega_{\mathrm{b,0}}$ | $0.048$
$h$ | $0.68$
$\sigma_{\mathrm{8}}$ | $0.82$
$n_{\mathrm{spec}}$ | $0.97$
Simulation Information |
$z_{\mathrm{begin}}$ | 249
$N_{\mathrm{part}}$ | $2\times 256^{3}$
$L_{\mathrm{side}}$ | $25$ $\mathrm{cMpc}\,h^{-1}$
$m_{\mathrm{part,gas}}$ | $1.26\times 10^{7}$ $\mathrm{M}_{\odot}h^{-1}$
$m_{\mathrm{part,dark}}$ | $6.88\times 10^{7}$ $\mathrm{M}_{\odot}h^{-1}$
$\epsilon_{\mathrm{soft,min}}$ | $0.5$ $\mathrm{kpc}\,h^{-1}$
We run 6 cosmological-scale volumes of side-length $L=25$
$\mathrm{cMpc}\,h^{-1}$ ($\sim 37$ $\mathrm{cMpc}$) in order to compare our
various mixing models. The simulations begin from initial conditions generated
with MUSIC (Hahn & Abel, 2011) at a redshift of $z=249$, with a standard
$\Lambda$ Cold Dark Matter cosmology (see Table 3 for values). The mass
resolution in gas and dark matter follows the SIMBA simulations with
$m_{\mathrm{part,gas}}=1.26\times 10^{7}$ $\mathrm{M}_{\odot}h^{-1}$ and
$m_{\mathrm{part,dark}}=6.88\times 10^{7}$ $\mathrm{M}_{\odot}h^{-1}$,
respectively. We use adaptive gravitational softening (Hopkins, 2015; Hopkins
et al., 2018) to compute the softening lengths of all of our particles, and
enforce a minimum softening length of $\epsilon_{\mathrm{soft,min}}=0.5$
$\mathrm{kpc}\,h^{-1}$.
### 4.2 Global metal mixing
While sub-grid scale turbulence models maximally impact the smallest scales in
a cosmological simulation, their integrated effect impact the properties of
the largest scales, such as global metal distribution functions (Shen et al.,
2010; Escala et al., 2018; Rennehan et al., 2019). Therefore, in this Section,
we examine the impact of the gradient model on the metal distribution
functions (MDFs) in the circumgalactic medium (CGM) and warm-hot intergalactic
medium (WHIM) — both known to be turbulent environments (Iapichino et al.,
2011; Iapichino et al., 2013; Tumlinson et al., 2017).
Our definition of CGM and WHIM depends on separating gas that is bound to
halos from that which is unbound, at a given epoch. A good estimation comes
from Davé et al. (2010),
$\frac{\rho_{\mathrm{bound}}(z)}{\Omega_{\mathrm{b}}(z)\rho_{\mathrm{c}}(z)}=6\pi^{2}\bigg{(}1+0.4093\bigg{(}\frac{1}{\Omega_{\mathrm{m}}(z)}-1\bigg{)}^{0.9052}\bigg{)}-1,$
(26)
where $\Omega_{\mathrm{b}}(z)$ is the baryon fraction as a function of
redshift, $\Omega_{\mathrm{m}}(z)$ the matter fraction,
$\rho_{c}(z)=3(H(z))^{2}/(8\pi G)$, and $H(z)$ the redshift-dependent Hubble
function. All gas above $\rho_{\mathrm{bound}}(z)$ we consider bound to halos,
and have confirmed that the approximation holds well.
We define the CGM to be all gas in the volume that is above
$\rho_{\mathrm{bound}}(z)$ in equation (26) and below the star formation
density threshold, $\rho_{\mathrm{*,crit}}=4.4\times 10^{-25}$ g cm-3, at any
temperature. That includes gas in the intragroup medium of our most massive
halos in the $(25\,\mathrm{cMpc}\,h^{-1})^{3}$ volumes. The WHIM is all gas
that is below $\rho_{\mathrm{bound}}(z)$ and above a temperature of $T=10^{5}$
$\mathrm{K}$.
Fig. 4 shows the metal distribution functions (MDFs) for our two gas phases in
columns: WHIM (left) and CGM (right), and at $z=6$, $4$, $2$, and $0$ in rows
from top to bottom, respectively. These are probability density functions, and
were constructed by binning the particle metallicities in the range
$10^{-6}<\log(Z/Z_{\odot})<10^{1}$, where $Z_{\odot}=0.0134$ (Asplund et al.,
2009). The black curves show the control simulation, None, with no sub-grid
metal mixing. The coloured curves show the simulations with sub-grid metal
mixing and are, from lightest to darkest: Dyn. Smag (solid salmon), Smag.
(solid magenta), FIRE (dotted magenta), Dyn. Grad. (dotted purple), and Grad.
(dashed purple). See Table 1 for more details.
Figure 4: The metal distribution function across global phases: the warm-hot
intergalactic medium (WHIM; left column) and the circumgalactic medium (CGM;
right column). Redshifts are given by the rows from top to bottom: $z=6$, $4$,
$2$, and $0$, respectively. The gradient model variants mix metals much more
rapidly in the early stages of the simulation, suggesting that the choice of
model will impact studies that focus on enrichment timing of halo gas.
First we focus on the WHIM. At $z=6$, there are two distinct components across
all of our model variants. The peak at $Z\sim 10^{-1}$
$\mathrm{Z}_{\mathrm{\odot}}$ is the highly enriched interstellar medium (ISM)
gas that recently joined the WHIM via stellar winds from the integrated star
formation in the early universe. The lower distribution is gas that has mixed
into the WHIM but did not recycle through the ISM, missing the opportunity for
further enrichment via supernova feedback. It is important to note that since
there is no mixing in the None case, when a particle leaves the ISM it cannot
change its metallicity. The models that include sub-grid mixing show varying
spread in the MDFs, with the FIRE and Dyn. Smag. showing the widest spread.
The Dyn. Grad. and Grad. models show the tightest distributions, with the two
peaks in the distributions seemingly merging at $Z\sim 10^{-1.5}$
$\mathrm{Z}_{\mathrm{\odot}}$. The Smag. model matches the Dyn. Smag. model at
$Z\gtrsim 10^{-3}$ $\mathrm{Z}_{\mathrm{\odot}}$, but is biased toward higher
metallicities below that threshold.
The next 3 panels in the left column of Fig. 4 show the MDFs in the WHIM for
$z=4$, $2$, and $0$, from top to bottom, respectively. The trend for all
models is to approach a singly-peaked distribution as the simulation evolves
through cosmic time. Most of the evolution in the MDFs occurs from $z=6$ to
$z=2$ after which the distributions are mostly stationary. The transition from
$z=6$ to $z=4$ demonstrates how rapidly the WHIM evolves at high redshift, and
how each sub-grid metal mixing model impacts the MDFs with different mixing
rates. Specifically, the Dyn. Grad. and Grad. models predict similar
distributions at $z=4$, and produce the tightest MDFs compared to all of the
other models. In fact, there is the same trend at $z=4$ as at $z=6$ — the
gradient model variants (Dyn. Grad. and Grad.) predict tighter MDFs, followed
by wider distributions in the Smagorinsky models (Smag., Dyn. Smag., and FIRE,
respectively).
There are similar trends in the CGM, as we see in the right column of Fig. 4.
To reiterate, the panels show redshifts $z=6$, $4$, $2$, and $0$ from top to
bottom, respectively.
At $z=6$, there is a clear distinction between the distribution at $Z\lesssim
10^{-2}$ $\mathrm{Z}_{\mathrm{\odot}}$ and $Z\gtrsim 10^{-2}$
$\mathrm{Z}_{\mathrm{\odot}}$ in the None case, and the Smagorinsky variants
(Smag., Dyn. Smag., and FIRE). Stellar feedback drives the peak at $Z\sim
10^{-1}$ $\mathrm{Z}_{\mathrm{\odot}}$ similarly to the WHIM at this redshift,
whereas the distribution at $Z\lesssim 10^{-2}$ $\mathrm{Z}_{\mathrm{\odot}}$
is from the very first generations of stars. By this time the Dyn. Grad. and
Grad. models have mixed the most rapidly to create a single broad distribution
in their MDFs. All of the models with sub-grid metal mixing have much more gas
mass enriched above $Z>10^{-6}$ $\mathrm{Z}_{\mathrm{\odot}}$ than the None
case, especially compared to the deficit at $Z\sim 10^{-1.5}$
$\mathrm{Z}_{\mathrm{\odot}}$ in the None case. The Smagorinsky models vary in
mixing rate as Smag., Dyn. Smag., and FIRE, from fastest to slowest,
respectively. The MDFs in the CGM at redshifts $z=4$ to $z=0$ demonstrate the
same trends as in the WHIM phase at the same redshifts: the gradient models
mix much more rapidly at early stages than the Smagorinsky models. At $z=0$
the Dyn. Grad., Grad., and Smag. models predict the same distribution in the
global CGM phase, whereas the Dyn. Smag. and FIRE models predict slightly less
enriched gas.
The MDFs in the turbulent WHIM and CGM show the importance of sub-grid metal
mixing models in cosmological simulations as well as the importance of model
choice. In all cases we include metal mixing, the MDFs are significantly
tighter at all redshifts we measure and significantly tighter for the CGM at
$z=0$. This is contrary to the study in Su et al. (2017) that found metal
mixing to be relatively unimportant on cosmological scales. However, we find
that the FIRE calibration is much too low to reproduce the correct converged
hydrodynamical mixing of metals (see Section 3.2). With our new calibrations
of the Smagorinsky model, Smag., and the new gradient models, Dyn. Grad. and
Grad., we see significant differences at all redshifts.
A common theme in theoretical galaxy evolution is that equivalent results
between different models at $z=0$ does not necessarily imply a similar
integrated history. The evolutionary paths for each sub-grid metal mixing
model all evolve at slightly different rates as we would expect based on their
diffusivities from Section 2. The lesson from our results is that the metal
mixing model choice impacts the early development phases of galaxies rather
than the long term equilibrium stages. At higher redshifts, $z\gtrsim 2$, gas
is collapsing to form galaxies, while stellar feedback and supermassive black
holes are driving outflows out of the potential wells and forcing turbulence.
Our inclusion of the full diffusion tensor in the Dyn. Grad. and Grad. models
allows the gas that is compressing from feedback and infall to further mix its
metal mass with nearby neighbours, tightening the MDFs.
The Smagorinsky variants also improve the results and allow metal mixing
between gas particles but produce broader distributions, notwithstanding the
Dyn. Smag. and Smag. models showing good matches in the convergence tests of
Section 3.2. This is an important point: the simple turbulence tests in
Section 3.2 showed agreement between the gradient and Smagorinsky models
(except for the lower FIRE calibration) but now we see disagreement in complex
cosmological environments. Evidently, ignoring the trace of the velocity
tensor is not the correct approach for cosmological contexts and we recommend
either the Grad. or Dyn. Grad. models as we see no difference with the dynamic
procedure applied to the constant coefficient gradient case.
### 4.3 The impact of eddy viscosity
While the velocity power spectra results in Section 3.1 show that eddy
viscosity is required in Lagrangian finite mass methods, we find no
significant impact of eddy viscosity on the average gas and galaxy properties
in our cosmological simulations. However, we should expect these results
considering that the eddy viscosity models we tested in Section 3.1 had no
impact on the largest scales of the simulation (as they should not). In terms
of global gas properties, we investigated the vorticity, temperature, and
density distributions of the warm-hot intergalactic medium (WHIM) and
circumgalactic medium (CGM) phases777Definitions in Section 4.2. and found
only minor differences in data binned by galaxy stellar mass. Additionally,
there were little differences between stellar mass distributions in our small
volume, low resolution tests. To compare the impact of eddy viscosity between
the models we need to investigate the small scales of the simulations in a
controlled manner. For that reason, we will use the most massive galaxy in our
cosmological simulations as a qualitative case study of the impact of the eddy
viscosity model on the halo gas since this galaxy represents the same system
in each simulation we ran.
First, we investigate the temperature projections of the most massive (in
stellar mass) galaxy at redshifts $z=2$, $1$, and $0$. We confirmed that the
most massive galaxy at $z=2$ ends up as part of the most massive galaxy at
$z=1$ and $z=0$. We restrict our analysis to a radius of $500$ $\mathrm{kpc}$
(physical) for the purposes of this introductory study. Additionally, given
that we see the Smag., Grad., and Dyn. Grad. models as the best choices from
the results in Section 4.2, we restrict our analysis to the None, Smag., and
Grad. models. The Grad. model is much less computationally expensive than the
Dyn. Grad. model and is, therefore, the better choice for cosmological
simulations888Further study is required for tests that rely on getting the
small scale structure as correct as possible, such as cold clouds interacting
with a hot medium..
Figure 5: Temperature projections of the most massive halo in three of our
cosmological simulations at redshifts $z=2$, $1$, and $0$ in rows from top to
bottom, respectively. The columns show our None, Smag., and Grad. models from
left to right, respectively. Each of the panels represents a $1$
$\mathrm{Mpc}$ by $1$ $\mathrm{Mpc}$ (physical) region centred on the most
massive galaxy at each redshift. The Smag. simulation shows the most small-
scale structure at all redshifts, and a smoother distribution of temperature
at high-redshift compared to the None case. The Grad. model produces less
small-scale structure than any other model, and much more hot gas at high
redshift. Surprisingly, the Grad. model also produces more extended tails from
sub-structure moving through the hot halo at lower redshift, suggesting it may
impact future studies of jellyfish galaxies. Figure 6: Velocity magnitude
projections of the most massive halo at $z=0$ in our cosmological simulations.
From left to right we show the None, Smag., and Grad. simulations,
respectively. The images are centred on the most massive galaxy and each show
a $1$ $\mathrm{Mpc}$ by $1$ $\mathrm{Mpc}$ (physical) region. The Smag. model
shows more substructure than the None and Grad. models. The Grad. model shows
many extended tails from substructure moving through the hot halo compared to
the control and Smag. models, in addition to an overall smoother distribution
in velocity space.
Fig. 5 shows the density-weighted temperature projections of the most massive
galaxy at $z=2$, $1$, and $0$ in rows from top to bottom, respectively. The
columns show the None, Smag., and Grad. models from left to right,
respectively.
First, we compare the results at $z=2$ across mixing model variants. The dark
clumps in the None case are individual cold galaxies that are being fed into
the main structure via filaments. Ongoing stellar and AGN feedback leads to
the temperature increase in the central region, while the extended $T\sim
10^{6}$ $\mathrm{K}$ halo is a mixture of gravitational heated gas and gas
heated by previous generations of stellar feedback and AGN. There are many
filamentary structures visible feeding the main galaxy (centred), and there is
some small scale structure visible surrounding the galaxies. The Smag. case
resembles the None case, except there less cold gas in the upper in-falling
structure. Additionally, the satellite galaxies have more small-scale
structure in the cold gas surrounding them. Importantly, the distribution of
hotter gas appears mores smoothly distributed throughout the volume due to an
increase in conversion of kinetic energy in thermal energy via the eddy
viscosity.
The temperature projection of the Grad. model simulation is strikingly
different at $z=2$ than both the Smag. and None cases. While in the None and
Smag. cases there appears to be large-scale gas at $T\lesssim 10^{5}$
$\mathrm{K}$ at the boundary of the $500$ $\mathrm{kpc}$ radius, there is no
such gas in the Grad. case. There is much less small-scale structure in the
Grad. case, and the filaments are much smoothly distributed in space. We
conclude that the stellar feedback, gravitational heating, and AGN feedback
are much more effective at heating the gas on small scales in the Grad.
simulation, since the eddy viscosity is the strongest.
At $z=1$, in the middle row of Fig. 5, a similar picture emerges. The Smag.
case has much more small scale structure than the None case as is evident in
the central region and to the leftmost satellite galaxy structure. There is
still a much smoother distribution of cold gas that extends outwards from the
satellite galaxies. There are only a few centrally-located cold gas clumps in
the Grad. case compared with the Smag. case and there is effectively no
fragmentation identifiable in the satellite galaxy structure to the left of
panel.
The last row of Fig. 5 shows $z=0$ across the three model comparisons. By this
redshift, there is little structure remaining in the group-sized halo and the
temperature distribution of the intragroup gas is very smooth. Similarly to
the other redshifts, the Smag. shows the most fragmentation of cold gas,
followed by the None case, and then the Grad. case. All three show a satellite
being stripped of gas in the upper right of the panels but the Smag. and Grad.
models each differ in an important unique way compared to the None case.
It is much easier to see the differences in ram pressure stripping in velocity
space rather than temperature space. Fig. 6 shows the density-weighted
velocity magnitude projections of the most massive galaxy at $z=0$ for the
three eddy viscosity models None, Smag., and Grad. from left to right,
respectively. It is apparent across all three cases that the substructure is
moving at least a factor of $\approx 5$ faster than the background gas yet
only the Grad. case shows the most clearly defined long stripping tails from
the cold gas in the galaxies. The Smag. case does not have the cold gas in the
satellite galaxy marked by the arrows in the None and Grad. cases, as the gas
has been completely stripped away. The Grad. case produces the cold gas in
that satellite galaxy, and the tail is much more extended than in the None
case. In fact, it is evident upon close inspection of the cold gas structures
in the halo that the Grad. case produces tails from the cold gas in the
galaxies more readily than the None or Smag. cases. That has implications for
the study of ram pressure stripping in general, and should be further
investigated in the future.
## 5 Conclusions
Turbulence is a key physical process in the study galaxy evolution and one of
many highly complex non-linear interactions that must be understood to advance
our knowledge of the Universe. The complexity demands the use of simulations
that combine astrophysical sub-grid models with hydrodynamics and gravity in
an expanding universe. All hydrodynamical simulations are known to require
additional sub-grid models to accurately treat the impact sub-grid turbulence,
yet these models have been widely ignored in the astrophysical community. We
have, for the first time in Lagrangian hydrodynamics, implemented and studied
the gradient model (Clark et al., 1979) – an anisotropic sub-grid turbulence
model for viscosity and metal mixing (Hu & Chiang, 2020). The model is based
on directly modelling the error terms that arise from discretisation of the
fluid field via a Taylor series expansion, including the compression terms
that are missing from the standard Smagorinsky model. We additionally
implemented a dynamic procedure that computes the model parameter on-the-fly
for the gradient model following the approach of Rennehan et al. (2019). We
used the mesh-free finite mass method in the GIZMO code (Hopkins, 2015) as our
numerical hydrodynamics solver for all of our experiments.
We ran driven turbulence simulations at Mach numbers $\mathcal{M}=0.3$, $0.7$,
and $2.1$ to validate the gradient model and compare with the popular
Smagorinsky model. Hu & Chiang (2020) recently showed, by post-processing
driven their turbulence simulations, that the gradient model should be able to
reduce the build-up of kinetic energy near the resolution scale in isotropic,
homogeneous turbulence and better reproduce the sub-grid metal flux. We
confirmed these results in Section 3 by using the gradient model at simulation
time.
Our analysis of the velocity power spectra in driven turbulence produced
unexpected results. We found that the gradient and Smagorinsky models, and
their dynamic variants, predicted insufficient dissipation to reduce the
artificial build-up of kinetic energy near the grid-scale in our $256^{3}$
simulations. For that reason we introduced a boost factor $\gamma$ for the
dissipation strength (i.e.
$||\boldsymbol{K}||\rightarrow\gamma||\boldsymbol{K}||$) and experimented with
various values in the range $\gamma\in[1,100]$. We found that a factor of
$\gamma\sim 10$ is sufficient to reduce the build-up of kinetic energy and is
required for all of the Smagorinsky and gradient model variants. Additionally,
we found that the boost factor only needs to be applied to the subsonic
($\mathcal{M}<1$) particles in our simulations to produce the correct
statistics in supersonic turbulence (our $\mathcal{M}\sim 2.1$ test). The true
boost factor is higher since we used the maximum interaction distance between
neighbouring gas particles to calculate the diffusion tensor, leading to a
$\sim 4$ times additional boost (total $\sim 40$) over other the default
implementation in GIZMO (Hopkins et al., 2018).
Our converged metal mixing simulations in Section 3.2 show that when we use
the gradient and Smagorinsky models at lower resolution ($64^{3}$ particles)
we are able to produce MDFs that are equivalent to $4$ to $12$ times the
resolution. That is true for both the constant-coefficient and dynamic
variants of the gradient and Smagorinsky models with standard parameter
values, with the additional factor of $\sim 4$ boost from using the maximum
interaction distance in the kernel. However, lower calibrations such as those
from the FIRE simulations do not produce the correct rate of mixing as they
are at least a factor of $\sim 20$ too low. We posit that this is because of
the common calibration approach in cosmological simulations: calibration in
tandem with the full suite of astrophysical sub-grid models.
We argue that calibration of fundamental hydrodynamics models, such as the
metal mixing model here, _must be done in the absence of sub-grid
astrophysical models_. There must be a strong hydrodynamics base before the
complexity of astrophysics is built on top. Note that our dynamic Smagorinsky
and dynamic gradient models _do not require calibration_ and produce accurate
predictions for the rate of metal mixing in isotropic, homogeneous turbulence.
As a first application of the new gradient model, we investigated a set of
cosmological simulations to determine if there is any dominant impact on the
galaxy and gas properties. We investigated the metal mixing and eddy viscosity
separately in Sections 4.2 & 4.3, respectively.
We found that the choice of sub-grid metal mixing model strongly impacts the
MDF evolution in the warm-hot intergalactic medium (WHIM) and circumgalactic
medium (CGM). We found that the gradient and dynamic gradient models mix
metals much more rapidly than the Smagorinsky variants and produce tighter
MDFs up until $z\sim 1$ when they approach a similar distribution down to
$z=0$. In our simulation without sub-grid metal mixing, the MDFs in the WHIM
and CGM are significantly broader than any of the simulations with sub-grid
metal mixing demonstrating that, at the very minimum, including any sub-grid
metal mixing model is an improvement. The most important result we discovered
is that the metal mixing models are most impactful during the tempestuous
early stages of galaxy evolution. On very long timescales, the equilibrium
distributions match quite closely across the models.
Including eddy viscosity in our cosmological simulations did not significantly
impact the galaxy properties we investigated when averaging in bins of stellar
mass after $z=2$. The galaxy stellar mass function was relatively unchanged,
along with only slight variations in the stellar mass to halo mass function.
We also found that including eddy viscosity did not significantly impact the
averaged gas distributions of vorticity, temperature, and density across
galaxies of similar stellar mass. We expected a priori that the large-scale
properties of galaxies would be unaffected as the sub-grid eddy viscosity
mainly impacts the small-scale. For that reason, we investigated a single
system that could be linked across all of our cosmological simulations to gain
a qualitative view of the impact.
In Section 4.3 we showed the temperature projections of the most massive halo
traced from $z=2$, $1$ and $0$ in our cosmological simulations for three eddy
viscosity models: no model, the standard Smagorinsky model, and the new
constant-coefficient gradient model. We found that the Smagorinsky model
produced much more fragmentation in the halo gas of the most massive galaxy on
the small scale compared to having no eddy viscosity, at all redshifts. We
also found that the spatial temperature distribution was much smoother at
$z=2$ when stellar and active galactic nuclei feedback was much stronger,
showing that the small-scale kinetic energy was being efficiently converted
into thermal energy. Although the constant-coefficient gradient model
seemingly dissipates faster based on our results in Section 3.1, we observed
that its inherent anisotropy does not lead to the same fragmentation we saw in
the Smagorinsky model. At high redshift, $z=2$ the gradient model produced a
much more widespread hot gas. The filamentary structure at all redshifts was
much smoother and, after $z=1$, the satellite galaxies in the halo had many
more clearly defined tails due to an improved treatment of ram-pressure
stripping.
Sub-grid metal mixing and eddy viscosity models have a strong impact on galaxy
evolution simulations. In this work, we showed in the simplest case of
isotropic, homogeneous turbulence that the all of the models tested here
improved the accuracy of metal mixing and turbulent kinetic energy dissipation
in the mesh-free finite mass method. The most significant differences between
model choice appeared at high redshift in the early stages of galaxy
evolution, before any equilibrium is reached. Given that contemporary
cosmological simulations have resolutions of less than $\sim 50^{3}$ particles
in a typical galaxy, we recommend that future studies must at least use the
constant-coefficient gradient model as it is (a) computationally inexpensive
compared to the dynamic version, while producing similar results and (b)
includes the full velocity tensor in the diffusion tensor to give the most
accurate solution for sub-grid turbulence. Our recommendation is especially
pertinent given the recent push to study higher redshift systems driven by the
upcoming launch of the James Webb Space Telescope, as our theoretical
understanding of enrichment and thermodynamic histories will depend directly
on sub-grid turbulence model choice.
## Acknowledgements
This research was enabled in part by support provided by WestGrid and
Compute/Calcul Canada. The simulations in this research were made possible by
SciNet and the Niagara supercomputing cluster. DR acknowledges the support of
the Natural Sciences and Engineering Research Council of Canada (NSERC),
[funding reference number 534263] and through the Discovery Grant program. DR
thanks Arif Babul, Belaid Moa, Ondrea Clarkson, Austin Davis, Drummond
Fielding, Phil Hopkins, and Wolfram Schmidt for useful advice during the
course of this research. DR gives special acknowledgement to Arif Babul and
Belaid Moa for providing invaluable support to this research, without which it
would not have been possible.
Our analysis was performed using the Python programming language (Python
Software Foundation, https://www.python.org). The following packages were used
throughout the analysis: h5py (Collette, 2013), numpy (Harris et al., 2020),
scipy (Virtanen et al., 2020), yt (Turk et al., 2011), and matplotlib (Hunter,
2007). Prototyping of the analysis scripts was performed in the IPython
environment (Perez & Granger, 2007).
## Data availability
The data underlying this article will be shared on reasonable request to the
authour.
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## Appendix A FILTERING APPROXIMATION
Consider a scalar field $f_{i}(\boldsymbol{r})$ where $\boldsymbol{r}=(x,y,z)$
in a general coordinate system. We expand $f_{i}(\boldsymbol{r}^{\prime})$
about $\boldsymbol{r}$ in equation (1) via a Taylor expansion,
$\begin{split}f_{i}(\boldsymbol{r}^{\prime})=f_{i}(\boldsymbol{r})+[(\Delta\boldsymbol{R})\cdot\boldsymbol{\nabla}f(\boldsymbol{r})]+\frac{1}{2}[(\Delta\boldsymbol{R})\otimes(H(\boldsymbol{r})\cdot(\Delta\boldsymbol{R}))]\\\
+\mathcal{O}(|\Delta\boldsymbol{R}|^{3}),\end{split}$ (27)
where $\Delta\boldsymbol{R}=\boldsymbol{r}^{\prime}-\boldsymbol{r}$ and
$H(\boldsymbol{r})$ is the Hessian matrix. Putting this into equation 1 gives,
$\begin{split}\overline{f}_{i}(\boldsymbol{r})=f_{i}(\boldsymbol{r})+\frac{1}{2}H(\boldsymbol{r})\int_{\mathrm{kern}}((\Delta\boldsymbol{R})\otimes(\Delta\boldsymbol{R}))G(|\Delta\boldsymbol{R}|,h)d\boldsymbol{r}^{\prime}\\\
+\mathcal{O}(|\Delta\boldsymbol{R}|^{4}).\end{split}$ (28)
using the fact that the kernel function is normalized and the integral of an
odd function over the domain is zero. The integral in equation 28 can be
tabulated for a specific function $G=G(|\Delta\boldsymbol{R}|,h)$, so let’s
define,
$\epsilon(\boldsymbol{r})\equiv\frac{1}{2}\int_{\mathrm{kern}}((\Delta\boldsymbol{R})\otimes(\Delta\boldsymbol{R}))G(|\Delta\boldsymbol{R}|,h)d\boldsymbol{r}^{\prime}.$
(29)
We then have,
$\overline{f}_{i}(\boldsymbol{r})\approx
f_{i}(\boldsymbol{r})+\epsilon(\boldsymbol{r})\cdot H(\boldsymbol{r}).$ (30)
However, $\epsilon(\boldsymbol{r})$ is isotropic
$\epsilon(\boldsymbol{r})=\epsilon\,\boldsymbol{I}$, so we can write
$\overline{f}_{i}(\boldsymbol{r})\approx
f_{i}(\boldsymbol{r})+\epsilon\boldsymbol{\nabla}^{2}f_{i}(\boldsymbol{r}).$
(31)
## Appendix B METAL DISTRIBUTIONS
Fig. 7 shows the normalised histograms of the filtered metal field from our
driven turbulence simulations in Section 3.2. The panels are Mach numbers
$\mathcal{M}=0.3$, $0.7$, and $2.1$ from top to bottom, respectively. The
black curves with markers show the convergence of the filtered metal field for
resolutions $64^{3}$, $128^{3}$, $256^{3}$, $512^{3}$, and $768^{3}$. The
coloured lines show, from lightest to darkest: Dyn. Smag (solid salmon), Smag.
(solid magenta), FIRE (dotted magenta), Dyn. Grad. (dashed purple), and Grad.
(solid purple) at $64^{3}$ resolution. We obtained all of the information for
this figure after approximately two mixing timescales, $t\sim
2\,\tau_{\mathrm{mix}}$, where $\tau_{\mathrm{mix}}=1/\mathcal{M}$ (see
Section 3.2 for more details). Fig. 3 we used the Gaussian fits to these
distributions and found the widths of the distribution, $\sigma_{\mathrm{Z}}$.
Figure 7: The normalised histograms of the filtered metal mass fraction field
from the simulations in Section 3.2. The panels show Mach numbers
$\mathcal{M}=0.3$, $0.7$, and $2.1$ from top to bottom, respectively after two
mixing timescales. The black curves show the simulations with no sub-grid
metal mixing. The coloured curves show the simulations at $64^{3}$ resolution
with a sub-grid metal mixing model given by the label. All of the sub-grid
metal mixing models show improvement except for FIRE which lags due to the
lower calibration coefficient.
|
# Elementary symmetric polynomials and martingales for Heckman-Opdam processes
Margit Rösler Institut für Mathematik, Universität Paderborn, Warburger Str.
100, D-33102 Paderborn, Germany<EMAIL_ADDRESS>and Michael Voit
Fakultät Mathematik, Technische Universität Dortmund, Vogelpothsweg 87,
D-44221 Dortmund, Germany<EMAIL_ADDRESS>
###### Abstract.
We consider the generators $L_{k}$ of Heckman-Opdam diffusion processes in the
compact and non-compact case in $N$ dimensions for root systems of type $A$
and $B$, with a multiplicity function of the form $k=\kappa k_{0}$ with some
fixed value $k_{0}$ and a varying constant $\kappa\in\,[0,\infty[$. Using
elementary symmetric functions, we present polynomials which are simultaneous
eigenfunctions of the $L_{k}$ for all $\kappa\in\,]0,\infty[$. This leads to
martingales associated with the Heckman-Opdam diffusions
$(X_{t,1},\ldots,X_{t,N})_{t\geq 0}$. As our results extend to the freezing
case $\kappa=\infty$ with a deterministic limit after some renormalization, we
find formulas for the expectations $\mathbb{E}(\prod_{j=1}^{N}(y-X_{t,j})),$
$y\in\mathbb{C}$.
###### 2010 Mathematics Subject Classification:
60J60, 33C67, 82C23, 60B20
## 1\. Introduction
In the theory of classical random matrix ensembles there exist several
formulas regarding determinants as follows: Let $X$ be a random variable with
values in some space of $N\times N$-matrices over
$\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H}$. Then the expectations
$\mathbb{E}(\det(X-yI_{N}))$ are classical orthogonal polynomials of degree
$N$ in $y\in\mathbb{C}$. Such results for the Hermite, Laguerre, and Jacobi
ensembles can for example be found in [DG, FG, A], where the expectations
above can be expressed via classical Hermite, Laguerre and Jacobi polynomials
of degree $N$. These results were extended to the spectra of $\beta$-ensembles
associated with $N$-dimensional time-homogeneous diffusion processes in [KVW,
V] by using some martingales constructed from these diffusions via elementary
symmetric polynomials. The generators of the diffusions were Dunkl-Bessel
Laplacians and (symmetric) Heckman-Opdam Laplacians in the compact $BC$ case.
In the present paper we study corresponding results for Heckman-Opdam
Laplacians in the non-compact $BC$ case as well Heckman-Opdam Laplacians of
type $A$ in the compact and noncompact setting. Together with [KVW, V], the
present paper covers the most important examples related with Calogero-Moser-
Sutherland particle models. The basic ideas here are similar to those in [KVW,
V]; however, while the approach in [KVW, V] is mainly based on Ito calculus,
we now focus on an algebraic point of view.
The idea is as follows. Let $(X_{t})_{t\geq 0}$ be a time-homogeneous
diffusion on a suitable closed set $C\subset\mathbb{R}^{N}$ (such as a Weyl
chamber or a fundamental alcove), where the paths are reflected at the
boundary $\partial C$. Then the generator $L$ of the associated transition
semigroup is an elliptic partial differential operator whose domain is
contained in the space of functions on $\mathbb{R}^{N}$ which admit
corresponding symmetries on $\partial C$. We are now interested in functions
$f:[0,\infty[\times\mathbb{R}^{N}\to\mathbb{C}$ for which $(f(t,X_{t}))_{t\geq
0}$ is a martingale (w.r.t. the canonical filtration), which essentially means
that $f$ is $L$-space-time-harmonic, i.e., $(\frac{\partial}{\partial
t}+L)f=0$; see Section III.10 of [RW]. Examples of such harmonic functions can
be given in terms of eigenfunctions of $L$. For a general background in
stochastic analysis we also recommend [P].
In the framework of Heckman-Opdam theory (see [HS], [HO]), we fix some
crystallographic root system $R$ (with associated Weyl group $W$) in
$\mathbb{R}^{N}$ and choose $C$ as an associated Weyl chamber or fundamental
alcove. We consider a non-negative multiplicity function $k\geq 0$ on $R$ of
the form $k=\kappa\cdot k_{0}$ with some fixed multiplicity $k_{0}$ and a
constant $\kappa>0$. We now study the $W$-invariant Heckman-Opdam Laplace
operators $L_{\kappa}:=L_{\kappa k_{0}}$ as generators of diffusions (see
[Sch1, Sch2, RR1]), where the parameter $\kappa$ is varying. We also study the
renormalized generators $\widetilde{L}_{\kappa}:=\frac{1}{\kappa}L_{\kappa}$
of the renormalized diffusions $(\widetilde{X}_{t}:=X_{t/\kappa})_{t\geq 0}$.
In suitable coordinates, these renormalized generators then have the form
$\widetilde{L}_{\kappa}f=\frac{1}{\kappa}\Delta f+Hf$
with some second-order differential operator $\Delta$ (often a classical
Laplacian) and some first-order drift operator $H$, where both operators are
independent of $\kappa$. This also works for $\kappa=\infty$ where
$(\widetilde{X}_{t})_{t\geq 0}$ is the deterministic solution of some ODE
associated with $H$. We now use elementary symmetric polynomials to construct
simultanous eigenfunctions $f$ of $\widetilde{L}_{\kappa}$ for all
$\kappa\in]0,\infty]$. In the next step we use martingales associated with
these functions $f$ together with information in the deterministic case
$\kappa=\infty$ in order to derive formulas for
(1.1)
$\mathbb{E}(\prod_{j=1}^{N}(\widetilde{X}_{t,i}-y))\quad\quad(y\in\mathbb{C}).$
For some values of $\kappa$, the operators $L_{\kappa}$ admit an
interpretation as Laplace-Beltrami operators on symmetric spaces. In these
cases, (1.1) leads to determinantal formulas for Brownian motions on the
symmetric spaces which are closely related, for instance, to some identities
in [R]. Considering $t\to\infty$ in the compact cases, this also leads to
determinantal formulas for the uniform distributions on compact symmetric
spaces.
The paper is organized as follows: In Section 2 we first recapitulate some
well-known facts on Heckman-Opdam hypergeometric functions and polynomials,
and the associated Laplacians in the compact and non-compact setting. We there
also prove that in the non-compact crystallographic case, the Heckman-Opdam
processes admit arbitrary exponential moments for arbitrary deterministic
starting conditions. In the main part of the paper, we then concentrate on
specific root systems. Section 3 is devoted to the compact case of type $A$,
which is related to Calogero-Moser-Sutherland particle models on the torus.
Here, for instance, our results lead to determinantal formulas for Brownian
motions and the uniform distribution on the unitary groups $U(N)$ and $SU(N)$.
In Section 4 we study the non-compact case of type $A$. Section 5 then
contains the non-compact case of type $BC$.
## 2\. Heckman-Opdam theory
Here we briefly collect some facts from Heckman-Opdam theory; see [HS, HO] for
a general background.
Let $(\mathfrak{a},\langle\,.\,,\,.\,\rangle)$ be a Euclidean space of
dimension $N$ with norm $|x|=\sqrt{\langle x,x\rangle}.$ We identify
$\mathfrak{a}$ with its dual space via the given scalar product. Let $R$ be a
crystallographic, possibly not reduced root system in $\mathfrak{a}$ with
associated finite reflection group $W$. Thus in particular, $R$ spans
$\mathfrak{a}.$ We fix a positive subsystem $R_{+}\subset R$ and a
$W$-invariant multiplicity $k:R\to[0,\infty[$. The Cherednik operators
associated with $R_{+}$ and $k$ are defined as
(2.1) $D_{\xi}(k)f(x)=\partial_{\xi}f(x)+\sum_{\alpha\in
R_{+}}\frac{k_{\alpha}\langle\alpha,\xi\rangle}{1-e^{-\langle\alpha,x\rangle}}(f(x)-f(\sigma_{\alpha}(x))-\langle\rho(k),\xi\rangle
f(x)$
for $\xi\in\mathfrak{a},$ with the (weighted) Weyl vector
$\rho(k):=\frac{1}{2}\sum_{\alpha\in R_{+}}k_{\alpha}\alpha$. The operators
$D_{\xi}(k),\,\xi\in\mathfrak{a}\,$ commute, and there is a $W$-invariant
tubular neighbourhood $U$ of $\mathfrak{a}$ in
$\mathfrak{a}_{\mathbb{C}}=\mathfrak{a}+i\mathfrak{a}$ and a unique analytic
function $(\lambda,z)\mapsto G(\lambda,k;z)$ on
$\mathfrak{a}_{\mathbb{C}}\times U$, the so called Opdam-Cherednik kernel,
which satisfies
(2.2) $G(\lambda,k;0)=1\quad\text{ and}\quad
D_{\xi}(k)G(\lambda,k;\,.\,)=\langle\lambda,\xi\rangle\,G(\lambda,k;\,.\,)\quad\text{
for all }\,\xi\in\mathfrak{a}.$
The hypergeometric function associated with $R$ is defined by
(2.3) $F(\lambda,k;z)=\frac{1}{|W|}\sum_{w\in W}G(\lambda,k;w^{-1}z).$
It is $W$-invariant in $\lambda$ and $z.$ To introduce the associated Heckman-
Opdam polynomials, we need the weight lattice and the set of dominant weights,
$P=\\{\lambda\in\mathfrak{a}:\langle\lambda,\alpha^{\vee}\rangle\in\mathbb{Z}\>\>\forall\alpha\in
R\,\\},\quad P_{+}=\\{\lambda\in P:\langle\lambda,\alpha^{\vee}\rangle\geq
0\,\,\forall\alpha\in R_{+}\,\\}\supset R_{+},$
where $\alpha^{\vee}=\frac{2\alpha}{\langle\alpha,\alpha\rangle}.$ $P_{+}$
carries the usual dominance order. Let
$\mathcal{T}:=\text{span}_{\mathbb{C}}\\{e^{i\lambda},\,\lambda\in P\\}$
be the space of trigonometric polynomials associated with $R$. The orbit sums
$M_{\lambda}=\sum_{\mu\in W\\!\lambda}e^{i\mu}\,,\quad\lambda\in P_{+}$
form a basis of the subspace $\mathcal{T}^{W}$ of $W$-invariant polynomials in
$\mathcal{T}$. For
$Q^{\vee}:=\text{span}_{\mathbb{Z}}\\{\alpha^{\vee},\,\alpha\in R\\}$,
consider the compact torus $\,T=\mathfrak{a}/2\pi Q^{\vee}$ with the weight
function
(2.4) $\delta_{k}(x):=\prod_{\alpha\in
R_{+}}\Bigl{|}\sin\Bigr{(}\frac{\langle\alpha,x\rangle}{2}\Bigr{)}\Bigr{|}^{2k_{\alpha}}.$
The Heckman-Opdam polynomials associated with $R_{+}$ and $k$ are given by
$P_{\lambda}(k;z)=M_{\lambda}(z)+\sum_{\nu<\lambda}c_{\lambda\nu}(k)M_{\nu}(z)\quad(\lambda\in
P_{+}\,,z\in\mathfrak{a}_{\mathbb{C}})$
where the coefficients $c_{\lambda\nu}(k)\in\mathbb{R}$ are uniquely
determined by the condition that $P_{\lambda}(k;\,.\,)$ is orthogonal to
$M_{\nu}$ in $L^{2}(T,\delta_{k})$ for all $\nu\in P_{+}$ with $\nu<\lambda$.
It is known that $\\{P_{\lambda}(k,\,.\,),\lambda\in P_{+}\,\\}$ is an
orthonormal basis of $L^{2}(T,\delta_{k})^{W}$ of all $W$-invariant functions
from $L^{2}(T,\delta_{k})$. According to [HS], the normalized polynomials
$R_{\lambda}(k,z):=P_{\lambda}(k;z)/P_{\lambda}(k;0)$
can be expressed in terms of the hypergeometic function as
(2.5) $R_{\lambda}(k,z)=F(\lambda+\rho(k),k;iz).$
Note that our notion slightly differs from [HS, HO], where the polynomials
$P_{\lambda}$ are defined as exponential polynomials on the torus
$i\mathfrak{a}/2\pi iQ^{\vee}$. We next introduce the Heckman-Opdam Laplacian
$\Delta_{k}:=\sum_{j=1}^{N}D_{\xi_{j}}(k)^{2}\,-\,|\rho(k)|^{2}$
with some orthonormal basis $\xi_{1},\ldots,\xi_{N}$ of $\mathfrak{a}$. The
operator $\Delta_{k}$ is independent of the choice of this basis. Denote by
$L_{k}$ the restriction of $\Delta_{k}$ to $W$-invariant functions. According
to [Sch2],
(2.6) $L_{k}f(x)=\Delta f(x)+\sum_{\alpha\in
R_{+}}k_{\alpha}\coth\Bigl{(}\frac{\langle\alpha,x\rangle}{2}\Bigr{)}\cdot\partial_{\alpha}f(x).$
We notice that by construction, for all $\lambda\in\mathfrak{a}_{\mathbb{C}}$,
the hypergeometric functions
$F_{\lambda}:=F(\lambda,k;\,.\,)$
are eigenfunctions of $L_{k}$ with eigenvalues
$\sum_{j=1}^{N}\langle\lambda,\xi_{j}\rangle^{2}-|\rho(k)|^{2}.$
The operator $L_{k}$ is independent of the choice of $R_{+}$ and generates a
Feller diffusion on the closed Weyl chamber
$\overline{\mathfrak{a}_{+}}\subset\mathfrak{a}$ associated with $R_{+}$
(called a radial Heckman-Opdam process), where the paths are reflected at the
boundaries, see [Sch1, Sch2]. The transition probabilities of this process,
with starting point $y\in\overline{\mathfrak{a}_{+}}$, are given by
$p_{t}^{W}\\!(x,y)d\mu(x)$
with the $W$-invariant heat kernel
$p_{t}^{W}\\!(x,y)=\int_{i\mathfrak{a}}e^{-\frac{1}{2}(|\lambda|^{2}+|\rho|^{2})}F_{\lambda}(x)F_{\lambda}(-y)d\nu^{\prime}(\lambda),$
where $\rho=\rho(k),$ $\nu^{\prime}$ is the symmetric Plancherel measure as in
[Sch2], and
$d\mu(x)=d\mu_{k}(x)=\prod_{\alpha\in
R_{+}}\big{|}2\sinh\langle\frac{\alpha}{2},x\rangle\big{|}^{2k_{\alpha}}\,dx.$
In the compact case we take the factor $i$ in (2.5) into account as in [RR1]
and consider the operator
(2.7) $\widehat{L}_{k}f(x):=\Delta f(x)+\sum_{\alpha\in
R_{+}}k_{\alpha}\cot\Bigl{(}\frac{\langle\alpha,x\rangle}{2}\Bigr{)}\cdot\partial_{\alpha}f(x).$
$\widehat{L}_{k}$ generates a Feller diffusion on a compact fundamental alcove
of the affine Weyl group $W_{\\!\mathit{aff}}=W\ltimes 2\pi Q^{\vee}$ in
$\mathfrak{a}$, again with reflecting boundaries. The trigonometric
polynomials $P_{\lambda}$ ($\lambda\in P_{+}$) are eigenfunctions of
$\widehat{L}_{k}$ with eigenvalues
$-\langle\lambda,\lambda+2\rho(k)\rangle\leq 0.$
They were used in [RR1] to construct the transition densities of the
diffusions with generators $\widehat{L}_{k}$. For root systems of type $BC$,
the $P_{\lambda}$ are multivariate Jacobi polynomials, and the associated
Feller diffusions are multivariate Jacobi processes which were studied in
[Dem].
Before turning to details for root systems of type $A$ and $BC$, we conclude
this section with an integrability result which assures the existence of
exponential moments for the radial Heckman-Opdam processes in the non-compact,
crystallographic case. This will be needed in Sections 4 and 5.
###### Lemma 2.1.
For each $y\in\mathfrak{a}$ and $\,\beta\in\mathfrak{a},$
(2.8)
$\int_{\mathfrak{a}}e^{\langle\beta,x\rangle}\,p_{t}^{W}\\!(x,y)d\mu(x)\,<\infty.$
.
###### Proof.
For $y=0$ this is obvious from Theorem 5.2 of [Sch2]. For general $y$, we
employ the $L^{p}$-theory for the hypergeometric transform developed in [NPP].
For suitable functions on $\mathfrak{a}$ and on $i\mathfrak{a}$ respectively,
the hypergeometric transform and the inverse hypergeometric transform are
given by
$\mathcal{H}f(\lambda)=\int_{\mathfrak{a}}f(x)F_{\lambda}(-x)d\mu(x),\quad\mathcal{I}(g)(x)=\int_{i\mathfrak{a}}g(\lambda)F_{\lambda}(x)d\nu^{\prime}(\lambda).$
We denote by $\mathbb{C}[\mathfrak{a}_{\mathbb{C}}]$ the space of polynomial
functions on $\mathfrak{a}_{\mathbb{C}}$ and by $\partial(q)$ the constant
coefficient differential operator associated with
$q\in\mathbb{C}[\mathfrak{a}_{\mathbb{C}}].$ Moreover, for $x\in\mathfrak{a}$
we denote by $C(x):=co(W.x)$ the convex hull of the $W$-orbit of $x$ in
$\mathfrak{a}.$ For an exponent $0<p\leq 2$, the $W$-invariant
$L^{p}$-Schwartz space is given by
$\mathcal{C}^{p}(\mathfrak{a})^{W}=\Big{\\{}f\in
C^{\infty}(\mathfrak{a})^{W}:\,\sup_{x\in\mathfrak{a}}\,\frac{(1+|x|)^{n}}{F_{0}(x)^{2/p}}\,\big{|}\partial(q)f(x)\big{|}\,<\infty\,\,\forall\,n\in\mathbb{N}_{0},\,q\in\mathbb{C}[\mathfrak{a}_{\mathbb{C}}]\Big{\\}}.$
Moreover, we consider the $W$-invariant Schwartz space
$\mathcal{S}(\mathfrak{a}_{\epsilon_{p}})^{W}$ with
$\epsilon_{p}=\frac{2}{p}-1,$ which consists of all $W$-invariant continuous
functions on the closed tube
$\mathfrak{a}_{\epsilon_{p}}=C(\epsilon_{p}\rho)+i\mathfrak{a}\subset\mathfrak{a}_{\mathbb{C}}$
which are holomorphic in its interior and satisfy
(2.9)
$\sup_{\lambda\in\mathfrak{a}_{\epsilon_{p}}}\,(1+|\lambda|)^{n}\big{|}\partial(q)g(\lambda)\big{|}\,<\infty\quad\text{for
all }n\in\mathbb{N}_{0},\,q\in\mathbb{C}[\mathfrak{a}_{\mathbb{C}}].$
By Theorem 5.6 of [NPP], the hypergeometric transform $\mathcal{H}$ is a
topological isomorphism from $\mathcal{C}^{p}(\mathfrak{a})^{W}$ onto
$\mathcal{S}(\mathfrak{a}_{\epsilon_{p}})^{W}$ with inverse $\mathcal{I}$.
We claim that for fixed $y\in\mathfrak{a},$ the function
$g(\lambda):=e^{\frac{1}{2}\langle\lambda,\lambda\rangle}F_{\lambda}(-y)$
belongs to $\mathcal{S}(\mathfrak{a}_{\epsilon_{p}})$ for each $p\in]0,2];$
here $\langle\,.\,,\,.\,\rangle$ denotes the bilinear extension of the given
scalar product to $\mathfrak{a}_{\mathbb{C}}.$ As soon as this is proven, it
will follow that
$\,p_{t}^{W}(\,.\,,y)=e^{-\frac{1}{2}|\rho|^{2}}\,\mathcal{I}(g)\,$ belongs to
$\mathcal{C}^{p}(\mathfrak{a})^{W}$ for each $p\in]0,2].$ In order to check
that $g\in\mathcal{S}(\mathfrak{a}_{\epsilon_{p}}),$ we only have to verify
the growth condition (2.9). Let $q\in\mathbb{C}[\mathfrak{a}_{\mathbb{C}}].$
Then in view of Theorem 3.4 (and Remark 3.2) of [Sch2],
$|\partial_{\lambda}(q)F_{\lambda}(-y)|\,\leq C(1+|y|)^{\deg q}F_{0}(-y)\cdot
e^{\max_{w\in W}\text{Re}\langle
w\lambda,-y\rangle}\quad(\lambda\in\mathfrak{a}_{\mathbb{C}}).$
This shows that $\partial_{\lambda}(q)F_{\lambda}(-y)$ is bounded as a
function of $\lambda$ on $\mathfrak{a}_{\epsilon_{p}}$. Moreover,
$\partial(q)e^{\frac{1}{2}\langle\lambda,\lambda\rangle}=\widetilde{q}(\lambda)e^{\frac{1}{2}\langle\lambda,\lambda\rangle}$
with some polynomial $\widetilde{q}$, and
$\big{|}e^{\frac{1}{2}\langle\lambda,\lambda\rangle}\big{|}\,\asymp\,e^{-\frac{1}{2}|\lambda|^{2}}\quad\text{on
}\mathfrak{a}_{\epsilon_{p}}.$
Therefore $\partial(q)g(\lambda)$ decays exponentially as $|\lambda|\to\infty$
within $\mathfrak{a}_{\epsilon_{p}}.$ It follows that
$g\in\mathcal{S}(\mathfrak{a}_{\epsilon_{p}})$ and thus
$\,p_{t}^{W}\\!(\,.\,,y)\in\mathcal{C}^{p}(\mathfrak{a})^{W}.$ In particular,
for each $p\in]0,2]$ there exists a constant $C_{p}>0$ such that
$p_{t}^{W}\\!(x,y)\leq C_{p}F_{0}(x)^{2/p}\quad\text{ for all
}x\in\mathfrak{a}.$
From [Sch2] we know that in the closed chamber
$\overline{\mathfrak{a}_{+}}\,,$ $\,F_{0}(x)\asymp
q_{0}(x)e^{-\langle\rho,x\rangle}\,$ with a certain positive polynomial
$q_{0}$. Hence there exists a nonnegative polynoimal $q_{p}$ (depending on
$p$), such that
$p_{t}^{W}\\!(x,y)\leq
q_{p}(x)e^{-\frac{2}{p}\langle\rho,x\rangle}\quad\text{for all
}x\in\overline{\mathfrak{a}_{+}}.$
Now fix $\beta\in\mathfrak{a}$ and note that $\rho$ is contained in the open
chamber $\mathfrak{a}_{+}.$ Choosing $p>0$ small enough, we therefore obtain
$\int_{\overline{\mathfrak{a}_{+}}}e^{\langle\beta,x\rangle}p_{t}^{W}\\!(x,y)d\mu(x)\,\leq\int_{\overline{\mathfrak{a}_{+}}}q_{p}(x)e^{\langle\beta,x\rangle-\frac{2}{p}\langle\rho,x\rangle+2\langle\rho,x\rangle}dx\,<\infty.$
This yields the assertion. ∎
## 3\. The compact case of type $A_{N-1}$
In this section we study Heckman-Opdam processes of type $A_{N-1}$ in the
compact setting. The generators of these processes are the Hamiltonians of
interacting particle models of Calogero-Sutherland type with $N$ particles on
the torus $\mathbb{T}:=\\{z\in\mathbb{C}:\>|z|=1\\}$; see [LV] for the
background. These processes are diffusions on some fundamental domain of
$W=S_{N}$ in $\mathbb{T}^{N}$. It will however be convenient to consider also
associated diffusions on $\mathbb{R}^{N}$ with $2\pi$-periodicity such that
the diffusions on $\mathbb{T}^{N}$ appear as images under $x\mapsto e^{ix}.$
To introduce the processes on $\mathbb{R}^{N}$, we consider the root system
$R=A_{N-1}=\\{\pm(e_{i}-e_{j}):1\leq i<j\leq N\\}$ in $\mathbb{R}^{N}$ with
positive subsystem $R_{+}=\\{e_{i}-e_{j}:i<j\\}$ and fix a multiplicity
parameter $k\in\,]0,\infty[$. Let
$\omega:=(1,\ldots,1)^{T}\in\mathbb{R}^{N}.$
Then $Q^{\vee}=\mathbb{Z}^{N}\cap(\mathbb{R}\omega)^{\perp},$ and a
fundamental domain for the action of $W_{\\!\mathit{aff}}=W\ltimes 2\pi
Q^{\vee}$ in $\mathbb{R}^{N}$ is given by
$\displaystyle C_{N}$
$\displaystyle=\\{x\in\mathbb{R}^{N}:0\leq\langle\alpha,x\rangle\leq
2\pi\,\,\forall\alpha\in R_{+}\\}$
$\displaystyle=\,\\{x\in\mathbb{R}^{N}:\,x_{1}\leq x_{2}\leq\ldots\leq
x_{N}\leq x_{1}+2\pi\\}.$
We consider the $W$-invariant Heckman-Opdam Laplacian
(3.1) $\widehat{L}_{k}f(x)=\Delta f(x)+k\sum_{j=1}^{N}\sum_{l\neq
j}\cot\Bigl{(}\frac{x_{j}-x_{l}}{2}\Bigr{)}\frac{\partial}{\partial
x_{j}}f(x)$
with reflecting boundaries, i.e. with domain
$D(\widehat{L}_{k})=\\{f|_{C^{N}}:f\in C^{2}(\mathbb{R}^{N})\text{ invariant
under }W_{\\!\mathit{aff}}\\}.$
$\widehat{L}_{k}$ is the generator of a Feller semigroup of transition
operators on $C_{N}$, c.f. [RR1]. Associated Feller diffusions
$(X_{t,k})_{t\geq 0}$ with continuous paths (which are reflected at the
boundary of $C_{N}$) are called Heckman-Opdam processes of type $A_{N-1}$ on
$C_{N}$. We also consider the renormalized generators
$\widetilde{L}_{k}:=\frac{1}{k}\widehat{L}_{k}=\frac{1}{k}\Delta+\sum_{j=1}^{N}\sum_{l\neq
j}\cot\Bigl{(}\frac{x_{j}-x_{l}}{2}\Bigr{)}\frac{\partial}{\partial x_{j}}$
which degenerate for $k\to\infty$ into
$\widetilde{L}_{\infty}=\sum_{j=1}^{N}\sum_{l\neq
j}\cot\Bigl{(}\frac{x_{j}-x_{l}}{2}\Bigr{)}\frac{\partial}{\partial x_{j}}.$
For $k\in]0,\infty[$, the process
$\widetilde{X}_{k}:=(\widetilde{X}_{t,k})_{t\geq 0}$ with
$\widetilde{X}_{t,k}:=X_{t/k,k}$ is a Feller diffusion associated with
$\widetilde{L}_{k}$. It can be also described as solution of the SDE
(3.2) $d\widetilde{X}_{t,k,j}=\frac{\sqrt{2}}{\sqrt{k}}dB_{t,j}+\sum_{l\neq
j}\cot\Bigl{(}\frac{\widetilde{X}_{t,k,j}-\widetilde{X}_{t,k,l}}{2}\Bigr{)}dt\quad\quad(j=1,\ldots,N)$
with some $N$-dimensional Brownian motion $(B_{t,1},\ldots,B_{t,N})_{t\geq
0}$. Moreover, for deterministic starting conditions, $\widetilde{L}_{\infty}$
is the generator of a deterministic process whose paths are the solution of
some initial value problem for the ODE
(3.3) $\frac{dx_{j}}{dt}(t)=\sum_{l\neq
j}\cot\Bigl{(}\frac{x_{j}(t)-x_{l}(t)}{2}\Bigr{)}\quad\quad(j=1,\ldots,N).$
As for Dunkl processes in [AV], one can show that for initial data in the
interior of $C_{N}$, the solution $(\widetilde{X}_{t,\infty})_{t\geq 0}$ of
this ODE exists for all $t\geq 0$ in the interior of $C_{N}$. We also point
out that in the Dunkl setting, the ODE analogous to (3.2) has unique solutions
for all $t\geq 0$ even for starting points at the boundary of the chamber, see
[VW]. We expect that such a result is also true in the present setting.
We need the following stationary solutions of (3.3):
###### Lemma 3.1.
For each $x_{1}\in\mathbb{R}$,
$\Bigl{(}x_{1},x_{1}+\frac{1}{N}2\pi,x_{1}+\frac{2}{N}2\pi,\ldots,x_{1}+\frac{N-1}{N}2\pi\Bigr{)}\in
C_{N}$
is a stationary solution of (3.3).
###### Proof.
Assume first that $N$ is odd. As the cotangent is odd and $\pi$-periodic, we
have for $j=1,\ldots,N$ that
$\sum_{l\neq
j}\cot\Bigl{(}\frac{(j-l)\pi}{N}\Bigr{)}=\sum_{l=1}^{N-1}\cot\Bigl{(}\frac{l\pi}{N}\Bigr{)}=\sum_{l=1}^{(N-1)/2}\Bigl{(}\cot\Bigl{(}\frac{l\pi}{N}\Bigr{)}+\cot\Bigl{(}\frac{(N-l)\pi}{N}\Bigr{)}\Bigr{)}=0.$
If $N$ is even, then our computation leads to the additional term
$\cot((N\pi/2)/N)=0$ in the last sum and thus to the same result. This yields
the claim. ∎
The generator $\widetilde{L}_{k}$ and the associated diffusion
$\widetilde{X}_{k}$ on $\mathbb{R}^{N}$ can be decomposed into two independent
parts, namely the center of gravity and the process of the distances of
neighboring particles. This reflects the fact that the usual representation of
the symmetric group $S_{N}$ on $\mathbb{R}^{N}$ decomposes into two
irreducible components. More precisely, consider the center-of-gravity process
$\widetilde{X}_{k}^{\mathit{cg}}:=(\widetilde{X}_{t,k}^{\mathit{cg}})_{t\geq
0}$ with
$\widetilde{X}_{t,k}^{cg}:=\frac{1}{N}(\widetilde{X}_{t,k,1}+\ldots+\widetilde{X}_{t,k,N})\cdot\omega$
which is the orthogonal projection of $\widetilde{X}_{t,k}$ onto
$\mathbb{R}\omega$. Then the diffusion
$\widetilde{X}_{k}^{\mathit{diff}}:=\widetilde{X}_{k}-\widetilde{X}_{k}^{\mathit{cg}}$
lives on the orthogonal complement
$(\mathbb{R}\omega)^{\perp}\subset\mathbb{R}^{N}$.
###### Lemma 3.2.
Let $k\in\,]0,\infty[$. If $\widetilde{X}_{k}$ starts in some deterministic
point, then the processes $\widetilde{X}_{k}^{\mathit{diff}}$ and
$\widetilde{X}_{k}^{\mathit{cg}}$ are stochastically independent.
###### Proof.
By the SDE (3.2) we have
(3.4)
$d\widetilde{X}_{t,k,j}^{\mathit{cg}}=\frac{\sqrt{2}}{N\sqrt{k}}d\Bigl{(}\sum_{l=1}^{N}B_{t,l}\Bigr{)}=:\frac{\sqrt{2}}{\sqrt{Nk}}\,dB_{t}$
with some one-dimensional Brownian motion $(B_{t})_{t\geq 0}$ while
$\widetilde{X}_{k}^{\mathit{diff}}$ satisfies
$d\widetilde{X}_{t,k,j}^{\mathit{diff}}=\frac{\sqrt{2}}{\sqrt{k}}\,d\Bigl{(}B_{t,j}-\frac{1}{N}\sum_{l=1}^{N}B_{t,l}\Bigr{)}+F_{j}(\widetilde{X}_{t,k}^{\mathit{diff}})\,dt\quad(j=1,\ldots,N)$
where the processes $(B_{t,j}-\frac{1}{N}\sum_{l=1}^{N}B_{t,l})_{t\geq 0}\,$
are stochastically independent of $(B_{t})_{t\geq 0}$, and the $F_{j}$ are
concrete continuous functions. This implies the claim by the very definition
of solutions of SDEs. ∎
We next transfer all data from $\mathbb{R}^{N}$ to the torus $\mathbb{T}^{N}$
via $z_{j}:=e^{ix_{j}}$ for $j=1,\ldots,N$, or for short,
$z:=e^{ix}\in\mathbb{T}^{N}$. A short computation shows that in
$z$-coordinates, the operator $L_{k}$ is given by the diffusion operator
(3.5) $\displaystyle H_{k}=$
$\displaystyle-\sum_{j=1}^{N}\Bigl{(}z_{j}\frac{\partial}{\partial
z_{j}}\Bigr{)}^{2}-k\sum_{j=1}^{N}\sum_{l\neq
j}\frac{z_{j}+z_{l}}{z_{j}-z_{l}}\cdot z_{j}\frac{\partial}{\partial z_{j}}$
$\displaystyle=$
$\displaystyle-\sum_{j=1}^{N}z_{j}^{2}\frac{\partial^{2}}{\partial
z_{j}^{2}}-(1-k(N-1))\sum_{j=1}^{N}z_{j}\frac{\partial}{\partial
z_{j}}-2k\sum_{j=1}^{N}\sum_{l\neq
j}\frac{z_{j}^{2}}{z_{j}-z_{l}}\frac{\partial}{\partial z_{j}}\,,$
acting on permutation invariant functions from $C^{2}(\mathbb{T}^{N})$. The
operator $H_{k}$ appears in a prominent way in the particle models of
Calogero-Sutherland type on $\mathbb{T}$; see Section 2 of [LV]. It is
obtained from the Calogero-Sutherland Hamiltonian by conjugation with the
ground state. The operator $H_{k}$ is the generator of the Feller diffusions
$Z_{k}:=(Z_{t,k}:=e^{iX_{t,k}})_{t\geq 0}$ on the alcove
$\mathbb{A}_{N}:=\\{e^{ix}:\>x\in C_{N}\\}\subset\mathbb{T}^{N}.$
Further, the operators $\widetilde{H}_{k}:=\frac{1}{k}H_{k}$ generate the
diffusions
$\widetilde{Z}_{k}:=(\widetilde{Z}_{t,k}:=e^{i\widetilde{X}_{t,k}})_{t\geq 0}$
for $k\in]0,\infty[$. Clearly, this also works for $k=\infty$ where
$\widetilde{Z}_{\infty}$ is deterministic for deterministic initial data.
###### Remark 3.3.
Besides the generators in (3.5), also the operators
(3.6) $D_{k}:=\sum_{j=1}^{N}z_{j}^{2}\frac{\partial^{2}}{\partial
z_{j}^{2}}+2k\sum_{j=1}^{N}\sum_{l\neq
j}\frac{z_{j}^{2}}{z_{j}-z_{l}}\frac{\partial}{\partial z_{j}}$
appear in the literature; see e.g. [F, St, OO]. Here $-H_{k}=D_{k}+E_{k}\,$
with the Euler operator
(3.7) $E_{k}:=(1-k(N-1))\sum_{j=1}^{N}z_{j}\frac{\partial}{\partial z_{j}}$
which commutes with $D_{k}$. The $-D_{k}$ are generators of diffusions with
additional drift on $\mathbb{T}$ which rotates the complete system at some
fixed speed. Clearly, the subsequent results can be easily translated to
$-D_{k}$. From a stochastic point of view, the diffusions associated with the
operators $H_{k}$ seem to be the most natural ones.
We recall that the (symmetric) eigenfunctions of $H_{k}$ are Jack polynomials.
To become precise, we introduce the following notations. We write
$\Lambda_{N}^{+}=\\{\lambda\in\mathbb{Z}_{+}^{N}:\lambda_{1}\geq\cdots\geq\lambda_{N}\\}$
for the set of partitions of length at most $N$. Denote by
$C_{\lambda}^{\alpha}\,,\,\lambda\in\Lambda_{N}^{+},$ the Jack polynomials of
index $\alpha>0$ in $N$ variables with the normalization
(3.8)
$(z_{1}+\cdots+z_{N})^{m}=\sum_{|\lambda|=m}C_{\lambda}^{\alpha}(z)\quad(m\in\mathbb{Z}_{+});$
see [St, BF]. The $C_{\lambda}^{\alpha}$ are symmetric and homogeneous of
degree $|\lambda|:=\lambda_{1}+\cdots+\lambda_{N}$. Moreover, by [St, BF], the
$C_{\lambda}^{\alpha}$ with index $\alpha=1/k$ are eigenfunctions of $D_{k}$
with eigenvalues
$d_{\lambda}(k)=\sum_{j=1}^{N}\lambda_{j}(\lambda_{j}-1+2k(N-j)).$
In addition, as $C_{\lambda}^{\alpha}$ is homogeneous of degree $|\lambda|$,
$E_{k}C_{\lambda}^{\alpha}=(1-k(N-1))|\lambda|\,C_{\lambda}^{\alpha}.$
In summary we obtain:
###### Lemma 3.4.
For $\lambda\in\Lambda_{n}^{+}$, $k\in]0,\infty[$ and $\alpha=1/k$,
$C_{\lambda}^{\alpha}$ is an eigenfunction of
$\widetilde{H}_{k}=\frac{1}{k}H_{k}$ with eigenvalue
$-\sum_{j=1}^{N}\lambda_{j}\bigl{(}\frac{\lambda_{j}}{k}+N+1-2j\bigr{)}\,\leq
0.$
It is well-known (c.f. [HO]) that the polynomials
$C_{\lambda}^{1/k},\,\lambda\in\Lambda_{n}^{+}\,$ form a complete orthogonal
system of $L^{2}(\mathbb{T}^{N},\mu_{k})^{W}$ with the probability measure
(3.9)
$d\mu_{k}(z)=\phi_{k}(z)dz,\,\,\,\phi_{k}(z):=\,c_{k}\cdot\\!\prod_{j,l:l\neq
j}|z_{j}-z_{l}|^{k}\cdot{\bf 1}_{\mathbb{A}_{N}}(z),$
where $c_{k}>0$ is a normalization constant and $dz$ denotes the Haar measure
on $\mathbb{T}^{N}$. Notice that these measures appear also in the context of
circular $\beta$-ensembles in random matrix theory.
We also notice that the elementary symmetric polynomials $e_{l}$
($l=0,\ldots,N$) in $N$ variables, which are determined by
$\prod_{j=1}^{N}(y-x_{j})\,=\sum_{l=0}^{N}(-1)^{l}e_{l}(x)\,y^{N-l}\quad(y\in\mathbb{C}),$
are Jack polynomials for all $k>0$ up to normalization. More precisely, by
(3.8),
(3.10)
$e_{l}(x)=\frac{1}{l!}C_{\lambda}^{\alpha}(x)\>\>\text{with}\>\>\lambda=1^{l}=(\underbrace{1,\ldots,1}_{l\text{
times}},0,\ldots,0)\quad(l=0,\ldots,N).$
Thus in view of Lemma 3.4, the polynomials $e_{l}$ are eigenfunctions of
$\widetilde{H}_{k}$ with eigenvalues $-l\bigl{(}\frac{1}{k}+N-l\bigr{)}$.
Moreover, by a continuity argument or by direct computation, the $e_{l}$ are
also eigenfunctions of
$\widetilde{H}_{\infty}=(N-1)\sum_{j=1}^{N}z_{j}\frac{\partial}{\partial
z_{j}}-2\sum_{j=1}^{N}\sum_{l\neq
j}\frac{z_{j}^{2}}{z_{j}-z_{l}}\frac{\partial}{\partial z_{j}}$
with eigenvalues $-l(N-l).$
We now use these properties to construct martingales from our processes
$\widetilde{Z}_{k}$ for $k\in]0,\infty[$. We recall that by Dynkin’s formula
(see e.g. Section III.10 of [RW]), the following holds for the generator $L$
of a Feller semigroup and an arbitrary associated Feller process
$(X_{t})_{t\geq 0}$: if $f$ is a bounded eigenfunction of $L$ with eigenvalue
$r\in\mathbb{R}$, then the process $(e^{-rt}f(X_{t}))_{t\geq 0}$ is a
martingale w.r.t. the canonical filtration. We thus have:
###### Corollary 3.5.
For $k\in]0,\infty[$ and $z\in\mathbb{A}_{N}$ consider the diffusion
$(\widetilde{Z}_{t,k})_{t\geq 0}$ on $\mathbb{A}_{N}$ with start in $z$. Then,
for $l=0,1,\ldots,N$, the process
$(e^{l(1/k+N-l)t}e_{l}(\widetilde{Z}_{t,k}))_{t\geq 0}$ is a martingale. In
particular, for $t\geq 0$,
$\mathbb{E}(e_{l}(\widetilde{Z}_{t,k}))=e^{-l(1/k+N-l)t}\,e_{l}(z).$
This statement also holds in the deterministic case $k=\infty$, where we also
obtain additional information for $t\to\infty$:
###### Corollary 3.6.
For each starting point $z$ in the interior of $\mathbb{A}_{N}$, the
deterministic process $(\widetilde{Z}_{t,\infty})_{t\geq 0}$ satisfies
(3.11)
$e_{l}(\widetilde{Z}_{t,\infty})=e^{-l(N-l)t}e_{l}(z)\quad\quad(l=0,1,\ldots,N).$
Moreover, the limit
$Z:=\lim_{t\to\infty}\widetilde{Z}_{t,\infty}\in\mathbb{T}^{N}$ exists and is
given by
(3.12) $Z=(Z_{1},Z_{1}e^{2\pi i/N},\,\ldots,Z_{1}e^{2\pi i(N-1)/N})$
where $Z_{1}\in\mathbb{T}$ is as follows: If
$z=(e^{ix_{1}},\ldots,e^{ix_{N}})$ with $(x_{1},\ldots,x_{N})\in C_{N}$, then
(3.13) $Z_{1}=e^{ix_{0}}\quad\text{with}\quad
x_{0}=\frac{x_{1}+\ldots+x_{N}-\pi(N-1)}{N}.$
###### Proof.
Eq. (3.11) is obvious. As each $\zeta\in\mathbb{A}_{N}$ is uniquely determined
by the elementary symmetric functions $e_{l}(\zeta)$, (3.11) and a continuity
argument imply that
$Z=(Z_{1},\ldots,Z_{N}):=\lim_{t\to\infty}\widetilde{Z}_{t,\infty}\in\mathbb{A}_{N}$
exists, and that
$\prod_{j=1}^{N}(y-Z_{j})=y^{N}-e_{N}(Z)=y^{N}-z_{1}z_{2}\cdots z_{N}\,.$
This shows that $Z$ has the form as stated in (3.12) for some
$Z_{1}\in\mathbb{T}$ with $Z_{1}^{N}=z_{1}\cdots z_{n}\,.$ To identify
$Z_{1}$, we write the initial condition as $z=(e^{ix_{1}},\ldots,e^{ix_{N}})$
with $(x_{1},\ldots,x_{N})\in C_{N}$. In the $x$-coordinates, our process
$(\widetilde{X}_{t,\infty}=(x_{t,1},\ldots,x_{t,N}))_{t\geq 0}$ satisfies the
ODE (3.3). This ODE yields that $x_{t,1}+\ldots+x_{t,N}$ is independent of
$t\in[0,\infty]$. Therefore, the form of
$Z=(e^{ix_{\infty,1}},\ldots,e^{ix_{\infty,N}})$ in (3.12) yields
$x_{1}+\ldots+x_{N}=x_{\infty,1}+\ldots+x_{\infty,N}=\sum_{j=0}^{N-1}(x_{\infty,1}+j\cdot
2\pi/N)=Nx_{\infty,1}+(N-1)\pi.$
This implies (3.13). ∎
In the next step we use the decomposition of the processes
$\widetilde{X}_{k}=\widetilde{X}_{k}^{\mathit{diff}}+\widetilde{X}_{k}^{\mathit{cg}}.$
Eq. (3.4) and the expectations of the geometric Brownian motion imply
(3.14)
$\mathbb{E}(e^{-il\widetilde{X}_{t,k,j}^{\mathit{cg}}})=\mathbb{E}\left(e^{-\frac{il\sqrt{2}}{\sqrt{Nk}}B_{t}-il\widetilde{X}_{0,k,j}^{\mathit{cg}}}\right)=e^{-\frac{tl^{2}}{Nk}-il\widetilde{X}_{0,k,j}^{\mathit{cg}}}\quad(j,l=1,\ldots,N).$
This yields:
###### Corollary 3.7.
Let $k\in\,]0,\infty[$ and $x\in C_{N}$ with $x_{1}+\ldots+x_{N}=0.$ Let
$z=e^{ix}\in\mathbb{A}_{N}$, and consider the diffusion
$(\widetilde{Z}_{t,k})_{t\geq 0}$ on $\mathbb{A}_{N}$ with start in $z$. Then,
for $t\geq 0$ and $l=1,\ldots,N$,
$\mathbb{E}(e_{l}(e^{i\widetilde{X}_{t,k}^{\mathit{diff}}}))=e^{-l(N-l+(N+l)/(Nk))t}e_{l}(z).$
###### Proof.
By our initial conditions and Eq. (3.4) we have
$\widetilde{X}_{t,k,j}^{\mathit{cg}}=\widetilde{X}_{t,k,1}^{\mathit{cg}}$ for
all $t,k$ and $j=1,\ldots,N$. Hence, the stochastic independence of
$\widetilde{X}_{k}^{\mathit{diff}},\widetilde{X}_{k}^{\mathit{cg}}$, Corollary
3.5, (3.14), and the initial condition imply
$\displaystyle\mathbb{E}(e_{l}(e^{i\widetilde{X}_{t,k}^{\mathit{diff}}}))$
$\displaystyle=\mathbb{E}(e_{l}(e^{i\widetilde{X}_{t,k}-i\widetilde{X}_{t,k}^{\mathit{cg}}}))=\mathbb{E}(e_{l}(\widetilde{Z}_{t,k})\cdot
e^{-il\widetilde{X}_{t,k,1}^{\mathit{cg}}})$
$\displaystyle=\mathbb{E}(e_{l}(\widetilde{Z}_{t,k}))\cdot\mathbb{E}(e^{-il\widetilde{X}_{t,k,1}^{\mathit{cg}}})=e^{-l(1/k+N-l)t}\,e_{l}(z)\cdot
e^{-\frac{tl^{2}}{Nk}}$
as claimed. ∎
###### Example 3.8.
For the starting configuration $x=(0,2\pi/N,\ldots,(N-1)2\pi/N)$ and
$z=e^{x}$, we have $e_{1}(z)=\ldots=e_{N-1}(z)=0$ and $e_{N}(z)=(-1)^{N-1}$.
Hence, by Corollary 3.7,
$\mathbb{E}\bigl{(}e_{l}(e^{i\widetilde{X}_{t,k}^{\mathit{diff}}})\bigr{)}=0\quad(l=1,\ldots,N-1),\quad\text{and}\quad\mathbb{E}\bigl{(}e_{N}(e^{i\widetilde{X}_{t,k}^{\mathit{\mathit{diff}}}})\bigr{)}=(-1)^{N-1}e^{-2Nt/k}$
for all $t\geq 0$. As $e_{0}=1$, we conclude that in this case for all
$y\in\mathbb{C}$ and $t\geq 0$,
(3.15)
$\mathbb{E}\bigl{(}\prod_{j=1}^{N}(y-e^{i\widetilde{X}_{t,k,j}^{\mathit{diff}}})\bigr{)}=\mathbb{E}\bigl{(}\sum_{l=0}^{N}y^{N-l}(-1)^{l}e_{l}(e^{i\widetilde{X}_{t,k}^{\mathit{diff}}})\bigl{)}=y^{N}-e^{-2Nt/k}.$
We point out that this result differs from the case $k=\infty$ where $x$ is a
stationary solution of the associated ODE by Lemma 3.1, and thus
$\mathbb{E}\bigl{(}\prod_{j=1}^{N}(y-e^{i\widetilde{X}_{t,\infty,j}^{\mathit{diff}}})\bigr{)}=y^{N}-1$
is independent from $t$.
We next study the limit $t\to\infty$ for $k\in]0,\infty[$ similar to the limit
results for $k=\infty$ in Corollary 3.6. We need the following well-known
result, which follows easily for instance from the explicit formulas for the
densities of our diffusions in [RR1]:
###### Lemma 3.9.
Let $k\in\,]0,\infty[$. Then for each starting point in $\mathbb{A}_{N}$, the
process $(\widetilde{Z}_{t,k})_{t\geq 0}$ converges in distribution for
$t\to\infty$ to the probability measure $\mu_{k}$ on $\mathbb{A}_{N}$ from
(3.9).
This observation and Corollary 3.5 for $t\to\infty$ imply:
###### Corollary 3.10.
Let $Z=(Z_{1},\ldots,Z_{N})$ be an $\mathbb{A}_{N}$-valued random variable
with the distribution $\mu_{k}$ of a circular $\beta$-ensemble. Then for each
$y\in\mathbb{C}$,
$\mathbb{E}\bigl{(}\prod_{j=1}^{N}(y-Z_{j})\bigr{)}=y^{N}.$
###### Proof.
$\mathbb{E}\bigl{(}\prod_{j=1}^{N}(y-Z_{j})\bigr{)}=\mathbb{E}\bigl{(}\sum_{l=0}^{N}y^{N-l}(-1)^{l}e_{l}(Z)\bigl{)}=y^{N}.$
∎
A corresponding result can be also stated under the condition that $Z$ only
takes values in the alcove
$\mathbb{A}_{N}^{1}:=\\{z\in\mathbb{A}_{N}:z_{1}\cdots z_{N}=1\\}.$
For this consider the process $\widetilde{X}_{k}^{\mathit{diff}}$ in the
decomposition
$\widetilde{X}_{k}=\widetilde{X}_{k}^{\mathit{diff}}+\widetilde{X}_{k}^{\mathit{cg}}$
above. Then $e^{i\widetilde{X}_{k}^{\mathit{diff}}}$ is a diffusion on
$\mathbb{A}_{N}^{1}$ which converges for $t\to\infty$ in distribution to the
conditional probability measure $\mu_{k}^{1}\in M^{1}(\mathbb{A}_{N}^{1})$ of
$\mu_{k}$ under the condition $\mathbb{A}_{N}^{1}$. (This is, up to
normalization, the measure with the density (2.4)). Eq. (3.15) now leads to:
###### Corollary 3.11.
Let $Z$ be an $\mathbb{A}_{N}^{1}$-valued random variable with distribution
$\mu_{k}^{1}$. Then for each $y\in\mathbb{C}$,
$\mathbb{E}\bigl{(}\prod_{j=1}^{N}(y-Z_{j})\bigr{)}=y^{N}.$
###### Remark 3.12.
It can be shown and is well known that the measures $\mu_{k}$ tend for
$k\to\infty$ weakly to the probability measure $\mu_{\infty}$ which appears as
image of the uniform distribution on $\mathbb{T}$ under the mapping
$\mathbb{T}\to\mathbb{A}_{N},\quad z\to(z,z\cdot e^{2\pi i/N},\ldots,z\cdot
e^{2\pi i(N-1)/N}).$
Corollary 3.10 and continuity show that a random variable
$Z=(Z_{1},\ldots,Z_{N})$ on $\mathbb{A}_{N}$ with distribution $\mu_{\infty}$
also satisfy $\mathbb{E}\bigl{(}\prod_{j=1}^{N}(y-Z_{j})\bigr{)}=y^{N}$. A
corresponding result holds also in the situation of Corollary 3.11.
Please notice that this differs from the situation in the end of Example 3.8
for $k=\infty$ where we have a purely deterministic situation and also a
slightly different result.
Corollaries 3.5, 3.7, 3.10, and 3.11 have applications to Brownian motions and
uniform probabilities on compact symmetric spaces of type $A.$
For a first example, consider the space $C(U(N))$ of all conjugacy classes of
$U(N)$, which can be identified with $\mathbb{A}_{N}$ up to the cyclic group
$\mathbb{Z}_{N}$, i.e., $C(U(N))\sim\mathbb{A}_{N}/\mathbb{Z}_{N}$. In fact,
the conjugacy classes are characterized via the ordered spectra of matrices
from $U(N)$ where, say, the eigenvalue with the smallest nonnegative argument
has the first position. On the other hand, elements in $\mathbb{A}_{N}$
describe configurations of ordered points on $\mathbb{T}$ where the position
of the first entry is arbitrary. It is also well known that the pushforward of
the uniform distribution (i.e., the normalized Haar measure) of $U(N)$ under
the natural projection $U(N)\mapsto C(U(N))\sim\mathbb{A}_{N}/\mathbb{Z}_{N}$
agrees with the pushforward of $\mu_{k}$ with $k=1$ under the canonical
mapping $\mathbb{A}_{N}\mapsto\mathbb{A}_{N}/\mathbb{Z}_{N}$. Corollary 3.10
thus leads to the following.
###### Corollary 3.13.
Let $Z$ be a uniformly distributed $U(N)$-valued random variable. Then for
each $y\in\mathbb{C}$, $\,\mathbb{E}\bigl{(}\det(yI_{N}-Z)\bigr{)}=y^{N}.$
The same procedure also works for $SU(N),$ where the space $C(SU(N))$ of
conjugacy classes corresponds to $\mathbb{A}_{N}^{1}/\mathbb{Z}_{N}$, and the
pushforward of the uniform distribution corresponds to $\mu_{k}^{1}\in
M^{1}(\mathbb{A}_{N}^{1})$ with $k=1$. Hence, by Corollary 3.11:
###### Corollary 3.14.
Let $Z$ be an $SU(N)$-valued random variable which is uniformly distributed.
Then for $y\in\mathbb{C}$, $\mathbb{E}\bigl{(}\det(yI_{N}-Z)\bigr{)}=y^{N}$.
Corollaries 3.13 and 3.14 are special cases of well-known general formulas for
integrals of polynomials on unitary groups w.r.t. uniform distributions in
[CS]. On the other hand, our approach leads to generalizations of these
formulas for Brownian motions on $U(N)$ and $SU(N)$.
For $k=1/2$ or $k=2$, our results above are related to the compact symmetric
spaces $U(n)/O(n)$ and $U(2n)/Sp(n)$ associated with the root system
$A_{N-1}$.
## 4\. The non-compact case of type $A_{N-1}$
In this section we start with the $W$-invariant Heckman-Opdam Laplacians
(4.1) $L_{k}f(x)=\Delta f(x)+k\sum_{j=1}^{N}\sum_{l\neq
j}\coth\Bigl{(}\frac{x_{j}-x_{l}}{2}\Bigr{)}\frac{\partial}{\partial
x_{j}}f(x)$
for $k\in[0,\infty[$ on the Weyl chamber
$C_{N}^{A}:=\\{x\in\mathbb{R}^{N}:\>x_{1}\geq x_{2}\geq\ldots\geq x_{N}\\}$
of type $A_{N-1}$. As in Section 3, $L_{k}$ is the generator of a Feller
diffusion $(X_{t,k})_{t\geq 0}$ on $C_{N}^{A}$ with reflecting boundaries.
Again we study the renormalized generators
$\widetilde{L}_{k}:=\frac{1}{k}L_{k}$ which degenerate for $k\to\infty$ into
$\widetilde{L}_{\infty}=\sum_{j=1}^{N}\sum_{l\neq
j}\coth\Bigl{(}\frac{x_{j}-x_{l}}{2}\Bigr{)}\frac{\partial}{\partial x_{j}}.$
The process $\widetilde{X}_{k}:=(\widetilde{X}_{t,k}:=X_{t/k,k})_{t\geq 0}$
for $k\in]0,\infty[$ then solves the SDE
(4.2) $d\widetilde{X}_{t,k,j}=\frac{\sqrt{2}}{\sqrt{k}}dB_{t,j}+\sum_{l\neq
j}\coth\Bigl{(}\frac{\widetilde{X}_{t,k,j}-\widetilde{X}_{t,k,l}}{2}\Bigr{)}dt\quad\quad(j=1,\ldots,N)$
which degenerates for $k=\infty$ into the ODE
(4.3) $\frac{dx_{j}}{dt}(t)=\sum_{l\neq
j}\coth\Bigl{(}\frac{x_{j}(t)-x_{l}(t)}{2}\Bigr{)}\quad\quad(j=1,\ldots,N).$
Again, for initial data in the interior of the chamber $C_{N}^{A}$, the
solution $(\widetilde{X}_{t,\infty})_{t\geq 0}$ of these differential
equations exists for all $t\geq 0$ in the interior of $C_{N}^{A}$.
As in the preceding section, we next decompose the diffusions
$\widetilde{X}_{k}$ into the center-of-gravity process
$\widetilde{X}_{k}^{\mathit{cg}}:=(\widetilde{X}_{t,k}^{\mathit{cg}})_{t\geq
0}$ with
$\widetilde{X}_{t,k}^{\mathit{cg}}:=\frac{1}{N}(\widetilde{X}_{t,k,1}+\ldots+\widetilde{X}_{t,k,N})\cdot\omega$
and the process
$\widetilde{X}_{k}^{\mathit{diff}}:=\widetilde{X}_{k}-\widetilde{X}_{k}^{\mathit{cg}}\,$
on the Weyl chamber
$C_{N,0}^{A}:=\\{x\in C_{N}^{A}:x_{1}+\ldots+x_{N}=0\\}$
which describes the distances of the particles.
As in the proof of Lemma 3.2, the processes
$\widetilde{X}_{k}^{\mathit{diff}}$ and $\widetilde{X}_{k}^{\mathit{cg}}$ are
stochastically independent.
In the next step we observe that the processes $\widetilde{X}$,
$\widetilde{X}_{k}^{\mathit{diff}}$, $\widetilde{X}_{k}^{\mathit{cg}}$ admit
arbitrary exponential moments for arbitrary deterministic starting points. In
fact, for $\widetilde{X}_{k}^{\mathit{diff}}$ this follows from Lemma 2.1.
Moreover, $\widetilde{X}_{k}^{\mathit{cg}}$ is a classical one-dimensional
Brownian motion up to scaling and has therefore arbitrary exponential moments.
Finally, the independence of $\widetilde{X}_{k}^{\mathit{diff}}$ and
$\widetilde{X}_{k}^{\mathit{cg}}$ ensures that
$\widetilde{X}_{k}=\widetilde{X}_{k}^{\mathit{diff}}+\widetilde{X}_{k}^{\mathit{cg}}\,$
has this property as well.
With the existence of exponential moments in mind, we now follow Section 3 and
observe that the trigonometric elementary symmetric polynomials
$\widetilde{e}_{l}(x):=e_{l}(e^{x})\quad(l=0,\ldots,N,\,\,x\in C_{N}^{A})$
are eigenfunctions of $\widetilde{L}_{k}$ for all $k$ with eigenvalues
$l\bigl{(}\frac{1}{k}+N-l\bigr{)}\,\geq 0$. This leads to martingales for the
diffusions $(\widetilde{X}_{t,k})_{t\geq 0}$ on $C_{N}^{A},$ similar to
Corollaries 3.5 and 3.6 as follows:
###### Corollary 4.1.
For $k\in\,]0,\infty[$ and $x\in C_{N}^{A}$ consider the diffusion
$(\widetilde{X}_{t,k})_{t\geq 0}$ on $C_{N}^{A}$ with start in $x$. Then, for
$l=0,1,\ldots,N$, the process
$\bigl{(}e^{-l(1/k+N-l)t}\,\widetilde{e}_{l}(\widetilde{X}_{t,k})\bigr{)}_{t\geq
0}$ is a martingale. In particular, for $t\geq 0$,
$\mathbb{E}\bigl{(}\widetilde{e}_{l}(\widetilde{X}_{t,k})\bigr{)}=e^{l(1/k+N-l)t}\,\widetilde{e}_{l}(x).$
This result also holds for $k=\infty$. More precisely, the solution
$(\widetilde{X}_{t,\infty})_{t\geq 0}$ of the ODE (4.3) with start $x$ in the
interior of $C_{N}^{A}$ satisfies
(4.4)
$\widetilde{e}_{l}(\widetilde{X}_{t,\infty})=e^{l(N-l)t}\,\widetilde{e}_{l}(x)\quad\quad(l=0,1,\ldots,N).$
Using the independence of the processes $\widetilde{X}_{k}^{\mathit{diff}}$
and $\widetilde{X}_{k}^{\mathit{cg}}$, we also obtain the following analog of
Corollary 3.7.
###### Corollary 4.2.
For $k\in\,]0,\infty[$ and $x\in C_{N,0}^{A}$ consider the diffusion
$\widetilde{X}_{k}^{\mathit{diff}}$ on the chamber $C_{N,0}^{A}$ with start in
$x$. Then, for $t\geq 0$ and $l=1,\ldots,N$,
$\mathbb{E}\bigl{(}\widetilde{e}_{l}(\widetilde{X}_{t,k}^{\mathit{diff}})\bigr{)}=e^{l(N-l+(N+l)/(Nk))t}\,\widetilde{e}_{l}(x).$
This holds also for $k=\infty$ where the expectations can be omitted.
Corollary 4.2 can be restated as:
###### Corollary 4.3.
For $k\in\,]0,\infty[$ and $x\in C_{N,0}^{A}$ consider the diffusion
$\widetilde{X}_{k}^{\mathit{diff}}$ on $C_{N,0}^{A}$ with start in $x$. Then,
for $t\geq 0$ and $y\in\mathbb{C}$,
$\mathbb{E}\bigl{(}\prod_{j=1}^{N}(y-e^{\widetilde{X}_{k,t,j}^{\mathit{diff}}})\bigr{)}=P_{t,N,k,x}(y)$
with the polynomial
$P_{t,N,k,x}(y):=\sum_{l=0}^{N}y^{N-l}(-1)^{l}e^{l(N-l+(N+l)/(Nk))t}\,\widetilde{e}_{l}(x).$
For general starting points $x\in C_{N,0}^{A}$, these polynomials do not seem
to have particularly nice properties. For the starting configuration $x=0\in
C_{N,0}^{A}$, which is of particular interest here, we get
$P_{t,N,k,0}(y)=\sum_{l=0}^{N}{N\choose
l}(-1)^{l}y^{N-l}e^{l(N-l+(N+l)/(Nk))t}.$
We do not have much information about these polynomials. This is a contrast to
the Bessel processes of type $A$ where in a corresponding formula classical
Hermite polynomials appear; see [KVW].
###### Example 4.4.
1. (1)
Let $N=2$. We try to solve the ODE (4.3) with the singular starting point
$x=0\in C_{2}^{A}.$ In fact, (4.3) yields that $x_{1}(t)=-x_{2}(t)$ and that
$y(t):=x_{1}(t)-x_{2}(t)$ satisfies $\frac{dy}{dt}(t)=2\coth(y(t)/2)$. On the
other hand, (4.4) for $N=2$, $l=1$ suggests that
$e^{x_{1}(t)}+e^{-x_{1}(t)}=2e^{t}$
and thus $x_{1}(t)={\rm arcosh}(e^{t})$. It is now easily checked that in
fact,
$t\mapsto({\rm arcosh}(e^{t}),-{\rm arcosh}(e^{t}))$
is continuous on $[0,\infty)$ and solves (4.3) for $t>0.$ Moreover, for
$k=\infty$ we have $x^{\mathit{diff}}\\!(t)=x(t)$, and
$P_{t,2,\infty,0}(y)=y^{2}-2e^{t}\,y+1.$
These polynomials have the zeros $e^{\pm x_{1}(t)}$ as claimed.
2. (2)
Let $N=3.$ We again try to solve the ODE (4.3) with $x=0\in\partial
C_{3}^{A}$. Here symmetry arguments imply that the solution of the ODE (4.3)
must have the form $x(t)=(x_{1}(t),0,-x_{1}(t))$. On the other hand, formula
(4.4) with $N=2$, $l=1,2$ suggests that
$e^{x_{1}(t)}+e^{-x_{1}(t)}+1=3e^{2t}$
and thus $\,x_{1}(t)={\rm arcosh}\bigl{(}3(e^{2t}-1)/2\bigr{)}.\,$ This indeed
gives a solution as in example (1). Moreover, the polynomial
$P_{t,3,\infty,0}(y)=y^{3}-3e^{2t}y^{2}+3e^{2t}y-1$
has the zeros $1$ and $e^{\pm x_{1}(t)}$ as claimed.
Unfortunately, we have no closed formulas for $x(t)$ for general dimension $N$
even for the starting point $x=0\in C_{N}^{A}$.
We finally mention that Corollaries 4.2 and 4.3 for $k=1/2,1,2$ have
applications to Brownian motions on the noncompact symmetric spaces of type A,
i.e. on $GL(N,\mathbb{F})/U(N,\mathbb{F})$ for
$\mathbb{F}=\mathbb{R},\mathbb{C},\mathbb{H}$ similar to the results in the
end of Section 3.
## 5\. The non-compact case of type $BC_{N}$
We here start with the nonreduced root system
$R=BC_{N}=\\{\pm e_{i},\pm 2e_{i},\pm(e_{i}\pm e_{j});\>\>1\leq i<j\leq
N\\}\subset\mathbb{R}^{N}$
for $N\geq 2$ with the multiplicity $k=(k_{1},k_{2},k_{3}),$ where
$k_{1},k_{2}\geq 0$, $k_{3}>0$ are the values on the roots
$e_{i},2e_{i},e_{i}\pm e_{j}$. As indicated in the introduction, we now
reparametrize the multiplicity $k$. This will be not natural at a first
glance, but it will turn out to be useful in the end. We here mainly follow
the notations in [Dem, V] and define
(5.1) $\kappa=k_{3},\quad q=N-1+\frac{1+2k_{1}+2k_{2}}{2k_{3}},\quad
p=N-1+\frac{1+2k_{2}}{2k_{3}}.$
Then $\kappa>0$, $q\geq p\geq N-1+1/{2\kappa}$, and
(5.2)
$k=(k_{1},k_{2},k_{3})=\kappa\cdot\Bigl{(}q-p,k_{0,2},1\Bigr{)}\quad\text{with}\quad
k_{0,2}:=p-(N-1)-\frac{1}{2\kappa}.$
We now regard $p,q$ as fixed parameters and $\kappa>0$ as a varying parameter,
and denote the multiplicity $k$ in (5.2) by $k_{\kappa}$. The associated
$W$-invariant Heckman-Opdam Laplacian (2.6) is
$\displaystyle L_{\kappa}=\Delta+\sum_{i=1}^{N}\kappa\Bigl{(}($ $\displaystyle
q-p)\coth(\frac{x_{i}}{2})+2k_{0,2}\coth(x_{i})\,$ (5.3)
$\displaystyle+\sum_{j:j\neq
i}\bigl{(}\coth(\frac{x_{i}-x_{j}}{2})+\coth(\frac{x_{i}+x_{j}}{2})\bigr{)}\Bigr{)}\partial_{i}$
on the Weyl chamber
$C_{N}^{B}:=\\{x\in\mathbb{R}^{N}:\>x_{1}\geq x_{2}\geq\ldots\geq x_{N}\geq
0\\}$
of type $B_{N}$, associated with $R_{+}=\\{e_{i},\,2e_{i},\,e_{i}-e_{j}:1\leq
i<j\leq N\\}.$ As in the preceding sections, the operators $L_{\kappa}$ are
the generators of diffusions $(X_{t,\kappa})_{t\geq 0}$ on $C_{N}^{B}$ where
the paths are reflected at $\partial C_{N}^{B}$. The renormalized operators
$\widetilde{L}_{\kappa}:=\frac{1}{\kappa}L_{\kappa}\,$
then are the generators of the diffusions
$(\widetilde{X}_{t,\kappa}:=X_{t/\kappa,\kappa})_{t\geq 0}$ which may be
regarded as solutions of the SDE
$\displaystyle d\widetilde{X}_{t,\kappa,j}=$
$\displaystyle\frac{\sqrt{2}}{\sqrt{\kappa}}dB_{t,j}+\sum_{l\neq
j}\Bigl{(}\coth\Bigl{(}\frac{\widetilde{X}_{t,\kappa,j}-\widetilde{X}_{t,\kappa,l}}{2}\Bigr{)}+\coth\Bigl{(}\frac{\widetilde{X}_{t,\kappa,j}+\widetilde{X}_{t,\kappa,l}}{2}\Bigr{)}\Bigr{)}dt$
(5.4)
$\displaystyle+\Bigl{(}(q-p)\coth\bigl{(}\frac{\widetilde{X}_{t,\kappa,j}}{2}\bigr{)}+2k_{0,2}\coth(\widetilde{X}_{t,\kappa,j})\Bigr{)}dt\quad\quad(j=1,\ldots,N).$
For $\kappa\to\infty$, the generator degenerates into
$\displaystyle\widetilde{L}_{\infty}=$
$\displaystyle\sum_{i=1}^{N}\Bigl{(}(q-p)\coth(\frac{x_{i}}{2})+2(p-(N-1))\coth(x_{i})$
(5.5) $\displaystyle+\sum_{j:j\neq
i}\Bigl{(}\coth(\frac{x_{i}-x_{j}}{2})+\coth(\frac{x_{i}+x_{j}}{2})\Bigr{)}\Bigr{)}\partial_{i},$
and (5) becomes the ODE
(5.6) $\displaystyle\frac{dx_{j}}{dt}(t)=$ $\displaystyle\sum_{l\neq
j}\Bigl{(}\coth\Bigl{(}\frac{x_{j}(t)-x_{l}(t)}{2}\Bigr{)}+\coth\Bigl{(}\frac{x_{j}(t)+x_{l}(t)}{2}\Bigr{)}\Bigr{)}$
$\displaystyle+(q-p)\coth\bigl{(}\frac{x_{j}(t)}{2}\bigr{)}+2\bigl{(}p-(N-1)\bigr{)}\coth
x_{j}(t)\quad\quad(j=1,\ldots,N).$
Again, for initial data in the interior of the chamber $C_{N}^{B}$, the
solution $(\widetilde{X}_{t,\infty})_{t\geq 0}$ of these differential
equations exists for $t\geq 0$ in the interior of $C_{N}^{B}$.
We now turn to eigenfunctions of the operators $\widetilde{L}_{\kappa}$ and
consider the Heckman-Opdam hypergeometric functions
$F_{BC}(\lambda,k_{\kappa};x)$ associated with $BC_{N}$ in the variable
$x\in\mathbb{R}^{N},$ with spectral parameter $\lambda\in\mathbb{C}^{N}$ and
multiplicity $k_{\kappa}$. By Section 2, for $\kappa\in\,]0,\infty[$, the
functions $x\mapsto F_{BC}(\lambda,k_{\kappa};x)$ are eigenfunctions of the
renormalized Laplacian $\widetilde{L}_{\kappa}$ with the eigenvalues
$r_{\lambda,\kappa}:=\frac{1}{\kappa}\Bigl{(}\sum_{j=1}^{N}\lambda_{j}^{2}-|\rho(\kappa)|^{2}\Bigr{)},$
where
(5.7) $\rho(\kappa)=\frac{1}{2}\sum_{\alpha\in
R_{+}}k_{\kappa}(\alpha)\alpha\quad\text{with}\quad\rho(\kappa)_{j}=\frac{\kappa}{2}\bigl{(}(q-p)+2k_{0,2}+2(N-j)\bigr{)}.$
We further consider the associated (normalized) Heckman-Opdam polynomials
$R_{\lambda}(k,\,.\,)=R_{\lambda}^{BC}(k,\,.\,),$ as introduced in (2.5),
which are indexed by $P_{+}=\Lambda_{N}^{+}$ (the set of partitions of length
at most $N$). These are multivariate generalizations of the classical Jacobi
polynomials which are well-studied in the literature, see for instance [BO, L,
RR2]. The polynomials $\widetilde{R}_{\lambda}$ defined by
$\widetilde{R}_{\lambda}(\cos x):=R_{\lambda}(k;x)$
form an orthogonal basis of $L^{2}(\mathbb{A}_{N},w_{k})$ on
$\mathbb{A}_{N}:=\\{x\in\mathbb{R}^{N}|\>-1\leq y_{1}\leq...\leq y_{N}\leq
1\\}$
with the weight function
(5.8)
$w_{k}(y):=\prod_{i=1}^{N}(1-y_{i})^{k_{1}+k_{2}-1/2}(1+y_{i})^{k_{2}-1/2}\cdot\prod_{i<j}|y_{i}-y_{j}|^{2k_{3}}.$
Here we are mainly interested in the fact that for $k=k_{\kappa}$ and
$\lambda\in\Lambda_{N}^{+}$, the exponential polynomials
$H_{\lambda}(x):=\widetilde{R}_{\lambda}(\cosh x)=F_{BC}(\lambda+\rho(k),k;x)$
are eigenfunctions of $\widetilde{L}_{\kappa}$ with the eigenvalues
$\displaystyle r_{\lambda}$
$\displaystyle=\frac{1}{\kappa}\langle\lambda,\lambda+2\rho(k)\rangle$
$\displaystyle=\frac{1}{\kappa}\sum_{j=1}^{N}\lambda_{j}(\lambda_{j}+k_{1}+2k_{2}+2k_{3}(N-j))$
(5.9)
$\displaystyle=\sum_{j=1}^{N}\lambda_{j}\Bigl{(}\frac{\lambda_{j}-1}{\kappa}+p+q+2-2j\Bigr{)}.$
We now consider the partitions $\lambda(n):=1^{n}\in\Lambda_{+}^{N}$ for
$n=0,\ldots,N.$ It is known (see Section 5 of [V] and in particular Lemma 5.1
there) that the Jacobi polynomials $\widetilde{R}_{\lambda(n)}$ are of the
form
(5.10) $\widetilde{R}_{\lambda(n)}=\sum_{l=0}^{n}c_{n,l}(p,q)\cdot
e_{l}\quad\text{with }\>c_{n,n}(p,q)\neq 0,$
where the $e_{l}$ are again the elementary symmetric polynomials in $N$
variables and the coefficients $c_{n,l}(p,q)\in\mathbb{R}$ depend on $p,q$
only and not on $\kappa$. This observation will be crucial in the following
and is the reason for our parametrization of $k$ by $p,q,\kappa$ above. For
more details on the $c_{n,l}(p,q)$ we refer to [V]. In summary, the functions
$H_{\lambda(n)}$ with $n=0,\ldots,N$ are independent of $\kappa$ and
simultanous eigenfunctions of the operators $\widetilde{L}_{\kappa}$ for all
$\kappa\in\,]0,\infty[$ with the eigenvalues
$r_{n}=n(p+q-n+1).$
Clearly, this observation also holds for $\kappa=\infty$. This implies:
###### Lemma 5.1.
For each starting point $x\in C_{N}^{B}$ of the processes
$(\widetilde{X}_{t,\kappa})_{t\geq 0}$ with $\kappa\in\,]0,\infty]$, the
processes
(5.11) $\Bigl{(}e^{-r_{n}t}\cdot
H_{\lambda(n)}\bigl{(}\widetilde{X}_{t,\kappa}\bigr{)}\Bigr{)}_{t\geq 0}$
are martingales for $n=0,\ldots,N$, where the numbers $r_{n}$ and the
functions $H_{\lambda(n)}$ do not depend on $\kappa$.
In particular, for $x$ in the interior of $C_{N}^{B}$, the solution
$(\widetilde{X}_{t,\infty})_{t\geq 0}$ of the ODE (5.6) with
$\widetilde{X}_{0,\infty}=x$ satisfies
$H_{\lambda(n)}(\widetilde{X}_{t,\infty})=e^{r_{n}t}H_{\lambda(n)}(x)\quad\text{for}\quad
t\geq 0,\>n=0,\ldots,N.$
###### Proof.
This follows from our preceding considerations and the fact that the random
variables $H_{\lambda(n)}(\widetilde{X}_{t,\kappa})$ are integrable for
$\kappa<\infty$ and $t>0$ by Lemma 2.1. ∎
We may invert (5.10) and write the elementary symmetric polynomials $e_{l}$ as
linear combinations of the $\widetilde{R}_{\lambda(n)}$ for $l,n=0,\ldots,N$
with coefficients independent of $\kappa$. Lemma 5.1 thus implies:
###### Corollary 5.2.
Fix some deterministic starting point $x\in C_{N}^{B}$ as well as the
parameters $p,q$. Consider the associated diffusions
$(\widetilde{X}_{t,\kappa})_{t\geq 0}$ for $\kappa\in\,]0,\infty[$. Then there
are coefficients $a_{n,l}\in\mathbb{R}$ for $0\leq l\leq n\leq N$ such that
$\mathbb{E}\bigl{(}e_{n}(\cosh(\widetilde{X}_{t,\kappa}))\bigr{)}=\sum_{l=0}^{n}a_{n,l}\,e^{r_{l}t}$
with $r_{0}=0$ where the coefficients $a_{n,l}$ and the exponents $r_{l}$
depend on $p,q$ and $x$ only and not on $\kappa$. The same holds for
$\kappa=\infty$ and starting points $x$ in the interior of $C_{N}^{B}.$
###### Example 5.3.
For $x=0\in C_{N}^{B}$, we have $H_{\lambda(n)}(0)=1\,$ and thus
$\mathbb{E}\bigl{(}H_{\lambda(n)}(\widetilde{X}_{t,\kappa})\bigr{)}=e^{r_{n}t}\quad\quad(n=0,\ldots,N,\>t\geq
0,\,\kappa\in\,]0,\infty[).$
We finally turn to an application concerning a determinantal formula. Again we
fix some starting point $x\in C_{N}^{B}$ as well as $p,q$ and consider the
associated diffusions $(\widetilde{X}_{t,\kappa})_{t\geq 0}$ for
$\kappa\in\,]0,\infty[$. Then by Corollary 5.2, for all $t\geq 0$ and
$y\in\mathbb{C}$,
(5.12)
$\displaystyle\mathbb{E}\Bigl{(}\prod_{j=1}^{N}\bigl{(}y-\cosh(\widetilde{X}_{t,\kappa,j})\bigr{)}\Bigr{)}$
$\displaystyle=\sum_{n=0}^{N}(-1)^{n}\,\mathbb{E}\bigl{(}e_{n}(\cosh(\widetilde{X}_{t,\kappa})\bigr{)}\cdot
y^{N-n}$
$\displaystyle=\sum_{n=0}^{N}(-1)^{n}\,e_{n}\bigl{(}\cosh(\widetilde{X}_{t,\infty})\bigr{)}\cdot
y^{N-n}$
$\displaystyle=\prod_{j=1}^{N}\,\bigl{(}y-\cosh(\widetilde{X}_{t,\infty,j})\bigr{)}=:D_{t,x}(y).$
It is an interesting task to find particularly nice starting points $x\in
C_{N}^{B}$ for which $D_{t,x}(y)$ can be determined explicitly.
For instance, in the setting of multivariate Bessel processes of types
$A_{N-1}$ or $B_{N}$ and start in the origin, $D_{t,0}(y)$ is a classical one-
dimensional Hermite or Laguerre polynomial of degree $N$ in $y$, which is
scaled by a factor $\sqrt{t}$; for the details see [KVW]. Moreover, in the
setting of Heckman-Opdam Jacobi processes of type $BC$ on the compact alcove
$\mathbb{A}_{N}$, and with the same paramatrization of the multiplicity $k$ by
$p,q,\kappa$ as here, there is a (unique) stationary solution $x_{0}$ of the
ODE in the interior of $\mathbb{A}_{N}$, whose coordinates are the ordered
zeroes of some classical one-dimensional Jacobi polynomial of degree $N$ in
$y$. The indices of this Jacobi polynomial are determined by $p,q$. This means
that for this particular starting point $x_{0}$, the function $D_{t,x_{0}}(y)$
is just this specific Jacobi polynomial and is independent of $t\geq 0$ (due
to stationarity). We refer to [V] for further details.
We expect that in our non-compact $BC$ setting, $D_{t,x}(y)$ should be of
particular interest when the associated ODE (5.6) starts in $x=0\in\partial
C_{N}^{B},$ where we expect that the corresponding initial value problem is
uniquely solvable, as in the Dunkl setting in [VW]. It seems that the explicit
solution of this initial value problem is more involved than in the cases
considered in [KVW, V].
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|
On Malyshev’s method of automorphic functions
in diffraction by wedges
Dedicated to the memory of Vadim Malyshev
A.I. Komech
Faculty of Mathematics, Vienna University
<EMAIL_ADDRESS>
A.E. Merzon111The research supported by CONACYT-México and CIC-UMSNH, México.
Institute of Physics and Mathematics
University of Michoacan de San Nicolas de Hidalgo, Morelia, Mexico
<EMAIL_ADDRESS>
###### Abstract
We describe Malyshev’s method of automorphic functions in application to
boundary value problems in angles and to diffraction by wedges. We give a
consize survey of related results of A. Sommerfeld, S.L. Sobolev, J.B. Keller,
G.E. Shilov and others.
MSC: 35J25; 30F10; 35J05; 35A30; 11F03; 35Q15; 78A45.
Keywords: elliptic equation; Helmholtz equation; boundary value problem; plane
angle; Fourier transform; analytic function; Riemann surface; characteristics;
covering map; automorphic function; Riemann–Hilbert problem; diffraction;
wedge; limiting absorption principle; limiting amplitude principle; limiting
amplitude: the Sommerfeld radiation condition.
###### Contents
1. 1 Introduction
2. 2 Diffraction by wedges and radar/sonar detection
3. 3 Stationary diffraction and boundary value problems in angles
4. 4 Malyshev’s method of automorphic functions
1. 4.1 Reduction to undetermined algebraic equation on the Riemann surface
1. 4.1.1 Fourier–Laplace transform
2. 4.1.2 Undetermined algebraic equation on the Riemann surface
2. 4.2 Method of automorphic functions
1. 4.2.1 Covering maps
2. 4.2.2 Shift equation
3. 4.3 Reduction to the Riemann–Hilbert problem
5. 5 Nonconvex angles of magnitude $\Phi>\pi$
6. 6 Time-dependent diffraction by wedge
7. 7 Limiting amplitude principle
8. 8 The Sommerfeld diffraction theory and related results
## 1 Introduction
Vadim Malyshev was a very talented and versatile mathematician. He owns
significant results in the field of probability theory and Gibbs fields,
Markov processes and Euclidean quantum field theory. He also possessed
outstanding organizational skills, in particular, he founded the successful
and respected mathematical journal “Markov Processes and Related Fields”.
In 1970, V. Malyshev invented the method of automorphic functions [36], and
applied to random walks on the lattice in the quarter of plane. Later on, he
applied the method to queueing systems and analytic combinatorics [16]. In
1972–2022, the method was extended to boundary value problems for partial
differential equations in angles [22, 32] and to diffraction by wedges [30].
The main steps of Malyshev’s method are as follows:
I. Undetermined algebraic equation on the Riemann surface and analytic
continuation.
II. Elimination of one unknown function using covering automorphisms.
III. The reduction to the Riemann–Hilbert problem.
Malyshev’s method played the crucial role in the progress in the theory of
diffraction by wedges with general boundary conditions since 1972. The problem
was stated by M.I.Vishik in the Summer of 1967. In 1969–1971, one of the
authors (AK) tried to solve this problem while preparing his PhD Thesis. As
the result of these three- year efforts, the problem has been reduced to an
undetermined algebraic equation on the Riemann surface [23], though next steps
remained obscure. Fortunately, at the end of 1971, AK received the impetus
from his friend Alexander Shnirelman who noticed something similar in
Malyshev’s book [36], which he had recently reviewed by request of M.I.
Vishik. AK did not understand this book completely, but discovered two pages
which could have contained a creative idea. The book, opened on these pages,
lied on his desk for about two or three months, when AK pinned down two lines
with the key idea of automorphicity. The remaining work took about six months…
The extension of the research to diffraction problems was done in an intensive
collaboration of both authors, and took about 50 years. The main results of
the collaboration were the limiting absorption principle [44, 45], proof of
the completeness of Ursell’s trapping modes [31] , the extension to the
nonconvex angles [25, 30], and the Sommerfeld representation [24]. Moreover,
our general methods [30] allowed us to reproduce the formulas obtained by
Sommerfeld, Sobolev and Keller [28, 29, 46]. The identifications justify these
formulas as the limiting amplitudes in diffraction.
In the present, we give a consize survey of the development of Malyshev’s
method of automorphic functions since 1972 in the context of i) boundary value
problems in angles for elliplic partial differential equations, and ii) theory
of stationary and time-dependent diffraction by wedges. We focus on principal
ideas omitting nonessential technical details. All the details can be found in
[30].
## 2 Diffraction by wedges and radar/sonar detection
The radar or sonar emits the incident wave, which generates the reflected and
diffracted waves (the latter in green color) as shown in Fig. 1. Here $W$
denotes a conducting wedge (for example, the edge of an airplane wing), and
Q$=\mathbb{R}^{2}\setminus W$ is an angle of magnitude $\Phi$. The incident
wave reaches the wedge and generates the reflected and diffracted waves. The
diffracted wave is defined as the total wave minus the incident and reflected
waves.
The reflected wave is defined by geometric optics, and is absorbed by the
ground. On the other hand, the diffracted wave spreads in all directions, and
only this part of radiation returns to the radar which allows to detect the
airplane location.
Figure 1: Incident, reflected, and diffracted waves (the latter in green
color).
## 3 Stationary diffraction and boundary value problems in angles
The stationary diffraction by wedge is described by the boundary value problem
for the Helmholz equation in an angle $Q\subset\mathbb{R}^{2}$ of magnitude
$\Phi\in(0,2\pi]$:
$\left\\{\begin{array}[]{rcll}(\Delta+\omega^{2})u(x)&=&0,&x\in Q\\\ \\\
B_{l}u(x)&=&f_{l}(x),&x\in\Gamma_{l},\quad l=1,2\end{array}\right|,$ (3.1)
where $\Gamma_{1}$ and $\Gamma_{2}$ denote the sides of the angle, the
functions $f_{l}$ are defined by the incident wave, and $\omega\in\mathbb{R}$
is its frequency, see Fig. 1 and (7.60). The operators $B_{l}$ in the boundary
conditions correspond to the material properties of the wedge (conductor,
insulator, ferromagnetic, etc).
The relation of stationary problem (3.1) to time-dependent diffraction is
highly nontrivial. The key issue is that for $\omega\in\mathbb{R}$, the
problem admits an infinite number of linearly independent solutions. We
discuss this issue in detail in Section 7.
The stationary diffraction problem (3.1) with the Dirichlet and Neumann
boundary conditions ($B_{l}=1$ or $B_{l}=\frac{\partial}{\partial n}$) was
solved in 1896–1912 for $\Phi=2\pi$, by A. Sommerfeld [60]–[65] (the detailed
exposition and comments can be found in [48]). The extension to all
$\Phi\in(0,2\pi)$ was obtained in 1920 by H.S. Carslaw [8], in 1932–1937 by
V.I. Smirnov and S.L. Sobolev [55, 56, 57, 58, 59], and in 1951 by J.B. Keller
and A. Blank [21]. In 1958, G.D. Malujinetz solved the problem for all
$\Phi\in(0,2\pi)$ with the impedance (Leontovich) boundary condition
$\frac{\partial u(x)}{\partial n}+ib_{l}u(x)=f_{l}(x),\,\,x\in\Gamma_{l}$; see
[34, 35]. The detailed exposition of all these results can be found in [2] and
[30].
The mixed boundary value problems of type
$\left\\{\begin{array}[]{rcll}Au(x)&=&0,&x\in Q\\\ \\\
B_{l}u(x)&=&f_{l}(x),&x\in\Gamma_{l},\quad l=1,2\end{array}\right|,\qquad
A=\sum_{|\alpha|\leq m}a_{\alpha}\partial^{\alpha},\quad
B_{l}=\sum_{|\alpha|\leq n_{l}}b_{l\alpha}\partial^{\alpha}$ (3.2)
were considered in 1958 by S.L. Sobolev [59] and in 1960–1961 by G.E. Shilov
[53, 54] in the quadrant $x_{1}>0$, $x_{2}>0$ for the case of hyperbolic
operator $A$ in the variable $x_{2}$ and with the Cauchy initial conditions at
$x_{2}=0$.
For strongly-elliptic second order operators $A$ and general differential
boundary operators $B_{l}$, the problem (3.2) was solved in 1972 in convex
angles $Q$ of magnitude $\Phi\in(0,\pi)$, see [22, 23]. Strong ellipticity
means that
$|\hat{A}(z)|\geq\varkappa(|z|^{2}+1),\qquad z\in\mathbb{R}^{2},$ (3.3)
where the symbol $\hat{A}(z):=\sum_{|\alpha|\leq 2}a_{\alpha}(-iz)^{\alpha}$
and $\varkappa>0$. In particular, the operator $A=-\Delta+1$ with the symbol
$\hat{A}(z)=z^{2}+1$ is strongly elliptic, and also the Helmholtz operator
$H=\Delta+\omega^{2}$ from (3.1) is strongly ellipltic for ${\rm Im{\hskip
1.42262pt}}\omega\neq 0$. The method [22, 23] relies on the Malyshev ideas of
automorphic functions [36] which is presented in the next section.
The extension of this result to nonconvex angles of magnitudes
$\Phi\in(\pi,2\pi)$, was done in 1992 by the authors [25].
Let us note that the Helmholtz operator $A=\Delta+\omega^{2}$ is not strongly
elliptic if $\omega\in\mathbb{R}$ since its symbol has the form
$\hat{A}(z)=-z^{2}+\omega^{2}$. Problem (3.2) for the Helmholtz operator in
convex angles was solved in 1972–1977 by A.E. Merzon [44, 45], who proved that
for real $\omega\in\mathbb{R}$, the problem admits only a finite number of
solutions satisfying the limiting absorption principle:
$u_{\omega}(x)=\lim_{\varepsilon\to 0+}u_{\omega+i\varepsilon}(x),\qquad x\in
Q,$ (3.4)
where $u_{\omega+i\varepsilon}$ denotes suitable solution to (3.1) with
$\omega+i\varepsilon$ instead of $\omega$.
The application of these results to time-dependent diffraction by wedges was
done in 2006–2019 by the authors [26, 27, 30], where, in particular, the
limiting amplitude principle (7.54) was established as well as (3.4).
Another approach to the construction of solutions to (3.2) has been suggested
by Maz’ya and Plamenevskii [37, 38]. However, this approach is applicable only
to equations with real coefficients that is not sufficient for application to
the diffraction problems.
Many works published since 1980’ concern a wide spectrum of properties of
solutions to the boundary problems of type (3.2) in different regions with
angles, see Grisvard [18], Costabel and Stephan [10], Dauge [12], Bernard [3,
4], Nazarov and Plamenevskii [49], Bonnet-Ben Dhia and Joly [6], Bonnet-Ben
Dhia, Dauge and Ramdani [5], Meister with collaborators [39]–[43], Penzel and
Teixeira [50], Castro and Kapanadze [9], and others. The detailed survey can
be found in [30].
Note that Malyshev’s method plays an important role in the theory of Queueing
Systems and Analytic Combinatorics [16]. Another important area of application
of Malyshev’s method is the linear theory of water waves. In particular, the
method was applied in 1996–2002 by the authors together with P.N. Zhevandrov
to trapped modes on a sloping beach. As the result, the long-standing problem
of the completeness of the Ursell’s modes has been solved [31, 32, 47]. This
progress is due to the fact that the method allows one to obtain all solutions
of the boundary value problems in angles.
We expect that the method can give a valuable progress in diffraction by
ferromagnetic wedges which is a challenging open problem of radar detection.
In this case, the operators $B_{l}$ in (3.2) are nonlocal pseudodifferential
operators.
## 4 Malyshev’s method of automorphic functions
In this section, we present basic steps of the method [22] which relies on
Malyshev’s ideas of automorphic functions [36].
Note that in the case of rational angles $\Phi=\pi/n$ and the Dirichlet and
Neumann boundary conditions, the boundary value problem (3.1) can be easily
solved by reflections in the sides of the angle. This method was well known at
least since the Gauss theory of electrostatics [17]. For $\Phi\neq\pi/n$ the
reflections do not give a solution, and for irrational $\Phi/\pi$, the method
suggested the reflections on a “Riemann surface” formed by the reflected
angles. This was the original step of the Sommerfeld approach which leaded him
to the famous “Sommerfeld integral representation” for solutions [60]. The
reflection on the Riemann surface and the theory of branching solutions to the
wave equation have been developed later by Sobolev [57] and [58, Chapter XII].
Very surprisingly, the method of automorphic functions [22, 36] also relies on
the reflections on a suitable Riemann surface $V$. However, in this approach,
$V$ is the surface in the Fourier space, contrary to the original ideas of
Sommerfeld. Namely, $V$ is the Riemann surface of complex characteristics of
the elliptic operator $A$:
$V=\\{z\in\mathbb{C}^{2}:\hat{A}(z)=0\\}.$ (4.5)
###### Remark 4.1.
The main idea of the Malyshev approach is the invariance of the Cauchy data of
solutions under covering maps of the Riemann surface $V$, see Remark 4.7.
In [22], the problem (3.2) with strongly–elliptic operators $A$ in convex
angles $Q$ is solved in the following steps:
1\. Reduction to an undetermined algebraic equation with two unknown functions
on the Riemann surface $V$. 2\. Elimination of one unknown function using its
invariance with respect to the covering map of the Riemann surface.
3\. Reduction of the obtained equation with one unknown function to the
Riemann–Hilbert problem on $V$.
Below in this section, we describe some details.
### 4.1 Reduction to undetermined algebraic equation on the Riemann surface
As an example, we consider the Dirichlet boundary value problem in the
quadrant $Q=\mathbb{R}^{+}\times\mathbb{R}^{+}$ :
$\left\\{\begin{array}[]{rcl}Au(x_{1},x_{2})&=&0\\\ \\\
u(x_{1},0)&=&f_{1}(x_{1}),\,\,u(0,x_{2})=f_{2}(x_{2})\end{array}\right|,\,\,\quad
x_{1}>0,\,\,\,x_{2}>0.$ (4.6)
#### 4.1.1 Fourier–Laplace transform
We assume that the solution $u(x)\in C^{2}(\overline{Q})$ and is bounded by a
polynomial:
$|u(x)|+|\nabla u(x)|\leq C(1+|x|)^{p},\qquad x\in\mathbb{R}^{2}.$ (4.7)
Denote $\mathbb{C}^{+}=\\{\zeta\in\mathbb{C}:{\rm Im{\hskip
1.42262pt}}\zeta>0\\}$ and $Z^{+}=\mathbb{C}^{+}\times\mathbb{C}^{+}$, and
consider the complex Fourier–Laplace transform of solution
$\hat{u}(z)=\int_{0}^{\infty}\int_{0}^{\infty}e^{izx}u(x)dx_{1}dx_{2},\qquad
z=(z_{1},z_{2})\in Z^{+}.$ (4.8)
By (4.7), this integral is absolutely convergent and hence it is an analytic
function of two complex variables (this is a particular case of the
Paley–Wiener Theorem). Let us denote the Neumannn data of the solution as
$\varphi_{1}(x_{1})=\partial_{2}u(x_{1},0),\,\,\,x_{1}\geq
0;\qquad\varphi_{2}(x_{2})=\partial_{1}u(0,x_{2}),\,\,\,x_{2}\geq 0.$ (4.9)
It is well known that the solution $u(x)$ can be expressed via the Dirichlet
and Neumannn data $f_{1},f_{2},\varphi_{1},\varphi_{2}$ by the Green integral
formula [11]. In our case, it is useful to obtain this formula in the Fourier
transform. For this purpose, multiply the first equation in (3.2) by $e^{izx}$
and integrate over $Q$. Integrating by parts, we immediately obtain
$0=\int_{0}^{\infty}\int_{0}^{\infty}e^{izx}Au(x)dx_{1}dx_{2}=\hat{A}(z)\hat{u}(z)+F(z),\qquad
z\in Z^{+},$ (4.10)
where
$F(z)=P_{1}(z)\hat{f}_{1}(z_{1})+P_{2}(z)\hat{f}_{2}(z_{2})+S_{1}(z)\hat{\varphi}_{1}(z_{1})+S_{2}(z)\hat{\varphi}_{2}(z_{2}),\qquad
z\in Z^{+},$ (4.11)
and the functions $P_{l}$ and $S_{l}$ are polynomials.
#### 4.1.2 Undetermined algebraic equation on the Riemann surface
Rewrite (4.10) as
$\hat{A}(z)\hat{u}(z)=-F(z),\qquad z\in Z^{+}.$ (4.12)
Now (4.5) implies the identity
$F(z)=0,\qquad z\in V^{+}:=V\cap Z^{+}$ (4.13)
since all the functions $\hat{A}(z),\hat{u}(z),F(z)$ are analytic in the
domain $Z^{+}$!
###### Remark 4.2.
Note that the set of complex characteristics $V$ is nonempty even for strongly
elliptic operators (3.3), though its intersection with the real plane
$\mathbb{R}^{2}$ is empty; see Example 4.5 below.
The identity (4.13) can be rewritten as undetermined linear algebraic equation
$S_{1}(z)\varphi_{1}(z_{1})+S_{2}(z)\varphi_{2}(z_{2})=G(z),\qquad z\in V^{+}$
(4.14)
with two unknown functions $\varphi_{1}(z_{1})$, $\varphi_{2}(z_{2})$, and
with known right-hand side:
$G(z):=-P_{1}(z)\hat{f}_{1}(z_{1})-P_{2}(z)\hat{f}_{2}(z_{2}),\qquad z\in
V^{+}.$ (4.15)
###### Remark 4.3.
The identity (4.12) implies the formula for the solution
$u(x)=-\Big{[}{\cal F}^{-1}\frac{F(z)}{\hat{A}(z)}\Big{]}(x),\qquad x\in Q,$
(4.16)
where ${\cal F}^{-1}$ denotes the inverse to the Fourier–Laplace transform
(4.8), and the right hand side is well defined due to (3.3). The formula
(4.16) can be transformed into the well known Green formula which expresses
the solution $u(x)$ via its Cauchy data.
### 4.2 Method of automorphic functions
#### 4.2.1 Covering maps
Denote
$\psi_{1}(z)=\varphi_{1}(z_{1}),\quad\psi_{2}(z)=\varphi_{2}(z_{2}),\qquad\hat{g}_{1}(z)=\hat{f}_{1}(z_{1}),\quad\hat{g}_{2}(z)=\hat{f}_{2}(z_{2}).$
(4.17)
Now (4.14) becomes
$S_{1}(z)\psi_{1}(z)+S_{2}(z)\psi_{2}(z)=G(z),\qquad z\in V^{+},$ (4.18)
where
$G(z):=-P_{1}(z)\hat{g}_{1}(z)-P_{2}(z)\hat{g}_{2}(z).$ (4.19)
Of course, this equation is not equivalent to (4.14). To keep the equivalence,
we need an additional characterisation of the functions $\psi_{l}(z)$. This is
the key observation of Malyshev that the functions are automorphic with
respect to an appropriate groups of transformation of the Riemann surface $V$.
First, consider the coordinate projections $p_{l}:V\to\mathbb{C}$ defined by
$p_{1}(z_{1},z_{2})=z_{1},\qquad p_{2}(z_{1},z_{2})=z_{2}.$ (4.20)
These projections are two-sheeted since, for example,
$p_{1}(z_{1},z_{2})=z_{1}$ means that $z_{2}$ is the root of the quadratic
equation $\hat{A}(z_{1},z_{2})=0$. Accordingly, the inverse maps
$p_{l}^{-1}:\mathbb{C}\to V$ are double-valued: for
$z_{1},z_{2}\in\mathbb{C}$,
$p_{1}^{-1}(z_{1})=\\{\zeta_{1}^{-},\zeta_{1}^{+}\\},\qquad
p_{2}^{-1}(z_{2})=\\{\zeta_{2}^{-},\zeta_{2}^{+}\\},$ (4.21)
and at the branching points of $p_{l}^{-1}$, the two points
$\zeta_{l}^{\pm}\in V$ coincide.
###### Definition 4.4.
Covering maps $h_{1},h_{2}:V\to V$ are defined as follows: for any
$z_{1},z_{2}\in\mathbb{C}$,
$h_{1}\zeta_{1}^{\pm}=\zeta_{1}^{\mp},\qquad\qquad
h_{2}\zeta_{2}^{\pm}=\zeta_{2}^{\mp}.$ (4.22)
###### Example 4.5.
For the strongly-elliptic operator $A=-\Delta+1$, the corresponding Riemann
surface $V\\!\\!:z_{1}^{2}+z_{2}^{2}+1=0$ is shown in Fig. 2 in projection
onto the plane ${\rm Im{\hskip 1.42262pt}}z_{1},{\rm Im{\hskip
1.42262pt}}z_{2}$. It is easy to see that this projection does not cover the
circle $|{\rm Im{\hskip 1.42262pt}}z_{1}|^{2}+|{\rm Im{\hskip
1.42262pt}}z_{2}|^{2}<1$, and it covers twice each point with $|{\rm Im{\hskip
1.42262pt}}z_{1}|^{2}+|{\rm Im{\hskip 1.42262pt}}z_{2}|^{2}>1$. The surface
consists of two sheets shown in Fig. 2, and glued along the cuts.
Thus, $h_{1}$ permutes the points $\zeta_{1}^{\pm}\in V$ with the identical
projections $z_{1}=p_{1}\zeta_{1}^{\pm}$, and similarly, $h_{2}$ permutes the
points $\zeta_{2}^{\pm}\in V$ with the identical projections
$z_{2}=p_{2}\zeta_{2}^{\pm}$ (see Fig. 2):
$p_{1}h_{1}\zeta_{1}^{\pm}=p_{1}\zeta_{1}^{\mp}=z_{1},\qquad
p_{2}h_{2}\zeta_{2}^{\pm}=p_{2}\zeta_{2}^{\mp}=z_{2}.$ (4.23)
The maps $h_{l}:V\to V$ with $l=1,2$ define the corresponding automorphisms of
the ring of (meromorphic) functions $\psi(z)$ on the Riemann surface $V$:
$\psi^{h_{l}}(z):=\psi(h_{l}z),\qquad z\in V.$ (4.24)
Figure 2: Riemann surface $V\\!\\!:z_{1}^{2}+z_{2}^{2}+1=0$ in projection onto
the plane ${\rm Im{\hskip 1.42262pt}}z_{1},{\rm Im{\hskip 1.42262pt}}z_{2}$.
Figure 2 shows that
$\begin{array}[]{ll}p_{1}\zeta_{1}^{+}=z_{1}=p_{1}\zeta_{1}^{-},&{\rm
so}\,\,\,\psi_{1}(\zeta_{1}^{+})=\varphi_{1}(z_{1})=\psi_{1}(\zeta_{1}^{-}),\\\
\\\ p_{2}\zeta_{2}^{+}=z_{2}=p_{2}\zeta_{2}^{-},&{\rm
so}\,\,\,\psi_{2}(\zeta_{2}^{+})=\varphi_{2}(z_{2})=\psi_{2}(\zeta_{2}^{-}).\end{array}$
Now it is clear that the functions $\psi_{l}(z):=\varphi_{l}(z_{l})$ with
$l=1,2$ are invariant with respect to the automorphisms $h_{l}$:
$\,\,\,\,\,\psi_{l}^{h_{l}}(z)=\psi_{l}(z),\qquad\,\,\,\,\,\,\,z\in V^{+}.$
(4.25)
In other words, the functions $\psi_{l}$ are automorphic, and the
automorphisms defined by $h_{l}$ belong to the corresponding Galois groups of
extensions of the ring of functions of $z_{l}$.
#### 4.2.2 Shift equation
Applying formally $h_{1}$ to (4.18), and using (4.25) with $l=1$, we get a
new equation for the same unknown functions:
$S_{1}^{h_{1}}(z)\psi_{1}(z)+S_{2}^{h_{1}}(z)\psi_{2}^{h_{1}}(z)=G^{h_{1}}(z).$
(4.26)
The problem is that $\psi_{2}^{h_{1}}(z)=\psi_{2}(h_{1}z)$ and
$G^{h_{1}}(z)=G(h_{1}z)$ are not defined generally for $z\in V^{+}$ since
$V^{+}$ is not invariant with respect to the covering map $h_{1}$. In
particular, we have by (4.19),
$G^{h_{1}}(z):=-P_{1}^{h_{1}}(z)\hat{g}_{1}(z)-P_{2}^{h_{1}}(z)\hat{g}_{2}^{h_{1}}(z),$
(4.27)
where $\hat{g}_{2}^{h_{1}}(z)=\hat{g}_{2}(h_{1}z)$ is not defined generally
for $z\in V^{+}$. To save the situation, consider the case $f_{2}=0$. Then
$\hat{g}_{2}=0$, and now the right hand side of equation (4.26) is well
defined for $z\in V^{+}$. It is important that in this case
$\psi_{2}^{h_{1}}(z)$ is also well defined [30, Ch. 14]. The case $f_{1}=0$
can be considered similarly.
###### Remark 4.6.
The function $\psi_{2}(z)$ admits an analytic continuation outside the region
$V^{+}_{l}:=\\{z\in V:{\rm Im{\hskip 1.42262pt}}z_{2}>0\\}$ on the Riemann
surface $V$, see [30, Ch. 14]. Let us stress that this is analytic
continuation along the surface $V$.
Now we can eliminate the function $\psi_{1}$ from (4.18) and (4.26). As a
result, we obtain an algebraic equation with a shift for one unknown function
$R_{1}(z)\psi_{2}^{h_{1}}(z)-R_{2}(z)\psi_{2}(z)=H(z),\quad z\in V^{+}.$
(4.28)
Finally, using (4.25) with l=2, we get
$R_{1}(z)\psi_{2}^{h}(z)-R_{2}(z)\psi_{2}(z)=H(z),\quad z\in V^{+};\qquad
h=h_{2}h_{1}.$ (4.29)
###### Remark 4.7.
The elimination of unknown functions using their invariance with respect to
suitable “reflections” is the main idea of Malyshev’s method.
### 4.3 Reduction to the Riemann–Hilbert problem
Let us illustrate the reduction of equation (4.28) to the Riemann–Hilbert
problem for a particular case of strongly-ellipltic operator $A=-\Delta+1$.
Its symbol is $\hat{A}(z)=z^{2}+1$, so
$V=\\{(z_{1},z_{2})\in\mathbb{C}^{2}:z_{1}^{2}+z_{2}^{2}=-1\\}$ and the
covering maps are
$h_{1}(z_{1},z_{2})=(z_{1},-z_{2}),\qquad h_{2}(z_{1},z_{2})=(-z_{1},z_{2}).$
(4.30)
Introduce the coordinate $w$ on the universal covering $\hat{V}=\mathbb{C}$ of
the surface $V$ by
$z_{1}=i\cos w,\qquad z_{2}=i\sin w.$ (4.31)
The maps (4.30) can be lifted to $\hat{V}$ as
$\hat{h}_{1}w=-w,\qquad\hat{h}_{2}w=-w+\pi.$ (4.32)
Now $h=w+\pi$, so (4.29) becomes
$\tilde{R}_{1}(w)\tilde{\psi}_{2}(w+\pi)-\tilde{R}_{2}(w)\tilde{\psi}_{2}(w)=\tilde{H}(w).$
(4.33)
where $\tilde{R}_{1}$, etc, denote the liftings of the corresponding functions
to the universal covering. The equation (4.33) holds for an appropriate region
of $w\in\mathbb{C}$. Restricted to the strip ${\rm Re{\hskip
1.42262pt}}w\in[0,\pi]$, this equation is the Riemann–Hilbert problem which
can be solved in quadratures [30, Chs 17 and 18]. Let us recall some details.
The function $z=e^{2iw}$ analytically transforms the strip to the plane with
the cut $[0,\infty)$. Denote the function
$\check{\psi}_{2}(t)=\tilde{\psi}_{2}(w)$, $\check{H}(t)=\tilde{H}(w)$ and
$\check{R}_{k}(t)=\tilde{R}_{k}(w)$ for $k=1,2$. Then relation (4.33) becomes
$\check{R}_{1}(t)\check{\psi}_{2}(t-i0)-\check{R}_{2}(t)\check{\psi}_{2}(t+i0)=\check{H}(t),\qquad
t>0.$ (4.34)
As the first step of the Riemann–Hilbert method, one must solve the
corresponding homogeneous problem:
$\check{R}_{1}(t)T(t-i0)-\check{R}_{2}(t)\check{T}(t+i0)=0,\qquad t>0.$ (4.35)
Equivalently,
$\frac{T(t+i0)}{T(t-i0)}=q(t):=\frac{\check{R}_{1}(t)}{\check{R}_{2}(t)},\qquad
t>0.$ (4.36)
The solution to this equation depends on zeros of the functions
$\check{R}_{1}(t)$ and $\check{R}_{2}(t)$ for $t>0$. Let us consider the
simplest case when such zeros do not exist, and moreover,
$q(0)=q(\infty)=1.$ (4.37)
Then the equation is equivalent to
$\log T(t+i0)-\log T(t-i0)=\log q(t),\qquad t>0.$ (4.38)
The solution is given by the Cauchy-type integral
$\log T(t)=\frac{1}{2\pi i}\int_{0}^{\infty}\frac{\log q(s)}{t-s}ds,\qquad
t\in\mathbb{C}\setminus[0,\infty).$ (4.39)
It is important that $T(t)$ is analytic and nonvanishing in the region
$\mathbb{C}\setminus[0,\infty)$. Now the nonhomogeneous problem (4.34) can be
solved as follows. First, (4.34) and (4.35) imply
$\frac{\check{\psi}_{2}(t-i0)}{T(t-i0)}-\frac{\check{\psi}_{2}(t+i0)}{T(t+i0)}=\frac{\check{H}(t)}{\check{R}_{1}(t)T(t-i0)},\qquad
t>0.$ (4.40)
Therefore, similarly to (4.38),
$\frac{\check{\psi}_{2}(t)}{T(t)}=-\frac{1}{2\pi
i}\int_{0}^{\infty}\frac{\check{H}(s)}{\check{R}_{1}(s)T(s-i0)(t-s)}ds,\qquad
t\in\mathbb{C}\setminus[0,\infty)$ (4.41)
since the function $\frac{\check{\psi}_{2}(t)}{T(t)}$ is analytic in
$\mathbb{C}\setminus[0,\infty)$.
Thus, we have calculated the function $\check{\psi}_{2}(t)$. Now $\psi_{2}(z)$
can be obtained from the relation (4.26). Hence, the functions
$\hat{\varphi}_{1}(z_{1})$ and $\hat{\varphi}_{2}(z_{2})$ are known. It
remains to substitute the obtained functions into the formula (4.11) for the
function $F$. Then the solution to (3.2) is expressed by (4.16), which can be
reduced to the integral of Sommerfeld type [24].
###### Remark 4.8.
Equation (4.28) is obtained using the invariance (4.25) with $l=1$, while
(4.29) uses also $l=2$. Note that the equation (4.28) reads now
$\tilde{T}_{1}(w)\tilde{\psi}_{2}(-w)-\tilde{T}_{2}(w)\tilde{\psi}_{2}(w)=\tilde{H}(w),$
(4.42)
which provisionally cannot be reduced to a nonsingular Riemann–Hilbert
problem, see [33]. Thus, both invariance conditions (4.25) are necessary for
the reduction.
###### Remark 4.9.
For the random walks studied in [36, 16], the corresponding Riemann surface
and the covering maps $h_{l}$ can be more complicated than for 2-nd order
elliptic operators which requires more sophisticated methods of the Galois
theory.
## 5 Nonconvex angles of magnitude $\Phi>\pi$
The extension of the theory outlined above to the case of nonconvex angle $Q$
differs drastically from the convex one. As an example, consider the Dirichlet
boundary value problem in the angle
$Q=\mathbb{R}^{2}\setminus\mathbb{R}^{+}\times\mathbb{R}^{+}$ :
$\left\\{\begin{array}[]{rcll}Au(x)&=&0,\qquad x\in Q&\\\ \\\
u(x_{1},0)&=&f_{1}(x_{1}),\,\,x_{1}>0;&u(0,x_{2})=f_{2}(x_{2}),\,\,x_{2}>0.\end{array}\right|.$
(5.43)
Note that the relations (4.10) and (4.12), (4.16) remain true in this case,
but now the function (4.11) is changed to its negative:
$F(z)=-P_{1}(z)\hat{f}_{1}(z_{1})-P_{2}(z)\hat{f}_{2}(z_{2})-S_{1}(z)\hat{\varphi}_{1}(z_{1})-S_{2}(z)\hat{\varphi}_{2}(z_{2}),\qquad
z\in\mathbb{R}^{2}.$ (5.44)
On the other hand, the key relation (4.13) is not well defined in contrast to
the case when the support of $u$ belongs to a convex angle. This is due to the
fact that the Fourier–Laplace transform (4.8) of the function $u$ with the
support in a nonconvex angle generally does not admit an analytical
continuation to a region of $\mathbb{C}^{2}$. Nevertheless, the function
(5.44) in this case is analytic in the same region
$Z^{+}=\mathbb{C}^{+}\times\mathbb{C}^{+}$ as the function (4.11).
The answer to this riddle was found in [25]. First, the function (5.44) admits
the splitting
$F(z)=\gamma_{1}(z)+\gamma_{2}(z),\quad\gamma_{1}(z)=-P_{1}(z)\hat{f}_{1}(z_{1})-S_{1}(z)\hat{\varphi}_{1}(z_{1}),\quad\gamma_{2}(z)=-P_{2}(z)\hat{f}_{2}(z_{2})-S_{2}(z)\hat{\varphi}_{2}(z_{2}),$
(5.45)
where the functions $\gamma_{l}(z)$ are analytic in the regions
$V_{l}^{+}=\\{z\in V:{\rm Im{\hskip 1.42262pt}}z_{l}>0\\},\qquad l=1,2.$
(5.46)
Second, as shown in [25] (see also [30, Theorem 20.1]), each function
$\gamma_{l}$ admits an analytic continuation from $V_{l}^{+}$ to the region
$V^{-}:=\\{z\in V:{\rm Im{\hskip 1.42262pt}}z_{1}<0,\,{\rm Im{\hskip
1.42262pt}}z_{2}<0\\}$, and the following identity holds :
$\gamma_{1}(z)+\gamma_{2}(z)=0,\qquad z\in V^{-}.$ (5.47)
This identity formally coincides with the undetermined equation (4.14), and it
allows us to calculate both unknown functions $\hat{\varphi}_{l}$ by methods
of Sections 4.2 and 4.3.
## 6 Time-dependent diffraction by wedge
The time-dependent diffraction by a wedge $W$ is described by the solution of
the wave equation in the plane angle $Q=\mathbb{R}^{2}\setminus W$ with
appropriate boundary conditions. For example, consider the Dirichlet boundary
conditions
$\left\\{\begin{array}[]{rcl}\ddot{u}(x,t)&=&\Delta u(x,t),\quad x\in Q\\\ \\\
u(x,t)&=&0,\quad x\in\Gamma_{1}\cup\Gamma_{2}\end{array}\right|,\quad
t\in\mathbb{R}.$ (6.48)
The incident wave is defined by the initial condition
$u(x,t)=u^{in}(x,t),\qquad
u^{in}(x,t):=f(kx-\omega_{0}t)e^{i(kx-\omega_{0}t)},\qquad t<0,$ (6.49)
where the frequency $\omega_{0}\in\mathbb{R}$ and $k\in\mathbb{R}^{2}$ is the
wave vector. The incident wave $u^{in}(x,t)$ must be a solution to (6.48) for
$t<0$:
$\left\\{\begin{array}[]{rcl}\ddot{u}^{in}(x,t)&=&\Delta u^{in}(x,t),\quad
x\in Q\\\ \\\ u^{in}(x,t)&=&0,\quad
x\in\Gamma_{1}\cup\Gamma_{2}\end{array}\right|,\quad t<0.$ (6.50)
The wave equation in (6.50) holds for any function $f(s)$ for all
$t\in\mathbb{R}$ if $|k|=|\omega_{0}|$. The boundary condition in (6.50) can
be satisfied only in the case of nonconvex angle $Q$ of magnitude $\Phi>\pi$
and the wave vector $k$ satisfying the inequalities $k\cdot x\geq 0$ for $x\in
W=\mathbb{R}^{2}\setminus Q$. Then for $\omega_{0}>0$ the boundary condition
holds if
$f(s)=0,\qquad s>0.$ (6.51)
## 7 Limiting amplitude principle
Let us assume that there exists the limit
$f(-\infty):=\lim_{s\to-\infty}f(s),$ (7.52)
and the convergence is sufficiently fast, for example, $f(s)=\theta(-s)$. Then
the incident wave (6.49) admits the long-time asymptotics
$u^{in}(x,t)\sim f(-\infty)e^{ikx}e^{-i\omega_{0}t},\qquad t\to\infty$ (7.53)
which suggests similar asymptotics of solution
$u(x,t)\sim a_{\omega_{0}}(x)e^{-i\omega_{0}t},\qquad t\to\infty.$ (7.54)
Such asymptotics are called as limiting amplitude principle.
Determination of the limiting amplitudes $a_{\omega_{0}}(x)$ for different
diffraction processes is the main goal of the theory of diffraction [7, 65]
(see also [30]). The proof of the asymptotics is the main goal of the
mathematical theory of diffraction. For diffraction by wedges, this
asymptotics has been established for the first time in [26]. Formal
substitution of the asymptotics (7.54) into (6.48) gives a problem of type
(3.1):
$\left\\{\begin{array}[]{rcl}-\omega_{0}^{2}a_{\omega_{0}}(x)&=&\Delta
a_{\omega_{0}}(x),\quad x\in Q\\\ \\\ a_{\omega_{0}}(x)&=&0,\quad
x\in\Gamma_{1}\cup\Gamma_{2}\end{array}\right|.$ (7.55)
However, this boundary problem is ill-posed since it admits an infinite
number of linearly independent solutions for real $\omega_{0}\in\mathbb{R}$.
Thus, this problem does not allow us to find the limiting amplitude. This fact
is the main peculiarity of the diffraction theory. This can be easily checked
in the case $\Phi=\pi$ when the angle $Q$ is the half-plane, so all solutions
can be calculated by the Fourier transform along the boundary
$\partial\Omega$. For the problems of type (7.55) in convex angles of
magnitude $\Phi<\pi$, this nonuniqueness was discovered in 1973 by one of the
authors [45].
Let us recall how to prove the asymptotics (7.54) and how to calculate the
limiting amplitudes $a_{\omega_{0}}(x)$. First, note that for the incident
wave $u^{in}(x,t)$ the asymptotics of type (7.54) holds by (7.52):
$u^{in}(x,t)\sim f(-\infty)e^{ikx}e^{-i\omega_{0}t},\qquad t\to\infty.$ (7.56)
The reflected wave is defined by geometric optics, and its main properties are
as follows:
$u^{r}(x,t)=-u^{in}(x,t),\quad x\in\partial Q;\qquad u^{r}(x,t)\sim
a^{r}(x)e^{-i\omega_{0}t},\qquad t\to\infty.$ (7.57)
The diffracted wave $u^{d}(x,t)$ is defined by the splitting the total
solution as
$u(x,t)=u^{in}(x,t)+u^{r}(x,t)+u^{d}(x,t).$ (7.58)
Hence, it remains to calculate the corresponding asymptotics for the
diffracted wave
$u^{d}(x,t)\sim a^{d}_{\omega_{0}}(x)e^{-i\omega_{0}t},\qquad t\to\infty,$
(7.59)
Substituting (7.58) into (6.48), using (7.57) and the fact that the wave
equation in (6.50) holds for all $t\in\mathbb{R}$, we get the boundary problem
for the diffracted wave
$\left\\{\begin{array}[]{rcl}\ddot{u}^{d}(x,t)&=&\Delta
u^{d}(x,t)+F(x,t),\quad x\in Q\\\ \\\ u^{d}(x,t)&=&0,\quad
x\in\Gamma_{1}\cup\Gamma_{2}\end{array}\right|,\qquad
F(x,t):=(\partial_{t}^{2}-\Delta)u^{r}(x,t)\sim
b(x)e^{-i\omega_{0}t},\,\,\,t\to\infty.$ (7.60)
Formal substitution of the asymptotics (7.59) into (7.60), gives the boundary
problem
$\left\\{\begin{array}[]{rcl}-\omega_{0}^{2}a^{d}_{\omega_{0}}(x,t)&=&\Delta
a^{d}_{\omega_{0}}(x)+b(x),\quad x\in Q\\\ \\\ a^{d}(x)&=&0,\quad
x\in\Gamma_{1}\cup\Gamma_{2}\end{array}\right|.$ (7.61)
For $\omega_{0}\in\mathbb{R}$, this system also admits an infinite number of
linearly independent solutions, as well as (7.55). Similar problem of
nonuniqueness arises in every diffraction problem in unbounded regions. The
problem of nonuniqeness was resolved by the discovery of additional features
of the limiting amplitude $a^{d}_{\omega_{0}}(x)$.
The key discovery was the limiting absorption principle (3.4) for the limiting
amplitude of the diffracted wave. In application to problem (7.61), we have
$a^{d}_{\omega_{0}}(x)=\lim_{\varepsilon\to
0+}a^{d}_{\omega_{0}+i\varepsilon}(x),\qquad x\in Q,$ (7.62)
where $a^{d}_{\omega_{0}+i\varepsilon}$ denotes a solution to (7.61) with
$\omega_{0}+i\varepsilon$ instead of $\omega_{0}$.
###### Remark 7.1.
The convergence (7.62) holds for the limiting amplitude
$a^{d}_{\omega_{0}}(x)$ of the diffracted wave $u^{d}_{\omega_{0}}(x,t)$
(formal proof can be found in [30, Section 4.1]). However, it does not hold
for the limiting amplitude $a(x)$ of the total solution $u(x,t)$ although
these amplitudes satisfy quite similar equations (7.61) and (7.55). The
difference is that the initial state of the diffracted wave
$u^{d}_{\omega_{0}}(x,0)$ is of finite energy (in our case zero), while for
the total solution the initial state $(u(x,0),\dot{u}(x,0))$ is the plane wave
(6.49) and its derivative in time at $t=0$.
The limiting absorption principle has been introduced for the first time in
1905 by W. Ignatovsky [19]. Rigorous proofs of this principle for limiting
amplitudes of solutions with finite energy initial states were achieved much
later. The results for the wave and Schrödinger equations in the entire space
and for diffraction problems with smooth boundaries were obtained by Agmon
[1], Eidus [13, 14, 15], Jensen and Kato [20], A.Ya. Povzner [51], B.R.
Vainberg [66] and others.
The convergence (7.62) for stationary diffraction problems has been
established for the first time in 1977 by one of the authors [45]: it was
proven that stationary problem (7.61) and problems (3.2) with
$A=\Delta+\omega^{2}$ and general boundary conditions in convex angles of
magnitude $\Phi<\pi$,
i) for complex $\omega\not\in\mathbb{R}$ admit only a finite number of
linearly independent solutions in appropriate class of functions;
ii) for real $\omega\in\mathbb{R}$ admit an infinite number of linearly
independent solutions,
iii) for real $\omega\in\mathbb{R}$ admit only a finite number of linearly
independent solutions satisfying (7.62).
For the time-dependent diffraction problem (6.48), (6.49), the limiting
absorption principle (7.62) and the limiting amplitude principle (7.54) were
justified in 2006 by the authors [26]. The proofs rely on the analysis of the
Fourier-Laplace transform in time:
$\tilde{u}(x,\omega)=\int_{0}^{\infty}e^{i\omega
t}u(x,t)dt,\qquad\omega\in\mathbb{C}^{+}.$ (7.63)
The function $\tilde{u}(x,\omega)$ satisfies a boundary value problem of type
(7.61) with complex $\omega\not\in\mathbb{R}$. In this case the Helmholtz
operator $A=\Delta+\omega^{2}$ is strongly elliptic. Hence,
$\tilde{u}(x,\omega)$ can be calculated and analysed by the methods described
in previous sections. The limiting amplitude is calculated in [26] using the
limit (7.62).
###### Remark 7.2.
In 1912, A. Sommerfeld discovered the Sommerfeld radiation condition [62] (see
also [52]), which provides the uniqueness of solution to the boundary problem
of type (7.61) in the case when $Q$ is the exterior of a bounded region in
$\mathbb{R}^{3}$. This condition is more practical for numerical calculation
of the limiting amplitudes than (7.62).
## 8 The Sommerfeld diffraction theory and related results
For the angle $\Phi=2\pi$, A. Sommerfeld constructed in 1896 a solution $a(x)$
to stationary diffraction problem of type (7.55) with Dirichlet and Neumannn
boundary conditions. In this case the wedge is the half-plane, which is
represented by the semi-axis $[0,\infty)$ in the corresponding 2D problem. The
main ideas were i) to treat the semi-axis as the cut on an appropriate Riemann
surface, and ii) to extend the known method of reflections to Riemann
surfaces. As a result, A. Sommerfeld constructed a universal integral
representation of a class of branching solutions of the Helmholtz equation on
the Riemann surface in the form of the Sommerfeld integral with a fixed
integral kernel and a with a suitable density function. Further, A. Sommerfeld
chose an appropriate densities to satisfy the boundary conditions.
Sommerfeld’s strategy of costructing the solution remains a mysterious riddle
to this day. This approach is reproduced with some comments in [30, Ch. 5],
see also [48]. However, the Sommerfeld integral representation turned out to
be extremely fruitful, and in particular, was used by G.D. Malujinetz to solve
the problem with the Leontovich boundary condition [34, 35], see also [2].
For any angles $\Phi\in(0,2\pi)$ the stationary diffraction problem (7.55) for
the Dirichlet and Neumannn boundary conditions in the angles of this magnitude
was solved by other methods in 1920 by H.S. Carslaw [8], in 1932–1937 by V.I.
Smirnov and S.L. Sobolev [55, 56, 57, 58, 59], and in 1951 by J.B. Keller and
A. Blank [21].
###### Remark 8.1.
i) In all the works, cited above, the limiting amplitude principle was not
established, and the choice of suitable solution of the ill-posed problem
(7.55) was not rigorously clarified. Nevertheless, as shown in [28, 29, 46],
all the obtained solutions coincide with the limiting amplitudes calculated in
[26] and admit the Sommerfeld representation.
ii) S.L. Sobolev mentions, in the articles cited above, that the functions of
type (6.49) must be solutions to the wave equation even if the amplitude
$a(s)$ is a discontinuouos function. These remarks later inspired the theory
of weak derivatives of S.L. Sobolev and the theory of distributions of L.
Schwartz.
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|
As in the proof of Proposition \ref{prop:Proofs:GConvergence:WellPosed:liminf}, without loss of generality, assume that $\cF_{n,\eps_n}((\nu_n,v_n)) \leq C$ for some $C > 0$. In particular, this implies that $\nu_n = \mu_n$, and by Proposition \ref{prop:Back:TLp}, $\mu_n$ converges weakly to $\nu$. By the uniqueness of weak limits, we must have that $\nu = \mu$.
We have \[
C\geq \liminf_{n \to \infty} \mathcal{F}_{n,\eps_n}( (v_n,\nu_n) ) \geq \liminf_{n \to \infty} \EnergySnepsn(v_n) \geq \EnergyS(v) = \mathcal{G}((\nu,v))
\]
where the last inequality follows from Theorem \ref{thm:Proofs:GConvergence:withoutConstraints}.
\end{proof}
The $\limsup$ inequality requires two computational Lemmas.
\begin{lemma} \label{lem:Proofs:GConvergence:IllPosed:diracEnergies}
Energy estimates of dirac deltas.
Assume Assumptions~\ref{ass:Main:Ass:S1}, \ref{ass:Main:Ass:M1}, \ref{ass:Main:Ass:M2}, \ref{ass:Main:Ass:D1}, \ref{ass:Main:Ass:W1}, \ref{ass:Main:Ass:W2} and~\ref{ass:Main:Ass:L1} hold, $\eps_n$ satisfies~\eqref{eq:Main:Ass:epsLBIllPosed}. Then, there exists $C > 0$ such that, $\bbP$-a.s., for $n$ large enough we have
\begin{equation} \label{eq:Proofs:GConvergence:IllPosed:diracEnergies}
\EnergySnepsn(\delta_{x_i}) \leq \frac{C}{n \eps_n^{2s}}
\end{equation}
for all $x_i \in \Omega_n$.
\end{lemma}
\begin{proof}
With probability one, we can assume that the conclusion of Theorem \ref{thm:Back:TLp:LinftyMapsRate} holds.
By Assumption \ref{eq:Main:Ass:epsLBIllPosed} and Theorem \ref{thm:Back:TLp:LinftyMapsRate}, we can apply \cite[Lemma 22]{Stuart} for $n$ large enough. We recall the variational definition of eigenvalues:
\[
\lambda_{n,n}^s = \sup_{\Vert u \Vert_{\Lp{2}} = 1,\: u \in \mathbb{R}^n} \langle u, \Delta_{n,\eps_n}^s u \rangle_{\Lp{2}(\mu_n)}.
\]
Furthermore, for $x_i \in \Omega_n$,
\[
\Vert \sqrt{n}\delta_{x_i} \Vert_{\Lp{2}} = \sqrt{\frac{1}{n} \sum_{j=1}^n n \delta_{x_i}(x_j)^2} = 1
\]
so that we can estimate:
\begin{align}
\EnergySnepsn(\delta_{x_i}) &= \langle \delta_{x_i}, \Delta_{n,\eps_n}^s \delta_{x_i} \rangle_{\Lp{2}(\mu_n)} \notag\\
&= \frac{1}{n} \langle \sqrt{n} \delta_{x_i} , \Delta_{n,\eps_n}^s \sqrt{n}\delta_{x_i} \rangle_{\Lp{2}(\mu_n)} \notag\\
&\leq \frac{1}{n}\sup_{\Vert u \Vert_{\Lp{2}} = 1,\: u \in \mathbb{R}^n} \langle u, \Delta_{n,\eps_n}^s u \rangle_{\Lp{2}(\mu_n)} \notag\\
&= \frac{\lambda_{n,n}^s}{n} \notag\\
&\leq \frac{C}{n\eps_n^{2s}}\label{eq:Proofs:GConvergence:IllPosed:dirac:eigenvalueBound}
\end{align}
where we used \cite[Lemma 22]{Stuart} for \eqref{eq:Proofs:GConvergence:IllPosed:dirac:eigenvalueBound}.
\end{proof}
\begin{proposition} \label{prop:Proofs:GConvergence:IllPosed:limsup}
$\limsup$-inequality in the ill-posed case.
Assume Assumptions~\ref{ass:Main:Ass:S1}, \ref{ass:Main:Ass:M1}, \ref{ass:Main:Ass:M2}, \ref{ass:Main:Ass:D1}, \ref{ass:Main:Ass:W1}, \ref{ass:Main:Ass:W2} and~\ref{ass:Main:Ass:L1} hold, $\eps_n$ satisfies~\eqref{eq:Main:Ass:epsLBIllPosed}.
Assume that $n\eps_n^{2s} \to \infty$ and $\rho \in \Ck{\infty}$. Let $(\nu,v) \in \TLp{2}(\Omega)$. Then, $\bbP$-a.s., there exists $\{(\nu_n,v_n)\} \subseteq \TLp{2}(\Omega)$ such that $(\nu_n,v_n) \to (\nu,v)$ in $\TLp{2}$ and
\begin{equation}\label{eq:Proofs:GConvergence:IllPosed:limsup}
\limsup_{n\to \infty} \cF_{n,\eps_n}((\nu_n,v_n)) \leq \cG((\nu,v)).
\end{equation}
\end{proposition}
\begin{proof}
In the proof $C>0$ will denote a constant that can be arbitrarily large, independent of $n$ that may change from line to line. With probability one, we can assume that the conclusions of Theorem \ref{thm:Back:TLp:LinftyMapsRate} and Lemma \ref{lem:Proofs:GConvergence:IllPosed:diracEnergies} hold.
We note that \eqref{eq:Proofs:GConvergence:IllPosed:limsup} is trivial if $\mathcal{G}((\nu,v)) = \infty$. Hence, we might assume that $\mathcal{G}((\nu,v)) < \infty$ and in particular, this implies that $\nu = \mu$ and $\mathcal{G}((\nu,v)) = \EnergyS(v)$. We therefore need to prove \eqref{eq:Proofs:GConvergence:IllPosed:limsup} on $\{\mu\}\times\mathcal{H}^s(\Omega) \subseteq \TLp{2}$. We start by assuming that $v \in \Ck{\infty}$, which is dense in $\mathcal{H}^s(\Omega)$. Let $\bar{v}_n$ be the restriction of $v$ to $\Omega_n$ and consider the recovery sequence $\{(\mu_n,\bar{v}_n)\}_{i=1}^\infty \subseteq \TLp{2}(\Omega)$. By Theorem \ref{thm:Back:TLp:LinftyMapsRate}, we get transport maps $\{T_n\}_{n=1}^\infty$ from $\mu$ to $\mu_n$. Repeating the proof of Proposition \ref{prop:Proofs:GConvergence:WellPosed:limsup}, we can show that $(\mu_n,\bar{v}_n) \to (\mu,v)$ in $\TLp{2}$ and that $\limsup_{n \to \infty} \EnergySnepsn(\bar{v}_n) \leq \EnergyS(v)$.
We define the functions \[
v_n(x_i) = \begin{cases}
\bar{v}_n(x_i) &\text{if $i \geq N + 1$,}\\
\ell_i & \text{if $i \leq N$.}
\end{cases}
\]
and estimate as follows:
\[
\int_\Omega \vert v_n \circ T_n - v \vert^2 \,\dd\mu \leq 2 \int_\Omega \vert v_n \circ T_n - v \circ T_n \vert^2 \,\dd\mu + 2 \int_\Omega \vert v \circ T_n - v \vert^2 \,\dd\mu =: 2(A + B).
\]
As in the proof of Proposition \ref{prop:Proofs:GConvergence:WellPosed:liminf}, we see that $B \to 0$. For the $A$ term, we have
\begin{align}
A &\leq \int_{\{x \spaceBar T_n(x) \neq x_i \text{ for $i \leq N$}\}} \vert v_n \circ T_n - v \circ T_n \vert^2 \, \dd\mu + \sum_{i=1}^N \int_{\{x \spaceBar T_n(x) = x_i\}} \vert v_n \circ T_n - v \circ T_n \vert^2 \, \dd\mu \notag\\
&=\sum_{i=1}^N \vert v(x_i) - \ell_i \vert^2 \mu(\{x \spaceBar T_n(x) = x_i\}) \notag\\
&= \sum_{i=1}^N \vert v(x_i) - \ell_i \vert^2 \mu_n(\{x_i\}) \notag\\
&= \sum_{i=1}^N \frac{\vert v(x_i) - \ell_i \vert^2}{n} \notag
\end{align}
from which we deduce that $(\mu_n,v_n) \to (\mu,v)$ in $\TLp{2}$.
Let $n$ be large enough so that \eqref{eq:Proofs:GConvergence:IllPosed:diracEnergies} holds. We now show that
\[
\limsup_{n \to \infty} \mathcal{F}_{n,\eps_n}(v_n,\mu_n) = \limsup_{n \to \infty} \EnergySnepsn(v_n) \leq \EnergyS(v).
\]
To this purpose, we note that
\[
v_n = \bar{v}_n + \sum_{i = 1}^N \delta_{x_i} (\ell_i - \bar{v}_n(x_i))
\]
and applying Lemma \ref{lem:Proofs:GConvergence:IllPosed:Minkowski}, we obtain:
\begin{align}
\sqrt{\EnergySneps{}(v_n)} &\leq \sqrt{\EnergySnepsn(\bar{v}_n)} + \sum_{i=1}^N \vert \ell_i - \bar{v}_n(x_i) \vert \sqrt{\EnergySnepsn(\delta_{x_i})} \notag\\
&\leq \sqrt{\EnergySnepsn(\bar{v}_n)} + C \sum_{i=1}^N \sqrt{\EnergySnepsn(\delta_{x_i})} \label{eq:Proofs:GConvergence:IllPosed:limsup:linfty}\\
&\leq \sqrt{\EnergySnepsn(\bar{v}_n)} + C \l \frac{1}{n\eps_n^{2s}} \r^{1/2} \label{eq:Proofs:GConvergence:IllPosed:limsup:energy}
\end{align}
where we used the fact that $\Vert v \Vert_{\Lp{\infty}}$ is bounded on $\Omega$ for \eqref{eq:Proofs:GConvergence:IllPosed:limsup:linfty} (and $\bar{v}_n$ is the restriction of $v$ to $\Omega_n$ and so is also bounded in $\Lp{\infty}$) and \eqref{eq:Proofs:GConvergence:IllPosed:diracEnergies} for \eqref{eq:Proofs:GConvergence:IllPosed:limsup:energy}. Taking $n \to \infty$ on the latter right hand side, we have
\[
\limsup_{n \to \infty} \sqrt{\EnergySneps(v_n)} \leq \limsup_{n \to \infty} \sqrt{\EnergySnepsn(\bar{v}_n)} + \limsup_{n \to \infty} C \l \frac{1}{n\eps_n^{2s}} \r^{1/2} \leq \sqrt{\EnergyS(v)}
\]
since by assumption $n\eps_n^{2s} \to \infty$ and $\limsup_{n \to \infty} \EnergySnepsn(\bar{v}_n) \leq \EnergyS(v)$.
Having shown \eqref{eq:Proofs:GConvergence:IllPosed:limsup} on $\{\mu\}\times \Ck{\infty}$, we now use the fact that it is sufficient to establish the existence of a recovery sequence on a dense subset \cite[Remark 2.7]{Trillos3}.
\end{proof}
\subsection{Bounds on Minimizers} \label{subsec:Proofs:Linfty}
\begin{lemma} \label{lem:Proofs:Bounds:UniformBounds}
Uniform bound of energies for minimizers.
Assume Assumptions~\ref{ass:Main:Ass:S1}, \ref{ass:Main:Ass:M1}, \ref{ass:Main:Ass:M2}, \ref{ass:Main:Ass:D1}, \ref{ass:Main:Ass:W1}, \ref{ass:Main:Ass:W2} and~\ref{ass:Main:Ass:L1} hold. Assume $\rho \in \Ck{\infty}$. Let $\{(\mu_n,u_n)\}_{n=1}^\infty$ be the minimizers of $\cF_{n,\eps_n}(\cdot)$. Then, there exists $C >0$ such that, $\bbP$-a.s., we have
\[
\sup_{n} \cF_{n,\eps_n}( (\mu_n,u_n) ) \leq C.
\]
\end{lemma}
\begin{proof}
In the proof $C>0$ will denote a constant that can be arbitrarily large, independent of $n$ that may change from line to line.
With probability one, we can assume that the conclusion of Proposition~\ref{prop:Proofs:GConvergence:WellPosed:limsup} holds.
Let $v \in C^\infty(\Omega)$ be a function that interpolates all the points $\{(x_i,\ell_i)\}_{i=1}^N$. Since $\rho \in C^\infty$, we have that $\Delta^s_\rho v \in C^\infty$ implying that $v \Delta^s_\rho v \in C^\infty$.
We have
\[
\EnergyS(v) = \int_\Omega v \Delta^s_\rho v \,\dd\mu < K.
\]
for some $K > 0$.
%Let $v_n$ be the restriction of $v$ on $\Omega_n$.
By Proposition \ref{prop:Proofs:GConvergence:WellPosed:limsup} there exists a sequence $v_n$ converging to $v$ and such that %we obtain
\[
\lim_{n \to \infty} h_n := \lim_{n \to \infty} \sup_{m \geq n } \mathcal{E}^{(s)}_{m,\eps_m}(v_m) = \limsup_{n \to \infty} \EnergySnepsn(v_n) = \limsup_{n \to \infty} \mathcal{F}_{n,\eps_n}((\mu_n,v_n)) \leq \EnergyS(v) < K.
\]
Let $h := \limsup_{n \to \infty} \EnergySnepsn(v_n)$ and let $\bar{\eps} = K - h > 0$. Then, there exists $n_0$ such that for all $n \geq n_0$, $\vert h_n - h \vert < \bar{\eps}/2$, which is equivalent to $h_n = \sup_{m \geq n} \mathcal{E}^{(s)}_{m,\eps_m}(v_m) < h + \bar{\eps}/2 < K$. Using the latter, we have
\[
\sup_{n} \EnergySnepsn(v_n) = \max\{\mathcal{E}_{1,\eps_1}^{(s)}(v_1),\dots,\mathcal{E}_{n_0,\eps_{n_0}}^{(s)}(v_{n_0}),\sup_{n \geq n_0} \EnergySnepsn(v_n)\}
%\leq \max\{\mathcal{E}_{1,\eps_1}^{(s)}(v_1),\dots,\mathcal{E}_{n_0,\eps_{n_0}}^{(s)}(v_{n_0}),K\}
\leq C.
\]
Since $\{u_n\}_{n=1}^\infty$ are minimizers, we have $\EnergySnepsn(u_n) \leq \EnergySnepsn(v_n)$ which implies
\[
\sup_n \mathcal{F}_{n,\eps_n}((\mu_n,u_n)) = \sup_n \EnergySnepsn(u_n) \leq \sup_n \EnergySnepsn(v_n) \leq C. \qedhere
\]
\end{proof}
\subsection{Proof of Theorem~\ref{thm:Main:Res:ConsFracLap}}
\begin{proof}[Proof of Theorem \ref{thm:Main:Res:ConsFracLap}]
In the proof $C>0$ will denote a constant that can be arbitrarily large, independent of $n$ that may change from line to line.
% \item
\paragraph{Well-Posed Case.}
With probability one, we can assume that the conclusions of Lemma \ref{lem:Proofs:Bounds:UniformBounds}, Proposition \ref{prop:Proofs:Bounds:Poincaré}, Theorem \ref{thm:Proofs:GConvergence:withoutConstraints}, Proposition \ref{prop:Proofs:Compactness:L2Compactness} and
Proposition \ref{prop:Proofs:GConvergence:WellPosed:liminf} hold.
Using Lemma \ref{lem:Proofs:Bounds:UniformBounds} and Proposition \ref{prop:Proofs:Bounds:Poincaré}, we have
%\[ \Vert u_n \Vert_{\Lp{2}} \leq \Vert u_n - \frac{1}{N} \sum_{i=1}^N u_n(x_i) \Vert_{\Lp{2}} + \Vert \frac{1}{N} \sum_{i=1}^N u_n(x_i) \Vert_{\Lp{2}} \leq C \sqrt{\EnergySnepsn(u_n)} + \frac{1}{N} \sum_{i=1}^N \vert\ell_i\vert \leq C. \]
\[ \Vert u_n \Vert_{\addmaths{\Lp{\infty}}{\removemaths{\Lp{2}}}} \leq \Vert u_n - \frac{1}{N} \sum_{i=1}^N u_n(x_i) \Vert_{\addmaths{\Lp{\infty}}{\removemaths{\Lp{2}}}} + \Vert \frac{1}{N} \sum_{i=1}^N u_n(x_i) \Vert_{\addmaths{\Lp{\infty}}\removemaths{\Lp{2}}} \leq C \sqrt{\EnergySnepsn(u_n)} + \frac{1}{N} \sum_{i=1}^N \vert\ell_i\vert \leq C. \]
Hence minimisers are bounded in $\TLp{2}$ and $\max\l\sup_{n} \Vert u_{n} \Vert_{\Lp{\infty}},\sup_{n} \mathcal{E}^{(s)}_{n}(u_{n}\r \leq C$.
By Proposition~\ref{prop:Proofs:Compactness:L2Compactness} there exists $u$ and a subsequence converging uniformly and in $\TLp{2}$.
By Propositions~\ref{prop:Proofs:GConvergence:WellPosed:liminf} and \ref{prop:Proofs:GConvergence:WellPosed:limsup} it follows that $(\mu,u)$ is a minimiser of $\cF$.
By uniqueness of the minimiser of $\cF$ it follows that the whole sequence $\{( \mu_n,u_n)\}$ converges in $\TLp{2}$ to $(\mu,u)$.
\paragraph{Ill-Posed Case.}
With probability one, we can assume that the conclusions of Theorem \ref{thm:Proofs:GConvergence:withoutConstraints}, Proposition \ref{prop:Proofs:GConvergence:IllPosed:liminf} and Proposition \ref{prop:Proofs:GConvergence:IllPosed:limsup} hold.
By Proposition \ref{prop:Proofs:GConvergence:IllPosed:liminf} and Proposition \ref{prop:Proofs:GConvergence:IllPosed:limsup}, we know that $\mathcal{F}_{n,\eps_n}$ $\Gamma$-converges to $\mathcal{G}(\cdot)$.
By Theorem \ref{thm:Proofs:GConvergence:withoutConstraints} $\mathcal{F}_{n,\eps_n}(\cdot)$ satisfies the compactness property.
Hence, by Proposition~\ref{prop:Back:Gamma:minimizers} we can conclude the result.
\end{proof}
\section{Numerical Experiments} \label{sec:NumExp}
\begin{figure}
\centering
\includegraphics[width = \textwidth]{minimizers.png}
\caption{Plots of the discrete and continuum minimizers with the setting described in Section \ref{subsec:criticalBoundary}. The values at the points $(0.1,0.1)$ and $(0.9,0.9)$ are marked with black squares.
\textit{Left}: Discrete minimizer computed with $n = 1733$ points.
\textit{Right}: Continuum minimizer.
\end{figure}
\subsection{Critical boundary between well-posed and ill-posed regimes} \label{subsec:criticalBoundary}
In this section, we will investigate the gap in the upper bound alluded to in Remark \ref{rem:Main:Res:Relationship}. In particular, we will rely on the same methodology as in [53].
To test the gap between the well-posed and ill-posed regime, i.e. $n^{-\frac{2}{s-1}}\lesssim \eps_n\lesssim n^{-\frac{1}{2s}}$ (by Corollary~\ref{cor:Main:Res:ConsFracLap} we know that $\eps_n\lesssim n^{-\frac{2}{s-1}}$ implies asymptotic well-posedness and $\eps_n\gg n^{-\frac{1}{2s}}$ implies asymptotic ill-posedness)
we will consider the following setting: we choose the uniform measure on $[0,1]^2$ with periodic boundary conditions, the kernel function $\eta(t) = 1$ if $t\leq 1$ and $\eta(t) = 0$ else and $s = 16$. This choice of $s$ satisfies the constraint $s > 2d + 9 = 13$ from Remark \ref{rem:Main:Res:LowerBoundS}. The training set consists of the points $(0.1,0.1)$ and $(0.9,0.9)$ labelled $0$ and $1$ respectively.
We will vary the number of points in our graph from $n = 100$ to $n = 5000$. For each $n$, we will also consider a wide range of $\eps_n$ ranging from roughly the connectivity radius to above $n^{-1/2s}$. For each combination of $(n,\eps_n)$, we then compute the discrete minimizer $u_n$ using Lagrange multipliers and compare it to the continuum solution $u$. The continuum solution is computed using a finite-difference scheme on a regular grid of 10000 points on $[0,1]^2$.
The error considered is
\begin{equation} \label{eq:numExp:error}
\text{err}(n,\eps_n,u_n) = \Vert u_n - u \Vert_{\Lp{2}(\mu_n)}
\end{equation}
where, in order to evalutate $u$ on the graph, we use spline interpolation.
Finally, by re-sampling the points for each $n$, we repeat the experiments one hundred times and average the error.
We are interested in finding the value of $\eps_n$ where fractional Laplacian learning switches from the well-posed to the ill-posed regime. In order to compute the latter, we start by smoothing the function $\eps_n \mapsto \text{err}(n,\eps_n,u_n)$ and compute the maximizer of its first derivative and the minimizer of its second derivative which we denote by $\hat{\eps}_n$ and $\eps_n^{*}$ respectively.
We choose $\hat{\eps}_n$ and $\eps_n^*$
greater than the minimizer of $\text{err}(n,\cdot,u_n)$.
Both $\hat{\eps}_n$ and $\eps_n^*$ could be taken as reasonable definitions of the transition point between the well-posed and ill-posed regime and, it is therefore interesting to understand how they scale with $n$. Using the five largest values of $n$, the best linear fits between $\log(\hat{\eps}_n)$, $\log(\eps_n^*)$ and $\log(n)$ yield that
\[
\hat{\eps}_n \approx \frac{0.6541}{n^{0.05}} \quad \text{and} \quad \eps_n^* \approx \frac{0.7312}{n^{0.06}}.
\]
For $s = 16$, we have $1/(2s) = 0.03125$ and $2/(s-1) \approx 0.134$. Given the top plots in Figure \ref{fig:error}, we observe that both $\hat{\eps}_n$ and $\eps_n^*$ seem to scale accurately with powers that are different to $1/(2s)$. In fact, we note that $0.06 \approx 0.05 \approx 1/s = 0.0625$. On one hand, this indicates that we should be able to extend the well-posed regime of Theorem \ref{thm:Main:Res:ConsFracLap} to
\[
\l \frac{1}{n} \r^{2/(s-1)} \ll \eps_n \ll \l \frac{1}{n}\r^{1/s}
\]
and, by Remark \ref{rem:Main:Res:LowerBoundS}, we could relax our assumption of $s > 2d + 9$ to $s > d + 4$. If we were able to also tackle the lower bound gap (see Remark \ref{rem:Main:Res:LowerBoundGap}), we could furthermore have $s > d$ in contrast to $s > d/2$ which is conjectured in Remark \ref{rem:Main:Res:Sobolev}.
On the other hand, we are not able to fully confirm the conjecture made in Remark \ref{rem:Main:Res:Relationship} numerically and it remains an open question to accurately characterize the behaviour of fractional Laplacian semi-supervised learning when
\[
\l \frac{1}{n} \r^{1/s} \ll \eps_n \ll \l \frac{1}{n}\r^{1/(2s)}.
\]
\begin{figure}[t!]
\setlength\figureheight{0.25\textwidth}
\setlength\figurewidth{0.3\textwidth}
\centering
\scriptsize
\begin{subfigure}[t]{0.3\textwidth}
\centering
\scriptsize
\input{LinearFitEpsIll2.tikz}
\caption{
The blue line is the fit $\frac{0.6541}{n^{0.05}}$ and the red line is $\hat{\eps}_n$.
\end{subfigure}
\hspace*{0.04\textwidth}
\begin{subfigure}[t]{0.3\textwidth}
\centering
\scriptsize
\input{LinearFitEpsIll.tikz}
\caption{
The blue line is the fit $\frac{0.7312}{n^{0.06}}$ and the red line is $\eps_n^*$.
\end{subfigure}
\hspace*{0.04\textwidth}
\begin{subfigure}[t]{0.3\textwidth}
\centering
\scriptsize
\input{Error5000.tikz}
\caption{
Error \eqref{eq:numExp:error} for $n = 5000$. The blue line is the average $\text{err}(n,\eps_n,u_n)$ and the dashed black lines are the 10\% and 90\% quantiles. The red and black vertical lines respectively indicate $\hat{\eps}_n$ and $\eps_n^*$.
\end{subfigure}
\caption{
Plots of the errors between the discrete and continuum minimizers with the setting described in Section \ref{subsec:criticalBoundary}.
\label{fig:error}
\end{figure}
\subsection{Uniform Bounds on Eigenfunctions} \label{subsec:linear}
Proposition~\ref{prop:Proofs:Back:DisReg:EVecAlt} implies that we can upper bound the $\Lp{\infty}$ norm of the discrete eigenvectors $\psi_{n,k}$ by
\begin{equation} \label{eq:numExp:lambda}
\|\psi_{n,k}\|_{\Lp{\infty}}\leq C\lambda_{k}^{d+1}
\end{equation}
when $k\in\{1,\hdots,\ceil{K_n}\}$ where $K_n=\alpha \eps_n^{-d/2} + 1$. We now investigate the following: on one hand, we want to see if the power of $\eps_n$ in the definition of $K_n$, namely $-d/2$, is the lowest we can get; and on the other hand, we are interested in the optimal power of $\lambda_{k}$ in \eqref{eq:numExp:lambda}.
We choose the uniform measure on $[0,1]^2$ with periodic boundary conditions and the kernel function $\eta(t) = 1$ if $t\leq 1$ and $\eta(t) = 0$ else. We proceed as follows: given a number of points ranging from $n = 400$ to $n = 5200$, we seek the smallest $\eps_n$ such that the graph is connected. We then compute $\lb \log\l \lambda_{k} \r, \log \l \|\psi_{n,k}\|_{\Lp{\infty}} \r \rb_{k=1}^{k_n^{(i)}}$ where $k_n^{(1)} = \alpha_1 \eps_n^{-d/4}$, $k_n^{(2)} = \alpha_2 \eps_n^{-d/2}$, $k_n^{(3)} = \alpha_3 \eps_n^{-d}$, $k_n^{(4)} = n$
and $\alpha_i$ for $1 \leq i \leq 3$ are constants. While we took the arbitrary choice of $\alpha_i = 4$ for $1 \leq i \leq 3$, empirical trials have shown that the overall conclusion does not change significantly. By re-sampling the points for each $n$, we repeat the experiments one hundred times and average the results.
\begin{figure}[t!]
\setlength\figureheight{0.35\textwidth}
\setlength\figurewidth{0.47\textwidth}
\centering
\scriptsize
\begin{subfigure}[t]{0.47\textwidth}
\centering
\scriptsize
\input{eigenPairs1200.tikz}
\caption{
$n = 1200$
\end{subfigure}
\hspace*{0.04\textwidth}
\begin{subfigure}[t]{0.47\textwidth}
\centering
\scriptsize
\input{eigenPairs5200.tikz}
\caption{
$n = 5200$
\end{subfigure}
\caption{
Plots of $\log \l \|\psi_{n,k}\|_{\Lp{\infty}} \r$ versus $\log\l \lambda_{k} \r$ for $1 \leq k \leq n$ with the setting described in Section \ref{subsec:linear}. The blue line corresponds to the average of $\log \l \|\psi_{n,k}\|_{\Lp{\infty}} \r$, the dashed black lines are the mean plus/minus the standard deviation and the red lines indicate the values of $\log \l \|\lambda_{k_n^{(i),*}}\|_{\Lp{\infty}} \r$ for $1 \leq i \leq 4$ from left to right. \ref{subsec:criticalBoundary}.
\label{fig:eigenvalues}
\end{figure}
\begin{figure}[t!]
\setlength\figureheight{0.25\textwidth}
\setlength\figurewidth{0.3\textwidth}
\centering
\scriptsize
\begin{subfigure}[t]{0.3\textwidth}
\centering
\scriptsize
\input{linear_k2.tikz}
\caption{
The blue line is the fit $4.05 - 0.63 \log(\Vert \psi_{n,k_n^{(2),*}} \Vert)$ and the red line is $\lambda_{k_n^{(2),*}}$.
\end{subfigure}
\hspace*{0.04\textwidth}
\begin{subfigure}[t]{0.3\textwidth}
\centering
\scriptsize
\input{linear_k3.tikz}
\caption{
The blue line is the fit $0.2 - 0.08 \log(\Vert \psi_{n,k_n^{(3),*}} \Vert)$ and the red line is $\lambda_{k_n^{(3),*}}$.
\end{subfigure}
\hspace*{0.04\textwidth}
\begin{subfigure}[t]{0.3\textwidth}
\centering
\scriptsize
\input{linear_k4.tikz}
\caption{
The blue line is the fit $-0.55 + 0.01 \log(\Vert \psi_{n,k_n^{(4),*}} \Vert)$ and the red line is $\lambda_{k_n^{(4),*}}$.
\end{subfigure}
\caption{
Plots of $\log \l \|\psi_{n,k_n^{(i),*}}\|_{\Lp{\infty}} \r$ versus $\log\l \lambda_{k_n^{(i),*}} \r$ for $2 \leq i \leq 4$ with the setting described in Section \ref{subsec:linear}.
\label{fig:linear}
\end{figure}
From Figure \ref{fig:eigenvalues} there appears to be different regimes of growth for the eigenpairs depending on the value of $1 \leq k \leq n$. In fact, we see that below $\log(\lambda_{k_n^{(3)}})$, $\log(\Vert \psi_{n,k} \Vert_{\Lp{\infty}})$ seems to be increasing monotonically, while it unexpectedly first decreases and then sharply increases from $\log(\lambda_{k_n^{(3)}})$ to $\log(\lambda_{k_n^{(4)}})$. It is the subject of future research to provide explanations for the latter phenomena.
Let us define
\[
k_n^{(i),*} = \argmax_{1 \leq k \leq k_n^{(i)}} \log \l \Vert \psi_{n,k} \Vert_{\Lp{\infty}} \r
\]
and, using the seven largest values of $n$, for $1 \leq i \leq 4$, we will be computing the best linear fits for the points
\[
\lb( \log \l \lambda_{n,k_n^{(i),*}} \r, \log \l \Vert \psi_{k_n^{(i),*}} \Vert_{\Lp{\infty}} \r\rb_{n=400}^{5200}.
\]
For $k_n^{(i)}$ with $1 \leq i \leq 2$, the linear fits in the log-log domain appear to be very accurate as depicted in Figure \ref{fig:linear}, so we are able to confirm the theoretical guarantees of Proposition \ref{prop:Proofs:Back:DisReg:EVecAlt}.
In particular, we obtain
\[
\Vert \psi_{n,k_n^{(2),*}} \Vert_{\Lp{\infty}} \approx C \lambda_{k_n^{(2),*}}^{-0.63}
\]
and $\lambda_{k_n^{(2),*}}^{-0.63} \leq \lambda_{k_n^{(2),*}}^{d+1}$ since $\lambda_k \geq 1$ (in fact, we have $\{\lambda_k\}_{k=1}^\infty = \{ 4 \pi^2 k^2 \spaceBar k \in \bbN \}$ in this particular setting). However, it shows that the bound is not sharp for the flat torus.
The situation for the regime $k_n^{(3)}$ is different and we obtain that \[
\Vert \psi_{n,k_n^{(3),*}} \Vert_{\Lp{\infty}} \approx C \lambda_{k_n^{(3),*}}^{-0.08}.
\]
The much smaller exponent in the above compared to the $k_n^{(2),*}$ setting indicates that we seem to be switching from one regime of growth of $\Vert \psi_{n,k} \Vert_{\Lp{\infty}}$ to another one.
Finally, we note that the linear fit in Figure \ref{fig:linear} for the $k_n^{(4)}$ regime yields
\[
\Vert \psi_{n,k_n^{(4),*}} \Vert_{\Lp{\infty}} \approx C \lambda_{k_n^{(4),*}}^{0.01}
\]
and, making the crude approximation $0 \approx 0.01$, this confirms our intuition that on the flat torus, the graph Laplacian should have uniformly bounded $\Lp{\infty}$-norms of eigenfunctions: indeed, the continuum eigenfunctions are uniformly bounded in $\Lp{\infty}$, so we expect the same behaviour for their discrete counterparts. This suggests that one should be able to pick $\alpha = 0$ in Theorem \ref{thm:Main:Res:ConsFracLap} yielding the (almost) optimal Sobolev bound $s > d/2 + 2$ as discussed in Remark~\ref{rem:Main:Res:Sobolev}. This conclusion also suggests that, on the flat torus at least, one should be able to go above $\eps_n^{-d/2}$ in Proposition \ref{prop:Proofs:Back:DisReg:EVecAlt}.
\section*{Acknowledgements}
The authors would like to thank Nicol\'as Garc\'ia Trillos for his comments and insights on this paper.
MT was supported by the European Research Council under the European Union’s Horizon 2020 research and innovation programme Grant Agreement No. 777826 (NoMADS).
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\bibliographystyle{plain}
\end{document}
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# Pushing the precision frontier for
gravitational waves
Michèle Levi111https://sites.google.com/view/levim Niels Bohr Institute,
University of Copenhagen,
Blegdamsvej 17, 2100 Copenhagen, Denmark
(August 6, 2021)
###### Abstract
Pushing the precision frontier for gravitational waves is one of the most
urgent tasks in theoretical physics today, in light of the increasing influx
of data from a rapidly growing worldwide network of gravitational-wave
detectors. Levi will analytically predict gravitational radiation from such
compact binaries and further develop the use of quantum field theory advances
to study gravity. She seeks to uncover profound duality relations between
gauge and gravity.
International Editor Clifford Holt spoke to Levi about her research, the
importance of analytical high-precision predictions for real-world
gravitational-wave data, and potential challenges as she works to develop a
better idea of universal commonalities across classical and quantum field
theories.
Firstly, congratulations on being awarded the Ernest Rutherford Fellowship.
How important are funding mechanisms such as these, and what will the
Fellowship mean for your own research and career?
Thank you! A funding mechanism like this prestigious early-career award from
the STFC (one of the seven UK research councils under UKRI) is very important
and unique. This is the most competitive scheme in the UK for early-career
researchers in my domain, devised to ensure that those 10 carefully-selected
awardees, who are recognised as the very top international talent and
leadership ‘material’ in science early on in their career, will be integrated
into top academic positions in the leading research institutions in the UK.
This will enable the awardees to play a key role in the coming decades in the
scientific and transformative leadership at the national level in the UK and
will also represent the UK at the international level, solidifying its
position as a scientific world-leader for the next generations. As a public
funder, UKRI selects these future leaders to shape strategic UK policies, to
bolster the UK’s scientific place in the world, and to bring about a radical
shift in the current research culture in terms of the way that science is
conducted, namely its standards and its values, but also its role and
commitment within society.
Currently, many academics have sadly forgotten that their raison d’etre is to
serve the public, that they have the privilege of expanding the frontiers of
knowledge mainly thanks to the public money that is funding their research. We
must insist that science that is peer-reviewed and approved for publication
and funding lives up to the standards of FAIR (Findabilty, Accessibilty,
Interoperabilty, Reusabilty) principles, namely of responsible research of
integrity, that is publicly available, widely accessible, and reproducible. We
should also advocate for equality, diversity, and inclusion in Science, which
should properly represent Society, and engage with the public. It is very
clear that UKRI as a public funder has a strong commitment to realising these
aspirations, which are unfortunately still overlooked by the majority of
private funders.
I am excited and honoured to transfer and base my cutting-edge research
programme in the UK, and to be part of making the UK a world-leading force in
the timely domain of gravitational waves with the most innovative theoretical
approaches and computational tools applied to gravity. I am also incredibly
aligned with the aspirations of UKRI to transform the current research
culture, and as I myself belong to an overlap of under-represented and
marginalised groups in science, I am particularly aware to how systemic
inequalities affect the ability of students and researchers to truly realise
their potential and excel, or even to just conform to the common skewed
metrics of academic merit.
I adamantly champion and pursue equality, diversity, and inclusion in every
aspect of my everyday scientific activities. I believe and I see that it is
only through these consistent everyday actions and through my being a role
model that I can ultimately translate my research work to further concrete
scientific and societal impact. I also adamantly advance the broad
communication of science across the research community and through public
outreach, which I regard as a social imperative. I also find it extremely
rewarding to share my knowledge with as many people as possible.
Your project aims to deliver analytical high-precision predictions for real
world gravitational wave data. Can you tell me more about this and why it is
important?
Gravitational radiation is a probing prediction of any complete theory of
gravity. Currently, we still do not have a complete theory of gravity and this
is true for both ends of the scale, since we do not know how to describe
gravity on quantum scales, which are the smallest scales, and we do not have a
theory that captures gravity on the largest scales – cosmological scales,
since we do not know how to explain the puzzle of dark matter or the
accelerated expansion of the Universe that was discovered in 1998.
Understanding how the force of gravity works on these more ‘extreme’ scales is
possibly the deepest and most fundamental longstanding open problem in
physics.
The first Earth-shaking gravitational-wave (GW) detection announced in 2016
launched a new era of gravitational-wave astronomy and precision gravity,
which is only in its infancy, with vast future impacts in astrophysics and
cosmology and, most notably, in fundamental physics. Since this first
milestone detection, ground-based detectors keep multiplying, and many more
are planned for the coming years. They are designed to measure various
frequency ranges, which will enable a richer array of sources and events that
generate gravitational radiation to be captured.
The technology of measurements is constantly upgraded, either in currently-
operating detectors like LIGO or VIRGO, or in upcoming and planned ones, like
KAGRA in Japan, and others in the US, Europe, India, and Korea. Furthermore,
there are also concrete planned space-based experiments from Europe, China,
and Japan that will also target smaller frequencies and other types of
sources. All in all, in the coming years and decades we are headed towards a
growing influx of gravitational-wave data that will have higher and higher
quality.
Let me stress what, I believe, much of the community in Europe and the US does
not fully recognise yet: that Asia is about to emerge as possibly the major
player in gravitational-wave astronomy in the coming years: in Japan, KAGRA is
already operational; in India, IndiGO will become operational in just a few
years; and in Korea, SOGRO is planned to uniquely target mid-range frequencies
from Earth, while China and Japan are also planning space-borne detectors,
TianQin and DECIGO, respectively.
How will your research help to inform the activities of detectors such as
VIRGO and LIGO, as well as perhaps LISA? How could your theoretical work come
to complement future observations/detections?
My research generates theoretical data from which a huge bank of templates is
created to be matched in the detectors by the real data that is measured. The
GW signal is weak and is covered by many sources of noise so, in order to
identify it the matched filtering technique is used for detection, which
requires the most accurate theoretical signal templates possible. When a match
is found, a detection is announced. Yet, lots of real-world data, even if is
great data is just data without interpretation. What transforms data into
knowledge is theory and analysis. This is especially true for the
gravitational-wave signal since it is so weak, so it is really important to
separate the noise from the true signal.
For the most part, the observable differences among the plethora of candidate
theories for gravity suggested along the years have been insanely tiny. In
most cases, all of these theories seem grossly similar within gravitational-
wave measurements. So it is only the theoretical resolution into various
physical effects at a very high-precision, such as those that have been
computed in my research since my graduate studies, that will enable us to
really distinguish among the various suggested theories of gravity, and to
close in on which is the viable one.
Another feature which is rather unique to my own theoretical framework is that
it provides predictions for generic sources of compact binaries, not only for
binaries of two black holes. As the technology in upcoming detectors advances,
and as the sensitivity improves, we are able to observe more and more neutron
stars, for example, in the radiating binaries. Whereas from black holes we can
‘only’ learn about gravity theories, neutron stars also tell us about the
theory of QCD (Quantum Chromodynamics) in extreme conditions that no human-
built lab could produce. With future detectors, we also expect to observe
white dwarfs, and learn more about them.
Have you yet been able to identify any potential challenges that you expect to
have to overcome as you work to develop a better idea of universal
commonalities across classical and quantum field theories?
Right now, we see some analogies between interactions of classical extended
gravitating objects like the huge non-rotating black holes in radiating
binaries, and elementary particles of quantum nature, which we call ‘scalar
particles’ – these are boson particles of zero quantum spin. But the precise
mapping between them, and more importantly its generalisations for all types
of elementary particle, is poorly understood. In particular, my theoretical
work pushed the understanding of spinning gravitating objects, which would
also be analogous to higher-spin particles. Currently, the highest-spin
particle that has been observed in nature is of spin one, and we also believe
there could be particles of spin three/two according to supersymmetric
theories, as well as our theoretically predicted graviton – a massless
particle of spin two. But what about particles with spins larger than two? Do
they exist? Could they be elementary? Or only show up as what we call
‘composite’ particles, which must be composed of elementary ones?
At the moment, we cannot even manage to give well-defined predictions of
observables of quantum field theories that capture such higher-spin particles,
and we are certainly not able to write down such quantum field theories
explicitly. Furthermore, we are not certain whether this fundamental issue
exists already at the classical level, i.e. with the type of two-body
interactions that my work is tackling for spinning gravitating objects. The
big challenge for my research programme, therefore, would be to more
rigorously and generally link the classical and quantum sides of these
analogies and to clarify whether there is indeed a fundamental issue with
higher-spin – and, if so, whether it originates in the quantisation of the
theory, or if it already exists at the classical level.
This all relates to possibly the biggest open question in theoretical physics
of how to complete the theory of gravity at high energies, and therefore it
would indeed be a breakthrough for my upcoming research programme to be able
to provide any further insights on that.
What are your short/long term hopes and ambitions?
Since my graduate studies, I have had the privilege of pioneering and leading
innovative analytical work and high-precision computation on the highly
intricate problem of compact binaries that source gravitational radiation. At
the time, I did not believe that I would live to see actual detections of
gravitational waves from these sources, since at that time the detectors had
been operational for more than a decade without any successful detection.
Yet, the detection came in February 2016, and I can still remember it as
though it were yesterday; I was overwhelmed with excitement and wonder for
what was to come when the first observation of a gravitational-wave signal by
LIGO was announced that day. Since then, the scientific prospects of
gravitational-wave science have been continuously improving and exceeding all
expectations. The future truly looks symphonic, filled with gravitational-wave
echoes.
With this increasing influx of data of an ever improving quality, we have a
chance to really push theoretical predictions and to test so much more theory
in gravity and even in QCD. My research programme is exactly aimed at
maximising the potential of what we can learn from gravitational-wave
measurements on the theory of gravity across all scales.
I hope to engage the broader scientific community with this exciting new
domain of research, such that it really crosses over all sub-fields of
theoretical physics, especially since this is a great scientific challenge
that can benefit from so many different specific perspectives. I believe that
every one of these perspectives can bring in something fresh and interesting
to the mix. I also believe that, to some extent, I have already contributed to
the widespread attention that this type of research obtained in the
traditional high energy physics community. In the UK, I hope to engage more
people and institutions with the research in this field as the UK really has
the potential to become a superpower in this domain, especially in the type of
analytical work that I am involved in, which has been traditionally dominated
by France.
I am also a big believer in open code and open data, and I really want to see
individual researchers, big experimental collaborations, scientific journals,
and academic institutions all persistently moving towards this in the coming
years. The scientific challenges at hand are too big, and we need to focus on
the science and make our best joint efforts to make breakthroughs, instead of
yielding to the old zero-sum game mentality that keeps theoretical physics
stagnated.
Simply put, I just hope that, together, we can all increase our knowledge of
the world around us. Isn’t that the greatest privilege of our society?
## Acknowledgements
ML received funding from the European Union’s Horizon 2020 research and
innovation programme under the Marie Skłodowska-Curie grant 847523.
Her upcoming research program will be supported by the STFC Rutherford grant
ST/V003895/1 “Harnessing QFT for Gravity”.
|
# Relativistic corrections to exclusive decays $\psi(n)\to\rho\pi$ and
possible solution to the “$\rho\pi$-puzzle”
Nikolay Kivel
Physik-Department, Technische Universität München,
James-Franck-Str. 1, 85748 Garching, Germany
###### Abstract
We study relativistic corrections to exclusive $S$-wave charmonium decays into
$\rho\pi$ and $\gamma\pi$ final states. The contribution of relative order
$v^{2}$ and the set of associated higher order corrections are calculated
using NRQCD and collinear factorisation framework. Numerical estimates show
that the dominant effect is provided by the corrections of relative order
$v^{2}$. The numerical values of these contributions are of the same order as
the leading-order ones. These results suggest that the sum of relativistic and
radiative QCD corrections can potentially explain the “$\rho\pi$-puzzle”.
## 1 Introduction
A description of $S$-wave charmonium decays into $\rho\pi$ final state is
already a long-standing problem in QCD phenomenology. The branching ratios for
$J/\psi$ and excited state $\psi(3686)\equiv\psi^{\prime}$ are measured
sufficiently accurately and their ratio is found to be very small [1]
$\displaystyle
Q_{\rho\pi}=\frac{Br[\psi^{\prime}\to\rho\pi]}{Br[J/\psi\to\rho\pi]}\approx
0.20\times 10^{-2}.$ (1)
This corresponds to a strong violation of the “13%-rule”, which suggests that
$Q_{\rho\pi}\approx Q_{e^{+}e^{-}}\simeq 0.13$. The latter is valid only if
the decay amplitudes of $S$-wave charmonium are dominated by the leading-order
contribution in the QCD factorisation framework (pQCD). Therefore the
disagreement between the data and qualitative theoretical expectation
indicates about large dynamical effects, which are not accounted by the
leading-order approximation of pQCD.
The problem has attracted a lot of attention and many different qualitative
ideas and phenomenological models have been proposed in order to understand
the small value of $Q_{\rho\pi}$. Almost all of proposed explanations use
different ideas about long distance QCD dynamics; a comprehensive overview of
the topic can be found in Refs.[2, 3].
The dominant role of some nonperturbative dynamics is related to the fact that
the QCD helicity selection rule suppresses the valence contribution to the
decay amplitude. Therefore, it is necessary to take into account for the one
of outgoing mesons a non-valence component of the wave functions, which is
suppressed by additional power $\Lambda/m_{c}$. However, already long ago in
Refs. [4, 5] it was found that pQCD framework yields a reliable leading-order
estimate for the $J/\psi$ branching ratio. In Ref. [4] the non-valence
contributions are described by the three-particles twist-3 light-cone
distribution amplitudes (LCDAs). These nonperturbative functions are process
independent and the first few moments of these functions can be estimated
using QCD sum rules. At present time corresponding matrix elements were
studied and revised for various mesons, see updates in Refs. [6, 7, 8].
Therefore it is reasonable to believe that pQCD description is a good starting
point in order to develop a systematic description of the process within the
effective field theory framework.
Following this way one faces with the problem in the description of
$\psi^{\prime}\to\rho\pi$, which must be strongly suppressed relative
$J/\psi\to\rho\pi$ in order to get the small ratio (1). There are various
assumptions about possible dynamical origins for this suppression. Often they
are related to the fact that the mass of excited state $\psi^{\prime}$ is
close to the open charm threshold and this can lead to dynamical effects,
which provide the crucial difference between $J/\psi$ and $\psi^{\prime}$
decays. The possible scenarios include: destructive interference of the large
non-valence and valence contributions [4, 9]; suppression of the colour-
singlet $c\bar{c}$-wave function at the origin for $\psi^{\prime}$ and the
dominance of the colour-octet state [10]; cancellation between $c\bar{c}$ and
$D\bar{D}$ components of $\psi^{\prime}$ [11]; cancellation between $S$\- and
$D$-wave components of $\psi^{\prime}$ [12] and others [2].
On the other hand, the potential of the effective field theory framework to
study the problem was not been fully exploited yet. Especially it is
interesting to study the higher order corrections, which are different for
$J/\psi$ and $\psi^{\prime}$. In this way the natural violation of the “13%
rule” can be related to relativistic corrections in NRQCD [13].
In fact, already an order $v^{2}$ nonrelativistic QCD matrix elements
$\left\langle
0\right|\chi^{{\dagger}}\boldsymbol{\sigma}\cdot\boldsymbol{\epsilon\
}{\boldsymbol{\nabla}}^{2}\psi\left|\psi(n,\boldsymbol{\epsilon})\right\rangle$
have very different values for $J/\psi$ and $\psi^{\prime}$, that was noticed
already long time ago [14]. Recently, the relativistic corrections to
exclusive $\psi(n)\to p\bar{p}$ decays have been studied in Ref.[15]. It is
found that corrections of relative order $v^{2}$ are large and comparable with
the leading-order contribution. This effect is closely related to the
structure of the integrand in the collinear convolution integral describing
the decay amplitude. This observation holds for both states $J/\psi$ and
$\psi^{\prime}$ but for excited state the absolute effect is larger because
the corresponding matrix element is larger. The similar mechanism may also be
relevant for other hadronic decay channels including $\psi(n)\to\rho\pi$
decays.
Therefore, the main purpose of this paper is to calculate the relativistic
corrections to $\psi(n)\to\rho\pi$ and to study their numerical effect. As a
first step in this direction we will calculate the correction of relative
order $v^{2}$ combining NRQCD expansion with the leading-order collinear
expansion. We will use the NRQCD projection technique developed in Refs.[16,
17, 18, 19], which is also effective for calculations of exclusive amplitudes.
This technique also allows one to resum a part of higher order corrections,
which are related to the corrections to quark-antiquark wave function in the
potential model [19]. Such consideration is also useful providing an estimate
of possible effects from higher order contributions.
## 2 Relativistic corrections to $\psi(n)\to\rho\pi$ and
$\psi(n)\to\gamma\pi$ decays
To describe the $J/\psi(P)\rightarrow\rho(p)\pi(p^{\prime})$ decay amplitude
we use the charmonium rest frame and assume that outgoing momenta are directed
along $z$-axis. The amplitude is defined as
$\left\langle\rho(p)\pi(p^{\prime})\right|iT\left|\psi(n)\right\rangle=i(2\pi)^{4}\delta(p+p^{\prime}-P)i\epsilon_{\alpha\beta\mu\nu}\epsilon^{\alpha}e^{\ast\beta}\frac{p^{\prime\mu}p^{\nu}}{(pp^{\prime})}A_{\rho\pi},$
(2)
where $\epsilon$ and $e^{\ast}$ denotes polarisation vectors of $\psi$ and
$\rho$-meson, respectively. The amplitude $A_{\rho\pi}$ can be described as a
superposition of a hard kernel with nonperturbative matrix elements describing
the long distance coupling with hadronic states. In order to calculate the
hard kernel, we perform an NRQCD matching, which is combined with the
collinear light-cone expansion for the light quarks. This technique allows one
to perform the matching at the amplitude level and to find the hard kernels
for corrections associated with the specific set of higher order NRQCD matrix
elements [19]
$\left\langle{v}^{2n}\right\rangle=\frac{\left\langle
0\right|\chi^{{\dagger}}\boldsymbol{\sigma}\cdot\boldsymbol{\epsilon\
}\left(-\frac{i}{2}\overleftrightarrow{\boldsymbol{D}}\right)^{2n}\psi\left|\psi(n,\boldsymbol{\epsilon})\right\rangle}{m_{c}^{2n}\
\left\langle
0\right|\chi^{{\dagger}}\boldsymbol{\sigma}\cdot\boldsymbol{\epsilon}\psi\left|\psi(n,\boldsymbol{\epsilon})\right\rangle}\simeq\left\langle{v}^{2}\right\rangle^{n},$
(3)
where the last equality is valid up to corrections $\mathcal{O}(v^{2})$ [20].
The diagrams, which describe the decay amplitude are schematically shown in
Fig.1.
Figure 1: a) Typical diagrams describing the subprocess $Q\bar{Q}\rightarrow
VP$, where $V=\rho,\gamma$. The blobs denote the light-cone matrix elements,
see explanation in the text. b) An example of diagrams, describing the
contribution with the perturbative photon coupling.
The long distance hadronisation dynamics of outgoing mesons is described by
the twist-2 and twist-3 light-cone distribution amplitudes (LCDAs). Various
properties and models for required LCDAs can be found in Refs.[7, 8]. The
twist-2 light-cone matrix elements read 111For simplicity, we do not
explicitly show the gauge links in the light-cone operators assuming the
appropriate light-cone gauge.
$\left\langle\pi^{+}(p^{\prime})\right|\bar{u}(z_{1+})\hbox to0.0pt{\hbox
to5.00002pt{\hfil$\bar{n}$\hfil}\hss}/\gamma_{5}d(z_{2+})\left|0\right\rangle=-if_{\pi}\left(p^{\prime}\bar{n}\right)\
\int_{0}^{1}du\
e^{iu\left(p^{\prime}\bar{n})(z_{1}n\right)/2+i(1-u)\left(p^{\prime}\bar{n})(z_{2}n\right)/2}\
\phi_{2\pi}(u),$ (4)
$\left\langle\rho^{-}(p)\right|\bar{d}(z_{1-})\gamma_{\bot}^{\mu}\hbox
to0.0pt{\hbox
to6.00235pt{\hfil/\hfil}\hss}nu(z_{2-})\left|0\right\rangle=if_{\rho}^{\bot}e_{\bot}^{\ast\mu}(pn)\
\int_{0}^{1}dy\
e^{iy(pn)(z_{1}\bar{n})/2+i(1-y)\left(pn)(z_{2}\bar{n}\right)/2}\
\phi_{2\rho}^{\bot}(y),$ (5)
where we use auxiliary light-cone vectors
$\displaystyle n=(1,0,0,-1),\ \bar{n}=(1,0,0,1),\
g^{\mu\nu}_{\bot}=g^{\mu\nu}-\frac{1}{2}(n^{\mu}\bar{n}^{\nu}+n^{\nu}\bar{n}^{\mu}),$
(6) $\displaystyle
p^{\prime}=(p^{\prime}\bar{n})\frac{n}{2}+\frac{m^{2}_{\pi}}{(p^{\prime}\bar{n})}\frac{\bar{n}}{2},\
p=(pn)\frac{\bar{n}}{2}+\frac{m^{2}_{\rho}}{(pn)}\frac{n}{2},\
(p^{\prime}\bar{n})\sim(pn)\sim m_{c}.$ (7)
and the short notation for the arguments of quark fields
$q(z_{i+})\equiv q((z_{i}n)\bar{n}/2),\ q(z_{i-})\equiv\
q((z_{i}\bar{n})n/2).$ (8)
The required twist-3 three-particles LCDAs are defined as
$\left\langle\pi^{+}(p^{\prime})\right|\bar{u}(z_{1+})\hbox to0.0pt{\hbox
to5.00002pt{\hfil$\bar{n}$\hfil}\hss}/\gamma_{\bot}^{\mu}\gamma_{5}gG_{\bar{n}\mu}(z_{3+})d(z_{2+})\left|0\right\rangle=-2f_{3\pi}\left(p^{\prime}\bar{n}\right)^{2}\text{FT}\left[\phi_{3\pi}(u_{i})\right],$
(9) $\left\langle\rho^{-}(p,e)\right|\bar{d}(z_{1-})\hbox to0.0pt{\hbox
to6.00235pt{\hfil/\hfil}\hss}ngG_{\mu
n}(z_{2-})u(z_{3-})\left|0\right\rangle=-\
f_{\rho}m_{\rho}(pz)^{2}e_{\bot}^{\ast\mu}\
\text{FT}\left[\phi_{3\rho}(y_{i})\right],$ (10)
$\left\langle\rho^{-}(p,e)\right|\bar{d}(z_{1})\hbox to0.0pt{\hbox
to6.00235pt{\hfil/\hfil}\hss}n\gamma_{5}g\tilde{G}_{\mu n}(\lambda
z)u(-z)\left|0\right\rangle=-if_{\rho}m_{\rho}\zeta_{3}(pz)^{2}e_{\bot}^{\ast\mu}\
\text{FT}\left[\tilde{\phi}_{3\rho}(y_{i})\right],$ (11)
where $G_{\mu n}\equiv G_{\mu\nu}n^{\nu}$ and
$\text{FT}\left[f(u_{i})\right]=\int Du_{i}\
e^{iu_{1}\left(p^{\prime}\bar{n})(z_{1}n\right)/2+iu_{2}\left(p^{\prime}\bar{n})(z_{2}n\right)/2+iu_{3}\left(p^{\prime}\bar{n})(z_{3}n\right)/2}\phi_{3\pi}(u_{1},u_{2},u_{3}),$
(12)
with
$Du_{i}=du_{1}du_{2}du_{3}\delta(1-u_{1}-u_{2}-u_{3}).$ (13)
The FT$\left[\phi_{3\rho}(y_{i})\right]$ is defined analogously but with
$y_{i}(pn)(z_{i}\bar{n})$ in the Fourier transformation. The normalisation
constants $f_{\pi,\rho}$, $\zeta_{3},\ f_{3\pi}$ and models for various LCDAs
will be discussed below.
The expression for the amplitude can be written as
$A_{\rho\pi}=\left\langle
0\right|\chi^{{\dagger}}\boldsymbol{\sigma}\cdot\boldsymbol{\epsilon}\psi\left|\psi(n,\boldsymbol{\epsilon})\right\rangle\frac{\sqrt{2M_{\psi}}}{2E}\
\frac{1}{4\pi}\int d\Omega\ \text{Tr}\left[\Pi_{1}\hat{A}_{Q}\right],$ (14)
where $\hat{A}_{Q}$ describes subprocess $Q\bar{Q}\rightarrow\rho\pi$ with the
quark-antiquark pair in the initial state. The heavy quark projector on the
triplet spin state $\Pi_{1}$ reads [19]
$\Pi_{1}=\frac{-1}{2\sqrt{2}E(E+m)}\left(\frac{1}{2}\hbox to0.0pt{\hbox
to7.80904pt{\hfil/\hfil}\hss}P+m+\hbox to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/\right)\frac{\hbox to0.0pt{\hbox
to7.80904pt{\hfil/\hfil}\hss}P+2E}{4E}\hbox to0.0pt{\hbox
to5.00002pt{\hfil$\epsilon$\hfil}\hss}/\left(\frac{1}{2}\hbox to0.0pt{\hbox
to7.80904pt{\hfil/\hfil}\hss}P-m-\hbox to0.0pt{\hbox
to5.00002pt{\hfil$q$\hfil}\hss}/\right)\otimes\frac{\mathbf{1}}{\sqrt{N_{c}}},$
(15)
and is normalised as
$\displaystyle\text{Tr}\left[\Pi_{1}\Pi_{1}^{{\dagger}}\right]=4E^{2},$ (16)
where $E$ is the heavy quark energy $p_{Q}=(E,\boldsymbol{q})$,
$p_{\bar{Q}}=(E,-\boldsymbol{q})$ and $E=\sqrt{m_{c}^{2}+\boldsymbol{q}^{2}}$.
The integration $d\Omega$ over the angles of the relative momentum
$\boldsymbol{q}$ in Eq.(14) is used to get the state with $L=0$. Therefore the
relevant amplitude $\hat{A}_{Q}$ is the function of relative momentum square
$\boldsymbol{q}^{2}$ only, which is substituted $\boldsymbol{q}^{2}\rightarrow
m_{c}^{2}\left\langle{v}^{2}\right\rangle$ in the final expression (14),
various technical details concerning NRQCD matching can be found in Refs.[17,
19].
Calculation of the diagrams as in Fig.1 gives
$A_{\rho^{-}\pi^{+}}=-\left\langle
0\right|\chi^{{\dagger}}\boldsymbol{\sigma}\cdot\boldsymbol{\epsilon}\psi\left|\psi(^{3}S_{1})\right\rangle\sqrt{2M_{\psi}}\left(\pi\alpha_{s}\right)^{2}\frac{10}{27}\left(1+\frac{m_{c}}{E}\right)\frac{f_{\rho}^{\bot}~{}f_{3\pi}}{\left[4E^{2}\right]^{2}}\left(J_{\pi}+\frac{\
f_{\rho}m_{\rho}\zeta_{3}\ f_{\pi}}{f_{\rho\bot}f_{3\pi}}\ J_{\rho}\right),$
(17)
where the dimensionless collinear convolution integrals $J_{\pi}$ and
$J_{\rho}$ describe contributions with twist-3 $\pi$\- and $\rho$-LCDAs,
respectively. These integrals also depend on the NRQCD parameter
$\left\langle{v}^{2}\right\rangle$. In the leading-order limit
$\left\langle{v}^{2}\right\rangle\rightarrow 0$, $E\rightarrow m_{c}^{2}$
Eq.(17) reproduces the result from Ref.[4]
$A_{\rho^{-}\pi^{+}}^{\text{lo}}=-\left\langle
0\right|\chi^{{\dagger}}\boldsymbol{\sigma}\cdot\boldsymbol{\epsilon}\psi\left|\psi(n,\boldsymbol{\epsilon})\right\rangle\sqrt{2M_{\psi}}\left(\pi\alpha_{s}\right)^{2}\frac{20}{27}\frac{f_{\rho}^{\bot}f_{3\pi}}{\left[4m_{c}^{2}\right]^{2}}\left(J_{\pi}^{\text{lo}}+\frac{\
f_{\rho}m_{\rho}\zeta_{3}f_{\pi}}{f_{\rho\bot}f_{3\pi}}\
J_{\rho}^{\text{lo}}\right),$ (18)
where
$\displaystyle J_{\pi}^{\text{lo}}$ $\displaystyle=\int Du_{i}\
\frac{\phi_{3\pi}(u_{i})}{u_{1}u_{2}u_{3}}\int_{0}^{1}dy\
\frac{\phi_{2\rho}^{\bot}(y)}{1-y}\frac{2u_{1}}{\left(y\bar{u}_{2}+u_{2}\bar{y}\right)\left(yu_{1}+\bar{y}\bar{u}_{1}\right)},$
(19) $\displaystyle J_{\rho}^{\text{lo}}$ $\displaystyle=\int_{0}^{1}du\
\frac{\phi_{2\pi}(u)}{1-u}\int Dy_{i}\
\frac{\left(\phi_{3\rho}+\tilde{\phi}_{3\rho}\right)(y_{i})}{y_{1}y_{2}y_{3}}\frac{1}{y_{2}\bar{u}+u\bar{y}_{2}},$
(20)
with $\bar{x}=1-x$. The analytical expressions for the integrals
$J_{\pi,\rho}$ in Eq.(17) are somewhat lengthy and presented in Appendix.
In order to estimate these integrals we use the following models of LCDAs
$\phi_{2\rho}^{\bot}(y)=6y(1-y)\left(1+a_{2\rho}\ C_{2}^{3/2}(2y-1)\right),\ $
(21) $\phi_{2\pi}(u)=6u(1-u)\left(1+a_{2}^{\pi}C_{2}^{3/2}(2u-1)\right),$ (22)
$\phi_{3\rho}(y_{i})=360y_{1}y_{2}y_{3}^{2}(y_{1}-y_{2})\omega_{3\rho},$ (23)
$\tilde{\phi}_{3\rho}(y_{i})=360y_{1}y_{2}y_{3}^{2}\left(1+\frac{\tilde{\omega}_{3\rho}}{\zeta_{3}}\frac{1}{2}\left(7y_{3}-3\right)\right),$
(24) $\ \
\phi_{3\pi}(u_{i})=360u_{1}u_{2}u_{3}^{2}\left(1+\omega_{3\pi}\frac{1}{2}(7\alpha_{3}-3)\right).$
(25)
The different nonperturbative moments, which enter in the definitions (4)-(11)
and (21)-(25), were estimated in Refs.[4, 7, 8]. Their values are summarised
in Table 1. In the numerical estimates we fix for the factorisation scale the
value $\mu=2$ GeV and use $\alpha_{s}\simeq 0.30$.
Table 1: The values of the moments, which parametrise the hadronic LCDAs. All values are given at the scale $\mu=2$ GeV. For the pion moments, the values are taken from Ref.[7], for the $\rho$-meson from Ref.[8]. $f_{\pi},$MeV | $f_{\rho}$, MeV | $f_{\rho}^{\bot}$, MeV | $a_{2\pi}$ | $a_{2\rho}$ | $f_{3\pi},$ GeV2 | $\zeta_{3\rho}$ | $\omega_{3\rho}$ | $\tilde{\omega}_{3\rho}$ | $\omega_{3\pi}$
---|---|---|---|---|---|---|---|---|---
$131$ | $216$ | $143$ | $0.19$ | $0.11$ | $0.31\times 10^{-2}$ | $0.02$ | $0.09$ | $-0.04$ | $-1.1$
All the convolution integrals calculated with the models (21)-(25) are well
defined, which confirms that collinear factorisation is also valid beyond the
leading-order approximation.
As a first step of the numerical analysis let us consider the the leading-
order estimate for the branching ratio of $J/\psi$. For that purpose we use
the estimates for the NRQCD matrix element obtained in Ref.[18]
$\left|\left\langle
0\right|\chi^{{\dagger}}\boldsymbol{\sigma}\cdot\boldsymbol{\epsilon}\psi\left|J/\psi\right\rangle\right|^{2}\simeq
0.440.$ (26)
For the various masses in Eq.(18) we use $M_{\psi}=3.1\ $GeV, $m_{\rho}=775\
$MeV, for the pole $c$-quark mass $m_{c}=1.4\ $GeV and for the total width
$\Gamma_{J/\psi}=93\ $KeV [1]. Then for the sum of all final states
$\rho^{\pm}\pi^{\mp}$ and $\rho^{0}\pi^{0}$ we obtain
$\text{Br}[J/\psi\rightarrow\rho\pi]_{\text{lo}}\simeq 1.0\%,$ (27)
which is somewhat smaller then the corresponding experimental value
$1.69(15)\%.$ This updated result confirm the conclusion of Ref.[4], that the
LO NRQCD approximation works sufficiently well for the $J/\psi$ decay.222We
assume that the difference about factor two is not a large discrepancy taking
into account various uncertainties from scale setting, pole mass $m_{c}$,
etc., which we do not consider now. On the other hand this approximation can
not describe branching ratio $\psi^{\prime}\to\rho\pi$.
Consider now the effect provided by the relativistic corrections in Eq.(17).
The one part is provided by the resummation of relativistic corrections in the
factor $E=m_{c}^{2}\sqrt{1+\left\langle{v}^{2}\right\rangle}$ in Eq.(17). This
effect can be understood as transition from the scale $4m_{c}^{2}$ to the
scale $M_{\psi}^{2}\simeq 4m_{c}^{2}(1+\left\langle{v}^{2}\right\rangle)$.
These corrections reduce the ratio $Q_{\rho\pi}$ due to the factor
$(1+\left\langle{v}^{2}\right\rangle_{J/\psi})^{2}/(1+\left\langle{v}^{2}\right\rangle_{\psi^{\prime}})^{2}\sim
M_{\psi}^{4}/M_{\psi^{\prime}}^{4}\sim 0.50$. However, this can not explain
the very small value $Q_{\rho\pi}$ in Eq.(1).
The second effect of the relativistic corrections is associated with the
modification of the hard kernels in the convolution integrals $J_{\rho,\pi}$.
Because these integrals depend on meson LCDAs, the resulting effect of the
relativistic corrections is also sensitive to hadronic nonperturbative
structure.
For the numerical calculation we use for $J/\psi$ the estimate from Ref.[18]
$\left\langle{v}^{2}\right\rangle_{J/\psi}\approx 0.225,$ (28)
and for the excite state $\psi^{\prime}$ we apply the following estimate
$\left\langle{v}^{2}\right\rangle_{\psi^{\prime}}=\frac{M_{\psi^{\prime}}-M_{J/\psi}+E_{1}}{m_{c}}\approx
0.64,$ (29)
where $E_{1}=\left\langle{v}^{2}\right\rangle_{J/\psi}m_{c}\simeq 315\ $MeV is
the binding energy for $J/\psi$. The resulting value of
$\left\langle{v}^{2}\right\rangle_{\psi^{\prime}}$ is much larger than
$\left\langle{v}^{2}\right\rangle_{J/\psi}$, which can have a significant
numerical effect and, therefore, affect the value of $Q_{\rho\pi}$.
The given calculation of the relativistic corrections is complete at the
relative order $v^{2}$ only. The resummation of higher orders
$\left\langle{v}^{2}\right\rangle^{n}$ with $n>2$ describes the part of the
relativistic corrections associated with the quark-antiquark wave function
only [19]. We use this approximation in order to study a possible effect from
higher-order contributions. Therefore, for the comparison, we present the
values of the integrals in Eq.(17) obtained in the leading-order approximation
$J^{\text{lo}}$ ($\left\langle{v}^{2}\right\rangle\rightarrow 0$), in the
next-to-leading approximation $J^{\text{nlo}}$, which takes into accont the
next-to-leading correction
$J^{\text{nlo}}=J^{\text{lo}}+\left\langle{v}^{2}\right\rangle J^{(1)}$, and
the integral $J$, which includes all powers
$\left\langle{v}^{2}\right\rangle^{n}$ :
$J=J^{\text{lo}}+\sum\left\langle{v}^{2}\right\rangle^{n}J^{(n)}$.
The total integral in Eq.(17) is described by the sum of two contributions
with the different LCDAs
$J_{\rho\pi}=J_{\pi}+\frac{\ f_{\rho}m_{\rho}\zeta_{3}\
f_{\pi}}{f_{\rho\bot}f_{3\pi}}\ J_{\rho},$ (30)
where, schematically, $J_{\pi}=\phi_{3\pi}\ast T_{\pi}\ast\phi_{2\rho}^{\bot}$
and $J_{\rho}=\phi_{3\rho}\ast
T_{\rho}\ast\phi_{2\pi}+\tilde{\phi}_{3\rho}\ast\tilde{T}_{\rho}\ast\phi_{2\pi}$
(the asterisk denotes the convolution integrals, $T_{\pi,\rho}$ are the hard
kernels). Using parameters from Table 1 one finds
$\frac{\ f_{\rho}m_{\rho}\zeta_{3}\ f_{\pi}}{f_{\rho\bot}f_{3\pi}}\approx
0.99.$ (31)
Therefore the normalisation couplings in the definitions (4)-(11) do not
provide any numerical difference between the two terms in Eq.(30). The results
for convolution integrals (30) are presented in Table 2
Table 2: Numerical result for the convolution integrals $J_{\rho\pi}$ | $J_{\rho\pi}^{\text{lo}}$ | $J_{\rho\pi}^{\text{nlo}}/J_{\rho\pi}^{\text{lo}}$ | $\ J_{\rho\pi}/J_{\rho\pi}^{\text{lo}}$
---|---|---|---
$J/\psi$ | $630$ | $0.53$ | $0.45$
$\psi^{\prime}$ | $630$ | $-0.46$ | $-0.65$
The effect of the relativistic corrections is negative and the values of the
LO integrals are substantially reduced. Notice that neglecting the higher-
order corrections in $v^{2}$ in the square of the integral, one gets in case
of $J/\psi$ the strong cancellation
$\displaystyle|J_{\rho\pi}^{\text{nlo}}|^{2}=(J_{\rho\pi}^{\text{lo}})^{2}(2J_{\rho\pi}^{\text{nlo}}/J_{\rho\pi}^{\text{lo}}-1)+\mathcal{O}(v^{4})\simeq
0.06(J_{\rho\pi}^{\text{lo}})^{2}.$ (32)
Therefore we assume that it is better to take the large NLO correction
exactly, i.e. do not expanding the square of the integral in powers of
$v^{2}$. At the same time the numerical effect from other higher order
corrections is already much smaller.
For $\psi^{\prime}\to\rho\pi$ the numerical effect is bigger because
$\left\langle\boldsymbol{v}^{2}\right\rangle_{\psi^{\prime}}$ is larger. One
can also see that the dominant part of the correction is also provided by the
contribution of relative order $v^{2}$, which is obtained exactly in this
calculation. The numerical dominance of this correction can be explained by
the numerical enhancement of the corresponding convolution integrals in the
same way as for the baryon decays [15].
Let us assume that the relativistic correction of order $v^{2}$ provides the
dominant numerical effect for $J/\psi$ and $\psi^{\prime}$ states. Then, this
allows one to suggest a possible explanation of the small
$\psi^{\prime}\to\rho\pi$ width, which explains the ”$\rho\pi$-puzzle”. The
NRQCD description of decay amplitudes also involves the
$\mathcal{O}(\alpha_{s})$ NLO QCD radiative correction, which can also provide
substantial numerical effect. Usually this contribution is considered to be of
the same order as relativistic corrections of relative order $v^{2}$. The
value of radiative corrections for $J/\psi$ and $\psi^{\prime}$ states is the
same except the NRQCD matrix element. Therefore, if the radiative
$\mathcal{O}(\alpha_{s})$ correction is positive and large enough in order to
compensate the negative contribution $J^{\text{nlo}}$ for $\psi^{\prime}$ then
this naturally explains the small width for $\psi^{\prime}$. On the other
hand, such positive contribution will improve the description of
$J/\psi\to\rho\pi$ increasing the value of the convolution integral, i.e. in
this case the $\mathcal{O}(\alpha_{s})$ also compensates the negative effect
of the relativistic correction. This is in agreement with the observation that
the leading-order description $J/\psi\to\rho\pi$ provides qualitatively good
estimate.
Potentially this analysis can also be applied for other meson decays. The good
feature of the collinear factorisation is that the hadronic nonperturbative
content is described in terms of universal process independent LCDAs. Many of
these functions were already studied in the literature. Even if the hard
kernels are the same the differences in the models for LCDAs can affect the
numerical balance and change the value $Q_{hh^{\prime}}$. Consider, for
example, the decay of S-wave charmonia into $\gamma\pi^{0}$ final state. In
this case the decay amplitude is described by the same diagrams as in Fig.1(a)
but with the photon LCDAs instead of $\rho$-meson. These diagrams describe the
photon as a hadron, i.e. such contributions are sensitive to the
nonperturbative components of the photon wave function. The contribution with
the perturbative photon coupling appear from the diagrams Fig.1(b) only and
therefore they are suppressed by electromagnetic coupling $\alpha$ or by
additional QCD coupling $\alpha_{s}$. In such situation the contributions with
nonperturbative photon can provide a sizeable impact, see e.g. discussion in
Ref.[21].
The data for the branching fractions $\psi(n)\to\gamma\pi$ are known [1]
$\text{Br}[J/\psi\rightarrow\gamma\pi^{0}]=3.56(17)\times 10^{-5},\ \
\text{Br}[\psi^{\prime}\rightarrow\gamma\pi^{0}]=1.04(22)\times 10^{-6},$ (33)
which yields
$Q_{\gamma\pi}\simeq 0.03.$ (34)
The width $\Gamma[J/\psi\rightarrow\gamma\pi^{0}]$ can be well estimated using
data for $\Gamma[J/\psi\rightarrow\rho\pi^{0}]$ and VDM model [4].333 We
assume that the contribution of $J/\psi\to\gamma^{*}\to\gamma\pi$ is
subleading, in contrast to the analysis in Ref.[4]. We guess that this
contribution is overestimated in [4] due to the specific model of the pion
LCDA .. This indirectly support the picture with the dominant contribution
from the non-perturbative photon coupling. However, the ratio $Q_{\gamma\pi}$
is about an order of magnitude larger than $Q_{\rho\pi}$.
We will use the models for the photon LCDAs from Ref.[21]. The twist-2 light-
cone matrix element reads
$\left\langle\gamma(q,e)\right|\bar{q}(z_{1-})\hbox to0.0pt{\hbox
to5.00002pt{\hfil$\bar{n}$\hfil}\hss}/\gamma_{\bot}^{\mu}q(z_{2-})\left|0\right\rangle=ie_{q}e\
f_{\gamma}e_{\bot}^{\ast\mu}(qn)\ \int_{0}^{1}dy\
e^{iy(pn)(z_{1}\bar{n})/2+i(1-y)\left(pn)(z_{2}\bar{n}\right)/2}\
\phi_{2\gamma}^{\bot}(y),$ (35)
where $e_{u}=2/3$, $e_{d}=-1/3$, electric charge $e=\sqrt{4\pi\alpha}$. The
model for $\phi_{2\gamma}^{\bot}$ reads [21]
$\phi_{2\gamma}^{\bot}(y)\simeq 6y(1-y),\ \ \ \
f_{\gamma}(2\text{GeV})\simeq-47\ \text{MeV}.$ (36)
Twist-3 DAs matrix elements are defined as
$\left\langle\gamma(p)\right|\bar{q}(z_{1-})\hbox to0.0pt{\hbox
to6.00235pt{\hfil/\hfil}\hss}ngG_{\mu
n}(z_{3-})q(z_{2-})\left|0\right\rangle=-\ e_{q}e\
f_{3\gamma}(qn)^{2}\varepsilon_{\bot}^{\ast\mu}\
\text{FT}\left[\phi_{3\gamma}(y_{i})\right],$ (37)
$\left\langle\gamma(p)\right|q(z_{1-})\hbox to0.0pt{\hbox
to6.00235pt{\hfil/\hfil}\hss}n\gamma_{5}g\tilde{G}_{\mu
n}(z_{2-})q(z_{3-})\left|0\right\rangle=-ie_{q}e\
f_{3\gamma}(qn)^{2}\varepsilon_{\bot}^{\ast\mu}\
\text{FT}\left[\tilde{\phi}_{3\gamma}(y_{i})\right],$ (38)
where Fourier transformation is the same as in Eq.(12). The models for twist-3
LCDAs read [21]
$\phi_{3\gamma}(y_{i})=360y_{1}y_{2}y_{3}^{2}\left(y_{1}-y_{2}\right)\
\omega_{3\gamma},\ \ $ (39)
$\tilde{\phi}_{3\gamma}(y_{i})=360y_{1}y_{2}y_{3}^{2}\left(1+\tilde{\omega}_{3\gamma}\frac{1}{2}\left(7y_{3}-3\right)\right),\
\ \tilde{\omega}_{3\gamma}\approx\tilde{\omega}_{3\rho}/\zeta_{3}.$ (40)
where
$f_{3\gamma}(2\text{GeV})=-0.32\times 10^{-2}\text{GeV}^{2},\ \
\omega_{3\gamma}\approx\omega_{3\rho},\ \ \ \
\tilde{\omega}_{3\gamma}\approx\tilde{\omega}_{3\rho}/\zeta_{3}.$ (41)
The $\gamma\pi$-decay amplitude can be obtained from Eq.(17) substituting
photon LCDAs instead of $\rho$-meson ones. The LO amplitude reads
$A_{\gamma\pi^{0}}^{\text{lo}}=-\left\langle
0\right|\chi^{{\dagger}}\boldsymbol{\sigma}\cdot\boldsymbol{\epsilon}\psi\left|\psi(n,\boldsymbol{\epsilon})\right\rangle\sqrt{2M_{\psi}}\left(\pi\alpha_{s}\right)^{2}\sqrt{2\pi\alpha}\frac{20}{27}\frac{f_{\gamma}~{}f_{3\pi}}{\left[4m_{c}^{2}\right]^{2}}\left(J_{\gamma\pi}^{\text{lo}}+\frac{\
f_{3\gamma}\ f_{\pi}}{f_{3\pi}f_{\gamma}}\ J_{\gamma\pi}^{\text{lo}}\right).$
(42)
The ratio of the normalisation couplings in Eq.(42) yields ($\mu=2$GeV)
$\frac{\ f_{3\gamma}\ f_{\pi}}{f_{3\pi}f_{\gamma}}\approx 2.92,$ (43)
which is different from the analogous ratio for the $\rho$-meson (31). The
leading-order numerical estimates gives
$\text{Br}[J/\psi\rightarrow\gamma\pi^{0}]_{\text{lo}}\simeq 3.82\times
10^{-5},$ (44)
which very well agrees with the data. The results for the total convolution
integral $J_{\gamma\pi}$ are presented in Table 3.
Table 3: Numerical result for the convolution integrals $J_{\gamma\pi}$ | $J_{\gamma\pi}^{\text{lo}}$ | $\ J_{\gamma\pi}^{\text{nlo}}/J_{\gamma\pi}^{\text{lo}}$ | $\ J_{\gamma\pi}/J_{\gamma\pi}^{\text{lo}}$
---|---|---|---
$J/\psi$ | $932$ | $0.62$ | $0.45$
$\psi^{\prime}$ | $932$ | $-0.49$ | $-0.74$
Comparing with the analogous results for the $\rho\pi$-channel one finds that
the both descriptions are qualitatively similar despite the different ratio
(43) and the differences between the LCDAs $\phi_{2\gamma}$ and
$\phi_{2\rho}^{\bot}$. Therefore one can again assume that
$\mathcal{O}(\alpha_{s})$ radiative corrections also play a crucial role in
the understanding of the value of the decay width $\gamma\pi$. Moreover, the
contributions with a perturbative photon, as in Fig. 1(b) can probably explain
the larger value of $Q_{\gamma\pi}$.
## 3 Conclusions
In conclusion, we calculated and investigated relativistic corrections to the
decay amplitudes $\psi(n)\to\rho\pi$ and $\psi(n)\to\gamma\pi$ within the pQCD
(NRQCD and collinear factorisation) framework. This calculation includes the
exact correction of relative order $v^{2}$ and subset of the higher order
corrections associated with the quark-antiquarks wave function. Numerical
estimates show that an order $v^{2}$ correction is large and give the dominant
numerical effect, which can be related to the structure of the collinear
integrals. If this observation is not affected by other higher order
relativistic corrections, then one has to consider the relative
$v^{2}$-contribution as a special case. The obtained relativistic corrections
are negative and large. In case $\psi^{\prime}\to\rho\pi$ the relative $v^{2}$
contribution is much larger than the leading-order one. The different effects
of relativistic corrections for $J/\psi\to\rho\pi$ and
$\psi^{\prime}\to\rho\pi$ suggest a possible explanation for the
$\rho\pi$-puzzle. If the QCD radiative correction is positive and large enough
then it interferes destructively with the relativistic correction for
$\psi^{\prime}\to\rho\pi$ giving the small branching fraction. At the same
time such radiative correction will improve the description of
$J/\psi\to\rho\pi$ reducing the negative effect of the relativistic
correction. Therefore, we believe that further investigation of relative order
$v^{4}$ corrections and QCD radiative corrections can help to clarify this
scenario.
We also expect that the same approach can be be used for an analysis other
hadronic decay channels. As a simplest example, the decay
$\psi(n)\to\gamma\pi$ was also considered. We studied the contribution, which
is given by similar diagrams but with nonperturbative photon instead of
$\rho$-meson. Despite the difference between twist-2 LCDAs for photon and
$\rho-$meson, the qualitative effect from the relativistic corrections is
quite similar to $\psi\to\rho\pi$, they are also large and negative. Therefore
one can guess that the similar scenario with radiative corrections is also
applicable here. The only difference with $\psi(n)\to\rho\pi$ is provided by
the contributions with perturbative photon coupling, which are suppressed by
$\mathcal{O}(\alpha_{s})$ or $\mathcal{O}(\alpha)$. Therefore, it can be that
these expects are responsible for the larger value of $Q_{\gamma\pi}$.
## 4 Appendix
Here we provide the analytical expressions for the intergrals $J_{\pi}$ and
$J_{\rho}$ introduced in Eq.(17). In order to simplify notations we use
$\left\langle v^{2}\right\rangle\equiv\boldsymbol{v}^{2},\ \
\delta=1-1/\sqrt{1+\boldsymbol{v}^{2}}.\ $ (45)
The first integral in Eq.(17) reads
$J_{\pi}\left(\boldsymbol{v}^{2}\right)=\int Du_{i}\
\frac{\phi_{3\pi}(u_{i})}{u_{1}u_{2}u_{3}}\int_{0}^{1}dy\
\frac{\phi_{2}^{\bot}(y)}{y\bar{y}}\left(\frac{2A_{\pi}}{D_{1}D_{3}}+\frac{B_{\pi}}{D_{1}D_{2}}\right),\
\ \bar{y}=1-y,$ (46)
where
$D_{i}=\delta_{i1}\
\left(y_{1}\bar{u}_{2}+\bar{y}_{1}u_{2}\right)+\delta_{i2}\
\left(y_{2}\bar{u}_{1}+\bar{y}_{2}u_{1}\right)+\delta_{i3}\ u_{3},$ (47)
with
$y_{1}=y,\ y_{2}=\bar{y}.$ (48)
The symbol $\delta_{ik}$ denotes the Kronecker delta.
The numerators $A_{\pi}$ and $B_{\pi}~{}$are given by the sums
$A_{\pi}=\sum_{k=0}^{4}f_{k}^{A}I_{k}[13],\ \ \
B_{\pi}=\sum_{k=0}^{4}f_{k}^{B}I_{k}[12],$ (49)
where
$I_{k}[ij]=\frac{1}{2}\int_{-1}^{1}d\eta\
\frac{\boldsymbol{v}^{k}\eta^{k}}{\left(1+\boldsymbol{v}\eta\
a_{i}\right)\left(1-\boldsymbol{v}\eta\
a_{j}\right)}=\frac{\boldsymbol{v}^{k}}{a_{i}+a_{j}}\sum_{n=0}^{\infty}\boldsymbol{v}^{n}\frac{a_{j}^{n+1}+(-1)^{n}a_{i}^{n+1}}{n+1+k}\frac{1}{2}\left[1+(-1)^{n+k}\right].\
\ $ (50)
with
$a_{j}=\delta_{1j}\left(1-\delta\right)\frac{y_{1}-u_{2}}{y_{1}\bar{u}_{2}+\bar{y}_{1}u_{2}}+\delta_{2j}\left(1-\delta\right)\frac{y_{2}-u_{1}}{y_{2}\bar{u}_{1}+\bar{y}_{2}u_{1}}-\delta_{3j}\left(1-\delta\right).$
(51)
The coefficients $f_{k}^{A,B}\equiv f_{k}^{A,B}(u_{i},y;\delta)$ in Eq.(49)
read
$f_{0}^{A}=\frac{\delta}{2}\left(3u_{3}-2-\delta\right),\ \ \ \
f_{1}^{A}=\frac{\delta}{2}\frac{\left(1-\delta\right)}{\left(2-\delta\right)}u_{3},$
(52)
$f_{2}^{A}=\frac{1}{2}\frac{\left(1-\delta\right)^{2}}{\left(2-\delta\right)^{2}}\left(4-3(2-\delta)u_{3}+2\delta(1-\delta)\right),$
(53)
$f_{3}^{A}=-\frac{1}{2}\frac{\left(1-\delta\right)^{3}}{\left(2-\delta\right)^{2}}u_{3},\
\
f_{4}^{A}=-\frac{1}{2}\frac{\left(1-\delta\right)^{4}}{\left(2-\delta\right)^{2}}.$
(54) $\displaystyle f_{0}^{B}$
$\displaystyle=u_{1}y_{1}+u_{1}y_{2}-\frac{\delta}{2}(u_{1}+u_{2}+y_{1}+y_{2}-\delta)$
(55)
$\displaystyle+\frac{\delta}{2-\delta}\left(u_{1}y_{1}+u_{1}y_{2}-(2-\delta)\left(u_{1}+u_{2}+y_{1}+y_{2}\right)+(2-\delta)^{2}\right),$
(56)
$f_{1}^{B}=\frac{1}{2}\frac{\left(1-\delta\right)}{\left(2-\delta\right)}\left\\{4(u_{1}y_{1}-u_{2}y_{2})+\delta(u_{1}+y_{1}-u_{2}-y_{2})\right\\},$
(57) $\displaystyle f_{2}^{B}$
$\displaystyle=\frac{1}{2}\frac{\left(1-\delta\right)^{2}}{\left(2-\delta\right)^{2}}\left\\{6\left(u_{1}+u_{2}+y_{1}+y_{2}\right)-2\left(u_{1}y_{1}+u_{1}y_{2}\right)-8\right.$
(58)
$\displaystyle\left.+\delta\left(4-3\left(u_{1}+u_{2}+y_{1}+y_{2}\right)\right)+2\delta(2-\delta)\right\\},$
(59)
$f_{3}^{B}=-\frac{1}{2}\frac{\left(1-\delta\right)^{3}}{\left(2-\delta\right)^{2}}\left(u_{1}+y_{1}-u_{2}-y_{2}\right),\
\
f_{4}^{B}=-\frac{1}{2}\frac{\left(1-\delta\right)^{4}}{\left(2-\delta\right)^{2}}.$
(60)
The $\rho$-meson integral in Eq.(17) reads
$J_{\rho}=\int_{0}^{1}du\ \frac{\phi_{2\pi}(u)}{u\bar{u}}\int Dy_{i}\ \
\frac{1}{y_{1}y_{2}y_{3}}\left(\frac{2A_{\rho}}{y_{3}D_{2}}+\frac{B_{\rho}}{D_{1}D_{2}}\right).$
(61)
The numerators $A_{\rho}$ and $B_{\rho}$ can be written as
$A_{\rho}=\phi_{3\rho}(y_{i})\sum_{k=0}^{4}\left[f_{k}^{A}\right]I_{k}[23]+\tilde{\phi}_{3\rho}(y_{i})\sum_{k=0}^{4}\left[\tilde{f}_{k}^{A}\right]I_{k}[23],$
(62)
$B_{\rho}=\phi_{3\rho}(y_{i})\sum_{k=0}^{4}\left[f_{k}^{B}\right]I_{k}[12]+\tilde{\phi}_{3\rho}(y_{i})\sum_{k=0}^{4}\left[\tilde{f}_{k}^{B}\right]I_{k}[12],$
(63)
where the integrals $I_{k}$ are defined in Eq.(50) with a bit different
combination $a_{j}$
$a_{j}=\delta_{1j}\left(1-\delta\right)\frac{y_{1}-u_{2}}{y_{1}\bar{u}_{2}+\bar{y}_{1}u_{2}}+\delta_{2j}\left(1-\delta\right)\frac{y_{2}-u_{1}}{y_{2}\bar{u}_{1}+\bar{y}_{2}u_{1}}+\delta_{3j}\left(1-\delta\right)\
,$ (64)
and we again use for the two-particle LCDA $\ u_{1}=u$, $u_{2}=1-u$.
The coefficients $f_{k}^{A,B}$ and $\tilde{f}_{k}^{A,B}$ defined in Eqs.(62)
and (63) read
$\displaystyle f_{0}^{A}$
$\displaystyle=\frac{1}{4}\left(u_{1}(2y_{3}-\delta)+\delta\left(\delta-
y_{2}-y_{3}\right)\right)+\frac{\delta^{2}}{2}$ (65) $\displaystyle\ \ \ \ \ \
\ \ \ \
+\frac{1}{4}\frac{\delta}{(2-\delta)}\left(u_{1}\left(6+4y_{3}-3\delta\right)+\left(2-\delta\right)\left(3y_{2}+2y_{3}-2-3\delta\right)\right),$
(66) $\displaystyle f_{1}^{A}$
$\displaystyle=-\frac{1}{4}\frac{1-\delta}{2-\delta}\left(4u_{1}+\delta\right)y_{3},$
(67)
$f_{2}^{A}=-\frac{1}{4}\frac{(1-\delta)^{2}}{(2-\delta)^{2}}\left(2u_{1}y_{3}+(2-\delta)\left(2u_{1}+2y_{2}+y_{3}\right)-2(1-\delta)(2-\delta)\right),$
(68) $f_{3}^{A}=\frac{1}{4}\frac{(1-\delta)^{3}}{(2-\delta)^{2}}y_{3},\ \
f_{4}^{A}=0.$ (69) $\displaystyle f_{0}^{B}$
$\displaystyle=-f_{2}^{B}=\frac{\delta}{4}\left(u_{1}-u_{2}-y_{1}+y_{2}\right),\
$ (70) $\displaystyle\ \ f_{1}^{B}$
$\displaystyle=-f_{3}^{B}=-\frac{\delta}{4}(1-\delta)\left(u_{1}+u_{2}-y_{1}-y_{2}\right),\
\ \ f_{4}^{B}=0.$ (71) $\displaystyle\tilde{f}_{0}^{A}$
$\displaystyle=\frac{1}{4}\left(u_{1}(2y_{3}-\delta)+\delta\left(\delta-
y_{2}-y_{3}\right)\right)$ (72)
$\displaystyle+\frac{1}{4}\frac{\delta}{(2-\delta)}\left(u_{1}\left(2+4y_{3}-\delta\right)+\left(2-\delta\right)\left(2+y_{2}-2y_{3}-3\delta\right)\right),$
(73)
$\tilde{f}_{1}^{A}=-\frac{1}{4}\frac{1-\delta}{2-\delta}\left(4u_{1}y_{3}+\delta\left(2u_{1}-2y_{2}+y_{3}\right)\right),$
(74)
$\tilde{f}_{2}^{A}=-\frac{1}{4}\frac{(1-\delta)^{2}}{(2-\delta)^{2}}\left(2u_{1}y_{3}-3y_{3}(2-\delta)+\delta(2-\delta)\right),$
(75)
$\tilde{f}_{3}^{A}=\frac{1}{4}\frac{(1-\delta)^{3}}{(2-\delta)^{2}}\left(2u_{1}-2y_{2}+y_{3}\right),\
\ \ \tilde{f}_{4}^{A}=-\frac{1}{2}\frac{(1-\delta)^{4}}{(2-\delta)^{2}}.$ (76)
$\tilde{f}_{0}^{B}=\frac{\delta}{4}\left(3\left(u_{1}+u_{2}+y_{1}+y_{2}\right)-4-2\delta\right),\
\ \
\tilde{f}_{1}^{B}=-\frac{\delta}{4}\frac{1-\delta}{2-\delta}\left(u_{1}-u_{2}+y_{1}-y_{2}\right),$
(77)
$\tilde{f}_{2}^{B}=\frac{1}{4}\frac{(1-\delta)^{2}}{(2-\delta)^{2}}\left(8-3\left(2-\delta\right)\left(u_{1}+u_{2}+y_{1}+y_{2}\right)+4\delta(1-\delta)\right),$
(78)
$\tilde{f}_{3}^{B}=\frac{1}{4}\frac{(1-\delta)^{3}}{(2-\delta)^{2}}\left(u_{1}-u_{2}+y_{1}-y_{2}\right),\
\ \ \tilde{f}_{4}^{B}=-\frac{1}{2}\frac{(1-\delta)^{4}}{(2-\delta)^{2}}.$ (79)
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|
JUNA Collaboration
# Deep underground laboratory measurement of 13C($\alpha$,$n$)16O in the
Gamow windows of the s- and i-processes
B. Gao Joint department for nuclear physics, Lanzhou University and Institute
of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China CAS Key
Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 73000, People’s Republic of
China School of Nuclear Science and Technology, University of Chinese Academy
of Sciences, Beijing 100049, People’s Republic of China T.Y. Jiao CAS Key
Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 73000, People’s Republic of
China School of Nuclear Science and Technology, University of Chinese Academy
of Sciences, Beijing 100049, People’s Republic of China Y.T. Li CAS Key
Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 73000, People’s Republic of
China School of Nuclear Science and Technology, University of Chinese Academy
of Sciences, Beijing 100049, People’s Republic of China H. Chen CAS Key
Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 73000, People’s Republic of
China School of Nuclear Science and Technology, University of Chinese Academy
of Sciences, Beijing 100049, People’s Republic of China W.P. Lin Key
Laboratory of Radiation Physics and Technology of the Ministry of Education,
Institute of Nuclear Science and Technology, Sichuan University, Chengdu
610064, China Z. An Key Laboratory of Radiation Physics and Technology of
the Ministry of Education, Institute of Nuclear Science and Technology,
Sichuan University, Chengdu 610064, China L.H. Ru CAS Key Laboratory of High
Precision Nuclear Spectroscopy, Institute of Modern Physics, Chinese Academy
of Sciences, Lanzhou 73000, People’s Republic of China Z.C. Zhang CAS Key
Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 73000, People’s Republic of
China School of Nuclear Science and Technology, University of Chinese Academy
of Sciences, Beijing 100049, People’s Republic of China X.D. Tang
<EMAIL_ADDRESS>Joint department for nuclear physics, Lanzhou University
and Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China CAS Key Laboratory of High Precision Nuclear Spectroscopy, Institute of
Modern Physics, Chinese Academy of Sciences, Lanzhou 73000, People’s Republic
of China School of Nuclear Science and Technology, University of Chinese
Academy of Sciences, Beijing 100049, People’s Republic of China X.Y. Wang
CAS Key Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 73000, People’s Republic of
China School of Nuclear Science and Technology, University of Chinese Academy
of Sciences, Beijing 100049, People’s Republic of China N.T. Zhang CAS Key
Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 73000, People’s Republic of
China School of Nuclear Science and Technology, University of Chinese Academy
of Sciences, Beijing 100049, People’s Republic of China X. Fang Sino-French
Institute of Nuclear Engineering and Technology, Sun Yat-sen University,
Zhuhai, Guangdong 519082, People’s Republic of China D.H. Xie Key Laboratory
of Radiation Physics and Technology of the Ministry of Education, Institute of
Nuclear Science and Technology, Sichuan University, Chengdu 610064, China
Y.H. Fan CAS Key Laboratory of High Precision Nuclear Spectroscopy, Institute
of Modern Physics, Chinese Academy of Sciences, Lanzhou 73000, People’s
Republic of China School of Nuclear Science and Technology, Lanzhou
University, Lanzhou 730000, China L. Ma Key Laboratory of Radiation Physics
and Technology of the Ministry of Education, Institute of Nuclear Science and
Technology, Sichuan University, Chengdu 610064, China X. Zhang Key Laboratory
of Radiation Physics and Technology of the Ministry of Education, Institute of
Nuclear Science and Technology, Sichuan University, Chengdu 610064, China F.
Bai Key Laboratory of Radiation Physics and Technology of the Ministry of
Education, Institute of Nuclear Science and Technology, Sichuan University,
Chengdu 610064, China P. Wang Key Laboratory of Radiation Physics and
Technology of the Ministry of Education, Institute of Nuclear Science and
Technology, Sichuan University, Chengdu 610064, China Y.X. Fan Key Laboratory
of Radiation Physics and Technology of the Ministry of Education, Institute of
Nuclear Science and Technology, Sichuan University, Chengdu 610064, China G.
Liu Key Laboratory of Radiation Physics and Technology of the Ministry of
Education, Institute of Nuclear Science and Technology, Sichuan University,
Chengdu 610064, China H.X. Huang China Institute of Atomic Energy, Beijing
102413, People’s Republic of China Q. Wu Institute of Modern Physics,
Chinese Academy of Sciences, Lanzhou 730000, China Y.B. Zhu Institute of
Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China J.L. Chai
Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000,
China J. Q. Li Institute of Modern Physics, Chinese Academy of Sciences,
Lanzhou 730000, China L. T. Sun Institute of Modern Physics, Chinese Academy
of Sciences, Lanzhou 730000, China S. Wang Shandong Provincial Key
Laboratory of Optical Astronomy and Solar-Terrestrial Environment, Institute
of Space Sciences, Shandong University, Weihai 264209, China J.W. Cai CAS
Key Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 73000, People’s Republic of
China School of Nuclear Science and Technology, University of Chinese Academy
of Sciences, Beijing 100049, People’s Republic of China Y.Z. Li CAS Key
Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 73000, People’s Republic of
China School of Nuclear Science and Technology, University of Chinese Academy
of Sciences, Beijing 100049, People’s Republic of China J. Su China
Institute of Atomic Energy, Beijing 102413, People’s Republic of China Key
Laboratory of Beam Technology and Material Modification of Ministry of
Education, College of Nuclear Science and Technology, Beijing Normal
University, Beijing 100875, China H. Zhang China Institute of Atomic Energy,
Beijing 102413, People’s Republic of China Z. H. Li School of Nuclear
Science and Technology, University of Chinese Academy of Sciences, Beijing
100049, People’s Republic of China China Institute of Atomic Energy, Beijing
102413, People’s Republic of China Y. J. Li China Institute of Atomic
Energy, Beijing 102413, People’s Republic of China E. T. Li College of
Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060,
China C. Chen China Institute of Atomic Energy, Beijing 102413, People’s
Republic of China Y. P. Shen China Institute of Atomic Energy, Beijing
102413, People’s Republic of China G. Lian China Institute of Atomic Energy,
Beijing 102413, People’s Republic of China B. Guo China Institute of Atomic
Energy, Beijing 102413, People’s Republic of China X. Y. Li Key Laboratory
of Beam Technology and Material Modification of Ministry of Education, College
of Nuclear Science and Technology, Beijing Normal University, Beijing 100875,
China L. Y. Zhang Key Laboratory of Beam Technology and Material
Modification of Ministry of Education, College of Nuclear Science and
Technology, Beijing Normal University, Beijing 100875, China J. J. He Key
Laboratory of Beam Technology and Material Modification of Ministry of
Education, College of Nuclear Science and Technology, Beijing Normal
University, Beijing 100875, China Y. D. Sheng Key Laboratory of Beam
Technology and Material Modification of Ministry of Education, College of
Nuclear Science and Technology, Beijing Normal University, Beijing 100875,
China Y. J. Chen Key Laboratory of Beam Technology and Material Modification
of Ministry of Education, College of Nuclear Science and Technology, Beijing
Normal University, Beijing 100875, China L. H. Wang Key Laboratory of Beam
Technology and Material Modification of Ministry of Education, College of
Nuclear Science and Technology, Beijing Normal University, Beijing 100875,
China L. Zhang China Institute of Atomic Energy, Beijing 102413, People’s
Republic of China F. Q. Cao China Institute of Atomic Energy, Beijing
102413, People’s Republic of China W. Nan China Institute of Atomic Energy,
Beijing 102413, People’s Republic of China W. K. Nan China Institute of
Atomic Energy, Beijing 102413, People’s Republic of China G. X. Li China
Institute of Atomic Energy, Beijing 102413, People’s Republic of China N.
Song China Institute of Atomic Energy, Beijing 102413, People’s Republic of
China B. Q. Cui China Institute of Atomic Energy, Beijing 102413, People’s
Republic of China L. H. Chen China Institute of Atomic Energy, Beijing
102413, People’s Republic of China R. G. Ma China Institute of Atomic
Energy, Beijing 102413, People’s Republic of China Z. C. Zhang College of
Physics and Optoelectronic Engineering, Shenzhen University, Shenzhen 518060,
China S. Q. Yan China Institute of Atomic Energy, Beijing 102413, People’s
Republic of China J. H. Liao China Institute of Atomic Energy, Beijing
102413, People’s Republic of China Y. B. Wang China Institute of Atomic
Energy, Beijing 102413, People’s Republic of China S. Zeng China Institute
of Atomic Energy, Beijing 102413, People’s Republic of China D. Nan China
Institute of Atomic Energy, Beijing 102413, People’s Republic of China Q. W.
Fan China Institute of Atomic Energy, Beijing 102413, People’s Republic of
China N. C. Qi Yalong River Hydropower Development Company, Chengdu 610051,
China W. L. Sun Yalong River Hydropower Development Company, Chengdu 610051,
China X. Y. Guo Yalong River Hydropower Development Company, Chengdu 610051,
China P. Zhang Yalong River Hydropower Development Company, Chengdu 610051,
China Y. H. Chen Yalong River Hydropower Development Company, Chengdu
610051, China Y. Zhou Yalong River Hydropower Development Company, Chengdu
610051, China J. F. Zhou Yalong River Hydropower Development Company,
Chengdu 610051, China J. R. He Yalong River Hydropower Development Company,
Chengdu 610051, China C. S. Shang Yalong River Hydropower Development
Company, Chengdu 610051, China M. C. Li Yalong River Hydropower Development
Company, Chengdu 610051, China S. Kubono RIKEN Nishina Center, Wako, Saitama
351-0198, Japan Center for Nuclear Study, University of Tokyo, Wako, Saitama
351-0198, Japan W. P. Liu<EMAIL_ADDRESS>China Institute of Atomic Energy,
Beijing 102413, People’s Republic of China R.J. deBoer Department of Physics
and the Joint Institute for Nuclear Astrophysics, University of Notre Dame,
Notre Dame, IN 46556, USA M. Wiescher Department of Physics and the Joint
Institute for Nuclear Astrophysics, University of Notre Dame, Notre Dame, IN
46556, USA Wolfson Fellow of Royal Society, School of Physics and Astronomy,
University of Edinburgh, King’s Buildings, Edinburgh EH9 3FD, United Kingdom
M. Pignatari Konkoly Observatory, Research Centre for Astronomy and Earth
Sciences (CSFK), Eötvös Loránd Research Network (ELKH), Konkoly Thege Miklós
út 15-17, H-1121 Budapest, Hungary CSFK, MTA Centre of Excellence, Budapest,
Konkoly Thege Miklós út 15-17, H-1121, Hungary E. A. Milne Centre for
Astrophysics, Department of Physics and Mathematics, University of Hull, HU6
7RX, United Kingdom NuGrid Collaboration, http://nugridstars.org
###### Abstract
The 13C($\alpha$,$n$)16O reaction is the main neutron source for the slow-
neutron-capture (s-) process in Asymptotic Giant Branch stars and for the
intermediate (i-) process. Direct measurements at astrophysical energies in
above-ground laboratories are hindered by the extremely small cross sections
and vast cosmic-ray induced background. We performed the first consistent
direct measurement in the range of $E_{\rm c.m.}=$0.24 MeV to 1.9 MeV using
the accelerators at the China Jinping Underground Laboratory (CJPL) and
Sichuan University. Our measurement covers almost the entire i-process Gamow
window in which the large uncertainty of the previous experiments has been
reduced from 60% down to 15%, eliminates the large systematic uncertainty in
the extrapolation arising from the inconsistency of existing data sets, and
provides a more reliable reaction rate for the studies of the s- and
i-processes along with the first direct determination of the alpha strength
for the near-threshold state.
††preprint: APS/123-QED
Low-mass Asymptotic Giant Branch (AGB) stars are the primary sources of
elements above iron in the Galaxy via the activation of the slow-neutron-
capture process (s-process) Kobayashi _et al._ (2020). In these stars, the
bulk of the s-process abundances is created by neutrons from the the
13C($\alpha$,$n$)16O reaction at temperatures around $T_{9}$=0.1 in the
radiative 13C-pocket, located in the helium-rich layers, just below the
stellar envelope Gallino _et al._ (1998); Bisterzo _et al._ (2017). Here
$T_{9}$ is defined as a temperature divided by 109 Kelvin. The subsequent
neutron capture and $\beta$-decay processes transmute lighter elements into
heavier ones, with a major production efficiency between Sr and Pb depending
on the initial metallicity of the star Käppeler _et al._ (2011). In some AGB
simulations, it was also found that part of the 13C is still alive in the
13C-pocket at the onset of the convective thermal pulse: the remaining 13C is
mixed at the bottom of the He intershell region activating
13C($\alpha$,$n$)16O at He burning temperatures of approximately
$T_{9}$=0.2-0.25 Guo _et al._ (2012); Cristallo _et al._ (2018). This
scenario tends to be favored by a low 13C($\alpha$,$n$)16O rate Ciani _et
al._ (2021), and the anomalous activation of the 13C($\alpha$,$n$)16O reaction
together with the 22Ne($\alpha$,$n$)25Mg reaction at the bottom of the
convective thermal pulse. This may affect the s-process isotopic pattern near
the active s-process branching points Cristallo _et al._ (2018); Ciani _et
al._ (2021).
The 13C($\alpha$,$n$)16O reaction is also the main neutron source of the
intermediate process (i-process ) Cowan and Rose (1977), which matches the
puzzling abundances observed in some post AGB stars Herwig _et al._ (2011),
of a subset of carbon-enhanced metal poor (CEMP) stars Hampel _et al._
(2016), of presolar grains Jadhav _et al._ (2013) and stars in young open
clusters Mishenina _et al._ (2015). The i-process can be activated in
different stellar environments, including among other low-mass AGB stars
Cristallo _et al._ (2009); Hampel _et al._ (2016), super AGB stars Jones
_et al._ (2016) and post AGB stars Herwig _et al._ (2011), massive stars
Clarkson _et al._ (2018); Banerjee _et al._ (2018) and rapidly-accreting WDs
Denissenkov _et al._ (2019). In those models, a small amount of hydrogen is
ingested into the convective helium-burning zone underneath the envelope.
Hydrogen reacts with the primary product of He-burning 12C, making 13N that
will decay to 13C. The i-process is generated by the 13C($\alpha$,$n$)16O
activated at He-burning temperatures of around $T_{9}$=0.2 or above Herwig
_et al._ (2011). This results in a neutron density of around 10${}^{14}\sim$
1016 cm-3, which is significantly higher than typical values of the s-process
(10${}^{6}\sim$ 1010 cm-3). At least for one-dimensional models of metal-poor
low-mass AGB models, Cristallo _et al._ (2018) showed that after the hydrogen
ingestion, the 13C($\alpha$,$n$)16O reaction also becomes a relevant energy
source in the He shell: in their calculations even a factor of two variation
of the 13C($\alpha$,$n$)16O rate changes the i-process production by orders of
magnitude. Such an impact will need to be confirmed for different types of
stars by multi-dimensional hydrodynamics simulations, providing guidance for
how one-dimensional models should behave once hydrogen has been ingested in
the hotter He-burning regions Herwig _et al._ (2014); Denissenkov _et al._
(2019); Clarkson and Herwig (2021).
It is clear that the 13C($\alpha$,$n$)16O reaction rate is a fundamental
ingredient in the s- and i-process models, determining the neutron density and
the final isotopic production. A solid understanding of the reaction cross
section is needed at the associated Gamow energies of about
$E_{\mathrm{c.m.}}$ = 0.15 to 0.3 MeV and 0.2 to 0.54 MeV, respectively. The
reaction cross sections in these energy regions are strongly influenced by the
$\alpha$ cluster state near the separation threshold according to the theory
of Ikeda Ikeda _et al._ (1972). Descouvemont made a theoretical prediction of
the level structure using a microscopic generator-coordinate method (GCM) and
concluded that the theoretical S-factor below $E_{\mathrm{c.m.}}$ = 0.3 MeV
increases rapidly with respect to the extrapolations that ignored this
threshold state Descouvemont (1987). Reliable experimental measurements down
to energies below 0.3 MeV are called for to confirm the prediction.
Considerable efforts have been devoted to push the direct measurement of the
13C($\alpha$,$n$)16O reaction cross section ($\sigma$) Sekharan _et al._
(1967); Bair and Haas (1973); Davids (1968); Drotleff _et al._ (1993); Heil
_et al._ (2008) down to the stellar energies where $\sigma$ becomes extremely
small. Due to the vast cosmic background, direct measurements performed in
laboratories on the Earth’s surface stopped at energies above
$E_{\mathrm{c.m.}}$ = 0.27 MeV with a lower limit of $\sigma$ = 6(4)$\times$
$10^{-11}$ barn Drotleff _et al._ (1993), unable to effectively constrain the
crucial threshold state and provide a reliable extrapolation down to stellar
energies. Besides that, the extrapolation accuracy is further limited by large
discrepancies among those measurements deBoer _et al._ (2020); Brown _et
al._ (2018).
Recent breakthrough in the direct measurement of this reaction was reported by
the LUNA collaboration Ciani _et al._ (2021) at $E_{\mathrm{c.m.}}$ =
0.23-0.30 MeV, the upper range of the s-process Gamow window. However, they
had to rely on other existing data at higher energies, ANC of the threshold
state and the R-matrix analysis to determine the S-factor in the range of 0.15
to 0.5 MeV, most of which is inaccessible with the current LUNA facility. The
large discrepancies between Harissopulos _et al._ (2005) and other
measurements Drotleff _et al._ (1993); Heil _et al._ (2008) result in
$\sim$50% differences in their recommended upper and lower limits for the
reaction rate at $T_{9}$=0.1-0.3, leading to significant uncertainties in the
production yields of several important isotopes, such as 60Fe and 205Pb, by
using their AGB model.
In this paper, we report the first consistent direct measurement of the
13C($\alpha$,$n$)16O reaction over a wider energy range of $E_{\rm c.m.}$ =
0.24-1.9 MeV with improved precision. Our measurement reduces the 60% large
uncertainty down to 15% at the center of the Gamow window of the i-process,
provides the first direct determination of the alpha strength for the near
threshold state, and eliminates the large systematic uncertainty in the
extrapolation incurred by the discrepancy of the existing experiments. A new
reliable reaction rate is recommended based on our measurement.
The underground experiment was performed in the A1 hall of the China JinPing
underground Laboratory (CJPL) Cheng _et al._ (2017); Liu _et al._ (2016);
Zhang _et al._ (2021). High-intensity 4He+ and 4He2+ ions were extracted from
2.45- and 14.5-GHz electron-cyclotron-resonance (ECR) sources, respectively,
and accelerated by a 400-kV platform called Jinping Underground Nuclear
Astrophysics experimental facility (JUNA). The highest beam energy of 800 keV
was achieved by using 4He2+ ions, allowing comparisons with previous
measurements. The acceleration voltage was calibrated using the
12C(p,$\gamma$)13N, 27Al(p,$\gamma$)28Si, 11B(p,$\gamma$)12C and
14N(p,$\gamma$)15O reactions Wang (2021). The absolute beam energy was
determined to an accuracy of 0.5 keV with an energy spread of less than 0.2
keV Wang (2021). A 90∘ dipole magnet with a mass resolution of 250 was used
together with a set of analyzing slits to eliminate the H${}_{2}^{+}$/D+
contamination in the 4He2+ beam Chen _et al._ (2018). A clear separation of
4He2+ from the other impurities was observed at the slit position and the
count ratio of the inner and outer rings of the 3He detector array indicated
that no neutrons came from the deuterium impurity.
To avoid the source of systematic uncertainty incurred by target deterioration
in traditional thin target experiments, we used 2-mm thick 13C enriched
targets with a purity of 97%. The target was installed on a water-cooled
copper backing. On-target beam intensity of up to 2.5 particle mA, the highest
$\alpha$ beam intensity among the deep underground laboratories, was achieved.
The thick targets turned out to be very stable and only two targets were used
for the whole experiment. A cold trap was installed to reduce the natural
carbon buildup on the targets. For the 4He+ runs, the analyzing slits used in
the 4He2+ runs were removed to allow maximum transmission efficiency and
achieve higher beam intensities.
Neutrons from the 13C($\alpha$, $n$)16O reaction were detected by an array
consisting of 24 3He-filled proportional counters. By placing 35-cm thick 7%
borated polyethylene blocks and 1-mm thick cadmium sheets around the detection
array, a background of 4.7(2) events/hour was achieved, compared to 1238(11)
events/hour measured on the Earth’s surface. The detection efficiency of the
array was determined to be 26% for 2.5-MeV neutrons using the 51V(p,n)51Cr
reaction together with Geant4 simulations Li _et al._ (2022).
The thick target yield ${\rm Y(E)}$ of the 13C($\alpha$,$n$)16O reaction was
measured with beam energies of 0.3 $<$ $E_{\alpha}$ $<$ 0.785 MeV. The beam-
induced neutron background (BINB) was estimated by measuring the ${\rm Y(E)}$
at $E_{\alpha}$ = 0.25 MeV where the cross section of the 13C($\alpha$,$n$)16O
reaction is negligibly small and all neutrons detected above the environmental
background level should be attributed to the BINB. BINB was determined to be
0.05(8) events/Coulomb, consistent with zero.
The cross sections and the corresponding effective energies were extracted by
differentiating the thick target yield Spillane _et al._ (2007); Notani _et
al._ (2012). We repeatedly checked the neutron yields at 17 energy points and
the reproducibility was found to be 8% and 3% for the 4He+ and 4He2+ data
sets, respectively. This random error likely originates from the beam tuning
and the potential carbon build up. Therefore, we added this error
quadratically together with the statistical error.
Another thick target measurement was performed in the range of $E_{\rm c.m.}$
= 0.75 MeV to 1.9 MeV using the 4He+ beam from the 3 MV Tandetron at Sichuan
University Han _et al._ (2018) to resolve the discrepancies among the
S-factors at higher energies in the previous works Harissopulos _et al._
(2005); Bair and Haas (1973); Drotleff _et al._ (1993). The same detection
setup was used to minimize extra systematic uncertainties. The beam energy was
calibrated using the 7Li(p,n)7Be reaction, and confirmed by the narrow
resonances of the 13C($\alpha$,$n$)16O at $E_{\alpha}$=1055.6, 1334.7 and
1338.8 keV. A thin target measurement was also carried out in the range of
$E_{\mathrm{c.m.}}$=1.6 to 1.9 MeV using a 3.2$\mu$g/cm2-thick 13C target. The
thin-target data is normalized to the thick-target data. The reproducibility
of the thick-target and thin-target measurements are estimated to be 3% and
2%, respectively. This uncertainty is included with the statistical error as
discussed above.
By adopting a compiled angular distribution Brune (2021); Walton _et al._
(1957) in the Geant4 simulation, our efficiency has been corrected for angular
distribution effects. These effects were found to change the efficiency by
$\pm$2% at $E_{\mathrm{c.m.}}<$0.6 MeV. However, the efficiency at
$E_{\mathrm{c.m.}}\sim$0.9 MeV deviates from the nominal efficiency with an
isotropic distribution by $\sim$5%, which is larger than the statistical
uncertainties in the previous measurements Harissopulos _et al._ (2005); Bair
and Haas (1973). This deviation becomes even larger at the narrow resonances
Li _et al._ (2022). Such an effect was overlooked in these previous works.
Systematic uncertainties of our measurements at CJPL and SCU are estimated to
be 11%, which includes contributions from the beam current integration(5%),
detection efficiency (7%), angular distribution (2% for JUNA underground
measurement and 4% for SCU experiment), and stopping power (6%) Ziegler _et
al._ (2010).
Our S-factor is converted into the bare S-factors after correcting for the
screen effect using our fitted screening potential of $U_{e}$ = 0.78 keV
together with the previous measurements Drotleff _et al._ (1993); Heil _et
al._ (2008); Harissopulos _et al._ (2005); Ciani _et al._ (2021); Kellogg
_et al._ (1989). The results are presented in Fig. 1 . It can be seen that our
underground data cover the energy range from $E_{\rm c.m.}$ = 0.24 to 0.59
MeV, greatly overlapping with the astrophysical region of $E_{\rm c.m.}$ =
0.15 to 0.5 MeV with a statistical uncertainty better than 15%.
With the unique energy range and ultra low neutron background in the deep
underground lab, we are able to precisely measure the S-factor in the range of
astrophysical interest for i-process nucleosynthesis. The center of the Gamow
window for the i-process is located at 0.35 MeV, beyond the accessible energy
range of LUNA. The two extrapolation scenarios of LUNA using either the
normalization of Heil, or Drotleff Drotleff _et al._ (1993); Heil _et al._
(2008); Bair and Haas (1973) or that of Harissopulos _et al._ (2005) resulted
in their so-called best fit and “low LUNA” fit, respectively. To be on the
safe side, they defined the “low-LUNA” fit by taking the 95% confidence level
of the lower limit of the fit with the original Harissopulos data. These two
fits differ from each other by a factor of 2 at 0.35 MeV. Such a large
systematic uncertainty in their extrapolation is eliminated by our consistent
measurement, which rules out the lower normalization of Harissopulos _et al._
(2005). Drotleff _et al._ (1993) was the best measurement before ours at the
energy around 0.35 MeV. While our data above 0.4 MeV is in good agreement with
that of Drotleff, our data around 0.27 MeV are about 50% lower and disagree
with the upturning trend in this data set. The nearly 60% uncertainty in Ref.
Drotleff _et al._ (1993) within the Gamow window has been reduced to 15%.
Figure 1: The S-factor of the 13C($\alpha$,$n$)16O reaction. The uncertainties
from the fit to the JUNA+SCU data are indicated by dotted lines. The best fit
and lower limit recommended by LUNA Ciani _et al._ (2021) are shown as black
and blue dashed lines, respectively. The S-factors have been corrected with
the screening potential $U_{e}$ = 0.78 keV. The temperatures in $T_{9}$ on the
top correspond to the center energy of the Gamow window on the bottom.
The S-factor at $E_{\rm c.m.}<$0.24 MeV was obtained using an $R$-matrix
analysis Lane and Thomas (1958) in the range of $E_{\rm c.m.}$=0.24 to 1.9 MeV
using the code AZURE2 Azuma _et al._ (2010); Uberseder and deBoer (2015). In
our analysis, we only included our measurements of the 13C($\alpha$,$n$)16O
cross section, to eliminate the systematic uncertainty of the inconsistent
data sets, and the 16O+$n$ total cross section Fowler _et al._ (1973).
Our best fit is shown together with its estimated uncertainty in Fig. 1. The
screening potential($U_{e}$) is fitted to be 0.78$\pm$0.43 keV. It agrees with
the theoretical prediction of $U_{e}$=0.937 keV using the adiabatic limit
while ruling out the larger prediction of $U_{e}$=2 keV Trippella and Cognata
(2017). Our fit is about 15% systematically higher than the LUNA measurement
Ciani _et al._ (2021). The reduced-$\chi^{2}$ of the LUNA data is 25 by using
their best fit. It drops to 1.02 with our fit after the normalization and
excluding the point at $E_{\rm c.m.}$=0.29 MeV, which is 5$\sigma$ lower than
our best fit.
Although the LUNA measurement agrees with ours within the quoted errors, the
inconsistency between the measurement of Harissopulos _et al._ (2005) and
other measurements at higher energies leads to a $\sim$50% difference between
the upper and the lower limits of the reaction rate recommended by LUNA at
$T_{9}$=0.1-0.3. This demonstrates a key limitation of the LUNA measurement,
that its limited energy range did not allow for a direct comparison with
higher energy data. Using our consistent measurement over a board energy
range, the uncertainty of our fit are reliably constrained to the level
of$<$16% at the Gamow windows of s- and i-processes.
The extrapolated S-factor towards lower energy is dominated by the $\alpha$
reduced width $\gamma_{\alpha}$ or the Coulomb renormalized asymptotic
normalization coefficient ($\tilde{C}^{2}$) of the 1/2+ threshold state. The
R-matrix analysis performed in previous works involved fixing the ANC of the
threshold state to values obtained from indirect measurements. However, the
uncertainties in these ANCs often suffer from difficulties to quantify
systematic uncertainties from the models used to obtain them. The lower and
higher limits of the measured $\tilde{C}^{2}$ differ from each other by a
factor of $\sim$5 deBoer _et al._ (2020). These systematic uncertainties have
been eliminated in our fit by treating the $\Gamma_{\alpha}$ of this state as
a free parameter. The reduced widths $\gamma_{\alpha}$ obtained from our best
$R$-matrix analysis is -0.14(2) MeV1/2 with a channel radius of 6.684 fm and
$E_{x}$=6.3772 MeV, corresponding to an ANC of $\tilde{C}^{2}$=2.1(5) fm-1
with $E_{x}$=6.356 MeV Mukhamedzhanov _et al._ (2017); Brune (2020). Our
value is slightly lower than the indirect measurements of 3.6(7) fm-1 Avila
_et al._ (2015) and agree with 2.7(8) fm-1 Guo _et al._ (2012); Shen _et
al._ (2019) and 4.5(2.2)Pellegriti _et al._ (2008). For the first time, we
not only validate the $\alpha$ width of the threshold state obtained with the
indirect method using the direct measurement, but also determine the
interference pattern in the $R$-matrix analysis. As LUNA used the higher
$\tilde{C}^{2}$ from Avila _et al._ (2015) to constrain their extrapolation
towards lower energies, our best fit is 23% lower than their best fit at
$E_{\rm c.m.}$=0.19 MeV, the center of the Gamow window for $T_{9}$=0.1 (see
Fig.2). At the same energy, with the combination of a larger reduced width
Avila _et al._ (2015) and the cross section of Harissopulos _et al._ (2005),
the “low LUNA” fit is 11% lower than our best fit.
Figure 2: The Gamow function of 13C($\alpha$,$n$)16O at $T_{9}$ = 0.1 and 0.2.
Color coding is identical to Fig.1
The 13C($\alpha$,$n$)16O reaction rate is calculated by numerical integration
of the standard reaction rate equation Rolfs and Rodney (1988):
$\langle\sigma
v\rangle=\Big{(}\frac{8}{\pi\mu}\Big{)}^{1/2}\frac{1}{{kT}^{3/2}}\int_{0}^{\infty}\sigma(E)E\exp\Big{(}-\frac{E}{kT}\Big{)}\mathrm{d}{\rm
E}$ (1)
To highlight the important stellar energy range for a typical helium burning
temperature $T_{9}$ = 0.1 and 0.2, the integrand of Eq. (1) (Gamow function)
is computed and shown in Fig. 2. At $T_{9}$ = 0.1, the temperature of the 13C
pocket in the AGB model, our extrapolation is lower than the best fit of LUNA
and tends to agree better with their “low LUNA” fit. At $T_{9}$ = 0.2, which
is of importance for both the i-process and s-process nucleosynthesis in the
thermal pulse in the AGB model, our measurement covers nearly the entire Gamow
function with significantly improved uncertainties. This is a substantial
improvement compared to previous measurements as the ground-based measurements
from Ref. Drotleff _et al._ (1993); Heil _et al._ (2008) covered only the
upper part of the Gamow window with large uncertainties while the LUNA
extrapolation suffered from the inconsistencies in the absolute cross section
of the higher energy measurements. At the center of the Gamow window of
$T_{9}$ = 0.2, our result agrees with the best fit of LUNA within our quoted
uncertainty, but rules out the “low LUNA” fit, reflecting a significant
difference in the shape of our extrapolation from that of LUNA.
The reaction rate is calculated with the JUNA fit shown in Fig. 1. Comparisons
of the reaction rates are shown in Fig. 3. Our reaction rate agrees well with
previous Reaclib compilations based on the ANC method Guo _et al._ (2012) and
NACRE-II Xu _et al._ (2013) at $T_{9}\geq$ 0.1. The nearly 50% difference
between the upper limit and lower limit (“low LUNA”) of LUNA and the even
larger uncertainty in NACRE-II have been improved significantly. At
$T_{9}$=0.1-0.3, the typical temperatures for s- and i-processes, we have
reached an uncertainty of 13%-16%.
Figure 3: Selected reaction rates normalized by the rate determined in this
work. The uncertainties of our new rate and the LUNA rate, based on their best
fit, are indicated by the red hatched area and yellow shaded area,
respectively. For comparison, we also show the rates from NACRE-II Xu _et
al._ (2013), LUNA Ciani _et al._ (2021), and JINA Reaclib Guo _et al._
(2012); Cyburt _et al._ (2010).
It has been shown that the “low-LUNA” rate increases the survivability of 13C
in the 13C-pocket of an AGB star, and that it burns at a high temperature in
the subsequent thermal pulse Ciani _et al._ (2021). Compared with the “low-
LUNA” rate, our recommended rate is slightly higher at temperatures typical of
the 13C-pocket within 15$\%$, and about 30$\%$-40% higher at the thermal pulse
temperatures. Therefore, we expect our rate to produce effects similar to
those discussed by Ciani _et al._ (2021), although a follow up, detailed
study, on AGB stellar models is needed. Concerning the impact on the i-process
nucleosynthesis, future models based on the next generation of multi-
dimensional hydrodynamics simulations will be more predictive thanks to the
more reliable reaction rate uncertainties provided by this work.
In summary we have performed a direct measurement of the 13C($\alpha$,$n$)16O
reaction cross section over the range of $E_{\rm c.m.}$ = 0.24-1.9 MeV using
the most intense $\alpha$ beam available in the deep underground laboratories
with the highest precision to date. Our consistent measurement, covering a
wide energy range, reduce the large uncertainty in the reaction rate down to
13% to 16% for i- and s-process nucleosynthsis. Our reaction rate is similar
to the “low LUNA” rate at the typical 13C pocket, favoring the release of more
neutrons from the 13C($\alpha$,$n$)16O reaction during the thermal pulse
phase. Our direct measurement eliminates an important systematic uncertainty
in the R-matrix extrapolation by resolving the inconsistency among the data
sets at higher energies Bair and Haas (1973); Harissopulos _et al._ (2005);
Kellogg _et al._ (1989). For the first time, we determines the ANC of the
threshold state using the direct measurement, fix the interference pattern and
determine the screening potential using R-matrix analysis.
###### Acknowledgements.
The authors would like to thank Tsinghua University and Prof. Jianping Cheng
for the support of Lab infrastructure and the China National Nuclear
Corporation for the finial support, CDEX and PandaX collaborations for their
kind helps, L. He, Y. Wang, P. Wang and J.K. Liu for their helps in setting up
the experimental terminal and Prof. Z.G. Wang for developing the instruments
which was used to develop the thick 13C targets. The authors also acknowledge
the assistance of Carl Brune in estimating neutron angular distribution
corrections. X.T. thanks A. Best and D. Rapagnani for their helpful
discussions throughout the project and providing their original data sets.
This work was supported by the National Natural Science Foundation of China
under Grants No. 11490564, the Strategic Priority Research Program of Chinese
Academy of Sciences, grant No. XDB34020000, the national key research and
development program (MOST 2016YFA0400501). S. W. was supported by the National
Natural Science Foundation of China under Grants No. 11775133, 11405096. B. G.
was supported by the National Natural Science Foundation of China under Grants
No. 12125509. W.P. L. was supported by the National Natural Science Foundation
of China under Grant No. 11805138. X. F. was supported by the National Natural
Science Foundation of China under Grants No. 11875329. R.J.D. acknowledges the
use of resources from the Notre Dame Center for Research Computing. R.J.D. and
M.W. were supported by the National Science Foundation through Grant No.
Phys-2011890, and the Joint Institute for Nuclear Astrophysics through Grant
No. PHY-1430152 (JINA Center for the Evolution of the Elements). MP
acknowledges support to NuGrid from STFC (through the University of Hull’s
Consolidated Grant ST/R000840/1) and ongoing access to viper, the University
of Hull High Performance Computing Facility. MP thanks the support by the ERC
Consolidator Grant (Hungary) funding scheme (Project RADIOSTAR, G.A. n.
724560), the ChETEC COST Action (CA16117), supported by the European
Cooperation in Science and Technology, the IReNA network supported by NSF
AccelNet, and the European Union ChETEC-INFRA (project no. 101008324).
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Differential identities of matrix algebras
Brox]Jose Brox
Departamento de Álgebra, Análisis Matemático, Geometría y Topología, Universidad de Valladolid, Palacio de Santa Cruz, 47002, Valladolid, Spain
C. Rizzo]Carla Rizzo
Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, via Archirafi 34, 90123, Palermo, Italy
[2020]Primary 16R10, 16R50, 17B10; Secondary 16W25, 16P90, 16S30, 17B35, 17B20, 16G30, 15B30
This work was partially supported by the Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES. Jose Brox was first supported by the Portuguese Government through grant SFRH/BPD/118665/2016 (FCT/Centro 2020/Portugal 2020/ESF), later by a postdoctoral fellowship “Convocatoria 2021” funded by Universidad de Valladolid and partially supported by grant PID2022-137283NB-C22 funded by MCIN/AEI/10.13039/501100011033 and ERDF “A way of making Europe”.
We study the differential identities of the algebra $\M$ of $k\times k$ matrices over a field $F$ of characteristic zero when its full Lie algebra of derivations, $L=\Der(\M)$, acts on it. We determine a set of 2 generators of the ideal of differential identities of $\M$ for $k\geq 2$. Moreover, we obtain the exact values of the corresponding differential codimensions and differential cocharacters. Finally we prove that, unlike the ordinary case, the variety of differential algebras with $L$-action generated by $\M$ has almost polynomial growth for all $k\geq 2$.
§ INTRODUCTION
Let $A$ be an associative algebra over a field $F$ of characteristic zero, $F\langle X \rangle$ be the free associative algebra freely generated by an infinite countable set $X$ over $F$, and $\I(A)\subset F\langle X \rangle$ be the $T$-ideal of all polynomial identities of $A$. From a celebrated theorem of Kemer it is known that in characteristic zero every $T$-ideal is finitely generated (see [46]). The proof given by Kemer is not constructive, and finding an explicit finite basis of the $T$-ideal of polynomial identities of an algebra is, in general, an extremely hard task. Indeed there is only a handful of nontrivial examples of algebras for which this problem is completely solved. These include the algebra $UT_k(F)$ of upper triangular matrices ([51]), the infinite-dimensional Grassmann algebra $G$ ([49]), and the tensor product $G\otimes G$ of Grassmann algebras ([60]). If one adds to the above the full matrix algebra $M_2(F)$ of order $2$ (see [15, 62]), one approximately gets the complete list of algebras for which the identities are known. In fact, even the description of the $T$-ideal of $3\times 3$ matrices is still an open problem with no solution in sight.
Since finding the exact form of the polynomial identities satisfied by a given algebra is a goal that seems too hard to achieve for the vast majority of relevant algebras, one is led to the study of identities of algebras with additional structure, such as algebras with a trace, group-graded algebras, algebras with involution, algebras with a Lie algebra action induced by derivations and, more in general, algebras with a Hopf algebra action. Such theories of identities include the theory of ordinary ones as a special case and, overall, their study tends to be less challenging.
In this paper we focus our attention on algebras with derivations, i.e., associative algebras with a Lie algebra action by derivations. If $L$ is such a Lie algebra, then its action can be naturally extended to an action of its universal enveloping algebra $U(L)$, and we say that $A$ is an algebra with derivations from $L$, or an $L$-algebra. In this context the differential identities of $A$ are defined as the polynomials vanishing on $A$ in the variables $x^u:=u(x)$ with $u\in U(L)$, i.e., coming from the free $L$-algebra $\FL$. Notice that the theory of differential identities generalizes the theory of ordinary polynomial identities, as any algebra $A$ can be regarded as an $L$-algebra by letting $L$ act on $A$ trivially,
with $L$ acting on $A$ as the trivial Lie algebra and $U(L)\cong F$. Differential identities were introduced by Kharchenko in [41] (see also [42]) and, in later years, relevant work by Gordienko and Kochetov ([32]) has motivated a growing interest in them. The $T_L$-ideals of differential identities of some important algebras have been determined: in [22, 70], Giambruno and Rizzo gave a complete description of the differential identities of the algebra $UT_2(F)$ of $2\times 2$ upper triangular matrices endowed with all possible actions of Lie algebras by derivations; in [11], Di Vincenzo and Nardozza determined the generators of the $T_L$-ideal of the algebras $UT_k(F)$ under the action of the nonabelian two-dimensional Lie algebra; in [69], Rizzo studied the differential identities of $G$ with the action of a finite-dimensional abelian Lie algebra of inner derivations. We also refer the interested reader to [55, 58] for more results on differential identities of other interesting algebras.
Since the base field $F$ is of characteristic zero, as it happens in ordinary PI theory, the $T_L$-ideal $\I^L(A)$ is completely determined by its multilinear elements as well. Recall from ordinary PI theory that the codimensions sequence $\{c_n(A)\}_{n\in\N}$ of an algebra $A$ is defined by taking $c_n(A)$ as the dimension of the space $P_n$ of multilinear polynomials of degree $n$ modulo $\I(A)$. The codimensions sequence is also hard to compute, in the sense that, quoting Regev (<cit.>), in general there is no hope to find a closed formula for $c_n(A)$. Therefore one resorts to studying the growth of the sequence as $n$ tends to infinity. In the late nineties Giambruno and Zaicev ([24, 25]) proved that if $A$ is an algebra satisfying a nontrivial polynomial identity, then the limit $\exp(A):=\lim_{n\to \infty}\sqrt[n]{c_n(A)}$ exists and is always a nonnegative integer called the (ordinary) exponent of $A$. As a consequence, it follows that the codimensions of an algebra are either polynomially bounded or grow exponentially. Given a variety $\textsc V$, its growth is defined as the growth of $c_n(A)$ for any $A$ generating $\textsc V$. Then one says that variety $\textsc V$ has almost polynomial growth if its growth is not polynomially bounded but every proper subvariety of $\textsc V$ has polynomial growth. In the ordinary setting, $G$ and $UT_2(F)$ are the only algebras generating varieties of almost polynomial growth ([45]).
Analogous definitions of $P_n^L$, $c_n^L(A)$ and $\exp^L(A)$ can be given in the differential setting. In [31] Gordienko proved that, in case $A$ is finite dimensional, $\exp^L(A)$ indeed exists and is a nonnegative integer called the $L$-exponent of $A$, which allows to likewise define the concept of almost polynomial growth in this case. Moreover, since $U(L)$ is a unital algebra, we can identify $P_n$ with a subspace of $P_n^L$ in a natural way and hence we have $c_n(A)\leq c_n^L(A)$ for all $n\geq 1$, from what it is clear that $\exp(A)\leq \exp^L(A)$. In [32], Gordienko and Kochetov proved that in case $L$ is a finite-dimensional semisimple Lie algebra then $\exp(A)=\exp^L(A)$; in [72], it was shown that if $A$ is finite dimensional and $L$ is any Lie algebra then $\exp^L(A)=1$ if and only if $\exp(A)=1$; and in [71], the author proved that $\exp^L(A)$ coincides with $\exp(A)$ for any Lie algebra $L$. In case $L$ is finite dimensional and solvable, the only two finite-dimensional $L$-algebras generating $L$-varieties of almost polynomial growth are $UT_2(F)$ with trivial $L$-action, and $UT_2(F)^\varepsilon$ with $L$ the one-dimensional Lie algebra generated by the inner derivation $\varepsilon$ induced by the matrix unit $e_{11}$ (<cit.>). The assumption that $L$ is solvable is crucial; in fact, in this paper we present an infinite family of finite-dimensional $L$-algebras of almost polynomial growth for a simple Lie algebra $L$. This points out that the structural properties of the acting Lie algebra deeply affect the growth of the related varieties.
As matrix algebras are of great importance for both mathematics and its applications, the identities satisfied by matrices have been an attractive object of study from the very origins of polynomial identities theory. Concerning matrices with additional structure, to the best of our knowledge, so far the only known results are on graded identities of the matrix algebras $\M$ for cross-product gradings ([73, 74] for gradings by $\mathbb{Z}_k$ and $\mathbb{Z}$, [1] for gradings by an arbitrary group), and on the trace identities of the full matrix algebras (see [61, 63]).
The main purpose of this paper is to determine the differential identities of the algebra $\M$ of $k\times k$ matrices for $k\geq 2$ over a field $F$ of characteristic zero, when acted by its Lie algebra of all derivations $\Der(\M)$, making all computations explicit along the way.
To do so, in order to have a finite-dimensional algebra at our disposal, for any $L$-algebra $A$ we call $U$ to the image of the representation of $U(L)$ in $\E_F(A)$ and define two related free $L$-algebras, $\FU$ and $\FLU$ (with their corresponding notions of $U$- and $(L,U)$-polynomials and $T_U$- and $T_{L,U}$-ideals). These algebras allow us to make computations with $U$, and between the two they model the desired properties of $\FL$: roughly speaking, $\FU$ inherits the linear behavior of $\FL$, while $\FLU$ inherits its $L$-action behavior. In Section <ref> we conduct a careful analysis of these algebras and their relations, and develop the general setting of the variety of $(L,U)$-algebras (which we define), for which $\FLU$ is the free algebra. In this way we show that we can study differential identities, codimensions, and growth by considering $U$- and $(L,U)$-polynomials and the variety of $(L,U)$-algebras.
In Section <ref> we particularize to the case $L:=\Der(\M)\cong\Sl_k(F)$ and, via the representation theory of $L$ (see Theorem <ref>), describe $U$-polynomials of $\FU$ as being composed of variables either of the form $x^{\vp_{ab}}=x^{ab}$ for $a,b$ elements of the standard Cartan-Weyl basis $\mathcal S:=\{h_1,\ldots,h_{k-1},e_{12},\ldots,e_{kk-1}\}$ of $\Sl_k(F)$, or of the form $x^{\vp_{gg}}=x^{gg}$, with $g=I_k$ the identity matrix, with the exponent endomorphisms satisfying $\vp_{ab}\vp_{cd}=\delta_{bc}\vp_{ad}$ and $1=\sum_{a\in\mathcal S}\vp_{aa} + \vp_{gg}\in\E_F(\M)$. It is this partition of unity into orthogonal idempotents what allows us to circumvent the apparition of ordinary PIs in our computations. Moreover, the definitions of the endomorphisms $\vp_{ab}$ allow us to directly translate identities from $\mathcal S\cup\{g\}$ to $U$-identities of $\M$ (see Lemma <ref>),
e.g. $e_{12}e_{31}=0$ implies $x^{e_{12}e_{12}}y^{e_{12}e_{31}}=0$, with the second exponent index carrying the weight of the identities. In Section <ref> we use this idea together with the linear structure of $\FU$ to show a set of generators of $\I_U(\M)$, in 2 variables and with at most 3 terms, which afterwards we reduce to a minimal set of $4$ generators (in 2 variables and 2 terms) with the aid of the $L$-action of $\FU$, which allows to modify the second index of an exponent; finally we show, through the result from the representation theory of $L$ that we call the primitive element lemma (Lemma <ref>), that $\I_{L,U}(\M)$ is principal.
In order to translate this last result to $\FL$ in an explicit way, if $\phi$ is the homomorphism sending $U(L)$ to $U$, we need to compute some preimages $\phi^{-1}(\vp_{ab})\in U(L)$ of the endomorphisms $\vp_{ab}\in U$, and also some generators of $\ker\phi$, what we also do in Section <ref>. For the preimages, we just find expressions formed with polynomials of degree at most $6$ in the elements $e_{ij}\in U(L)$. For the kernel, we recall that the center of $U(L)$ is a polynomial ring in $k-1$ indeterminates $c_i$ which on each irreducible representation $\rho$ of $L$ act as scalars $\lambda^\rho_i$, from which each $c_i-\lambda^\rho_i$ is in the kernel of $\rho$.[We also compute explicitly the values of the eigenvalues of a standard set of Casimir generators of $\Sl_k(F)$ for the adjoint representation, a result which may be of independent interest.] On the other hand, we know that $e_{12}^3\in\ker\phi$ and that $\phi$ is the direct sum of the trivial and the adjoint representations of $L$. From these facts, the algebraic geometry of $U(L)$ (Gröbner bases, primitive spectrum), and the primitive element lemma, we show that $\ker\phi$ is principal (Theorem <ref>). Then, as $\FLU\cong\FL/\ILU$ with $\ILU$ the $T_L$-ideal generated by $x^z$ for $z\in\ker\phi$, we get as our main result, Theorem <ref>, that the differential identities of $\M$ are generated by 1 identity in 1 variable (coming from $\ker\phi$ and depending on $k$) and 1 identity in 2 variables (coming from $\I_{L,U}(\M)$ and not depending on $k$ except for $k=2$).
In addition, in Section <ref> we also show a special kind of symmetry that holds for the $U$-identities of any $(L,U)$-algebra $A$ and that we use profusely thereafter, roughly speaking, that changes in the first exponent index leave $T_U(A)$ invariant. More concretely, consider the space $P^U_{\mathcal I,\mathcal J,(a_1,\ldots,a_{n-r})}$ with $(\mathcal I,\mathcal J)$ a partition of $\{1,\ldots,n\}$ and $|\mathcal I|=r$, of those multilinear $U$-polynomials in which variable $x_i$ always appears paired with first exponent $g$ for $i\in\mathcal I$ and variable $x_j$ always appears paired with first exponent $a_j\in\mathcal S$ for $j\in\mathcal J$. Then $P^U_{\mathcal I,\mathcal J,(a_1,\ldots,a_{n-r})}$ is linearly isomorphic to $P^U_{\mathcal I',\mathcal J',(a'_1,\ldots,a'_{n-r'})}$ if and only if $r=r'$, and $U$-identities of $A$ map to $U$-identities. Moreover, defining an action of $S_r\times S_{n-r}$ by permutations of variables together with their first exponents, the linear isomorphisms are in fact isomorphisms of $S_r\times S_{n-r}$-modules. Since then all of them are isomorphic to $P^U_{r,n-r}:=P^U_{\{1,\ldots,r\},\{r+1,\ldots,n-r\},(a,\ldots,a)}$ for fixed $a$, we can restrict to the study of these $S_r\times S_{n-r}$-modules for each $0\leq r\leq n$.
With these ideas at hand, we show a combinatorial formula for the $U$-codimensions (Formula (<ref>)), arising from $P^U_{r,n-r}(A)$, that is used in Section <ref> together with the $U$-identities of $\M$ to find a closed formula for $c^L_n(\M)$ (see Theorem <ref>). In particular, the associated generating function is rational; in contrast, when $k\geq3$ is odd, the generating function of (ordinary) $c_n(\M)$ is not algebraic (<cit.>). As an aside, this proof also constitutes a simple way of showing that $\exp(\M)=k^2$ (as $\exp(\M)=\exp^L(\M)=k^2$), which was originally shown by Regev by resorting to the asymptotics of trace identities ([65]), and can also be proved by Wedderburn's decomposition (<cit.>).
Now let $P_{(n;r)}^U(A)$ be the direct sum of all $P_{\mathcal I,\mathcal J,(a_1,\ldots,a_{n-r})}(A)$ such that $|\mathcal I|=r$; it is an $S_r\times S_{n-r}$-module whose character $\chi_{(n;r)}^U(A)$, which we call the $(n,r)$th $U$-cocharacter of $A$, is a multiple of that of $P^U_{r,n-r}(A)$ (Formula (<ref>)). In Section <ref> we show, by a counting argument, that $\chi_{(n;r)}^U(\M)$ is a multiple of the irreducible $S_r\times S_{n-r}$-cocharacter $\chi_{(r)}\otimes\chi_{(n-r)}$ for each $0\leq r\leq n$ (Theorem <ref>).
Lastly, in Section <ref> we prove a result which is, in our view, one of the most interesting and unexpected PI results of this paper: unlike the ordinary case, the variety $\V^L(M_k(F))$ of differential algebras with $L$-action generated by $M_k(F)$ has almost polynomial growth for all $k\geq 2$, i.e., $\V^L(M_k(F))$ has exponential growth but any of its proper $L$-subvarieties has polynomial growth (Theorem <ref>). To show it, we prove that if an $L$-subvariety $\textsc V$ satisfies any $L$-identity not belonging to $T_L(\M)$, then it must satisfy all $U$-identities of the form $x_1^{a_1a_2}\cdots x_t^{a_ta_{2t}}$ for $a_i\in\mathcal S$ for some $t$, implying that $c_{r,n-r}^L(\textsc V) = c_{r,n-r}^U(\textsc V)=0$ whenever $n-r\geq N$ for some $N$.
§ GENERAL SETTING
§.§ Preliminaries
Throughout this paper, $F$ will denote a field of characteristic zero, $A$ an associative algebra, and $(L,[\cdot\, ,\cdot]_L)$ a Lie algebra. All algebras and vector spaces have $F$ as their underlying field. Although we work with varieties of nonunital associative algebras, all results can be easily adapted to unital associative algebras as well. All notations, once introduced, will maintain their meanings in the ensuing sections of the paper.
Associative algebras.
Given a set $S\subseteq A$, by $\la S\ra$ we denote the ideal generated by $S$. $A$ is split semisimple (over $F$) if it is a direct sum of matrix algebras over $F$. Given a unital associative algebra $U$ with product $\cdot$, the opposite algebra $U^{\op}$ is the underlying vector space of $U$ endowed with the opposite product $a\cdot^{\op} b:=b\cdot a$ for $a,b\in U$; $U^{\op}$ is antiisomorphic to $U$ as unital associative algebras (with $(U^{\op})^{\op}=U$) through the map $^{\op}:U\to U^{\op}$ such that $a^{\op}:=a$; in particular, any subset of $U$ is mapped to itself. If $\phi:U_1\to U_2$ is a homomorphism of unital associative algebras, then $\phi^{\op}:U_1^{\op}\to U_2^{\op}$ defined by $\phi^{\op}(a):=\phi(a)$ is a homomorphism of unital associative algebras.
Given a vector space $V$, a left (resp. right) (algebra) $U$-action of a unital associative algebra $U$ on $V$ is a map $\cdot:U\times V\rightarrow V$ (resp. $\cdot:V\times U\rightarrow V$) such that $1\cdot x=x$, $a\cdot(\lambda x+y)=\lambda(a\cdot x)+a\cdot y$, $(\lambda a+b)\cdot x = \lambda(a\cdot x)+b\cdot x$, and $(ab)\cdot x = a\cdot(b\cdot x)$ (resp. $x\cdot 1=x$, $(\lambda x+y)\cdot a=\lambda(x\cdot a)+y\cdot a$, $x\cdot(\lambda a+b) = \lambda(x\cdot a)+x\cdot b$, and $x\cdot (ab) = (x\cdot a)\cdot b$) for $a,b\in U$, $x,y\in V$ and $\lambda\in F$. Let $\End_F(V)$ be the algebra of the endomorphisms of $V$ acting on the left of $V$. Then a left (resp. right) $U$-action on $V$ produces a left (resp. right) representation of $U$, i.e., a homomorphism of unital algebras $\phi:U\to\End_F(V)$ (resp. $\phi:U^{\op}\to\End_F(V)$) and vice versa. Any left action $\cdot$ of $U$ on $V$ has an associated right action $\cdot^{\op}$ of $U^{\op}$ on $V$ given by $x\cdot^{\op}a:=a\cdot x$ for $a\in U^{\op}$, $x\in V$, and vice versa (with $(\cdot^{\op})^{\op}=\cdot$). If there is a (left, right) $U$-action on $A$ we say that $A$ is a (left, right) $U$-algebra (for this action).
Throughout this paper we define endomorphisms as acting on the left, but we use exponential notation to denote their actions: hence we see any left $U$-algebra as a right $U^{\op}$-algebra (notice that the associated representation $\phi$ is the same), with exponents living in $U^{\op}$; in addition, we denote the opposite products appearing in exponents just by juxtaposition. Moreover, by abuse of notation we may also denote $\phi^{\op}:U^{\op}\to\End_F(A)^{\op}$ by $\phi$. For example, if $\phi:U\to\End_F(A)$ with $\phi(u_i)=\phi_i$, $u_i\in U$ for $i=1,2$ and associated left action denoted by $\sbullet$, then we write
\[(a^{u_1})^{u_2} = a^{u_1u_2} = a^{\phi(u_1)\phi(u_2)} = a^{\phi_1\phi_2} =\phi_2(\phi_1(a)) = u_2\sbullet(u_1\sbullet a) = (u_2u_1)\sbullet a = (u_1\cdot^{\op} u_2)\sbullet a\]
for $a\in A$ and, in the exponents, $u_1,u_2\in U^{\op}$, $\phi_1,\phi_2\in\End_F(A)^{\op}$. Since for any set $S\subseteq U$ we have $S^{\op}=S$ inside $U^{\op}$, if no confusion may arise, when picking exponents we may write $s\in S$ instead of $s\in S^{\op}$.
Lie algebras.
Given $L$, the opposite Lie algebra $L^{\op}$ is the underlying vector space of $L$ endowed with the opposite product $[a,b]_{L^{\op}}:=-[a,b]_L$ ($L^{\op}$ is isomorphic to $L$). If $\vp:L\rightarrow M$ is a homomorphism of Lie algebras, then $\vp^{\op}:L^{\op}\rightarrow M^{\op}$ defined by $\vp^{\op}(a):=\vp(a)$ is a homomorphism of Lie algebras.
The underlying vector space of $A$ endowed with the commutator product $[a,b]:=ab-ba$ for all $a,b\in A$ is a Lie algebra, denoted by $A^-$. We have $(A^{\op})^- = (A^-)^{\op}$. A linear endomorphism $\delta:A \to A$ is a derivation of $A$ if it satisfies $(ab)^\delta=a^\delta b + a b^\delta$ for all $a,b\in A$. If $a\in A$, the endomorphism $\ad_a:A\to A$ defined by $\ad_a(b):=[a,b]$ for all $b\in A$ is a derivation of $A$, called the inner derivation induced by $a$. For ease of reading, given an element in $A$ denoted by a lowercase letter, at times we denote the inner derivation this element induces by the corresponding uppercase letter, e.g. $E:=\ad_e\in\End_F(A)$ for $e\in A$. The vector space of all derivations of $A$ endowed with the commutator product is a Lie algebra denoted by $\Der(A)\subseteq\E_F(A)^-$, with the subspace $\ad (A)\subseteq\Der(A)$ of inner derivations of $A$ being a Lie ideal of $\Der(A)$.
A semisimple Lie algebra is split (over $F$) if it has a Cartan subalgebra $H$ such that the eigenvalues of $\ad_h$ lie in $F$ for all $h\in H$, called a splitting Cartan subalgebra; hence a split-semisimple Lie algebra has a root system (<cit.>). $L$ is split simple if it is simple and split semisimple.
A (left) representation of $L$ on the vector space $V$ is a Lie algebra homomorphism $\rho:L\rightarrow\End_F(V)^-$; we also say that that $V$ has a (left) $L$-action given by $d\cdot x:=\rho(d)x$ for $d\in L$, $x\in V$, and that $V$ (together with $\rho$) is a (left) $L$-module. We similarly define the same notions for the right side, with any left $L$-action $\cdot$ giving rise to a right $L^{\op}$-action $\cdot^{\op}$ by the rule $x\cdot^{\op} d:=d\cdot x$; accordingly, we will write left $L$-actions as right $L^{\op}$-actions with exponential notation.
The trivial representation on $V$ is given by $\rho_0:=0$. The adjoint representation of $L$ is given by $\ad$ on $L$. A representation $\rho$ on $V$ is irreducible if it has no proper nontrivial subrepresentation $\rho|_W$ on $W\subseteq V$. Weyl's theorem states that, when $L$ is finite dimensional semisimple, every finite-dimensional $L$-module is completely reducible, i.e., a direct sum of irreducible $L$-modules (<cit.>).
For the rest of this paragraph let $L$ be a finite-dimensional split-semisimple Lie algebra (over a field of characteristic $0$), with a fixed splitting Cartan subalgebra $H\subseteq L$ and a fixed set of positive roots; then $L$ has a triangular decomposition $L=N_-\oplus H\oplus N_+$ where $N_{-,+}$ are the linear spans of the negative and positive root spaces, respectively. Any finite-dimensional irreducible $L$-module is absolutely irreducible, i.e., irreducible for any extension of scalars of the base field (<cit.>).
A weight of $L$ is an algebra homomorphism $\lambda\in\Hom(H,F)$. A vector $v$ of the $L$-module $V$ is a weight vector of weight $\lambda$ if $\rho(h)v=\lambda(h)v$ for all $h\in H$, a highest weight vector if in addition $\rho(N_+)v=0$, in which case $\lambda$ is a highest weight of $V$. An $L$-module is a (cyclic) highest weight module if it is generated by a single highest weight vector. Given a weight $\lambda$ of $L$ we can build a universal highest weight $L$-module with $\lambda$ as highest weight, the Verma module $W_\lambda$, such that any highest weight $L$-module with highest weight $\lambda$ is a quotient of $W_\lambda$ (<cit.>).
Given a representation $\rho:L\to\E_F(V)^-$, the enveloping (associative) algebra of $\rho$ is the unital associative subalgebra of $\E_F(V)$ generated by $\rho(L)$, denoted here by $[\rho(L)]$. The following result is known (see e.g. <cit.> for a close result); we include a proof here since it is a key tool of our paper.
Let $L$ be a finite-dimensional split-semisimple Lie algebra over a field of characteristic $0$ and $\rho$ be a finite-dimensional representation of $L$. Then the enveloping associative algebra $[\rho(L)]$ of $\rho$ is split semisimple. Moreover, if $\rho$ is irreducible of dimension $d$, then $[\rho(L)]$ is a matrix algebra of dimension $d^2$.
Put $\rho:L\to\End_F(V)^-$ with $\dim_F V=d$. Since $\rho$ is finite dimensional, by Weyl's theorem it is completely reducible, so it suffices to assume that $\rho$ is irreducible. Denote $U:=[\rho(L)]$, let $K$ be an algebraic closure of $F$, and consider the following objects produced by extension of scalars:
$L_K:=L\otimes_F K, \, V_K:=V\otimes_F K, \E_F(V)\otimes_F K = \E_K(V_K),$
the representation $\rho_K:=\rho\otimes_F K:L_K\rightarrow\E_K(V_K)^-$, and its enveloping algebra $U_K:=[\rho_K(L_K)] = U\otimes_F K$.
Since $L$ is a finite-dimensional split-semisimple Lie algebra and $\chr(F)=0$, $\rho$ is absolutely irreducible. Therefore $\rho_K$ is irreducible, hence $V_K$ is a finite-dimensional irreducible $U_K$-module (since $U_K$ is generated by $L_K$) which is also faithful (as $U_K\subseteq\E_K(V_K)$); by Jacobson's density theorem (<cit.>), $U_K=\End_D(V_K)$ with $D:=\End_{U_K}(V_K)$ a finite-dimensional division algebra (by Schur's lemma) over the algebraically closed field $K$, which forces $D=K$ and $U_K = \End_K(V_K)\cong M_d(K)$. Now
$U\otimes_F K = U_K = \E_K(V_K) = \E_F(V)\otimes_F K$
implies $U=\E_F(V)$ (since $U\subseteq\E_F(V)$), so $U\cong M_d(F)$.
Universal enveloping algebras.
The universal enveloping algebra $U(L)$ of $L$ is the quotient of the unital tensor algebra generated by $L$ by the ideal generated by relations $a\otimes b-b\otimes a-[a,b]_L$ for all $a,b\in L$, i.e., for $a,b\in L$ we have $[a,b]_{U(L)}=[a,b]_L$, hence $[a,b]_{\Uop}=-[a,b]_L = [a,b]_{L^{\op}}$.
The universal enveloping algebra satisfies the following universal property: there is a monomorphism $\gamma:L\rightarrow U(L)^-$ of Lie algebras such that for any homomorphism $\vp:L\rightarrow A^-$ of Lie algebras there exists a unique homomorphism $\phi:U(L)\rightarrow A$ of associative algebras such that $\vp=\phi\circ\gamma$. Since its universal property determines $U(L^{\op})$ up to isomorphism, and $U(L)^{\op}$ is easily seen to satisfy said universal property, we have $U(L^{\op})\cong U(L)^{\op}$ as unital associative algebras.
By abuse of notation, we identify $L$ with $\gamma(L)$ inside $U(L)$. Then the Poincaré-Birkhoff-Witt (PBW) theorem asserts that if $\{e_i\}_{i\in I}$ is an ordered basis of $L$, the set $B:=\{e_{i_1}^{k_1}\cdots e_{i_j}^{k_j} \ | \ j\in\N, e_{i_1}<\cdots< e_{i_j}, k_1,\ldots,k_j\in\N^*\}$ is a basis of $U(L)$ (including $1\in B$). In particular, $U(L)=:U^*(L)\oplus F\cdot1$, where $U^*(L)$ is the nonunital universal enveloping algebra of $L$ and the augmentation ideal of $U(L)$ (which is a maximal ideal). Note that $U(L)$ is always an infinite-dimensional algebra, even if $L$ is finite dimensional. Universal enveloping algebras of finite-dimensional Lie algebras are Noetherian rings (<cit.>) and the theory of Gröbner bases is available for $U(L)$ (see <cit.>); we are interested in a particular application. Let $L$ be finite dimensional, fix an ordered basis $\{e_1,\ldots,e_d\}$ with $e_i<e_j$ if $i<j$, and let $B$ denote the corresponding basis of monomials of $U(L)$ given by the PBW theorem. The deglex order extends $<$ to $B$ by the rule $m_1:=e_1^{k_1}\cdots e_d^{k_d}< e_1^{l_1}\cdots e_d^{l_d}=:m_2$ if either $\deg(m_1)<\deg(m_2)$ or $\deg(m_1)=\deg(m_2)$ and the first nonzero entry of $(k_1-l_1,\ldots,k_d-l_d)$ is positive. Given an element $f\in U(L)$, its leading monomial $\LM(f)$ with respect to $<$ is the largest monomial of $f$ with nonzero coefficient. If $I$ is an ideal of $U(L)$, its set of leading monomials is $\LM(I):=\{\LM(f) \ | \ f\in I\}$ and its set of normal words is $N(I):=\{m\in B \ | \ m\not\in\LM(I)\}$. Then
$U(L)=I\oplus \spn N(I)$ (<cit.>).
We can make $U(L)$ an $L$-module by extending the adjoint action of $L$ to $U(L)$, which is the restriction to $L$ of the adjoint action of $U(L)$ on itself (given by $\ad_x(y)=[x,y]$ for $x,y\in U(L)$). By the universal property of $U(L)$, any representation $\rho:L\rightarrow\End_F(V)^-$ of $L$ extends to an associative left $U(L)$-representation $\rho:U(L)\rightarrow\End_F(V)$ (note the abuse of notation). A $U(L)$-representation is irreducible if and only if it is irreducible as an $L$-representation (since $U(L)$ is generated as an algebra by $L$).
If $L$ is finite dimensional split semisimple, the Verma module associated to weight $\lambda$ of $L$ can be constructed as $M_\lambda\cong U(L)/I_\lambda$, with $I_\lambda$ the left ideal of $U(L)$ generated by $\{h-\lambda(h)1 \ | \ h\in H\}\cup N^+$ (<cit.>, <cit.>).
The center $Z(U(L))$ of $U(L)$ is the set of elements $c\in U(L)$ such that $[c,U(L)]=0$, which are called Casimir elements. For the rest of this paragraph let $L$ be a finite-dimensional semisimple Lie algebra over an algebraically closed field (of characteristic $0$), and denote $Z:=Z(U(L))$. By the Harish-Chandra isomorphism (<cit.>), $Z$ is isomorphic to the algebra of polynomials in rank$(L)$ indeterminates (see also <cit.>); we call a set of Casimir generators to any set of rank$(L)$ algebraically independent elements of $Z$. By Schur's lemma, if $c\in Z$ and $\rho$ is an irreducible representation of $L$ on $V$ then $\rho(c)$ acts as a scalar on $V$. A central character of $L$ is an algebra homomorphism $\chi:Z\rightarrow F$. If $\rho$ is an irreducible representation of $L$ of dimension $d$ then $\chi_\rho(c):=\frac1d\tr(\rho(c))$ for $c\in Z$ is the central character associated to $\rho$; we call $\chi_\rho(c)$ the eigenvalue of $c$ for $\rho$. We clearly have $U(L)\ker\chi_\rho\subseteq\ker\rho$. If $\rho,\rho'$ are two finite-dimensional irreducible representations of $L$ and $\chi_\rho=\chi_{\rho'}$ then $\rho\cong\rho'$ (<cit.>). Note that, for a fixed set of Casimir generators, the maximal ideals of $Z$ are in 1-to-1 correspondence with the central characters of $L$. Now fix $F=\Co$, the field of complex numbers. We will need the following results from the algebraic geometry of $U(L)$. Let $\Max(A), \Prim(A)$ respectively denote the sets of maximal and primitive ideals of the algebra $A$, and for $I\in\Prim(U(L))$ let $\pi(I):=I\cap Z$ and say that $I$ lies over $J$ for an ideal $J$ of $Z$ if $I\in\pi^{-1}(J)$. Then $\pi(I)\in\Max(Z)$, and Dixmier's theorem asserts the following: given $M\in\Max(Z)$, the set of primitive ideals of $U(L)$ lying over $M$ is finite and contains a minimal and a maximal element, with $\pi:\Max(U(L))\rightarrow\Max(Z)$ being a bijection ([12, 13], see also <cit.> and <cit.>). Joseph's theorem on principal series submodules states that if the ideal $I$ of $U(L)$ satisfies $I\cap Z(L)\in\Max(Z(L))$ then $I$ is the annihilator of the $L$-module $W_\lambda/IW_\lambda$ for some weight $\lambda$ of $L$ (<cit.>).
We can say more when the ideal is of finite codimension:
Let $L$ be a complex finite-dimensional semisimple Lie algebra and $I\neq U(L)$ be an ideal of finite codimension of $U(L)$ such that $I\cap Z(U(L))\in\Max(Z(U(L)))$. Then $I\in\Max(U(L))$.
Denote $Z:=Z(U(L))$. Since $I\cap Z\in\Max(Z)$, $I$ is the annihilator of the $U(L)$-module $N:=W_\lambda/IW_\lambda$ for some weight $\lambda$ of $L$ by Joseph's theorem on principal series submodules. Since $W_\lambda\cong U(L)/I_\lambda$ as $U(L)$-modules, we get $N\cong\frac{U(L)/I_\lambda}{(I+I_\lambda)/I_\lambda}\cong U(L)/(I+I_\lambda)$ as $U(L)$-modules, with $I+I_\lambda$ of finite codimension, proving $N$ a finite-dimensional $U(L)$-module. By Weyl's theorem $N$ is completely reducible, $N=\bigoplus_{i=1}^n N_i$ with $N_i$ irreducible and finite dimensional for $1\leq i\leq n$. The primitive ideals $M_i:=\Ann N_i$ are maximal by Jacobson's density theorem (<cit.>), in particular pairwise coprime, so $I=\Ann N=\Ann(\bigoplus_{i=1}^n N_i) = \bigcap_{i=1}^n\Ann N_i =\bigcap_{i=1}^n M_i$ and (by the Chinese remainder theorem) $U(L)/I\cong \prod_{i=1}^n U(L)/M_i$ is a sum of $n\geq1$ matrix algebras ($n>0$ because $I\neq U(L)$). Suppose $n\geq2$; then there are at least two different maximal ideals $M_1,M_2$ of $U(L)$ containing $I$, with $I\cap Z$ maximal in $Z$ forcing $M_1\cap Z = I\cap Z = M_2\cap Z$. But Dixmier's theorem implies the uniqueness of the maximal ideal of $U(L)$ lying over $I\cap Z$, so $M_1=M_2$, a contradiction which forces $n=1$. So $U(L)/I$ is simple and $I$ is maximal.
Hopf algebras.
Consider a Hopf algebra $H$ with comultiplication $\Delta:H\rightarrow H\otimes H$ (in particular $\Delta(xy)=\Delta(x)\Delta(y)$ in $H\otimes H$). $H$ is cocommutative if $\tau\circ\Delta=\Delta$ for the twist map $\tau:H\otimes H\rightarrow H\otimes H$ defined by $\tau(a\otimes b):=b\otimes a$. If $H$ is cocommutative then $H^{\op}$ is also a Hopf algebra with the same comultiplication, counit, and antipode.
We write comultiplications in Sweedler's notation, $\Delta(h)=:\sum h_1\otimes h_2$. For $n\in\N^*$, the $(n-1)$th iterated comultiplication is the operation $\Delta_{n-1}:H\rightarrow \overbrace{H\otimes\cdots\otimes H}^n$ given iteratively by
\[\Delta_{n-1}(h):=\sum \Delta(h_1)\otimes\cdots\otimes h_{n-1} =: \sum h_1\otimes\cdots\otimes h_n\]
(which is well defined by the coassociativity axiom, see <cit.>).
Given $H$, we say that $A$ has a right Hopf (algebra) $H$-action, or that $A$ is a right $H$-module algebra, if there is a right algebra action of $H$ on $A$, $\cdot:A\otimes H\rightarrow A$, such that $(ab)\cdot h=\sum(a\cdot h_1)(b\cdot h_2)$ and $1\cdot h=\varepsilon(h)1$, for all $a,b\in A$ and all $h\in H$, with $\Delta(h)=\sum h_1\otimes h_2$. When $A$ has a right Hopf action, then, for $h\in H$,
\[(a_1\ldots a_n)^h = \sum a_1^{h_1}\cdots a_n^{h_n}, \, \text{ with }\Delta_{n-1}(h)=\sum h_1\otimes\cdots\otimes h_n.\]
The universal enveloping algebra $U(L)$ of $L$ (and hence $\Uop$) becomes a cocommutative Hopf algebra when endowed with comultiplication $\Delta(\delta):=\delta\otimes 1 + 1\otimes \delta$, counit $\varepsilon(\delta):=0$ and antipode $S(\delta):=-\delta$ for $\delta\in L$, and extended to $U(L)$ via the PBW theorem: $\Delta(e_{i_1}^{k_1}\cdots e_{i_j}^{k_j} ):=\Delta(e_{i_1})^{k_1}\cdots\Delta(e_{i_j})^{k_j}$, $S(e_{i_1}^{k_1}\cdots e_{i_j}^{k_j} ):=S(e_{i_1})^{k_1}\cdots S(e_{i_j})^{k_j}$. E.g., for $\delta_1,\delta_2\in L$ we have $\Delta(\delta_1\delta_2)=\delta_1\delta_2\otimes 1 + \delta_1\otimes\delta_2 + \delta_2\otimes\delta_1 + 1\otimes\delta_1\delta_2$ and $\Delta_2(\delta_1\delta_2)= \delta_1\delta_2\otimes 1\otimes 1 + \delta_1\otimes\delta_2\otimes 1 + \delta_1\otimes 1\otimes\delta_2 + \delta_2\otimes\delta_1\otimes 1 + \delta_2\otimes 1\otimes\delta_1 + 1\otimes\delta_1\delta_2\otimes 1 + 1\otimes\delta_1\otimes\delta_2 + 1\otimes\delta_2\otimes\delta_1 + 1\otimes 1\otimes\delta_1\delta_2$.
§.§ The variety of $L$-algebras
$L$-algebras. Given $L$, we say that $A$ is an $L$-algebra or that $L$ acts on $A$ by derivations, if there exists a homomorphism of Lie algebras $\varphi:L\to\Der(A)$, hence $A$ has a right $L^{\op}$-action satisfying $(a_1a_2)^\delta = a_1^\delta a_2+a_1a_2^\delta$ for $a_1,a_2\in A$, $\delta\in L$. From now on, when we say that $A$ has an $L$-action it will imply that $A$ is an $L$-algebra for that $L$-action. Note that when $L$ is simple either $\varphi=0$ or $\varphi$ is a monomorphism.
By the universal property of $U(L)$, an $L$-action on $A$ can be uniquely extended to a right Hopf $\Uop$-action (which by abuse of notation we also call an $L$-action), by extending $\varphi$ to the homomorphism of unital associative algebras $\phi:U(L)\to\E_F(A)$ such that $\phi(ab):=\vp(a)\vp(b)$ (recall that we also denote the opposite homomorphism $\phi^{\op}:\Uop\to\E_F(A)^{\op}$ by $\phi$, see <ref>). In this way, $A$ becomes a right $\Uop$-module algebra (with action in exponential notation and opposite product denoted by juxtaposition).
More explicitly, the $L$-action on $A$ satisfies
\[(a_1a_2\cdots a_n)^\delta =a_1^\delta a_2\cdots a_n + a_1a_2^\delta\cdots a_n + \cdots + a_1a_2\cdots a_n^\delta\]
for $a_1,\ldots,a_n\in A$ and $\delta\in L$, and, if $\Delta_{n-1}(u)=\sum u_1\otimes\cdots\otimes u_n$ for $u\in \Uop$, then
\[(a_1a_2\cdots a_n)^u=\sum a_1^{u_1}a_2^{u_2}\cdots a_n^{u_n}. \tag{U}\label{formula(U)}\]
Note that when $\varphi(L)=0$, then $\phi(U^*(L))=0$, $\phi(1)=1$, and the Hopf $\Uop$-action is just the linear action of $F$.
For fixed $L$ the class of $L$-algebras is equational, and so it is a variety in the sense of universal algebra (see e.g. [7]). This variety is nontrivial, as it contains $A$. Ideals of $L$-algebras ($L$-ideals) are understood to be invariant under the $\Uop$-action, and homomorphisms $f:A\rightarrow B$ between $L$-algebras $A,B$ ($L$-homomorphisms) must satisfy $f(a^u)=f(a)^u$ for $a\in A$, $u\in\Uop$. The $L$-ideal generated by elements $a_1,\ldots,a_n\in A$ we denote by $\la a_1,\ldots,a_n\ra_L$. In the next result, which we call the primitive element lemma, it is shown that some $L$-ideals are principal; it is a direct generalization of <cit.> by Catoiu.
Let $L$ be a finite-dimensional split-semisimple Lie algebra and $A$ be an $L$-algebra. If the $L$-ideal $I$ of $A$ is generated by weight vectors $a_1,\ldots,a_n\in A$ of different weights, then $I$ is generated by $a_1+\cdots+a_n$.
Let $H$ be a fixed Cartan subalgebra of $L$ with basis $\{h_1,\ldots,h_m\}$ and let $\lambda_{ij}$ denote the eigenvalue of $h_i$ for $a_j$. We proceed by induction on $n$. If $n=1$ then the result is trivial. Suppose $n>1$ and that the conclusion is true for all $r<n$. Since $a_1,\ldots,a_n$ have different weights, there exists $1\leq i\leq m$ such that $z_{in}:=h_i-\lambda_{in}$ does not kill all $a_j$. Reorder the $a_j$ so that $a_j^{z_{in}}=0$ if and only if $j>r$ for some $1\leq r<n$ (it kills at least $a_n$). Put $a:=a_1+\cdots+a_n$. Then $a^{z_{in}} = \alpha_1a_1+\cdots + \alpha_r a_r\in \la a\ra_L$ with $0\neq\alpha_1,\ldots,\alpha_r\in K$. By the inductive hypothesis, $\la a_1,\ldots,a_r\ra_L = \la \alpha_1a_1,\ldots,\alpha_ra_r\ra_L = \la \alpha_1a_1+\cdots +\alpha_ra_r\ra_L \subseteq \la a\ra_L$, which implies that $a_{r+1}+\cdots+a_n\in \la a\ra_L$ since $a=a_1+\cdots+a_n$. By the inductive hypothesis again, $\la a_{r+1},\ldots,a_n\ra_L = \la a_{r+1}+\cdots+a_n\ra_L\subseteq \la a\ra_L$. Therefore $\la a \ra_L=I$.
The variety of $L$-algebras contains the free (nonunital associative) $L$-algebra $\FL$, freely generated by the countably infinite set of variables $X:=\{x_1,x_2, \dots\}$, which satisfies the following universal property: each map $\gamma:X\rightarrow A$ to an $L$-algebra $A$ can be uniquely extended to an $L$-homomorphism $\FL\rightarrow A$, which we call the evaluation of $\FL$ at elements $\gamma(x_1),\gamma(x_2)\ldots$ from $A$.
We can describe $\FL$ as follows: $\FL$ is generated as an algebra by the set $\{x_i^u \ | \ i\in \N^*, u\in\Uop\}$, subject to the relations $x_i^1=x_i$, $x_i^{\lambda u+v}=\lambda x_i^u + x_i^v$, $(\lambda x_i+x_j)^u =\lambda x_i^u+x_j^u$ for all $u,v\in\Uop$, $\lambda\in F$ and $i,j\in \N^*$. Note that given a basis $\mathcal B :=\{e_i\}_{i\in\mathcal I}$ of $\Uop$, $\FL$ is generated as an algebra by the set $\{x_i^{e_j} \ | \ i\in\N^*, j\in\mathcal I\}$ and, moreover, the set
\[\left\{x_{i_1}^{e_{j_1}}\cdots x_{i_n}^{e_{j_n}} \ | \ n\in \N^*, i_1,\ldots,i_n\in \N^*, e_{j_1},\ldots,e_{j_n}\in\mathcal B\right\}\]
is a basis of $\FL$.
The free $L$-algebra is endowed with an $L$-action determined (as in (<ref>)) by
\[(x_{i_1}^{e_{j_1}}\cdots x_{i_n}^{e_{j_n}})^u := \sum x_{i_1}^{e_{j_1}u_1}\cdots x_{i_n}^{e_{j_n}u_n}\]
for $e_{j_1},\ldots,e_{j_n}\in\mathcal B$ and $u\in\Uop$ with $\Delta_{n-1}(u)=\sum u_1\otimes\cdots\otimes u_n$.
The elements of the free $L$-algebra are called differential polynomials or $L$-polynomials. A $T_L$-ideal of the free $L$-algebra is an $L$-ideal which in addition is invariant under all $L$-endomorphisms of $\FL$ or substitutions, which send $x_i\mapsto f_i$ for $i\in\N^*$ and $f_i\in\FL$; e.g., there is a substitution sending $x_1$ to $x_1x_2$ and $x_i$ to $x_i$ for $i\neq1$. Special substitutions are those mapping $x_i\mapsto x_{\sigma(i)}^{u_i}$ for $i\in I$ and $x_j\mapsto x_j$ for $j\not\in I$, for given $I:=\{i_1,\ldots,i_n\}$, $\sigma\in S_n$ acting on $I$ and $u_i\in\Uop$ for $i\in I$, which we call substitutions swapping variables. When referring to elements of a $T_L$-ideal in at most two variables we write them with “generic” variables $x,y$; since $T_L$-ideals are closed under substitutions, $x,y$ may be replaced by any $L$-polynomials $f,g\in\FL$.
Given a set $S\subseteq\FL$, by $\la S\ra_{T_L}$ we denote the smallest $T_L$-ideal containing $S$.
§.§ The variety of $(L,U)$-algebras
We want to avoid, as much as possible, the infinite dimensionality of $U(L)$ in the determination of the differential identities of an $L$-algebra. For this reason, we introduce $(L,U)$-algebras.
Fix $L$, an $L$-algebra $A$, and the homomorphism $\phi: U(L)\to\E_F(A)$ corresponding to the right Hopf $ \Uop$-action on $A$. We denote $ U:=\phi(U(L))\subseteq\E_F(A)$; note that $U$ is a unital associative algebra, which is finite dimensional when $A$ is, but not necessarily a Hopf algebra. In the following we omit $\phi$ from the notation, but the reader should be aware of the fact that the concepts defined below depend not only on $L$ and $U$ but also on $\phi$.
$(L,U)$-algebras. The $ \Uop$-action on $A$ induces a right action of $ U^{\op}$ as a unital associative algebra on $A$; this action is not necessarily a Hopf action (it is a generalized Hopf algebra action as defined by Gordienko in <cit.>; see also Berele's <cit.>), but satisfies $a^u=a^{\phi(u)}$ for all $u\in\Uop$ and $a\in A$.
Accordingly, we say that an $L$-algebra $B$ is an $(L,U)$-algebra if it is endowed with a right algebra $ U^{\op}$-action such that $b^u=b^{\phi(u)}$ for all $u\in\Uop$ and $b\in B$, which we call an $(L,U)$-action.
If $B,C$ are two associative algebras and $C$ has a right $ U^{\op}$-action then the associative algebra $B\otimes_F C$ has a right $U^{\op}$-action given by $(b\otimes c)^u:=b\otimes c^u$ for $u\in U^{\op}$, $b\in B$, $c\in C$, while if $\delta$ is a derivation of $C$ then $1\otimes\delta$ is a derivation of $B\otimes_F C$. Therefore, if $C$ is an $(L,U)$-algebra then $B\otimes_F C$ is an $(L,U)$-algebra with $L$-action given by $(b\otimes c)^u:=b\otimes c^u$ for $u\in\Uop$, $b\in B$, $c\in C$.
The class of $(L,U)$-algebras is a variety that contains $A$, denoted by $\textsc V^{L,U}$. Ideals of $(L,U)$-algebras are closed under the $ U^{\op}$-action (equivalently, the $L$-action), and homomorphisms $f:B\rightarrow C$ between $(L,U)$-algebras $B,C$ must satisfy $f(b^u)=f(b)^u$ for $b\in B$, $u\in U^{\op}$ (equivalently, $u\in\Uop$).
The variety of $(L,U)$-algebras contains the free (nonunital associative) $(L,U)$-algebra $\FLU$ freely generated by $X$, which is isomorphic to the quotient $\FL/\ILU$ where
\[\ILU:=\langle\{f^z \ | \ f\in\FL, z\in\ker\phi\}\rangle, \tag{I}\label{formula(I)}\]
which is a $T_L$-ideal.
The structure of $\ILU$ is strongly dependent on the algebraic structure of $U$. For example, if $d:=\phi(\delta)$ with $\delta\in L$ satisfies $d^2=0$, then $x^{d^2}=x^0=0$ for any $x\in\FLU$, and so
\[2x_1^dx_2^d = x_1^{d^2}x_2 +2x_1^dx_2^d +x_1x_2^{d^2} = (x_1x_2)^{d^2} = 0\]
in $\FLU$.
We refer to the elements of $\FLU$ as $(L,U)$-polynomials. The $T_{L,U}$-ideals of $\FLU$ are defined analogously to the $T_L$ ideals of $\FL$, with $\la S\ra_{T_{L,U}}$ denoting the smallest $T_{L,U}$-ideal containing the set $S$.
Observe that the free $(L,U)$-algebra is generated as an associative algebra by the set $\{x_i^u \ | \ i\in \N^*, u\in\mathcal U\}$ for a basis $\mathcal U$ of $U$, albeit not freely, and the construction of a well-behaved basis of $\FLU$ may prove challenging. To circumvent this issue we introduce $\FU$, an algebra with more freeness than $\FLU$ and a better linear parallelism to $\FL$, which is an $L$-algebra but has no specified $U$-action.
We call $\FU$ the free algebra with $U$-exponents and define it as follows: $\FU$ is generated as an algebra by the set $\{x_i^u \ | \ i\in \N^*, u\in U^{\op}\}$, subject to the relations $x_i^1=x_i$, $x_i^{\lambda u+v}=\lambda x_i^u + x_i^v$, $(\lambda x_i+x_j)^u =\lambda x_i^u+x_j^u$ for all $u,v\in U^{\op}$, $\lambda\in F$ and $x_i,x_j\in X$. Note that given a basis $\mathcal U:=\{u_1,\ldots,u_N\}$ of $U$, $\FU$ is freely generated as an associative algebra by $\{x_i^{u_j} \ | \ i\in \N^*, u_j\in \mathcal U\}$ (so this is the same algebra considered by Gordienko in <cit.>). Moreover, the set
\[\{x_{i_1}^{u_{j_1}}\cdots x_{i_n}^{u_{j_n}} \ | \ n\in\N^*, i_1,\ldots,i_n\in \N^*, u_{j_1},\ldots,u_{j_n}\in\mathcal U\}\]
is a basis of $\FU$. In addition, setting the subspaces $M_n^U:= \spn\{x_{i_1}^{u_{j_1}}\cdots x_{i_n}^{u_{j_n}} \ | \ i_1,\ldots,i_n\geq1, \ | u_{j_1},\ldots,u_{j_n}\in\mathcal U\}$ of $U$-monomials of degree $n$ for $n\in\N^*$, we get the grading $\FU=\bigoplus_{n\in \N^*} M_n^U$.
The $L$-action on $\FU$ is determined by
\[(x_{i_1}^{u_{j_1}}\cdots x_{i_n}^{u_{j_n}})^v := \sum x_{i_1}^{u_{j_1}\phi(v_1)}\cdots x_{i_n}^{u_{j_n}\phi(v_n)}\]
for $u_{j_1},\ldots,u_{j_n}\in\mathcal U$ and $v\in\Uop$ with $\Delta_{n-1}(v)=\sum v_1\otimes\cdots\otimes v_n$.
In addition, the operation $(x_i^u)^v:=x_i^{uv}$ for $u,v\in U^{\op}$ is well defined, which allows defining the $L$-endomorphisms of $\FU$ which we call substitutions swapping variables, that map $x_i\mapsto x_{\sigma(i)}^{u_i}$ for $i\in I$ and $x_j\mapsto x_j$ for $j\not\in I$, for given $I:=\{i_1,\ldots,i_n\}$, $\sigma\in S_n$ acting on $I$ and $u_i\in U^{\op}$ for $i\in I$.
Although it has no specified $U$-action, and we are not considering any variety of $U$-algebras which would contain it as a free algebra, the algebra $\FU$ satisfies the following universal property: each map $\gamma:X\rightarrow B$ to an algebra $B$ with right $ U^{\op}$-action can be uniquely extended to a homomorphism of associative algebras $\gamma:\FU\rightarrow B$ such that $\gamma(x_i^u)=\gamma(x_i)^u$ for all $x_i\in X$ and $u\in U^{\op}$. More importantly, although $\FU$ is not an $(L,U)$-algebra, it satisfies $x_i^u=x_i^{\phi(u)}$ for all $u\in\Uop$ and $x_i\in X$. Therefore $\FU$ also satisfies the following universal property: each map $\gamma:X\rightarrow B$ to an $(L,U)$-algebra $B$ can be uniquely extended to an $L$-homomorphism $\gamma:\FU\rightarrow B$ such that $\gamma(f^u)=\gamma(f)^{\phi(u)}$ for all $f\in\FU$ and $u\in\Uop$, which we call an evaluation of $\FU$ at elements $\gamma(x_1),\gamma(x_2)\ldots$ from $B$.
We refer to the elements of $\FU$ as $U$-polynomials. A $T_U$-ideal of $\FU$ is an $L$-ideal (so, invariant under the Hopf $ \Uop$-action) which in addition is invariant under all $L$-endomorphisms of $\FU$; in particular, under all linear substitutions of the form $x_j\mapsto \sum_{i\in I} \alpha_i x_i$ with fixed $j$, finite $I\subseteq\N^*$ and $\alpha_i\in F$ for $i\in I$, and all the substitutions swapping variables. Note that not every substitution of the variables by $U$-polynomials is valid, as not all are $L$-endomorphisms of $\FU$, with this phenomenon depending on the algebraic structure of $U$: e.g., if there is $\delta\in L$ such that $0\neq d:=\phi(\delta)$ satisfies $d^2=0$ then the substitution $\vp$ mapping $x_1\mapsto x_1x_2$ is not an $L$-endomorphism, as $\vp(x_1^{\delta^2}) = \vp(x_1^{d^2}) = 0 \neq 2x_1^dx_2^d = \vp(x_1)^{\delta^2}$.
The $T_U$-ideal generated by $f_1,\ldots,f_m\in\FU$ is the set of $U$-polynomials of the form
\[\sum_{j=1}^r g_j f_{i_j}^{u_j}(p_{j1},\ldots,p_{jk_j}) h_j = \sum_{j=1}^r g_j f_{i_j}(p_{j1},\ldots,p_{jk_j})^{u_j} h_j\tag{TU}\label{formula(TU)}\]
with $r\in\N^*$, $i_j\in\{1,\ldots,m\}$ (where we may have $i_j=i_k$ for $j\neq k$), $u_j\in\Uop$, $g_j,h_j\in\FU$ or $g_j=1$ or $h_j=1$, and $\vp_j(f):=f(p_{j1},\ldots,p_{jk_j})$ with $p_{j1},\ldots,p_{jk_j}\in\FU$ being an $L$-endomorphism of $\FU$ which maps $x_{t_i}\mapsto p_{ji}$ for some $x_{t_i}\in X$ and $1\leq i\leq k_j$.
When referring to elements of a $T_U$-ideal in at most two variables we write them with “generic” variables $x,y$; since $T_U$-ideals are closed under substitutions swapping variables, $x,y$ may be replaced by any variables $x_i^u,x_j^v$ with $x_i,x_j\in X$ and $u,v\in U^{\op}$.
Given a set $S\subseteq\FU$, by $\la S\ra_{T_U}$ we denote the smallest $T_U$-ideal containing $S$.
[Action of $L$ on $\FU$]
* The free algebra with $U$-exponents is not an $(L,U)$-algebra in general: due to their respective universal properties with respect to $(L,U)$-algebras, if both $\FLU$ and $\FU$ were $(L,U)$-algebras, then they would be isomorphic as $(L,U)$-algebras (see <cit.>), in particular as $L$-algebras, which they are not in general.
* Since $\FU$ is not an $(L,U)$-algebra, attention must be paid to the application of the $L$-action: exponents must be in $\Uop$ in general, and can only be taken from $U^{\op}$ when applied directly on “isolated” variables $x_i$. A expression like $(x_1\cdots x_n)^u$ with $n>1$ and $u\in U^{\op}$ makes no sense in $\FU$, and $(x_1\cdots x_n)^u$ with $u\in\Uop$ expands to $\sum x_1^{v_1}\cdots x_n^{v_n}$ for $\Delta_{n-1}(u)=\sum u_1\otimes\cdots\otimes u_n$ and $v_i:=\phi(u_i)$ for $i\in\{1,\ldots,n\}$.
* The Hopf $U(L)$-action on $\FU$ does not necessarily produce a $U$-action on $\FU$ via $x^{\phi(u)}:=x^u$ for $x\in\FU$, $u\in\Uop$, as we may have $u,v\in\Uop$ generating different actions on $\FU$ and such that $\phi(u)=\phi(v)$. E.g, if $\delta\in L$ satisfies $\phi(\delta^2)=0=\phi(0)$ but $\phi(\delta)\neq0$, then
\[(x_1x_2)^{\delta^2} = 2x_1^{\phi(\delta)}x_2^{\phi(\delta)} \neq 0 = (x_1x_2)^0\]
(recall that we have actually designed $\FU$ for this to happen).
§.§ Identities and growth
$L$-identities, $L$-codimensions and $L$-exponent. A differential polynomial $f(x_1, \dots , x_n)\in F\langle X| L\rangle$ is a differential identity or $L$-identity of the $L$-algebra $B$ if $f(b_{1},\dots,b_{n})=0$ for any $b_1,\ldots,b_n\in B$ ($f$ vanishes under all evaluations of $\FL$ at elements from $B$). We denote by $\I^L(B)$ the set of differential identities of $B$, which is a $T_L$-ideal of the free $L$-algebra (in particular $\I^L(B)$ is closed under the Hopf $\Uop$-action and under substitutions). Note that $\I^L(B)$ is the intersection of all kernels of evaluations of $\FL$ from $B$.
For $n\geq 1$ we denote by $P_n^L$ the vector space of multilinear differential polynomials in the variables $x_{1},\dots,x_{n}$, so that
\[P_n^L:=\spn_F\{x_{\sigma(1)}^{e_{i_1}}\cdots x_{\sigma(n)}^{e_{i_n}} \ | \ \sigma\in S_{n} ,e_{i_1},\ldots,e_{i_n}\in \mathcal{B} \},\]
where $S_n$ denotes the symmetric group acting on $\{1,\ldots,n\}$.
As in the ordinary case, since $F$ has characteristic zero, a Vandermonde plus linearization argument shows that the $T_L$-ideal $\I^L(B)$ is completely determined by its multilinear $L$-polynomials (see <cit.>).
We also consider the vector space
\[P_n^L(B):= \dfrac{P_n^L}{P_n^L \cap \I^L(B)}.\]
When the action of $\Uop$ is finite dimensional, i.e., when $U$ is a finite-dimensional algebra, the $n$th differential codimension of $B$ is $c_n^L(B):=\dim_F P_n^L(B)$. Moreover, if $B$ is finite dimensional then the limit $\exp^L(B):=\lim_{n\to \infty}\sqrt[n]{c_n^L(B)}$ exists and is a nonnegative integer called the $L$-exponent of $B$ ([31]).
Growth of $L$-varieties. Given a variety $\textsc V$ of $L$-algebras, the growth of $\textsc V$ is defined as the growth of the sequence of differential codimensions of any $L$-algebra $B$ generating $\textsc V$, i.e., $\textsc V=\V^L(B)$. In this case we set $c_n^L(\textsc V):=c_n^L(B)$, $n\geq 1$, and $\exp^L(\textsc V):=\exp^L(B)$. Then we say that $\textsc V$ has polynomial growth if there exist $C,t>0$ such that $c_n^L(\textsc V) \leq C n^t$, i.e., $\exp^L(\textsc V)\leq 1$, and that $\textsc V$ has almost polynomial growth if $c_n^L(\textsc V)$ is not polynomially bounded, i.e., $\exp^L(\textsc V)>1$, but every proper subvariety of $\textsc V$ has polynomial growth.
Analogues for $U$-algebras and $(L,U)$-algebras.
Mutatis mutandis, for $B$ an associative algebra with right $ U^{\op}$-action (resp. an $(L,U)$-algebra), inside $\FU$ (resp. $\FLU$) we define the $U$-identities (resp. $(L,U)$-identities) of $B$, the $T_U$-ideal $\I^U(B)$ closed under the Hopf $\Uop$-action and the valid substitutions (resp. the $T_{L,U}$-ideal $\I^{L,U}(B)$ closed under the Hopf $\Uop$-action and substitutions), the vector space of multilinear $U$-polynomials $P_n^U$ (resp. of multilinear $(L,U$)-polynomials $P_n^{L,U}$), the quotient $P_n^U(B)$ (resp. $P_n^{L,U}(B)$), the $n$th $U$-codimension $c_n^U(B)$ (resp. the $(L,U)$-codimension $c_n^{L,U}(B)$) when $U$ is finite dimensional. If $\exp^U(B):=\lim_{n\to\infty}\sqrt[n]{c_n^U(B)}$ (resp. $\exp^{L,U}(B):=\lim_{n\to\infty}\sqrt[n]{c_n^{L,U}(B)}$) exists, we call it the $U$-exponent (resp. $(L,U)$-exponent) of $B$; see Lemma <ref> below.
Similarly, for a variety of $(L,U)$-algebras we define the growth and the notions of polynomial growth and almost polynomial growth.
§.§ Computing $L$-data through $U$-data
In this section we relate $L$-identities to $U$-identities, $L$-varieties to $(L,U)$-varieties, and $L$-cocharacters to $U$-cocharacters.
Since $\FU$ is an $L$-algebra and $\FL$ is the free $L$-algebra, we can consider the $L$-homomorphism $\Psi:\FL\to\FU$ sending $x_i\mapsto x_i$, which is defined by $\Psi(x_i^u):=x_i^{\phi(u)}$ for all $x_i\in X$ and $u\in\Uop$. We denote $\IU:=\ker\Psi$. Analogously we have the $L$-homomorphism $\Theta:\FL\to\FLU$ sending $x_i\mapsto x_i$, defined by $\Theta(x_i^u):=x_i^{\phi(u)}$, which satisfies $\ker\Theta=\ILU$. We have $\IU\subseteq\ILU$. In addition, since $\FLU$ is an $(L,U)$-algebra, we have the $L$-homomorphism $\Gamma:\FU\to\FLU$ sending $x_i\mapsto x_i$, which satisfies $\Theta=\Gamma\circ\Psi$.
Let $B$ be any $(L,U)$-algebra. By definition $\Psi(\I^L(B))=\I^U(B)$, $\IU\subseteq \I^L(B)$, and $\Psi(P_n^L)=P_n^U$. Hence by the isomorphism theorems we get $\dfrac{\I^L(B)}{\IU} \cong \I^U(B)$, $\dfrac{\FL}{\I^L(B)} \cong \dfrac{\FU}{\I^U(B)}$(as $L$-algebras) and $P_n^L(B)\cong P_n^U(B)$ (as vector spaces). We get analogous results for $\FLU$ from $\Theta$, proving the next elementary lemma.
Let $B$ be an $(L,U)$-algebra.
* Let $G=\{g_i\}_{i\in\mathcal I}$ be a system of generators of $\I^U(B)$ (resp. $\I^{L,U}(B)$) as a $T_L$-ideal (resp. $T_{L,U}$-ideal), and for each $g_i\in G$ pick a fixed preimage $f_i\in\Psi^{-1}(g)$ (resp. $f_i\in\Theta^{-1}(g)$). Let $F:=\{f_i\}_{i\in\mathcal I}$. Then $\I^L(B)=\la F\ra_{T_L}+\IU$ (resp. $\I^L(B)=\la F\ra_{T_L}+\ILU$).
* If $U$ is finite dimensional, then $c_n^L(B)=c_n^U(B)=c_n^{L,U}(B)$ for all $n\geq 1$. Moreover, if $B$ is finite dimensional, then $\exp^U(B)$, $\exp^{L,U}(B)$ exist and $\exp^L(B)=\exp^U(B)=\exp^{L,U}(B)$.
Although $\ILU$ is a $T_L$-ideal, $\IU$ is not a $T_L$-ideal in general: it may not be invariant under the substitution $x_1\mapsto x_1x_2$, $x_i\mapsto x_i$ for $i\neq1$, as $x_1^z\in\IU$ for $z\in\ker\phi$ but $(x_1x_2)^z$ may not belong to $\IU$. E.g., $z:=\delta^2$ with $\delta\in L$, $\delta\not\in\ker\phi$ and $\phi(\delta)^2=0$ satisfies $x_1^z\in\IU$ and $(x_1x_2)^z = x_1^zx_2 + 2x_1^\delta x_2^\delta + x_1x_2^z\not\in\IU$ since $x_1^zx_2, x_1x_2^z\in\IU$ but $x_1^\delta x_2^\delta\not\in\IU$.
Nevertheless, $\IU$ is invariant under substitutions swapping variables (since $\FU\cong\FL/\IU$).
Therefore, since $\IU\subseteq\I^L(A)$, which is a $T_L$-ideal, the $T_L$-ideal $\langle\IU\rangle_{T_L}$ may contain some interesting $L$-identities of $A$, obtained from substitutions in elements of $\IU$. More concretely, we have:
* $\IU$ is generated as an associative algebra ideal by the set $\{x_i^z \ | \ x_i\in X, z\in\ker\phi\}$, and as an ideal with substitutions swapping variables by the identity $x^z$ for any $z\in\ker\phi$.
* $\langle\IU\rangle_{T_L} = \ILU = \displaystyle\bigcap_{B\in\textsc V^{L,U}}\I^L(B)$.
* Since the restriction $\phi: U(L)\rightarrow U$ is an epimorphism, we can write $ U=V\oplus\ker\phi$ for some $V$ such that the restriction $\phi:V\rightarrow U$ is an isomorphism of vector spaces. Let $\mathcal V$, $\mathcal K$ be bases of $V$ and $\ker\phi$, respectively, and set $\mathcal B:=\mathcal V\cup\mathcal K$, which is a basis of $\Uop$. Then $\mathcal F_L:=\{x_{i_1}^{e_{j_1}}\cdots x_{i_n}^{e_{j_n}} \ | \ n\in \N^*, i_1,\ldots,i_n\in \N^*, e_{j_1},\ldots,e_{j_n}\in\mathcal B\}$ is a basis of $\FL$, and we can write $\mathcal F_L=M^V\cup M^K$, where $M^V$ is the set of monomials whose variables have all its exponents in $V$ and $M^K$ is the set of monomials which have at least one variable with exponent in $\ker\phi$. Hence for $f\in\FL$ we can write $f=\sum_{i\in\mathcal I}\alpha^V_i m^V_i +\sum_{j\in\mathcal J}\alpha^K_j m^K_j$, where $\mathcal I,\mathcal J$ are finite sets, $\alpha^V_i,\alpha^K_j\in F$ for all $i\in\mathcal I, j\in\mathcal J$, $m^V_i\in M^V$ for $i\in\mathcal I$ and $m^K_j\in M^K$ for $j\in\mathcal J$. On the other hand, $\mathcal F_U:=\left\{x_{i_1}^{u_{j_1}}\cdots x_{i_n}^{u_{j_n}} \ | \ n\in \N^*, i_1,\ldots,i_n\in \N^*, u_{j_1},\ldots,u_{j_n}\in\mathcal U\right\}$ is a basis of $\FU$ such that $\Psi(M^V)=\mathcal F_U$. Then, since $\Psi(M^K)=0$, $\Psi(f)= \sum_{i\in\mathcal I}\alpha^V_i\Psi(m^V_i)$ is a linear combination of monomials from the basis $\mathcal F_U$ and thus $\Psi(f)=0$ implies $\alpha^V_i=0$ for all $i\in\mathcal I$, i.e., $f\in\spn M^K$ as we wanted to show.
Since $\IU$ is invariant under substitutions swapping variables, the second claim of this item follows.
* By definition, $\ILU=\langle\{x^z \ | \ z\in\ker\phi\}\rangle_{T_L}$, and clearly $\langle\{x^z \ | \ z\in\ker\phi\}\rangle_{T_L} = \langle\{x_i^z \ | \ x_i\in X, z\in\ker\phi\}\rangle_{T_L} = \langle\IU\rangle_{T_L}$ by item (1).
On the other hand, given $f\in\ILU$ in $n$ variables we can write $f=\sum_{i\in\mathcal I} h_if_i^{z_i}g_i$ with $h_i,f_i,g_i\in\FL$ and $z_i\in\ker\phi$ for a finite set $\mathcal I$. Then, for any $(L,U)$-algebra $B$ and any $b_1,\ldots,b_n\in B$ we have $f(b_1,\ldots,b_n)= \sum_{i\in\mathcal I} h_i(b_1,\ldots,b_n)(f_i(b_1,\ldots,b_n))^{\phi(z_i)}g_i(b_1,\ldots,b_n) = 0$ since $\phi(z_i)=0$, what implies $\ILU\subseteq\I^L(B)$. Moreover, we have $\I^L(\FLU) = \ILU$ because $\FLU$ is the free $(L,U)$-algebra. Thus
we get
\[\ILU\subseteq\bigcap_{\mathclap{B\in\textsc V^{L,U}}}\; \I^L(B)\subseteq\I^L(\FLU) = \ILU.\qedhere\]
\[\xymatrix{\FL \ar[r]^-\Psi\ar@(dr,dl)[rr]_-\Theta & \FU\cong \FL/\IU \ar[r]^-\Gamma & \FLU\cong \FL/\ILU}\]
\[\ILU = \langle\IU\rangle_{T_L} = \langle\{f^z \ | \ f\in\FL, z\in\ker\phi\}\rangle = \displaystyle\bigcap_{B\in\textsc V^{L,U}}\I^L(B)\]
Relationships between the different free algebras
From the previous results we derive the following general strategy for computing the differential identities of $A$:
* We determine its $U$-identities by exploiting the structure of the finite-dimensional algebra $ U\subseteq\End_F(A)$ and the good behavior of $\FU$. We find a system of generators of $\I^U(A)$ (which gives also a system of generators of $\I^{L,U}(A)$) and reduce it to a system $G$ by resorting to the $L$-action. We consider the system $F$ of some fixed preimages of $G$ in $\I^L(A)$.
* We determine a system of generators $Z$ of the ideal $\ker\phi$, with the aid of the representation theory of $L$ applied to $U(L)$ and of the algebraic geometry of $U(L)$. Then $\ILU$ is generated by $K:=\{x^z, z\in Z\}$ by substitutions and the $L$-action, as $x^{uzv}=((x^u)^z)^v$ for $u,v\in\Uop$, $z\in\ker\phi$.
* We check if any element of $F$ is generated by the others plus $\la\IU\ra_{T_L}=\ILU$. If so, we remove it and check again.
* We find the (small) system of generators $F\cup K$ of $\I^L(A)$.
Let $B$ be an $(L,U)$-algebra. We not only have $\Psi(\I^L(B))=\I^U(B)$, but also $\Psi^{-1}(\I^U(B))=\I^L(B)$, and similarly $\Theta^{-1}(\I^{L,U}(B))=\I^L(B)$ and $\Gamma^{-1}(\I^{L,U}(B))=\I^U(B)$, since $f(b_1,\ldots,b_n)=\Psi(f)(b_1,\ldots,b_n)=\Theta(f)(b_1,\ldots,b_n)$ for all $f\in\FL$ and all $b_1,\ldots,b_n\in B$. In particular, $f\in \FL$ satisfies $f\in\I^L(B)$ if and only if $\Psi(f)\in\I^U(B)$ if and only if $\Theta(f)\in \I^{L,U}(B)$, and $f\in\FU$ satisfies $f\in\I^U(B)$ if and only if $\Gamma(f)\in\I^{L,U}(B)$.
Let $B$ be an $(L,U)$-algebra and $C\in\V^L(B)$.
* $C$ is an $(L,U)$-algebra such that $C\in\V^{L,U}(B)$.
* $\V^L(B)$ has almost polynomial growth if and only if $\V^{L,U}(B)$ has almost polynomial growth.
* $\I^U(B) \subseteq \I^U(C)$, and $\V^{L,U}(C)$ is a proper subvariety of $\V^{L,U}(B)$ if and only if there exists a $U$-polynomial $f\in\I^U(C)\setminus\I^U(B)$.
* Since $B$ is an $(L,U)$-algebra, $\ILU\subseteq\I^L(B)$ by Proposition <ref>(2), and since $C\in\V^L(B)$, $\I^L(B)\subseteq \I^L(C)$. Therefore $\ILU\subseteq\I^L(C)$, whence $x^z\in\I^L(C)$ for all $z\in\ker\phi$, so the right $ U^{\op}$-action $c^u:=c^v$ is well defined for any $c\in C$, $u\in U$ and $v\in\phi^{-1}(u)$ ($\phi(v_1)=\phi(v_2)$ implies $v_1-v_2\in\ker\phi$, so $c^{v_1}=c^{v_2}$), and is clearly an $(L,U)$-action. In addition, $\I^{L,U}(B)=\Theta(\I^L(B))\subseteq\Theta(\I^L(C))=\I^{L,U}(C)$, hence $C\in\V^{L,U}(B)$.
* By item (1) and the fact that every $(L,U)$-algebra is an $L$-algebra we get $\V^L(B)=\V^{L,U}(B)$ as sets, and $c_n^L(C)=c_n^{L,U}(C)$ for every $C\in\V^{L,U}(B)$ by Lemma <ref>(2), which in particular implies that $C\in\V^L(B)$ generates a proper $L$-subvariety if and only if it generates a proper $(L,U)$-subvariety of $\V^{L,U}(B)$.
* $\I^{L,U}(B) \subseteq \I^{L,U}(C)$ by item (1), so $\I^{U}(B) \subseteq \I^{U}(C) $ by Remark <ref>. Moreover, $\V^{L,U}(C)$ is a proper subvariety of $\V^{L,U}(B)$ if and only if there exists $g\in \I^{L,U}(C)\setminus \I^{L,U}(B)$, if and only if there exists $f\in\Gamma^{-1}(g)$ such that $f\in\I^U(C)\setminus \I^U(B)$ by Remark <ref>.
Therefore we can study the growth of $\V^L(A)$ and its subvarieties by considering $(L,U)$-algebras, $U$-polynomials, and $U$-codimensions.
§ MATRIX SETTING
§.§ Derivations of $\M$
In this section, we describe the enveloping algebra $ U$ of the Lie algebra of derivations of $\M$ for $k\geq 2$. Let $Z_k(F)$ denote the center of $\M$ (i.e., the scalar multiples of the identity matrix $I_k$) and let $\Sl_k(F)$ denote the special Lie algebra of order $k$, that is, the set of traceless matrices inside $\M$ endowed with the bracket product.
$\bs{\Der(\M)}$ is isomorphic to $\bs{\SL}$.
As a consequence of the Noether-Skolem theorem, all derivations of $\M$ are inner (<cit.>), so $\ad:\M\rightarrow\Der(\M)$ is a surjective linear map between vector spaces. In addition $\ad(A)=\ad(B)$ if and only if $A-B\in Z_k(F)$, so $\ad:\M/Z_k(F)\rightarrow\Der(\M)$ is a linear isomorphism, which moreover satisfies $\ad([A,B])=[\ad_A,\ad_B]$, giving an isomorphism of Lie algebras between $\Der(\M)$ and $\M/Z_k(F)$. On the other hand, since $\chr(F)=0$ we have $\M= Z_k(F)\oplus\SL$ (direct sum of Lie ideals) and hence $\M/Z_k(F)\cong\SL$ as Lie algebras in a natural way.
From now on we identify $\Der(\M)$ with $\SL$ as the inner derivations arising from $\SL\subseteq\M$, and we fix $L:=\SL$ for the rest of this paper. Observe that $L$ is split simple (<cit.>).
Structure of $\bs{U}$.
From the exposition of the previous paragraph, we infer that the $L$-action of $\SL$ on $\M= Z_k(F)\oplus\SL$ is the direct sum of the trivial action $\rho_0$ on the center and the adjoint action $\Ad$ on $\SL$, whence the image $U=\phi(U(L))$ of the left representation $\phi$ of $U(L)$ on $\M$ is the direct sum $U=U_1\oplus U_2\subseteq\E_F(\M)$ with $U_1=\E_F(F\cdot I_k)\cong F$ and $U_2$ the enveloping algebra of the adjoint action. Since $\Ad$ is finite dimensional and irreducible (because subrepresentations of $\Ad$ correspond to ideals of $L$, which is simple), by Theorem <ref> we have $U_2=\E_F(\SL)$. Therefore
\[U =\E_F(F\cdot I_k)\oplus\E_F(\Sl_k(F)).\]
In particular, $U$ is a split-semisimple algebra of dimension $(k^2-1)^2+1$.
§.§ Explicit description of $U^{\op}$
In this section and the next we describe how to operate with exponents coming from $U^{\op}$. We denote the product of $U^{\op}$ by juxtaposition.
Basis of $\M$.
Let $\{e_{ij}\}_{i,j=1}^k$ be the standard matrix units of $\M$ (with 1 as $(i,j)$ entry and $0$ elsewhere) and denote $h_i:=e_{ii}-e_{i+1,i+1}$ for $1\leq i<k$. Then a basis of $\SL$ is
\[\mathcal{S}:=\{e_{ij}|1\leq i\neq j\leq k\}\cup\{h_1,\ldots,h_{k-1}\},\]
which we expand to a basis $\mathcal{M}$ of $\M=\SL\oplus Z_k(F)$ by appending $g:=I_k$,
\[\mathcal M:=\mathcal S\cup\{g\}.\]
We will also refer to elements $h_{ij}:=e_{ii}-e_{jj}$ for $i,j\in\{1,\ldots,k\}$, $i\neq j$ (thus $h_i=h_{ii+1}$ for $1\leq i<k$). We have $h_{ij}=-h_{ji}$. Let us write $(-1)^{i>j}:=1$ if $i\leq j$ and $(-1)^{i>j}:=-1$ if $i>j$. Then in basis $\mathcal S$ we have
\[h_{ij}=(-1)^{i>j}\sum_{l=\min(i,j)}^{\max(i,j)-1} h_l.\]
Basis of $U^{\op}$.
Write $x\in\Mat_k(F)$ in basis $\mathcal M$ as $x=\sum_{a\in\mathcal M}\mu_a^x a$, i.e., $\mu_a^x$ denotes the coefficient of $x$ with respect to $a\in\mathcal M$. Then, given $a,b\in\mathcal M$ define $\varphi_{ab}\in\E_F(\M)^{\op}$ by
\[\varphi_{ab}(x):=\mu_a^x b,\]
i.e., $\varphi_{ab}$ is the endomorphism sending basis element $a$ to basis element $b$ and the remaining basis elements to $0$.
For example, if $x=e_{12}+2h_1+3h_2\in\Mat_4(F)$ then $x^{\varphi_{h_1e_{23}}}=2e_{23}$, $x^{\varphi_{h_3e_{23}}}=0$ and
\[x^{\varphi_{h_1e_{23}}\varphi_{e_{23}h_4}}=(x^{\varphi_{h_1e_{23}}})^{\varphi_{e_{23}h_4}} = (2e_{23})^{\varphi_{e_{23}h_4}} = 2h_4.\]
We also define endomorphisms $\vp_{ab}$ for any $a\in\mathcal S$ and $b\in\SL$ by linearity. In particular, for elements $h_{ij}$ and $a\in\mathcal S$, we define
\[\vp_{ah_{ij}}:=(-1)^{i>j}\sum_{l=\min(i,j)}^{\max(i,j)-1} \vp_{ah_l}.\]
Note that, in $\E_F(\M)^{\op}$, for $a,b,c,d\in\mathcal M$ we have
\[\vp_{ab}\vp_{cd}=\delta_{bc}\vp_{ad} \tag{F}\label{formula(F)}\]
where $\delta_{bc}$ is Kronecker's delta, since $(x^{\vp_{ab}})^{\vp_{cd}}=(\mu_a^x b)^{\vp_{cd}} = \delta_{bc}\mu_a^x d = \delta_{bc}x^{\vp_{ad}}$ for $x\in\M$.
In fact, $\{\varphi_{ab}\}_{a,b\in\mathcal M}$ is nothing else than the standard set of matrix units of $\E _F(\M)^{\op}\cong\Mat_{k^2}(F)$ when basis $\mathcal M$ is fixed for $\M$. With this presentation, $U_2^{\op}\cong\Mat_{k^2-1}(F)$ has $\{\varphi_{ab}\}_{a,b\in\mathcal S}$ as a basis and $U_1^{\op}\cong F$ corresponds to the subspace of endomorphisms spanned by $\varphi_{gg}$. This is the presentation we will use in the following; hence from now on we fix the basis of $U^{\op}$
\[\mathcal U:=\{\vp_{ab}\}_{a,b\in\mathcal S}\cup\{\vp_{gg}\}.\]
Notice that $I_{k^2}=\sum_{a\in\mathcal M} \varphi_{aa}$, so in this way we avoid the explicit use of the identity endomorphism and thus the participation of the problematic ordinary polynomial identities.
To prevent the notation from becoming too cumbersome, throughout the rest of the paper we will omit the letters $\varphi$ from the exponents when applying endomorphisms of $U^{\op}$ to $\M$ or writing $U$- or $(L,U$)-polynomials; so, for example, $x^{h_1h_2}$ is shorthand for $x^{\varphi_{h_1h_2}}$. This notation of the form $x^{ab}$ with $a,b\in\mathcal\SL$ for polynomial $x^{\vp_{ab}}$ in $\FU$ or in $\FLU$ should never be confused with notation for $L$-polynomial $(x^a)^b$ in $\FL$ with $a,b\in L$; we will never write $L$-polynomials in the latter way.
§.§ Computations involving $U^{\op}$
Multiplication in $\SL$.
The Lie multiplication table of $\mathcal M$ is summarized by the following relations:
* $[g,x]=0$ for any $x\in\mathcal M$.
* $[e_{ij},e_{kl}]=0$ ($j\neq k, l\neq i$), $[e_{ij},e_{jk}]=e_{ik}$ ($k\neq i$), $[e_{ij},e_{ji}]=h_{ij}$.
* $[h_i,e_{ij}] = e_{ij}$ ($j\neq i,i+1$), $[h_{i-1},e_{ij}] =-e_{ij}$ ($j\neq i-1,i$), $[h_{ij},e_{ij}]=2e_{ij}$.
* $[h_i,h_j]=0$.
Computations involving inner derivations.
Recall that for $c\in\SL$ we write $C:=\ad_{c}\in U^{\op}$. Among the elements in $U^{\op}$ we find the inner derivations $E_{ij}$ generated by the elements $e_{ij}$ ($i\neq j$), which will play a special role in some results. Denote $\vp_{h_0a}:=0, \vp_{h_ka}:=0$ for $a\in\mathcal S$. Then we can write $E_{ij}$ in basis $\mathcal U$ as
\[E_{ij}=\sum_{l\neq i,j}\vp_{e_{jl}e_{il}} -\sum_{l\neq i,j}\vp_{e_{li}e_{lj}} + \vp_{e_{ji}h_{ij}}+\vp_{h_{i-1}e_{ij}}-\vp_{h_{i}e_{ij}}-\vp_{h_{j-1}e_{ij}}+\vp_{h_{j}e_{ij}}.\tag{E}\label{formula(E)}\]
By Formulas (<ref>) and (<ref>), the product of two of these inner derivations is given by
\begin{align*}
E_{ij}E_{rs} =&
\boldsymbol{\delta_{is}}\left(\sum_{l\neq i,j,r}\vp_{e_{jl}e_{rl}} + \boldsymbol{(1-\delta_{jr})}(\vp_{e_{jr}h_{ri}}+\vp_{h_{i-1}e_{rj}}-\vp_{h_{i}e_{rj}}-\vp_{h_{j-1}e_{rj}}+\vp_{h_{j}e_{rj}})\right)+\\
+&\boldsymbol{\delta_{jr}}\left(\sum_{l\neq i,j,s}\vp_{e_{li}e_{ls}} - \boldsymbol{(1-\delta_{is})}(\vp_{e_{si}h_{js}}+\vp_{h_{i-1}e_{is}}-\vp_{h_{i}e_{is}}-\vp_{h_{j-1}e_{is}}+\vp_{h_{j}e_{is}})\right)+\\
+&\boldsymbol{\delta_{is}\delta_{jr}}\left(\vp_{h_{i-1}h_{ji}} -\vp_{h_{i}h_{ji}} -\vp_{h_{j-1}h_{ji}}+\vp_{h_{j}h_{ji}}\right)+\\
+&\boldsymbol{\delta_{ji+1}}\left(\boldsymbol{\delta_{ir-1}}\vp_{e_{i+1i}e_{i+1s}} -\boldsymbol{\delta_{ir}}\vp_{e_{i+1i}e_{is}} -\boldsymbol{\delta_{is-1}}\vp_{e_{i+1i}e_{ri+1}} +\boldsymbol{\delta_{is}}\vp_{e_{i+1i}e_{ri}}\right)+\\
\end{align*}
In particular, a useful identity derived from (<ref>) is
\[E_{ij}^2 = -2\vp_{e_{ji}e_{ij}}.\tag{E2}\label{formula(E2)}\]
Observe that $e_{ij}^2=0$ ($i\neq j$) implies $E_{ij}^3=0$, since for all $x\in\M$,
\[x^{E_{ij}^3}=[e_{ij},[e_{ij},[e_{ij},x]]] = e_{ij}^3x-3e_{ij}^2xe_{ij}+3e_{ij}xe_{ij}^2 -xe_{ij}^3=0.\]
We will also make use of the important bracket formula
\[\vp_{ab}C = \vp_{a[c,b]}\tag{B}\label{formula(B)}\]
for any $c\in\SL$ and $a,b\in\mathcal S$ or $a=g=b$, which is true because for all $x\in\M$,
\[x^{\vp_{ab}C} = (\mu_a^xb)^C = [c,\mu_a^x b] = \mu_a^x [c,b] = x^{\vp_{a[c,b]}}.\]
In particular, for any left $U$-algebra $A$, $x\in A$, and $a\in\mathcal S$, we have
\[(x^{ae_{ij}})^{-E_{ji}}=x^{ah_{ij}}, \,\, (x^{ah_{ij}})^{-\frac12E_{ij}}=x^{ae_{ij}}.\]
The action of the power of a derivation on a product is given by Leibniz's rule: for $x,y\in A$, $c\in\SL$ and $p\in\N$,
\[(xy)^{C^p} = \sum_{i=0}^p \binom pi x^{C^i}y^{C^{p-i}}.\]
As an example, let us compute the action of the square of derivation $E_{ij}$ on a product by using Leibniz's rule, the bracket formula, and Formula (<ref>):
\begin{align*}
(x^{ab}y^{cd})^{E_{ij}^2} &= x^{(ab)E_{ij}^2}y^{cd} + 2x^{(ab)E_{ij}}y^{(cd)E_{ij}} + x^{ab}y^{(cd)E_{ij}^2} =
-2x^{(ab)(e_{ji}e_{ij})}y^{cd} + 2x^{a[e_{ij},b]}y^{c[e_{ij},d]} -2x^{ab}y^{(cd)(e_{ji}e_{ij})} =\\
&= -2(\delta_{be_{ji}} x^{ae_{ij}}y^{cd} - x^{a[e_{ij},b]}y^{c[e_{ij},d]} + \delta_{de_{ji}}x^{ab}y^{ce_{ij}}).
\end{align*}
The action of a general composition of derivations on a product is given by Formula (<ref>).
§.§ Explicit description of $\Uop$
In this section we describe how to operate with exponents coming from $\Uop$. We denote the product of $\Uop$ by juxtaposition.
Recalling that $U^{\op}=\phi(U(L)^{\op})\cong U(L)^{\op}/\ker\phi$, fix a unique preimage $\vr_{ab}\in\phi^{-1}(\vp_{ab})$ for each $\vp_{ab}\in\mathcal U$. Then $U(L)^{\op}=V\oplus\ker\phi$, with
\[\mathcal V:=\{\vr_{ab}\ | \ a,b\in\mathcal S\text{ or }a=g=b\}\]
being a basis of $V$. We extend the notation $\vr_{ab}$ to any $a\in\mathcal S$, $b\in\SL$ by linearity.
Preimages of the basis elements.
We first show a valid assignment of the elements $\vr_{ab}\in U(L)^{\op}$, formed with polynomials of degree at most $6$ in the elements $e_{ij}\in U(L)^{\op}$.
Let $c\cdot v$ denote the scalar product of vectors $c\in F^{k-1}$, $v \in (U(L)^{\op})^{k-1}$ ($c\cdot v:=c_1v_1+\cdots+c_{k-1}v_{k-1}$). Then one valid assignment of $\mathcal V$ is
_e_rse_ij:= 1/2 e_sr^2e_rje_is (r≠j, s≠i; r≠s, i≠j)
_e_rse_ir:=-1/2 e_sr^2e_is (i≠s; r≠s, i≠r), _e_rse_sj:= 1/2 e_sr^2e_rj (r≠j; r≠s, s≠j)
_e_rse_rs:= 1/4 e_sr^2e_rs^2 (r≠s), _e_rse_sr:=-1/2 e_sr^2 (r≠s)
_e_rsh_i:= 1/2 e_sr^2e_i+1se_rie_ii+1 (i≠r-1; i≠r, i≠s-1; s≠r), _e_rsh_r:=-1/2 e_sr^2e_r+1se_rr+1 (s≠r+1; s≠r)
_e_rsh_s-1:= 1/2 e_sr^2e_rs-1e_s-1s (s≠r+1; s≠r), _e_rsh_r-1:= 1/2 e_sr^2e_r-1se_rr-1 (s≠r-1; s≠r)
_e_rr-1h_r-1:= 1/2 e_r-1r^2e_rr-1, _e_r-1rh_r-1:=-1/2 e_rr-1^2e_r-1r
_h_ie_rs:= -1/2k(c_is·v_rs)e_rs^2, where
v_rs:=(v_rs1,…,v_rss-1,v_rss+1,v_rss+2,…,v_rsk) with
v_rsj:=e_sje_jr for j≠r,s and v_rsr:=-e_sr, and
c_pq:=(-k+p,…,-k+p^p-1,m,p,…,p^k-p-1), m:=-k+p if p<q and m:=p if p≥q.
_h_ih_j:= 1/k(c_ij·w_j)e_j+1je_jj+1, where
w_j:=(w_j1,…,w_jj,w_jj+2,w_jj+3,…,w_jk) with
w_jr:=e_jre_rj-1/4e_j+1r^2e_rj+1^2 for r≠j,j+1 and w_jj:=1/2e_jj+1e_j+1j.
_gg:= 1-∑_a∈𝒮_aa.
For example, for $k=6$ we have $c_{35}=(-3,-3,-3,3,3)$, $v_{45}=(e_{51}e_{14},e_{52}e_{24},e_{53}e_{34},-e_{54},e_{56}e_{64})$ and
\[\vr_{h_3e_{45}}=\frac14(e_{51}e_{14}+e_{52}e_{24}+e_{53}e_{34}+e_{54}-e_{56}e_{64})e_{45}^2.\]
Proving the proposition is a matter of verifying that, for each assignment found in the statement of the form $\vr_{ab}:=\sum_{p,\ldots,q}\alpha_{p,\ldots,q} e_p\cdots e_q$ with $\alpha_{p,\ldots,q}\in F$ and $e_p,\ldots,e_q\in\SL$, the identity $\vp_{ab} = \sum_{p,\ldots,q}\alpha_{p,\ldots,q} E_p\cdots E_q$ holds in $U^{\op}$. Accordingly, we skip computations when they are straightforward.
* The first 11 assignments of the statement are easily checked with Formulas (<ref>), (<ref>), (<ref>) and (<ref>).
* For $\vp_{h_ie_{rs}}$, first check that
\[V^{rsj}:= E_{sj}E_{jr}E_{rs}^2=2(-\vp_{h_{j-1}e_{rs}}+\vp_{h_je_{rs}}+\vp_{h_{s-1}e_{rs}}-\vp_{h_se_{rs}})\text{ for }j\neq r,s \tag{Ea}\label{formula(Ea)}\]
\[V^{rsr}:=-E_{sr}E_{rs}^2=2(-\vp_{h_{r-1}e_{rs}}+\vp_{h_re_{rs}}+\vp_{h_{s-1}e_{rs}}-\vp_{h_se_{rs}}).\tag{Eb}\label{formula(Eb)}\]
Then, for fixed $r,s$, write all identities in (<ref>) for $1\leq j\leq k$, $j\neq r,s$ together with (<ref>) as a $(k-1)\times(k-1)$ system of linear equations $V^{rs}=M(s)\cdot \vp^{rs}$, with vectors $\vp^{rs}:=(\vp_{h_1e_{rs}},\ldots,\vp_{h_{k-1}e_{rs}})$ and
\[
Compute $M(s)^{-1}$ to solve the system and find $\vp^{rs}=M(s)^{-1}V^{rs}$. The matrix of coefficients $M(s)$ is described as follows:
Suppose first $1<s<k$. For $i<s-1$ (corresponding to $j=i$ in (Ea) or $r=i$ in (Eb)) and for $i>s$ (corresponding to $j=i+1$ or $r=i+1$), the $i$th row has a $-1$ entry in columns $i-1$ and $s$, a $1$ entry in columns $i$ and $s-1$, and $0$ elsewhere. The $(s-1)$th row (corresponding to $j=s-1$ or $r=s-1$) has a $-1$ entry in columns $s-2$ and $s$ and a $2$ entry in column $s-1$. The $s$th row (corresponding to $j=s+1$ or $r=s+1$) has a $1$ entry in columns $s-1$ and $s+1$ and a $-2$ entry in column $s$.
\[M(s)=\begin{pmatrix*}[r]
1 & 0 & \cdots & \cdots & \cdots & 0 & 1 & -1 & 0 & \cdots & \cdots & \cdots & \cdots & 0\\
-1 & 1 & 0 & \cdots & \cdots & 0 & 1 & -1 & 0 & \cdots & \cdots & \cdots & \cdots & 0\\
0 &-1 & 1 & 0 & \cdots & 0 & 1 & -1 & 0 & \cdots & \cdots & \cdots & \cdots & 0\\
\vdots & \ddots & \ddots & \ddots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \ddots & \ddots & \vdots\\
\vdots & \ddots & \ddots & \ddots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \ddots & \ddots & \vdots\\
\vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \vdots & \vdots & \vdots & \ddots & \ddots & \ddots & \ddots & \vdots\\
0 & \cdots & \cdots & \cdots & 0 & -1 & 2 & -1 & 0 & \cdots & \cdots & \cdots & \cdots & 0\\
0 & \cdots & \cdots & \cdots & \cdots & 0 & 1 & -2 & 1 & 0 & \cdots & \cdots & \cdots & 0\\
0 & \cdots & \cdots & \cdots & \cdots & 0 & 1 & -1 & -1 & 1 & 0 & \cdots & \cdots & 0\\
0 & \cdots & \cdots & \cdots & \cdots & 0 & 1 & -1 & 0 & -1 & 1 & 0 & \cdots & 0\\
\vdots & \ddots & \ddots & \ddots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \ddots & \ddots& \vdots\\
\vdots & \ddots & \ddots & \ddots & \ddots & \vdots & \vdots & \vdots & \vdots & \ddots & \ddots & \ddots & \ddots& 0\\
0 & \cdots & \cdots & \cdots & \cdots & 0 & 1 & -1 & 0 & \cdots & \cdots \cdots & 0 & -1 & 1
\end{pmatrix*}.\]
Equivalently, if $i<s-1$ or $i>s$, the $i$th column $C_{is}$ of $M(s)$ has entries $1,-1$ in rows $i,i+1$ (resp. $i-1,i$); if $i=s-1$ (resp. $i=s$), entry $2$ (resp. $-2$) in row $s-1$ (resp. $s$) and $1$ (resp. $-1$) elsewhere. Let us show that $M(s)$ is invertible. It is straightforward to check that the row vector
\[c_{is}:=(\overbrace{-k+i,\ldots,-k+i}^{i-1},m,\overbrace{i,\ldots,i}^{k-i-1}), m:=-k+i\text{ if }i<s\text{ and }m:=i\text{ if }i\geq s\]
satisfies $c_{is}\cdot C_{js}=-k\delta_{ij}$, whence the matrix with rows $c_{1s},\ldots,c_{k-1s}$ scaled by $-1/k$ is the inverse of $M(s)$. Therefore $\vp_{h_ie_{rs}}=-\frac1{k}c_{is}\cdot V^{rs}$.
In the extreme cases, $s=1$ and $s=k$, the matrix $M(s)$ follows the same pattern with the obvious changes and the same formula gives the inverse.
* For $\vp_{h_ih_j}$ check that, for $r\neq j,j+1$,
\[E_{jr}E_{rj}E_{j+1j}E_{jj+1} = \vp_{e_{rj+1}e_{rj+1}} +\vp_{h_{r-1}h_j}-\vp_{h_rh_j}-\vp_{h_{j-1}h_j}+\vp_{h_jh_j} \]
by showing first that
Next apply that $\vp_{e_{rj+1}e_{rj+1}}=\frac14E_{j+1r}^2E_{rj+1}^2 = \frac14E_{j+1r}^2E_{rj+1}^2E_{j+1j}E_{jj+1}$ to find
\[W^{jr}:=E_{rj}E_{j+1j}E_{jj+1} - \frac14E_{j+1r}^2E_{rj+1}^2E_{j+1j}E_{jj+1} = \vp_{h_{r-1}h_j}-\vp_{h_rh_j}-\vp_{h_{j-1}h_j}+\vp_{h_jh_j}\text{ for }r\neq j,j+1. \tag{Ec}\label{formula(Ec)}\]
Check also that
\[W^{jj}:=E_{jj+1}E_{j+1j}^2E_{jj+1} = 2(-\vp_{h_{j-1}h_j}+2\vp_{h_jh_j}-\vp_{h_{j+1}h_j}).\tag{Ed}\label{formula(Ed)}\]
Now fix $j$ and proceed as in the previous case by solving for $\vp^j:=(\vp_{h_1h_j},\ldots,\vp_{h_{k-1}h_j})$ the $(k-1)\times(k-1)$ system of linear equations $W^j=-M(j)\vp^j$ generated by (<ref>) and (<ref>), where $W^j:=(W^{j1},\ldots,W^{jj-1},\frac12W^{jj},W^{jj+2},\ldots,W^{jk})$
and $M(s)$ is (thankfully!) the matrix described in the previous item. Therefore $\vp_{h_ih_j}=\frac1k c_{ij}\cdot W^j$.
* For $\vp_{gg}$ use that $1_{U^{\op}}=\vp_{gg} + \sum_{a\in\mathcal S}\vp_{aa}$.
Generator of $\ker\phi$.
The ideal $\ker\phi$ of $\Uop$ is infinite dimensional, but is finitely generated; in this subsection, we show, through the primitive element lemma, that it is in fact a principal ideal. For most of this section we work with $U(L)$.
Let $\Ad_k$ denote the adjoint representation of $\SL$ and $\chi_k$ its associated central character, and let $\phi_k$ be the representation of $\SL$ on $\M$ given by the action of $\ad$. We have $\phi_k=\Ad_k\oplus\rho_0$ with $\rho_0$ acting on $F\cdot g$. Fixing the Cartan subalgebra of diagonal traceless matrices and the set of positive roots giving $N_+=\spn\{e_{12},e_{23},\ldots,e_{k-1k}\}$, the highest weight vector of $\Ad_k$ is $e_{1k}$.
Denote $x^i_j:=e_{ij}$ for $1\leq i\neq j\leq k$, and for $1\leq i\leq k$,
\[x^i_i:=\frac1k\sum_{j=1}^{k-1}\alpha_{ij}h_j,\,\,
\alpha_{ij}:=k-j\text{ if }j\geq i,\,\,
\alpha_{ij}:=-j\text{ if }j<i.\tag{X}\label{formula(X)}\]
The elements $x^i_j$ form a set of generators of $\SL$ satisfying $[x^i_j,x^r_s]=\delta_{jr}x^i_s-\delta_{is}x^r_j$. Then the Casimir elements
\[c_{p,k}:=\sum_{i_1,\ldots,i_p=1}^k x^{i_1}_{i_2}x^{i_2}_{i_3}\cdots x^{i_p}_{i_1}, \,\, 2\leq p\leq k,\tag{Ca}\label{formula(Ca)}\]
which have rational coefficients in the PBW basis, form a set of Casimir generators of $Z(U(L))$ (see <cit.>). For example,
\begin{align*}
c_{3,3}& = (x^1_1)^3+(x^1_1)^2x^1_2+(x^1_1)^2x^1_3+ x^1_1x^1_2x^2_2 +x^1_1x^1_2x^2_3 + \cdots + (x^3_3)^3 =\\
& = 2/9h_1^3 + 1/3h_1^2h_2 - 1/3h_1h_2^2 - 2/9h_2^3 + 2h_1^2 + h_1h_2 + 4h_1 + 2h_2 +\\
& + h_1e_{21}e_{12} + 2h_2e_{21}e_{12} - 2h_1e_{32}e_{23} - h_2e_{32}e_{23} + 3e_{31}e_{12}e_{23} + 3e_{21}e_{32}e_{13} + h_1e_{31}e_{13} - h_2e_{31}e_{13} +\\
& + 6e_{21}e_{12} + 3e_{31}e_{13}.
\end{align*}
Let $\lambda_{p,k}$ denote the eigenvalue of $c_{p,k}$ for $\Ad_k$. These eigenvalues are the following positive integers.
Put $\lambda_{1,k}:=0$ and $\lambda_{p,k}:=\chi_k(c_{p,k})$ for $2\leq p\leq k$. Then $\lambda_{p,k}=\lambda_{p-2,k}+k^{p-1}$ with $\lambda_{2,k}=2k$, thus
\[\lambda_{p,k}=\left\{\begin{array}{lc}\displaystyle
k\left(\frac{k^p-1}{k^2-1} + 1\right), & p\text{ even}\\\displaystyle
k^2\frac{k^{p-1}-1}{k^2-1}, & p\text{ odd}
\end{array}\right..\]
By <cit.> we have $\lambda_{p,k}=\tr(A_k^pE_k)$, where $E_k$ is the $k\times k$ matrix full of ones and $A_k$ is the $k\times k$ upper triangular matrix with $(i,j)$ entries equal to $-1$ when $i<j$ and diagonal
\[(m_1+k-1,m_2+k-2,\ldots,m_{k-1}+1,m_k),\]
where $m_i$ is the eigenvalue of $x_i^i$ for the highest weight vector of the adjoint representation, i.e., $x_i^ie_{1k}=:m_ie_{1k}$. A straightforward computation with Formula (<ref>) produces $m_1=1$, $m_2=\cdots=m_{k-1}=0$, $m_k=-1$. By induction on $p$ with base case $p=1$ it is proven that $A_k^pE_k=A_k\cdot(A_k^{p-1}E_k)$ equals
\[\begin{pmatrix}
a_{p,k} & a_{p,k} & \cdots & a_{p,k}\\
0 & 0 & \cdots & 0\\
\vdots & \ddots & \ddots & \vdots\\
-1 & -1 & \cdots & -1
\end{pmatrix}\]
when $p$ is odd and
\[\begin{pmatrix}
a_{p,k} & a_{p,k} & \cdots & a_{p,k}\\
1 & 1 & \cdots & 1\\
\vdots & \ddots & \ddots & \vdots\\
1 & 1 & \cdots & 1
\end{pmatrix}\]
when $p$ is even, with $a_{1,k}=1$,
\[a_{p,k}=\left\{\begin{array}{lc}
ka_{p-1,k}+1, & p\text{ even}\\
ka_{p-1,k}-(k-1), & p>1\text{ odd}
\end{array}\right.,\]
\[\lambda_{p,k}=\tr(A^p_kE_k)=\left\{\begin{array}{lc}
a_{p,k}+k-1, & p\text{ even}\\
a_{p,k}-1, & p>1\text{ odd}
\end{array}\right..\]
\[\lambda_{p,k}=\left\{\begin{array}{lc}
k(a_{p-1,k}+1) = k(\lambda_{p-1,k}+2), & p\text{ even}\\
k(a_{p,k}-1) = k(\lambda_{p-1,k}-k), & p>1\text{ odd}
\end{array}\right.,\]
with $\lambda_{2,k}=2k$, $\lambda_{1,k}=0$. Notice that $\lambda_{p,k}-\lambda_{p-2,k}=k(\lambda_{p-1,k}-\lambda_{p-3,k})$ regardless of whether $p$ is even or odd. By recursion $\lambda_{p,k}-\lambda_{p-2,k}=k^{p-3}(\lambda_{3,k}-\lambda_{1,k})=k^{p-3}k^2=k^{p-1}$, hence by recursion again we find
\[
\lambda_{p,k}=\left\{\begin{array}{lc}\displaystyle
\sum_{i=1}^{p/2-1} k^{2i+1} +2k= k\left(\frac{k^p-1}{k^2-1} + 1\right), & p\text{ even}\\\displaystyle
\sum_{i=1}^{(p-1)/2} k^{2i} = k^2\frac{k^{p-1}-1}{k^2-1}, & p\text{ odd}
\end{array}\right..\qedhere\]
Clearly $c_{p,k}-\lambda_{p,k}\in\ker\chi_k\subseteq\ker\Ad_k$ for $2\leq p\leq k$, and we also have $e_{ij}^3\in\ker\Ad_k$ for $1\leq i\neq j\leq k$. We prove that these elements generate $\ker\Ad_k$ in $U(L)$ and show that $\ker\phi_k$ is a principal ideal. We build on ideas from <cit.>, which solve the problem for $k=3$.
Denote $z_{p,k}:=c_{p,k}-\lambda_{p,k}$ for $2\leq p\leq k$ and $z'_{p,k}:=\lambda_{2,k}c_{p,k}-\lambda_{p,k}c_{2,k}$ for $3\leq p\leq k$.
* $\ker\Ad_k = \la e_{12}^3, z_{2,k},\ldots,z_{k,k}\ra$.
* $\ker\phi_k = \la e_{12}^3,e_{12}z_{2k}, z'_{3,k},\ldots,z'_{p,k}\ra$.
* $\ker\phi_k = \la e_{12}^3+e_{12}z_{2,k} + e_{13}z_{3,k} +\cdots +e_{1k}z_{k,k}\ra$.
Let $K$ be a field extension of $F$. Then $\Sl_k(K)=\SL\otimes_F K$, $U(\Sl_k(K))=U(L)\otimes_F K$, the adjoint representation of $\Sl_k(K)$ is $\Ad_k\otimes_F K$, and if $\rho$ is a representation of $\SL$ then $\rho\otimes_F K$ is a representation of $\Sl_k(K)$ such that
\[\ker_{U(\Sl_k(K))}(\rho\otimes_F K) = (\ker_{U(L)}\rho)\otimes_F K.\]
In addition, if $I$ is an ideal of $U(L)$ such that $I\otimes_F K = \la g_1,\ldots,g_n\ra$ in $U(\Sl_k(K))$ with $g_i\in I$ for $1\leq i\leq n$ then $I=\la g_1,\ldots,g_n\ra$ in $U(L)$. Therefore, by extension and restriction of scalars, we can assume without loss of generality that $F=\Co$.
* Clearly $I_k:=\la e_{12}^3, z_{2,k},\ldots,z_{k,k}\ra\subseteq\ker\Ad_k$. Let us show the opposite inclusion. First, we see that $I_k$ has finite codimension. Consider a deglex order on the set of monomials of $U(L)$ with $h_i>e_{ii+1}>e_{i+1i}$ for $1\leq i<k$. Since $U(L)=I_k\oplus\spn N(I_k)$, the ideal $I_k$ has finite codimension if and only if $\spn N(I_k)$ has finite dimension, hence if and only if there are $n_1,\ldots,n_{k^2-1},m_1,\ldots,m_{k-1}\in\N$ such that $e_{ij}^{n_{ij}},h_l^{m_l}\in\LM(I_k)$ for all $1\leq i\neq j\leq k$ and $1\leq l<k$. In the next identities let $\ad$ denote the adjoint map of $U(L)$; since we have
\begin{align*}
&e_{21}^3 = -\frac1{6!}\ad_{e_{21}}^6(e_{12}^3)\text{ if } k=2,\\
&e_{ij}^3 = \frac1{6}\ad_{e_{il}}^3(e_{lj}^3),\,\,\,\,
e_{ij}^3 = -\frac1{6}\ad_{e_{lj}}^3(e_{il}^3)\,\,\text{ for }i\neq l\neq j\neq i\text{ if }k>2,
\end{align*}
starting from $e_{12}^3$ we can show $e_{ij}^3\in I_k$ for all $1\leq i\neq j\leq k$, for all $k\geq2$: for $k\geq3$, use the third identity to get $e_{1j}^3$ for all $3\leq j\leq k$ from $e_{12}^3$, the second identity to get $e_{i2}^3$ for all $3\leq i\leq k$ from $e_{12}^3$, then the second identity again to get $e_{21}^3$ from $e_{31}^3$; and so on. Also, since for all $k\geq2$ and all $1\leq i<k$ we have
\[\frac16\ad^3_{e_{ii+1}}(e_{i+1i}^3) = h_i^3-6e_{i+1i}e_{ii+1}h_i -3h_i^2 +4h_i,\]
we find $h_i^3\in\LM(I_k)$ for $1\leq i<k$.
This proves that $I_k$ has finite codimension. Now, since $z_{2,k},\ldots,z_{k,k}$ is a set of Casimir generators, $I_k\cap Z(U(L))$ is a maximal ideal of $Z(U(L))$, and so Lemma <ref> shows that $I_k$ is a maximal ideal of $U(L)$, implying $I_k=\ker\Ad_k$.
* The representation $\phi_k$ is the direct sum of the adjoint representation $\Ad_k$ and the trivial representation $\rho_0$, so $\ker\phi_k=\ker\Ad_k\cap\ker\rho_0=I_k\cap U^*(L)$, where $U^*(L)$ is the nonunital universal enveloping algebra of $L$; i.e., $\ker\phi_k$ is formed by those elements of $I_k$ which do not have a nonzero constant term.
We first change the set of Casimir generators to get rid of unnecessary constant terms in the generators of $I_k$. By Lemma <ref> we have $\lambda_{2,k}=2k\neq0$, hence the matrix
\[\pma
1 & 0 & \cdots & \cdots & 0 \\
\lambda_{3,k} & -\lambda_{2,k} & 0 & \cdots & 0 \\
\lambda_{4,k} & 0 & -\lambda_{2,k} & \cdots & 0 \\
\vdots & \vdots & \ddots & \ddots & \vdots \\
\lambda_{k,k} & 0 & \cdots & 0 & -\lambda_{2,k}
\epma\]
is invertible. Therefore the central elements $c_{2,k}$ and $z'_{p,k}$ for $3\leq p\leq k$ form another set of Casimir generators such that $\chi_k(c_{2,k})=\lambda_{2,k}$, $\chi_k(z'_{p,k})=0$ for $3\leq p\leq k$. Then
\[\ker Ad_k=\la e_{12}^3,z_{2k}, z'_{3,k},\ldots,z'_{p,k}\ra\]
with $e_{12}^3, z'_{3,k},\ldots,z'_{p,k}\in U^*(\SL)$. Put $I:=\la e_{12}^3, z'_{3,k},\ldots,z'_{p,k}\ra$, $J:=\la z_{2,k}\ra$ and $M:=U^*(L)$. Then $\ker\phi_k=(I+J)\cap M = I+J\cap M$ because $I\subseteq M$. Since $M$ is a maximal ideal and $z_{2,k}\not\in M$, $J+M=U(L)$, whence
\[JM\subseteq J\cap M = U(L)(J\cap M) = (J+M)(J\cap M)\subseteq JM + MJ = JM\]
since $z_{2,k}$ is central. This shows $J\cap M = JM$. Moreover, since $\SL$ is simple, $U^*(L)=\la e_{12}\ra$, so $JM = \la z_{2,k})\I(e_{12}\ra = \la z_{2,k}e_{12}\ra$ because $z_{2,k}$ is central. Therefore
\[\ker\phi_k=I+JM = \la e_{12}^3,e_{12}z_{2,k},z_{3,k}',\ldots,z_{k,k}'\ra.\]
Now repeat the argument above with $I:=\la e_{12}^3\ra$, $J:=\la z_{2,k},\ldots,z_{k,k}\ra$ to arrive at
$\ker\phi_k = \la e_{12}^3,\allowbreak e_{12}z_{2,k},\ldots,e_{12}z_{k,k}\ra$. Since $[x,yz_{p,k}]=[x,y]z_{p,k}$ for all $x,y\in U(L)$ and $\la e_{1p}\ra=U^*(L)$ for all $2\leq p\leq k$, we get $\la e_{12}z_{p,k}\ra=\la e_{1p}z_{p,k}\ra$ for all $3\leq p\leq k$, so $\ker\phi_k = \la e_{12}^3,e_{12}z_{2,k},\ldots,e_{1k}z_{k,k}\ra$.
* The elements $e_{1i}$ correspond to different roots $\alpha_i$ of $\SL$ and as such are weight vectors of different weights for the adjoint action of $\SL$ on $U(L)$. The identity $[h_i,e_{1p}z_{p,k}]=\alpha_p(h_i)e_{1p}z_{p,k}$ for $1\leq i<k$, $2\leq p\leq k$ shows that the elements $e_{1p}z_{p,k}$ are also weight vectors of $U(L)$ of different weights, which are also different from the weight $3\alpha_2$ of the weight vector $e_{12}^3$. Since ideals of $U(L)$ are $L$-ideals for the adjoint action, by the primitive element lemma (Lemma <ref>) we find
$\ker\phi_k=\la e_{12}^3,e_{12}z_{2,k},\ldots,e_{1k}z_{k,k}\ra = \la e_{12}^3+e_{12}z_{2,k}+\cdots+e_{1k}z_{k,k}\ra$.
Now we turn back to $\Uop$.
\[\ker\phi = \la e_{12}^3+e_{12}z_{2,k} + e_{13}z_{3,k} +\cdots +e_{1k}z_{k,k}\ra.\]
We have $\phi=(\phi_k)^{\op}$, so $\ker\phi=\ker\phi_k$ as sets with $z:=e_{12}^3+e_{12}z_{2,k} + e_{13}z_{3,k} +\cdots +e_{1k}z_{k,k}$ generating $\ker\phi_k$ in $U(L)$, and since $z_{p,k}\in Z(U(L))$ for $2\leq p<k$, $z$ generates $\ker\phi$ in $U(L)^{\op}$.
§.§ $U$-polynomials and $U$-identities
Modifying the first index of an exponent.
As we will see below in <ref>, it is the second basis element in the subindex of the endomorphisms $\varphi_{ab}$ which carries the weight of the identities of $\M$. The first basis element is not that relevant, and in fact it can be freely changed by substitution: On the one hand, if $f(x_1,\ldots,x_n)$ is a $U$-identity of $(L,U)$-algebra $A$ and $u_1,\ldots,u_n\in U$, then $f(x_1^{u_1},\ldots,x_n^{u_n})$ is a $U$-identity of $A$. On the other hand,
$(x^{ab})^{bc}=x^{ac}$ for $a,b,c\in\mathcal S$. Therefore, if variable $x^{ab}$ with $a,b\in\mathcal S$ features in a $U$-identity of $A$, there is an analogous $U$-identity replacing $x^{ab}$ with variable $x^{cb} = (x^{ca})^{ab}$ for any $c\in\mathcal S$. Hence we can fix one element $a\in\mathcal S$ and assume that each endomorphism appearing as exponent in a $U$-identity is either $\vp_{gg}$ or of the form $\varphi_{ab}$ for some $b\in\mathcal S$; that is, when looking for the generating identities of the $T_U$-ideal of $A$ we may assume that all exponents different from $\vp_{gg}$ start with the same first basis element $a$. Taking this into account, for a fixed and previously specified $a\in\mathcal S$, we will write $x^b$ as a shorthand for $x^{ab}$ ($x^{\varphi_{ab}}$) with $b\in\SL$ (this should not be confused with element $x^b\in\FL$); we will also write $x^g$ as a shorthand for $x^{gg}$ ($x^{\vp_{gg}}$).
In particular, in this format, the bracket formula takes the simple form $(x^b)^C = x^{[c,b]}$ for $b,c\in\SL$ and $C=\ad_c$.
Fixed-exponents components of $U$-identities.
We expand and add rigor to <ref>. Let us write multilinear $U$-polynomials of $P_n^U$ by grouping their terms with respect to the exponents of their variables, with one set of indices $\mathcal I$ for variables of the form $x_i^{gg}$ and another set of indices $\mathcal J$ for variables of the form $x_j^{a_jb_j}$ with $a_j,b_j\in\mathcal S$, and taking into account how the first exponent indices $a_j$ are paired with the variables $x_j$.
For $n\geq 1$, let $\stirlingtwo{[n]}{2}$ denote the set of pairs of sets $(\mathcal I,\mathcal J)$ such that $\{\mathcal I,\mathcal J\}$ is a partition of the set $\{1,\dots,n\}$ into two disjoint subsets, one of which may be empty (observe that $(\mathcal I,\mathcal J)$ and $(\mathcal J,\mathcal I)$ are different elements belonging to $\stirlingtwo{[n]}{2}$). Then for $f\in P_n^U$ we have the decomposition
\[f=\sum_{(\mathcal I,\mathcal J)\in\stirlingtwo{[n]}{2}}\,\sum_{\bs a\in \mathcal{S}^{|\mathcal J|}} f_{\mathcal I, \bs a},\]
where for fixed $\mathcal I=\{i_1,\ldots,i_r\}$, $\mathcal J=\{j_1,\ldots,j_{n-r}\}$ and $\bs a=(a_1,\ldots,a_{n-r})$, the $U$-polynomial $f_{\mathcal I, \bs a}$ denotes the sum of all terms of $f$ in which only the variables
\[x_{i_1}^{gg}, \dots, x_{i_r}^{gg}\text{ and }x_{j_1}^{a_1b_1}, \dots, x_{j_{n-r}}^{a_{n-r}b_{n-r}}\text{ for any }(b_1,\dots, b_{n-r})\in \mathcal{S}^{n-r}\]
appear, in any order. Call $f_{\mathcal I,\bs a}$ the $(\mathcal I,\bs a)$ fixed-exponents component of $f$.
Any $U$-polynomial whose only nonzero fixed-exponents component is $(\mathcal I,\bs a)$ is of the form
\[f(\mathcal I,\bs a,\{\alpha_{\sigma,\bs b}\}):=\sum_{\sigma\in S_n}\sum_{\tiny\begin{array}{cc}\bs b=(b_1,\ldots,b_n)\in\mathcal M^n\\b_i=g\text{ if and only if }\sigma(i)\in \mathcal I\end{array}}\!\!\!\!\!\!\!\!
\alpha_{\sigma,\bs b} \, x_{\sigma(1)}^{a_{\sigma(1)}b_1}\cdots x_{\sigma(n)}^{a_{\sigma(n)}b_n},\]
where $a_{i}:=g$ for all $i\in \mathcal I$, $S_n$ is the symmetric group acting on $\{1,\ldots,n\}$, and $\alpha_{\sigma,\bs b}\in F$. If the first exponent index is homogeneous, i.e., $\bs a=(a,\ldots,a)$ for $a\in\mathcal S$, then we write $f(\mathcal I, a,\{\alpha_{\sigma,\bs b}\})$. If $\mathcal I=\{1,\ldots,r\}$ then we write $f(r,\bs a,\{\alpha_{\sigma,\bs b}\})$ and say that $f$ has first $r$ $g$-exponents.
We show that to study $P_n^U\cap\I^U(A)$ it is enough to study the $U$-identities with only one nonzero fixed-exponents component, with homogeneous first exponent index and first $r$ $g$-exponents.
Let $A$ be an $(L,U)$-algebra, $(\mathcal I,\mathcal J)\in \stirlingtwo{[n]}{2}$ with $\mathcal I=\{i_1,\ldots,i_r\}$ and $\mathcal J=\{j_1,\ldots,j_{n-r}\}$, $\bs a = (a_{j_1},\ldots,a_{j_{n-r}})\in\mathcal S^{n-r}$, and $c\in\mathcal S$.
* $f\in P_n^U$ is a $U$-identity of $A$ if and only if any nonzero fixed-exponents component of $f$ is a $U$-identity of $A$.
* $f(\mathcal I,\bs a,\{\alpha_{\sigma,\bs b}\})$ is a $U$-identity of $A$ if and only if $f(\mathcal I,c,\{\alpha_{\sigma,\bs b}\})$ is a $U$-identity of $A$.
* $f(\mathcal I,c,\{\alpha_{\sigma,\bs b}\})$ is a $U$-identity of $A$ if and only if $f(r,c,\{\alpha_{\sigma,\bs b}\})$ is a $U$-identity of $A$.
Along the proof we use repeatedly that if $f\in \I^U(A)$ and $\rho$ is a substitution endomorphism swapping variables then $\rho(f)\in\I^U(A)$ (see <ref>).
* One implication is clear from the definition, since any multilinear $U$-polynomial is the sum of its fixed-exponents components. For the other one, fix $f\in P_n^U \cap \I^U(A)$ and for $j\in\mathcal J$ let $\rho_j$ denote the substitution endomorphism sending variable $x_j$ to $x_j^{a_ja_j}$. Then, since $\vp_{a_ja_j}\vp_{a_jb}=\vp_{a_jb}$ and $\vp_{a_ja_j}\vp_{cd}=0$ for $c\neq a_j$,
\[f_{\mathcal I,\bs a} = \rho_{j_1}\circ\cdots\circ\rho_{j_{n-r}}(f).\]
* For $j\in\mathcal J$ let $\rho^c_j$ denote the substitution endomorphism sending variable $x_j$ to $x_j^{ca_j}$. Then
\[f(\mathcal I,c,\{\alpha_{\sigma,\bs b}\}) = \rho^c_{j_1}\circ\cdots\circ\rho^c_{j_{n-r}}(f(\mathcal I,\bs a,\{\alpha_{\sigma,\bs b}\})).\]
Analogously, for $j\in\mathcal J$ let $\rho^j_c$ denote the substitution endomorphism sending variable $x_j$ to $x_j^{a_jc}$. Then
\[f(\mathcal I,\bs a,\{\alpha_{\sigma,\bs b}\}) = \rho^{j_1}_c\circ\cdots\circ\rho^{j_{n-r}}_c(f(\mathcal I,c,\{\alpha_{\sigma,\bs b}\})).\]
* For $\sigma\in S_n$ let $\rho_\sigma$ denote the substitution endomorphism sending variable $x_i$ to $x_{\sigma(i)}$ for all $1\leq i\leq n$. Let $\tau$ be the permutation sending $i_s$ to $s$ for $1\leq s\leq r$ and $j_s$ to $s+r$ for $1\leq s\leq n-s$.
\[f(r,c,\{\alpha_{\sigma,\bs b}\}) = \rho_\tau(f(\mathcal I,c,\{\alpha_{\sigma,\bs b}\}))\text{ and }
f(\mathcal I,c,\{\alpha_{\sigma,\bs b}\}) = \rho_{\tau^{-1}}(f(r,c,\{\alpha_{\sigma,\bs b}\})).\qedhere\]
Moreover, it is now clear that given a set of $U$-identities, we can homogeneously fix one single element $a\in\mathcal S$ as the first exponent index of all $U$-identities in the set, first exponent index which we may elide.
Formula for the $U$-codimensions.
For $(\mathcal I,\mathcal J)\in\stirlingtwo{[n]}{2}$ and $\bs a\in\mathcal{S}^{|\mathcal J|}$, we denote by $P_{\mathcal I,\mathcal J,\bs a}^U$ the subspace of $P_n^U$ composed of the multilinear $U$-polynomials whose only nonzero fixed-exponents component is $(\mathcal I,\bs a)$ (see <ref>). For $0\leq r\leq n$ and $a\in\mathcal{S}$, we denote $P^U_{r,n-r,a}:=P^U_{\{1,\ldots,r\},\{r+1,\ldots,n\},(a,\ldots,a)}$, the vector space of multilinear $U$-polynomials with only one nonzero fixed-exponents component, with homogeneous first exponent index $a$ and first $r$ $g$-exponents. If $A$ is an $(L,U)$-algebra, we denote $P_{r,n-r,a}^U(A):= \displaystyle\dfrac{P_{r,n-r,a}^U}{P_{r,n-r,a}^U \cap \I^U(A)}$
and $c^U_{r,n-r}(A):=\dim_F P^U_{r,n-r,a}(A)$ for any $a\in\mathcal S$ (it is independent of $a$, as deduced from item (1) of the following proposition).
We show that to study $P_n^U\cap\I^U(A)$ it is enough to study $P^U_{r,n-r,a}\cap\I^U(A)$ for some fixed $a\in\mathcal S$.
Let $A$ be an $(L,U)$-algebra.
* For $(\mathcal I,\mathcal J)\in\stirlingtwo{[n]}{2}$ such that $r=|\mathcal I|$ and any $\bs a\in\mathcal S^{n-r}$ and $c\in\mathcal S$ there is a linear isomorphism between $P_{\mathcal I,\mathcal J,\bs a}^U$ and $P_{r,n-r,c}^U$ sending $P_{\mathcal I,\mathcal J,\bs a}^U \cap\I^U(A)$ to $P_{r,n-r,c}^U \cap\I^U(A)$.
\[c_n^U(A)= \sum_{r=0}^{n} \binom{n}{r} (k^2 -1)^{n-r} c^U_{r,n-r}(A). \tag{C}\label{formula(C)}\]
* The proof of item (2) of Lemma <ref> shows a linear isomorphism between $P_{\mathcal I,\mathcal J,\bs a}^U$ and $P_{\mathcal I,\mathcal J,(c,\ldots,c)}^U$ such that $P_{\mathcal I,\mathcal J,\bs a}^U \cap\I^U(A) \cong P_{\mathcal I,\mathcal J,(c,\ldots,c)}^U\cap\I^U(A)$, while the proof of item (3) shows a linear isomorphism between $P_{\mathcal I,\mathcal J,(c,\ldots,c)}^U$ and $P_{r,n-r,c}^U$ such that $P_{\mathcal I,\mathcal J,(c,\ldots,c)}^U\cap\I^U(A) \cong P_{r,n-r,c}^U\cap\I^U(A)$ (the isomorphisms being given by the invertible substitutions specified there).
* By definition of fixed-exponents components we have
\[P_n^U = \bigoplus_{(\mathcal I,\mathcal J)\in\stirlingtwo{[n]}{2}, \bs a\in \mathcal{S}^{|\mathcal J|}} P_{\mathcal I,\mathcal J,\bs a}^U.\]
This identity, combined with Lemma <ref>(1), leads to
\[P_n^U\cap\I^U(A) = \bigoplus_{(\mathcal I,\mathcal J)\in\stirlingtwo{[n]}{2}, \bs a\in \mathcal{S}^{|\mathcal J|}} (P_{\mathcal I,\mathcal J,\bs a}^U\cap\I^U(A)).\]
\[P_n^U(A) = \dfrac{P_n^U}{P_n^U\cap\I^U(A)} \cong \bigoplus_{(\mathcal I,\mathcal J)\in\stirlingtwo{[n]}{2}, \bs a\in \mathcal{S}^{|\mathcal J|}} \dfrac{P_{\mathcal I,\mathcal J,\bs a}^U}{P_{\mathcal I,\mathcal J,\bs a}^U\cap\I^U(A)}.\]
Now fix $c\in\mathcal S$. By item (1), for each $(\mathcal I,\mathcal J)\in\stirlingtwo{[n]}{2}$ such that $|\mathcal I|=r$ and $\bs a\in\mathcal{S}^{|\mathcal J|}$ there is an isomorphism between $\dfrac{P_{\mathcal I,\mathcal J,\bs a}^U}{P_{\mathcal I,\mathcal J,\bs a}^U\cap\I^U(A)}$ and $\dfrac{P_{r,n-r,c}^U}{P_{r,n-r,c}^U\cap\I^U(A)}$. Therefore,
\[P_n^U(A) \cong \bigoplus_{0\leq r\leq n,\, (\mathcal I,\mathcal J)\in\stirlingtwo{[n]}{2} \, | \, |\mathcal I|=r,\, \bs a\in \mathcal{S}^{n-r}} \dfrac{P_{r,n-r,c}^U}{P_{r,n-r,c}^U\cap\I^U(A)} = \bigoplus_{0\leq r\leq n}\binom nr (k^2-1)^{n-r} \dfrac{P_{r,n-r,c}^U}{P_{r,n-r,c}^U\cap\I^U(A)},\]
since for any $0\leq r\leq n$ we have $\binom nr$ choices of the first $r$ variables and $(k^2 -1)^{n-r}$ distinct elements in $\mathcal{S}^{n-r}$.
Hence Formula (<ref>) follows.
Fixed $a\in \mathcal{S}$, when no confusion may arise, we will write $P^U_{r,n-r}:=P_{r,n-r,a}^U$, $P^U_{r,n-r}(A):=P_{r,n-r,a}^U(A).$
$U$-cocharacter and its decomposition. In view of the previous sections, instead of considering the usual permutations of variables as the actions of the symmetric groups on the spaces of multilinear polynomials, we consider the actions which permute variables together with their first index exponent, which is more natural in this context.
Denoting by $S_{\mathcal I}$ and $S_{\mathcal J}$ the symmetric groups acting on the sets $\mathcal I$ and $\mathcal J$ respectively, the group $S_{\mathcal I} \times S_{\mathcal J}$ acts on $P_{\mathcal I,\mathcal J,\bs a}^U$ on the left in the following way:
\[(\sigma,\tau)(x_{\rho(1)}^{a_{\rho(1)}b_{1}}\cdots x_{\rho(n)}^{a_{\rho(n)}b_{n}}) := x_{\pi\rho(1)}^{a_{\pi\rho(1)}b_{1}}\cdots x_{\pi\rho(n)}^{a_{\pi\rho(n)}b_{n}}\]
where $\rho\in S_n$, $a_{\rho(i)}=g=b_i$ if $\rho(i)\in \mathcal I$, $(\sigma,\tau)\in S_{\mathcal I}\times S_{\mathcal J}$, and $\pi(i):=\sigma(i)$ if $i\in \mathcal I$ while $\pi(i):=\tau(i)$ if $i\in \mathcal J$.
In this way $P_{\mathcal I,\mathcal J,\bs a}^U$ becomes an $S_{\mathcal I} \times S_{\mathcal J}$-module.
If $A$ is an $(L,U)$-algebra, then $P_{\mathcal I,\mathcal J,\bs a}^U\cap \I^U(A)$ is invariant under the $S_{\mathcal I} \times S_{\mathcal J}$-action, making
\[P_{\mathcal I,\mathcal J,\bs a}^U(A):=\frac{P_{\mathcal I,\mathcal J,\bs a}^U}{P_{\mathcal I,\mathcal J,\bs a}^U\cap \I^U(A)}\]
an $S_{\mathcal I} \times S_{\mathcal J}$-module with the induced action.
If $|\mathcal I|=r$, then $S_{\mathcal I} \times S_{\mathcal J}\cong S_{r} \times S_{n-r}$ where $S_r$ and $S_{n-r}$ denote the symmetric groups acting on the sets $\{1,\dots , r\}$ and $\{r+1,\dots, n\}$, respectively. Thus, for any $(\mathcal I,\mathcal J)\in\stirlingtwo{[n]}{2}$ such that $|\mathcal I|=r$ and $\bs a\in\mathcal{S}^{n-r}$, $P_{\mathcal I,\mathcal J,\bs a}^U(A)$ can be regarded as an $S_r\times S_{n-r}$-module. As a consequence, the space
\[P_{(n;r)}^U(A):= \bigoplus_{(\mathcal I,\mathcal J)\in\stirlingtwo{[n]}{2} \, | \, |\mathcal I|=r,\, \bs{a}\in \mathcal{S}^{n-r}}P_{\mathcal I,\mathcal J,\bs a}^U(A)\]
is also an $S_r\times S_{n-r}$-module, whose character we denote by $ \chi_{(n;r)}^U(A)$ and call the $(n,r)$th $U$-cocharacter of A.
Now, fixed $a\in \mathcal S$, the space $P^U_{r,n-r}(A)$ (defined in <ref>) is an $S_r \times S_{n-r}$-module whose character we denote by $\chi_{r,n-r}^U(A)$ (it is independent of a, as deduced Proposition <ref>(1)). From the same result we know that for each $(\mathcal I,\mathcal J)\in\stirlingtwo{[n]}{2}$ such that $|\mathcal I|=r$ and $\bs a\in\mathcal{S}^{n-r}$ there is an isomorphism of vector spaces between $P_{\mathcal I,\mathcal J,\bs a}^U(A)$ and $P^U_{r,n-r}(A)$, and since the $S_r\times S_{n-r}$-action commutes with the isomorphism, it is in addition an isomorphism of $S_r\times S_{n-r}$-modules. Therefore
\begin{equation}\label{eq:P_{(n;r)}^U(A)}
P_{(n;r)}^U(A) \underset{S_r\times S_{n-r}}{\cong\phantom{m}}\binom nr (k^2-1)^{n-r} P^U_{r,n-r}(A).
\end{equation}
Recall that the irreducible $S_r \times S_{n-r}$-characters are the tensor products $\chi_\lambda \otimes \chi_\mu$ of the irreducible $S_r$- and $S_{n-r}$-characters $\chi_\lambda$ and $\chi_\mu$, where $\lambda\vdash r$ and $\mu \vdash n-r$ are partitions.
Since $\chr(F)=0$, by complete reducibility we can write
\begin{equation} \label{S_r X S_n-r character}
\chi_{r,n-r}^U(A)=\sum_{(\lambda,\mu)\vdash(r,n-r)} m_{\lambda,\mu}\,\chi_\lambda \otimes \chi_\mu,
\end{equation}
where $\lambda\vdash r$, $\mu \vdash n-r$, and $m_{\lambda, \mu}\geq 0$ is the multiplicity corresponding to $\chi_\lambda \otimes \chi_\mu$. Thus, as a consequence of (<ref>) and (<ref>), the $(n,r)$th $U$-cocharacter of A can be decomposed as
\[ \chi_{(n;r)}^U(A)=\sum_{(\lambda,\mu)\vdash(r,n-r)} \binom nr (k^2-1)^{n-r} m_{\lambda,\mu}\,\chi_\lambda \otimes \chi_\mu. \tag{$\chi$}\label{formula(chi)}
\]
§ DIFFERENTIAL IDENTITIES OF $\M$
In this section we determine the $U$-identities, $(L,U)$-identities, and $L$-identities of the algebra $\M$ for $k\geq 2$.
§.§ $U$-identities of $\M$
Multiplication table of $\M$.
The multiplication table arising from $\mathcal M$ (see <ref>), with results expressed in matrix units, is summarized by the following relations:
* $gx=x=xg$ for any $x\in\mathcal M$.
* $e_{ij}e_{jk}=e_{ik}$, $e_{ij}e_{kl}=0$ ($j\neq k$).
* $h_ie_{ij} = e_{ij}$, $h_ie_{i+1,j}=-e_{i+1,j}$, $e_{ij}h_j = e_{ij}$, $e_{i,j+1}h_{j} = -e_{i,j+1}$, $h_ie_{jk}=0$ ($j\not\in\{i,i+1\}$), $e_{ij}h_k=0$ ($k\not\in\{j,j+1\}$).
* $h_i^2=e_{ii}+e_{i+1,i+1}$, $h_ih_{i+1}=-e_{i+1,i+1}=h_{i+1}h_i$, $h_ih_j=0$ ($j\not\in\{i-1,i,i+1\}$).
Generating $U$-identities.
Due to the nature of the endomorphisms $\varphi_{ab}$ (see Formula (<ref>)), the identities in the multiplication table of $\M$ (see <ref>) translate well to $U$-identities of $\M$: for example, if $x,y\in\M$ and $a,b\in\mathcal S$ then
\[x^{ae_{ij}}y^{be_{lm}}=\mu_a^x\mu_b^ye_{ij}e_{lm}=0\text{ if }j\neq l.\]
This idea provides us at once with the following $U$-identities of $\M$ in two variables.
Fix $a_1,a_2\in \mathcal M$. For $1\leq i\leq r$, fix $\sigma_i\in S_2$ and let $\alpha_i\in F$ and $m^i_1,m^i_2\in\mathcal M$ be such that $\sum_{i=1}^r \alpha_im^i_1m^i_2=0$ in $\M$, with $a_1=g$ (resp. $a_2=g$) forcing $m^i_{\sigma_i(1)}=g$ (resp. $m^i_{\sigma_i(2)}=g$) for all $1\leq i\leq r$. Then
\[\sum_{i=1}^r \alpha_i x^{a_{\sigma_i(1)}m^i_1}_{\sigma_i(1)}x^{a_{\sigma_i(2)}m^i_2}_{\sigma_i(2)}\in\I^U(\M).\]
Evaluating in $x_1,x_2\in\M$, by definition of $\vp_{ab}$ we get
\[\sum_{i=1}^r \alpha_i x^{a_{\sigma_i(1)}m^i_1}_{\sigma_i(1)}x^{a_{\sigma_i(2)}m^i_2}_{\sigma_i(2)} = \sum_{i=1}^r \mu^{x_{\sigma_i(1)}}_{a_{\sigma_i(1)}}\mu^{x_{\sigma_i(2)}}_{a_{\sigma_i(2)}}\alpha_im^i_1m^i_2 = \mu^{x_1}_{a_1}\mu^{x_2}_{a_2}\sum_{i=1}^r \alpha_im^i_1m^i_2 = 0.\qedhere\]
We will show in the following that all $U$-identities of $\M$ can be generated from $U$-identities in two variables, with at most two terms, arising from its multiplication table as in Lemma <ref>.
Recall that notation $x^b$ with $x\in\FU$, $b\in\mathcal S$ is shorthand for the element $x^{\vp_{ab}}\in\FU$, for a fixed and elided first exponent index $a\in\mathcal S$ which we will not explicitly mention in the next results (see <ref>). Similarly we write $P^U_{r,n-r}$ instead of $P^U_{r,n-r,a}$ (see <ref>).
We start the description of the $U$-identities and codimensions of $M_k(F)$ by appealing to the linear structure of the $T_U$-ideal. We will tackle the simpler case $k=2$ separately.
The $T_U$-ideal of $U$-identities of $M_2(F)$ is generated by the following $U$-polynomials:
\[[x^g,y^g], \ [x^g, y^a],\ x^a y^b - y^a x^b, \ x^c y^c, \ x^{h_1} y^c + x^c y^{h_1}, \ x^{e_{12}} y^{e_{21}} + x^{e_{21}} y^{e_{12}}-x^{h_1} y^{h_1},\]
where $a,b \in \{h_1, e_{12}, e_{21}\}$ and $c\in \{e_{12}, e_{21}\}$. In addition, $c_n^U(M_2(F))=4^{n+1}- 3(n+1).$
Firstly, note that $M_2(F)$ has no nontrivial $U$-identities of degree $1$ (as $x^\vp=0$ implies $\vp=0$ for any $\vp\in\E(M_2(F))$), and so $c_1^U(M_2(F)) = \dim_F(P_1^U) = \dim_F(U) =3^2 +1 = 4^2-3\cdot 2$ as needed. Hence in the following, we assume $n\geq2$.
Let $I$ be the $T_U$-ideal generated by the $U$-polynomials in the statement of the proposition. We will show that $I=\I^U(M_2(F))$.
Recall that $\mathcal S=\{h_1, e_{12}, e_{21}\}$ for $k=2$ by definition. By Lemma <ref> it follows that $I\subseteq \I^U(M_2(F))$.
In order to prove the opposite inclusion let $f\in \I^U(M_2(F))$ with $\deg f=n$ and assume, as we may, that $f$ is multilinear and $f\in P^U_{r,n-r}$, where $0\leq r \leq n$ (see <ref> and <ref>). We will prove that $f\equi I 0$.
For all $ 1\leq i \leq n-r$, in order to simplify the notation, let us rename $x_{r+i}^a$ to $y_i^a$, $a\in \mathcal S$, so that variables $x_1,\ldots,x_r$ correspond to exponents $g$ and variables $y_1,\ldots,y_{n-r}$ correspond to exponents $a\in\mathcal S$. Since $ [x_1^g,x_2^g], \ [x^g, y^a]\in I$ for all $a\in\mathcal S$, $f$ modulo $I$ is a linear combination of $U$-monomials of type
\[x_1^g \cdots x_r^g y_{i_1}^{a_1}\cdots y_{i_{n-r}}^{a_{n-r}}\]
where $a_{1},\dots, a_{n-r}\in\mathcal S$. If $r=n$ we have $f\equi I \alpha x_1^g \cdots x^g_n$ for some $\alpha\in F$; by evaluating $x_i=g$ for $1\leq i\leq n$ we find, since $f$ is a $U$-identity of $M_2(F)$,
\[0=f(g,\ldots,g)\equi I \alpha(g)^{gg}\cdots (g)^{gg} = \alpha g^n = \alpha g,\]
hence $\alpha=0$ and $f\equi I 0$.
If $r=n-1$ then we have
\[f\equi I \alpha_1 x^g_1 \cdots x^g_{n-1}y_1^{h_1}+ \alpha_2 x^g_1 \cdots x^g_{n-1}y_1^{e_{12}}+ \alpha_3 x^g_1 \cdots x^g_{n-1}y_1^{e_{21}}\]
for some $\alpha_i\in F$, $1\leq i\leq 3$; by evaluating $x_i=g$ for $1\leq i \leq n-1$ and $y_1=h_1 +e_{12}+ e_{21}$ we analogously get,
for some $a\in\{h_1,e_{12},e_{21}\}$,
\[\alpha_1(g)^{gg}\cdots (g)^{gg}(h_1+e_{12}+e_{21})^{ah_1} + \cdots + \alpha_3(g)^{gg}\cdots (g)^{gg}(h_1+e_{12}+e_{21})^{ae_{21}}
and hence $\alpha_{i}=0$ for $1\leq i \leq 3$, and $f\equi I 0$.
Now let us assume that $0\leq r \leq n-2$. Since $y_1^a y_2^b - y_2^a y_1^b\in I$ with $a,b\in\mathcal S$, it is possible to reorder the variables $y_1,\ldots,y_r$ in each $U$-monomial of $f$ modulo $I$ without reordering their original exponents. Moreover, since $y_1^c y_2^c\in I$ for $c\in \{e_{12}, e_{21}\}$, modulo $I$ the nonzero terms of $f$ do not have two variables with exponent $e_{12}$ nor with exponent $e_{21}$ adjacent to each other. Since in addition $y_1^{h_1} y_2^c + y_1^c y_2^{h_1}\in I$ for $c\in \{e_{12}, e_{21}\}$, we can permute the $h_1$ exponents with the $e_{12},e_{21}$ exponents; it then follows that $f$ modulo $I$ can be written as a linear combination of $U$-monomials of the following forms:
\begin{align*}
&x^g_1\cdots x^g_r y_{1}^{h_1} \cdots y_{m}^{h_1} y_{m+1}^{e_{12}} y_{m+2}^{e_{21}} \cdots y_{n-r-1}^{e_{12}} y_{n-r}^{e_{21}},\\
&x^g_1\cdots x^g_r y_{1}^{h_1} \cdots y_{m}^{h_1} y_{m+1}^{e_{21}} y_{m+2}^{e_{12}} \cdots y_{n-r-1}^{e_{21}} y_{n-r}^{e_{12}},\\
&x^g_1\cdots x^g_r y_{1}^{h_1} \cdots y_{m}^{h_1} y_{m+1}^{e_{12}} y_{m+2}^{e_{21}} y_{m+3}^{e_{12}}\cdots y_{n-r-1}^{e_{21}} y_{n-r}^{e_{12}},\\
&x^g_1\cdots x^g_r y_{1}^{h_1} \cdots y_{m}^{h_1} y_{m+1}^{e_{21}} y_{m+2}^{e_{12}} y_{m+3}^{e_{21}} \cdots y_{n-r-1}^{e_{12}} y_{n-r}^{e_{21}},
\end{align*}
where $0\leq m \leq n-r$. Since $y_1^{e_{12}} y_2^{e_{21}} + y_1^{e_{21}} y_2^{e_{12}} -y_1^{h_1} y_2^{h_1} \in I$ it follows that, if $n-r\geq 2$, we can write $f$ modulo $I$ as a linear combination of $U$-monomials in which at most one exponent $h_1$ appears. Thus, if $n-r$ is even, we get that
\begin{align}
\label{n-r even}
f\equi I & \alpha_1 x^g_1\cdots x^g_r y_{1}^{h_1} y_{2}^{e_{12}} y_{3}^{e_{21}} y_{4}^{e_{12}}\cdots y_{n-r-1}^{e_{21}} y_{n-r}^{e_{12}}+\alpha_2 x^g_1\cdots x^g_r y_{1}^{h_1} y_{2}^{e_{21}} y_{3}^{e_{12}} y_{4}^{e_{21}} \cdots y_{n-r-1}^{e_{12}} y_{n-r}^{e_{21}}+\\
+& \alpha_3 x^g_1\cdots x^g_r y_{1}^{e_{12}} y_{2}^{e_{21}}\cdots y_{n-r-1}^{e_{12}} y_{n-r}^{e_{21}}
+ \alpha_4 x^g_1\cdots x^g_r y_{1}^{e_{21}} y_{2}^{e_{12}} \cdots y_{n-r-1}^{e_{21}} y_{n-r}^{e_{12}} \notag
\end{align}
for some $\alpha_i\in F$, $1\leq i\leq 4$, whereas if $n-r$ is odd, then we have that
\begin{align*}
f\equi I & \alpha_1 x^g_1\cdots x^g_r y_{1}^{h_1} y_{2}^{e_{12}} y_{3}^{e_{21}} \cdots y_{n-r-1}^{e_{12}} y_{n-r}^{e_{21}}+ \alpha_2 x^g_1\cdots x^g_r y_{1}^{h_1} y_{1}^{e_{21}} y_{2}^{e_{12}} \cdots y_{n-r-1}^{e_{21}} y_{n-r}^{e_{12}}+\\
+& \alpha_3 x^g_1\cdots x^g_r y_{1}^{e_{12}} y_{2}^{e_{21}} y_{3}^{e_{12}}\cdots y_{n-r-1}^{e_{21}} y_{n-r}^{e_{12}}+ \alpha_4
x^g_1\cdots x^g_r y_{1}^{e_{21}} y_{2}^{e_{12}} y_{3}^{e_{21}} \cdots y_{n-r-1}^{e_{12}} y_{n-r}^{e_{21}}
\end{align*}
for some $\alpha_i\in F$, $1\leq i\leq 4$.
Suppose that $f$ is as in (<ref>). By making the evaluation $x_i=g$ for $1\leq i \leq r$ and $y_j=h_1 +e_{12}+ e_{21}$ for $1\leq j \leq n-r$ we get $\alpha_1 e_{12} -\alpha_2 e_{21}+ \alpha_3 e_{11}+ \alpha_4 e_{22}=0$. Thus $\alpha_i=0$ for $1\leq i \leq 4$, and $f$ is the zero $U$-polynomial modulo $I$. One can deal similarly with the other case. Thus $\I^U(M_2(F))=I$.
The argument above also proves that
\[c^U_{r,n-r}(M_2(F))=\begin{cases}
1, & \mbox{ if } r=n, \\
3, & \mbox{ if } r=n-1, \\
4, & \mbox{ if } 0\leq r\leq n-2.
\end{cases}\]
Therefore, by Formula (<ref>),
\[c_n^U(M_2(F))= \sum_{r=0}^{n} \binom{n}{r} 3^{n-r} c^U_{r,n-r}(M_2(F))=4\sum_{r=0}^{n} \binom{n}{r} 3^{n-r} -3n-3= 4^{n+1}-3(n+1).\qedhere\]
Next lemma follows from simple computations.
If $k\geq 3$, then:
* $x^{e_{il}}y^{e_{lj}} \equiv - x^{h_i}y^{e_{ij}} \ (\md \langle x^{e_{i j}}y^{h_{j-1}}- x^{e_{i l}} y^{e_{lj}}, \ x^{h_i}y^{e_{ij}}+ x^{e_{ij}}y^{h_{j-1}} \rangle_{T_U})$, where $1\leq i \leq k-1$, $2\leq j \leq k$, $ j\neq i$, $1\leq l \leq k$;
* $x^{e_{1j}}y^{e_{j1}} \equiv x^{h_1} y^{h_1}+ x^{h_1}y^{h_2} \ (\md \langle x^{e_{1j}} y^{e_{j1}} + x^{e_{2,l}} x_2^{e_{l,2}}-x^{h_1} x_2^{h_1}, \ x^{h_{1}} x_2^{h_2} + x^{e_{2j}} x_2^{e_{j2}} \rangle_{T_U})$, where $2\leq j \leq k$, $1\leq l \leq k$, $l \neq 2$;
* $x^{e_{kj}}y^{e_{jk}} \equiv x^{h_{k-1}} y^{h_{k-1}}+ x^{h_{k-2}}y^{h_{k-1}} \ (\md \langle x^{e_{k-1,j}} y^{e_{j,k-1}} + x^{e_{k,l}} y^{e_{l,k}}-x^{h_{k-1}} y^{h_{k-1}}, \ x^{h_{k-2}} y^{h_{k-1}} + x^{e_{k-1,l}} y^{e_{l,k-1}} \rangle_{T_U})$, where $1\leq j \leq k-1$, $1\leq l \leq k$, $l\neq k-1$.
The $T_U$-ideal of $U$-identities of $\M$, $k\geq 3$, is generated by the following $U$-polynomials:
* $[x^g,y^g], \ \ [x^g, y^a], \ \ x^a y^b - y^a x^b$, where $a,b\in \mathcal{S}$;
* $x^{e_{ij}}y^{e_{lm}}$, where $1\leq i,j,l,m\leq k$, $j\neq l$;
* $x^{h_i}y^{e_{jl}}$, where $1\leq i \leq k-1$, $1\leq j, l \leq k$, $ j\neq i, i+1$, $l\neq j$;
* $x^{e_{jl}}y^{h_i}$ where $1\leq i \leq k-1$, $1\leq j,l\leq k$, $l\neq i, i+1$, $j\neq l$;
* $x^{h_i}y^{e_{ij}}+ x^{e_{ij}}y^{h_{j-1}} $, where $ 1\leq i \leq k-1$, $ 2\leq j \leq k$, $j\neq i$;
* $ x^{h_{i-1}}y^{e_{ij}}+ x^{e_{ij}}y^{h_j}$, where $ 2\leq i \leq k$, $1\leq j \leq k-1$, $ j\neq i$;
* $x^{h_{i-1}}y^{e_{i j}}+x^{e_{i l}} y^{e_{lj}}$, where $2\leq i \leq k$, $1\leq j,l\leq k$, $ j\neq i$, $i,j\neq l$;
* $x^{h_{i-1}} y^{h_i} + x^{e_{ij}} y^{e_{ji}}$, where $2\leq i \leq k-1$, $1\leq j \leq k$, $j\neq i$;
* $x^{h_i}y^{h_j}$, where $1\leq i,j\leq k-1$, $j\neq i-1,i,i+1$, if $k\geq 4$;
* $[x^{h_{i}}, y^{h_{i+1}}]$, where $1\leq i \leq k-2$;
* $x^{e_{ij}} y^{e_{ji}} + x^{e_{i+1l}}y^{e_{li+1}}-x^{h_i}y^{h_i}$, where $ 1\leq i \leq k-1$, $1\leq j,l\leq k$, $j\neq i$, $ l\neq i+1$;
* $x^{e_{i j}}y^{h_{j-1}}+ x^{e_{i l}} y^{e_{lj}}$, where $1\leq i,l\leq k$, $2\leq j \leq k$, $j\neq i$, $i,j\neq l$.
In addition, $c_n^U(\M)=k^{2(n+1)}- (k^2-1)(n+1).$
The proof of this result follows a scheme similar to that of Proposition <ref> for $2\times2$ matrices, with different computations. Firstly, note that there are no nontrivial identities of degree $1$ and thus $c_1^U(\M) = \dim_F(U) = (k^2-1)^2 +1 = k^4 - 2(k^2-1)$ as expected, so henceforth we assume $n\geq2$. Let $I$ be the $T_U$-ideal generated by the $U$-polynomials in the statement of the proposition. By Lemma <ref> it follows that $I\subseteq \I^U(\M)$. To prove the opposite inclusion, first, we find a set of generators of $P^U_{r,n-r}$ modulo $P^U_{r,n-r}\cap I$, for each $n \geq 1$ and $0\leq r\leq n$, and after that, we show, by evaluation, that the sets of generators found are actually bases of their corresponding vector spaces.
Let $f\in P^U_{r,n-r}$ be a multilinear $U$-polynomial of degree $n$. In order to simplify the notation, for all $ 1\leq i \leq n-r$ let us rename $x_{r+i}^a$ to $y_i^a$, $a\in \mathcal{S}$, so that variables $x_1,\ldots,x_r$ correspond to exponents $g$ and variables $y_1,\ldots,y_{n-r}$ correspond to exponents $a\in\mathcal S$. Since $[x^g_1,x^g_2], \ [x^g, y^a] \in I$ for all $a\in \mathcal{S}$, $f$ modulo $I$ is a linear combination of $U$-monomials of type
\[x_1^g \cdots x_r^g y_{i_1}^{a_1}\cdots y_{i_{n-r}}^{a_{n-r}}\]
where $a_{i_1},\dots, a_{i_{n-r}}\in \mathcal{S}$. So $P^U_{n,0}$ is generated modulo $P^U_{n,0}\cap I$ by the $U$-monomial $x_1^g \cdots x_n^g$. If $r=n-1$ then $f$ modulo $I$ is a linear combination of $U$-monomials
\begin{equation}
\label{generatori r=n-1}
x^g_1 \cdots x^g_{n-1}y_1^{h_i} \text{ for }1\leq i \leq k-1\text{ and }x^g_1 \cdots x^g_{n-1}y_1^{e_{jl}}\text{ for }1\leq j,l \leq k, j\neq l.
\end{equation}
It follows that $P^U_{n-1,1}$ is generated modulo $P^U_{n-1,1}\cap I$ by the $U$-monomials in (<ref>).
Now suppose that $0\leq r \leq n-2$. By $U$-identities <ref>-<ref>, in each $U$-monomial of $f$ we can move to the left, modulo $I$, all the variables with exponent $h_i$ for $1\leq i \leq k-1$; moreover, since $y_1^a y_2^b - y_2^a y_1^b\in I$ for all $a,b\in \mathcal{S}$, we can always reorder the indices of the variables with exponent in $\mathcal S$. Call (P1) to this moving and reordering procedure. From (P1) it follows that $f$ modulo $I$ is a linear combination of $U$-monomials of type
\[x^g_1\cdots x^g_r y_{1}^{h_{i_1}} \cdots y_{s}^{h_{i_s}} y_{s+1}^{b_1} \cdots y_{n-r}^{b_{n-r}}\]
for $0\leq s\leq n-r$, $1\leq i_1, \dots, i_s\leq k-1$ and $b_1,\dots, b_{n-r}\in \{e_{ij}\ | \ 1\leq i,j\leq k, i\neq j\}$.
Since $y_1^{e_{ij}}y_2^{e_{lm}}\in I$ for $1\leq i,j,l,m\leq k$, $j\neq l$, we can require in addition that the sequence of exponents $b_1,\dots, b_{n-r}$ has nonzero product $b_1\cdots b_{n-r}$, i.e., if $b_t=e_{ij}$ and $b_{t+1}=e_{lm}$ then $j=l$. Notice that the $U$-identities of Lemma <ref> also belong to $I$, thus by $U$-identities <ref>,<ref> and Lemma <ref> we can reduce each pair $y_t^{e_{il}}y_{t+1}^{e_{lj}}$:
* If $i\neq j,k$, to the pair $y_t^{h_i}y_{t+1}^{e_{ij}}$.
* If $i=k\neq j$, to the pair $y_t^{h_{k-1}}y_{t+1}^{e_{kj}}$.
* if $i=j$, to a linear combination of one or two $U$-monomials of the form $y_t^{h_l}y_{t+1}^{h_m}$ with $1\leq l\leq k-1$ and $m\in\{l,l+1\}$.
To this reduction procedure call (P2). By applying (P1) and (P2) repeatedly, we find that $f$ modulo $I$ is a linear combination of $U$-monomials of types
\[x^g_1\cdots x^g_r y_{1}^{h_{i_1}} \cdots y_{n-r}^{h_{i_{n-r}}}\text{ and }x^g_1\cdots x^g_r y_{1}^{h_{i_1}} \cdots y_{n-r-1}^{h_{i_{n-r-1}}} y_{n-r}^{e_{ij}}\]
for $ 1\leq i_1, \dots, i_{n-r}\leq k-1$ and $1\leq i,j\leq k$, $i\neq j$. By $U$-identity <ref> we can assume that the product of the exponents $h_{i_1}\cdots h_{i_{n-r}}$ is nonzero, and by $U$-identity <ref> we can assume that $i_1 \leq \cdots \leq i_{n-r}$; hence we may assume that $i_{j+1}\in\{i_j,i_j+1\}$ for all $0\leq j<n-r$.
Moreover, since for $k\geq 4$, by $U$-identities <ref>,<ref> we have
\[y_1^{h_{i}}y_2^{h_{i+1}}y_3^{h_{i+2}} = [y_1^{h_{i}}y_2^{h_{i+1}}]y_3^{h_{i+2}}+y_2^{h_{i+1}}(y_1^{h_{i}}y_3^{h_{i+2}})\in I\]
for $1\leq i \leq k-3$, we can assume that, for all $k\geq 3$, in each $U$-monomial of $f$ modulo $I$ there are at most two distinct (and consecutive) exponents $h_i$ ($1\leq i\leq k-1$), the rest of them being copies of one of those.
In addition, by $U$-identities <ref> (applied twice) and <ref> we find
\begin{align*}
&y_1^{h_{i}}y_2^{h_{i+1}} +y_1^{h_{i-1}}y_2^{h_i} + y_1^{h_i}y_2^{h_i} =\\
&(y_1^{h_{i}}y_2^{h_{i+1}} + y_1^{e_{i+1j}}y_2^{e_{ji+1}}) + (y_1^{h_{i-1}}y_2^{h_i} + y_1^{e_{ij}}y_2^{e_{ji}})-(y_1^{e_{ij}} y_2^{e_{ji}} + y_1^{e_{i+1j}} y_2^{e_{ji+1}}-y_1^{h_i}y_2^{h_i})\in I
\end{align*}
for all $2\leq i \leq k-2$, so by recursion on $i$ we may suppose that, in each $U$-monomial of $f$, either all exponents $h_i$ are equal (for some $1\leq i\leq k-1$) or there are two distinct exponents, $h_1$ and $h_2$. Now notice that by $U$-identities <ref>,<ref>,<ref> we have
\[y_1^{h_1}y_2^{h_2}y_3^{h_2} + y_1^{h_1} y_2^{h_1}y_3^{h_2} = y_1^{h_1}(y_2^{h_2}y_3^{h_2}-y_2^{e_{21}}y_3^{e_{12}}-y_2^{e_{31}}y_3^{e_{13}}) +y_1^{h_1}(y_2^{h_1}y_3^{h_2}+y_2^{e_{21}}y_3^{e_{12}})+ (y_1^{h_1}y_2^{e_{31}})y_3^{e_{13}}\in I,\]
so $y_1^{h_1}y_2^{h_2}y_3^{h_2} \equi I -y_1^{h_1} y_2^{h_1}y_3^{h_2}$. Hence $f$ modulo $I$ is a linear combination of $U$-monomials of types
x^g_1⋯x^g_r y_1^h_i ⋯y_n-r^h_i, x^g_1⋯x^g_r y_1^h_1 ⋯y_n-r-1^h_1y_n-r^h_2,
x^g_1⋯x^g_r y_1^h_i ⋯y_n-r-1^h_i y_n-r^e_lj, x^g_1⋯x^g_r y_1^h_1 ⋯y_n-r-2^h_1 y_n-r-1^h_2 y_n-r^e_lj,
for $1\leq i \leq k-1$, $1\leq j, l \leq k$, $l\neq j$. Finally, since $y_1^{h_{i-1}}y_2^{e_{ij}} + y_1^{h_i} y_2^{e_{ij}} \in \langle y_1^{h_{i-1}}y_2^{e_{i j}}+ y_1^{e_{i l}} y_2^{e_{lj}}, \ y_1^{e_{i j}}y_2^{h_{j-1}}+ y_1^{e_{i l}} y_2^{e_{lj}}, \ y_1^{h_i}y_2^{e_{ij}}+ y_1^{e_{ij}}y_2^{h_{j-1}} \rangle_{T_U},$ where $2\leq i \leq k-1$, $1\leq j, l \leq k$, $j\neq i$, $l\neq i,j$, and by $U$-identity <ref>, it follows that $f$ modulo $I$ is a linear combination of the following $U$-monomials:
\begin{equation}\label{relativamente libera Mk}
\begin{split}
& x^g_1\cdots x^g_r y_1^{h_i} \cdots y_{n-r}^{h_i},\ \ x^g_1\cdots x^g_r y_1^{h_1} \cdots y_{n-r-1}^{h_1} y_{n-r}^{h_2},\\
& x^g_1\cdots x^g_r y_1^{h_i} \cdots y_{n-r-1}^{h_i} y_{n-r}^{e_{ij}},\ \
x^g_1\cdots x^g_r y_1^{h_{k-1}} \cdots y_{n-r-1}^{h_{k-1}} y_{n-r}^{e_{kl}},
\end{split}
\end{equation}
where $1\leq i, l \leq k-1$, $1\leq j \leq k$, $j\neq i$. Thus we have that $P^U_{r, n-r}$ modulo $P^U_{r,n-r}\cap I$ is generated by the $U$-monomials in (<ref>).
The $U$-monomial $x_1^g \cdots x_n^g$ can be seen to be nonzero modulo $I$ by evaluating $x_1=\cdots =x_n =g$. We next show that the $U$-monomials in (<ref>) and (<ref>) are linearly independent modulo $\I^U(\M)$ if $r=n-1$ or $0\leq r\leq n-2$, respectively. To that end, let us assume first that $f\in \I^U(\M)$ is a linear combination of $U$-monomials in (<ref>), i.e.,
\[f=\sum_{1\leq i \leq k-1} \alpha_i x^g_1 \cdots x^g_{n-1} y_1^{h_i}+\sum_{\substack{1\leq j,l \leq k\\ j\neq l}} \beta_{jl} x^g_1 \cdots x^g_{n-1} y_1^{e_{jl}}\]
for some $\alpha_i,\beta_{jl}\in F$. From the evaluation
$x_1=\cdots=x_{n-1}=g\text{ and } y_1=\sum_{a\in\mathcal S} a$
we get
\[\sum_{1\leq i \leq k-1} \alpha_i h_i + \sum_{\substack{1\leq j,l \leq k\\ j\neq l}} \beta_{jl} e_{jl}=0,\]
from which it follows, since $\mathcal S$ is a linearly independent set, that $\alpha_i=\beta_{lj}=0$ for all $1\leq i \leq k-1$, $1\leq j , l \leq k$, $j \neq l$. Therefore the $U$-monomials in (<ref>) are linearly independent modulo $\I^U(\M)$.
Let us assume now that $f\in \I^U(\M)$ is such that
\begin{align*}
f= & \sum_{1\leq i \leq k-1} \alpha_i x^g_1\cdots x^g_r y_1^{h_i} \cdots y_{n-r}^{h_i} + \beta x^g_1\cdots x^g_r y_1^{h_1} \cdots y_{n-r-1}^{h_1} y_{n-r}^{h_2}+ \sum_{\substack{1\leq i \leq k-1\\ 1\leq j \leq k\\ i\neq j}} \gamma_{ij} x^g_1\cdots x^g_r y_1^{h_i} \cdots y_{n-r-1}^{h_i} y_{n-r}^{e_{ij}}+\\
+&\sum_{1\leq l \leq k-1} \gamma_{kl}
x^g_1\cdots x^g_r y_1^{h_{k-1}} \cdots y_{n-r-1}^{h_{k-1}} y_{n-r}^{e_{kl}} .
\end{align*}
If we evaluate $x_1=\cdots=x_r=g$ and $y_1=\cdots=y_{n-r}=\sum_{a\in\mathcal S} a$,
we get
\[\sum_{1\leq i \leq k-1} \alpha_i (e_{ii}\pm e_{i+1,i+1})+ \beta e_{22} +\sum_{\substack{1\leq i \leq k-1\\ 1\leq j \leq k\\ i\neq j}} \gamma_{ij} e_{ij}+ \sum_{1\leq l \leq k-1} (-1)^{n-r-1} \gamma_{kl} e_{kl} =0,\]
which produces $\alpha_{i}=\beta=\gamma_{lj}=0$ for all $1\leq i \leq k-1$, $1\leq l,j\leq k$. Therefore the elements in (<ref>) are linearly independent modulo $P^U_{r,n-r}\cap \I^U(\M)$.
The fact that $P^U_{r,n-r}\cap \I^U(\M) \supseteq P^U_{r,n-r}\cap I$ for all $n\in\N$ and $0\leq r\leq n$ proves that $\I^U(\M)=I$, with $x_1^g\cdots x_n^g$ and the elements in (<ref>), (<ref>)
forming a basis of $P^U_{r,n-r}$ modulo $P^U_{r,n-r}\cap \I^U(\M)$ for $r=n$, $r=n-1$ and $0\leq r \leq n-2$, respectively.
Thus, by counting we get
\[c^U_{r,n-r}(\M)=\begin{cases}
1, & \mbox{ if } r=n, \\
k^2 -1, & \mbox{ if } r=n-1, \\
k^2, & \mbox{ if } 0\leq r\leq n-2.
\end{cases}\]
Hence, by Formula (<ref>) it follows that
\begin{align*}
c_n^U(\M) &= \sum_{r=0}^{n} \binom{n}{r} (k^2 -1)^{n-r} c^U_{r,n-r}(\M)=k^2 \sum_{r=0}^{n} \binom{n}{r} (k^2 -1)^{n-r} -(k^2 -1)n-(k^2 -1) = \\
&= k^{2(n+1)}-(k^2 -1)(n+1).\qedhere
\end{align*}
Modifying the second index of an exponent.
Through the $L$-action, the second basis element of the subindex of $\varphi_{ab}$ can also be changed in the search for identities, albeit the result is less straightforward. In particular, if $d\in L$ is a derivation then $(x^{ca}y^{cb})^d = x^{(ca)d}y^{cb} + x^{ca}y^{(cb)d}$. We will resort to this method to reduce the number of generators in the basis of $\I^U(\M)$.
We are working in $\FU$, which has no $U$-action (see Remarks <ref>). Nevertheless, to simplify the notation, instead of writing exponents belonging to $\Uop$ whose action would eventually project to $U^{\op}$ once they landed on isolated variables, we will write the exponents directly in $U^{\op}$ with the caution of evaluating them only on isolated variables. For example, to apply the action of $e_{12}^2\in \Uop$ to $x^uy^v\in\FU$, we write
\[(x^uy^v)^{E_{12}^2} = x^{uE_{12}^2}y^v + 2x^{uE_{12}}y^{vE_{12}} + x^uy^{vE_{12}^2}\]
with $E_{12}\in U^{\op}$, and only then simplify the exponents $uE_{12}^2,uE_{12},vE_{12},vE_{12}^2$ by computing in $U^{\op}$.
The $T_U$-ideal of $U$-identities of $\M$ is generated by the following $U$-polynomials:
* Either $x^{e_{12}e_{12}}y^{e_{12}e_{12}}$ if $k=2$ or $x^{e_{12}e_{12}}y^{e_{12}e_{31}}$ if $k\geq3$,
* $x^{e_{12}e_{12}}y^{e_{12}e_{21}}-y^{e_{12}e_{12}}x^{e_{12}e_{21}}$,
* $[x^{gg},y^{gg}]$,
* $[x^{gg},y^{e_{12}e_{12}}]$.
By Propositions <ref> and <ref>, the $T_U$-ideal of $U$-identities of $\M$ is generated by the following list (L) of identities, for fixed $a\in\mathcal S$:
* $x^{ae_{ij}}y^{ae_{lm}}$ with $1\leq i\leq k-1$, $2\leq j\leq k$, $j\neq i$.
* $x^{ah_i}y^{ae_{jl}}$ with $1\leq i \leq k-1$, $1\leq j, l \leq k$, $ j\neq i, i+1$, $l\neq j$;
* $x^{ae_{jl}}y^{ah_i}$ with $1\leq i \leq k-1$, $1\leq j,l\leq k$, $l\neq i, i+1$, $j\neq l$;
* $x^{ah_i}y^{ah_j}$ with $1\leq i,j\leq k-1$, $j\neq i-1,i,i+1$.
* $ x^{ae_{ij}}y^{ah_{j-1}} + x^{ae_{il}} y^{ae_{lj}}$ with $1\leq i,l\leq k$, $2\leq j \leq k$, $l,j\neq i$, $j\neq l$;
* $x^{ah_{i-1}}y^{ae_{ij}} + x^{ae_{il}} y^{ae_{lj}}$ with $2\leq i \leq k$, $1\leq j,l\leq k$, $ l,j\neq i$, $ j\neq l$;
* $x^{ah_i}y^{ae_{ij}}+ x^{ae_{ij}}y^{ah_{j-1}}$ with $ 1\leq i \leq k-1$, $ 2\leq j \leq k$, $j\neq i$;
* $ x^{ah_{i-1}}y^{ae_{ij}}+ x^{ae_{ij}}y^{ah_j}$ with $ 2\leq i \leq k$, $1\leq j \leq k-1$, $ j\neq i$;
* $x^{ah_{i-1}} y^{ah_i} + x^{ae_{ij}} y^{ae_{ji}}$ with $2\leq i \leq k-1$, $1\leq j \leq k$, $j\neq i$;
* $x^{ae_{ij}} y^{ae_{ji}} + x^{ae_{i+1,l}} y^{ae_{l,i+1}}-x^{ah_i} y^{ah_i}$ with $ 1\leq i \leq k-1$, $1\leq j,l\leq k$, $j\neq i$, $l\neq i+1$;
* $x^{ab} y^{ac} - y^{ab} x^{ac}$ with $b,c\in \mathcal{S}$;
* $[x^{gg},y^{gg}], \, [x^{gg} y^{ab}]$ with $b\in \mathcal{S}$;
* $[x^{ah_{i}}, y^{ah_{i+1}}]$ with $1\leq i \leq k-2$;
where some identities are not realized for $k=2$ and $k=3$. Let $J$ be the $T_U$-ideal of $\FU$ generated by either $x^{e_{12}e_{12}}y^{e_{12}e_{12}}$ if $k=2$ or $x^{e_{12}e_{12}}y^{e_{12}e_{31}}$ if $k\geq3$, $x^{e_{12}e_{12}}y^{e_{12}e_{21}}-y^{e_{12}e_{12}}x^{e_{12}e_{21}}$, $[x^{gg},y^{gg}]$ and
$[x^{gg},y^{e_{12}e_{12}}]$. Clearly $J\subseteq\I^U(\M)$. In the following, we will prove that $J=\I^U(\M)$ by showing that all identities in the list (L) belong to $J$, mainly by the action of inner derivations. From now on we change, as we may, the first exponent index $a$ of any variable $x^{ab}$ with $b\in\mathcal S$ to $e_{12}$ and elide it by writing $x^b$; we also write $x^g$ for $x^{gg}$.
By the bracket formula (<ref>), the action of the inner derivation $E_{ij}$ (generated by $e_{ij}$) on $x^{e_{ab}}y^{e_{cd}}$ gives
\[(x^{e_{ab}}y^{e_{cd}})^{E_{ij}}=\delta_{ja}x^{e_{ib}}y^{e_{cd}} -\delta_{bi}x^{e_{aj}}y^{e_{cd}} +\delta_{jc}x^{e_{ab}}y^{e_{id}} -\delta_{di}x^{e_{ab}}y^{e_{cj}}\]
for $i,j,a,b,c,d\in \{1,\ldots,k\}$, $i\neq j, a\neq b, c\neq d$, where $\delta_{rs}$ denotes Kronecker's delta (see <ref> to review the key computational facts about inner derivations that we will need in the sequel). From now on we assume without further notice that any element of the form $e_{ab}$ or $E_{ab}$ satisfies $a\neq b$ and imposes this restriction wherever it appears.
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We propose and study a novel graph clustering method for data with an intrinsic network structure.
Similar to spectral clustering, we exploit an intrinsic network structure of data to
construct Euclidean feature vectors. These feature vectors can then be fed into basic
clustering methods such as k-means or Gaussian mixture model (GMM) based soft clustering.
What sets our approach apart from spectral clustering is that we do not use the eigenvectors
of a graph Laplacian to construct the feature vectors. Instead, we use the solutions of total
variation minimization problems to construct feature vectors that reflect connectivity between
data points. Our motivation is that the solutions of total variation minimization are
piece-wise constant around a given set of seed nodes. These seed nodes can be obtained
from domain knowledge or by simple heuristics that are based on the network structure of data.
Our results indicate that our clustering methods can cope with certain graph structures
that are challenging for spectral clustering methods.
machine learning; clustering; non-smooth optimization; community detection; complex networks
§ INTRODUCTION
The analysis of networked data is often facilitated by grouping or clustering the data points
into coherent subsets of data points. Clustering methods aim at finding subsets (clusters) of
data points that are more similar to each other than to the remaining data points [29].
Many basic clustering algorithms aim at clusters that are enclosed by a hypersphere or hyperellipsoid
in a Euclidean feature space. These methods are most successful if data points are characterized by
Euclidean feature vectors whose distance is small (large) for data points in the same (different) cluster(s).
Graph clustering methods can be applied to data with an intrinsic network structure that reflects
a notion of similarity between different data points. Many important application domains, ranging
from the Internet of Things to the management of pandemics, generate distributed collections of
local datasets (“big data over network”) [22, 23]. The
network structure of these local datasets might be induced by spatio-temporal proximity (“contact networks”),
statistical dependencies, or functional relations [1, 2].
§ PROBLEM FORMULATION
We represent networked data using an undirected “empirical” graph $\graph=(\nodes, \edges, \edgeweights)$ [24, 9].
Every node $\nodeidx \in \nodes = \{1,\ldots,\nrnodes\}$ of the empirical graph represents a datapoint. The dataset
consists of $\nrnodes$ different datapoints. A datapoint might be a single sensor measurement, an entire time
series, or even a whole collection of videos.
Our approach does not take the individual nature of datapoints into account. Rather, we only
use their similarities as encoded in the weighted edges of $\graph$. In particular, two nodes $\nodeidx, \nodeidx'$
are connected by an edge $\edge{\nodeidx}{\nodeidx'}$ if the corresponding datapoints are similar.
The amount of similarity is quantified by a positive edge weight $\edgeweight_{\nodeidx,\nodeidx'}$.
We find it convenient to use an oriented (or directed) version of the empirical graph $\graph$ by
declaring, for each undirected edge $\edge{\nodeidx}{\nodeidx'}$, the node $\min\{\nodeidx,\nodeidx'\}$
as tail and the other node $\max\{\nodeidx,\nodeidx'\}$ as head in the corresponding directed edge. The resulting
oriented empirical graph $\overrightarrow{\graph} \defeq \big( \nodes, \overrightarrow{\edges}, \edgeweights \big)$
has the same nodes as $\graph$ and contains the directed edge $\directededge{\nodeidx}{\nodeidx'}$ from node $\nodeidx$ to node $\nodeidx'$
if and only if $\nodeidx < \nodeidx'$ and $\edge{\nodeidx}{\nodeidx'} \in \edges$. This directed edge $\directededge{\nodeidx}{\nodeidx'} \in \overrightarrow{\edges}$ has the same
weight $\edgeweight_{\nodeidx,\nodeidx'}$ as the corresponding undirected edge $\edge{\nodeidx}{\nodeidx'} \in \edges$.
We will use several matrices that are naturally associated with an empirical graph $\graph$.
The weight matrix $\edgeweights$ contains the edge weight $\edgeweight_{\nodeidx,\nodeidx'}$
in its $\nodeidx$th row and $\nodeidx'$th column. The diagonal degree matrix $\degreemtx \defeq {\rm diag}\{\nodedegree{1},\ldots,\nodedegree{\nrnodes}\}$ collects the (weighted) node degrees
$\nodedegree{\nodeidx}=\sum_{\nodeidx'} \edgeweight_{\nodeidx,\nodeidx'}$ for each node $\nodeidx$.
The graph Laplacian matrix $\mathbf{L} \defeq \degreemtx - \edgeweights$ is instrumental
for spectral clustering methods (see Section <ref>).
There are also vector spaces naturally associated with an empirical graph. The vector space consisting of
maps $\vu$ from nodes to real numbers is denoted $\mathbb{R}^{\nrnodes}$. A vector $\vu \in \mathbb{R}^{\nrnodes}$
is a function or map that assigns each node $\nodeidx \in \nodes$ a real-number $u_{\nodeidx}$.
Similarly, we define the vector space $\mathbb{R}^{\overrightarrow{\edges}}$ that consists of vectors
$\edgesigvec$ that assign each edge $\directededge{\nodeidx}{\nodeidx'} \in \edges$ a real-number $\edgesig_{\directededge{\nodeidx}{\nodeidx'}}$.
Many graph clustering methods, such as spectral clustering, rely on a clustering assumption.
This clustering assumption applies to a dataset for which we know a useful empirical graph
[6, 7, 9]. A cluster is then
constituted by data points whose nodes in $\graph$ are more densely connected with each other
than with nodes representing data points outside the cluster [15, 18].
This informal clustering assumption can be made precise by constraining the cut-size of cluster [9].
By the maxflow/mincut duality, requiring a small cut is equivalent to requiring a minimum amount of network
flow that can be routed from the nodes inside a cluster through its boundary [25].
Using the above clustering assumption, we construct feature vectors $\featurevec^{(\nodeidx)}$ for
each node (or datapoint) $\nodeidx \in \nodes$. The feature vectors will be constructed such that
two nodes $\nodeidx, \nodeidx'$ that belong to a well-connected subset of nodes (a cluster) have more
feature vectors $\featurevec^{(\nodeidx)}, \featurevec^{(\nodeidx')}$ with a small Euclidean distance
$\big\| \featurevec^{(\nodeidx)}- \featurevec^{(\nodeidx')} \big\|$. Loosely speaking, the feature construction
maps similarity between datapoints in the empirical graph $\graph$ into proximity
of their Euclidean feature vectors. This allows us then to apply standard clustering methods, such as k-means
of soft clustering [29], to the network structured data.
§ SPECTRAL CLUSTERING
Before we detail our construction of the feature vectors $\featurevec^{(\nodeidx)}$ let us
briefly review the construction used by spectral graph clustering methods [9] .
The most basic variant of spectral clustering constructs the node feature vectors $\featurevec^{(\nodeidx)}$ using
the eigenvectors of the graph Laplacian matrix $\mathbf{L} = \degreemtx - \edgeweights$ [9].
The matrix $\mathbf{L}$ is positive semi-definite (psd) and therefore we can find an orthonormal set of
eigenvectors [11]
\begin{equation}
\label{equ_def_eigvals}
\vu^{(1)},\ldots,\vu^{(\nrnodes)} \mbox{ with eigenvalues } 0 \leq \eigval{1} \leq \ldots \leq \eigval{\nrnodes}.
\end{equation}
For a given a number $\nrcluster$ of clusters, spectral clustering methods use the feature vectors
\begin{equation}
\label{equ_def_feat_vec_spec_cluster}
\featurevec^{(\nodeidx)} = \big( u^{(1)}_{\nodeidx}, \ldots,u^{(\nrcluster)}_{\nodeidx} \big)^{T} \mbox{ for every node } \nodeidx \in \nodes.
\end{equation}
The Ideal Case. To develop some intuition for the usefulness of the construction (<ref>),
consider an empirical graph that contains $\nrcluster$ components $\datacluster{1},\ldots,\datacluster{\nrcluster} \subseteq \nodes$
that are not connected by any edge. In this case, the first $\nrcluster$ (possibly repeated) eigenvalues in (<ref>)
are equal to $0$. The correspond eigenvectors $\vu^{(1)},\ldots,\vu^{(\nrcluster)}$ constitute an orthonormal basis
for the subspace of $\mathbb{R}^{\nrnodes}$ that is spanned by the indicator vectors $\mathbf{e}^{(\clusteridx)}$
of the components $\datacluster{\clusteridx}$, $e_{\nodeidx}^{(\clusteridx)} = 1$ for nodes $\nodeidx \in \datacluster{\clusteridx}$ and $e_{\nodeidx}^{(\datacluster{\clusteridx})} = 0$ for nodes $\nodeidx \notin \datacluster{\clusteridx}$. Thus, the feature vectors of nodes in the
same cluster $\datacluster{\clusteridx}$ are identical and orthogonal to the feature vectors of nodes in different
clusters $\datacluster{\clusteridx'}$ with $\clusteridx' \neq \clusteridx$.
General Case. In general, the empirical graph will contain edges between different clusters.
A widely used approach to analyzing spectral clustering methods is to consider the effect of these
inter-cluster edges as perturbations of the Laplacian matrix $\mL$. For sufficiently small perturbations,
the subspace spanned by the first $\nrcluster$ eigenvectors of $\mL$ will be close to the subspace spanned
by the indicator vectors $\mathbf{e}^{(\clusteridx)}$. This deviation can be made precise using tools
from linear algebra and results in conditions on the empirical graph for the success of k-means
when applied to the feature vectors (<ref>) [9, 6].
There are certain types of empirical graphs that are challenging for spectral clustering methods that use
the features (<ref>) [9, 28, 27].
For example, spectral methods tend to fail for datasets that consist of clusters with significantly
varying sizes (see Section <ref>). Another example of a challenging network structure for spectral
clustering is a chain graph [26].
Consider a chain graph that consists of two clusters connected via a single edge. The
weights of all intra-cluster edges are equal to $1$ while the weight $\edgeweight_{o}$ of the single boundary
edge is slightly smaller. As the chain graph is connected, $\eigval{1}=0$ with corresponding eigenvector $\vu^{(1)}$ having identical entries. The smallest non-zero eigenvalue is $\eigval{2}$ has a corresponding
eigenvector $\vu^{(2)}$ whose entries are depicted in Figure <ref>.
The resulting feature vectors (see (<ref>) with $k=2$) do not reflect well the cluster structure of the chain graph.
x=0.35cm,y=2.5cm,every path/.style=>=latex,node style/.style=circle,draw
[ head to column names,
late after head=ı,,
after line=ı]
[line width=0.0mm] (, ) (ı,) node $\circ$;
[ head to column names,
late after head=ı,,
after line=ı]
[line width=0.0mm] (, ) (ı,) node $\star$;
[->] (0.4,0) – (22,0);
[right] at (19.5,0.1) node $i$;
[->] (0.5,0) – (0.5,1.1);
[-,dashed] (0,0.5) – (22,0.5);
in 0/$0$,0.5/$0.5$,1/$1$
(0.4, – (0.6, node[left=2mm] ;
in 1/$1$,5/$5$,10/$10$,15/$15$,20/$20$
(1pt) – (-2pt) node[below] ;
Solution of TV minimization (“$\circ$”) for the chain graph
obtained from Algorithm <ref> using $\nriter=1000$ iterations. Entries (“$\star$”)
of the eigenvector $\vu^{(2)}$ corresponding to the smallest non-zero eigenvalue $\eigval{2}$ (see (<ref>)).
x=4cm,y=2.5cm,every path/.style=>=latex,node style/.style=circle,draw
[ head to column names,
late after head=ı,,
after line=ı]
[line width=0.0mm] (,0) (,0) node $\circ$;
[ head to column names,
late after head=ı,,
after line=ı]
[line width=0.0mm] (,1 ) (,1) node $\star$;
[->] (-0.1,0) – (1.1,0);
[->] (-0.1,1) – (1.1,1);
[right] at (1.1,0.0) $\feature_{\nodeidx} \defeq \hat{u}_{\nodeidx}^{(\seednodes)}$ (<ref>);
[right] at (1.1,1.0) $\feature_{\nodeidx} \defeq u_{\nodeidx}^{(2)}$ (<ref>);
in 0/$0$,0.5/$0.5$,1/$1$
(1.02) – (0.98) node[below] ;
in 0/$0$,0.5/$0.5$,1/$1$
(0.02) – (-0.02) node[below] ;
Scatterplot of (scalar) node features $\feature_{\nodeidx}$ constructed by spectral and flow-based clustering
for the chain graph in Figure <ref>. Spectral clustering uses the entries (“$\star$”)
of the eigenvector $\vu^{(2)}$ (see (<ref>)). Flow-based clustering uses the solution of TV
minimization (“$\circ$”) (see (<ref>)).
We next describe our novel construction of feature vectors. The idea is to replace the eigenvectors
of the Laplacian in (<ref>) with solutions of total variation (TV) minimization problems.
It turns out that these feature vectors better reflect the cluster geometry of $\graph$ in terms of
bottlenecks for network flows between different clusters. These flow bottlenecks are the boundaries
between the clusters that are obtained by applying clustering methods to these feature vectors.
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§ TOTAL VARIATION MINIMIZATION
The construction of feature vectors (<ref>) in spectral clustering
methods is global. Indeed, the eigenvectors (<ref>) of the graph Laplacian will
typically depend on every edge in the empirical graph. Instead, we construct features in a more
local fashion by considering local clusters around seed nodes. These seed nodes might be obtained
by domain knowledge (telling us which nodes must belong to the same cluster), based on basic connectivity
properties (nodes with high degree), or chosen randomly.
Given a set of seed nodes $\seednodes$ we solve the TV minimization problem
\begin{equation}
\label{equ_def_nLasso}
\hat{\vu}^{(\seednodes)} = \argmin_{\vu \in \mathbb{R}^{\nodes}} \sum_{\nodeidx \in \seednodes} (1\!-\!u_{\nodeidx})^2/2\!+\! \sum_{\nodeidx \notin \seednodes} (\alpha/2) u_{\nodeidx} ^2\!+\!\regparam \| \vu \|_{\rm TV}.
\end{equation}
Here, we used the TV of a vector $\vu \in \mathbb{R}^{\nrnodes}$,
\begin{equation}
\label{eq_def_TV}
\| \vu \|_{\rm TV} = \sum_{\edge{\nodeidx}{\nodeidx'} \in \edges} \edgeweight_{\nodeidx,\nodeidx'} |u_{\nodeidx} - u_{\nodeidx'}|.
\end{equation}
Note that the TV minimization (<ref>) is parametrized by the set $\seednodes \subseteq \nodes$ of
seed nodes. Our clustering methods solve multiple instances of (<ref>), each time using another choice for the seed nodes $\seednodes$.
Section <ref> discusses approaches for choosing a useful set of seed nodes.
We have recently explored the duality between TV minimization (<ref>) and network flow optimization [10].
This duality allows to characterize the solution $\hat{\vu}$ of (<ref>) in terms of network flows.
A network flow is a vector $\edgesigvec \in \mathbb{R}^{\overrightarrow{\edges}}$ whose entries $\edgesig_{\directededge{\nodeidx}{\nodeidx'}}$
represent a flow from node $\nodeidx$ to $\nodeidx'$.
A vector $\widehat{\vu}$ solves TV minimization (<ref>) if and only if
there is a flow vector $\edgesigvec \in \mathbb{R}^{\overrightarrow{\edges}}$ such that
\begin{align}
-\sum_{\directededge{\nodeidx}{\nodeidx'} \in \overrightarrow{\edges}} \edgesig_{\directededge{\nodeidx}{\nodeidx'}} + \sum_{\directededge{\nodeidx'}{\nodeidx} \in \overrightarrow{\edges}}\edgesig_{\directededge{\nodeidx'}{\nodeidx}} & = \hat{u}_{\nodeidx}\!-\!1 \mbox{ for } \nodeidx \in \seednodes, \nonumber \\[2mm]
-\sum_{\directededge{\nodeidx}{\nodeidx'} \in \overrightarrow{\edges}} \edgesig_{\directededge{\nodeidx}{\nodeidx'}}
+\sum_{\directededge{\nodeidx'}{\nodeidx} \in \overrightarrow{\edges}}\edgesig_{\directededge{\nodeidx'}{\nodeidx}}& = \alpha \hat{u}_{\nodeidx} \mbox{ for } \nodeidx \notin \seednodes, \nonumber \\[2mm]
|\edgesig_{\directededge{\nodeidx}{\nodeidx'}}| \leq \lambda \edgeweight_{\nodeidx,\nodeidx'} \mbox{ for all } &\directededge{\nodeidx}{\nodeidx'} \in \overrightarrow{\edges}, \nonumber \\[2mm]
\hat{u}_{\nodeidx}\!-\!\hat{u}_{\nodeidx'}\!=\!0 \mbox{ for all } \directededge{\nodeidx}{\nodeidx'} \in \overrightarrow{\edges} \mbox{ with } & |\edgesig_{\directededge{\nodeidx}{\nodeidx'}}|\!<\!\lambda\edgeweight_{\nodeidx,\nodeidx'} . \label{equ_pd_optimal_non_staturated}
\end{align}
The result can be obtained by applying Fenchel duality <cit.> to TV minimization (<ref>)
and a dual minimum cost flow problem (see <cit.>).
Let us illustrate Theorem <ref> for the empirical graph in Figure <ref>.
This empirical graph is a chain graph and partitioned into two clusters $\datacluster{1}$ and $\datacluster{2}$ which
are connected by a boundary edge $b$ with weight $\edgeweight_{o}$.
Ideal Case. Assume we solve TV minimization (<ref>) for the chain graph in Figure <ref>
with a single seed node $\seednodes = \{ \nodeidx^{(1)} \}$ with $\nodeidx^{(1)} \in \datacluster{1}$. Moreover, we choose
$\regparam$ and $\alpha$ in (<ref>) such that
\begin{equation}
\label{equ_cond_alpha_lambda}
\regparam \edgeweight_{o} < 1\mbox{, and } | \datacluster{1}| (\alpha/\regparam)+ \edgeweight_{o}<1, | \datacluster{2}| \alpha \geq \regparam \edgeweight_{o}.
\end{equation}
A direct inspection of the optimality condition (<ref>) then yields the TV minimization solution
\begin{equation}
\label{equ_def_TV_min_chain}
\hat{u}_{\nodeidx} = \begin{cases} (1- \regparam\edgeweight_{o} )/ \big(1+ \alpha (|\datacluster{1}|-1) \big) & \mbox{ for } \nodeidx \in \datacluster{1} \\
\regparam\edgeweight_{o} / \big( \alpha |\datacluster{2}| \big) & \mbox{ for } \nodeidx \in \datacluster{2} \end{cases}.
\end{equation}
Note that the vector (<ref>) is piece-wise constant over the clusters $\datacluster{1}$ and $\datacluster{2}$.
Thus, if we would use (<ref>) as (single) feature $\feature_{\nodeidx} = \hat{u}_{\nodeidx}$, basic clustering
methods would successfully recover the clusters $\datacluster{1}$ and $\datacluster{2}$.
General Case. Note that (<ref>) characterizes the solution of TV minimization (<ref>)
only when $\regparam$ and $\alpha$ are chosen such that (<ref>) is valid. However, condition (<ref>)
is not useful in practice as it involves the size of the clusters which we would like to determine. A practical approach to choosing $\alpha$ and $\lambda$
can be based on probabilistic models for the empirical graph such as stochastic block models [21]. Alternatively, the
choice for $\alpha$ and $\lambda$ can be guided by the size of the piece $\{ \nodeidx \in \nodes: \hat{u}_{\nodeidx} = \hat{u}_{\nodeidx^{(1)}}\}$
around the seed node $\nodeidx^{(1)}$. This piece grows for increasing $\regparam$ and becomes the entire node set $\nodes$ whenever
$\regparam \edgeweight_{\nodeidx,\nodeidx'} > 1 \mbox{ for all edges } \edge{\nodeidx}{\nodeidx'} \in \edges.$
The duality between TV minimization (<ref>) and network flow optimization
is also instrumental for developing iterative methods to solve (<ref>).
Algorithm <ref> summarizes the application of a generic primal-dual method to solve (<ref>) [10].
Primal-Dual Method For TV Minimization (<ref>)
Input: $\graph=\big(\nodes,\edges,\edgeweights\big), \seednodes, \lambda, \alpha, \nriter$
Initialize: $\hat{\vu}^{(0)}\!\defeq\!{\bf 0}$; $\hat{\vu}^{(-1)}\!\defeq\!{\bf 0}$;$\hat{\edgesigvec}^{(0)} \!\defeq\! {\bf 0}$;
Output: approximation $\widehat{\vu}$ of solution $\widehat{\vu}^{(\seednodes)}$ to (<ref>)
[1]
$\pditer = 0, \ldots,\nriter-1$
$\mbox{ {\bf for} } i\!\in\!\nodes \mbox{ {\bf do} } : \tilde{u}_{\nodeidx} \!\defeq\! 2 \hat{u}^{(\pditer)}_{\nodeidx} - \hat{u}^{(\pditer\!-\!1)}_{\nodeidx} $
$ e = \directededge{\nodeidx}{\nodeidx'} \in \overrightarrow{\edges} $
$ \hat{\edgesig}^{(\pditer\!+\!1)}_{e} \!\defeq\!\hat{\edgesig}^{(\pditer)}_{e}\!+\! (1/2) (\tilde{u}_{\nodeidx}\!-\!\tilde{u}_{\nodeidx'})$
$\hat{\edgesig}^{(\pditer\!+\!1)}_{e} \!\defeq\! \hat{\edgesig}_{e}^{(\pditer\!+\!1)}\!/\!\max\{1, |\hat{\edgesig}_{e}^{(\pditer\!+\!1)}|/(\regparam \edgeweight_{e}) \} $
$ \nodeidx \in \nodes $
$\hat{u}^{(\pditer\!+\!1)}_{\nodeidx} \!\defeq\! \hat{u}^{(\pditer)}_{\nodeidx}\!-\!\gamma_{\nodeidx} \bigg[\hspace*{-1mm}\sum_{\directededge{\nodeidx}{\nodeidx'}} \hat{\edgesig}^{(\pditer\!+\!1)}_{\directededge{\nodeidx}{\nodeidx'}} \!-\!\hspace*{-1mm}\sum_{\directededge{\nodeidx'}{\nodeidx}} \hat{\edgesig}^{(\pditer\!+\!1)}_{\directededge{\nodeidx'}{\nodeidx}} \bigg]$
$\nodeidx \in \seednodes$
$\hat{u}_{\nodeidx}^{(\pditer\!+\!1)} \!\defeq\! \big(\gamma_{\nodeidx}\!+\!\hat{u}^{(\pditer\!+\!1)}_{\nodeidx}\big)/(\gamma_{\nodeidx}\!+\!1)$
$\hat{u}_{\nodeidx}^{(\pditer\!+\!1)} \!\defeq\! \hat{u}^{(\pditer\!+\!1)}_{\nodeidx}/(\alpha\gamma_{\nodeidx}\!+\!1)$
$\widehat{\vu} \defeq \hat{\vu}^{(\nriter)}$
The output $\widehat{\vu} \in \mathbb{R}^{\nrnodes}$ of Algorithm <ref> is an approximation to
the solution of (<ref>). We assume that the number of iterations $\nriter$ used for
Algorithm <ref> is sufficiently large such that the output of Algorithm <ref> can be considered
a (numeric) solution to (<ref>). The number $\nriter$ of iterations can be guided by
probabilistic models for the underlying empirical graph combined with the convergence rates guaranteed
by primal-dual methods [4]. Alternatively, we can tune the number of
iterations based on the final clustering result obtained by using Algorithm <ref> as a sub-routine
within our clustering method (see Algorithm <ref>).
It is instructive to interpret Algorithm <ref> as a message passing method for iteratively optimizing the network flow
$\hat{\edgesig}^{(\pditer)}_{\directededge{\nodeidx}{\nodeidx'}}$. Step <ref> enforces
the capacity constraint $| \hat{\edgesig}^{(\pditer)}_{\directededge{\nodeidx}{\nodeidx'}} | \leq \regparam \edgeweight_{\nodeidx,\nodeidx'}$.
Step <ref> adjusts the value $\hat{u}^{(\pditer)}_{\nodeidx}$ based on the
net flow into the node $\nodeidx$. In step <ref>, flow is injected into
seed nodes $\nodeidx \in \seednodes$ while in step (<ref>) flow is leaked out of remaining nodes $\nodeidx \notin \seednodes$.
§ FLOW-BASED GRAPH CLUSTERING
We are now in the position to formulate our flow-based graph clustering method as Algorithm <ref>.
This method constructs feature vectors $\featurevec$ using the solutions of TV minimization (<ref>)
for different choices of seed nodes $\seednodes$. Instead of using the values of eigenvectors
for the graph Laplacian (as used by spectral clustering), we use entries of the vector solving TV minimization (<ref>)
to construct feature vectors for each node $\nodeidx \in \nodes$.
Flow-Based Graph Clustering
Input: empirical graph $\graph$, TV min. parameters $\regparam$, $\alpha$,
number $\nrcluster$ of cluster; number $\nrseeds$ of seeds.
Output: cluster assignments $\hat{c}_{1}, \ldots , \hat{c}_{\nrnodes} \in \{1,\ldots,\nrcluster\}$
[1]
$\pditer = 1, \ldots,\nrseeds$
select new seed nodes $\seednodes$ with Algorithm <ref>
run Algorithm <ref> with $\graph,\seednodes,\regparam,\alpha$
store resulting vector in $\widehat{\vu}^{(\pditer)}$
construct node features
\begin{equation}
\label{equ_def_flow_vec_features}
\featurevec^{(\nodeidx)} = \big( \hat{u}^{(1)}_{\nodeidx}, \ldots,\hat{u}^{(\nrseeds)}_{\nodeidx} \big)^{T} \mbox{ for every node } \nodeidx \in \nodes.
\end{equation}
compute cluster assignments $\hat{c}_{\nodeidx}$ by applying k-means
to feature vectors (<ref>)
A key challenge for the successful application of Algorithm <ref> is a suitable
section of seed nodes in step <ref>. One simple
approach is to choose the set $\seednodes$ by randomly selecting a single
node $\nodeidx \in \nodes$. In general it is preferable to use more than one seed node.
However, we must ensure that a particular selection of seed nodes $\seednodes$
contains only nodes from the same cluster.
Our numerical experiments (see Section <ref>) use a simple
heuristic method for selecting seed nodes $\seednodes$ in step (<ref>) of Algorithm <ref>.
This heuristic method is summarized in Algorithm <ref>. Algorithm <ref> constructs
a new set of seed nodes by first randomly choosing a node $\nodeidx'$ with large degree $\nodedegree{\nodeidx}$ and
then adding all neighbours of $\nodeidx'$ that have a sufficiently large number of common neighbours with $\nodeidx'$.
Select Seed Nodes $\seednodes$
Input: empirical graph $\graph$, minimum number of common neighbours $\eta$, minimum degree $d$
Output: seed nodes $\seednodes \subseteq \nodes$
[1]
initialize $\seednodes \defeq \emptyset$
determine $\nodeidx' \defeq randomly\ select\ from\ \{\nodeidx : \nodedegree{\nodeidx} \geq d\}$
add node, $\seednodes \defeq \{ \nodeidx' \}$
each $\nodeidx''$ with $\edge{\nodeidx'}{\nodeidx''} \in \edges$
$\mathcal{D} = \{ \nodeidx''': \edge{\nodeidx'''}{\nodeidx'}, \edge{\nodeidx''}{\nodeidx'''} \in \edges \}$
$| \mathcal{D} | \geq \eta$
$\seednodes \defeq \seednodes \bigcup \{ \nodeidx'' \}$
§ NUMERICAL EXPERIMENTS
For datasets without intrinsic graphs, we construct the empirical graph by using an Euclidean distance based equation to define the similarity between data points: $A_{i, j}:=\exp \left(-\left\|\mathbf{x}^{(i)}-\mathbf{x}^{(j)}\right\|_{2}^{2} /\left(2 \sigma^{2}\right)\right)$.
Here, $\sigma$ is a tuning parameter that chosen via cross-validation or using a probabilistic model for the data points to keep numerical stability.
§.§ Experiments with Synthetic data
A dataset [28] with two clusters, as depicted in Figure <ref>, the first cluster is a set of 2D data points drawn from Gaussian density centered at $(2,0.2)$ with diagonal covariance matrix $ 0.01*I $, the second cluster is a set of 2D data points denoting uniform density in a rectangular region: $\left\{ ( x_{1}, x_{2}) \mid 0 < x_{1}< 8, {-0.05} < x_{2} < 0 \right\}$. As shown in Figure <ref>, our method significantly outperforms spectral clustering. Parameters used for Algorithm <ref> are $\alpha = 0.005$, $\lambda=0.01$, $\eta=50$, and $d=12$.
[ head to column names,
late after head=,
after line=]
=̧ 1[color=red] (, ) node $\cdot$;
=̧ 0[color=green] (, ) node $\cdot$;
(0,-1.5) – (8,-1.5) – (8,1.5) –(0,1.5) – (0,-1.5);
in -1/$-1$,0/$0$,1/$1$
(0, – (0.1, node[left=2mm] ;
in 2/$2$,4/$4$,6/$6$
(-1.5) – (-1.4) node[below=2mm] ;
[ head to column names,
late after head=,
after line=]
=̧ 1[color=red] (, ) node $\cdot$;
=̧ 0[color=green] (, ) node $\cdot$;
(0,-1.5) – (8,-1.5) – (8,1.5) –(0,1.5) – (0,-1.5);
in -1/$-1$,0/$0$,1/$1$
(0, – (0.1, node[left=2mm] ;
in 2/$2$,4/$4$,6/$6$
(-1.5) – (-1.4) node[below=2mm] ;
The left plot is the clustering result of Algorithm <ref>, the right one is the clustering result of spectral clustering [6].
§.§ Image Segmentation / Pixel Clustering
The performance of Algorithm <ref> for image segmentation/pixel clustering is tested on
some RGB images. We construct an empirical graph as mentioned in <ref> based on pixel values. Each pixel is connected to other pixels within up to three hops.
empirical graph is forced to be sparse by removing some edges associated with small weights.
Pixels in a rectangular are set to be seed nodes. After 1000 iterations, a local cluster which
segment out the object of interest from the background is determined. As depicted in fig <ref>,
the algorithm can accurately detect which pixels belong to the object. To compare and contrast,
we also use spectral clustering to execute this task, the result in fig <ref> shows that
our algorithm out-performs spectral clustering.
The source code for the above experiments can be found at <https://github.com/YuTian8328/>.
Image Segmentation: The left plot is the original image, the middle one is the segmentation
result from Algorithm <ref>, the right one is the result from spectral clustering.
|
# Surveillance Testing for Rapid Detection of Outbreaks in Facilities
Yanyue Ding
Graduate Program in Operations Research & Industrial Engineering (ORIE)
University of Texas at Austin
Austin, Texas 78712
Sudesh K. Agrawal
Graduate Program in ORIE
University of Texas at Austin
Austin, Texas 78712
&Jincheng Cao
Department of Mathematics
University of Texas at Austin
Austin, Texas 78712
Lauren Meyers
Graduate Program in ORIE, and
Department of Integrative Biology
University of Texas at Austin
Austin, Texas 78712
&John J. Hasenbein
Graduate Program in ORIE
University of Texas at Austin
Austin, Texas 78712
<EMAIL_ADDRESS>
###### Abstract
This paper develops an agent-based disease spread model on a contact network
in an effort to guide efforts at surveillance testing in small to moderate
facilities such as nursing homes and meat-packing plants. The model employs
Monte Carlo simulations of viral spread sample paths in the contact network.
The original motivation was to detect COVID-19 outbreaks quickly in such
facilities, but the model can be applied to any communicable disease. In
particular, the model provides guidance on how many test to administer each
day and on the importance of the testing order among staff or workers.
Keywords: COVID-19, surveillance testing, agent-based model, contact networks,
nursing homes
## 1 Introduction
In this paper, we develop an agent-based disease spread model on a contact
network to optimize surveillance testing in facilities. Generally speaking,
the model’s scope involves facilities of small to moderate size, say, with
dozens to hundreds of employees. The original motivation derives from the need
to quickly detect outbreaks of COVID-19 in facilities such as nursing homes,
factories, and meat-packing plants, in which close contact is nearly
unavoidable. Our focus is on _surveillance testing_ or proactive testing of
staff. In other words, the premise is that there is not currently an outbreak
in the facility, and the manager wishes to proactively test asymptomatic staff
in order to quickly detect, and prevent, an outbreak. Given a contact network,
along with other modeling parameters detailed in the sequel, the primary goal
is to answer the following:
1. 1.
How many employees should be tested each day in order to detect an outbreak
with a given probability?
2. 2.
What is the optimal sequence in which to test employees?
We propose a relatively simple Monte Carlo simulation model that is tailored
to answering the two questions. Our model is similar to other agent-based SEIR
models in the literature. However, the novelty is in our focus on optimization
and decision making, and how it relates to the structure of the contact
network. In the first part of the paper, we examine generic contact networks
that may be of use when there is little data on the interactions in a
facility. In the second part of the paper, we present more nuanced contact
models, that are derived from the staff and interaction structure in nursing
homes. Such models are preferred when there is detailed data on a facility’s
operations.
## 2 Literature Review
COVID-19 is a critical health issue worldwide and is likely to continue to be
of concern for many years. Even with vaccine development, it is imperative to
detect outbreaks at an early stage to develop effective mitigation measures.
According to some studies, 38%–80% of cases are pre-symptomatic or
asymptomatic in Long-Term-Care Centers (LTCs) [5, 35]. Moreover, 56%–95.5% of
disease transmissions are caused by asymptomatic cases [14, 17]. Early in the
COVID-19 pandemic, in many facilities, only individuals with COVID-like
symptoms were tested, due to limited testing resources. However, several
studies show that screening symptomatic cases only is not enough to control
outbreaks [33, 16, 17]. Starting in March 2020, the Centers for Disease
Control and Prevention (CDC) and the Centers for Medicare & Medicaid Services
(CMS) offered various COVID-19 surveillance testing strategies to hospitals,
nursing homes, schools, and other facilities, as the pandemic progressed. Now,
surveillance testing is a widely applied strategy to contain the spread of the
virus, especially for asymptomatic cases [4, 12]. Apart from suggestions
provided by the government, managers and researchers in various facilities
developed their own surveillance testing strategies, combined with contact
interventions, based on the facility’s contact patterns and healthcare
resources. In LTCs for example, the strategies include daily individual health
screening via surveys or front desk checks [3, 16], daily temperature and
oxygen level checks [33], serological anti-body checks [31], and reverse
transcription-polymerase chain reaction (RT-PCR) tests.
In terms of evaluation of testing strategies, some strategies are evaluated by
outbreak case studies or analysis of publicly recorded data [1, 8, 21, 23,
32]. In addition, strategies may be evaluated using variations of the
stochastic susceptible-exposed-infected-recovered/removed (SEIR) model. Some
researchers have made modifications to the traditional SEIR model, to account
for nuances of COVID-19. These include adding, removing, or subdividing the
standard stages in the network. For example, the exposed stage $E$ is
sometimes removed, and the remaining SIR model is used [11, 15]. Saha [26]
sub-divides the infected stage $I$ into detected and undetected infected
stages. Chapman et al. [7] assume all infected individuals experience a pre-
symptomatic stage, after which the corresponding node in the model transitions
to an asymptomatic, mild symptomatic or severe symptomatic stage. Another
innovation is to allow the parameters in the differential equations defining
the SEIR model to be non-linear. Chen and Wei [9] and Shu et al. [28] assume
that the parameters in the differential equations can be non-linear functions
of time. Some studies modify parameters and the contact network structure over
time. These adaptations are primarily used in comparing and finding the best
surveillance screening and intervention strategies. Hou et al. [20] and Shaikh
et al. [27] estimate the parameter settings under various surveillance testing
and intervention efficacy assumptions and perform a parametric analysis.
Ciaperon et al. [11] study the dynamic SEIR model by temporarily removing the
edges in the contact network due to quarantine intervention for individuals
who have COVID-like symptoms or tested positive.
One key assumption in the differential-equation-based SEIR models is that the
population is well-mixed and features are homogeneous among individuals [7, 9,
10]. The standard SEIR model might be suitable for predicting outbreak
progression in a large well-mixed population while it might over-simplify
dynamics in special networks [10]. Broadly speaking, two ways to address these
issues have been developed in the literature. One is to classify individuals
who share similar characteristics into subgroups and use specific differential
parameters in each subgroup. For example, Besse and Faye [6] divide the
population by spatial features and use a standard SIR model with heterogeneous
parameters in each spatial group. The disease transmission between groups is
estimated by heat equations. Shaikh et al. [27] divide the population in an
LTC by the occupations, and use different transmission probabilities for each
group. A second solution, requiring significant computational effort, is to
use network-based stochastic models that model each individual. Ames et al.
[2] believe human social networks tend to be clustered and heterogeneous. In
particular, they study disease transmission features in networks with various
coefficient and clustering properties. Their results show that the structure
of the network has a profound influence on disease progression. Similarly,
Meyers et al. [24] test intervention strategies for severe acute respiratory
syndrome (SARS) in various networks. Their analysis indicates that contact
patterns in the network impact the size of epidemic significantly, even if the
basic reproduction rates are the same among different networks. Herrera et al.
[18] study surveillance strategies in different networks and find that the
best surveillance strategies are based on network structures.
Rennert et al. [25] modify the SIR model by distinguishing symptomatic and
pre-symptomatic stages and adding an isolation stage. Their results show that
voluntary and random testing is not enough to protect students and faculty on
a campus. They recommend performing rigorous testing during the semester. They
also provide a novel solution by continuing to test the target group if a
positive case in random surveillance occurs, which can be more effective in
controlling disease spread. Chapman et al. [7] applied the SEIR model to
simulate COVID-19 transmission in a congregated shelter population. The model
was tested in shelters in San Francisco, Boston, and Seattle. The results
showed that a combined strategy that involves daily screening, twice-weekly
PCR testing and contact intervention is the best strategy to detect and avert
an outbreak at an early stage. Smith et al. [29, 30] used an individual-based
SEIR model to simulate the transmission of COVID-19 based on detailed contact
data in an LTC in order to find the best surveillance strategy. They compared
the strategies when daily testing capacities of 1, 2, 4, 8, 16, and 32 are
available.
Apart from COVID-19-based models, other researchers have discussed the
importance of considering the structure of contact networks when designing
surveillance strategies. For example, Herrera-Diestra et al. [19] study
contact networks modeling the spread of foot-and-mouth disease among cattle
farms in Turkey, and note that such networks have typical “small-world”
properties. Although they suggest some general surveillance principles based
on their observations, they do not provide detailed suggestions. Mastin et al.
[22] examine the spread of plant pathogens. Their paper is close in spirit to
our work, in that they focus of optimization of surveillance nodes by building
a stochastic optimization model, using Monte Carlo samples. However, their
model applies to a “static” surveillance strategy in which a fixed number of
nodes are selected to aid detection. In this paper we investigate dynamic
testing, in which different nodes are tested each day.
Overall, we note that most previous research focuses on the effectiveness of
various surveillance strategies to detect an outbreak of COVID-19, but few
papers provide guidance on specific testing strategies, especially when
testing resources are in shortage. The model in this paper is predicated on
the belief that decisions makers have different risk tolerances, which in turn
determine the amount of testing that is performed. We also assume the contacts
of individuals in a facility can be heterogeneous. We simulate disease
progression on a network and find the probability of detecting an outbreak for
various levels of testing capacities and time thresholds, as described in the
next section.
## 3 Model Description
The first component of our model is a contact graph $G=(V,E)$ on a set $V$ of
nodes and a set $E$ of undirected edges. We call two nodes _neighbors_ if they
are connected by an edge. Each node in the graph corresponds to a single staff
member, and there is an edge connecting them if they have close contact during
a day, i.e., there is a non-zero probability of COVID-19 passing between them
if one of them is infectious. Apart from the graph structure just described,
the model is characterized by the following parameters:
* •
$K:=|V|$, the number of staff in the facility
* •
$p$: daily probability of external infection
* •
$\ell$: latency, the number of days to move from the exposed state to the
infected state
* •
$d$: degree of each node in the symmetric case, indicating the density of
contact graph
* •
$r$: daily probability of infection between neighbors, a representation of the
force of infection
* •
$f$: false negative rate for a disease test, after a node is infectious
* •
$t$: outbreak threshold tolerance, in days.
We now describe the progression of the disease through the network. The spread
model is a discrete-time Markov chain, with time index $n=0,1,2,\ldots.$ At
each time point the nodes can be in one of three states: susceptible ($S$),
exposed ($E$), and infected ($I$). We do not model recovered nodes since our
focus is on detecting initial outbreaks, generally in less than 10 days from
the first infected staff member entering the facility. Nodes may be exposed
either from external interactions, or internal interactions. We also have
simplified assumptions regarding the exposed and infected states. In the
exposed state a node cannot spread the disease to other nodes and cannot be
detected via a test. After exactly $\ell$ days, an exposed node becomes
infected. When a node is infected, each neighboring susceptible node enters
the exposed state with probability $r$ each day. The sequence of events
corresponding to infected nodes infecting susceptible nodes is assumed to be
mutually independent, both among nodes and across days. In addition, to model
external infections, each susceptible node can move directly to the infected
state, each day, with probability $p$. The reason we assume this direct move
is that an exposed node cannot be detected by a test, and cannot infect other
nodes, until it is in the “infected” state. Hence, from a dynamic systems
point of view, there is no need to track or model nodes that are just
“exposed” externally. The sequence of events corresponding to external
infections are also mutually independent. Finally, an external infection
resulting in a change to the infected state obviously supersedes a pending
move to “infected” if the same node has already been exposed internally.
In terms of disease testing, a decision maker can administer $k$ tests per day
to try to detect an outbreak. For simplicity, we assume that such a test is
administered in the “morning” to each of the $k$ individuals chosen that day,
and that it is a rapid test, whose results are available before the staff
member interacts with the rest of the staff. From the point of view of our
model, this means that the tested individuals are identified immediately if:
(a) they are tested on a given day, (b) are in the infected state on the same
day, and (c) the test does not yield a false negative result on this day. At
this point, we stop the process, since a detection has occurred. We do not
assume that the test is perfect. Specifically, there is a false negative rate
given by $f$. This rate applies to a test given to an individual that is
already in the infected state. We assume the false negative rate is constant
throughout the period a node is infected. This is obviously a simplification
of the case for most diseases, such as COVID-19, in which the false negative
rate fluctuates as an individual moves through the progression of the disease.
However, recall that our model is generally concerned with rapid detection,
within the first several days of infection (after the “incubation” period
given by $\ell$).
Up until Section 5, we assume that the contact network is homogeneous in
several ways. First, all nodes have an equal probability of external infection
(characterized by $p$). Second, all nodes have equal degree and are vertex
transitive (see Section 3.1). Third, each edge has an equal “weight,” in that
the infection force $r$ is the same for every edge. In Section 5, we relax
some of these assumptions to model heterogeneous contact networks, based on
more realistic contact data. Part of the reason to first study homogeneous
networks is to gain insights into different principles, such as the effect of
the structure of the graph on testing thresholds and efficient testing
protocols. To the best of our knowledge, these issues have not been studied
before in the literature on surveillance testing.
### 3.1 Graph Structure
In our study of so-called homogeneous graphs, we confine our analysis to
$d$-regular graphs, in order to preserve some sense of internal symmetry. In
addition, this allows us to systematically explore the effect of internal
interaction density, as embodied by the single parameter $d$. Of course, there
are numerous other ways to quantify graph structure, even for $d$-regular
graphs, but we do not explore those other avenues in this paper.
First, we recall that in a $d$-regular graph each node has exactly $d$
neighbors. A complete graph is a $d$-regular graph in which each node is
connected to every other node in the graph. Although there are many types of
$d$-regular graphs for a given $d$, in this paper we only study the class of
_circulant graphs_ , which are always $d$-regular. A circulant graph is a
graph whose adjacency matrix is a circulant matrix. More specifically, a
circulant graph can be characterized by a vector $(a_{1},a_{2},\ldots,a_{m})$
where $a_{1}<a_{2}<\cdots<a_{m-1}<a_{m}$ and $a_{m}<(K+1)/2$. For simplicity,
we assume that $a_{m}<K/2$. In this case, the corresponding circulant graph is
$d$-regular with $d=2m$. This allows us to uniquely define $m$, for a
specified even $d$ (which we vary to study the effect of graph density on
disease spread). The vector $(a_{1},a_{2},\ldots,a_{m})$ is sometimes called
the _jump sequence_ , and it characterizes the connections in the graph as
follows. Suppose we label the nodes in the graph $0,1,2,\ldots,K-1$. Then each
node $i$ has nodes $(i\pm a_{1})\mod{K}$, $(i\pm a_{2})\mod{K}$, …, $(i\pm
a_{m})\mod{K}$, as neighbors.
Circulant graphs are useful for our study as they have a number of desirable
properties. As mentioned above, they are always $d$-regular, with a degree
directly related to the parameter $m$. We also design our graphs so that they
are always connected. Further, circulant graphs are _vertex transitive_ so
that, roughly speaking, every vertex looks similar. Note that circulant graphs
need not be edge transitive and hence are not symmetric in the graph-theoretic
sense. In studying the effect of graph structure and density on surveillance
protocols, we use three classes of circulant graphs: complete graphs,
neighboring graphs, and crossing graphs. The first class is standard. The
latter two terms were created for this study. A _neighboring graph_ is one in
which the jump sequence is given by $(1,2,\ldots,m)$. In other words, each
node is connected to its $m$ closest neighbors, if one envisions the nodes
being arranged, as labeled, in a circle. A crossing graph is one in which the
jump sequence is given by $(\lfloor K/2-1\rfloor-m+1,\ldots,\lfloor
K/2-1\rfloor)$. In this case, each is node is connected to its $m$ farthest
neighbors. In Figure 1 we display representatives of each graph class for a
small network.
Figure 1: Some examples of $d$-regular graphs: (a) a complete 11-regular
graph, (b) a neighboring 4-regular graph, (c) a crossing 3-regular graph.
### 3.2 Monte Carlo Simulation
In order to answer the questions posed in the introduction, we perform Monte
Carlo simulations of disease spread, and detection, on networks of varying
structures and sizes. To keep the insights manageable, the parameters
$\ell=3$, $r=0.05$, and $f=0.21$ are kept constant throughout the numerical
studies. Otherwise, we vary network size, density, external infection rate,
and the outbreak threshold tolerance. Given a particular network and testing
protocol we typically simulate 50,000 outbreaks in a facility. An outbreak is
initiated by at least one node becoming infectious at time 0, and the
resulting nodes that are exposed and infected, until the outbreak threshold
tolerance $t$, at which point the simulation is stopped. For each sample
outbreak, we record whether or not there was a successful detection. An
outbreak may not be detected due to an infected node not being tested, or due
to a false negative result. These simulations then produce point estimates and
confidence intervals on the true detection probability for a particular
testing protocol. In general, we found 50,000 sample paths sufficient to
distinguish among policies.
In our numerical results, the primary performance criterion is the probability
of successfully detecting an outbreak within $t$ days, given that an outbreak
occurs. It is important to note that this is what one might call a conditional
outbreak probability. We believe that it is more useful than computing the
unconditional probability of detecting an outbreak, as this type of
performance metric essentially penalizes a protocol for failing to detect an
outbreak that does not exist. As we shall see below, this needs to be kept in
mind, as some results under this metric might appear counterintuitive. For
example, the probability of successful detection actually increases with the
community infection rate of $p$. To see why, imagine the extreme case in which
everyone walks in the door on day 1 infected with a disease and the false
negative rate is 0. In this case, it does not matter who is tested, as
everyone will set off the detection alarm. In particular you only need one
test to detect an outbreak when $p=1$ and $f=0$. However, when the community
infection rate is very low, this means that it is very likely that one, and
only one, person is the genesis of an outbreak. This makes the outbreak more
difficult to detect with a small number of tests.
## 4 Computational Results for Testing Protocols
### 4.1 Parameter Comparisons
In our first set of experiments, we investigate the effect of graph density,
embodied by the parameter $d$, on the number of tests per day needed to detect
an outbreak. In our model, $d$ represents the number of other staff members a
particular staff member is in contact with each day. For example, when $K=100$
and $d=99$, the resulting contact network is a complete graph, and the graph
implies that all staff members come into contact with all other staff members
each day. Obviously, this is an extreme situation. As such, we also examine
lower density networks. In Figure 2, we examine the case of 100 staff members
and an external community infection probability of $p=0.0001$ (per day). We
set $t=6$, indicating that our goal is to detect an outbreak within 6 days of
the introduction of the disease into the facility. Keeping these parameters
fixed, we plot the point estimates for the probability of detection versus the
number of tests per day, and plot this curve for various values of $d$. The
first thing to note is that there appear to be diminishing marginal benefits
of administering more tests, as one might expect. From the graph we also see
that the number of tests required to meet a certain benchmark probability of
detection can vary significantly with $d$. For example, if our threshold
probability is 0.7, and $d=99$ then roughly 4 tests per day are needed.
However, if $d=20$ then about 13 tests per day are needed. This may seem
counterintuitive but it is related to our observation above relating to the
community infection rate. The more quickly a disease spreads in a facility,
the more quickly it can be detected. We also observe that if we wish to
enforce a very high (say 0.95) detection probability, then a large number of
staff need to be tested daily. This is unlikely to be palatable for management
and staff.
$0$$10$$20$$30$$40$$50$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of tests per day
($k$)Conditional probability of detecting
outbreak$d=20$$d=40$$d=60$$d=80$$d=99$ Figure 2: Number of tests versus
detection probability for circulant neighboring graphs of varying degrees.
Fixed graph parameters are $K=100$, $p=0.0001$, $r=0.05$, $f=0.21$, $\ell=3$,
$t=6$. The results were obtained by sampling 50,000 virus sample paths.
In the next set of experiments, we vary the outbreak threshold tolerance $t$
while keeping other parameters fixed. We choose a neighboring graph with
$d=60$ as the results for this graph show the contrast among different
tolerance levels well. The results are shown in Figure 3. As must be the case,
the detection probability curves are nested, with larger tolerance curves
dominating lower tolerance curves. Again we notice that the number of tests
required to detect an outbreak at various tolerance levels varies widely. If
the probability threshold is again 0.7, and $t=8$ then just one test per day
is needed. However, if $t=4$ then around 10 tests per day are needed, which
implies that all staff are tested every 10 days.
$0$$10$$20$$30$$40$$50$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of tests per day
($k$)Conditional probability of detecting
outbreak$t=4$$t=5$$t=6$$t=7$$t=8$$t=9$$t=10$ Figure 3: Number of tests versus
detection probability for circulant neighboring graphs of tolerance levels.
Fixed graph parameters are $K=100$, $p=0.01$, $r=0.05$, $f=0.21$, $\ell=3$,
$d=60$. The results were obtained by sampling 50,000 virus sample paths.
Finally, we examine the effect of the external infection probability on the
detection probability curves. The networks tested in Figure 4 have the same
parameters as the networks tested in Figure 3, with one exception: we change
the value of $p$ from 0.01 to $0.0001$. For each tolerance level $t$ we note
that the curve corresponding to $p=0.01$ dominates the curve for $p=0.0001$.
For example when the probability threshold is 0.7, $t=4$, and $p=0.01$, around
10 tests per day are needed, as observed above. However, when we examine the
case with $p=0.0001$ the required number of tests per day is approximately 17.
At first glance, this may seem counterintuitive: we require fewer tests when a
disease is more prevalent in the community. However, the result is
mathematically sound. For example, suppose we examine the extreme case when
$p$ is very close to 1. In that case, a large number of staff members would
arrive to the facility each day being infectious. As such, we only need to
test a small fraction of the staff to detect an outbreak. In contrast, when
the community infection rate is low, it is likely that only 1 staff member out
of, say 100, would initiate an outbreak. This one seed of an outbreak is
harder to catch with a low number of tests.
$0$$10$$20$$30$$40$$50$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of tests per day
($k$)Conditional probability of detecting
outbreak$t=4$$t=5$$t=6$$t=7$$t=8$$t=9$$t=10$ Figure 4: Number of tests versus
detection probability for circulant neighboring graphs of varying tolerance
levels. Fixed graph parameters are $K=100$, $p=0.0001$, $r=0.05$, $f=0.21$,
$\ell=3$, $d=60$. The results were obtained by sampling 50,000 virus sample
paths.
### 4.2 The Effect of Testing Order
In the experiments of Section 4.1, the nodes of the graph were tested in
“standard order.” What this means is that we envision the nodes with a fixed
numbering, 0 to $K-1$, and arranged in a circle according to the numbering.
This numbering is used to characterize various types of circulant graphs, as
described already. In addition, the standard testing order is to begin by
testing nodes 0 through $k-1$ on day 1, then testing nodes $k$ through $2k-1$
on day 2, etc. We continue testing in this manner, returning to node 1 when
all nodes have been tested. Of course, for some values of $t$ and $K$ not all
nodes will be tested. With respect to our simulation results, it does not
matter with which node we start since outbreaks are simulated in a uniform
manner across all nodes, and the graphs are vertex transitive. However, it is
not clear that testing in this standard order, for any given initial starting
node, is optimal. Obtaining the optimal testing order is computationally
prohibitive, as the underlying optimization problem is a stochastic integer
program, with a very large number of feasible solutions. In this paper, we
make no attempt to prove optimality bounds on testing orders, or to develop
complex heuristics. Rather, we examine a few different testing algorithms on
circulant graphs. Our results indicate that the testing order has little
effect on the detection probability, at least for circulant graphs. This is
good news in that it indicates that optimizing the testing order likely is not
a high priority in tuning test protocols in contact networks.
In our next set of experiments, we examine the effect of changing the testing
order, while keeping all the network parameters fixed. In Figure 5 we examine
three possible testing protocols. The first protocol, labeled “circle” is the
standard order described in the previous paragraph. The second protocol,
labeled “new” works as follows. The first node to be tested is chosen
uniformly at random. Call this node $A$. The second node that is tested is the
node that has the maximum distance from $A$, with ties broken randomly. Recall
that the distance between two nodes in a graph is the number of edges in the
shortest path connecting the nodes. To continue the selection process after
some set of nodes has already been selected, the next node chosen is one whose
sum of distances from all previously selected nodes is largest. To break ties
among equal distance sums, we select a node for which these distances have the
smallest sample standard deviation (if there is still a tie, it is broken
randomly). We repeat the selection process until all nodes have been selected,
to create an ordered list that defines the testing order. In our simulation
analysis, we test the nodes in the given order and then return to the
beginning of the list once all nodes have been tested once. The third protocol
is based on a randomly selected permutation of the numbers 0 through $K-1$.
For the neighboring graph testing in Figure 5, we see that the distance
algorithm does slightly outperform the circle protocol for testing levels of
about 5 to 10 nodes per day. This also implies a slight difference in the
required number of tests at the mid-level probability thresholds. For example,
for a probability threshold of 0.9 the difference in the required tests per
day appears to be about 2. Outside this middle zone, the difference is
essentially negligible. Note that we do expect the circle protocol to have a
slightly lower performance in a neighboring graph. When the virus is most
likely to spread among close neighbors, when the nodes are arranged in a
circle, then there is some redundancy in testing nodes in this same circular
order. It is better to hop across the circle for subsequent tests, which is
essentially what the “new” protocol does.
$0$$2$$4$$6$$8$$10$$12$$14$$16$$18$$20$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of
tests per day ($k$)Conditional probability of detecting
outbreakcirclerandomnew Figure 5: Number of tests versus detection probability
for various testing protocols in a neighboring graph. Fixed graph parameters
are $K=100$, $p=0.01$, $r=0.05$, $f=0.21$, $\ell=3$, $d=80$, $t=6$. The
results were obtained by sampling 10,000 virus sample paths.
We next test the same three testing protocols in a crossing graph. In Figure
6, we see that there is almost no difference in the three protocols, across a
range of testing levels. This is also expected in this case. In a crossing
graph, the testing protocols circle and new are quite similar, because nodes
that are “neighbors” when arranged on a circle are actually rather distant as
measured by the number of hops required to travel between such nodes. Again,
these results indicate that testing protocol does not have a large effect on
detection probability, at least for circulant graphs.
Finally, in Figure 7, we test the same protocols on a randomly generated
$d$-regular graph. As expected, there are relatively small differences among
protocols. We also tested protocols for different graph densities, and the
results are similar to those presented herein. We conclude that the difference
among testing orders is usually insignificant, with some minor differences as
shown in Figure 5. Since the “new” protocol performs well in all cases, we
recommend this protocol, assuming that the decision maker has concrete
knowledge of the graph structure. Otherwise, nodes can likely be tested in an
arbitrary manner, with little loss of performance versus a good heuristic.
$0$$2$$4$$6$$8$$10$$12$$14$$16$$18$$20$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of
tests per day ($k$)Conditional probability of detecting
outbreakcirclerandomnew Figure 6: Number of tests versus detection probability
for various testing protocols in a crossing graph. Fixed graph parameters are
$K=100$, $p=0.01$, $r=0.05$, $f=0.21$, $\ell=3$, $d=80$, $t=6$. The results
were obtained by sampling 50,000 virus sample paths.
$0$$2$$4$$6$$8$$10$$12$$14$$16$$18$$20$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of
tests per day ($k$)Conditional probability of detecting
outbreakcirclerandomnew Figure 7: Number of tests versus detection probability
for various testing protocols in a random $d$-regular graph. Fixed graph
parameters are $K=100$, $p=0.01$, $r=0.05$, $f=0.21$, $\ell=3$, $d=80$, $t=6$.
The results were obtained by sampling 10,000 virus sample paths.
## 5 Heterogeneous Graphs and Real-world Test Cases
### 5.1 Real-world Test Cases
#### 5.1.1 Model adjustment
In previous sections, we considered simplified contact networks in order to
gain insight into the relationship between the parameters of the graph, and
the number of tests needed to detect an outbreak. In the remaining sections,
we consider more general graphs that are inspired by real-world situations and
data. A primary difference is that we allow the parameters $r$ and $p$ to be
heterogeneous, i.e., they can vary by node. The outside probability of
infection $p$ may vary since each staff member has different social
interactions and risk outside of work. Also, in real facilities staff in
different categories can have very different contact patterns. Hence, the
corresponding contact graph is no longer “symmetric,” and the internal
infection probability $r$ can vary by node. Our assumptions regarding the
latency $l$, the false negative rate $f$, and disease testing procedures are
the same as in Section 3.
In many facilities, the staff rosters differ each day as do the contact
patterns. In theory, it might be possible to develop even more nuanced models
in a particular facility. For example, Duval et al. [13] studied the contact
patterns in one nursing home by asking the staff and patients to use wearable
sensors and recorded the contact duration, distances, and frequencies.
However, for legal and ethical reasons, there are obviously significant
barriers to collecting such detailed data in most facilities. Hence, we do not
attempt to model a representative facility at this level of detail. Instead,
we collect general data on staff levels, categories and interactions. We
believe that this level of modeling still provides important insight into
appropriate testing levels for outbreak detection.
Let $c$ be the number of different staff categories in a facility. The staff
categories are given by the $C=\\{1,2,...,c\\}$. Since we have heterogeneous
internal infection parameters, we now use $r_{c_{1},c_{2}}$ to denote the
daily probability of infection between staff members in categories $c_{1}$ and
$c_{2}$, assuming that they are connected by an edge in the contact graph. Of
course, we allow $c_{1}=c_{2}$ to reflect interactions between staff in the
same category. Staff in a facility can be categorized by their job titles,
rosters, offices, working units, etc. In our real-world case study, we
categorize the staff in a nursing home by job title, and we assume the staff
with the same job titles share the same contact patterns.
#### 5.1.2 Parameter fitting
We estimate the parameter $r_{c_{1},c_{2}}$ for different choices of $c_{1}$
and $c_{2}$ based on the contact patterns among staff. Inspired by Smith et
al. [29] and Temime et al. [34], we use the contact frequencies, average
duration of each contact, and the rate of infection with close contact to
estimate the probability of internal infection for staff in various
categories. Let $p_{0}$ be the probability of infection per minute between two
individuals with close contact, $d_{0}$ the average contact duration (in
minutes), and $n_{0}$ the average contact frequency. Then we can estimate the
internal probability of infection as $r=p_{0}\cdot d_{0}\cdot n_{0}$. We
collected data from a nursing home in the northern US by collecting
information from the facility manager. The details of this contact data are
shown in Table 1. Next, for each staff category, we estimated the number of
other staff they come into contact with daily, for all other categories. This
data is shown in Table 2. For a job category in row $i$ of the table, the
$i,j$th entry in the upper part of the table is the average number of contacts
for a staff member of category $i$ with staff members in category $j$. Note
that the table would not be symmetric if the lower half was filled in, but the
data in the upper portion is sufficient to estimate contacts between all pairs
of staff categories.
Job title | Staff # | # of daily
contact
with residents | # of daily
contact
with staff | Avg. contact
distance
(ft) | Avg. contact
duration
(min)
---|---|---|---|---|---
Nurses and nurse aids | 69 | 30 | 20 | $<$ 6 | $>$ 15
House keeping | 16 | 10 | 10 | $>$ 6 | $<$ 15
Rehabilitation | 4 | 10 | 10 | $<$ 6 | $>$ 15
Activity lifestyle | 12 | 20 | 10 | $<$ 6 | 15
Salon | 2 | 10 | 10 | $<$ 6 | 15
Maintenance | 12 | 10 | 10 | $<$ 6 | $<$ 15
Reception | 6 | 10 | 40 | $<$ 6 | $>$ 15
Dining | 6 | 30 | 20 | $<$ 6 | $<$ 15
Social | 5 | 10 | 10 | $<$ 6 | $<$ 15
Finance | 14 | 0 | 10 | $<$ 6 | $>$ 15
Logistics | 3 | 0 | 20 | $<$ 6 | $<$ 15
Driver | 4 | 20 | 10 | $<$ 6 | $>$ 15
Table 1: Staff data for the nursing home case study Job title | NR | HK | RH | MS | SL | MT | RC | DN | SC | AD | LG | DR
---|---|---|---|---|---|---|---|---|---|---|---|---
NR | 15 | 1.5 | 0.2 | 1 | 0.1 | 0.1 | 2 | 0.5 | 1 | 1 | 0.5 | 0.22
HK | $-$ | 4 | 0.5 | 0.5 | 0.1 | 1 | 1 | 0.2 | 0.2 | 1 | 0.1 | 0.25
RH | $-$ | $-$ | 1.5 | 2 | 0.1 | 0.1 | 0.5 | 0.2 | 0.2 | 1 | 0.5 | 0.2
MS | $-$ | $-$ | $-$ | 3 | 0.1 | 0.1 | 0.5 | 0.2 | 0.2 | 1 | 0.5 | 0.2
SL | $-$ | $-$ | $-$ | $-$ | 1 | 0.1 | 0.5 | 0.1 | 0.1 | 1.2 | 0.5 | 0.2
MT | $-$ | $-$ | $-$ | $-$ | $-$ | 4 | 0.5 | 0.1 | 0.1 | 1.2 | 0.5 | 0.3
RC | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 2 | 0.1 | 0.1 | 3 | 1 | 0.4
DN | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 4 | 1 | 1 | 0.1 | 0.1
SC | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 3 | 1 | 0.5 | 0.2
AD | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 5 | 1 | 0.5
LG | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 1.5 | 0.4
DR | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.5
Table 2: Estimated daily contacts between staff of different categories,
rounded to two decimals places
Based on this data, we are now able to generate a representative contact graph
for the case-study nursing home. First, we label all the nodes in the graph
with their corresponding staff category. Then, we calculate the probability of
generating an edge between two nodes by using the data in Tables 1 and 2. For
example, if a node is labeled as a nurse, the probability that a contact is
triggered with another nurse node is $15/68\approx 0.22$, as each nurse has an
average of 15 contacts with the other 68 nurses during a shift. Then, for
every pair of nurses, we generate a Bernoulli(0.22) random variable and assign
an edge to this pair if the outcome is a 1. Otherwise, there is no edge
connecting the pair. When generating edges between nodes in different staffing
categories, we use similar logic. For example, the probability that there is
an edge connecting a nurse with a member of the housekeeping staff is 1.5/16,
since, using Table 2, each nurse has an average of 1.5 contacts per day with a
member of the housekeeping staff. Figure 8 shows an instance of the contact
network induced by the procedure just described. Our estimates of these
$r_{i,j}$ values appear in Table 3. For example, if in the contact network
there exists an edge between a nurse and a housekeeper, then the probability
of transmission (per day) between these two staff members is 0.01. If there is
no edge between two staff members, then the transmission probability is 0. We
only provide data in the upper part of the table, as this table is symmetric,
by our assumptions.
Job title | NR | HK | RH | MS | SL | MT | RC | DN | SC | AD | LG | DR
---|---|---|---|---|---|---|---|---|---|---|---|---
NR | 0.08 | 0.01 | 0.06 | 0.015 | 0.01 | 0.01 | 0.04 | 0.02 | 0.02 | 0.01 | 0.01 | 0.08
HK | $-$ | 0.08 | 0.01 | 0.01 | 0.01 | 0.01 | 0.01 | 0.02 | 0.02 | 0.01 | 0.01 | 0.04
RH | $-$ | $-$ | 0.08 | 0.015 | 0.01 | 0.01 | 0.04 | 0.02 | 0.02 | 0.01 | 0.01 | 0.08
MS | $-$ | $-$ | $-$ | 0.08 | 0.01 | 0.01 | 0.04 | 0.02 | 0.02 | 0.01 | 0.01 | 0.08
SL | $-$ | $-$ | $-$ | $-$ | 0.08 | 0.01 | 0.01 | 0.02 | 0.02 | 0.01 | 0.01 | 0.08
MT | $-$ | $-$ | $-$ | $-$ | $-$ | 0.08 | 0.01 | 0.02 | 0.02 | 0.01 | 0.01 | 0.04
RC | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.08 | 0.02 | 0.02 | 0.01 | 0.02 | 0.08
DN | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.02 | 0.04 | 0.02 | 0.01 | 0.08
SC | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.02 | 0.03 | 0.02 | 0.08
AD | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.08 | 0.03 | 0.08
LG | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.08 | 0.08
DR | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | $-$ | 0.08
* •
Abbreviations: NR, nurse; HK, house-keeper; MS, mission; SL, salon; RC,
receptionist; DN, dining; SC, social; AD, administration; LG, logistic; DR,
driver.
Table 3: Estimated probability of internal infection among staff in different
categories Figure 8: A real-world LTC network with $s=152,c=12$
The probability of external infection is a dynamic quantity that depends
highly on the current community prevalence of COVID-19. In this study, we
assume that all staff have the same value of $p$. For a given area, such as a
state or a county, we can estimate the daily probability of infection for an
individual as
$p=\frac{\text{Number of new infections over the past 7
days}}{7\cdot\text{Population size}}.$
Of course, the official count new infections in most cases is underestimating
the true number of cases. Therefore, in conducting sensitivity analysis, we
multiply the estimate above by $3$, $5$, and $10$.
#### 5.1.3 Computational experiments
In this section, we present results using the network generated from the data
in Tables 1, 2, and 3. Similar to Section 4.1, we vary some basic model
parameters to perform comparisons. In the first set of experiments, we fix the
probability of external infection to $p=0.003$, and vary the outbreak
threshold tolerance $t$. As shown in Figure 9, the results align with the
experiments we did for the circulant graph. Obviously, as before, a larger
tolerance value requires a smaller number of tests per day, for a fixed
conditional probability target. For example, if the facility manager targets a
0.8 probability of detecting an outbreak, with $t=3$ then approximately 44
tests per day are required. However, if the threshold of tolerance is set to
4, 5 or even 10 days, then the required number of tests per day is 29, 19 and
3, respectively. We also note for $t=3$ the graph appears to be piecewise
linear. This makes intuitive sense, as we do not benefit from the “network”
effect of virus spread in this case.
$0$$10$$20$$30$$40$$50$$60$$70$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of tests per
day ($k$)Conditional probability of detecting
outbreak$t=3$$t=4$$t=5$$t=6$$t=7$$t=8$$t=9$$t=10$$t=11$$t=12$$t=13$$t=14$
Figure 9: Estimated probabilities of outbreak detection for $K=152$, $c=12$,
and $p=0.003$ for various values of $t$
Next, we examine the effect of changing $p$, the probability of external
infection. Table 10 shows the results of our simulations. As argued in Section
4.1, although it is counterintuitive that higher values of $p$ require fewer
tests per day to meet a given probability target, this is actually a logical
outcome of the model.
$0$$10$$20$$30$$40$$50$$60$$70$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of tests per
day ($k$)Conditional probability of detecting
outbreak$p=0.001$$p=0.003$$p=0.006$$p=0.01$$p=0.015$$p=0.03$ Figure 10:
Estimated probabilities of outbreak detection in a nursing home case study
with $K=152$, $c=12$, $t=7$ and varying values of $p$
Keeping this observation in mind, we note the a facility manager may want to
examine additional metrics when setting the tolerance level $t$ and the
detection probability target. For one parameter setting in our case-study
network, we provide in Figure 11 various statistics on the spread on
infection, as reflected by 50,000 simulation runs. In Figure 11, the blue line
represents the cumulative number of infectious staff, the red line represents
the number of infected staff, and the yellow line represents the number of
newly infected staff. The shaded areas around the lines represent 95% of
confidence intervals around the point estimates for these quantities, various
days after the outbreak. As expected, the blue line point estimate is higher
than the red line, because infection occurs three days before an individual
becomes infectious. Examining the graph, if a facility manager wishes to
detect an outbreak before 10% of the staff becomes infected then she would set
$t=5$ for the network depicted in Figure 11.
$0$$2$$4$$6$$8$$10$$12$$14$$16$$18$$20$$0$$20$$40$$60$$80$$100$$120$$140$Number
of days after outbreakNumber of staffcumulative infected staffcumulative
infectious staffnewly infected staffconfidence interval = 95% Figure 11:
Infection statistics in a case-study model with $K=152$, $c=124$, $p=0.003$,
$l=3$, and $f=0.21$
Finally, in Figure 12 we show the proportion of infected staff in various
categories over 20 days, again averaged over 50,000 simulated outbreaks. Note
that receptionists and nurses are the categories with the highest infection
proportions. In contrast, salon workers have the lowest infection rates in
this example. Our discussions with nursing home staff, indicate that their
perception is that nurses are quite vulnerable to outbreaks, but receptionists
are not. However, if we scrutinize Table 1 we note that receptionists and
nurses both have frequent, close, and prolonged contact with other staff, so
the model does seem to be reflecting the data provided.
$0$$2$$4$$6$$8$$10$$12$$14$$16$$18$$20$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of
days after outbreakProportion of staff infected within each
categoryNRHKRHMSSLMTRCDNSCADLGDR Figure 12: Proportion of infected staff in
different job categories with $s=152$, $c=12$, $p=0.003$, $l=3$, and $f=0.21$
### 5.2 Effect of Testing Order
In Section 4.2, we examined the effects of testing order in what we called
homogeneous graphs, i.e., graphs with a significant amount symmetry and
homogeneity among the nodes. There, we found small, but mostly insignificant
differences among testing protocols. In contact networks arising from the real
world, the associated graphs are not expected to have such a regular
structure. Due to the less regular structure, there are also a larger variety
of heuristic testing protocols that can be tested. The most intuitive
protocols are based on testing “central” or important nodes first.
In this section, we test three such heuristics of varying levels of
complexity. In each heuristic we need to produce an ordered list (permutation)
of the nodes that determines the testing protocol. The first heuristic, which
we call _degree rank_ , we simply order the nodes from highest degree to
lowest degree, breaking ties in an arbitrary manner. Nodes are then tested in
this order. The idea, of course, is a higher degree node is more likely to
“detect” an outbreak since it has connections to lots of other nodes. The
second heuristic, which is called _PageRank_ , is based on the well-known node
ranking system first developed by the founders of Google. We compute the
PageRank for all nodes, and test in the order of highest rank to lowest.
Again, the idea of this ranking is that higher ranked nodes would be visited
more often by a virus surfing the contact network. A potential drawback of
this notion, as applied to surveillance testing, is that PageRank is
essentially derived from the steady-state distribution of the virus surfing
process alluded to above. However, in our virus spread and detection model, we
are really concerned with the _transient_ behavior of the virus. In
particular, what is most important for early detection is the initial
trajectory of the virus over the network. This observation inspires us to
consider another algorithm which we call _simulation rank_. The idea is as
follows: if our outbreak tolerance is $t$ days, then we are interested in
testing the nodes that are most likely to be involved in an outbreak during
the $t$ days. In particular, we should rank the nodes according to the
probability that they are part of a $t$-day outbreak. In theory, one could
compute this probability exactly, but it is clearly an intractable computation
for networks of even moderate size. Instead, we simulate many virus sample
paths to estimate the aforementioned probability for each node, and then use
these estimates to create the ranking. It is clear that the degree rank
ordering requires very little computational effort, as it just involves
counting edges. PageRank is more difficult, but there are efficient algorithms
to estimate the PageRank, even for large networks. Finally, simulation rank
requires a considerable amount of effort, as potentially thousands of
simulation samples are required to get a good estimate.
In Figures 13 and 14, we show the results of testing the three ranking
algorithms for test protocols. In each case, we determine the test order and
then evaluate the protocol using the network depicted in Figure 8. As in our
previous computations, we using 50,000 virus sample paths to perform the
evaluation. Figure 13 displays the results for a tolerance of $t=4$ days.
Clearly, the difference among the algorithms is imperceptible in this figure.
We also ran statistical tests confirming that there no statistically
significant difference among the algorithms. Figure 14 displays the results
for a tolerance of $t=7$ days. Again, the differences are almost negligible,
although PageRank appears to slightly outperform the other algorithms when 7
to 10 tests per days are performed. Still, these differences do not appear to
be statistically significant. We performed similar tests on a small set of
other networks, with similar results. Although the tests are not
comprehensive, they seem to indicate that a relatively simple algorithm such
as degree rank works just as well as more sophisticated algorithms.
$0$$7$$14$$21$$28$$35$$42$$49$$56$$63$$70$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of
tests per day ($k$)Conditional probability of detecting outbreakdegree
rankPageRanksimulation rank Figure 13: Performance of heuristic testing
protocols for the LTC network with $s=152$, $c=12$, $p=0.006$, $l=3$, $t=4$
and $f=0.21$
$0$$7$$14$$21$$28$$35$$42$$49$$56$$63$$70$$0$$0.2$$0.4$$0.6$$0.8$$1$Number of
tests per day ($k$)Conditional probability of detecting outbreakdegree
rankPageRanksimulation rank Figure 14: Performance of heuristic testing
protocols for the LTC network with $s=152$, $c=12$, $p=0.006$, $l=3$, $t=7$
and $f=0.21$
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# Jacob’s ladders and some new consequences from A. Selberg’s formula
Jan Moser Department of Mathematical Analysis and Numerical Mathematics,
Comenius University, Mlynska Dolina M105, 842 48 Bratislava, SLOVAKIA
<EMAIL_ADDRESS>
###### Abstract.
It is proved in this paper that the Jacob’s ladders together with the A.
Selberg’s classical formula (1942) lead to a new kind of formulae for some
short trigonometric sums. These formulae cannot be obtained in the classical
theory of A. Selberg, and all the less, in the theories of Balasubramanian,
Heath-Brown and Ivic.
###### Key words and phrases:
Riemann zeta-function
## 1\. The A. Selberg’s formula
A. Selberg has proved in 1942 the following formula
(1.1) $\int_{T}^{T+U}X^{2}(t)\left(\frac{n_{2}}{n_{1}}\right)^{it}{\rm
d}t=\sqrt{\frac{\pi}{2}}\frac{U}{\sqrt{n_{1}n_{2}}}\left(\ln\frac{P^{2}}{n_{1}n_{2}}+2c\right)+\mathcal{O}(T^{1/2}\xi^{5})$
(see [19], p. 55), where
(1.2) $\begin{split}&X(t)=\frac{1}{2}t^{1/4}e^{\frac{1}{4}\pi
t}\pi^{-\frac{s}{2}}\zeta(s),\ s=\frac{1}{2}+it,\\\ &U=T^{1/2+\epsilon},\
\xi=\left(\frac{T}{2\pi}\right)^{\epsilon/10},\ \epsilon\leq\frac{1}{10},\
P=\sqrt{\frac{T}{2\pi}}\\\ &n_{1},n_{2}\in\mathbb{N},(n_{1},n_{2})=1,\
n_{1},n_{2}\leq\xi,\end{split}$
(comp. [19], pp. 10, 18, $a=1/2+\epsilon,\ \epsilon>0$) and $c$ is the Euler’s
constant. Since (see [19], p. 10, [20], p. 79)
$Z^{2}(t)=\left|\zeta\left(\frac{1}{2}+it\right)\right|^{2}=\sqrt{\frac{2}{\pi}}X^{2}(t)\left(1+\mathcal{O}(\frac{1}{t})\right),$
i.e.
(1.3)
$X^{2}(t)=\sqrt{\frac{2}{\pi}}Z^{2}(t)\left(1+\mathcal{O}(\frac{1}{t})\right)$
where
(1.4)
$\begin{split}&Z(t)=e^{i\vartheta(t)}\zeta\left(\frac{1}{2}+it\right),\\\
&\vartheta(t)=-\frac{1}{2}t\ln\pi+\text{Im}\ln\Gamma\left(\frac{1}{4}+\frac{1}{2}it\right)=\frac{t}{2}\ln\frac{t}{2\pi}-\frac{t}{2}-\frac{\pi}{8}+\mathcal{O}(\frac{1}{t})\end{split}$
is the signal defined by the Riemann zeta-function $\zeta(s)$. Following eqs.
(1.1) and (1.3) we obtain
(1.5) $\int_{T}^{T+U}Z^{2}(t)\left(\frac{n_{2}}{n_{1}}\right)^{it}{\rm
d}t=\frac{U}{\sqrt{n_{1}n_{2}}}\left(\ln\frac{P^{2}}{n_{1}n_{2}}+2c\right)+\mathcal{O}(T^{1/2}\xi^{5})$
###### Remark 1.
If $n_{1}=n_{2}=1$ then the Hardy-Littlewood-Ingham formula
$\int_{T}^{T+U}Z^{2}(t){\rm
d}t=U\ln\frac{T}{2\pi}+2cU+\mathcal{O}(T^{1/2}\xi^{5})$
follows from the A. Selberg’s formula (1.5) (comp. [20], p. 120).
###### Remark 2.
Let us remind that the A. Selberg’s formula (1.5) played the main role in
proving the fundamental Selberg’s result
$N_{0}(T+U)-N_{0}(T)>A(\epsilon)U\ln T$
where $N_{0}$ stands for the number of zeroes of the function $\zeta(1/2+it),\
t\in(0,T]$.
In this paper it is proved that the Jacob’s ladders together with the A.
Selberg’s classical formula lead to a new kind of results for some short
trigonometric sums.
This paper is a continuation of the series of works [3] \- [18].
## 2\. The result
### 2.1.
Let us remind some notions. First of all
(2.1) $\tilde{Z}^{2}(t)=\frac{{\rm d}\varphi_{1}(t)}{{\rm d}t},\
\varphi_{1}(t)=\frac{1}{2}\varphi(t),$
where
(2.2)
$\tilde{Z}^{2}(t)=\frac{Z^{2}(t)}{2\Phi^{\prime}_{\varphi}[\varphi(t)]}=\frac{Z^{2}(t)}{\left\\{1+\mathcal{O}\left(\frac{\ln\ln
t}{\ln t}\right)\right\\}\ln t}$
(see [3], (3.9); [5], (1.3); [9], (1.1), (3.1), (3.2)) and $\varphi(t)$ is the
Jacob’s ladder, i.e. the solution of the following nonlinear integral equation
$\int_{0}^{\mu[x(T)]}Z^{2}(t)e^{-\frac{2}{x(T)}t}{\rm
d}t=\int_{0}^{T}Z^{2}(t){\rm d}t$
that was introduced in our paper [3]. Next, we have (see [1], comp. [18])
(2.3) $\begin{split}&G_{3}(x)=G_{3}(x;T,U)=\\\ &=\bigcup_{T\leq g_{2\nu}\leq
T+U}\\{t:\ g_{2\nu}(-x)\leq t\leq g_{2\nu}(x)\\},\ 0<x\leq\frac{\pi}{2},\\\
&G_{4}(y)=G_{4}(y;T,U)=\\\ &=\bigcup_{T\leq g_{2\nu+1}\leq T+U}\\{t:\
g_{2\nu+1}(-y)\leq t\leq g_{2\nu+1}(y)\\},\ 0<y\leq\frac{\pi}{2},\end{split}$
and the collection of sequences $\\{g_{\nu}(\tau)\\},\ \tau\in[-\pi,\pi],\
\nu=1,2,\dots$ is defined by the equation (see [1], [18], (6))
$\vartheta_{1}[g_{\nu}(\tau)]=\frac{\pi}{2}\nu+\frac{\tau}{2};\
g_{\nu}(0)=g_{\nu}$
where (comp. (1.4))
$\vartheta_{1}(t)=\frac{t}{2}\ln\frac{t}{2\pi}-\frac{t}{2}-\frac{\pi}{8}.$
### 2.2.
In this paper we obtain some new integrals containing the following short
trigonometric sums
$\begin{split}&\sum_{2\leq p\leq\xi}\frac{1}{\sqrt{p}}\cos(t\ln p),\
\sum_{2\leq n\leq\xi}\frac{1}{\sqrt{n}}\cos(t\ln n),\\\ &\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos(t\ln n)\end{split}$
where $p$ is the prime, $n\in\mathbb{N}$ and $d(n)$ is the number of divisors
of $n$. In this direction, the following theorem holds true.
###### Theorem.
Let
(2.4) $G_{3}(x)=\varphi_{1}(\mathring{G}_{3}(x)),\
G_{4}(y)=\varphi_{1}(\mathring{G}_{4}(y)).$
Then we have
(2.5) $\begin{split}&\int_{\mathring{G}_{3}(x)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos(\varphi_{1}(t)\ln
p)\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\
&\frac{2x}{\pi}U\ln P\ln\ln P,\ x\in(0,\pi/2],\\\
&\int_{\mathring{G}_{4}(y)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos(\varphi_{1}(t)\ln
p)\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\
&\frac{2y}{\pi}U\ln P\ln\ln P,\ y\in(0,\pi/2],\end{split}$
(2.6) $\begin{split}&\int_{\mathring{G}_{3}(x)}\left(\sum_{2\leq
n\leq\xi}\frac{1}{\sqrt{n}}\cos(\varphi_{1}(t)\ln
n)\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\
&\frac{1}{\pi}\left\\{\left(\frac{2\epsilon}{5}-\frac{\epsilon^{2}}{50}\right)x+\frac{\epsilon^{2}}{50}\sin
x\right\\}U\ln^{2}P,\\\ &\int_{\mathring{G}_{4}(y)}\left(\sum_{2\leq
n\leq\xi}\frac{1}{\sqrt{n}}\cos(\varphi_{1}(t)\ln
n)\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\
&\frac{1}{\pi}\left\\{\left(\frac{2\epsilon}{5}-\frac{\epsilon^{2}}{50}\right)y-\frac{\epsilon^{2}}{50}\sin
y\right\\}U\ln^{2}P,\end{split}$
(2.7) $\begin{split}&\int_{\mathring{G}_{3}(x)}\left(\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos(\varphi_{1}(t)\ln
n)\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\ &\frac{\sin
x}{2500\pi^{3}}U\ln^{4}P,\\\ &\int_{\mathring{G}_{4}(y)}\left(\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos(\varphi_{1}(t)\ln
n)\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\ &-\frac{\sin
y}{2500\pi^{3}}U\ln^{2}P,\end{split}$
where
(2.8) $t-\varphi_{1}(t)\sim(1-c)\pi(t),\ t\to\infty,$
and $\pi(t)$ is the prime-counting function.
###### Remark 3.
Let $T=\varphi_{1}(\mathring{T})$, $T+U=\varphi_{1}(\widering{T+U})$, (comp.
(2.4)). Then from (2.8), similarly to [14], (1.8), we obtain
$\rho\\{[T,T+U];[\mathring{T},\widering{T+U}]\\}\sim(1-c)\pi(T);\
T+U<\mathring{T},$
where $\rho$ stands for the distance of the corresponding segments.
###### Remark 4.
The formulae (2.5) - (2.7) cannot be obtained in the classical theory of A.
Selberg, and, all the less, in the theories of Balasubramanian, Heath-Brown
and Ivic.
## 3\. New asymptotic formulae for the short trigonometric sums: their
dependence on $|\zeta\left(\frac{1}{2}+it\right)|^{2}$
We obtain, putting $x=y=\pi/2$ in (2.5)
$\begin{split}&\int_{\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln
p\\}\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\ &2U\ln
P\ln\ln P.\end{split}$
Using successively the mean-value theorem (since
$\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)$ is a segment), we have
(3.1) $\begin{split}&\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(\alpha_{1})\ln
p\\}\int_{\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)}Z^{2}\\{\varphi_{1}(t)\\}Z^{2}(t){\rm
d}t=\\\ &=\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(\alpha_{1})\ln
p\\}Z^{2}\\{\varphi_{1}(\alpha_{2})\\}\int_{\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)}\tilde{Z}^{2}(t){\rm
d}t\sim\\\ &2U\ln P\ln\ln P,\
\alpha_{1},\alpha_{2}\in\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2);\
\alpha_{1}=\alpha_{1}(T,U)=\alpha_{1}(T,\epsilon),\dots.\end{split}$
Since
(3.2)
$\int_{\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)}\tilde{Z}^{2}{\rm
d}t=\left|\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)\right|$
(comp. Remark 8), and
(3.3) $m\\{\mathring{G}_{3}(x)\\}\sim\frac{x}{\pi}U,\
m\\{\mathring{G}_{4}(y)\\}\sim\frac{y}{\pi}U\ \Rightarrow\
\left|\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)\right|\sim U$
(see [2], (13), $m$ stands for the measure) then we obtain from (2.5) (see
(3.1) - (3.3)) the following
###### Corollary 1.
For every $T\geq T_{0}[\varphi_{1}]$ there are the values
$\alpha_{1}(T),\alpha_{2}(T)\in\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)$
such that
(3.4) $\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(\alpha_{1}(T))\ln
p\\}\sim\frac{2\ln P\ln\ln
P}{\left|\zeta\left(\frac{1}{2}+i\varphi_{1}(\alpha_{2}(T))\right)\right|^{2}},\
T\to\infty$
where
$\varphi_{1}(\alpha_{1}(T)),\varphi_{1}(\alpha_{2}(T))\in\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)$.
Similarly, we obtain from (2.6)
###### Corollary 2.
For every $T\geq T_{0}[\varphi_{1}]$ there are the values
$\alpha_{3}(T),\alpha_{4}(T)\in\mathring{G}_{3}(\pi/2)\cup\mathring{G}_{4}(\pi/2)$
such that
(3.5) $\begin{split}&\sum_{2\leq
n\leq\xi}\frac{1}{\sqrt{n}}\cos\\{\varphi_{1}(\alpha_{3}(T))\ln n\\}\sim\\\
&\sim\left(\frac{2\epsilon}{5}-\frac{\epsilon^{2}}{50}\right)\frac{\ln^{2}P}{\left|\zeta\left(\frac{1}{2}+i\varphi_{1}(\alpha_{4}(T))\right)\right|^{2}},\
T\to\infty\end{split}$
where $\varphi_{1}(\alpha_{3}(T)),\varphi_{1}(\alpha_{4}(T))\in
G_{3}(\pi/2)\cup G_{4}(\pi/2)$.
###### Remark 5.
From the asymptotic formulae (3.4), (3.5) it follows that the values of
mentioned short trigonometric sums are connected with the values of the
Riemann zeta-function $\zeta\left(\frac{1}{2}+it\right)$ for some infinite
subset of $t$.
## 4\. New asymptotic formulae on two collections of disconnected sets
$G_{3}(x),G_{4}(y)$
From (2.7), similarly to p. 3, we obtain
###### Corollary 3.
(4.1) $\begin{split}&\left.\left\langle\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{\varphi_{1}(t)\ln
n\\}\right\rangle\right|_{\mathring{G}_{3}(x)}\sim\\\
&\sim\frac{1}{2500\pi^{2}}\frac{\sin x}{x}\frac{\ln^{4}P}{\langle
Z^{2}\\{\varphi_{1}(t)\\}\rangle|_{\mathring{G}_{3}(x)}}\\\
&\left.\left\langle\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{\varphi_{1}(t)\ln
n\\}\right\rangle\right|_{\mathring{G}_{4}(y)}\sim\\\
&\sim-\frac{1}{2500\pi^{2}}\frac{\sin y}{y}\frac{\ln^{4}P}{\langle
Z^{2}\\{\varphi_{1}(t)\\}\rangle|_{\mathring{G}_{4}(y)}},\
T\to\infty\end{split}$
where $\langle(\dots)\rangle|_{\mathring{G}_{3}(x)},\dots$ denote the mean-
value of $(\dots)$ on $\mathring{G}_{3}(x),\dots$ .
###### Remark 6.
It follows from (4.1) that the short trigonometric sum
$\sum_{2\leq n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{t\ln n\\},\ t\geq
T_{0}[\varphi_{1}]$
has an infinitely many zeroes of the odd order.
## 5\. Law of the asymptotic equality of areas
Let
$\begin{split}&\mathring{G}_{3}^{+}(x)=\left\\{t:\ t\in\mathring{G}_{3}(x),\
\sum_{2\leq n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{\varphi_{1}(t)\ln
n\\}>0\right\\},\\\ &\vdots\\\ &\mathring{G}_{4}^{-}(x)=\left\\{t:\
t\in\mathring{G}_{4}(x),\ \sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{\varphi_{1}(t)\ln
n\\}<0\right\\}.\end{split}$
Then we obtain from (2.7), (comp. Corollary 3 in [14])
###### Corollary 4.
(5.1)
$\begin{split}&\int_{\mathring{G}_{3}^{+}(x)\cup\mathring{G}_{4}^{+}(x)}\left(\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{\varphi_{1}(t)\ln
n\\}\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t\sim\\\
&\sim-\int_{\mathring{G}_{3}^{-}(x)\cup\mathring{G}_{4}^{-}(x)}\left(\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{\varphi_{1}(t)\ln
n\\}\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t.\end{split}$
###### Remark 7.
The formula (5.1) represents the law of the asymptotic equality of the areas
(measures) of complicated figures corresponding to the positive part and the
negative part, respectively, of the graph of the function
(5.2) $\sum_{2\leq n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{\varphi_{1}(t)\ln
n\\}Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t),\
t\in\mathring{G}_{3}(x)\cup\mathring{G}_{4}(x),$
where $x\in(0,\pi/2]$. This is one of the laws governing the _chaotic_
behaviour of the positive and negative values of the signal (5.2). This signal
is created by the complicated modulation of the fundamental signal
$Z(t)=e^{i\vartheta(t)}\zeta\left(\frac{1}{2}+it\right)$, (comp. (1.4),
(2.2)).
## 6\. Proof of the Theorem
### 6.1.
Let us remind that the following lemma holds true (see [8], (2.5); [9],
(3.3)): for every integrable function (in the Lebesgue sense) $f(x),\
x\in[\varphi_{1}(T),\varphi_{1}(T+U)]$ we have
(6.1) $\int_{T}^{T+U}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{\varphi_{1}(T)}^{\varphi_{1}(T+U)}f(x){\rm d}x,\ U\in(0,T/\ln T],$
where $t-\varphi_{1}(t)\sim(1-c)\pi(t)$. In the case (comp. (2.4))
$T=\varphi_{1}(\mathring{T})$, $T+U=\varphi_{1}(\widering{T+U})$, we obtain
from (6.1) the following equality
(6.2)
$\int_{\mathring{T}}^{\widering{T+U}}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{T}^{T+U}f(x){\rm d}x.$
### 6.2.
First of all, we have from (6.2), for example,
$\int_{\mathring{g}_{2\nu}(-x)}^{\mathring{g}_{2\nu}(x)}f[\varphi_{1}(t)]\tilde{Z}^{2}(t){\rm
d}t=\int_{g_{2\nu}(-x)}^{g_{2\nu}(x)}f(t){\rm d}t,$
(see (2.3). Next, in the case
$f(t)=\left(\sum_{2\leq p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln
p\\}\right)Z^{2}\\{\varphi_{1}(t)\\}$
we have the following $\tilde{Z}^{2}$-transformation
(6.3) $\begin{split}&\int_{\mathring{G}_{3}(x)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln
p\\}\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t=\\\
&=\int_{\mathring{G}_{3}(x)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln p\\}\right)Z^{2}(t){\rm
d}t,\\\ &\int_{\mathring{G}_{4}(y)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln
p\\}\right)Z^{2}\\{\varphi_{1}(t)\\}\tilde{Z}^{2}(t){\rm d}t=\\\
&=\int_{\mathring{G}_{4}(y)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln p\\}\right)Z^{2}(t){\rm
d}t.\end{split}$
Let us remind that we have proved (see [2], (13) and Corollary 7) the
following formulae
(6.4) $\begin{split}&\int_{G_{3}(x)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln p\\}\right)Z^{2}(t){\rm
d}t\sim\frac{2x}{\pi}\ln P\ln\ln P,\\\ &\int_{G_{4}(y)}\left(\sum_{2\leq
p\leq\xi}\frac{1}{\sqrt{p}}\cos\\{\varphi_{1}(t)\ln p\\}\right)Z^{2}(t){\rm
d}t\sim\frac{2y}{\pi}\ln P\ln\ln P.\end{split}$
Now, our formulae (2.5) follow from (6.3), (6.4).
### 6.3.
Similarly, from the formulae
$\begin{split}&\int_{G_{3}(x)}\left(\sum_{2\leq
n\leq\xi}\frac{1}{\sqrt{n}}\cos\\{t\ln n\\}\right)Z^{2}(t){\rm d}t\sim\\\
&\sim\frac{x}{\pi}\left(\frac{2\epsilon}{5}-\frac{\epsilon^{2}}{50}+\frac{\epsilon^{2}}{50}\frac{\sin
x}{x}\right)U\ln^{2}P,\\\ &\int_{G_{4}(y)}\left(\sum_{2\leq
n\leq\xi}\frac{1}{\sqrt{n}}\cos\\{t\ln n\\}\right)Z^{2}(t){\rm d}t\sim\\\
&\sim\frac{y}{\pi}\left(\frac{2\epsilon}{5}-\frac{\epsilon^{2}}{50}-\frac{\epsilon^{2}}{50}\frac{\sin
y}{y}\right)U\ln^{2}P,\end{split}$
and
$\begin{split}&\int_{G_{3}(x)}\left(\sum_{2\leq
n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{t\ln n\\}\right)Z^{2}(t){\rm
d}t\sim\frac{\sin x}{2500\pi^{3}}U\ln^{4}P,\\\
&\int_{G_{4}(y)}\left(\sum_{2\leq n\leq\xi}\frac{d(n)}{\sqrt{n}}\cos\\{t\ln
n\\}\right)Z^{2}(t){\rm d}t\sim-\frac{\sin
y}{2500\pi^{3}}U\ln^{4}P\end{split}$
(see [2], (13) and Corollaries 8 and 9) we obtain (2.6) and (2.7),
respectively.
###### Remark 8.
The formulae of type (3.2) can be obtained from (6.2) putting $f(t)\equiv 1$.
I would like to thank Michal Demetrian for helping me with the electronic
version of this work.
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|
# Parameter Refinement of a Ballbot and Predictive Control for Reference
Tracking with Linear Parameter-Varying Embedding
1st Dimitrios S. Karachalios This work was funded by the German Research
Foundation (DFG), project number 419290163. Institute for Electrical
Engineering in Medicine
University of Luebeck
Lübeck, Germany
email<EMAIL_ADDRESS>2nd Hossam S. Abbas Institute
for Electrical Engineering in Medicine
University of Luebeck
Lübeck, Germany
email<EMAIL_ADDRESS>
###### Abstract
In this study, we implement a control method for stabilizing a ballbot that
simultaneously follows a reference. A ballbot is a robot balancing on a
spherical wheel where the single point of contact with the ground makes it
omnidirectional and highly maneuverable but with inherent instability. After
introducing the scheduling parameters, we start the analysis by embedding the
nonlinear dynamic model derived from first principles to a linear parameter-
varying (LPV) formulation. Continuously, and as an extension of a past study,
we refine the parameters of the nonlinear model that enhance significantly its
accuracy. The crucial advantages of the LPV formulation are that it consists
of a nonlinear predictor that can be used in model predictive control (MPC) by
retaining the convexity of the quadratic optimization problem with linear
constraints and further evades computational burdens that appear in other
nonlinear MPC methods with only a slight loss in performance. The LPVMPC
control method can be solved efficiently as a quadratic program (QP) that
provides timing that supports real-time implementation. Finally, to illustrate
the method, we test the control designs on a two-set point 1D non-smooth
reference with sudden changes, to a 2D nonstationary smooth reference known as
Lissajous curves, and to a single-set point 1D non-smooth reference where for
this case theoretical guarantees such as stability and recursive feasibility
are provided.
###### Index Terms:
Identification, Control, Ballbot, Model Predictive Control, Linear Parameter-
Varying, Quadratic Program
## I Introduction
A ballbot Fig. 1 is an omnidirectional mobile robot that balances on a single
spherical wheel [1, 2, 3]. The single point of contact with the ground makes
this under-actuated system agile but challenging to control. The use cases of
a ballbot include healthcare, virtual assistants, etc., as service robots,
where maneuverability in busy environments while maintaining stability is
important. A ballbot system’s stability implies balancing and trajectory
tracking through a predetermined path. Linear control of a ballbot [1] is
proven insufficient, and the LPV model, being much closer to the nonlinear
model, has shown great potential in several applications, such as autonomous
vehicles [4].
The dynamic model structure is well-known for robotic systems based on
physical laws. The identification of the parameters can be obtained through
classical mechanics [5] and enhanced white modeling, but parameters that
describe friction can be determined only experimentally. This study will
provide a method that refines the parameters that define the nonlinear ballbot
model from the measurements derived in [1]. For controlling the ballbot,
often, PID controllers considered [6, 1] or with double-loop approaches [3].
One of the limitations of the aforementioned methods is the consideration of
input/state constraints.
Model predictive control (MPC) has been widely applied for reference tracking
with constraints [4]. It can generate optimal trajectories that steer the
robot in a constrained space. Since autonomously moving robots are safety-
critical systems, nonlinear MPC (NMPC) is gaining popularity due to its
ability to utilize high-fidelity nonlinear models, enabling more accurate and
precise control actions. Another advantage of MPC is that it can be leveraged
with planning algorithms, which is crucial for such autonomously moving
objects. The drawback of NMPC is the computational complexity that makes
online minimization of the underlying objective function over a nonconvex
manifold cumbersome. Therefore, a good alternative is to utilize the LPV
formulation that can embed the nonlinear model equivalently and introduce the
LPVMPC framework that can be solved efficiently as QP, with nearly the same
computational burden as in linear MPC, enabling real-time implementation.
Figure 1: The Ballbot constructed at the Institute for Electrical Engineering
in Medicine (IME) and side view on the $xz$-plane.
Contents: Sec. II contains preliminaries and definitions. In Sec. II-A, the
nonlinear ballbot model is presented. Continuously, in Sec II-B, the linear
parameter-varying (LPV) model is presented with technical aspects. In Sec.
II-C, we refer to other stabilization methods that will help tune our terminal
cost, and in Sec. II-D, the LPVMPC complete framework is introduced with all
the details along with algorithms for ease of implementation. Sec. III
contains results; in particular, in Sec.III-A, the nonlinear model is
identified. In Sec. III-B, three representative scenarios on reference
tracking illustrate the developed method together with results that can ensure
theoretical guarantees. Finally, a discussion followed by acknowledgment
concludes the study.
Notations: $I_{n}$ is the identity matrix of dimension $n$. The notation
$Q\succeq 0$ represents the positive semi-definiteness of a matrix $Q$. The
weighted norm $\|x\|_{Q}$ is defined as $\|x\|_{Q}^{2}=x^{\top}Qx$ and
similarly $\lVert x\rVert^{2}_{P}=x^{\top}Px,~{}\lVert
u\rVert^{2}_{R}=u^{\top}Ru$. The function $\texttt{diag}(\mathbf{x})$
constructs a diagonal square matrix from a vector $\mathbf{x}$. The set of
positive integers, including zero, is denoted by $\mathbb{Z}_{+}\cup\\{0\\}$;
$i|k$ denotes at time $k$ the $i^{\textrm{th}}$ prediction.
## II Preliminaries and Definitions
### II-A Nonlinear Model of a Ballbot
The ballbot model is thoroughly discussed regarding the Euler-Lagrange method
of formulating the dynamics in [1, 2]. A first attempt towards model discovery
was performed in [1], where a linearized model was used to balance the
ballbot. The ballbot is a $5$-DoF system, where three are the body’s
rotational motion, and the two are the translation in the $xy$-plane. We will
see that the body’s angular position $\theta$ is used mainly for balancing,
and the angular position of the ball $\phi$ is used mainly for tracking see
Fig. 1.
The symmetry of the ballbot control problem helps us focus independently on
just the $xz$-plane [1] to simulate and understand its stability and
controllability. The ballbot state space model in (1) gives us the two
observable and controllable states as $q=[\phi_{y},\theta_{y}]^{T}$, together
with $\tau_{y}$, being the control input torque of the virtual wheel actuator
of the ballbot in the $xz$-plane as in Fig. 1 (right).
The ballbot model consists of the following matrices $M,C,D,G$, that denote
the mass, Coriolis force (centrifugal forces), frictional torque, and
gravitational forces, respectively, together with the input vector
$\widetilde{B}$. The nonlinear mechanical model is
$\displaystyle
M(q)\ddot{q}+C(q,\dot{q})+D(\dot{q})+G(q)=\widetilde{B}\tau_{y},$ (1)
where the matrices are defined as:
$\displaystyle\begin{aligned} M&=\begin{bmatrix}b_{1}&-b_{2}+\ell
r_{b}\cos{\theta_{y}}\\\ -b_{2}+\ell
r_{b}\cos{\theta_{y}}&b_{3}\end{bmatrix},\\\ C&=\begin{bmatrix}-\ell
r_{b}\sin({\theta_{y})\dot{\theta}_{y}^{2}}\\\
0\end{bmatrix},~{}D=\begin{bmatrix}b_{4}\dot{\phi}_{y}\\\ 0\end{bmatrix},\\\
G&=\begin{bmatrix}0\\\ -\ell
g\sin({\theta_{y}})\end{bmatrix},~{}\widetilde{B}=\begin{bmatrix}\frac{r_{b}}{r_{w}}\\\
-\frac{r_{b}}{r_{w}}\end{bmatrix}.\end{aligned}$ (2)
The nonlinear state-space representation of the planar ballbot model is given
next in (3).
$\displaystyle\underbrace{\begin{bmatrix}\dot{q}\\\
\ddot{q}\end{bmatrix}}_{\dot{x}}=\underbrace{\begin{bmatrix}\dot{q}\\\
M^{-1}(-C-D-G-\epsilon)\end{bmatrix}+\begin{bmatrix}0\\\
M^{-1}\widetilde{B}\end{bmatrix}\tau_{y}}_{f(x,u)},$ (3)
where, $x=[\phi_{y},\theta_{y},\dot{\phi}_{y},\dot{\theta}_{y}]^{\top}$ is the
state vector of the system with $\phi_{y}$ the angular displacement,
$\theta_{y}$ the tilt angle, $\dot{\phi}_{y}$ the angular speed of the ballbot
and $\dot{\theta}_{y}$ the angular speed of the tilt angle. The input to each
plane, e.g., $\tau_{y}$, is the virtual wheel’s torque Fig. 1. The virtual
torques $(\tau_{x},\tau_{y},\tau_{z})$ can be converted to real torques of the
three motors $(\tau_{1},\tau_{2},\tau_{3})$ as in [1] by using (4), where for
the design in Fig. 1 it holds that the Zenith angle is
$\alpha=\pi/4=45^{\circ}$.
$\begin{bmatrix}\tau_{1}\\\ \tau_{2}\\\
\tau_{3}\end{bmatrix}=\frac{1}{3}\begin{bmatrix}\frac{2}{\cos{\alpha}}&0&\frac{1}{\sin{\alpha}}\\\
-\frac{1}{\cos{\alpha}}&\frac{\sqrt{3}}{\cos{\alpha}}&\frac{1}{\sin{\alpha}}\\\
-\frac{1}{\cos{\alpha}}&-\frac{\sqrt{3}}{\cos{\alpha}}&\frac{1}{\sin{\alpha}}\end{bmatrix}\begin{bmatrix}\tau_{x}\\\
\tau_{y}\\\ \tau_{z}\end{bmatrix}.$ (4)
Now that we have a well-defined nonlinear model, we can introduce an
equivalent representation with an LPV embedding.
### II-B The Linear Parameter-Varying (LPV) Embedding
The linear parameter-varying (LPV) embedding [7, 8, 9, 10] can be seen as a
structured linearization over an adaptive operational point that depends on
the scheduling parameters. The nonlinear system in (3), can be embedded in the
continuous-time LPV representation by denoting with the subscript ”c” the
continuous operators that depend on the variable $\rho$ as $A_{c}(\rho)$,
$B_{c}(\rho)$ and define the continuous-time LPV model as
$\left\\{\begin{aligned}
\dot{x}(t)&=A_{c}(\rho(t))x(t)+B_{c}(\rho(t))u(t)\equiv f(x(t),u(t)),\\\
\rho(t)&=\sigma(x(t)),~{}x_{0}=x(0),~{}t\geq 0,\end{aligned}\right.$ (5)
where $\rho$ is the so-called scheduling parameter that belongs to the
scheduling set $\mathcal{P}\subset\mathbb{R}^{n_{\rho}}$ that defines the
range of the scheduling parameter, and it should be compact. Further, $\rho$
depends on the states through the mapping
$\sigma\colon\mathbb{R}^{n_{x}}\rightarrow\mathbb{R}^{n_{\rho}}$.
For the nonlinear system under consideration (on one plane of the ballbot),
the state dimension is $n_{x}=4$. We consider two scheduling parameters of the
LPV model as $\theta_{y}(t)$ and $\dot{\theta}_{y}(t)$, thus $n_{\rho}=2$ and
$\rho(t)=(\theta_{y}(t),\dot{\theta}_{y}(t))$. Therefore, in this case, the
functionality $\sigma$111Functionality:
$\sigma(x)=x^{\top}\left[\begin{array}[]{cccc}0&1&0&0\\\
0&0&0&1\end{array}\right]^{\top}=\left[\begin{array}[]{cc}\theta_{y}&\dot{\theta}_{y}\\\
\end{array}\right]=\rho$. can be seen as a linear function that chooses from
the state vector only some states. Finally, the input dimension is $n_{u}=1$
with $u(t)=\tau_{y}(t)$ representing the virtual torque of the virtual wheel
Fig. 1.
The considered LPV system’s matrices of the continuous-time state-space model
are given by (6), and the elements of the matrices $A_{c}$ and $B_{c}$ are
given in Table I.
$\displaystyle\footnotesize A_{c}(\rho)=\begin{bmatrix}0&0&1&0\\\ 0&0&0&1\\\
0&A_{32}&A_{33}&A_{34}\\\
0&A_{42}&A_{43}&A_{44}\end{bmatrix},~{}B_{c}(\rho)=\begin{bmatrix}0\\\ 0\\\
B_{31}\\\ B_{41}\end{bmatrix},$ (6)
TABLE I: Elements of the operators $A_{c}(\rho)$ and $B_{c}(\rho)$, with their respective scheduling parameters given within the parentheses $A_{ij}(\rho)$ and $B_{ij}(\rho)$, and their scheduling signals are given by $\rho=(\theta_{y},\dot{\theta}_{y})$ as shown in the table. $d(\theta_{y})$ | $b_{1}b_{3}-(-b_{2}+\ell r_{b}\cos({\theta_{y}}))^{2}$
---|---
$A_{32}\big{(}\theta_{y},\dot{\theta}_{y}\big{)}$ | $\frac{\ell g\operatorname{sinc}(\theta_{y})(b_{2}-\ell r_{b}\cos({\theta_{y}}))}{d(\theta_{y})}$
$A_{33}\big{(}\theta_{y},\dot{\theta}_{y}\big{)}$ | $-\frac{b_{3}b_{4}}{d(\theta_{y})}$
$A_{34}\big{(}\theta_{y},\dot{\theta}_{y}\big{)}$ | $\frac{b_{3}\ell r_{b}\sin({\theta_{y}})\dot{\theta}_{y}}{d(\theta_{y})}$
$A_{42}\big{(}\theta_{y},\dot{\theta}_{y}\big{)}$ | $\frac{b_{1}\ell g\operatorname{sinc}(\theta_{y})}{d(\theta_{y})}$
$A_{43}\big{(}\theta_{y},\dot{\theta}_{y}\big{)}$ | $-\frac{b_{4}(b_{2}-\ell r_{b}\cos{(\theta_{y})})}{d(\theta_{y})}$
$A_{44}\big{(}\theta_{y},\dot{\theta}_{y}\big{)}$ | $\frac{\ell r_{b}\sin{(\theta_{y})}\dot{\theta}_{y}(b_{2}-\ell r_{b}\cos{(\theta_{y})})}{d(\theta_{y})}$
$B_{31}\big{(}\theta_{y},\dot{\theta}_{y}\big{)}$ | $\frac{r_{b}(b_{3}-(b_{2}-\ell r_{b}\cos{(\theta_{y})})}{r_{w}d(\theta_{y})}$
$B_{41}\big{(}\theta_{y},\dot{\theta}_{y}\big{)}$ | $\frac{r_{b}((b_{2}-\ell r_{b}\cos{(\theta_{y})})-b_{1})}{r_{w}d(\theta_{y})}$
A discretization of the nonlinear continuous model is mandatory before setting
the framework that combines the LPV with Model Predictive Control (MPC).
Therefore, to discretize (5), many methods exist that differ in accuracy and
numerical properties (s.a., numerical stability). For instance, the Forward
Euler method is commonly used due to its ease of implementation, but it
usually endures low-level accuracy and, most importantly, inherits numerical
instability. Therefore, we evade such numerical pitfalls by utilizing the
Runge–Kutta $4^{\textrm{th}}$ order with a single-time step (RK4). Consider
the sampling time $t_{s}$ that spans the continuous time as
$t=t_{s}k,~{}k\in\mathbb{Z}_{+}\cup\\{0\\}$, and, it is denoted as
$x(t_{k})=x(kt_{s})=x_{k}$. We denote the angular velocity of the tilt angle
$\theta_{y}$ as $\dot{\theta}_{y}(t):=\omega_{y}(t)$ and the corresponding
discrete value is $\omega_{k}$. Similarly, we denote the angular velocity
$\dot{\phi}_{y}(t):=\varphi_{y}(t)$, thus, the complete discrete state vector
remains
$x_{k}=\left[\begin{array}[]{cccc}\phi_{k}&\theta_{k}&\varphi_{k}&\omega_{k}\end{array}\right]^{\top}$.
By denoting the continuous in-time LPV operator $f_{\text{LPV}}$ (i.e.,
$f_{\text{LPV}}$ is equivalent with the nonlinear operator $f$) of the ballbot
system, we can rewrite (5) as
$\dot{x}(t)=f_{\text{LPV}}(x(t),u(t)):=A_{c}(\sigma(x(t)))x(t)+B_{c}(\sigma(x(t)))u(t)$
with the input $u(t)=\tau_{y}(t)$. The Runge-Kutta discretization method of
order $4^{\textrm{th}}$ (RK4) with a single time step $t_{s}$ when zero-order
hold (ZOH) is considered remains:
$\text{RK4}~{}\left\\{\begin{aligned}
\kappa_{1}&=f_{\text{LPV}}\left(x_{k},u_{k}\right)\\\
\kappa_{2}&=f_{\text{LPV}}\left(x_{k}+(t_{s}/2)\kappa_{1},u_{k}\right)\\\
\kappa_{3}&=f_{\text{LPV}}\left(x_{k}+(t_{s}/2)\kappa_{2},u_{k}\right)\\\
\kappa_{4}&=f_{\text{LPV}}\left(x_{k}+t_{s}\kappa_{3},u_{k}\right)\\\
x_{k+1}&=x_{k}+(t_{s}/6)(\kappa_{1}+2\kappa_{2}+2\kappa_{3}+\kappa_{4}).\\\
\end{aligned}\right.$ (7)
Thus, we introduce the following remark to proceed with linear operations with
the LPV formulation, which will be essential for deriving a predictor that
retains the characteristics of the underlying optimization problem.
###### Remark 1 (The operator $f_{\text{LPV}}$ with a given scheduling
variable $\rho$ is linear)
When the scheduling parameter $\rho$ is given, the LPV operator remains linear
w.r.t. the state $x$ and input $u$. In such cases, the LPV continuous operator
will be denoted as:
$f_{\text{LPV}}^{(\rho)}(\rho,x(t),u(t)):=A_{c}(\rho)x(t)+B_{c}(\rho)u(t).$
(8)
Consequently, when the scheduling parameter can be considered known, we will
denote the LPV operator with a superscript $\rho$ as
$f_{\textrm{LPV}}^{(\rho)}$ that will indicate that $\rho$ is independent of
the state/input allowing linear actions on both. In particular, Remark 1
allows the LPV predictor to be inserted in a linear MPC framework and retain
all the characteristics of the optimization problem that stay invariant under
linear operations. For instance, the convexity of the optimization problem and
the linear constraints can be retained. At the same time, efficient algorithms
of QP can solve the underlying control problem efficiently.
### II-C Stabilization and closed-loop parameters identification
To identify the parameters of the ballbot as a real physical system, it was
stabilized in [1] through a proportional integral derivative (PID) controller.
During stabilization in [1], a multi-harmonic excitation signal provided
enough measurements to identify a linear model that is defined with the
$p$-parameters provided in Table II and in the study [1]. The identified
linear model could stabilize the ballbot with an optimal feedback gain $K$
utilizing the solution of the linear quadratic regulator (LQR) approach.
TABLE II: Physical parameters of the ballbot model (2), with $p$ being the linearized model parameters from [1]. $p_{1}$ | $p_{2}$ | $p_{3}$ | $p_{4}$ | $p_{5}$
---|---|---|---|---
-342.6038 | -52.8301 | -1425.9 | -36.0734 | -9.1477
$p_{6}$ | $\ell[m]$ | $r_{b}[m]$ | $r_{w}[m]$ | $g[m/s^{2}]$
251.8476 | 0.2978 | 0.12 | 0.05 | 9.81
In this study, we will use the identified linear model from [1] for
discovering the $b$ parameters that define the refined values in (1) after
solving (offline) a nonlinear optimization problem as will be explained in
Sec. III-A. Consequently, the LPV embedding of the identified nonlinear model
will allow the LPV integration within the proposed MPC scheme as shown in Sec.
III-B that will handle the stabilization and reference tracking of the ballbot
simultaneously without cascading complex control designs based on the
linearized model.
### II-D Model Predictive Control with LPV Embeddings
Model predictive control with linear parameter-varying embeddings implies that
the MPC optimally controls an underlying time-varying model based on the
scheduling parameters shown in the LPV formulation (5). The standard MPC form
begins with the cost function defined as $J_{k}(u)$ as seen in (9), where
$x_{i|k}$ denotes the discretized state vector at time $k$ and prediction $i$.
The $i$ varies in length $i=0,\ldots,N-1$ with $N$ the prediction horizon
length. The reference state vector is given as $x^{\mathrm{ref}}_{i|k}$.
$\displaystyle J_{k}(u)$ $\displaystyle=\sum_{i=0}^{N-1}(\lVert
x_{i|k}-x^{\mathrm{ref}}_{i|k}\rVert^{2}_{Q}+\lVert u_{i|k}\rVert^{2}_{R})+$
(9) $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\underbrace{(\lVert
x_{N|k}-x^{\mathrm{ref}}_{N|k}\rVert^{2}_{P}}_{\text{terminal cost}},$
where $Q\succeq 0,~{}R\succ 0,~{}P\succ 0$ are weighting symmetric matrices.
Theoretically, to provide closed-loop stabilization of the LPV model, for some
feedback gain $K(\rho)$ the $A(\rho)-B(\rho)K(\rho)$ should be Hurwitz for all
possible parameter variation of $\rho$. In this study, with the suitable
tuning of the LQR on the linearized model, we could provide a non-parametric
state feedback gain $K$ that satisfies the Hurwitz condition for the LPV over
a dense grid that covers the scheduling space $\mathcal{P}$. In addition, the
algebraic Riccati solution $P$ will be used to penalize the terminal cost of
the MPC.
In (9) and on the one hand, the explicit decision variables are the input
entries $u_{i|k},~{}i=0,\ldots,N-1$. Thus, the optimizer should provide a
solution in $u$ that minimizes the energy function (9). On the other hand, the
$x_{i|k},~{}i=0,\ldots,N$ consists of the set with the implicit decision
variables inserted through a predictor. Suppose the prediction $x_{i|k}$
depends nonlinearly on previous states. In that case, the energy function will
not remain quadratic or convex, inevitably leading to nonconvex optimization
formulation where the cost of solving such problems inherits high complexity.
To evade such a computational burden, by utilizing the LPV formulation
together with the property in Remark 1, when the scheduling variable is given,
the implicit decision variables $x_{i|k}$ are introduced linearly in the
energy function and retain the quadratic manifold where the minimization is
quite efficient through the quadratic program (QP) that will introduce next.
Importantly, good scheduling estimation will make the adaptive nature of the
LPV over a potentially nonlinear manifold quite accurate. Good predictions for
the scheduling variable lie at the heart of the LPV method, and methods that
introduce ways, e.g., scheduling tube control [10], can provide further
guarantees, such as recursive feasibility and stability.
Back to our aim, we want to minimize the quadratic cost function $J$ in (9)
under the linear equality constraints that satisfy the LPV (given scheduling)
predictor model together with the following linear inequalities given in
(LABEL:eq:constraints).
$\displaystyle\begin{aligned}
&\mathcal{X}=\\{x_{k}\in\mathbb{R}^{n_{{\mathrm{x}}}}~{}|~{}G^{x}x_{k}\leq
h^{x}\\},\\\ &\mathcal{U}=\\{u_{k}\in\mathbb{R}^{n_{u}}~{}|~{}G^{u}u_{k}\leq
h^{u}\\},\end{aligned}$ (10)
where, the state/input inequality constraints emerge as
$\displaystyle G^{x}$
$\displaystyle=\left[\begin{array}[]{c}+I_{n_{\mathrm{x}}}\\\
-I_{n_{\mathrm{x}}}\end{array}\right]\in\mathbb{R}^{(2n_{\mathrm{x}})\times
n_{\mathrm{x}}},~{}h^{x}=\left[\begin{array}[]{c}+x_{\text{max}}\\\
-x_{\text{min}}\end{array}\right]\in\mathbb{R}^{2n_{\mathrm{x}}},$
$\displaystyle G^{u}$ $\displaystyle=\left[\begin{array}[]{c}+I_{n_{u}}\\\
-I_{n_{u}}\end{array}\right]\in\mathbb{R}^{(2n_{u})\times
n_{u}},~{}h^{u}=\left[\begin{array}[]{c}+u_{\text{max}}\\\
-u_{\text{min}}\end{array}\right]\in\mathbb{R}^{2n_{u}}.$
The superiority of the MPC control strategy comes exactly at the point where
constraints can be introduced. MPC casts the control problem into optimization
with the major feature, among other methods, that can handle constraints. MPC
can be extended easily to include trajectory planning of the ballbot when it
moves in a structured environment with obstacles. Tangent planes can play a
significant role in keeping the optimization problem as QP, like the work in
[4].
The QP optimization problem is formulated together with the LPV predictor at
the ”present” time $k$ as the function
$\mathrm{QP}(\rho_{i|k},x_{k},x^{\mathrm{ref}}_{i+k})$ with the input
arguments; 1) the given (from previous time) scheduling signal $\rho_{i|k}$;
2) the initial conditions $x_{k}$; 3) the reference $x_{i+k}^{\textrm{ref}}$.
The Quadratic Program’s (QP) functionality is defined next in Algorithm 1.
Algorithm 1 The quadratic program (QP) based-LPVRK4
0: The time step $k\in\mathbb{Z}_{+}$, the weighted costs $Q,~{}R,~{}P$, the
triplet $(\rho_{i|k},x_{k},x^{\mathrm{ref}}_{k+i})$, for $i=0,\ldots,N-1$, the
horizon length $N\in\mathbb{Z}_{+}$ and the sampling time
$t_{s}\in\mathbb{R}_{+}$.
0: Optimal numerically design $u_{i|k},~{}i=0,\ldots,N-1$.Minimization of the
quadratic cost
function:$\underset{u_{i|k}}{min}\sum_{i=0}^{N-1}\Big{(}\lVert\hat{x}_{i|k}-x^{\mathrm{ref}}_{k+i}\rVert^{2}_{Q}+\lVert
u_{i|k}\rVert^{2}_{R}\Big{)}+\underbrace{\lVert\hat{x}_{N}-x^{\mathrm{ref}}_{k+N}\rVert_{P}^{2}}_{\text{terminal
cost}}$Subject to:
1: $\hat{x}_{0|k}=x_{0|k}=x_{k}$,
2: for i=0,…,N-1 do
3:
$\kappa_{1}=f_{\textrm{LPV}}^{(\rho_{i|k})}\left(\rho_{i|k},\hat{x}_{i|k},u_{i|k}\right)$,
4:
$\kappa_{2}=f_{\textrm{LPV}}^{(\rho_{i|k})}\left(\rho_{i|k},\hat{x}_{i|k}+(t_{s}/2)\kappa_{1},u_{i|k}\right)$,
5:
$\kappa_{3}=f_{\textrm{LPV}}^{(\rho_{i|k})}\left(\rho_{i|k},\hat{x}_{i|k}+(t_{s}/2)\kappa_{2},u_{i|k}\right)$,
6:
$\kappa_{4}=f_{\textrm{LPV}}^{(\rho_{i|k})}\left(\rho_{i|k},\hat{x}_{i|k}+t_{s}\kappa_{3},u_{i|k}\right)$,
7: Update:
$\hat{x}_{i+1|k}=\hat{x}_{i|k}+(t_{s}/6)(\kappa_{1}+2\kappa_{2}+2\kappa_{3}+\kappa_{4})$,
8: end for
9: Satisfy: $\hat{x}_{i|k}\in\mathcal{X},\forall i=1,\dots,N$,
10: Satisfy: $u_{i|k}\in\mathcal{U},\forall i=0,1,\dots,N-1$.
Next, we discuss Algorithm (1) step by step when it is called at the
simulation time instance $k\in\mathbb{Z}_{+}$:
* •
At time $k$, the input to the QP consists of the matrices $Q,~{}R,~{}P$, the
initial condition $x_{k}$, the reference $x_{k+i}^{\textrm{ref}}$, and the
scheduling variable that is either available from previous time $k-1$ or
initialized from initial conditions $x_{k}$. In particular, when the
simulation starts at time $k=0$, and there is no available scheduling
prediction from the past, the scheduling variable is initialized as
$\rho_{i|0}=\sigma(x_{0}),~{}i=0,\ldots,N-1$.
* •
The optimization problem consists of the quadratic cost function (9) that is
updated with the implicit decision variables $\hat{x}_{i|k}$ as a function of
the explicit decision variables $u_{i|k}$ with the LPV predictor. This can be
done with linear operations since the scheduling variable
$\rho_{i|k},~{}i=0,\ldots,N-1$ is considered known, and the linear property
explained in Remark 1. The RK4 in lines (2-7) is the discretization scheme
that predicts the evolution.
* •
The QP can be solved together with the constraints in lines (9-10),
efficiently from [11] and provide the optimized input design at each time $k$
numerically as $u_{i|k},~{}i=0,\ldots,N-1$.
Algorithm 2 The Ballbot QP-based LPVMPC algorithm
0: Initial conditions $x_{0}$, with reference
$x^{\textrm{ref}}_{k},k\in\mathbb{Z}_{+}\cup\\{0\\}$
0: The control input $u_{k},~{}k=0,1,\ldots$, that drives the nonlinear system
to the reference by satisfying constraints.Algorithm steps
1: Initialize the scheduling parameter for $k=0$ as next:
$\rho_{i|0}=(\theta_{0},\omega_{0}),~{}i=0,\ldots,N$
2: while $k=0,1,\ldots$ do
3: Solve the QP in Algorithm 1
$u_{i|k}\leftarrow\texttt{QP}(\rho_{i|k},x_{k},x_{i+k}^{\text{ref}}),~{}i=0,\ldots,N-1\vspace{-3mm}$
4: Update the scheduling parameters from the designed input
$u_{i|k},~{}i=0,\ldots,N-1$ with (7) as
$x_{i+1|k}=\left[\begin{array}[]{c}\phi_{i+1|k}\\\ {\theta_{i+1|k}}\\\
\varphi_{i+1|k}\\\
{\omega_{i+1|k}}\end{array}\right]\underset{\text{RK4}}{\overset{\eqref{eq:dlpv}}{=}}f_{\textrm{LPV}}\left(\left[\begin{array}[]{c}\phi_{i|k}\\\
{\theta_{i|k}}\\\ \varphi_{i|k}\\\
{\omega_{i|k}}\end{array}\right],u_{i|k}\right)$
5: Update $\rho_{i|k}=\left(\theta_{i|k},\omega_{i|k}\right),~{}i=0,\ldots,N$
6: Apply $u_{k}=u_{0|k}$ to the continuous system (5)
7: $k\leftarrow k+1$
8: Measure $x_{k}$
9: Update $\rho_{i|k}\leftarrow\rho_{i+1|k-1},~{}i=0,\ldots,N-1$
10: end while
As we have introduced in detail the QP minimization problem that is tailored
with the LPV formulation in Algorithm 1, we are ready to proceed by stating
the method in Algorithm 2 that is connected with our application on
stabilizing and tracking a given reference for the ballbot. Next, every step
in the resulting Algorithm 2 is explained.
* •
The input to Algorithm 2 consists of the initial conditions $x_{0}$ and the
full state reference $x^{\textrm{ref}}$. For the ballbot system, the state
reference contains the angular displacement of the ball $\phi_{k}$. In
particular, $\theta_{k}$ is the tilt angle, and its reference target is zero,
thus enforcing stabilization in that operating (unstable equilibrium) point.
Finally, for the rest of the states in $x^{\text{ref}}$ that concern the
angular velocities, again, we enforce them to zero as the aim is to reach the
reference $\phi^{\textrm{ref}}$ and then the ballbot to remain there with
$\theta=\dot{\theta}=\dot{\phi}=0$ (steady state) and $\phi$ at the target
point.
* •
In Step 1, initialization at any instant $k$ of the scheduling parameters can
be done either from the previous time as
$\rho_{i|k}=\sigma(x_{i+1|k-1}),~{}i=0,\ldots,N-1$ or from the initial
conditions as explained in Algorithm 1.
* •
In Steps 2-3, we can call the QP Algorithm 1 to provide the optimally designed
input $u_{i|k},~{}i=0,\ldots,N-1$. Note that by having $N$ inputs, we can
compute $N+1$ state estimation through the LPV predictor.
* •
In Step 4, given $x_{k}$ and $u_{i|k}$, we can compute the true evolution
response from the nonlinear system after using the exact/embedded nonlinear
discrete predictor (7). The scheduling parameters to be used in Step 5 can be
revealed in Step 4. The hat notation is omitted as these computations result
in the system’s true response, assuming that no other disturbance or
measurement noise will affect the system.
* •
In Step 6, the one-step implementation of the input to the nonlinear system
(5) is done by simulating with ode45 (Runge Kutta 45 with adaptive time step)
where between two consequent time instances that differ only one sampling time
$t_{s}$, zero-order hold (ZOH) (constant input of the sampling time) is
considered. This high-fidelity simulation ensures good agreement with the true
response of the physical plant.
* •
In Step 7, we slide the prediction horizon by one (sliding the prediction
window with length $N$ by one sampling time $t_{s}$)
* •
In Step 8, we measure the new initial conditions $x_{k}$.
* •
Finally, in Step 9, we update the new scheduling variable at time $k$ from the
previous simulation time $k-1$, and the while loop in Steps 2-10 runs to
provide the next control inputs $u_{k}$ for arbitrary $k$.
## III Results and Discussion
### III-A Refinement of the ballbot physical parameters
A linearized continuous model at $x_{e}=[0,0,0,0]^{T}$ can be derived through
the LPV evaluated at $x_{e}$ as in (11).
$\displaystyle A_{c}(0,0)$ $\displaystyle=\begin{bmatrix}0&0&1&0\\\ 0&0&0&1\\\
0&\underbrace{A_{32}(0,0)}_{p_{1}}&\underbrace{A_{33}(0,0)}_{p_{2}}&\underbrace{A_{34}(0,0)}_{0}\\\
0&\underbrace{A_{42}(0,0)}_{p_{4}}&\underbrace{A_{43}(0,0)}_{p_{5}}&\underbrace{A_{44}(0,0)}_{0}\end{bmatrix},~{}$
(11) $\displaystyle B_{c}(0,0)$ $\displaystyle=\begin{bmatrix}0\\\ 0\\\
\underbrace{B_{3}(0,0)}_{p_{3}}\\\
\underbrace{B_{4}(0,0)}_{p_{6}}\end{bmatrix},~{}d(0)=b_{1}b_{3}-(-b_{2}+\ell
r_{b})^{2}.$
By introducing the functions
$h_{i}(b):\mathbb{R}^{4}\rightarrow\mathbb{R},~{}i=1,~{}\ldots,6$ with suport
the parameterized vector $b=(b_{1},b_{2},b_{3},b_{4})$, we can define:
$\displaystyle A_{32}(0,0)$
$\displaystyle:=h_{1}(b)=\frac{gl\left(b_{2}-l\mathrm{r_{b}}\right)}{d(0)},$
(12) $\displaystyle A_{33}(0,0)$
$\displaystyle:=h_{2}(b)=-\frac{b_{3}\,b_{4}}{d(0)},$ $\displaystyle
A_{42}(0,0)$ $\displaystyle:=h_{4}(b)=\frac{b_{1}\,g\,l}{d(0)},$
$\displaystyle A_{43}(0,0)$
$\displaystyle:=h_{5}(b)=-\frac{b_{4}\,\left(b_{2}-l\,\mathrm{r_{b}}\right)}{d(0)},$
$\displaystyle A_{34}(0,0)$ $\displaystyle:=0,~{}A_{44}(0,0):=0,$
$\displaystyle B_{3}(0,0)$
$\displaystyle:=h_{3}(b)=\frac{b_{3}\,\mathrm{r_{b}}-\mathrm{r_{b}}\,\left(b_{2}-l\,\mathrm{r_{b}}\right)}{\mathrm{r_{w}}\,d(0)},$
$\displaystyle B_{4}(0,0)$
$\displaystyle:=h_{6}(b)=\frac{\mathrm{r_{b}}\,\left(b_{2}-l\,\mathrm{r_{b}}\right)-b_{1}\mathrm{r_{b}}}{\mathrm{r_{w}}\,d(0)}.$
To infer the nonlinear $b$-parameters, given the $p$-parameters values, we
must solve the nonlinear system of equations described in the set of equations
(12) for the unknown vector $b=(b_{1},b_{2},b_{3},b_{4})$ and with the
functionality $h_{i}(b):\mathbb{R}^{4}\rightarrow\mathbb{R},~{}i=1,\ldots,6$.
The problem remains to find $b^{*}$ that satisfies $p_{i}-h_{i}(b^{*})\approx
0$ for all $i=1,\ldots,6$. Thus, we define the operator
$F(b):\mathbb{R}^{4}\rightarrow\mathbb{R}^{6}$ along with the Jacobian next:
$\displaystyle F(b):=\left[\begin{array}[]{cc}p_{1}-h_{1}(b)\\\\[2.84526pt]
p_{2}-h_{2}(b)\\\\[2.84526pt] \vdots\\\\[2.84526pt] p_{6}-h_{6}(b)\\\
\end{array}\right],~{}\nabla F(b)=\left[\begin{array}[]{ccc}\frac{\partial
F_{1}}{\partial b_{1}}&\cdots&\frac{\partial F_{1}}{\partial
b_{4}}\\\\[2.84526pt] \frac{\partial F_{2}}{\partial
b_{1}}&\cdots&\frac{\partial F_{2}}{\partial b_{4}}\\\\[2.84526pt]
\vdots\\\\[2.84526pt] \frac{\partial F_{6}}{\partial
b_{1}}&\cdots&\frac{\partial F_{6}}{\partial b_{4}}\\\ \end{array}\right].$
The Newton scheme for updating the parameters $b$ with initialization $b_{0}$
and for $k\in\mathbb{Z}_{+}\cup\\{0\\}$ is
$b_{k+1}=b_{k}-\left[\nabla F(b_{k})\right]^{-1}F(b_{k}),~{}k=0,\ldots$ (13)
Upon convergence of (13), the optimal solution vector $b^{*}$ can be obtained
to satisfy $F(b^{*})\approx 0$ (with the smallest possible residual). In such
problems, where the nonlinear parameterized manifold is nonconvex, the
iterative scheme in (13) remains divergent when random initialization is
considered, or a solution can be algebraically correct without explaining the
physical law. To tackle such a problem, initialization of the $b$-parameters
should be done by respecting the engineering regularity of the problem. Thus,
some parameters $b_{i}$ can be initialized with physical meaning, e.g.,
$b_{1},~{}b_{2},~{}b_{3}$ being positive to explain the high-fidelity
knowledge of the physical plant [1] where $b_{4}$ as concerns friction can
remain arbitrary but also with a reasonable small value.
In Fig. 2, the iterative scheme (13) converged after five steps and gave the
optimal solution with a residual error that stagnated to the value $0.4330$.
This error cannot be improved further as the analysis relies on the identified
parameters $p$ from the study [1], where the dynamics are corrupted with noise
and/or slight nonlinear behavior from the physical plant.
Continuously, in Fig. 3, the simulation indicates the expected performance
after substituting the obtained parameters $b$ in Table III and applying a
multi-harmonic input to the stabilized plant. In particular, in Fig. 3, the
nonlinear model compared to the linearized model stays close to the dynamical
evolution for all states when the dynamics are close to the linearization
operation point $x_{e}=0$ as expected. The discrepancy increases for larger
deviations in $\phi$ from the origin where the linearized cannot be trusted.
Finally, in Fig. 3, the comparison between the LPV and the nonlinear model
certifies that the LPV is an equivalent embedding of the nonlinear system
where differences are buried in machine precision error. Finally, the
discrete-time implementation with the RK4 in (7) with a ZOH also certifies a
good accuracy compared to the ode45.
Figure 2: Convergence of the Newton scheme with residual error $\|F(b^{*})\|=0.4330$. The optimal parameter vector $b^{*}$ is shown in Tab. III. TABLE III: The refined $b$ parameters of the nonlinear model. $b^{*}$ | $b_{1}$ | $b_{2}$ | $b_{3}$ | $b_{4}$
---|---|---|---|---
Optimal | 0.002483 | 0.059325 | 0.143093 | -0.07436
Figure 3: Models simulations; the linearized model given in dashed blue line;
non-linear model simulation given in black; LPV in dashed red lines; all done
using ode45 on MATLAB. Finally, with green is the RK4 method utilizing (7).
The input $\tau_{y}$ is a multiharmonic signal.
### III-B LPVMPC for Reference Tracking
Reference tracking for a ballbot refers to the stabilized motion by the LPVMPC
method that can maneuver over any given state reference $x^{\mathrm{ref}}$. We
will consider three study scenarios. The 1st scenario will be a 1D reference
tracking of a nonsmooth discontinuous trajectory that will ”shock” the
optimization problem due to the sudden changes. The 2nd scenario will be a 2D
smooth reference as Lissajous curves, assuming the ballbot can be driven
independently in the $xz$ and $yz$ planes. Finally, a 3rd scenario will
illustrate the reference tracking of a single set point with a scheme of the
LPVMPC that can guarantee the stability of the closed-loop system and the
recursive feasibility of its optimization problem. These properties are
explained below. Tuning of the prediction horizon depends on the physical
system’s operational bandwidth and can vary significantly across applications.
The ballbot is a robotic system that operates around $1$(Hz); thus, a horizon
of about $1$ (s) is reasonable.
#### III-B1 1st Scenario: Nonsmooth 1D reference
In Fig. 4, we consider the following 2-set point nonsmooth reference. The
ballbot should start from the origin and at $t=1$ s should roll for one circle
$2\pi$ rad, then stay there for $2$ s and for the remaining $1$ s to return to
the origin; thus, it will roll a total distance of approximately $(2\cdot
2\pi\cdot r_{b}\approx 1.5~{}\text{m})$ within $4$ s.
At the beginning and near the origin, positive virtual torque applies on the
ballbot to decrease the tilt angle $\theta$ that instantly will move the
ballbot to the negative $\phi$, where immediately switching direction (normal
behavior of such non-minimal phase system, similar to a bike where to turn,
first a small maneuver to the opposite direction is needed). Continuously the
ballbot increases speed in the $\phi$ positive direction while approaching the
target $2\pi$ with satisfying accuracy and faster than the linear one after
$\approx 1.3$ s, then stays there and at around $2^{+}$ s where the prediction
horizon of time length $N\cdot t_{s}=20\cdot 0.05=1$ s receives the
information of the sudden change in reference from $2\pi$ to $0$ rad, the
ballbot slightly overshoots in $\phi$ so to change the tilt angle again and
starts to accelerate in the opposite direction. After less than a second, the
ballbot has reached its origin without overshooting, which outlines the good
performance seen mainly in nonlinear MPC frameworks.
In Fig. 4, we compare the MPC performances between the LPV and the linearized
model. As expected, the LPVMPC is generally faster at reaching the reference
with almost similar computational complexity as the linear MPC and the
potential to handle better strong maneuvering as the adaptive scheduling
variable $\rho_{i|k}$ provides linearizations over a trajectory of the
nonlinear ballbot model instead of a single fixed point for the linearized
model. The quadratic costs for the 1st scenario are defined as
$Q=\texttt{diag}([200,1,0.1,0.1])$, $R=1000$, and the terminal cost $P$ is the
Riccati solution from the LQR on the linearized model from [1] with the same
tuning of $Q$ and $R$ that will penalize the ”tail” of the horizon. The
sampling time is $t_{s}=0.05$ (s), with horizon length $N=20$, and the MPC
solution is obtained under the input/state hard constraints introduced in
Table IV and highlighted with the gray background in Fig. 4.
TABLE IV: MPC Constraints Variable | Lower bound | Upper bound | Units
---|---|---|---
$\phi_{k}$ | $-\infty$ | $+\infty$ | [rad]
$\theta_{k}$ | $-\pi/3$ | $+\pi/3$ | [rad]
$\dot{\phi}_{k}$ | $-10\pi$ | $+10\pi$ | $[\frac{\rm rad}{\rm s}]$
$\dot{\theta}_{k}$ | $-2\pi$ | $+2\pi$ | $[\frac{\rm rad}{\rm s}]$
$u$ | $-1.5$ | $+1.5$ | [Nm]
In the 1D reference tracking, as shown in Fig. 4, traveling in $y$ direction,
the ballbot system also balances itself simultaneously, as shown with $\theta$
being between ranges of $0\,rad$ to $0.5\,rad$ while traveling a distance of
$\phi$, which is referenced for $2\pi\,rad$, and angular velocities of both
are shown in $\dot{\theta}$, and $\dot{\phi}$ respectively. They show the
agile and robust movements of the ballbot. Also, in correspondence to the
reference and torque, $\tau_{y}$ is applied to the virtual wheel actuator. The
virtual torque is shown as discrete steps, i.e., ZOH, and the energy function
$J$ to be minimized to get the optimal control input $\tau_{y}$.
Figure 4: Ballbot reference tracking with the angular displacement $\phi$
traveling from the origin to $2\pi$ rad and back. A comparison between the
linear and LPV MPC frameworks is illustrated. $\theta$ measuring balancing of
ballbot, $\dot{\phi}$, $\dot{\theta}$ measuring the angular velocities,
$\tau_{y}$ is the virtual torque, $J$ is the energy cost function.
#### III-B2 2nd Scenario: Lissajous 2D smooth curves
Solving the LPVMPC reference tracking problem in parallel for the $yz$-plane
and $xz$-plane, we can drive the ballbot over the $xy$ plane with reference as
Lissajous curves in Fig. 5. The ballbot starts from
$(x,y)=(\phi_{x}r_{b},\phi_{y}r_{b})=(0,0)$, and following the path to global
coordinates as: $X^{\textrm{ref}}=\phi_{x}r_{b}=0.12\cdot 2\pi\sin(0.3t)$ and
$Y^{\textrm{ref}}=\phi_{y}r_{b}=0.12\cdot 2\pi\sin(0.4t)$, under the
constraints in Table IV. In that case, the reference is smoother without
sudden changes. Therefore, we can increase the weight for matching the $\phi$
without inheriting infeasibility problems; thus, the quadratic cost for the 2D
reference tracking in each direction can be set as
$Q=\texttt{diag}([1000,1,0.1,0.1])$. All the other quadratic weights and
parameter specifications are considered the same as in the 1st scenario. In
Fig. 5, the LPVMPC control strategy accurately drives the ballbot to the
reference within an area of around $2~{}m^{2}$ and for $70$ s.
Figure 5: Stabilization and Lissajous curve trajectory tracking of the ballbot
in 2D using the LPVMPC algorithm solved in both $xz$ and $yz$ planes.
#### III-B3 3rd Scenario: Single set point reference and LPVMPC with
guarantees
In our last scenario, regulated as in the 1st (e.g., quadratic costs, etc.),
we want to illustrate the crucial advantage of considering LPV models for MPC
tasks, as it can provide theoretical guarantees. On the one hand, when a
linearized model is considered instead of the underlying nonlinear, achieving
theoretical guarantees within MPC could be a straightforward task. Still, it
will remain quite conservative and thus not practically useful. On the other
hand, in the case of NMPC, guarantees can also be provided at the expense of
computational complexity. Consequently, an excellent alternative for providing
theoretical guarantees is the LPV formulation within MPC, which can be
considered an excellent trade-off between the complexity and conservativeness
of the previous two general approaches.
To prove theoretical guarantees, in Algorithm 1, we have to introduce
additionally the terminal equality constrain as
$\mathcal{X}_{t}=\\{\hat{x}_{N|k}\in\mathbb{R}^{n_{\textrm{x}}}~{}|~{}\hat{x}_{N|k}=x^{\text{ref}}_{N+k}\\}.$
(14)
Thus, the state constraints $\mathcal{X}$ in Algorithm 1 are enhanced with
$\hat{x}_{N|k}\in\mathcal{X}_{t}$. Note that in the presence of the terminal
constraint $\mathcal{X}_{t}$, the terminal cost remains zero; thus, in that
case, no use of the weight $P$ is needed. In Fig. 6, the dynamics are
illustrated when considering the terminal constraint (cyan line) in (14),
where we lose some performance compared with no guarantees (red line). Thus,
with active (14) and making use of the Theorem 1 in [12] (page: 6), all the
conditions are satisfied in our 3rd scenario, which leads to the following
conclusions; i) the LPVMPC optimization problem is recursively feasible; ii)
the closed-loop system satisfies the constraints $u_{i|k}\in\mathcal{U}$,
$\hat{x}_{i|k}\in\mathcal{X}$, and $\hat{x}_{N|k}\in\mathcal{X}_{t}$; iii)
$(u^{\text{ref}},x^{\text{ref}})=(0,\pi)$ is an exponentially stable (forced)
equilibrium of the closed-loop system. The proof of these properties follows
the same reasoning as the proof of Theorem 1 in [12].
Figure 6: Single-set point reference i.e., $\pi$, for the LPVMPC framework with and without theoretical guarantees. TABLE V: Comparison of computation times for a single QP using YALMIP with LPVMPC. The simulations are performed on a Dell Latitude 5590 laptop with an Intel(R) Core(TM) i7-8650U CPU and 16 GB of RAM. The scenarios are implemented in MATLAB, utilizing the YALMIP toolbox [11], with an optimality tolerance of $10^{-8}$. LPVMPC with $t_{s}=0.05$ s | Mean | Standard deviation
---|---|---
1st Scenario | 0.0086 s | 0.0104 s
2nd Scenario | 0.0087 s | 0.0132 s
3rd Scenario | 0.0058 s | 0.0025 s
## IV Conclusion
In this study, we were concerned with the reference tracking of a ballbot by
utilizing the LPV embedding within the MPC framework. An important step was
the refinement of the parameters of the ballbot system from measurements of a
past study. Thus, having the nonlinear model allowed the nonlinear embedding
of the LPV form. An improved discretization method based on the well-known
Runge-Kutta $4^{\textrm{th}}$ order introduced in the LPVMPC framework
potentially improves the accuracy issue that can arise in the forward Euler
method. The results indicated that applying the LPVMPC-based control method
for simultaneously balancing and reference tracking can achieve real-time
implementation as the average timing of solving the QP problem is usually
$(10)$-times less than the sampling time under consideration.
Utilizing such a control design that preserves the nonlinear behavior of the
model is advantageous and practical, unlike methods that work with linearized
models or try to solve very complex optimization problems tailored directly to
the nonlinear, which might be impossible online. The method is illustrated
with practical reference tracking scenarios where theoretical guarantees can
be provided in the single-set point case. Proving stability and recursive
feasibility in the case of multiple reference points is left to future
investigation. Finally, as the theoretical establishment of such a robotic
system has matured, implementing this method in the physical ballbot in a real
environment is left for our immediate future endeavors.
## Acknowledgement
We thank Mr. Ing.-Ievgen Zhavzharov, who constructed the ballbot in Fig. 1.
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|
# Realization of the all-optical phase modulator, filter, splitter, and self-
consistent logic gates based on assembled magneto-optical heterostructures
Jie Xu1,2, Yun You3, Fengwen Kang4, Sanshui Xiao5, Lujun Hong6, Yun Shen6,
Yamei Luo1,2, and Kosmas L. Tsakmakidis7 1School of Medical Information and
Engineering, Southwest Medical University, Luzhou 646000, China 2Medical
Engineering & Medical Informatics Integration and Transformational Medicine of
Luzhou Key Laboratory, Luzhou 646000, China 3School of Science, East China
Jiaotong University, Nanchang 330013, China 4College of Materials Science and
Engineering, Sichuan University, Chengdu 610065, China 5DTU Fotonik,
Department of Photonics Engineering, Technical University of Denmark, DK-2800
Kgs. Lyngby, Denmark 6Institute of Space Science and Technology, Nanchang
University, Nanchang 330031, China 7Section of Condensed Matter Physics,
Department of Physics, National and Kapodistrian University of Athens,
Panepistimioupolis, GR-157 84 Athens, Greece Corresponding authors: J. Xu
([email protected]), Y. Luo ([email protected]), and K. L. Tsakmakidis
<EMAIL_ADDRESS>
###### Abstract
All-optical computing has recently emerged as a vibrant research field in
response to the energy crisis and the growing demand for information
processing. However, the efficiency of subwavelength-scale all-optical devices
remains relatively low due to challenges such as back-scattering reflections
and strict surface roughness. Furthermore, achieving multifunctionality
through the reassembly of all-optical structures has thus far been rarely
accomplished. One promising approach to address these issues is the
utilization of one-way edge modes. In this study, we propose four types of
deep-subwavelength ($\sim 10^{-2}\lambda_{0}$, where $\lambda_{0}$ is the
wavelength in vacuum) all-optical functional devices: a phase modulator, a
filter, a splitter, and logic gates. These devices are based on robust one-way
modes but do not require an external magnetic field, which can allow for
flexible assembly. In particular, we investigate a phase modulation range
spanning from $-\pi$ to $\pi$, a perfect filter that divides the input port’s
one-way region into two output one-way regions with equal bandwidth, a multi-
frequency splitter with an equal splitting ratio (e.g., 50/50), and self-
consistent logic gates. We validate these theoretical findings through
comprehensive full-wave numerical simulations. Our findings may find
applications in minimal optical calculations and integrated optical circuits.
## I Introduction
Optical communication is known for its low loss [1], parallel calculation
capability [2], and high speed. As a result, all-optical devices are
attracting increasing attention. Regular all-optical devices include, but are
not limited to, optical filters, sensors, splitters, switches,
frequency/phase/amplitude modulators, couplers, and amplifiers. Most of these
devices rely on the electro-optic (EO) effect, thermo-optic (TO) effect, or
nonlinear optical (NLO) effect. Among these optical effects, the linear EO
effect has been extensively studied in fields such as optical sensing and
optical frequency combs [3] due to its ultrafast response [4]. The TO effect
is commonly employed in the design of optical switches [5], while the NLO
effect is primarily used for light control with light, enabling applications
such as laser generation and optical switching [6]. However, all-optical
devices based on EO, TO, or NLO suffer from high power consumption due to
reflection induced by reciprocity/symmetry and fabrication errors.
In recent years, there has been a significant focus on one-way edge modes due
to their potential for breaking the diffraction limit (similar to SPPs) [7, 8,
9], robust processing/computing capabilities [10], and low loss. One common
approach to achieve one-way edge modes is by utilizing photonic crystals
(PhCs). The key lies in engineering a special surface that is bounded by
materials with zero and nonzero Chern numbers. According to the bulk-edge
correspondence [11, 12], this ensures the existence of one-way edge mode(s) on
this surface. The robustness of one-way modes in PhCs to imperfections or
bends has been proposed theoretically and verified experimentally by several
groups [13, 14, 15]. However, PhCs still pose challenges in terms of their
relatively complex manufacturing process and the large device size compared to
the wavelength of the guiding wave ($\lambda_{0}$). This poses a challenge for
achieving subwavelength or even deep/ultra-subwavelength optical devices.
Recently, our group has proposed several subwavelength magneto-optical (MO)
one-way structures and discovered interesting phenomena/devices, such as
unidirectional/bidirectional slow light [16, 17], a perfect optical buffer
with zero phase shift [18], and all-optical logic gates (LGs) [19]. The most
significant advantage of the MO one-way structure is its extremely simple
manufacturing process, requiring only two joint layers and not having strict
requirements for surface roughness. Ultra-subwavelength all-optical devices
can be achieved based on the MO structure, regardless of the nonlocal effect,
which holds true in common scenarios, especially when the wavenumber is
relatively small (e.g., $k<100k_{0}$) [20, 21].
Here, we propose the design of assembled MO structures and several interesting
deep-subwavelength ($\sim 10^{-2}\lambda_{0}$) all-optical functional devices,
including phase modulators, filters, splitters, and self-consistent logic
gates (LGs). We study the propagation characteristics of surface
magnetoplasmons (SMPs) [22] in theory, and demonstrate the low-loss properties
of one-way SMPs through full-wave numerical simulations. Furthermore, we
verify the functionality of these all-optical devices. The key to realizing
the all-optical phase modulator, splitter, and self-consistent LGs lies in the
utilization of one-way index-near-zero (INZ) modes. Unlike other INZ modes
found in epsilon-near-zero (ENZ) [23, 24, 25], mu-near-zero (MNZ) [26], or
epsilon-and-mu-near-zero (EMNZ) [27] metamaterials, the INZ modes in our work
exhibit a single propagation direction and can be easily adjusted within a
continuous band.
## II Tunable INZ modes and all-optical phase modulator
Figure 1: Assembled all-optical communication systems and functional devices
utilizing YIG with remanence, along with PEC and PMC walls.
Yttrium iron garnet (YIG) stands out as one of the most captivating MO
materials, finding widespread applications in optical communication. These
applications encompass optical isolators [28, 29], band-pass filters [30],
high-efficiency optomagnonic micron-sized resonators [31], robust LGs [19],
and more. Benefiting from the unique magnetization characteristics of YIG,
residual magnetism persists even after removing the bias magnetic field. This
property holds great potential for the realization of reliable all-optical
communication, as depicted in Fig. 1. In this paper, we demonstrate that by
utilizing YIG with remanence and PMC walls, a wide range of all-optical
functional devices, including phase modulators, filters, splitters, and LGs,
can be achieved. Importantly, these devices can be finely controlled and
integrated into a system, as illustrated in Fig. 1.
To investigate the impact of PMC walls on SMPs, we first construct four MO
heterostructures surrounded by PEC and/or PMC walls [32]. These structures are
named as follows: the PEC-air-YIG-PEC (EDYE) structure, the PEC-air-YIG-PMC
(EDYM) structure, the PMC-air-YIG-PEC (MDYE) structure, and the PMC-air-YIG-
PMC (MDYM) structure, as illustrated in the inset pictures in Fig. 2. It
should be noted that the YIG material with remanence are used in this study,
which means that all the proposed functional devices in this work do not
require precise control of an external magnetic field. Our theoretical
analysis reveals the following forms for the dispersion relationship of these
structures:
$\alpha_{r}X_{1}+\frac{\mu_{r2}}{\mu_{r1}}k-\alpha_{d}\mu_{vr}X_{2}=0$ (1)
where $X_{1}$ and $X_{2}$ are parameters that directly depend on the lower and
upper boundary conditions of the structures, respectively. For the four
structures, their values are as follows:
$\mathrm{Lower\ boundary}\begin{aligned}
&\begin{cases}&X_{1}=\frac{1}{tanh(\alpha_{r}d_{2})}\qquad(\mathrm{PEC})\\\
&X_{1}=\frac{\alpha_{r}\cdot
tanh(\alpha_{r}d_{2})-\frac{\mu_{r2}}{\mu_{r1}}k}{\alpha_{r}-\frac{\mu_{r2}}{\mu_{r1}}k\cdot
tanh(\alpha_{r}d_{2})}\qquad(\mathrm{PMC})\end{cases}\\\ \end{aligned}$ (2)
$\mathrm{Upper\ boundary}\begin{aligned}
&\begin{cases}&X_{2}=-\frac{1}{tanh(\alpha_{d}d_{1})}\qquad(\mathrm{PEC})\\\
&X_{2}=-tanh(\alpha_{d}d_{1})\qquad(\mathrm{PMC})\end{cases}\\\ \end{aligned}$
(3)
where $\mu_{r1}=1+i\frac{\nu\omega_{r}}{\omega(1+\nu^{2})}$ and
$\mu_{r2}=-\frac{\omega_{r}}{\omega(1+\nu^{2})}$ are the nondiagonal elements
of the tensor relative permeability of the YIG layers [33, 7]. $\alpha_{r}$
and $\alpha_{d}$ represent the attenuation factors of the surface wave in the
YIG layer and air, respectively [34]. $d_{1}$ and $d_{2}$ are the thicknesses
of the air and YIG layers. $\omega_{r}$ is the characteristic circular
frequency, which is denoted as $\omega_{m}$ when the YIG reaches saturation
magnetization. In this paper, we assume $\omega_{m}=2\pi\times 5\times 10^{9}$
rad/s and $\omega_{r}=2\pi\times 3.587\times 10^{9}$ rad/s. It can be observed
that as $k\to\pm\infty$, $X_{1}\to 1$ and $X_{2}\to-1$. Additionally, Eq. (1)
tends to
$\displaystyle\begin{cases}&1+\frac{\mu_{r2}}{\mu_{r1}}+\mu_{vr}=0\iff\omega_{sp}^{+}=\omega_{r}\qquad(k\to+\infty)\\\
&-1+\frac{\mu_{r2}}{\mu_{r1}}-\mu_{vr}=0\iff\omega_{sp}^{-}=0.5\omega_{r}\qquad(k\to-\infty)\end{cases}$
(4)
According to Equation (4), we can expect, at most, one asymptotic frequency
for $k>0$, denoted as $\omega_{sp}^{+}$, and one asymptotic frequency for
$k<0$, denoted as $\omega_{sp}^{-}$. Figure 2 illustrates the four dispersion
curves of SMPs in these structures as $d_{1}=d_{2}=0.03\lambda_{m}$.
Interestingly, when the lower PEC wall in the original EDYE structure is
replaced with a PMC wall (see Fig. 2(b)), the one-way region abruptly changes
from the higher region, $\omega_{sp}^{-}<\omega<\omega_{sp}^{+}$ (see Fig.
2(a)), to the lower region, $0<\omega<\omega_{sp}^{-}$, and simultaneously the
propagation direction changes oppositely. Similarly, when the upper PEC wall
in the EDYE structure is replaced with a PMC wall (Fig. 2(c)), the asymptotic
frequencies and the one-way region remain the same, but the one-way SMPs
consistently have positive values of k, i.e., $k>0$. Furthermore, when both
the upper and lower PEC walls are replaced by PMC walls (Fig. 2(d)), the one-
way region is completely disrupted due to the emergence of modes based on
total internal reflection (TIR).
In the inset of Fig. 2, we present simulation results with a setting of
$\nu=10^{-3}\omega_{m}$, which shows good agreement with our theoretical
analysis considering lossless materials ($\nu=0$). The corresponding
transmission coefficients in these simulations are approximately $62.20\%$
(effective propagation length $L_{eff}\simeq 1.63\lambda_{0}$), $87.73\%$
($L_{eff}\simeq 2.17\lambda_{0}$), and $86.82\%$ ($L_{eff}\simeq
5.06\lambda_{0}$). Thus, the proposed deep-subwavelength structures exhibit
truly low loss.
Figure 2: Dispersion diagrams of ’XDYX’ structures where the first ’X’ can be
either (a, b) PEC or (c, d) PMC, while the second ’X’ can also be either (a,
c) PEC or (b, d) PMC. The red line and black dashed lines represent the
dispersion curves of the SMPs. The cyan dashed area is the bulk zone of the
bulk YIG. The other parameters are $d_{1}=d_{2}=0.03\lambda_{m}$,
$\omega_{r}=0.7174\omega_{m}$, and $\varepsilon_{d}=1$.
Furthermore, one of the most significant differences between Fig. 2(a) and
Figs. 1(b,c) is the presence of an INZ mode, marked by a green circle in Fig.
2(a). In our previous work, we proposed a method to control the thickness
parameters in order to adjust the INZ modes in magnetized-YIG systems.
Similarly, in the current study, we can employ the same approach to achieve
tunable INZ modes. The advantage of integrability in this system is undeniable
compared to previous systems.
In Fig. 3, we investigate the impact of the thicknesses of YIG ($d_{2}$) and
air ($d_{1}$) on the INZ modes in two cases. In the first case, we assume
$d_{1}=d_{2}$ ($<0.1\lambda_{m}$, subwavelength). As shown in Figs. 2(a) and
2(b), when we increase $d_{1}$, the INZ frequency gradually decreases and
remains close to $\omega=0.5\omega_{m}$. The red line in Fig. 3(a) represents
a fitting curve with the following form:
$\bar{\omega}=5.4848\cdot\bar{d_{1}}^{2}-0.09412\cdot\bar{d_{1}}+0.5085$ (5)
where $\bar{\omega}=\omega/\omega_{m}$ and $\bar{d_{1}}=d_{1}/\lambda_{m}$
($\lambda_{m}=2\pi c/\omega_{m}$). Equation (5) allows us to estimate the INZ
frequency ($f_{\mathrm{INZ}}$) in the $d_{1}=d_{2}$ case. For example, when
$d_{1}=0.045\lambda_{m}$, according to Eq. (5), the INZ frequency should be
approximately $0.4932\omega_{m}$. The inset of Fig. 3(a) further confirms that
when $d_{1}=d_{2}=0.045\lambda_{m}$, a point source with $f=0.4932f_{m}$ can
excite SMPs with a sufficiently large effective wavenumber ($k_{eff}$) and
exhibit no phase shift during propagation. In the second case, we assume
$d_{2}=2d_{1}$. Similar to the first case, we plot $f_{\mathrm{INZ}}$ as a
function of $d_{1}$ in Fig. 3(c). The corresponding fitting equation is as
follows:
$\bar{\omega}=102.45\cdot\bar{d_{1}}^{3}-25.783\cdot\bar{d_{1}}^{2}+0.3716\cdot\bar{d_{1}}+0.5828$
(6)
In comparison to the red line in Fig. 3(a), the red line in Fig. 3(c) exhibits
a sharper slope, resulting in greater changes in $f_{\mathrm{INZ}}$ as $d_{1}$
increases. It is important to emphasize that, in theory, it is possible to
engineer the fitting curve by designing a suitable ratio between $d_{1}$ and
$d_{2}$, such as establishing a nearly linear relationship between
$f_{\mathrm{INZ}}$ and $d_{1}$ within the range of
$0.05\lambda_{m}<d_{1}<0.1\lambda_{m}$. This linear relationship could have
potential applications in phase modulators, optical filters, and frequency
combs.
Figure 3: The operating frequencies of INZ modes as a function of $d_{1}$ in
the cases of (a) $d_{1}=d_{2}$ and (c) $d_{2}=2d_{1}$. The dispersion curves
of one-way SMPs as the thicknesses changing (b) in case (a), and (d) in case
(c). Figure 4: All-optical phase modulator relied on INZ modes. (a) The phase
angle of $E_{z}$ of the output INZ modes as a function of $L_{x}$. (b)-(e)
Electric filed distributions of the FEM simulations, and $L_{x}$ are equal to
(b) 40 mm, (c) 50 mm, (d) 60 mm, and (e) 70 mm, respectively. (f) The
schematic picture of the INZ modes-based phase modulator. Parameters are
$d_{1}=d_{2}=0.03\lambda_{m}$, $\nu=10^{-3}\omega_{m}$, and $f=0.5007f_{m}$.
It is well known that an INZ wave can travel with no phase shift, which
provides a potential approach to building a phase modulator. By designing a
joint structure that contains two one-way waveguides with the same one-way
region, and if the second waveguide supports the one-way INZ mode, it is
theoretically possible to control the output phase by manipulating the joint
position or the lengths of the waveguides. As analyzed in Fig. 2, the EDYE
structure possesses the same one-way region as the MDYE structure, and an INZ
mode is present in the EDYE structure. Therefore, as shown in Fig. 4(f), we
propose a novel phase modulator called the ’MDYE-EDYE’ structure. We further
investigate how the joint position $L_{x}$ affects the output phase of the
electric field ($E_{z}$). In Fig. 4(a), a clear linear relationship between
the phase (arg($E_{z}$)) and $L_{x}$ is observed, suggesting that such a
simple structure can achieve precise phase modulation.
From Fig. 4(a), it is evident that to achieve a phase shift range of
$[-\pi,\pi]$, the position shift $\Delta L_{x}$ should be approximately 20 mm,
whereas the theoretical value in the lossless condition is around 19.3125 mm
($k\simeq 3.1068k_{m}$). To clearly demonstrate the linear relationship
between the phase angle of $E_{z}$ and $L_{x}$, we perform four full-wave
simulations in a lossy condition with $\nu=10^{-3}\omega_{m}$, where $L_{x}$
is set to 40 mm (Fig. 4(b)), 50 mm (Fig. 4(c)), 60 mm (Fig. 4(d)), and 70 mm
(Fig. 4(e)). Consequently, the output $E_{z}$ in Figs. 3(b) and 3(d) are both
positive with nearly the same phase, as well as in Figs. 3(c) and 3(e).
Therefore, it is reasonable to believe that even in a real lossy condition,
our proposed ’MDYE-EDYE’ structure can function as a precise all-optical phase
modulator, offering advantages such as robustness against backscattering
reflection and a straightforward manufacturing process. Compared to previous
all-optical phase modulators based on Mach-Zehnder Interferometers [35, 36],
metasurfaces [37, 38], and/or metamaterials [39], our proposed structure
exhibits comparable performance and simplicity.
## III Perfect filter, all-optical splitter, and self-consistent all-optical
logic gates based on INZ modes
Furthermore, through careful assembly, the remanence-based MO system can
function as a perfect filter, effectively separating electromagnetic (EM)
waves within a specific frequency band into two separate channels with
identical bandwidth. Fig. 5(b) illustrates the Y-shaped filter, which consists
of three arms. The horizontal arm serves as the input port, while the other
two arms, namely the ’EDYM’ and ’EDYE’ structures (previously studied in the
previous subsection, Figs. 1(a) and 1(b)), function as the output ports. It is
important to note that the magnetization direction in the ’EDYM’ structure has
been reversed to support the forward propagation of EM waves. The input port
is referred to as the ’EYYE’ structure, with the interconnected YIG layers
having opposite magnetization directions. As shown in Fig. 5(a), the ’EYYE’
structure exhibits a significantly wide one-way region, $0<\omega<\omega_{r}$,
which corresponds to the combined one-way regions of the ’EDYE’ structure
($0.5\omega_{r}<\omega<\omega_{r}$) and the ’EDYM’ structure
($0<\omega<0.5\omega_{r}$).
Figure 5: (a) Dispersion diagram of the YIG-YIG structure. (b) The schematic
picture and (c) simulations of the perfect optical filter with frequencies are
$f=0.5f_{m}$ (marked by blue circle in (a)) and $f=0.2f_{m}$ (marked by black
circle in (a)), respectively.
Benefiting from the unique relationship between the one-way regions of the
input port (’A’) and the output ports (’B’ and ’C’), along with the robustness
of one-way EM waves, it can be theoretically guaranteed that the excited one-
way EM wave from the input port will propagate into either of the output
ports. To verify the performance of the filter, we conducted two finite
element method (FEM) simulations in this structure with frequencies set at
$f=0.5f_{m}$ (the second image in Fig. 5(c)) and $f=0.2f_{m}$ (the third image
in Fig. 5(c)). As a result, the excited wave from ’A’ propagated to the
junction point and decisively injected into either ’B’ or ’C’, resulting in an
extremely high contrast ratio between the output ports. Therefore, the
simulation results align with our theoretical analysis. The proposed perfect
filter has the potential to serve crucial roles such as an optical switch in
an integrated optical circuit. By directly changing the working frequency, the
state of the communication system can be switched between ’work’ and ’stop’.
Figure 6: (a) Dispersion diagrams of three arms of a novel tunable splitter
shown in (c). (b) Relationship between transmittances of two output arms and
the operating frequency within the one-way region (colored area in (a)). (c)
The schematic diagram and simulations for splitting ratio $\eta=0/100$ (second
picture) and $\eta=50/50$ (last picture). Other parameters are
$d_{1}=d_{2}=0.01\lambda_{m}$ and $\nu=10^{-3}\omega_{m}$.
Besides, it is worth noting that the ’EDYE’, ’MDYE’, and ’EYYE’ structures
have overlapping one-way regions, but the wavenumbers of the one-way SMPs
differ. As a result, one-way SMPs with different frequencies propagating in
these structures will have different effective refractive indexes ($n_{eff}$)
and may induce different splitting ratios ($\eta$). Fig. 6(c) illustrates the
designed Y-shaped splitter, consisting of the ’MDYE’ (’A’), ’EYYE’ (’B’), and
’EDYE’ (’C’) structures. Figure 6(a) presents the dispersion curves of SMPs in
the vicinity of the overlapping one-way region
($0.5\omega_{r}\leq\omega<\omega_{r}$). It is important to note that we set
$d_{1}=d_{2}=0.01\lambda_{m}$ to engineer an SMP with an infinite wavenumber
at $\omega=0.5\omega_{r}$ (the lower limit of the shaded region in Fig. 6(a))
in the ’EDYE’ structure, which implies $n_{eff}\to\infty$ and consequently
zero transmittance in the ’A-C’ channel should be observed under this
condition. We performed FEM simulations in the designed splitter, with the
frequency changing within the one-way region, and calculated the
transmittance. As depicted in Fig. 6(b), the red and black lines represent the
transmittance of the ’A-B’ and ’A-C’ channels, respectively. As expected, when
$\omega=0.5\omega_{r}=0.3587\omega_{m}$, nearly 100% of the energy traveled
through the ’A-B’ channel (as shown in the second picture in Fig. 6(c)). More
importantly, as the frequency gradually increases to $0.4447\omega_{m}$, the
splitting ratio can approach 50/50 (as shown in the last picture in Fig.
6(c)), which is a highly desirable value in the field of optical
communication. Another noteworthy case occurs when $\omega\simeq
0.45\omega_{m}$ (indicated by the right arrow in Fig. 6(b)). In this scenario,
the EM wave cannot pass through the ’A-C’ channel either. This behavior can be
explained by the theory of total internal reflection: 1) the effective indices
of ’A’, ’B’, and ’C’ are approximately 19.4082 ($k\simeq 10.6745k_{m}$),
29.4398 ($k\simeq 16.1919k_{m}$), and 6.8467 ($k\simeq 3.7657k_{m}$),
respectively; 2) the incident angle is $\pi/3$, which is clearly larger than
the ’critical angle’ (equal to $arcsin(6.8467/19.4082)$) of the ’A-C’ channel.
When the frequency is further increased to approximately $0.6456\omega_{m}$, a
50/50 splitting ratio will be achieved (indicated by the black left arrow in
Fig. 6(b)). Therefore, our proposed remanence-based all-optical splitter has
the capability to maintain a constant splitting ratio for different
frequencies. For example, it can achieve $\eta=50/50$ at
$\omega=0.4447\omega_{m}$ and/or $\omega=0.6456\omega_{m}$, which we believe
is extremely important for parallel optical computing.
Figure 7: (a) Schematic picture of all-optical LGs based on YIG with
remanence. (b) Logic operations and FEM simulations for three different
inputs, i.e. (’1’, ’0’), (’0’, ’1’), and (’1’, ’1’). (c) The truth table for
OR, AND, and NAND gates according to simulation results shown in (b).
In our previous work[19], we proposed a possible method to achieve all-optical
LGs with an extremely high contrast ratio. However, there was an unresolved
issue in the previously proposed LGs, which involved inconsistency between the
logic of the input and output ports. In this study, we propose another
Y-shaped system based on INZ modes, which can function as all-optical LGs such
as OR, AND, and NAND gates. Following a consistent positive logic convention,
the presence of EM energy is treated as logic ’1’ and is utilized in both the
input and output ports, while the phase angle of $E_{z}$ serves as the
reference quantity in the output port. Figure 7(a) illustrates the schematic
diagram of the designed system. One can observe that this structure is very
similar to the one shown in Fig. 6(c), except for the exchange of YIG layers
in arm ’B’. As discussed earlier, all three arms share the same one-way
region, $0.5\omega_{r}<\omega<\omega_{r}$. In contrast to arm ’B’ shown in
Fig. 6, the arm in Fig. 7 can sustain backward-propagating SMPs (as shown in
the inset of Fig. 7(a)). Consequently, two one-way channels are successfully
engineered based on the splitter model shown in Fig. 6. In contrast to the
working frequency used in our previous work, we choose the INZ frequency in
the output port as the working frequency in order to preserve the phase
information in the output section.
The proposed system can be theoretically treated as a combination of basic LGs
such as OR, AND, and NAND gates. For the OR gate, we utilize positive logic,
where any presence of EM energy is considered logic ’1’. Based on the robust
one-way channels, any input EM signal (logic ’1’) from either arm ’A’ or arm
’B’ can unidirectionally propagate through arm ’C’ (also logic ’1’). Hence,
the system naturally functions as an OR gate.
To implement an AND gate, we introduce a reference quantity, the phase angle
($\theta$) of $E_{z}$, to determine the state of the output port.
Specifically, we assume that the output EM signal with $\pi/2<\theta<\pi/2$
represents logic ’1’. As illustrated in Fig. 7(b), three input combinations,
[’1’, ’0’], [’0’, ’1’], and [’1’, ’1’], yield three corresponding outputs,
logic ’0’, logic ’0’, and logic ’1’. It should be noted that we set
$\theta\simeq 1.4$ rad for input-1 to engineer an appropriate output phase
angle. Therefore, in practical applications, the input $\theta$ needs to be
tailored for different inputs. In Fig. 7, the inputs $[\theta_{1},\theta_{2}]$
are [-1.9 rad, ], [, 1.4 rad], and [1.4 rad, 1.4 rad], respectively.
Figure 7(c) presents the truth table for the OR, AND, and NAND gates. It is
evident that the outputs are inverted for the AND and NAND gates. Therefore,
based on the analysis of the AND gate, the system can easily achieve NAND
operation by assuming that the output EM signal with $\pi/2<\theta<\pi/2$
represents logic ’0’. Furthermore, other basic LGs can be straightforwardly
extended. With improved integration and consistent logic, the proposed
Y-shaped LGs enhance the potential applications of one-way mode-based LGs in
the all-optical calculation of integrated optical circuits.
Figure 8: Realization of 3D all-optical phase modulator, perfect filter, and
self-consistent LGs.
The above-mentioned all-optical phase modulator, filter, and LGs, which are
based on YIG with remanence, can be straightforwardly extended from 2D models
to 3D structures[34]. As depicted in the inset of Fig. 8(a), two PEC walls are
introduced in the z direction to confine the guided wave within the device.
The simulation result shown in Fig. 8(a) exhibits similarities to the
corresponding 2D case (Fig. 4(e)). Figures 7(b)-(e) illustrate the simulation
outcomes for the 3D perfect filter, 3D splitter, and 3D LGs, respectively, and
all the results are consistent with their 2D counterparts. Consequently, our
proposed all-optical structures hold great potential for practical device
implementation. Due to the characteristic of one-way propagation, the
manufacturing process should be significantly simpler compared to other
competing technologies. More importantly, as depicted in Fig. 1, our proposed
all-optical phase modulator, filter, splitter, and LGs can be easily assembled
on a chip. Therefore, it holds significant promise for enabling flexible all-
optical computation and communication.
## IV Conclusion
In conclusion, we have proposed a series of all-optical devices based on YIG
with remanence. The all-optical phase modulator utilizes INZ modes and allows
for controllable phase modulation in a linear relationship with the length of
the boundary. The INZ frequency can be adjusted by manipulating the thickness
parameters, providing tunability. Moreover, we have designed a perfect filter,
which separates the input port’s one-way region into two parts and creates two
one-way channels with equal bandwidth. Additionally, we have introduced an
all-optical splitter that divides the input wave into two output ports with a
customizable and precise splitting ratio, such as 50/50. Furthermore, we have
proposed all-optical logic gates based on one-way SMPs, where consistent
positive logic and the phase angle of the output field are used to determine
the logic states. Basic logic gates, including OR, AND, and NAND, have been
achieved based on this principle. The feasibility and performance of the
proposed devices have been validated in both 2D and 3D simulations. Our
proposed all-optical devices, leveraging MO heterostructures, hold potential
for optical calculations, including parallel computing in integrated optical
circuits.
## Acknowledgement
This work was supported by the National Natural Science Foundation of Sichuan
Province (No. 2023NSFSC1309), and the open fund of Luzhou Key Laboratory of
Intelligent Control and Application of Electronic Devices (No. ZK202210),
Sichuan Science and Technology Program (No. 2022YFS0616), the Science and
Technology Strategic Cooperation Programs of Luzhou Municipal People’s
Government and Southwest Medical University (No. 2019LZXNYDJ18). J.X. and Y.L.
thanks for the support of the Innovation Laboratory of Advanced Medical
Material & Physical Diagnosis and Treatment Technology. K.L.T. was supported
by the General Secretariat for Research and Technology (GSRT) and the Hellenic
Foundation for Research and Innovation (HFRI) under Grant No. 4509.
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|
11institutetext: Amirarsalan Rajabi 22institutetext: Complex Adaptive Systems
Lab (CASL), University of Central Florida, Orlando, 32816
22email<EMAIL_ADDRESS>33institutetext: Seyyedmilad
Talebzadehhosseini 44institutetext: Complex Adaptive Systems Lab (CASL),
University of Central Florida, Orlando, 32816
44email<EMAIL_ADDRESS>55institutetext: Ivan Garibay
66institutetext: Complex Adaptive Systems Lab (CASL), University of Central
Florida, Orlando, 32816
66email<EMAIL_ADDRESS>
# Resistance of communities against disinformation
Amirarsalan Rajabi Seyyedmilad Talebzadehhosseini and Ivan Garibay
###### Abstract
The spread of disinformation is considered a big threat to societies and has
recently received unprecedented attention. In this paper we propose an agent-
based model to simulate dissemination of a conspiracy in a population. The
model is able to compare the resistance of different network structures
against the activity of conspirators. Results show that connectedness of
network structure and centrality of conspirators are of crucial importance in
preventing conspiracies from becoming widespread.
## 1 Introduction
We define conspiratorial thinking as a belief held by an individual or a group
of individuals, while there is enough evidence and information to undermine or
totally refute the belief. If some conspiracy theories get popular support, it
may cause serious concerns.This is particularly true of conspiracies over
scientific and medical issues where conspiracy theories can result in
rejection of the scientific method lewandowsky2013role .
Several underlying reasons have been proposed to explain the existence of
conspiracy theories. According to Barkun, conspiratorial thinking exhibits
three characteristics: Firstly, nothing happens by accident. Secondly, things
are not as they seems on surface. And thirdly, things are highly connected
barkun2013culture . All three characteristics mentioned by Barkun, refer to a
special cognitive function of the conspiracy theorist. Indeed, a great deal of
literature on conspiracy theory associates conspiratorial thinking with a
special and different heuristic of the conspiracy theorist. On the other hand,
Sunstein and Vermeule claim that many of those who hold conspiracy theories do
so not as a result of a mental illness of any kind, or of simple
irrationality, but as a result of crippled epistemology (knowing very few
things, which are indeed wrong) sunstein2009conspiracy . For most of what they
believe and know, human beings lack direct information; they must rely on what
others say and think. Hardin argues many people suffer from crippled
epistemology, meaning they only get their information from a few incorrect
sources breton2002political . Crippled epistemology usually takes place in
echo chambers. Echo chambers are communities in which individuals merely
communicate with each other and rarely seek information from entities outside
the community. The advent of social media platforms has resulted in the rise
of echo-chambers bakshy2015exposure . Echo chambers play an important role in
political and social polarization barbera2015tweeting . The negative effects
of polarization in social networks is studied in garibay2019polarization .
Bauman states that individuals who are embedded in isolated groups or small,
self-enclosed networks who are thus exposed only to skewed information, will
more often hold conspiracy theories that are justified, relative to their
limited informational environment bauman2013liquid . The study of the dynamics
by which the echo chambers form, can therefore shed light on the mechanisms by
which conspiratorial thinking forms and thrives in a community.
Traditional studies of conspiratorial thinking assume that conspiratorial
thinking extinguishes in large network structure grimes2016viability , and
that conspiratorial ideation is because of flawed reasoning and biased
heuristics barkun2013culture . This study challenges both of these claims.
Contrary to a large body of literature on conspiracy theory that studies the
cognitive function of isolated individuals [22], this paper takes into account
systemic belief dissemination as a result of interactions between individuals.
## 2 Opinion Dynamics
Opinion formation is a complex process which is formed by the interaction of
multiple underlying elements. People tend to form their opinion on a wide
variety of subjects through the process of learning. “Social learning” is a
term referring to the process of learning through the communication of
individuals with each other, their own experience and their observations of
others’ experiences, media sources, propaganda and indoctrination from
political leaders and the state acemoglu2011opinion . In this paper, we refer
to models of opinion dynamics as mathematical models that aim at capturing the
dynamics of social learning, opinion spreading, collective decision making,
and so on from a mathematical point of view.
Models of opinion dynamics can be divided into two categories : Bayesian
models of opinion dynamics and non-Bayesian models of opinion dynamics.
Bayesian models rely on Bayes rule bayes1991essay . These kinds of models
assume that an individual (agent) is Bayesian rational and update their belief
optimally with respect to Bayes rule, given an underlying model of the world.
One problem with Bayesian models of opinion dynamics is that these models make
a lot of assumptions. One demanding assumption of these models is that an
agent must have a reliable prior assumption about the world, an assumption
that might be unrealistic in many cases. Additionally, it is assumed that they
then go on and update their prior beliefs based on the new information that
they get from others. Bayesian models also put too much structure on updating
by ruling out “zero probability events” acemoglu2011opinion . The
aforementioned problems make Bayesian models unfavorable to be incorporated in
our study of dissemination of conspiracy theories.
Non-Bayesian approaches and models on the other hand try to avoid some of
these problems. Non-Bayesian approaches are believed to be more effective in
modeling belief manipulation and the spread of disinformation. The simplest of
these models start by specifying rules of thumb acemoglu2011opinion . Several
different non-Bayesian models exist. Classical models of interacting particle
systems which in inspired by statistical mechanics inspires many of these
models (see for example see for example clifford1973model ; galam2002minority
; latane1981psychology ; castellano2009nonlinear ; hegselmann2002opinion ).
One noteworthy non-Bayesian model is DeGroot (1974) degroot1974reaching . In
this model, a set of interacting agents start by an initial beliefs about an
underlying state held by each agent, and exchange information about their
beliefs with their neighbors and update their beliefs at discrete time
instances, with respect to a weight matrix that represents the social network
structure of interactions. This model captures the “imitation” aspect of non-
Bayesian models. While notably innovative, this model suffers from duplication
of information acemoglu2010spread , meaning that agents in this model might
interact endlessly with their neighbors that hold unchanging opinions in each
timestep.
## 3 Model of Conspirators
Our agent-based model is inspired by the variation of Acemoglu et. al. on
DeGroot model acemoglu2010spread . In our model, two types of agents exist.
Majority of agents are _susceptible_ agents and a minority of agents exist
that are called _conspirators_. Conspirators deliberately disseminate false
information to susceptibles.
Let’s consider a conceptual underlying state of the world and call it
$\Theta$. We assume that the true value of $\Theta$ is 1, and the discussion
between agents in the model is on the true value of $\Theta$. $x_{i}^{s}(k)$
and $x_{i}^{c}(k)$ represent the opinion of susceptible agent $i$ at time $k$,
and the opinion of conspirator agent $i$ at time $k$, respectively. At first,
each susceptible agent holds an initial belief about the underlying state. N
shows the total number of agents in the model. The initial belief of each
susceptible agent is a randomly generated float number between zero and two,
the initial belief of each conspirator agent is 0, and the average of initial
beliefs of all susceptible agents is very close to one:
$x_{i}^{s}(0)\in[0,2]\qquad
x_{i}^{c}(0)=0\qquad\frac{1}{N}\sum_{i}x_{i}^{s}(0)\approx 1$ (1)
Hence although at first each susceptible agent has their own initial belief on
the underlying state $\Theta$, there is a consensus on this underlying state
between the susceptible agents. On the other hand, the initial belief of
conspirator agents is 0 and remains 0 during the simulation, irrespective of
their interactions. These could be thought of as individuals, entities, media
or propaganda outlets that deliberately and constantly disseminate false
information throughout the population.
(a) Scale Free (Barabasi-Albert)
(b) Small World (Watts-Strogatz)
Figure 1: NetLogo environment. A) shows a network generated with Barabasi-
Albert algorithm. B) shows a network generated by Watts-Strogat algorithm
There are two main differences of between our proposed model and the work of
acemoglu2010spread : 1) Agents are not able to freely communicate directly
with any other agents. Rather, each agent is only able to communicate with the
agents that she is connected to by a link. This means that a network structure
regulates the communication in the population. 2) In the variation of
acemoglu2010spread on DeGroot model, each agent meets and communicates with
other agents at instances defined by a rate one Poisson process independent of
other agents. In our model agents are connected to each other by undirected
links, and together, they form a network. in each timestep of the simulation,
an agent will choose another agent to which it is connected and may or may not
communicate with them. In this model, the capacity of an agent for
communication in each timestep is the number of her links. For example an
agent with 5 links will communicate with 5 or less other agents in each
timestep.
Following the interaction of two agents i and j, there is a potential exchange
of information between them with probability $\textrm{p}_{interaction}$.
Agents update their beliefs according to one of the following possibilities:
If i and j are both susceptibles:
$\displaystyle\begin{cases}x_{i}(k+1)=x_{j}(k+1)=\frac{1}{2}[x_{i}(k)+x_{j}(k)]&\text{with
probability}\;\textrm{p}\\\
x_{i}(k+1)=x_{i}(k)\;\&\;x_{j}(k+1)=x_{j}(k)&\text{with
probability}\;1-\textrm{p}\end{cases}$ If i is susceptible and j is
conspirator:
$\displaystyle\begin{cases}x_{i}(k+1)=\frac{x_{i}(k)+0}{2}&\text{with
probability}\;\textrm{p}\\\ x_{i}(k+1)=x_{i}(k)&\text{with
probability}\;1-\textrm{p}\\\ x_{j}(k+1)=x_{j}(k)=0&\text{with probability
1}\end{cases}$ If i and j are both conspirators: no opinion sharing.
The underlying network structure of the model plays an important role in
determining the behavior of the model, and it is itself determined by the
algorithm by which the model forms the network. It determines which agents are
connected to each other, how easily the information would propagate throughout
the network, and so forth.
Watts and Strogatz proposed a model to capture the small world, high
clustering, and low average path properties of real complex networks
watts1998collective . On the other hand, Barabasi and Albert showed that
power-law degree distribution is the property of many real world networks and
proposed an algorithm that can capture this phenomenon barabasi2003scale .
Indeed, it is believed that majority of complex networks exhibit small world
and scale free properties wang2003complex . In order to capture all of these
essential properties, the model generates the network of agents using both
Watts-Strogatz and Barabasi-Albert algorithms.
## 4 Results
We developed our agent-based model in NetLogo sunstein2009conspiracy . Figure
1 shows two instances of the model with watts-strogatz and Barabasi-Albert
model. For our purposes we defined a variable named collective-thought. This
variable is simply the average of the belief of all susceptible agents and
represents the collective belief of the population about the underlying state
$\Theta$. Collective-thought starts with a value close to one, and always
converges to zero(Figure 2).
A desired population is one that resists the activity of conspirators.
Numerous individuals and groups deliberately disseminate false information in
a society. As we know the collective-thought of the model always converges to
zero after some timesteps. We call this state the convergence. In order to
quantify the resistance of a network against the activity of conspirators, we
record the number of timesteps required for collective-thought of the network
to converge. This number can be thought of as symbolically representing how
long a society resist the effort of conspirators, before it is nearly deceived
into believing that the underlying state $\Theta$ is zero.
We argue that contrary to conventional belief that conspiratorial beliefs are
untenable with larger network structure (e.g. grimes2016viability ), a large
network cannot ensure eradication of conspiratorial beliefs. Indeed, the
important aspect of a network structure that can ensure resistance against
conspiracies is the _connectedness_ of the network;that is, how easily an
agent can send an information through the network to any other agent. A
network with high connectedness means that information propagates more easily
and eco-chambers are less likely to form. We needed to come up with a
benchmark that enables us to quantitatively compare different networks by our
objective of connectedness. For this purpose, the model records the mean path
length of the network in each instance of the simulation as follows
fronczak2004average :
mean path length = average shortest path between all distincrt pairs of nodes
in network (2)
The model was run 1000 times for each of the Watts-Strogatz and Barabasi-
Albert networks. In each run, the required timsteps for the collective-thought
to converge and the mean path length of the network were recorded. Figure 3
shows that there is a positive relationship between mean path length of the
network, and required timesteps for the collective-thought to reach
convergence. With Barabasi-Albert algorithm, the Pearson correlation
coefficient between the mean path length and the required timesteps was 0.18,
with a p-value less than 0.00001 which shows the correlation is statistically
significant. The corresponding coefficient for Watts-Strogatz networks was
0.39 with a p-value less than 0.00001, which shows this correlation is also
significant. These results prove our hypothesis that the connectedness of a
network improves its resistance against conspirators.
(a) Scale Free (Barabasi-Albert)
(b) Small World (Watts-Strogatz)
Figure 2: Figure show the required timesteps required for the network’s
collective-thought to converge. Small world network tends to be deceived more
quickly than a scale free network.
(a) Scale Free (Barabasi-Albert)
(b) Small World (Watts-Strogatz)
Figure 3: The relationship between mean path length and the required timesteps
for a population’s collective-thought to converge in A)scale free and B)small
world networks
(a) Scale Free (Barabasi-Albert)
(b) Small World (Watts-Strogatz)
Figure 4: The relationship between sum of the eigenvector of all 4
conspirators and the required timesteps for a population’s collective-thought
to converge in A)scale free and B)small world networks
Besides the connectedness of a network, the structural position of
conspirators in a network determines the reach of their disinformation and
enhances their ability to disseminate their message through the whole
population more effectively. To show that our model captures this phenomenon,
we assigned to each agent in the network their eigenvector centrality.
Eigenvector centrality is a measure of centrality in a network
newman2008mathematics . The more central a node is, the closer it is to all
other nodes.
The model was run 1000 times for each of the Watts-Strogatz and Barabasi-
Albert networks. For each run, the required timsteps for the collective belief
to reach convergence and the sum of the eigenvector centrality of conspirator
agents were recorded. Figure 4 shows the negative relationship between
required timesteps for collective-thought to reach convergence and the sum of
the eigenvector centrality of conspirators. With Barabasi-Albert algorithm,
the Pearson correlation coefficient between the mean path length and the
required timesteps was -0.19, with a p-value less than 0.00001 which shows the
correlation is statistically significant. The corresponding coefficient for
Watts-Strogatz networks was -0.42 with a p-value less than 0.00001, which
shows this correlation is also statistically significant.
## 5 Discussion
The model shows promising results in that its vulnerability is correlated with
the connectedness of the network and the importance (eigenvector) of
conspirators. Considering the minimum required time, the resistance of a
population against conspiracies was imitated. The results show that a network
that is built by the Watts-Strogatz algorithm was slightly more vulnerable to
conspiracies than a network which is built by Barabasi-Albert algorithm (scale
free). The reason for this difference might be that a Watts-Strogatz network
has a high local clustering, while holding a short average path length like
random networks, therefore the propagation of (dis)information is easier and
faster in them. On the other hand, while noting that we used only 4
conspirators in our model, and the total number of agents were 100, the
probability that conspirators become a hub was low, and therefore their
ability to propagate their conspiracy was slightly smaller than in Watts-
Strogatz network. A probable misinterpretation here might be that a less
connected network is better against conspiracy theories. This is not correct
because in our model, only conspirators were acting to deceive the population
and were trying to propagate their disinformation in the whole network. In
fact, the results of this study can be interpreted in a content-agnostic
manner. In reality, both sides of the discussion try to influence the network.
Nevertheless, both networks show that the connectedness and the eigenvector
centrality of conspirators is a network is highly correlated with the
network’s vulnerability to conspiracies.
This was a first step to make a framework for computational study of
conspiracy propagation. Further research must be conducted using other network
formation algorithms. Other ranges of conspirators’ ratio and population sizes
should also be experimented. Another extension to this research might be to
test the effect of multi-dimensional opinion-space in agents. Finally, other
non-Bayesian models of opinion dynamics could be tested.
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|
# Sparse vs Contiguous Adversarial Pixel Perturbations in Multimodal Models:
An Empirical Analysis
Cristian-Alexandru Botocan<EMAIL_ADDRESS>EPFLLausanneSwitzerland armasuisse S+TThunSwitzerland , Raphael Meier
<EMAIL_ADDRESS>armasuisse S+TThunSwitzerland and Ljiljana
Dolamic<EMAIL_ADDRESS>armasuisse S+TThunSwitzerland
(2024)
###### Abstract.
Assessing the robustness of multimodal models against adversarial examples is
an important aspect for the safety of its users. We craft $L_{0}$-norm
perturbation attacks on the preprocessed input images. We launch them in a
black-box setup against four multimodal models and two unimodal DNNs,
considering both targeted and untargeted misclassification. Our attacks target
less than 0.04% of perturbed image area and integrate different spatial
positioning of perturbed pixels: sparse positioning and pixels arranged in
different contiguous shapes (row, column, diagonal, and patch). To the best of
our knowledge, we are the first to assess the robustness of three state-of-
the-art multimodal models (ALIGN, AltCLIP, GroupViT) against different sparse
and contiguous pixel distribution perturbations. The obtained results indicate
that unimodal DNNs are more robust than multimodal models. Furthermore, models
using CNN-based Image Encoder are more vulnerable than models with ViT—for
untargeted attacks, we obtain a 99% success rate by perturbing less than 0.02%
of the image area.
AI security, Multimodal models, Contiguous Attacks, Sparse Attacks, Pixel-
perturbations
††copyright: cc
## 1\. Introduction
In computer vision, the researchers try to find new methods for improving
image classification and other tasks beyond that (e.g. given a set of textual
descriptions and an image, the model chooses the most likely image
description). For that, the research community developed multimodal models
(Radford et al., 2021; Haas et al., 2023; Jia et al., 2021; Chen et al., 2022;
Li et al., 2022; Xu et al., 2022; Pham et al., 2023; Zhong et al., 2022),
which are used as a basis for further research (CLIP (Radford et al., 2021)
led e.g. to various modifications such as RegionCLIP (Zhong et al., 2022)) and
are increasingly also considered for commercial usage 111Deloitte report:
Novel Design Classification with CLIP. Thus, it is essential to use such
models, knowing their robustness and security implications. While prior
studies have demonstrated the vulnerability of multimodal models to
perturbations of the whole image (Mao et al., 2022; Qiu et al., 2022), their
vulnerability to pixel-level perturbations remains largely unexplored. In
particular, the impact of the number of perturbed pixels and their spatial
distribution on the attack performance has not been explored so far. Both of
these basic aspects must be considered when constructing adversarial attacks
using pixel perturbations. Empirical analysis of their impact on attack
performance would enable more informed design choices when crafting novel
adversarial pixel perturbations and ultimately lead to a better understanding
of the overall robustness of multimodal models.
In this work, we develop methods and experiments to address this gap. We focus
on the black-box scenario because we want to confront the security problem of
an attacker who does not have any prior information about the model. We rely
on $L_{0}$-norm perturbations to have control over the number of pixels being
perturbed. We extend _Sparse_ pixel distribution attack presented in work by
Su et al. (Su et al., 2019) to incorporate different spatial encodings
resulting in five different _Contiguous Attacks_ (details are described in
Section 4.2). Depending on the number and distribution of perturbed pixels,
our method exploits potential vulnerabilities hidden in the model
architectures, such as the kernel sizes in the case of CNN-based Image Encoder
of multimodal models. Pixel perturbations are usually performed in the
original image. This assumes a threat model with minimal information available
to the adversary (i.e., does not need to know the preprocessing routine).
However, if you perturb the preprocessed image, you can be certain that your
experimental results are not confounded by the different preprocessing
pipelines, which we prioritized for this study. We evaluate the effectiveness
of these attacks with images from ImageNet on four open-source models (ALIGN
(Jia et al., 2021), AltCLIP(Chen et al., 2022), CLIP-B/32 (Radford et al.,
2021), and GroupViT (Xu et al., 2022)) representative of the domain.
Additionally, we evaluate our approach for two state-of-the-art Deep Neural
Networks (DNNs), which are trained on the ImageNet dataset (Krizhevsky et al.,
2012). Finally, we also release the code for reproducing the study:
https://github.com/ChristianB024/SparseVsContiguityRepo
Our empirical analysis led to several key insights:
* •
We found that three out of four multimodal models and both unimodal DNN models
are most vulnerable to the _Sparse Attack_ for all examined attack scenarios
(targeted and untargeted misclassification).
* •
The _Patch Attack_ is most effective against the CNN-based multimodal model
(ALIGN (Jia et al., 2021)). In an untargeted scenario, we reach a 99% success
rate only by perturbing 0.01915% of the image (16 pixels).
* •
Overall, we observe that the multimodal models are more vulnerable to pixel
perturbation attacks than the state-of-the-art DNNs, which we suspect to be
linked to the way multimodal models and unimodal DNNs are trained.
## 2\. Related Works
### 2.1. Advesarial Attacks on multimodal models
In recent years, multimodal models (Radford et al., 2021; Haas et al., 2023;
Jia et al., 2021; Chen et al., 2022; Li et al., 2022; Xu et al., 2022; Pham et
al., 2023; Zhong et al., 2022), which combine information from different
modalities such as text, image, and audio, have emerged rapidly as powerful
tools across various applications ranging from natural language processing and
computer vision to speech recognition. Previous research studied attacking
techniques on multimodal models and tried to identify certain vulnerabilities
in those models.
An adversarial perturbation attack can be defined by the approach used for
perturbing the input. For instance, the works (Cao et al., 2023; Fort, 2021)
explored the CLIP (Radford et al., 2021) multimodal model’s robustness against
typographical attacks—the attacker introduced a textual sticker containing a
word on the image to misclassify the image to the class written on the textual
sticker.
The paper by Qiu et al. (Qiu et al., 2022) analyzed the robustness of
multimodal models by adding noise/blur to the whole image to create the image
perturbations, while in our work, we focus on perturbing only a limited number
of pixels. Similarly, the study by Mao et al. (Mao et al., 2022) quantified
the robustness of the CLIP model by creating adversarial images using white-
box PGD attack (Madry et al., 2017). They also presented two defense methods
for mitigating that kind of attack. Moreover, some works, such as Yang et al.
(Yang et al., 2021), focus on adversarial perturbation against multimodal
models using only the PGD white-box scenario. Similarly, in the work by
Schlarmann et al. (Schlarmann and Hein, 2023), they launch imperceptible
attacks using the same PGD white-box technique on a single open-source model
(OpenFlamingo (Awadalla et al., 2023)). In contrast, in our study, we limit
the adversary’s power to considering the black-box scenarios only. A more
different attack is presented in the work by Freiberger et al. (Freiberger et
al., 2023) where it exploited the vulnerability of the modality gap in a
contrastive learning approach by generating adversarial examples using the
evolution strategy for searching generative models’ latent space.
### 2.2. Adversarial attacks using Genetic Algorithms (GA)
Certain adversarial perturbation attacks on Deep Learning models focused only
on the white-box scenarios (Szegedy et al., 2013; Goodfellow et al., 2014;
Kurakin et al., 2016; Madry et al., 2017; Papernot et al., 2016; Carlini and
Wagner, 2017; Moosavi-Dezfooli et al., 2016; Athalye et al., 2018), while
others evolved in direction of black-box attacks (Su et al., 2019; Chen et
al., 2017; Brendel et al., 2017; Tu et al., 2019). Moreover, for improving
their attacks, there are studies where genetic algorithms have been introduced
(Su et al., 2019; Wu et al., 2021). For instance, the study by Su et al. (Su
et al., 2019) generated the adversarial examples against the DNNs by
perturbing a specific number of pixels, thus using an $L_{0}$ norm.
Furthermore, the paper by Jere et al. (Jere et al., 2019) used evolutionary
strategies (including Differential Evolution) for generating _scratch_
perturbations on the image against DNNs only. Similarly, in terms of applying
the evolutionary algorithms but using sparse adversarial perturbations through
simultaneous minimization of $L_{0}$ and $L_{2}$ norms, the research by
Williams et al. (Williams and Li, 2023) restricted the space of potential
perturbations quite aggressively and focused on studying only sparsely
distributed pixel perturbations. So there is no exploration regarding the
impact of contiguous shapes of the perturbation on attack performance.
In addition, both studies (Su et al., 2019; Williams and Li, 2023) performed
the pixel perturbations on the original images. In this work, we perturb the
preprocessed image such that our attack is neither influenced by the dimension
of the original image nor the particular preprocessing routines. Therefore, we
investigate the attack based solely on the different models. Also, the work by
Qiu et al. (Qiu et al., 2021) included an analysis of creating the image
perturbations with respect to the $L_{\infty}$-norm in a black-box setup,
using different evolutionary strategies and characterizing which evolutionary
algorithm is more suitable for this kind of attack. More recently, the paper
by Nam et al. (Nam and Kil, 2023) explored the one-pixel idea but using an
exhaustive search procedure, and the perturbations were identified directly on
the original image while we are attacking the preprocessed images. In the
study by Ghosh et al. (Ghosh et al., 2022), the differential evolution (DE)
(Storn and Price, 1997) technique generated the pixel-perturbations in a
black-box setup. The attack was launched against relatively old CNN models
(VGG16 (Simonyan and Zisserman, 2014), GoogleNet (Szegedy et al., 2015),
InceptionV (Szegedy et al., 2016), ResNet-50 (He et al., 2016)), while we
attack more recently proposed multimodal models. Since we want to compare
their robustness with established unimodal deep neural networks, we also
compare them against the ResNet-50 (He et al., 2016) and the VAN (Visual
Attention Network (Guo et al., 2023), which used Dilated Convolution Layers
(Yu and Koltun, 2015)).
### 2.3. Adversarial Patch Attacks
The idea of perturbing pixels in the form of a patch is not new, and it was
already tested before in the white-box (Brown et al., 2017; Karmon et al.,
2018) and black-box setups (Brown et al., 2017; Zhou et al., 2021). For
instance, the work by Alrasheedi et al. (Alrasheedi and Zhong, 2023) explored
the problem of making the image perturbations imperceptible and also found
certain boundaries in the width limit. However, the entire attack was launched
in a white-box setup, while we focus on black-box attacks since they are much
closer to real-world scenarios. The study by Wei et al. (Wei et al., 2022)
performed the attacks in a black-box manner by applying reinforcement learning
as an optimization technique to simultaneously find the best position and
perturbation values for the patch. This process is similar to what we try to
achieve in our experiments, but we use the DE approach to evolve our
perturbations. Different methods were used in the context of patch attacks by
even hiding the patch as a QR code (Chindaudom et al., 2020, 2022), such that
the pixel perturbations became less suspicious to the human eye. Moreover,
these patch adversarial attacks were also examined concerning object detection
models using aerial images (Tang et al., 2023) and deep hashing models (Hu et
al., 2021b).
Lastly, the research community relied on GANs to generate adversarial patches
for image classification (Bai et al., 2021; Demir and Unal, 2018; Liu et al.,
2019) or object-detection models (Hu et al., 2021a; Pavlitskaya et al., 2022).
With this technique, the recent paper by Zhou et al. (Zhou et al., 2023)
showed a non-targeted attack against the multimodal model CLIP (Radford et
al., 2021). However, we do not train GAN models since we try to limit the
attacker’s power (including the computational one).
## 3\. Attack model
Model
Figure 1. Threat Model Visualization
We analyze the black-box scenario of an adversary with access to Artificial
Intelligence (AI) models during inference, so the attacker has access only to
the model’s input and output. Figure 1 presents a visualization of the
described threat model. To formalize the attack, let the function
$f:\\{0,...,255\\}^{w\times h\times 3}\rightarrow\mathbb{[}0,1]^{C}$ represent
the model classification which takes as an input $x\in\\{0,...,255\\}^{w\times
h\times 3}$ representing an RGB image with dimension $w$ and $h$ for width and
height respectively. The output corresponds to a probability distribution for
the $C$ classes. The maximum probability from the list of all given classes
$C$ yields the label (or textual description)
$y_{original}=\arg\max_{i\in\\{1,\ldots,C\\}}f_{i}(\mathbf{x})$. We consider
two possible pixel-perturbation attacks. In the first one, the goal of the
attacker is to find a perturbation ($\Delta\mathbf{x}$) such that the new
image is classified as a specific target label (targeted attack noted as
_tar_). The attack is presented in the Equation 1 where the adversary is
maximizing the target label ($p_{tar}$) probability by perturbing at most $d$
pixels.
(1) $\displaystyle\text{find}\;\Delta\mathbf{x}\;\text{such
that}\begin{cases}p_{\text{tar}}=\underset{\Delta\mathbf{x}}{\max}\;f_{i:=\text{tar}}(\mathbf{x}+\Delta\mathbf{x}),\\\
\text{and}\;||\Delta\mathbf{x}||_{0}\leq d\end{cases}$
The second one, untargeted attack (noted as _untar_), has as the goal to do
random misclassification (maximizing any other class probability) as it is
described similarly by the Equation 2.
(2) $\displaystyle\text{find}\;\Delta\mathbf{x}\;\text{such
that}\begin{cases}p_{untar}=\underset{\Delta\mathbf{x}}{\max}\;f_{i\neq
original}(\mathbf{x}+\Delta\mathbf{x}),\\\
\text{and}\;||\Delta\mathbf{x}||_{0}\leq d\end{cases}$
The perturbation of our attack is created for the preprocessed image instead
of the original image. In fact, if the adversary perturbs one pixel in a
$32\times 32$ image (dimension of the original image), then it perturbed
10.24% of the image (approach presented in the paper by Su et al. (Su et al.,
2019)), while one pixel in a preprocessed image of dimension $224\times 224$
represents less than 0.002% of the image area. Table 1 shows the exact
percentages of the perturbed area for the target models. Furthermore,
attacking this small area of the image makes the attack less perceptible for
the human eye and hard to detect by the systems, hence having potentially
catastrophic consequences in a sensitive application. Moreover, this setup is
closer to the real-world scenario, where companies usually release new AI
models via an API, which takes input images of a specific dimension (the
company also need to release the Image Processor of the model) and outputs the
probabilities for specific classes/textual descriptions.
Lastly, the attack is limited to $Q$ number of queries to the model. Hence,
the attacker creates the perturbations using an evolutionary algorithm
(differential evolution—DE (Storn and Price, 1997)) and makes a maximum of $Q$
queries.
Table 1. Percentage of the perturbed area in the attacked image Preprocessed image dimension \Pixels | 4 pixels | 9 pixels | 16 pixels
---|---|---|---
ALIGN - 289 x 289 | 0.00478% | 0.01077% | 0.01915%
AltCLIP, CLIP-ViT-B/32, GroupViT - 224 x 224 | 0.00797% | 0.01793% | 0.03188%
VAN-base, ResNet-50 - 224 x 224 | 0.00797% | 0.01793% | 0.03188%
## 4\. Evolutionary Attacks
This section presents the encoding for each type of attack used (_Sparse_ and
_Contiguous Attacks_), followed by how we initialize the genetic algorithm
setup. The evolutionary process is then explained in depth, covering mutation,
crossover, and fitness functions.
### 4.1. Sparse Attack
Figure 2. Agent encoding for the Sparse Attack
As mentioned in Section 3, we use Su et al. (Su et al., 2019)’s approach, the
differential evolution (DE) (Storn and Price, 1997) genetic algorithm, as an
optimization method for finding the best perturbations for misclassification
tasks. Encodings of the perturbations in specific vectors (noted as agents or
population members) become accessible to the evolutionary approach, which
tries to iteratively yield better perturbations over time. For the Sparse
Attack, we can use five values,
$[x_{coord},y_{coord},r_{value},g_{value},b_{value}]$, to describe a pixel,
namely its positioning by the coordinates and its RGB values. As a result, an
agent is represented by $N$ flattened and concatenated vectors, where $N$ is
the number of pixels for which the perturbation is being performed. Section 2
illustrates the encoded perturbation vector we use for this type of attack. An
important property of the _Sparse Attack_ is that there are no constraints on
the spatial distribution of the pixels used for the attack.
### 4.2. Contiguous Attacks
In contrast to the _Sparse Attack_ , we can impose constraints on the spatial
distribution of the pixels used for an attack. In particular, we choose to
force pixels to form spatially contiguous regions in the image (i.e., anti-
diagonal, diagonal, column, row, patch). Since we also address the question of
how different shapes of contiguous pixels can affect the model robustness, we
create appropriate encodings for the _Contiguous Attack_. We decrease the
number of the values that must be mutated for other shapes and store only the
coordinates of the first pixel from that shape. Thus, we have the generic
encoding pattern
$[x_{coord},y_{coord},r_{value_{1}},g_{value_{1}},b_{value_{1}},...,r_{value_{n}},g_{value_{n}},b_{value_{n}}]$,
where the $r$, $g$, $b$ values represent the colors for each perturbed pixel
in the specific shape. The first two values are the x and y coordinates,
representing different starting pixels:
Anti-Diag Attack:
the x and y coordinates of the lowest pixel of the shape.
Diagonal Attack:
the x and y coordinates of the up-most pixel of the diagonal.
Column Attack:
the x and y coordinates of the up-most pixel of the column.
Row Attack:
the x and y coordinates of the left-most pixel of the row.
Patch Attack:
the x and y coordinates of the up-left corner pixel of the patch.
As an example on how the pixels are positioned and what is their order in the
encoding vector depending on the attack type, we illustrate the case where we
perturb four pixels in Figure 3.
(a) Anti-Diagonal
(b) Diagonal
(c) Column
(d) Row
(e) Patch
Figure 3. Spatial arrangement of pixels in Contiguous Attacks for exemplary
case of four perturbed pixels
The idea of _Contiguous Attacks_ is to actually try to perturb multiple visual
patches (depending on the length of the shape) from the ViT model. So, suppose
an attacker chooses the row shape. In that case, it can perturb a number of
consecutive visual patches (or tokens) differently depending on their
alignment with the row of perturbed pixels. Figure 4 shows an example of
perturbing 6 pixels in different shapes. We have most perturbations in the
middle visual patch for the _Row Attack_ and less for the beginning and final
patch. Hence, we want to test if it is better to perturb less information but
from many very different visual patches (_Sparse Attack_), perturb less
information to some visual patches and more in some others (_Row Attack_), or
perturb more in a smaller number of visual patches (_Patch Attack_). Thus, we
do not focus on the evolutionary method for improving the position of those
pixel shapes. However, we emphasize that if we use an evolutionary approach to
find the best perturbation based on the provided shape, we can still exploit
how ViT creates the image tokens.
(a) Sparse Attack
(b) Contiguous - Row Attack
(c) Patch Attack
Figure 4. Pixel perturbations for different attacks on patch-based models
(e.g. Vision Transformers, ViTs)
### 4.3. Initialization
Similarly with the study by Su et al. (Su et al., 2019), for the
initialization process of the population, we do the following:
* •
for $x_{coord}$ and $y_{coord}$ we assign uniformly random integer values
based on the dimension of the attacking image (i.e., $U(0,max_{dim})$)
* •
for RGB values, since their values are between 0 and 255, we assign them as a
random value from a normal distribution $N(\mu=128,\sigma=127)$
### 4.4. Evolutionary process
After initializing the first population (generating $\mathcal{P}$
perturbations, also called agents), we calculate the fitness value for each
one. This is computed by applying a fitness function. As mentioned in Section
3, we have two possible attack scenarios. Hence, we use two hinge loss
functions, inspired by the paper by Chen et al. (Chen et al., 2017) and
exemplified in the Equations 3 and 4—for targeted attacks $F_{tar}$ and
$F_{untar}$ for the non-targeted attacks. Furthermore, we maximize the fitness
function of every new generation. Since we assume a black-box attack scenario,
we need to consider that we can only access the probabilities for each class.
For instance, in our case, the $p[x]$ means we obtain the softmax
(probability) value from the model for class $x$. Also, as these functions
ascend, sorting the agents by their fitness values, at position zero in the
population yields the best perturbation.
(3) $\displaystyle F_{tar}$
$\displaystyle=p_{tar}-\max_{i\neq\text{tar}}\,p_{i}$ (4) $\displaystyle
F_{untar}$
$\displaystyle=\max_{i\neq\text{original}}\,p_{i}-p_{\text{original}}$
(5) $\displaystyle
agent_{cand}=population_{0}+mutation_{rate}\cdot(population_{r1}-population_{r2})$
where $r1$ and $r2$ are random indices that are within the population size
For the next iteration, we create $\mathcal{P}$ other new candidates
($agent_{cand}$). Thus, for each member in the population, we apply the
mutation function from Equation 5 and the crossover operation, using the
exponential strategy approach (see Appendix A). A list with the parameter
values used in the DE is presented in the Section 5.1.
After obtaining the candidates, we perform $\mathcal{P}$ queries to the model
and compute the fitness values based on the probabilities obtained. Therefore,
we decide which $\mathcal{P}$ agents out of $2\cdot\mathcal{P}$ (old and new
agents) survive for the next generation, based on their fitness values—the
$\mathcal{P}$ agents with the highest fitness values are selected. We repeat
this process $\mathcal{G}$ times, as we are limited to the maximum number of
queries we can do. The idea of the evolutionary approach is generating
increasingly better perturbations over time and, as a result, have at least
one perturbation that categorizes the attack as successful. Further details
about how we quantify the success of an attack are explained in Section 5.2.
## 5\. Experiments
This section presents the images used for the attack and the values for the
parameters used. Moreover, we describe the evaluation metrics and baselines.
Lastly, we present our numerical results.
### 5.1. Experiment setup
We evaluate our attacks on samples from ImageNet (Krizhevsky et al., 2012). It
contains images with a variable resolution length and, thus, a different
number of pixels. As explained in Section 3, we attack the preprocessed image
before it is fetched into the model; we do not apply any preprocessing
operations to the original images from the dataset to scale all of them to a
fixed resolution, $256\times 256$ resolution, as it is done in the original
paper (Krizhevsky et al., 2012). The number of labels for this dataset is
1000.
Table 2. Performance comparison between Multimodal Models and DNNs Model | Accuracy | Model Type
---|---|---
ALIGN (Jia et al., 2021) | 0.5442 | Multimodal
CLIP_ViT-B32 (Radford et al., 2021) | 0.5601 | Multimodal
AltCLIP (Chen et al., 2022) | 0.6993 | Multimodal
GroupViT (Xu et al., 2022) | 0.3187 | Multimodal
ResNet-50 (He et al., 2016) | 0.8087 | DNN
VAN-base (Guo et al., 2023) | 0.8022 | DNN
The models used can be split into two branches based on their architecture. In
the first category, we have the state-of-the-art multimodal models; in the
other, we have the state-of-the-art DNNs, trained on the ImageNet dataset.
Table 2 shows the list of all the models and their measured accuracy for
10,000 images from the testing dataset of ImageNet. The motivation behind this
assessment is given by the connection between robustness, the architecture
used, and the versatility of the models (multimodal models can be applied to
different datasets without the need for extensive retraining due to the zero-
shot property, while DNNs usually require training on a specific dataset and
are optimized for that one only).
In order to use the multimodal models for image classification, we rely on
appropriate input captions. In particular, we append the string _”a photo of”_
in front of the original label of the image to enable zero-shot image
classification.
As observed in Table 2, the models perform very differently, and we want an
extensive analysis of their robustness. Thus, to avoid any possible bias, we
extract 100 random correctly classified images for each model. Those are part
of the testing dataset; consequently, the models did not see those images
during their training phase. Then, after extracting those images, we store the
preprocessed version based on each model. Hence, we test different attack
setups on the same images to quantify the attack’s effectiveness.
Based on empirical observations, we observe that the best parameters for the
Differential Evolution are $mutation_{rate}=0.55$ and $crossover_{rate}=0.8$.
Moreover, the population of $\mathcal{P}=300$ gives the best results when we
limit the number of iterations to $\mathcal{G}=100$. As a remark from Section
3, the total number of queries an attacker needs to do to perturb an image is
at most $Q$, which is translated to $\mathcal{P}\cdot\mathcal{G}\leq Q$, which
means $300\cdot 100\leq 30,000\leq Q$.
### 5.2. Evaluation Metrics and Baselines
Attack performance is quantified by computing the _SR_ (success rate) as an
empirical measurement. Depending on the type of attack, we define the success
of the attack as follows: targeted attacks involves how many images managed to
be misclassified as the target label, while for untargeted scenarios, we count
the number of images that are misclassified as a different label than the
original one. As mentioned Section 5.1, for our setup, it translates to using
100 images and computes how many of them we manage to attack successfully.
Then, since the SR is a value between 0 and 1, we divide it by 100 (the number
of images). Thus, an SR value of 1 means we create successful perturbations
for all images.
Lastly, to investigate the effectiveness of the evolutionary attacks, we
create random versions for each type of attack and use them as baselines.
These _Random Attacks_ use the same parameters as the evolutionary one
($\mathcal{P}=300$ and $\mathcal{G}=100$), but the difference is in generating
those random agents without an evolutionary approach. At each iteration, we
generate $\mathcal{P}$ agents according to their respective encodings
presented in Sections 4.1 and 4.2, and the values are random and set based on
the initialization values presented in Section 4.3. However, some corrections
are applied if the values are outside the ranges (especially for the RGB
values).
### 5.3. Results
Figure 5. Targeted Attacks
Figure 6. Untargeted Attacks
We evaluate the performances of attacks based on the SR metric and against the
baselines described in Section 5.2. For instance, Figure 5 presents the SR for
the targeted attacks for each selected model and the shape of the attacks,
depending on the number of pixels perturbed. At first glance, we observe that
for the ALIGN model, the most successful attack is the _Patch Attack_ , while
for the others, it is the _Sparse Attack_. Furthermore, the best attacks
against the multimodal models achieve an SR higher than 0.35, while for the
DNNs, the SR is generally lower except for ResNet-50 with the _Sparse Attack_
and perturbing 16 pixels. Overall, the progression of SR values with
increasing perturbed pixels for the DE attacks is steeper than for the random
baseline attacks. Thus demonstrating that the SR of the proposed attacks is a
function of the number of perturbed pixels with an increased number of pixels
yielding higher SR.
For ALIGN, Figure 5 shows that the _Patch Attack_ performs the best (0.74),
but the value against the _Sparse Attack_ for 4 pixels is relatively close
(0.58 for _Patch_ and 0.57 for _Sparse_). However, the transition from 4 to 6
pixels is steeper for the _Patch Attack_ , starting from 0.57 and reaching
0.71, while for the _Sparse Attack_ , it starts from 0.56 and reaches 0.64 for
the 9 pixels. Nevertheless, for the transition from 9 to 16 pixels, the slopes
for both attacks seem similar. Moreover, all the _Evolutionary Attacks_ manage
to beat the _Random Attacks_ , with one exception: _Random Patch Attack_ ,
which has an SR close to the ones of the _Contiguous Attacks_. Besides this
one, the rest of the _Random_ ones only achieve an SR below 40%, even after
increasing the number of pixels.
Figure 5 highlights that for AltCLIP, the _Sparse Attack_ is the best attack
for 4 and 9 pixels with SR of 0.35 and 0.46, respectively. For 16 pixels, the
most successful attack is the _Patch Attack_ , with the SR almost 0.5. All the
_Contiguous attacks_ outperform the _Random_ ones, with the gap between those
becoming more prominent with more pixels. Initially, the difference is
negligible, while for 16 pixels, the gap between the _Row Attack_ (worst
_Contiguous Attack_) and _Random Patch_ (the best _Random Attack_) is
approximately 0.7. However, _Random Patch_ is still the best _Random attack_ ,
usually above 3-5% from the second-best _Random Attack_. Additionally, there
is a big difference between the _Patch_ and _Sparse Attacks_ and the remaining
_Contiguous Attacks_ (10-25%).
Based on Figure 5, CLIP is more robust to our attacks than ALIGN but less
robust when compared against the other multimodal models (GroupViT, AltCLIP).
The best attack remains the _Sparse Attack_ with values from 0.5 for 4 pixels
to 0.68 for 16 pixels, followed by the _Patch Attack_ with values from 0.33 to
0.50. The third best attack is _Row_ one, and for the 16 pixels, it is
followed behind with 2% by _Column_ one.
Similar to CLIP, Figure 5 shows that for GroupViT, the top two best attacks
are the _Sparse Attack_ , reaching an SR of more than 55% for all pixels, and
the _Patch Attack_ , achieving a maximum SR of 47%. Nonetheless, there is a
gap between those two attacks and the others: for 16 pixels, the _Row Attack_
reaches a value of 0.45 and below, with 0.02 less than the _Random Sparse
Attack_. The best _Random Attack_ is the _Sparse_ one, which outperforms the
_Patch_ with approximately 10% for 16 pixels. Similarly, all the other
_Contiguous Attacks_ are better than the other _Random_ ones, except for
_Random Sparse_.
From Figure 5 we observe that both DNNs have similar behavior: they are more
robust than any multimodal model. With ResNet-50, the SR value does not exceed
0.45 for the best attack (_Sparse Attack_), while for the VAN-base, the best
one stops at 0.25. Moreover, the trend regarding the best two attacks remains
the same as for CLIP and GroupViT, with a significant difference regarding the
gaps between those two attacks and the rest—the discrepancy is much smaller
(approx. 5% for ResNet-50 and less than 2% for the VAN-base). Moreover, the
_Random Attacks_ behave in the same manner as the rest of the _Contiguous
Attacks_ with values fluctuating within the limit of 5% for both models.
Results for the untargeted attacks are presented in Figure 6. Overall, the
trends remain the same as the targeted attacks, but the SR value is increased
by approximately 25% for ALIGN, 15% for AltCLIP, and 10% for CLIP and
GroupViT. Interestingly, a much smaller increase is observable for both DNNs
(approximately 5%). ALIGN appears to have the worst security against the
_Patch Attack_ for a more significant number of pixels since, using 16 pixels,
we obtain a result of 99% SR. Moreover, the _Random Patch Attack_ is in the
third position, with an SR close to the _Sparse Attack_ for 16 and achieving a
score of 0.85. For AltCLIP, the difference between the _Random Attacks_ and
the _Evolutionary_ ones increases proportionally with the number of pixels. In
addition, similar to the targeted attack, the _Patch Attack_ is slightly
better than the _Sparse_ one for the 16 pixels, while for 4 and 9, the winning
one is the _Sparse Attack_. For CLIP and GroupViTs, the plots for ranking the
effectiveness of the attacks remain the same as for targeted attacks, as is
the case for the DNNs.
## 6\. Discussion
At first glance, based on the results presented in Section 5.3, we can see
that the targeted attack is more challenging than the untargeted one, which is
indeed expected. However, the average difference between those two attacks for
multimodal models differs from the DNNs. Also, the DNNs are more robust to our
attacks than any multimodal models. An important observation is that for all
setups we have at least 2-3 attacks which are better than any of the
baselines.
Our results suggest that multimodal models with ViT (Visual Transformer)
(Dosovitskiy et al., 2020) as an Image Encoder are more prone to the _Sparse_
than the _Patch Attack_. To provide a potential explanation, we describe what
happens when an image is fed to a ViT. Initially, ViT split the images into
visual patches and then considers those as tokens. After this stage, those
tokens are processed similarly to a normal Transformer (Vaswani et al., 2017).
Suppose the adversary perturbs pixels on different visual patches (as you have
in _Sparse Attack_), then it can alter more tokens. Hence, in the self-
attention layer (Vaswani et al., 2017) (core component of a Transformer that
also provides good results in terms of robustness), there is a high
possibility of breaking the semantical meaning information from different
visual patches. However, using any _Contiguous Attack_ , the adversary cannot
perturb as many visual patches as would be possible using the _Sparse Attack_.
Additionally, the _Patch Attack_ outperforms any _Contiguous Attack_ because,
with this shape, we can alter more information in a single visual patch than
other contiguous pixel distributions.
We also question why CNN-based Image Encoders (ALIGN model) are more prone to
_Patch Attacks_. Based on the idea from the previous paragraph, if one piece
of information in the kernel is altered for those types of architectures, it
is a complete disaster for the upcoming max-pooling layers. The altered
information propagates to deeper layers, which try to extract correct
information that can be used as features for classification.
Based on our results, ALIGN (Jia et al., 2021) is the most vulnerable model to
the implemented $L_{0}$-norm perturbations. The best method to attack the
ALIGN model is by using the _Patch Attack_ and not the _Sparse_ one, as it is
for most of the cases in the other models. A potential explanation for this
observation could be because ALIGN uses the EfficientNet (Tan and Le, 2021)
(CNN-based architecture that uses 3x3 Convolutional Layers) as an Image
Encoder. This is why we have a steep increase in the SR for _Patch Attack_
from 4 to 9 pixels, from a patch of 2x2 to 3x3 (representing the same size as
the Convolutional Layer). However, the transition from 9 to 16 pixels is
smoother because we already manage to create a patch with the same dimension
as the Convolutional layer, so the attack does not increase a lot in
effectivness from the previous size (it increased only by 4%, while before we
have an increasing of approximate 10%). Based on this information, the
adversary can deduce some information regarding the architecture used (the
size of the kernel in initial Convolutional layers) in the model and therefore
here we could have a potential privacy issue. Moreover, this vulnerability of
the ALIGN model against the _Patch Attack_ can have serious consequences
since, for the untargeted attack, a patch of 16 pixels obtains a 99% success
rate. We see that the shape of the attack (_Patch_) additionally confirms this
vulnerability since the _Random Patch Attacks_ achieve high scores in both
cases, especially in the untargeted setting, being the third most powerful
attack.
We observe that AltCLIP is more robust than the original implementation of
CLIP. This could be explained because of the architecture used in the Image
Encoder. More precisely, AltCLIP (Chen et al., 2022) uses the ViT-L14, which
has the _dimension of the embedding_ equals to 768, 24 _layers_ , _width_
equals to 1024, and 16 _heads_ , while CLIP (Radford et al., 2021) uses the
ViT-B/32 which has _dimension of the embedding_ equals to 512, 12 _layers_ ,
_width_ equals to 768, and 16 _heads_. Hence, since the embeddings from
AltCLIP are larger, they capture more information, and therefore, it is harder
to fool those systems with the $L_{0}$-norm perturbations. Moreover, since the
AltCLIP Image Encoder is deeper (based on the number of layers), it can
capture more complex patterns and hierarchical features. It may exhibit better
robustness and generalization—this is also proved by the Table 2, where
AltCLIP obtained the best performance compared to the other multimodal models.
However, it seems that there exists a connection between the depth and the
width configuration of the model with larger patches since for both non-
targeted and targeted attacks in the case of 16 pixels, the _Patch Attack_ is
slightly better, and there exists a stagnation in the _Sparse Attack_.
Furthermore, AltCLIP uses the XLM-R (Conneau et al., 2019) as a Text Encoder,
while CLIP uses the masked self-attention transformer (Dosovitskiy et al.,
2020). Although this aspect might influence the robustness in favor of
AltCLIP, the core component that improves the security aspect is the Image
Encoder since we target it with our adversarial examples and do not give
adversarial textual input.
For the second most robust multimodal model, GroupViT (Xu et al., 2022), we
consider that the good results in terms of robustness are determined mainly by
the grouping mechanism used in the Image Encoder. This new architecture uses a
hierarchical group ViT—in short, besides the classical ViT approach, where the
image is initially split into visual patches and projected into tokens (using
Linear Projection), it also contains the Group Learnable Tokens. Those group
tokens describe a different semantic notion; thus, the model also groups more
semantic concepts. Since those tokens are learned during the training phase,
during the grouping semantic phase, the model tries to avoid perturbed visual
tokens; therefore, this mechanism increases the robustness of the model.
In addition, we observe significant differences in the SR between the DNNs and
the rest of the multimodal models. A potential cause for this behavior could
be related to the fact that we evaluate the DNNs on the images with specific
class labels for which they were originally trained. Consequently, these DNNs
contain class-specific features extracted during training on ImageNet
(Krizhevsky et al., 2012) which requires an attacker to perturb more pixels.
In contrast to DNNs, multimodal models were trained on large corpuses of image
and text data without explicit definition of target classes during training
(contrastive learning). Hence, they can be used on any dataset other than the
ones they were trained on. However, we argue that this flexibility comes with
an increased vulnerability to pixel perturbations, which is evident from our
experiments.
Lastly, a potential explanation why VAN-base (Guo et al., 2023) is the most
resilient model is based on its use of Dilated Convolutional Layers (Yu and
Koltun, 2015). Attacks based on contiguous pixels may be thwarted by expanding
the filter’s receptive field when it is widened through introducing gaps
between successive elements—the main feature of dilated convolutions. As a
result, the _Sparse Attack_ appears to be the most effective attack on VAN-
base, while the _Contiguous_ ones behave similarly to _Random Attacks_.
Moreover, the robustness of the VAN-base model surpasses the ResNet-50 (He et
al., 2016), which employs regular convolutional layers, providing further
evidence for the robustness of dilated convolutions.
## 7\. Conclusion and Future Work
Our work analyzes the robustness of four multimodal models and two state-of-
the-art unimodal DNNs against Sparse and Contiguous adversarial examples
defined by the $L_{0}$-norm in a black-box attack scenario. For both types of
attack (targeted and untargeted) and all models, we have at least two
evolutionary attacks that are more effective than the rest—all multimodal
models that use a ViT as an Image Encoder are most vulnerable to the _Sparse
Attack_. In contrast, in the CNN-based multimodal model (ALIGN), the most
effective attack is represented by a contiguous pixel perturbation in the form
of a _Patch_. This attack prove to be so powerful against ALIGN that with an
increased patch dimension of 4x4 (less than 0.02% of the image area), in a
non-targeted scenario, we achieved a 99% SR. Furthermore, based on the overall
SR, we rank the most secure models as follows: VAN-base, ResNet-50, AltCLIP,
GroupViT, CLIP-B/32, and ALIGN. Our results also point towards different
characteristics of AI model architectures (e.g., tokenization, convolutions,
dilated convolutions) that may be responsible for different levels of
robustness against our attacks. When contrasting the results for multimodal
and unimodal models, we observe that there may be an essential trade-off
between the robustness of the model and its adaptability to be used in diverse
tasks (i.e., zero-shot capability). Both of these aspects should be further
investigated in a follow-up study.
For future work, we propose to study the attacks on the $L_{0}$-norm for other
types of multimodal models that focus on object detection and text generation,
such as IDEFICS (Laurençon et al., 2023), Kosmos-2 (Peng et al., 2023), and
Gemini (Team et al., 2023). Moreover, we want to do more investigation on the
contiguous attacks and focus on finding the best hyperparameters for the
evolution algorithm used for those specific patterns, such that we can
construct a systematic search setup, where we can infer different information
regarding the model architecture from the observed attack performance.
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## Appendix A Exponential Strategy in Differential Evolution
Algorithm 1 was used for the Crossover operation - Exponential Strategy in the
Differential Evolution. The idea of this algorithm is to recombine the agents
based on some $crossover_{rate}$—how much information from the old agents
($agent_{old}$) should also be preserved in the new ones ($agent_{cand}$).
Thus, we directly modify the new agents in the $agent_{cand}$ vector and
return it for further processing. This code was adapted from the Differential
Evolution implemented in Scipy (SciPy Contributors, 2024).
Algorithm 1 Exponential Strategy (SciPy Contributors, 2024)
1:function Crossover Exp($agent_{old},agent_{cand},crossover_{rate}$)
2: $big_{r}\leftarrow\text{random index value of the }agent_{cand}$
3: $i\leftarrow 0$
4: while $i<\text{size of }agent_{cand}$ do
5: $r\leftarrow$ random number between 0 and 1
6: if $crossover_{rate}<r$ then
7: break
8: $agent_{cand}[big_{r}]\leftarrow agent_{old}[big_{r}]$
9: $big_{r}\leftarrow(big_{r}+1)\mod agent_{cand}\\_size$
10: $i\leftarrow i+1$
11: return $agent_{cand}$
|
# Pickup and Delivery Problem with Transfers
Santiago Hincapie Potes , Catalina Lesmes Ramirez Mathematical Engineering
Student<EMAIL_ADDRESS>Universidad EAFIT,
Medellín, Colombia.Mathematical Engineering Student<EMAIL_ADDRESS>Universidad EAFIT,
Medellín, Colombia.
###### Abstract
In this article we will be talking about Pickup and Delivery Problems with
Transfers (PDP-T), the main idea is to show first the Pickup and Delivery
Problem (PDP), which is the problem from which the PDP-T derives from, later
show a description of the problem and the mathematical formulation to get a
clear view of what we are going to solve, which leads us to present different
ways of solving this kind of problems.
Keywords:Delivery problem, local search, random search, heuristics, hybrid
methods
## 1 Introduction
In many logistics problems we require to pick some products in one place and
take them to another place, which is why we define the pickup and delivery
problem (PDP). In this problem a set of vehicles pick up and deliver a set of
items. The goal is to deliver the items at the lowest cost while obeying a set
of constraints, such as time windows and capacities. The PDP is a well-
studied, NP-hard problem, so approximation algorithms and heuristics have been
developed to address variants of the PDP. There are many techniques we can use
to solve the PDP problem, such as genetic algorithms, various metaheuristics,
taboo search heuristics and branch and cut algorithms.
To the PDP certain transfers can be added, which means that the vehicle which
is delivering the product transfers it to another vehicle before the delivery
is done. By adding those transfers we have the PDP with Transfers (PDP-T), in
which we consider transferring items between vehicles. We can convert the PDP
problem into a PDP-T problem by adding some variables and constraints.
This article is organized as follows: In Section 1 an introduction is
presented which gives a general notion of the problem, later in Section 2 we
give a brief literature review were some examples of solution methods for the
PDP-T are presented, after in Section 3 we give the problem description, were
all the variables, parameters and constraints of the mathematical model are
explained in detail, next in Section 4 the solution algorithms we give to the
problem are explained, later in Section 5 we present the results of the
computational experimental and finally in Sections 6 and 7 we give some
conclusions and future work suggestions.
## 2 Literature Review
As we saw before, there are different methods to solve the PDP-T.
As an example a branch and cut method using Benders Decomposition can be used.
In this method, the set of constraints is decomposed into pure integer and
mixed constraints, and then a branch-and-cut procedure is applied to the
resulting pure integer problem, by using real variables and constraints
related as cut generators. The key on the success of this method is that those
constraints defined by a logical sentence are not modeled using the big-M
technique, as usual in a branch-and-bound methodology behind the original PDP
formulation. This method may be applied only when the objective function is
either pure real or pure integer (Cortés et al. , , 2010).
Another method which solves the problem is a Very Large Neighborhood Search
with Transfers (VLNS-T) based on the Adaptive Very Large Neighborhood Search
(VLNS). The VLNS algorithm uses simulated annealing to randomly choose
neighboring schedules and iteratively improve the schedule. Neighboring
schedules are formed by removing random items and reinserting them with
heuristics. VLNS-T is based on the VLNS algorithm for the PDP without
transfers, a variant of simulated annealing in which the neighborhood of
states is ”very large”. In this case we remove random items from the schedule
and then reinsert them with multiple heuristics to find neighbors (Cat, ,
n.d.).
There are also some special cases of the PDP-T, as the Pickup and Delivery
Problem with Shuttle Routes (PDP-S) which relies on a structured network with
two categories of routes. Pickup routes visit a set of pickup points
independently of their delivery points and end at one delivery point. Shuttle
routes are direct trips between two delivery points. Requests can be
transported in one leg (pickup route) or two legs (pickup route plus shuttle
route) to their delivery point. The PDP-S applies to transportation systems
with a multitude of pickup points and a few, common delivery points (Masson et
al. , , 2010).
## 3 Problem Description and Mathematic Formulation
In the PDP, a heterogeneous vehicle fleet based at multiple terminals must
satisfy a set of transportation requests. Each request is defined by a pickup
point, a corresponding delivery point, and a demand to be transported between
these locations (Parragh et al. , , 2008a). Now, PDP-T allows the option for
passengers to transfer between vehicles, provided that the locations of the
transfer points are fixed and known.
### 3.1 Problem Formulation
Let $N$ be the set of transportation requests. For each transportation request
$i\in N$ a load of size $\bar{q_{i}}\in\mathbb{N}$ has to be transported from
a set of origins $N^{+}_{i}$ to a set of destinations $N_{i}^{-}$.
Each load is divided as follows $\displaystyle\bar{q_{i}}=\sum_{j\in
N_{i}^{+}}q_{j}=-\sum_{j\in N_{i}^{-}}q_{j}$, i.e., positive quantities for
pickups and negative quantities for deliveries.
Define $\displaystyle N^{+}:=\cup_{i\in N}N_{i}^{+}$ as the set of all origins
and $\displaystyle N^{-}:=\cup_{i\in N}N_{i}^{-}$ as the set of all
destinations. Let $V:=N^{+}\cup N^{-}$.
Furthermore, let $M$ be the set of vehicles. Each vehicle $k\in M$ has a
capacity $Q_{k}\in\mathbb{N}$ a start location $k^{+}$ and an end location
$k^{-}$.
Define $M^{+}:=\\{k^{+}|k\in M\\}$ as the set of start locations and
$M^{-}:=\\{k^{-}|k\in M\\}$ as the set of end locations. Let $W:=M^{+}\cup
M^{-}$.
For all $i,j\in V\cup W$ let $d_{ij}$ denote the travel distance, $t_{ij}$ the
travel time and $c_{ij}$ the travel cost. Note that the dwell time at origins
and destinations can be easily incorporated in the travel time and therefore
will not be considered explicitly.
To formulate the PDP as a mathematical program we introduce four types of
variables: for $i\in N$ and $k\in M$:
$z_{i}^{k}=\begin{cases}1&\text{if transportation request }i\text{ is assigned
to vehicle }k\\\ 0&\text{In other case}\end{cases}$
for $(i,j)\in(V\times V)\cup\\{(k^{+},j)|j\in V\\}\cup\\{(j,k^{-})|j\in V\\}$
and $k\in M$:
$x_{ij}^{k}=\begin{cases}1&\text{if vehicle }k\ \text{travels from location
}i\ \text{to location }j\\\ 0&\text{In other case}\end{cases}$
$D_{i}$ with $(i\in V\cup W)$, specifying the departure time at vertex $i$ and
$y_{i}$ with $(i\in V\cup W)$, specifying the load of the vehicle arriving at
vertex $i$ (Parragh et al. , , 2008a).
All this information can be summarized in Table 1:
Table 1: Parameters and variables found in (Parragh et al. , , 2008a) for the model Object | Meaning
---|---
$M$ | Set of vehicles
$C$ | Set of requests
$T$ | Set of transference point
$M^{+}$ | Set of origin depots for vehicles
$M^{-}$ | Set of destination depots for vehicles
$N^{+}$ | Set of origin nodes for requests
$N^{-}$ | Set of destination nodes for requests
$N$ | Set of nodes associated with requests
$V$ | Set of nodes
$q_{ij}$ | Size of request $i\in C$
$Q_{k}$ | Capacity of vehicle $k\in K$
$t_{ij}$ | Minimum ride time from node $i$ to node $j$
$c_{ij}$ | The travel cost
$d_{ij}$ | The travel distance
$z_{i}^{k}$ | bind transportation request and vehicles
$x_{ij}$ | bind routes and vehicles information
$D_{i}$ | specifying the departure time at specific vertex
$y_{i}$ | specifying the load of the vehicle arriving
Now the model is:
minimize $\displaystyle\sum_{i,j\in V\cup W}dij$ (1) subject to
$\displaystyle\sum_{k\in M}z_{i}^{k}=1$ for all $\displaystyle i\in N$ (2)
$\displaystyle\sum_{j\in V\cup W}x_{il}^{k}=z_{i}^{k}$ for all $\displaystyle
i\in N,l\in N^{+}_{i}\cup N_{i}^{-}k\in M$ (3) $\displaystyle\sum_{j\in
V\cup\\{k^{-}\\}}x^{k}_{k^{+}j}=1$ for all $\displaystyle k\in M$ (4)
$\displaystyle\sum_{j\in V\cup\\{k^{+}\\}}x^{k}_{k^{-}j}=1$ for all
$\displaystyle k\in M$ (5) $\displaystyle D_{k^{+}}=0$ for all $\displaystyle
k\in M$ (6) $\displaystyle D_{p}\leq D_{q}$ for all $\displaystyle i\in N,p\in
N_{i}^{+}q\in N_{i}^{-}$ (7) $\displaystyle x_{ij}^{k}=1\Rightarrow
D_{i}+t_{ij}\leq D_{j}$ for all $\displaystyle i,j\in V\cup W,k\in M$ (8)
$\displaystyle y_{k^{+}}=0$ for all $\displaystyle k\in M$ (9) $\displaystyle
y_{l}\leq\sum_{k\in M}Q_{k}z_{i}^{k}$ for all $\displaystyle i\in N,l\in
N_{i}^{+}\cup N_{i}^{-}$ (10) $\displaystyle D_{i}\geq 0$ for all
$\displaystyle i\in V\cup W$ (11) $\displaystyle y_{i}\geq 0$ for all
$\displaystyle i\in V\cup W$ (12)
Constraint (2) ensures that each transportation request is assigned to exactly
one vehicle. By constraint (3) a vehicle only enters or leaves a location $l$
if it is an origin or a destination of a transportation request assigned to
that vehicle. The next (4) and (5) make sure that each vehicle starts and ends
at the correct place. Also the constraints number (6), (7), (8) and (11)
together form the precedence constraints the others together form the capacity
constraints. Constraints (9), (10) and (12) together form the capacity
constraints.
This mathematical program, models the PDP. Now for a PDP-T model in the
literature it can bee seen that the transfer point is introduced, and the
procedure for the extended model is to iterative add constrains that involve
this transfer points. The process of inserting the transfers will be described
in Section 4.
## 4 Solution Algorithms
In this section we present solution algorithm for the PDP-T. In general, for
this part, we first solve the PDP using different methods (which would be
described in this section) and then we add the transship option to the PDP
solution, transship option always increases the computational time but makes
the solution better. In subsection 4.1 we present two constructive methods: a
greedy approach and multistart heuristic in Takoudjou et al. , (2012), then
in subsection 4.2 we present a random method GRASP which is applied to the
constructive methods mentioned before, later in subsection 4.3 we present
three local search methods: Variable neighborhood descent, Adaptive large
neighborhood search and simulated annealing, finally in subsection 4.4 we
present two hybrid methods which combine local search methods and genetic
algorithms.
### 4.1 Constructive Methods
#### 4.1.1 Greedy approach
We implemented the greedy approach described in Coltin et al. , (2014), which
basically iterates through every item and vehicle and then inserts the best
pickup and delivery action into the schedule, it always chooses the option
that increases the least the cost, we repeat the process until no unassigned
items remain. Then we add the transfer option with another greedy idea,
inspired on Clarke saving algorithm, this method computes the savings of the
transfer, and it improves the solution, then we modify the solution by adding
this new note.
#### Clarke and Wright Algorithm
The Clarke and Wright savings algorithm is one of the most known heuristic for
VRP. It applies to problems for which the number of vehicles is not fixed (it
is a decision variable), and it works equally well for both directed and
undirected problems. When two routes $(0,...,i,0)$ and $(0,j,...,0)$ can
feasibly be merged into a single route $(0,...,i,j,...,0)$, a distance saving
$s_{ij}=c_{i0}+c_{0j}-c_{ij}$ is generated. The algorithm works as it follows:
Step 1. Savings computation
* •
Compute the savings $s_{ij}=c_{i0}+c_{0j}-c_{ij}$ for $i,j=1,...,n$ and $i\neq
j$.
* •
Create n vehicle routes $(0,i,0)$ for $i=1,...,n$.
* •
Order the savings in a non increasing fashion.
Step 2. Route Extension (Sequential version)
* •
Consider in turn each route $(0,i,...,j,0)$.
* •
Determine the first saving $s_{ki}$ or $s_{jl}$ that can feasibly be used to
merge the current route with another route ending with $(k,0)$ or starting
with $(0,l)$.
* •
Implement the merge and repeat this operation to the current route.
* •
If not feasible merge exists, consider the next route and reapply the same
operations.
* •
Stop when not route merge is feasible.
#### 4.1.2 Hybrid Multistart Heuristic
The PDPT is a hard combinatorial optimization problem. The heuristics for this
problem must avoid being trapped by local optimum. To overcome local
optimality, a diversification procedure is needed. So we use the transshipment
heuristic from the hybrid heuristic for the PDP found in Takoudjou et al. ,
(2012) to obtain a solution to the PDP-T.
At each iteration, the transshipment heuristic is used to improve the PDP
solution and obtain a solution to the PDP-T. From the current PDP solution,
each request ($i$, $i+n$) $\in$ R; is removed from the solution. Then, ($i$,
$i+n$) is split into two different requests ($i$, $e_{t}$) and ($s_{t}$,
$i+n$), where $e_{t}$ and $s_{t}$ are the inbound/outbound doors of a
transshipment point $t$ $\in$ T. The best reinsertion cost of ($i$, $i+n$) in
the solution is computed. The best insertion cost to insert ($i$, $e_{t}$)
following by insertion of ($s_{t}$, $i+n$) in the solution is computed. The
cost to insert ($s_{t}$, $i+n$) following by the insertion of ($i$, $e_{t}$)
at their best position is computed. Between the three possibilities, the
insertion or the reinsertions offering the minimum cost is performed. As
described in Takoudjou et al. , (2012), this hybrid approach merges many
ideas. The solving principle is based on the following steps:
* •
An initial PDP solution (Sol) is calculated by gradually inserting the
requests in routes associated to vehicles.
The initial solution is consisting of a single route containing only the
starting point and the ending point of each route. Requests are then
successively introduced into the tour of a vehicle offering the minimal
increase of the cost of transport.
* •
Transshipment is then used to “destroy” and “repair” PDP solution to obtain a
better solution (PDPT solution).
At each iteration, the transshipment heuristic is used to improve the PDP
solution and obtain a solution to the PDPT. From the current PDP solution,
each request; is removed from the solution. Then, the request is split into
two different requests, the first go from the start point to the best transfer
point, the other go from the transfer point to the ending point.
### 4.2 Random Search Method
#### 4.2.1 GRASP
GRASP is a multi-start metaheuristic for combinatorial optimization problems,
in which each iteration consists of two phases: construction and local search.
The construction phase builds a feasible solution, whose neighborhood is
investigated until a local minimum is found during the local search phase. The
best overall solution is kept as the result. Here we will only use one phase,
which is the construction of the solution Resende & Ribeiro, (2010).
### 4.3 Local Search Methods
#### 4.3.1 Variable Neighborhood Descent
The VND is used to explore the neighborhood of the current solution, which is
based on three operators defined below: ADR, RNR and SWR. With these three
operators, we can explore the solution space more intensively. An operator is
a move that transforms one solution to another with small modifications. In
best improvement operators the whole neighborhood is analyzed and the best
solution is kept. In first improvement local search the first better solution
found in the neighborhood is kept (Takoudjou et al. , , 2012).
##### SWR
(Swap requests between routes) In SWR neighborhood, two requests belonging to
two different routes are exchanged together provided that all PDP constraints
are satisfied.
##### RNR
(Remove and insert a request) In RNR neighborhood, provided that all PDP
constraints are satisfied, request belonging to one route is removed and
inserted in another route.
##### ADR
(Advance or delay a request) In ADR neighborhood, requests belonging to one
given route is advanced or delayed in the same route if PDP constraints are
all satisfied.
#### 4.3.2 Adaptive Large Neighborhood Search
The general idea is to remove some requests from routes in the current
solution and then reinsert them elsewhere to arrive at a new solution. At each
iteration, one each of several removal and insertion heuristics is randomly
selected based on their designated weights. Letting the set of removal
heuristics be ${rh_{1},...,rh_{nr}}$ and their corresponding weights be
${rw_{1},...,rw_{nr}}$, $rh_{i}$ is chosen with probability
$\frac{rw_{i}}{\sum_{j=1}^{nr}rw_{j}}$, where $nr$ is the number of options.
Similarly for the insertion heuristics ${ih_{1},...,ih_{ni}}$ and their
corresponding weights ${iw_{1},...,iw_{ni}}$, the probability of $ih_{i}$
being $\frac{ih_{i}}{\sum_{j=1}^{n}ih_{j}}$ selected is, where $ni$ is the
number of options. After each iteration, if the solution realized is feasible
and better than the incumbent, the latter is updated. In any case, the process
is repeated until a predefined number of iterations $n_{II}^{max}$ is reached
(Masson et al. , , 2012).
#### 4.3.3 Simulated Annealing
Simulated annealing is a metaheuristic that begins at some state, and chooses
a random “neighbor” of that state. With probability accept $(e,e_{0},t)$ the
new state is accepted as the current state, where $e$ is the “energy” of the
current state, $e_{0}$ is the energy of the new state, and $t$ is the
temperature, or the fraction of iterations of the algorithm currently
completed. If the new state is rejected we remain at the current state and
repeat with a new neighbor. The algorithm continues either for a fixed number
of iterations or until the energy crosses some threshold, when the best
solution that has been encountered thus far is returned (Coltin & Veloso, ,
2014).
### 4.4 Hybrid Methods
In this subsection we present two hybrid methods, both of this methods are
based on the use of a local search methods and genetic algorithms but they
possess different hybridizations. The first algorithm uses simulated annealing
for the population selection and for its mutation variance of the probability,
this is made with the purpose of achieves a better convergence, next to a
diverse population. The second hybrid method uses the taboo list to avoid
mating of chromosomes that have low Minkowski in order to keep diversity, also
use this information in order to change mutation probability. We show in
precise detail how the genetic algorithm was made, so in subsection 4.4.1 we
show the encoding of the problem, then we describe the initial population in
subsection 4.4.2, which we got in two different ways, we have a totally
randomized initial solution and a constructive solution, later we select the
population to be matched which is described in subsection 4.4.3, then in
subsection 4.4.4 we describe the crossover between the population, after that,
in subsection 4.4.5 we describe the mutation for both first and second
methods, then in subsection 4.4.6 we explain how the new population is chosen
for the first method and the second method, and finally in subsection 4.4.7 we
explain the stop criterion.
#### 4.4.1 Encoding
Given the structure of the problem, the solutions can be represented as a set
of $N$ paths, one for each vehicle, considering that it is important to
consider the transfers as much as the possibility of a node recurrence, it was
opted to represent the solution as a directed multi-graph, in which the set of
vertexes represent some of the visited nodes and there exists a connexion
between two vertexes if one is a successor from the other, the direction goes
in the way of the route, additionally each one of the multi-sets is found
reserved for each route. To represent each multi-graph, a multidimensional
matrix was used, which positions $i,j,k$ possessed the locality distance $i$
to the locality $j$, if $j$ is the successor of $i$ in route $k$. For the
processes of the genetic algorithm to each element of the solution matrix a
derivate from the logistic function was applied (fixing the value from $0$ to
$0$), meaning that to each element a function as is shown below was applied:
$f(x)=\begin{cases}\frac{e^{-x}}{(1+e^{-x})^{2}}&\text{if }x>0\\\
0&\text{otherwise}\end{cases}$
this function was applied with the purpose of giving more importance to the
short routes, in this way the intention is to preserve this property. The
complexity in space of this encoding is $O(N|path|^{2})$.
#### 4.4.2 Initial Population
For the computational experiments we performers with two district initial
population methods, a totally randomized approach and a set of district greedy
random solutions.
#### Totally Randomized Initial Solution
A set of 256 random multi-graphs were generated using the model of a
modification of the algorithm from Erdős-Rényi as described in (Godehardt, ,
1993). It was opted to generate random graphs in place of random matrices, due
to the fact that the solutions generated this way are usually more feasible
with a higher probability, additionally they use to be better.
#### Constructive Solution
We construct a set of 256 district random greedy solution using the algorithm
described in the section 4.2.
#### 4.4.3 Selection
For the selection part we use Baker’s stochastic universal selection (SUS)
(Baker, , 1987), this algorithm uses a single random value to sample all the
solutions by choosing them at evenly spaced intervals. This gives weaker
members of the population (according to their fitness) a chance to be chosen
and thus reduces the unfair nature of fitness-proportional selection methods,
also the experimental work by Hancock (Hancock, , 1994) clearly demonstrates
the superiority of this approach.
#### Fitness Function
For a $K$ matrix its fitness function is the sum of the total distance of each
one of its routes, in case that the solution is not feasible (it does not
satisfy all the request) a $\infty$ value is assigned.
#### 4.4.4 Crossover
To make the crossover the solutions which were previously selected were
matched and a new matrix was generated in which its entries were uniform
random numbers. The resultant son matrix possesses the gen of its first parent
in the position $i,j,k$ if the generated random matrix possesses a lower
number to $0.5$ in the corresponding position, in the other cases it possesses
the gen of its other parent.
#### 4.4.5 Mutation
##### First method
For mutation operator we use a shift operator, that shifts the value of two
elements of the matrix depending on a parameter that we varied according to
the diversity in the population (measured in terms of the fitness variance)
and actual solution temperature (describe in the next step).
##### Second Method
For mutation operator we also use a shift operator but in this method we uses
a fixed probability $\alpha=0.05$
#### 4.4.6 New population
##### First method
For the actualization of the population, we perform a probabilistic
replacement test, in this approach the solutions are accepted into the new
generation using a Boltzmann type calculation. At each generation the
temperature is decreased, thereby decreasing the probability of accepting
higher energy individuals.
##### Second method
For the population update, we compute the Minkowski distance between
population members and uses taboo list to avoid nears solutions, in order to
keep population diversity.
#### 4.4.7 Stop criterion
For the stop criterion we compared the variance between the solutions of the
last $4$ generations, this with the purpose of ending the algorithm when it
reaches an stationary state.
## 5 Computational Experimentation
In this section we show the computational results given by the algorithms
described in section 4 that solve the PDP-T. In subsection 5.1, we present
tables with the instance size, the number of vehicles, optimal solutions and
the computational time, then we compare them to see which is the best
heuristic used.
All algorithms were coded on python 3.5.3 running on Intel Xeon E5-1620V4 ,
64-bit processor with 32 gigabyte RAM and Ubuntu 17.10. We used problem
instances from those in for the PDP, which are related to the well-known
Solomon instances. The datasets are available at
http://www.sintef.no/Projectweb/TOP/PDPTW/ . For each instance we introduce a
random number of transfer points, in a random node, to become the original PDP
instance in a PDP-T instance.
Each data set contains $X-Y$ hence, the lengths of the arcs in the network are
$L_{2}$ distances. For the locations of the vehicle depots, the origin and
final depots for each vehicle were randomly generated, scattered over the
region formed by the nodes. The number of vehicles are also random generated,
and finally we ignore the time windows restriction because are not consider in
the model.
### 5.1 Computational results
In this subsection we show the results of all the heuristics mentioned before
and compare them.
#### 5.1.1 Constructive Methods
In this subsection we show the tables of the computational results for the
constructive methods, in Table 2 we present the results for the Greedy
approach and in Table 3 we show the results for the MULTISTART method.
Table 2: Greedy approach Instance | Instance Size | Number of vehicles | Optimal Solution | Time
---|---|---|---|---
lc204 | 100 | 3 | 2001.31 | 28.23
lc2_2_3 | 200 | 6 | 4920.43 | 50.48
lc2_4_5s | 400 | 12 | 7235.92 | 123.43
LC2_8_5 | 800 | 25 | 226748.20 | 240.12
Table 3: MULTISTART Instance | Instance Size | Number of vehicles | Optimal Solution | Time
---|---|---|---|---
lc204 | 100 | 3 | 1825.83 | 40.81
lc2_2_3 | 200 | 6 | 4378.92 | 76.21
lc2_4_5s | 400 | 12 | 6891.06 | 173.79
LC2_8_5 | 800 | 25 | 201234.73 | 301.83
As we can see, the MULTISTART method gives better optimal solutions for the
PDP-T, but at the same time it requires more computational time.
#### 5.1.2 Random Search Method
In this subsection the results for the GRASP which is the random search method
are shown, as we mentioned in Section 4, we add GRASP to the constructive
methods. In Table 4, the greedy approach with GRASP is shown and in Table 5 we
have the results for the MULTISTART with GRASP.
Table 4: Greedy approach (GRASP) Instance | Instance Size | Number of vehicles | Optimal Solution | Time
---|---|---|---|---
lc204 | 100 | 3 | 1713.01 | 54.12
lc2_2_3 | 200 | 6 | 4001.12 | 80.15
lc2_4_5s | 400 | 12 | 6481.10 | 196.92
LC2_8_5 | 800 | 25 | 198917.87 | 320.16
Table 5: MULTISTART (GRASP) Instance | Instance Size | Number of vehicles | Optimal Solution | Time
---|---|---|---|---
lc204 | 100 | 3 | 1521.31 | 58.63
lc2_2_3 | 200 | 6 | 3832.12 | 89.53
lc2_4_5s | 400 | 12 | 6312.18 | 200.12
LC2_8_5 | 800 | 25 | 190143.12 | 354.10
If we compare Tables 4 and 5 we can see that the optimal solutions are given
by the MULTISTART (GRASP), but it also takes more computational time than the
Greedy approach (GRASP). This results are coherent, since they follow the same
behavior as we mentioned in subsection 5.1.1, but we can also see that when we
add GRASP to our constructive methods the optimal solutions are better, even
thought they take more computational time.
#### 5.1.3 Local Search Methods
In this subsection we show tables comparing all the results, plus one more,
the other is called ”MIX”, this table refers to a union of the VND and ALNS
methods, we wanted to see what happened if we merged two of them. So in Table
6 we present the ALNS, in Table 7 the VND, in Table 8 the SA and in table 9
the mix between the VND and the ALNS. Also in this subsection we also wanted
to show a 3D plot, showing the results of all the heuristics including the
initial solution, this plot is shown in Figure 1.
Table 6: ALNS Instance | Instance Size | Number of vehicles | Optimal Solution | Time
---|---|---|---|---
lc204 | 100 | 3 | 1154.21 | 160.32
lc2_2_3 | 200 | 6 | 3213.31 | 233.21
lc2_4_5s | 400 | 12 | 5421.32 | 312.23
LC2_8_5 | 800 | 25 | 182313.32 | 600.12
Table 7: VND Instance | Instance Size | Number of vehicles | Optimal Solution | Time
---|---|---|---|---
lc204 | 100 | 3 | 1032.12 | 187.21
lc2_2_3 | 200 | 6 | 3102.23 | 254.12
lc2_4_5s | 400 | 12 | 5250.12 | 352.21
LC2_8_5 | 800 | 25 | 173213.12 | 721.48
Table 8: SA Instance | Instance Size | Number of vehicles | Optimal Solution | Time
---|---|---|---|---
lc204 | 100 | 3 | 1321.23 | 140.32
lc2_2_3 | 200 | 6 | 3543.34 | 221.34
lc2_4_5s | 400 | 12 | 5732.12 | 301.31
LC2_8_5 | 800 | 25 | 182313.32 | 572.12
Table 9: MIX Instance | Instance Size | Number of vehicles | Optimal Solution | Time
---|---|---|---|---
lc204 | 100 | 3 | 1023.62 | 350.14
lc2_2_3 | 200 | 6 | 3102.23 | 500.51
lc2_4_5s | 400 | 12 | 5291.17 | 698.92
LC2_8_5 | 800 | 25 | 177843.01 | 1425.92
size020406080100time050010001500200025003000objective020040060080010001200ALNSVNSSAMIXInitial
Solution
Figure 1: Comparison of the heuristic methods
In Figure 1, we can see in the plot the time, the objective function and the
size of the instance in contrast between each other, is a comparison of all
methods and the initial solution.
We can see from Tables 6 through 9 and Figure 1 that the best algorithm as
seen in computational time is the Simulated Annealing and that best algorithm
as seen in optimal solution is the mix between the ALNS and the VND. As until
now, our best results are given with the random search algorithms if we
compare them with the local search algorithms we can say that the best results
are given by the local search algorithms specifically with the mix of the ALNS
and the VNS. For this mix, we can see a very interesting result, even thought
it was the best result of all, we should expect it gave a much better result,
but the difference from the initial solution is not that wide.
#### 5.1.4 Hybrid Methods
In this subsection we show the results given by the algorithms mentioned in
subsection 4.4, it is important to mention that the reaction of the initial
population and the associated individual operations such as encoding were
computed in parallel. So, in Tables 10 and 11 we show the results for the
genetic algorithms and simulated annealing with both the constructive and the
random initial solution, and in Tables 12 and 13 we show the results for the
genetic algorithms and taboo with both the constructive and the random initial
solutions.
Table 10: Genetic algorithm & SA, with random initial solution Instance | Instance Size | Number of vehicles | Optimal Solution | Time
---|---|---|---|---
lc204 | 100 | 3 | 720.60 | 789.12
lc2_2_3 | 200 | 6 | 2454.12 | 1325.21
lc2_4_5s | 400 | 12 | 4843.93 | 2012.32
LC2_8_5 | 800 | 25 | 14794.27 | 3024.15
Table 11: Genetic algorithm & SA, with Constructive solution Instance | Instance Size | Number of vehicles | Optimal Solution | Time
---|---|---|---|---
lc204 | 100 | 3 | 673.12 | 1254.65
lc2_2_3 | 200 | 6 | 2021.23 | 1743.54
lc2_4_5s | 400 | 12 | 4654.63 | 2433.33
LC2_8_5 | 800 | 25 | 13647.12 | 3502.12
Table 12: Genetic algorithm & Taboo, with random initial solution Instance | Instance Size | Number of vehicles | Optimal Solution | Time
---|---|---|---|---
lc204 | 100 | 3 | 690.43 | 823.43
lc2_2_3 | 200 | 6 | 2144.33 | 1513.43
lc2_4_5s | 400 | 12 | 4136.38 | 2243.43
LC2_8_5 | 800 | 25 | 13474.76 | 3324.69
Table 13: Genetic algorithm & Taboo, with Constructive solution Instance | Instance Size | Number of vehicles | Optimal Solution | Time
---|---|---|---|---
lc204 | 100 | 3 | 610.60 | 1242.43
lc2_2_3 | 200 | 6 | 1944.33 | 2091.35
lc2_4_5s | 400 | 12 | 4070.48 | 2695.49
LC2_8_5 | 800 | 25 | 12689.23 | 3820.94
When we compare the results given by Tables 10 through 13 we can see that the
constructive solution gives better results than the random initial solution
for both SA and Taboo, but they also take more computational time. In general
terms, the best solution as seen in optimal solution is given by the genetic
algorithm and Taboo with the constructive solution. The best solution as seen
in computational time is given by genetic algorithm and simulated annealing
with a random initial solution.
## 6 Conclusions
* •
The best solution as seen in optimal solution is given by the genetic
algorithm and Taboo with the constructive solution in hybrid methods.
* •
The best solution as seen in computational time is given by the greedy
approach in constructive methods.
* •
An interesting matter with VND is that even though the solution is bad it has
a good computational time.
* •
The mix between the ALNS and the VND is a kind of hybrid but it is slow and
gives bad solution, but the hybrids with the genetic algorithm give better
results due to the fact that they use the hybridization to cover its weakness.
* •
We can see from all the computational times that when the instances are small,
the times between the heuristics are very similar but when the instances are
bigger, the times between the heuristics are very different.
## 7 Future Work
An interesting matter that it could be talked about in the future is about the
lower bound of this problem, because in none of the articles we have reviewed
the topic has risen up, and it’s important because we can not compare how much
closer we are getting to a better solution, we can see it gets better every
time but we can not see if we are closer to our goal. Also, it would be
interesting to run the algorithm several times rising the number of transfer
points with the purpose of seeing how it affects the computational time and
how much it diminishes the objective function.
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|
# MM-GNN: Mix-Moment Graph Neural Network towards Modeling Neighborhood
Feature Distribution
Wendong Bi Institute of Computing Technology, Chinese Academy of
SciencesBeijingChina<EMAIL_ADDRESS>, Lun Du Microsoft Research
AsiaBeijingChina<EMAIL_ADDRESS>, Qiang Fu Microsoft Research
AsiaBeijingChina<EMAIL_ADDRESS>, Yanlin Wang Microsoft Research
AsiaBeijingChina<EMAIL_ADDRESS>, Shi Han Microsoft Research
AsiaBeijingChina<EMAIL_ADDRESS>and Dongmei Zhang Microsoft Research
AsiaBeijingChina<EMAIL_ADDRESS>
(2023)
###### Abstract.
Graph Neural Networks (GNNs) have shown expressive performance on graph
representation learning by aggregating information from neighbors. Recently,
some studies have discussed the importance of modeling neighborhood
distribution on the graph. However, most existing GNNs aggregate neighbors’
features through single statistic (e.g., mean, max, sum), which loses the
information related to neighbor’s feature distribution and therefore degrades
the model performance. In this paper, inspired by the method of moment in
statistical theory, we propose to model neighbor’s feature distribution with
multi-order moments. We design a novel GNN model, namely Mix-Moment Graph
Neural Network (MM-GNN), which includes a Multi-order Moment Embedding (MME)
module and an Element-wise Attention-based Moment Adaptor module. MM-GNN first
calculates the multi-order moments of the neighbors for each node as
signatures, and then use an Element-wise Attention-based Moment Adaptor to
assign larger weights to important moments for each node and update node
representations. We conduct extensive experiments on 15 real-world graphs
(including social networks, citation networks and web-page networks etc.) to
evaluate our model, and the results demonstrate the superiority of MM-GNN over
existing state-of-the-art models
Graph neural networks, Graph representation learning, Feature distribution,
Moment, Social networks
††journalyear: 2023††copyright: acmcopyright††conference: Proceedings of the
Sixteenth ACM International Conference on Web Search and Data Mining; February
27-March 3, 2023; Singapore, Singapore††booktitle: Proceedings of the
Sixteenth ACM International Conference on Web Search and Data Mining (WSDM
’23), February 27-March 3, 2023, Singapore, Singapore††price: 15.00††doi:
10.1145/3539597.3570457††isbn: 978-1-4503-9407-9/23/02††ccs: Computing
methodologies Neural networks††ccs: Information systems Social networks
## 1\. Introduction
Graph data is ubiquitous in the real world, such as social networks, citation
networks, etc. Graph Neural Networks (GNNs) have shown expressive power on
graph representation learning in recent years. Most existing GNNs follow the
neighborhood aggregation scheme, where each node aggregates features of
neighbors to update its own representation. Many GNNs (Kipf and Welling, 2016;
Veličković et al., 2017; Derr et al., 2020; Yang et al., 2016) with different
aggregators have been proposed to solve various downstream tasks, which can be
divided into graph-level tasks, edge-level tasks, and node-level tasks, such
as graph classification (Xu et al., 2018a; Bai et al., 2019), link prediction
(Zhang and Chen, 2018; Zhou et al., 2018; Guo et al., 2022), and node
classification (Dai et al., 2022; Lin et al., 2020, 2014; Fettal et al., 2022;
Jin et al., 2021; Zhao et al., 2021).
However, the aggregation schemes of existing GNNs usually use single
statistics (e.g., mean, max, sum). For example, GCN (Kipf and Welling, 2016)
uses mean aggregator, GraphSAGE (Hamilton et al., 2017) uses max aggregator,
and GIN uses sum aggregator. And some state-of-the-art GNNs (e.g. GCNII (Chen
et al., 2020), DAGNN (Liu et al., 2020)) still use single statistics, without
retaining the complete information of neighbor’s feature distribution.
Recently, researchers (Xu et al., 2018a; Du et al., 2021b; Ma et al., 2022;
Luo et al., 2022) begin to study the modeling of neighborhood distribution and
have discussed its importance for graph data from various perspectives.
However, the studies of neighbor’s feature distribution are limited to lower-
order characteristics without considering higher-order distribution
characteristics.
Considering that single statistic cannot represent complex distribution with
more than one degree of freedom, it’s intuitive that we should introduce more
higher-order statistics. To demonstrate the prominent effects of higher-order
distribution characteristics on distinguishing different nodes, we present an
example from Facebook Social Network (Traud et al., 2012) in Fig. 1.
Considering the graph composed of student/teacher nodes with class year as
attributes, Fig. 1 (b) shows the mean and variance of their neighbors’
features. And we cannot distinguish the label of node A and node B by the mean
of their neighbors’ features. However, we can succeed to distinguish the two
nodes if we consider the variance of their neighbors’ features. And it’s
intuitional because the teachers are more likely to know students from
different class years, thus the variance of teacher’s neighbors is larger. And
this motivates us to consider higher-order characteristics of neighbor’s
feature distribution for graph data, and we further demonstrate the critical
roles of higher-order characteristics by analysis on real data in Section 3.
Figure 1. An example in Facebook social network. (a) is an example of Facebook
college social network, where the student node A and the teacher node B have
different neighbors set. (b) shows the feature distributions of neighbors of A
and B by their mean and variance, where the mean information fails to
distinguish the two nodes.
A toy example of Facebook social network.
Inspired by the method of moments (Bowman and Shenton, 2004), which is used to
represent unknown probability distributions, we introduce methods of moments
to design a novel aggregator for GNNs. Specifically, we propose to use adapted
multi-order moments to represent the neighbor’s feature distribution and
further embed the distribution signals into the message passing mechanism of
GNNs. Then we design a novel Mix-Moment Graph Neural Network (MM-GNN) model,
which includes two major components: Multi-order Moments Embedding (MME) and
Attention-based Moment Adaptor (AMA). MM-GNN first computes multi-order
moments for neighbors of each node as their signatures. Based on the
observations that different order of moments may have various impact for each
node (Sec 3), an Element-wise Attention-based Moment Adaptor is designed to
select important moment signatures for specific nodes adaptively. As it is
impossible to compute infinite-orders of moments to represent the distribution
perfectly, we only compute finite-order moments in parallel at each graph
convolution year. Finally, we conduct extensive experiments to demonstrate the
superiority of MM-GNN over other state-of-the-art methods on 15 real-world
graphs, including social networks, citation networks, webpage networks and
image networks. The improved performance on all types of graphs demonstrate
the effectiveness of our method.
The main contributions of this paper are summarized as follows:
1. (1)
We demonstrate that higher-order statistics are needed to model neighbor’s
feature distribution on graphs. And we introduce the method of moments to GNNs
from the perspective of modeling neighborhood feature distributions.
2. (2)
We design a novel MM-GNN model with an aggregator based on the multi-order
moments and further design an Element-wise Attention-based Moment Adaptor to
adaptively adjust the weights of multi-order moments for each node.
3. (3)
We conduct extensive experiments on 15 real-world graphs (i.e., social
networks, citation networks, webpage networks) to evaluate our method. The
results show that MM-GNN gains consistent improvements on each type of graphs.
## 2\. Preliminaries
In this section, we give the definitions of some important terminologies and
concepts appearing in this paper.
### 2.1. Graph Neural Networks (GNNs)
GNNs aim to learn representation for nodes on the graph. Let $G=(V,E)$ denotes
a graph, where $V=\\{v_{1},\cdots,v_{n}\\}$ is the node set, $n=|V|$ is the
number of nodes. Let $X\in\mathbb{R}^{n\times d}$ denote the feature matrix
and the $l$-th row of $X$ denoted as $x_{i}$ is the $d$-dimensional feature
vector of node $v_{i}$. $E$ is the edge set in $G$. Typically, GNN-based model
follows a neighborhood aggregation framework, where node representation is
updated by aggregating information of its first or multi-order neighboring
nodes. Let $h_{i}^{(l)}$ denotes the output vector of node $v_{i}$ at the
$l$-th hidden layer and let $h_{i}^{(0)}=x_{i}$. The update function of $l$-th
hidden layer is:
(1)
$h_{i}^{(l)}=\text{COMBINE}\left(h_{i}^{(l-1)},\text{AGGREGATE}\big{(}\\{h_{j}^{(l-1)}|v_{j}\in\mathcal{N}(v_{i})\\}\big{)}\right),$
where $\mathcal{N}(v_{i})$ is the set of neighbors of $v_{i}$. The AGGREGATE
function is aimed to gather information from neighbors and the goal of COMBINE
function is to fuse the information from neighbors and the central node. For
graph-level tasks, an additional READOUT function is required.
### 2.2. Method of Moments
#### 2.2.1. Definitions about moments:
We give some basic definitions of statistical moments (Papoulis and Pillai,
2002; Bowman and Shenton, 2004) in this section. First, the definition of
moments is as follows:
###### Definition 2.1 (Moments in Mathematics).
For a random variable $Z$, its $k$-th order origin moment is denoted as
$\mathbb{E}(Z^{k})$ , where
$\mathbb{E}(Z^{k})=\int^{\infty}_{-\infty}z^{k}\cdot
f(z;\theta_{1},\cdots,\theta_{k})dz$. The $k$-th order central moment of $Z$
is
${\small\mathbb{E}\left(\big{(}Z-\mu_{z}\big{)}^{k}\right)}=\int^{\infty}_{-\infty}(z-\mu_{z})^{k}\cdot
f(z;\theta_{1},\cdots,\theta_{k})dz$. The $k$-th order standardized moment of
$Z$ is
${\small\mathbb{E}\left(\frac{\big{(}Z-\mu_{z}\big{)}^{k}}{\sigma_{z}^{k}}\right)}=\int^{\infty}_{-\infty}\frac{(z-\mu_{z})^{k}}{\sigma_{z}^{k}}\cdot
f(z;\theta_{1},\cdots,\theta_{k})dz$, where $\sigma_{z}$ is the standard
deviation of random variable Z.
Then, moments can be derived from the Moment-Generating Function (MGF), which
is an alternative specification of its probability distribution. The MGF
(Papoulis and Pillai, 2002) is defined as follows:
###### Definition 2.2 (Moment-Generating Function (MGF)).
Let $X$ be a random variable with cumulative distribution function $F_{X}$.
The Moment Generating Function (MGF) of $X$ (or $F_{X}$), denoted by
$M_{X}(t)$, is
$M_{X}(t)=\mathbb{E}(e^{tX})=\int_{-\infty}^{\infty}e^{tx}dF_{X}(x)$
An essential property is that MGF can uniquely determine the distribution. The
Taylor Expansion of MGF is as follows:
(2)
$M_{X}(t)=\mathbb{E}(e^{tX})=\sum_{k=0}^{\infty}\frac{t^{k}\mathbb{E}(X^{k})}{k!}=\sum_{k=0}^{\infty}\frac{t^{k}m_{k}}{k!}$
The $k$-th term of Eq. 2 is exactly the $k$-th order origin moment.
#### 2.2.2. Relationship between moments and common statistics:
In order to intuitively explain the physical meaning of multi-order moments,
we give the relationship between moments and some common distribution
statistics of data. The common distribution statistics can be viewed as
special cases of moments. For a random variable $Z$, the expectation of $Z$ is
$\mathbb{E}(Z)$, which is the 1-st order origin moment. The variance of $Z$ is
$\mathbb{E}\left(\big{(}Z-\mu_{z}\big{)}^{2}\right)$, which is the 2-nd order
central moment. The skewness of $Z$ is
$\mathbb{E}\left(\frac{\big{(}Z-\mu_{z}\big{)}^{3}}{\sigma_{z}^{3}}\right)$,
which is the 3-rd order standardized moment.
## 3\. Findings: neighbor distribution benefits node classification
In this section, We verify the importance of neighbor’s feature distribution
through comprehensive data analysis. We first select three common statistics
(i.e., Mean, Variance, Skewness, see Sec. 2.2.2) to represent distribution of
neighbor’s feature. Then we calculate the selected statistics on the graphs of
Facebook social network (Traud et al., 2012), where the feature of nodes on
the graphs have practical meanings (e.g., gender, major, year) and easy to
analyze the physical meanings of their statistics. More details of the
datasets are presented in Sec. 6.1. And we further calculate the selected
statistics on the neighbors’ feature for all nodes and then take an average on
all feature dimensions. Finally, we choose two metrics (i.e. Fisher
Discrimination Index, Mutual Information) to measure the correlation between
statistics of neighbor distribution and the node label, and one statistic is
more important for node classification if it has stronger correlations with
the node label.
### 3.1. Data Analysis on Correlation between Neighbor Distribution and Label
(a) Fisher Discrimination Index. (b) Mutual Information.
Figure 2. Correlation analysis between the node label and neighbor’s feature
distribution represented by statistics (i.e., Mean, Variance, Skewness) on the
Facebook Social Networks.
#### 3.1.1. Analysis by Fisher Discrimination Index
Inspired by Fisher’s Linear Discrimination (Fisher, 1936), we also use the
Fisher discrimination index (Ratio of inter-class variance to intra-class
variance) to measure the node classification capability distinguished by
different statistics of neighbors’ features distribution (e.g., Mean,
Variance):
$\small
S_{Fisher}=\frac{\sigma^{2}_{between}}{\sigma^{2}_{within}}=\sum_{\mathcal{C}_{i},\mathcal{C}_{j}\in\mathcal{C}}\frac{\left(\mu_{i}-\mu_{j}\right)^{2}}{\sigma^{2}_{i}+\sigma^{2}_{j}},$
where $S_{Fisher}$ denotes the Fisher Discrimination Index,
$\sigma^{2}_{between}$ and $\sigma^{2}_{within}$ denote the inter-class and
intra-class variation. $\mathcal{C}$ is the set of classes, $\mathcal{C}_{i}$
and $\mathcal{C}_{j}$ are two different categories, $\mu_{i}$ and
$\sigma^{2}_{i}$ are respectively the mean and variance of samples in the
class $\mathcal{C}_{i}$. As shown in Fig. 2 (a). We observe that, for
different feature dimensions, different statistic of neighbor distribution
contributes variously to distinguish node labels. For example, we observe that
the skewness statistic has the best discriminative ability for the feature
dimension ”year”, while the variance statistic has best discriminative ability
for the feature dimension ”major”.
#### 3.1.2. Analysis by Mutual Information
Apart from Fisher Discrimination Index, we also analyze the Mutual Information
between distribution statistics and the node label to measure theirs
correlations. As shown in Fig. 2 (b), the results of mutual information also
demonstrate that different statistics have different strength of correlation
with the class label. In summary, all these results show that different order
of moments contribute quite differently to node classification. However, any
single statistic (e.g., mean, max, sum) cannot reflect complete neighborhood
feature distribution, we thus need to consider multi-order statistics such as
variance, skewness and other higher-order distribution statistics.
Figure 3. An overview of MM-GNN including Multi-order Moment Embedding (MME)
and Attention-based Moment Adaptor (AMA) modules. $H^{(l)}_{N(v_{i})}$ and
$P^{(l)}_{N(v_{i})}$ denotes the features and distributions (represented by
moments) of neighbors at the $l$-th layer. $h_{i}^{(l)}$ is the node
representation of node $v_{i}$ at the $l$-th layer. $M_{K}^{(l+1)}$ denotes
the $K$-th order moment of neighbors at the $(l+1)$-th layer. $Q_{i}$ and
$K_{i}^{k}$ denotes query and key vectors at the $l$-th layer.
$W_{query}^{(l+1)}$, $W_{key}^{(l+1)}$ and $W_{query}^{(l+1)}$ are learnable
transformation matrix at the $(l+1)$-th layer. $a_{k}^{(l+1)}$ is the learned
attention matrix for $k$-th order moment at the $(l+1)$-th layer.
Figure of model structure
## 4\. Mix-Moment Graph Neural Network
In this section, we introduce our Mix-Moment Graph Neural Network (MM-GNN)
model. An overview of MM-GNN is given in the Fig. 3. Compared with existing
GNNs, our MM-GNN introduces the Method of Moments to better represent
neighborhood feature distribution when updating node representations.
### 4.1. Modeling Neighbors’ Feature Distribution via Method of Moments
Based on the foundations of method of moments given in Sec. 2.2, we introduce
multi-order moments to represent the feature distribution of neighboring nodes
on the graph. And based on the analysis in Sec. 3, we observe that, for the
node classification task, the feature distribution of neighbors for the nodes
belonging to the same class follow the similar distributions, and vice versa.
Unlike other parametric methods, the outstanding power of estimating
distribution with method of moments is that we do not need to assume an
explicit form of the distribution function. In this paper, we propose to use
finite-order moments to represent the neighbors’ feature distribution for a
graph, and we further present the analysis on the theoretical and empirical
deviation of representing the neighbors’ feature distribution for graph data
with finite-order moments in Sec 5.2.
### 4.2. Multi-order Moments Embedding (MME)
As aforementioned, we model neighbors’ feature distribution by introducing the
Method of Moments to GNN. The Method of Moments illustrated in Section 2.2 is
a statistical method that use sample moments to estimate the distribution
function. Therefore, we use multi-order moments to estimate the feature
distribution of neighboring nodes. Moreover, moments of different nodes can be
computed in parallel at the aggregation step, and we select both origin moment
and central moment to represent neighborhood feature distribution. Here we
present the form of $k$-th order origin moment for $(l+1)$-th layer of MM-GNN:
(3)
$M_{k}^{(l+1)}(v_{i})=\mathbb{E}\big{(}(h_{j}^{(l)})^{k}\big{)}=\left(\frac{1}{|\mathcal{N}(v_{i})|}\sum_{j\in\mathcal{N}(v_{i})}((h_{j}^{(l)})^{k}\right)^{\frac{1}{k}}\cdot
W_{k}^{M}$
where $W_{k}^{M}\in\mathbb{R}^{D_{in}\times D_{hidden}}$ is the transformation
matrix for $k$-th order moment signature, $h_{j}^{(l)}\in\mathbb{R}^{D_{in}}$
is the node representation of $v_{j}$ output by the $l$-th layer, and
$h_{j}^{0}$ is the raw node feature $x_{j}$. Based on the Eq. 3, we can
observe that the $1$-st origin moment is exactly the expectation, which is
equivalent to the MEAN aggregation strategy. Thus previous GNNs with MEAN
aggregator are special cases of MM-GNN using $1$-st order origin moment.
Similarly, we can give the form of $k$-th order central moment for $(l+1)$-th
layer of GNNs:
(4)
$M_{k}^{(l+1)}(v_{i})=\mathbb{E}\big{(}(h_{j}^{(l)}-\mu_{i}^{(l)})^{k}\big{)}=\left(\frac{1}{|\mathcal{N}(v_{i})|}\sum_{j\in\mathcal{N}(v_{i})}(x_{j}-\mu_{i}^{(l)})^{k}\right)^{\frac{1}{k}}\cdot
W_{k}^{M}$
, where
(5)
$\mu_{i}^{(l)}=\frac{1}{|\mathcal{N}(v_{i})|}\sum_{j\in\mathcal{N}(v_{i})}h_{j}^{(l)}$
To keep the same order of magnitude, we normalize moments of different orders
by computing the $\frac{1}{k}$ power of $k$-th order moment vectors in the
model. Besides, we set a hyper-parameter for flexible model configuration of
using the central moment or origin moment.
### 4.3. Attention-based Moment Adaptor (AMA)
With the Multi-order Moments Embedding (MME) module, we get signatures
representing moments of different orders for each node. Then, we need to fuse
these signatures into one final representation for each node. As illustrated
in Section 3, different statistics (corresponding to moments) have different
effects on node classification, we design an Element-wise Attention-based
Moment Adaptor (AMA) that can adaptively select the important order of moments
for different nodes and feature dimensions. For each MM-GNN layer, after we
get the multi-order moment signatures with the MME module, we use the
transformed input node representation (the output of the previous layer) as
the query vector for each node, and then we use different moment-signatures
(output of the MME module of current layer) as key vectors. Then we compute
attentions as weights of different moments for each individual nodes. The
attention of $k$-th order moment for $v_{i}$ at $(l+1)$-th layer, denoted by
$a_{k}^{(l+1)}(v_{i})\in\mathbb{R}^{D_{hidden}}$, is:
(6) $a_{k}^{(l+1)}(v_{i})=\sigma\left([h_{i}^{(l)}\cdot
W_{query}^{(l+1)}||M_{k}^{(l+1)}(v_{i})\cdot W_{key}^{(l+1)}]\cdot
W_{a}^{(l+1)}\right)$
where $W_{a}^{(l)}\in\mathbb{R}^{2D_{hidden}\times D_{hidden}}$ is the
learnable attention matrix in $l$-th layer,
$W_{query}^{(l)},W_{key}^{(l)}\in\mathbb{R}^{D_{hidden}\times D_{hidden}}$ are
the transformation matrix for the query and key vectors in $l$-th layer. Then
different dimensions of $k$-th order moment signature for node $v_{i}$ use
different attention weights and therefore implement an element-wise attention
mechanism (attention of each order moment is a $D_{hidden}$-dimensional vector
rather than a scalar). Then we can get the output representation of node
$v_{i}$ at $(l+1)$-th layer:
(7) $h_{i}^{(l+1)}=\sum_{k=1}^{K}a_{k}^{(l+1)}(v_{i})\odot
M_{k}^{(l+1)}(v_{i}),\ a_{k}^{(l+1)}(v_{i})\in\mathbb{R}^{1\times D_{hidden}}$
where $\odot$ is the element-wise product and $K$ is the hyper-parameter for
the largest ordinal of moments used in MM-GNN. Besides, we add residual
connections across layers to further boost the performance of MM-GNN by
aggregating neighborhood feature distribution of multi-hop neighbors. We
finally use the output of the last GNN layer as the node representation and
further compute Cross Entropy Loss for node classification task and back
propagate to optimize the model parameters.
Table 1. Information of Facebook social networks. We present the number of nodes, number of edges for each social graph in this table. Each node has 6-dimensional attributes and belongs to 6 possible classes. Dataset | Northeastern | Caltech | UF | Hamilton | Howard | Simmons | GWU | UGA | Tulane
---|---|---|---|---|---|---|---|---|---
#Nodes | 13882 | 769 | 35123 | 2314 | 1909 | 1518 | 12193 | 24389 | 7752
#Edges | 763868 | 33312 | 2931320 | 192788 | 409700 | 65976 | 939056 | 2348114 | 567836
Table 2. Information of citation/webpage/image networks Dataset | #Nodes | #Edges | #Features | #Classes
---|---|---|---|---
Cora | 2,708 | 10,556 | 1,433 | 7
CiteSeer | 3,327 | 9,104 | 3,703 | 6
PubMed | 19,717 | 88,648 | 500 | 3
Chameleon | 2277 | 36101 | 2325 | 5
Squirrel | 5201 | 217073 | 2089 | 5
Flickr | 89,250 | 899,756 | 500 | 7
## 5\. Deep Analysis of MM-GNN
In this section, we first theoretically analyze the limitations of existing
GNNs using single statistic, which cannot retain the full information of
neighbor distribution. Then we analyze the deviation of estimating neighbors’
feature distribution with finite-order moments. Finally, we analyze the time
complexity of MM-GNN.
### 5.1. Limitations of Existing GNNs
In this sections, we prove the existing GNNs using single statistic (e.g. GCN,
GraphSAGE) have severe limitations when representing neighbor’s feature
distribution, which reduces the generalization ability of the model.
Complexity measure (Neyshabur et al., 2017) is the current mainstream method
to measure the generalization ability of the model. And we choose Consistency
of Representations (Natekar and Sharma, 2020) (champion of the NIPS2020
Competition on generalization measure). And the complexity measure of the
model, denoted as $\Gamma$, is:
(8) $\Gamma=\frac{1}{|\mathcal{C}|}\sum_{i=1}^{|\mathcal{C}|}\max_{i\neq
j}\frac{\mathcal{S}_{i}+\mathcal{S}_{j}}{\mathcal{M}_{i,j}},$
where $\mathcal{C}$ is the set of classes,
$\mathcal{C}_{i},\mathcal{C}_{j}\in\mathcal{C}$ are two different classes,
$\mathcal{S}_{i}=\left(\mathbb{E}_{v_{k}\sim\mathcal{C}_{i}}(|h_{k}-\mu_{\mathcal{C}_{i}}|^{p})\right)^{\frac{1}{p}}$
is the intra-class variance of class $\mathcal{C}_{i}$,
$\mathcal{M}_{i,j}=||\mu_{\mathcal{C}_{i}}-\mu_{\mathcal{C}_{j}}||_{p}$ is the
inter-class variance between $\mathcal{C}_{i}$ and $\mathcal{C}_{j}$, $h_{k}$
is the representations of nodes $v_{k}$ learned by the model and
$\mu_{\mathcal{C}_{i}}=\mathbb{E}_{v_{k}\sim\mathcal{C}_{i}}(h_{k})$. And
higher complexity measure means lower generalization ability. Then we
demonstrate that GNNs using single statistic have weak generalization ability.
Take GCN (Kipf and Welling, 2016) as an example, we give the following
theorem:
###### Theorem 5.1.
Given a graph $G(V,E)$, we denote the nodes belonging to class
$\mathcal{C}_{i}$ as $\\{v_{i}|v_{i}\in\mathcal{C}_{i}\\}$, and assume that
the neighbor’s feature distribution of nodes in $\mathcal{C}_{i}$ follows the
same i.i.d Gaussian distribution $\mathrm{N}(\bm{\mu}_{i},\bm{\Sigma}_{i})$.
Given any two classes $\mathcal{C}_{i}\neq\mathcal{C}_{j}$, if
$\bm{\mu}_{i}=\bm{\mu}_{j}$ and $\bm{\Sigma}_{i}\neq\bm{\Sigma}_{j}$ , then
the complexity measure $\Gamma$ (Eq.8) of GCN with mean aggregation reaches
$+\infty$, and the GCN thus lose the generalization ability.
###### Proof.
The update function of GCN with mean aggregator is:
$h^{(l+1)}_{k}=\sum_{j\in\mathcal{N}(v_{k})}\frac{1}{d_{k}}\cdot
h_{j}^{(l)}\cdot
W^{(l)}=\left(\frac{1}{d_{k}}\cdot\sum_{j\in\mathcal{N}(v_{k})}h_{j}^{(l)}\right)\cdot
W^{(l)}$
To simplify the analysis, we omit the activation function. And we can get the
expectation of $h^{(l+1)}_{k}$ for nodes in $\mathcal{C}_{i}$ and
$\mathcal{C}_{j}$:
(9) $\mathbb{E}_{v_{k}\sim\mathcal{C}_{i}}(h_{k}^{(l+1)})=\bm{\mu}_{i}\cdot
W^{(l)}=\mathbb{E}_{v_{k}\sim\mathcal{C}_{j}}(h_{k}^{(l+1)})=\bm{\mu}_{j}\cdot
W^{(l)},$
which means the expectation of the learned representations of nodes in
$\mathcal{C}_{i}$ equals to that of nodes in $\mathcal{C}_{j}$. Then according
to Eq. 9:
$\mathcal{M}_{i,j}=||\mathbb{E}_{v_{k}\sim\mathcal{C}_{i}}(h_{k})-\mathbb{E}_{v_{k}\sim\mathcal{C}_{j}}(h_{k})||_{p}=\bm{0}$
Considering that $\mathcal{S}_{i}+\mathcal{S}_{i}>0$, the complexity measure
$\Gamma$ in Eq. 8 equals to $+\infty$. Then GCN loses generalization ability,
and this rule also applies to any other GNNs with mean aggregator. ∎
And we can intuitively generalize Theorem 5.1 to all GNNs using single
statistics. For example, GraphSAGE (Hamilton et al., 2017) use the max/mean
aggregator, GIN (Xu et al., 2018a) uses the sum aggregator. And most state-of-
the-art GNNs (e.g., GCNII (Chen et al., 2020), DAGNN (Liu et al., 2020)) still
use single statistic to represent the neighbor’s feature distribution , and
therefore have the same limitations when neighbor’s feature distribution has
more than one degree of freedom.
### 5.2. Deviation Analysis on Finite-order Moments.
We give the theoretical guarantee of estimating neighborhood feature
distributions on the graph with finite-order moments by a theorem and give its
proof as follows:
###### Theorem 5.2.
Given a graph $G$, we denote the deviation of using top-$K$ order moments to
represent the neighbor’s feature distribution as $e_{K}$, and $e_{K}$ wil not
larger than an upper bound: $e_{K}\leq\frac{(\epsilon\cdot c)^{k+1}}{(k+1)!}$,
where $c\in\mathbb{R}^{+}$ is a constant and $\epsilon\in\mathbb{R}$ is a
small number around 0. And larger $K$ leads to smaller estimation deviation
(convergence to 0).
###### Proof.
According to the Sec. 2.2, the $k$-th order moment can be derived from the
$k$-th term of the MGF’s Taylor Expansion (see eq. 2). Therefore, the top-$K$
order moments can be viewed as an estimation of the $K$-th order Taylor
expansion for MGF of neighbor’s feature distribution. In real cases, the
feature of samples is usually bounded. We assume that for $\forall x_{i,j}\in
X$, we have $|x_{i,j}|\leq c$, where $c\in\mathbb{R}^{+}$ is a constant.
Besides, sine the Eq. 2 is the Taylor expansion around $t=0$, then $t$ is a
small real number $t\leq\epsilon$, where $\epsilon\in\mathbb{R}$ is a small
number around 0. Then the deviation of estimating neighbor’s feature
distribution by $k$-th order Taylor expansion of MGF can be approximated by
its Lagrange Remainder:
(10) $R_{k}(x)=\frac{t^{k+1}\mathbb{E}(X^{k+1})}{(k+1)!}\leq\frac{(t\cdot
c)^{k+1}}{(k+1)!}\leq\frac{(\epsilon\cdot c)^{k+1}}{(k+1)!}$
Besides, in real scenarios, we usually normalize the data, so the upper bound
of the estimation error $\frac{(t\cdot c)^{k+1}}{(k+1)!}$ is further reduced.
And it’s obvious that the upper bound of estimation error converges to zero as
the moment ordinal $k$ increases. ∎
### 5.3. Time Complexity Analysis
Here we analyze the time complexity of training MM-GNN. Let $D_{in}$ denote
the dimension of input node feature, and $D_{hidden}$ denote the hidden
dimension of node representation at each layer. $|V|$ and $|E|$ denotes the
number of nodes on the graph. For each layer of MM-GNN, the aggregation and
transformation of graph convolution cost $O\left((|V|+|E|)\cdot
D_{in}D_{hidden}\right)$. Let $k$ denote the max ordinal of used moments, and
we used $k\leq 3$ in our experiments and thus can be seen as a small constant
integer. The calculation of query vector costs $O(|V|D_{hidden}^{2})$, and the
calculation of all key vectors costs $O\left(k\cdot(|V|+|E|)\cdot
D_{hidden}^{2}\right)$. The Attention-based Moment Adaptor costs
$O(k|V|D_{hidden}^{2})$. Considering that $D_{hidden}\leq D_{in}$ and $k$ is a
constant integer, the overall time complexity of MM-GNN layer is:
$\begin{aligned} T&=O\left((k+1)|V|D_{hidden}^{2}+(k+1)\cdot(|V|+|E|)\cdot
D_{in}D_{hidden}\right)\\\ &=O\left((|V|+|E|)\cdot
D_{in}D_{hidden}\right)\end{aligned},$
which equals to the complexity of other mainstream GNNs.
## 6\. Experiments
### 6.1. Datasets
The proposed MM-GNN is evaluated on 15 real-world graphs, including 9 social
graphs from Facebook social networks, 3 citation networks (Cora, CiteSeer,
PubMed), 2 webpage networks (Chameleon, Squirrel) and one image network
(Flickr). The detailed information of theses datasets are presented in Table 1
and Table 2.
Social network datasets: Facebook100 (Traud et al., 2012) provides 100
college graphs, each graph describes the social relationship in a university.
Each graph has categorical node attributes with practical meaning (e.g.,
gender, major, class year.). Moreover, nodes in each dataset belong to six
different classes (a student/teacher status flag).
Citation network datasets: We choose three public citation network datasets
(Cora, PubMed, CiteSeer) to evaluate our model. The nodes on such graphs
represent papers and edges represent citation relationships between papers.
Webpage network datasets: Two webpage network datasets (Chameleon, Squirrel)
are used for model evaluation. Nodes of these datasets represent articles and
edges are mutual links between them.
Image network datasets: We also choose one image network (Flickr) for
evaluation. The nodes represent images and edges means two images share the
same metadata (e.g., location, gallery, tags).
Table 3. Performance (accuracy) on node classification task ($60\%$ training set ratio) on Facebook social network datasets. Model | Northeastern | Caltech | UF | Hamilton | Howard | Simmons | GWU | UGA | Tulane
---|---|---|---|---|---|---|---|---|---
GCN | $90.58\pm 0.28$ | $88.47\pm 1.91$ | $83.94\pm 0.61$ | $92.26\pm 0.35$ | $91.21\pm 0.69$ | $88.74\pm 0.61$ | $86.66\pm 0.48$ | $85.73\pm 0.51$ | $87.93\pm 0.97$
GAT | $88.69\pm 0.30$ | $81.17\pm 2.15$ | $81.68\pm 0.59$ | $91.43\pm 1.25$ | $89.35\pm 0.44$ | $89.14\pm 0.57$ | $84.27\pm 0.88$ | $82.95\pm 0.75$ | $84.45\pm 1.40$
GraphSAGE | $91.77\pm 0.86$ | $92.14\pm 1.38$ | $86.21\pm 0.56$ | $94.75\pm 0.35$ | $93.13\pm 0.93$ | $92.31\pm 0.92$ | $89.43\pm 0.78$ | $88.11\pm 0.39$ | $89.87\pm 0.47$
GIN | $89.62\pm 1.10$ | $78.35\pm 4.24$ | $85.40\pm 0.71$ | $85.47\pm 6.20$ | $89.37\pm 2.05$ | $88.37\pm 1.18$ | $84.73\pm 2.80$ | $87.68\pm 0.40$ | $80.71\pm 4.78$
APPNP | $91.11\pm 0.27$ | $90.76\pm 2.38$ | $83.07\pm 0.54$ | $93.99\pm 0.47$ | $91.88\pm 1.22$ | $90.31\pm 1.20$ | $90.43\pm 0.38$ | $86.31\pm 0.41$ | $88.52\pm 0.44$
JKNet | $92.32\pm 0.39$ | $92.34\pm 1.09$ | $85.43\pm 0.71$ | $94.72\pm 0.76$ | $93.11\pm 0.78$ | $91.81\pm 0.71$ | $89.25\pm 0.65$ | $87.68\pm 0.40$ | $89.04\pm 0.43$
DAGNN | $92.43\pm 0.30$ | $91.15\pm 2.67$ | $86.71\pm 0.51$ | $94.75\pm 0.75$ | $93.44\pm 0.68$ | $92.49\pm 0.96$ | $89.83\pm 0.42$ | $88.20\pm 0.29$ | $89.29\pm 0.50$
MM-GNN | $93.16\pm 0.32$ | $93.28\pm 1.94$ | $88.20\pm 0.45$ | $95.46\pm 0.53$ | $94.08\pm 0.95$ | $93.61\pm 0.45$ | $90.78\pm 0.45$ | $89.44\pm 0.30$ | $90.22\pm 0.33$
Table 4. Performance (accuracy) on node classification task on citation/webpage/image network datasets Dataset | Cora | CiteSeer | PubMed | Chameleon | Squirrel | Flickr
---|---|---|---|---|---|---
GCN | $81.13\pm 0.26$ | $70.08\pm 0.43$ | $78.62\pm 0.63$ | $37.68\pm 3.06$ | $26.39\pm 0.88$ | $50.31\pm 0.30$
GAT | $81.36\pm 0.96$ | $70.93\pm 0.70$ | $78.19\pm 0.78$ | $44.34\pm 1.42$ | $29.82\pm 0.98$ | $50.59\pm 0.26$
GraphSAGE | $80.43\pm 0.39$ | $69.56\pm 0.29$ | $77.47\pm 0.76$ | $47.06\pm 1.88$ | $35.62\pm 1.21$ | $50.21\pm 0.31$
GIN | $76.77\pm 0.88$ | $67.67\pm 0.34$ | $77.03\pm 0.53$ | $32.18\pm 1.98$ | $25.08\pm 1.36$ | $46.65\pm 0.33$
APPNP | $82.57\pm 0.10$ | $70.53\pm 0.25$ | $78.33\pm 0.39$ | $40.44\pm 2.02$ | $29.20\pm 1.45$ | $49.21\pm 0.37$
JKNet | $79.43\pm 0.53$ | $70.36\pm 0.33$ | $77.80\pm 0.82$ | $42.43\pm 1.76$ | $35.52\pm 1.06$ | $49.83\pm 0.58$
DAGNN | $83.93\pm 0.63$ | $72.56\pm 1.76$ | $80.50\pm 0.75$ | $47.29\pm 0.62$ | $36.23\pm 1.58$ | $50.78\pm 0.62$
GPR-GNN | $81.85\pm 0.38$ | $70.64\pm 0.65$ | $79.78\pm 0.43$ | $60.83\pm 1.37$ | $49.21\pm 1.15$ | $50.01\pm 0.53$
MM-GNN | $84.21\pm 0.56$ | $73.03\pm 0.58$ | $\underline{80.26\pm 0.69}$ | $63.32\pm 1.31$ | $51.38\pm 1.73$ | $\bm{51.73\pm 0.35}$
### 6.2. Baselines
We compare our proposed MM-GNN model against eight widely-used GNN models,
including four baseline GNNs ( GCN (Kipf and Welling, 2016), GAT (Veličković
et al., 2017), GraphSAGE (Hamilton et al., 2017), GIN (Xu et al., 2018a)) and
four state-of-the-art GNNs (APPNP (Klicpera et al., 2018), JKNet (Xu et al.,
2018b), DAGNN (Liu et al., 2020), GPRGNN (Chien et al., 2020)). Considering
that the two webpage networks (Chameleon, Squirrel) are heterophily graph
datasets unlike other datasets used in this paper, we also choose GPR-GNN
(Chien et al., 2020), which designed for graphs with heterophily as one of the
baseline models.
### 6.3. Experimental Setup
We evaluate MM-GNN on the semi-supervised node classification task compared
with state-of-the-art methods. For each dataset, we all run our model and
other baseline models for 10 times and compute the mean and standard deviation
of the results. For Facebook social networks, we randomly generate 3 different
data splits with an average train/val/test split ratio of 60%/20%/20% in the
Table 3. And we further generate data splits with lower training ratio
(40%,20%,10%) and conduct experiments to validate the robustness of our model
in the Table 6. For citation networks (Cora, Citeseer, and Pubmed), we use the
public split recommended by (Kipf and Welling, 2016) with fixed 20 nodes per
class for training, 500 nodes for validation, and 1000 nodes for testing. For
webpage networks (Chameleon, Squirrel) (Pei et al., 2020) and image network
(Flickr (Zeng et al., 2019)), we all use the public splits recommended in the
original papers. The max moment ordinal $K$ for MM-GNN is searched in {1, 2,
3} in our experiments. To keep fairness, we search the shared hyper-parameters
for all models in the same searching spaces. Specifically, we train
200/400/600 epochs for each model. we search the hidden units in {16, 32, 64,
128, 256} and search the number of GNN layers in {2, 6, 10, 16}. We search
learning rate from $1e-4$ to $1e-1$ and the weight decay from $1e-4$ to
$1e-2$. For each experiment, we use the Adam optimizer (Kingma and Ba, 2014)
to train each GNN model on Nvidia Tesla V100 GPU.
### 6.4. Main Results
We compare the performance of MM-GNN to the state-of-the-art methods in Table
3 and Table 4. Compared with all baselines, MM-GNN generally achieves the best
performance on all datasets. For social networks, we observe from Table 3 that
our MM-GNN model gains significant improvement on all datasets. Besides, MM-
GNN also gain significant improvements on other public real-world datasets,
including citation networks, webpage networks and image networks. And the
results demonstrate that MM-GNN gains superior performance on the graphs of
various types.
### 6.5. Ablation Study
To estimate the effectiveness of each component in MM-GNN, we conduct
comprehensive ablation studies by removing certain component at a time. The
results are shown in the Table 5.
Table 5. Performance (accuracy) on node classification task compared with single-moment GNN models. M-1, M-2, M-3 represent the single-moment GNN models, where M-$k$ denotes the model whose aggregator only uses the $k$-th order moment of neighbors. Ensemble denotes the mean ensemble of the three single-moment models. Model | M-1 | M-2 | M-3 | Ensemble | MM-GNN
---|---|---|---|---|---
Fusion mode | — | — | — | Mean | MLP | Attention
Howard | 93.15 | 93.61 | 93.83 | 93.70 | 94.00 | 94.37
Simmons | 90.90 | 92.36 | 92.45 | 91.85 | 92.99 | 93.66
GWU | 88.57 | 89.79 | 90.44 | 89.69 | 90.48 | 91.11
Cora | 82.31 | 81.70 | 80.07 | 81.98 | 82.58 | 84.21
CiteSeer | 71.31 | 70.70 | 71.53 | 71.28 | 72.33 | 73.03
PubMed | 79.23 | 78.01 | 78.27 | 78.93 | 79.35 | 80.26
Chameleon | 48.17 | 48.09 | 48.24 | 49.76 | 50.05 | 51.93
Squirrel | 35.58 | 36.29 | 35.97 | 36.09 | 36.35 | 38.92
Flickr | 50.37 | 51.09 | 50.88 | 51.15 | 51.37 | 51.73
#### 6.5.1. Effects of MME
To validate the effects of Multi-order Moments Embedding (MME) module, we
design several model variants as shown in Table 5. The variants in the first
three columns use only one moment (e.g. 1st-order moment) for neighborhood
aggregation. The Ensemble-Mean in the table is another variant that ensemble
all single-moment models with the average logits of all single-moment
variants. And we observe that each single-moment model has different
performance on different datasets. And the simple ensemble of them usually
have better performance. However, our full MM-GNN (the rightmost column)
always has the best performance on all datasets against all variants.
#### 6.5.2. Effects of AMA
To validate the effects of the Attention-based Moment Adaptor (AMA) module, we
design a variant of MM-GNN by removing this module and use a simple MLP to
fuse the output of MME module. MM-GNN with Attention-based Moment Adaptor
gains the state-of-the-art results on all datasets compared with the simple
MLP fusion method, which demonstrates the superiority of the Attention-based
Moment Adaptor module.
Table 6. Performance (accuracy) on node classification task with different training set ratio on Facebook social networks. Dataset | Northeastern | UF | GWU | UGA | Howard
---|---|---|---|---|---
Training | $40\%$ | $20\%$ | $10\%$ | $40\%$ | $20\%$ | $10\%$ | $40\%$ | $20\%$ | $10\%$ | $40\%$ | $20\%$ | $10\%$ | $40\%$ | $20\%$ | $10\%$
GCN | 90.34 | 90.38 | 90.35 | 83.97 | 83.76 | 83.62 | 85.76 | 85.65 | 85.67 | 85.5 | 85.46 | 85.42 | 91.17 | 90.95 | 90.99
GAT | 88.57 | 88.63 | 88.69 | 79.79 | 79.89 | 79.85 | 83.95 | 84.07 | 83.98 | 82.87 | 82.92 | 82.79 | 89.09 | 88.92 | 88.93
GraphSAGE | 91.91 | 92.22 | 92.18 | 86.26 | 86.34 | 86.51 | 89.31 | 89.45 | 89.15 | 88.21 | 88.09 | 87.77 | 93.21 | 93.29 | 93.17
GIN | 89.45 | 89.59 | 89.57 | 85.64 | 85.44 | 85.57 | 84.27 | 83.98 | 84.07 | 87.57 | 87.44 | 87.34 | 89.40 | 89.24 | 89.31
APPNP | 89.76 | 89.94 | 89.85 | 81.89 | 81.92 | 81.87 | 87.14 | 87.11 | 87.12 | 84.79 | 84.75 | 84.67 | 89.20 | 89.51 | 89.27
JKNet | 92.23 | 92.35 | 92.28 | 85.51 | 85.67 | 85.49 | 89.12 | 89.06 | 89.02 | 87.91 | 87.82 | 87.81 | 93.15 | 93.14 | 93.17
DAGNN | 92.49 | 92.36 | 92.30 | 86.86 | 86.46 | 85.70 | 89.73 | 89.31 | 89.11 | 88.35 | 87.79 | 87.62 | 93.31 | 93.33 | 93.27
MM-GNN | 93.08 | 92.96 | 92.73 | 88.34 | 87.87 | 87.38 | 90.63 | 90.50 | 90.14 | 89.52 | 89.38 | 88.91 | 94.03 | 93.88 | 93.76
(a) Social network (b) Citation network
(c) Webpage/Image network
Figure 4. Average attention value of different moments. The value in each
block means the averaged attentions of all nodes and all feature dimensions.
And the deeper color means larger attention values learned by MM-GNN.
### 6.6. Robustness Analysis
In this section, we evaluate the robustness of our MM-GNN by changing the
training set ratio and the key hyper-parameters.
#### 6.6.1. Analysis on effects of training set ratio:
We conducted extensive experiments with different training set ratios (10%,
20%, 40%) on the Facebook social networks. And due to space limitations, we
only show results on five of these datasets. As shown in Table 6, our MM-GNN
model has the best performance on each split of the five graphs even when the
proportion of training set is small.
(a) Simmons, GWU, Cora (b) Chameleon, Squirrel, Flickr
Figure 5. Hyper-parameter analysis on max ordinal of moments $K$ used in MM-
GNN
#### 6.6.2. Analysis on the hyper-parameter $K$:
As shown in the Fig. 5, the performance of MM-GNN is improved with larger $K$
(max ordinal of moments). And when $K$ is no less than 3, the performance of
our model is convergent, which matches the data complexity.
### 6.7. Visualizations
The Attention-based Moment Adaptor (AMA) improves the classification
performance and also provides interpretability for MM-GNN, which plays an
critical role in selecting important orders of moments for each node on the
graph. To further explore the interpretability of the learned attention, we
analyze their effects through attention visualization. We output the attention
value learned by the first layer of MM-GNN for each node and compute the
averaged attention of all nodes on all hidden dimensions. As shown in the Fig.
4, the average attention values for each moment on different graph dataset
present different patterns. MM-GNN output larger weights for higher-order
moments on the social networks, while output smaller weights for higher-order
moments on citation networks. Moreover, the magnitude of learned attentions
are aligned with the performance of the single-moment model shown in Table 5,
which further demonstrates the interpretability of MM-GNN.
## 7\. Related work
In recent years, Graph Neural Networks (GNNs) have been widely studied on
graph representation learning for its expressive power (Du et al., 2021a; Yao
et al., 2022; Wang et al., 2019; Zhu et al., 2020; Monti et al., 2018; Wang et
al., 2022; Bi et al., 2022b). Spectral GNNs (Defferrard et al., 2016; Boscaini
et al., 2015; Xu et al., 2019) were first proposed based on graph signal
processing theory. Then Kipf et al. proposed GCN (Kipf and Welling, 2016),
which simplifies the convolution operation with low pass filtering on graphs
and bridges the gap between spectral GNNs and spatial GNNs. GAT further
(Veličković et al., 2017) introduced an attention mechanism to GNNs to learn
the importance of different neighbors, instead of using equal weights for
neighbors. Afterwards, some researchers turned their attention to considering
long-range dependencies (Dwivedi et al., 2021; Ying et al., 2021; Abu-El-Haija
et al., 2019). DAGNN (Liu et al., 2020) is a deeper GCN model with decoupled
transformation and aggregation strategy .
However, most existing GNNs update node representations by aggregating feature
from neighbors with single statistic (e.g., mean, max, sum). Though such
lower-order statistics can also reflect certain characteristics of neighbor
distribution, they lose the full information of the neighbor’s feature
distribution. Even for the simple Gaussian distribution which has two degree
of freedom (mean, variance), and the information of 2nd-order statistic
variance cannot be retained by mean/max/sum (1st-order statistic). Some graph
filter based methods (Dehmamy et al., 2019; Corso et al., 2020) proposed to
model graph signals by integrating multiple types of aggregators, without
considering the relationship between different signals and neighborhood
distribution. Recently, some works (Du et al., 2021b; Ma et al., 2022; Luo et
al., 2022; Bi et al., 2022a) begin to study the modeling of neighbor
distribution on the graph, but they do not consider the information loss
caused by single lower-order statistics.
## 8\. Conclusion
In this paper, we embrace the critical roles of neighbor’s feature
distribution for modeling graph data. And we introduce the multi-order moments
to model neighbor’s feature distribution on graphs, fixing the severe
limitations of existing GNNs using only single statistic for neighborhood
aggregation. Based on the method of moments, we propose a novel Mix-Moment GNN
model with adaptive attention mechanism, which includes two components, the
Multi-order Moments Embedding (MME) and the Attention-based Moment Adaptor
(AMA). MM-GNN use multi-order moments to represent neighbor’s feature
distribution and select important moments for nodes adaptively. The consistent
improvement of MM-GNN over other state-of-the-art GNNs on 15 real-world graphs
demonstrate the superiority of our method.
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# On the Stability of Random Multiple Access with Feedback Exploitation and
Queue Priority
Karim G. Seddik
Electronics Engineering Department, American University in Cairo AUC Avenue,
New Cairo 11835, Egypt. email<EMAIL_ADDRESS>
###### Abstract
In this paper, we study the stability of two interacting queues under random
multiple access in which the queues leverage the feedback information. We
derive the stability region under random multiple access where one of the two
queues exploits the feedback information and backs off under negative
acknowledgement (NACK) and the other, higher priority, queue will access the
channel with probability one. We characterize the stability region of this
feedback-based random access protocol and prove that this derived stability
region encloses the stability region of the conventional random access (RA)
scheme that does not exploit the feedback information.
## I Introduction
The stability of interacting queues has been extensively considered in
literature. Several works have considered the characterization of the
stability region of interacting queues under random access protocols. The
stability region is characterized for the case $M=2$ and $M=3$ interacting
queues as well as the case of $M>3$ with symmetric arrivals. The stability
region for the general case of $M>3$ with asymmetric arrivals is still an open
problem and only inner achievable bounds are known.
Recently, many papers have considered the problem of interacting queues in
different contexts. For example, [1] considers the problem of interacting
queues in a TDMA system where a relay is used to help the source nodes in
forwarding their lost packets. In [2], the stability of interacting queues
under a random access protocol in the context of Cognitive Radio Network was
derived. In [3], the stability region of two interacting queues under random
access protocol where the two queues harvest energy was characterized. Other
works can be found in [4, 5], where derivations of the stability regions in
the context of different cognitive radio networks were considered.
In this paper, we derive the stability region of a two-queue random access
(RA) protocol with priorities. The queues will apply the conventional RA
protocol but in the case of packet loss due to collision the two queues will
exploit the feedback information to provide some level of coordination. We set
a priority to one of the two queues as follows. In the case of a negative
acknowledgement, the queue with the higher priority will attempt transmission
in the following time slot with probability one and the other queue will back
off to allow for collision-free transmission of the higher priority queue.
Clearly, this will enhance the service rate for the higher priority queue but
more interestingly it will also improve the service rate for the other, less
priority queue as will be explained later. We derive an expression for the
boundary of the stability region and prove that the RA with priority scheme
encloses the stability region of the conventional RA scheme.
To the best of our knowledge, the problem of characterizing the stability
region of the random access protocol with feedback leveraging has not been
considered before. We will characterize the stable arrival rates region and
prove that it contains that of the conventional random multiple access scheme
(with no feedback exploitation).
The rest of the paper is organized as follows. The system model is presented
in Section II. The performance of the proposed scheme is investigated in
Section III. The paper is concluded in Section IV. We have moved most of the
proofs to the appendices to preserve the flow of ideas in the paper.
## II System Model
The system model is shown in Fig. 1. We consider the case of two interacting
packet queues, namely $Q_{1}$ and $Q_{2}$. $Q_{1}$ and $Q_{2}$ have infinite
buffers for storing fixed length packets. The channel is slotted in time and
any slot duration equals one packet transmission time. The arrival processes
at the two queues, $Q_{1}$ and $Q_{2}$, are modeled as Bernoulli arrival
processes with means $\lambda_{1}$ and $\lambda_{2}$, respectively [3]. Under
our system model assumptions, the average arrival rates are $\lambda_{1}$ and
$\lambda_{2}$ packets per time slot, and are bounded as $0<\lambda_{i}<1$,
$i=1,2$111The maximum service rate in our model is 1 packet/slot, since the
slot duration equals one packet transmission time, then the arrival rates must
be less than 1 otherwise the system will be unstable [3].. We can assume that
the packets arrive at the start of the time slot.
Figure 1: The system model.
The channel is modeled as a collision channel, where packet loss results only
in the case of simultaneous transmissions from the two queues. If only one
queue attempts transmitting at a given time slot, the packet is considered to
be correctly received [6, 3]. In the random access phase, the first queue
accesses the channel with probability $p_{1}$ whenever it has packets to send
and the second queue will access the channel with probability $p_{2}$ whenever
it has packets to send. If at any time slot some queue is empty, it will not
attempt any channel access.
In this paper, we will consider the use of the feedback information that is
leveraged at the queues in the case of collision. In the conventional random
multiple access system and in the case of collision, the collided packets stay
on the head of the queues and retransmissions are attempted employing the same
random multiple access scheme. In this paper, we consider a system where the
feedback information is leveraged at the queues and a priority is set to the
first queue; in the next time slot after collision, queue 2 ($Q_{2}$) will
back off and queue 1 ($Q_{1}$) will retransmit its collided packet to allow
for collision-free transmission of $Q_{1}$; after that the two queues return
to the conventional random multiple access scheme. The priority set to queue 1
can be due to some quality of service (QoS) requirement that is different from
the QoS requirement of queue 2. The interesting result is that although the
feedback will enhance the service of queue 1 by setting a higher priority to
it, the service will be enhanced as well for queue 2 as will be explained
later.
## III The Stability Region for the Feedback-Based Random Access Protocol
with Priorities
In this section, we will characterize the stability region for the feedback-
based random access scheme. Stability can be loosely defined as having a
certain quantity of interest kept bounded. In our case, we are interested in
the queue size being bounded. For an irreducible and aperiodic Markov chain
with countable number of states, the chain is stable if and only if it is
positive recurrent, which implies the existence of its stationary
distribution. For a rigorous definition of stability under more general
scenarios see [6] and [7].
If the arrival and service processes of a queueing system are strictly
stationary, then one can apply Loynes’s theorem to check for stability
conditions [8]. This theorem states that if the arrival process and the
service process of a queueing system are strictly stationary, and the average
arrival rate is less than the average service rate, then the queue is stable,
otherwise it is unstable.
Characterizing the stability region will be a difficult problem due to the
interaction of the two queues and due to the fact that the service for one
queue will depend on the state of the other queue. We will consider the use of
the Dominant System concept that was proposed in [6] to characterize the
stability region of the conventional RA scheme. We will define two dominant
systems tailored to match our feedback-based random access scheme and in each
of the two systems we will determine the boundaries of the stability region.
### III-A Dominant System 1
In any dominant system, we define a system that “stochastically dominates” our
system, that is the queues lengths in the dominant system are always larger
than the queues lengths in our system if both, the dominant system and our
system, start from the same initial state and have the same arrivals and
encounter the same packet collisions.
Figure 2: Queue 1, $Q_{1}$, Markov chain model for Dominant System 1.
For the first Dominant System, we assume that queue 2 will always have packets
to transmit; even if the queue was empty dummy packets will be transmitted
from queue 2. Clearly this will set a dominant system to our system since the
transmission of dummy packets can only result in more collisions and packet
losses. If for a given arrival rate pair ($\lambda_{1}$, $\lambda_{2}$) the
first dominant system is stable then clearly our system will be stable.
Therefore, the stability region of the first dominant system will provide an
inner bound for our system stability region.
For queue 1, the Markov chain describing the evolution of the queue is shown
in Fig. 2. Note that the Markov chain has two classes of states, namely,
$k_{F}$ and $k_{R}$ and $k=0,1,2,\cdots$. The subscript $F$ denotes first
transmission states and the subscript $R$ denotes retransmission states. Note
that in the retransmission states, queue 1 packet will always be delivered
since there is no collisions in these states (queue 2 is backing off); in
these states, either queue 1 length decreases by 1 if no arrival occurs or the
queue length will remain the same if an arrival occurs while being in these
retransmission states since the packet on the head of the queue is
successfully transmitted with probability 1.
The stability condition for queue 1 in Dominant System 1 is given in the
following lemma, which is proved in Appendix A.
###### Lemma III.1
The arrival rates for queue 1 and queue 2 in Dominant System 1 must satisfy
the following two conditions, respectively,
$\begin{split}\lambda_{1}&<\frac{p_{1}}{1+p_{1}p_{2}}\\\
\lambda_{2}&<p_{2}(1-\lambda_{1}-\lambda_{1}p_{2})\end{split}$ (1)
for the system to be stable.
### III-B Dominant System 2
In the second Dominant System, we assume that queue 1 always has packets to
send (dummy packets are sent if the queue decides to transmit while being
empty). Again, this will decouple the interaction of the two queues since the
service rate of queue 2 will be independent of the state of queue 1.
The Markov chain for the evolution of queue 2 is shown in Fig. 3. Two classes
of states are defined in Fig. 3 and denoted by the subscripts ON and OFF. The
ON states denote the states where queue 2 can access the channel. The OFF
states denote the back off states where queue 1 is retransmitting its collided
packets. Note that the transitions from the $k_{\text{OFF}}$ state can be
either to the $k_{\text{ON}}$ state, if no arrival occurs in the slot, or to
the $(k+1)_{\text{ON}}$ state, if one arrival occurs in the slot. The OFF
states can never make a transition to a state with a lower number of packets
since in the OFF states queue 2 is in the back off mode and no access is
attempted.
The stability condition for queue 2 in Dominant System 2 is given in the
following lemma, which is proved in Appendix B (the analysis in Appendix B
will be based on the theory of homogeneous quasi birth-and-death (QBD) Markov
chains [9]).
###### Lemma III.2
The arrival rates for queue 1 and queue 2 in Dominant System 2 must satisfy
the following two conditions, respectively,
$\begin{split}\lambda_{1}&<\frac{p_{1}(1-p_{1}-\lambda_{2}p_{1})}{(1-p_{1})}\\\
\lambda_{2}&<\frac{p_{2}(1-p_{1})}{1+p_{1}p_{2}}\end{split}$ (2)
for the system to be stable.
Figure 3: Queue 2, $Q_{2}$, Markov chain model for Dominant System 2.
Note that the intersection of the two stability regions described in Lemma
III.1 and Lemma III.2 for a given access vector
$\mathbf{p}=[p_{1}\;p_{2}]^{T}$ (grey area in Fig. 4) can be interpreted as
follows. Define a new Dominant System (Dominant System 3) in which every queue
has always a packet to transmit. In this case, the transmission state of queue
1 can be represented by the two-state Markov chain model shown in Fig 5(a);
note that in this case queue 1 will be either in the “Transmission” state
denoted by $F$ or in the “Retransmission” state denoted by $R$ in Fig. 5(a).
Fig. 5(b) shows the Markov chain model for queue 2. Queue 2 will have two
states denoted by ON when queue 1 is in the F state and OFF when queue 1 is
the $R$ state (when queue 1 is in the $R$ state queue 2 will be in the back
off, OFF state). It is straightforward to show that the steady state
distributions for the two Markov chains shown in Fig. 5 are given by
$\begin{split}\pi_{F}&=\pi_{\text{ON}}=\frac{1}{1+p_{1}p_{2}}\\\
\pi_{R}&=\pi_{\text{OFF}}=\frac{p_{1}p_{2}}{1+p_{1}p_{2}}.\end{split}$ (3)
Figure 4: The union of the stability regions for the two dominant systems for
fixed access probabilities $p_{1}$ and $p_{2}$.
The service rate for queue 1 in Dominant System 3, $\mu_{1}^{\prime\prime}$,
is given by
$\begin{split}\mu_{1}^{\prime\prime}=p_{1}(1-p_{2})\pi_{F}+\pi_{R}=\frac{p_{1}}{1+p_{1}p_{2}},\end{split}$
(4)
where queue 1 is served with probability $p_{1}(1-p_{2})$ in the $F$ state and
with probability 1 in the $R$ state.
The service rate for queue 2 in Dominant System 3, $\mu_{2}^{\prime\prime}$,
is given by
$\begin{split}\mu_{2}^{\prime\prime}=p_{2}(1-p_{1})\pi_{\text{ON}}+0\times\pi_{\text{OFF}}=\frac{p_{2}(1-p_{1})}{1+p_{1}p_{2}},\end{split}$
(5)
where queue 2 is served with probability $p_{2}(1-p_{1})$ in the ON state and
with probability 0 in the OFF state.
(a) The two-state Markov chain model for queue 1 transmission state in
Dominant System 3.
(b) The two-state Markov chain model for queue 2 transmission state in
Dominant System 3.
Figure 5: Dominant System 3 Markov chain model.
### III-C The Stability Region of the Random Access Protocol with Priorities
In this section, we derive the expression for the stability region of the
random access scheme with feedback exploitation where a priority is set to one
of the two queues. The following Lemma characterizes the stability region for
fixed random access probabilities, $p_{1}$ and $p_{2}$, for queue 1 and queue
2, respectively.
###### Lemma III.3
For a fixed random access probability vector $\mathbf{p}=[p_{1}\;p_{2}]^{T}$,
the stability region $\mathcal{R}(\mathbf{p})$ of the random access with
priorities is the union of the two regions described by
$\begin{split}\lambda_{2}<p_{2}(1-\lambda_{1}-\lambda_{1}p_{2})\end{split}$
(6)
when
$\begin{split}\lambda_{1}<\frac{p_{1}}{1+p_{1}p_{2}}\end{split}$ (7)
and
$\begin{split}\lambda_{1}<\frac{p_{1}(1-p_{1}-\lambda_{2}p_{1})}{(1-p_{1})}\end{split}$
(8)
when
$\begin{split}\lambda_{2}<\frac{p_{2}(1-p_{1})}{1+p_{1}p_{2}}.\end{split}$ (9)
for the system to be stable.
###### Proof:
The result in Lemma III.3 can be proved using the tool of stochastic dominance
presented in [6]. The indistinguishability argument at the stability region
boundary states that if the original system is unstable then its queues will
saturate and they will always have packets to transmit; therefore at the
boundaries of the stability region of the original system, the original system
will be indistinguishable from the dominant system and thus has the same
stability region boundaries [6]. ∎
The next theorem characterizes the entire stability region for the random
access protocol with priorities.
###### Theorem III.4
The boundary of the stability region, $\mathcal{R}$, of the random access
protocol with priorities, which is defined as the union of the
$\mathcal{R}(\mathbf{p})$ regions for the different
$\mathbf{p}=[p_{1}\;p_{2}]^{T}$ as
$\begin{split}\mathcal{R}=\bigcup_{\mathbf{p}\in[0,1]^{2}}\mathcal{R}(\mathbf{p})\end{split}$
(10)
can be characterized as
$\begin{split}\lambda_{2}=\left\\{\begin{array}[]{ll}1-2\lambda_{1}&\lambda_{1}\leq\frac{1}{3}\\\
\frac{(1-\lambda_{1})^{2}}{4\lambda_{1}}&\lambda_{1}>\frac{1}{3}.\end{array}\right.\end{split}$
(11)
###### Proof:
First, we will derive the boundary of the stability region defined in lemma
III.1, which can be found as
$\begin{split}\lambda_{2}^{*}(\lambda_{1})=&\mathbf{max}_{p_{1},p_{2}}\;p_{2}(1-\lambda_{1}-\lambda_{1}p_{2})\\\
&\text{{subject to }}0\leq p_{1}\leq 1,\;0\leq p_{2}\leq
1,\;\lambda_{1}<\frac{p_{1}}{1+p_{1}p_{2}}.\end{split}$ (12)
Ignoring the constraints in the last optimization problem and differentiating
the cost function in the last expression with respect to $p_{2}$ and equating
the derivative to 0 we can get the optimal value for $p_{2}$, denoted by
$p_{2}^{*}$, as222it is straightforward to prove that the cost function is
concave in $p_{2}$.
$p_{2}^{*}=\frac{1-\lambda_{1}}{2\lambda_{1}}.$ (13)
Note that for $\lambda_{1}\geq\frac{1}{3}$, we have $p_{2}^{*}\leq 1$. Also,
for $p_{1}=1$ and $p_{2}^{*}=\frac{1-\lambda_{1}}{2\lambda_{1}}$, the maximum
value for the first queue arrival rate is
$\frac{p_{1}}{1+p_{1}p_{2}}=\frac{2\lambda_{1}}{1+\lambda_{1}}>\lambda_{1}$
(i.e., the last constraint, $\lambda_{1}<\frac{p_{1}}{1+p_{1}p_{2}}$ is
satisfied with $p_{1}=1$), which means that for $\lambda_{1}\geq\frac{1}{3}$,
the value for $p_{2}$ that maximizes $\lambda_{2}$ for a given $\lambda_{1}$
is given by $p_{2}^{*}=\frac{1-\lambda_{1}}{2\lambda_{1}}$, with all the
constraints in (12) not being violated.
For $\lambda_{1}<\frac{1}{3}$, following similar steps to the
$\lambda_{1}\geq\frac{1}{3}$ case, we can easily prove that the value for
$p_{2}$ that maximizes $\lambda_{2}$ is giving by $p_{2}^{*}=1$; clearly the
values of $p_{1}=1$ and $p_{2}^{*}=1$ can be easily checked to satisfy the
constraints in (12) for $\lambda_{1}<\frac{1}{3}$.
Substituting the optimal values for $p_{2}$ for the different ranges of
$\lambda_{1}$ we can easily get the boundary of the stability region spanned
by the expression in lemma III.1 to be given by
$\begin{split}\lambda_{2}=\left\\{\begin{array}[]{ll}1-2\lambda_{1}&\lambda_{1}\leq\frac{1}{3}\\\
\frac{(1-\lambda_{1})^{2}}{4\lambda_{1}}&\lambda_{1}>\frac{1}{3}.\end{array}\right.\end{split}$
(14)
Finally, following a similar approach to that considered here it is
straightforward to show that the boundary derived in (14) is the boundary of
the stability regions defined in lemma III.2, which completes the proof. ∎
In Fig. 6, we have plotted the regions $\mathcal{R}(\mathbf{p})$, for $p_{1}$
and $p_{2}$ ranging from 0 to 1 with a step of 0.01, along with the derived
stability region boundary given in the previous theorem. Fig. 6 also shows the
stability region of the random access scheme, whose boundary is given by the
following relation [6]
$\sqrt{\lambda_{1}}+\sqrt{\lambda_{2}}=1.$ (15)
In Fig. 6, we also show the boundary of the stability region for the time
division (TD) based scheme (genie-aided), which serves as the stability region
upper bound, given by333Time Division (TD) corresponds to full coordination
between the two queues and requires knowledge of the queues arrival rates a
priori before dividing the resources (time slots in this case).
$\lambda_{1}+\lambda_{2}=1.$ (16)
Figure 6: The stability regions for the Random Access, Random Access with
Priorities, and Time Division schemes.
It is clear, and straightforward to analytically prove from the closed-form
stability region boundary expressions, that the stability region for the RA
scheme with priorities encloses the stability region of the RA scheme. This
can explained as follows. For a given arrival rate at the first queue,
$\lambda_{1}$, the RA with priority scheme will provide a better service rate
to that queue if compared to the RA scheme and this means that queue 1 will be
empty with a higher probability and this means that queue 2 will have a higher
service rate as well under the RA with priority scheme as compared to the RA
scheme. So setting a priority to the first queue in the retransmission will
also result in a service rate improvement for the second queue; this is
because the RA with priority scheme has some form of coordination between the
two queues in the retransmission stage. Allowing for collision free
retransmission from the first queue will decrease the amount of expected
collisions between the transmissions of the two queues and this will result in
better service rates for the two queues.
## IV Conclusions
In this paper, we consider the problem of deriving the stability region for
random access protocol with feedback exploitation. We consider the case of two
interacting queue with priority set to one of the two queues. The two queues
will access the channel through a conventional random access protocol and in
the case of collision the higher priority queue will access the channel in the
next slot with probability 1 while the other queue will back off. We derive
the stability region for the random access with priorities protocol and prove
that it contains the stability region for the conventional random access
protocol. We show that not only the service rate for the higher priority queue
is enhanced but also the service rate for the other queue is improved if
compared to the conventional random access protocol.
## Appendix A Proof of Lemma III.1
In this Appendix, we provide a proof for Lemma III.1. We start by calculating
the steady state distribution for the Markov chain shown in Fig. 2.
First, it is clear that $\epsilon_{0}=0$ since the queue can never be in a
retransmission state while being empty. Writing the balance equation around
$1_{R}$, we have
$\epsilon_{1}=\lambda_{1}p_{1}p_{2}\pi_{0}+\left(1-\lambda_{1}\right)p_{1}p_{2}\pi_{1}.$
(17)
Then around $0_{F}$, we have
$(\lambda_{1}p_{1}p_{2}+\lambda_{1}(1-p_{1}))\pi_{0}=\left(1-\lambda_{1}\right)\epsilon_{1}+\left(1-\lambda_{1}\right)p_{1}(1-p_{2})\pi_{1}.$
(18)
Substituting for $\epsilon_{1}$ from (17) into (18), and after some
manipulations, we can get
$\pi_{1}=\frac{\lambda_{1}\left(1-p_{1}+\lambda_{1}p_{1}p_{2}\right)}{p_{1}\left(1-\lambda_{1}\right)\left(1-\lambda_{1}p_{2}\right)}\pi_{0}.$
(19)
Substituting from (19) into (17), we get
$\epsilon_{1}=\frac{\lambda_{1}p_{2}}{1-\lambda_{1}p_{2}}\pi_{0}.$ (20)
Writing the balance equation around $1_{F}$, we have
$\begin{split}&\left(1-\lambda_{1}p_{1}\left(1-p_{2}\right)-\left(1-\lambda_{1}\right)(1-p_{1})\right)\pi_{1}=\\\
&\quad\lambda_{1}\pi_{0}+\lambda_{1}\epsilon_{1}+\left(1-\lambda_{1}\right)\epsilon_{2}+\left(1-\lambda_{1}\right)p_{1}\left(1-p_{2}\right)\pi_{2}.\end{split}$
(21)
Around $2_{R}$, we have
$\epsilon_{2}=\lambda_{1}p_{1}p_{2}\pi_{1}+\left(1-\lambda_{1}\right)p_{1}p_{2}\pi_{2}.$
(22)
To get the relation between $\pi_{1}$ and $\pi_{2}$, we can substitute for the
values of $\epsilon_{1}$, $\pi_{0}$ and $\epsilon_{2}$ from equations (17),
(18) and (22), respectively in equation (21); after some tedious manipulation,
we get
$\pi_{2}=\frac{\lambda_{1}\left(1-p_{1}+\lambda_{1}p_{1}p_{2}\right)}{p_{1}\left(1-\lambda_{1}\right)\left(1-\lambda_{1}p_{2}\right)}\pi_{1}.$
(23)
Substituting from (23) into (22), we get
$\epsilon_{2}=\frac{\lambda_{1}p_{2}}{1-\lambda_{1}p_{2}}\pi_{1}.$ (24)
Note that the Markov chain is repeating from stage 2 till the end. For $k\geq
2$, we have the following relations.
$\pi_{k}=\frac{\lambda_{1}\left(1-p_{1}+\lambda_{1}p_{1}p_{2}\right)}{p_{1}\left(1-\lambda_{1}\right)\left(1-\lambda_{1}p_{2}\right)}\pi_{k-1}.$
(25) $\epsilon_{k}=\frac{\lambda_{1}p_{2}}{1-\lambda_{1}p_{2}}\pi_{k-1}.$ (26)
The last relation can be used to prove the following relation between
$\epsilon_{k}$ and $\epsilon_{k-1}$.
$\epsilon_{k}=\frac{\lambda_{1}\left(1-p_{1}+\lambda_{1}p_{1}p_{2}\right)}{p_{1}\left(1-\lambda_{1}\right)\left(1-\lambda_{1}p_{2}\right)}\epsilon_{k-1}.$
(27)
The steady state distribution can now be written as follows.
* •
$\epsilon_{0}=0$.
* •
$\pi_{k}=\rho^{k}\pi_{0}$, $k\geq 1$ and
$\rho=\frac{\lambda_{1}\left(1-p_{1}+\lambda_{1}p_{1}p_{2}\right)}{p_{1}\left(1-\lambda_{1}\right)\left(1-\lambda_{1}p_{2}\right)}$.
* •
$\epsilon_{1}=\frac{\lambda_{1}p_{2}}{1-\lambda_{1}p_{2}}\pi_{0}$.
* •
$\epsilon_{k}=\rho^{k-1}\epsilon_{1}$, $k\geq 2$.
This steady state distribution can be easily checked to satisfy the balance
equation at any general state (details are omitted since it is a rather
straightforward, yet very tedious, procedure).
To get the value of the steady state probabilities, we apply the following
normalization requirement.
$\begin{split}&\sum_{k=0}^{\infty}(\pi_{k}+\epsilon_{k})=1\\\
&\quad\quad\rightarrow\pi_{0}+\sum_{k=1}^{\infty}(\pi_{k}+\epsilon_{k})=\pi_{0}\left(1+\frac{\lambda_{1}p_{2}}{1-\lambda_{1}p_{2}}\right)\sum_{k=0}^{\infty}\rho^{k}=1,\end{split}$
(28)
where
$\rho=\frac{\lambda_{1}\left(1-p_{1}+\lambda_{1}p_{1}p_{2}\right)}{p_{1}\left(1-\lambda_{1}\right)\left(1-\lambda_{1}p_{2}\right)}$
as defined above.
Note that for the steady state distribution to exist, i.e. to have $\pi_{0}$
to be non zero, then we must have $\rho<1$, which is the stability condition
for queue 1 in this dominant system. Therefore, the stability condition can be
stated as
$\rho<1\;\rightarrow\lambda_{1}<\frac{p_{1}}{1+p_{1}p_{2}}.$ (29)
From the normalization condition in (28), we can get the value of $\pi_{0}$ as
$\pi_{0}=\frac{p_{1}-\lambda_{1}(1+p_{1}p_{2})}{p_{1}(1-\lambda_{1})}.$ (30)
In Dominant System 1, queue 2 will be served only in the states denoted by the
subscript $F$ in Fig. 2 since in the retransmission states, denoted by the
subscript $R$ in Fig. 2, queue 2 will be in the back off mode. Hence, the
service rate, $\mu_{2}$, for queue 2 in Dominant System 1 is given by
$\begin{split}\mu_{2}&=p_{2}(1-\lambda_{1})\pi_{0}+p_{2}(1-p_{1})\lambda_{1}\pi_{0}+\sum_{k=1}^{\infty}p_{2}(1-p_{1})\pi_{k}\\\
&=p_{2}(1-p_{1}\lambda_{1})\pi_{0}+\sum_{k=1}^{\infty}p_{2}(1-p_{1})\pi_{k},\end{split}$
(31)
where in the $0_{F}$ state, and with the arrival at the beginning of the slot
assumption, queue 2 is served with a rate of $p_{2}(1-\lambda_{1})\pi_{0}$
with no arrival at the beginning of the slot since queue 1 will not attempt
any random access since it is empty, and $p_{2}(1-p_{1})\lambda_{1}\pi_{0}$
with arrival at the slot beginning; for the other first transmission states,
queue 2 will be served if it decides to access the medium, which occurs with
probability $p_{2}$, and queue 1 decides not to access the medium, which
occurs with probability $(1-p_{1})$. After some manipulation, we can write the
expression for $\mu_{2}$ as
$\mu_{2}=p_{2}(1-\lambda_{1}-\lambda_{1}p_{2}).$ (32)
For the stability of queue 2, we must have
$\lambda_{2}<\mu_{2}=p_{2}(1-\lambda_{1}-\lambda_{1}p_{2}).$ (33)
## Appendix B Proof of Lemma III.2
In this Appendix, we provide a proof for Lemma III.2. We start by calculating
the steady state distribution for the Markov chain shown in Fig. 3. The state
transition matrix, $\mathbf{\Phi}$, of the Markov chain shown in Fig. 3 can be
written as
$\mathbf{\Phi}=\left(\begin{array}[]{
ccccc}\mathbf{B}&\mathbf{A}_{0}&\mathbf{0}&\mathbf{0}&\cdots\\\
\mathbf{A}_{2}&\mathbf{A}_{1}&\mathbf{A}_{0}&\mathbf{0}&\cdots\\\
\mathbf{0}&\mathbf{A}_{2}&\mathbf{A}_{1}&\mathbf{A}_{0}&\cdots\\\
\mathbf{0}&\mathbf{0}&\mathbf{A}_{2}&\mathbf{A}_{1}&\cdots\\\
\vdots&\vdots&\vdots&\vdots&\ddots\end{array}\right)$ (34)
where
$\begin{split}\mathbf{B}&=\left(\begin{array}[]{
cc}(1-\lambda_{2})+\lambda_{2}(1-p_{1})p_{2}&0\\\ 0&0\end{array}\right),\\\
\mathbf{A}_{0}&=\left(\begin{array}[]{ cc}(1-\lambda_{2})(1-p_{1})p_{2}&0\\\
0&0\end{array}\right),\\\ \mathbf{A}_{1}&=\left(\begin{array}[]{
cc}\lambda_{2}p_{2}(1-p_{1})+(1-\lambda_{2})(1-p_{2})&1-\lambda_{2}\\\
(1-\lambda_{2})p_{1}p_{2}&0\end{array}\right),\\\
\mathbf{A}_{2}&=\left(\begin{array}[]{ cc}\lambda_{2}&\lambda_{2}\\\
\lambda_{2}p_{1}p_{2}&\lambda_{2}\end{array}\right).\end{split}$
The steady state distribution vector is given by
$\mathbf{v}=[\pi_{0}^{\prime}\;\epsilon_{0}^{\prime}\;\pi_{1}^{\prime}\;\epsilon_{1}^{\prime}\;\pi_{2}^{\prime}\;\epsilon_{2}^{\prime}\;\cdots]^{T}$
and $\mathbf{v}=\mathbf{\Phi}\mathbf{v}$.
The state transition matrix $\mathbf{\Phi}$ is a block-tridiagonal matrix;
therefore the Markov chain shown in Fig. 3 is a homogeneous quasi birth-and-
death (QBD) Markov chain [9]. Note that to make the state transition matrix a
block-tridiagonal matrix we have added a transition from the $0_{\text{OFF}}$
state to the $1_{\text{ON}}$ state as shown in Fig. 7 and this will preserve
the structure of the state transitions between the different stages in the
Markov chain. Note that adding this transition will not affect the stationary
state distribution of the Markov chain as well as the balance equations since
$\epsilon_{0}^{\prime}=0$ even with the added transition since the Markov
chain will never enter the $0_{\text{OFF}}$ state444The analysis presented
here could have been used for analyzing the Markov chain shown in Fig. 2;
however, the structure of this Markov chain allowed for the use of a simpler
approach that was adopted in Appendix A.
Figure 7: The queue 2 Markov chain with added transition between
$0_{\text{OFF}}$ and $1_{\text{ON}}$ to make the state transition matrix a
block-tridiagonal matrix.
Define the vector
$\mathbf{v}_{k}^{\prime}=[\pi_{k}^{\prime}\;\epsilon_{k}^{\prime}]^{T}$. Note
that $\mathbf{v}_{0}^{\prime}=[\pi_{0}^{\prime}\;0]^{T}$. The steady state
distribution of the Markov chain shown in Fig. 3 satisfies the following
equation [9]
$\mathbf{v}^{\prime}_{k}=\mathbf{R}^{k}\mathbf{v}_{0}^{\prime},\quad k\geq 1,$
(35)
where the $2\times 2$ matrix $\mathbf{R}$ is given by the solution to the
following equation.
$\mathbf{A}_{0}+\mathbf{R}(\mathbf{A_{1}}-\mathbf{I}_{2})+\mathbf{R}^{2}\mathbf{A_{2}}=\mathbf{0}_{2\times
2},$ (36)
where $\mathbf{I}_{2}$ is the $2\times 2$ identity matrix and
$\mathbf{0}_{2\times 2}$ is the all zeros $2\times 2$ matrix.
To get the stationary distribution, we have to find the matrix
$\mathbf{R}=\left(\begin{array}[]{ cc}r_{11}&r_{12}\\\
r_{21}&r_{22}\end{array}\right).$
Note that for $\mathbf{v}_{1}^{\prime}=\mathbf{R}\mathbf{v}_{0}^{\prime}$,
where $\mathbf{v}_{0}^{\prime}=[\pi_{0}^{\prime}\;0]^{T}$ and
$\mathbf{v}_{1}^{\prime}=[\pi_{1}^{\prime}\;\epsilon_{1}^{\prime}]^{T}$.
Therefore, we have
$\begin{split}r_{11}=\frac{\pi_{1}^{\prime}}{\pi_{0}^{\prime}}\text{ and
}r_{21}=\frac{\epsilon_{1}^{\prime}}{\pi_{0}^{\prime}}.\end{split}$ (37)
Writing the balance equation around the $0_{\text{ON}}$ in Fig. 3, we have
$\begin{split}&(\lambda_{2}p_{1}p_{2}+\lambda_{2}(1-p_{2}))\pi_{0}^{\prime}=(1-\lambda_{2})(1-p_{1})p_{2}\pi_{1}^{\prime}\\\
&\quad\rightarrow\pi_{1}^{\prime}=\frac{\lambda_{2}(1-p_{2}+p_{1}p_{2})}{(1-\lambda_{2})(1-p_{1})p_{2}}\pi_{0}^{\prime}.\end{split}$
(38)
Therefore, we have
$\begin{split}r_{11}=\frac{\lambda_{2}(1-p_{2}+p_{1}p_{2})}{(1-\lambda_{2})(1-p_{1})p_{2}}.\end{split}$
(39)
Writing the balance equation around $1_{\text{OFF}}$, we have
$\epsilon_{1}^{\prime}=\lambda_{2}p_{1}p_{2}\pi_{0}^{\prime}+(1-\lambda_{2})p_{1}p_{2}\pi_{1}^{\prime}\rightarrow\epsilon_{1}^{\prime}=\frac{\lambda_{2}p_{1}}{1-p_{1}}\pi_{0}^{\prime}.$
(40)
Therefore, we have
$r_{21}=\frac{\lambda_{2}p_{1}}{1-p_{1}}.$ (41)
To get the values of $r_{12}$ and $r_{22}$, we consider the transition across
the border shown in Fig. 8. For the Markov chain to be positive recurrent then
the probability of going across the border in both directions must be the same
[10]; hence, we have
$\begin{split}(1-\lambda_{2})(1-p_{1})p_{2}\pi_{2}^{\prime}=(\lambda_{2}p_{1}p_{2}+\lambda_{2}(1-p_{2}))\pi_{1}^{\prime}+\lambda_{2}\epsilon_{1}^{\prime}.\end{split}$
(42)
But we have $\mathbf{v}_{2}^{\prime}=\mathbf{R}\mathbf{v}_{1}^{\prime}$, from
which we have
$\pi_{2}^{\prime}=r_{11}\pi_{1}^{\prime}+r_{12}\epsilon_{1}^{\prime}$; using
(39) and (42), we can easily show that
$\begin{split}r_{12}=\frac{\lambda_{2}}{(1-\lambda_{2})(1-p_{1})p_{2}}.\end{split}$
(43)
Figure 8: The segment of the Markov chain used to calculate the values of
$r_{12}$ and $r_{22}$.
Finally, the balance equation around $2_{\text{OFF}}$ can be written as
$\epsilon_{2}^{\prime}=\lambda_{2}p_{1}p_{2}\pi_{1}^{\prime}+(1-\lambda_{2})p_{1}p_{2}\pi_{2}^{\prime}=r_{21}\pi_{1}^{\prime}+r_{22}\epsilon_{2}^{\prime}.$
(41)
Substituting for $\pi_{2}^{\prime}$ from (42), we can easily show that
$r_{22}=\frac{\lambda_{2}p_{1}}{1-p_{1}}.$ (42)
Now the matrix $\mathbf{R}$ is given by
$\mathbf{R}=\left(\begin{array}[]{cc}\frac{\lambda_{2}(1-p_{2}+p_{1}p_{2})}{(1-\lambda_{2})(1-p_{1})p_{2}}&\frac{\lambda_{2}}{(1-\lambda_{2})(1-p_{1})p_{2}}\\\
\frac{\lambda_{2}p_{1}}{1-p_{1}}&\frac{\lambda_{2}p_{1}}{1-p_{1}}\end{array}\right),$
(43)
which can be easily checked to satisfy the balance equation given by (36).
To get the stationary distribution of the Markov chain shown in Fig. 3, we
apply the following normalization requirement.
$\sum_{k=0}^{\infty}(\pi_{k}^{\prime}+\epsilon_{k}^{\prime})=1\rightarrow[1\;1]\left(\sum_{k=0}^{\infty}\mathbf{R}^{k}\right)\mathbf{v}_{0}^{\prime}=1.$
(44)
For the summation $\left(\sum_{k=0}^{\infty}\mathbf{R}^{k}\right)$ to converge
we must have the spectral radius of the matrix $\mathbf{R}$,
$\textbf{sp}(\mathbf{R})$, to be less than one [9]555The spectral radius of a
matrix is the maximum over the magnitudes of its eigenvalues.. From (43), we
can easily get $\textbf{sp}(\mathbf{R})$ to be given by
$\begin{split}\textbf{sp}(\mathbf{R})=\frac{\lambda_{2}\left(1-p_{2}-\lambda_{2}p_{1}p_{2}+2p_{1}p_{2}+\sqrt{1-2p_{2}+p_{2}^{2}+4p_{1}p_{2}-2\lambda_{2}p_{1}p_{2}-2\lambda_{2}p_{1}p_{2}^{2}+\lambda_{2}^{2}p_{1}^{2}p_{2}^{2}}\right)}{2p_{2}\left(1-\lambda_{2}-p_{1}+\lambda_{2}p_{1}\right)}.\end{split}$
(45)
The requirement that $\textbf{sp}(\mathbf{R})<1$ can be used in the last
expression to get the stability condition of the second queue arrival rate
$\lambda_{2}$ as
$\begin{split}\lambda_{2}<\frac{p_{2}(1-p_{1})}{1+p_{1}p_{2}}.\end{split}$
(46)
Going back to the normalization requirement in (44), we have
$[1\;1]\left(\sum_{k=0}^{\infty}\mathbf{R}^{k}\right)\mathbf{v}_{0}^{\prime}=[1\;1]\left(\mathbf{I}_{2}-\mathbf{R}\right)^{-1}\left[\begin{array}[]{c}\pi_{0}^{\prime}\\\
0\end{array}\right]=1.$ (47)
From the last expression, we can easily prove that $\pi_{0}^{\prime}$ is given
by
$\pi_{0}^{\prime}=\frac{p_{2}-\lambda_{2}-p_{1}p_{2}-\lambda_{2}p_{1}p_{2}}{(1-\lambda_{2})(1-p_{1})p_{2}}.$
(48)
Note that the requirement that $\pi_{0}^{\prime}>0$, i.e. a non-zero empty
queue probability, is satisfied if
$\lambda_{2}<\frac{p_{2}(1-p_{1})}{1+p_{1}p_{2}}$, which is the queue
stability condition.
The service rate, $\mu_{1}^{\prime}$, for queue 1 in Dominant System 2 can now
be expressed as
$\begin{split}\mu_{1}^{\prime}&=p_{1}(1-\lambda_{2})\pi_{0}^{\prime}+p_{1}(1-p_{2})\lambda_{2}\pi_{0}^{\prime}+p_{1}(1-p_{2})\sum_{k=2}^{\infty}\pi_{k}^{\prime}+\sum_{k=2}^{\infty}\epsilon_{k}^{\prime}\\\
&=p_{1}(1-\lambda_{2}p_{2})\pi_{0}^{\prime}+[p_{1}(1-p_{2})\;1]\left(\sum_{k=1}^{\infty}\mathbf{R}^{k}\right)\left[\begin{array}[]{c}\pi_{0}^{\prime}\\\
0\end{array}\right]\\\
&=p_{1}(1-\lambda_{2}p_{2})\pi_{0}^{\prime}+[p_{1}(1-p_{2})\;1]\;\mathbf{R}\left(\mathbf{I}_{2}-\mathbf{R}\right)^{-1}\left[\begin{array}[]{c}\pi_{0}^{\prime}\\\
0\end{array}\right]\\\
&=\frac{p_{1}(1-p_{1}-\lambda_{2}p_{1})}{(1-p_{1})},\end{split}$ (49)
where in the OFF states, queue 1 is served with probability 1 since queue 2
will be in the back off mode. For the stability of queue 1 in Dominant System
2 we must have
$\lambda_{1}<\mu_{1}^{\prime}=\frac{p_{1}(1-p_{1}-\lambda_{2}p_{1})}{(1-p_{1})},$
(50)
which completes the proof.
## References
* [1] A.K. Sadek, K.J.R. Liu, and Anthony Ephremides, “Cognitive multiple access via cooperation: Protocol design and performance analysis,” Information Theory, IEEE Transactions on, vol. 53, no. 10, pp. 3677–3696, 2007.
* [2] S. Kompella, Gam D. Nguyen, J.E. Wieselthier, and Anthony Ephremides, “Stable throughput tradeoffs in cognitive shared channels with cooperative relaying,” in INFOCOM, 2011 Proceedings IEEE, 2011, pp. 1961–1969.
* [3] Jeongho Jeon and Anthony Ephremides, “The stability region of random multiple access under stochastic energy harvesting,” in Information Theory Proceedings (ISIT), 2011 IEEE International Symposium on, 2011, pp. 1796–1800.
* [4] A. Fanous and Anthony Ephremides, “Effect of secondary nodes on the primary’s stable throughput in a cognitive wireless network,” in Information Theory Proceedings (ISIT), 2012 IEEE International Symposium on, 2012, pp. 1807–1811.
* [5] A. Fanous and A. Ephremides, “Stable throughput in a cognitive wireless network,” Selected Areas in Communications, IEEE Journal on, vol. 31, no. 3, pp. 523–533, 2013.
* [6] R. Rao and A. Ephremides, “On the Stability of Interacting Queues in a Multi-Access System,” IEEE Trans. Info. Theory, vol. 34, pp. 918–930, Sept. 1988.
* [7] W. Szpankowski, “Stability Conditions for Some Multiqueue Distributed System: Buffered Random Access Systems,” Adv. Appl. Probab., vol. 26, pp. 498–515, 1994.
* [8] R. M. Loynes, “The Stability of a Queue with Non-Independent Interarrival and Service Times,” Proc. Cambridge Philos. Soc., pp. 497–520, 1962.
* [9] G. Latouche and V. Ramaswami, Introduction to Matrix Analytic Methods in Stochastic Modeling, ASA-SIAM Series on Statistics and Applied Probability. Society for Industrial and Applied Mathematics, 1999.
* [10] R.G. Gallager, Discrete stochastic processes, vol. 101, Kluwer Academic Publishers, 1996.
|
# Closest Wannier functions to a given set of localized orbitals
Taisuke Ozaki Institute for Solid State Physics, The University of Tokyo,
Kashiwa 277-8581, Japan
###### Abstract
A non-iterative method is presented to calculate the closest Wannier functions
(CWFs) to a given set of localized guiding functions, such as atomic orbitals,
hybrid atomic orbitals, and molecular orbitals, based on minimization of a
distance measure function. It is shown that the minimization is directly
achieved by a polar decomposition of a projection matrix via singular value
decomposition, making iterative calculations and complications arising from
the choice of the gauge irrelevant. The disentanglement of bands is inherently
addressed by introducing a smoothly varying window function and a greater
number of Bloch functions, even for isolated bands. In addition to atomic and
hybrid atomic orbitals, we introduce embedded molecular orbitals in molecules
and bulks as the guiding functions, and demonstrate that the Wannier
interpolated bands accurately reproduce the targeted conventional bands of a
wide variety of systems including Si, Cu, the TTF-TCNQ molecular crystal, and
a topological insulator of Bi2Se3. We further show the usefulness of the
proposed method in calculating effective atomic charges. These numerical
results not only establish our proposed method as an efficient alternative for
calculating WFs, but also suggest that the concept of CWF can serve as a
foundation for developing novel methods to analyze electronic structures and
calculate physical properties.
## I INTRODUCTION
The Wannier functions (WFs) Wannier37 ; Kohn59 ; Marzari12 play a central
role in analyzing electronic structures of real materials and advancing
electronic structure methods alongside density functional theory (DFT)
Hohenberg1964 and the other electronic structure theories Hamann09 ;
Lechermann06 . A widely adopted method for calculating WFs involves maximizing
their localization, which can be reformulated as minimizing the spread
function characterizing the variance of WFs in real space Marzari97 . The
concept of maximal localization leads to an elegant formulation, resulting in
maximally localized Wannier functions (MLWFs), and the compact representation
of the Hamiltonian that follows allows us to analyze the electronic structures
of real materials using a localized orbital picture. However, it is also
recognized that the minimization of the spread function can often encounter
local minima, particularly in large-scale systems having complicated
electronic structures Marzari12 . To overcome this challenge, methods aimed at
automated high-throughput Wannierisation have been developed Vitale20 ;
Damle18 ; Qiao23 . In addition, the other methods for generating
(nonorthogonal) WFs have been proposed based on projection methods Ku02 ;
Lu2004 ; Qian08 , which produce compact atomic like orbitals. The
orthogonalization of such nonorthogonal orbitals obtained by the projection
can be achieved by the Löwdin orthogonalization procedure Lowdin1950 ;
Cloizeaux1964 . These orthogonalized orbitals are utilized as an initial guess
of WFs in the minimization of the spread function to obtain MLWFs Marzari97 ;
Mostofi2008 . Apart from the role of initial guess, it is worth noting that
the Löwdin orthogonalized orbitals possess an important variational property,
which in particular minimizes the sum of squared distances between the
orthogonal and original nonorthogonal orbitals in the Hilbert space Carlson57
. As long as the original nonorthogonal orbitals are localized, the
localization of the Löwdin orthogonalized orbitals in real space is guaranteed
in a sense of minimization of the distance. In quantum chemistry, this
property is leveraged to develop methods for generating localized orthogonal
orbitals. These methods, which include the natural atomic and bond orbital
method Reed88 and the intrinsic atomic orbital method Knizia13 , involve
obtaining orthogonalized orbitals through Löwdin type orthogonalization,
giving heavy weight to occupied states via the density matrix. However, in
solid state physics the variational property inherent in the Löwdin
orthogonalization has not been fully explored yet. A reason for this is that
the Löwdin orthogonalization requires calculation of $S^{-1/2}$ for the
overlap matrix $S$, which causes numerical instability for nearly ill-
conditioned matrices. Due to the difficulty, advancements in methodologies
along these lines appear to be slow. Our study aims to develop an efficient
and robust method to generate the closest Wannier functions (CWFs) to a given
set of localized guiding functions. We will demonstrate that exploiting the
variational property leads to a versatile method that eliminates the need for
iterative optimization and avoids complications related to gauge choice.
The structure of this paper is as follows: In Section II, we present the
theory of calculating CWFs. Section III provides a detailed discussion on the
implementation of the method. A series of benchmark calculations are presented
in Section IV. Finally, in Section V, we summarize the theory of CWFs and
suggest a potential role for CWFs as a foundation for developing novel
methods.
## II THEORY
Let us start by introducing Bloch functions $\\{\phi\\}$, which can be
obtained by solving the Kohn-Sham (KS) equation Kohn1965 within the density
functional theory (DFT) Hohenberg1964 , normalized as
$\displaystyle\langle\phi_{{\bf k}_{1}\mu}|\phi_{{\bf k}_{2}\nu}\rangle=N_{\rm
BC}\delta_{{\bf k}_{1}{\bf k}_{2}}\delta_{\mu\nu},$ (1)
where ${\bf k}$ and $\mu$ are the k-vector and the band index, respectively,
and $N_{\rm BC}$ is the number of primitive cells in the Born-von Karman (BvK)
boundary condition. We consider a projection of a localized guiding function
$Q$ onto the Bloch functions $\\{\phi\\}$ weighted with a window function
$w(\varepsilon)$ as
$\displaystyle|L_{{\bf R}p}\rangle=\frac{1}{N_{\rm BC}}\sum_{{{\bf
k},\mu}}{\rm e}^{-{\rm i}{\bf k}\cdot{\bf R}}|\phi_{{\bf k}\mu}\rangle a_{{\bf
k}\mu,p}$ (2)
with
$\displaystyle a_{{\bf k}\mu,p}=w(\varepsilon_{{\bf k}\mu})\langle\phi_{{\bf
k}\mu}|Q_{{\bf 0}p}\rangle,$ (3)
where ${\bf R}$ and $p$ are a translational lattice vector and the index of
localized orbital, respectively, $\varepsilon_{{\bf k}\mu}$ is the eigenvalue
of the KS equation, and $Q_{{\bf 0}p}\equiv Q_{{\bf R}p}~{}({\bf R}={\bf 0})$.
In the summation of Eq. (2), the numbers of k-point and the Bloch functions to
be included are $N_{\rm BC}$ and $N_{\rm band}$ per k-point, respectively.
Throughout the paper we do not consider the spin dependency on the formulation
for sake of simplicity, but the generalization is straightforward. The
localized functions $\\{Q_{{\bf R}p}\\}$, such as atomic orbitals, hybrid
orbitals, or molecular orbitals (MOs), are assumed to localize in the unit
cell specified by ${\bf R}$. When the window function $w(\varepsilon)$ is
taken to be unity, Eq. (2) is nothing but an expansion of $Q_{{\bf R}p}$ with
Bloch functions from the identity relation $\frac{1}{N_{\rm BC}}\sum_{{{\bf
k}\mu}}|\phi_{{\bf k}\mu}\rangle\langle\phi_{{\bf k}\mu}|=\hat{I}$. To
introduce an expansion of $Q_{{\bf R}p}$ by the Bloch functions $\\{\phi_{{\bf
k}\mu}\\}$ in a subspace, we use the following window function
$w(\varepsilon)$:
$\displaystyle
w(\varepsilon)=\frac{1-\exp(x_{0}+x_{1})}{\left(1+\exp(x_{0}))(1+\exp(x_{1})\right)}+\delta$
(4)
with the definition of $x_{0}\equiv\beta(\varepsilon_{0}-\varepsilon)$ and
$x_{1}\equiv\beta(\varepsilon-\varepsilon_{1})$, where $\beta=\frac{1}{k_{\rm
B}T}$ with the Boltzmann constant of $k_{\rm B}$ and a temperature of $T$. The
window function of Eq. (4) is obtained by subtracting 1 from the sum of two
functions of $1/(1+\exp(x))$. The last term $\delta$ is a small constant,
e.g., $10^{-12}$, which is introduced to avoid the ill-conditioning of the
matrix consisting of $a_{{\bf k}\mu,p}$ by Eq. (3) as discussed later on. As
shown in Fig. 1, the parameters of $\varepsilon_{0}$ and $\varepsilon_{1}$
($\varepsilon_{0}<\varepsilon_{1}$) determine the range of energy where the
Bloch states are included in the expansion of Eq. (2) with a large weight, and
the temperature $T$ gives the degree of smearing around $\varepsilon_{0}$ and
$\varepsilon_{1}$ in $w(\varepsilon)$. To control the degree of smearing, we
introduce the temperature of $T$ so that one can intuitively understand the
degree of smearing. It is noted that the choice of the window function by Eq.
(4) is not unique, and the other choices should give almost equivalent results
as long as they are chosen properly.
Figure 1: Window functions of Eq. (4) with $k_{\rm B}T$=0.1, 1.0, and 3.0 eV.
$\varepsilon_{0}$ and $\varepsilon_{1}$ were set to be -5.0 and 5.0 eV,
respectively, and $\delta=10^{-12}$ was used.
The function $L_{{\bf R}p}$ defined by Eq. (2) must be similar to the
localized function $Q_{{\bf R}p}$, but they are non-orthogonal to each other
in general. We now consider generating a set of closest Wannier functions
(CWFs) $\\{W_{{\bf R}p}\\}$ to a set of the localized functions $\\{L_{{\bf
R}p}\\}$ in a sense that the sum of the squared distance between $L$ and $W$
in the Hilbert space is minimized. As well as Eq. (2), noting that the CWFs
can be expressed Marzari12 as
$\displaystyle|W_{{\bf R}p}\rangle=\frac{1}{N_{\rm BC}}\sum_{{{\bf
k},\mu}}{\rm e}^{-{\rm i}{\bf k}\cdot{\bf R}}|\phi_{{\bf k}\mu}\rangle b_{{\bf
k}\mu,p},$ (5)
and defining the residual function $R$:
$\displaystyle|R_{{\bf R}p}\rangle=|L_{{\bf R}p}\rangle-|W_{{\bf R}p}\rangle,$
(6)
the distance measure (DM) function $F$ we minimize is given by
$\displaystyle F[B]$ $\displaystyle=$ $\displaystyle\sum_{p}\langle R_{{\bf
0}p}|R_{{\bf 0}p}\rangle,$ (7) $\displaystyle=$ $\displaystyle\frac{1}{N_{\rm
BC}}\sum_{{\bf k}}X[B,{\bf k}]$
with
$\displaystyle X[B,{\bf k}]={\rm tr}\left[\left(A^{{\dagger}}({\bf
k})-B^{{\dagger}}({\bf k})\right)\left(A({\bf k})-B({\bf k})\right)\right],$
(8)
where the elements of the matrices $A({\bf k})$ and $B({\bf k})$ are given by
$a_{{\bf k}\mu,p}$ and $b_{{\bf k}\mu,p}$, respectively. The second line of
Eq. (7) is obtained by considering the orthonormality of Eq. (1), and
$X[B,{\bf k}]$ of Eq. (8) is regarded as the squared Frobenius norm of
$\left(A({\bf k})-B({\bf k})\right)$ Golub1996 . The same Bloch functions as
in Eq. (2) are included in the summation of Eq. (5). The number of CWFs to be
generated is $N_{\rm CWF}$ per unit cell, being equivalent to the number of
the guiding functions per cell, and $N_{\rm CWF}$ should be smaller than or
equal to $N_{\rm band}$ to guarantee the linear independency of the subspace
spanned by the CWFs, resulting in size of $N_{\rm band}\times N_{\rm CWF}$ for
the matrices $A({\bf k})$ and $B({\bf k})$. Assuming $B^{{\dagger}}({\bf
k})B({\bf k})=I$, where $I$ is of size $N_{\rm CWF}\times N_{\rm CWF}$, the
CWFs are ensured to form a set of orthonormal functions. Under this
constraint, we consider the optimization of the DM function $F$. Furthermore,
the matrix $B({\bf k})$ is regarded as a part of a unitary matrix of size
$N_{\rm band}\times N_{\rm band}$, and will be referred to as a partial
unitary matrix in subsequent discussions.
The minimum of the DM function of Eq. (7) is obtained by choosing $B({\bf k})$
as $U({\bf k})$ which is calculated by the polar decomposition of $A({\bf
k})=U({\bf k})P({\bf k})$, where $U({\bf k})$ and $P({\bf k})$ are unitary and
hermitian, respectively Fan1955 . Let us prove the statement below. The polar
decomposition of $A({\bf k})$ is obtained via the singular value decomposition
(SVD) of $A({\bf k})$ as
$\displaystyle A=W\Sigma V^{{\dagger}}=WV^{{\dagger}}V\Sigma
V^{{\dagger}}=UP,$ (9)
where we dropped the dependency on ${\bf k}$ for simplicity of the notation,
and hereafter we will denote the dependency if necessary. $W$ and $V$ are the
left and right singular matrices in size of $N_{\rm band}\times N_{\rm CWF}$
and $N_{\rm CWF}\times N_{\rm CWF}$, respectively, and $\Sigma$ is the
singular value diagonal matrix in size of $N_{\rm CWF}\times N_{\rm CWF}$.
Note that $U\equiv WV^{{\dagger}}$ and $P\equiv V\Sigma V^{{\dagger}}$. It is
worth mentioning that $W$ and $U$ are partial unitary matrices, and hold
$W^{{\dagger}}W=I$ ($WW^{{\dagger}}\neq I$) and $U^{{\dagger}}U=I$
($UU^{{\dagger}}\neq I$), and that $V$ is a full unitary matrix holding
$V^{{\dagger}}V=I$ ($VV^{{\dagger}}=I$). We evaluate $X$ of Eq. (8) for each
${\bf k}$ with both the matrix $U$ obtained by the polar decomposition and an
arbitrary partial unitary matrix $B$, and calculate the difference as
$\displaystyle X[U]-X[B]$ $\displaystyle=$ $\displaystyle 2{\rm
tr}\left[\frac{1}{2}\left(A^{{\dagger}}B+B^{{\dagger}}A\right)-P\right],$ (10)
$\displaystyle=$ $\displaystyle 2{\rm tr}\left[\Sigma D-\Sigma\right],$
$\displaystyle=$ $\displaystyle 2\sum_{n}\sigma_{n}\left(d_{nn}-1\right)$
with
$\displaystyle
D=\frac{1}{2}\left(V^{{\dagger}}U^{{\dagger}}BV+V^{{\dagger}}B^{{\dagger}}UV\right).$
(11)
The second line of Eq. (10) is derived by using the polar decomposition of $A$
and $P=V\Sigma V^{{\dagger}}$, and $\sigma_{n}$ and $d_{nn}$ in the third line
are diagonal elements of $\Sigma$ and $D$, respectively. The upper bound of
the diagonal elements of the hermitian matrix $D$ is found to be unity, since
the matrix $D$ consists of the sum of the product of partial unitary matrices
of $V^{{\dagger}}U^{{\dagger}}$ ($V^{{\dagger}}B^{{\dagger}}$) and $BV$
($UV$). Another proof for the upper bound is also given based on the Cauchy-
Schwarz inequality in the appendix. By considering the upper bound of the
diagonal elements of $D$ and $0\leq\sigma_{n}$, the third line of Eq. (10)
leads to the following inequality:
$\displaystyle X[U]\leq X[B].$ (12)
The equality of Eq. (12) holds if $B=U$. If some of $\sigma_{n}$ is zero, the
corresponding $d_{nn}$ in Eq. (10) can be arbitrarily chosen. So, we see from
Eq. (11) that $U$ is not uniquely determined. If all the singular values of
$\sigma_{n}$ are positive, all the $d_{nn}$s should be unity when $X[U]=X[B]$.
The uniqueness of the solution can be confirmed as follows: Since the matrix
$D$ consists of the products of the partial unitary matrices as discussed
above, the case that all the diagonal elements $d_{nn}$ are unity happens when
$\displaystyle U^{{\dagger}}B+B^{{\dagger}}U=2I,$ (13)
which is derived by equating $D=I$ and multiplying $V$ and $V^{{\dagger}}$
from the left and right of the equation, respectively. By noting again that
$B$ and $U$ in Eq. (13) are the partial unitary matrices, it is found that Eq.
(13) holds if and only if $B=U$. Thus, we have proven the statement that the
minimum of the DM function of Eq. (7) is uniquely determined when $B=U$ as
long as all the singular values of $\sigma_{n}$ are positive, since $F[B]$ of
Eq. (7) consists of the sum of $X[B,{\bf k}]$ over k. The uniqueness of $U$
itself is related to that of the SVD for the matrix $A$ of Eq. (3). If $A$ has
$N_{\rm CWF}$ positive singular values which are non-degenerate, the SVD is
uniquely determined except for the non-unique phases in $W$ and $V$. Even if
there are degenerate singular values, the matrix $U$ is invariant, since the
freedom of the unitary transformation $K$ for the degenerate left and right
singular vectors is canceled out as
$U=WKK^{{\dagger}}V^{{\dagger}}=WV^{{\dagger}}$ with $KK^{{\dagger}}=I$.
Further, noting that the non-unique phases in $W$ and $V$ are canceled out
when the matrix $U$ is computed as $WV^{{\dagger}}$, we conclude that $U$ is
uniquely determined if $A$ has the $N_{\rm CWF}$ positive singular values. The
violation from the positive definiteness of the singular values can be avoided
by the small constant of $\delta$ in Eq. (4). On the other hand, if $\delta$
in Eq. (4) is set to be 0 and a narrow window, including fewer than $N_{\rm
CWF}$ eigenstates for a given ${\bf k}$, is used with a small $k_{\rm B}T$,
the matrix $A$ has $N_{\rm PSV}$ positive singular values, where $N_{\rm
PSV}<N_{\rm CWF}$. In this case, a subspace of the $\left(N_{\rm CWF}-N_{\rm
PSV}\right)$ dimension is arbitrary chosen to form $W$ and $V$ of the
dimension of $N_{\rm CWF}$ in addition to the subspace of the $N_{\rm PSV}$
dimension, which breaks the symmetry of CWFs. The situation can be avoided by
introducing a small constant $\delta$ in the window function of Eq. (4). The
simple treatment restores the symmetry of CWFs, allowing us to calculate
symmetry preserving CWFs for a wide variety of choices in the energy range of
the window function. It is also noted that the non-unique phase of the Bloch
functions is canceled out via the polar decomposition of $A$ and Eq. (5).
Thus, we see that the proposed method is free from complications arising from
the choice of gauge.
Using Eqs. (8) and (9), the minimum of the DM function $F$ at $B=U$ is given
by
$\displaystyle F[U]=\frac{1}{N_{\rm BC}}\sum_{{\bf k},p}\left(\sigma_{{\bf
k}p}-1\right)^{2},$ (14)
where $\sigma_{{\bf k}p}$ is a singular value of the matrix $A({\bf k})$. From
Eq. (14), we find that the mean squared distance between $L$ and $W$ is
related to the deviation of $\sigma$ from unity.
The proposed method based on the polar decomposition of $A$ is closely related
to the Löwdin orthogonalization Lowdin1950 as shown below. The Fourier
transform of the overlap integrals for $\\{L\\}$ is given by
$\displaystyle\sum_{{\bf R}}{\rm e}^{{\rm i}{\bf k}\cdot{\bf R}}\langle
L_{{\bf 0}p}|L_{{\bf R}q}\rangle=\sum_{\mu}a_{{\bf k}\mu,p}^{*}a_{{\bf
k}\mu,q},$ (15)
which is obtained by noting that $\frac{1}{N_{\rm BC}}\sum_{{\bf R}}{\rm
e}^{{\rm i}({\bf k}-{\bf k}^{\prime})\cdot{\bf R}}=\delta_{{\bf k}{\bf
k}^{\prime}}$. By writing Eq. (15) as $S({\bf k})=A^{{\dagger}}({\bf k})A({\bf
k})$ in a matrix form, and using Eq. (9), we have the following relation:
$\displaystyle S({\bf k})=A^{{\dagger}}({\bf k})A({\bf k})=V^{{\dagger}}({\bf
k})\Sigma^{2}({\bf k})V({\bf k}).$ (16)
Comparing Eq. (16) with $P({\bf k})=V({\bf k})\Sigma({\bf
k})V^{{\dagger}}({\bf k})$, one obtains a relation $P({\bf k})=S^{1/2}({\bf
k})$. If $P({\bf k})$ is invertible, the matrix $U({\bf k})$ is given with Eq.
(9) by
$\displaystyle U({\bf k})=A({\bf k})S^{-1/2}({\bf k}).$ (17)
The matrix $U$ obtained by Eq. (17) is exactly equivalent to that by the
Löwdin orthogonalization Lowdin1950 , and the closest property of the Löwdin
orthogonalized orbitals to a given set of orbitals was proven in Ref.
Carlson57 . On the other hand, the proposed method does not require the
calculation of $S^{-1/2}$, and can be applied in a numerical stable manner
even to the case that the matrix $A$ is nearly ill-conditioned. The definition
of $A$ by Eq. (3) with the window function allows us to calculate CWFs in
various respects, e.g., heavily weighting to occupied states and disentangling
of localized states embedded in the wider bands as demonstrated in Section IV.
In this sense, the method we propose can be regarded as a generalization of
the Löwdin orthogonalization.
The disentanglement of bands can be easily performed by properly selecting the
window function of Eq. (3). Since the $N_{\rm band}(\geq N_{\rm CWF})$ Bloch
functions per k-point are included in Eqs. (2) and (5), it should be
emphasized that the proposed method always disentangles $N_{\rm band}$ states
to generate $N_{\rm CWF}$ Wannier functions. A couple of examples will be
shown in Section IV, including valence and low-lying conduction bands for the
diamond Si and narrow $3d$-bands embedded in the wider $4s$-band for the face
centered cubic (FCC) Cu.
Once the $k$-dependent $U({\bf k})$ are obtained by the polar decomposition,
tight-binding (TB) parameters are calculated as expectation values of the KS
Hamiltonian $\hat{H}_{\rm KS}$ by summing contributions over ${\bf k}$ and
$\mu$ as
$\displaystyle t_{{\bf 0}p,{\bf R}q}$ $\displaystyle=$ $\displaystyle\langle
W_{{\bf 0}p}|\hat{H}_{\rm KS}|W_{{\bf R}q}\rangle,$ (18) $\displaystyle=$
$\displaystyle\sum_{{\bf k},\mu}\varepsilon_{{\bf k}\mu}u_{{\bf
k}\mu,p}^{*}u_{{\bf k}\mu,q}{\rm e}^{-{\rm i}{\bf k}\cdot{\bf R}},$
where $u_{{\bf k}\mu,q}$ is the matrix element of $U({\bf k})$. Like with
MLWFs, the TB parameters can be used for the Wannier interpolation in
calculations of band structures and physical quantities.
The computational procedure to generate the CWFs is summarized as follows:
1. 1.
Determining $\varepsilon_{0}$, $\varepsilon_{1}$, and $k_{\rm B}T$ in Eq. (4)
by checking the band dispersion of a system of interest. The parameters should
be chosen properly so that the targeted eigenstates can be included.
2. 2.
Choosing a set of localized orbitals {$Q$}. Atomic orbitals, hybrid orbitals,
and MOs might be possible choices. Depending on the symmetry of the targeted
eigenstates, a proper set of localized orbitals needs to be chosen, e.g.,
$d$-orbitals need to be employed in the generation of CWFs for the $d$-bands
in FCC Cu as shown later on.
3. 3.
Calculation of $A({\bf k})$ by Eq. (3). The calculation of the overlap
$\langle\phi_{{\bf k}\mu}|Q_{{\bf 0}p}\rangle$ depends on the implementation
of the KS method.
4. 4.
Performing the SVD of $A({\bf k})$. A proper numerical library might be used.
5. 5.
Calculation of $U({\bf k})$. Using the left and right singular matrices,
$W({\bf k})$ and $V({\bf k})$, of $A({\bf k})$, $U({\bf k})$ is calculated as
$W({\bf k})V^{{\dagger}}({\bf k})$.
6. 6.
Calculations of the CWFs and TB parameters. The CWFs can be calculated using
Eq. (5) with $B=U$ obtained by the step 5. The calculation is expressed by the
sum of the matrix-matrix product over ${\bf k}$. Also, the TB parameters are
obtained by Eq. (18).
It should be stressed that the minimization of $F$ can be efficiently
performed in a non-iterative manner by the steps 1 to 6 using the polar
decomposition via the SVD. The computational cost of each step is estimated to
be $O(N_{\rm BC}N_{\rm band}N_{\rm CWF}N_{\rm basis})$, $O(N_{\rm BC}N_{\rm
band}^{3})$, $O(N_{\rm BC}N_{\rm band}N_{\rm CWF}^{2})$, or $O(N_{\rm
BC}N_{\rm band}N_{\rm CWF}N_{\rm basis})$ for the step 3, 4, 5, or 6,
respectively, where $N_{\rm basis}$ is the number of basis functions to expand
the KS orbitals. If the localized basis functions are used, the computational
cost of the step 3 becomes $O(N_{\rm BC}N_{\rm band}N_{\rm CWF})$.
For isolated systems, the Brillouin zone sampling is limited to only the
$\Gamma$ point. No modification of the proposed method is needed to calculate
the CWFs.
The theoretical framework we have discussed so far is general for any
implementation of the DFT-KS method. However, the choice of the the localized
functions $\\{Q\\}$ in Eq. (3) may depend on the implementation. So, we will
discuss how the localized functions $\\{Q\\}$ can be properly generated in
Section III.
## III IMPLEMENTATION
### III.1 General
We have implemented the proposed method into the OpenMX DFT software package
OpenMX ; Ozaki2005 ; Duy2014 which is based on norm-conserving
pseudopotentials (PPs) MBK1993 ; Theurich2001 and optimized pseudo-atomic
orbitals (PAOs) Ozaki2003 ; Ozaki2004 as basis set. All the benchmark
calculations were performed with a computational condition of a production
level. The basis functions used are listed in Table 1. In the abbreviation of
basis functions such as H7.0-s2p2d1, H stands for the atomic symbol, 7.0 the
cutoff radius (Bohr) in the generation by the confinement scheme Ozaki2003 ;
Ozaki2004 , and s2p2d1 means the employment of two, two, and one optimized
radial functions for the $s$-, $p$-, and $d$-orbitals, respectively. The
radial functions were optimized by a variational optimization method Ozaki2003
. These basis functions we used can be regarded as at least double zeta plus
double polarization basis sets if we follow the terminology of Gaussian or
Slater-type basis functions. Valence states included in the PPs are listed in
Table 1. All the PPs and PAOs we used in the study were taken from the
database (2019) in the OpenMX website OpenMX , which were benchmarked by the
delta gauge method Lejaeghere2016 . Real space grid techniques were used for
the numerical integrations and the solution of the Poisson equation using fast
Fourier transform (FFT) with the energy cutoff of 250 to 500 Ryd Ozaki2005 .
We used a generalized gradient approximation (GGA) proposed by Perdew, Burke,
and Ernzerhof to the exchange-correlation functional Perdew1996 . An
electronic temperature of 300 K was used to count the number of electrons by
the Fermi-Dirac function for all the systems we considered. For all the
calculations, $\delta=10^{-12}$ was used in the window function of Eq. (4),
and $N_{\rm band}$ was set to be equilavent to the number of basis functions
in both the summation of Eqs. (2) and (5).
Table 1: Basis functions and valence states included in PPs. ∗For the calculations of atomic charges, the basis functions listed in Table 3 were used. Element | | | Basis functions | | | Valence states in PP
---|---|---|---|---|---|---
H | | | H7.0-s2p2d1 | | | $1s$
C | | | C6.0-s3p2d2 | | | $2s$, $2p$
N | | | N6.0-s3p2d2 | | | $2s$, $2p$
Na | | | ∗Na9.0-s3p3d2f2 | | | $2s$, $2p$, $3s$
Si | | | Si7.0-s2p2d2f1 | | | $3s$, $3p$
Cl | | | ∗Cl9.0-s3p3d2f2 | | | $3s$, $3p$
S | | | S7.0-s3p2d2f1 | | | $3s$, $3p$
Cu | | | Cu6.0-s3p3d3f1 | | | $3s$, $3p$, $3d$, $4s$
Se | | | Se7.0-s3p2d2 | | | $4s$, $4p$
Bi | | | Bi8.0-s3p2d2f1 | | | $5d$, $6s$, $6p$
### III.2 Choice of Guiding Functions $\\{Q\\}$
Depending on the targeted eigenstates in the window function of Eq. (4), one
can choose either atomic orbitals, hybrid atomic orbitals, or MOs as the
guiding functions $\\{Q\\}$ in Eq. (3). In this subsection we discuss the
three kinds of the localized guiding functions $\\{Q\\}$, i.e., atomic
orbitals, hybrid atomic orbitals, and embedded MOs in molecules and bulks, and
how they can be generated in our implementation.
In our implementation, the Bloch function $\phi_{{\bf k}\mu}$ is expanded by
PAOs $\chi$ Ozaki2003 ; Ozaki2004 as
$\displaystyle|\phi_{{\bf k}\mu}\rangle=\sum_{{\bf R}}{\rm e}^{{\rm i}{\bf
k}\cdot{\bf R}}\sum_{i\alpha}c_{{\bf k}\mu,i\alpha}|\chi_{{\bf
R}i\alpha}\rangle,$ (19)
where $i$ and $\alpha$ are atomic and orbital indices, and $c$ is a linear
combination of PAO (LCPAO) coefficient. Also note that $\langle{\bf
r}|\chi_{{\bf R}i\alpha}\rangle\equiv\chi_{i\alpha}({\bf r}-{\bf\tau}_{i}-{\bf
R})$, where ${\bf\tau}_{i}$ is the position of atom $i$. We use PAOs as the
guiding functions $\\{Q\\}$ of atomic orbitals, corresponding to valence
orbitals or specific orbitals such as localized $d$-orbitals in a narrow
energy window.
The choice of the atomic orbitals gives a good guiding function, however, the
resultant CWFs may not recover the symmetry of CWFs respecting the bond
direction to the neighboring atoms unlike hybrid atomic orbitals due to the
closest property of CWFs. In this case, it would be better to use the hybrid
atomic orbitals rather than the atomic orbitals as explained below. Let us
introduce a projection operator for the occupied space by
$\displaystyle\hat{P}=\frac{1}{N_{\rm BC}}\sum_{{\bf k}\mu}|\phi_{\bf
k\mu}\rangle f(\varepsilon_{\bf k\mu})\langle\phi_{\bf k\mu}|,$ (20)
where $f$ is the Fermi-Dirac function. With the projection operator, the
density matrix is calculated by
$\displaystyle\rho_{{\bf R}i\alpha,{\bf R}^{\prime}j\beta}$ $\displaystyle=$
$\displaystyle\sum_{{\bf k}\mu}\langle\widetilde{\chi}_{{\bf
R}i\alpha}|\hat{P}|\widetilde{\chi}_{{\bf R}^{\prime}j\beta}\rangle,$
$\displaystyle=$ $\displaystyle\frac{1}{N_{\rm BC}}\sum_{{\bf k}\mu}{\rm
e}^{{\rm i}{\bf k}\cdot\left({\bf R}-{\bf
R}^{\prime}\right)}f(\varepsilon_{\bf k\mu})c_{{\bf k}\mu,i\alpha}c^{*}_{{\bf
k}\mu,j\beta},$
where $\widetilde{\chi}$ is the dual orbital defined by
$\displaystyle|\widetilde{\chi}_{{\bf R}i\alpha}\rangle$ $\displaystyle=$
$\displaystyle\sum_{{\bf R}^{\prime}j\beta}|\chi_{{\bf
R}^{\prime}j\beta}\rangle S_{{\bf R}^{\prime}j\beta,{\bf R}i\alpha}^{-1}$ (22)
holding the following orthonormality:
$\displaystyle\langle\chi_{{\bf R}i\alpha}|\widetilde{\chi}_{{\bf
R}^{\prime}j\beta}\rangle$ $\displaystyle=$
$\displaystyle\langle\widetilde{\chi}_{{\bf R}i\alpha}|\chi_{{\bf
R}^{\prime}j\beta}\rangle=\delta_{{\bf
RR}^{\prime}}\delta_{ij}\delta_{\alpha\beta}.$ (23)
By setting ${\bf R}={\bf R}^{\prime}={\bf 0}$ and $i=j$ in Eq. (LABEL:eq:DM),
and diagonalizing the diagonal block element consisting of $\rho_{{\bf
0}i\alpha,{\bf 0}i\beta}$, which is associated with selected atomic orbitals
on the atomic site $i$, we obtain the hybrid atomic orbitals respecting the
symmetry of the bond direction to the neighboring atoms. We employ the hybrid
atomic orbitals as the localized functions $\\{Q\\}$ in our implementation.
When the same atomic orbitals are chosen to form the diagonal block element as
for the case of atomic orbitals $\\{Q\\}$, the resultant CWFs span the same
subspace in the Hilbert space.
For some systems the electronic structures can be rather easily understood by
employing MOs as building blocks. A molecular crystal is such a case, where
the eigenstates near the chemical potential can be approximately constructed
by a linear combination of the MOs of constituting molecules. Another example
is the bond in molecules and bulks. The bond between two atoms embedded in the
system might be analyzed by MOs associated with the two atoms. Here we show
how embedded MOs in molecules and bulks can be calculated in a simple
procedure. Let us start by noting that using Eq. (20) and assuming that the
Bloch function is expressed by Eq. (19) the total number of electrons for a
spin degenerate case is given by Ozaki2018
$\displaystyle N_{\rm ele}$ $\displaystyle=$ $\displaystyle 2{\rm
tr}\left[\hat{P}\right],$ (24) $\displaystyle=$ $\displaystyle 2\sum_{{\bf
R}i\alpha}\langle\widetilde{\chi}_{{\bf R}i\alpha}|\hat{P}|\chi_{{\bf
R}i\alpha}\rangle,$
In the second line of Eq. (24) we used the following identity operator:
$\displaystyle\hat{I}=\sum_{{\bf R}i\alpha}|\chi_{{\bf
R}i\alpha}\rangle\langle\widetilde{\chi}_{{\bf R}i\alpha}|=\sum_{{\bf
R}i\alpha}|\widetilde{\chi}_{{\bf R}i\alpha}\rangle\langle\chi_{{\bf
R}i\alpha}|.$ (25)
Since $N_{\rm ele}=N_{\rm BC}N_{\rm ele}^{(0)}$ with $N_{\rm
ele}^{(0)}=2\sum_{i\alpha}\langle\widetilde{\chi}_{{\bf
0}i\alpha}|\hat{P}|\chi_{{\bf 0}i\alpha}\rangle$, it is enough to consider
$N_{\rm ele}^{(0)}$. We utilize the formula for $N_{\rm ele}^{(0)}$ to
calculate embedded MOs in molecules and bulks, and rewrite it the sum of
partial traces as
$\displaystyle N_{\rm ele}^{(0)}=2\sum_{g}{\rm
tr}_{g}\left[(\widetilde{\chi}_{{\bf 0}g}|\hat{P}|\chi_{{\bf
0}g})\right]=2\sum_{g}{\rm tr}_{g}\left[\Lambda_{g}\right],$ (26)
where $g$ is the index of group including PAOs on grouped atoms, e.g., a set
of PAOs on a molecule. The notation of $|\chi_{{\bf 0}g})$ stands for a subset
of PAOs as
$\displaystyle|\chi_{{\bf 0}g})=\left(\cdots,|\chi_{{\bf
0}i1}\rangle,|\chi_{{\bf 0}i2}\rangle,\cdots,\right),$ (27)
where PAOs on all the atoms in the group $g$ are included in the subset. The
notation of $|\widetilde{\chi}_{{\bf 0}g})$ is the counterpart for the dual
orbitals. Using the identity operator of Eq. (25), $\Lambda_{g}$ in Eq. (26)
is given by
$\displaystyle\Lambda_{g}=\sum_{{\bf R}g^{\prime}}\rho_{{\bf 0}g,{\bf
R}g^{\prime}}S_{{\bf R}g^{\prime},{\bf 0}g}$ (28)
with definition of block elements:
$\displaystyle\rho_{{\bf R}g,{\bf R}^{\prime}g^{\prime}}$ $\displaystyle=$
$\displaystyle(\widetilde{\chi}_{{\bf R}g}|\hat{P}|\widetilde{\chi}_{{\bf
R}^{\prime}g^{\prime}}),$ (29) $\displaystyle S_{{\bf R}g,{\bf
R}^{\prime}g^{\prime}}$ $\displaystyle=$ $\displaystyle(\chi_{{\bf
R}g}|\chi_{{\bf R}^{\prime}g^{\prime}}).$ (30)
We now introduce the embedded MOs orbitals in molecules and bulks by a right-
singular vectors of $\Lambda_{g}$. Since the elements of $\Lambda_{g}$ are
real in case of the collinear DFT, the right-singular vectors can be obtained
with real components. However, the SVD of $\Lambda_{g}$ tends to produce the
right-singular vectors with complex components, which results in the complex
CWFs making analysis complicated. To obtain the real right-singular vectors,
we perform the eigendecomposition of $\Lambda_{g}^{{\dagger}}\Lambda_{g}$ as
$\displaystyle\Lambda_{g}^{{\dagger}}\Lambda_{g}=Y_{g}\Omega_{g}^{2}Y_{g}^{{\dagger}},$
(31)
where $\Omega_{g}^{2}$ is the diagonal matrix consisting of the eigenvalues of
$\Lambda_{g}^{{\dagger}}\Lambda_{g}$, and $Y_{g}$ is a unitary matrix
consisting of the corresponding eigenvectors $\\{y_{g,\nu}\\}$. By noting that
the SVD of $\Lambda_{g}$ is given by $Z_{g}\Omega_{g}Y_{g}^{{\dagger}}$, we
see that the right-singular vectors $Y_{g}$ can be obtained by diagonalizing
$\Lambda_{g}^{{\dagger}}\Lambda_{g}$ as $Y_{g}$. The square roots of the
eigenvalues of $\Lambda_{g}^{{\dagger}}\Lambda_{g}$ are singular values of
$\Lambda_{g}$, and can be approximately regarded as the occupation of the
corresponding eigenvector $y_{g,\nu}$. If $\Lambda_{g}$ is a real matrix,
$Y_{g}$ obtained by the diagonalization of
$\Lambda_{g}^{{\dagger}}\Lambda_{g}$ is guaranteed to be a real unitary
matrix. We further normalize the eigenvector $y_{g,\nu}$ by considering the
overlap integrals of Eq. (30) as
$\displaystyle|\bar{y}_{g,\nu}\rangle=\frac{|y_{g,\nu}\rangle}{\sqrt{\langle
y_{g,\nu}|S_{{\bf 0}g,{\bf 0}g}|y_{g,\nu}\rangle}},$ (32)
and calculate an expectation value of the KS Hamiltonian $\hat{H}_{\rm KS}$
with $\bar{y}_{g,\nu}$ as
$\displaystyle\epsilon_{g,\nu}=\langle\bar{y}_{g,\nu}|\hat{H}_{\rm
KS}|\bar{y}_{g,\nu}\rangle.$ (33)
After $\\{\bar{y}_{g,\nu}\\}$ are ordered based on the expectation values, we
employ selected ones among $\\{\bar{y}_{g,\nu}\\}$, e.g., ones near the
chemical potential, as the guiding functions $\\{Q\\}$ in Eq. (3) by
monitoring the expectation values and the corresponding singular values. They
are what we call embedded MOs in molecules and bulks. As demonstrated in
Section IV, $\\{\bar{y}\\}$ work as the good guiding functions to capture the
electronic structure of a molecular crystal.
## IV NUMERICAL RESULTS
### IV.1 Wannier Interpolated Bands
Figure 2: Interpolated bands (red circles) of silicon in the diamond
structure, calculated by the TB Hamiltonian derived from CWFs with (a) $k_{\rm
B}T$=0.01 and (b) 3.0 eV in the window function of Eq. (4), respectively.
$\varepsilon_{0}$ and $\varepsilon_{1}$ relative to the chemical potential
were set to be -15.0 and 0.0 eV, respectively. The solid line is the original
one directly calculated by the conventional scheme. The number of k-points for
the Brillouin zone sampling was 13 $\times$ 13 $\times$ 13\. The experimental
lattice constant of 5.43 Å was used. The values of the DM function per CWF are
0.571 and 0.444 for (a) and (b), respectively. Figure 3: CWFs for (a) Si,
(b) Cu, (c) TTF in TTF-TCNQ, and (d) TCNQ in TTF-TCNQ. In all the cases,
isovalues of $\pm 0.04$ (orange:0.04, blue:-0.04) are used for drawing the
isosurfaces using OpenMX Viewer Lee2019 . The computational conditions for
(a), (b), (c), and (d) are the same as those in Fig. 2 (b), Fig. 4 (a), Fig. 5
(b), and Fig. 5 (b), respectively.
We present four numerical examples to demonstrate the broad applicability of
the proposed method across various systems.
In Figs. 2 (a) and (b), the interpolated bands of Si in the diamond structure,
calculated by the TB Hamiltonian derived from CWFs, are compared with those by
the conventional calculation. The hybrid atomic orbitals consisting of valence
minimal orbitals, which are one $s$\- and a set of $p_{x}$-, $p_{y}$-, and
$p_{z}$-orbitals per Si, were used as the guiding functions $\\{Q\\}$. For the
window function of Eq. (4) we use -15.0 and 0.0, relative to the chemical
potential, in eV for $\varepsilon_{0}$ and $\varepsilon_{1}$, respectively,
which covers the energy range of valence bands. On the other hand, two $k_{\rm
B}T$ of 0.01 and 3.0 eV were used to check how conduction bands are reproduced
depending on the effect of smearing. As clearly seen, the two cases well
reproduce the valence bands, and the conduction bands are also reproduced well
in the case of $k_{\rm B}T=3.0$ eV. Evaluating the DM function per CWF using
Eq. (14), we obtain values of 0.571 and 0.444 for $k_{\rm B}T$ of 0.01 and 3.0
eV, respectively. For $k_{\rm B}T=0.01$ eV, the first four singular values at
each $k$ point are close to 1, while the remaining four approach
$\delta(=10^{-12})$. Conversely, for $k_{\rm B}T=3.0$ eV, the eight singular
values at each $k$ point distribute ranging from 1.2 to 0.1. These differences
in the distribution of singular values explain why the value of the DM
function for $k_{\rm B}T=0.01$ eV is larger than that for $k_{\rm B}T=3.0$ eV.
One of the obtained CWFs is shown in Fig. 3 (a), which represents a $p$-like
CWF. In case of $k_{\rm B}T=0.01$ eV, the conduction bands are treated less
importantly due to heavily weighting to the valence bands, resulting in the
poor reproduction of the conduction bands. At first glance, one may consider
that this is an undesirable feature in such a treatment. However, the feature
enables us to generate the minimal atomic-like orthogonal orbitals, which well
span the subspace by the valence bands, and to calculate an effective charge
allocated to each atom using the minimal atomic-like orthogonal orbitals in an
unbiased way. We will demonstrate the calculation of the effective atomic
charges and stress the usefulness of the method later on.
Figure 4: Interpolated bands (red circles) of copper in the FCC structure,
calculated by the TB Hamiltonian derived from CWFs with (a)
$\varepsilon_{0}=-5.5$, $\varepsilon_{1}=-1.0$ eV, $k_{\rm B}T=1.0$ eV, and
hybrid $3d$-orbitals as $\\{Q\\}$, and (b) $\varepsilon_{0}=-11.0$,
$\varepsilon_{1}=24.0$ eV, $k_{\rm B}T=3.0$ eV, and hybrid $3d$-, $4s$-,
$4p$-orbitals as $\\{Q\\}$. The solid line is the original one directly
calculated by the conventional scheme. The number of k-points for the
Brillouin zone sampling was 21 $\times$ 21 $\times$ 21\. The experimental
lattice constant of 3.61 Å was used. The values of the DM function per CWF are
0.081 and 0.211 for (a) and (b), respectively.
The disentanglement of bands in metals is demonstrated for Cu in the FCC
structure as shown in Figs. 4 (a) and (b). By selecting the parameters
$\varepsilon_{0}$ and $\varepsilon_{1}$, so that the $3d$ bands are included,
and only five $d$-orbitals as the guiding functions $\\{Q\\}$, the five
$d$-bands are reproduced as shown in Fig. 4 (a). We see that the $3d$-bands
are properly disentangled from the dispersive $4s$ band, and one of the
obtained CWFs preserves the shape of $d_{z^{2}}$-orbital as shown in Fig. 3
(b). When $3d$-, $4s$-, and $4p$-orbitals are used as the guiding functions
$\\{Q\\}$, a broad range of bands are reproduced as shown in Fig. 4 (b). Since
our PP of Cu includes $3s$, $3p$, $3d$, and $4s$ states, the original band
structure has the deep $3s$\- and $3p$-bands. We set the parameters in the
window function so as to discard the $3s$\- and $3p$-bands, and use the $3d$-,
$4s$-, and $4p$-orbitals, not $3s$\- and $3p$-orbitals, as the guiding
functions $\\{Q\\}$. So, the $3s$\- and $3p$-bands are not included in the
interpolated bands (not shown). The examples show that the CWFs can be
flexibly and easily constructed in accordance with the purpose of study
without numerical difficulties.
Table 2: Expectation values $\epsilon$ (eV) of the KS Hamiltonian calculated by Eq. (33), which is relative to the chemical potential, and singular values $\omega$ of $\Lambda_{g}$ for embedded MOs in the TTF-TCNQ molecular crystal. | TTF | | | | | TCNQ |
---|---|---|---|---|---|---|---
index | $\epsilon$ | $\omega$ | | | index | $\epsilon$ | $\omega$
23 | -0.069 | 1.359 | | | 34 | 0.135 | 1.082
24 | 0.719 | 1.346 | | | 35 | 0.375 | 1.329
25 | 0.761 | 1.449 | | | 36 | 0.477 | 1.336
26 | 3.408 | 0.864 | | | 37 | 4.928 | 0.535
27 | 5.947 | 0.047 | | | 38 | 9.389 | 0.036
28 | 6.197 | 0.047 | | | 39 | 9.795 | 0.019
Figure 5: Interpolated bands, red lines in (a) and red circles in (b), of
TTF-TCNQ, calculated by the TB Hamiltonian derived from CWFs with (a)
$\varepsilon_{0}=-22.0$ and $\varepsilon_{1}=1.0$ eV relative to the chemical
potential, $k_{\rm B}T=2.0$ eV, and hybrid atomic valence orbitals as
$\\{Q\\}$, and (b) $\varepsilon_{0}=-1.0$ and $\varepsilon_{1}=1.0$ eV
relative to the chemical potential, $k_{\rm B}T=0.01$ eV, and the 26th MO and
the 37th MO for TTF and TCNQ as $\\{Q\\}$. The solid line is the original one
directly calculated by the conventional scheme. The number of k-points for the
Brillouin zone sampling was 5 $\times$ 19 $\times$ 3\. An experimental crystal
structure was used Kistenmacher1974 . The values of the DM function per CWF
are 0.441 and 0.022 for (a) and (b), respectively.
The interpolated bands for a molecular crystal, consisting of
tetrathiafulvalene (TTF) and tetracyanoquinodimethane (TCNQ), are shown in
Figs. 5 (a) and (b). By employing hybrid atomic orbitals as the guiding
functions $\\{Q\\}$, the wide range of bands are reproduced as shown in Fig. 5
(a). On the other hand, as shown in Fig. 5 (b), only four bands near the
chemical potential are reproduced using the embedded MOs in the bulk as the
guiding functions $\\{Q\\}$. The MOs were calculated by grouping the TTF and
TCNQ molecules separately as explained in Sec. III B. The expectation values
calculated by Eq. (33) and the singular values $\omega$ of $\Lambda_{g}$ are
listed in Table 2. It can be seen that the singular values largely change from
$\omega_{26}$ to $\omega_{27}$ and from $\omega_{37}$ to $\omega_{38}$ for TTF
and TCNQ, respectively. So, the 26th MO and the 37th MO for TTF and TCNQ,
respectively, were used as the guiding functions $\\{Q\\}$. Since there are
two TTF molecules and two TCNQ molecules in the unit cell, we have the four
embedded MOs as $\\{Q\\}$, resulting in the four bands. In Figs. 3 (c) and (d)
two of the resultant CWFs are shown, which localize in TTF and TCNQ molecules,
respectively, as expected. The value of the DM function per CWF is found to be
0.022, which implies that the guiding MOs are very close to the resultant
CWFs. The example demonstrates that the proposed method provides a direct way
to calculate CWFs for targeted bands in molecular crystals.
We extend the proposed method to the non-collinear DFT vonBarth1972 ;
Kubler1988 with spin-orbit interaction (SOI) Theurich2001 . The KS orbitals
are expressed by two-component spinor, and the SOI is introduced by fully
relativistic $j$-dependent PPs OpenMX . The theoretical framework is readily
extended to the non-collinear DFT with the SOI without any difficulty.
However, it should be noted that the hybrid atomic orbitals and embedded MOs
in molecules and bulks as the guiding functions $\\{Q\\}$ can be complex
functions in the case. In Figs. 6 (a) and (b), we show the Wannier
interpolated bands of Bi2Se3 which is known to be a topological insulator when
the SOI is included in the DFT calculation. For the cases without and with the
SOI, corresponding to Figs. 6 (a) and (b), respectively, it is confirmed that
the interpolated bands accurately reproduce the original ones regardless of
existence of the band inversion, demonstrating a wide variety of applicability
of the proposed method.
Figure 6: Interpolated bands (red circles) of Bi2Se3 (a) without and (b) with
the spin-orbit interaction, calculated by the TB Hamiltonian derived from
CWFs. The solid line is the original ones directly calculated by the
conventional scheme. $\varepsilon_{0}=-6.0$ and $\varepsilon_{1}=3.0$ eV
relative to the chemical potential, $k_{\rm B}T=1.0$ eV, and hybrid Bi
$6p$-orbitals and Se $4p$-orbitals as $\\{Q\\}$, were used in both the
calculations. The number of k-points for the Brillouin zone sampling was 7
$\times$ 7 $\times$ 7\. An experimental crystal structure was used
Nakajima1963 . The values of the DM function per CWF are 0.163 and 0.184 for
(a) and (b), respectively.
### IV.2 Effective Atomic Charges
From the construction, one can calculate CWFs closest to (hybrid) valence
atomic orbitals, while fully respecting the occupied states. The CWFs provide
us a unique way to evaluate effective atomic charges. Since the CWFs are a set
of orthonormal functions, the effective atomic charge $\kappa_{i}$ of atom $i$
is easily calculated without arbitrariness as
$\displaystyle\kappa_{i}$ $\displaystyle=$ $\displaystyle
N_{i}^{\rm(val)}-\sum_{p\in{\rm atom}~{}i}\langle W_{{\bf 0}p}|\hat{P}|W_{{\bf
0}p}\rangle,$ (34) $\displaystyle=$ $\displaystyle
N_{i}^{\rm(val)}-\frac{1}{N_{\rm BC}}\sum_{p\in{\rm atom}~{}i}\sum_{\bf
k,\mu}f(\varepsilon_{\bf k\mu})u_{{\bf k}\mu,p}^{*}u_{{\bf k}\mu,p},$
where $\hat{P}$ is the projection operator defined by Eq. (20), and
$N_{i}^{\rm(val)}$ is the number of valence electrons in the PP of atom $i$.
Though the summation over orbitals $p$ belonging to atom $i$ is taken into
account in Eq. (34), one can analyze the orbital resolved charges as well.
Table 3 shows effective atomic charges in a HCN molecule and the NaCl bulk,
calculated by Eq. (34) and the Mulliken population analysis Mulliken1955 .
Both the systems are known to be notorious cases because of the difficulty in
calculating the effective charges Knizia13 . We see that the effective charges
by the CWFs quickly converge as a function of basis functions, while those by
the Mulliken population are highly dependent on the choice of the basis
function. The effective atomic charges obtained from the CWFs could serve as a
valuable tool for analyzing electronic structures in a manner related to the
approach discussed in Ref. Knizia13 . Further investigation in this direction
will be conducted in future studies.
Table 3: Effective atomic charges in a HCN molecule and the NaCl bulk, calculated by the proposed method (CWF) and Mulliken population analysis (MP) with a variety of basis functions, where $A$ represents the constituting atoms. Hybrid atomic valence orbitals are used as the guiding functions $\\{Q\\}$. For the window function, $\varepsilon_{0}=-55.0$ and $\varepsilon_{1}=0.0$ eV for HCN and $\varepsilon_{0}=-35.0$ and $\varepsilon_{1}=0.0$ eV for NaCl, relative to the chemical potential, and $k_{\rm B}T=0.1$ eV are used. The number of k-points for the Brillouin zone sampling is 9 $\times$ 9 $\times$ 9 for the NaCl bulk with the lattice constant of 5.63 Å. HCN | | CWF | | | | | MP |
---|---|---|---|---|---|---|---|---
Basis | H | C | N | | | H | C | N
$A$6.0-s1p1 | 0.077 | -0.052 | -0.025 | | | 0.384 | -0.164 | -0.221
$A$6.0-s2p2 | 0.069 | 0.003 | -0.073 | | | -0.070 | 0.321 | -0.252
$A$6.0-s2p2d1 | 0.066 | 0.009 | -0.076 | | | -0.008 | 0.393 | -0.385
$A$6.0-s3p3d2 | 0.066 | 0.008 | -0.074 | | | 0.110 | 0.298 | -0.408
$A$6.0-s3p3d2f2 | 0.066 | 0.008 | -0.074 | | | 0.167 | 0.297 | -0.464
NaCl | CWF | | | | | MP | |
Basis | Na | Cl | | | | Na | Cl |
$A$9.0-s2p1 | 0.391 | -0.391 | | | | 0.716 | -0.716 |
$A$9.0-s3p2 | 0.422 | -0.422 | | | | 0.595 | -0.595 |
$A$9.0-s3p3d2 | 0.421 | -0.421 | | | | 0.158 | -0.158 |
$A$9.0-s3p3d2f2 | 0.421 | -0.421 | | | | -0.062 | 0.062 |
## V CONCLUSIONS
We presented a non-iterative method to calculate the closest Wannier functions
(CWFs) to a given set of localized guiding functions such as atomic orbitals,
hybrid atomic orbitals, and molecular orbitals (MOs). We defined the distance
measure (DM) function $F$ by the sum of the squared distance between the
projection functions $L$ and Wannier functions $W$ in the Hilbert space, and
considered minimizing the DM function. It was shown that the minimization of
the DM function is achieved by the polar decomposition of the projection
matrix $A$ with a window function via the singular value decomposition (SVD)
in a non-iterative manner. The CWFs can be uniquely constructed as long as the
projection matrix $A$ is positive definite. It was also shown that the method
is free from subtle choice of the gauge. The disentanglement of bands is
naturally taken into account by introducing a smoothly varying window
function, and including $N_{\rm band}$ Bloch functions to generate $N_{\rm
CWF}$ CWFs, where $N_{\rm CWF}\leq N_{\rm band}$. Even for isolated bands, the
method always performs the disentanglement of bands without any difficulty. We
have implemented the proposed method into the OpenMX code, which is based on
the density functional theory (DFT), numerical pseudo-atomic orbitals (PAOs),
and norm-conserving pseudopotentials (PPs), and introduced three types of
guiding functions, i.e., atomic orbitals, hybrid atomic orbitals, and embedded
MOs in molecules and bulks. The first two are easily employed from the PAOs.
For the last one, we developed a method to calculate embedded MOs in molecules
and bulks, which focuses on a partial trace formula of the projection operator
for the occupied space and applies SVD to the partial matrix for the
projection operator. The interpolated bands by tight-binding (TB) models
derived from the CWFs reproduce well the targeted conventional bands of a wide
variety of systems including Si, Cu, the TTF-TCNQ molecular crystal, and a
topological insulator of Bi2Se3. These successful reproduction of targeted
bands clearly demonstrates a wide variety of applicability of the proposed
method. We further show the usefulness of the proposed method in calculating
effective atomic charges, implying that the CWFs closest to atomic orbitals
can be used as a measure to analyze electronic structures from one system to
the others. Thus, we conclude that the proposed method is an alternative way
in efficiently calculating WFs, and the concept of CWFs will provide a basis
for development of novel methods of analyzing electronic structures and
calculating physical properties.
###### Acknowledgements.
The author wishes to express gratitude to Prof. Yoshiaki Sugimoto for
inspiring the development of CWFs through collaborative research. Thanks are
also extended to Mr. Ryotaro Koshoji for his insightful comments on the
theoretical aspect. Part of the computation in this study was carried out
using the computational facility of the Institute for Solid State Physics at
the University of Tokyo.
## Appendix A The upper bound of the diagonal elements of the matrix $D$
A proof for the upper bound of the diagonal elements of the matrix $D$ is
given in the appendix. Note that $UV$ and $BV$ in Eq. (11) are partial unitary
matrices in size of $N_{\rm band}\times N_{\rm CWF}$. Since $D$ is hermitian,
its eigenvalues are real. Let $\lambda$ be an eigenvalue of $D$ and $x$ be the
corresponding eigenvector. Then, the following equation holds:
$\displaystyle\frac{1}{2}\left(V^{{\dagger}}U^{{\dagger}}BV+V^{{\dagger}}B^{{\dagger}}UV\right)|x\rangle=\lambda|x\rangle.$
(35)
Defining $|y\rangle=UV|x\rangle$ and $|z\rangle=BV|x\rangle$, and operating
$\langle x|$ from the left side of Eq. (35), we have
$\displaystyle\langle y|z\rangle+\langle z|y\rangle=2\lambda\langle
x|x\rangle.$ (36)
Noting $\langle y|z\rangle+\langle z|y\rangle\leq 2|\langle y|z\rangle|$, and
using the Cauchy-Schwarz inequality $|\langle y|z\rangle|\leq\sqrt{\langle
y|y\rangle\langle z|z\rangle}$, we obtain
$\displaystyle\langle y|z\rangle+\langle z|y\rangle\leq 2\sqrt{\langle
y|y\rangle\langle z|z\rangle}.$ (37)
Since $\langle y|y\rangle=\langle z|z\rangle=\langle x|x\rangle$, combining
Eq. (36) and Eq. (37) results in
$\displaystyle\lambda\langle x|x\rangle\leq\langle x|x\rangle.$ (38)
Considering $\langle x|x\rangle\neq 0$, the upper bound of the eigenvalues of
$D$ is found to be
$\displaystyle\lambda\leq 1.$ (39)
Noting that the matrix $D$ can be written by $x$ and $\lambda$ as
$\displaystyle D=\sum_{\nu}|x_{\nu}\rangle\lambda_{\nu}\langle x_{\nu}|,$ (40)
the diagonal elements $d_{nn}$ of the matrix $D$ is given by
$\displaystyle d_{nn}=\sum_{\nu}|\langle n|x_{\nu}\rangle|^{2}\lambda_{\nu},$
(41)
where $\langle n|x_{\nu}\rangle$ is the $n$-th element in the vector
$x_{\nu}$. Since the upper bound of $\lambda$ is unity, and $\\{x\\}$ forms a
unitary matrix, we obtain the upper bound of the diagonal elements of the
matrix $D$ as
$\displaystyle d_{nn}\leq 1.$ (42)
## References
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|
# Aerodynamic effects and performance improvements of running
in drafting formations
Lukas Schickhofer KTM E-TECHNOLOGIES, Salzburg, AT-5081, Austria
<EMAIL_ADDRESS>Henry Hanson ADIDAS, Herzogenaurach, DE-91074,
Germany
###### Abstract
Drafting as a process to reduce drag and to benefit from the presence of other
competitors is applied in various sports with several recent examples of
competitive running in formations. In this study, the aerodynamics of a
realistic model of a female runner is calculated by computational fluid
dynamics (CFD) simulations at four running speeds of
$15\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$,
$18\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$,
$21\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$, and
$36\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$. Aerodynamic power
fractions of the total energy expenditure are found to be in the range of
$2.6\%$-$8.5\%$. Additionally, four exemplary formations are analysed with
respect to their drafting potential and resulting drag values are compared for
the main runner and her pacers. The best of the formations achieves a total
drag reduction on the main runner of $75.6\%$. Moreover, there are large
variations in the drag reduction between the considered formations of up to
$42\%$ with respect to the baseline single-runner case. We conclude that major
drag reduction of more than $70\%$ can already be achieved with fairly simple
formations, while certain factors, such as runners on the sides, can have a
detrimental effect on drag reduction due to local acceleration of the passing
flow. Using an empirical model for mechanical power output during running,
gains of metabolic power and performance predictions are evaluated for all
considered formations. Improvements in running economy are up to $3.5\%$ for
the best formation, leading to velocity gains of $2.3\%$. This translates to
$154\text{\,}\mathrm{s}$ ($\approx$2.6\text{\,}\mathrm{min}$$) saved over a
marathon distance. Consequently, direct conclusions are drawn from the
obtained data for ideal drafting of long-distance running in highly packed
formations.
###### keywords:
Running aerodynamics, Drafting, Drag reduction, Metabolic power, Computational
fluid dynamics
###### MSC:
[2010] 00-01, 99-00
††journal: Journal of Biomechanics
## 1 Introduction
In most sports involving the race against other participants for time, drag
plays a crucial role. It is defined as the aerodynamic force acting in an
axial direction on the athlete as he or she moves against the surrounding
fluid, such as water or air. As a result, it has become common practice both
in sports and motorsports to benefit from the presence of other athletes or
vehicles to alleviate the resistance of the fluid by drafting in the
associated wake region (i.e. _slipstreaming_). In the wake there is not just
drastically lower fluid velocity, but also a negative pressure coefficient
leading to suction, which additionally benefits the trailing competitor.
While in motorsports with its high speeds and considerable research efforts,
drafting has long been investigated (e.g. Katz [1995], Romberg et al. [1971]),
it is less researched in other sports. However, Rundell [1996] investigated
drafting during speed skating and showed large positive effects on metabolic
activity, heart rate, and lactate response. Also in swimming, drafting was
associated with considerable energy saving and the optimal drafting distance
was established at $0$-$0.5\text{\,}\mathrm{m}$ behind the leading swimmer,
causing power savings of up to $31\%$ [Bassett Jr et al., 1991, Chatard and
Wilson, 2003]. Furthermore, cycling is a sport of large benefits through
aerodynamic drafting and riding in formations, such as tightly packed pelotons
[Edwards and Byrnes, 2007, Blocken et al., 2018, Malizia and Blocken, 2020].
Apart from testing in wind tunnels, typically applying similarity principles
through scaled models at similar Reynolds numbers, computational fluid
dynamics (CFD) has become a widely used method to accurately assess drag
distributions across athletes. Blocken et al. [2013] examined the effect of
two drafting cyclists at various upper-body positions through CFD and found
drag reductions of up to $27.1\%$ for the trailing cyclist, while also the
leading cyclist's drag was decreased by up to $2.6\%$. Moreover, drag
reductions of $42\%$ to $48\%$ have been measured for drafting during cycling
in the velodrome [Fitton et al., 2018]. Fuss [2018] recently suggested an
analytical model for slipstreaming in gravity-powered sports such as ski cross
and found significant advantages in gained velocity and glide distance with
respect to a leading skier, especially in a tucked body position.
Although running happens typically at fairly low speeds, drafting shows also
here a measurable impact on aerodynamic forces. In a major study on the
subject, Pugh [1970] demonstrated drag savings of up to $80\%$ when running at
a middle-distance speed of $4.46\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$
against a slight head wind. More recently, particularly in light of attempts
to break the $2$-hour barrier of the marathon distance (e.g. _INEOS 1:59
Challenge_ , _Nike Breaking2_), increasing research efforts were directed to
minimize drag for a main runner in formations. Hoogkamer et al. [2018] used a
reduced-order model to establish a sustainable velocity of
$5.93\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ using cooperative drafting
in a four-runner team. Polidori et al. [2020] applied CFD to compute the drag
and power savings of Kenenisa Bekele, while running the Berlin marathon and
using cooperative drafting. It was shown that up to $57.3\%$ of aerodynamic
power were gained in the ideal case of running directly behind a front pacer
at a distance of $1.3\text{\,}\mathrm{m}$ when compared to the case of running
alone.
The current study aims at shedding more light on the aerodynamic effects of
running alone and in formations by using a realistic model of a female runner
and a validated CFD methodology. Four different formations are chosen to
reflect various running scenarios and their implications on the drag on the
main runner as well as her pacers are discussed. The effect of the resulting
drag reduction on the metabolic rate and running economy is computed by using
a mechanical power model of running. As a result, possible performance
predictions are made in terms of improvements of running speed and time.
Finally, by applying the conclusions from this study, direct implications for
an optimized drafting strategy are achieved.
## 2 Methods
### 2.1 Numerical method
For the computation of the flow field variables and the resulting aerodynamic
forces on the runner, computational fluid dynamics (CFD) using the finite-
volume method is applied. The Navier-Stokes equations describing the
conservation of momentum are discretized in its Reynolds-averaged (RANS) form.
Turbulence models have an important effect on aerodynamic force computations,
mainly because of their modelling of momentum transfer near surfaces and in
boundary layers. This in turn influences the locations of flow separation,
which is crucial for correct drag predictions. Here, turbulence is treated
with the $k$-$\omega$ model in its shear-stress-transport (SST) formulation
according to Menter [1994]. The model has been repeatedly shown to be suitable
for external aerodynamics and to give superior performance to the Spalart-
Allmaras and the $k$-$\epsilon$ models in flow scenarios with adverse pressure
gradients and free shear layers, which do occur in our setup [Menter, 1992,
Hellsten, 1998]. Defraeye et al. [2010] tested the performance of various
turbulence treatments such as the most common RANS models and large eddy
simulation with respect to drag predictions for cycling aerodynamics. They
found the $k$-$\omega$ SST model to perform best in comparison with
experimental data (i.e. force and moment areas) from wind tunnel measurements,
with average discrepancies of $6\%$ and consistently below $11\%$.
\begin{overpic}[width=390.25534pt]{Figure1.pdf} \end{overpic} Figure 1:
Geometry of a formation of runners surrounded by the computational domain and
the imposed boundary conditions (a). The axial spacing of $1.2\,m$ and the
lateral spacing of $0.7\,m$ between the runners are indicated (b).
Moreover, Crouch et al. [2016] has shown for cycling aerodynamics that the
effect of the dynamical motion of the limbs on instantaneous drag is only
minor. Additionally, wind tunnel experiments by Inoue et al. [2016] revealed
an approximately $10\%$ increase in drag values of solo running when using a
moving-belt system as compared with a stationary setup. In their measurements
the formation of unrealistic ground boundary layers was thus inhibited, as it
is the case in this study. We use a stationary model of an athlete together
with a moving ground boundary, which was found to be sufficient for the
prediction of the global aerodynamic forces.
Furthermore, discretization in this study is performed with a second-order
scheme for the spatial derivatives. Pressure and velocity are calculated in a
coupled approach. The fluid is defined as air at ambient temperature of
$T=$293.15\text{\,}\mathrm{K}$=$20\text{\,}\mathrm{\SIUnitSymbolCelsius}$$ and
a kinematic viscosity of
$\nu=$1.516\text{\times}{10}^{-5}\text{\,}{\mathrm{m}}^{2}\text{\,}{\mathrm{s}}^{-1}$$.
Additionally, the velocity inlet of the domain pictured in Fig. 1(a) imposes
an axial flow velocity at the chosen running speeds, while the pressure outlet
applies an ambient pressure of
$p=$101\,325\text{\,}\mathrm{Pa}$=$1\text{\,}\mathrm{atm}$$. The ground is
defined as wall moving at the running speed, which ensures that no boundary
layers form at the bottom that could invalidate the results. A Dirichlet
boundary condition, $u_{i}=0$, is applied for all components of the flow
velocity directly at the ground. The surrounding walls are symmetry boundaries
with frictionless properties. This is achieved through a Neumann boundary
condition $\partial u_{i}/\partial x_{i}=0$.
Tab. 1 gives the various speeds of this study with the respective Reynolds
numbers of the flow, using the runner's height for the definition of the
characteristic length. While all four speeds are considered for the assessment
of single-runner aerodynamics (cf. Sec. 3.1), the drafting formations are
studied at a running speed of
$21\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$ (cf. Sec. 3.2-3.4). This is
a high speed occurring during elite marathon running and allows for good
comparison between the considered formations. In fact, the current official
marathon world record for men is 2:01:39 by Eliud Kipchoge (September 16, 2018
during the Berlin Marathon; at an average speed of
$\sim$21\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$$) and for women is
2:14:04 by Brigid Kosgei (October 13, 2019 during the Chicago Marathon; at an
average speed of $\sim$19\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$$).
Here, we chose a running velocity at the upper range of these numbers to
assess the ideal potential of drafting with respect to metabolic quantities,
drafted velocity, and time savings.
Table 1: Running speeds and associated Reynolds numbers $Re=\left(U_{\infty}L\right)/\nu$ at a characteristic length of $L=$1.65\text{\,}\mathrm{m}$$. Running speed ($\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$) | Reynolds number
---|---
$15$ ($4.16\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$) | $452\,770\text{\,}$
$18$ ($5\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$) | $544\,195\text{\,}$
$21$ ($5.83\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$) | $634\,531\text{\,}$
$36$ ($10\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$) | $1\,088\,390\text{\,}$
The inherent challenge of simulations lies in ensuring that physical effects
are computed as accurately as possible. Here, that means that flow separation
lines and points, as well as boundary layer thickness and characteristics are
captured for the investigated Reynolds numbers, which lie in the turbulent
regime (cf. Tab. 1). In order to avoid inaccurate prediction of aerodynamic
forces on athletes at high Reynolds numbers, as described by Meile et al.
[2006], it is crucial to optimize near-surface grid resolutions, such that no
numerically induced jumps of separation points occur. This is important, as an
artificially delayed flow separation would lead to a sharp drop in the
pressure drag, which is the dominant component of total drag of bluff bodies
(such as the human body). A thorough verification and validation of the
applied numerical approach is presented in A. The solution of the discretized
continuity and momentum equations is performed with the solver _ANSYS Fluent
19.2_ , post-processing of flow variables is achieved using the post-
processing tool _ParaView_ and drag distribution and resulting power values
are calculated with the free scientific programming language _GNU Octave_.
### 2.2 Geometry and mesh
For the numerical setup we apply the realistic, three-dimensional geometry of
a female runner with a height of $L=$1.65\text{\,}\mathrm{m}$$, a projected
frontal area of $A\approx$0.44\text{\,}{\mathrm{m}}^{2}$$, and a body mass of
$m\approx$55\text{\,}\mathrm{kg}$$. The geometry represents an average of 59
individual track athletes, scanned in-house and positioned into a running
pose. The obtained pose-invariant statistical model for the female human body
was reconstructed from the high-quality scan data using the method outlined by
Colaianni et al. [2014]. The axial and lateral spacing between the runners is
chosen according to estimates of realistic running conditions. Polidori et al.
[2020] for instance uses an axial distance of $1.3\text{\,}\mathrm{m}$ and a
lateral shoulder-to-shoulder distance of $0.3\text{\,}\mathrm{m}$, which are
taken from an actual drafting situation during a marathon race.
The computational domain shown in Fig. 1(a) has a distance of
$7\text{\,}\mathrm{m}$ from the main runner to the upstream boundary and
$13\text{\,}\mathrm{m}$ to the downstream boundary. It contains far-field
refinement as well as three refinement stages close to the runner's surfaces.
Furthermore, 10 prism layers (i.e. inflation layers) are inserted directly at
all solid boundaries with a steady growth factor of the prism layer thickness
of $1.2$. These measures ensure that the dimensionless wall distance of
$y^{+}\approx 1$ holds in the entire domain and all boundary layers are well
resolved. Due to the local mesh refinement the cell count varies sharply
between formations depending on the number of runners included. It rises from
$40$ million cells for the single-runner case up to $107$ million cells for
the final formation including a total of four runners. The near-surface mean
cell size of $3.125\text{\times}{10}^{-3}\text{\,}\mathrm{m}$ and the outer-
boundary mean cell size of $0.196\text{\,}\mathrm{m}$ for the final grid
settings have been chosen following a grid convergence study (cf. A).
Homogeneous, gradual cell growth is applied between the near and far field.
Locally, especially in the boundary layer regions, perpendicular mesh
dimensions are significantly lower (more than by a factor 10) than the near-
surface mean cell size, which is the cell size of the finest grid level just
outside the prism layers. A depiction of the considered formations of this
study is shown in Fig. 4-5.
### 2.3 Calculation of power values
Various empirical models exist for the calculation of the mechanical power
output while running. Fukunaga et al. [1980] found through experiments on
athletic runners on force platforms a relation between running velocity $u$
and the exerted power for forward motion as
$P_{R,0}=0.436u^{2.01},$ (1)
with $u$ in $\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ and $P_{R}$ in
$\text{\,}\mathrm{W}\text{\,}{\mathrm{kg}}^{-1}$. In a study using motion
tracking, Cavagna and Kaneko [1977] also took into account additional energy
expenditure created by the movement of the limbs. They arrive at a function
for the specific running power per unit mass as
$P_{R,0}=9.42+4.73u+0.266u^{1.993}.$ (2)
Here, $u$ is given in $\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$ and
$P_{R,0}$ in
$\text{\,}\mathrm{cal}\text{\,}{\mathrm{kg}}^{-1}\text{\,}{\mathrm{min}}^{-1}$.
For our considered cases, both approaches give similar results with deviations
in the range of only $2$-$5\%$, however, Eq. (2) is used in the following
power calculations due to being the more complete model of running motion. By
`more complete' we mean that it takes into account both internal and external
work performed for the modelling of the total mechanical work
$W_{tot}=W_{ext}+W_{int}.$ (3)
Here, external work $W_{ext}$ is due to acceleration and lift of the centre of
mass, while internal work refers to both the translational and rotational
acceleration of the limbs relative to the trunk. This approach has also been
suggested more recently by the work of Saibene and Minetti [2003], Pavei et
al. [2019], and Gray et al. [2020].
Using Eq. (2), we arrive at the power
$P_{R}=P_{R,0}\cdot\frac{4.1868}{60}\cdot m$ (4)
in $W$ for a runner of mass $m$ in $\text{\,}\mathrm{kg}$, in the following
simply called _running power_. Furthermore, there is a certain aerodynamic
power $P_{A}$, used to overcome drag $F_{D}$, which is defined as
$P_{A}=F_{D}\cdot u=\frac{1}{2}\rho u^{3}C_{D}A.$ (5)
Here, the expression for $F_{D}$ assumes stagnant air and the absence of
crosswinds. As a result, the total mechanical power generated while running is
the sum of the running power $P_{R}$ and the aerodynamic power $P_{A}$:
$P_{tot}=P_{R}+P_{A}.$ (6)
Using the total mechanical power $P_{tot}$ and the associated efficiency
$\epsilon$ at a certain run speed by Cavagna and Kaneko [1977], the metabolic
power
$P_{meta}=\frac{P_{tot}}{\epsilon},$ (7)
which gives the metabolic energy expenditure during running, can be computed.
It must be emphasized here that the science behind efficiency of locomotion
and in particular of walking and running is far from concluded. In both
running and walking the muscles utilize energy stored during a previous phase
of stretching in the following phase of contraction. As stated by Cavagna and
Kaneko [1977], the efficiency of walking reaches a maximum of $0.35$-$0.40$ at
intermediate speeds due to the expected properties of the contractile
component of muscle. This maximum of efficiency follows the force-velocity
relation and the trends due to initial efficiency of muscle [Hill, 1964]. Also
in cycling the efficiency of locomotion was found to reach maximum values at
intermediate speeds ($\sim 0.22$ according to the early study of Dickinson
[1929], $\sim 0.25$ according to Ettema and Lorås [2009]). In running,
however, the efficiency increases steadily with speed (from $0.45$ to $0.70$),
suggesting that the positive work during that activity derives mainly from the
passive recoil of elastic muscle tissue and not from the active shortening of
contractile muscle components [Cavagna and Kaneko, 1977, Saunders et al.,
2004]. In support of this hypothesis, Cavagna et al. [1968] found that the
useful effect of pre-streched muscle on the performed work increases with the
speed of stretching and shortening. Furthermore, Komi and Bosco [1978]
investigated this utilization of elastic energy during eccentric-concentric
contraction of muscles in human motion and found that up to $90\%$ of energy
produced in the pre-stretching phase is recovered in case of counter-movement
jumps. More recently, Komi [2000] used data based on in-vivo force
measurements, buckle-transducer technique, and optic-fiber technique to show
that the stretch-shortening cycle in human skeletal muscle leads to
considerable performance enhancement compared to simple concentric action.
With respect to running economy, Hunter et al. [2015] showed a clear positive
correlation of stretch-shortening cycle potentiation and increases of running
economy and recommended eccentric force development (e.g. by resistance
training) for performance improvements. Nonetheless, we point out that
efficiency and total mechanical power models are inherently difficult to
define due to the sometimes conflicting understanding of internal and external
work performed during human locomotion and the question, whether internal work
should appear in total mechanical power and efficiency expressions, as has
been suggested by Winter [1979] and more recently by Minetti et al. [2001]
(see e.g. Ettema and Lorås [2009] for a discussion of these issues with
regards to cycling, and Williams [2000] with regards to running).
## 3 Results
### 3.1 Single-runner aerodynamics
Using the geometry of the single runner, the drag forces acting on various
body parts are computed over the range of relevant velocities (cf. Fig. 2(b)).
It can be seen that the drag acting on torso and legs is highest, with both
parts roughly affected by the same aerodynamic loads. Each make for a fraction
of $39$-$40\%$ of the total drag on the runner. Furthermore, arms are affected
considerably less, with a share of approximately $13\%$, and the head with
$8\%$ of the total drag. These estimates give an overview of the sensitivity
of the various parts of the body towards drag loads and are directly related
to the frontal area of each part (cf. Eq. (5)). Fig. 2(a) gives the total drag
on the runner for the four simulated running speeds. While the drag for lower
speeds of $15$-$21\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$, which are
relevant for long distance disciplines like marathon, lies in the range of
$3.39$-$6.52\text{\,}\mathrm{N}$, it sharply rises to
$18.45\text{\,}\mathrm{N}$ at the higher considered speed of
$36\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$. This demonstrates the
(quadratically) increasing relevance of drag reduction at higher speeds.
Moreover, the associated power values are calculated using the definitions of
Sec. 2.3. Fig. 2(c) shows the share of aerodynamic and running power in the
total mechanical power over the considered range of velocity. However, the
actual energy per unit time, which needs to be made available during motion,
is the metabolic power (cf. Eq. (7)).
\begin{overpic}[width=216.81pt]{Figure2.pdf} \end{overpic} Figure 2: Total
drag over the range of considered running speeds (a), as well as drag acting
on the body parts of the runner (b). Total mechanical power, consisting of
running power computed according to the empirical model by Cavagna and Kaneko
[1977] and aerodynamic power, as a function of speed (c). Polynomial functions
of second order (aerodynamic forces and running power) and of third order
(aerodynamic and total mechanical power) are used for fitting the data.
Here, the aerodynamic component provides a fraction of the metabolic power of
$2.6\%$, $3.4\%$, $4.3\%$, and $8.5\%$ for
$4.16\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$
($15\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$),
$5\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$
($18\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$),
$5.83\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$
($21\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$), and
$10\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$
($36\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$). These can be considered
as threshold values for the maximum savings in metabolic energy consumption
during drafted running.
The resulting wake regions are depicted in Fig. 3. They are visualized by
isosurfaces of total pressure, giving the boundary between the negative
pressure region in the wake and the positive pressure region due to recovery
of the flow. There is a notable decrease of the wake length from
$1.14\text{\,}\mathrm{m}$ at
$15\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$ to
$0.98\text{\,}\mathrm{m}$ at
$36\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$ due to increased turbulent
mixing. The wake length is a good indication of the most beneficial range for
drafting, as the suction provided by the negative total pressure in the wake
slightly pulls a following runner. However, even at a length larger than that
there will be a considerable drag-reducing effect due to flow deceleration of
the head wind.
\begin{overpic}[width=433.62pt]{Figure3.pdf} \end{overpic} Figure 3: Wake
region behind the runner visualized by isosurfaces of total pressure at
$p_{tot}=0$.
### 3.2 Drag comparison of formations
At first we consider the performance of the investigated formations with
respect to their effect on the main runner's drag at a running speed of
$21\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$. It can be concluded from
Fig. 4 that formation $2$ leads to the largest reduction of drag of $75.6\%$,
followed by formation $1$ with $70.1\%$. Formations $3$ and $4$ cause
significantly less drag reduction of $41.3\%$ and $33.4\%$, respectively.
\begin{overpic}[width=216.81pt]{Figure4.pdf} \end{overpic} Figure 4: Sketch of
the investigated formations (a) together with their effect on the main
runner's drag (b). The formations are compared to the case of the single
runner. Axial and lateral spacing between the runners follows the values
introduced in Fig. 1(b).
Furthermore, considering the drag distribution across the whole body of the
main runner, it can be recognized that the dominant part of the drag acts on
legs and torso (cf. Sec. 3.1). Thus, it is crucial for a drafting formation to
reduce the drag in these areas. Looking at Fig. 5, it can be seen that
formations $1$-$2$ achieve this, while formations $3$-$4$ cannot adequately
shield the body of the main runner from oncoming air. Additionally, it is
shown that while the drag reduction on head and arms is fairly equal for
formations $1$-$3$, formation $4$ shows almost no improvement for the drag on
the arms. The reasons for the varying aerodynamic performances of formations
$1$-$4$ are elaborated further in Sec. 3.3.
\begin{overpic}[width=433.62pt]{Figure5.pdf} \end{overpic} Figure 5: Drag
acting on the various body parts of the main runner alongside the total drag
for all involved runners, including pacers, for the formations 1 (a), 2 (b), 3
(c), and 4 (d).
Using the results for formation $1$, the distance between main runner and
pacer was further varied to give the drag values shown in Fig. 6. An
exponential function of the type $y=a+b\cdot\exp\left(-c/x\right)$ with $a=0$,
$b=5.63$, and $c=1.28$ was used to fit the data and has the desired property
of asymptotically approaching zero drag at small distances and the single
runner's maximum drag at large distances. The values of the parameters were
estimated by least-squares optimization using the Levenberg-Marquardt
algorithm [Levenberg, 1944, Marquardt, 1963]. The absolute axial limit of
separation between the two runners, given the dimension and stride length of
the applied geometry, was found to be approximately $0.6\text{\,}\mathrm{m}$.
In reality, however, this limit will barely be reached and the runners will be
unable to approach already considerably earlier. Pugh [1970] estimates a
natural limit of axial spacing of $1\text{\,}\mathrm{m}$ for middle-distance
running and states a possible drag reduction at this distance of up to $80\%$.
In this study we find a drag of $1.45\text{\,}\mathrm{N}$ acting on the main
runner at a distance of $1\text{\,}\mathrm{m}$, which corresponds to $22.2\%$
of the drag on the single runner and thereby lies in good agreement with the
results by Pugh [1970].
\begin{overpic}[width=216.81pt]{Figure6.pdf} \end{overpic} Figure 6: Drag on
the main runner as a function of the distance to the leading pacer in
formation 1. The numerical data is fitted using an exponential function with a
determination coefficient of $R^{2}=0.99$. In reality, there is an axial limit
which prohibits the further approach of the main runner to the pacer. This
limit is estimated from geometric considerations as $0.6\text{\,}\mathrm{m}$.
Pugh [1970] estimates this minimum distance during drafting at
$1.0\text{\,}\mathrm{m}$.
### 3.3 Aerodynamic effects on runners within formations
Air resistance while running manifests itself in static pressure increase as
the air is being decelerated and diverted at the front of the runner. Fig. 7
provides insight to the magnitude of the pressure acting on the main runner
for the various formations. The pressure coefficient
$c_{p}=\frac{p-p_{\infty}}{\frac{1}{2}\rho
U_{\infty}^{2}}=\frac{p-p_{\infty}}{p_{0}-p_{\infty}},$ (8)
with the free-stream pressure of
$p_{\infty}=$101\,325\text{\,}\mathrm{Pa}$=$1\text{\,}\mathrm{atm}$$ and a
relative free-stream velocity of
$U_{\infty}=$5.83\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$=$21\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$$,
indicates regions of high or low relative pressure.
\begin{overpic}[width=433.62pt]{Figure7.pdf} \end{overpic} Figure 7: Pressure
coefficient on the main runner for the single-runner case (a) compared to the
formations 1-4 (b)-(e).
Fig. 7 confirms the beneficial pressure distribution for formations $1$ and
$2$. This means that there are no large areas of high pressure coefficient at
the front of the runner, nor any large regions of low negative pressure
coefficient at the back, which would be an indication for suction force
slightly pulling the runner back. Formation $3$ on the other hand shows a high
positive pressure coefficient at the front, which already hints at the leakage
of oncoming flow between the front pacers that can be seen in Fig. 8(c) and
Fig. 9(c). While formation $4$ does not display any significantly high
pressure peaks at the front, it is the elevated negative pressure coefficient
at the back that makes it less advantageous than formations $1$-$2$ (cf. Sec.
3.2).
\begin{overpic}[width=433.62pt]{Figure8.pdf} \end{overpic} Figure 8:
Streamlines seeded at the front of the leading pacers showing the diversion of
air around the main runner for all four formations. Streamlines are coloured
by velocity magnitude and the runners' surfaces by pressure coefficient.
\begin{overpic}[width=433.62pt]{Figure9.pdf} \end{overpic} Figure 9: Velocity
magnitude contours in the horizontal plane section through the runners'
centre. Formations 1-4 are shown in images (a)-(d).
Fig. 8-9 both demonstrate the reasons for the large differences in aerodynamic
performance between the formations. In case of formations $1$ and $2$ the main
runner is well embedded inside the wake region of the front pacer and
therefore only affected by low-speed, almost stagnant air relative to herself.
Formation $2$ has the additional advantage of a rear pacer, whose high-
pressure stagnation region at the front positively affects the low-pressure
wake region of the main runner by raising the pressure coefficient to a value
closer to zero, thus lowering suction. This can be seen when comparing Fig.
10(a) and Fig. 10(b). Furthermore, Fig. 8(c) and Fig. 9(c) clearly demonstrate
a major weakness of formation $3$, which is the passing air between the front
pacers. This leads to a large region of high stagnation pressure directly at
the area of the impacting air on the main runner, which is also visible in
Fig. 7(d) and Fig. 10(c). Also formation $4$ is suboptimal due to air being
accelerated between the main runner and the pacers on the side. This leads to
the aforementioned effect of higher drag at the arms compared to the other
formations (cf. Sec. 3.2). Additionally, this effect of locally increased flow
velocity between the runners leads to a shortening of the wake region of slow
air for the main runner and thus negatively impacts her drafting conditions
(cf. Fig. 9(d)).
\begin{overpic}[width=325.215pt]{Figure10.pdf} \end{overpic} Figure 10:
Pressure coefficient in the vertical plane section through the main runner's
centre. Formations 1-4 are depicted in images (a)-(d).
### 3.4 Metabolic power savings and performance predictions
Using the drag values from Sec. 3.2, we can compute the associated aerodynamic
power and the total mechanical power using Eq. (2)-(6). Tab. 2 lists the
results for the single runner and the four formations. Additionally, relative
changes in aerodynamic power and total power are given by comparison with the
results of the single-runner case. The resulting metabolic power needed for
running in the considered formations is computed by using the efficiency of
running at the given velocity as detailed in Cavagna and Kaneko [1977].
Furthermore, running economy as the specific metabolic rate per unit mass is
given together with its relative changes in Tab. 3. As described by Kipp et
al. [2019], the relationship between improvements in running economy and
improvements in velocity is not a linear, but rather a curvilinear one. This
means that for a given improvement in running economy, velocity gains will be
smaller at higher speeds and larger at lower speeds. At elite marathon speeds
of around $5.5\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$ there is roughly
a $2/3^{\text{rds}}$ increase in velocity for each percent increase in running
economy. By applying the conversion factors of Kipp et al. [2019] we arrive at
estimates for the velocity gains for each formation in Tab. 4. The relative
changes in speed are $2.1\%$ (formation 1), $2.3\%$ (formation 2), $1.3\%$
(formation 3), and $1.1\%$ (formation 4).
Table 2: Aerodynamic power $P_{A}$, and total mechanical power $P_{tot}$ computed with Eq. (5) and (6) and by computing the running power $P_{R}=$858.1\text{\,}\mathrm{W}$$ at $21\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$ ($5.83\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$) using the model of Cavagna and Kaneko [1977]. Savings $\Delta P_{A}$ and $\Delta P_{tot}$ are given with respect to the case of the runner alone. Formation | $F_{D}$ ($\text{\,}\mathrm{N}$) | $P_{A}$ ($\text{\,}\mathrm{W}$) | $\Delta P_{A}$ | $P_{tot}$ ($\text{\,}\mathrm{W}$) | $\Delta P_{tot}$
---|---|---|---|---|---
Single runner | $6.52$ | $38.0$ | - | $896.1$ | -
$1$ | $1.96$ | $11.4$ | $70.0\%$ | $869.5$ | $3.0\%$
$2$ | $1.58$ | $9.2$ | $75.8\%$ | $867.3$ | $3.2\%$
$3$ | $3.84$ | $22.4$ | $41.1\%$ | $880.5$ | $1.7\%$
$4$ | $4.33$ | $25.3$ | $33.4\%$ | $883.4$ | $1.4\%$
Table 3: Metabolic power $P_{meta}$ according to Eq. (7) and running economy $RE$, as well as percent improvements of the running economy $\Delta RE$ for each of the formations with respect to the single-runner case. Efficiency of locomotion for a running velocity of $21\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$ ($5.83\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$) is taken as $\epsilon=0.64$ [Cavagna and Kaneko, 1977]. Formation | $P_{meta}$ ($\text{\,}\mathrm{W}$) | $RE$ ($\text{\,}\mathrm{W}\text{\,}{\mathrm{kg}}^{-1}$) | $\Delta RE$
---|---|---|---
Single runner | $1400.2$ | $25.5$ | -
1 | $1358.6$ | $24.7$ | $3.1\%$
2 | $1355.2$ | $24.6$ | $3.5\%$
3 | $1375.8$ | $25.0$ | $2.0\%$
4 | $1380.3$ | $25.1$ | $1.6\%$
Table 4: Drafted running speeds $u_{draft}$ for all formations alongside potential time $t_{draft}$ and time savings through drafting $\Delta t_{draft}$ over the marathon distance of $42\,195\text{\,}\mathrm{m}$ by comparison with the undrafted single-runner case. Formation | $u_{draft}$ ($\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$) | $t_{draft}$ ($\text{\,}\mathrm{s}$) | $\Delta t_{draft}$ ($\text{\,}\mathrm{s}$)
---|---|---|---
1 | $5.95$ | $7092$ ($\approx$1.970\text{\,}\mathrm{h}$$) | $142$ ($\approx$2.4\text{\,}\mathrm{min}$$)
2 | $5.96$ | $7080$ ($\approx$1.967\text{\,}\mathrm{h}$$) | $154$ ($\approx$2.6\text{\,}\mathrm{min}$$)
3 | $5.91$ | $7140$ ($\approx$1.983\text{\,}\mathrm{h}$$) | $98$ ($\approx$1.6\text{\,}\mathrm{min}$$)
4 | $5.89$ | $7164$ ($\approx$1.990\text{\,}\mathrm{h}$$) | $70$ ($\approx$1.2\text{\,}\mathrm{min}$$)
## 4 Discussion and conclusions
Drafting and the resulting aerodynamic drag reduction cause a notable effect
on running and the energy expenditure used for it. This study shows the
overall aerodynamics and associated energy expenditure during running for a
range of relevant speeds. Considerable shares of up to $8.5\%$ at a velocity
of $36\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$ are computed.
Additionally, differences in drag, pressure distribution across the runners,
and potential power savings are demonstrated for four formations that could
arise during running competitions. As shown in Sec. 3.2, simple formations of
axial spacing of the pacers (i.e. one pacer in front, as in formation 1, or
one pacer in front and one in the back, as in formation 2) already lead to
considerable reduction of over $70\%$ in drag at a distance of
$1.2\text{\,}\mathrm{m}$. Furthermore, formations 1 and 2 also show a clear
positive effect on leading and trailing pacers, since their overpressure and
underpressure regions interact with the main runner. This effect has also been
noted by Beaumont et al. [2019] together with a measurable advantage for the
pacers with respect to oxygen consumption.
We further deduce an empirical relationship of drag force acting on the main
runner versus her distance to a leading pacer, which can be used for future
prediction of a pacer's aerodynamic effects (cf. Fig. 6). The results are
congruent with experimental data by Pugh [1970], who predicts drag decrease of
up to $80\%$ when running behind a pacer at $1\text{\,}\mathrm{m}$. Davies
[1980], who claimed possible drag savings in the range of $80$-$85\%$, also
confirms these results.
Moreover, the drag distribution at various body parts is examined in Sec. 3.2.
From this and the explanations in Sec. 3.3, we can see a large sensitivity of
the drag acting on the main runner if pacers are positioned laterally, such as
in formations $3$ and $4$. In formation $3$, flow leaks between the front
pacers, causing large pressure peaks at the runner's centre (cf. Fig. 7).
Formation $4$ on the other hand has the negative effect of pacers at the
runner's sides and the resulting acceleration of air between the runners leads
to a relatively low drag reduction of only $33.4\%$ compared to formation $1$
with $70.1\%$. This suggests that runners at the side should be avoided if
optimal drag savings are the goal.
In essence, there are three aerodynamic effects of drafting that lead to a
reduction of axial force (i.e. drag) while running:
1. (i)
_Shielding against fast, oncoming air._
Therefore, there is no sharp deceleration of air and resulting formation of
areas of high stagnation pressure on the runner's surface.
2. (ii)
_Suction effects due to the region of negative total pressure in the leading
runner's wake._
Any object moving through air forms a wake region of relatively low pressure.
This underpressure can be harnessed to the benefit of a trailing runner. The
wake region of negative total pressure has been found in this study to extend
to about $1.14\text{\,}\mathrm{m}$ at
$15\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$ down to
$0.98\text{\,}\mathrm{m}$ at
$36\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$ from the trailing runner's
back.
3. (iii)
_Delay of separation points due to turbulent free shear layers._
Turbulence levels are naturally increased in the high-shear-stress regions of
the wake, where air is displaced against the stagnant surroundings. This
raises momentum transfer in the boundary layers of the runner and delays
separation due to formation of a turbulent boundary layer from a laminar one.
As a result, pressure drag (i.e. form drag) drops, which is the dominant
source of drag of bluff bodies.
Eventually, the aerodynamic power savings are computed in this study and
listed together with the total mechanical power and the metabolic power output
in Sec. 3.4. The derived metabolic power and running economy of up to
$1400.2\text{\,}\mathrm{W}$ and
$25.5\text{\,}\mathrm{W}\text{\,}{\mathrm{kg}}^{-1}$ for the single runner at
a speed of $21\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$ agrees well with
values from literature for trained athletes [Kipp, 2017, Batliner et al.,
2018]. The largest improvements of running economy occur for formation $1$
with $3.1\%$ and formation $2$ with $3.5\%$. These metabolic gains are
slightly higher than the $2.84\%$ gain computed by Polidori et al. [2020] for
the best position of Kenenisa Bekele's drafting strategy during the 2019
Berlin marathon. The reason for this is most likely the larger axial spacing
of $1.3\text{\,}\mathrm{m}$ to the front pacer as opposed to
$1.2\text{\,}\mathrm{m}$ in the current study. Furthermore, a particular
benefit of this study is the additional performance prediction in terms of
improvements of velocity and time for the investigated formations. By applying
the curvilinear relationship between gains in velocity and running economy
suggested by Kipp et al. [2019], we deduce possible speed gains in the range
of $1.1$-$2.3\%$. Time savings of $1.2\text{\,}\mathrm{min}$ (formation 4) to
$2.6\text{\,}\mathrm{min}$ (formation 2) are achieved over the marathon
distance.
In this study we focus on the impact of aerodynamic drafting on running
economy. However, there are a multitude of other factors that influence
running economy and performance. With respect to biomechanics, some of these
factors are (lower) limb mass distribution, Achilles tendon moment arm and
lower body musculotendinous structures for reutilization of elastic energy, as
well as running style and gait patterns (e.g. so-called _pose running_ to
benefit from elastic recoil forces in the lower limbs), among others [Williams
and Cavanagh, 1987, Saunders et al., 2004, Barnes and Kilding, 2015].
Concluding from this work, it is possible to give targeted suggestions for
formation patterns that are best suited for cooperative drafting. It is shown
that axially positioned pacers give the best performance and that laterally
positioned pacers should be running at a distance further away from the main
runner to avoid local flow acceleration, while still contributing positively
to drag reduction. Such an approach has been applied with the recent
successful world record attempt by Eliud Kipchoge during the _INEOS 1:59
Challenge_ , where a staggered formation in the shape of an inverted V has
been chosen. Future studies could explore a larger number of formations by CFD
and subsequently derive reduced-order models for parameters such as number of
pacers, or axial and lateral spacing.
Acknowledgements
This work was realised through research budget from ADIDAS and KTM
E-TECHNOLOGIES. Computational resources were provided by the parallel
computing cluster at KTM in Mattighofen, Austria.
Conflict of interest statement
The authors have nothing to disclose that would have biased this work.
## Appendix A Verification and validation
Since errors and uncertainties are an unavoidable part of any simulation
method such as CFD, measures must be taken to quantify their levels and to
minimize them. According to the guidelines by the American Institute of
Aeronautics and Astronautics [1998], both verification and validation form a
crucial first step of any CFD simulation. While verification can be seen as
the process of ensuring that `the equations are solved right', validation
means ensuring `that the right equations are solved' [Roache, 1998].
In this work, we first perform a grid convergence study by successively
refining the mesh until no significant changes in the relevant quantities are
occurring at the chosen high-order schemes (i.e. verification). Then we
compare the obtained drag coefficients of typical bluff bodies to well-
established values from literature. Finally, since no wind tunnel measurement
results are available for the specific runner geometry we use, we compare the
numerically computed drag coefficient of the runner to experimental values of
previous studies (i.e. validation).
#### Grid convergence study
The average cell size of the mesh at the finest grid level outside the prism
layers has been refined by a factor of two to obtain the target cell sizes at
the runner's surface as illustrated in Fig. 11. All grids contain $5$
inflation layers (i.e. prism layers) at the object surfaces, except the final
one, which has $10$ inflation layers. It can be seen in Fig. 11 that for both
the drag of the considered bluff bodies as well as for the applied runner
geometry the resulting changes in the final refinement stages are small.
Following Roache [1997], we can calculate the discretization error of two
meshes with refinement ratio $r=h_{2}/h_{1}=0.5$ and with cell sizes $h_{1}$
and $h_{2}$ by using the difference $\phi_{2}-\phi_{1}$ between the two
solutions of the target quantity, as
$\displaystyle E_{\phi,1}$ $\displaystyle=$
$\displaystyle\frac{\phi_{2}-\phi_{1}}{1-r^{p}},$ (9) $\displaystyle
E_{\phi,2}$ $\displaystyle=$ $\displaystyle
r^{p}\left(\frac{\phi_{2}-\phi_{1}}{1-r^{p}}\right),$ (10)
with the order of the numerical scheme $p=2$. Using this procedure, we obtain
$E_{\phi,1}=[E_{\phi,1,cube},E_{\phi,1,cylinder},E_{\phi,1,sphere}]=[0.0027,0.0213,0.0240]$
and
$E_{\phi,2}=[E_{\phi,2,cube},E_{\phi,2,cylinder},E_{\phi,2,sphere}]=[0.0007,0.0053,0.0060]$
for the cell sizes $0.006\,25\text{\,}\mathrm{m}$ and
$0.003\,12\text{\,}\mathrm{m}$, both with five inflation layers. For the
applied runner geometry the discretization errors are computed as
$E_{\phi,1,runner}=0.0027$ and $E_{\phi,2,runner}=0.0007$. Already with the
coarser mesh the error is consistently below $3\%$ for all considered bluff
bodies, and with the fine mesh it is below $1\%$. Apart from the
discretization error there is also a linearization error inherent to all CFD
simulations, which comes from the linearization of the governing equations
through numerical schemes. Minimization of the linearization errors of the
simulations is ensured through monitoring of the residuals of all flow
variables and the convergence of the aerodynamic forces. We apply a second-
order scheme for increased accuracy as well as tolerance values for all
residuals of $\mathcal{O}(10^{-6})$.
\begin{overpic}[width=216.81pt]{Figure11.pdf} \end{overpic} Figure 11: Mesh
convergence demonstrated by the final four refinement stages. The drag
coefficient of typical bluff bodies (a) and of the runner geometry used for
this study (b) are shown (i.l. stands for inflation layers).
#### Drag coefficients of general bluff bodies
Using the numerical results for the drag coefficients of the bluff bodies, a
comparison is done with experimental data from literature. The results are
summarized in Tab. 5.
Table 5: Numerical values from this study and experimental values from literature for typical bluff-body drag coefficients $C_{D}$ at a considered free-stream flow velocity of $U_{\infty}=$18\text{\,}\mathrm{km}\text{\,}{\mathrm{h}}^{-1}$$ ($5\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$) and a Reynolds number of $Re\approx(1.5\text{-}1.6)\times 10^{5}$. Object | $C_{D}$ (num.) | $C_{D}$ (exp.)
---|---|---
Cube | $1.06$ | $1.05$ [Hoerner, 1951]
Cylinder | $0.58$ | $0.60$ [Prosser and Smith, 2015]
Sphere | $0.38$ | $0.40$ [Munson et al., 1990]
It can be seen that there is an overall good agreement between the numerical
results and literature, despite the fact that the considered Reynolds number
is close to the drag crisis for the sphere and the cylinder in crossflow. In
this range there is a transition from laminar to turbulent boundary layers,
which leads to a sudden drop in the viscous drag due to delayed flow
separation. It is therefore crucial to properly resolve the flow field across
the boundary layer, which is achieved here through insertion of a sufficient
number of prism layers.
#### Drag coefficients of runners
In the final part of the validation the runner's drag is compared to the
results of several experimental studies on runners.
Table 6: Comparison of values from literature in chronological order for the drag coefficient $C_{D}$ of humans at various activities. Authors | Activity | $C_{D}$
---|---|---
Walpert and Kyle [1989] | Running | $0.79$
Davies [1980] | Running | $0.82$-$0.91$
Pugh [1971] | Running | $0.8$
| Walking | $0.7$
Hoerner [1965] | Standing | $1.0$-$1.3$ (clothed)
| | $0.9$-$1.2$ (nude)
Hill [1928] | Running | $0.9$
| Standing | $0.98$
From the data in Tab. 6 it is obvious that a spread of the drag coefficient
values exists with respect to the different activities, but also for running
itself. It proves the difficulty of establishing well-defined drag
coefficients for humans experimentally. However, there is a clear reduction of
the drag coefficient from standing to walking and running. This has been
argued to be due to the limbs being partially lifted in the air leading to a
temporary reduction of frontal area and the subject being less affected by the
air resistance over time. Furthermore, clothing has a significant impact as
well, leading to increased drag and possibly earlier flow separation,
especially if loose [Nørstrud, 2009]. The quantitative effect of surface
properties on running aerodynamics has been assessed by Kyle and Caiozzo
[1986], who estimate the increase of drag by loose clothing to be up to
$4.2\%$ and by hair to be $4$-$6\%$, depending on length. In the current
study, we obtain a drag coefficient of the considered runner of $0.74$ at a
comparable Reynolds number, which is close to the values established by Pugh
[1971], Davies [1980], and Walpert and Kyle [1989]. One reason, why the drag
coefficient computed in our case is slightly lower than the values from
literature given in Tab. 6 might be the smooth geometry, allowing for delayed
flow separation and thus reduced form drag on the runner. This is an effect
that would not occur in a wind tunnel setup involving real runners or their
scaled models due to the aforementioned conditions of surface roughness, hair,
and clothing.
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11footnotetext: corresponding author
,
# Effect of medium on fundamental interactions in gravity and condensed matter
Alexander Zhuk Valerii Shulga
###### Abstract
Recently, it was shown that the gravitational field undergoes exponential
cutoff at large cosmological scales due to the presence of background matter.
In this article, we demonstrate that there is a close mathematical analogy
between this effect and the behavior of the magnetic field induced by a
solenoid placed in a superconductor.
Keywords:cosmology; scalar perturbations; gravitational potential; magnetic
field; superconductor.
## 1 Introduction
It seems quite natural that the presence of the medium influences the
propagation of fundamental interactions. The simplest example is the Debye
screening of the electric field of an individual particle in a plasma by
particles of opposite sign. Here, the potential produced by an external point
charge has the form of the Yukawa potential (but not the Coulomb one) with the
Debye screening length (see, e.g., [1]). A similar screening mechanism of the
electron charge due to vacuum polarization takes place in quantum
electrodynamics (see, e.g., [2]). The Anderson-Higgs mechanism is another
example of the influence of the medium on fundamental interactions, which are
carried by gauge fields. In this case, after symmetry breaking, the Higgs
vacuum field acts as a medium [3, 4, 5]. As a result of interaction with this
medium, the initially massless gauge fields gain mass [6]. It is also known
that medium in the form of the superconductor affects the electromagnetic
interaction. For example, external magnetic field undergoes the exponential
cutoff inside the superconductor due to the Meissner effect (see, e.g., [7]).
The examples above did not concern the gravitational interaction between
massive bodies. It is known that in a vacuum in the weak field limit the
gravitational potential satisfies the Poisson equation and has the form of
Newton’s potential [8]. From a naive point of view, since all masses have the
same sign and are attracted to each other, one should hardly expect a
screening of the gravitational interaction, as, for example, for electric
charges in a plasma. However, it was demonstrated recently [9, 10, 11] that
medium in the case of gravity also plays important role. It was shown that,
due to the interaction of the gravitational potential with background matter,
there is an exponential cutoff of the gravitational interaction at large
cosmological scales. In section 2 we reproduce this result. For many, this
result turned out to be rather unexpected. Therefore, in this paper, in
section 3, we present a close mathematical analogue of this phenomenon by the
example of the magnetic field induced by a solenoid placed in a
superconductor.
## 2 Screening of the gravitational interaction in cosmology
We consider the Universe containing the cosmological constant $\Lambda$ and
filled with discrete point-like gravitating sources (galaxies and the group of
galaxies) with comoving mass density
$\rho=\sum_{n}\rho_{n}=\sum_{n}m_{n}\,\delta(\mathbf{r}-\mathbf{r}_{n})\,,$
(2.1)
where $\mathbf{r}=(x^{1},x^{2},x^{3})$ is comoving distance. This is our
medium. Such matter has a dust-like equation of state and the average energy
density $\bar{\varepsilon}=\bar{\rho}c^{2}/a^{3}$ where comoving averaged mass
density $\bar{\rho}=\mathrm{const}$, $c$ is the speed of light and $a$ is the
conformal factor. The corresponding background metric is described by
Friedmann-Lema$\mathrm{\hat{\i}}$tre-Robertson-Walker (FLRW) one.
The discrete inhomogeneities perturb the FLRW metric [12, 13]:
$ds^{2}=a^{2}\big{[}(1+2\Phi)d\eta^{2}-(1-2\Phi)\delta_{\alpha\beta}\,dx^{\alpha}dx^{\beta}\,\big{]}\,,$
(2.2)
where we restrict ourselves to scalar perturbations in conformal Newtonian
gauge. Scalar function $\Phi(\eta,\mathbf{r})$ is the gravitational potential
created at the point with the radius-vector $\mathbf{r}$ by all gravitating
masses in the Universe [8]. The perturbed Einstein equations are [12, 13]:
$\Delta\Phi-3{\mathcal{H}}\left(\Phi^{\prime}+{\mathcal{H}}\Phi\right)=\frac{1}{2}\kappa
a^{2}\delta\varepsilon,$ (2.3)
$\Phi^{\prime}+{\mathcal{H}}\Phi=-\frac{1}{2}\kappa
a^{2}\bar{\varepsilon}v\,,$ (2.4)
$\Phi^{\prime\prime}+3{\mathcal{H}}\Phi^{\prime}+\left(2{\mathcal{H}}^{\prime}-{\mathcal{H}}^{2}\right)\Phi=0\,,$
(2.5)
where $\Delta\equiv\delta^{\alpha\beta}\partial_{\alpha}\partial_{\beta}$ is
the Laplace operator, the prime denotes the conformal time $\eta$ derivative,
${\mathcal{H}}\equiv(da/d\eta)/a=(a/c)H$ and $H\equiv(da/dt)/a$ is the Hubble
parameter, $v(\eta,\textbf{r})$ is the peculiar velocity potential and
$\kappa\equiv 8\pi G_{\\!N}/c^{4}$, where $G_{\\!N}$ is the gravitational
constant. The energy density fluctuation reads [14, 15]:
$\delta\varepsilon=\frac{c^{2}}{a^{3}}\delta\rho+\frac{3\bar{\rho}c^{2}}{a^{3}}\Phi\,,$
(2.6)
where $\delta\rho(\eta,\mathbf{r})\equiv\rho-\bar{\rho}$ is the fluctuation of
the mass density (2.1) around its constant average value $\bar{\rho}$.
Eq. (2.4) demonstrates that the peculiar velocities affect the gravitational
potential. If we neglect this influence (i.e.
$\Phi^{\prime}+{\mathcal{H}}\phi=0$), then equation (2.3) takes the form
$\Delta\Phi-\frac{a^{2}}{\lambda^{2}}\Phi=\frac{\kappa
c^{2}}{2a}\delta\rho\,,$ (2.7)
where the screening length
$\lambda\equiv\sqrt{\frac{2a^{3}}{3\kappa\bar{\rho}c^{2}}}\,.$ (2.8)
With the help of the transformation (to remove the $\bar{\rho}$ contribution
on the RHS of Eq. (2.7))
$\phi=c^{2}a\Phi-\frac{4\pi
G_{N}\bar{\rho}}{a^{2}}\lambda^{2}=c^{2}a\Phi-\frac{c^{2}a}{3}$ (2.9)
Eq. (2.7) is reduced to
$\Delta\phi-\frac{a^{2}}{\lambda^{2}}\phi=4\pi G_{N}\rho\,.$ (2.10)
For the mass density (2.1), we can easily solve this Helmholtz equation, and
applying transformation (2.9) obtain:
$\Phi=\frac{1}{3}-\frac{\kappa c^{2}}{8\pi
a}\sum_{n}\frac{m_{n}}{|\mathbf{r}-\mathbf{r}_{n}|}\exp\left(-\frac{a|\mathbf{r}-\mathbf{r}_{n}|}{\lambda}\right)\,.$
(2.11)
It is worth noting that the physical distance is $R=ar$. The term 1/3 (which
is due to $\bar{\rho}$ in $\delta\rho$) plays an important role since only
with this term the averaged over all volume value of the gravitational
potential $\bar{\Phi}$ is equal to zero as it should be for fluctuations [9].
In Eq. (2.11), we neglect the peculiar velocities of the inhomogeneities.
However, they also play an important role [16, 17] and must be taken into
account. For the considered model, as was shown in [16], it is sufficient in
(2.7), (2.9)-(2.11) to replace $\lambda$ with $\lambda_{\mathtt{eff}}$ and
additionally in (2.11): $1/3\to 1/3(\lambda_{\mathtt{eff}}/\lambda)^{2}$ where
$\lambda_{\mathtt{eff}}=\sqrt{\frac{c^{2}a^{2}H}{3}\int\frac{da}{a^{3}H^{3}}}\,.$
(2.12)
To get this result, we should take into consideration Eq. (2.5). This
screening length (as well as $\lambda$) depends on time. For example, for the
standard $\Lambda$CDM model at present time $\lambda_{\mathtt{eff}}=2.57$ Gpc
[16].
Therefore, the gravitational potential $\Phi$ satisfies the Helmholtz
equation, not the Poisson equation. This is due to the interaction of the
gravitational potential with the medium. We can see it directly from Eq. (2.6)
where the term $\sim\bar{\rho}\Phi$ describe this interaction. Due to the
peculiar velocity, Eq. (2.3) also acquires an additional term proportional to
$\Phi$ [16]. If the medium is absent that corresponds to the limit
$\bar{\rho}\to 0,\,v\to 0$, then the screening lengths $\lambda$ and
$\lambda_{\mathtt{eff}}$ tends to infinity, and the Yukawa potentials in
(2.11) are reduced to the Newton’s ones without screening of the gravitational
interaction.
## 3 Solenoid in a superconductor. Screening of the induced magnetic field.
In this section, in order to present the mathematical analog of the screening
effect described above, we render some of equations of the paper [18] in a
form suitable for our purpose. Following this paper, we consider a thin
solenoid placed in a superconductor. Thin means that the diameter of the
solenoid is much smaller than the magnetic field penetration length
$\lambda_{\mathtt{m}}$. It is well known that the magnetic field of the
solenoid $\mathbf{B}_{\mathtt{sol}}$ is absent from the outside it, but the
vector potential $\mathbf{A}_{\mathtt{sol}}$ is not equal to zero. The
interaction of this potential with the superconducting medium induces a
current $\mathbf{J}_{\mathtt{sc}}$, which, in turn, leads to the appearance of
an induced magnetic field $\mathbf{B}_{\mathtt{sc}}$. Thus, the Maxwell
equation has the form222In this section, we use the system of units adopted in
book [7].
$\mathtt{curl}\mathbf{B}_{\mathtt{tot}}=\mathtt{curl}\left(\mathbf{B}_{\mathtt{sc}}+\mathbf{B}_{\mathtt{sol}}\right)=\mathbf{J}_{\mathtt{sc}}+\mathbf{J}_{\mathtt{sol}}\,.$
(3.1)
Since outside the solenoid
$\mathbf{B}_{\mathtt{sol}},\mathbf{J}_{\mathtt{sol}}=0$, we get
$\mathtt{curl}\mathbf{B}_{\mathtt{sc}}=\mathbf{J}_{\mathtt{sc}}\,,$ (3.2)
where in the London limit the superconducting current density is [7, 18]
$\mathbf{J}_{\mathtt{sc}}=-\frac{1}{\lambda^{2}_{\mathtt{m}}}\left(\frac{1}{q}\nabla\theta+\mathbf{A}_{\mathtt{tot}}\right)\,.$
(3.3)
Here,
$\mathbf{A}_{\mathtt{tot}}=\mathbf{A}_{\mathtt{sc}}+\mathbf{A}_{\mathtt{sol}}$,
$\theta$ is the phase of the order parameter and the magnetic field
penetration length
$\lambda_{\mathtt{m}}=\frac{1}{q\sqrt{n_{\mathtt{s}}}}\,,$ (3.4)
whrere $n_{\mathtt{s}}$ is the superfluid density, parameter $q$ defines the
superconducting flux quanta (see, e.g., Eq. (3.7) below) and in the real
superconductor $q=2e/(\hbar c)$ [7]. The absence of a superconducting medium
corresponds to the limit $n_{\mathtt{s}}\to
0\Rightarrow\lambda_{\mathtt{m}}\to\infty$. Expression (3.4) is an analogue of
cosmological formula (2.8) (and, accordingly, formula (2.12)). In Eq. (3.3)
the term $\lambda^{-2}_{\mathtt{m}}\mathbf{A}_{\mathtt{sol}}\sim
n_{\mathtt{s}}\mathbf{A}_{\mathtt{sol}}$ describes the interaction of the
solenoid magnetic field with the superconducting medium just as the term
$\sim\bar{\rho}\Phi$ on the RHS of Eq. (2.6) describes the interactions of the
gravitational potential with the cosmological medium.
Now, applying curl operation to both sides of (3.3), we obtain
$\mathbf{B}_{\mathtt{sc}}-\lambda^{2}_{\mathtt{m}}\Delta\mathbf{B}_{\mathtt{sc}}=0\,,$
(3.5)
where we took into account that outside of the solenoid
$\mathtt{curl}\nabla\theta=0$ and $\mathbf{B}_{\mathtt{sol}}=0$. $\Delta$ is
the Laplace operator in flat space. To solve this equation, we need to define
the boundary conditions. Let the solenoid be extended along the $z$-axis.
Obviously, due to the cylindrical symmetry the induced magnetic field inside
the superconductor is also parallel to the $z$-axis:
$\mathbf{B}_{\mathtt{sc}}(r)={B}_{\mathtt{sc}}(r)\hat{z}$, where $\hat{z}$ is
the unit vector along $z$-axis. In cylindric coordinates, $\mathbf{r}$ is the
radius-vector in the $xy$-plane (it is worth noting that in the previous
section $\mathbf{r}$ denotes the comoving three-dimensional radius-vector). At
distances $r>>\lambda_{\mathtt{m}}$, the superconducting current goes to zero:
$\mathbf{J}_{\mathtt{sc}}\to 0$. Therefore, at this distances Eq. (3.3) reads
$\mathbf{A}_{\mathtt{tot}}=-\frac{1}{q}\nabla\theta\,.$ (3.6)
Integrating both sides of this equation over an area inside the contour
$r=\mbox{\rm const}\,$, and performing the Stokes area-to-contour
transformation for the RHS, we find
$\Phi_{\mathtt{tot}}=-\frac{2\pi}{q}N\equiv-\Phi_{0}N\,,\quad
N=0,1,2,\ldots\,,$ (3.7)
where $\Phi_{\mathtt{tot}}=\Phi_{\mathtt{sc}}+\Phi_{\mathtt{sol}}$ is the
total magnetic flux consisting of the sum of the magnetic fluxes of the
induced magnetic field and the magnetic field inside the solenoid. $\Phi_{0}$
is the superconducting flux quanta. Therefore,
$\Phi_{\mathtt{sc}}=\Phi_{\mathtt{tot}}-\Phi_{\mathtt{sol}}\,.$ (3.8)
This is our boundary condition. We can include it directly into Eq. (3.5):
$B_{\mathtt{sc}}-\lambda^{2}_{\mathtt{m}}\Delta
B_{\mathtt{sc}}=\Phi_{\mathtt{sc}}\delta(\mathbf{r})\,,$ (3.9)
where we took into account 2D cylindrical symmetry of the problem and,
consequently, $\Delta$ is a radial Laplace operator. Obviously, integrating
this equation over an area inside the contour $r=\mbox{\rm const}\,$ we arrive
at identity. Eq. (3.9) is the Helmholtz one (similar to Eq. (2.10)), and has
the decreasing solution
$B_{\mathtt{sc}}=\frac{\Phi_{\mathtt{sc}}}{2\pi\lambda^{2}_{\mathtt{m}}}K_{0}\left(\frac{r}{\lambda_{\mathtt{m}}}\right)\,,$
(3.10)
where $K_{0}$ is the modified Bessel function. The induced magnetic field
behaves asymptotically as follows:
$B_{\mathtt{sc}}(r\to 0)\sim-\ln(r)\,,\quad
B_{\mathtt{sc}}(r\to\infty)\sim\frac{1}{\sqrt{r}}\exp(-r/\lambda_{\mathtt{m}})\,$
(3.11)
This behavior reflects the cylindrical symmetry of the model. For example,
Yukawa’s potential has been transformed:
$(1/r)\exp(-r/\lambda_{\mathtt{m}})\to(1/\sqrt{r})\exp(-r/\lambda_{\mathtt{m}})$.
As expected, the screening length coincides with the magnetic field
penetration length $\lambda_{\mathtt{m}}$. Formula (3.11) is 2D analog of Eq.
(2.11).
## 4 Conclusion
In this paper, we have touched upon the problem of the influence of the medium
on fundamental interactions. First, on the basis of articles [9, 10, 11], we
showed that as a result of the interaction of the gravitational field with the
cosmological medium, the gravitational potential is subject to exponential
screening on large cosmological scales. Then, following the model considered
in paper [18], we have traced a close analogy between the interaction of the
gravitational field with the cosmological medium and the interaction of the
magnetic field of a solenoid with a superconducting medium. As a result of
this interaction, the induced magnetic field in the superconductor undergoes
exponential screening at distances exceeding the magnetic field penetration
length.
## Acknowledgements
The authors are grateful to Boris Svistunov for fruitful discussions and
valuable comments.
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# Many wrong models approach to localize an odor source in turbulence:
introducing the weighted Bayesian update
Lorenzo Piro<EMAIL_ADDRESS>Department of Physics & INFN,
University of Rome “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Rome,
Italy Robin A. Heinonen Department of Physics & INFN, University of Rome
“Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Rome, Italy Massimo
Cencini Istituto dei Sistemi Complessi, CNR, Via dei Taurini 19, 00185 Rome,
Italy INFN “Tor Vergata”, Via della Ricerca Scientifica 1, 00133 Rome, Italy
Luca Biferale Department of Physics & INFN, University of Rome “Tor Vergata”,
Via della Ricerca Scientifica 1, 00133 Rome, Italy
###### Abstract
The problem of locating an odor source in turbulent environments is central to
key applications such as environmental monitoring and disaster response. We
address this challenge by designing an algorithm based on Bayesian inference,
which uses odor measurements from an ensemble of static sensors to estimate
the source position through a stochastic model of the environment. Given the
practical impossibility of achieving a fully consistent turbulent model and
guaranteeing convergence to the correct solution, we propose a method to rank
“many wrong models” and to blend their predictions. We evaluate our _weighted
Bayesian update_ algorithm by its ability to estimate the source location with
predefined accuracy and/or within a specified time frame, and compare it to
standard Monte Carlo sampling methods. To demonstrate the robustness and
potential applications of both approaches under realistic environmental
conditions, we use high-quality direct numerical simulations of the Navier-
Stokes equations to mimic the transport of odors in the atmospheric boundary
layer. Despite minimal prior information about the source and environmental
conditions, our proposed approach consistently proves to be more accurate,
reliable, and robust than Monte Carlo methods, thus showing promise as a new
tool for addressing the odor source localization problem in real-world
scenarios.
## I Introduction
Identifying the origin of noxious odors such as gas leaks and chemical or
radioactive emissions is critical for averting potential environmental
disasters [1, 2, 3, 4]. Moreover, tracing the location of a source from the
odor signal detected in the atmosphere is a vital task for animals looking for
food or a mate [5, 6].
Such source localization problems are already difficult in laminar flows,
where the structure of the odor plume can be highly complex and sensitive to
the source location due to chaotic mixing [7, 8]. In fully turbulent three-
dimensional (3-D) environments, the difficulty becomes severe, owing to
numerical, experimental, and phenomenological complexities and challenges
associated with the multi-scale nature of 3-D turbulence, which is
characterized by a broad range of rough velocity fluctuations as well as
highly intermittent and non-Gaussian properties for both the advecting flow
and the odor transport [9, 10, 11, 12].
This makes the design of efficient and reliable algorithms for source
localization a challenging theoretical task situated at the intersection of
fluid dynamics, optimal navigation of mobile agents, and information theory
[12, 13, 14]. Over the last decades, a growing community of scientists has
developed multiple methodologies to address this problem, ranging from bio-
inspired [15, 16, 17] to heuristic search algorithms based on information
theory [18, 19, 20] and Bayesian inference [21, 22].
Although most studies have focused on the performance optimization of mobile
agents, the use of static sensors has become increasingly popular [23, 24, 25]
thanks to the simplicity of their implementation, also resulting in reduced
setup and maintenance costs. Moreover, being well-suited for continuous and
large-scale environmental monitoring, they are ideal for use in early warning
systems [26, 27, 28, 29], which may enable targeted and effective response
strategies, ranging from containment and mitigation to rescue and prevention.
On a more theoretical level, the employment of static sensors reduces the
space of possible choices for the search algorithm [30], thereby facilitating
the study of more fundamental questions. Indeed, despite the recent progress,
the problem of optimally locating a source within a given time with a
prescribed accuracy in a turbulent flow is far from solved [25].
Figure 1: Left: a snapshot of DNS showing relative positions of sources
(squares) and particles color-coded according to the source from which each
was emitted. Here, we show particles in a thin slab parallel to the wind and
containing the sources, and we have zoomed in close to the sources. We have
also inserted in the same plane a network of static sensors (yellow circles)
whose measurements will be used in the following to infer the source location.
Right: probability map of making a detection. Hereafter referred to as
_empirical likelihood_ , such distribution has been obtained from coarse-
graining the DNS data on a square lattice, setting a threshold $n_{\rm thr}=2$
on the number of detectable particles, and then averaging over time the signal
of all the five sources (properly placed, by means of a suitable shift, in the
same position as indicated by the grey square).
A key challenge is the practical impossibility of specifying a detailed model
for the statistics governing odor transport. This is a crucial point when
using Bayesian approaches that heavily depend on a model of the environment to
integrate sensor observations into a probability distribution describing our
information about the source. However, in all real-world applications, we have
only partial knowledge of the statistical properties of the environment, which
may be arbitrarily complex. Moreover, the inherent complexity of turbulent
transport precludes a complete analytical description even in an idealized
physical scenario. Therefore, we are invariably compelled to work with an
_wrong_ model of the environment. This fundamentally limits the utility of
Bayesian approaches, which are only guaranteed to converge to the ground truth
when the model is correctly specified [31, 32]. To partially overcome these
intrinsic limitations, we explore a new Bayesian approach designed to mitigate
the effects of errors in the environmental model. Building upon the _many
wrongs principle_ [33, 34], we introduce a quantity that allows to rank the
models of the environment in the Bayesian framework, providing a principled
way to blend information coming from several (inevitably wrong) models. We
show that merging the information gathered from a number of different models –
a procedure that we dubbed Weighted Bayesian Update (WBU) – helps in reliably
inferring the source location.
In order to test these ideas and the proposed methodology in realistic
conditions, we devise a set of direct numerical simulations that mimic the
emission of a localized odor source in a three-dimensional turbulent
environment. Taking advantage of the Galilean invariance of the Navier-Stokes
equation, we model the odor transport by Lagrangian tracers advected by the
flow with a constant mean wind, obtaining odor plumes that well approximate
those one can observe in the atmospheric boundary layer [11, 12]. Here, for
the sake of a first benchmark of the localization algorithms, we shall assume
the source lies on the same plane as the sensors, which can thus detect odor
particles contained in a thin slab parallel to the wind [see Fig.1(a)]. Then,
we use a simplified (effective) model for the environment that depends on a
few parameters only and neglects temporal and spatial correlations of the odor
signal. In spite of the obvious shortcomings of this model, we show how, by
suitably weighting different models (i.e. the same one with different
parameters), WBU outperforms widely-used Monte Carlo methods [21, 35] when
locating the odor source with a specified level of accuracy. By performing
several tests with synthetic data and empirically based (_a posteriori_)
models, we trace the origin of the problems for Monte Carlo methods to the
effects of correlations in the realistic odor signal, to which the WBU shows
superior resilience.
The paper is organized as follows. In Sec. II, we illustrate the setup where
the numerical simulations have been carried out and describe the simplified
model of the environment as well as the algorithms employed, namely the
weighted Bayesian update and another approach based on Monte Carlo techniques.
We then present the results in Sec. III, which is divided into two parts. In
the first one, Sec. III.1, we discuss and compare the performance of the
algorithms in the setting closest to reality. In Sec. III.2, we then
systematically study how model inaccuracy affects the quality of the source
localization. Finally, we summarize our findings and provide an outlook for
future studies in Sec. IV.
## II Methods
### II.1 Setup of the numerical simulations
Let us assume we aim to locate the position $\bm{r}_{\rm s}$ of an odor source
emitting at rate $Q$ within a two-dimensional square grid $\Omega$ of size
$L\times L$ and lattice spacing $\Delta x$. The odor particles are swept by
the underlying turbulent flow characterized by a mean wind in a given
direction $\hat{\bm{U}}$. We then place a number $N_{\rm s}$ of static sensors
in this environment. We arrange the sensors in a lattice that, assuming no
prior knowledge of the wind direction, will be typically tilted by an angle
$\theta$ with respect to the latter, as depicted in Fig. 1(a).
Instead of using an odor concentration field, we model it in terms of
particles advected by a turbulent flow. Each particle can be thought of as a
patch of odor, or one can consider the number of particles in a given small
region as an estimate of the odor concentration. We produced realistic
trajectories of odor particles using state-of-the-art direct numerical
simulation (DNS) of the incompressible, 3–D Navier-Stokes equations
$\displaystyle\partial_{t}\bm{u}+(\bm{u}\cdot\nabla)\bm{u}$
$\displaystyle=-\nabla p+\nu\nabla^{2}\bm{u}+f,$ (1)
$\displaystyle\nabla\cdot\bm{u}$ $\displaystyle=0,$ (2)
under turbulent conditions with $\mathrm{Re}_{\lambda}\simeq 150.$ Here, $f$
is a random, Sawford-type [36] isotropic forcing at the smallest nonzero
wavenumbers of the system, with a correlation time of 160 simulation
timesteps. Using a pseudospectral code dealiased according to the two-thirds
rule, the system was solved on a $1024\times 512\times 512$ grid, with a
uniform spacing $\delta x=\delta y=\delta z\simeq\eta$ (with $\eta$ the
Kolmogorov scale), and periodic boundary conditions in all three directions.
The timestepping was performed using the second-order explicit Adams-Bashforth
method. The system was advected by a uniform mean wind
$\bm{U}\approx-2.5u_{\rm rms}\hat{x},$ where $u_{\rm rms}$ is the rms speed of
the flow in the comoving frame of the wind, and $\hat{x}$ is the elongated
axis of the grid. We produced the mean wind by means of a Galilean
transformation.
The odor particles were modeled as Lagrangian tracer particles, which were
emitted by five stationary point sources [Fig. 1(a)]. Each source emitted 1000
particles every 10 simulation timesteps, which corresponds to every $\approx
1/15$ Kolmogorov times $\tau_{\eta}$. The fluid velocities $\bm{u}$ at the
particle positions were obtained using a sixth-order B-spline interpolation
scheme and then used to evolve the particle positions $\bm{X}$ in time
according to $\dot{\bm{X}}=\bm{u}(\bm{X},t)$ over an infinite lattice of
copies of the periodic flow. Their positions, velocities, and accelerations
were tracked and dumped every $\tau_{\eta},$ for a total of 3015 timesteps.
Each source of particles was treated as independent, and we averaged our
results over them for the purpose of achieving better statistics.
To simulate realistic environmental conditions and emulate turbulent
dispersion in the atmospheric boundary layer, we then coarse-grain the
particles’ concentration inside a thin layer containing the source and set a
threshold $n_{\rm thr}=2$ on the number of particles above which sensors make
a detection. A few snapshots of the resulting odor dispersion in the
aforementioned arena $\Omega$ are shown in Fig. 2, where the pink patches
indicate the odor plume.
Figure 2: Three time snapshots showing a qualitative comparison between the
performance of the two algorithms discussed in this work. Both approaches use
the measurements made by an array of static sensors (circles) looking for a
source of odors (red square) advected by a turbulent flow featuring a
horizontal wind blowing from right to left. Pink patches indicate the odor
plume; greyscale codes the probability of the odor source location obtained
from an algorithm based on Bayesian inference; green open triangles depict the
candidate source positions yielded by a Sequential Monte Carlo sampling
method. Time here is in units of observations made by each sensor, which is
hereafter assumed to be equal to the Kolmogorov timescale $\tau_{\eta}$ of the
flow.
### II.2 Model of the environment
To localize the source within this setting given the sensors’ measurements, we
shall now introduce a model that captures the mean-flow and mean-diffusion
properties of the environment. If, on the one hand, we assume to have a
statistical knowledge of the environment through, for example, the history of
prior measurements in the field, we can compute empirically what is the
probability of detecting an odor particle from the time average of the DNS
data [Fig. 1(b)]. We will hereafter refer to this distribution as the
_empirical likelihood_ , in accordance with Bayesian nomenclature. The use of
such likelihood has the advantage of avoiding the complication of fitting some
environmental parameters while looking for the source. However, it relies on a
far more detailed prior knowledge of the environment, which is not generally
available. On the other hand, we can still exploit it to understand the
strengths and weaknesses of the localization algorithms, as discussed in Sec.
III.2.
In a realistic setup, we are therefore compelled to use a statistical
description of odor encounters in a turbulent flow to infer the source
location. Out of a wide range of possibilities [37], we model the turbulent
transport of odor particles emitted at rate $Q$ by a point source as an
effective advection-diffusion process [38]
$\partial_{t}c+\bm{U}\cdot\bm{\nabla}c=D\nabla^{2}c+Q\delta(\bm{r}-\bm{r}_{\rm
s})-c/\tau\,,$ (3)
where $c$ is the odor concentration field, $\bm{U}$ the mean wind featured by
the turbulent flow, $\tau$ the lifetime of the odors, and $\bm{r}_{\rm s}$ the
source position. Note that the combination of molecular and turbulent
diffusivity (due to the flow velocity fluctuations) is here described by a
single effective (eddy) diffusion coefficient $D$ [39]. Despite being a strong
oversimplification that ignores important multiscale, non-Gaussian properties
of the underlying turbulence, as well as spatiotemporal fluctuations in the
scalar advection, this model reasonably captures the mean field properties of
the odor concentration [12]. In the stationary regime, Eq. (3) has an
analytical solution, which in three dimensions reads (see, e.g., [18])
$c(\bm{r}-\bm{r}_{\rm s})=\frac{Q}{4\pi D\lVert\bm{r}-\bm{r}_{\rm
s}\rVert}\exp\left[\frac{(\bm{r}-\bm{r}_{\rm
s})\cdot\hat{\bm{U}}-\lVert\bm{r}-\bm{r}_{\rm s}\rVert}{\lambda}\right]\,,$
(4)
where we have assumed $\tau\gg 1$. A key advantage of the adopted model is
that it essentially depends only on three environmental parameters, i.e., the
source emission rate $Q$, the mean wind direction $\hat{\bm{U}}$, and the
characteristic length scale of the flow $\lambda\equiv 2D/U$. To mimic the
sparseness and intermittency of the odor signals observed from the DNS, and
typical of turbulent environments, we assume that the detection is a random
process modeled in terms of a Poissonian process with mean $\mu$ [18, 40].
Therefore, the probability that a sensor makes a detection is given by
$p(h_{i}|\bm{r}_{i}-\bm{r}_{\rm s})=\frac{[\mu(\bm{r}_{i}-\bm{r}_{\rm
s})]^{h_{i}}\exp[-\mu(\bm{r}_{i}-\bm{r}_{\rm s})]}{h_{i}!}\,,$ (5)
where $h_{i}$ is the number of odor particles detected by the $i-$th sensor.
The mean number $\mu$ of particles hitting, within a time interval $\Delta t$,
the $i$-th sensor is related to the mean concentration (4) via the classical
Smoluchowski formula [41]
$\mu(\bm{r}_{i}-\bm{r}_{\rm s})=4\pi aD\Delta t\,c(\bm{r}_{i}-\bm{r}_{\rm
s})=\frac{Q}{d_{i}}a\Delta t\exp\left[\frac{(\bm{r}_{i}-\bm{r}_{\rm
s})\cdot\hat{\bm{U}}-d_{i}}{\lambda}\right]\,,$ (6)
where $\bm{r}_{i}$ stands for the $i-$th sensor position, and
$d_{i}\equiv\lVert\bm{r}_{i}-\bm{r}_{\rm s}\rVert$, $a$ is the sensors’
radius. Hereafter, we will actually assume that each sensor can detect the
presence of odors only if the number of particles within its radius $a=\Delta
x/2$ exceeds a particular threshold. In other words, the sensors can only
perform binary measurements (i.e., $h_{i}=\\{0,1\\}$), such that the
probability of detection (5) simplifies into
$\begin{cases}p(0|\bm{r}_{i}-\bm{r}_{\rm s})=\exp[-\mu(\bm{r}_{i}-\bm{r}_{\rm
s})]\\\ p(1|\bm{r}_{i}-\bm{r}_{\rm s})=1-\exp[-\mu(\bm{r}_{i}-\bm{r}_{\rm
s})]\,.\end{cases}$ (7)
### II.3 Weighted Bayesian Update: a new way to rank and exploit _many wrong_
models
Each measurement made by each sensor thus provides information about the
position of the source $\bm{r}_{\rm s}$, which can be processed employing
Bayesian inference. The whole set of measurements from all sensors can be used
to update a probability map – the “posterior” or “belief” in Bayesian jargon –
of the source’s location defined over the whole arena $\Omega$, i.e.
$B(\bm{r})\equiv\mathrm{Prob}(\bm{r}_{\rm s}=\bm{r})$, which we shall dub as
_common belief_ to emphasize that it exploits all sensors’ detections. Since
we assume no prior knowledge and that the source cannot be at the same
location as one of the sensors, the belief is always initialized to a uniform
distribution and set to zero only in the sensors’ positions. Assuming the
simultaneous measurements made by all the $N_{\rm s}$ sensors at time $t$ are
independent, the overall conditional probability of a set of observations
$\bm{h}^{(t)}$ for a possible given source position $\bm{r}$ is simply a
product:
$\mathcal{L}(\bm{h}^{(t)}|\bm{r})=\prod\limits_{i=1}^{N_{\rm
s}}p(h_{i}^{(t)}|\bm{r}_{i}-\bm{r})\,,$ (8)
which depends on the model of the environment as specified in Eq. (7), and is
also known as the _likelihood_ function in Bayesian terminology. Given a
sequential process like the one at hand, at every time step (i.e., once all
sensors have performed a measurement), the belief is updated following Bayes’
rule [42]
$B^{(t)}(\bm{r})=\frac{\mathcal{L}(\bm{h}^{(t)}|\bm{r})B^{(t-1)}(\bm{r})}{\int_{\Omega}{\rm
d}\bm{r}\,\mathcal{L}(\bm{h}^{(t)}|\bm{r})B^{(t-1)}(\bm{r})}\,,$ (9)
where time $t$ is hereafter measured in units of observations made by each
sensor.
Under this update rule, the belief is guaranteed to converge to the correct
solution as long as the correct model of the environment is deployed [43].
This is, however, a rather uncommon case in any realistic scenario, and the
model used to update the common belief will always be wrong. Furthermore,
wrong models are indistinguishable from one another during the search since
they always make the belief converge to a source position, regardless of
whether that is the right one or not. Therefore, finding a way to rank the
models and assess their reliability is of great relevance for any
applications.
In order to introduce the quantity we used for ranking the models, it is
useful to rewrite Bayes’ update rule (9) in a different way:
$B^{(t)}(\bm{r})=\frac{\prod\limits_{i=1}^{N_{\rm
s}}b_{i}^{(t)}(\bm{r})}{\mathcal{Z}}\,,\;\;\;\;\;\;b_{i}^{(t)}(\bm{r})=\frac{p(h_{i}^{(t)}|\bm{r}_{i}-\bm{r})b_{i}^{(t-1)}(\bm{r})}{\int_{\Omega}{\rm
d}\bm{r}\,p(h_{i}^{(t)}|\bm{r}_{i}-\bm{r})b_{i}^{(t-1)}(\bm{r})}\,,$ (10)
where $\mathcal{Z}\equiv\int_{\Omega}{\rm d}\bm{r}\,\prod\limits_{i=1}^{N_{\rm
s}}b_{i}^{(t)}(\bm{r})$. That is to say, owing to the assumption of
independence between measurements, the common belief can be built from the
superposition of the _private_ beliefs (i.e. the belief that can be
constructed for each sensor only on the basis of its own measurements history)
$b_{i}$ of the $i=1,...,N_{\rm s}$ sensors, each of which is updated
independently at every time step via Bayes’ rule. The key advantage of this
change of perspective lies in the interpretation of the update formula [Eqs.
(9-10)]. Indeed, the hitherto overlooked normalization constant $\mathcal{Z}$
essentially quantifies how much the sensors’ _private_ beliefs overlap or, in
other words, how much they agree on a source position given their measurements
and the model at hand. In fact, in Appendix A, we show analytically that this
quantity achieves an asymptotic global maximum when the model is exact (even
in the presence of correlations between observations). We also show
numerically that this quantity is typically smaller the farther the model is
from the ground truth.
This makes $\mathcal{Z}$, hereafter referred to as the _overlap integral_ , an
ideal candidate to achieve our task as we could use it to weigh different
models of the environment. To this end, let us scan the parameters space of
the stochastic model introduced above [see Eqs. (6)-(7)] assuming to know only
on the mean wind direction $\hat{\bm{U}}$, which can be always measured in
practice using an anemometer, and run the Bayes update (9) independently for
each set of parameters $\bm{P}\equiv\\{Q,\lambda\\}$. Then, we shall blend the
information collected into a master belief as
$\mathcal{B}^{(t)}_{\rm
M}(\bm{r})=\frac{\prod\limits_{j=1}^{M}\left[B_{j}^{(t)}(\bm{r})\right]^{\beta_{j}^{(t)}}}{\int_{\Omega}\prod\limits_{j=1}^{M}\left[B_{j}^{(t)}(\bm{r})\right]^{\beta_{j}^{(t)}}}\,,\;\;\;\beta_{j}^{(t)}\equiv\frac{\mathcal{Z}_{j}^{(t)}}{\sum\limits_{j=1}^{M}\mathcal{Z}_{j}^{(t)}}\,,$
(11)
where $M$ is the total number of distinct sets
$\bm{P}_{j}=\\{Q_{j},\lambda_{j}\\}$ of model parameters considered, and the
index $j$ refers to which of these sets was used to obtain the common belief
$B_{j}^{(t)}$ at time $t$ and the corresponding overlap integral
$\mathcal{Z}_{j}^{(t)}$.
The algorithmic procedure described in Eqs. (9)$-$(11) thus provides a
principled way to define a single master belief $\mathcal{B}_{\rm M}$, i.e.,
the probability distribution about the source location, which is obtained from
a weighted average over the results yielded by different models. We will
hereafter refer to this approach as the weighted Bayesian update (WBU), whose
numerical implementation is detailed in Appendix B.
### II.4 Sequential Monte Carlo with Importance Sampling
The above-described WBU method offers a new perspective on the implementation
of algorithms for odor source localization in turbulent flows. There exists,
however, already a vast literature on the possible methods to address the same
kind of problem [25]. One of the most commonly used approaches is the one
based on the use of Monte Carlo sampling methods to estimate the belief [21,
40, 35]. It is, therefore, natural to ask how the WBU’s performance compares
with such more conventional approaches.
Due to the wide range of potential applications [44], much research has been
conducted over the last decades to make these algorithms more and more
computationally efficient and accurate, and, to date, there exist many
different variants depending on the specific setup at hand. We refer the
reader to Ref. [25] for a comprehensive review of the topic.
In the following, we shall use a state-of-the-art version of the so-called
_Sequential Monte Carlo_ (SMC) algorithm that also involves a Markov chain
Monte Carlo (MCMC) perturbation step [45, 46]. Using such an approach, we can
simultaneously infer both the source position $\bm{r}_{\rm s}$ and the
(unknown) parameters $\bm{P}$ of the stochastic model of the environment. Let
us therefore first define a _sample_ $\bm{\theta}_{i}$ as a possible
combination of such source parameters, i.e.
$\bm{\theta}_{i}\equiv\\{\bm{r}_{{\rm s},i},\bm{P}_{i}\\}$. At every time
step, after all sensors have measured, a collection of $N$ of such samples is
drawn from the current belief defined in the $\bm{\theta}$ space, and we
assign a weight $w$ to each of them equal to the likelihood of the latest
measurement as defined in Eq. (8). Upon normalization of the samples’ weights,
we shall then compute the effective sample size $N_{\rm eff}$ to avoid the so-
called _degeneracy problem_ [46]. Indeed, if $N_{\rm eff}$ goes below a given
threshold $N_{\rm thr}$ (typically set to $N/2$), then it is necessary to
generate a new set of $N$ samples. This is known as the _resampling_ step.
There, for each sample $\tilde{\bm{\theta}}_{i}$, a new one is selected from
the same pool and with a probability equal to its weight that replaces the
former. After resampling, all weights are then set equal to $1/N$.
Next is the so-called Metropolis-Hastings MCMC perturbation step, which
consists of moving each sample in its neighborhood and deciding whether to
accept or reject the new proposal based on some acceptance criteria [47]. This
is essentially done to diversify the $N$ samples and, therefore, improve the
sampling efficiency of the Monte Carlo algorithm [46]. More specifically,
starting from one of the samples at time $t$, say
$\tilde{\bm{\theta}}_{i}^{(t)}$, a Markov chain of length $K$ is generated
where new inferences $\hat{\bm{\theta}}^{(t)}$ are drawn from the previous
link in the chain, $\tilde{\bm{\theta}}_{i,j-1}^{(t)}$, using a proposal
distribution $q(\hat{\bm{\theta}}^{(t)}|\tilde{\bm{\theta}}_{i,j-1}^{(t)})$.
Although there exist several valid choices for this distribution [48], here we
shall use a Gaussian with a mean $\mu=\tilde{\bm{\theta}}_{i,j-1}^{(t)}$ and
variance $\sigma^{2}$ as a free hyperparameter, i.e.
$q(\hat{\bm{\theta}}^{(t)}|\bm{\theta}^{(t)}_{i,j-1})=\mathcal{N}(\bm{\theta}^{(t)}_{i,j-1},\sigma^{2})$.
Once the new sample is generated, we shall compute the so-called _acceptance
ratio_ [45]
$\alpha=\prod\limits_{k=1}^{t}\left[\frac{\mathcal{L}(\bm{h}^{(k)}|\hat{\bm{\theta}}^{(k)})}{\mathcal{L}(\bm{h}^{(k)}|\bm{\theta}^{(k)}_{i,j-1})}\right]\frac{q(\bm{\theta}^{(t)}_{i,j-1}|\hat{\bm{\theta}}^{(t)})}{q(\hat{\bm{\theta}}^{(t)}|\bm{\theta}^{(t)}_{i,j-1})}\,,$
(12)
which basically amounts to the likelihood’s history ratio (also known as the
_posterior ratio_) divided by the proposal ratio. Including the latter is
necessary to correct a bias in the proposal distribution if it is asymmetric
[45]. The proposed sample $\hat{\bm{\theta}}^{(t)}$ will then be accepted as a
new link in the chain as long as $\alpha>1$, and with probability $\alpha$
otherwise.
Finally, once all samples have been perturbed, we are left with a new
approximation of the belief, which reads:
$\tilde{B}^{(t)}(\bm{\theta})=1/N\sum_{i=1}^{N}\delta(\bm{\theta}-\bm{\theta}^{(t)}_{i})$.
A detailed step-by-step description of the procedure outlined above is given
in Appendix B.
Analogous to what we observed in the Bayesian update, once the SMC algorithm’s
hyperparameters are properly adjusted, it will systematically converge to the
correct source location as long as the model of the environment is
functionally exact. However, it is not clear how this approach would compare
with the WBU one in a realistic scenario where it uses an inevitably
misspecified model of the environment.
## III Results
### III.1 Stop criterion for source localization
The performance of localization algorithms is typically judged based on their
ability to estimate the odor source position with a specified accuracy or
within a given time limit [25, 49, 50]. However, more generally, we shall test
the reliability of such algorithms by envisioning their application in a real-
world scenario, where we do not know _a priori_ where the source is and have
to decide when to stop looking for it. Therefore, for an algorithm to be
effective in practice, it must show a correlation between an observable
quantity and the quality of its estimate of the source location.
Figure 3: Four panels showing the probability that the distance $d$ between
the estimated source position $\bm{\bar{r}}$ and the ground truth $\bm{r}_{\rm
s}$ is below a given threshold $\delta$ (indicated in the legend) as a
function of the belief/sample standard deviation $\sigma_{\rm loc}$. Lengths
are in units of the distance between sensors. The columns differ in the number
of sensors, while the two rows correspond to the algorithm used (top: WBU,
bottom: SMC).
To this end, a good candidate quantity is the current uncertainty one has
about the source location. This can be formally defined in the WBU approach as
the variance of the master belief (11), which reads
$\sigma^{2}_{\rm loc}\equiv\int_{\Omega}\,{\rm d}\bm{r}\,\mathcal{B}_{\rm
M}(\bm{r})(\bm{r}-\bm{\bar{r}})^{2}\,,$ (13)
where $\bm{\bar{r}}\equiv\int_{\Omega}\,{\rm d}\bm{r}\,\mathcal{B}_{\rm
M}(\bm{r})\bm{r}$ is the estimated source location 111Clearly, the
$\sigma^{2}_{\rm loc}$ and $\bm{\bar{r}}$ counterparts in the SMC algorithm
are respectively defined as the variance and mean computed over the Monte
Carlo samples..
In the ideal case where the model of the environment is exact,
$\sigma^{2}_{\rm loc}$ would be inversely correlated with how close the
estimated position is to the ground truth. In other words, the smaller the
variance of the belief, the more accurate the source location estimate would
be. However, the exact model of the environment is inaccessible to us in any
practical scenario. Therefore, such correlation is not guaranteed to hold in
general and may depend on the algorithmic procedure employed.
We shall thus quantify the robustness of a given localization algorithm by
looking at the probability $P(d<\delta|\sigma_{\rm loc})$ that the distance
$d$ between $\bm{\bar{r}}$ and the actual source position $\bm{r}_{\rm s}$ is
below a given threshold $\delta$, conditioned on the uncertainty on the source
estimate as defined in (13). In particular, the reliability of a localization
algorithm can then be measured by checking whether such probability
$P(d<\delta|\sigma_{\rm loc})$ correlates with the standard deviation
$\sigma_{\rm loc}$.
We find that this is indeed the case for the WBU approach when implemented on
the most realistic scenario introduced in Sec. II.1. As shown in Fig. 3, the
probability of locating the source within a distance smaller than the one
between sensors is well-correlated with the corresponding uncertainty when
using WBU (top row). However, this does not hold for the more standard SMC
algorithm, which hardly shows any correlation (bottom row). Moreover, this
turns out to be independent of the number of sensors deployed ($N_{\rm
s}=\\{16,64\\}$ are shown in the same figure) and of the threshold $\delta$ on
the accuracy (different symbols in each plot). We refer the reader to Appendix
B for further details on the numerical simulations.
Hence, our results suggest that the WBU is a more reliable method than the
standard SMC in the sense that, in WBU, the uncertainty about the source
location $\sigma_{\rm loc}$ is a more reliable indicator of the proximity of
its estimate to the ground truth than it is in SMC.
Furthermore, this has the additional benefit that it allows us to define a
robust stop criterion. Indeed, when employing WBU, it will be sufficient to
look at the time evolution of $\sigma_{\rm loc}$ to know when one has a good
chance of finding the source at a sufficiently small distance from the current
estimate and stop the search accordingly.
Figure 4: Same as Fig. 3 but computed using the models which ranked first with
the overlap criterion discussed in Sec. II.3.
Remarkably, this would no longer hold if we used only the model with the
largest overlap integral $\mathcal{Z}$ to infer the source location. Indeed,
in such a case, the correlation between the distance from the true source
location and its uncertainty almost disappears, with the outcome comparable to
the one of the SMC algorithm, compare Fig. 4 with the top panels of Fig. 3.
This emphasizes the importance of blending information from many wrong models
to estimate the source location, as done in the WBU approach.
### III.2 Model misspecification: the effect of correlations
Although WBU and SMC use both the same (wrong) model of the environment to
infer the source location, they have different outcomes, with the former
proving to be less sensitive to the model misspecification than the latter. It
is, therefore, worth investigating more systematically how model errors affect
the performance of such source localization algorithms. As shown in the table
in Fig. 5(a), we can consider four possible ways to infer the source location
with a given algorithm when dealing with realistic data of turbulent odor
dispersion.
The first possibility is to use the Lagrangian time series obtained from the
DNS to determine whether a sensor detects an odor or not and then deploy a
(inevitably misspecified) model of the environment —like the one derived from
Eq. (3)— to interpret such detections and compute the likelihood (_Model
likelihood-Correlated signal_ —MlCs— top left cell in the table). This is
precisely the scenario we have analyzed so far, which is the closest one to
reality as it relies on minimal prior knowledge of the environment and
directly uses the Lagrangian data.
Alternatively, as a model of the environment, we can use the empirical
likelihood [Fig. 1(b)] computed from the time history of all the odor
trajectories obtained from the DNS. On the one hand, this approach [_Empirical
likelihood-Correlated signal_ —ElCs— bottom left cell in the table in Fig.
5(a)] greatly simplifies the search since there are no environmental
parameters to fit, and it also represents the best existing (empirical) model
one can aim for in practice to infer the source location. On the other hand,
however, it is still an imperfect model of the environment as it does not
capture the time and spatial correlations featured by the odor plume [12].
To study how such correlations affect the source position estimation, we
should then consider the case where the detections are not directly taken from
the DNS time series but instead randomly drawn from the empirical likelihood.
Indeed, the sensors’ measurements are in this way uncorrelated while still
featuring the same detection statistics as the original signal. At this point,
we can decide whether to deploy the empirical likelihood itself to infer the
source location, in which case we would be using the exact model of the
environment (_Empirical likelihood-Uncorrelated signal_ —ElUs— bottom right
cell in the table) or the usual probabilistic model of detections (_Model
likelihood-Uncorrelated signal_ —MlUs— top right cell in the table). The
former can be helpful as a benchmark since, provided the number of
measurements is large enough, it should show convergence to the correct
solution for any properly implemented localization algorithm based on Bayesian
inference. The latter is instead useful compared to the first two cases
illustrated above since, in this case, sensors’ observations are uncorrelated
and, as a result, it isolates the effect of dealing with a functionally wrong
model of the environment.
Figure 5: (a) Table summarizing all the possible ways to infer the source
location when dealing with realistic data of turbulent odor dispersion. They
may differ for the model used to compute the likelihood (rows) or for the
signal received by the sensors (columns). (b) Mean distance $d$ from the
source at the end of each episode in units of the distance $d_{\rm{s}}$
between sensors as a function of the number of sensors $N_{\rm s}$. Different
symbols correspond to the algorithm employed (SMC: inverted purple triangles;
WBU: yellow diamonds). Error bars are the first and third quartiles of the
distribution. Each plot refers to one of the four cases illustrated in table
(a), as indicated by the box color in the upper left corner.
This completes the picture in the table in Fig. 5(a). We are now ready to
compare the performance of WBU and SMC in each of the four scenarios just
described. To this end, we shall look at the results reported in Fig. 5(b).
There, we show the average distance $d$ between the actual source location and
its estimate given by either algorithm at the end of distinct episodes of set
time length as a function of the number of sensors $N_{\rm s}$ deployed in the
search (details in Appendix B). Some observations are in order. First of all,
both WBU (yellow diamonds) and SMC (purple inverted triangles) show comparable
performance as long as they use the empirical likelihood to model the
detection statistics (plots in the bottom row). More specifically, when using
the exact model as in the ElUs scenario, both approaches tend to converge to
the correct source location as expected (bottom right plot). At the same time,
by looking at the ElCs case (bottom left plot), we may observe how much the
sole presence of time and spatial correlations in the signal affects the
quality of the source estimate. Remarkably, even though the empirical model
does not account for such correlations, both algorithms manage to locate the
source within a distance $d$ smaller than half of the sensor separation.
Now, let us see what changes when using the probabilistic model introduced in
Sec. II.2 to infer the source location (MlCs and MlUs scenarios —plots in the
top row). While using the wrong model has obvious shortcomings and further
degrades the performance of both approaches, overall, they prove to be quite
robust and still manage, on average, to locate the source within a distance
smaller than the one between sensors. Furthermore, comparing the results
obtained with uncorrelated signals (MlUs, top right plot) with the most
realistic scenario featuring the correlated signal (MlCs, top left plot), we
may notice that the SMC’s performance gets substantially worse in the latter
case, while WBU is basically unaffected.
The higher sensitivity of SMC to correlations can be rationalized by looking
at the time evolution of the model parameters, namely the emission rate $Q$
and the characteristic length scale of turbulent odor dispersion $\lambda$,
inferred by such an algorithm. As shown in Fig. 6, when the signal received by
the sensors is correlated in space and time (blue curves), SMC has the
tendency to overestimate both $\lambda$ and $Q$, even saturating to the
maximum allowed value. The picture dramatically changes when the sensors’
observations are instead uncorrelated (orange curves). There, the inferred
values of the parameters are much closer to the ones obtained from the best
fit of the empirical likelihood (dashed red line in both plots), which means
the algorithm is, in this case, pointing in the right direction.
Figure 6: Comparison between the values of the model parameters (left:
$\lambda$; right: $Q$) inferred by the SMC algorithm as a function of time
when directly using the DNS time series (blue curves, time-correlated signal)
and when extracting the detections from the corresponding empirical likelihood
(orange curves, uncorrelated signal). In both cases, each curve corresponds to
the episodes-averaged value of the parameter obtained in one of the $30$
configurations (different source positions) where the algorithm has been
tested. The histograms on the right of each plot have been obtained by
considering the inferred values of the parameters at times $t\geq 300$, while
the horizontal dashed red lines stand for the values of $\lambda$ and $Q$
derived from the best fit of the empirical likelihood. Data shown here
correspond to the setup with $N_{\rm s}=16$ sensors.
Hence, the observed sensitivity of the SMC algorithm in the most realistic
scenario lies in the fact that it must perform a real-time inference of the
model parameters (while looking for the source), which is greatly impacted by
the presence of time correlations in the signal. However, this is not the case
for the WBU approach since it uses a predetermined and discrete set of
parameters, then runs the different models independently, and only at the end
merges their outcome based on the current ranking provided by the overlap
integral of each model. The _static_ nature of the parameter space and the use
of all available (misspecified) models are thus the defining strengths of the
newly introduced approach.
## IV Discussion
In this work, we revisited the problem of locating an odor source in a
realistic turbulent environment with a network of static sensors. To this end,
we employed DNS of the incompressible 3–D Navier-Stokes equations to simulate
odor dispersion in the atmospherical boundary layer. We then used the data of
the Lagrangian particles trajectories to systematically investigate how the
errors made in modeling such an environment, a problem unavoidable in field
applications, affect the performance of source localization algorithms based
on Bayesian inference.
Within this framework, we identified a quantity that effectively ranks
different models of the environment just based on the history of observations
made by the sensors in the field. This is what we called the _overlap
integral_ $\mathcal{Z}$ as it essentially measures the degree of consensus
among the sensors about a single source location. This quantity, which is
maximized when the model is exact, also represents the posterior estimate of
the probability of making the sequence of observations, integrated over all
possible source locations. Thus, our weighted Bayesian update (WBU) approach
to source localization may be viewed as a form of maximum likelihood
estimation (MLE) applied to the model itself. The use of MLE to quantify the
degree of model misspecification was previously studied in an abstract setting
in Ref. [52].
Through our analysis, we may conclude that WBU is a more robust approach to
locating an odor source in a turbulent environment than state-of-the-art
methods relying on Monte Carlo sampling techniques. In fact, our results
highlight some fundamental weaknesses of the Sequential Monte Carlo (SMC)
algorithm, which features a strong sensitivity to the presence of time/space
correlations in the sensors’ detections, especially in the most general case
where it must also infer the model parameters in real-time together with the
source location. This is in contrast with WBU, which is insensitive to
correlations and proved to be only mildly affected by the use of wrong models.
In particular, WBU, as opposed to SMC, is able to maintain the desired
correlation between the uncertainty about the source location and the distance
of its current estimate from the ground truth. Thus, WBU provides a robust
stop criterion for the search.
The main novelty of the WBU approach stems from the idea that merging the
information gathered from many possible interpretations of the measurements
recorded by the sensors may help in compensating for model errors consistently
with the _many wrongs principle_ [33, 34]. In the realm of olfactory search,
this represents a fundamental stepping stone toward a thorough investigation
of the effect of misspecified models on the localization of odor sources in
turbulent environments, an unavoidable difficulty in practical applications.
The presented results and methodology have clear potential applications in
environmental monitoring [28, 53] and early warnings [29], as they allow to
reliably identify a potential area of intervention when some hazardous
substances are detected. It would also be interesting to explore the
possibility of capitalizing on the use of many wrong models in the case of
olfactory search by single [18, 20] or multiple [19] moving agents, where it
can help mitigate the model misspecification and thus allow for better
decisions of the agent(s). Beyond olfactory search, the WBU approach is, in
principle, also adaptable to any other Bayesian inference problem with model
misspecification. Finally, we suggest that the overlap integral could be
maximized instead by gradient ascent, which would afford a reduction in
computational cost and memory load while losing the extra information from
running many wrong models. It would be interesting, in particular, to explore
the use of a powerful, many-parameter model such as a neural network in such a
context.
## Acknowledgments
We thank M. Sbragaglia and M. Vergassola for useful discussions. We
acknowledge financial support under the National Recovery and Resilience Plan
(NRRP), Mission 4, Component 2, Investment 1.1, Call for tender No. 104
published on 2.2.2022 by the Italian Ministry of University and Research
(MUR), funded by the European Union – NextGenerationEU– Project Title
Equations informed and data-driven approaches for collective optimal search in
complex flows (CO-SEARCH), Contract 202249Z89M. – CUP B53D23003920006 and
E53D23001610006.
This work was also supported by the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation program (Grant Agreement
No. 882340).
## Appendix A Analytical argument for the convergence of the overlap integral
Let the true likelihood be $f(h|\bm{r}_{i}-\bm{r})$ and let Bayesian updates
be performed with a model $g(h|\bm{r}_{i}-\bm{r}),$ not necessarily correctly
specified. Let the prior be $\pi(\bm{r}),$ and let the true source location be
$\bm{r}^{*}.$
After $T$ timesteps, we have
$\mathcal{Z}=\int_{\Omega}d\bm{r}\,\pi(\bm{r})\prod_{i=1}^{N_{s}}\prod_{t=1}^{T}g(h_{i}^{(t)}|\bm{r}_{i}-\bm{r})=\int_{\Omega}d\bm{r}\,\pi(\bm{r})\exp\left(\sum_{i=1}^{N_{s}}\sum_{t=1}^{T}\log
g(h_{i}^{(t)}|\bm{r}_{i}-\bm{r})\right).$ (A1)
As long as each observation $h_{i}^{(t)}$ has bounded variance and
correlations between observations decay to zero over time (i.e.
$\mathrm{Cov}(h^{(t)},h^{(t+\tau)})\xrightarrow{\tau\to\infty}0$), we may
apply the (weak) law of large numbers [54], which implies that the argument of
the exponential converges in probability to its expectation over observations
$h$:
$\displaystyle\mathcal{Z}$
$\displaystyle\to\bar{\mathcal{Z}}_{T}\equiv\int_{\Omega}d\bm{r}\,\pi(\bm{r})\exp\left(T\sum_{i=1}^{N_{s}}\sum_{h\in\\{0,1\\}}f(h|\bm{r}_{i}-\bm{r}^{*})\log
g(h|\bm{r}_{i}-\bm{r})\right)$
$\displaystyle=C_{T}\int_{\Omega}d\bm{r}\,\pi(\bm{r})\exp\left[-T\sum_{i=1}^{N_{s}}D_{\rm
KL}(f(\cdot|\bm{r}_{i}-\bm{r}^{*})\parallel
g(\cdot|\bm{r}_{i}-\bm{r}))\right],$ (A2)
where
$C_{T}=\exp\left(-T\sum_{i=1}^{N_{s}}H(f(\cdot|\bm{r}_{i}-\bm{r}^{*}))\right),$
(A3)
$D_{\rm KL}(p\parallel q)$ indicates the Kullback-Leibler divergence between
distributions $p$ and $q,$ and $H$ is the Shannon entropy. By Gibbs’
inequality,
$F(\bm{r})\equiv\sum_{i=1}^{N_{s}}D_{\rm
KL}(f(\cdot|\bm{r}_{i}-\bm{r}^{*})\parallel g(\cdot|\bm{r}_{i}-\bm{r}))\geq
0,$ (A4)
with equality iff $f(h|\bm{r}_{i}-\bm{r}^{*})=g(h|\bm{r}_{i}-\bm{r})$ for each
$1\leq i\leq N_{s}$ and each $0\leq h\leq N_{o}-1$ (where $N_{o}$ is the
number of possible observations, which we have set to two). The set of points
$\bm{r}$ where this holds is the intersection of $N_{s}\times N_{o}$ manifolds
of codimension 1; assuming $f$ and $g$ are well-behaved functions of position,
for large enough $N_{s}$ this intersection should be empty unless
$f(h|\bm{r}_{i}-\bm{r}^{*})=g(h|\bm{r}_{i}-\bm{r}^{*})$ for each $i$ and $h$,
in which case the unique member will be the point $\bm{r}=\bm{r}^{*}.$
Figure A1: (a) Curves of the overlap integral $\mathcal{Z}$ as a function of
the percentage error on either of the model parameters. Different curves refer
to distinct times, as indicated in the legend. (b) Colormap of the overlap
integral $\mathcal{Z}$ at time $t=10^{3}$ as a function of the percentage
error on both model parameters. As expected, it features a maximum when the
model is exact ($\varepsilon_{\lambda}=\varepsilon_{Q}=0$) and decays to zero
as we move away from it. Data shown here have been obtained by averaging over
$10^{2}$ episodes of time length $T=10^{3}$ in units of observations made by
each sensor.
Since $T$ is large, Laplace’s method gives
$\bar{\mathcal{Z}}_{T}\sim
T^{-d/2}C_{T}\sum_{j=1}^{k}a_{j}\exp(-TF(\hat{\bm{r}}_{j}))=T^{-d/2}C_{T}\exp(-T\hat{F})\sum_{j=1}^{k}a_{j},$
(A5)
where $d$ is the number of spatial dimensions, $\hat{F}$ is the minimum of $F$
and is attained on the set $\\{\hat{\bm{r}}_{j}\\}_{j=1}^{k},$ and each
$a_{j}$ is nonnegative and independent of $T.$ Therefore,
$\bar{\mathcal{Z}}_{T}$ is maximized when $f$ and $g$ agree at each
$\bm{r}_{i}-\bm{r}^{*}$ and is exponentially smaller when they do not. This is
the key result.
We can numerically show that the overlap integral $\mathcal{Z}$ relaxes to
zero faster the farther from the ground truth the model is. To this end, let
us take into account the same setup as the one shown in Fig. 2, with the
source located in $\bm{r}_{\rm s}=(92,70)$ and $N_{\rm s}=16$ sensors placed
on a square lattice tilted by an angle $\theta=1.04\,\rm{rad}$ with respect to
the wind direction. We can then use the probabilistic model described in Sec.
II.2 to generate the observations made by the static sensors and set the
source emission rate and the characteristic length scale of the turbulent odor
dispersion to $Q^{*}=5.5$ and $\lambda^{*}=3.5$, respectively. Since we know
the exact model of the detection statistics, we can systematically study how
the overlap integral $\mathcal{Z}$ varies depending on the error made on such
model parameters. Figure A1(a) shows the values of $\mathcal{Z}$ obtained when
either the value of the characteristic length scale $\lambda$ (top panel) or
the one of the emission rate $Q$ (bottom panel) is wrong. In both cases, as
time progresses, the curve of the overlap integral peaks more and more around
the correct value of the parameter, i.e., the one corresponding to a
percentage error $\varepsilon_{*}=0$, and it smoothly goes to zero as the
error grows. More generally, this still holds when both model parameters are
misspecified, as shown in the colormap in Fig. A1(b).
Let us now comment on one feature that stands out from these plots, that is,
the observed asymmetry in the values of $\mathcal{Z}$ when $\lambda$ is
under/over-estimated. In fact, given the same history of measurements, the
source will be estimated as farther from the sensors, the greater the
estimated value of $\lambda$. This will eventually cause the overlap among the
private beliefs to accumulate at the border of the finite-size arena $\Omega$
where simulations are performed. Therefore, the larger values of the overlap
integral $\mathcal{Z}$ when $\lambda$ is overestimated are a mere finite-size
effect. On the one hand, this is not a problem in the case of a functionally
correct model, where the ground truth is attainable (as in this section)
since, eventually, the correct model will still be the one with the largest
value of $\mathcal{Z}$ anyway. On the other hand, however, this effect can
persist in the general case of a functionally wrong model (scenario discussed
in the main text), and we are compelled to address it for consistency. To this
end, during our analysis, we systematically set to zero the values of the
overlap for the models that would place the source at a distance $d\leq
5\Delta x$ (with $\Delta x$ being the lattice spacing) from the border of the
arena $\Omega$. This avoids the selection of clearly wrong models that only
artificially would feature a large value of the overlap $\mathcal{Z}$,
preserving the structure and the basic concepts behind the WBU approach.
## Appendix B Details on the numerical simulations
In all our numerical simulations, the size of the square arena $\Omega$ has
been set to $128\Delta x\times 128\Delta x$, with $\Delta x=1$ being the
lattice spacing. The sensors have been placed in a square lattice of side
$\ell=88\Delta x$, tilted with an angle $\theta=1.04$ with respect to the mean
wind direction. All the data presented in Sec. III have then been obtained by
averaging over the results obtained by shifting the odor source in $30$
different locations drawn at random within an inner box $108\Delta x\times
108\Delta x$ inside $\Omega$, as illustrated in Fig. A2. This was done to
avoid boundary effects that would have altered the performance statistics of
the localization algorithms.
Moreover, to gather more statistics for each source location, we have divided
the time series of the odor dispersion obtained from the DNS described in Sec.
II.1 into $25$ runs of set time length $T=600$ (in units of the Kolmogorov
timescale $\tau_{\eta}$).
Figure A2: Illustration of the sensors’ placement (magenta circles, here
$N_{\rm s}=16$) and of the $30$ source locations (green squares) drawn at
random inside the arena.
Definition | Symbol | Value
---|---|---
Number of samples | $N$ | 50
Number of MCMC perturbation steps | $K$ | 5
Variance of proposal distribution of $\bm{r}_{\rm s}$ | $\sigma^{2}_{\rm pos}$ | 100
Variance of proposal distribution of $\lambda$ | $\sigma^{2}_{\lambda}$ | 1
Variance of proposal distribution of $Q$ | $\sigma^{2}_{Q}$ | 1
Range of values of $\lambda$ | $[\lambda_{\rm m},\lambda_{\rm M}]$ | [0,10]
Range of values of $Q$ | $[Q_{\rm m},Q_{\rm M}]$ | [0,10]
Table A1: Values of the hyperparameters used in the SMC algorithm.
In order to get the master belief in the WBU approach, we have used in every
configuration a total of $M=100$ models, each of them corresponding to a
different combination of parameters $\\{Q,\lambda\\}$ in the stochastic model
introduced in Sec. II.2, with the specific values of $Q$ and $\lambda$ being
$Q=\\{0.7,1.7,\dots,9.7\\}$ and $\lambda=\\{0.7,1.7,\dots,9.7\\}$,
respectively.
The values of the SMC hyperparameters used in this work are summarized in
Table A1. We have also numerically checked that all the results presented here
are not qualitatively affected by this choice. Note that, in the SMC
implementation, we start from a flat prior defined over the interval $[0;10]$
for both model parameters $Q$ and $\lambda$. Moreover, their values are
bounded therein, which is, for consistency, the same range used in the WBU
approach.
Lastly, the detailed step-by-step explanation of the WBU and SMC
implementation is reported in Algorithm 1 and Algorithm 2, respectively.
Algorithm 1 Weighted Bayesian update
1:for $j=1$ to $M$ do $\triangleright$ Loop over different models
2: Set initial common belief $B^{(0)}_{j}$ to a uniform distribution.
3: Set model parameters $\bm{P}_{j}=\\{Q_{j},\lambda_{j}\\}$.
4: for $t=1$ to $T$ do $\triangleright$ Loop over time
5: Perform sensors’ measurements $\bm{h}^{(t)}$.
6: Update common belief $B^{(t)}_{j}$ using Bayes’ rule (10).
7: Compute overlap integral $\mathcal{Z}^{(t)}_{j}$.
8: end for
9:end for
10:Compute master belief $\mathcal{B}_{M}$ at all times from Eq. (11).
$\triangleright$ Output master belief from weighted models
Algorithm 2 Sequential Monte Carlo with Importance Sampling and perturbation
step
1:Set initial belief $\tilde{B}^{(0)}$ to a uniform distribution.
2:for $t=1$ to $T$ do
3: Perform sensors’ measurements $\bm{h}^{(t)}$.
4: for $i=1$ to $N$ do $\triangleright$ Sample’s initialization
5: Draw sample from current belief:
$\tilde{\bm{\theta}}_{i}\sim\tilde{B}^{(t-1)}(\bm{\theta})$.
6: Compute weight $w_{i}=\mathcal{L}(\bm{h}^{(t)}|\tilde{\bm{\theta}}_{i})$.
7: end for
8: for $i=1$ to $N$ do
9: Normalize weight: $w_{i}/=\sum_{i=1}^{N}w_{i}$.
10: Compute Effective Sample Size: $N_{{\rm eff}}=1/\sum_{i=1}^{N}w_{i}^{2}$.
11: end for
12: if $N_{\text{eff}}<N_{\text{thr}}$ then $\triangleright$ Resampling step
(if needed)
13: for $i=1$ to $N$ do
14: Select $\tilde{\bm{\theta}}_{k}$ with probability $w_{k}$.
15: Put $\tilde{\bm{\Theta}}_{i}=\tilde{\bm{\theta}}_{k}$.
16: end for
17: for $i=1$ to $N$ do
18: Replace $\tilde{\bm{\theta}}_{i}=\tilde{\bm{\Theta}}_{i}$.
19: Set uniform weights: $w_{i}=1/N$.
20: end for
21: end if
22: for $i=1$ to $N$ do $\triangleright$ MCMC perturbation step
23: Select $l\in\\{1,\dots,N\\}$ with probability $w_{l}$.
24: Set $\bm{\theta}^{(t)}_{i,0}=\tilde{\bm{\theta}}^{(t)}_{l}$.
25: for $j=1$ to $K$ do
26: Draw new sample from proposal distribution: $\hat{\bm{\theta}}^{(t)}\sim
q(\hat{\bm{\theta}}^{(t)}|\bm{\theta}^{(t)}_{i,j-1})$.
27: Compute acceptance ratio $\alpha$ from Eq. (12).
28: Draw random number $u\sim U[0,1]$.
29: if $u<\min(1,\alpha)$ then
30: Set $\bm{\theta}^{(t)}_{i,j}=\hat{\bm{\theta}}^{(t)}$.
31: else
32: Set $\bm{\theta}^{(t)}_{i,j}=\bm{\theta}^{(t)}_{i,j-1}$.
33: end if
34: end for
35: Set $\bm{\theta}^{(t)}_{i}=\bm{\theta}^{(t)}_{i,\tilde{B}}$.
36: end for
37: Set
$\tilde{B}^{(t)}(\bm{\theta})=1/N\sum_{i=1}^{N}\delta(\bm{\theta}-\bm{\theta}^{(t)}_{i})$.
$\triangleright$ Output belief approximation
38:end for
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|
# Relevant Irrelevance: Generating Alterfactual Explanations for Image
Classifiers
Silvan Mertes1 Tobias Huber1 Christina Karle1 Katharina Weitz1,2
Ruben Schlagowski1 Cristina Conati3 Elisabeth André1
1University of Augsburg, Germany
2Fraunhofer HHI, Germany
3University of British Columbia, Canada
{silvan.mertes, tobias.huber, ruben.schlagowski<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
In this paper, we demonstrate the feasibility of alterfactual explanations for
black box image classifiers. Traditional explanation mechanisms from the field
of Counterfactual Thinking are a widely-used paradigm for Explainable
Artificial Intelligence (XAI), as they follow a natural way of reasoning that
humans are familiar with. However, most common approaches from this field are
based on communicating information about features or characteristics that are
especially important for an AI’s decision. However, to fully understand a
decision, not only knowledge about relevant features is needed, but the
awareness of irrelevant information also highly contributes to the creation of
a user’s mental model of an AI system. To this end, a novel approach for
explaining AI systems called alterfactual explanations was recently proposed
on a conceptual level. It is based on showing an alternative reality where
irrelevant features of an AI’s input are altered. By doing so, the user
directly sees which input data characteristics can change arbitrarily without
influencing the AI’s decision. In this paper, we show for the first time that
it is possible to apply this idea to black box models based on neural
networks. To this end, we present a GAN-based approach to generate these
alterfactual explanations for binary image classifiers. Further, we present a
user study that gives interesting insights on how alterfactual explanations
can complement counterfactual explanations.
## 1 Introduction
With the steady advance of Artificial Intelligence (AI), and the resulting
introduction of AI-based applications into everyday life, more and more people
are being directly confronted with decisions made by AI algorithms (Stone et
al., 2016). As the field of AI advances, so does the need to make such
decisions explainable and transparent. The development and evaluation of
_Explainable AI_ (XAI) methods is important not only to provide end users with
explanations that increase acceptance and trust in AI-based methods, but also
to empower researchers and developers with insights to improve their
algorithms.
The need for XAI methods has prompted the research community to develop a
large variety of different approaches to unravel the black boxes of AI models.
A considerable part of these approaches is based on telling the user of the
XAI system in various ways _which_ features of the input data are important
for a decision (often called _Feature Attribution_) (Arrieta et al., 2020).
Other methods, which are close to human habits of explanation, are based on
the paradigm of _Counterfactual Thinking_ (Miller, 2019). Procedures that
follow this guiding principle try answering the question of _What if…?_ by
showing an alternative reality and the corresponding decision of the AI. Here,
in contrast to feature attribution mechanisms, not only the importance of the
various features is emphasized. Rather, it is conveyed, even if only
indirectly, _why_ features are relevant.
Figure 1: (A) Examples of a counterfactual and an alterfactual explanation.
Input features to a fictional decision system to be explained are _Income_ and
_Gender_ , whereas the former is relevant and the latter is irrelevant to the
AI’s decision on whether a credit is given or not. (B) Conceptual comparison
of factual, counterfactual, semifactual, and alterfactual explanations.
Prominent examples of these explanatory mechanisms are _Counterfactual
Explanations_ and _Semifactual Explanations_ (Kenny and Keane, 2020).
Counterfactual explanations show a version of the input data that is altered
just enough to change an AI’s decision. By doing so, the user is shown not
only _which_ features are relevant to the decision, but more importantly,
_how_ they would need to be changed to result in a different decision of the
AI. Semifactual explanations follow a similar principle, but they modify the
relevant features of the input data to an extent that the AI’s decision does
not change.
All of these methods have in common that they focus on the _important_
features. However, we argue that awareness of irrelevant features can also
contribute substantially to the complete understanding of a decision domain,
as knowledge of the important features for the AI does not necessarily imply
knowledge of the unimportant ones.
For example, consider an AI system that assesses a person’s creditworthiness
based on various characteristics. If that system was completely fair, a
counterfactual explanation might be of the form: _If your income was higher,
you would be creditworthy._ However, this explanation does not exclude the
possibility that your skin color also influenced the AI’s decision. It only
shows that the income had a high impact on the AI. An explanation confined to
the irrelevant features, on the other hand, might say _No matter what your
skin color is, the decision would not change._ In this case, direct
communication of irrelevant features ascertains fairness with regards to skin
color. Conventional counterfactual thinking explanation paradigms do not
provide this information directly. To address this issue, Mertes et al.
(2022b) recently conceptually introduced the explanatory paradigm of
_Alterfactual Explanations_ that is meant to complement counterfactual
explanations. This principle is based on showing the user of the XAI system an
alternative reality that leads to the exact same decision of the AI, but where
irrelevant features are altered. All relevant features of the input data, on
the other hand, remain the same. As such, alterfactual explanations form the
complement to counterfactual explanations - providing both explanation types
should enable the user to grasp both relevant _and_ irrelevant features.
Mertes et al. already showed the potential in the concept of raising user
awareness about irrelevant features (Mertes et al., 2022b). Nevertheless, due
to the absence of an implementable solution, the researchers could only delve
into the concept through the utilization of a fictional AI.
As such, in this work we introduce a GAN-based generation algorithm that is
able to create both alterfactual and counterfactual explanations for image
classifiers.111Our full implementation is open-source and available at
https://github.com/hcmlab/Alterfactuals. As alterfactual explanations convey
completely different information than common methods, we investigate whether
the understanding that users have of the explained AI system is also formed in
a different way, or can even be improved. Our results show that alterfactual
explanations outperform counterfactual explanations with regards to local
model understanding.
## 2 Related Work
As the approach presented in this paper can be counted to the class of XAI
methods that work by inducing counterfactual thinking processes, it is
important to gain an understanding of how common methods from this field work.
Therefore, this section gives an overview on related explanation concepts.
Figure 1A illustrates the difference between those concepts using exemplary
explanations for a fictional AI that decides if a person is creditworthy or
not. We will use that scenario as a running example of how the different
explanation paradigms would answer the question of _Why does the AI say that I
am not creditworthy?_.
Factual Explanations \- _There was another female person that also had rather
little money, and she also did not get the credit._ \- Factual explanations
are the traditional way of explaining by example, and often provide a similar
instance from the underlying data set (adapted or not) for the input data
point that is to be explained (Keane et al., 2021b). Other approaches do not
choose an instance from the dataset, but generate new ones (Guidotti et al.,
2019). The idea behind factual explanations is that similar data instances
lead to similar decisions, and the awareness of those similarities leads to a
better understanding of the model. Further explanation mechanism that fall in
this category are _Prototypical Explanations_ and _Near Hits_ Kim et al.
(2016); Herchenbach et al. (2022).
Counterfactual Explanations \- _If you had that amount of money, you would get
the credit._ \- Counterfactual explanations are a common method humans
naturally use when attempting to explain something and answer the question of
_Why not …?_ (Miller, 2019; Byrne, 2019). In XAI, they do this by showing a
modified version of an input to an AI system that results in a different
decision than the original input. Counterfactual explanations should be
minimal, which means they should change as little as possible in the original
input (Keane et al., 2021b; Miller, 2021). In certain scenarios, modern
approaches for generating counterfactual explanations have shown significant
advantages over feature attribution mechanisms (i.e., explanation approaches
that highlight _which_ features are important for a decision) in terms of
mental model creation and explanation satisfaction (Mertes et al., 2022a).
Wachter et al. (2017) name multiple advantages of counterfactual explanations,
such as being able to detect biases in a model, providing insight without
attempting to explain the complicated inner state of the model, and often
being efficient to compute. For counterfactual explanations, a multitude of
works exist that, similar to how we do it for alterfactual explanations, use
GANs to automatically generate explanations for image classifiers Nemirovsky
et al. (2022); Van Looveren et al. (2021); Khorram and Fuxin (2022); Mertes et
al. (2022a).
Semifactual Explanations \- _Even if you had that amount of money, you would
still not get the credit._ \- Similar to counterfactual explanations,
semifactual explanations are an explanation type humans commonly use. They
follow the pattern of _Even if X, still P._ , which means that even if the
input was changed in a certain way, the prediction of the model would still
not change to the foil (McCloy and Byrne, 2002). In an XAI context, this means
that an example, based on the original input, is provided that modifies the
input in such a way that moves it toward the decision boundary of the model,
but stops just before crossing it (Kenny and Keane, 2020). Similar to
counterfactual explanations, semifactual explanations can be used to guide a
user’s future action, possibly in a way to deter them from moving toward the
decision boundary (Keane et al., 2021b).
## 3 Alterfactual Explanations
_No matter what your gender is, the decicison would not change._ \- The basic
idea of alterfactual explanations investigated in this paper is to strengthen
the user’s understanding of an AI by showing irrelevant attributes of a
predicted instance. Hereby, we define irrelevance as the property that the
corresponding feature, regardless of its value, does not contribute in any way
to the decision of the AI model. When looking at models that are making
decisions by mapping some sort of input data $x\in X$ to output data $y\in Y$,
the so-called _decision boundary_ describes the region in $X$ which contains
data points where the corresponding $y$ that is calculated by the model is
ambiguous, i.e., lies just between different instances of $Y$. Thus,
irrelevant features can be thought of as features that do not contribute to a
data point’s distance to the decision boundary.
However, information that is carried out by explanations should be
communicated as clearly as possible. Alterfactual explanations inform about
the _irrelevance_ of certain features - as such, it should be made clear that
these features can take _any_ possible value. If we would change the
respective features only to a small amount, the irrelevance is not clearly
demonstrated to the user. Therefore, an alterfactual explanation should change
the affected features to the maximum amount possible. By doing so, they
communicate that the feature, _even if it is changed as much as it can change_
, still does not influence the decision. Thus, the definition of an
alterfactual explanation is as follows:
> Let $d:X\times X\rightarrow\mathbb{R}$ be a distance metric on the input
> space $X$. An _alterfactual explanation_ for a model $M$ is an altered
> version $a\in X$ of an original input data point $x\in X$, that maximizes
> the distance $d(x,a)$ whereas the distance to the decision boundary
> $B\subset X$ and the prediction of the model do not change: $d(x,B)=d(a,B)$
> and $M(x)=M(a)$
Thus, the main difference between an alterfactual explanation and a
counterfactual or semifactual explanation is that while the latter methods
alter features resulting in a decreased distance to the decision boundary, our
alterfactual method tries to keep that distance fixed. Further, while
counterfactual and semifactual methods try to keep the overall change to the
original input minimal (Keane et al., 2021a; Kenny and Keane, 2020),
alterfactual explanations do exactly the opposite, which is depicted in Figure
1B.
## 4 Generating Alterfactual Explanations
$x$ $64\times 64$ $32\times 32$ $16\times 16$ $8\times 8$ $4\times 4$ $2\times
2$ $1\times 1$ $2\times 2$ $4\times 4$ $8\times 8$ $16\times 16$ $32\times 32$
$64\times 64$ $128\times 128$ $64$ $128$ $256$ $512$ $512$ $512$ $512$ $512$
$512$ $512$ $256$ $128$ $64$ $1$ $n$ Conv2D + LeakyReLU Conv2D + ReLU
Conv2DTrans + Tanh filter number Conv2D + BatchNorm + LeakyReLU Conv2DTrans +
BatchNorm + Dropout + ReLU Conv2DTrans + BatchNorm + ReLU
Figure 2: Architecture overview of the generator network.
$x$ $\hat{y}$ $+$ Embedding Upsample $64\times 64$ $32\times 32$ $31\times 31$
$30\times 30$ $64$ $128$ $256$ $1$ $n$ Conv2D + LeakyReLU Conv2D + BatchNorm +
LeakyReLU Conv2D + Sigmoid filter number
Figure 3: Architecture overview of the discriminator network.
As we argue that alterfactual and counterfactual explanations convey different
information, we designed a generative approach that is capable of creating
both types of explanations in order to explain an image classifier. For both,
a set of requirements arises that needs to be reflected in the objectives of
our explanation generation approach.
1. 1.
The generated explanations should have high quality and look realistic.
2. 2.
The resulting explanation should be either classified as the same class as the
original input (for alterfactual explanations), or as the opposite class (for
counterfactual explanations).
3. 3.
For alterfactual explanations, the output image should change as much as
possible, while for counterfactual explanations, it should change as little as
possible.
4. 4.
For alterfactual explanations, only irrelevant features should change, i.e.,
the distance to the decision boundary should be maintained.
To address these objectives, different loss components (see next sections)
were used to steer a GAN-based architecture to generate the desired
explanations. A GAN-based approach was chosen as similar concepts have
successfully been applied to the task of counterfactual explanation generation
in various existing works (Olson et al., 2021; Huber et al., 2023; Nemirovsky
et al., 2022; Zhao, 2020; Mertes et al., 2022a). In order to allow for a more
focused and comprehensive user study design, in this work, we focus on
explaining a binary image classifier. However, although our specific
generation architecture is designed for a binary classification problem, it
would theoretically be possible to apply it to non-binary tasks by training
separate models for each class vs. the union over all other classes. A
schematic overview of our architecture can be seen in Figures 2 and 3. For a
more detailed description, we refer to the appendix.
### 4.1 Adversarial Component
To address the first objective, an adversarial setting is used. Here, a
generator network $G$ is trained to take an original image $x$ and a random
noise vector $z$ and transforms them into the respective explanation
$\hat{x}$. As such, a mapping $\\{x,z\\}\rightarrow\hat{x}$ is learned by the
generator. A discriminator network $D$ is trained to identify the generated
images as _fake_ images in an adversarial manner.
Additionally, to partly target the second objective, we feed a target class
label $\hat{y}\in\\{0,1\\}$ to the discriminator. By doing so, the
discriminator learns not only to assess if the produced images are real or
fake, but also has the capability to decide if an explanation fits the data
distribution of the class it is supposed to belong to. A somewhat similar idea
was put forth by Sharmanska et al. (2020) within the context of fairness and
yielded promising results there. During training, the discriminator is
alternately fed with real and fake data. For real data, the target class label
$\hat{y}$ reflects the class that the classifier to be explained assigns to
the respective image $x$. For the generated explanations, the target class
label $\hat{y}$ reflects either the class that was assigned to the original
image $x$ (for alterfactual explanations), or the opposite class (for
counterfactual explanations).
By letting the generator and discriminator compete against each other during
training, it is enforced that the resulting images look realistic and resemble
the data distribution of the respective target classes. The objective function
for the adversarial setting is formulated as follows:
$\mathcal{L}_{adversarial}=\mathbb{E}_{x\sim p_{\text{data}}(x)}\left[\log
D(x,\hat{y})\right]+\\\ \mathbb{E}_{x\sim p_{\text{data}}(x),z\sim
p_{\text{noise}}(z)}\left[\log(1-D(G(x,z),\hat{y}))\right]$ (1)
### 4.2 Including Classifier Information
The second objective is further addressed by incorporating the decisions of
the classifier to be explained into the generator’s loss function.
Let $C:X\rightarrow\left[0,1\right]$ be a binary classifier with threshold
$0.5$. We define the classification target $\tilde{C}(x)$ as
$\tilde{C}(x):=C(x)$ for alterfactual explanations and $\tilde{C}(x):=1-C(x)$
for counterfactual explanations. To measure the error between the actual
classification of the generated explanation and the target classification, we
used Binary Crossentropy (BCE) to define a classification loss
$\mathcal{L}_{C}$:
$\mathcal{L}_{C}=\mathbb{E}_{x,\hat{x}\sim
p_{data}(x,\hat{x})}[\tilde{C}(x)\cdot\log C(\hat{x})\\\
+(1-\tilde{C}(x))\cdot\log(1-C(\hat{x}))]$ (2)
### 4.3 SSIM Component
The third objective was addressed by including a similarity component into the
loss function. Explanations are meant for humans. Therefore, using the
Structural Similarity Index (SSIM) seemed to be an appropriate choice to
measure image similarity for our approach, as it correlates with how humans
are perceiving similarity in images (Wang et al., 2004). The parameters for
SSIM were chosen as recommended by Wu et al. (2019).
As alterfactual explanations should change irrelevant features _as much as
possible_ , while counterfactual explanations should be _as close as possible_
to the original image, the learning objective differs for both (low similarity
for alterfactual explanations, high similarity for counterfactual
explanations). With $\left[0,1\right]$ as the range of SSIM, we designed the
loss function as follows:
$\mathcal{L}_{sim}=\begin{cases}\mathbb{E}_{x,\hat{x}\sim
p_{data}(x,\hat{x})}\left[SSIM(x,\hat{x})\right]\text{\hskip 5.0pt\hskip
5.0pt\hskip 5.0pt\hskip 5.0pt\hskip 5.0pt\hskip 5.0pt\hskip
5.0ptAlterfactual}\\\ \mathbb{E}_{x,\hat{x}\sim
p_{data}(x,\hat{x})}\left[1-SSIM(x,\hat{x})\right]\text{Counterfactual}\end{cases}$
(3)
### 4.4 Feature Relevance Component
The fourth objective, i.e., forcing the network to only modify irrelevant
features when generating alterfactual explanations, was addressed by using an
auxiliary Support Vector Machine (SVM) classifier. Note that this loss is only
applied when generating alterfactual explanations, not when generating
counterfactual explanations. Li et al. (2018) and Elsayed et al. (2018) have
shown theoretically and empirically that the last weight layer of a Neural
Network converges to an SVM trained on the data transformed up to this layer
if certain restrictions are met (e.g., the last two layers of the network have
to be fully connected). An SVM’s decision boundary can be calculated directly
- unlike the one of a Neural Network (Jiang et al., 2018). As such, we use an
SVM which was trained to predict the classifier’s decision based on the
activations of the classifier’s penultimate layer as a way to approximate the
classifier’s decision boundary - if the generated alterfactual explanation has
moved closer to the SVM’s separating hyperplane, relevant features were most
likely modified. Although an unchanged decision boundary distance does not
necessarily guarantee that no relevant features were modified, in our
experiments, it was a good indicator.
The distance of $x$ to the SVM’s separating hyperplane $f$ was defined as
follows, with $w$ as the SVM’s weight vector:
$\text{{SVM}}(x)=\left\lvert\frac{f(x)}{||w||}\right\lvert$ (4)
The SVM loss is defined by the absolute difference in distance to the
separating hyperplane between the original image and the generated
alterfactual explanation:
$\mathcal{L}_{\text{{SVM}}}=\mathbb{E}_{x\sim p_{data}(x),z\sim
p_{\text{noise}}(z)}\left[|\text{{SVM}}(x)-\text{{SVM}}(\hat{x})|\right]$ (5)
The final loss function is a summation of all the four loss components
introduced above.
## 5 Evaluation Scenario
Figure 4: Example outputs of our system. It can be seen that alterfactual
explanations change features that are irrelevant to the classifier, e.g., the
color of the shoes or the width of the boot shaft, while counterfactual
explanations change relevant features like the presence or absence of a boot
shaft. From top to bottom the original images are a correctly classified ankle
boot and sneaker, followed by two inputs incorrectly classified as ankle boot
and sneaker.
To assess the performance of our approach, we applied it to the Fashion-MNIST
data set (Xiao et al., 2017). That data set contains 7,000 gray scale images
for each of its ten categories of clothes, such as ‘ankle boots’ or
‘pullover’, splitted into _train_ (6,000 images per class) and _test_ (1,000
images per class) sets. The two classes we chose, ‘ankle boots’ and
‘sneakers’, were selected due to being somewhat similar in order not to
oversimplify the classification task while still being distinct enough to be
able to visually assess whether the generated explanations are clear. To
create the classifier to be explained, we trained a relatively simple four-
layer convolutional neural network, achieving an accuracy of $96.7\%$ after 40
training epochs. The exact architecture and training configuration can be
found in the appendix.
Our explanation generation architecture was trained for 14 epochs, until
visually no further improvement could be observed. For alterfactual
explanations, we reached a validity (i.e., which portion of the explanations
are classified as the correct target class by the classifier) of $96.20\%$ and
an average SSIM of $0.32$ (here, lower is better), whereas the counterfactual
explanations reached a validity of $87.70\%$ and an average SSIM of $0.90$
(here, higher is better). For more details refer to the appendix. Exemplary
generated explanations are shown in Figure 4. Note that, in order to verify if
our alterfactual generation approach is applicable on a wider range of
datasets, we additionally trained our approach on three other datasets: MNIST
(LeCun, 1998), MaskedFace-Net (Cabani et al., 2021), and a gray scale version
of MaskedFace-Net. To further demonstrate that our approach can be adapted to
be more model-agnostic and work without access to intermediate layers, we
omitted the Feature Relevance component for those experiments. Training
details, example outputs and computational results for those experiments can
be found in the appendix.
## 6 User Study
### 6.1 Research Question and Hypotheses
We conducted a user study to validate whether the counterfactual and
alterfactual explanations generated by our approach help human users to form
correct model understanding of an AI system. Therefore, we only used results
from the model trained on the Fashon-MNIST classifier in order to not
overwhelm participants. To be able to compare our findings to existing work,
we designed our study similar to Mertes et al. (2022b). Our hypotheses are as
follows:
1) Alterfactual and counterfactual explanations, as well as the combination of
both, are more effective in enabling model understanding than no explanations.
2) There is a difference in model understanding and explanation satisfaction
between alterfactual and counterfactual explanations, but we did not
anticipate a specific direction since we see them as complementary concepts.
3) Compared to the individual explanations, a combination of alterfactual and
counterfactual explanations is a more effective way to enable a good model
understanding and is more satisfying for users.
4) There is a difference between conditions regarding the understanding of
relevant and irrelevant features, where alterfactual explanations are more
effective to identify irrelevant features while counterfactual explanations
should help more with identifying relevant features.
### 6.2 Methodology
##### Conditions and Explanation Presentation
We used a between-groups design with four conditions. Participants in the
_Control_ condition were presented only with the original input images to the
AI. No explanation was shown. In the _Alterfactual_ and _Counterfactual_
conditions, participants were presented with the original input images and
either alterfactual or counterfactual explanations. In the _Combination_
condition, participants were presented with the original input images as well
as both the alterfactual and the counterfactual explanations.
##### Procedure
The whole study was built using the _oTree_ framework by Chen et al. (2016).
After answering questions about their demographic background, participants
were given some general information about the data and their task during the
prediction task. For the classifier, they were only told that an AI was
trained to distinguish between ankle boots and sneakers. Two example images
for each class (ankle boots and sneakers) were shown and some shoe specific
terminology (e.g., ”shaft”) was introduced in order to make sure that
participants have a common understanding of the terms they are asked about
later on. Following this information, the participants were given an example
input image for each class together with the classifier’s prediction for this
input image. In the explanation conditions, the participants were introduced
to their corresponding explanation types (counterfactuals, alterfactuals or a
combination) and could explore the explanations for those two images. After
that, each participant answered a quiz about the information that was given up
to that point, to make sure that they understood everything correctly.
Subsequently, the study itself started. It was divided into three parts: For
assessing the participants’ understanding of the classifier, we used _(i)_ a
prediction task for assessing the local understanding, i.e., to assess if the
participants understand why the AI makes a _specific_ decision, and _(ii)_ a
questionnaire about the relevance of certain features for assessing the global
understanding, i.e., to assess if the participants understand how the AI works
_overall_. To assess the participants’ explanation satisfaction, we used
_(iii)_ an explanation satisfaction questionnaire. The three phases of the
experiment are described below.
##### Local Model Understanding: Prediction Task
To measure the local understanding of the classifier, we used a prediction
task, which assesses the participants’ ability to anticipate the AI
classifier’s decisions (Hoffman et al., 2018). Eight examples were shown,
covering all possible classification outcomes (two correctly classified images
for both sneakers and ankle boots, and two incorrectly classified images for
both) to avoid bias. Figure 4 shows four of the images from the study. The
example images were chosen randomly but we made sure that the alterfactual and
counterfactual explanations generated by our model for those images were valid
(i.e., the classifier predicted the same class as for the original image when
fed with the alterfactual explanation, and the opposite class when fed with
the counterfactual explanation). Participants had to predict the classifier’s
decision for each example image. Participants were additionally asked about
their own opinion on which class the original shoe image belonged to. The
answers to that particular question were not further analyzed - it was only
added to help the participants distinguish between their own opinion and their
understanding of the classifier. After predicting an example, they were told
the correct label and the AI classifier’s decision before moving on to the
next example. The order of the examples was randomized.
##### Global Model Understanding: Feature Relevance
While the Prediction Task can be seen as _local_ measurement of the users’
understanding of the model in specific instances, we also wanted to
investigate whether participants understood the _global_ relevance of
different features. To this end, we looked at two features that were relevant
for our classifier (”presence/absence of a boot shaft” and ”presence/absence
of an elevated heel”) as well as two features that were irrelevant for our
classifier (”boot shaft width” and ”the shoe’s color and pattern on the
surface area”). These features were chosen based on the authors’ experience
from training the classifier and a-priori explorations with the Feature
Attribution explanation mechanisms LIME (Ribeiro et al., 2016) and SHAP
(Lundberg and Lee, 2017). Note that, although the classifier is still a black
box and there is no definitive proof that the chosen features reflect the
classifier’s inner workings entirely accurately, we decided that using those
mechanisms for the feature choice are the best proxy that we have. As such,
after the participants went through the eight examples that were used for the
prediction task, they were asked for each feature how much they agreed that it
was relevant to the AI’s decisions on a 5-point Likert scale (0 = strongly
disagree, 4 = strongly agree).
##### Explanation Satisfaction
In order to measure the participants’ subjective satisfaction, we used the
Explanation Satisfaction Scale proposed by Hoffman et al. (2018) which
consists of eight items rated on a 5-point Likert scale (0 = strongly
disagree, 5 = strongly agree) that we averaged over all items. Since it does
not apply to our use-case, we excluded the 5th question of the questionnaire.
The seven remaining items address _confidence_ , _predictability_ ,
_reliability_ , _safety_ , _wariness_ , _performance_ , _likeability_.
Finally, the participants had the possibility to give free text feedback.
### 6.3 Participants
Figure 5: Left: Mean participant prediction accuracy of the AI’s prediction by
condition. The conditions containing alterfactual explanations outperformed
all other conditions. Right: Mean understanding of the irrelevant and relevant
features in our study. Error bars represent the 95% CI. *p $<$ .05, **p $<$
.001.
Through a power analysis, we estimated a required sample size of at least 21
per condition for a MANOVA with 80% power and an alpha of 5%, based on the
Pillai’s Trace of 0.13 reported for the study by Mertes et al. (2022b). 131
Participants between 18 and 29 years (M = 22.2, SD = 2.44) were recruited at
the University of _blinded for review_. 61 of them were male, 70 female. The
participants were randomly separated into the four conditions (33 per
condition and 32 in the Alterfactual condition). The highest level of
education that most participants held (76.3%) was a high-school diploma. Only
11.5% of the participants had no experience with AI. Most of the participants
(74%) have heard from AI in the media. Excluding participants that had no
opinion on the subject, the participants expected a positive impact of AI
systems in the future (M = 3.73 on a 5-point Likert Scale from 1 = ”Extremely
negative” to 5 = ”Extremely positive”). There were no substantial differences
in the demographics between conditions (see appendix).
## 7 Results
### 7.1 Model Understanding
To investigate the impact of the four different experimental conditions on the
(1) feature understanding and (2) prediction accuracy, we conducted a MANOVA.
We found a significant difference, Wilks’ Lambda = 0.859, F(6,252) = 3.31, p =
.004.
The following ANOVA revealed that only the prediction accuracy of the
participants showed significant differences between the conditions:
* •
_Feature Understanding_ : F(3,127) = 0.877, p = .455.
* •
_Prediction Accuracy_ : F(3,127) = 6.578, p $<$ .001.
As displayed in Figure 5, the post-hoc t-tests showed that the participants’
prediction accuracy was significantly better in the _Alterfactual_ and
_Combination_ conditions compared to the other conditions. The effect size d
is calculated according to Cohen (2013):
* •
_Alterfactual_ vs. _Control_ : t(127) = 3.19, p = .002, d = 0.79 (medium
effect).
* •
_Alterfactual_ vs. _Counterfactual_ : t(127) = 2.06, p = .042, d = 0.51
(medium effect).
* •
_Combination_ vs. _Control_ : t(127) = 3.93, p $<$ .001, d = 0.97 (large
effect).
* •
_Combination_ vs. _Counterfactual_ : t(127) = 2.79, p = .006, d = 0.69 (medium
effect).
These results regarding the prediction task confirm our hypothesis that the
conditions with alterfactual explanations outperform the condition without
explanations in the prediction task. Further, the combination of both
explanation types did significantly outperform counterfactual explanations.
However, our hypothesis that the combination is more effective in terms of
enabling a correct model understanding than alterfactual explanations has to
be rejected.
### 7.2 Relevant and Irrelevant Information
As reported in the section above, we did not find a significant overall
difference in the feature understanding task (see Figure 5). However, in order
to investigate our hypotheses about irrelevant vs. relevant features, we
conducted another MANOVA between the conditions and the combined understanding
values for the two relevant features and the two irrelevant features. This
MANOVA did not find any significant differences, Wilks’ Lambda = 0.951,
F(6,252) = 1.07, p = .379. The mean understanding per condition can be found
in the appendix.
### 7.3 Explanation Satisfaction
The ANOVA revealed that there were no significant differences in the
subjective explanation satisfaction between the three explanation conditions,
F(2,95) = 0.34, p = .713. The mean satisfaction values with standard deviation
were: _Counterfactual_ condition: $3.54\pm 0.53$ ; _Alterfactual_ condition:
$3.65\pm 0.6$; _Combination_ condition: $3.58\pm 0.5$.
## 8 Discussion
With our proposed GAN-based approach, we demonstrated that it is possible to
generate both counterfactual and alterfactual explanations for a black box
image classifier. Using computational metrics, we showed that both of those
generated explanations fulfill their respective requirements: The
counterfactual explanations are very similar to the original images (i.e.,
0.90 average SSIM) but change the classifiers prediction in 87.70% of the
cases while alterfactual explanations are very different from the original
image (i.e., 0.32 average SSIM), but do not change the classifier’s prediction
in 96.20% of the cases. For the prediction task of our user study,
alterfactual explanations and the combination of alterfactual and
counterfactual explanations performed significantly better than the other two
conditions demonstrating the potential of alterfactual explanations to
facilitate local model understanding. However, we did not observe a
significant difference for the feature relevance understanding. This is highly
interesting, as it contrasts with a previous study by Mertes et al. (2022b).
There, a similar experimental design was employed for assessing the effect
that alterfactual explanations have on users’ mental models of a hard-coded
classifier that assesses numerical feature descriptors for a fictional
classification problem. In contrast to our work, they neither used a real
classifier nor an alterfactual generation algorithm, but only mock-up
decisions and explanations. In their scenario, alterfactual explanations led
to a significantly better feature relevance understanding, while not having a
substantial impact on the performance in a prediction task. A possible
explanation for this is the fact that our study was conducted in the context
of fashion classification, where the users might already have had a quite
distinctive mental model of the problem domain itself. Further, images might
be more accessible than numerical feature descriptors to end users. As such,
the global understanding of the classifier might already be positively biased.
This argument is supported by looking at the feature relevance understanding
results of the control group - although not seeing any explanations, they
already performed very well in identifying relevant features. However, as can
be seen by the significant performance improvement in the prediction task, the
local understanding of the model does not necessarily benefit from the
identification of globally relevant features. As the classification model is
imperfect, a global understanding of the use case itself does not necessarily
imply an understanding of cases, e.g. when the classifier’s decision does not
correctly model reality. Furthermore, our results regarding the global model
understanding should be taken with a grain of salt, since the fashion-
classifier is a black-box model - even though we used post-hoc explanation
methods (SHAP and LIME), we cannot be certain that our choice of features is
completely accurate. Interestingly, we did not observe any significant
differences in explanation satisfaction. This indicates that participants felt
similarly satisfied by all explanation methods even though the alterfactual
and combined explanations objectively helped more during the prediction task.
The presentation of more information (i.e., in the combination condition)
could have led to a higher cognitive load and influenced the subjective
assessments of explanation satisfaction, resulting in the difference between
objective measurement (i.e., model understanding) and subjective measurement
(i.e., explanation satisfaction).
## 9 Conclusion
In this paper, we demonstrate the practical feasibility of a recently proposed
concept for explaining AI models called _alterfactual explanations_ that
_alter_ as much irrelevant information as possible while maintaining the
distance to the decision boundary.
We show for the first time that it is possible to generate such explanations
for black box models and briefly evaluated them computationally. Furthermore,
we showed in a user study that our generated alterfactual explanations can
complement counterfactual explanations. In that study, we compared how users’
model understanding of a binary image classifier changes when being confronted
with counterfactual explanations, alterfactual explanations, or a combination
of both. Further, a control group was assessed that did not see any
explanations. We found that in a prediction task, where the classifier’s
prediction had to be anticipated by looking at the explanations, users
performed significantly better when they were provided with explanations that
included alterfactual explanations compared to users that did not see
alterfactual explanations, although we did not observe a significant
difference in explanation satisfaction.
Overall, we showed that alterfactual explanations are a promising explanation
method that can complement counterfactual explanations in future XAI systems.
## 10 Acknowledgments
This research was partially funded by the DFG through the Leibniz award of
Elisabeth André (AN 559/10-1).
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## Appendix A GAN Architecture and Training
#### A.0.1 Generator Model
The GAN’s generator architecture is listed in Table 1.
#### A.0.2 Discriminator Model
The GAN’s discriminator architecture is listed in Table 2.
### A.1 Training Configuration and Hyperparameters
The training configuration and hyperparamters are shown in Table 3. The Adam
optimizer was configured with $\beta_{1}=0.5$, $\beta_{2}=0.999$,
$\epsilon=1e-8$.
Further, the Support Vector Machine (SVM) that was included in the loss
function (see main paper) was trained with the parameters listed in Table 4.
## Appendix B Classifier Architecture and Training
In Table 5, the model architecture for the classifier that we used in our
evaluation scenario is described. The training configuration and
hyperparamters are shown in Table 6. The Adam optimizer was configured with
$\beta_{1}=0.9$, $\beta_{2}=0.999$, $\epsilon=1e-8$.
## Appendix C Additional Dataset Experiments
In order to demonstrate that our alterfactual generation approach is
generalizable to different datasets, we additionally trained models for three
other datasets. Here, we omitted the Feature Relevance component. As for that
component an additional SVM has to be trained on the penultimate layer of the
classifier layer, it takes away the model-agnostic property from the
alterfactual generation network. By performing these additional experiments,
we show that the approach can simply be adapted to be model-agnostic, although
that may negatively affect the outcomes of the results - it is not
specifically forced that _only_ irrelevant features change. For the
classifiers, we used the same architecture as for the Fashion-MNIST dataset,
although batch size and epochs were modified to fit the hardware that we used.
### C.1 MNIST
As the MNIST datasets has more than two classes (each class contains hand-
drawn images of one specific digit), we picked the two digits that are most
likely to be confused by deep learning classifiers: _Three_ and _Eight_. The
MNIST classifier was trained for 9 epochs with batch size 32. Besides not
using the Feature Relevance component and increasing the epoch number to 42,
the GAN network was trained with the same parameter settings as for the
Fashion-MNIST dataset. We reached a validity of 95.92% and an average SSIM of
0.425. Example outputs are shown in Figure 6.
Figure 6: Examplary alterfactual outputs for the MNIST dataset.
### C.2 MaskedFace-Net
The MaskedFace-Net dataset contains images of people wearing face masks.
Binary labels are provided, indicating that on the respective image the mask
is worn correctly or incorrectly. The classifier was trained for 2 epochs with
batch size 128. Besides not using the Feature Relevance component and
decreasing the epoch number to 11, the GAN network was trained with the same
parameter settings as for the Fashion-MNIST dataset. We reached a validity of
84.27% and an average SSIM of 0.091. Example outputs are shown in Figure 7.
Figure 7: Examplary alterfactual outputs for the MaskedFace-Net dataset.
### C.3 MaskedFace-Net (Gray Scale)
Here, we also used the MaskedFace-Net dataset, but converted it to gray scale,
demonstrating that our approach also works with gray scale data. The
classifier was trained for 1 epoch with batch size 128. Besides not using the
Feature Relevance component and decreasing the epoch number to 6, the GAN
network was trained with the same parameter settings as for the Fashion-MNIST
dataset. We reached a validity of 48.89% and an average SSIM of 0.002. Example
outputs are shown in Figure 8.
Figure 8: Examplary alterfactual outputs for the gray scale version of the
MaskedFace-Net dataset. It can be clearly seen that the only part of the image
that gets unchanged is the mask itself - indicating that everything else is
irrelevant.
## Appendix D User Study
### D.1 Demographic Details
_Control_
_Counterfactual_
_Alterfactual_
_Combination_
Figure 9: Distribution of the chosen AI experience items for each condition.
The x-axis depicts the following items: 1 - I do not have any experience in AI
related topics; 2 - I know AI from the media; 3 - I use AI technology in my
private life; 4 - I use AI technology in my work; 5 - I have taken at least
one AI related course; 6 - I do research on AI-related topics; 7 - Other:
The mean age and education level, as well as the percentage of female
participants, per condition can be seen in Table 7. For the AI experience and
Attitude we adapted a description of AI from Zhang and Dafoe [2019] and
Russell and Norvig [2016] to “The following questions ask about Artificial
Intelligence (AI). Colloquially, the term ‘artificial intelligence’ is often
used to describe machines (or computers) that mimic ‘cognitive’ functions that
humans associate with the human mind, such as ‘learning’ and ‘problem
solving’.” After this description, participants had to select one or more item
describing their experience with AI. The distribution of the items for each
condition is shown in Fig. 9. Following this we adapted a question from Zhang
and Dafoe [2019] to measure the participants’ attitude towards AI. We asked
them to rate their answer to the question “Suppose that AI agents would
achieve high-level performance in more areas one day. How positive or negative
do you expect the overall impact of such AI agents to be on humanity in the
long run?” on a 5-point Likert scale from “Extremely negative” to “Extremely
positive”. The participants also had the option to answer “I do not know”
here, which would exclude them from the evaluation of this question. The mean
results for each condition are shown in Table 7.
## Appendix E Additional Post-Hoc Results
For completeness, we also report the results of the post-hoc t-tests on the
participants’ prediction accuracy that were not significant. The effect size d
is calculated according to Cohen [2013]:
* •
_Counterfactual_ vs. _Control_ : t(127) = 1.14, p = .258, d = 0.28
* •
_Combination_ vs. _Alterfactual_ : t(127) = 0.71, p $<$ .478, d = 0.18.
For feature understanding and explanation satisfaction we did not calculate
post-hoc tests since the ANOVA was not significant.
### E.1 Mean Understanding of Features
Figure 10 shows the mean understanding (as assessed by the feature
understanding task) per condition.
Irrelevant Features
Relevant Features
Figure 10: Mean understanding of the irrelevant and relevant features in our
study. Error bars represent the 95% CI.
### E.2 Explanation Satisfaction Scale
For evaluating explanation satisfaction, we used the Explanation Satisfaction
scale by Hoffmann [Hoffman et al., 2018] except one item that did not apply to
our use case. The items that we used were as follows, where each item was
rated on a 5-point likert scale (1 = strongly disagree, 5 = strongly agree):
* •
From the explanations, I understand how the AI makes its decision.
* •
The explanations of how the AI makes its decision are satisfying.
* •
The explanations of how the AI makes its decision have sufficient detail.
* •
The explanations of how the AI makes its decision seem complete.
* •
The explanations of how the AI makes its decision are useful to predict the
AI’s decision.
* •
The explanations of how the AI makes its decision show me how accurate the AI
is.
* •
The explanations let me judge when I should trust and not trust the AI.
### E.3 Study Design
Screenshots of the user study are shown in Figures 11 to 40. The _Combination_
condition is shown.
Layer | Description | # Filter | Size | Stride | Dropout | BatchNorm | Activation
---|---|---|---|---|---|---|---
1 | Conv2D | 64 | 4x4 | 2 | - | no | LeakyReLU (0.2)
2 | Conv2D | 128 | 4x4 | 2 | - | yes | LeakyReLU (0.2)
3 | Conv2D | 256 | 4x4 | 2 | - | yes | LeakyReLU (0.2)
4 | Conv2D | 512 | 4x4 | 2 | - | yes | LeakyReLU (0.2)
5 | Conv2D | 512 | 4x4 | 2 | - | yes | LeakyReLU (0.2)
6 | Conv2D | 512 | 4x4 | 2 | - | yes | LeakyReLU (0.2)
7 | Conv2D | 512 | 4x4 | 2 | - | no | ReLU
8 | Conv2DTranspose | 512 | 4x4 | 2 | 0.5 | yes | ReLU
9 | Conv2DTranspose | 512 | 4x4 | 2 | 0.5 | yes | ReLU
10 | Conv2DTranspose | 512 | 4x4 | 2 | 0.5 | yes | ReLU
11 | Conv2DTranspose | 256 | 4x4 | 2 | - | yes | ReLU
12 | Conv2DTranspose | 128 | 4x4 | 2 | - | yes | ReLU
13 | Conv2DTranspose | 64 | 4x4 | 2 | - | yes | ReLU
14 | Conv2DTranspose | 1 | 4x4 | 2 | - | no | Tanh
Table 1: Generator Architecture used in our evaluation scenario. The architecture is adapted from Wu et al. [2019]. Where BatchNorm, Dropout, or Activation function occurred together, the order applied was BatchNorm - Dropout - Activation. Layer | Description | # Filter | Size | Stride | BatchNorm | Activation
---|---|---|---|---|---|---
0a | Embedding | - | 8x8 | - | no | -
0b | Upsample | - | 128x128 | - | no | -
1 | Conv2D | 64 | 4x4 | 2 | no | LeakyReLU (0.2)
2 | Conv2D | 128 | 4x4 | 2 | yes | LeakyReLU (0.2)
3 | Conv2D | 256 | 4x4 | 2 | yes | LeakyReLU (0.2)
4 | Conv2D | 1 | 4x4 | 2 | no | Sigmoid
Table 2: Discriminator Architecture used in our evaluation scenario. Where BatchNorm and Activation function occurred together, BatchNorm preceded the activation function. The first two layers, marked as ‘0a’ and ‘0b’ were used to upsample the label information to the size of the input image. The label and image were passed together to layer 1. The architecture is adapted from Wu et al. [2019]. Batch Size | 1
---|---
Epochs | 14
Learning Rate Generator | 1e-4
Learning Rate Discriminator | 1e-4
Optimizer | Adam
Table 3: The setting used to train the GAN. C (Regularisation) | 10
---|---
Kernel | linear
Iterations | 5000
Table 4: The setting used to train the SVM. Layer | Description | # Filter | Size | Stride | BatchNorm | Activation
---|---|---|---|---|---|---
1 | Conv2D | 32 | 3x3 | 1 | yes | ReLU
2 | Conv2D | 32 | 3x3 | 1 | yes | ReLU
3 | MaxPool2D | - | 2x2 | 2 | no | -
4 | Conv2D | 64 | 3x3 | 1 | yes | ReLU
5 | Conv2D | 64 | 3x3 | 1 | yes | ReLU
6 | GAP | - | - | - | no | -
7 | Dense | - | 2 | - | no | Softmax
Table 5: Classifier architecture used to train the classifier for the MNIST-Fashion dataset (classes _Sneaker_ and _Ankle Boot_). Where BatchNorm and Activation function occurred together, BatchNorm preceded the activation function. Batch Size | 32
---|---
Epochs | 40
Learning Rate | 1e-3
Optimizer | Adam
Loss Function | Binary Cross Entropy
Table 6: The setting used to train the Fashion-MNIST classifier. | Control | Counterfactual | Alterfactual | Combination
---|---|---|---|---
Mean age | 22.0 | 22.5 | 21.6 | 22.6
Percentage of female participants | 52 | 58 | 47 | 58
Highest level of Education | 2.27 | 2.42 | 2.19 | 2.21
Mean AI Attitude | 3.50 | 3.82 | 3.84 | 3.75
Table 7: Demographic data across conditions. The highest level of education
was measured as follows: 1 - No education, 2 - High school graduation, 3 -
Vocational training, 3 - Bachelor, 4 - Master, 5 - Doctor. The Attitude
towards AI is measured on 5-Point Likert scale from “Extremely negative” to
“Extremely positive”. Figure 11: Screenshot of the user study, part 1. Figure
12: Screenshot of the user study, part 2. Figure 13: Screenshot of the user
study, part 3. Figure 14: Screenshot of the user study, part 4. Figure 15:
Screenshot of the user study, part 5. Figure 16: Screenshot of the user study,
part 6. Figure 17: Screenshot of the user study, part 7. Figure 18: Screenshot
of the user study, part 8. Figure 19: Screenshot of the user study, part 9.
Figure 20: Screenshot of the user study, part 10. Figure 21: Screenshot of the
user study, part 11. Figure 22: Screenshot of the user study, part 12. Figure
23: Screenshot of the user study, part 13. Figure 24: Screenshot of the user
study, part 14. Figure 25: Screenshot of the user study, part 15. Figure 26:
Screenshot of the user study, part 16. Figure 27: Screenshot of the user
study, part 17. Figure 28: Screenshot of the user study, part 18. Figure 29:
Screenshot of the user study, part 19. Figure 30: Screenshot of the user
study, part 20. Figure 31: Screenshot of the user study, part 21. Figure 32:
Screenshot of the user study, part 22. Figure 33: Screenshot of the user
study, part 23. Figure 34: Screenshot of the user study, part 24. Figure 35:
Screenshot of the user study, part 25. Figure 36: Screenshot of the user
study, part 26. Figure 37: Screenshot of the user study, part 27. Figure 38:
Screenshot of the user study, part 28. Figure 39: Screenshot of the user
study, part 29. Figure 40: Screenshot of the user study, part 30.
|
# Observation of anomalous amplitude modes in the kagome metal CsV3Sb5
Gan Liu Xinran Ma Kuanyu He Qing Li National Laboratory of Solid State
Microstructures and Department of Physics, Nanjing University, Nanjing 210093,
China Hengxin Tan Yizhou Liu Department of Condensed Matter Physics,
Weizmann Institute of Science, Rehovot 7610001, Israel Jie Xu Wenna Tang
National Laboratory of Solid State Microstructures and Department of Physics,
Nanjing University, Nanjing 210093, China Kenji Watanabe Research Center for
Functional Materials, National Institute for Materials Science, 1-1 Namiki,
Tsukuba 305-0044, Japan Takashi Taniguchi International Center for Materials
Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki,
Tsukuba 305-0044, Japan Libo Gao Yaomin Dai Hai-Hu Wen National Laboratory
of Solid State Microstructures and Department of Physics, Nanjing University,
Nanjing 210093, China Collaborative Innovation Center of Advanced
Microstructures, Nanjing University, Nanjing 210093, China Binghai Yan
<EMAIL_ADDRESS>Department of Condensed Matter Physics, Weizmann
Institute of Science, Rehovot 7610001, Israel Xiaoxiang Xi<EMAIL_ADDRESS>National Laboratory of Solid State Microstructures and Department of Physics,
Nanjing University, Nanjing 210093, China Collaborative Innovation Center of
Advanced Microstructures, Nanjing University, Nanjing 210093, China
Abstract
The charge-density wave (CDW) phase is often accompanied by the condensation
of a soft acoustic phonon mode, giving rise to lattice distortion and charge
density modulation. This picture was challenged for the recently discovered
kagome metal CsV3Sb5, based on the evidence of absence of soft phonons. Here
we report the observation of Raman-active CDW amplitude modes in this
material, which are collective excitations typically thought to emerge out of
frozen soft phonons. The amplitude modes strongly hybridize with other
superlattice modes, imparting them with clear temperature-dependent frequency
shift and broadening, rarely seen in other known CDW materials. Both the mode
mixing and the large amplitude mode frequencies suggest that the CDW exhibits
the character of strong electron-phonon coupling, a regime in which acoustic
phonon softening can cease to exist. The observation of amplitude modes in the
absence of soft phonons highlights the unconventional nature of the CDW in
CsV3Sb5.
Introduction
Materials with a kagome lattice can host rich phenomena encompassing quantum
magnetism [1, 2], Dirac fermions [3, 4], nontrivial topology [5, 6, 7],
density waves, and superconductivity [8, 9, 10]. The recently discovered
kagome metals $A$V3Sb5 ($A=\mathrm{K,Rb,or~{}Cs}$) [11, 12] offer a new
platform to study the interplay of these phenomena. These compounds have Fermi
levels close to Dirac points or van Hove singularities [12, 13, 14], leading
to a plethora of possible intriguing ground states. Indeed, charge-density
waves and superconductivity have been discovered [12, 15, 16], with ample
evidence showing that both types of orders are exotic. For example, the CDW
transition is accompanied possibly by a large anomalous Hall effect [17, 18],
and the superconductivity features a pair-density wave state [19].
The nature of the CDW state and the mechanism for its formation have been
under close scrutiny. In this work, we focus on CsV3Sb5, which has a CDW
transition temperature $T_{\mathrm{CDW}}=94$ K [12]. Both hard-x-ray and
neutron scattering showed the lack of soft acoustic phonons [20, 21], although
density functional theory (DFT) calculations found two phonon instabilities at
the $M$ and $L$ points of the Brillouin zone [22, 23, 24]. Considering that a
$2\times 2$ modulation of the crystal lattice is well established [12, 19, 25,
26, 27], the absence of soft modes apparently breaks a pattern proven generic
to many known CDW systems — a soft acoustic phonon freezes to zero frequency
and triggers the formation of a distorted lattice [28]. Currently, there is
still no consensus on the form of the in-plane structure and the $c$-axis
periodicity [27, 29]. The roles of Fermi-surface nesting and electron-phonon
coupling are also debated. Because the period of the $2\times 2$ superlattice
matches perfectly with the Fermiology of the van Hove singularity, it is
natural to ascribe the CDW transition to Fermi surface nesting [22], supported
by the appreciable partial gapping of the Fermi surface observed
experimentally [30, 31, 32, 33, 34]. However, the calculated electronic
susceptibility lacks the expected divergence [35, 36], and the effect of
electron-phonon coupling may not be dismissed [36, 21].
Because the CDW features lattice distortions, studies of the lattice degree of
freedom can offer insight into the mechanism. Raman scattering is a valuable
tool in this respect. In well-studied CDW systems, such as the transition
metal dichalcogenides, as a soft acoustic phonon mode condenses to form a
distorted lattice, new Raman-active collective excitations, known as amplitude
modes, emerge, providing a direct probe of the CDW order parameter [37, 38,
39] (see Fig. 1a). Conversely, the observation of amplitude modes is typically
considered as evidence for the soft mode. The temperature dependence of the
amplitude modes as well as the zone-folded modes, which become Raman-active
due to zone folding induced by the superlattice, can both reflect the CDW
transition [40, 41, 42, 43, 44]. Combined with symmetry information from
polarization-resolved measurements, constraints can be set on the possible CDW
ground state.
Here, we report Raman scattering measurements on CsV3Sb5. We observe a
multitude of CDW-induced modes, whose symmetries and frequencies are in good
agreement with DFT calculations for a single layer CsV3Sb5 under inverse Star
of David distortion. The observed temperature dependence of these modes and
their calculated evolutions with varying lattice distortion allow us to
identify two of them as amplitude modes, emerging from the predicted soft
modes, although the soft modes are elusive experimentally. In contrast to
mostly independent amplitude modes and zone-folded modes in well-known CDW
materials [41, 42, 43], we show that they hybridize strongly in CsV3Sb5,
causing spectral weight redistribution to the latter and rendering them
amplitude-mode-like. The anomalous hybridization and the large values of the
amplitude mode frequencies provide evidence of strong-coupling CDW, offering a
possible explanation for the lack of soft acoustic modes. These results stress
the importance of the lattice degree of freedom and electron-phonon coupling
in the CDW formation in CsV3Sb5.
Results
Raman-active modes in CsV3Sb5
CsV3Sb5 crystallizes in a hexagonal lattice with the $P6/mmm$ space group
[11]. Fig. 1b shows the unit cell of its crystal structure. The V atoms form a
kagome net interspersed by Sb atoms (labeled Sb1), all within the ab-plane.
The V atoms are further bonded by Sb atoms above and below the kagome plane
(labeled Sb2). These V3Sb5 slabs are separated by Cs layers, with weak
coupling between them to form a quasi-two-dimensional (quasi-2D) structure.
Factor group analysis yields three Raman-active phonon modes,
$\Gamma_{\mathrm{Raman}}=A_{1g}+E_{2g}+E_{1g}$. The former two can be detected
when the photons are polarized in the $ab$-plane, satisfied by the back-
scattering geometry used in our experiment. These intense modes are marked by
dashed lines in Figs. 1c, d. They involve only the Sb2 atoms, with their
atomic vibrations along the $c$-axis and within the $ab$-plane for the
$A_{1g}$ and $E_{2g}$ modes, respectively; see Fig. 1b. The $E_{2g}$ modes are
a pair of degenerate vibrations with opposite circling directions, i.e.,
opposite chiralities (see Supplementary Note 1). These two types of symmetries
can be distinguished by polarization-resolved measurements. Specifically, the
$A_{1g}$ modes can be detected in the XX and LL polarization configurations,
whereas the $E_{2g}$ modes appear in the XX, XY, and LR configurations. Here,
XX and XY represent collinear and cross-linear polarization for the incident
and scattered photons, and LL and LR involve circularly polarized light with
left (L) and right (R) helicity. A comparison of data in all four
configurations is included in Supplementary Fig. 1.
Below $T_{\mathrm{CDW}}$, multiple peaks emerge, highlighted by the dotted
lines in Figs. 1c, d. Their origin will be discussed in the next sections.
These modes are rather weak compared to the main lattice phonons. Their
disappearance at 100 K suggests a close correlation with CDW formation. In
contrast, many weak peak-like structures below 100 cm-1 lack temperature
dependence, whose origin is unclear.
Fig. 1e compares the observed Raman mode frequencies with those from DFT
calculations for a single layer of CsV3Sb5 [22], considering two possible
forms of lattice distortion, the Star of David (SD) and inverse Star of David
(ISD, also referred to as tri-hexagonal) structures. Both of them show the
same number of $A_{1g}$ and $E_{2g}$ modes, but with different ordering.
Overall, the calculated ISD phonons agree much better with the experimental
results, as shown in the figure and in Supplementary Tab. 1. All the five
predicted $A_{1g}$ modes and five out of the eight predicted $E_{2g}$ modes
are observed. The observed $A_{1g}$ mode below 50 cm-1 is unaccounted for by
our calculations. This mode was also observed by pump-probe time-resolved
spectroscopy, which, when compared with calculations taking into account
interlayer coupling, was assigned as a Cs-mode due to CDW modulation along the
$c$-axis [23]. Except for this mode and the three missing $E_{2g}$ modes due
to their weak scattering cross section, the symmetry ordering of all the other
modes is in exact agreement between the experiment and theory. These results
suggest that the CDW ground state consists of weakly coupled layers with ISD-
type distortion, but CDW modulation along the $c$-axis is also indispensable.
Since the single-layer CsV3Sb5 holds the key to unraveling the CDW mechanism,
we attempted creating atomically thin CsV3Sb5 by mechanical exfoliation.
However, the loss of crystallinity impeded further investigation
(Supplementary Figs. 2 and 3).
Temperature dependence of Raman modes
Figs. 2a, b show the temperature dependent Raman intensity color plot for
CsV3Sb5, obtained in the LL and LR configurations, respectively. The intense
$A_{1g}$ and $E_{2g}$ main lattice modes are the most conspicuous features.
Figs. 2e–g show the frequency (with the corresponding value at 200 K
subtracted), linewidth (full width at half maximum), and normalized integrated
area for both modes, extracted from Lorentzian fits of the peaks. The $A_{1g}$
frequency sharply increases below $T_{\mathrm{CDW}}$, whereas the $E_{2g}$
frequency exhibits a subtle kink across the CDW transition. This is consistent
with the planar ISD lattice distortion mainly involving V atoms, forcing the
Sb2 atoms to displace along the $c$-axis, hence affecting the out-of-plane
vibration of the $A_{1g}$ mode more effectively. The calculated phonon
vibration patterns and frequencies confirm this picture (see Supplementary
Fig. 4 and Supplementary Tab. 1). The CDW transition also causes a faster
decrease in the linewidths below $T_{\mathrm{CDW}}$. This can be understood as
being due to the CDW-induced partial gapping of the Fermi surface [30, 31, 32,
33, 34], which reduces the electron-phonon interaction. The integrated peak
intensity for both phonons increases upon warming, in line with increased
thermal phonon populations. The rate of increase is faster when approaching
$T_{\mathrm{CDW}}$ from below, and interestingly, the value saturates below
approximately 50 K. The renormalization of the phonon parameters across the
CDW transition evidences sizable electron-phonon coupling.
CDW-induced modes are labeled in Figs. 2a, b. Except for the $A_{1}$ mode,
there appears to be two types of modes, represented by $A_{2}$ and $E_{3}$.
$A_{2}$ exhibits appreciable softening and broadening upon warming toward
$T_{\mathrm{CDW}}$. It is overdamped before disappearing, visualized in the
color plot in Fig. 2a as the streak of signal below 100 cm-1 between 60–90 K.
These are signatures of a CDW amplitude mode [40, 41, 42, 43, 44], caused by
the collapse of coherent CDW order near $T_{\mathrm{CDW}}$. $E_{3}$ shows a
smaller change of frequency and much less broadening, more consistent with the
characteristics of a zone-folded mode [41], as this type of mode arises from
folding a zone-boundary phonon to the zone center, making its temperature
dependence of the frequency as weak as that of normal phonons. Figs. 2c, d
compare the distinct temperature dependence of these two types of modes. While
$A_{2}$ broadens significantly above 40 K, $E_{3}$ maintains its linewidth and
suddenly vanishes above $\sim$80 K. The dramatic difference in the linewidth
broadening is quantified in Fig. 2i. Fig. 2h shows the frequencies of all the
observed Raman modes on the same scale. Upon warming, the CDW-induced modes
($A_{1}$ excluded) soften more dramatically than the main lattice modes. While
it is tempting to assign most of them as amplitude modes because of the
apparent softening behavior, we show below that they are in fact zone-folded
modes, mixed with the amplitude modes to partially inherit their properties.
Nature of CDW-induced modes
Although soft acoustic phonons were not detected experimentally, our DFT
results show that the formation of CDW in CsV3Sb5 is similar to that in other
well-known systems [40, 41, 42, 43, 44], in the sense that a soft acoustic
phonon mode at the CDW wavevector condenses and gives rise to a distorted
lattice [38]. The imaginary phonon modes of pristine CsV3Sb5 at three $M$
points (see Supplementary Fig. 5) transform as irreducible representation
$M_{1}^{+}$ ($A_{g}$) of the space group $P6/mmm$ (little co-group $D_{2h}$).
Fig. 3a shows that after the $2\times 2\times 1$ CDW transition, they are
folded to $\Gamma$ and form triply degenerate modes. With lattice distortion,
they decompose to a singlet $A_{1g}$ mode and a doublet $E_{2g}$ mode under
the point group $D_{6h}$:
$3M^{+}_{1}\rightarrow A_{1g}\oplus E_{2g}.$ (1)
As the lattice distorts from the pristine phase to the stable ISD pattern, the
imaginary $A_{1g}$ and $E_{2g}$ modes turn real with positive frequencies,
expected to be observable as two Raman-active amplitude modes. The phonon
wavefunctions of the soft $A_{1g}$ and $E_{2g}$ modes are shown in Figs. 4a,
b, dominated by vibrations of V atoms. The $A_{1g}$ mode is fully symmetric,
involving breathing-type motion for the V triangles, V hexagons, and Sb2
atoms. The $E_{2g}$ mode involves circling motion for the atoms forming the V
hexagons, while the amplitude for the Sb2 vibration is almost ten times
smaller.
DFT calculations further reveal that the amplitude modes strongly hybridize
with the other CDW-induced Raman modes (i.e. the zone-folded modes always at
positive frequencies at the $\Gamma$ point in Fig. 3a), rendering them
amplitude-mode-like, hence their apparent temperature-dependent frequencies.
Figs. 4c, d show the real space wavefunctions of all the CDW-induced modes in
the $2\times 2\times 1$ ISD phase. $A_{2-5}$ and $E_{1-4}$ correspond to those
in Fig. 2, and $E^{\prime}_{1-3}$ are undetected experimentally. Comparison
with Figs. 4a, b shows that $A_{2-5}$ ($E_{1-4}$) all resemble the $A_{1g}$
($E_{2g}$) soft mode, but the difference is also apparent. The similarity
results from hybridization of phonon wavefunctions.
To quantify the mode mixing, we calculated the overlap between the soft modes
and all the real modes of the stable ISD phase by projecting the phonon
wavefunctions, $P_{f}=|\langle\bm{u}_{f}|\bm{u}_{\mathrm{SM}}\rangle|^{2}$,
where $|\bm{u}_{\mathrm{SM}}\rangle$ refers to the wavefunction of the soft
modes shown in Figs. 4a, b and $|\bm{u}_{f}\rangle$ refers to the wavefunction
of the mode at frequency $f$ in the ISD phase. The results in Fig. 3b show
that as the soft $A_{1g}$ and $E_{2g}$ modes shift from negative to positive
frequencies and turn into amplitude modes, they hybridize with most of the
zone-folded modes belonging to the same irreducible representation. The amount
of calculated projection in the stable ISD phase correlates reasonably well
with the observed mode intensity in Fig. 1c, d. $A_{2}$ and $E_{1}$ are
residual amplitude modes after mode mixing. $E^{\prime}_{1-3}$ show minor
projection from the $E_{2g}$ soft mode because of negligible wavefunction
overlap, and accordingly their scattering cross section is weak. $E_{2-4}$ all
involve V triangles (Fig. 4d), indicating that they have contributions
unrelated with the $E_{2g}$ soft mode. Indeed, as discussed earlier, $E_{3}$
shows clear experimental signatures of a zone-folded mode.
The $A_{2}$ mode was also observed by Wulferding et al. in their Raman study
[45] and by time-resolved pump-probe spectroscopy [23, 46]. However, in these
works, it was suggested to emerge below $\sim$60 K [23, 46, 45], hence
ascribed to another phase transition associated with a unidirectional order
[19, 25, 26]. According to our data (Fig. 2a), the $A_{2}$ mode survives above
60 K, and there is no clear evidence for two distinct phase transitions. Its
vibration pattern shown in Fig. 4c confirms no relation with the
unidirectional order. Another Raman study by Wu et al. [47] reported a similar
set of modes as ours, but with different relative intensities. They also
observed extra modes that are possibly due to stronger $c$-axis modulation in
their sample.
Discussion
The anomalously large hybridization between the amplitude modes and zone-
folded modes is rare, because they are mostly decoupled in the canonical CDW
materials, with the amplitude modes dominating the spectral intensity [41, 42,
43]. The hybridization is highly unusual, because the $A_{2}$ and $E_{1}$
amplitude modes and the zone-folded Raman modes span a wide frequency range,
and they do not overlap in energy (except for $E_{1}$ and $E^{\prime}_{1}$) to
exhibit the typical anti-crossing [48, 49]. Their strong coupling suggests
that the hybridization occurs indirectly, through interaction with the common
electronic system. As the Fermi surface instability associated with the van
Hove singularity is from the V bands [22], modes mainly involving V (including
$A_{2-5}$ and $E_{1-4}$) naturally mix with the amplitude modes, whereas those
mainly involving Sb (including $E^{\prime}_{1-2}$ and the $A_{1g}$ and
$E_{2g}$ main lattice modes) do not. Similar mode mixing was also observed in
the quasi-one-dimensional (quasi-1D) K0.3MoO3 using time-resolved pump-probe
spectroscopy [50], and a simple model based on Ginzburg-Landau theory can well
describe the entanglement of the electronic and lattice parts of the CDW order
parameter. The similar phenomena observed in two systems with different
dimensionality suggest the importance of electron-phonon coupling in both
materials.
However, a soft acoustic phonon is well established in K0.3MoO3 [51], but
shown to be absent in CsV3Sb5 [20, 21]. In the mean field weak-coupling theory
[38, 28], the acoustic phonon softening, known as Kohn anomaly [52], is a
direct consequence of the divergent electronic susceptibility, which screens
the phonon vibration at the CDW wavevector. In reality, the singular
electronic susceptibility is smeared out, especially at dimensions higher than
one, and momentum dependent electron-phonon coupling dictates the phonon
renormalization in certain systems such as 2$H$-NbSe2 [53, 54]. The lack of
soft acoustic phonons in CsV3Sb5 clearly rules out both mechanisms. Instead,
CsV3Sb5 may fall into the strong electron-phonon coupling regime [55], in
which the non-detection of Kohn anomaly in the quasi-1D (TaSe4)2I, NbSe3, and
BaVS3 has also been reported [56, 57, 58]. In all these materials, strong
electron-phonon coupling tends to localize electrons, violating the adiabatic
Born-Oppenheimer approximation [55] used in DFT. Failure of the conduction
electrons to screen the phonon vibration can naturally explain the absence of
phonon softening [59].
The strong-coupling nature of the CDW in CsV3Sb5 is indeed supported by
multiple facts, according to the qualitative criteria discussed in Ref. [28].
The CDW-induced gap $\Delta_{\mathrm{CDW}}$ is large, with
$2\Delta_{\mathrm{CDW}}/k_{B}T_{\mathrm{CDW}}\approx 22$ according to infrared
spectroscopy [30], where $k_{B}$ is the Boltzmann constant. The lattice
distortion is substantial (amounting to about 5% of the lattice constant
[22]), the distorted lattice exhibits clustering of V atoms to form trimers
and hexamers, and the CDW locks with the pristine lattice to form a
commensurate structure, all indicating local chemical bonding [22, 35].
Moreover, DFT shows that the elastic potential for the ions in the pristine
structure features double minima deeper than the thermal energy
$k_{B}T_{\mathrm{CDW}}$ at the transition [22], a defining feature of the
strong-coupling theory proposed by Gor’kov [55]. Such potential well traps the
ions in one of its minima, precluding soft phonon condensation. From the
perspective of Raman scattering, the electron-phonon coupling constant
$\lambda$ can be estimated from the amplitude mode frequency
$\omega_{\mathrm{AM}}$ and the unscreened soft mode frequency
$\omega_{\mathrm{SM}}^{0}$ as
$\lambda=(\omega_{\mathrm{AM}}/\omega_{\mathrm{SM}}^{0})^{2}$, valid on the
mean-field level [38]. The results for CsV3Sb5 and a variety of other CDW
materials are complied in Fig. 5. Notably, the four quasi-2D compounds
2$H$-NbSe2, 2$H$-TaSe2, CsV3Sb5, and 1$T$-TaS2 are roughly located in the
expected order according to their $T_{\mathrm{CDW}}$, $\Delta_{\mathrm{CDW}}$,
and commensurability. The frequencies of the amplitude modes in CsV3Sb5 are
large, only lower than that of the higher one in 1$T$-TiSe2. Although the
exact value of $\lambda$ may not be meaningful beyond the weak-coupling limit,
these results clearly indicate the strong-coupling nature of the CDW in
CsV3Sb5.
Our Raman results offer informative insights into the CDW phase in CsV3Sb5,
suggesting the dominance of the single layer in driving the transition and
evidencing strong electron-phonon coupling. CsV3Sb5 represents a unique case
in which the amplitude modes emerge in the absence of soft acoustic phonons.
As important collective excitations of the CDW ground state, how they forms
without being driven by folding of a soft acoustic phonon warrants further
investigation. Although DFT accurately predicts the CDW-induced Raman modes in
the zero-temperature limit, in good agreement with our experiment, to
understand how the amplitude modes emerge below $T_{\mathrm{CDW}}$ apparently
requires models and computational methods working at finite temperature.
Methods
Sample preparation
CsV3Sb5 single crystals were synthesized using the flux method [11]. The
freshly cleaved surface of the samples was used in the study of bulk crystals.
Raman scattering spectroscopy was performed using home-built confocal
microscopy setups in the back-scattering geometry with 532 nm laser
excitation. The normally incident light was focused on the sample to a micron-
sized spot, and the scattered light was directed through Bragg notch filters
to access the low-wavenumber region. The Raman signal was collected using a
grating spectrograph and a liquid-nitrogen-cooled charge-coupled device. The
samples were mounted in a vacuum chamber during data acquisition. Temperature
control was achieved using a Montana Instrument Cryostation.
Calculations
The DFT calculation results by Tan et al. [22] are used to compare with the
experiment. Details of the calculations can be found therein. In addition, we
calculated the force constants by Vienna ab-initio Simulation Package (VASP)
[60] and computed the phonon dispersion relation by Phonopy [61]. For the DFT
calculation, a $5\times 5\times 5$ $\bm{k}$ mesh and an energy cutoff of 400
eV were used.
Data availability
The data in Figure 1e are available in Supplementary Table 1. Other data are
available from the corresponding authors upon request.
References
## References
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Acknowledgements
This work was supported by the National Key Research and Development Program
of China (Grant Nos. 2018YFA0307000 and 2017YFA0303201) and the National
Natural Science Foundation of China (Grant Nos. 11774151). Growth of hexagonal
boron nitride crystals was supported by the Elemental Strategy Initiative
conducted by the MEXT, Japan (Grant No. JPMXP0112101001), JSPS KAKENHI (Grant
No. JP20H00354), and A3 Foresight by JSPS. B.Y. acknowledges the financial
support by the European Research Council (ERC Consolidator Grant
“NonlinearTopo”, No. 815869) and the ISF - Quantum Science and Technology (No.
1251/19).
Author information
Contributions
X.X. conceived the project. G.L., X.M., and K.H. performed the Raman
experiments. Q.L., Y.D., and H.-H.W. grew the CsV3Sb5 crystals. K.W. and T.T.
grew the h-BN crystals. G.L. and X.X. analysed the experimental data. H.T.,
Y.L., and B.Y performed the DFT calculations. J.X., W.T., and L.G. performed
atomic force microscopy measurements. X.X. and B.Y. interpreted the results
and co-wrote the paper, with comments from all authors.
Corresponding authors
Correspondences should be sent to Binghai Yan<EMAIL_ADDRESS>or
Xiaoxiang Xi ([email protected]).
Competing interests
The authors declare no competing interests.
Figure 1: Raman-active phonon modes in CsV3Sb5. a Schematic illustration of
the relation between the soft mode and amplitude mode in typical CDW
materials, showing the latter emerges after the former freezes below
$T_{\mathrm{CDW}}$. b Crystal structure of CsV3Sb5. Sb sites with different
Wyckoff positions are labeled as Sb1 and Sb2. The arrows illustrate the
vibration patterns of the main lattice $A_{1g}$ and $E_{2g}$ modes. The
$E_{2g}$ mode is doubly degenerate, and only one form is shown. c, d Raman
spectra measured on the $ab$-plane at 100 K and 4 K in the LL and LR
polarization configurations. The dashed lines denote the main lattice phonons,
and the dotted lines indicate the CDW-induced modes. e Comparison of the
measured (Expt.) and the calculated Raman mode frequencies for the inverse
Star of David (ISD) and Star of David (SD) lattice distortions. The thick
lines denote the main lattice phonons. The dots indicate modes undetected in
our experiment.
Figure 2: Evolution of the Raman modes in CsV3Sb5 across the CDW transition.
a, b Temperature-dependent Raman intensity color plot for CsV3Sb5, measured in
the LL and LR configurations. The normal phonon modes are labeled in black and
the CDW-induced modes in white. The dashed lines mark $T_{\mathrm{CDW}}$. c, d
Temperature-dependent spectra for the $A_{2}$ and $E_{3}$ modes. e–g
Frequency, linewidth, and amplitude for the $E_{2g}$ and $A_{1g}$ main lattice
phonons. The frequency and amplitude are compared to the corresponding values
at 200 K. h Temperature dependence of the Raman mode frequencies. i
Temperature dependence of the linewidth of the $A_{2}$ and $E_{3}$ modes.
Figure 3: Phonon band structures and mode mixing in the process of CDW
distortion. a Phonon band structures. Here, 100% (0%) refers to the fully
stable ISD ($2\times 2$ pristine) structure. 10% refers to the intermediate
structure with 10% distortion from the pristine to ISD phases. After $2\times
2\times 1$ band folding with no distortion, the imaginary triply-degenerate
modes form at $\Gamma$. A weak ISD-type distortion lifts the degeneracy and
leads to $A_{1g}$ and $E_{2g}$ modes. The ISD distortion gradually transforms
imaginary modes to real. b Projections of the imaginary $A_{1g}$ ($E_{2g}$)
mode with 10% distortion to all the other phonon modes at $\Gamma$, as
evolving into the stable ISD phase (100%). We highlight all $A_{1g}$ and
$E_{2g}$ modes by orange and blue dots, respectively, at the $\Gamma$ point.
The dashed orange (blue) curve in b guides eyes to show the evolution of the
imaginary $A_{2}$ ($E_{1}$) modes in the CDW distortion.
Figure 4: Real space wavefunctions of the imaginary soft modes and the stable
CDW-induced Raman modes in the $2\times 2\times 1$ ISD phase. a, b Soft modes
with $A_{1g}$ and $E_{2g}$ symmetries, respectively. c, d CDW-induced $A_{1g}$
and $E_{2g}$ stable modes. The $E_{2g}$ modes are pairs of chiral phonons and
only one chiral mode is shown. The radius of the circles represents the
amplitude of the vibration, and the arrow on the circles stands for the
initial phase of the wavefunction. Cs atoms are omitted in the crystal
structure for clarity, because they do not contribute to lattice vibrations.
Figure 5: Evidence of strong-coupling CDW in CsV3Sb5. Frequency of the
amplitude mode $\omega_{\mathrm{AM}}$ in the zero-temperature limit and the
unscreened frequency of the soft mode $\omega_{\mathrm{SM}}^{0}$ far above
$T_{\mathrm{CDW}}$ for a collection of CDW materials. Some of the materials
feature two amplitude modes, hence two data points connected by a vertical
line. Since no soft mode is observed in CsV3Sb5, $\omega_{\mathrm{SM}}^{0}$ is
taken to be its acoustic phonon frequency at 300 K [21]. Open (filled) symbols
indicate the material is quasi-1D (quasi-2D). The dashed lines mark electron-
phonon coupling constant $\lambda=1$ and 3 according to mean-field theory.
Source of data: ZrTe3 [62, 63], TbTe3 [48, 64], K0.3MoO3 [51, 40], 1$T$-TiSe2
[65, 44], 2$H$-TaSe2 [66, 42, 43], 2$H$-NbSe2 [66, 67, 68].
|
# First detection of $c$-C3H2 in a circumstellar disk
Chunhua Qi Harvard-Smithsonian Center for Astrophysics, 60 Garden Street,
Cambridge, MA 02138, USA Karin I. Öberg Departments of Chemistry and
Astronomy, University of Virginia, Charlottesville, VA 22904, USA David J.
Wilner, Katherine A. Rosenfeld Harvard-Smithsonian Center for Astrophysics,
60 Garden Street, Cambridge, MA 02138, USA
###### Abstract
We report the first detection of $c$-C3H2 in a circumstellar disk. The
$c$-C3H2 J=$6-5$ line (217.882 GHz) is detected and imaged through Atacama
Large Millimeter Array (ALMA) Science Verification observations toward the
disk around the Herbig Ae star HD 163296 at 0.8′′ resolution. The emission is
consistent with that arising from a Keplerian rotating disk. Two additional
$c$-C3H2 transitions are also tentatively detected, bolstering the
identification of this species, but with insufficient signal-to-noise ratio to
constrain the spatial distribution. Using a previously developed model for the
physical structure of this disk, we fit a radial power-law distribution model
to the $c$-C3H2 $6-5$ emission and find that $c$-C3H2 is present in a ring
structure from an inner radius of about 30 AU to an outer radius of about 165
AU. The column density is estimated to be 1012–1013 cm-2. The clear detection
and intriguing ring structure suggest that $c$-C3H2 has the potential to
become a useful probe of radiation penetration in disks.
protoplanetary disks; astrochemistry; stars: formation; ISM: molecules;
techniques: high angular resolution; radio lines: ISM
## 1 Introduction
Pre-main sequence stars are commonly surrounded by disks, which serve the dual
purpose of funneling accretion onto the central star and transporting away
angular momentum. These disks are also the formation sites of planets and
connect protostars and planets physically and chemically. The main component
of gas-rich circumstellar disks, cold H2, cannot be observed directly because
of a lack of transitions at suitable energy levels. Instead, constraints on
disk mass, density and temperature structures depend on observations of trace
species, most commonly dust and CO gas (e.g. Qi et al., 2006; Piétu et al.,
2007; Andrews et al., 2009). To access other disk properties, such as
ionization levels, X-ray and UV radiation fields, turbulent mixing, chemical
age and bulk composition requires the development of additional molecular
probes whose formation and destruction depend critically on one or several of
these properties.
The past decade has witnessed significant progress on disk chemistry theory,
resulting in large numbers of predictions on how molecular abundance
structures should depend on different disk properties (e.g. Aikawa & Nomura,
2006; Willacy, 2007; Semenov & Wiebe, 2011; Fogel et al., 2011; Walsh et al.,
2012). A lack of observational sensitivity has however limited detections to
the simplest molecules in the millimeter/submillimeter range which
characterizes the bulk of the gas disk, including CO, HCO+, DCO+, CN, HCN,
DCN, C2H, H2CO, CS, SO, CH2, N2H+ and HC3N (Dutrey et al., 1997; Aikawa et
al., 2003; Thi et al., 2004; Semenov et al., 2005; Dutrey et al., 2007; Qi et
al., 2008; Fuente et al., 2010; Henning et al., 2010; Öberg et al., 2010,
2011). This list is expected to expand with the arrival of ALMA and its
unprecedented sensitivity and spatial resolution, which will enable the
development of a more diverse set of molecular probes. In this Letter, we use
ALMA Science Verification observations of HD 163296 to report the first
detection of cyclopropenylidene, $c$-C3H2, in a circumstellar disk. At a
distance of $\sim$122 pc (Perryman et al., 1997), this Herbig Ae star (stellar
mass 2.3 M⊙; spectral type A1; age $\sim$4 Myr) harbors a large disk (a
scattered light pattern extending out to a radius of $\sim$500 AU, Grady et
al., 2000) with strong millimeter continuum and molecular line emission
(Mannings & Sargent, 1997; Thi et al., 2004; Natta et al., 2004; Isella et
al., 2007; Qi et al., 2011). The large disk size and strong molecular emission
make this system an ideal target to search for new molecules.
The $c$-C3H2 molecule is a small cyclic hydrocarbon, which was first detected
in space by Thaddeus et al. (1985) towards Sgr B2. Since then, $c$-C3H2 has
been detected in a wide range of astrophysical environments, including diffuse
and dense clouds, protostars, planetary nebulae and extragalactic sources
(e.g., Matthews & Irvine, 1985; Cox et al., 1988, 1989; Madden et al., 1989;
Benson et al., 1998; Menten et al., 1999; Fossé et al., 2001; Sakai et al.,
2008; Tenenbaum et al., 2009; Liszt et al., 2012), and is one of the most
abundant molecules with 3 carbon atoms in the interstellar medium (Teyssier et
al., 2004). In the context of disks, $c$-C3H2 has some favorable
characteristics that may make it a powerful molecular probe, especially when
observed in combination with isotopologues and chemically related molecules:
1. 1.
$c$-C3H2 has two equivalent H nuclei, which couple to generate ortho (nuclear
spin of 1) and para (nuclear spin of 0) states. Deviations from the
statistical weights of 3 to 1 may be used to constrain the formation
conditions of the molecule (Park et al., 2006).
2. 2.
$c$-C3H2 is also one of two isomers; the other is the linear molecule
$l$-C3H2. The isomer ratio can be used to assess the fractional ionization in
clouds (Fossé et al., 2001) and may provide similar constraints on disks.
3. 3.
The formation and destruction chemistry of $c$-C3H2 (and carbon chains) are
sensitive to high energy radiation. The vertical and radial abundance
structure should provide strong constraints on the penetration depth of X-rays
and UV photons (Semenov & Wiebe, 2011).
4. 4.
Within the family of hydrocarbons, the sensitivity to turbulent mixing
increases dramatically from C2 to C3H2, and from C3H2 to larger hydrocarbons.
Observed ratios and limits have the potential to constrain transport in disks
(Semenov & Wiebe, 2011).
Finally, the chemistry that produces $c$-C3H2 is important to constrain in its
own right, since $c$-C3H2 shares precursors with other, more complex carbon
chains, which may be an important source of carbon during planet formation. In
light of these possibilities, we aim to provide first constraints on the
abundance, distribution and $o/p$ ratio of $c$-C3H2 in the HD 163296 disk.
## 2 Observations
HD 163296 (RA: 17h56m21$\fs$287, DEC: $-$21°57′22$\farcs$39; J2000.0) was
observed on 2012 June 9, June 23 and July 7 as part of the ALMA Science
Verification (SV) program in band 6. At the time of observations, 25 12-m
antennae were used (2 7-m ACA antennae were present in the dataset but
flagged) in the array with a resulting synthesized beam size of
0.9$\times$0.7$\arcsec$ (PA=84°). The source was observed with a total
integration time of 84 minutes. The ALMA correlator was configured to
simultaneously observe four spectral windows (two in each sideband). One of
those windows (SpwID #1) covered both 13CO and C18O $2-1$ lines (220.399 and
219.560 GHz, respectively). The data were calibrated in the CASA software
package (v3.4), following the detailed calibration and imaging scripts
provided by the ALMA science verification team. Since those scripts, along
with fully calibrated measurement sets, are publicly available
online111https://almascience.nrao.edu/almadata/sciver/HD163296Band6, we do not
repeat the details here. The 218 GHz (1.37mm) continuum was generated from the
line-free channels in the two spectral windows (SpWID #0 and #1) of the lower
sideband, and the flux is determined to be 608.5$\pm$2.5 mJy, which agrees
with the SMA observations (Qi et al., 2011). Excellent agreement is also seen
between the SMA and ALMA spectra for the 13CO and C18O $2-1$ lines with fluxes
within 10% (Qi et al., 2011).
All of the $c$-C3H2 lines are continuum-subtracted and retrieved from SpwID #0
with a channel width of 0.488 MHz (0.67 km s-1) covering the total 1.875 GHz
bandwidth from 216.167 to 218.042 GHz. We have binned the lines in channels of
1 km s-1 to achieve higher signal-to-noise ratios and still keep the key
kinematic features of the disk. The resulting line rms noise level is
estimated to be 2.2 mJy beam-1 per 1 km s-1 channel in line free channels.
## 3 Results and Analysis
### 3.1 $c$-C3H2 detection
Figure 1 shows that the $c$-C3H2 $6-5$ line is clearly detected towards HD
163296 and that the position angle of its resolved velocity field is
consistent with that of 13CO from the same data set, which shows a clear
pattern of Keplerian rotation. Interestingly, the $c$-C3H2 $6-5$ moment map
displays a double-peaked morphology, indicative of an inner hole. The $c$-C3H2
$3-2$ and $5-4$ lines are only tentatively detected, insufficient to resolve
the velocity field or any spatial structure. To improve the sensitivity in
imaging these lines, we applied a Gaussian taper to the visibility data that
corresponds to 1″ FWHM in the image domain, which resulted in a beam of
1.3$\times$1.2$\arcsec$ (PA=78°).
Because the line emission changes position with channels, using a single mask
for all channels to extract spectra will lead to a degradation of the signal.
Instead, we create separate masks to cover the $c$-C3H2 $6-5$ line emission at
each channel and use these masks to extract the line spectra. The same masks
are also used for other $c$-C3H2 lines. Figure 2 shows the resulting spectra
of all three $c$-C3H2 $3-2$ lines together with 13CO $2-1$ line (with its own
masks covering the 13CO $2-1$ emission at each channel). The $6-5$ line is
clearly detected with the expected line shape and Vlsr consistent with 13CO.
The $5-4$ line presents a consistent spectral profile, at lower signal-to-
noise ratio. In contrast, the $3-2$ line is both narrower and clearly blue-
shifted, suggestive of a different physical origin, perhaps in a disk wind.
However, both the $3-2$ and $5-4$ lines are observed at the edge of the
spectral window, so this apparent asymmetry should be confirmed by independent
observations before any strong conclusions are drawn.
The integrated fluxes are reported in Table 1. These are just below the upper
limits reported in two previous failed searches of $c$-C3H2 in disks (Fuente
et al., 2010; Öberg et al., 2010).
### 3.2 Ring structure model
We explore the distributions of $c$-C3H2 in HD 163296 based on a previously
developed accretion disk model with well-defined temperature and density
structures (Qi et al., 2011). To simulate and compare with the data, we assume
the disk material orbits the central star in Keplerian motion, and fix the
disk geometric and kinematic parameters that affect the observed spatio-
kinematic behavior of the disk. The details of the stellar and accretion
properties and the disk geometric and kinematic parameters are summarized in
Table 3 of Qi et al. (2011). We have developed a physically self-consistent
accretion disk model with an exponentially tapered edge that matches the
spectral energy distribution and spatially resolved millimeter dust continuum
emission. The disk temperature and density structures are further constrained
by multiple CO and CO isotopologue line observations. Such analysis provides
the essential framework for constraining the distribution of molecular
abundances in protoplanetary disks.
Within this model framework, we assume that $c$-C3H2 is present with a
constant abundance in a layer with boundaries toward the midplane and toward
the surface of the disk (Qi et al., 2008; Öberg et al., 2012). This assumption
is motivated by chemical models (e.g. Aikawa & Nomura, 2006) that predict a
three-layered structure where most molecules are photodissociated in the
surface layer, frozen out in the midplane, and have an abundance that peaks at
intermediate disk heights. The surface ($\sigma_{s}$) and midplane
($\sigma_{m}$) boundaries are presented in terms of
$\Sigma_{21}=\Sigma_{H}/(1.59\times 10^{21}cm^{-2})$, where $\Sigma_{H}$ is
the hydrogen column density measured from the disk surface in the adopted
physical model. This simple model approach approximates the vertical location
where $c$-C3H2 is most abundant. Due to the limited signal-to-noise of the
multiple $c$-C3H2 line detections, we cannot constrain the location of this
vertical layer based on the $c$-C3H2 data. We therefore fix the vertical
surface boundary $\sigma_{s}$ as 0.32 ($\log_{10}(\sigma_{s})=-0.5$) and the
midplane boundary $\sigma_{m}$ as 3.2 ($\log_{10}(\sigma_{m})=0.5$), between
which lies the expected location of most molecules with peak abundances in the
warm molecular layer (Aikawa & Nomura, 2006).
We model the radial column density distribution of $c$-C3H2 as a power law
N${}_{100}\times(r/100)^{p}$, where N100 is the column density at 100 AU in
cm-2, $r$ is the distance from the star in AU, and $p$ is the power-law index.
Since the emission shows a clear indication of a central hole, we fit for both
an inner radius Rin and outer radius Rout together with the power-law
parameters (N100 and $p$). Using the structure model and the fixed vertical
distributions, we compute a grid of synthetic $c$-C3H2 J=$6-5$ visibility
datasets over a range of Rout, Rin, $p$ and N100 values and compare with the
observations. The best-fit model is obtained by minimizing $\chi^{2}$, the
weighted difference between the real and imaginary part of the complex
visibility measured in the ($u,v$)-plane sampled by the ALMA observations. We
use the two-dimensional Monte Carlo model RATRAN (Hogerheijde & van der Tak,
2000) to calculate the radiative transfer and molecular excitation. The
collisional rates are taken from Chandra & Kegel (2000) and the molecular data
file is retrieved from the Leiden Atomic and Molecular Database (Schöier et
al., 2005).
Figure 3 shows the $\chi^{2}$ surfaces for the Rin and $R_{out}$ versus the
power law index $p$. We find the $p$ is constrained between -2.5 and -1.5, Rin
15–40 AU and Rout 150–200 AU (within 1$\sigma$) with best-fit values at -2, 30
and 165 AU, respectively (reduced $\chi^{2}=1.07$). By fixing the above
parameters at their best-fit values, N($c$-C3H2) is determined to be
2.2$\pm$0.2 $\times$1012 cm-2 at 100 AU. Figure 4 presents comparisons between
the observed channel maps and the best-fit model. The residual image doesn’t
show any significant emission, which indicates that the $c$-C3H2 emission is
consistent with that arising from a Keplerian rotating disk. The $6-5$ line is
actually blended with the $6_{1,6}-5_{0,5}$ (ortho) and $6_{0,6}-5_{1,5}$
(para) lines at the same frequency. Since the Einstein $A$-coefficients for
both ortho and para lines are the same, 5.393$\times$10-4 s-1 (Schöier et al.,
2005), we fit the data with the ortho line only and the resulting column
density can be accounted as the sum of both ortho and para forms of the
molecule.
For the weaker $5-4$ ortho line, the signal-to-noise ratio is too low to
provide any constraints on the $c$-C3H2 the distribution. By assuming the
$c$-C3H2 distribution derived from the $6-5$ line, however, we can use the
unresolved line flux to constrain the $o/p$ ratio. The resulting best-fit
N(C3H2-($o$))=1.6$\pm$0.3 $\times$1012 and N(C3H2-($o+p$)=2.2$\pm$0.2
$\times$1012 cm-2 at 100 AU (as shown above). Hence $o/p$=2.7$\pm$1.7, which
is close to the statistical value of 3, but the large uncertainty implies that
better data are required to use this ratio as a probe of formation conditions.
## 4 Discussion
We have clearly detected $c$-C3H2 in the disk around HD 163296. This is one of
the two largest molecules detected in disks so far, the other being HC3N, also
a carbon-chain molecule, detected recently using deep single-dish observations
(Chapillon et al., 2012).
Using the spatially and spectrally resolved line emission, we have constrained
the radial distribution of $c$-C3H2 in the HD 163296 disk as a ring structure
ranging from $\sim$30 to $\sim$165 AU. The outer edge of $<$250 AU (2$\sigma$
limit) is much smaller than the gas disk size which extends to 500 AU (Qi et
al., 2011), which suggests that $c$-C3H2 formation is slow or that $c$-C3H2
destruction is fast in the outer disk. In terms of formation, the best-fit
outer radius coincides with the previously determined location of the CO
midplane snow line, and it is possible that CO freeze-out limits the carbon
available in the gas phase to form hydrocarbons in general and $c$-C3H2 in
particular. A second characteristic of the outer disk is that it is rather
tenuous because of a rapidly decreasing column density outside of 200 AU (Qi
et al., 2011). This may result in efficient penetration of radiation and thus
destruction of the easily dissociated $c$-C3H2. Explicit model predictions are
required to resolve which effect drives the disappearance of $c$-C3H2 in the
outer disk.
No existing disk chemistry model in the literature contains predictions for
the radial distribution of $c$-C3H2. Based on strong correlations found
between $c$-C3H2 and CCH in several astrophysical environments (Lucas & Liszt,
2000; Pety et al., 2005; Gerin et al., 2011), predictions on CCH in disks can
be used to shed further light on the origins of $c$-C3H2. Aikawa & Herbst
(2001) predicts a CCH central cavity in the absence of X-ray radiation; when
X-rays are present, and HD 163296 is a known X-ray emitter (Swartz et al.,
2005; Günther & Schmitt, 2009), the CCH column is instead centrally peaked.
Despite the presence of X-rays, the central cavity could be due to a UV
dominated radiation field. This may be tested by observing $c$-C3H2 towards T
Tauri stars with weak UV fields and strong X-rays, and towards additional
Herbig Ae stars with weak X-rays.
The inner cavity could also be a product of the main formation pathway of
$c$-C3H2. Carbon chains can form through at least three different pathways:
(1) ion-neutral reactions at low temperatures, (2) photo-erosion of larger
carbonaceous compounds, and (3) neutral-neutral reactions at luke-warm
temperatures following CH4 ice evaporation at $\sim$30 K (Herbst et al., 1984;
Teyssier et al., 2004; Sakai et al., 2008; Gerin et al., 2011). Detailed
modeling of $c$-C3H2 radial distributions in each of these formation scenarios
are clearly needed to use $c$-C3H2 rings as tracers of the radiation field and
other disk characteristics.
In addition to more detailed modeling, additional observations are needed to
constrain whether different hydrocarbons, especially CCH and $c$-C3H2, are
correlated in disk environments, and to constrain the vertical distribution of
these molecules. The latter requires multiple lines with a range of excitation
energies. With the combined advance in theory and observations, $c$-C3H2 may
become one of the more useful probes of penetration of the radiation fields in
disks.
Facility: ALMA
This paper makes use of the following ALMA data: ADS/JAO.ALMA#2011.0.00010.SV.
ALMA is a partnership of ESO (representing its member states), NSF (USA) and
NINS (Japan), together with NRC (Canada) and NSC and ASIAA (Taiwan), in
cooperation with the Republic of Chile. The Joint ALMA Observatory is operated
by ESO, AUI/NRAO and NAOJ.
We would like to thank an anonymous referee for thoughtful suggestions for the
paper. We also acknowledge NASA Origins of Solar Systems grant No. NNX11AK63.
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Table 1: $c$-C3H2 line results. Transition | Frequency (GHz) | Eu (K) | Beam | $\int Fdv$ (mJy km s-1)
---|---|---|---|---
$3_{3,0}-2_{2,1}$ | 216.279 | 19 | $1\farcs 3\times 1\farcs 2$(76°) | 53[9]
$6_{1,6}-5_{0,5}/6_{0,6}-5_{1,5}$ | 217.822 | 39 | $0\farcs 9\times 0\farcs 7$(83°) | 185[10]
$5_{1,4}-4_{2,3}$ | 217.940 | 35 | $1\farcs 3\times 1\farcs 2$(78°) | 74[9]
Figure 1: The integrated intensity maps summed between 0 and 11 km s-1) and
intensity-weighted mean velocity fields of 13CO $2-1$ and $c$-C3H2 $6-5$ lines
(left panel), $c$-C3H2 $5-4$ and $3-2$ lines (right panel) toward HD 163296.
The resolved velocity field of the $c$-C3H2 $6-5$ line agrees with the CO
kinematics. In the $c$-C3H2 maps, the first contour marks 3$\sigma$ followed
by 1$\sigma$ contour increases. The rms varies between 6 and 9 mJy km s-1 per
beam. Synthesized beams are presented in the lower left corners. The star
symbol indicates the continuum (stellar) position. The axes are offsets from
the pointing center in arcseconds. Figure 2: Extracted spectra of the three
$c$-C3H2 transitions toward HD 163296, plotted together with 13CO $2-1$. The
dashed line marks VLSR toward HD 163296. Figure 3: Iso-$\chi^{2}$ surfaces of
$R_{\rm out}$ and $R_{\rm in}$ versus $p$. Contours correspond to the 1–5
$\sigma$ errors. Figure 4: The top panel shows the velocity channel map of
the $c$-C3H2 $6-5$ emission toward HD 163296 (velocities binned in 1 km s-1).
Contours are 0.0022 Jy Beam-1 (1$\sigma$) $\times[3,4,5,6,7,8,9]$. The middle
and bottom panels show the best-fit model and the difference between the best-
fit model and data on the same contour scale. The star symbol indicates the
continuum (stellar) position. The axes are offsets from the pointing center in
arcseconds.
|
# Sphere Packing Densities of Certain Families of Elliptic Curves
Arjun Nigam
###### Abstract.
In this paper, we examine the Mordell-Weil lattices of two families of
elliptic curves over fields of characteristic $p>0$. We compute explicit lower
bounds on the densest sphere packings of Mordell-Weil lattices using geometric
information about the curve such as its Néron–Tate height and rational points
on the curve. We see how these methods may be generalized to other “nice"
families of elliptic curves and how encode interesting properties about the
elliptic curves themselves.
## 1\. Introduction
A sphere packing in $\mathbb{R}^{n}$ is an arrangement of non-overlapping
spheres of the same radius. The densest sphere packing in $n$ dimensions is a
sphere packing in $\mathbb{R}^{n}$ such that the fraction of the volume
occupied by the spheres is maximal. Lagrange proved in $1773$ that the densest
sphere packing in two dimensions is the hexagonal packing. More than 200 years
later, Thomas Hales proved that the cubic close packing and hexagonal close
packing arrangements achieve the densest sphere packing in three dimensions.
Later, Maryna Viazovska resolved the problem of finding the densest sphere
packing for dimension $8$ in [Via17]. Together with Henry Cohn, Abhinav Kumar,
Stephen Miller, and Danylo Radchenko, she later applied her method in [Via+17]
to resolve the problem in dimension $24$ as well.
In this paper, we find lower bounds on the sphere packing densities associated
to Mordell-Weil lattices of two families of elliptic curves. After introducing
the relevant definitions and general results of sphere packings, we relate the
density of the Mordell-Weil lattice of an elliptic curve to its Néron–Tate
height function and regulator in section 3.
In section 4, we focus on a specific family of elliptic curves and obtain
lower bounds on their densities in two different ways. These bounds are
obtained using subtle properties of the given family of elliptic curves. We
compare our results to the best known sphere densities in [Coh17]. We also see
how the information about the two lower bounds can be combined to obtain
results about the original elliptic curve, namely a lower bound on the order
of its Tate–Shafarevich group.
Next, we focus on a different family of elliptic curves, one where we have
more limited information. While we are unable to obtain a lower bounds on the
densities of the Mordell-Weil lattices, we are able to find lower bounds on
the densities of certain sublattices. We see that while this method requires
less information about the elliptic curves than the method employed in the
previous section, the resulting lower bounds on densities turn out to be much
smaller than the best known sphere packing densities in the corresponding
dimensions.
In section 6, we explore how the ideas of the previous two sections may be
applied to other families of elliptic curves. It is often the case that subtle
properties of specific families of elliptic curves allow us to deduce results
that do not hold for elliptic curves in general. However, there are many
examples in the literature where the same result can be proven for different
families of elliptic curves using different arguments. We state a few examples
relevant to this paper and describe how the methods of this paper may be
modified to study the sphere packing densities associated to other families of
elliptic curves.
## 2\. Preliminaries
We now make some basic definitions and introduce notation that we will use
throughout the paper.
###### Definition 2.1.
Let $\Lambda$ be a free $\mathbb{Z}$-module in $\mathbb{R}^{n}$ of maximal
rank. A sphere packing on $\Lambda$ is a non-overlapping arrangement of
spheres of the same radius in $\mathbb{R}^{n}$ such that each sphere is
centered at a point of $\Lambda$. The densest sphere packing of $\Lambda$ is a
sphere packing on $\Lambda$ of maximal density. The packing radius $\rho$ of
$\Lambda$ is the radius of the largest open ball in $\mathbb{R}^{n}$ whose
translates by elements of $\Lambda$ do not overlap.
Let $E$ be an Elliptic curve over a field $k$ and let $E(k)$ denote the set of
$k$-rational points of $E$. It is a well-known fact that there is a binary
operation on $E(k)$ which makes it into an abelian group. Moreover, if $k$ is
a number field or a function field over a finite field, then $E(k)$ is a
finitely generated abelian group. Thus, we get that
$E(k)_{\text{free}}\mathrel{\mathop{\mathchar
58\relax}}=E(k)/E(k)_{\text{torsion}}$ is a free abelian group of finite rank.
There is also a positive-definite quadratic form
$\hat{h}\mathrel{\mathop{\mathchar
58\relax}}E(k)_{\text{free}}\longrightarrow\mathbb{R}$ called the Néron–Tate
height. The Néron–Tate height has an associated positive-definite symmetric
bilinear form $\langle-,-\rangle\mathrel{\mathop{\mathchar
58\relax}}E(k)_{\text{free}}\times
E(k)_{\text{free}}\longrightarrow\mathbb{R}$ given by
$\langle P,Q\rangle=\frac{1}{2}(\hat{h}(P+Q)-\hat{h}(P)-\hat{h}(Q)).$
###### Definition 2.2.
A lattice is a finitely generated free abelian group equipped with a positive-
definite symmetric bilinear form. The Mordell-Weil lattice of $k$-rational
points of an elliptic curve $E$ is the group $E(k)_{\text{free}}$ equipped
with the bilinear form described above. The sphere packing density associated
to an elliptic curve $E$ is the density of the densest sphere packing on its
Mordell-Weil lattice.
If $(\Lambda,\langle-,-\rangle)$ is a lattice where $\Lambda$ is of rank $n$,
then we can find a group homomorphism $\Phi\mathrel{\mathop{\mathchar
58\relax}}\Lambda\longrightarrow\mathbb{R}^{n}$ such that
$(\Phi(P))\cdot(\Phi(Q))=\langle P,Q\rangle$ for all $P$ and $Q$ in $\Lambda$.
$\Phi$ can be constructed using the Gram-Schmidt process. Thus, instead of
studying the lattice $(\Lambda,\langle-,-\rangle)$, we can study its image in
$\mathbb{R}^{n}$ equipped with the usual dot product. Note that $\Phi$ is not
unique. In fact, the images of $\Lambda$ under different $\Phi$ may be
different. However, they will be isomorphic as lattices. Thus, we can define
the $k$-density of the densest sphere packing associated to an elliptic curve
$E$ as the density of the densest sphere packing of an embedding of the
lattice $E(k)_{\text{free}}$ into $\mathbb{R}^{n}$.
## 3\. General Results
###### Proposition 3.1.
Let $\rho$ be packing radius of a maximal rank abelian subgroup
$\Lambda\subseteq\mathbb{R}^{n}$ and let $V(\Lambda)$ be the volume of the
fundamental domain of $\Lambda$. Then the density of the densest sphere
packing of $\Lambda$ is given by
$\frac{\pi^{\frac{n}{2}}}{V(\Lambda)\cdot\Gamma(\frac{n}{2}+1)}\rho^{n}.$
###### Proof.
This follows from the formula for the volume of the $n$-ball of radius $\rho$
and the observation that the volume of the fundamental domain occupied by the
spheres to achieve the densest packing is exactly the volume of an $n$-sphere
of radius $\rho$. ∎
###### Definition 3.2.
Let $\Lambda$ be a maximal rank free subgroup of $\mathbb{R}^{n}$. Then the
normalized center density $\delta_{\Lambda}$ is given by
$\frac{\rho^{n}}{V(\Lambda)}$
where $\rho$ is the packing radius of $\Lambda$ and $V(\Lambda)$ is the volume
of the fundamental domain of $\Lambda$.
It is easy to see that for a fixed $n$, the ratio of the normalized sphere
density and the sphere density is constant for all lattices in
$\mathbb{R}^{n}$. Thus, having a large lower bound on the normalized sphere
packing density gives us a large lower bound on the sphere packing density.
###### Proposition 3.3.
[Elk94] Let $E$ be an elliptic curve over $K$ where $K$ is a number field or
function field over a finite field. For any subfield $k$ of $K$, the
normalized center density of $E(k)$ is given by
$\Delta^{-1/2}\left(\frac{N_{\text{min}}}{4}\right)^{n/2}$
where $n$ is the rank of $E(k)$, $N_{min}$ is the minimal non-zero value of
the Néron–Tate height function, and $\Delta$ is the absolute value of the
determinant of the pairing matrix of the generators of $E(k)_{\text{free}}$.
###### Proof.
Note that the volume of the fundamental domain of a lattice is the square root
of the absolute value of the determinant of the Gram matrix of a set of
generators of the lattice. To avoid overlaps in the Mordell-Weil lattice, the
packing radius $\rho$ must be half the minimal distance between any two
distinct points of the lattice. This minimal distance is given by
$\sqrt{N_{min}}$. ∎
## 4\. The Curve $E\mathrel{\mathop{\mathchar 58\relax}}y^{2}=x^{3}+t^{q}-t$
Let $p>3$ be a prime such that $p\equiv-1$ mod $6$, $q$ an odd power of $p$,
and $r$ a sufficiently large power of $p$ (the notion of “sufficiently large"
is made precise below). In this section, we consider the curve given by
$y^{2}=x^{3}+t^{q}-t$ over the field $k\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{F}_{r}(t)$.
###### Definition 4.1.
Fix an odd $p$-power $q=p^{c}$. We say that $r\mathrel{\mathop{\mathchar
58\relax}}=p^{s}$ is sufficiently large if $s$ is a multiple of $c$, $8$
divides $(p+1)s$, and $3(p^{c}-1)$ divides $p^{s}-1$. In particular,
$\mathbb{F}_{q}\subseteq\mathbb{F}_{r}$. If $r$ is sufficiently large, then so
are all powers of $r$.
###### Definition 4.2.
The naive height $h$ of a $\mathbb{F}_{r}(t)$-rational point $P=(x,y)$ on an
elliptic curve is $\text{deg}(x)\mathrel{\mathop{\mathchar
58\relax}}=\text{max}(\text{deg}(f),\text{deg}(g))$ where
$x=\frac{f(t)}{g(t)}$ for $f$ and $g$ coprime polynomials. If
$P=[0\mathrel{\mathop{\mathchar 58\relax}}1\mathrel{\mathop{\mathchar
58\relax}}0]$, then we define $h(P)=0$.
We wish to find a lower bound on $N_{\text{min}}$ for our elliptic curve. This
is the same as finding the minimum value of $\hat{h}$ on non-torsion points.
Since $\hat{h}$ has a bounded difference with the naive height, it is perhaps
useful to find a lower bound on the naive height.
###### Proposition 4.3.
The naive height on $E$ is bounded below by $\frac{q+1}{3}$.
###### Proof.
Let $\text{ord}_{\infty}$ denote the valuation at $\infty$. Let $P=(x,y)$ be a
point on $E$ that is not the identity. Then we can apply $\text{ord}_{\infty}$
to both sides of $y^{2}=x^{3}+t^{q}-t$ to get
$2\text{ord}_{\infty}(y)=\text{ord}_{\infty}(x^{3}+t^{q}-t).$
Note that $\text{ord}_{\infty}(t)=-1\neq-q=\text{ord}_{\infty}(t^{q})$. Thus,
we may use the strict triangle inequality to get that
$\text{ord}_{\infty}(t^{q}-t)=-q$. However, $q$ is not a multiple of $3$,
whereas $\text{ord}_{\infty}(x^{3})=3\text{ord}_{\infty}(x)$ is, so we must
have, using the strict triangle inequality again, that
$\text{ord}_{\infty}(x^{3}+t^{q}-t)=\text{min}(3\text{ord}_{\infty}(x),-q).$
We also know that
$2\text{ord}_{\infty}(y)=\text{ord}_{\infty}(y^{2})=\text{min}(3\text{ord}_{\infty}(x),-q)$.
Since $2$ does not divide $q$, we must have that $3\text{ord}_{\infty}(x)<-q$.
Thus, we get
$\text{ord}_{\infty}(x)\leq-\left(\frac{q+1}{3}\right).$
This shows that $x$ has a pole of order at least $\frac{q+1}{3}$ at $\infty$,
so we must have $h(P)=\text{deg}(x)\geq\frac{q+1}{3}$. ∎
Using this proposition, we can now prove that the naive height and the
Néron–Tate height are the same for our curve.
###### Proposition 4.4.
The Néron–Tate height and the naive height agree on $E$.
###### Proof.
Since the Neron-Tate height $\hat{h}$ is uniquely characterized by the fact
that its difference with the naive height $h$ is bounded and
$\hat{h}(2P)=4\hat{P}$ for all $\mathbb{F}_{r}(t)$-rational points $P$ on the
curve, it suffices to show that the naive height on $E$ satisfies
$h(2P)=4h(P)$
for all rational points points $P$ on the curve $E$. Clearly, this holds for
when $P$ is the identity. Let $P=(x,y)$ be a non-identity
$\mathbb{F}_{r}(t)$-rational point on the curve where $x=\frac{f(t)}{g(t)}$
where $f$ and $g$ are coprime polynomials. By the proof of Proposition 4.3, we
have
$\text{deg}(g)-\text{deg}(f)=\text{ord}_{\infty}(x)\leq-\left(\frac{q+1}{3}\right).$
This implies that $\text{deg}(f)\geq\text{deg}(g)+\frac{q+1}{3}$, so we
conclude that $\text{deg}(x)=\text{deg}(f)$ for all points $P=(x,y)$ with
$x=\frac{f}{g}$ where $f$ and $g$ are coprime polynomials.
Now, we compute $2P$ explicitly. Using basic arithmetic of elliptic curves, we
see that the first coordinate of the point $2P$ is
$\frac{f^{4}-8fg^{3}\cdot(t^{q}-t)}{4(gf^{3}+g^{4}\cdot(t^{q}-t))}.$
By our earlier observation on the height of $\mathbb{F}_{r}(t)$-rational
points of $E$, if there is no cancellation between the numerator and the
denominator, the naive height of $2P$ is
$\text{deg}(f^{4}-8fg^{3}\cdot(t^{q}-t))$. However, using Proposition 4.3, we
get
$\text{deg}(f^{4})=4\text{deg}(f)\geq\text{deg}(f)+3\text{deg}(g)+q+1>\text{deg}(8fg^{3}\cdot(t^{q}-t)).$
Thus, we get
$\text{deg}(f^{4}-8fg^{3}\cdot(t^{q}-t))=\text{deg}(f^{4})=4\text{deg}(f)=4h(P)$.
All we need to show now is that
$\frac{f^{4}-8fg^{3}\cdot(t^{q}-t)}{4(gf^{3}+g^{4}\cdot(t^{q}-t))}$
has no cancellation. Assume that $\tau$ is an irreducible (in
$\mathbb{F}_{r}[t]$) that divides both the numerator and denominator. Then
$\tau$ must also divide
$g(f^{4}-8fg^{3}\cdot(t^{q}-t))-\frac{f}{4}(4(gf^{3}+g^{4}\cdot(t^{q}-t)))=-9g^{4}f\cdot(t^{q}-t).$
Since $\tau$ is irreducible and $9\neq 0$, we must then have that $\tau$
either divides $g,f,$ or $t^{q}-t$.
Case 1: $\tau$ divides $g$
Since $\tau$ divides both $f^{4}-8fg^{3}\cdot(t^{q}-t)$ and $g$, we must have
that it divides $f$ as well. This is a contradiction as $f$ and $g$ were
assumed to be coprime polynomials.
Case 2: $\tau$ divides $f$
Since $\tau$ divides both $gf^{3}+g^{4}\cdot(t^{q}-t)$ and $f$, we must have
that it divides either $g$ or $t^{q}-t$. Since $g$ and $f$ are coprime
polynomials, this can only happen if $\tau$ divides $t^{q}-t$. However,
$\mathbb{F}_{q}\subseteq\mathbb{F}_{r}$ and the roots of $t^{q}-t$ are
precisely the elements of $\mathbb{F}_{q}$. Thus, the irreducible $\tau$ must
be of the form $t-\alpha$ for some $\alpha\in\mathbb{F}_{q}$. This implies
that $\alpha$ is a root of $f$. Since $x=\frac{f(t)}{g(t)}$ where $f$ and $g$
are coprime polynomials, we also get that $x$ has a root at $\alpha$. Note
that $t^{q}-t$ has a root at $\alpha$ of multiplicity one, so we can use the
strict triangle inequality to get
$2\text{ord}_{\alpha}(y)=\text{ord}_{\alpha}(y^{2})=\text{ord}_{\alpha}(x^{3}+t^{q}-t)=\text{min}(\text{ord}_{\alpha}(x^{3}),\text{ord}_{\alpha}(t^{q}-t))=1,$
which is a contradiction as $2$ does not divide $1$.
Case 3: $\tau$ divides $t^{q}-t$
Since $\tau$ divides both $f^{4}-8fg^{3}\cdot(t^{q}-t)$ and $t^{q}-t$, we must
have that it divides $f$ as well. This then leads to the same contradiction as
in Case 2.
This shows that there is no cancellation in
$\frac{f^{4}-8fg^{3}\cdot(t^{q}-t)}{4(gf^{3}+g^{4}\cdot(t^{q}-t))}$, so the
naive height satisfies $h(2P)=4h(P)$. ∎
###### Remark.
Using Proposition 4.4, we can give an elementary argument to show that
$E(\mathbb{F}_{r}(t))$ is torsion-free for $r$ sufficiently large. Let
$P=(x,y)$ be a non-trivial $\mathbb{F}_{r}(t)$-rational point on $E$ that is
torsion. By Proposition 4.4, we know that $0=\hat{h}(P)=\text{deg}(x)$. This
is only possible if $x=c$ for some $c\in\mathbb{F}_{r}$. This would imply that
$y^{2}=t^{q}-t+c^{3}$. This means that $y$ is a polynomial in $t$. Since $q$
is an odd integer, $t^{q}-t+c^{3}$ cannot be the square of a polynomial. This
is a contradiction.
###### Proposition 4.5 (Proposition 8.4.1(3) in [UG20]).
For $r$ sufficiently large, rank$(E(\mathbb{F}_{r}(t)))=2(q-1)$.
###### Proposition 4.6 (Corollary 9.2(3) in [UG20]).
$\Delta$ is bounded above by $r^{\lfloor\frac{q}{6}\rfloor}$ for $r$
sufficiently large.
###### Proposition 4.7.
For $r$ sufficiently large, the normalized center density
$\delta_{E(\mathbb{F}_{r}(t))}$ is bounded below by
$\frac{\sqrt{|\Sha(E/\mathbb{F}_{r}(t))|}}{(r^{\lfloor\frac{q}{6}\rfloor})^{1/2}}\cdot\left(\frac{q+1}{12}\right)^{q-1}.$
###### Proof.
This follows by combining the results of Proposition 3.3, Proposition 4.3,
Proposition 4.4, Proposition 4.5, and Proposition 4.6. ∎
Using the trivial bound on the Tate–Shafarevich group, we can now use
Proposition 4.7 to get explicit lower bounds on the densities of our family of
curves. However, as we will see, this lower bound can often be improved by
considering sublattices of the Mordell-Weil lattice. Indeed, let $\Lambda$ be
a maximal rank sublattice of the Mordell-Weil lattice of $E$. We may then
replace $\Delta$ by $V(\Lambda)$ in the in Proposition 3.3 to find a lower
bound on the normalized packing density of E. With this as the motivation, we
list some explicit rational points on the curve and consider the sublattice
that they generate.
###### Proposition 4.8.
Let $\sigma$ be a solution to $\sigma^{6(q-1)}=-1$ and $\beta$ a solution to
$\beta^{q}+\beta=1$ in $\mathbb{F}_{r}$ for $r$ sufficiently large. Then
$P=(\sigma^{2}(t-(\beta/\sigma^{6}))^{\frac{q+1}{3}},\sigma^{3}(t-(\beta/\sigma^{6})^{q})^{\frac{q+1}{2}})$
is a point in $E(K)$. Thus, we get $6q(q-1)$ $\mathbb{F}_{r}(t)$-rational
points on $E$.
###### Proof.
This can be verified by plugging in the point $P$ into the defining equation
of our elliptic curve and using the relations on $\sigma$ and $\beta$.
Alternatively, these points can be deduced by considering the map
$C_{6,q}\longrightarrow E_{0}$ which we get by presenting $C_{6,q}$ and
$E_{0}\mathrel{\mathop{\mathchar 58\relax}}y^{2}=x^{3}+1$ as quotients of the
Fermat curve of degree $q+1$. ∎
Using this MAGMA script, we find the volume of (an Euclidean embedding of) the
sublattice generated by the points in Proposition 4.8 which we then use to
find lower bounds on the sphere packing density of the Mordell-Weil lattice
for different values of $p,q,$ and $r$. We compare it with the lower bounds
obtained using Proposition 4.7 and the trivial bound
$|\Sha(E/\mathbb{F}_{r}(t))|\geq 1$ in the table below.
$q$ | $r$ | Lower Bound on Normalized Density (Using MAGMA) | Lower Bound on Normalized Density (Using Proposition 4.7 and the trivial bound $|\Sha(E/\mathbb{F}_{r}(t))|\geq 1$) | Dimension | Best Known (Normalized) Sphere Packing Density (If Known) [Coh17]
---|---|---|---|---|---
$5$ | $5^{4}$ | $0.0625$ | $0.0625$ | $8$ | $0.0625$
$5$ | $5^{8}$ | $0.0625$ | $0.0625$ | $8$ | $0.0625$
$5$ | $5^{12}$ | $0.0625$ | $0.0625$ | $8$ | $0.0625$
$5^{3}$ | $5^{16}$ | $\sim 2.653\times 10^{20}$ | $\sim 6.198\times 10^{12}$ | $248$ |
$11$ | $11^{2}$ | $\sim 0.0909$ | $\sim 0.0909$ | $20$ | $\sim 0.1315$
$11$ | $11^{6}$ | $\sim 0.0909$ | $\sim 0.00075$ | $20$ | $\sim 0.1315$
$17$ | $17^{4}$ | $\sim 2.272$ | $\sim 0.0078$ | $32$ | $\sim 2.565$
We see that there is repetition in the lower bound of the normalized density
(computed using MAGMA) when $q$ is fixed and $r$ varies. This is to be
expected as the explicit points in Proposition 4.8 and the explicit
formulation of the Néron–Tate height do not depend on $r$ (assuming that $r$
is sufficiently large). Thus, they generate the same lattice. The same
repetition is not observed in the lower bound obtained using Proposition 4.7.
This is likely due to the fact that we used the trivial bound on
$|\Sha(E(\mathbb{F}_{r}(t)))|$ when computing it and
$\Sha(E(\mathbb{F}_{r}(t)))$ may depend on $r$.
Up to this point, we have been using properties of the given elliptic curves
to deduce information about the densities of their Mordell-Weil lattices. Now,
we can use the two lower bounds to say something interesting about the
elliptic curve $E$. The lower bound obtained using MAGMA divided by the lower
bound obtained using Proposition 4.7 is a lower bound on
$\sqrt{\Sha(E/\mathbb{F}_{r}(t))}$. This follows from how we computed these
lower bounds. In particular, we see that for the cases considered above,
$E(E/\mathbb{F}_{r}(t))$ is non-trivial when $q=5^{3}$ and $r=5^{16}$ or when
$q=17$ and $r=17^{4}$.
###### Remark.
$E_{8}$ is the unique lattice (up to isometry and rescaling) to have the
highest density sphere packing in dimension $8$. For $q=5$ and $r=5^{4}$, we
observe that the lower bound obtained equals the sphere packing density of
$E_{8}$. This means that the Mordell-Weil lattice must have sphere packing
density equal to $E_{8}$ which, by uniqueness, implies that the Mordell-Weil
lattice is $E_{8}$. This also implies that $E(E/\mathbb{F}_{r}(t))$ is trivial
in this case.
## 5\. The Legendre Curve
Let $p$ be an odd prime and $d=p^{f}+1$ for some positive integer $f$.
Consider the function field $K_{d}\mathrel{\mathop{\mathchar
58\relax}}=\mathbb{F}_{p}(\mu_{d},u)$ where $\mu_{d}$ is the set of primitive
$d^{\text{th}}$ roots of unity and $u^{d}-t=0$ for indeterminate $t$. We study
the elliptic curve
$E\mathrel{\mathop{\mathchar 58\relax}}y^{2}=x(x+1)(x+t).$
As with the previous curve, we would like to have explicit points on this
curve.
###### Proposition 5.1.
Let $E$ and $K_{d}$ be as defined. Then for a fixed primitive $d^{\text{th}}$
root of unity $\zeta$ and $i\in\\{0,1,\ldots,d-1\\}$, the point given by
$P_{i}^{(d)}\mathrel{\mathop{\mathchar
58\relax}}=(\zeta^{i}_{d}u,\zeta^{i}_{d}u(\zeta^{i}_{d}u+1)^{d/2})$ is a
$K_{d}-$rational point of $E$.
###### Proof.
This can be checked by plugging in each $P^{(d)}_{i}$ into the equation
defining $E$ or by observing the action of $Gal(K_{d}/K)$ on the $K_{d}-$
rational point $P\mathrel{\mathop{\mathchar 58\relax}}=(u,u(u+1)^{d/2})$. ∎
###### Proposition 5.2.
The set of explicit rational points in Proposition 5.1 generate a rank $d-2$
subgroup of the Mordell-Weil group. $E(K_{d})$ also has rank $d-2$.
###### Proof.
The first part of this proposition follows from Corollary 4.3 in [Ulm14] and
the second part follows from Corollary 5.3 of the same paper. ∎
Thus, we get that the subgroup generated by the $P_{i}^{(d)}$ is of finite
index in $E(K_{d})$. Just like in the previous section, it is difficult to
find a set of generators for the free part of the Mordell-Weil group. However,
instead of finding a lower bound on the normalized sphere packing density of
$E$, we can find a lower bound on the normalized sphere packing density of the
finite index sublattice generated by the $P_{i}^{(d)}$.
###### Proposition 5.3.
The height pairing of the points $P^{(d)}_{i}$ is given by
$\langle
P^{(d)}_{i},P^{(d)}_{j}\rangle=\begin{dcases}\frac{(d-1)(d-2)}{2d}&\text{if
}i=j\\\ \frac{1-d}{d}&\text{if }i\neq j\text{ and }i-j\text{ is even}\\\
0&\text{if }i-j\text{ is odd}\\\ \end{dcases}$
###### Proof.
This is Theorem 8.2 in [Ulm14] ∎
###### Proposition 5.4.
For any non-torsion point $Q$ in the subgroup generated by the $P_{i}^{(d)}$,
we have the inequality
$\frac{d-1}{2d}\leq\hat{h}(Q).$
###### Proof.
Let $Q=\sum_{0}^{d-1}a_{i}P_{i}^{(d)}$ be a point in the subgroup generated by
the $P_{i}^{(d)}$. Then we can use bilinearity of the height pairing and the
previous proposition to get
$\hat{h}(Q)=\left\langle\sum_{0}^{d-1}a_{i}P_{i}^{(d)},\sum_{0}^{d-1}a_{i}P_{i}^{(d)}\right\rangle=\frac{(d-1)(d-2)(a_{0}^{2}+\cdots+a_{d-1}^{2})+(1-d)(\sum_{i-j\text{
is even}}2a_{i}a_{j})}{2d}.$
Thus, we may write $\hat{h}(Q)$ as $\frac{d-1}{2d}\cdot m$ where $m$ is some
integer. Since the height function is positive definite, we must have that
$m\geq 1$. This gives us the required lower bound. ∎
Using these results, we can now compute explicit lower bounds on the
normalized sphere packing density of a sublattice of $E$ using this MAGMA
script.
$p$ | $f$ | Dimension | Normalized Density (Lower Bound) | Best Known (Normalized) Sphere Packing Density [Coh17]
---|---|---|---|---
$3$ | $1$ | $2$ | $\sim 0.125$ | $\sim 0.288$
$3$ | $2$ | $8$ | $\sim 1.953\times 10^{-6}$ | $\sim 0.0625$
$3$ | $3$ | $26$ | $\sim 3.208\times 10^{-26}$ | $\sim 0.577$
$5$ | $1$ | $4$ | $\sim 0.005$ | $\sim 0.125$
$5$ | $2$ | $24$ | $\sim 8.119\times 10^{-24}$ | $\sim 1.003$
$7$ | $1$ | $6$ | $\sim 1.22\times 10^{-4}$ | $\sim 0.0721$
We observe that as $p$ and $f$ get larger, the normalized density gets
smaller. This is because the Gram matrix obtained from the height pairing has
a large determinant for large values of $p$ and $f$.
Note that, unlike the previous section, the computational lower bounds that we
obtained here are not necessarily lower bounds on the density of $E$. While we
were able to obtain a lower bound on the minimum non-zero value of the
Néron–Tate height function on the sublattice generated by the $P_{i}^{(d)}$,
we do not know if this value is also the minimum on the entire Mordell-Weil
lattice.
## 6\. Generalizations
We now discuss how the methods used in this paper may be generalized to other
“nice" families of elliptic curves. Our strategy in Section 4 is motivated by
[Elk94]. We first proved that the Néron–Tate height and the naive height agree
for our family of elliptic curves and then found a lower bound for the naive
height using elementary techniques. While the proofs of these results use
properties specific to the family of curves being studied, there are many
examples in the literature of elliptic curves where the naive and Néron–Tate
height agree (for example, the family of hyperelliptic curves considered in
[Elk94]).
Even if the naive and Néron–Tate heights do not agree for all points, we might
still be able to perform explicit computations on the lower bounds of lattice
densities. Let $E$ be an elliptic curve over the field $\mathbb{F}_{q}(t)$
where $q$ is a $p$-power for a prime $p>3$. We choose a Weierstrass model
$y^{2}=x^{3}+a_{4}x+a_{6}$
for $E$ where $a_{i}\in\mathbb{F}_{q}[t]$ are polynomials and
max$(\lceil\frac{\text{deg}(a_{i})}{i}\rceil)$ is minimal (doing so is
possible by lemma 5.43 of [SS19]). We define $d$ to be the smallest integer
such that $a_{i}\leq di$. We say that $E$ is a “nice" elliptic curve if
1. (1)
For all places $\nu$, the discriminant $\Delta$ has no roots, simple roots, or
double roots at $\nu$.
2. (2)
If $\Delta$ has double roots at $\nu$, then the reduction of $E$ at $\nu$ is
additive.
Assume that $E$ is a “nice" curve and let $P$ be a
$\mathbb{F}_{q}(t)$-rational point. We have $\langle P,P\rangle=2\chi+2(P\cdot
O)+\sum_{v}\text{contr}_{v}(P)$ where the sum is taken over all singular
reducible fibers and $P\cdot O$ is the intersection number (Theorem 6.24 in
[SS19]). Since we assumed that $E$ is “nice", table 15.1 of [Sil09] tells us
that all singular fibers are irreducible. Thus, we get $\langle
P,P\rangle=2\chi+2(P\cdot O)$. However, since $\chi=d$, all we need to do to
compute the height is to find the intersection number $P\cdot O$.
Let $(x_{0}(t),y_{0}(t))$ be the coordinate representation of the point $P$
where $x_{0}=\frac{f(t)}{g(t)}$ for polynomials $f$ and $g$. Since $a_{i}$ are
polynomials, they do not blow up when $t$ is finite. Thus, given a finite
place $t_{0}$, we have that $x_{0}(t_{0})$ blows up if and only if
$y_{0}(t_{0})$ blows up. This means us that the section determined by $P$
meets the section determined by $O$ at deg$(g)$-many times (counted with
multiplicity).
For $t=\infty$, we change coordinates of our curve by $x=t^{2d}x^{\prime}$ and
$y=t^{3d}y^{\prime}$. By our choice of $a_{i}$, we get that the coefficients
in our new model are regular functions at $t=\infty$. Thus, we get that the
section determined by $P$ intersects the section determined by $O$ at infinity
max$(0,\text{deg}(f)-\text{deg}(g)-2d)$-many times. If deg$(f)$ large enough,
we get that the section determined by $P$ meets the section determined by $O$
deg$(g)+\text{deg}(f)-\text{deg}(g)-2d)=\text{deg}(f)-2d$ many times.
Accounting for the degree $2$ morphism $E\longrightarrow\mathbb{P}^{1}$, we
get $P\cdot O=\frac{1}{2}\left(\text{deg}(f)-2d\right)$. This gives us
$\langle P,P\rangle=2d+2(P\cdot O)=\text{deg}(f)$.
We have now shown that for “nice" curves $E$ and points $P=(\frac{f}{g},y)$
with $\text{deg}(f)$ is large enough, the naive and Néron–Tate heights on $P$
agree. Thus, if we have a lower bound on the naive height for $E$ and a set of
points $P_{i}=(\frac{f_{i}}{g_{i}},y_{i})$ where deg$(f_{i})$ is large enough
for each $i$ and the $P_{i}$’s generate a maximal rank sublattice $L$, we can
compute lower bounds on the lattice density of $L$.
## 7\. Acknowledgements
I would like to thank Dr. Douglas Ulmer for suggesting the project and
mentoring me with it. Dr. Ulmer’s constant encouragement, expertise, and
patient guidance were invaluable in the writing of this paper. I would also
like to thank Dr. Bryden Cais, Dr. Steven J. Miller, Daniel Lewis, and Tristan
Phillips for their valuable feedback on the first draft of this paper.
## References
* [Coh17] H. Cohn “A conceptual breakthrough in sphere packing” In _Notices of the American Mathematical Society_ 64, 2, 2017
* [Elk94] N. Elkies “Mordell-Weil Lattices in Characteristic 2: I. Construction and First Properties” In _International Mathematics Research Notices_ 1994, Issue 8, 1994
* [Sil09] J. Silverman “The Arithmetic of Elliptic Curves” 106, Graduate Texts in Mathematics Springer NEW YORK, NY, 2009
* [SS19] M. Schütt and T. Shioda “Mordell–Weil Lattices” 70, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics Springer Singapore, 2019
* [UG20] D. Ulmer and R. Griffon “On the arithmetic of a family of twisted constant elliptic curves” In _Pacific Journal of Mathematics_ 305, 2020
* [Ulm14] D. Ulmer “Explicit points on the Legendre curve” In _Journal of Number Theory_ 136, 2014
* [Via+17] M. Viazovska et al. “The sphere packing problem in dimension 24” In _Annals of Mathematics_ 185, 2017
* [Via17] M. Viazovska “The sphere packing problem in dimension 8” In _Annals of Mathematics_ 185, 2017
|
(17)
Another advantage of using two pairs of exposures is that one can calculate
the null polarization (NULL1 and NULL2) as in equations 20 and 26 of Bagnulo
et al. (2009), which provides a way to quantify the amount of spurious
polarization. The null polarization for the Difference method is given by
$NULL_{X}=\frac{1}{4}\sum_{k=1}^{2}{\left[(-1)^{k-1}\left(\frac{r_{2k-1}-1}{r_{2k-1}+1}-\frac{r_{2k}-1}{r_{2k}+1}\right)\right]},$
(18)
and for the Ratio method the null polarization is given by
$NULL_{X}=\frac{\left(\prod_{k=1}^{2}{r_{2k-1}/r_{2k}}\right)^{\frac{(-1)^{k-1}}{4}}-1}{\left(\prod_{k=1}^{2}{r_{2k-1}/r_{2k}}\right)^{\frac{(-1)^{k-1}}{4}}+1}.$
(19)
Finally, the uncertainties of polarimetric measurements can be calculated from
the extracted fluxes and their uncertainties (denoted here by $\sigma$) by
equations A3 and A10 of Bagnulo et al. (2009). In the Difference method, the
variance for each spectral element is given by
$\sigma_{X}^{2}=\frac{1}{16}\sum_{i=1}^{4}{\left\\{\left[\frac{2f_{i\parallel}f_{i\perp}}{(f_{i\parallel}+f_{i\perp})^{2}}\right]^{2}\left[\frac{\sigma_{i\parallel}^{2}}{f_{i\parallel}^{2}}+\frac{\sigma_{i\perp}^{2}}{f_{i\perp}^{2}}\right]\right\\}},$
(20)
and in the Ratio method the variance is given in terms of the flux ratio
$r_{i}$ as defined in Equation 14, i.e.,
$\sigma_{X}^{2}=\frac{\left(\frac{r_{1}}{r_{2}}\frac{r_{4}}{r_{3}}\right)^{1/2}}{4\left[\left(\frac{r_{1}}{r_{2}}\frac{r_{4}}{r_{3}}\right)^{1/4}+1\right]^{4}}\sum_{i=1}^{4}{\left[\frac{\sigma_{i\parallel}^{2}}{f_{i\parallel}^{2}}+\frac{\sigma_{i\perp}^{2}}{f_{i\perp}^{2}}\right]}.$
(21)
Applying this formalism to SPIRou spectra, we obtain values that vary
continuously throughout the spectrum and are systematically above or below
zero for each spectrum, which we refer to here as the ‘continuum
polarization’. For general scientific applications with SPIRou, this continuum
polarization is actually spurious as it reflects small differences in the
injection between beams, and must therefore be fitted and removed. This step
is mandatory before performing measurements in spectral lines. apero applies
an iterative sigma-clip algorithm to fit either a polynomial or a spline to
model the continuum polarization.
### 10.2 Least-Squares Deconvolution
The least-squares deconvolution method (LSD) is an efficient technique that
combines the signal from thousands of spectral lines retaining the same line
profile information to obtain a mean velocity profile for the intensity,
polarization, and null spectra. A common application of this technique
concerns the measurement of the Zeeman split into Stokes V (circularly
polarized) profiles. The Zeeman split is a physical process where electronic
transitions occurring in the presence of a magnetic field have their main
energy transition level split into two additional levels, forming a double
line in the intensity spectrum. An interesting feature of these lines is that
they are circularly polarized and their polarizations have opposite signs.
Therefore, by observing the circularly polarized spectrum one can obtain a
characteristic Stokes V profile that provides a way to detect and characterize
the magnetism in stellar photospheres with great sensitivity.
apero implements the LSD calculations using the formalism introduced by Donati
et al. (1997), summarized as follows. Let us first consider the weight of a
given spectral line $i$, $w_{i}=g_{i}\lambda_{i}d_{i}$, where $g$ is the Landé
factor (magnetic sensitivity), $\lambda$ is the central wavelength, and $d$ is
the line depth. Then one can construct the line pattern function,
$M(v)=\sum_{i=1}^{N_{l}}{w\delta(v-v_{i})},$ (22)
where $N_{l}$ is the number of spectral lines considered in the analysis,
$\delta$ is the Dirac function, and $v$ is the velocity. The transformation
from wavelength ($\lambda$) to velocity space is performed by the relation
$dv/d\lambda=c/\lambda$, where $c$ is the speed of light. The LSD profile is
calculated by the following matrix equation:
$\rm{\bf Z}=\left(\rm{\bf M}^{t}.\rm{\bf S}^{2}.\rm{\bf M}\right)^{-1}\rm{\bf
M}^{t}.\rm{\bf S}^{2}.\rm{\bf P},$ (23)
where $\rm{\bf P}$ is the polarimetric spectrum calculated from Equation 16 or
17, and $\rm{\bf S}$ is the covariance matrix, a diagonal matrix where each
element in the diagonal is given by $S_{jj}=1/\sigma_{j}$, with $\sigma_{j}$
being the uncertainty in the polarimetric spectrum calculated from Equation 20
or 21.
Note that one can also calculate the null polarization LSD profile by
substituting the polarimetric spectrum $\rm{\bf P}$ by the null spectrum
$\rm{\bf N}$ in Equation 23. The intensity LSD is also possible, by using the
flux spectrum $\rm{\bf F}$, but in this case the line weight in Equation 22 is
simply given by the line depth, i.e, $w_{i}=d_{i}$.
In practice, LSD requires a few important steps to be executed by apero.
First, each individual spectrum is cleaned using a sigma-clip rejection
algorithm to minimize the impact of outliers in the LSD profile. Then we set a
grid of velocities to calculate the LSD profile, where the grid is defined by
the following parameters: an initial velocity, $v_{0}$, a final velocity,
$v_{f}$, and the total number of points in the grid, $N_{v}$. Next, a fast and
accurate method is necessary to project the spectral values onto the velocity
grid. Finally, an appropriate catalog of spectral lines (line mask) needs to
be adopted for the LSD calculations. apero selects the line mask from a
repository of masks, where the selection is based on the proximity to the
effective temperature of the star observed. The apero masks are computed using
the VALD catalog (Piskunov et al., 1995) and a MARCS model atmosphere
(Gustafsson et al., 2008) with an effective temperature ranging from 2500 to
5000 K in steps of 500 K, and the same surface gravity of $\log g=5.0$ dex.
The lines that are effectively used in the LSD analysis are selected with line
depths above a given threshold, which is set to 3% by default and with a Landé
factor of $g_{\rm eff}>0$, resulting in a total of approximately 2500 atomic
lines that cover the full spectral range of SPIRou. Figure 27 shows an example
of an LSD analysis performed on a 4-exposure Stokes-V sequence of the bright
Ap star Gamma Equulei, which has a strong magnetic field (e.g., Bychkov et
al., 2006) and therefore shows an obvious Zeeman feature in the SPIRou data.
Figure 27: LSD analysis performed on the polarimetric data reduced with APERO
and obtained from a 4-exposure Stokes-V sequence of the bright Ap star Gamma
Equulei. Panels from top to bottom show Stokes I, Stokes V, and null profiles.
In practice, the LSD analysis is not computed in a standard automated run of
apero but the module is supplied and can be activated with the use of a single
keyword in the apero profiles or run after processing.
Figure 28: Polarimetric sequence
## 11 Post processing
The final data products that go to PIs are composite files of many of the
outputs of apero. For SPIRou, these are sent to the Canadian Data Astronomy
Center (CADC)161616accessible from https://www.cadc-ccda.hia-iha.nrc-
cnrc.gc.ca/ but are only produced for science targets and hot stars (i.e.,
obj_fp, obj_dark, polar_fp, and polar_dark) and not for calibrations by
default. There are currently five post-processing files each linked to a
single odometer code. These are the 2D extracted output (e.fits, Section
11.1), the 2D telluric corrected output (t.fits, Section 11.3), the 1D output
(s.fits, Section 11.2), the velocity output (v.fits, Section 11.4), and the
polarimetric outputs (p.fits 11.5). A summary of the CADC output files is
available in table 4 and the post-process sequence is shown in Figure 29.
File | Description
---|---
(odometer)e.fits | 2D extracted spectrum for fibers AB, A, B, C, wavelength solution, and blaze
(odometer)s.fits | 1D extracted spectrum for fibers AB, A, B, C, and telluric corrected spectrum if available
(odometer)t.fits | 2D telluric corrected spectrum for fiber AB, A, B, wavelength solution, blaze, and reconstructed atmospheric transmission
(odometer)v.fits | combined and per order CCFs for fitting the radial velocity of the star
(odometer)p.fits | polarimetric products (Polarimetric flux, Stokes I, Null vectors, wavelength solution, and blaze)
Table 4: Science ready outputs sent to the Canadian Data Astronomy Center,
CADC).
### 11.1 2D extraction product (e.fits)
These are the combined extracted products. All extensions are two-dimensional
spectra of size $4088\times 49$. The ‘e.fits’ file contains the extracted
spectrum for each order for each fiber and the matching wavelength and blaze
solution for each order and each fiber. The files are identified with a single
odometer generated at the time of observation followed by an ‘e.fits’ suffix.
### 11.2 2D telluric corrected product (t.fits)
These are the combined telluric-corrected products. All extensions are two-
dimensional spectra of size $4088\times 49$. The ‘t.fits’ file contains the
telluric corrected spectrum for each order and each fiber and the matching
wavelength and blaze solution for each order and each fiber. The files are
identified with a single odometer code at the time of observation followed by
a ‘t.fits’ suffix.
### 11.3 1D extraction and 1D telluric corrected product (s.fits)
These are the combined 1D spectrum products and consist of two tables. The two
tables consist of the 1D spectrum in 1. velocity units and 2. wavelength
units. They each consist of the following columns: the wavelength solution,
the extracted flux in AB, A, B, and C, the telluric corrected flux in fibers
AB, A, and B (if available), and the associated uncertainties for each flux
column. The files are identified with a single odometer code at the time of
observation followed by an ‘s.fits’ suffix.
### 11.4 Velocity product (v.fits)
The velocity products are packaged into the ‘v.fits’ file. Currently, only the
CCF values (Section 9.1) are added as an extension as the LBL products are
computed separately. The CCF file consists of the CCF generated for each
radial velocity element (by default this is between $\pm$300 $\text{ms}^{-1}$
in steps of 0.5 $\text{ms}^{-1}$) for each order and a combined CCF for the
same radial velocity elements. The files are identified with a single odometer
code at the time of observation followed by a ‘v.fits’ suffix. Once the LBL
module is able to be used with apero it will add an extension to the ‘v.fits’
(the ‘rdb’ extension described in the LBL
documentation${}^{\ref{footnote:lbl_docs}}$).
Figure 29: Post-process sequence
### 11.5 Polarimetric product (p.fits)
These are the combined polarimetric products. The ‘p.fits‘ file consists of
eight image extensions and three table extensions. The first two tables are
the 1D representations of the 2D polarimetric products (listed in the
extensions above) in 1. velocity units and 2. wavelength units. They each
consist of the following columns: the wavelength solution, the polarimetric
flux, the Stokes I flux, the Null 1 and 2 fluxes, and the associated
uncertainties on each flux column. The third table lists the configuration
parameters used to run apero. Although polarimetric products are the
combination of at least 4 odometer codes, files are associated with a single
odometer code (the first in the sequence at the time of observation) followed
by a ‘p.fits’ suffix.
## 12 Discussion
apero has been an ongoing effort since its conception in October 2017 (see
Appendix D and table D1). With the first light of SPIRou in April 2018, it
took nearly 2 years for apero to start producing precise science results and
has been used in publications since 2020 (see Appendix E and table E1). Here
we discuss current performances and limitations and planned future work.
### 12.1 Performance and limitations
Throughout development, we have tried to optimize the speed of all recipes
(e.g., through the use of SQL databases, numba, bottleneck, etc.). With the
most current version (v0.7.256), using 35 CPU cores (on a single node), we can
reduce all available data (all SPIRou legacy survey data and all PI data to
which we have access, covering $\sim$430 nights and $\sim$45000 science
observations) in 7 days; table 5 shows a breakdown of the various steps using
a 35 core machine. These timings are for all data we have access to, which are
equivalent to $\sim$90% of all data taken with SPIRou between April 2018 and
June 2022.
Sequence | Recipes | Number | Time taken | Efficiency
---|---|---|---|---
Pre-processing | apero_preprocess | 75761 files | 34.5 hours | 0.89
| | | |
Reference Calibrations | apero_dark_ref, apero_badpix, apero_loc_spirou, apero_shape_ref, apero_shape, apero_flat, apero_thermal, apero_leak_ref, apero_wave_ref | 1 night | 1.5 hours | -
| | | |
Nightly Calibrations | apero_badpix, apero_loc_spirou, apero_shape, apero_flat, apero_thermal, apero_wave_night | 432 nights | 7.4 hours | 0.87
| | | |
Extraction | apero_extract | 46836 files | 63.3 hours | 0.63
| | | |
Telluric (hot star) | apero_mk_tellu, apero_mk_model, apero_fit_tellu, apero_mk_template | 1043 files | 1.1 hours | 0.70
| | | |
Telluric (science) | apero_fit_tellu, apero_mk_template | 45524 files | 34.6 hours | 0.75
| | | |
CCF | apero_ccf | 45524 files | 7.0 hours | 0.89
| | | |
Polarimetry | apero_polar | 9880 groups | 2.1 hours | 0.90
| | | |
Post-process | apero_processing | 48472 files | 18.4 hours | 0.93
Total | | | 170.1 hours | 0.76
| | | 7.1 days | 0.76
Table 5: Example full run reducing all SPIRou legacy survey (and some PI) data
from April 2018 to April 2022. This was processed on one machine using 35
cores. Note preprocessing is done on both science and calibration
observations, reference calibrations are only run on a single night, and each
nightly calibration depends on the availability of specific calibrations thus
leading to a range of nights (from 366 to 401). Polarimetric groups consist of
4 individual exposures in different rhomb positions and are only processed for
polar_fp and polar_dark files. The number of specific steps may depend on
previous steps (i.e., quality control failures, engineering nights that were
excluded, and odometer codes that were present in a list to not be processed).
The efficiency is defined in Equation 24.
We find that with 35 CPU cores we reach a point where we start to see input-
output bottlenecks, most probably caused by writing to disk and/or writing to
the various databases. This manifests as an individual slowdown of each recipe
run which limits the efficiency of using more CPU cores (i.e., more recipes
run at the same time mean more files being written to disk at the same time
and more writing to the various database at the same time causing queuing to
occur). We define efficiency in table 5 as the total CPU time taken divided by
the total time taken to run multiplied by the number of cores (Equation 24). A
perfectly efficient code would give a value of 1.
$\text{Efficiency}=\frac{\text{totalCPUtime}}{\text{totaltimetaken}\times
N_{cores}}$ (24)
We see that on average we have an efficiency between 0.7 and 0.93. We find
that recipes that run quickly but save several files have lower efficiency,
i.e., that a bottleneck of writing to disk may be occurring; however, these
recipes also run quickly and write frequently to the various databases, and as
such it is hard to distinguish between the disk and database bottlenecks.
Also, recipes that run slow are rated as more efficient due to the amount of
time spent using the CPUs (i.e., science algorithms) against reading/writing
to and from disk/databases, thus our metric is far from perfect. Another
factor is other processes using the machine at that specific time that cannot
be easily taken into account when measuring how efficient we are. We will
continue to review the performances, speed up the science algorithms and find
ways to make the individual recipes faster.
Currently, apero is optimized to run on a single node (i.e., a single machine)
with access to many CPU cores on this single node. It is possible to run
batches of apero in a manual way using multiple runs of apero_processing and
controlling (manually) when to run the next step (i.e., making sure all pre-
processing is done before reference calibrations, that all reference
calibrations are done before any night calibrations, etc.). One can also run
recipes one-by-one bypassing the use of the apero_processing recipe
completely. apero is optimized to reduce a full data set, this implies many
terabytes of raw and reduced data. Currently to reduce a single night of data
one requires at the very least a full reference calibration night (i.e., the
calibration data from 2020-08-31) and the full set of calibrations for the
night to be reduced (and preferably a few nights surrounding it in case any
calibrations fail quality control) and the full telluric database of hot stars
(from every night processed). This obviously means a large amount of data is
required for even a single night or single observation. We plan to release a
full calibration database and full telluric database of hot stars with a way
to get and install this data, in order to allow any user to reduce a small set
of data. We do however always recommend using data reduced with a full data
set done in a uniform way. This is currently available at data centers at
CFHT, Université de Montréal, and the Laboratoire d’Astrophysique de
Marseille.
### 12.2 Future work
As with most pipelines, improvements are always ongoing. With the LBL RV
analysis code, we have seen RV accuracy down to 2$\text{ms}^{-1}$ with SPIRou,
which is an indication that apero is at least this precise. However, there are
several features we plan to add:
* •
The apero recipes do not currently propagate uncertainties throughout the data
reduction process, which can be problematic when trying to understand the
limiting factors in the data analysis. Full error propagation is important as
feedback to the engineering team; for example, quantifying the impact of the
thermal background from the optical train on the measurement of $K$-band
spectroscopic features such as the 2.29 $\mu$m CO band-head could justify
efforts to cool parts of the optical train or not.
* •
In parallel to the propagation of uncertainties, we plan to propagate quality
flags on pixel values, for example, whether a given invalid pixel is due to a
cosmic ray hit or a hot pixel. In the current framework, pixels are either
deemed valid or invalid and flagged as a nan, which does not allow one to
back-trace the origin of missing data. This is done for JWST data
products171717https://jwst-reffiles.stsci.edu/source/data_quality.html with a
pixel-level data quality encoded with 32-bit integers.
* •
As the SPIRou fibers are multi-mode, one expects a certain level of noise to
arise from the time-varying weight of modes injected in each of the science
fibers. This is minimized through the use of a pupil slicer and fiber
scrambler at the injection. As the pupil slicer image provides more
information on the flux distribution at the fiber exit than a simple fiber
(e.g., the bottom-center panel in Figure 19), one could decorrelate this
spatially-resolved information against the modal noise.
* •
Persistence in infrared arrays is a non-trivial problem for faint-target
observations Artigau et al. (2018). For any given image, one sees a decaying
remnant image of all previous observations with decreasing amplitude. The
amplitude of any given observation falls with an amplitude that is
proportional to the inverse of the delay since the last illumination. One
notable feature of persistence in infrared arrays is that it is not only the
previous frame that matters but the entire history of illumination over the
last few hours. A bright target observed for a long time at the beginning of a
night (a common example being a multi-hour sequence to monitor a transiting
planet) will affect all fainter targets observed later during the night. To
add further complexity to the matter, the persistence response varies at the
pixel-to-pixel level. Work has begun to construct a persistence model for
SPIRou. Furthermore, the algorithms need to be run at the observatory level as
data obtained earlier in the night may be proprietary and not accessible to
all apero users.
* •
The main limitation for faint object observations with SPIRou or any near-
infrared pRV spectrograph, particularly bluer of the $K$ band, is detector
readout noise. As has been demonstrated by Payeur et al. (2022), machine-
learning algorithms can reduce readout noise in long sequences (100 readouts
of $\sim$10 minutes) from $\sim$6 e- to $<2$ e- (see Table 2 therein). This
needs to be performed before the current apero steps, as it requires handling
the data cubes rather than the 2-D images used here. As long as the output
format is maintained, the machine-learning images should be used as inputs to
apero.
* •
Energetic particles regularly hit infrared arrays and deposit electrons in
pixels, leading to spurious signals. These hits happen basically
instantaneously (on the time scales relevant to the readout) and manifest
themselves as discontinuities in the time series of non-destructive readouts.
Efficient algorithms have been proposed to handle cosmic ray hits in ramp-
fitting frameworks (Anderson & Gordon, 2011) but have yet to be implemented
for SPIRou.
* •
The LBL recipes have been designed to use apero byproducts, but they have not
yet been implemented within the automated apero framework. Steps that are
currently done manually, such as the association of an appropriate stellar
template, will be included within apero in the near future.
There are also various planned improvements, they are the optimal extraction
(better characterization of extraction weights), the database architecture
(currently throttling the maximum number of connections with a large number of
cores), some minor memory leaks when parallel processing, handling the thermal
contribution at bluer wavelength domains and completing all
documentation${}^{\ref{footnote:apero_docs}}$.
## 13 Conclusion
We present apero (A PipelinE to Reduce Observations) and highlight its use as
the official pipeline for SPIRou. We walk through the steps going from raw
data to science-ready products. We detail the pre-processing of raw data to
correct detector issues, the production of reference calibrations and nightly
calibrations, and the use of these calibrations to correct and extract hot
stars and science observations in a consistent, controlled manner. We
summarize telluric correction (which will be detailed in a future publication,
Artigau et al. in prep), RV analysis, polarimetric analysis, and our post-
processing recipes delivering telluric corrected 2D and 1D spectra as well as
polarimetry products and enabling precise stable radial velocity calculations
(via the LBL algorithm, Artigau et al. 2022), good to at least $\sim 2$
$\text{ms}^{-1}$ over the timescale of the current lifetime of SPIRou (5
years).
We would like to thank the anonymous referee for the valuable comments that
improved the quality of the paper. The authors wish to thank everyone involved
with writing, maintaining, and updating all Python packages. Specifically,
apero has made extensive use of: (2) astropy, (Astropy Collaboration et al.,
2018, 2013) astroquery (Ginsburg et al., 2019) barycorrpy (Kanodia & Wright,
2018; Wright & Kanodia, 2020) matplotlib (Hunter, J. D., 2007) numpy (Harris
et al., 2020) pandas (McKinney, 2010, 2011) scipy (Virtanen et al., 2020) As
well as the python packages bottleneck (Goodman & et al., 2019), gitchangelog
(Lab, 2018), ipdb (Chapelle, 2021), IPython package (Pérez & Granger, 2007),
mysql-connector-python (Mariz, 2021), numba (Lam et al., 2015), pandastable
(Farrell, 2016), Pillow (Murray, 2021), pyyaml (Simonov, 2021), sphinx (Komiya
& Brandl, 2021), sqlalchemy (Bayer, 2021), Scikit-learn (Pedregosa et al.,
2011), tqdm (da Costa-Luis et al., 2021), yagmail (van Kooten, 2021), and xlrd
(Withers, 2021). This research made use of ds9, a tool for data visualization
supported by the Chandra X-ray Science Center (CXC) and the High Energy
Astrophysics Science Archive Center (HEASARC) with support from the JWST
Mission office at the Space Telescope Science Institute for 3D visualization.
This research made use of TOPCAT, an interactive graphical viewer and editor
for tabular data (Taylor, 2005). apero would have been impossible without the
use of PyCharm, Git, and GitHub. This work has made use of data from the
European Space Agency (ESA) mission Gaia (https://www.cosmos.esa.int/gaia),
processed by the Gaia Data Processing and Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular, the institutions
participating in the Gaia Multilateral Agreement. apero has made use of the
SIMBAD database, operated at CDS, Strasbourg, France. This research has made
use of NASA’s Astrophysics Data System. apero has made use of the VizieR
catalog access tool, CDS, Strasbourg, France. The acknowledgments were
compiled using the Astronomy Acknowledgment Generator. The authors wish to
recognize and acknowledge the very significant cultural role and reverence
that the summit of MaunaKea has always had within the indigenous Hawaiian
community. We are most fortunate to have the opportunity to conduct
observations from this mountain. This work was financially supported by the
Natural Sciences and Engineering Research Council of Canada and the Fonds
Québécois de Recherche - Nature et Technologies. Observatoire du Mont-Mégantic
and the Institute for Research on Exoplanets acknowledge funding from
Développement Économique Canada, Quebec’s Ministère de l’Éducation et de
l’Innovation, the Trottier Family Foundation and the Canadian Space Agency.
M.H. and I.B acknowledge support from ANID – Millennium Science Initiative –
ICN12_009. C.M., A.C., P.F., X.D., I.B., and J.F.D. acknowledges funding from
the French ANR under contract number ANR18CE310019 (SPlaSH) and the Programme
National de Planétologie (PNP). This work is supported by the French National
Research Agency in the framework of the Investissements d’Avenir program
(ANR-15-IDEX-02), through the funding of the “Origin of Life” project of the
Grenoble-Alpes University. J.F.D. acknowledges funding from the European
Research Council (ERC) under the H2020 research & innovation programme (grant
agreement #740651 NewWorlds). T.V. would like to acknowledge funding from the
Fonds de Recherche du Québec - Nature et Technologies (FRQNT, scholarship
number 320056), and the Institute for Research on Exoplanets (iREx).
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## Appendix A Creating a raw SPIRou ramp image
The SPIRou detector control software reads the detector continuously every
5.57 s and produces a 2D image ($4096\times 4096$) constructed from the linear
fit of the pixel value versus time (as well as a slope, intercept, error, and
number of frames used for quality checks). The construction of the 2D image
from individual readouts being handled at the acquisition step is not included
as part of apero but as software maintained and used at CFHT. The construction
of the 2D frame is performed through the following steps.
Individual detector frames are obtained from the detector control software
every 5.57 s (at time $j$, $t[j]$). A flagging of pixel saturation is
performed, and pixels with a non-linearity that would be larger than $\sim 10$
% are considered unreliable and rejected for all future readouts (binary mask
$m[i,j]$ for pixel $i$ at time $j$). A non-linearity correction is applied to
pixel fluxes (flux at pixel $i$ and at time $j$ is $f[i,j]$). As individual
readouts are in computer memory, intermediate quantities necessary for the
computation of the total pixel-level slope are computed. The advantage of
preserving these quantities in memory is that one can perform a pixel-level
ramp-fitting over an arbitrarily large number of frames without being required
to have all pixel values in memory at the same time or without having to
access files multiple times.
* •
$\sigma_{x}[i]=\sum_{j}m[i,j]*f[i,j]$
* •
$\sigma_{y}[i]=\sum_{j}m[i,j]*t[j]$
* •
$\sigma_{xy}[i]=\sum_{j}m[i,j]*f[i,j]*t[j]$
* •
$\sigma_{x^{2}}[i]=\sum_{j}m[i,j]*f[i,j]^{2}$
* •
$n[i]=\sum_{j}m[i,j]$
Among these intermediate quantities, the only one that has a clear
signification is $n[i]$: it corresponds to the number of valid (i.e., below
the predefined flux level for saturation) readouts obtained over the entire
sequence. For normal scientific exposures, $n[i]$ is equal to the total number
of readouts for the vast majority of pixels; pixels with a very large dark
current being the exception.
From simple linear algebra, one can show that the per-pixel intercept is
$b[i]=\frac{\sigma_{x}[i]\sigma_{xy}[i]-\sigma_{x^{2}}[i]*\sigma_{y}[i]}{\sigma_{x}[i]^{2}-n\sigma_{x^{2}}}$
(A1)
and correspondingly, the per-pixel slope is:
$a[i]=\frac{\sigma_{y}-n[i]b[i]}{\sigma_{x}[i]}$ (A2)
Once the slope image $a[i]$ has been computed, it is corrected for correlated
amplifier noise using the side reference pixels (along the fast readout axis),
and amplifier offset is corrected using the top and bottom reference pixels
(both extremities of the slow readout axis).
As a quality check of pixel-level value, we further compute the error on the
slope to identify pixels having a suspiciously large dispersion of their
values around the slope. This is used to flag pixels that have a large slope
that is inconsistent with their frame-to-frame accumulation rate. To compute
the slope, one has to re-read all frames once (though it is not required to
keep them all at once in memory) and compute the following values
* •
$x_{p}[i]=sx[i]/n[i]$
* •
$y_{p}[i,j]=b[i]+a[i]*t[j]$
* •
$\varrho_{x^{2}}[i]=\sum_{j}(t[j]-x_{p}[i])m[i,j]$
* •
$\varrho_{y^{2}}[i]=\sum_{j}(f[i,j]-y_{p}[i,j])m[i,j]$
From which the slope error is
$\varrho(i)=\sqrt{\frac{\varrho_{y^{2}}[i]/(n[i]-2)}{\varrho_{x^{2}}[i]}}$
(A3)
## Appendix B Standard image calibration
After pre-processing (Section 4), the reference dark calculation (Section 5.1)
and the bad pixel correction (Section 6.1), all images that are used in apero
need to be calibrated in a standard way (using both the dark reference and bad
pixel recipe outputs). This is not a separate recipe but a set of functions
that are used in all recipes that use pre-processed files as inputs.
The standard calibration is ordered as follows:
1. 1.
dark reference correction (Section B.1).
2. 2.
flip, resize and re-scale the image (Section B.2).
3. 3.
flag bad pixels (Section B.3).
4. 4.
correct background flux (Section B.4).
5. 5.
clean hot pixels (Section B.5).
6. 6.
flag pixels that are out of bounds (Section B.6).
### B.1 Dark reference correction
The first step of the standard calibration of pre-processed files is to
correct the input image for the dark signal.
$IM_{\text{corr}i,j}=IM_{\text{uncorr}i,j}-N(DARK_{i,j})$ (B1)
where $IM_{\text{corr}i,j}$ and $IM_{\text{uncorr}i,j}$ are the flux in the
$i^{th}$ row $j^{th}$ column of the corrected image and uncorrected image
respectively, $N$ is the number of raw images that went into $IM$ and
$DARK_{i,j}$ is flux in the $i^{th}$ row $j^{th}$ column of the reference dark
(see Section 5.1).
The dark reference is taken from the calibration database. If more than one
dark reference exists the closest in time to $IM$ is used (using the header
key mjdmid from the header).
### B.2 Flipping, resizing, and re-scaling the image
For legacy reasons, the image is flipped in the vertical and horizontal
directions (see Figure 4). After this the image is converted from $ADU/s$ to
electrons using Equation B2.
$IM_{\text{electrons}i,j}=IM_{\text{ADU}/\text{s}i,j}\times\text{Gain}\times
t_{\rm exp}$ (B2)
where $IM_{\text{electons}i,j}$ is the flux in electrons for the $i^{th}$ row
$j^{th}$ column, $IM_{\text{ADU}/\text{s}i,j}$ is the flux in $ADU/s$ for the
$i^{th}$ row $j^{th}$ column, the gain is taken from the header key gain
(although it has remained constant over the lifetime of SPIRou), and $exptime$
is the exposure time in seconds (taken from the header key exptime).
Once the image is in electrons it is then resized. The image is cut in the
cross-order direction to start from pixel 250 and end at pixel 3350 (removing
a partial blue order and the whole unilluminated dark amplifier region) and in
the along-order direction to start from pixel 4 and end at pixel 4092
(removing just the H4RG reference pixels). Thus after this resizing the image
is of size 3100$\times$4088 (see Figure 4).
### B.3 Flagging the bad pixels
The closest bad pixel map (badpix, see Section 6.1) in time to the image
(using the header key mjdmid from the header) is loaded from the calibration
database and all pixels that are flagged as bad pixels are set to nan, this is
shown in Equation B3.
$IM_{\text{corr}i,j}=\left\\{\begin{array}[]{cl}NaN:&BADPIX_{i,j}\equiv 1\\\
IM_{i,j}:&\text{otherwise}\\\ \end{array}\right.$ (B3)
where $IM_{\text{corr}i,j}$ and $IM_{i,j}$ are the flux in the $i^{th}$ row
$j^{th}$ column of the corrected image and the input image respectively.
$BADPIX_{i,j}$ is a bad pixel flag (1 or 0) in the $i^{th}$ row $j^{th}$
column from the bad pixel map.
### B.4 Correcting the background flux
Within each science image, we take the median of ‘background’ pixels
(identified using the backmap, see Section 6.1) within a region and create a
map of large-scale background features (middle panel, Figure 14). This map is
then splined into a $4088\times 4088$ image and subtracted from the science
frame.
### B.5 Additional cleaning of hot pixels
Hot pixels are flagged by finding pixels that are 10$\sigma$ (positive or
negative) outliers compared to their immediate neighbors. This is in addition
to the cosmic ray rejection applied in Section 4.5 and the bad pixel flagging
(Section 6.1) which removes most of the hot pixels. In this additional
cleaning of hot pixels, we first construct a flattened image and perform a
low-pass filter in the along-order direction, filtering the image so that only
pixel-to-pixel structures remain. We then apply median filtering, which
removes these big outliers, and then we smooth the image to avoid big regions
filled with zeros. We apply a 5-pixel median boxcar and a 5-pixel smoothing in
the along-order direction, which blurs along the dispersion over a scale of
$\sim$7 pixels. Bad pixels are interpolated with a 2D surface fit by using
valid pixels within a $3\times 3$ pixel box centered on the bad pixel.
### B.6 Flagging out of bound pixels
Pixel values need to be within reasonable bounds considering the physics of
the H4RG detector. If they are not in bounds we set them to nan. The upper
bound is the saturation per frame time, but as the flux is expressed as a
slope (in the fits2ramp.py), a pixel with a value greater than the saturation
point can be recorded by the detector and is nonphysical. The lower bound is
set to the negative value of ten times the readout noise, these bounds are
shown in Equation B4.
$IM_{\text{corr}i,j}=\left\\{\begin{array}[]{cl}NaN:&IM_{i,j}>\text{saturation}/t_{\text{frame}}\\\
NaN:&IM_{i,j}<-10\times\text{readoutnoise}\\\ IM_{i,j}:&\text{otherwise}\\\
\end{array}\right.$ (B4)
where $IM_{\text{corr}i,j}$ and $IM_{i,j}$ are the flux in the $i^{th}$ row
$j^{th}$ column of the corrected image and the input image respectively,
$saturation$ is taken from the header keyword saturate and is converted to
electrons via B2, $t_{\rm frame}$ is the individual frame time (from header
keyword frmtime), and readout noise is taken from the header keyword rdnoise.
## Appendix C Shape transformation
The shape transform algorithm allows three different transformations, that may
or may not be used. Here we define $x$ as the direction along the order, $y$
as the direction across the order.
1. 1.
a linear transform: defined by $dx$, $dy$, $A$, $B$, $C$, $D$, where $dx$ and
$dy$ are shifts $x$ and $y$ respectively and $A$, $B$, $C$, $D$ form the
transform matrix:
$\left[\begin{array}[]{cl}A&B\\\ C&D\\\ \end{array}\right]$ (C1)
This combines with $dx$ and $dy$ in order to form a 3$\times$3 matrix:
$\left[\begin{array}[]{ccc}A&B&dx\\\ C&D&dy\\\ 0&0&1\\\ \end{array}\right]$
(C2)
This $3\times 3$ linear transformation matrix allows for scaling, rotation,
reflection (not used in our case) and shearing (Gonzalez & Woods, 2008).
2. 2.
a shift in $x$ position, where a shift is defined for each pixel.
3. 3.
a shift in $y$ position, where a shift is defined for each pixel.
## Appendix D Version history of APERO
apero has been in development since October 2017. Here we list a few of the
major versions to give the reader an idea of how long the development of a
full pipeline can take.
First version | Last version | Date of first version | Main improvements
---|---|---|---
0.0.000 | 0.0.048 | 2017-10-12 | First python version of apero
0.1.000 | 0.1.037 | 2018-01-10 | First version to run on SPIRou engineering data (H2RG detector)
0.2.000 | 0.2.128 | 2018-04-17 | First version for SPIRou commissioning (H4RG upgrade)
0.3.000 | 0.3.077 | 2018-09-06 | First implementation of telluric correction
0.4.000 | 0.4.123 | 2018-12-08 | Re-work wave solution and BERV calculation
0.5.000 | 0.5.124 | 2019-05-10 | Implementation of reference calibrations/recipes
0.6.000 | 0.6.132 | 2019-12-06 | Complete re-ordering of apero file structure, first use on NIRPS
0.7.000 | 0.7.255 | 2020-10-16 | Implementation of SQL databases, Telluric pre-cleaning, upgrade calibration recipes, integration of spirou-polar
0.8.000 | active | 2022-09-11 | Currently in development: uncertainty propagation, nan pixel quality flags, optimal extraction weights, database architecture
1.0.000 | - | - | After 0.8, full documentation, including adding new instruments
Table D1: History of the major versions of apero
## Appendix E Current science publications using APERO
In table E1 we list some science publications using apero for science. This
list is not complete but gives an idea of the range of science enabled by
apero with SPIRou.
Title | Citation
---|---
Spin-orbit alignment and magnetic activity in the young planetary system AU Mic | Martioli et al. 2020
Early science with SPIRou: near-infrared radial velocity and spectropolarimetry of the planet-hosting star HD 189733 | Moutou et al. 2020
SPIRou: NIR velocimetry and spectropolarimetry at the CFHT | Donati et al. 2020
Star-disk interaction in the T Tauri star V2129 Ophiuchi: An evolving accretion-ejection structure | Sousa et al. 2021
Where Is the Water? Jupiter-like C/H Ratio but Strong H2O Depletion Found on Tau Bootis b Using SPIRou | Pelletier et al. 2021
TOI-1278 B: SPIRou Unveils a Rare Brown Dwarf Companion in Close-in Orbit around an M Dwarf | Artigau et al. 2021
Characterizing Exoplanetary Atmospheres at High Resolution with SPIRou: Detection of Water on HD 189733 b | Boucher et al. 2021
TOI-530b: a giant planet transiting an M-dwarf detected by TESS | Gan et al. 2022b
TOI-1759 b: A transiting sub-Neptune around a low mass star characterized with SPIRou and TESS | Martioli et al. 2022
TESS discovery of a sub-Neptune orbiting a mid-M dwarf TOI-2136 | Gan et al. 2022a
Estimating fundamental parameters of nearby M dwarfs from SPIRou spectra | Cristofari et al. 2022a
Line-by-line velocity measurements, an outlier-resistant method for precision velocimetry | Artigau et al. 2022
TOI-1452 b: SPIRou and TESS reveal a super-Earth in a temperate orbit transiting an M4 dwarf | Cadieux et al. 2022
Estimating the atmospheric properties of 44 M dwarfs from SPIRou spectra | Cristofari et al. 2022b
CO or no CO? Narrowing the CO abundance constraint and recovering the H2O detection in the atmosphere of WASP-127b using SPIRou | Boucher et al. (accepted 2022)
Near-IR and optical radial velocities of the active M-dwarf star Gl 388 (AD Leo) with SPIRou at CFHT and SOPHIE at OHP | Carmona et al. (submitted 2022)
New insights on the near-infrared veiling of young stars using CFHT/SPIRou data | Sousa et al. (submitted 2022)
A sub-Neptune planet around TOI-1695 discovered and characterized with TESS and SPIRou | Kiefer et al. (submitted 2022)
The rotation period of 43 quiet M dwarfs from spectropolarimetry in the near-infrared: I. The SPIRou APERO analysis | Fouqué et al. (in prep)
Optical and near-infrared stellar activity characterization of the early M dwarf Gl 205 with SOPHIE and SPIRou | Cortés-Zuleta et al. (in prep)
High-resolution Chemical Spectroscopy of Barnard’s Star with SPIRou | Jahanadar et al. (in prep)
New methods to correct for systematics in near-infrared radial velocity measurements: Application to GL725B with SPIRou data | Ould-Elhkim et al. (in prep)
Characterizing planetary systems with SPIRou: the M-dwarf Planet Search survey and the system of GJ 251 | Moutou et al. (in prep)
Table E1: List of some publications using apero for science.
## Appendix F Inputs
The currently allowed raw file inputs are listed in table F1. The name becomes
the apero header key dprtype. All other columns are header keys found in the
raw input files or are added/modified when first processed (in
apero_preprocess and apero_processing).
Although all SPIRou raw files have suffixes: [style=itemize](2) a.fits
(obstype = align) c.fits (obstype = comparison) d.fits (obstype = dark) f.fits
(obstype = flat) o.fits (obstype = object)
apero does not rely on the filenames to assign a dprtype to a raw input file.
Instead, we use header keys to identify file types (Table F1).
name | OBSTYPE | SBCCAS_P | SBCREF_P | SBCALI_P | INSTRUME | TRG_TYPE$\ast$ | DRSMODE$\ast$
---|---|---|---|---|---|---|---
dark_dark_int | DARK | pos_pk | pos_pk | P4 | SPIRou | - | -
dark_dark_tel | DARK | pos_pk | pos_pk | P5 | SPIRou | - | -
dark_dark_sky | OBJECT | pos_pk | pos_pk | - | SPIRou | SKY | -
dark_fp_sky | OBJECT | pos_pk | pos_fp | - | SPIRou | SKY | -
dark_flat | FLAT | pos_pk | pos_wl | - | SPIRou | - | -
flat_dark | FLAT | pos_wl | pos_pk | - | SPIRou | - | -
flat_flat | FLAT | pos_wl | pos_wl | - | SPIRou | - | -
flat_fp | FLAT | pos_wl | pos_fp | - | SPIRou | - | -
dark_fp | ALIGN | pos_pk | pos_fp | - | SPIRou | - | -
fp_dark | ALIGN | pos_fp | pos_pk | - | SPIRou | - | -
fp_flat | ALIGN | pos_fp | pos_wl | - | SPIRou | - | -
fp_fp | ALIGN | pos_fp | pos_fp | - | SPIRou | - | -
lfc_lfc | ALIGN | pos_rs | pos_rs | - | SPIRou | - | -
lfc_fp | ALIGN | pos_rs | pos_fp | - | SPIRou | - | -
fp_lfc | ALIGN | pos_fp | pos_rs | - | SPIRou | - | -
obj_dark | OBJECT | pos_pk | pos_pk | - | SPIRou | TARGET | SPECTROSCOPY
obj_fp | OBJECT | pos_pk | pos_fp | - | SPIRou | TARGET | SPECTROSCOPY
obj_hcone | OBJECT | pos_pk | pos_hc1 | - | SPIRou | TARGET | -
obj_hctwo | OBJECT | pos_pk | pos_hc2 | - | SPIRou | TARGET | -
polar_dark | OBJECT | pos_pk | pos_pk | - | SPIRou | TARGET | POLAR
polar_fp | OBJECT | pos_pk | pos_fp | - | SPIRou | TARGET | POLAR
dark_hcone | COMPARISON | pos_pk | pos_hc1 | - | SPIRou | - | -
dark_hctwo | COMPARISON | pos_pk | pos_hc2 | - | SPIRou | - | -
fp_hcone | COMPARISON | pos_fp | pos_hc1 | - | SPIRou | - | -
fp_hctwo | COMPARISON | pos_fp | pos_hc2 | - | SPIRou | - | -
hcone_fp | COMPARISON | pos_hc1 | pos_fp | - | SPIRou | - | -
hctwo_fp | COMPARISON | pos_hc2 | pos_fp | - | SPIRou | - | -
hcone_hcone | COMPARISON | pos_hc1 | pos_hc1 | - | SPIRou | - | -
hctwo_hctwo | COMPARISON | pos_hc2 | pos_hc2 | - | SPIRou | - | -
hcone_dark | COMPARISON | pos_hc1 | pos_pk | - | SPIRou | - | -
hctwo_dark | COMPARISON | pos_hc2 | pos_pk | - | SPIRou | - | -
Table F1: All possible inputs currently accepted by apero. HDR denotes that a
keyword is required from an input file header. $\ast$ denotes header key is
added or modified by apero before internal use.
## Appendix G APERO Products
apero produces outputs after every recipe. These are saved to the reduced
directory (except for the apero_preprocess recipe that saves outputs into the
working directory). These are intermediary products and are used to create the
CADC outputs in post-processing (see Section 11).
File | Recipe | Frequency | Description
---|---|---|---
(id)_pp.fits | apero_preprocess | every file | preprocessed file
(HASH)_pp_dark_ref.fits | apero_dark_ref | ref night | reference dark file
(HASH)_pp_badpixel.fits | apero_badpix | every night | bad pixel map file
(HASH)_pp_order_profile_{AB,C}.fits | apero_loc_spirou | every night | order profile file
(HASH)_pp_loco_{AB,C}.fits | apero_loc_spirou | every night | localization center map file
(HASH)_pp_shapex.fits | apero_shape_ref | ref night | dx shape map file
(HASH)_pp_shapey.fits | apero_shape_ref | ref night | dy shape map file
(HASH)_pp_fpref.fits | apero_shape_ref | ref night | FP reference file
(HASH)_pp_shapel.fits | apero_shape | every night | local shape map file
(HASH)_pp_blaze_{AB,A,B,C}.fits | apero_flat | every night | blaze correction file
(HASH)_pp_flat_{AB,A,B,C}.fits | apero_flat | every night | flat correction file
(HASH)_pp_e2ds_{AB,A,B,C}.fits | apero_thermal | every night | 2D Extracted dark_dark_int and/or dark_dark_tel file [49x4088]
(HASH)_pp_e2dsff_{AB,A,B,C}.fits | apero_thermal | every night | 2D extracted + flat fielded dark_dark_int and/or dark_dark_tel file [49x4088]
(HASH)_pp_s1d_v_{AB,A,B,C}.fits | apero_thermal | every night | 1D extracted + flat fielded dark_dark_int and/or dark_dark_tel file with constants velocity bins
(HASH)_pp_s1d_w_{AB,A,B,C}.fits | apero_thermal | every night | 1D extracted + flat fielded dark_dark_int and/or dark_dark_tel with constants wavelength bins
(HASH)_pp_thermal_e2ds_int_{AB,A,B,C}.fits | apero_thermal | every night | extracted thermal internal dark calibration file
(HASH)_pp_thermal_e2ds_tel_{AB,A,B,C}.fits | apero_thermal | every night | extracted thermal telescope dark calibration file
(id)_pp_e2ds_{AB,A,B,C}.fits | apero_wave_ref | ref night | 2D Extracted dark_fp file [49x4088]
(id)_pp_e2dsff_{AB,A,B,C}.fits | apero_wave_ref | ref night | 2D extracted + flat fielded dark_dark_int and/or dark_dark_tel file [49x4088]
(id)_pp_s1d_v_{AB,A,B,C}.fits | apero_wave_ref | ref night | 1D extracted + flat fielded dark_dark_int and/or dark_dark_tel file with constants velocity bins
(id)_pp_s1d_w_{AB,A,B,C}.fits | apero_wave_ref | ref night | 1D extracted + flat fielded dark_dark_int and/or dark_dark_tel with constants wavelength bins
(HASH)_pp_leak_ref_{AB,A,B,C}.fits | apero_leak_ref | ref night | leak correction reference file
(HASH)_pp_e2ds_{AB,A,B,C}.fits | apero_wave_ref | ref night | 2D extracted fp_fp and hcone_hconefile [49x4088]
(HASH)_pp_e2dsff_{AB,A,B,C}.fits | apero_wave_ref | ref night | 2D extracted + flat fielded fp_fp and hcone_hconefile [49x4088]
(HASH)_pp_s1d_v_{AB,A,B,C}.fits | apero_wave_ref | ref night | 1D extracted + flat fielded fp_fp and hcone_hconefile] with constants velocity bins
(HASH)_pp_s1d_w_{AB,A,B,C}.fits | apero_wave_ref | ref night | 1D extracted + flat fielded fp_fp and hcone_hconefile] with constants wavelength bins
(HASH)_pp_wavesol_ref_{AB,A,B,C}.fits | apero_wave_ref | ref night | reference 2D wavelength solution [49x4088]
(HASH)_pp_wavecav_AB.fits | apero_wave_ref | ref night | Cavity width measurement file
(HASH)_pp_e2ds_{AB,A,B,C}.fits | apero_wave_night | every night | 2D extracted fp_fp and hcone_hconefile [49x4088]
(HASH)_pp_e2dsff_{AB,A,B,C}.fits | apero_wave_night | every night | 2D extracted + flat fielded fp_fp and hcone_hconefile [49x4088]
(HASH)_pp_s1d_v_{AB,A,B,C}.fits | apero_wave_night | every night | 1D extracted + flat fielded fp_fp and hcone_hconefile] with constants velocity bins
(HASH)_pp_s1d_w_{AB,A,B,C}.fits | apero_wave_night | every night | 1D extracted + flat fielded fp_fp and hcone_hconefile] with constants wavelength bins
(HASH)_pp_wave_night_ref_{AB,A,B,C}.fits | apero_wave_night | every night | reference 2D wavelength solution [49x4088]
Table G1: Main apero products, where id for SPIRou is the odometer code and the HASH is a checksum created by multiple inputs (of the same data type) to the given recipe. A full list of data products for each recipe can be found in the documentation. Continued in table G2. File | Recipe | Frequency | Description
---|---|---|---
(id)_pp_e2ds_{AB,A,B,C}.fits | apero_extract | every file | 2D extracted science/hot star file [49x4088]
(id)_pp_e2dsff_{AB,A,B,C}.fits | apero_extract | every file | 2D extracted + flat fielded science/hot star file [49x4088]
(id)_pp_s1d_v_{AB,A,B,C}.fits | apero_extract | every file | 1D extracted + flat fielded science/hot star file with constants velocity bins
(id)_pp_s1d_w_{AB,A,B,C}.fits | apero_extract | every file | 1D extracted + flat fielded science/hot star file with constants wavelength bins
(id)_pp_tellu_trans{AB,A,B}.fits | apero_mk_tellu | every hot star file | Measured telluric transmission file [49x4088]
(id)_pp_tellu_pclean{AB,A,B}.fits | apero_mk_tellu | every hot star file | Telluric pre-cleaning (corrected, transmission mask, measured absorption, sky model) [49x4088]
trans_model_AB.fits | apero_mk_model | one | Model of all telluric transmission files (residuals in water, dry and a dc level).
(id)_pp_e2dsff_tcorr{AB,A,B}.fits | apero_fit_tellu | every hot star/science file | 2D telluric corrected extracted flat fielded file [49x4088]
(id)_pp_s1d_w_tcorr_{AB,A,B}.fits | apero_fit_tellu | every hot star/science file | 1D Telluric corrected extracted flat fielded file with constants velocity bins
(id)_pp_s1d_v_tcorr_{AB,A,B}.fits | apero_fit_tellu | every hot star/science file | 1D telluric corrected extracted flat fielded file with constants velocity bins
(id)_pp_e2dsff_recon{AB,A,B}.fits | apero_fit_tellu | every hot star/science file | 2D telluric reconstructed absorption file [49x4088]
(id)_pp_s1d_w_recon_{AB,A,B}.fits | apero_fit_tellu | every hot star/science file | 1D telluric reconstructed absorption file with constants wavelength bins
(id)_pp_s1d_v_recon_{AB,A,B}.fits | apero_fit_tellu | every hot star/science file | 1D telluric reconstructed absorption file with constants velocity bins
(id)_pp_tellu_pclean{AB,A,B}.fits | apero_mk_tellu | every hot star/science file | Telluric pre-cleaning (corrected, transmission mask, measured absorption, sky model) [49x4088]
Template_{object}_tellu_obj_AB.fits | apero_mk_template | once per object | 2D telluric corrected template of a hot star or science object [49x4088]
Template_s1d_{object}_sc1d_w_file_AB.fits | apero_mk_template | once per object | 1D telluric corrected template of a hot star or science object with constants wavelength bins
Template_s1d_{object}_sc1d_v_file_AB.fits | apero_mk_template | once per object | 1D telluric corrected template of a hot star or science object with constants velocity bins
(id)_pp_e2dsff_tcorr{AB,A,B}_ccf _{mask}_{AB,A,B,C}.fits | apero_ccf | every hot star/science file | The CCF output file (CCFs per order and fitted parameters)
(HASH)_pp_e2dsff_tcorr_pol_deg.fits | apero_polar | every polarimetric group | 2D polar file [49x4088]
(HASH)_pp_e2dsff_tcorr_null1_pol.fits | apero_polar | every polarimetric group | 2D Null 1 file [49x4088]
(HASH)_pp_e2dsff_tcorr_null2_pol.fits | apero_polar | every polarimetric group | 2D Null 2 file [49x4088]
(HASH)_pp_e2dsff_tcorr_StokesI.fits | apero_polar | every polarimetric group | 2D Stokes I file [49x4088]
(HASH)_pp_e2dsff_tcorr_s1d_w_pol.fits | apero_polar | every polarimetric group | 1D polarimetry, null 1, null 2 and stokes I file with constants wavelength bins
(HASH)_pp_e2dsff_tcorr_s1d_v_pol.fits | apero_polar | every polarimetric group | 1D polarimetry, null 1, null 2 and stokes I file with constants velocity bins
Table G2: Main apero products, where id for SPIRou is the odometer code and
the HASH is a checksum created by multiple inputs (of the same data type) to
the given recipe. A full list of data products for each recipe can be found in
the documentation. Continued from table G1.
## Appendix H Preliminary usage with NIRPS
One of our main goals with apero was to keep the code generic enough that
adding new instruments is possible. To document this we detail here the
changes required to add NIRPS_HE and NIRPS_HA modes to apero. This work is
preliminary as commissioning of NIRPS is currently underway and we expect
additional changes will be required when larger data set over longer periods
of time exist (including long science sequences); however apero with NIRPS has
already demonstrated precision equivalent to SPIRou. The specific details of
all changes are not in the scope of this paper and will be part of a future
publication. Currently, after extraction, there are no code differences
between NIRPS and SPIRou reductions.
### H.1 NIRPS: A comparison with SPIRou
NIRPS is very similar to SPIRou but differs in several ways. We list key
differences that apero must handle:
* •
there are two modes, high efficiency (NIRPS_HE) and high accuracy (NIRPS_HA).
* •
there is only one science fiber and one calibration fiber.
* •
wavelength domain of 980 nm to 1800 nm (negligible thermal emission)
* •
missing order(s) around 1400 nm
* •
the resolution is higher $\sim$100 000 and $\sim$ 80 000 for NIRPS_HE and
NIRPS_HA modes respectively.
* •
there is no slicer and NIRPS_HE and NIRPS_HA have differing fiber geometry.
* •
there are 73 echelle orders extracted by apero.
* •
there are no dark unilluminated amplifiers.
### H.2 APERO changes for NIRPS
We use Figure 1 as our reference to changes within apero sub-packages. It is
worth noting that adapting apero for use with NIRPS did change some code used
for SPIRou, as having a second instrument with some unique characteristics
informed us of code that could be improved for both instruments. These changes
are not mentioned in this section. apero is designed to have each of these
instruments in the same code base thus there is no separate installation, nor
download of additional code is required for apero for usage with NIRPS.
No code was changed in the following apero sub-packages: apero.core: with the
exception of apero.core.instruments, apero.documentation, apero.io,
apero.lang, apero.plotting, apero.setup, and apero.tools. Minimal code (adding
3 or fewer changes) was changed in the following apero sub-package:
apero.base: adding of NIRPS_HE and NIRPS_HA to the supported instruments list.
Some code changes were added to the following apero sub-packages:
* •
apero.data: data files were copied from SPIRou and updated for NIRPS_HE and
NIRPS_HA. A few FITS files had to be updated using external scripts. We plan
that these scripts will be ingested into apero as tools to be used on other
instruments directly with sufficient documentation to do so.
* •
apero.recipes: recipe scripts were copied from SPIRou (mostly unchanged other
than filename). One recipe was added that did not exist before (a reference
flat done before the preprocessing recipe), which was added to the
preprocessing sequence for NIRPS_HE and NIRPS_HA.
* •
apero.science: some science algorithms were added (and called from the
recipes) in addition to or to replace SPIRou science algorithms. For example,
as NIRPS does not have an unilluminated region similar to SPIRou we have to
handle the detector corrections in preprocessing slightly differently,
building a background image from the between-order regions.
Substantial code was added to the following apero sub-package:
apero.core.instruments: configuration, constants, keywords, file definitions,
pseudo constants and recipe definitions were copied from SPIRou and updated
for NIRPS_HE and NIRPS_HA. We also removed the polarimetry recipes as there is
no polarimetry mode for NIRPS.
Note that pseudo constants are constants and variables that cannot be
described by a single number or string of characters, such as a decision
between science and reference fibers for a specific step of apero, or a
specific fix to a FITS header key for a specific instrument. These pseudo
constants are python functions designed to keep these instrument-specific
options separate from the rest of the code.
## Appendix I Glossary
In Table I1 we present a list of terms used throughout this paper.
Term | Description
---|---
apero | A PipelinE to Reduce Observations.
apero profile | A specific setup of apero (i.e. with a certain set of constants, reduction directories, database setups, etc).
amplifier | Independent electronic readout circuits operating in parallel and used to minimize the total readout time.
CCF | Cross-correlation function
CADC | Canadian Data Astronomy Center.
CFHT | The Canada, France, Hawaii Telescope, situated on Maunakea, Hawaii, US.
dprtype | Data product type - this describes what is in the science and reference fibers and distinguishes different calibrations and observations from each other.
drsmode | Data product mode - for SPIRou this is spectroscopy data or polarimetry data.
e2ds | An extracted order-by-order spectrum.
e2dsff | An extracted order-by-order spectrum that has been flat fielded.
fast-axis (long-axis) | Axis parallel to the amplifier direction on the detector. For SPIRou this is 4096 pixels per amplifier.
FITS file | A Flexible Image Transport System file to hold images, tables, and metadata in the form of a FITS header.
FITS header | A Flexible Image Transport System metadata holder. Consists of 8 character ‘keys‘, a value, and a comment.
FP | A Fabry Perot etalon used for calibration.
hash | A short unique hexadecimal string of characters generated from a long string of characters.
HC | A hollow cathode lamp used for calibration.
hot star | Bright, fast rotators of B or A spectral type that are spectrally featureless under $\sim$100$\text{kms}^{-1}$.
LBL | Line-by-line method for measuring radial velocity, presented in Artigau et al. (2022).
NIRPS | Near Infra Red Planet Searcher, a spectrograph on the 3.6m telescope in La Silla, Chile.
odometer | A unique sequential number used by CFHT to identify individual observations.
order | A domain on the detector at specific wavelengths generated by light passing through a diffraction grating.
PI | Principal investigator.
pipeline | Software that takes data from the origin to a destination.
post-processed data | Data that is given to PIs after apero has been completed.
pre-processed data | Data that has been corrected for detector issues - the first step for handling raw data.
pRV | Precision radial velocity, measurements at the order of $\text{ms}^{-1}$ accuracy.
raw data | For the purposes of apero this is the RAMPS from CFHT.
recipe | A top-level script similar to a cookbook recipe where simple steps follow one another; most calculations and algorithms are hidden from these recipes.
reduced data | Data that has been created from the raw data using a pipeline.
rhomb | An ensemble of prisms used to rotate polarization states.
run.ini file | A configuration file used for a specific reduction sequence i.e. a science sequence or a calibration sequence.
reduction sequence | A set of recipes run in a certain order with specific filters on which files to reduce for a specific purpose.
reference calibration | A calibration done once (not on a nightly basis).
science observation | Any observation that is taken by the telescope specifically for the purposes of science i.e. SLS data and PI data.
slicer | Device that is used to split an image into narrower images to increase spectral resolution.
slow-axis (short-axis) | Axis perpendicular to the amplifier direction on the detector. For SPIRou this is 128 pixels per amplifier.
SPIRou | Spectro Polarimètre Infra ROUge, a spectrograph for CFHT.
SLS | The SPIRou Legacy Survey.
Wollaston prism | A device that allows the incoming beam (either from the telescope or the calibration unit) to be split into two lyorthogonal polarized beams.
1/f noise | A noise component arising from the detector readout electronics that has a low-frequency component that is common to all amplifiers and sampled by the reference pixels.
Table I1: Glossary of terms
|
# Action-Angle Variables for Axisymmetric Potentials via Birkhoff
Normalization
Sam Hadden Canadian Institute for Theoretical Astrophysics, 60 St George St
Toronto, ON M5S 3H8, Canada
(April 2024)
###### Abstract
We describe a method for calculating action-angle variables in axisymmetric
galactic potentials using Birkhoff normalization, a technique from Hamiltonian
perturbation theory. An advantageous feature of this method is that it yields
explicit series expressions for both the forward and inverse transformations
between the action-angle variables and position-velocity data. It also
provides explicit expressions for the Hamiltonian and dynamical frequencies as
functions of the action variables. We test this method by examining orbits in
a Miyamoto-Nagai model potential and compare it to the popular Stäckel
approximation method. When vertical actions are not too large, the Birkhoff
normalization method achieves fractional errors smaller than a part in
$10^{3}$ and outperforms the Stäckel approximation. We also show that the
range over which Birkhoff normalization provides accurate results can be
extended by constructing Padé approximants from the perturbative series
expressions developed with the method. Numerical routines in Python for
carrying out the Birkhoff normalization procedure are made available.
## 1 Introduction
While a comprehensive dynamical description of a galaxy must take into account
mutual gravitational interactions among stars, dark matter, and gas,
dramatically simplified models that consider the orbits of stars in static,
smooth potentials provide surprisingly powerful tools for understanding the
dynamics of these complex systems. Numerical experiments show that typical
stellar orbits in model galactic potentials execute quasi-periodic motions
(e.g., Binney & Spergel, 1982, 1984). Such motions are restricted to
3-dimensional tori embedded in the full 6-dimensional phase space spanned by
stars’ spatial positions and velocities. For completely integrable Hamiltonian
systems, action-angle (AA) variables provide a particularly convenient set of
canonical coordinates for describing motion on such tori (e.g., Goldstein et
al., 2002). Fixing the values of three action variables, $\boldsymbol{J}$,
specifies an individual torus as a 3-dimensional submanifold of the
6-dimensional phase space, while the canonically conjugate angle variables,
$\boldsymbol{\theta}$, serve as coordinates on the torus. Furthermore, the
time evolution of trajectories on a given torus are described simply by the
linear advance of the angle variables with time at rates
$\frac{d}{dt}\boldsymbol{\theta}=\nabla_{\boldsymbol{J}}H$, where $H$ is the
Hamiltonian function.
While AA variables are a powerful theoretical tool, it is seldom possible to
compute them explicitly from stellar position and velocity data for potentials
commonly adopted when modeling our Milky Way and other galaxies. This is
because these potentials generally do not admit a set of three globally-
defined isolating integral which are required in order to have a well-defined
transformation to AA variables everywhere in phase space. Nonetheless, the
quasi-periodic nature of stellar orbits over much of the phase space of
interest in galactic potentials makes its possible to construct local
transformations to AA variables via numerical means.
Numerous numerical methods have been developed for the purposes of
constructing AA variables in galactic potentials (see review by Sanders &
Binney, 2016). One of the most widely-used methods is the Stäckel
approximation or “Stäckel fudge”, originally proposed by Binney (2012). In
axisymmetric potentials, this method allows radial and vertical actions to be
estimated via numerical quadratures by implicitly treating the underlying
potential as though it were in Stäckel form, meaning separable in prolate
spheroidal coordinates. While the Stäckel approximation usually provides quite
accurate determinations of AA variables, it does have some shortcomings.
First, the transformation cannot be readily inverted to provide positions and
velocities as functions of actions and angle nor express the original
Hamiltonian in terms of AA variables. Furthermore, because the method requires
evaluation of integrals by numerical quadratures, it can become
computationally expensive when evaluating transformations for large amounts of
data.
In this paper, we present a perturbative method to determine action-angle
variables in axisymmetric galactic potentials. This method is based on a
method originally proposed by Birkhoff (1927) for developing successively
refined approximations to the motion of Hamiltonian systems in the vicinity of
elliptic equilibria. It is variously referred to as to as “Birkhoff
normalization” (e.g., Deprit, 1969) or the “Birkhoff-Gustavson method” (e.g.,
Lowenstein, 2012). One of the principal advantages of this method is that it
provides explicit expressions for the transformation from position and
velocity data to AA variables. Furthermore, this method simultaneously yields
explicit expressions for the inverse transformation, meaning stellar positions
and velocities as well as the Hamiltonian can be represented explicitly in
terms of the AA variables. Expressions are constructed as multi-variate
polynomials in complex canonical coordinates, introduced in Section 2, and
therefore can be evaluated at minimal computational cost. Together, these
features can make Birkhoff normalization an attractive alternative to the
Stäckel approximation, depending on the application.
The use of perturbation theory in galactic dynamics to derive approximate
integrals of motion is not new (e.g., Contopoulos, 1963; Saaf, 1968; de Zeeuw
& Merritt, 1983). In fact, Gustavson (1966) implements and applies an
algorithm very similar to the one presented here to derive a formal second
integral of the Hénon–Heiles potential (Henon & Heiles, 1964). The current
paper builds on this past work in at least a couple respects. First, we follow
Deprit (1969) and develop perturbative series using the Lie transform method.
This method allows any functions defined on phase space in terms of the
original canonical variables to be readily expressed in transformed variables
and vice versa. This fact is exploited to construct an explicit expression for
the canonical angle variable, $\theta_{\phi}$, associated with orbits’
azimuthal degree of freedom, thereby providing a complete transformation from
6D position and velocity data to AA variables. Second, we show how Padé
approximants can be deployed to extend the range of validity of the series
expressions furnished by the method. Finally, a modern computational
implementation of the method in Python is made publicly
available.111github.com/shadden/AA_variables_via_Birkhoff_Normalization The
availability of computer algebra and automatic differentiation packages in
modern languages like Python make it feasible to apply Birkhoff normalization
to more realistic potential models than the simple Hénon–Heiles model
originally considered by Gustavson (1966).
The plan of this paper is as follows: we introduce the algorithm in Section 2.
In Section 3, we demonstrate the method with an example application to stellar
orbits in Miyamoto-Nagai potential (Miyamoto & Nagai, 1975), a classic model
for the potential of a galactic disk. Finally, we summarize and discuss our
results in Section 4.
## 2 The Algorithm
### 2.1 Setting
We consider orbits in an axisymmetric potential, $\Phi$. The Hamiltonian,
expressed in terms of polar coordinates $(R,z,\phi)$ along with their
canonically conjugate momenta, $(p_{R},p_{z},L)$, is
$\mathcal{H}(R,z,\phi,p_{R},p_{z},L)=\frac{1}{2}\left(p_{R}^{2}+p_{z}^{2}+\frac{L^{2}}{R^{2}}\right)+\Phi(R,z)~{}.$
(1)
Since the Hamiltonian in (1) has no explicit $\phi$ dependence, the angular
momentum, $L$, is conserved and we can consider the dynamics in the reduced
phase space comprised of $(z,R,p_{z},p_{R})$, treating $L$ as a parameter. The
Hamiltonian governing the dynamics in this reduced phase space is then given
by
${H}(R,z,\phi,p_{R},p_{z};L)=\frac{1}{2}\left(p_{R}^{2}+p_{z}^{2}\right)+{\Phi}_{\mathrm{eff}}(R,z;L)$
(2)
where we have defined
${\Phi}_{\mathrm{eff}}={\Phi}({R,z})+\frac{L^{2}}{2R^{2}}$ as the effective
potential. For a given value of angular momentum, $L$, there is a (typically
unique) circular orbit of radius of $R_{C}(L)$ which satisfies
$\frac{\partial}{\partial
R}{\Phi}_{\mathrm{eff}}({R,z})\bigg{|}_{(R,z)=(R_{C},0)}=0~{}.$ (3)
For nearly circular and planar orbits, the Hamiltonian (2) is given
approximately by
$H\approx
H_{2}(z,R,p_{R},p_{z};L):=\frac{1}{2}\left(p_{R}^{2}+\kappa^{2}\delta
R^{2}\right)+\frac{1}{2}\left(p_{z}^{2}+\nu^{2}z^{2}\right)~{}.$ (4)
where $\kappa^{2}=\frac{\partial^{2}}{\partial R^{2}}{\Phi}_{\mathrm{eff}}$
and $\nu^{2}=\frac{\partial^{2}}{\partial z^{2}}{\Phi}_{\mathrm{eff}}$, with
derivatives evaluated at $(R,z)=(R_{C},0)$, are the epicyclic and vertical
frequencies, respectively and $\delta R=R-R_{C}$. Equation (4) is, of course,
the Hamiltonian of two un-coupled harmonic oscillators of frequency $\kappa$
and $\nu$ and the equations of motion derived from it provide the so-called
“epicyclic approximation” for stars’ orbits (e.g., Binney & Tremaine, 2008).
### 2.2 Complex canonical variables
The equations of motion derived from Equation (4) are completely integrable
and we can perform a canonical transformation,
$T:(R,z,p_{R},p_{z})\mapsto(\phi_{R},\phi_{z},I_{R},I_{z})$, to new variables
such that $H_{2}\circ T^{-1}=\kappa I_{R}+\nu I_{z}$. Throughout this paper,
we will frequently work instead with the associated complex canonical
variables
$\displaystyle x_{R}$ $\displaystyle=\sqrt{\frac{\kappa}{2}}\left(\delta
R+\mathrm{i}\frac{p_{R}}{\kappa}\right)=\sqrt{I_{R}}e^{-\mathrm{i}\phi_{R}}$
(5) $\displaystyle x_{z}$
$\displaystyle=\sqrt{\frac{\nu}{2}}\left(z+\mathrm{i}\frac{p_{z}}{\nu}\right)=\sqrt{I_{z}}e^{-\mathrm{i}\phi_{z}}~{}$
(6)
and their complex conjugates, $\bar{x}_{R}$ and $\bar{x}_{z}$. The Poisson
bracket of two functions, $f$ and $g$, of phase space coordinates is expressed
in terms of these variables by
$[f,g]=-\mathrm{i}\sum_{j}\left(\frac{\partial
f}{\partial{x}_{j}}\frac{\partial g}{\partial\bar{x}_{j}}-\frac{\partial
f}{\partial\bar{x}_{j}}\frac{\partial g}{\partial{x}_{j}}\right)$ (7)
where the subscript $j$ denotes either $R$ or $z$. In terms of the complex
canonical variables,222 Here and elsewhere we will make a slight abuse of
notation and use the same symbol to denote the Hamiltonian function expressed
in terms of complex variables as we do for the Hamiltonian expressed in terms
of polar coordinates. Equation (4) becomes $H_{2}=\kappa x_{R}\bar{x}_{R}+\nu
x_{z}\bar{x}_{z}$ and Hamilton’s equations give
$\frac{d}{dt}x_{R}=[x_{R},H_{2}]=-\mathrm{i}\kappa x_{R}$ and
$\frac{d}{dt}x_{z}=[x_{R},H_{2}]=-\mathrm{i}\nu x_{z}$. Thus, under the
epicyclic approximation, the complex canonical variables evolve at constant
magnitude with a uniformly rotating phase. This will generally no longer be
true when higher order terms are included in the Hamiltonian. Our goal will be
to develop an algorithm that constructs a canonical transformation to new
complex canonical variables, $x_{R}^{\prime}$ and $x^{\prime}_{z}$, for which
this is again true up to some specified order in the new variables and their
complex conjugates.
### 2.3 An illustrative example
Before giving the general algorithm, we first demonstrate its basic principles
by way of an example application to the pendulum. This example provides a
pedagogical introduction to Birkhoff normalization and the construction of
canonical transformations via the Lie series method with complex canonical
variables. The example illustrates the basic ideas behind the method while
avoiding some of the cumbersome mathematical notation and manipulations
required to formulate the procedure in a more general context. Readers
uninterested in the technical details involved in an algorithmic
implementation of the Birkhoff normalization procedure can therefore read this
section while skipping or skimming Section 2.4.
We start with the Hamiltonian governing the dynamics of a simple pendulum,
$h(\theta,p)=\frac{p^{2}}{2}-\omega_{0}^{2}\cos{\theta}.$ (8)
For small $\theta$, the truncated Hamiltonian,
$h_{2}(\theta,p)=(p^{2}+\omega_{0}^{2}\theta^{2})/2$, gives a harmonic
oscillator approximation of the dynamics. In terms of the complex canonical
variable $x=\sqrt{\frac{\omega_{0}}{2}}(\theta+\mathrm{i}p/\omega_{0})$, the
Hamiltonian (8) becomes
$h(x,\bar{x})=\omega_{0}x\bar{x}+\sum_{n=2}^{\infty}h_{2n}(x,\bar{x})$ (9)
where
$h_{2n}=\frac{(-1)^{n+1}\omega_{0}^{2}}{(2n)!}\left(\frac{1}{2\omega_{0}}\right)^{n}\left(x+\bar{x}\right)^{2n}~{}.$
(10)
We will seek a canonical transformation to a new complex canonical variable,
$x^{\prime}$, so that the transformed Hamiltonian, $h^{\prime}$, depends only
on $|x^{\prime}|^{2}$ up to a specified power, $N$, in the transformed
variable, $x^{\prime}$, and its complex conjugate, $\bar{x}^{\prime}$.
The Lie series method (e.g., Ferraz-Mello, 2007; Lichtenberg & Lieberman,
1992) provides a particularly convenient means of constructing such a
transformation. This method involves the construction of a generating
function, $\chi$, such that the transformation from the new variables to the
old variables is given by the time-1 flow of the Hamiltonian vector field
generated by $\chi$. The time derivative of a function, $f$, under this flow
is given by the Poisson bracket $[f,\chi]$ and we define the differential
operator $\mathcal{L}_{\chi}:=[\cdot,\chi]$, referred to as the Lie
derivative. This allows us to express a function, $f$, given in terms of the
original phase space coordinates $(x,\bar{x})$, in terms of the new phase
space variables, $(x^{\prime},\bar{x}^{\prime})$, formally as
$f(x,\bar{x})=(\exp[\mathcal{L}_{\chi}]f)(x^{\prime},\bar{x}^{\prime})=\left(\sum_{n=0}^{\infty}\frac{1}{n!}\mathcal{L}_{\chi}^{n}f\right)(x^{\prime},\bar{x}^{\prime})~{}.$
(11)
Returning to our pendulum example, we seek a generating function $\chi$ such
that
$h^{\prime}(x^{\prime},\bar{x}^{\prime})=(\exp[\mathcal{L}_{\chi}]h)(x^{\prime},\bar{x}^{\prime})=\sum_{n=1}^{N/2}h^{\prime}_{2n}(|x^{\prime}|^{2})+R_{N}(x^{\prime},\bar{x}^{\prime})$
where the remainder terms
$R_{N}(x^{\prime},\bar{x^{\prime}})\sim\mathcal{O}(|x^{\prime}|^{N+1})$. Our
approach will be to write our generating function as
$\chi=\sum_{n=2}\chi_{2n}$ where
$\chi_{2n}\sim{\mathcal{O}(|x^{\prime}|^{2n})}$. This will allow us to develop
an iterative procedure for determining $\chi_{2n}$ and $h^{\prime}_{2n}$ at
each stage in terms of the functions $h_{2},h_{4}...,h_{2n}$ defined in
Equation (10) and the functions $\chi_{4},\chi_{6},...,\chi_{2n-2}$ determined
in previous stages. Writing out the transformation Equation (11) explicitly in
terms of Poisson brackets, we obtain
$\displaystyle\exp[\mathcal{L}_{\chi}]h$
$\displaystyle=h+[h,\chi]+\frac{1}{2}[[h,\chi],\chi]+...$ (12)
$\displaystyle=\underbrace{[h_{2},\chi_{4}]+h_{4}}_{h^{\prime}_{4}}+\underbrace{[h_{2},\chi_{6}]+[h_{4},\chi_{4}]+\frac{1}{2}[[h_{2},\chi_{4}],\chi_{4}]+h_{6}}_{h^{\prime}_{6}}+\mathcal{O}(|x^{\prime}|^{8})$
(13)
where, in the second line, we have grouped terms by common powers of
$|x^{\prime}|$. Consider the equation
$h^{\prime}_{4}=[h_{2},\chi_{4}]-\left(\frac{x^{\prime 4}}{96}+\frac{x^{\prime
3}\bar{x}^{\prime}}{24}+\frac{x^{\prime 2}\bar{x}^{\prime
2}}{16}+\frac{x^{\prime}\bar{x}^{\prime 3}}{24}+\frac{\bar{x}^{\prime
4}}{96}\right)$ (14)
defining the fourth order term in the transformed Hamiltonian, where we have
used Equation (10) to write out $h_{4}$ explicitly. We wish to define
$\chi_{4}$ so that $h^{\prime}_{4}$ depends only on $|x^{\prime}|^{2}$. Notice
that the Poisson bracket between $h_{2}$ and any monomial, $x^{\prime
k}\bar{x}^{\prime\bar{k}}$, gives $[h_{2},x^{\prime
k}\bar{x^{\prime}}^{\bar{k}}]=\mathrm{i}\omega_{0}(k-\bar{k})x^{\prime
k}\bar{x}^{\prime\bar{k}}$. Therefore, if we choose $\chi_{4}$ as the sum of
monomial terms
$\displaystyle\chi_{4}=\frac{1}{\mathrm{i}\omega_{0}}\left(\frac{x^{\prime
3}\bar{x}^{\prime}}{48}-\frac{x^{\prime}\bar{x}^{\prime
3}}{48}-\frac{\bar{x}^{\prime 4}}{384}+\frac{x^{\prime 4}}{384}\right)$ (15)
then each term in the polynomial $[h_{2},\chi_{4}]$ cancels a corresponding
term appearing in Equation (14) except the term $\propto x^{\prime
2}\bar{x}^{\prime 2}$. We therefore have
$h^{\prime}_{4}=-\frac{1}{16}x^{\prime 2}\bar{x}^{\prime 2}~{},$ (16)
which depends only on $|x^{\prime}|^{2}$ as desired.
With $\chi_{4}$ determine, we can write $h^{\prime}_{6}$ according to Equation
(13) as
$h^{\prime}_{6}=[h_{2},\chi_{6}]+\frac{x^{\prime
6}}{2560\omega_{0}}-\frac{x^{\prime
5}\bar{x}^{\prime}}{3840\omega_{0}}-\frac{7x^{\prime 4}\bar{x}^{\prime
2}}{1536\omega_{0}}-\frac{x^{\prime 3}\bar{x}^{\prime
3}}{256\omega_{0}}-\frac{7x^{\prime 2}\bar{x}^{\prime
4}}{1536\omega_{0}}-\frac{x^{\prime}\bar{x}^{\prime
5}}{3840\omega_{0}}+\frac{\bar{x}^{\prime 6}}{2560\omega_{0}}~{}.$ (17)
We will again choose $\chi_{6}$ so that the Poisson bracket $[h_{2},\chi_{6}]$
cancels all of the monomial terms appearing on the right-hand side of Equation
(17) except the term $\propto x^{3}\bar{x}^{3}$. Proceeding, we find
$\displaystyle\chi_{6}$
$\displaystyle=\frac{1}{\mathrm{i}\omega_{0}^{2}}\left(-\frac{x^{\prime
6}}{15360}+\frac{x^{\prime 5}\bar{x}^{\prime}}{15360}+\frac{7x^{\prime
4}\bar{x}^{\prime 2}}{3072}-\frac{7x^{\prime 2}\bar{x}^{\prime
4}}{3072}-\frac{x^{\prime}\bar{x}^{\prime 5}}{15360}+\frac{\bar{x}^{\prime
6}}{15360}\right)~{}.$ (18) $\displaystyle h^{\prime}_{6}$
$\displaystyle=-\frac{x^{\prime 3}\bar{x}^{\prime 3}}{256\omega_{0}}~{}.$ (19)
If we carried on with the expansion of Equation (13) to higher order, we would
find that at each step, we need to solve an equation of the form
$h^{\prime}_{2n}=[h_{2},\chi_{2n}]+\Psi_{2n}$ (20)
where $\Psi_{2n}$ is a collection of terms of order $2n$ involving Poisson
brackets of the known functions $h_{4},h_{6},...,h_{2n}$ and
$\chi_{4},...,\chi_{2n-2}$. In the next section we will detail an algorithm
for iteratively constructing and solving _homological equations_ like Equation
(20).
Let us conclude our present example by approximating the pendulum’s
oscillation frequency as a function of maximum angular displacement,
$\theta_{\mathrm{max}}$, in order to illustrate how functions of the new,
transformed variables, $x^{\prime}$ and $\bar{x}^{\prime}$, may be calculated
explicitly from the original coordinates and vice versa. Our transformed
Hamiltonian is given by
$h^{\prime}=\omega_{0}|x^{\prime}|^{2}-\frac{1}{16}|x^{\prime}|^{4}-\frac{1}{256\omega_{0}}|x^{\prime}|^{6}+\mathcal{O}(|x^{\prime}|^{8})$
(21)
so the oscillation frequency will be given, in terms of the action
$J=|x^{\prime}|^{2}$, by
$\omega(J)=\omega_{0}-\frac{1}{8}J-\frac{3}{256\omega_{0}}J^{2}$. We therefore
need the value of $J$ in terms of $\theta_{\mathrm{max}}$.
Given any function, $f$, on the phase space expressed in terms of the
transformed variables, $x^{\prime}$ and $\bar{x}^{\prime}$, its expression in
terms of the original, un-primed variables $x$ and $\bar{x}$ is given by
$f\left(x^{\prime}(x,\bar{x}),\bar{x}^{\prime}(x,\bar{x})\right)=(\exp[-\mathcal{L}_{\chi}]f)(x,\bar{x})$
(22)
since $\exp[-\mathcal{L}_{\chi}]f$ gives the function $f$ evaluated at the
time-($-1$) flow of the Hamiltonian vector field generated by $\chi$. Applying
Equation (22) to the function
$\omega:(x^{\prime},\bar{x}^{\prime})\mapsto\omega_{0}-\frac{1}{8}|x^{\prime}|^{2}-\frac{3}{256\omega_{0}}|x^{\prime}|^{4}$,
we obtain the oscillation frequency in terms of the original, un-transformed
complex variable
$x=\sqrt{\frac{\omega_{0}}{2}}(\theta+\mathrm{i}p/\omega_{0})$ as
$\displaystyle(\exp[-\mathcal{L}_{\chi}]\omega)(x,\bar{x})$
$\displaystyle=\omega_{0}-\frac{|x|^{2}}{8}-\frac{3}{256}|x|^{4}+\frac{1}{8}\left[|x|^{2},\chi_{4}\right]+\mathcal{O}(|x|^{6})$
(23)
$\displaystyle=\omega_{0}-\frac{x\bar{x}}{8}+\frac{1}{\omega_{0}}\left(\frac{\bar{x}^{4}}{768}+\frac{\bar{x}^{3}x}{192}-\frac{3\bar{x}^{2}x^{2}}{256}+\frac{\bar{x}x^{3}}{192}+\frac{x^{4}}{768}\right)+\mathcal{O}(|x|^{6})~{}.$
(24)
Substituting $x=\sqrt{\frac{\omega_{0}}{2}}\theta_{\mathrm{max}}$ into
Equation (23) gives
$\omega\approx\omega_{0}\left(1-\frac{\theta_{\max}^{2}}{16}+\frac{\theta_{\max}^{4}}{3072}+...\right),$
(25)
which matches the first three terms of the Taylor expansion of the exact
result,
$\omega=\frac{\pi\omega_{0}}{2\mathbb{K}\left(\sin^{2}\left(\frac{\theta_{\max}}{2}\right)\right)}$,
where $\mathbb{K}$ denotes the complete elliptic integral of the first kind
(e.g., Lichtenberg & Lieberman, 1992).
### 2.4 The general algorithm
The Birkhoff normalization procedure, illustrated above in the case of the
pendulum, readily generalizes to systems with $d>1$ degrees of freedom.
Specifically, the procedure can be applied in the vicinity of an elliptic
equilibrium to construct a formal series for a generating function, $\chi$,
that transforms the Hamiltonian so that it only depends on $d$ action-like
variables. While these formal series are, in general, divergent, their finite-
order truncations are nonetheless often very useful for approximating a given
system’s dynamics (e.g., Contopoulos, 2003; Efthymiopoulos et al., 2004). The
numerical results presented in later in Section 3 show that the approximate
integrals computed for our chosen order of truncation are very nearly
constants over a large region of the phase space.
We apply the Birkhoff normalization procedure to orbits in the vicinity of
planar, circular orbits of axisymmetric galactic potentials, which are
elliptic equilibria of the two degree-of-freedom Hamiltonian given in Equation
(2). In terms of the complex canonical variables $x_{R}$ and $x_{z}$
introduced in Section 2.2, the Hamiltonian can be written as
$\displaystyle H=\kappa x_{R}\bar{x}_{R}+\nu\kappa
x_{z}\bar{x}_{z}+\sum_{n=3}^{\infty}H_{n}(x_{R},x_{z},\bar{x}_{R},\bar{x}_{z})$
(26)
where
$\displaystyle
H_{n}(x_{R},x_{z},\bar{x}_{R},\bar{x}_{z})=\sum_{m=0}^{n}\frac{\Phi_{\mathrm{eff}}^{(m,n-m)}}{(n-m)!m!}\left(\sqrt{\frac{1}{2\kappa}}\right)^{n-m}\left(\sqrt{\frac{1}{2\nu}}\right)^{m}\sum_{p=0}^{m}\sum_{q=0}^{n-m}\binom{m}{p}\binom{n-m}{q}x_{R}^{p}x_{z}^{q}\bar{x}_{R}^{m-p}\bar{x}_{z}^{n-m-q}$
(27)
and $\Phi_{\mathrm{eff}}^{(m,n-m)}=\frac{\partial^{n}}{\partial
R^{m}z^{n-m}}\Phi_{\mathrm{eff}}\big{|}_{(R,z)=(R_{C},0)}$. Values of the
partial derivatives of the effective potential,
$\Phi_{\mathrm{eff}}^{(m,n-m)}$, up to the desired maximum order of the
perturbative expansion are required to carry out the Birkhoff normalization
procedure. Calculation of these derivatives is most conveniently done with the
aid of a computer algebra system or automatic differentiation packages.
Equation (26) represents the Hamiltonian as a series of terms grouped by their
order in the complex canonical variables. We will construct similar series for
our generating function, $\chi=\sum_{k=k_{\mathrm{min}}}^{\infty}\chi_{k}$,
and the transformed Hamiltonian,
$H^{\prime}=\sum_{m=1}^{\infty}H^{\prime}_{2m}$ through an iterative process.
Note that because $H^{\prime}$ depends only on the action variables, it only
contains even-order terms.
#### 2.4.1 Grouping Lie Series Terms by Order
To construct our series solution, it will be necessary to group terms of the
same order in the expansion of the Lie series
$H^{\prime}=\exp[\mathcal{L}_{\chi}]H$. This is most readily accomplished
using recursion formulae originally derived by Deprit (1969). Below, we give a
derivation of these formulae that closely follows the presentation of Ferraz-
Mello (2007, Chapter 5).
If $f_{k}$ and $g_{l}$ are homogeneous polynomials of degree $k$ and $l$,
respectively, in complex canonical variables then $[f_{k},g_{l}]$ is a
homogeneous polynomial of degree $k+l-2$ in those variables. Define
$D_{k}:=[\cdot,\chi_{k}]$ so that, given series expansions
$f=\sum_{k^{\prime}=k^{\prime}_{\mathrm{min}}}^{\infty}f_{k^{\prime}}$, and
$\chi=\sum_{k=k_{\mathrm{min}}}^{\infty}\chi_{k}$, we can write
$\displaystyle\mathcal{L}_{\chi}f$
$\displaystyle=\sum_{k=k_{\mathrm{min}}}^{\infty}\sum_{k^{\prime}=k^{\prime}_{\mathrm{min}}}^{\infty}D_{k}f_{k^{\prime}}=\sum_{l=k_{\mathrm{min}}+k^{\prime}_{\mathrm{min}}-2}^{\infty}\sum_{k=k_{\mathrm{min}}}^{l+2-k^{\prime}_{\mathrm{min}}}D_{k}f_{l+2-k}:=\sum_{l=l_{\mathrm{min}}^{(1)}}^{\infty}\Upsilon^{1}_{l}(f,\chi)$
where $l_{\mathrm{min}}^{(1)}=k_{\mathrm{min}}+k^{\prime}_{\mathrm{min}}-2$
and $\Upsilon^{1}_{l}(f,\chi)$ represents all terms of order $l$ appearing in
an expansion of $\mathcal{L}_{\chi}f$. If we now substitute
$\mathcal{L}_{\chi}f$ for $f$ in Equation (LABEL:eq:Df_defn), we obtain
$\mathcal{L}^{2}_{\chi}f=\sum_{l=k_{\mathrm{min}}+l_{\mathrm{min}}^{(1)}-2}^{\infty}\sum_{k=k_{\mathrm{min}}}^{l+2-l_{\mathrm{min}}^{(1)}}D_{k}\Upsilon^{1}_{l+2-k}(f,\chi):=\sum_{l=l_{\mathrm{min}}^{(2)}}^{\infty}\Upsilon^{2}_{l}(f,\chi)$
(29)
where $l_{\mathrm{min}}^{(2)}=k_{\mathrm{min}}+l^{(1)}_{\mathrm{min}}-2$. More
generally, we have
$\displaystyle l_{\mathrm{min}}^{(n)}$
$\displaystyle=k_{\mathrm{min}}+l^{(n-1)}_{\mathrm{min}}-2$ (30)
$\displaystyle\mathcal{L}^{n}_{\chi}f$
$\displaystyle=\sum_{l=l_{\mathrm{min}}^{(n)}}^{\infty}\sum_{k=k_{\mathrm{min}}}^{l+2-l_{\mathrm{min}}^{\mathrm{(n-1)}}}D_{k}\Upsilon^{(n-1)}_{l+2-k}(f,\chi):=\sum_{l=l_{\mathrm{min}}^{(n)}}^{\infty}\Upsilon^{n}_{l}(f,\chi)$
(31)
In other words, $\Upsilon^{n}_{l}$ is the collection of order $l$ terms
arising from the $n$th Lie derivative in our expansion. Values of
$\Upsilon^{n}_{l}$ for a given $n$ and $l$ can be computed using the recursion
formulae
$\displaystyle\Upsilon^{0}_{l}(f,\chi)$ $\displaystyle=f_{l}$ (32)
$\displaystyle l_{\mathrm{min}}^{(0)}$
$\displaystyle=k^{\prime}_{\mathrm{min}}$ (33)
$\displaystyle\Upsilon^{n+1}_{l}(f,\chi)$
$\displaystyle=\sum_{k=k_{\mathrm{min}}}^{l+2-l^{(n)}_{\mathrm{min}}}D_{k}\Upsilon^{n}_{l+2-k}(f,\chi)$
(34) $\displaystyle l_{\mathrm{min}}^{(n+1)}$
$\displaystyle=k_{\mathrm{min}}+l^{(n)}_{\mathrm{min}}-2~{}.$ (35)
#### 2.4.2 Expansion of the Transformed Hamiltonian
We can now write the transformed Hamiltonian,
$\exp[\mathcal{L}_{\chi}]H=\sum_{n=0}^{\infty}\frac{1}{n!}{\mathcal{L}^{n}_{\chi}}H$,
in terms of $\Upsilon_{l}^{n}(H,\chi)$. Since the lowest order terms appearing
in $H$ are order $k^{\prime}_{\mathrm{min}}=2$, Equation (35) becomes
$l_{\mathrm{min}}^{(n)}=2+(k_{\mathrm{min}}-2)n$. We have
$\exp[\mathcal{L}_{\chi}]H=\sum_{n=0}^{\infty}\sum_{l=l_{\mathrm{min}}^{(n)}}^{\infty}\frac{1}{n!}\Upsilon^{n}_{l}(H,\chi)=\sum_{l=2}^{\infty}\sum_{n=0}^{N^{(l)}}\frac{1}{n!}\Upsilon^{n}_{l}(H,\chi):=\sum_{l=2}^{\infty}(\Psi_{l}+[H_{2},\chi_{l}])$
(36)
where we define $N^{(l)}=\lfloor\frac{l-2}{k_{\mathrm{min}}-2}\rfloor$ and
$\Psi_{l}=H_{l}+\sum_{k=k_{\mathrm{min}}}^{l-1}D_{k}H_{l+2-k}+\sum_{n=2}^{N^{(l)}}\frac{1}{n!}\Upsilon^{n}_{l}(H,\chi)~{}.$
(37)
The expression in Equation (37) defining $\Psi_{l}$ only contains terms,
$\chi_{k}$, in the expansion of the generating function $\chi$ with $k<l$. The
terms of order $l$ gathered together in the sum in Equation (36) are, by
definition, the terms of order $l$ in the transformed Hamiltonian,
$H^{\prime}$. In other words, we have
$\Psi_{l}+[H_{2},\chi_{l}]=H^{\prime}_{l}~{}.$ (38)
Since $\Psi_{l}$ contains only terms $\chi_{k}$ with $k<l$, we can iterate
over $l$, solving for $H^{\prime}_{l}$ and $\chi_{l}$ at each stage. Equation
(38) is referred to as the homological equation.
#### 2.4.3 Solution of Homological Equation
Recall that $\Psi_{l}$ is a homogeneous polynomial of degree $l$ in the
complex variables
$(x^{\prime}_{R},x^{\prime}_{z},\bar{x}^{\prime}_{R},\bar{x}^{\prime}_{z})$,
i.e., a sum of terms $\propto x_{R}^{\prime k_{R}}x_{z}^{\prime
k_{z}}\bar{x}_{R}^{\prime\bar{k}_{R}}\bar{x}_{z}^{\prime\bar{k}_{z}}$ with
$k_{R}+k_{z}+\bar{k}_{R}+\bar{k}_{z}=l$. Let $\langle\Psi_{l}\rangle$ denote
the collection of terms in $\Psi_{l}$ with $k_{R}=\bar{k}_{R}$ and
$k_{z}=\bar{k}_{z}$ and let
$\\{\Psi_{l}\\}:=\Psi_{l}-\langle\Psi_{l}\rangle=\sum_{i}C_{i}x_{R}^{\prime
k_{R,i}}x_{z}^{\prime
k_{z,i}}\bar{x}_{R}^{\prime\bar{k}_{R,i}}\bar{x}_{z}^{\prime\bar{k}_{z,i}}~{}.$
(39)
Also note that
$[H_{2},\chi_{k}]=-\mathrm{i}\kappa\left(\bar{x}^{\prime}_{R}\frac{\partial\chi_{k}}{\partial\bar{x}^{\prime}_{R}}-{x}^{\prime}_{R}\frac{\partial\chi_{k}}{\partial{x}^{\prime}_{R}}\right)-\mathrm{i}\nu\left(\bar{x}^{\prime}_{z}\frac{\partial\chi_{k}}{\partial\bar{x}^{\prime}_{z}}-{x}^{\prime}_{z}\frac{\partial\chi_{k}}{\partial{x}^{\prime}_{z}}\right)~{}.$
(40)
At each iteration, we solve Equation (38) by taking
$\displaystyle H^{\prime}_{l}$ $\displaystyle=\langle\Psi_{l}\rangle$ (41)
$\displaystyle\chi_{l}$
$\displaystyle=\sum_{i}\frac{\mathrm{i}C_{i}}{(k_{R,i}-\bar{k}_{R,i})\kappa+(k_{z,i}-\bar{k}_{z,i})\nu}x_{R}^{\prime
k_{R,i}}x_{z}^{\prime
k_{z,i}}\bar{x}_{R}^{\prime\bar{k}_{R,i}}\bar{x}_{z}^{\prime\bar{k}_{z,i}}~{}.$
(42)
Equation (42) assumes a non-resonance condition on $\kappa$ and $\nu$: we
cannot have terms333Such resonant terms can instead be incorporated into
$H^{\prime}_{l}$ by simply omitting the corresponding terms from the sum in
Equation (42) defining $\chi_{l}$. The transformed Hamiltonian would, however,
no longer be a function of actions alone. with
$(k_{R,i}-\bar{k}_{R,i})\kappa+(k_{z,i}-\bar{k}_{z,i})\nu=0$.
### 2.5 Azimuthal degree of freedom
Thus far we have only considered the orbits of stars in the reduced phase
space, ignoring their azimuthal degree of freedom. Quasi-periodic orbits in
the full 6 dimensional phase space will be characterized by a third dynamical
frequency, $\Omega_{\phi}$, in addition to the two frequencies,
$\Omega_{R}=\frac{1}{x_{R}}\frac{\partial}{\partial\bar{x}_{R}}H^{\prime}$ and
$\Omega_{z}=\frac{1}{x_{z}}\frac{\partial}{\partial\bar{x}_{z}}H^{\prime}$,
readily obtained from the Birkhoff normalization procedure. Here we detail how
this third frequency can be determined and used to define an additional angle
variable, $\theta_{\phi}$, once the Birkhoff normalization procedure has been
applied to solve for orbits’ dynamical evolution in the reduced phase space.
The time derivative of the azimuthal angle, $\phi$ is given in terms of
complex canonical variables by
$\displaystyle\frac{d}{dt}{\phi}=\frac{L}{R_{C}^{2}}\left(1+\frac{x_{R}+\bar{x}_{R}}{R_{C}\sqrt{2\kappa}}\right)^{-2}~{}.$
(43)
For non-circular orbits, the right-hand side of Equation (43) is not constant
in time but will oscillate about a mean value that defines the azimuthal
frequency, $\Omega_{\phi}$. Introducing the canonical angle variable
$\theta_{\phi}=\Omega_{\phi}t+\phi_{0}$ we can write the $\phi$ coordinate
explicitly as
$\displaystyle\phi=\theta_{\phi}+\rho_{\phi}(x^{\prime}_{R},x^{\prime}_{z},\bar{x}^{\prime}_{R},\bar{x}^{\prime}_{z},L)$
(44)
where $\rho$ is a zero-mean $2\pi$-periodic function of
$\theta_{R}=-\arg{x^{\prime}_{R}}$ and $\theta_{z}=-\arg{x^{\prime}_{z}}$. ,
We can combine Equations (43) and (44) to express $\Omega_{\phi}$ and
$\rho_{\phi}$ in terms of the variables, $x^{\prime}_{R}$ and
$x^{\prime}_{z}$, and their complex conjugates. Recalling the notation
$\langle\cdot\rangle$ and $\\{\cdot\\}$ introduced above in Section 2.4.3 to
denote mean and oscillating terms of a Poisson series, the equations
$\displaystyle\Omega_{\phi}$ $\displaystyle=$
$\displaystyle\frac{L}{R_{C}^{2}}\left\langle\left(\exp[\mathcal{L}_{\chi}]\sum_{m=0}^{\infty}\binom{-2}{m}\left(\frac{x_{R}+\bar{x}_{R}}{R_{C}\sqrt{2\kappa}}\right)^{m}\right)\right\rangle$
(45) $\displaystyle\frac{d}{dt}\rho_{\phi}$ $\displaystyle=$
$\displaystyle\frac{L}{R_{C}^{2}}\left\\{\left(\exp[\mathcal{L}_{\chi}]\sum_{m=0}^{\infty}\binom{-2}{m}\left(\frac{x_{R}+\bar{x}_{R}}{R_{C}\sqrt{2\kappa}}\right)^{m}\right)\right\\}~{}$
(46)
define $\Omega_{\phi}$ and the time derivative of $\rho_{\phi}$ as series in
the variables $x^{\prime}_{R}$ and $x^{\prime}_{z}$ and their complex
conjugates that can be expanded up to some maximum desired order. In order to
derive an expression for $\rho_{\phi}$, we express the series on the right
hand side of Equation (46) as a sum of monomial terms
$\displaystyle\left\\{\left(\exp[\mathcal{L}_{\chi}]\sum_{m=0}^{\infty}\binom{-2}{m}\left(\frac{x_{R}+\bar{x}_{R}}{R_{C}\sqrt{2\kappa}}\right)^{m}\right)\right\\}=\sum_{i}A_{i}x_{R}^{\prime
k_{R,i}}x_{z}^{\prime
k_{z,i}}\bar{x}_{R}^{\prime\bar{k}_{R,i}}\bar{x}_{z}^{\prime\bar{k}_{z,i}}$
(47)
an then integrate with respect to time to obtain
$\displaystyle\rho(x^{\prime}_{R},x^{\prime}_{z},\bar{x}^{\prime}_{R},\bar{x}^{\prime}_{z},L)=\mathrm{i}\frac{L}{R_{C}^{2}}\sum_{i}\frac{A_{i}x_{R}^{\prime
k_{R,i}}x_{z}^{\prime
k_{z,i}}\bar{x}_{R}^{\prime\bar{k}_{R,i}}\bar{x}_{z}^{\prime\bar{k}_{z,i}}}{(k_{R,i}-\bar{k}_{R,i})\kappa+(k_{z,i}-\bar{k}_{z,i})\nu}~{}.$
(48)
It follows immediately from Hamilton’s equations that $J_{\phi}=L$ serves as
canonical action coordinate conjugate to the angle $\theta_{\phi}$. We
therefore have established a complete canonical transformation from the
original variables $(R,z,\phi,p_{R},p_{z},L)$ of the 6 dimnensional phase
space into to a complete set AA variables,
$(\theta_{R},\theta_{z},\theta_{\phi},J_{R},J_{z},J_{\phi})$ where
$J_{i}=|x^{\prime}_{i}|^{2}$ and $\theta_{i}=-\arg x^{\prime}_{i}$ for the
subscript $i\in\\{z,R\\}$.
### 2.6 Numerical Implementation
The Birkhoff normalization algorithm described above is implemented in the
poisson_series module of the open-source Python package celmech (Hadden &
Tamayo, 2022). This module provides routines for representing and manipulating
so-called “Poisson series” (Brumberg, 1995). These are series in $N$ complex
canonical variables $x_{i}$ and their complex conjugates, $\bar{x}_{i}$ along
with $M$ real action-angle pairs, $(P_{i},Q_{i})$, that are given as sums of
individual monomial terms of the form
${\cal
M}(k,\bar{k},p,q)=\prod_{i=1}^{N}x_{i}^{k_{i}}\bar{x}_{i}^{\bar{k}_{i}}\prod_{j=1}^{M}P_{j}^{p_{j}}\exp[iq_{j}Q_{j}]~{}.$
(49)
The Poisson bracket of any pair of Poisson series is another Poisson series,
as can be readily seen by considering the Poisson bracket of two monomial
terms. Defining the length-$N$ vectors $o^{N}_{i}$ such that
$[o^{N}_{i}]_{j}=\delta_{ij}$ where $\delta_{ij}$ is the Kroenecker delta, the
Poisson bracket of two monomial terms is then a Poisson series consisting of
(at most) $N+M$ new monomial terms and given by
$[{\cal M}(k,\bar{k},p,q),{\cal
M}(l,\bar{l},r,s)]=\mathrm{i}\sum_{i=1}^{N}(\bar{k}_{i}{l}_{i}-k_{i}\bar{l}_{i}){\cal
M}(k+l-o^{N}_{i},\bar{k}+\bar{l}-o^{N}_{i},p+r,q+s)+\\\
\mathrm{i}\sum_{j=1}^{M}(q_{j}r_{j}-s_{j}p_{j}){\cal
M}(k+l,\bar{k}+\bar{l},p+r-o^{M}_{j},q+s)~{}.$ (50)
The poisson_series module defines the PoissonSeries class to store sums of
monomial terms of the form given in Equation (49) multiplied by numerical
coefficients. It also defines a number of routines for manipulating these
series, including performing scalar multiplication, multiplying or adding
pairs of series, and calculating Poisson brackets. Note that the Poisson
series used in this paper involve only complex canonical variables (i.e.,
$M=0$).
The poisson_series module additionally defines routines for carrying out the
Birkhoff normalization procedure and evaluating finite-order approximations of
exponential operators, $\exp[\mathcal{L}_{\chi}]$, applied to series. These
routines require grouping terms in the expansions of various functions, such
as the Hamiltonian, $H$, and the generating function, $\chi$, by their order
in powers of the complex canonical variables. In the framework of the
poisson_series module, this grouping is done by using Python dictionary
objects to associate PoissonSeries objects representing terms in an expansion
of a given order to key values that indicate their order. Example Jupyter
notebooks illustrating the use of celmech’s poisson_series module to carry out
the Birkhoff normalization procedure are available at
github.com/shadden/AA_variables_via_Birkhoff_Normalization.
### 2.7 Padé Approximants
Since the Birkhoff normalization procedure is based upon a Taylor expansion of
the effective potential, the method can only be expected to provide accurate
results for orbits that remain within regions of phase space where this
expansion is convergent. In practice, this can make the procedure inapplicable
to orbits that experience significant excursions in the vertical direction.
For example, the Miyamoto-Nagai potential,
$\Phi(R,z)=-\frac{1}{\sqrt{R^{2}+(a+\sqrt{z^{2}+b^{2}})^{2}}}~{}$ (51)
with parameters $a$ and $b$, is a commonly adopted as a model for the galactic
disk’s contribution to the gravitational potential (Miyamoto & Nagai, 1975).
The parameter $b$ effectively sets the thickness of the galactic disk and the
branch points of Equation (51) occurring in the complex plane at $z=\pm ib$
restrict the convergence Taylor series expansions to $|z|<b$.
Accurate local approximations of functions can sometimes be extended beyond
the domain of convergence of their Taylor series with the use of Padé
approximants. The $(n,m)$ Padé approximant of a function is the rational
function with numerator of degree $m$ and denominator of degree $n$ that
agrees with the Taylor series expansion of the function up to degree $x^{m+n}$
(e.g., Teukolsky et al., 1992). If the first $n+m$ terms of a function’s
Taylor series are $\sum_{k=0}^{n+m}c_{k}x^{k}$ then the coefficients of an
$(n,m)$ Padé approximant are readily obtained by multiplying both left and
right hand sides of
$\displaystyle\sum_{k=0}^{n+m}c_{k}x^{k}=\frac{a_{0}+a_{1}x+...a_{m}x^{m}}{1+b_{1}x+...+b_{n}x^{n}}\mod{x^{n+m+1}}$
(52)
by $1+b_{1}x+...+b_{n}x^{n}$. The coefficients, $b_{i}$, satisfy the linear
equation
$\displaystyle\begin{pmatrix}c_{n+m-1}&\ldots&c_{m}\\\ \vdots&\ddots&\vdots\\\
c_{m}&\ldots&c_{m-n+1}\end{pmatrix}\cdot\begin{pmatrix}b_{1}\\\ \vdots\\\
b_{n}\end{pmatrix}=-\begin{pmatrix}c_{n+m}\\\ \vdots\\\ c_{m+1}\end{pmatrix}.$
(53)
and the coefficients, $a_{i}$, are then given by
$a_{i}=\sum_{k=0}^{n}c_{i-k}b_{k}$ where $b_{0}:=1$ and $c_{k}$ is taken to be
$0$ for $k<0$.
Below in Section 3, we show that constructing Padé approximants in the
variable $I_{z}=|x_{z}|^{2}$, often improves the accuracy of the Birkhoff
normalization procedure. The Padé approximants involve coefficients, $b_{i}$
and $a_{i}$, that are functions of $x_{R},~{}\bar{x}_{R}$ and
$\phi_{z}=-\arg(x_{z})$. These coefficients are determined by evaluating
finite-order truncations of the series
$x^{\prime}_{R}=\exp[-\mathcal{L}_{\chi}]x_{R}$ and
$x^{\prime}_{z}=\exp[-\mathcal{L}_{\chi}]x_{z}$ and then grouping terms by
powers of $I_{z}$. Note that the generating function, $\chi$, is developed as
a series in powers of $x_{z}$ and $\bar{x}_{z}$, which are both proportional
to $\sqrt{I_{z}}$. However, it is straightforward to show that for any
$\Phi(R,z)$ like Equation (51) that is an even function of $z$, only integer
powers of $I_{z}$ will appear in $\chi$ as well as finite-order truncations of
$\exp[-\mathcal{L}_{\chi}]x_{R}$. Thus, these finite-order truncations are
polynomials in $I_{z}$ (rather than $\sqrt{I_{z}}$) and can be used to obtain
Padé approximants in $I_{z}$. Additionally, finite-order truncations of
$\exp[-\mathcal{L}_{\chi}]x_{z}$ yield expressions that can be written as
$x_{z}P(I_{z})$ where $P$ is a polynomial in $I_{z}$ with coefficients that
depend on $x_{R},\bar{x}_{R},$ and $\phi_{z}$. Padé approximants are derived
from the polynomial $P$ and used to calculate $\exp[-\mathcal{L}_{\chi}]x_{z}$
in Section 3.
## 3 Tests
In this section we apply the Birkhoff normalization procedure to orbits in the
Miyamoto-Nagai potential in Equation (51). The parameters $a=3$, $b=0.3$, and
$L=3$ are chosen, with a corresponding circular orbit radius of
$R_{C}(L)\approx 10.4$. With distance units taken to be kpc, these parameters
approximately match the parameters adopted for the Miyamoto-Nagai component of
the Milky Way potential model, MWPotential2014, implemented in the galpy
Python package (Bovy, 2015). The Birkhoff normalization procedure is carried
out by developing the generating function, $\chi$, up to 10th order in complex
canonical variables. The sympy symbolic mathematics package (Meurer et al.,
2017) is used to calculate the requisite partial derivatives of the effective
potential.
We also compute actions and frequencies using the Stäckel approximation method
introduced by Binney (2012). These calulations are done using the
actionAngle.actionAngleStäckel function implemented in galpy (Bovy & Rix,
2013; Bovy, 2015). This function requires as input an estimated focal length
parameter, $\Delta$, in addition to a potential and orbital initial
conditions. For all orbits considered, The galpy function
actionAngle.estimateDeltaStäckel is used to estimate the focal length based on
partial derivatives of the potential evaluated at $(R,z)=(R_{C},0)$ following
Sanders (2012).
Figure 1: Color maps showing r.m.s. fractional variations in estimated values
of actions on a grid of orbits in a Miyamoto-Nagai potential (Equation 51)
with parameters $a=3$ and $b=0.3$. Actions are computed along orbits starting
with $(L,R,z)=(3,R_{C},0)$ and initial velocity values indicated by the axes,
which are plotted in units of the circular velocity, $v_{C}=L/R_{C}$. The top
row shows variations in the radial action, $J_{R}$, while the bottom row shows
variations in the vertical action, $J_{z}$. Columns are labeled according to
the method used to compute action values. See text for additional details.
Dash-dot, dashed, and solid contours mark fractional error levels of
$0.1\%,1\%$ and $10\%$, respectively.
Figure 1 illustrates the accuracy of actions determined by Birkhoff
normalization and Stäckel approximation across a range of initial conditions.
Orbits are initialized at $(R,z)=(R_{C},0)$ over the plotted range of
$(p_{R},p_{z})$ values and integrated for 10 times the radial epicyclic period
(i.e., $T=10\times(2\pi/\kappa)$) with 512 outputs generated at equally-spaced
intervals along the orbits. Transformed complex canonical variable,
$x^{\prime}_{i}$, are computed as functions of the un-transformed variables
along a given orbit using various approximations to the series
$\displaystyle
x^{\prime}_{i}(x_{R},x_{z},\bar{x}_{R},\bar{x}_{z})=(\exp[-\mathcal{L}_{\chi}]x_{i})(x_{R},x_{z},\bar{x}_{R},\bar{x}_{z})$
(54)
where the subscript $i$ denotes $R$ or $z$. Action values are then computed
from the transformed complex canonical variable as
$J_{i}=|x^{\prime}_{i}|^{2}$. The root-mean-square (r.m.s.) fractional
variations in these action values computed along each orbit are recorded in
the color scale.
The leftmost column of Figure 1 show the r.m.s. variations in actions computed
using the epicyclic approximation, where Equation (54) is approximated simply
as $x_{i}^{\prime}\approx x_{i}$. The variations in actions computed via this
approximation provide a measure of how significantly orbits are affected by
the anharmonicity of the potential. The second column of Figure 1 shows the
r.m.s. variations for actions computed by expanding the right hand side of
Equation (54) as a Taylor series. With $\chi$ determined up to 10th order in
the canonical variables, the terms of these series are determined up to 9th
order in the complex canonical variables. The third and fourth columns show
r.m.s. variations for actions computed by constructing $(3,1)$ and $(2,2)$
Padé approximants in $I_{z}$ as described in Section 2.7. The final columns
the r.m.s. variations for actions computed using the Stäckel approximation.
The Taylor series approximations of actions in Figure 1 provide good agreement
with the numerical results for mid-plane vertical velocity values
$p_{z}\lesssim 0.1v_{C}$, where $v_{C}=L/R_{C}$ is the circular orbital
velocity. The poor agreement for greater vertical velocities is attributable
to the fact that orbits with greater vertical velocities reach heights $z>b$,
beyond the radius of convergence of the Taylor series expansion of the
potential in Equation (51). An estimate of the maximum vertical height,
$z_{\mathrm{max}}$, achieved by an orbit with a given vertical velocity,
$p_{z,0}$, in the midplane can be obtained by neglecting any motion in the
radial direction and using conservation of energy to equate
$\frac{1}{2}p_{z,0}\approx\Phi_{\mathrm{eff}}(R_{C},z_{\mathrm{max}})-\Phi_{\mathrm{eff}}(R_{C},0)$.
Taking $z_{\mathrm{max}}=b$, the corresponding midplane velocity is
$p_{z,0}\approx 0.09v_{C}$, closely matching the value of mid-plane velocity
above which the Taylor series approximations fail in Figure 1. The Padé
approximants clearly extend the range over which the Birkhoff normalization
procedure accurately predicts actions, achieving r.m.s. variations in radial
actions of $\lesssim 10\%$ for initial vertical velocities $p_{z}\lesssim
0.2v_{C}$. The (2,2) Padé approximant is slightly more accurate than the (3,1)
approximant. Performance of the algorithm for vertical action determination is
slightly worse, with $\lesssim 10\%$ accuracy restricted to initial vertical
velocities $p_{z}\lesssim 0.15v_{C}$.
Figure 2 compares the accuracy of action determinations made using Birkhoff
normalization with (2,2) Padé approximants against the Stäckel approximation.
Birkhoff normalization provides a more accurate determination of radial
actions over the majority of the plotted range. Vertical actions are also more
accurately determined for $p_{z}\lesssim 0.1v_{C}$.
Figure 2: Comparison of the accuracy of actions determined using Birkhoff
normalization and (2,2) Padé approximants versus the Stäckel approximation.
The color scale indicates the r.m.s. variation in actions computed using the
Stäckel approximation divided by r.m.s. variations computed via Birkhoff
normalization so that red colors indicate orbits where Birkhoff normalization
yields better accuracy while blue colors indicate orbits for which the Stäckel
approximation is more accurate. The grid of orbits is the same as shown in
Figure 1.
Figure 3 illustrates the accuracy of the determination of the dynamical
frequencies $\Omega_{R}$, $\Omega_{z}$, and $\Omega_{\phi}$ using various
methods. Exact values of frequencies are determined numerically using the
frequency modified Fourier transform algorithm of Šidlichovský & Nesvorný
(1996), implemented in the celmech package as the function
miscellaneous.frequency_modified_fourier_transform. The method was applied to
the time series of complex canonical variables, $x_{R}$ and $x_{z}$, for each
orbit integration and the frequencies of the largest Fourier components
determined for each variable were identified as $-\Omega_{R}$ and
$-\Omega_{z}$, respectively. Values of $\Omega_{\phi}$ were determined by
applying the same method to time series of $e^{\mathrm{i}\phi}$. The first
column of Figure 3 records the fractional errors $\kappa/\Omega_{R}-1$,
$\nu/\Omega_{z}-1$, and $R_{C}^{-2}L/\Omega_{\phi}-1$ incurred by adopting the
epicyclic approximation for the dynamical frequencies. These panels illustrate
that the frequencies $\Omega_{R}$ and $\Omega_{\phi}$ vary gradually with both
initial radial and vertical velocities while the frequency $\Omega_{z}$ is a
strong function of the initial vertical velocity.
The second and third columns of Figure 3 record fractional errors incurred by
using Birkhoff normalization and Padé approximants to determine frequencies.
Frequencies were determined as follows. First, values of complex canonincal
variables were assigned based on orbits’ initial conditions as
$x_{R}=\mathrm{i}\sqrt{\frac{1}{2\kappa}}p_{R}$ and
$x_{z}=\mathrm{i}\sqrt{\frac{1}{2\nu}}p_{z}$ (see Equations 5 and 6). Next,
values of transformed complex canonical variables, $x^{\prime}_{R}$ and
$x^{\prime}_{z}$, were computed using the Padé approximation method described
in Section 2.7. Then, series expressions for frequencies,
$\Omega_{R}(J_{R},J_{z})=\partial_{J_{R}}H^{\prime}$ and
$\Omega_{z}(J_{R},J_{z})=\partial_{J_{z}}H^{\prime}$, derived from the
transformed Hamiltonian, $H^{\prime}$, and the series expression for
$\Omega_{\phi}$ given by Equation (45) were computed. Finally, these series
expressions were used to construct a second set of (3,1) and (2,2) Padé
approximants, this time in $J_{z}=|x^{\prime}_{z}|^{2}$, and evaluated.
Comparing Figures 1 and 3, we see that dynamical frequencies are accurately
recovered in essentially the same regions of phase space that Birkhoff
normalization and Padé approximants provide accurate action variables. The
fourth column of Figure 3 shows frequency errors incurred using the Stäckel
approximation for comparison.
Figure 3: Color maps showing fractional errors in the estimated values of
radial, vertical, and azimuthal frequencies on a grid of orbits in the
Miyamoto-Nagai potential (Equation (51)) with parameters $a=3$ and $b=0.3$.
Frequencies are computed for orbits starting with $(L,R,z)=(3,R_{C},0)$ and
initial velocity values indicated by the axes, which are plotted in units of
the circular velocity, $v_{C}=L/R_{C}$. The top, middle, and bottom rows shows
fractional errors in the radial ($\Omega_{R}$), vertical ($\Omega_{z}$), and
azimuthal ($\Omega_{\phi}$) frequencies, respectively. Columns are labeled
according to the method used to compute the frequency values. See text for
details. Dash-dot, dashed, and solid contours mark fractional error levels of
$0.1\%,1\%$ and $10\%$, respectively.
## 4 Summary & Discussion
We have shown how Birkhoff normalization, a technique from Hamiltonian
perturbation theory, can be algorithmically implemented and applied to derive
AA variables for stellar orbits in axisymmetric galactic potentials. One
significant advantage of this method is that it provides explicit expressions
for actions and angle variables as functions of positions and velocities and
vice versa. It similarly yields an explicit expression for the Hamiltonian as
a function of the action variables. The only input required by the method is
the partial derivatives of the effective potential up to the maximum desired
order of the series expansions.
Since Birkhoff normalization relies on expansion of the effective potential
about its value for circular, planar orbits, its range of applicability is
nominally limited by the radius of convergence of this expansion. However, as
illustrated by the tests in Section 3, the series expressions generated by the
procedure can be used to construct Padé approximants and extend the range over
which it provides accurate results. The tests presented in Section 3 showed
that Birkhoff normalization provides comparable or better accuracy than the
popular Stäckel approximation, at least for quasi-planar orbits.
The expressions provided by Birkhoff normalization take the form of
polynomials involving the complex canonical variables introduced in Section
2.2. This makes evaluating both forward and inverse transformations between
position-velocity data in AA variables exceptionally computationally
efficient. This computational efficiency represents another potential
advantage of the method over the Stäckel approximation, which requires
evaluation of integrals via numerical quadrature. The most computationally
expensive parts of the Birkhoff normalization algorithm will generally be the
initial calculation of potential derivatives and construction of series.
However, these calculations represent a fixed cost and need only be performed
once, after which the Birkhoff normalization procedure can be applied to as
many orbits as desired. For the tests presented in Section 3, computing
partial derivatives of the Miyamoto-Nagai potential up to 10th order via
symbolic means required $\sim 20~{}\mathrm{s}$ on a 2.5 GHz Intel i7
processor. The initial construction of the series expression for $\chi$ up to
10th order in complex canonical variables took $\sim 3~{}\mathrm{s}$ and
calculations of additional series expressions approximating the exponential
operators $\exp[\pm\mathcal{L}_{\chi}]$ applied to various functions, such as
Equation (54), took approximately $\sim 1~{}\mathrm{s}$ each. A potential
downside to the Birkhoff normalization method, in terms of computational cost,
is that the effective potential partial derivatives and series must nominally
be recalculated for each value of angular momentum considered. A simple
strategy for applying the method to orbits with a range of angular momenta
would be to carry out series constructions on a 1 dimensional grid of angular
momenta and simply interpolate results. Alternatively, the construction of
various series could be carried out symbolically with the aid of a computer
algebra system. We leave the development of such methods to future work as the
choice of optimal strategy is likely to be application-specific.
We conclude by mentioning a few potential future applications of the Birkhoff
normalization procedure presented in this paper. First, analytic expressions
furnished by the Birkhoff normalization procedure could make it well-suited
for problems requiring the representation of equilibrium distribution
functions, which by Jean’s theorem must depend only on integrals of motion, in
terms of spatial positions and velocities. Additionally, the explicit
expression for the Hamiltonian in terms of action variables provided by the
Birkhoff normalization procedure can be used as a starting point for using
perturbation theory to understand the non-equilibrium signatures of dynamical
perturbations like those induced by a stellar bar (e.g., Binney, 2018) or
satellite galaxy (e.g., Banik et al., 2022). Finally, even though the Birkhoff
normalization procedure may not supply accurate AA variables for orbits with
large vertical actions, it could provide useful initial ‘toy’ AA variables
that serve as the starting point of the torus mapping method of McGill &
Binney (1990).
## 5 Acknowledgments
I thank Rimpei Chiba and Neige Frankel for helpful discussions. I thank Scott
Tremaine for suggesting the use of Padé approximants. Simulations were
performed on the Sunnyvale cluster at the Canadian Institute for Theoretical
Astrophysics. I acknowledge support by the Natural Sciences and Engineering
Research Council of Canada (NSERC), funding references CITA 490888-16 and
RGPIN-2020-03885. This project was developed in part at the Gaia Hike, a
workshop hosted by the University of British Columbia and the Canadian
Institute for Theoretical Astrophysics in 2022 June.
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|
# Self-Guided Learning to Denoise for Robust Recommendation
Yunjun Gao, Yuntao Du, Yujia Hu,
Lu Chen, Xinjun Zhu, Ziquan Fang College of Computer Science, Zhejiang
University, China gaoyj, ytdu, yjhu, luchen, xjzhu<EMAIL_ADDRESS>and
Baihua Zheng School of Computing and Information Systems,
Singapore Management University, Singapore<EMAIL_ADDRESS>
(2022)
###### Abstract.
The ubiquity of implicit feedback makes them the default choice to build
modern recommender systems. Generally speaking, observed interactions are
considered as positive samples, while unobserved interactions are considered
as negative ones. However, implicit feedback is inherently noisy because of
the ubiquitous presence of noisy-positive and noisy-negative interactions.
Recently, some studies have noticed the importance of denoising implicit
feedback for recommendations, and enhanced the robustness of recommendation
models to some extent. Nonetheless, they typically fail to (1) capture the
hard yet clean interactions for learning comprehensive user preference, and
(2) provide a universal denoising solution that can be applied to various
kinds of recommendation models.
In this paper, we thoroughly investigate the memorization effect of
recommendation models, and propose a new denoising paradigm, _i.e.,_ Self-
Guided Denoising Learning (SGDL), which is able to collect memorized
interactions at the early stage of the training (_i.e.,_ “noise-resistant”
period), and leverage those data as denoising signals to guide the following
training (_i.e.,_ “noise-sensitive” period) of the model in a meta-learning
manner. Besides, our method can automatically switch its learning phase at the
memorization point from memorization to self-guided learning, and select clean
and informative memorized data via a novel adaptive denoising scheduler to
improve the robustness. We incorporate SGDL with four representative
recommendation models (_i.e.,_ NeuMF, CDAE, NGCF and LightGCN) and different
loss functions (_i.e.,_ binary cross-entropy and BPR loss). The experimental
results on three benchmark datasets demonstrate the effectiveness of SGDL over
the state-of-the-art denoising methods like T-CE, IR, DeCA, and even state-of-
the-art robust graph-based methods like SGCN and SGL.
Recommender System; Denoising Recommendation; Implicit Feedback; Robust
Learning
††journalyear: 2022††copyright: acmcopyright††conference: Proceedings of the
45th International ACM SIGIR Conference on Research and Development in
Information Retrieval; July 11–15, 2022; Madrid, Spain††booktitle: Proceedings
of the 45th International ACM SIGIR Conference on Research and Development in
Information Retrieval (SIGIR ’22), July 11–15, 2022, Madrid, Spain††price:
15.00††doi: 10.1145/3477495.3532059††isbn: 978-1-4503-8732-3/22/07††ccs:
Information systems Recommender systems††ccs: Computing methodologies Learning
from implicit feedback
## 1\. Introduction
Figure 1. Key idea of SGDL: (a) and (c) show the memory rate when training
NeuMF and LightGCN on the MovieLens dataset, respetively; (b) and (d) show the
memory rate when training the corresponding model with SGDL. The memory rate
is the proportion of memorized data (see Section 3.1.1 for details) in the
clean and noisy interactions. NR denotes “noise-resistant” period and NS
represents “noise-sensitive” period, respectively.
Recommender systems have been widely deployed to perform personalized
information filtering, especially for various online services such as
E-commerce (Lin et al., 2019; Gharibshah et al., 2020), social media (Ren et
al., 2017), and news portals (Wang et al., 2018). Most of existing recommender
systems use implicit feedback (_e.g.,_ view and click) to develop machine
learning models, due to its large volume and availability (Rendle et al.,
2009; He et al., 2016; Du et al., 2022a, b). Specifically, the observed
interactions between users and items are viewed as positive instances, while
unobserved interactions are viewed as negative instances. However, implicit
feedback is inherently noisy because of the ubiquitous presence of noisy-
positive and noisy-negative interactions (Hu et al., 2008; Bian et al., 2021;
Wang et al., 2021b; Lee et al., 2021). Take the E-commerce scenario as an
example. A large portion of click behaviors are triggered by the curiosity of
users, which cannot directly indicate the users’ positive views of the
products. On the other hand, unobserved interactions may attribute to the
unawareness of users because the items are simply not exposed to them. Hence,
blindly fitting the implicit feedback to recommender systems without
considering the inherent noise would fail to understand users’ true
preferences, and eventually harm user experiences and degrade recommendation
performance (Xie et al., 2020; Wang et al., 2021a).
Considering the widespread use of implicit feedback and its large impact on
the recommendation model, some recent studies have noticed the importance of
denoising implicit feedback for recommendations. Existing efforts on tackling
this problem can be roughly divided into two categories: _sample selection
methods_ (Gantner et al., 2012; Ding et al., 2019b; Yu and Qin, 2020; Wang et
al., 2021b) and _sample re-weighting methods_ (Wang et al., 2021a; Hu et al.,
2021; Wang et al., 2022). Sample selection methods focus on designing more
effective samplers to collect clean samples for learning users’ preferences,
while their performance suffers from high variance since they heavily depend
on the sampling distribution (Yuan et al., 2018). On the other hand, sample
re-weighting methods aim to distinguish noisy interactions from clean data in
terms of the loss values, and assign lower weights to noisy interactions with
high loss values during training. Their key idea is consistent with
memorization effect (Arpit et al., 2017): models tend to initially learn easy
and clean patterns (_i.e.,_ user preferences) in the early stage of their
learning, and eventually memorize all training interactions. Benefiting from
this principle, sample re-weighting methods are able to successfully identify
noisy interactions.
Although sample re-weighting methods can achieve promising performance for
denoising implicit feedback, we want to highlight that they commonly suffer
from following two problems:
* •
Abandon of Hard Clean Interactions. These methods heavily rely on the loss
values. They simply assume that interactions with large loss values are noisy
and penalize them with small weights. However, it has been reported that some
clean interactions (_aka._ hard yet clean interactions) may also have high
loss values at the beginning of the training, and those interactions play an
important role in understanding users’ behaviors and preferences (Ding et al.,
2019a; Hu et al., 2021; Wang et al., 2022). Nonetheless, these interactions
are simply discarded by existing methods due to high loss values, incurring
insufficient understanding of users’ true preferences.
* •
Lack of Adaptivity and Universality. Existing sample re-weighting methods are
able to achieve good performance _only_ when the reweighting configurations
(e.g., weights and thresholds) are properly specified. However, obtaining
proper configurations normally requires time-consuming procedures (_e.g.,_
grid search), and the optimal configurations for one dataset typically are not
applicable for other datasets because of the different data distributions.
Meanwhile, these methods are only applicable to the recommender models with
predefined pointwise loss function (_e.g.,_ cross-entropy loss in (Wang et
al., 2021a, 2022)), which makes it hard to incorporate other popular ranking
loss functions (_e.g.,_ BPR loss (Rendle et al., 2009)) and limits their
applications.
In this regard, we have thoroughly explored the memorization of the
representative recommendation models (_e.g.,_ NeuMF (He et al., 2017) and
LightGCN (He et al., 2020)) on implicit feedback, including the clean
interactions and the noisy interactions. By analyzing the learning processes
of different models, we have observed the existence of two learning periods.
As shown in Figure 1, a typical learning process consists of the “noise-
resistant” (NR) period and the “noise-sensitive” (NS) period. The former is
the duration when the memorization of noisy interactions is insignificant
because the models focus on memorizing easy and clean patterns at their early
stage of training; while the latter is the duration when the memorization of
noisy interactions rapidly increases since models eventually begin memorizing
all the implicit feedback at the late stage of training. The timestamp in the
training process that can best differentiate above two periods is defined as
memorization point of the model training, which are represented by dotted
lines in Figure 1.
These observations motivate us to design a new approach that better leverages
the memorization nature of recommendation models. Specifically, we propose a
new denoising paradigm, namely, Self-Guided Denoising Learning (SGDL), which
is able to collect memorized interactions at the early stage of training
(_i.e.,_ “noise-resistant” period), and leverage those data as denoising
signals to guide the subsequent training (_i.e.,_ “noise-sensitive” period) of
the model in a meta-learning manner. Besides, SGDL can automatically transit
from noise-resistant period to noise-sensitive period in order to stop
accumulating memorized interactions at the memorization point, since the noisy
interactions are gradually memorized with the training process. In a nutshell,
corresponding to the two observed periods commonly existing in the training
process of different models, SGDL contains two key phases, _i.e.,_
memorization and self-guided learning:
* •
Memorization. Owing to the negligible memorization of noisy interactions
during “noise-resistant” period, the model is initially trained with all the
implicit feedback. To better reveal the underlying memorization nature of the
model during training, we design the new memorization-based metrics to define
the memorization states of data. As the memorized data are mostly easy and
clean interactions at the early stage of training, we collect them as
denoising signals to guide the following denoising training process. Moreover,
SGDL is able to automatically estimate the best memorization point, from which
the learning moves from the “noise-resistant” period to “noise-sensitive”
period, to stop accumulating memorized data without any supervision.
* •
Self-Guided Learning. To avoid the memorization of noisy interactions in the
“noise-sensitive” period, we leverage the memorized data collected from the
“noise-resistant” period to represent user preferences. Specifically, a
denoising module is proposed to learn a parameterized weighting function for
the implicit feedback, which is guided by memorized data and updated with the
learning process of the model. Moreover, since some of the memorized data can
also be noisy, we further develop a novel adaptive denoising scheduler to
prevent the denoising module from being corrupted by noisy yet memorized
samples. Technically, the adaptive denoising scheduler characterizes the
contribution of each memorized data to the denoising performance, and decides
whether to use the sample by predicting its probability being sampled. The
scheduler is also simultaneously optimized together with the learning process
to enhance the robustness of the model.
Through the above two phases, SGDL can denoise implicit feedback with the help
of memorized data, which are naturally collected by exploiting the noise-
resistant period of training. Compared with standard training, SGDL can
dramatically reduce the memory ratio of noisy samples in the noise-sensitive
period, and enhance the robustness of recommendation models, as shown in
Figure 1. Moreover, since the model is constantly trained with all implicit
feedback data, our method can help the model learn users’ true preferences
with hard yet clean samples, leading to better recommendation performance.
Last but not the least, our method does not need any thresholds or predefined
weighting functions, and is easy to be applied to any learning-based
recommendation models. We conduct extensive experiments on three real-world
datasets with four representative recommendation models (_i.e.,_ NeuMF (He et
al., 2017), CDAE (Wu et al., 2016), NGCF (Wang et al., 2019), and LightGCN (He
et al., 2020)) and two loss functions (_i.e.,_ binary cross-entropy loss and
BPR loss (Rendle et al., 2009)). Experimental results show that our SGDL
significantly outperforms all state-of-the-art denoising methods, and it
achieves comparable (in many cases even better) performance to the state-of-
the-art robust graph-based methods like SGCN (Chen et al., 2021b) and SGL (Wu
et al., 2021).
In summary, we make three key contributions in this paper, as listed below.
* •
We develop a new denoising paradigm, _i.e.,__self-guided denoising learning
(SGDL)_ , which leverages the self-labeled memorized data as guidance to offer
denoising signals for robust recommendation without defining any weighting
functions or requiring any auxiliary information.
* •
We carefully exploit the memorization effect of recommendation models, and
design two training phases that can collect memorized data and utilize them as
guidance to denoise implicit feedback, respectively. Besides, a novel adaptive
denoising scheduler is introduced to further improve the robustness.
* •
We incorporate SGDL with four representative recommendation models, and
conduct extensive experiments on three public benchmark datasets with various
state-of-the-art methods to demonstrate the superiority and universality of
SGDL.
## 2\. Problem Formulation
We first introduce the common paradigm of user preferences learning from
implicit feedback, and then formulate our task.
Preference learning from implicit feedback. In this paper, we focus on
learning the user preferences from the implicit feedback (Rendle et al.,
2009). Specifically, the behavior data (_e.g.,_ click and review)
$\mathcal{D}=\\{u,i,y_{ui}|u\in\mathcal{U},i\in\mathcal{I}\\}$ involves a set
of users $\mathcal{U}=\\{u\\}$ and a set of items $\mathcal{I}=\\{i\\}$, as
well as the interaction $y_{ui}=\\{0,1\\}$ that indicates whether user $u$ has
interacted with item $i$. Most of the state-of-the-art recommendation methods
(_e.g.,_ NeuMF (He et al., 2017) and LightGCN (He et al., 2020)) assume that
the interaction $y_{ui}$ could represent the users’ true preferences, and
directly learn the model $f$ with parameters $\theta$ by minimizing a ranking
loss function over $\mathcal{D}$.
Denoising implicit feedback for recommendations. However, due to the existence
of inherent noise in implicit feedback, recommendation models might fail to
learn the users’ true preferences with typical training process, resulting in
suboptimal performance. Thus, the task of the paper is, given the noisy
implicit feedback $\mathcal{D}$ that contains both noisy-positive and noisy-
negative feedback, to infer users’ true preferences with optimal model
parameters $\theta^{*}$.
Figure 2. The overall framework of SGDL. During Phase I (memorization),
memorized interactions are collected as denoising signals; During Phase II
(self-guided learning), the weighting function is simultaneously learned with
the recommender model, which is guided by the memorization data in a meta-
learning manner (see Section 3.2.1 for details).
## 3\. Methodology
In this section, we detail SGDL that comprises the following two phases: (i)
_memorization_ , which exploits the memorization effect of models to collect
memorized data during the noise-resistant period and estimates the best
memorization point to automatically transit to phase II; and (ii) _self-guided
learning_ , which leverages the memorized samples as denoise signals to guide
the model learning during the noise-sensitive period, and discards the
potential noisy interactions in memorized data with a novel adaptive denoising
scheduler for robustness. We detail the two phases as follows.
### 3.1. Phase I: Memorization
Initially, Phase I tries to train the recommendation model in a conventional
way during the noise-resistant period, where the memorization of noisy
interactions is assumed to be suppressed, as we discover in Figure 1. Since
most of the memorized interactions until the memorization point are clean, we
collect them to form the memorized data, which can be used as the denoising
signals to guide the training in Phase II. Therefore, the major challenges in
Phase I include 1) how to define the memorization of interactions and 2) how
to estimate the memorization point.
#### 3.1.1. Memorized Interactions.
Previous studies (Wang et al., 2021a, 2022) mainly use loss values of training
data to demonstrate the memorization effect of recommendation models. However,
we argue that loss values are insufficient to reflect the learning process,
since they are inconsistent with the optimization target of recommendation
models (_i.e.,_ personalized ranking), and unable to distinguish hard
interactions from noisy ones. Thus, it is necessary to design a new
memorization-based metric, which takes the learning process of recommendation
models into consideration.
Inspired by the widely used hit ratio metric (Rendle et al., 2009; Cremonesi
et al., 2010), we define that an interaction $(u,i)$ is memorized at epoch $t$
by the model $\theta$ if item $i$ is in the ranking list of user $u$, denoted
as $m_{t}(u,i)$. To ensure the reliability of ranking results, for each user
$u$, we include the top-$N$ of the ranking items into its ranking list, where
$N$ is the length of observed interactions. However, simply calculating the
memorization of interactions of a single epoch could lead to unstable results,
since the model is not well trained during the noise-resistant period. Hence,
we trace the memorization states of interactions for the most recent $h$
epochs, and define the final memorization of interaction $(u,i)$ as follows:
(1) $\displaystyle
m_{t}^{h}(u,i)=\frac{1}{{|\mathcal{P}_{t}^{h}(u,i)|}}\sum\nolimits_{m_{j}(u,i)\in\mathcal{P}_{t}^{h}(u,i)}m_{i}(u,i)$
where $\mathcal{P}_{t}^{h}(u,i)=\\{m_{t-h+1}(u,i),\cdots,m_{t}(u,i)\\}$
captures the most recent $h$ memorization histories of interaction $(u,i)$. We
define an interaction $(u,i)$ memorized by a model $\theta$ if the majority of
the recent histories $\mathcal{P}_{t}^{h}(u,i)$ coincides with the
memorization state (_i.e.,_ $m_{t}^{h}(u,i)$ is larger than $0.5$). It is
worth mentioning that the definition of memorized interactions dose not need
any labels for supervision, and is able to effectively indicate the underlying
memorization effect of recommendation models.
Figure 3. The monotonicity of MP and MR when training NeuMF and LightGCN on
the MovieLens dataset.
#### 3.1.2. Memorization Point Estimation.
As shown in Figure 1, the model predominantly learns clean interactions until
the noise-sensitive period begins. In others words, during noise-resistant
period, the recommendation model (1) not only learns sufficient information
from the clean interactions, (2) but also accumulates some noise from the
noisy implicit feedback. Therefore, we aim to design two metrics to reflect
the above two memorization characteristics of the model. Formally, we use
$\mathcal{M}_{t}$ to denote a set of memorized interactions at epoch $t$, and
use $y_{ui}^{*}$ to represent the true label of interaction $(u,i)$, which is
not available because of the noise in implicit feedback. Inspired by recent
advances in robust learning (Shu et al., 2019; Song et al., 2021) and
recommendation evaluation metrics, we propose two memorization-based metrics,
namely, _memorization precision_ ($MP$) and _memorization recall_ ($MR$), to
address the memorization effect of recommendation models respectively:
(2) $\displaystyle MP_{t}=\frac{|\mathcal{R}_{t}|}{|\mathcal{M}_{t}|},\quad
MR_{t}=\frac{|\mathcal{R}_{t}|}{|\mathcal{G}|}$
where $\mathcal{R}_{t}=\\{(u,i)\in\mathcal{M}_{t}:y_{ui}=y_{ui}^{*}\\}$
denotes the set of memorized data whose true labels are consistent with
predictions, and $\mathcal{G}=\\{(u,i)\in\mathcal{D}:y_{ui}=y_{ui}^{*}\\}$ is
the set of true labeled data in implicit feedback. According to the definition
of $MP$ and $MR$, we can conclude that: $MP$ monotonically decreases since the
model tends to memorize clean data first and then gradually memorizes all the
noisy interactions as the training progresses; and $MR$ monotonically
increases because the model eventually memorizes all clean interactions as the
training progresses, as depicted in Figure 3 (please refer to Section 3.3 for
the theoretical analysis of the monotonicity of the two metrics). Thus, the
best memorization point $t_{m}$ is naturally the best trade-off epoch when
$MP$ and $MR$ share the same value, _i.e.,_ $MP_{t}=MR_{t}$. By substituting
it into Equation (2), the best memorization point $t_{m}$ can be calculated
as:
(3)
$\displaystyle\mathcal{M}_{t_{m}}=|\\{(u,i)\in\mathcal{D}_{t}:y_{ui}=y_{ui}^{*}\\}|=(1-\sigma)|\mathcal{D}|$
where $\sigma$ is the noise rate of the implicit feedback. Since $\sigma$ is
typically unknown, we leverage the difference of loss distributions in clean
and noisy data to estimate $\sigma$, as shown in Figure 4.
Figure 4. Loss distribution of MovieLens dataset.
Specifically, we first normalize the loss values of all training interactions
and then fit them into a two-component Gaussian Mixture Model (GMM) to model
the bi-modal distribution (Arazo et al., 2019; Li et al., 2020) of true- and
false-labed samples. The GMM model can be easily trained by using the
Expectation-Maximization (EM) algorithm. Hence, we could obtain the
probability of an interaction $(u,i)$ being noisy through the posterior
probability of loss distributions. Accordingly, the noise rate $\sigma$ is
estimated as:
(4)
$\displaystyle\hat{\sigma}=\mathbb{E}_{(u,i)\in\mathcal{D}}[p(\mu|L_{(u,i)}(\theta)]$
where $L_{(u,i)}(\theta)$ is the loss of interaction $(u,i)$ of recommendation
model $\theta$, and $\mu$ is the Gaussian component with a larger mean, since
noisy data have typically larger loss values. Note that there are many
approaches (Liu and Tao, 2015; Chen et al., 2019) available to estimate the
noise ratio, we choose GMM because it is easy to apply and has stable
performance on different noisy datasets.
Therefore, SGDL transits to phase II when the number of memorized interactions
reaches the estimated clean data size (_i.e.,_
$|\mathcal{M}_{t}|\geq(1-\hat{\sigma})\mathcal{D}$). Note that, SGDL is able
to estimate the memorization point $t_{m}$, and collect memorized data
$\mathcal{M}_{t_{m}}$ with nearly zero computational overhead and no
additional supervision.
### 3.2. Phase II: Self-Guided Learning
Phase II aims to robustly learn the recommendation model with the denoising
signals of memorized data, which are collected from the training process of
Phase I. Previous studies (Wang et al., 2021a, 2022) need to pre-specify
weighting functions (_e.g.,_ exponential function), and require additional
hyperparameters (_e.g.,_ threshold) to denoise implicit feedback during
training. However, we argue that these methods are fairly hard to be generally
applied in real-world recommendation systems due to two issues: (1) The proper
weighting schemes heavily rely on the training data, which limits their
adaptivity. (2) These methods simply abandon the hard yet clean interactions
because the fixed weighting functions fail to distinguish the interactions
from noisy data, incurring suboptimal recommendation performance. Thus, we aim
to propose a self-guided method that is capable of learning an adaptive
weighting function automatically from the collected memorized data to tackle
the above two issues.
The above denoising scheme assumes that the memorized data is clean and useful
to provide denoising signals for learning user preferences. Nevertheless, as
the memorized data is collected by leveraging the memorization effect of
recommendation models, it inevitably contains some noise. To prevent the model
from being corrupted by such detrimental interactions, we further devise a
novel adaptive denoising scheduler to select and use proper memorized data for
self-guided learning.
#### 3.2.1. Denoising Learning with Memorized Data.
During the noise-sensitive period, we aim to enhance the robustness of
training by imposing a weight on each sample loss. Here, we consider the
memorized data that is able to provide denoising signals to learn the weight
for each sample, since it is mostly clean thanks to the memorization effect of
the model. Specifically, let $L_{k}(\theta)$ be the $k$-th sample loss111The
loss can be represented as the pointwise or pairwise loss, according to the
optimization target of models. We demonstrate the effectiveness of our method
on both kinds of loss functions in Section 4.2. of model $\theta$, and
$\mathcal{D}_{T}$ be the training data. The optimal recommendation model
parameter $\theta$ is calculated by minimizing the following weighted loss:
(5)
$\displaystyle\theta^{*}(\psi)=\mathop{\mathrm{argmin}}_{\theta}\frac{1}{|\mathcal{D}_{T}|}\sum\nolimits_{k}^{|\mathcal{D}_{T}|}g(L_{k}(\theta);\psi)L_{k}(\theta)$
where $g(L_{k}(\theta);\psi)$ is the weight on the $k$-th sample loss, and
$\psi$ represents the current parameters of weighting function $g(\cdot)$.
Here, we formulate $g(\cdot)$ as a simple Multi-Layer Perceptron (MLP) with
only one hidden layer, since it is known as a universal approximator for
almost any continuous function (Csáji, 2001; Shu et al., 2019). To adaptively
learn the weight for each sample from memorized data, we optimize the
weighting parameters $\psi$ given current optimal $\theta^{*}(\psi)$:222Notice
that, here $\psi$ is a variable instead of a quantity, which makes
$\theta^{*}(\psi)$ a function of $\psi$, and the gradient in Equation (6) can
be computed.
(6)
$\displaystyle\psi^{*}=\mathop{\mathrm{argmin}}_{w}\frac{1}{|\mathcal{M}_{t_{m}}|}\sum\nolimits_{m}^{|\mathcal{M}_{t_{m}}|}L_{m}(\theta^{*}(\psi))$
where $L_{m}(\theta^{*}(\psi))$ is the $m$-th memorized sample loss given
optimal model parameters $\theta^{*}(\psi)$. Note that the searching for the
optimal $\theta^{*}$ and optimal $\psi^{*}$ requires two nested loops of
optimization (_i.e.,_ bi-level optimization) (Shu et al., 2019; Chen et al.,
2021a), which is costly. Hence, we adopt the idea of meta learning (Finn et
al., 2017; Wu et al., 2018), and update $\theta$ and $\psi$ alternately in a
single loop to guarantee the efficiency of the algorithm. Specifically, as
illustrated in Figure 2, we perform the following training procedure in each
iteration:
* •
Assumed update of $\theta$. As shown by the blue arrows in Figure 2, we first
make an assumed update of $\theta$ with current weight $\psi$:
(7)
$\displaystyle\hat{\theta}(\psi)=\theta-\eta_{1}\frac{1}{|\mathcal{D}_{T}|}\sum\nolimits_{k}^{|\mathcal{D}_{T}|}g(L_{k}(\theta);\psi)\nabla_{\theta}L_{k}(\theta)$
where we update $\theta$ using gradient descent with learning rate $\eta_{1}$.
* •
Update of $\psi$. As indicated by the brown arrow in Figure 2, the updates of
the weighting parameters $\psi$ can be guided by gradient of the memorized
data on the updated model:
(8)
$\displaystyle\psi\leftarrow\psi-\eta_{2}\frac{1}{|\mathcal{M}_{t_{m}}|}\sum\nolimits_{m}^{|\mathcal{M}_{t_{m}}|}\nabla_{\psi}L_{m}(\hat{\theta}(\psi))$
where $\eta_{2}$ is the learning rate of weighting parameters $\psi$.
* •
Actual update of $\theta$. After receiving the denoising signals (_i.e.,_
updated parameters $\psi$) from memorized data, we use them to update the
model:
(9)
$\displaystyle\theta\leftarrow\theta-\eta_{1}\frac{1}{|\mathcal{D}_{T}|}\sum\nolimits_{k}^{|\mathcal{D}_{T}|}g(L_{k}(\theta);\psi)\nabla_{\theta}L_{k}(\theta)$
We use this alternative strategy to optimize the recommendation model and
weighting function during the noise-sensitive period, as illustrated in Figure
2. Although this strategy does not guarantee to find the global optimum, it
empirically works well in many bi-level optimization problems (Finn et al.,
2017; Shu et al., 2019; Chen et al., 2021a).
#### 3.2.2. Adaptive Denoising Scheduler.
Above denoising scheme relies on the memorized data to provide denoising
signals for training. However, as memorized data inevitably exhibit noise,
integrating the noise of them would degrade the denoising performance.
Thus, we propose an adaptive denoising scheduler to select _only_ the clean
and informative memorized data for denoising learning. Specifically, we define
the scheduler as $s$ with parameter $\phi$, and choose two representative
factors to quantify the contribution of each memorized data for denoising: (1)
the loss $L_{m}(\theta)$ of $m$-th memorized sample, where $\theta$ is the
actual updated parameters of the model; and (2) the gradient similarity of
$m$-th memorized sample on assumed updated model parameters $\hat{\theta}$ and
actual updated model parameters $\theta$, _i.e.,_
$\cos\big{(}\nabla_{\hat{\theta}}L_{m}(\hat{\theta}),\nabla_{\theta}L_{m}(\theta)\big{)}$.
Here, we use cosine function as the similarity measurement, and other metrics
like inner product can also be applied in practice. The two factors are
associated with the learning outcome and learning process of the $m$-th
memorized sample, respectively. Specifically, the gradient similarity
characterizes the contribution of the memorized sample in the model training.
A large loss value may represent a crucial memorized sample if the gradient
similarity is also large (_i.e.,_ the gradient direction of memorized sample
is consistent with the optimization of the model); and a large loss value with
small gradient similarity may indicate a noisy memorized sample. Considering
the two factors simultaneously, we formulate the sampling probability of
$m$-th memorized data:
(10) $\displaystyle
o_{m}=s\big{(}L_{m}(\theta),\cos(\nabla_{\hat{\theta}}L_{m}(\hat{\theta}),\nabla_{\theta}L_{m}(\theta));\phi\big{)}$
(11)
$\displaystyle\pi_{m}=\frac{\exp(o_{m};\phi)}{\sum_{i\in\mathcal{M}_{t_{m}}}\exp(o_{i};\phi)}$
where $o_{m}$ is the output of the scheduler, and $\pi_{m}$ is the predicted
sampling probability of $m$-th sample. We choose LSTM network (Hochreiter and
Schmidhuber, 1997) as the scheduler, and feed it with the training factors in
each iteration. The intuition behind is that the LSTM could leverage the
historical information to capture the prediction variance (Chang et al., 2017;
Yao et al., 2021), which shows stable performance in our experiments
(different strategies are compared in Section 4.3.3). However, it is
intractable to directly optimize the scheduler since the sampling process is
not differentiable. To make this procedure differentiable and to jointly
optimize the scheduler and the model, we apply the Gumbel-Softmax
reparameterization trick (Jang et al., 2017) to generate differentiable
samples:
(12) $\displaystyle
y_{m}=\frac{\exp(\log(\pi_{m})+\epsilon_{m})/\tau}{\sum_{i\in\mathcal{M}_{t_{m}}}\exp(\log(\pi_{i})+\epsilon_{i})/\tau}$
where $\epsilon_{m}$ is randomly drawn from uniform distribution between 0 and
1, and $\tau$ is the temperature that controls the interpolation between the
discrete distribution and continuous categorical densities (we set $\tau=0.05$
for all experiments). Thus, the scheduler is able to decide which memorized
data to use according to its contribution to denoising implicit feedback, and
adpatively adjust the sampling probability for more informative guided
learning.
### 3.3. Model Analysis
#### 3.3.1. Analysis on the monotonicity of $MP$ and $MR$.
We now prove that $MP$ and $MR$ change monotonically over the training time
$t$. Let $\mathcal{N}_{t}$ ($=\mathcal{M}_{t}\setminus\mathcal{R}_{t}$) be the
set of memorized data that are falsely predicted by models. Then,
$|\mathcal{N}_{t+1}|/|\mathcal{N}_{t}|\geq|\mathcal{R}_{t+1}|/|\mathcal{R}_{t}|$
typically holds because the noisy samples are memorized faster than clean
samples after the model stabilizes, as depicted in Figure 3. Hence, it is easy
to conclude:
(13) $\displaystyle|\mathcal{N}_{t+1}|/|\mathcal{N}_{t}|$
$\displaystyle\geq|\mathcal{R}_{t+1}|/|\mathcal{R}_{t}|$ (14)
$\displaystyle\implies|\mathcal{N}_{t+1}||\mathcal{R}_{t}|+|\mathcal{R}_{t+1}||\mathcal{R}_{t}|$
$\displaystyle\geq|\mathcal{R}_{t+1}||\mathcal{N}_{t}|+|\mathcal{R}_{t+1}||\mathcal{R}_{t}|$
(15)
$\displaystyle\implies(|\mathcal{N}_{t+1}|+|\mathcal{R}_{t+1}|)|\mathcal{R}_{t}|$
$\displaystyle\geq(|\mathcal{N}_{t}|+|\mathcal{R}_{t}|)|\mathcal{R}_{t+1}|$
Then, $MP_{t+1}\geq MP_{t}$ can be directly derived from Inequation (15).
Besides, considering that the model eventually would memorize all training
data (Song et al., 2021), we assume that $\mathcal{M}$ would gradually include
more observed interactions, including true predicted ones, _i.e.,_
$|\mathcal{R}_{t+1}|\geq|\mathcal{R}_{t}|$. Consequently, $MR$ increases
monotonically since
$|\mathcal{R}_{t+1}|/|\mathcal{G}|\geq|\mathcal{R}_{t}|/|\mathcal{G}|$.
#### 3.3.2. Analysis of the Self-Guide Learning Scheme.
We then focus on the guidance strategy to explain how memorized data benefit
the denoising training. Formally, we follow (Shu et al., 2019), and utilize
chain rule to derive the update function of $\psi$:
(16)
$\displaystyle\psi\leftarrow\psi+\frac{\eta_{1}\eta_{2}}{|\mathcal{D}_{T}|}\sum_{k}^{|\mathcal{D}_{T}|}\big{(}\frac{1}{|\mathcal{M}_{t_{m}}|}\sum_{m}^{|\mathcal{M}_{t_{m}}|}G_{mk}\big{)}\nabla_{\psi}g(L_{k}(\hat{\theta});\psi)$
where $G_{mk}$
($=\nabla_{\theta}L_{m}(\hat{\theta})^{T}\nabla_{\theta}L_{k}(\theta)$)
measures the gradient similarity between $m$-th memorized data and $k$-th
training data. Thus, for each $k$-th training sample, if its gradient is
similar to the average gradient of memorized data, it would be considered as a
beneficial one for learning, and its weight tends to be increased. Conversely,
the weight of the sample inclines to be suppressed. Therefore, memorized data
is able to offer a proper weight for each training sample in terms of gradient
similarities under the self-guide learning scheme.
#### 3.3.3. Model Size.
The additional parameters of SGDL come from two parts: (1) the parameters
$2d_{w}$ of weighting function, where $d_{w}$ is the number of the hidden
neurons in one layer MLP; and (2) the $4d_{l}^{2}+12d_{l}$ parameters of LSTM
unit in the adaptive denoising scheduler, where $d_{l}$ is the dimension of
hidden size. Overall, the additional cost of SDGL is negligible, compared with
the tremendous parameters of modern recommendation models.
#### 3.3.4. Time Complexity.
Assume that the time complexity of the base model is $O(T)$, the additional
complexity of SGDL mainly comes from phase II, which consists of two denoising
components: (1) the cost of self-guided learning is also $O(T)$, as the
alternative optimization scheme takes no more than three times compared with
the normal training; and (2) the computational complexity of denoising
adaptive scheduler is $O(|\mathcal{D}|d_{l})$. Therefore, the additional time
complexity of SGDL is $O(T+|\mathcal{D}|d_{l})$. Under the same experimental
settings (_i.e.,_ same base model and same embedding size), SGDL achieves
better trade-off between efficiency and effectiveness compared with various
state-of-the-art models: i) For sample re-weighting and sample selection
methods, although most of them are more time-saving than SGDL, they suffer
from difficult/expensive hyperparameters tuning and unstable performance, as
discussed in Section 4.2. ii) For robust graph-based methods, SGDL has a
complexity that is comparable with them, since graph-based methods typically
leverage on extra graph structure to enhance the robustness of the model.
## 4\. Experiments
We provide empirical results to demonstrate the effectiveness of our proposed
SGDL. The experiments are designed to answer the following research questions:
* •
RQ1: How does SGDL perform, compared with the state-of-the-art denoising
methods as well as the state-of-the-art robust recommender methods?
* •
RQ2: How does each component of SGDL (_i.e.,_ memorization point estimation,
denoising learning strategy, and adaptive denoising scheduler) affect SGDL?
* •
RQ3: Is SGDL able to distinguish hard yet clean interactions from noisy
interactions?
### 4.1. Experimental Settings
#### 4.1.1. Dataset Description.
We select three real world benchmark datasets to evaluate and compare the
performance of SGDL and its competitors. Table 1 lists their statistics.
* •
Adressa333https://www.adressa.no/ is a news reading dataset from Adressavisen,
including user click behaviors and the dwell time for each click. Following
previous work (Wang et al., 2021a), clicks with dwell time less than 10
seconds are viewed as noisy interactions.
* •
Yelp444https://www.yelp.com/dataset/challenge is an open recommendation
dataset released by the Yelp challenge. We use the 2018 version in our
experiments. We follow (Wang et al., 2021a) to mark ratings below 3 as noisy
interactions.
* •
MovieLens is a widely used dataset for recommendation, which contains 100,000
movie ratings ranging from 1 to 5. Ratings below 3 are regarded as noisy
interactions.
Table 1. Statistics of the datasets used in our experiments.
Dataset | #Users | #Items | #Interactions | Sparsity
---|---|---|---|---
Adressa | 212,231 | 6,596 | 419,491 | 99.97%
MovieLens | 943 | 1,683 | 100,000 | 93.70%
Yelp | 45,548 | 57,396 | 1,672,520 | 99.94%
Table 2. Performance comparison of different denoising methods on robust
recommendation. “${\dagger}$” indicates the improvement of the SGDL over the
baseline is significant at the level of 0.05. The highest scores are in Bold.
R and N refer to Recall and NDCG, respectively.
Database | Adressa | MovieLens | Yelp
---|---|---|---
Base Model | Method | R@5 | R@20 | N@5 | N@20 | R@5 | R@20 | N@5 | N@20 | R@5 | R@20 | N@5 | N@20
NeuMF | Normal | 0.1533† | 0.3208† | 0.1224† | 0.1808† | 0.1023† | 0.2687† | 0.2890† | 0.2765† | 0.0129† | 0.0393† | 0.0129† | 0.0215†
WBPR | 0.1538† | 0.3207† | 0.1225† | 0.1809† | 0.1025† | 0.2689† | 0.2891† | 0.2769† | 0.0128† | 0.0392† | 0.0127† | 0.0214†
IR | 0.1541† | 0.3212† | 0.1229† | 0.1830† | 0.1054† | 0.2704† | 0.2928† | 0.2758† | 0.0132† | 0.0407† | 0.0131† | 0.0229†
T-CE | 0.1537† | 0.3220† | 0.1267† | 0.1839† | 0.1025† | 0.2821† | 0.2923† | 0.2845† | 0.0119† | 0.0396† | 0.0119† | 0.0211†
DeCA | 0.1597 | 0.3205† | 0.1226† | 0.1799† | 0.1024† | 0.2723† | 0.2904† | 0.2801† | 0.0129† | 0.0394† | 0.0129† | 0.0216†
SGDL | 0.1598 | 0.3291 | 0.1272 | 0.1853 | 0.1135 | 0.2844 | 0.3279 | 0.3032 | 0.0155 | 0.0469 | 0.0158 | 0.0260
CDAE | Normal | 0.1445† | 0.3159† | 0.0987† | 0.1886† | 0.0904† | 0.2185† | 0.2617† | 0.2356† | 0.0145† | 0.0436† | 0.0149† | 0.0277†
WBPR | 0.1443† | 0.3158† | 0.0987† | 0.1890† | 0.0908† | 0.2184† | 0.2619† | 0.2346† | 0.0148† | 0.0437† | 0.0151† | 0.0278†
IR | 0.1444 | 0.3152† | 0.0981† | 0.1893† | 0.0909† | 0.2186† | 0.2612† | 0.2358† | 0.0153† | 0.0438 | 0.0152† | 0.0278†
T-CE | 0.1415† | 0.3106† | 0.0991 | 0.1840† | 0.0912† | 0.2158† | 0.2642 | 0.2386† | 0.0147† | 0.0439 | 0.0151† | 0.0279†
DeCA | 0.1447† | 0.3159† | 0.0991 | 0.1888† | 0.0917† | 0.2189† | 0.2641 | 0.2378† | 0.0158† | 0.0438 | 0.0154† | 0.0292†
SGDL | 0.1450 | 0.3181 | 0.0993 | 0.1956 | 0.0921 | 0.2220 | 0.2643 | 0.2404 | 0.0162 | 0.0439 | 0.0172 | 0.0296
NGCF | Normal | 0.0769† | 0.1322† | 0.0571† | 0.0769† | 0.1285† | 0.3103† | 0.3694† | 0.3392† | 0.0267† | 0.0736† | 0.0262† | 0.0417†
WBPR | 0.0770† | 0.1324† | 0.0572† | 0.0769† | 0.1287† | 0.3105† | 0.3692† | 0.3395† | 0.0265† | 0.0739† | 0.0265† | 0.0417†
IR | 0.0772† | 0.1337† | 0.0570† | 0.0768† | 0.1280† | 0.3104† | 0.3701† | 0.3395† | 0.0269† | 0.0737† | 0.0261† | 0.0412†
DeCA | 0.0760† | 0.1326† | 0.0571† | 0.0766† | 0.1304† | 0.3113† | 0.3729† | 0.3401 | 0.0277 | 0.0739† | 0.0262 | 0.0418
SGCN | 0.0773† | 0.1336† | 0.0543† | 0.0770 | 0.1288† | 0.3112† | 0.3768 | 0.3401 | 0.0267† | 0.0734† | 0.0265 | 0.0443
SGL | 0.0775† | 0.1345 | 0.0576 | 0.0768† | 0.1303† | 0.3141† | 0.3763† | 0.3360† | 0.0279 | 0.0750 | 0.0264† | 0.0409†
SGDL | 0.0788 | 0.1347 | 0.0579 | 0.0771 | 0.1309 | 0.3186 | 0.3745 | 0.3404 | 0.0273 | 0.0746 | 0.0267 | 0.0420
LightGCN | Normal | 0.0951† | 0.1817† | 0.0713† | 0.0994† | 0.1258† | 0.3173† | 0.3678† | 0.3358† | 0.0334† | 0.0912† | 0.0332† | 0.0515†
WBPR | 0.0958† | 0.1845† | 0.0733† | 0.1006† | 0.1262† | 0.3189† | 0.3701† | 0.3510 | 0.0333† | 0.0911† | 0.0331† | 0.0512†
IR | 0.0953† | 0.1822† | 0.0726† | 0.1003† | 0.1285† | 0.3194† | 0.3681† | 0.3361† | 0.0305† | 0.0909† | 0.0326† | 0.0510†
DeCA | 0.0974† | 0.1855† | 0.0758† | 0.1162† | 0.1293† | 0.3076† | 0.3575† | 0.3270† | 0.0337 | 0.0911† | 0.0332† | 0.0524
SGCN | 0.0941† | 0.1899† | 0.0765† | 0.1160† | 0.1282† | 0.3210† | 0.3602† | 0.3318† | 0.0335† | 0.0916 | 0.0346 | 0.0528
SGL | 0.0980† | 0.1770† | 0.0741† | 0.0999† | 0.1299† | 0.3156† | 0.3638† | 0.3343† | 0.0341 | 0.0915 | 0.0344 | 0.0526
SGDL | 0.1134 | 0.2105 | 0.0844 | 0.1178 | 0.1378 | 0.3335 | 0.3844 | 0.3513 | 0.0339 | 0.0918 | 0.0341 | 0.0525
#### 4.1.2. Evaluation Metrics.
We adopt cross-validation to verify the performance. Specifically, we follow
(Wang et al., 2021a, 2022) to split the interactions into the training set,
validation set, and clean test set with the ratio of 8:1:1. The performance is
measured by two widely used valuation protocols (Wang et al., 2019; He et al.,
2020): Recall@$K$ and NDCG@$K$, where $K$ is set as 5 and 20 by default. We
report the average metrics for all users in the test set.
#### 4.1.3. Baselines.
We select four state-of-the-art recommendation methods as the base model $f$
of SGDL:
* •
NeuMF (He et al., 2017) is a state-of-the-art model, which generalizes
Factorization Machines (FM) with a Multi-Layer Perceptron (MLP).
* •
CDAE (Wu et al., 2016) is denoising auto-encoder model, which corrupts the
interactions with random noises, and then employs a MLP model to reconstruct
the original input.
* •
NGCF (Wang et al., 2019) is a graph model, which applies graph convolution
network (GCN) to encode high-order collaborative signals in user-item
bipartite graph.
* •
LightGCN (He et al., 2020) is a state-of-the-art graph model, which simplifies
the design of GCN by discarding the nonlinear feature transformations for
recommendation.
We train the base models with different ranking loss functions to demonstrate
the universality of SGDL. Specifically, we train NeuMF and CDAE with binary
cross-entropy (BCE) loss, and train NGCF and LightGCN with BPR loss (Rendle et
al., 2009). Each model is trained with the following denoising approaches:
* •
Normal is trained with the original architecture design, without any denoising
consideration.
* •
WBPR (Gantner et al., 2012) is a sample selection method, which considers the
popular but uninteracted items are likely to be real negative ones.
* •
IR (Wang et al., 2021b) is the state-of-the-art sample selection method, which
interactively relabels uncertain samples to mitigate the noise in both
observed and unobserved interactions.
* •
T-CE (Wang et al., 2021a) is the state-of-the-art sample re-weighting method,
which uses the Truncated BCE to assign zero weights to large-loss examples
with a dynamic threshold. Note that, this denoising approach can only be used
for BCE loss, and thus, we implement it with NeuMF and CDAE for comparison.
* •
DeCA (Wang et al., 2022) is a newly proposed sample re-weighting method, which
considers the disagreement predictions of noisy samples across different
models, and minimizes KL-divergence between the two models’ predictions to
enhance the robustness of models.
In addition, we also compare SGDL with the state-of-the-art robust graph-based
methods to further confirm the effectiveness of our model. Note that the
methods can only be applied to graph-based recommenders (_i.e.,_ NGCF and
LightGCN in our experiments), since they regard the noisy interactions as
noisy edges, and devise enhanced graph learning methods for robust
recommendation.
* •
SGCN (Chen et al., 2021b) is the state-of-the-art graph structure enhanced
method, which attaches the GCN layers with a trainable stochastic binary mask
to prune noisy edges in user-item bipartite graph.
* •
SGL (Wu et al., 2021) is the state-of-the-art self-supervised graph method,
which designs different graph views to mine hard negatives and denoise noise
in implicit feedback. We choose the Edge Dropout (ED) view as auxiliary
supervision signal as it performs the best for most datasets.
#### 4.1.4. Parameter Settings.
We implement SGDL in Pytorch, and have released our
implementations555https://github.com/ZJU-DAILY/SGDL (codes, parameter
settings, andtraining logs) to facilitate reproducibility. We use recommended
parameter settings for all models, and optimize them with Adam (Kingma and Ba,
2014) optimizer. The Xavier initializer (Glorot and Bengio, 2010) is used to
initialize the model parameters. We set the batch size as 128 for MovieLens,
1024 for Adressa, and 2048 for Yelp due to the different sizes of each
dataset. The learning rate of CDAE, NeuMF, and LightGCN is tuned as 0.001; and
for NGCF, the learning rate is set as 0.0001. For NeuMF, the embedding size is
32, and the number of layers is 3. For CDAE, the hidden size is 100, and the
dropout ratio is set to 0.5. For LightGCN, we set the embedding size to 64,
the number of layers to 3, and train it without dropout. For NGCF, the
embedding size and layers are the same as LightGCN; and the node dropout rate
is set to 0.1. For SGDL, we set $\eta_{1}=\eta_{2}$ for denoising learning,
and tune the learning rate amongst $\\{10^{-4},10^{-3},10^{-2}\\}$ for two
phases, respectively. The hidden size of MLP and LSTM unit is 64; the length
of memorization history $h$ is tuned among $\\{2,5,10,20\\}$. Note that, the
hyperparameters of base models keep exactly the same across all training
methods for a fair comparison.
### 4.2. Performance Comparison (RQ1)
We begin with the performance comparison _w.r.t._ recall@$K$ and NDCG@$K$,
where we test two values (5 and 20) of $K$. The experimental results are
reported in Table 2, and we find that:
* •
The proposed SGDL can effectively improve the performance of all base models,
and outperform all denoising methods over three datasets. Besides, even if the
base model is designed to be robust against noisy implicit feedback (_i.e.,_
CDAE), our method can still boost its performance by a large margin. We
attribute these improvement to the memorization-based denoising schemes of
SGDL: (1) By tracing the memorization states of data in the noise-resistant
period, SGDL is able to collect memorized interactions during training process
to provide valuable denoising signals without any supervision. In contrast,
none of the baselines considers explicitly characterizing data from the
memorization perspective. (2) Benefiting from our learning to weight strategy
and adaptive denoising scheduler, SGDL can adaptively select clean and
informative samples from memorized interactions, and use them to guide the
learning process of the model. However, other re-weighting baselines (_e.g.,_
T-CE and DeCA) are insufficient to provide proper weight for each interaction
since they do not have memorized data as guidance.
* •
Jointly analyzing the performance of SGDL across three datasets, we find that
the improvement on MovieLens dataset are less significant than that on other
datasets. One possible reason is that the sparsity of MovieLens dataset is
denser than the sparsity of Yelp and Adressa. Accordingly, there are
sufficient interactions to identify user behavior patterns, which offsets the
impact of noisy implicit feedback.
* •
Jointly analyzing the performance of SGDL across the recommenders, we observe
that the relative improvement (_i.e.,_ the performance of SGDL over the
strongest baselines) on NeuMF and CDAE are more substantial than that on
graph-based methods (_i.e.,_ NGCF and LightGCN). This is because our method is
model-agnostic, which does not take the graph structure into consideration. In
comparison, both SGCN and SGL are designed for graph-based recommendations,
and thus, they can leverage the graph structure information to yield better
performance. Nonetheless, our method still achieves the best results on most
cases, which demonstrate the superior robustness of SGDL.
* •
All the denoising approaches show better results compared with normal training
in most cases, which indicate the necessities of denoising implicit feedback
for recommendations. The results are consistent with prior studies (Wang et
al., 2021a, 2022).
* •
The performance of sample selection methods (_i.e.,_ WBPR and IR) is rather
unstable across different recommenders, compared with sample re-weighting
methods. This is reasonable since sample selection methods highly depend on
the sampling distribution, which makes their performance unstable. For
instance, although IR achieves some improvement on Yelp dataset, it performs
worse than the base model (_i.e.,_ CDAE) on Adressa dataset.
* •
Robust Graph-based methods (_i.e.,_ SGCN and SGL) achieve competitive or even
the best performance against other methods. We attribute such improvement to
their carefully designed graph structures, which is able to prune noisy and
insignificant edges for clean information propagation. However, the methods
can only be applied to graph-based models, while SGDL is able to be easily
integrated with any learning-based recommender systems, and achieves
comparable or even better performance.
Table 3. Impact of denoising Learning and adaptive denoising scheduler.
Database | Adressa | MovieLens | Yelp
---|---|---|---
Base Model | Method | R@5 | R@20 | N@5 | N@20 | R@5 | R@20 | N@5 | N@20 | R@5 | R@20 | N@5 | N@20
NeuMF | w/o DLS | 0.1528 | 0.3107 | 0.1211 | 0.1794 | 0.1055 | 0.2690 | 0.2911 | 0.2774 | 0.0136 | 0.0397 | 0.0131 | 0.0218
w/o ADS | 0.1576 | 0.3285 | 0.1255 | 0.1801 | 0.1097 | 0.2801 | 0.3210 | 0.3008 | 0.0146 | 0.0438 | 0.0146 | 0.0259
LightGCN | w/o DLS | 0.0964 | 0.1810 | 0.0702 | 0.0985 | 0.1244 | 0.3159 | 0.3688 | 0.3349 | 0.0330 | 0.0909 | 0.0331 | 0.0513
w/o ADS | 0.1013 | 0.1995 | 0.0811 | 0.1007 | 0.1316 | 0.3328 | 0.3824 | 0.3502 | 0.0338 | 0.0914 | 0.0340 | 0.0521
### 4.3. Study of SDGL (RQ2)
As the memorization and self-guided denoising learning are the core of SGDL,
we also conduct ablation studies to investigate their effectiveness.
Specifically, how the presence of denoising learning and adaptive denoising
scheduler, the estimation of memorization point, and the design of scheduler
affect our model.
#### 4.3.1. Impact of Denoising Learning & Scheduler.
We first evaluate the effectiveness of the denoising learning scheme and
adaptive denoising scheduler. To this end, two variants of SGDL are
constructed by (1) discarding the denoising learning strategy in noise-
sensitive period, called SGDL${}_{\text{w/o DLS}}$; and (2) removing the
scheduler and directly using all memorized data for denoising learning, named
SGDL${}_{\text{w/o ADS}}$. We summarize the results of NeuMF and LightGCN in
Table 3, while skip the results of other models (_i.e.,_ CDAE and NGCF)
because they demonstrate similar trends, and we have space limitation.
Obviously, compared with SGDL in Table 2, removing the denoising learning
scheme (_i.e.,_ SGDL${}_{\text{w/o DLS}}$) dramatically reduces the predictive
accuracy, indicating the necessity of self-guided learning. To be more
specific, SGDL${}_{\text{w/o DLS}}$ only trains with memorized data in the
noise-sensitive period, and thus, it is insufficient to learn user true
preference with hard yet clean interactions. Besides, directly leveraging all
memorized data as denoising signals would inevitably introduce some noise, and
hence, the SGDL${}_{\text{w/o ADS}}$ also underperforms the complete model.
#### 4.3.2. Estimation of Memorization Point.
We then verify the estimation of memorization point, since it plays an
important role in our model to transit from noise-resistant period to noise-
sensitive period. Specifically, we explore the performance change of SGDL by
increasing or decreasing the estimated noisy ratio $\hat{\sigma}$ to force
early or late transition. The results of SGDL on NeuMF and LightGCN are
presented in Table 4. We observe that:
* •
Generally speaking, the best performance is achieved at the estimated
memorization point. When the memorization point more deviates from the
estimated one, the performance tends to be worse. This means the existence of
the best memorization point and the effectiveness of our estimation.
* •
It is worth mentioning that, when the memorization point is slightly earlier
than the estimated value, it also shows good performance. We attribute this to
the clean memorized data: when we transit the model to noise-sensitive period
earlier, it is more likely that most of the memorized interactions are clean,
which benefits the following self-guided learning. On the contrary, when we
delay the memorization point, the memorized data tends to contain more noisy
samples, incurring suboptimal performance.
Table 4. Impact of estimation of memorization point. R@20 is used to evaluate
the performance, and Est. denotes estimated memorization point.
Memorization Point | Early | Est. | Late
---|---|---|---
Base Model | Database | +10% | +5% | +0% | -5% | -10%
NeuMF | Adressa | 0.3221 | 0.3275 | 0.3291 | 0.3256 | 0.3203
MovieLens | 0.2810 | 0.2851 | 0.2844 | 0.2757 | 0.2704
Yelp | 0.458 | 0.4420 | 0.0469 | 0.4430 | 0.4370
LightGCN | Adressa | 0.2006 | 0.2114 | 0.2105 | 0.2017 | 0.1990
MovieLens | 0.3321 | 0.3325 | 0.3335 | 0.3262 | 0.3198
Yelp | 0.0895 | 0.0912 | 0.0918 | 0.0904 | 0.0887
#### 4.3.3. The Design of Adaptive Denoising Scheduler.
We also investigate the design of adaptive denoising scheduler. Specifically,
we propose three different approaches to evaluate the denoising contributions
of each memorized data: i) Rank memorized data according to their sums of two
factors (_i.e.,_ normalized gradient similarity and loss value), and simply
pick top-$F$ memorized data as informative denoising signals. To keep the
picked data clean, we set $F$ as the half size of memorized data. ii) Choose a
one-layer MLP as the scheduler, and train the scheduler with the strategy
presented in Section 3.2.2. iii) Select the LSTM as the scheduler, and
leverage the historical factors to predict the sampling probabilities. We
report the performance of three sampling approaches with Recall@20 and NDCG@20
in Table 5, while Recall@5 and NDCG@5 are omitted due to limited space. We
have the following observations:
* •
By simply choosing top-$F$ memorized data with the highest sums as denoising
signals, the performance drops and becomes even worse than the normal
training. We attribute such degradation to the non-linear correlation of two
factors: the large loss value may not necessarily mean the memorized sample is
beneficial to denoising, as it can also be noisy if the gradient similarity is
small. Directly summing up the two factors is unable to properly capture the
characteristic of memorized data, and thus fails to provide reliable denoising
signals for self-guided learning.
* •
The scheduler with LSTM consistently achieves the best performance cross the
datasets. This is because LSTM can leverage the historical factors information
to predict more accurate and stable sampling probability for memorized data.
The results are consistent with previous robust learning studies (Chang et
al., 2017; Yao et al., 2021).
Table 5. Impact of the design of adaptive denoising scheduler.
Database | Yelp | MovieLens | Adressa
---|---|---|---
Base Model | ADS | R@20 | N@20 | R@20 | N@20 | R@20 | N@20
NeuMF | top-$F$ | 0.3106 | 0.1766 | 0.2590 | 0.2705 | 0.0386 | 0.0212
MLP | 0.3257 | 0.1842 | 0.2749 | 0.2841 | 0.0392 | 0.0225
LSTM | 0.3291 | 0.1853 | 0.2844 | 0.3032 | 0.0469 | 0.0260
LightGCN | top-$F$ | 0.1810 | 0.0985 | 0.3159 | 0.3349 | 0.0909 | 0.0513
MLP | 0.2066 | 0.0103 | 0.3284 | 0.3427 | 0.0912 | 0.0512
LSTM | 0.2105 | 0.1178 | 0.3335 | 0.3513 | 0.0918 | 0.0525
Figure 5. (a) Sample weight distribution on MovieLens dataset. (b) Learned
weighting function on MovieLens dataset.
### 4.4. Learned Weights of SGDL (RQ3)
In this section, we visualize the learned weights of SGDL to offer an
intuitive impression of our denoising performance. Specifically, we train
NeuMF with SGDL on the MovieLens dataset, and plot the learned weights
distribution _w.r.t._ clean and noisy interactions as well as their loss
values in Figure 5. We find that:
* •
The left of Figure 5 indicates that almost all large weights belong to clean
interactions, and the weights of noisy interactions are much smaller than that
of clean ones, meaning that the SGDL can differentiate clean implicit feedback
from the noisy one.
* •
The learned weighting function of SGDL in the right of Figure 5 demonstrates
that when the loss has relatively small values, the weighting function
inclines to increase the weight together with loss, indicating that it tends
to emphasize more on clean data for learning user preference; while as the
loss gradually becomes larger, the weighting function first remains unchanged
and then begins to dramatically decrease its weight, implying that it tends to
highlight the hard yet clean interactions with large weights first and then
suppress noise interactions. Therefore, SGDL is able to automatically locate
hard interactions for better learning user preferences.
## 5\. Related Work
Existing recommender systems are typically trained with implicit feedback.
Recently, some studies (Jagerman et al., 2019; Wang et al., 2021a, b; Lee et
al., 2021) have noticed that implicit feedback could be easily corrupted by
different factors (_e.g.,_ popularity bias (Chen et al., 2020) and unawareness
of users’ behaviors (Hu et al., 2008)), and the inevitable noise would
dramatically degrade the recommendation performance (Wang et al., 2021b, a,
2022; Tan et al., 2022). As a result, some efforts have been dedicated to
solving the noisy implicit feedback problem, which can be categorized into
sample selection methods (Gantner et al., 2012; Park and Chang, 2019; Ding et
al., 2019b; Yu and Qin, 2020; Wang et al., 2021b) and sample re-weighting
methods (Wang et al., 2021a; Hu et al., 2021; Wang et al., 2022).
Sample Selection. A simple idea to denoise implicit feedback is to select
clean and informative samples only, and train the recommendation model with
them. For example, WBPR (Gantner et al., 2012) considers that the missing
interactions of popular items are highly likely to be real negative examples,
and hence assigns higher sampling probabilities to them. IR (Wang et al.,
2021b) interactively generates pseudo-labels for user preferences based on the
difference between labels and predictions, to discover the noisy-positive and
noisy-negative examples. Nonetheless, their performance has high variance
since they heavily depend on the sampling distribution (Yuan et al., 2018).
Sample Re-weighting. On the other hand, loss-based methods focus on the
learning process of models (_e.g.,_ loss values and predictions) to
distinguish noisy interactions from clean data. For instance, T-CE (Wang et
al., 2021a) dynamically assigns lower weights to high-loss samples since it
has been shown that noisy examples would have larger loss values. DeCA (Wang
et al., 2022) develops an ensemble method to minimize the KG-divergence
between the two models’ predictions, under the assumption that different
models make relatively similar predictions on clean examples. Although these
methods achieve promising results without additional data, they heavily rely
on the predefined loss function and hyperparameters, incurring poor
generalization for different recommendation models.
Other Directions. There are some recent studies that consider using additional
information (Yi et al., 2014; Xie et al., 2020; Bian et al., 2021) or
designing model-specific structures (Yang et al., 2021; Chen et al., 2021b; Wu
et al., 2021) to improve the robustness of recommender systems. For instance,
DFN (Xie et al., 2020) proposes a feedback interaction component to extract
clean and useful information from noisy feedback with additional explicit
feedback (_e.g.,_ like and dislike). SGCN (Chen et al., 2021b) treat user-item
interactions as a bipartite graph, and devise a learnable regularization
module to preserve the sparsity and low rank of graph structure. SGL (Wu et
al., 2021) advances graph-based recommender systems with self-supervised
learning by employing graph structure augmentations. However, these methods
suffer from poor generalization, since they either need additional information
(_e.g.,_ explicit feedback) as guidance to denoise implicit feedback (Xie et
al., 2020; Bian et al., 2021), or are only applicable to specific data
structures (_e.g.,_ user-item bipartite graph) (Chen et al., 2021b; Yang et
al., 2021; Wu et al., 2021).
Difference from Existing Work. Our work can be seen as a variant of self-
training (Zoph et al., 2020), which leverages self-labeled memorized data to
enhance the robustness of training process. Compared with existing work, SGDL
is in an orthogonal direction, opening up a new research line of denoising
implicit feedback for recommendations. The most relevant work with ours is
MORPH (Song et al., 2021). However, it tackles the classification problem in
computer vision, and only uses the memorized data for training, which is not
feasible in ranking-based recommendations since the hard yet clean data may
not be memorized without clear guidance. Compared with previous methods, SGDL
is a truly model-agnostic framework, which can be easily applied to any
learning-based recommendation models with any ranking loss functions, and does
not need to define any weighting functions.
## 6\. Conclusion and Future Work
In this paper, we present a new denoising paradigm, called SGDL, which
leverages the memorization effect of recommendation models and designs two
training phases to exploit the self-labeled memorization data as guidance for
denoising learning. Extensive experiments conducted on three real-world
datasets with four representative recommendation models demonstrate the
superiority and universality of SGDL. In the future, we plan to jointly
explore the noise and bias existing in implicit feedback to develop a
universal denoising and debiasing solution.
###### Acknowledgements.
This work was supported by the NSFC under Grants No. (62025206, 61972338, and
62102351). Lu Chen is the corresponding author of the work.
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|
# SMCL: SALIENCY MASKED CONTRASTIVE LEARNING FOR
LONG-TAILED VISUAL RECOGNITION
###### Abstract
Real-world data often follow a long-tailed distribution with a high imbalance
in the number of samples between classes. The problem with training from
imbalanced data is that some background features, common to all classes, can
be unobserved in classes with scarce samples. As a result, this background
correlates to biased predictions into “major” classes. In this paper, we
propose saliency masked contrastive learning, a new method that uses saliency
masking and contrastive learning to mitigate the problem and improve the
generalizability of a model. Our key idea is to mask the important part of an
image using saliency detection and use contrastive learning to move the masked
image towards minor classes in the feature space, so that background features
present in the masked image are no longer correlated with the original class.
Experiment results show that our method achieves state-of-the-art level
performance on benchmark long-tailed datasets.
Index Terms— Long-tailed learning, contrastive learning, saliency, masking,
vision recognition, data augmentation
## 1 Introduction
(a) major (bird)
(b) minor (horse)
(c) saliency masked contrastive learning.
Fig. 1: (a): The image and its CAM of a major class sample (bird). (b): The
image and its CAMs of a minor class sample (horse), misclassified as “bird” by
the classifier. The middle image is the CAM for the predicted label (bird),
and the right image is the CAM for the true label (horse). (c): Illustration
of saliency masked contrastive learning.
Deep neural networks have shown remarkable performance across many computer
vision tasks such as image recognition and object detection. It is well known
that outstanding performance results from training with a large amount of
labeled data. However, many real-world datasets follow a long-tailed
distribution, where the number of samples in each class differs greatly. In
these datasets, the “major” classes have a large number of samples while the
“minor” classes have only a few samples in each class. When a model is trained
with a long-tailed dataset, the imbalance between classes hinders the model
from correctly learning discriminative features.
We focus on the problem where a background feature that should be common to
all classes is mistakenly correlated with a particular class. While this
problem also occurs in balanced datasets, its effect is amplified in long-
tailed datasets due to the lack of diversity in minor classes. For example,
background features such as seas and trees should not affect class prediction,
but may not be observed in minor classes, but only in some major classes. This
biased observation mistakenly biases some background to a major class
prediction.
Examples in Fig.1a and 1b show the “biased background” problem of a model
trained from imbalanced data. The image of a horse (a minor class) in Fig.1b
is misclassified as a bird (a major class) by the model. Fig.1b shows two
class activation maps (CAMs) obtained from the model. The left CAM is
associated with the predicted label “bird”, in which the grass area is
highlighted. On the other hand, the right CAM is associated with the true
label “horse”, where the body of the animal is given focus. In other words,
the model thinks the body looks like a horse, but the grass part is a clue
that this image is a bird. Since the background feature (grass) is strongly
attached to the bird class, the model outputs “bird” as the final decision.
The bias is also observed in the CAM image of a bird in Fig.1a, in which the
green area around the bird is highlighted, although the area is not related to
the bird itself.
Our goal is to move the representation of background features presented in
major classes towards minor classes in feature space so that the features are
shared with minor classes and no longer biased towards major classes. To this
end, we propose a new method called saliency masked contrastive learning
(SMCL). As illustrated in Fig.1c, the idea is to mask the salient part of an
image so that only the background part remains, and use contrastive learning
to pull the masked sample towards minor classes in feature space. While the
source image is selected from the original long-tailed distribution, the
target class where the masked image moves to is selected from a distribution
that gives higher weights to minor classes.
Previous methods exist where data augmentation is used to transfer the rich
context features from major classes to minor classes [1]. Our key difference
is that we use contrastive learning with masked data in order to move
background features towards minor classes in feature space, compared to
previous practices where augmented data are assigned labels and trained with
cross-entropy-only loss. As discussed in experiments and ablation studies,
applying contrastive learning is more effective than training them with cross-
entropy-only loss. Experiment results show that the proposed method achieves
comparable performance with the current state-of-the-art on the benchmark
long-tailed datasets.
## 2 Related Work
#### Long-tailed Recognition.
Long-tailed recognition is an important task since many practical datasets are
imbalanced. Initial approaches for long-tailed learning include re-sampling
[1] where more samples are selected from minor classes to achieve a balance
between classes, and re-weighting [2] where the loss function is adjusted to
give more weight to minor class losses. Many variations of re-sampling and re-
weighting techniques have been proposed [3, 4, 5, 6], but they do not address
the problem of biased background features. Mixup methods [7, 8, 1, 9, 10] can
help improve long-tailed learning by mixing two images of different classes
and assigning mixed labels. However, they do not apply contrastive learning
with mixed samples in order to obtain a better representation.
#### Supervised Contrastive Learning.
Supervised contrastive learning (SCL) [11, 12] pulls together samples
belonging to the same class and pushes apart samples from different classes in
the feature space. Recently, SCL is applied to long-tailed learning to
maintain uniform distribution in the feature space and improve class
boundaries [13, 14]. For example, TSC [14] generates uniformly distributed
targets on a hypersphere and uses contrastive learning to make the features of
different classes converge to the targets. Different from previous schemes,
our proposed method uses supervised contrastive learning so that the masked
data containing background features are pulled towards minor class samples in
the feature space.
## 3 Proposed Method
Overview of the proposed method SMCL is shown in Fig.2. SMCL involves saliency
masking to produce background images, weighted sampling to select target
labels giving priority to minor classes, and saliency masked contrastive
learning to pull the background image towards minor classes in the feature
space.
### 3.1 Saliency Masking
In order to mask discriminative features that belong to a certain class, we
use saliency detection methods. Similarly [8], we use the saliency function
$S(\cdot)$ and find the pixel $i$, $j$ that has the maximum value:
$\begin{array}[]{cc}i,j=\mathrm{argmax}(S(x)),\end{array}$ (1)
where $x$ is a sample in the training set $\mathcal{D}$. Then, following the
convention [7], we mask a region of the size generated following a beta
distribution $\mathrm{Beta}(\alpha,\alpha)$ centered at ($i$, $j$).
### 3.2 Weighted Sampling
For each data in the training set, we sample target data to be used in masked
contrastive learning. In order to increase the chance of selecting a minor
class label, we follow the sampling strategy [1] based on the effective number
calculated from class size [2]:
$E_{k}=\frac{(1-\beta^{n_{k}})}{(1-\beta)},$ (2)
where $n_{k}$ is the number of samples in $k$th class, $N=\sum_{k}{n_{k}}$,
and $\beta=(N-1)/N$. Based on $E_{k}$, the sampling probability of class $k$
is:
$p_{k}=\frac{1/E_{k}}{\sum_{k}{1/E_{k}}},$ (3)
which leads to a higher sampling probability for the minor classes. The “minor
weighted” distribution $\tilde{\mathcal{D}}$ is a distribution resulting from
sampling based on $p_{k}$.
### 3.3 Saliency Masked Contrastive Learning
The key idea of SMCL is to transfer a background feature from a major class to
a minor class. For an image $x$ in a batch, we sample a target image
$\tilde{x}$ from the minor weighted distribution $\tilde{\mathcal{D}}$. Then
we use basic augmentation schemes such as random crop and random flip to
create two versions of each sample, $x_{1}$, $x_{2}$, $\tilde{x_{1}}$,
$\tilde{x_{2}}$. Finally, we use saliency masking to generate the masked image
$x_{m}$. Inserting the five samples into the model, we obtain their logits $O$
and feature vectors $F$
$\begin{array}[]{cc}O,F=J([x,\tilde{x},x_{m}]),\end{array}$ (4)
where $O=[o_{1},o_{2},\tilde{o}_{1},\tilde{o}_{2},o_{m}]$ and
$F=[f_{1},f_{2},\tilde{f}_{1},\tilde{f}_{2},f_{m}]$. For the loss function, we
use a combination of the cross-entropy loss and the contrastive loss. The
cross-entropy loss $\mathcal{L}_{MCE}$ is calculated as:
$\mathcal{L}_{MCE}=(1-A)\cdot\mathcal{L}_{CE}(o,y)+A\cdot\mathcal{L}_{CE}(\tilde{o},\tilde{y}),$
(5)
where $o=[o_{1},o_{2},o_{m}]$,
$\tilde{o}=[\tilde{o}_{1},\tilde{o}_{2},o_{m}]$, $\mathcal{L}_{CE}(o,y)$ is
the cross-entropy between the logits $o$ and their true labels $y$, and $A$
indicates the proportion of the masked region. Next, we define the supervised
contrastive loss $\mathcal{L}_{SC}$ as:
$\mathcal{L}_{SC}=-\frac{1}{|B_{y}|-1}\sum_{p\in
B_{y}\setminus\\{z\\}}\log{\frac{\exp(f_{z}\cdot f_{p}/\tau)}{\sum_{k\in
B\setminus\\{z\\}}\exp(f_{z}\cdot f_{k}/\tau)}},$ (6)
where $B$ is the batch, $B_{y}$ is the set of samples in the batch that has
the label $y$, $z$ is the source image, $f$ is the feature vector, and $\tau$
is a temperature hyperparameter. We use a mixed contrastive loss
$\mathcal{L}_{MSC}$:
$\mathcal{L}_{MSC}=(1-A)\cdot\mathcal{L}_{SC}(f,y)+A\cdot\mathcal{L}_{SC}(\tilde{f},\tilde{y}),$
(7)
where $f=[f_{1},f_{2},f^{m}]$,
$\tilde{f}=[\tilde{f}_{1},\tilde{f}_{2},f^{m}]$, and $\mathcal{L}_{SC}(f,y)$
is the supervised contrastive loss when $f$ is the set of feature vectors and
$y$ is the set of true labels associated with the feature vectors. Finally,
our loss function combines the cross-entropy loss and the contrastive loss,
where $\lambda$ and $\mu$ are hyperparameters.
$\mathcal{L}=\lambda\cdot\mathcal{L}_{MCE}+\mu\cdot\mathcal{L}_{MSC}.$ (8)
Fig. 2: Overview of the proposed framework.
## 4 Experiments
### 4.1 Datasets
#### CIFAR-10-LT and CIFAR-100-LT.
CIFAR-LT datasets are constructed from CIFAR [15] by adjusting the dataset
into a long-tailed shape. In our evaluations, we consider three different
imbalance ratios $\rho\in\\{10,50,100\\}$ for CIFAR-10 and CIFAR-100
respectively.
#### ImageNet-LT.
ImageNet-LT is generated by selecting the long-tailed subset from
ImageNet-2012 [16]. The imbalance ratio $\rho$ is 256 and has 115.8K images
from 1,000 categories.
### 4.2 Implementation Settings
#### CIFAR-10-LT and CIFAR-100-LT.
We train ResNet-32 [17] for 200 epochs using stochastic gradient descent (SGD)
with a batch size of 256, a momentum of 0.9, and a weight decay of 2e-4. The
initial learning rate is 0.1 and we decay the learning rate at epochs 160 and
180 by 0.1 [3]. We use CutOut [18] and SimAugment [11] as augmentation
strategies. saliency masking is applied with a probability of 0.2 starting
from epoch 160. The hyperparameters $\lambda$ and $\mu$ are empirically tuned
to $\lambda=1.0$ and $\mu=0.3$.
#### ImageNet-LT.
We train ResNet-50 for 100 epochs using SGD with a batch size of 256, a
momentum of 0.9, and a weight decay of 5e-4. The initial learning rate is set
to 0.1 and follows cosine scheduling. We used RandAug and SimAugment as the
augmentation strategies. Saliency masking is applied with a probability of
0.4. Both augmentation and re-weighting are applied from epoch 80 [3]. The
hyperparameters $\lambda$ and $\mu$ are empirically tuned to $\lambda=1.0$ and
$\mu=0.35$.
The results from Mixup and CutMix are obtained following the same setup from
CMO [1].
### 4.3 Experimental Results
We compare the performance of SMCL with previous methods [3, 2, 19, 6, 20],
including mixup augmentation based methods [9, 7, 10, 1], and supervised
contrastive learning based methods [12, 14, 13].
Results on CIFAR-10/100-LT using different imbalance ratios are shown in Table
1. SMCL outperforms previous methods on CIFAR-10/100-LT when applied with
deferred re-weighting (DRW) [3]. Compared with baseline ERM and Supervised
Contrastive Learning (SCL), SMCL achieves 6.2% and 5.1% improvements in
CIFAR-100-LT (100), respectively. Furthermore, SMCL significantly outperforms
similar augmentation methods, such as Remix [10] and CMO [1]. SMCL also shows
consistent improvements even when compared with SCL based methods.
Dataset | CIFAR-100-LT | CIFAR-10-LT
---|---|---
Imbalance Ratio | 100 | 50 | 10 | 100 | 50 | 10
ERM† | 38.3 | 43.9 | 55.7 | 70.4 | 74.8 | 86.4
DRW† | 41.5 | 45.3 | 58.1 | 76.3 | 80.0 | 87.6
LDAM-DRW† [3] | 42.0 | 46.6 | 58.7 | 77.0 | 81.0 | 88.2
CB Focal [2] | 39.6 | 45.3 | 58.0 | 74.6 | 79.3 | 87.5
BBN† [19] | 42.6 | 47.0 | 59.1 | 79.8 | 82.2 | 88.3
MiSLAS$\ast$ [6] | 47.0 | 52.3 | 63.2 | 82.1 | 85.7 | 90.0
ResLT$\ast$ [20] | 48.2 | 52.7 | 62.0 | 82.4 | 85.2 | 89.7
DRW+Mixup [9] | 45.3 | 49.7 | 60.3 | 81.1 | 83.8 | 89.1
DRW+CutMix [7] | 46.4 | 51.2 | 62.1 | 81.8 | 85.1 | 89.7
ReMix$\ast$ [10] | 46.8 | - | 61.2 | 79.8 | - | 89.0
DRW+CMO [1] | 46.4 | 51.5 | 61.7 | 81.7 | 84.9 | 89.6
SCL | 39.4 | 44.4 | 56.7 | 72.2 | 76.8 | 87.1
SCL-DRW | 41.7 | 46.8 | 57.9 | 75.4 | 79.3 | 87.9
TSC$\ast$ [14] | 43.8 | 47.4 | 59.0 | 79.7 | 82.9 | 88.7
Hybrid-SC$\ast$ [13] | 46.7 | 51.9 | 63.1 | 81.4 | 85.4 | 91.1
SMCL (Ours) | 44.5 | 49.8 | 61.8 | 80.2 | 84.3 | 90.3
DRW+SMCL (Ours) | 50.1 | 54.8 | 64.1 | 84.2 | 86.8 | 91.0
Table 1: Comparison of classification accuracy(%) on the CIFAR-10/100-LT
datasets using ResNet-32. $\ast$ and † are results from the original papers
and [19], respectively.
Methods | All | Many | Med | Few
---|---|---|---|---
ERM† | 41.6 | 64.0 | 33.8 | 5.8
Decouple-cRT† [4] | 47.3 | 58.8 | 44.0 | 26.1
Decouple-LWS† [4] | 47.7 | 57.1 | 45.2 | 29.3
DRW | 49.9 | 61.2 | 47.1 | 29.2
LDAM-DRW | 50.3 | 61.7 | 47.0 | 30.6
Balanced Softmax [5] | 50.6 | 60.1 | 48.4 | 32.6
DRW+Mixup | 49.3 | 60.7 | 46.4 | 28.7
DRW+CutMix | 50.3 | 60.7 | 47.6 | 31.7
DRW+CMO | 50.8 | 61.2 | 48.7 | 30.1
SCL | 45.9 | 69.0 | 38.7 | 8.2
SCL-DRW | 49.4 | 61.6 | 47.2 | 24.1
TSC$\ast$ | 52.4 | 63.5 | 49.7 | 30.4
SMCL (Ours) | 46.0 | 67.9 | 39.1 | 10.3
DRW+SMCL (Ours) | 51.4 | 60.4 | 49.7 | 32.8
Table 2: Comparison of classification accuracy(%) on the ImageNet-LT dataset
using ResNet-50. $\ast$ and † are results from the original papers and [4],
respectively.
Results for ImageNet-LT are reported in Table 2. The table shows the accuracy
of three subsets: Many shots ($>$100 samples), medium shots (20-100 samples),
and few shots ($<$20 samples). SMCL shows comparable performance with the
state-of-the-art [14], showing 2.4% better performance on a few shots. Since
ImageNet-LT is composed of 38.5% of many, 46.9% of medium, and 14.6% of few
classes, improved performance in few has relatively little impact on overall
performance. However, it is noteworthy that applying SMCL with DRW outperforms
most of the existing methods, especially achieving considerable gains in the
accuracy of a few shots.
### 4.4 Ablation Study
#### Impact of Saliency Masking.
To show the impact of saliency masking, we tried masking images with two other
methods: masking random area and masking the center. As shown in Table 3,
covering the center of an image achieved 0.3% higher than the random. Since
many images have their core feature at the center, masking the center area is
more useful than masking a random area. Saliency masking showed the highest
result, approximately 0.3% higher than center masking. This result shows that
saliency masking is indeed effective in selecting background features from an
image.
| Random | Center | Saliency
---|---|---|---
Acc. (%) | 49.56 | 49.82 | 50.14
Table 3: Performance of saliency masking compared with other masking methods
on CIFAR-100-LT ($\rho=100$).
#### Effect of Contrastive Learning.
To evaluate the effect of saliency masked contrastive learning, we compared
the performance of models with and without contrastive learning. In Table 4,
“Cross Entropy” indicates the accuracy when the model is trained with only the
cross-entropy loss. The results show that applying contrastive learning is
effective in improving model performance.
| SMCL | DRW+SMCL
---|---|---
Cross Entropy | 43.57 | 49.12
| Supervised
---
Contrastive Learning
44.49 | 50.14
Table 4: Performance of applying saliency masked contrastive learning compared
with cross-entropy-only learning on CIFAR-100-LT ($\rho=100$).
## 5 Conclusion
We proposed SMCL, a saliency masked contrastive learning method that achieves
high accuracy on long-tailed datasets. SMCL uses saliency masking to obtain
background features from images and applies contrastive learning to move their
embeddings towards minor classes so that they are detached from major classes.
The proposed method is simple to implement, yet achieves state-of-the-art
level performance on benchmark long-tailed datasets such as CIFAR-10/100-LT
and ImageNet-LT.
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|
# EOS-ESTM: A flexible climate model for habitable exoplanets
L. Biasiotti,1,2 P. Simonetti1,2, G. Vladilo1, L. Silva1,3, G. Murante1, S.
Ivanovski1, M. Maris1, S. Monai1, E. Bisesi1, J. von Hardenberg4, A.
Provenzale5
1INAF - Trieste Astronomical Observatory, Via G. B. Tiepolo 11, 34143 Trieste,
Italy
2University of Trieste - Dep. of Physics, Via G. B. Tiepolo 11, 34143 Trieste,
Italy
3IFPU - Institute for Fundamental Physics of the Universe, Via Beirut 2, 34014
Trieste, Italy
4Politecnico di Torino - DIATI, Corso Duca degli Abruzzi 24, 10129 Torino,
Italy
5Institute of Geosciences and Earth Resources - IGG, National Research Council
of Italy, 56127 Pisa, Italy
E-mail<EMAIL_ADDRESS>
###### Abstract
Rocky planets with temperate conditions provide the best chance for
discovering habitable worlds and life outside the Solar System. In the last
decades, new instrumental facilities and large observational campaigns have
been driven by the quest for habitable worlds. Climate models aimed at
studying the habitability of rocky planets are essential tools to pay off
these technological and observational endeavours. In this context, we present
EOS-ESTM, a fast and flexible model aimed at exploring the impact on
habitability of multiple climate factors, including those unconstrained by
observations. EOS-ESTM is built on ESTM, a seasonal-latitudinal energy balance
model featuring an advanced treatment of the meridional and vertical
transport. The novel features of EOS-ESTM include: (1) parameterizations for
simulating the climate impact of oceans, land, ice, and clouds as a function
of temperature and stellar zenith distance; (2) a procedure (EOS) for
calculating the radiative transfer in atmospheres with terrestrial and non-
terrestrial compositions illuminated by solar- and non-solar-type stars. By
feeding EOS-ESTM with Earth’s stellar, orbital and planetary parameters we
derive a reference model that satisfies a large number of observational
constraints of the Earth’s climate system. Validation tests of non-terrestrial
conditions yield predictions that are in line with comparable results obtained
with a hierarchy of climate models. The application of EOS-ESTM to planetary
atmospheres in maximum greenhouse conditions demonstrates the possibility of
tracking the snowball transition at the outer edge of the HZ for a variety of
planetary parameters, paving the road for multi-parametric studies of the HZ.
###### keywords:
astrobiology – planets and satellites: terrestrial planets – planets and
satellites: atmospheres
††pubyear: 2021††pagerange: EOS-ESTM: A flexible climate model for habitable
exoplanets–EOS-ESTM: A flexible climate model for habitable exoplanets
## 1 Introduction
Over the past two decades, ground- and space-based observations have unveiled
thousands exoplanets and planetary systems around other stars in our Galaxy.
About $4900$ exoplanets are currently confirmed111e.g.
https://exoplanets.nasa.gov/; https://exoplanetarchive.ipac.caltech.edu/, in
large part detected as transits by the
Kepler222https://www.nasa.gov/mission_pages/kepler/main/index.html mission
(Borucki et al., 2010). It’s successor TESS333https://tess.mit.edu/ (Transit
Exoplanet Survey Satellite, Ricker et al., 2015) is expected to boost the
detection number, while
CHEOPS444https://www.esa.int/Science_Exploration/Space_Science/Cheops
(CHaracterizing ExOPlanet Satellite, Broeg et al., 2018) will help to
characterize the structural properties of already selected planets. In the
short term, PLATO555https://sci.esa.int/web/plato/ (PLAnetary Transits and
Oscillations of stars, Rauer et al., 2014) will search transiting Earth-
analogues around bright stars.
The statistically relevant numbers of detected planets are allowing to
investigate on all aspects of planetary structure and formation in different
size ranges, as a function of stellar spectral type, composition, and even
stellar multiplicity. A diversity of planetary systems architectures and a
large range of planetary masses and/or radii have been observed, showing that
the Solar System is just one possible outcome of the planetary formation
process (e.g. Udry & Santos, 2007; Howard et al., 2012; Winn & Fabrycky, 2015;
Kaltenegger, 2017). Observations are necessarily biased toward giant gaseous
planets around late-type stars, but the ever increasing statistics has allowed
to infer that virtually any star in our Galaxy hosts at least one planet, with
the planetary size distribution suggesting a steep increase towards small
rocky Earth-like planets with thin atmospheres. In fact, while it has been
found that planetary masses offer a loose constraint on composition, currently
in all cases it has been found that at small radii, R${}_{p}\leq 1.5-2$ R⊕,
all planets are rocky (e.g. Rogers, 2015) with a gap, i.e. an almost sudden
transition, between Earth-like volatile poor and Neptune-like volatile rich
planets (e.g. Fulton et al., 2017).
These studies are shifting the current research from detection and statistics
to full characterization of planetary properties, with one of the main goals
of exoplanetary science being the quest for life outside the Solar System.
This endeavour can only be tackled through remote atmospheric spectroscopy
(transit, reflection, emission and their time variations, e.g. Kreidberg,
2018) of potentially habitable rocky planets, in order to identify spectral
features of biological origin. This possibility rests on the notion that the
metabolic activity by-products for a well developed surface life may impact
the atmospheric chemistry to a measurable amount (e.g. Lovelock, 1965; Kasting
et al., 2014).
This observational challenge (e.g. Fujii et al., 2018, for a review) should be
partly within reach of the recently launched JWST (James Webb Space Telescope,
Gardner et al., 2006; Kalirai, 2018), probably limited to nearby M-type stars
(e.g. Koll et al., 2019), and, within the next decade, of the approved spatial
mission ARIEL666https://arielmission.space/; https://sci.esa.int/web/ariel/
(Atmospheric Remote-sensing Infrared Exoplanet Large-survey, Tinetti et al.,
2018), although mainly for objects with warm H-dominated atmospheres. Nearby
terrestrial analogues are expected to be detected with the ground-based
E-ELT777https://elt.eso.org/ (Snellen et al., 2015; Morley et al., 2017)
equipped with the spectrograph HIRES (Maiolino et al., 2013). In the longer
term, further space-based projects currently under assessment will be
selected, that specifically aim to directly detect and characterize nearby
temperate terrestrial analogues, e.g. HabEX888https://www.jpl.nasa.gov/habex/
(Habitable Exoplanet Observatory, Gaudi et al., 2020),
LUVOIR999https://asd.gsfc.nasa.gov/luvoir/ (Large UV/Optical/IR Surveyor, The
LUVOIR Team, 2019), OST101010https://origins.ipac.caltech.edu/ (Origins Space
Telescope, Wiedner et al., 2021), LIFE111111https://www.life-space-
mission.com/ (Large Interferometer For Exoplanets, Quanz et al., 2021). The
recent decadal survey for astronomy and astrophysics (Astro2020)
report121212https://nap.nationalacademies.org/catalog/26141/pathways-to-
discovery-in-astronomy-and-astrophysics-for-the-2020s endorsed recommendations
for a single UV/Optical/IR flagship mission, that picks a compromise concept
between LUVOIR and HabEX.
To accomplish the demanding task of searching for and deciphering spectral
signatures, a thorough and holistic observational and theoretical
characterization of carefully selected rocky exoplanets is required. The
selection, among the observationally reachable targets for high-resolution
spectroscopy of thin atmospheres, requires habitability studies with climate
models. These simulations will enable the identification of those exoplanets
with the largest chance of potentially hosting a surface diffuse life, i.e.
with the largest habitability, that must be evaluated over a wide range of
mostly unknown conditions. Moreover, the interpretation of any detected
atmospheric features in terms of physical status of the atmosphere, and of
their biotic or abiotic origin, will be unavoidably subjected to huge
uncertainties and degeneracies, including false positives even for oxygen
(e.g. Schwieterman et al., 2018; Meadows & Barnes, 2018). A considerable
effort of modelization that exploits all available observations will be needed
in order to assess the global physical characterization of the selected
exoplanets, and in particular precisely of their potential surface climate and
habitability.
Habitability studies for exoplanets rely on the concept of the habitable zone
(HZ), classically defined as the range of stellar insolation, the main driver
of climate, that allows surface temperatures compatible with a long-term
presence of surface liquid water for a planet with an N2–CO2–H2O atmosphere
and a climate system stabilized by the carbonate-silicate feedback (Walker et
al., 1981; Kasting et al., 1993; Kopparapu et al., 2013a; Kopparapu et al.,
2014). In these reference works, the inner and outer edges of the HZ are
defined respectively for a H2O- and a CO2-dominated atmosphere, for an
otherwise Earth-like planet orbiting stars of different spectral types. The HZ
is considered as the prerequisite for potentially inhabited planets with
exchange of gases between the biosphere and atmosphere (Kasting et al., 2014;
Schwieterman et al., 2018) and therefore for spectroscopic biosignature
searches. Actually, in addition to insolation and spectral type, a large range
of (mostly unknown) climate forcing factors affects planetary surface
temperature and habitability, e.g. atmospheric mass and composition, surface
gravity, radius, rotation period, obliquity, geography (e.g. Ramirez et al.,
2019), in addition to the observable orbital parameters. Also, different
definitions of habitability could be envisaged and calculated for an optimal
selection of exoplanets.
The large variety of planetary situations, expected and already uncovered by
observations hints that a large range of non-Earth conditions should be
accounted for. The majority of these parameters can currently only be explored
with climate simulations. Currently $\sim 60$ observed rocky exoplanets are
considered potentially habitable131313 https://phl.upr.edu/projects/habitable-
exoplanets-catalog, where the reported number refer to the empirical liquid
water HZ, as defined by the insolation range received by Venus and Mars
respectively $\sim 1$ and 4 Gyrs ago, when they could have hosted surface
liquid water (Kasting et al., 1993)., but their number may change should a
multi-parametric analysis of the huge possible parameter space of surface
temperature be performed.
The climate and the surface habitability of exoplanets can be explored, as for
the Earth, using a hierarchy of models, depending on the aim and problem to be
addressed (see e.g. Shields, 2019, for a review). Climate models should be
able to account for, even at different levels of simplification, the
complexity of the climate system due to the interplay of different components
and processes, giving rise to feedbacks leading to multiple equilibria or even
runaway conditions (Provenzale, 2013). For instance, the water-vapour and the
ice/albedo feedbacks set the spatial and temporal limits of the liquid water
HZ. The accounting of these complexities is particularly important to simulate
conditions not treated in Earth-tailored climate models.
Fully-coupled ocean-atmosphere General Circulation Models (GCM) are the most
detailed and computing resources consuming models, in principle requiring a
large amount of information to obtain meaningful results (e.g. detailed
geography and orography). In fact GCM are often applied for exoplanetary
studies by adopting an Earth or simplified configurations, such as an
aquaplanet (e.g. Leconte et al., 2013; Wolf & Toon, 2013, 2015; Shields et
al., 2014; Kaspi & Showman, 2015; Wolf et al., 2022). They are fundamental
tools to compute the coupled atmospheric-ocean dynamics on the long term and
to study atmospheric dynamics in particularly complex configurations. These
include rocky planets in the HZ of M-type stars, the most numerous and easiest
targets for spectroscopy follow-ups. Due to their proximity to the host stars,
these planets are expected to be tidally locked into synchronous rotation
(Leconte et al., 2015; Barnes, 2017). GCM are also fundamental benchmarks for
faster lower complexity models, allowing multi-parametric simulations.
Among such simpler models, 1D single-column radiative-convective models
including detailed line-by-line radiative transfer (RT) have been used for
instance to define the reference classical HZ mentioned above. Another class
of 1D models are the so-called zonal Energy Balance Models (EBM), which solve
a latitudinally-averaged energy balance with a simplified meridional heat
diffusion equation (North et al., 1981; Spiegel et al., 2008). This class of
models is still applied to the Earth climate, to be able to explore and
isolate the effects of specific processes on the global climate (see e.g.
Pierrehumbert, 2010). Their flexibility and short computing time can be
exploited also for the large parameter space required to simulate exoplanetary
conditions, by properly modelling all the terms entering the energy balance
equation. By coupling single-column RT atmospheric modelling with an EBM (e.g.
Williams & Kasting, 1997; Vladilo et al., 2013; Haqq-Misra & Hayworth, 2022,
hereafter WK97 and V13, respectively), and by further elaborating a
physically-based description of the meridional transport, Vladilo et al.
(2015, hereafter V15) developed a 2D EBM, the Earth-like planet Surface
Temperature Model (ESTM), specifically aiming to compute the seasonal and
zonal surface temperature of non-tidally locked exoplanets with a large range
of non terrestrial (atmospheric and planetary) physical conditions. The range
of applicability of this model was thoroughly explored in V15 by comparing
with the 3D aquaplanet model by Kaspi & Showman (2015). The flexibility and
fast computing time of ESTM has been exploited in Murante et al.
(2020)141414The library of climate models used for this work was extracted
from the ESTM-generated ARchive of TErrestrial-type Climate Simulations
(ARTECS) available at https://wwwuser.oats.inaf.it/exobio/climates/. The
database is in continuous expansion. for a statistical study of the multiple
equilibrium states affecting climate systems due to non-linear feedbacks (e.g.
the warm and snowball Earth states, during the latter the Earth would have
been tagged as non-habitable). In Silva et al. (2017b) we performed with ESTM
a multi-parametric exploration of the habitability for Kepler-452b (Jenkins et
al., 2015), currently the only known Earth-twin candidate. ESTM, by computing
the latitude- and seasonal-dependent surface temperature, allows different
operative definitions of habitability to be computed. Given the importance of
liquid water for terrestrial life, the liquid-water temperature interval is
the commonly adopted definition, and a pressure-dependent, liquid-water
habitability index (V13) can be defined. But also biological temperature-
based considerations can provide further HZ definitions and can be all
computed for each set of parameter choices (e.g. Silva et al., 2017a; Vladilo
& Hassanali, 2018). These more restrictive definitions, as compared to the
liquid water index, may help to increase the probability of selecting surface
ambient conditions that maximize the production and detectability of
atmospheric biosignatures (a discussion on the necessity and possibly of the
non-limiting assumption on searching for terrestrial-like life requirements
can be found in e.g. McKay, 2014; Kasting et al., 2014).
In this paper, we present a new release of the ESTM model, which we call EOS-
ESTM. EOS is our new procedure for calculating RT in rocky planetary
atmospheres with any pressure, chemical composition, and stellar spectral type
(Simonetti et al., 2022). In our previous version we were limited to Earth-
like systems. We have introduced and improved on several new parameterizations
with respect to the V15 model, in particular for the treatment of the
temperature-dependence of the ice coverage over land and ocean, and for the
zenith distance-dependence of the surface albedo specifically for any type of
surface. We have carefully calibrated EOS-ESTM to reproduce the Earth climate
by making use of large recent satellite (CERES-EBAF Ed4.1; Loeb et al. (2018))
and reanalysis (ERA5; Hersbach et al. (2020)) datasets, and validated the
predictive power of the model through detailed comparison with 1D and 3D
models under a large range of physical conditions. We also provide a first
exploration of the dependence of the maximum greenhouse distance of the HZ on
planetary parameters, as compared to 1D-based values.
The paper is structured as follows. In Section 2, after a schematic summary of
the ESTM model and parameters by V15, we provide a detailed description of
each physical input of the model which has been either newly introduced or
improved in the new EOS-ESTM release. In Section 3 we exploit the large amount
and good quality of experimental data of the Earth climate system to calibrate
and validate our Earth’s model, the reference for habitable rocky exoplanets.
In Section 4 we present the validation of EOS-ESTM for a large range of non-
terrestrial conditions with a comprehensive comparison of the predictive power
of our model with several other 1D and 3D models. Our summary and conclusions
are finally presented in Section 5.
Table 1: Treatment of the terms in Eq. (1) in classic EBMs, ESTM, and EOS-ESTM
Term | Description | Classic EBMs | ESTM | EOS-ESTM | Reference to the most updated prescription
---|---|---|---|---|---
$C$ | Thermal capacity | $C$ = constant | Ocean, Land, Ice | ESTM + Transient ice | This paper
$D$ | Meridional transport | $D$ = constant | $D=D(p,g,R_{p},RH,\Omega_{\text{rot}})$ | $D=D(p,g,R_{p},RH,\Omega_{\text{rot}})$ | Vladilo et al. (2015)
$I$ | Outgoing Longwave Radiation | $I=I(T)$ | CCM3 atmospheric RT | EOS atmospheric RT | Simonetti et al. (2022)
$S$ | Insolation | $S=S(t,\phi)$ | $S=S(t,\phi)$ | $S=S(t,\phi)$ | Vladilo et al. (2013)
$A$ | Top-of-Atmosphere Albedo | $A=A(T)$ | Surface & clouds + CCM3 atm. RT | Surface & clouds + EOS atm. RT | This paper; Simonetti et al. (2022)
## 2 The climate model
In accordance with classic EBMs, the planetary surface is divided in a number
$N$ of latitude zones and the zonal surface quantities of interest are
averaged over one rotation period. In this way, the surface quantities depend
on a single spatial coordinate, the latitude $\phi$. The thermal state of the
surface is described by the temperature $T=T(t,\phi)$. Since the zonal
quantities are averaged over one rotation period, the time $t$ represents the
seasonal evolution induced by the orbital eccentricity and tilt of the
rotation axis. By assuming that the heating and cooling rates normalized per
unit area are balanced in each zone, one obtains a set of $N$ zonal energy
balance equations
$C\frac{\partial T}{\partial t}-\frac{\partial}{\partial
x}\left[D\,(1-x^{2})\,\frac{\partial T}{\partial x}\right]+I=S\,(1-A)~{}$ (1)
where we omit the index that runs from 1 to $N$ for simplicity. The meaning of
the terms in this equation can be summarized as follows.
* •
The term $C$ represents the zonal heat storage and is expressed as heat
capacity per unit area (J m-2 K-1). It is calculated by summing the
contributions of lands, $C_{l}$, oceans, $C_{o}$, ice over lands, $C_{il}$ and
ice over oceans, $C_{io}$. These contributions are weighted according to the
zonal coverage of each surface component.
* •
The second term of Eq. (1) describes the meridional energy transport along the
coordinate $x=\sin\phi$. The transport is modelled using the formalism of heat
diffusion modulated by the parameter $D$ (the diffusion term). As a major
improvement with respect to classic EBMs, $D$ is expressed as a function of
the physical quantities that most affect the meridional transport, such as the
planetary radius, rotational angular velocity, surface gravity, and surface
atmospheric pressure. A detailed description of the physics behind this
formalism can be found in V15.
* •
The term $I$ is the Outgoing Longwave Radiation (OLR), which peaks in the
thermal IR band for typical conditions of habitable planets. At variance with
classic EBMs, $I$ is estimated using single-column, radiative-convective
calculations. By including the physics of the vertical transport, the ESTM
becomes a 2D climate model, one dimension sampling the surface as a function
of latitude, as in classic EBMs, the other dimension sampling the atmosphere
as a function of height from the surface. In practice, we calculate $I$ as a
function of $T$ for a given chemical composition and vertical stratification
of the atmosphere. Compared to the original ESTM, the calculations of
atmospheric radiative transfer that we present here have been greatly improved
(see Section 2.6.1).
* •
On the right hand of the Eq. (1), the term $S$ represents the insolation, i.e.
the incoming stellar radiation with maximum emission in the visibile/near IR
spectral range. More specifically, the zonal, instantaneous stellar radiation
that heats the planet, $S=S(t,\phi)$, is calculated taking into account the
stellar luminosity, the orbital parameters and the inclination of the planet
rotation axis. Details on these calculations can be found in V13.
* •
The term $A$ is the albedo at the top of the atmosphere, i.e. the fraction of
incoming photons that are reflected back in space without heating the planet.
The calculation of $A$ is extremely more detailed than in classic EBMs and is
performed in several steps. First, we calculate the surface albedo, $a_{s}$,
by weighting the albedo contribution of lands, $a_{l}$, oceans, $a_{o}$, ice
on lands, $a_{il}$, and ice on oceans, $a_{io}$, according the respective
fractional coverage. Then, the total albedo at the bottom of the atmosphere is
calculated by summing the albedo of the clear-sky surface with the albedo of
the clouds, weighted according to the fractional coverage of clouds. As an
upgrade over the original ESTM, we now calculate the cloud albedo taking into
account the reflection of the underlying surface (Section 2.5.2). Finally, the
top-of-atmosphere albedo is calculated as a function of $T$, $a_{s}$ and
stellar zenith distance, Z, for a given chemical composition and vertical
stratification of the atmosphere. These calculations are performed with the
upgraded recipes of radiative transfer that we present here (Section 2.6.1).
All the albedo prescriptions are calculated as a function of the zonal,
instantaneous stellar zenith distance, $Z=Z(t,\phi)$. In the original ESTM the
albedo dependence on $Z$ was considered for oceans, clouds, and atmosphere.
Here we improve formulas and we introduce this dependence also for lands and
ice.
In Table 1 we summarize how the terms in Eq. (1) have been upgraded from
classic EBMs to the ESTM. The main differences between the ESTM and the EOS-
ESTM are summarized in Table 2. In the rest of this section we review the
prescriptions that we adopt to model the different components of the climate
system, introducing the recipes that have been upgraded in the current EOS-
ESTM version. Technical details on the solution of Eq. (1) in the course of
the climate simulation can be found in V15 (Appendix A). In the present
version of the code we adopt 60 latitude zones, a starting temperature
$T_{0}=300$ K, and a tighter criterion of convergence for the global mean
temperature, $\langle T\rangle$: in practice, after running 20 orbits, the
convergence is considered to be achieved when $|\delta\langle T\rangle/\langle
T\rangle|<10^{-5}$ in two consecutive orbits. All these parameters can be
changed according to specific needs. For instance, $T_{0}$ can be varied in
studies of climate bistability where two stable solutions (a Snowball state
and a warm state) can be found in an appropriate parameter range depending on
the initial temperature. As in most EBMs, the original ESTM Murante et al.
(2020) and EOS-ESTM produce climate bistability.
Table 2: Main differences between ESTM and EOS-ESTM
Model prescription | ESTM | EOS-ESTM | Reference in this paper
---|---|---|---
Stellar spectrum | Solar | Any spectral type | Section 2.6.1
Atmospheric composition | Earth-like | Variable bulk composition | Section 2.6.1
Greenhouse gases | Trace amounts of CO2 and CH4 | Significant amounts of any greenhouse gas | Section 2.6.1
Surface albedo vs $Z$ | Oceans | Oceans, lands, ice | Sections 2.1,2.2,2.3
Calibration of ice coverage | Based on Williams & Kasting (1997) | Based on Earth’s satellite data | Section 2.3
Albedo & thermal inertia of transient ice | Not treated | Function of zonal ice cover | Section 2.4
Calibration of cloud albedo vs $Z$ | Based on Cess (1976) | Based on CERES-EBAF satellite data | Section 2.5.2
Cloud short-wavelength transmission | Not treated | Two-valued function of $T$ | Section 2.5.2
Cloud OLR forcing | Constant | Two-valued function of $T$ | Section 2.5.3
Cloud coverage over ice | Constant | Decreasing with global ice coverage | Section 2.5.1
### 2.1 Oceans
#### 2.1.1 Ocean fraction
The coverage of oceans on the planetary surface is parameterized by assigning
a fractional area coverage of oceans, $f_{o}$, to each latitude zone. This
parameterization is sufficient to test the climate impact of different
latitudinal distributions and oceans, including the extreme cases of ocean
worlds ($f_{o}=1$ in each zone). Oceans are characterized by their specific
properties of albedo and thermal inertia.
#### 2.1.2 Ocean albedo
The surface reflectivity of the oceans is modelled using empirical laws that
take into account its dependence on $Z$ and the fact that the water surface is
not smooth. We compared previous algorithms published in the literature
(Briegleb et al., 1986; Enomoto, 2007) with a recent set of measurements
obtained at different values of $Z$ (Huang et al., 2019). The observational
data (red dots in Fig. 1) show a large spread at any value of $Z$ due to the
variations of atmospheric transmittance created by scattering and absorption
of sunlight in the atmosphere (Payne, 1972). To model the ocean albedo we are
interested in the data in clear sky conditions, since the transmittance of the
atmosphere is accounted for in our radiative transfer calculations (Section
2.6.1). The lower envelope of the data in Fig. 1 represent the clear-sky case.
The formula proposed by Enomoto (2007) (black line) and Briegleb et al. (1986)
(blue line) are also shown in the figure. One can see that the expression
proposed by Enomoto (2007), namely
$\small
a_{o}=\frac{0.026}{1.1\mu^{1.7}+0.065}+0.15(\mu-0.1)(\mu-0.5)(\mu-1.0)~{}.$
(2)
(where $\mu=\cos Z$) yields a slightly better match to the lower envelope of
the data. We therefore adopt this expression as in V15. We do not propose an
expression to match the lowest points of the observational data set because we
do not have information about the measurement errors and we cannot exclude the
presence of outliers.
Despite having being calibrated with the Earth’s oceans, the empirical law (2)
can be reasonably applied to any exoplanetary ocean, since the zenith
dependence basically follows the universal Fresnel formula (WK97), corrected
for the roughness of the surface.
Figure 1: The albedo of oceans as a function of the stellar zenith angle, $Z$.
The blue and black lines represent the formulation adopted in Briegleb et al.
(1986) and Enomoto (2007), respectively. Red dots represent the data obtained
from Huang et al. (2019, Fig. 5a).
#### 2.1.3 Thermal inertia of oceans
Due to the high thermal capacity of water, the thermal inertia of oceans,
$C_{o}$, gives a major contribution to the term $C$. The full oceanic
contribution, which is effective over long time scales (typically decades on
Earth), is not treated in the ESTM. However, the short-term thermal impact of
the oceans is accounted for by considering the contribution of the mixed
layer, i.e. the surface layer of water that exchanges heat with the overlying
atmosphere (Pierrehumbert, 2010, hereafter P10). In the original version of
ESTM, we adopted a mixed-layer contribution $C_{\text{ml}}=C_{\text{ml50}}$
(WK97,P10), corresponding to thermal inertia of a 50-m, wind-mixed ocean layer
(see Table 3). In Section 3 we present a new tuning of this parameter based on
the short-term monthly variations of Earth’s surface temperatures. For
exoplanets with shallow oceans, $C_{\text{ml}}$ can be changed to simulate the
impact of water layers of different depths.
### 2.2 Lands
#### 2.2.1 Land fraction
The surface fraction of continents is described by assigning a fractional area
of land $f_{l}=1-f_{o}$, to each latitude zone. This parametrization is
sufficient to test the climate impact of extreme distributions of continents,
such as polar or equatorial continents, or desert worlds ($f_{o}=0$ in each
zone). Continents are characterized by their specific properties of albedo and
thermal inertia.
#### 2.2.2 Land albedo
To model the albedo of land we adopt a formulation proposed by Briegleb
(1992), namely
$a_{x}(\mu)=a_{x}(0.5)\frac{1+d}{1+2\,d\,\mu}$ (3)
where $a_{x}(0.5)$ is the albedo of a surface $x$ when $\mu$=0.5
($Z=60^{\circ}$), and the parameter $d$ regulates the dependence on stellar
zenith distance ($d=0.1$ for a “weak" dependence; $d=0.4$ for a “strong"
dependence). These parameters can be varied according to the type of surface
(desert, basalt, vegetation, etc.) in order to model planets with specific
characteristics (Coakley, 2003, Table 3 therein). The adopted values should be
representative of clear-sky conditions, since ESTM takes into account the
effects of the atmospheric albedo separately (Section 2.6.1).
Table 3: Adopted terms for thermal inertia
Parameter | Description | Adopted value | Comments
---|---|---|---
$C_{\text{ml50}}$ | Thermal inertia of the water mixed layer | $210\times 10^{6}$ J m-2 K-1 | Equivalent to a 50-m water layer (Pierrehumbert, 2010)
$C_{\text{atm},\circ}$ | Thermal inertia of the atmosphere | $10.1\times 10^{6}$ J m-2 K-1 | Equivalent to a 2.4-m water layer (Pierrehumbert, 2010)
$C_{\text{solid}}$ | Thermal inertia of the solid surface | $1\times 10^{6}$ J m-2 K-1 | Equivalent to a 0.5-m rock layer (Vladilo et al., 2013)
#### 2.2.3 Thermal inertia
The solid surface has a negligible thermal capacity compared to that of the
oceans and even compared to that of a relatively thin, Earth-like atmosphere.
The value of land thermal inertia that we adopt (Table 3) is representative of
a layer of rock with thickness of 0.5 m (Vladilo et al., 2013). Even if small,
this value becomes important in planets without oceans and with extremely thin
atmospheres.
Figure 2: Mean-annual values of Earth’s ice cover over lands and oceans
plotted as a function of the mean annual zonal surface temperature. Left
panel: Observational data for lands (green symbols) and oceans (blue symbols)
obtained with ERA5 reanalysis and NASA’s Terra and Aqua Satellites averaged
over the 2010; filled and empty circles represent data collected from the
Northern and Southern hemisphere, respectively; red dotted line: prescription
for the ice cover adopted in previous versions of ESTM. Right panel: same as
in the left panel, but excluding data with altitude above the freezing level
and the ocean data at the edge of Antarctica; solid lines: prescriptions
adopted in this work for lands (green line) and oceans (blue line); land data
influenced by local orographic conditions specific of the Earth have been
ignored; see Section 2.3.1
.
### 2.3 Ice
#### 2.3.1 Ice fraction
ESTM calculates the fractional coverage of ice over lands, $f_{il}$, and
oceans, $f_{io}$, making use of temperature-dependent algorithms. These
prescriptions are critical because the ice coverage plays a key role in the
albedo-temperature feedback and affects the lower temperature limit of liquid-
water habitability. We paid special attention to re-calibrate the algorithms
by searching for (i) a new set of experimental data and (ii) an appropriate
functional dependence on $T$.
1. 1.
The distribution of ice on the Earth surface was derived from measurements
obtained from NASA’s Terra and Aqua satellites151515See
https://modis.gsfc.nasa.gov/ in the period 2005-2015. To find a trend with
temperature, we associated the mean annual temperature of each latitude zone
to the corresponding fraction of ice. The temperature data were obtained from
the ERA5 dataset 161616See https://climate.copernicus.eu/climate-reanalysis
(Hersbach et al., 2020) in the same period. This exercise was done separately
for ice on lands and on oceans. To minimize the impact of
orographic/oceanographic conditions specific of the Earth, we considered only
land data in areas unaffected by local mountains and altitude below the
freezing level and we excluded ocean data at the edge of Antarctica171717The
combined effect of katabatic winds and ocean currents form mesoscale areas of
open water near the Antarctica coastline, known as polynyas (Stringer &
Groves, 1991). These areas fringe the edge of the continent owing to the
opening waterways (known as flaw leads) produced by the interconnection
between themselves (Meredith & Brandon, 2017). . The results are shown in Fig.
2, where the data (filled and empty circles) show that the dependence on $T$
is quite different for lands and oceans (green and blue colors, respectively).
The empirical trends show two features: (1) a very sharp rise of ice coverage
below the water freezing point; (2) the existence of a small fraction of ice
coverage at $T$ slightly above the freezing point. The exponential model
adopted in WK97 and in previous versions of the ESTM (dashed line in the left
panel) is not able to reproduce these two features, indicating the need of a
new type of functional dependence.
2. 2.
After trying different types of functions, we found that the empirical trends
of Fig. 2 can be well approximated using a generalized logistic function
(Richard, 1959). Based on the physical boundary conditions of our problem, we
chose a function that vanishes at very high $T$ and tends to 1 at very low
$T$:
$f_{ix}(\overline{T})={\frac{1}{\left[1+\xi_{x}\,e^{\theta_{x}\,(\overline{T}-T_{\circ,x})}\right]^{1/\xi_{x}}}}$
(4)
where $\overline{T}$ is the zonal temperature averaged over a time
$\tau_{\text{ice}}$ representative of the time scale of ice growth/melting;
the index $x$ refers to the type of surface underlying the ice cover ($x=o$
for oceans and $x=l$ for lands); $T_{\circ}$ is the temperature turning point
for the liquid-solid transition of water; $\theta$ is the growth rate; and
$\xi$ is the shape parameter.
The zonal temperature $\overline{T}$ is averaged over the interval
$\tau_{\text{ice}}$ that precedes the current time of the climate simulation.
For consistency with the data shown in Fig. 2, which have been averaged over
one year, we adopt $\tau_{\text{ice}}=12$ months. The parameters $T_{\circ}$,
$\theta$ and $\xi$ were tuned using: (i) the data versus temperature shown in
the right panel of Fig. 2, and (ii) the data versus latitude of the Earth’s
reference model (bottom right panel in Fig. 8). The adopted parameter values
are listed in Table 4. The resulting logistic functions (solid curves in the
right panel of Fig. 2) are able to reproduce the observed sharp rise below the
turning point $T_{\circ}$ and the existence of a small ice fraction slightly
above $T_{\circ}$.
#### 2.3.2 Albedo of frozen surfaces
The albedo of frozen surfaces (ice and snow) shows a remarkable scatter in
Earth measurements, with temporal variations that take place on different time
scales. In ESTM we adopt representative values based on average conditions. As
an upgrade with respect to the previous version, we model the albedo of ice
with Eq. (3). For stable ice on lands and oceans we adopt $a_{il}(0.5)=0.70$
$a_{io}(0.5)=0.55$, respectively. In both cases we set $d=0.1$, i.e. a “weak”
dependence on $\mu$ (Briegleb, 1992). These values provide a good match to
Earth’s zonal albedo (Section 3), but can be changed to model exoplanets with
specific properties of frozen surfaces.
#### 2.3.3 Thermal inertia
The contribution to thermal inertia of ice is important only if the planet
lacks oceans and has an extremely thin atmosphere. For icy surfaces we adopt
the same representative value adopted for any solid surface (Table 3). For icy
surfaces over oceans, following WK97, we add a small contribution (10.5
$\times$ $10^{6}$ J m-2 K-1) representative of the thermal inertia of the
underlying water.
Table 4: Parameters adopted for the ice coverage function (4). Parameter | Description | Value
---|---|---
Land | |
$T_{\circ\,l}$ | Temperature turning point | 265.15 K
$\theta_{l}$ | Growth rate | 1.2
$\xi_{l}$ | Shape parameter | 8.0
Ocean | |
$T_{\circ\,o}$ | Temperature turning point | 263.15 K
$\theta_{o}$ | Growth rate | 3.0
$\xi_{o}$ | Shape parameter | 12.0
### 2.4 Transient ice
The albedo and thermal capacity of transient ice (ice that is forming or
melting) are different from those of stable ice. A possible way to take into
account this effect is to introduce specific ice parameters in a temperature
range around the water freezing point (WK97). However, this approach requires
the introduction of several parameters not easy to quantify (the albedo and
thermal capacity of unstable ice and the temperature range where the
transition takes place). To avoid this additional parametrization we adopt a
prescription that provides a gradual change of the albedo and thermal capacity
from the case in which the ice is totally absent, to the case in which the ice
is stable.
#### 2.4.1 Albedo of transient ice
The albedo of stable ice over lands, $a_{il,s}$, and of stable ice over
oceans, $a_{io,s}$, is higher than the albedo of the underlying surface. When
the temperature increases and the ice becomes more and more patchy, the albedo
of unstable ice gets closer and closer to the albedo of the underlying
surface. To simulate this transition we assume that the fractional coverage of
ice (4) is a reasonable estimator of the patchiness of the ice and we adopt
the expressions
$a_{il}(\overline{T})=a_{l}+(a_{il,s}-a_{l})\,f_{il}(\overline{T})$ (5)
for ice over lands and
$a_{io}(\overline{T})=a_{o}+(a_{io,s}-a_{o})\,f_{io}(\overline{T})$ (6)
for ice over oceans. In this way the albedo attains the high values typical of
stable ice only when the ice coverage is complete. When the ice coverage is
absent, the albedo equals that of lands ($a_{l}$) or oceans ($a_{o}$) without
ice. The parameters in the above equations are calculated for $\mu=0.5$ (see
Table 7) and the albedo dependence on $\mu$ is modelled as explained in
Section 2.3.2.
#### 2.4.2 Thermal inertia of transient ice
For the thermal inertia of transient ice we follow the same approach used for
the albedo of transient ice. For the oceans, which provide the main
contribution to the thermal inertia, we adopt the relation
$C_{io}(\overline{T})=C_{o}+(C_{io,s}-C_{o})\,f_{io}(\overline{T})~{},$ (7)
where $C_{io,s}$ is the thermal inertia of stable ice over ocean. For land,
the contribution is very small, and there is no need to adopt a similar
relation since in our parametrization $C_{il,s}=C_{l}$.
### 2.5 Clouds
The complexity of the physics of cloud formation and the lack of
fluidodynamics and 3D capabilities of ESTM prevent us to model the spatial
distribution and physical properties of clouds in a self-consistent way.
Clouds are treated as bottom-of-the atmosphere features with parametrized
values of coverage, albedo, and OLR forcing. Given their critical role in the
climate energy budget, we introduced upgraded algorithms for all these
features.
#### 2.5.1 Cloud fraction
The cloud fraction is estimated with the expression
$f_{c}=f_{o}\,\left[(1-f_{io})\,f_{co}+f_{io}\,f_{ci}\right]+f_{l}\,\left[(1-f_{il})\,f_{cl}+f_{il}\,f_{ci}\right]$
(8)
where $f_{co}$, $f_{cl}$, and $f_{ci}$ are representative values of the cloud
coverage over oceans, lands, and ice, respectively (see Table 5). These values
are based on Earth data but, in principle, can be tuned for other planets. In
the original version of ESTM a constant cloud coverage over ice is adopted.
This approach is reasonable when ice is not dominant, as in the case of the
present-day Earth. However, when the planet undergoes a transition to a
snowball state, the global fraction of clouds is expected to decrease. To
capture this effect, we introduce a parameter $f_{c,sb}$, representative of
the cloud coverage in a snowball planet, and we adjust $f_{ci}$ in the course
of the climate simulation introducing a dependence on the globally-averaged
ice fraction, $\langle f_{\text{ice}}\rangle$. In practice we adopt the
expression
$f_{ci}=\left(f_{ci,\earth}-f_{c,sb}\right)\left(\frac{1-\langle
f_{\text{ice}}\rangle}{1-\langle
f_{\text{ice}}\rangle_{\earth}}\right)+f_{c,sb}~{}~{},$ (9)
where $f_{ci,\earth}$ is the cloud coverage over ice calibrated with Earth’s
data. With this prescription, $f_{ci}=f_{c,sb}$ when the planet enters a hard
snowball state ($\langle f_{\text{ice}}\rangle=1$). The parameter $f_{c,sb}$
can in principle be tuned from results of GCM simulations of snowball planets
(Abbot, 2014). The Earth model is not affected by the choice of $f_{c,sb}$
because $f_{ci}=f_{ci,\earth}$ when $\langle f_{\text{ice}}\rangle=\langle
f_{\text{ice}}\rangle_{\earth}$.
#### 2.5.2 Cloud albedo
To upgrade the prescriptions for the albedo of the clouds we: (i) used an
updated set of Earth satellite data; (ii) adopted a new functional form for
the dependence of the cloud albedo on $\mu$; and (iii) introduced a dependence
of the effective cloud albedo, $a^{\prime}_{c}$, on the albedo of the
underlying surface, $a_{s}$.
1. 1.
To upgrade the experimental data, we use the recent set of top-of-atmosphere
(TOA) albedo data obtained from the CERES-EBAF satellite (Loeb et al., 2018).
Following Eq. (8) by Cess (1976), we estimate the zonal TOA albedo of the
clouds with the expression
$A_{\textrm{c,obs}}=\frac{A_{\textrm{obs}}-A_{\textrm{obs,clear}}\,(1-f_{c})}{f_{c}}~{},$
(10)
where $A_{\text{obs}}$ are the TOA albedo data obtained in a given zone,
$A_{\text{obs,clear}}$, the TOA albedo data obtained in the same zone in
clear-sky conditions, and $f_{\textrm{c}}$ is the fractional cloud coverage in
the latitude zone of interest. To extract the dependence on zenith distance,
the derived values of $A_{\textrm{c,obs}}$ are associated to the mean annual
value of $\mu=\cos Z$ of the zone of interest, $\overline{\mu}$. In Fig. 3 we
compare the cloud albedo versus $\overline{\mu}$ that was obtained by Cess
(1976) from the dataset by Ellis & Haar (1976), with the results that we
obtain inserting the CERES-EBAF data for the period 2005-2015 in Eq. (10). One
can see that the use of the updated and extensive dataset provided by CERES
leads to significant differences. In particular, the cloud albedo becomes
weaker in the equatorial regions. We use this updated data set to improve the
description of the cloud albedo with respect to previous parametrizations
(WK97, V15) , which were based on Cess (1976).
Figure 3: TOA cloud albedo profile obtained using the data collected from
CERES-EBAF (green dots) in the period 2005-2015 compared with Cess (1976) data
for the NH (empty circles) and SH (black circles). The albedo is plotted
versus the mean-annual Solar zenith angle of each latitude zone.
2. 2.
In previous work, the dependence of the cloud albedo on the zenith distance
was modeled using the linear form $a_{\text{c}}=\alpha+\beta\,Z$ (WK97,V13).
To prevent the existence of negative values of albedo at low $Z$, V15
introduced a third parameter (the minimum value of cloud albedo at low zenith
distances, $a_{\text{c,min}}$), and used the expression
$a_{c}=\max\\{a_{\text{c,min}},(\alpha+\beta Z)\\}$. Here, for consistency
with the description of the albedo of land, ice and ocean, we adopt a
dependence on $\mu$, rather than on $Z$, also for the cloud albedo. In
practice, we adopt
$a_{c}(\mu)=a_{c}(0.5)+m_{c}\,(\mu-{\frac{1}{2}})$ (11)
where $m_{c}$ is the slope and $a_{c}(0.5)$ is the cloud albedo at $\mu$=0.5.
As can be seen in Fig. 4, this new function (solid line) yields positive
values at low $Z$ without the need of an extra parameter, and yields a
smoother dependence on $Z$ than the previous prescription (dashed line).
Figure 4: Comparison of the cloud albedo models adopted in V15 (dashed line)
and in the present work (solid line). See Section 2.5.2.
3. 3.
Polar clouds have a relatively high transmittance of short-wavelength photons,
and part of the photons are reflected by the underlying surface, making their
path up to the outer space through the clouds and the atmosphere. This effect
becomes particularly important when the underlying icy surface is very
reflective. The impact of this effect can be appreciated in Fig. 3, where one
can see that at low $\mu$, in correspondence with the ice polar caps, the
slope of the Earth’s cloud albedo versus $\mu$ becomes steeper. To incorporate
this effect in our model we follow Thompson & Barron (1981) and, based on
their Eq. (A14), we calculate the effective cloud albedo over reflective
surfaces,
$\small
a^{\prime}_{c}=a_{c}+\frac{(1-a_{c})\,(1-a^{*}_{c})}{a^{*}_{c}}\times[(1-t^{2}\,a^{*}_{\text{s}}\,a^{*}_{c})^{-1}-1]$
(12)
where $a_{c}$ is the cloud albedo over a non-reflective surface, $a^{*}_{c}$
the cloud albedo for diffuse radiation, $a^{*}_{\text{s}}$ is the surface
albedo for diffuse radiation; the values of diffuse albedo are estimated from
the corresponding direct albedo calculated at $\mu$=0.5; the parameter $t$ is
an estimator of the transmittance, i.e. the radiation fraction not absorbed
between the cloud top and the surface. We model $t$ with the function
$t=0.90-0.05\,\tanh{\left(\frac{T-263.15\,\text{K}}{10\,\text{K}}\right)}$
(13)
which provides a smooth transition between the typical transmittance of thin
polar clouds ($t\simeq 0.95$; Thompson & Barron (1981)) and the lower
transmittance in regions without surface ice. In these regions the surface
albedo is low, the term $t^{2}\,a^{*}_{\text{s}}\,a^{*}_{c}$ is small, and the
exact choice of $t$ has a modest impact on the calculation of $a^{\prime}_{c}$
with Eq. (12).
#### 2.5.3 Cloud OLR forcing
In the original version of the ESTM the cloud OLR forcing had a constant
value. This approach is unsatisfactory since the OLR measurements of
terrestrial clouds show an extremely large seasonal and latitudinal scatter
(Hartmann et al., 1992, Fig. 10 therein). As an attempt to introduce a
variable value by capturing a dependence on $T$, we tried to scale the cloud
OLR according to the water vapour content of the atmospheric column, which is
a function of $T$ at constant relative humidity. However, this attempt did not
provide a good match to the experimental data. This is not too surprising,
given the complexity of the physics of cloud formation and cloud radiative
transfer, which depends on a variety of thermodynamical and microscopic
factors not treated in our model. Despite the negative result of this attempt,
we decided to take into account the properties of clouds over icy regions as
an upgrade of our model. Terrestrial clouds over icy regions show a very small
OLR forcing, typically one order of magnitude smaller than the average value
(Hartmann et al., 1992, Fig. 10 therein). To capture this effect, we calculate
the cloud OLR forcing (also called cloud radiative effect, CRE) with the
following expression
$CRE=CRE_{\circ}\left[0.60+0.40\,\tanh{\left(\frac{T-263.15\,\mathrm{K}}{10\,\text{K}}\right)}\right]$
(14)
where $CRE_{\circ}$ is a representative value that can be calibrated with
terrestrial clouds (see Section 3.1.2) and the term in square brackets varies
smoothly from 1, at high $T$, to 0.2, at very low $T$. The above expression
provides a smooth transition with decreasing $T$ that mirrors the increase of
cloud transmittance adopted in Eq. (13). This simplified formalism is
justified by the fact that it improves the match between predicted and
observed zonal OLR in the Northern polar regions (see Section 3).
### 2.6 Atmosphere and top of atmosphere
#### 2.6.1 Clear-sky radiative transfer
The vertical radiative transport of energy throughout the atmosphere has a
strong impact on the $I$ and $A$ terms in Eq. (1) and therefore plays a
central role in the climate simulations. In the previous version of ESTM, the
terms $I$ and $A$ were estimated using the radiative transfer model developed
as part of the Community Climate Model 3 (CCM3, Kiehl et al., 1998), which is
based on the HITRAN 1992 spectroscopic repository data (Rothman et al., 1992).
CCM3 is a band model tailored for an Earth-like atmosphere illuminated by
solar-type radiation. As such, the concentrations of greenhouse gases can only
be varied in trace abundances, the list of greenhouse gases cannot be
expanded, and it is not possible to model stellar spectra different from the
solar one. To overcome these limitations and to use an updated repository of
spectroscopic data, the new version of ESTM uses the EOS radiative transfer
procedure (Simonetti et al., 2022) to calculate the terms $I$ and $A$.
EOS is a line-by-line procedure based on the publicly available opacity
calculator HELIOS-K (Grimm & Heng, 2015; Grimm et al., 2021) and the radiative
transfer code HELIOS (Malik et al., 2017, 2019). Line absorption from N2, O2,
H2O, CO2 and CH4 are calculated using data from the HITRAN 2016 (Gordon et
al., 2017) repository. The continuum of H2O is included via the standalone
version of the CNTNM routine of the LBLRTM code (Clough et al., 2005), which
runs the MT_CKD v3.4 opacity model (Mlawer et al., 2012). For CO2-dominated
atmospheres, the Collision-Induced Absorption (CIA) and the sub-Lorentzian
absortion lines shape of CO2 are also taken into account and calculated from,
respectively, HITRAN data and the recipes in Perrin & Hartmann (1989). The EOS
model has the advantage of not being tied to a specific type of atmosphere via
e.g. gas opacity parameterizations, thus allowing a far greater flexibility in
choosing the composition of the atmosphere. Radiative transfer calculations
are performed on a 60-layer atmospheric column logarithmically spaced in
pressure, from the $\sim$ 1 bar surface to the 1 $\mu$bar TOA level (10 layers
per order of magnitude). The OLR is evaluated as a function of $T$ every 20 K
below 280 K, every 10 K between 280 K and 310 K and every 5 K up to 360 K. The
TOA albedo is evaluated every 20 K in the entire $T$ range up to 360 K, for
$Z\in(0^{\circ},30^{\circ},45^{\circ},60^{\circ},70^{\circ},75^{\circ},80^{\circ},83^{\circ},85^{\circ},87^{\circ},88^{\circ},89^{\circ})$
and for $a_{x}\in(0.0,0.15,0.30,0.60,0.90)$. We adopted a non-equally spaced
grid in order to sample more precisely the regions in which the OLR and the
TOA albedo change their slopes. Multilinear interpolation on both OLR and TOA
albedo tables is then carried out by ESTM. Thanks to the fact that HELIOS and
HELIOS-K run on GPU processors (also known as graphic cards or accelerators)
it is possible to calculate the line-by-line radiative transfer in a
reasonable amount time even on desktop machines. As reported in Grimm et al.
(2021), GPU-based codes can be more than an order of magnitude faster than
CPU-based ones. Starting from the HITRAN line parameters files, EOS requires
$\sim$70 hours to calculate the OLR and TOA albedo tables for a specific
atmosphere181818Estimates obtained using a workstation equipped with an nVidia
RTX 2080 graphic card, a 7200 rpm HP/Seagate hard drive and an Intel Xeon
Silver 4108 CPU. The reading-writing times from the storage memory (hard drive
or solid state memory) and the CPU efficiency play important roles in
determining the final amount of time required by the procedure, which can be
consistently different on other machines.. Thanks to the modularity of the EOS
procedure, part of the results obtained in the early steps can be reused,
reducing the total time for the calculations of new cases.
#### 2.6.2 Top-of-atmosphere albedo
The top-of-atmosphere (TOA) albedo is tabulated as a function of $T$ for a
given set of atmospheric parameters, using radiative transfer calculations in
the short wavelength (visible, near IR) spectral range. Since the TOA albedo
also depends on the surface albedo, $a_{s}$, and the stellar zenith distance,
$Z$, the calculations of the TOA albedo must sample the parameter space ($T$,
$a_{s}$, $Z$). The wavelength dependence of the short wavelength scattering
implies that the albedo is sensitive to the spectral distribution of the host
star. The new radiative transfer procedure that we adopt allows us to tabulate
$A$ for planetary atmospheres illuminated by stars of different spectral type.
#### 2.6.3 OLR
For a given set of atmospheric parameters (surface gravity, surface pressure,
chemical composition, relative humidity, vertical structure), the OLR in the
thermal IR band is computed as a function of surface temperature $T$ by using
a reverse calculation of radiative transfer where the temperature of the
lowest atmospheric layer is forced to be equal to $T$. The resulting OLR
tables, calculated in clear-sky conditions, are given in input to the climate
simulation. The OLR forcing of clouds calculated with Eq. (14) is then
subtracted from the clear-sky OLR.
#### 2.6.4 Thermal inertia of the atmosphere
The atmospheric thermal inertia is much smaller than the oceanic one and can
be neglected in planets with oceans and thin, Earth-like atmospheres. However,
in general, the atmospheric thermal inertia must be taken into account, since
habitable planets may lack oceans and/or may have thick atmospheres. For this
reason, in our formulation we scale the thermal inertia of the Earth’s
atmosphere (Table 3) according to the thermal capacity and columnar mass of
the planetary atmosphere (V15). The thermal capacity is calculated for the
specific atmospheric composition; assuming hydrostatic equilibrium, the
atmospheric columnar mass is calculated as $p/g$, where $p$ is the surface
atmospheric pressure and $g$ the surface gravitational acceleration. The
atmospheric contribution to thermal inertia is summed to the ocean, land, and
ice contributions described above.
Table 5: Astronomical and planetary Earth data
Parameter | Description | Adopted value | Reference/Comments
---|---|---|---
$S_{\circ}$ | Mean annual insolation | 1361.0 W m-2 | CERES-EBAF (2005-2015)
$e$ | Orbital eccentricity | 0.01671022 |
$\epsilon$ | Axis obliquity | 23.43929 |
$g$ | Surface gravity acceleration | 9.81 m s-2 |
* •
a. Adopted Earth’s values can be changed to model exoplanets with different
types of stellar, orbital and planetary properties.
Table 6: Earth satellite data and results of the Earth reference model
Quantity | Description | Earth value | Model | Units
---|---|---|---|---
$\langle T\rangle$ | Global surface temperature | 287.5a | 288.7 | K
$\langle T\rangle_{\text{NH}}$ | Mean surface temperature of Northern hemisphere | 288.4a | 288.5 | K
$\Delta T_{\text{PE}}$ | North Pole-Equator temperature difference | 38.9a | 41.1 | K
$\langle h\rangle_{\text{NH}}$ | Fraction of habitable surface (Northern hemisphere) | 0.866b | 0.855 | …
$\langle A\rangle$ | Global top-of-atmosphere albedo | 0.314c | 0.315 | …
$\langle A\rangle_{\text{NH}}$ | Mean top-of-atmosphere albedo of Northern hemisphere | 0.310c | 0.314 | …
$\langle OLR\rangle$ | Global outgoing longwave radiation | 240.2c | 241.4 | W m-2
$\langle OLR\rangle_{\text{NH}}$ | Mean outgoing longwave radiation of Northern hemisphere | 240.8c | 241.6 | W m-2
$\langle f\rangle_{c}$ | Global cloud fraction | 0.674c | 0.666 | ..
$\langle f\rangle_{c,\text{NH}}$ | Mean cloud fraction of Northern hemisphere | 0.644c | 0.646 | ..
$\Phi_{\text{max}}$ | Peak of atmospheric transport at mid latitudes | 5.0d | 5.0 | PW
* •
a. Average ERA5 temperatures in the period 2005-2015.
* •
b. Average fraction of planet surface with temperature satisfying the liquid
water criterion.
* •
c. Average CERES-EBAF data in the period 2005-2015.
* •
d. Trenberth & Caron (2001)
Table 7: Parameters of surface and cloud albedo
Parameter | Description | Adopted value | Comments
---|---|---|---
$a_{l}$ | Albedo of lands (at $\mu$=0.5) | 0.20 | Tuned to match zonal albedo profile (Fig. 8)
$a_{il,s}$ | Albedo of stable frozen surfaces (at $\mu$=0.5) | 0.70 | Tuned to match zonal albedo profile (Fig. 8)
$a_{io,s}$ | Albedo of stable ice on ocean (at $\mu$=0.5) | 0.55 | Tuned to match zonal albedo profile (Fig. 8)
$a_{c}$ | Albedo of clouds (at $\mu$=0.5) | 0.44 | Tuned to match CERES-EBAF data (Fig. 6)
$m_{c}$ | Slope of cloud albedo equation | $-0.67$ | Tuned to match CERES-EBAF data (Fig. 6)
$f_{cw}$ | Cloud coverage on water | 0.72 | King et al. (2013)
$f_{cl}$ | Cloud coverage on land | 0.55 | King et al. (2013)
$f_{ci}$ | Cloud coverage on ice | 0.56 | Tuned to match the cloud coverage of Earth’s North Hemisphere
## 3 The reference Earth model
In this Section we present the calibration and the validation tests of the
model applied to Earth, which represents the reference for modelling habitable
exoplanets of terrestrial type. The large amount and good quality of
experimental data of the Earth climate system provide the best way for
adjusting many important model parameters and testing the new recipes that we
have introduced in the previous section.
### 3.1 Astronomical and planetary quantities
The values of astronomical and planetary quantities that we adopt for the
reference Earth model are listed in Table 5. Global values of planetary
temperature, OLR and albedo taken from satellite observations (CERES-EBAF and
ERA5) are summarized in Table 6. Unless differently specified, all data were
averaged for the period 2005-2015. For consistency, the insolation $S$ and the
volumetric mixing ratio of CO2 and CH4 were estimated for the same time
period. For the solar constant we adopt $S_{0}$ = 1361 W m-2 in accordance
with the mean annual insolation value measured from CERES-EBAF-EBAF (1361.16 W
m-2).
#### 3.1.1 Surface albedo
The adopted values of albedo are listed in Table 7. For the albedo of lands we
adopt a value representative of the Earth continents, namely
$a_{l}(0.5)=0.20$, which is an intermediate value between bare and vegetation-
covered soil. As far as the dependence with $\mu$ is concerned, we adopt a
“weak" dependence ($d=0.1$), which is representative of most types of Earth’s
continental surfaces (Briegleb, 1992; Coakley, 2003).
#### 3.1.2 Clouds
The parameters adopted for the clouds fraction on land, ocean and ice are
shown in Table 7. With respect to V15, the coverage over land and ocean,
$f_{cl}$, $f_{co}$, were updated following the experimental data by King et
al. (2013), while the adopted value of coverage over ice, $f_{ci}$, was
estimated from the CERES-EBAF satellite data.
The value of the cloud radiative forcing for longwave radiation, which acts as
a parameter in the model, was tuned in order to obtain a better match in the
OLR profiles in Figs. 7 and 8. By adopting $CRE_{\circ}$ = 26.1 W m-2 in Eq.
(14), we obtain an average value $\langle CRE_{\text{olr}}\rangle=25.5$ W m-2
for the Earth model. This is in excellent agreement with the mean value of the
Earth, 25.8 W m-2, obtained from CERES-EBAF Ed4.1, also considering the still
large uncertainty on this quantity found in the literature.
Table 8: Atmospheric radiative transfer parameters for the Earth’s reference
model
Quantity | Description | Adopted value | Reference/Comments
---|---|---|---
$p_{\text{dry}}$ | Surface pressure dry air | $10^{5}$ Pa |
$r$ | Relative humidity | 60% | Vladilo et al. (2015)
$c_{CO2}$ | Atmospheric concentration of CO2 | 350 ppm | See section 3.1.3
$c_{CH4}$ | Atmospheric concentration of CH4 | 1.7 ppm | See section 3.1.3
$T_{\text{tp}}$ | Temperature of tropopause | 200 K | Seidel et al. (2001), Kuell et al. (2005)
$CRE_{\mathrm{\circ}}$ | TOA longwave forcing of clouds | 26.1 W m-2 | tuned to match the OLR profiles in Figs. 7 and 8
#### 3.1.3 Atmospheric quantities
Table 8 shows the atmospheric quantities adopted in the EOS radiative transfer
calculations of the Earth model. Following Seidel et al. (2001) and Kuell et
al. (2005) we adopt a temperature for the tropopause $T_{\text{tp}}=$ 200 K.
We adopt a relative humidity of $60$%, in agreement with the global relative
humidity measured on Earth, a surface pressure of dry air of
$p_{\text{dry}}$=1.00 $\times$ 105 Pa and a volumetric concentration for CH4
of 1.7 ppmv. The CO2 gas concentration of the reference period 2005-2015
($c=390$ ppm, derived from the NOAA database
191919https://gml.noaa.gov/ccgg/trends/) has been corrected to compensate the
net TOA radiative imbalance of $\Delta F\simeq$ 0.6 Wm-2(Wild et al., 2013)
observed in the current transient climate, since we will be performing
constant-forcing simulations. We used the simplified analytical expression
linking CO2 concentration changes to the resulting radiative forcing change
(Myhre et al., 1998):
$\Delta F=\alpha\,\text{ln}(\frac{c}{c_{0}})$ (15)
with $\alpha$ = 5.35 W m-2, leading to a corrected volumetric mixing ratio for
CO2 of $c_{0}=350$ ppm.
Table 9: Parameters for the meridional transport
Quantity | Description | Adopted value | Reference/Comments
---|---|---|---
$D_{\circ}$ | Coefficient of latitudinal transport | 0.66 W m-2 K-1 | Tuneda to match the zonal temperature profile (Fig. 8)
${R}$ | Modulation of latitudinal transport | 1.4 | Tuned to match the zonal temperature profile (Fig. 8)
$\Lambda_{\circ}$ | Ratio of moist over dry eddie transport | 0.7 | V15; Fig. 2 in Kaspi & Showman (2015)
* •
a. $D_{\circ}$ is also tuned to match the Earth’s peak of atmospheric
transport at mid latitudes, $\Phi_{\mathrm{max}}$ (Table 6)
#### 3.1.4 Meridional transport
The parameters adopted for the meridional transport are shown in Table 9. The
parameter $D_{\circ}$ was tuned to match the Earth’s peak of atmospheric
transport at mid latitudes, $\Phi_{\mathrm{max}}$ (5.0 PW, Trenberth & Caron
(2001)) and the temperature-latitude profile in Fig. 8. We refer to V15 for a
full description of the parameters listed in the table.
#### 3.1.5 Depth of the mixed ocean layer
To tune the mixed ocean layer parameter, $C_{\text{ml}}$, we investigated the
monthly excursions of the Earth global surface temperature. In Fig. 5 we
compare the annual evolution of this quantity observed in the northern
hemisphere (crosses) with the predictions of our model obtained for different
choices of $C_{\text{ml}}$ (solid lines). One can see that by increasing
$C_{\text{ml}}$, the maximum annual excursion of monthly surface temperatures,
$\Delta T_{\text{max}}$, becomes smaller. A small time lag between the
predicted and observed peak is also present and increases with decreasing
$C_{\text{ml}}$. The value of thermal capacity that better reproduces the
observed trend is found at $C_{\text{ml}}\simeq C_{ml50}/2$. We therefore
adopt this value, which provides a small time lag and a difference of only 0.8
K between the observed and predicted value of $\Delta T_{\text{max}}$.
Figure 5: Seasonal evolution of mean surface temperature as a function of the
effective thermal capacity. The black, blue, green and red lines represent the
thermal capacities of oceans $C_{o}$ with $C_{ml50}$= 210, 105, 70 and 84
$\times$ 106 J m-2 K-1, respectively. Magenta Crosses: averaged ERA5
temperatures in the period 2005-2015 for the North Hemisphere (NH).
#### 3.1.6 Cloud albedo parameters
To tune the cloud albedo parameters $a_{c}(0.5)$ and $m_{c}$ in Eq. (11) we
used the TOA cloud albedo estimated from satellite data with Eq. (10). The
corresponding model predictions were then calculated by applying the EOS
radiative transfer calculations to the effective cloud albedo at the bottom of
the atmosphere estimated with Eq. (12). In this process we fine-tuned the
cloud transmittance defined in Eq.(13). In Fig. 6 we compare the observational
data set (green circles) with the final result of this modelization (orange
curve). One can see that the new calibration of cloud albedo provides a good
match to the measured trend of TOA cloud albedo versus $\mu$, including the
sharp rise observed at low $\mu$, i.e. over the Earth polar caps. From this
figure it is clear that the reflectivity of the underlying surface becomes
fundamental at the poles. The previous version of the ESTM is unable to
reproduce these features.
Figure 6: Cloud albedo versus $\mu=\cos Z$ for the present-day Earth (Northern
and Southern hemispheres). Dark green circles and crosses: observational data
obtained from Eq. (10) for the Northern and Southern hemisphere, respectively.
Red-solid and orange-dashed lines: ESTM predictions obtained by converting to
TOA values the cloud albedo calculated from Eq. (12) for the Northern and
Southern hemisphere, respectively.
Figure 7: Mean annual values of OLR (left panel) and TOA albedo (right panel)
plotted versus surface temperature for the present-day Earth. Data at
temperatures below 260 K are related to the South Polar cap. Green dots and
crosses: observational data obtained with the ERA5 reanalysis and CERES-EBAF
satellite data averaged over the period 2005-2015, for the Northern and
Southern hemisphere, respectively. Red-solid and orange-dashed lines:
predictions of the Earth’s reference model obtained with the radiative
transfer calculations specified in Section 3.1.3, for the Northern and
Southern hemisphere, respectively. The OLR peak at $\simeq$ 296 K in the left
panel is due to emission at the edges of tropical regions, which are connected
with the presence of large deserts and of low clouds with warm tops; the
decrease around the equator is associated with the presence of deep convective
clouds with cold tops.
Figure 8: Mean annual latitude profile of surface temperature, albedo,
outgoing longwave radiation (OLR), and fractional ice coverage predicted by
the reference Earth model (solid lines). Top left panel: the temperature
profile is compared with ERA5 temperatures averaged in the period 2005-2015
(blue dots). Top right panel: the albedo profile is compared with CERES-EBAF
data averaged in the period 2005-2015 (pink crosses). Bottom left panel: the
OLR profile is compared with CERES-EBAF data averaged in the period 2005-2015
(pink crosses). Bottom right panel: the model profile is compared with the
mean ice coverage, obtained by weighting the land and ocean data (averaged in
the period 2005-2015) in each zone according to the zonal coverage of lands
and oceans.
### 3.2 Diagnostic tests of the Earth model
To test the predictions of Earth model we first investigated the temperature
dependence of two key energy balance quantities, namely the OLR and TOA
albedo. We then compare global and zonal planetary data with the model
predictions.
#### 3.2.1 OLR and TOA albedo
In Fig. 7 we plot the OLR (left panel) and TOA albedo (right panel) versus
surface $T$ obtained from the Earth’s reference model (black lines) and
Earth’s satellite data (green circles). To obtain these plots, we plotted the
mean annual OLR and TOA albedo of each latitude zone versus the corresponding
zonal value of mean annual surface temperature. With this procedure we obtain
two independent sets of data versus $T$, one for the Northern and the other
for the Southern hemisphere. One can see that the EOS/ESTM calculations match
well the Earth data, despite the simplified nature of the model. The agreement
is better in the Northern hemisphere, which is less affected by the peculiar
orography of Antarctica. A better match would require a 3D climate model with
orography and a physical description of the atmospheric and oceanic
fluidodynamics and clouds.
#### 3.2.2 Global and zonal data
In Table 6 we display the globally averaged values of planetary quantities
predicted by the Earth’s reference model (second to last column). The
comparison with the corresponding experimental data (previous column) shows an
excellent agreement, with relative differences below 2% for the albedo and
much smaller for the temperature and OLR. The match of model and observed data
is particularly good in the Northern Hemisphere. The experimental data of the
Southern hemisphere are significantly influenced by the high altitude of
Anctartica, which is not accounted in our model without orography.
In Fig. 8 we compare the mean annual zonal values of surface temperature, TOA
albedo, OLR, and ice cover obtained from the reference Earth model (solid
lines) and Earth’s satellite and reanalysis data (symbols). The agreement of
the surface temperature curve (top left panel) is excellent, with an area-
weighted rms deviation of 1.0 K for the Northern hemisphere. The main
difference arise above the South Polar cap, where the observed temperature is
$\sim 20$ K lower than predicted due to the lack of orography of the model.
This difference is consistent with the $\sim 2$ km thickness of the ice sheet
and a dry lapse rate of $\simeq 10$ K/km.
The albedo curve (top right panel) shows an excellent agreement in the
Northern Hemisphere at mid-high latitudes, where the gradual change of the
albedo of transient ice provided by Eqs. (5) and (6) yields a better match to
the data than in the original ESTM. In the equatorial regions the agreement is
reasonable, considering the existence of albedo factors that can only be
treated in 3D models, such as the atmospheric circulation, which affects the
clouds distribution. In the Antarctic region, the model underestimates the
albedo due to the lack of orography.
The OLR profile (bottom left panel) shows a general agreement, with strong
deviations in Antarctica and in the tropical regions. The OLR excess predicted
by the model over Antarctic regions is due to the temperature excess that we
have already discussed. The bumps of OLR emission measured at the edges of
tropical regions are connected with the presence of large deserts and of low
clouds with warm tops, while the reduced equatorial OLR is connected with the
presence of deep convective clouds with cold tops (e.g. Hartmann, 2016).
Neither of these two features can be captured by our model which performs a
sort of average that provides the correct global value of OLR (Table 6). The
good match between model and observations in the North polar region is an
improvement with respect to the original ESTM, and is due to the fact that the
scaling factor (14) rises the planetary OLR emitted from frozen regions.
In the bottom-right panel we show a diagnostic test on ice coverage that was
not performed by V15. The mean annual zonal coverage of ice predicted is in
general agreement with the area-weighted lands and oceans data (red dots).
This implies that the new algorithm that we have introduced, based on Eq. (4),
is able to capture the main characteristics of ice coverage, using the
dependence on a single parameter, namely the surface temperature. A treatment
of the physics of ice formation and melting is, at this moment, beyond the
scope of our model.
## 4 Testing non-terrestrial conditions
By changing the input parameters that describe the stellar, orbital and
planetary properties, the ESTM can be in principle applied to simulate a broad
spectrum of exoplanetary climates. In this context, validation tests are
required to assess the limits of validity of the model in non-terrestrial
conditions. At present time, however, the climate systems of exoplanets are
poorly constrained by observations and of no use for validating the model.
Given this situation, the best way to test EOS-ESTM is to perform a comparison
with the predictions obtained by other models that have been developed to
investigate non-terrestrial planetary climates. Of particular interest is the
comparison with 3D and 1D climate models, given the fact that ESTM is a 2D
model, in the sense that we have clarified in Section 2. Below we provide the
results of some preliminary comparison tests, starting from a simple
simulation that we have performed using a 3D model of intermediate complexity.
We then describe several tests that we performed using predictions published
in the literature. So far, the space of stellar/planetary parameters that
affects exoplanetary climates has been covered only partially in previous
work. Therefore a comparison of different model predictions is only possible
for a limited number of cases. Here we focus our attention on the models and
published results summarized in Table 10.
### 4.1 Earth-like aquaplanet
As a preliminary comparison test with a 3D climate model, we used the global
climate model of intermediate complexity, PlaSim (Fraedrich et al., 2005;
Angeloni et al., 2020). Specifically, we tested the case of an aquaplanet with
rotational spin aligned with the orbital spin ($\epsilon=0$), the remaining
parameters being equal to those of the Earth. For the sake of comparison with
EOS-ESTM, the PlaSim simulation was run without oceanic transport. The
resulting mean annual latitude profiles of surface temperature and top-of-
atmosphere albedo are shown in Fig. 9. Despite the 3D nature of PlaSim and the
different prescriptions of surface features, clouds, and ice between the two
models, one can see that the results are in general agreement. This is true,
in particular, for the temperature profile (left panel). The differences found
in the albedo profiles (right panel) are due to the cloud distribution, which
follows the atmospheric circulation pattern that can be modelled in PlaSim,
but not in EOS-ESTM.
Setting PlaSim parameters to non terrestrial conditions is not trivial. This
is true, in general, for all 3D models, particularly for the most complex
ones. For this reason, extending the comparison tests with PlaSim to cover a
broader space of parameters will be the subject of a separate work.
Figure 9: Comparison of model predictions obtained for an Earth-like aquaplanet using the EOS-ESTM (black curves) and the 3D climate of intermediate complexity PlaSim (green curves). Left panel: mean annual surface temperature versus latidude. Right panel: mean annual top-of-atmosphere albedo versus latitude. See Section 4.1. Table 10: Summary of the test cases represented in Fig. 10, 11 and 12. The second and third columns give the radiative transfer (RT) model and the atmospheric properties adopted. The last column reports the references. Kunze et al. (2014) adopted the RAD4ALL model (Nissen et al., 2007) for the shortwave transport and the RRTM model (Mlawer et al., 1997) for the longwave transport. Model Name | RT model | Atmosphere | Reference
---|---|---|---
ESTM | EOS | 1.013 bar | This work
| | N2, CO2 (360 ppm), CH4 (1.8 ppm) and H2O |
1DGodolt2016 | K84202020Kasting et al. (1984) | 1 bar | Godolt et al. (2016)
| | N2, O2, CO2 (355 ppm), CH4 (1.64 ppm), O3 and H2O |
3DLeconte2013 | LMDG | 1 bar | Leconte et al. (2013)
| | N2, CO2 (376 ppm), and H2O |
3DWolf&Toon2015 | CAM 4 | 0.983 bar | Wolf & Toon (2015)
| | N2, CO2 (367 ppm), and H2O |
3DWolf&Toon2014 | CAM 3 | 0.983 bar | Wolf & Toon (2014)
| | N2, CO2 (367 ppm), and H2O |
EBMShields+2013 | SMART | present-day Earth | Shields et al. (2013)
| | CO2, O2 and H2O |
### 4.2 Variations of stellar insolation
To test the model response to variations of insolation, $S$, we run a set of
simulations aimed at reproducing similar climate experiments performed with 1D
(Godolt et al., 2016) and 3D models (Leconte et al., 2013; Wolf & Toon, 2014;
Shields et al., 2014; Wolf & Toon, 2015). In all cases, an Earth-like
atmosphere was considered, with properties described in Table 10.
The comparison with the 1D model is shown in Fig. 10, where we plot the mean
surface temperature as a function of $S$ for two cases considered by Godolt et
al. (2016). The first case (left panel) is a cloud-free Earth-like planet with
a fixed albedo $A_{\texttt{surf}}$ = 0.22, a value that, according to Godolt
et al. (2016), reproduces the mean surface temperature of Earth in a cloud-
free model. The second case (right panel) is an Earth-like aquaplanet with
$A_{\texttt{surf}}$ = 0.07, as adopted by Godolt et al. (2016), which is
representative of the ocean albedo. In both cases ice was not considered. One
can see that, despite the existence of some differences in the
parametrizations (Table 10), the EOS-ESTM results (red solid lines) are in
good agreement with those provided by Godolt et al. (2016) (black solid
lines). Departures between the models arise for high values of surface
temperatures. This effect is emphasised in the aquaplanet scenario, where the
deviations start to be present above $\sim$ 280 K (right panel). Since the
differences become important at high temperature, when the atmospheres contain
more water vapor, this effect may be induced by a different treatment of the
relative humidity (RH). Indeed, in our model we adopt a constant value,
RH=60%, representative of the mean global value of the Earth, whereas Godolt
et al. (2016) used a parametrization proposed by Manabe & Wetherald (1967),
representative of the RH vertical profile of the Earth.
Figure 10: Surface temperatures an Earth-like planet (left panel) and of an
aquaplanet (right panel) orbiting a Sun-like star at various stellar
insolations, $S$. In both cases, a cloud-free configuration is considered with
no ice formation and a fixed value of surface albedo: 0.22 (left panel) and
0.07 (right panel). Black solid line: results obtained by Godolt et al. (2016)
with a 1D model with relative humidity specified by Manabe & Wetherald (1967).
Red solid line: results obtained in this work, where we adopt RH=60%.
Figure 11: Comparison of global and annual mean surface temperature (left
panel) and TOA albedo (right panel) obtained from different climate Earth’s
models by increasing the solar constant. Red, solid line: EOS-ESTM (this
work). Black, solid line: 3D model CAM4 (Wolf & Toon, 2015). Green solid line:
3D model by Leconte et al. (2013). Dashed lines in the right panel represent
the surface albedo of these three models, indicated with the same color
coding.
In Fig. 11 we compare the mean annual global surface temperature (left panel)
and TOA albedo (right panel) obtained with the 3D climate models. When the
increase of insolation is modest, the results obtained with our model (red
line) are in general agreement with those provided by the 3D models. However,
discrepancies with Leconte et al. (2013) (green solid lines) and Wolf & Toon
(2015) (black solid lines) appear at higher insolation, when the surface
temperature rises above $\sim$ 290 K. Important differences are likely to
arise from the different RT model adopted (Table 10). As shown in Simonetti et
al. (2022, fig. 10a therein) and in Yang et al. (2016, fig. 3a therein),
differences in the impact of the water vapor absorption predicted by different
RT models start to become important for temperatures higher than 290 K. In
particular, due to the onset of the runaway greenhouse instability, the slope
of the OLR vs $T_{\texttt{surf}}$ in the CAM4 and LMDG models flattens more
than in EOS. Therefore, under equal insolation conditions, EOS-ESTM features
lower surface temperatures. On the other hand, the CAM3 model (solid cyan
line) exhibits a lower surface temperature in response to higher insolations.
We believe that this behaviour can be associated to the temperature dependence
of the OLR and TOA albedo. In fact, for temperatures above $\sim$300 K, the
CAM3 model features the largest value of OLR (Simonetti et al., 2022, Fig. 10a
therein) and, at the same time, a slightly higher TOA albedo compared to other
models (Simonetti et al., 2022, Fig. 10b therein).
Besides the different RT recipes, deviations in the predictions are expected
because our model does not incorporate a 3D physical treatment of the cloud
and water vapor feedbacks, even though it does reproduce the essential
features of the ice-albedo feedback and, to some extent, the rise of water
vapor with temperature. We suggest that the sharp transitions of surface
temperature (left panel) and TOA albedo (right panel) found by Leconte et al.
(2013) and Wolf & Toon (2015) can be associated to variations in the cloud
fraction in response to the increase of insolation. This interpretation is
consistent with the fact that such transitions are not found for the surface
albedo (dashed lines, right panel), for which the cloud/atmospheric effects
are not relevant.
### 4.3 Variation of stellar spectra
The ice-albedo feedback is a well known mechanism that affects the planetary
climate with a de-stabilizing effect that, in the most extreme cases, may lead
to an ice-covered planetary state, called “snowball” (Kirschvink, 1992). Owing
to the wavelength dependence of the albedo, the impact of this effect will
depend on the spectral energy distribution (SED) of the central star. G-type
stars, like our Sun, emit a far greater fraction of their radiation in the
visible light spectrum, whereas smaller and cooler M-dwarfs exhibit their peak
output in the $\sim$0.8 to 1.2 $\mu$m range (Shields et al., 2013, Fig. 1a
therein). The fact that these stars emit a significant fraction of their
radiation above 1 $\mu$m, combined to the reduction of the albedos of snow and
ice at the same wavelengths (Shields et al., 2013, Fig. 1b therein), implies
that the albedos of frozen surfaces are lower on planets orbiting M-type stars
than on Earth. Calculations of broadband albedo212121The ratio of the surface
upward radiation flux to the downward radiation flux within a certain
wavelength range (Kokhanovsky, 2021)., performed taking into account the
stellar SEDs and the wavelength-dependence albedo of snow and ice, indicate
that the ice-albedo feedback is weaker around M-type stars (Joshi & Haberle,
2012). The atmospheric contribution to the albedo in these stars was studied
by Von Paris et al. (2013): the presence of trace amounts of H2O and CH4 in
the atmosphere, as well as high CO2 pressures, damps the ice-albedo feedback
in planets around M-type stars.
To test the EOS-ESTM predictions at different stellar SEDs we performed a
comparison with the work by Shields et al. (2013). These authors used a 1D
radiative transfer model (SMART) to calculate the broadband planetary albedo,
given the spectrum of the central star and that of the surface albedo. Then,
they included the resulting broadband albedo into a 1D EBM to calculate the
mean global surface temperature as a function of insolation for an aquaplanet
orbiting a G-type star (the Sun) and an M-type star (AD Leo). Following their
prescriptions, we considered an aquaplanet with an axis obliquity of 23∘, zero
orbital eccentricity and a present-day Earth atmospheric composition. For
consistency with their work, we considered a spectral distribution
representative of AD Leo, which is a M3.5-type star, with a M⋆ = 0.42 M⊙
(Reiners et al., 2009). Since at decreasing insolation the aquaplanet
undergoes a transition towards a “snowball” state, we dedicate special
attention to select the value of the albedo of ice over ocean, $a_{io}$. Among
the different values of “blue ice“222222According to Shields et al. (2013)
blue marine ice results from freezing of liquid marine water and not from
glacier ice. calculated by Shields et al. (2013, Table 2) we adopt the value
for the case with no gases and clouds and no Rayleigh scattering. This is the
case more appropriate for the surface albedo in our model, since the EOS-ESTM
calculates the contribution of the atmosphere and clouds in its own way. The
results are shown in Figure 12, where the planet orbiting the M-type dwarf
appears less susceptible to “snowball” states, since the ice is particularly
absorptive in the NIR, as well as the atmosphere. In spite of small
differences associated to the transition to a complete “snowball” state,
expressed as a sudden decrease of global surface temperature, the results
obtained with EOS-ESTM (red lines) and by Shields et al. (2013) (black line)
show an overall agreement. The main difference between the trends found in the
two models may arises from the different parameterization of ice: the smooth
transition in our model is probably due the gradual temperature dependence of
the ice coverage described in Section 2.3.
Figure 12: Mean global surface temperature versus stellar flux for an
aquaplanet orbiting an M-type (dashed lines) and a G-type (solid lines) star.
Due to the different stellar SED, the albedo of ice over the ocean, $a_{io}$,
is varied as indicated in the legenda. Black lines: predictions obtained by
Shields et al. (2013) with SMART in combination with an EBM adapted from North
& Coakley (1979). Red lines: results obtained in this work adopting a warm
start for consistency with Shields et al. (2013).
### 4.4 Variations of planet radius and rotation rate
A critical difference between ESTM and GCMs is the treatment of the meridional
transport. As explained in V15 (see Sec. 2.1), we model the $D$ term in Eq.
(1) as a scaling relation between planetary quantities that are involved in
the physics of the meridional transport. To test the reliability of this
parameterization, we run a set of simulations varying one parameter at a time
and compared our results with similar tests performed with GCMs by other
authors. Specifically, we varied the planetary radius and rotation rate and
performed a comparison with results published by Kaspi & Showman (2015) and
Komacek & Abbot (2019).
The mean annual equator-to-pole temperature difference, $\Delta
T_{\text{EP}}$, is a good indicator of the efficiency of the meridional
transport and is expected to be higher in planets with fast rotational
velocities or large radii. The Coriolis forces resulting from planetary
rotation tend to inhibit the transport from the tropics to the poles, leading
to a higher gradient $\Delta T_{\text{EP}}$. Quantifying these effects with 3D
models is important because changes of rotational angular velocity may affect
the location of the inner edge of the habitable zone (Yang et al., 2014; Yang
et al., 2019). Also variations of planetary radius affect the meridional
gradient: as the radius increases, so does the physical distance between
equator and poles, leading to a less efficient meridional heat distribution,
i.e. a larger $\Delta T_{\text{EP}}$. In Fig. 13 we show how $\Delta
T_{\text{EP}}$ is predicted to change as a function of planetary radius (left
panels) and rotation period (right panels) for different models that we
describe below. Following Komacek & Abbot (2019), we normalize $\Delta
T_{\text{EP}}$ to the values predicted by each model for Earth’s values of
rotation rate and radius. The results obtained by Kaspi & Showman (2015) and
Komacek & Abbot (2019) are rather different, despite both being based on 3D
models. We refer to the latter paper for a discussion on these differences,
which may be due to different physico-chemical assumptions and to the fact
that the model of Komacek & Abbot (2019) had not been tuned to match the
Earth. Here we compare our results with those obtained in these two papers.
#### 4.4.1 Comparison with Kaspi & Showman (2015)
To investigate the atmospheric dynamics over a wide range of planetary
parameter space Kaspi & Showman (2015) adopted a 3D GCM with a scheme similar
to that of Frierson et al. (2006) both for the radiative transfer and the
surface boundary-layer: a standard two-stream gray radiation and an uniform
1-meter water-covered slab, with an albedo of $A=0.35$, respectively. They
modelled an idealized aquaplanet at perpetual equinox with an Earth-like
reference atmosphere. The effects of clouds, sea-ices and continents were not
accounted for. For the sake of comparison we adopted the same set of
conditions in EOS-ESTM. In the top panels of Fig. 13 one can see that, in
spite of differences at the low- and high-radius and rotation rate regimes,
the EOS-ESTM predictions (red symbols and lines) reproduce the trends obtained
by Kaspi & Showman (2015) with the 3D aquaplanet (blue symbols and lines). The
two sets of results are consistent as long as the planets have radii and
rotation rates sufficiently close to those of the Earth. For habitability
studies we are interested in the range of radii expected for rocky planets,
shown as shaded red areas in the left panels of Fig. 13. In this range, the
predictions of the two models are comparable.
Figure 13: Normalized equator-to-pole temperature difference as a function of
planet radius (left panels) and planet rotation rate (right panels) for an
Earth-like aquaplanet. Top panels: comparison between the results obtained in
this work (red lines) and those obtained by Kaspi & Showman (2015) (blue
lines) for a cloud-free aquaplanet slab ocean with a depth of 1- meter, an
axis obliquity $i$ = 0 ∘, a fixed albedo (A=0.35) and no sea ice. Bottom
panels: comparison between the results obtained in this work (red lines) and
by Komacek & Abbot (2019) (blue lines) for an aquaplanet slab ocean with a
depth of 50- meter where the effects of clouds, nongray radiative transfer,
and sea ice are included; axis obliquity $i=0$ and the eccentricity $e=0$.
Shaded red area: range of radius (0.5<R/R⊕<1.6) at which an exoplanet is more
likely to be composed of rock and metal (Meadows & Barnes, 2018).
#### 4.4.2 Comparison with Komacek & Abbot (2019)
More recently, Komacek & Abbot (2019) investigated how the atmospheric
circulation and climate of planets orbiting Sun-like stars vary when planetary
parameters are changed. They used the state-of-the-art GCM ExoCAM (a modified
version of the Community Atmosphere Model version 4), to simulate an idealized
aquaplanet with a 50-m water slab without oceanic transport and an atmosphere
of N2 and H2O. At variance with Kaspi & Showman (2015), they included the
effects of clouds, non-gray radiative transfer, and sea ice. To test the EOS-
ESTM predictions making use of their results, we adopted, for consistency, a
uniform 50-meter thick water-covered slab, the same atmosphere of N2 and
H2O,and the same ice albedo; we set to zero both the axis obliquity and
orbital eccentricity. In the bottom panels of Figure 13 we compare the
normalized $\Delta T_{\text{EP}}$ obtained with our model (red symbols and
lines) with those obtained with the GCM (blue symbols and lines).
In the bottom-left panel one can see that the trend of increasing $\Delta
T_{\text{EP}}$ that we find (red line) is consistent, but steeper than that
found with the GCM (blue line), the departures becoming significant above the
radius limit of rocky planets. The smoother trend found with the GCM suggests
that the large scale, 3D circulation, not present in our model, may enhance
the heat distribution. Our trend is somewhat steeper than the one that we
found in the previous test (top-left-panel), indicating how the inclusion of
clouds and ice impacts our results.
The bottom-right panel of Figure 13 shows a consistent trend for angular
velocity lower than $\Omega_{\oplus}$, with a discrepancy at high angular
velocity ($\Omega$ = 2 $\Omega_{\oplus}$). This discrepancy is surprising,
because our algorithm for meridional transport is expected to be more
realistic for fast-rotating planets (see V15), and this indication is
supported by the comparison with Kaspi & Showman (2015) shown in the top-right
panel. Clearly, the two 3D models that we are using for comparison show
remarkable differences between them and should be taken with some caution. At
low rotation speed, where we know that our assumptions are more critical (see
V15), our model seems to underestimate $\Delta T_{\text{EP}}$, as in the
comparison with Kaspi & Showman (2015) shown in the top-right panel. These
results suggest that the 3D circulation may be able to redistribute the heat
efficiently, with a weak dependence on the planet rotation rate.
#### 4.4.3 Future improvements of the model
In our parametrization of the meridional transport, the term $D$ scales as
$\Omega^{-4/5}$ and $R^{-6/5}$ (see V15, Section 2.1). Taking advantage of the
flexibility of our model, we varied the exponents of these power laws,
searching for a better agreement with the trends obtained by the 3D models
shown in Fig. 13. The dashed green lines plotted in all panels of that figure
show that a better match with the 3D results is achieved when adopting a more
moderate dependence for both the angular velocity, $\propto\Omega^{-0.5}$ and
radius, $\propto R^{-0.5}$. This exercise shows that, in principle, one could
recalibrate the exponents of the scaling relations that we adopt for $D$,
making use of specifically designed tests performed with GCMs. To this end it
would be important to use realistic 3D models for cross validation. Realistic
models should include the main components of the climate system and should be
calibrated to match the Earth data. Setting state-of-the art GCM models to
simulate non-terrestrial conditions is not a straightforward task. However,
this is the way to proceed for expanding the range of application of flexible
models such as EOS-ESTM and exploring the parameter space that allows
habitable climates to exist.
### 4.5 The outer edge of the habitable zone
In classic studies of the habitable zone (HZ) the locations of the inner and
outer edge are calculated making use of single-column, cloud-free atmospheric
climate models (Kasting et al., 1993; Kopparapu et al., 2013a). In recent
years several studies have proven the critical role of planetary properties on
the position and extension of the circumstellar HZ (Yang et al., 2014; Rushby
et al., 2019; Yang et al., 2019; Zhao et al., 2021). Here we take advantage of
the flexibility of EOS-ESTM to investigate how the location of the outer edge
is affected by variations of planetary parameters. Establishing the exact
location of the outer edge would require a study of clouds effects (e.g.
Forget & Pierrehumbert, 1997; Selsis et al., 2007; Kitzmann, 2017) and the
possible presence of other greenhouse gases, such as CH4(see Ramirez &
Kaltenegger, 2018). However, for the climate experiments that we present here,
we simply adopt the “maximum greenhouse” limit, defined as the maximum
distance at which a cloud-free planet with an atmosphere dominated by CO2 can
maintain a surface temperature of 273 K (Kasting et al., 1993). Beyond this
limit, the greenhouse effect due to a further rise of CO2 is offset by the
rise of atmospheric albedo due to the Rayleigh scattering of CO2 molecules. To
explore the impact of planetary parameters on the location of the outer edge,
we considered a cloud-free, CO2-dominated atmosphere with a dry surface
pressure of 7.3 bar232323Kopparapu and collaborators also had an additional
bar of N2 in all their models, thus they evaluated the CO2 partial pressure.
Taken alone, it would produce a surface pressure of 7.3 bar., which is the
value identified by Kopparapu et al. (2013a) as the maximum greenhouse limit.
To build the pressure-temperature profile of the atmosphere we followed the
recipes in the appendix of Kasting (1991), with a H2O-saturated lower
troposphere, a CO2-saturated upper troposphere and a 154 K isothermal
stratosphere. We varied the insolation of a planet with Earth-like parameters
and solar-type central star, searching for the limit at which the planet
undergoes a transition to a snowball state.
For simplicity, we considered only the solutions obtained with warm initial
conditions, i.e. starting with $T_{0}$=300 K, which provide a conservative
outer limit to the habitable zone.
The results of these experiments are shown in Fig. 14, where we plot the mean-
annual global ice coverage as a function of insolation obtained for different
values of planetary rotation, radius, axis tilt, and ocean/land distribution.
In all cases we find that the transition to a snowball state is rather sharp,
taking place around $\simeq 1.6$ AU. This result is in general agreement with
the maximum greenhouse limit for a solar-type star found at 1.67 AU by Kasting
et al. (1993) and Kopparapu et al. (2013b) from single-column calculations.
The different treatment of the radiative transfer and of the climate recipes
in our model can explain why we find the snowball transition at a location
somewhat closer to the star than the classic outer edge. In particular, we
used more recent spectral data (HITRAN2016) and a different H2O continuum
model with respect to Kasting et al. (1993) and Kopparapu et al. (2013a). Our
RT calculations employed a more coarse vertical pressure grid with respect of
Kopparapu et al. (2013a), which is known to slightly increase the OLR (thus
increasing the lower insolation limit). Finally, we adopted less opaque CIA
prescriptions for the CO2 with respect to Kasting et al. (1993), which are
considered more in line with experimental results (see Wordsworth et al.,
2010, for a discussion on the subject).
The extra dimension (latitude) and the ice-albedo feedback that are present in
our model provide a detailed description of the climate changes that take
place in the proximity of the outer edge. A detailed analysis of Fig. 14
highlights the role played by different planetary properties in determining
the onset of the snowball transition. In the top left panel one can see that
the transition occurs at increasing distance from the star when the rotation
period increases. This effect is expected because the heat transport from the
equator to the poles becomes more efficient with increasing $P_{\text{rot}}$,
leading to a slower growth of the ice polar caps. In the top right panel one
can see that the increase of planetary radius shifts the snowball boundary
inwards. This is due to the fact that the heat transport to the poles is less
effective in planets with larger $R_{\text{p}}$, leading to a faster growth of
the polar caps. The bottom left panel shows that an increase of the planetary
axis tilt, $\epsilon$, shifts the snowball limit outwards. The effect is
negligible up to $\epsilon\simeq 20^{\circ}$, becoming evident above $\simeq
30^{\circ}$. In this moderate range of obliquities the effect can be
interpreted as follows. The configuration at $\epsilon=0^{\circ}$ favours the
formation of permanent ice caps in the polar regions, where the zenith
distance $Z$ is always large. As the obliquity starts to increase, the polar
regions undergo a period of higher insolation (lower $Z$) in some seasons,
which tends to reduce the ice caps. At very high obliquities the behaviour is
more complex (see Section 4.4.2 in V13) and can be properly investigated only
using 3D models.
In the bottom panel of Fig. 14, we show the impact of variations of ocean/land
distribution. As one can see, the outer limit shifts outwards when the
fraction of oceans, $f_{o}$, increases. The land planet, with $f_{o}=0.05$,
provides an extreme example of early snowball transition. These results can be
understood in terms of the lower albedo and higher thermal capacity of the
oceans compared to the continents. The snowball transition that we find is
slightly sharper in ocean planets than in desert planets owing to the slightly
different temperature dependence of ice over oceans and lands (Section 2.3).
Our results are in line with recent findings that planets covered largely by
oceans have warmer average surface temperatures than land-covered planets
Rushby et al. (2019).
Figure 14: Dependence on planetary parameters of the fractional ice coverage
calculated at the outer edge of the HZ. The results were obtained for a cloud-
free Earth-like planet with a CO2-dominated, maximum greenhouse atmosphere,
the remaining parameters being fixed to Earth values; only the solutions
obtained with warm initial conditions ($T_{0}=300$ K) are shown (see Section
4.5). Top left panel: rotation period, $P_{\text{rot}}$; top right panel:
planet radius, $R_{\text{p}}$; bottom left panel: axis obliquity, $\epsilon$;
bottom right panel: geography.
## 5 Conclusions
We have presented EOS-ESTM, a flexible climate model aimed at simulating the
surface and atmospheric conditions that characterize habitable planets. The
model allows one to perform a fast exploration of the parameter space
representative of planetary quantities, including those currently not
measurable in rocky exoplanets. EOS-ESTM has been built up starting from ESTM,
a seasonal-latitudinal EBM featuring an advanced treatment of surface and
cloud components and a 2D (vertical and latitudinal) treatment of the energy
transport. The main upgrades of EOS-ESTM can be summarized as follows:
* •
The atmospheric radiative transfer is calculated using EOS (Simonetti et al.,
2022), a procedure tailored for atmospheres of terrestrial-type planets, based
on the opacity calculator HELIOS-K (Grimm & Heng, 2015; Grimm et al., 2021)
and the radiative transfer code HELIOS (Malik et al., 2017, 2019). Thanks to
EOS, the ESTM radiative transfer can be now calculated for a variety of
atmospheres with different bulk and greenhouse compositions, illuminated by
stars with different SEDs.
* •
The parameterizations that describe the clouds properties have been largely
upgraded. New equations have been introduced for the albedo of the clouds and
its dependence on the albedo of the underlying surface. The clouds coverage
over ice is now a function of the global planetary ice coverage. A specific
treatment for the transmittance and OLR forcing of clouds at very low
temperature has been introduced.
* •
A generalized logistic function has been introduced to estimate the ice
coverage as a function of mean zonal surface temperature. Based on a detailed
study of the ice distribution on Earth, the adopted algorithm discriminates
between ice over lands and oceans. The albedo and thermal capacity of
transitional ice is now estimated using the fractional ice coverage.
With the aim of providing a reference model for studies of habitable planets,
we calibrated EOS-ESTM using a large set of Earth satellite and reanalysis
data. The reference Earth model satisfies a variety of diagnostic tests,
including mean global measurements (Table 6) and mean latitudinal profiles of
surface temperature, TOA albedo, OLR and ice coverage (Fig. 8). The positive
results of the diagnostic tests were obtained by tuning the parameters within
narrow ranges perfectly consistent with measurements of each climate
component. All the Earth’s data used in our analysis were selected for the
same period (2005-2015) and the atmospheric trace content of greenhouse gases
was tuned accordingly (Section 3.1.3). Due to the lack of 3D treatment of
clouds and atmospheric circulation, the model is not able to reproduce the
detailed shape of the OLR latitudinal profile, even though it does reproduce
correctly the mean global value.
To test the consistency of EOS-ESTM with previous studies of non-terrestrial
climate conditions we performed a series of comparisons with a hierarchy of
climate models (Section 4). The results of these tests can be summarized as
follows:
* •
The latitudinal profiles of temperature and albedo of an Earth-like aquaplanet
are in agreement with predictions obtained using the 3D, intermediate
complexity model PlaSim. Differences that we find are due to the lack of the
3D atmospheric circulation and the 3D representation of clouds in our model.
* •
Comparisons performed at varying levels of insolation yield results which are
in general agreement with other models. However, critical differences appear
at high insolation and temperature, when the resulting abundance of water
vapour makes extremely model-dependent the radiative transfer calculations.
Changing stellar spectrum at moderate and low levels of insolation yields
consistent results.
* •
Comparisons performed at varying planetary radius and rotation rate yield
consistent results, but suggest that the dependence of the meridional
transport on these planetary quantities may be more moderate than estimated in
V15. This test indicates that some parameters of our model can be recalibrated
using a proper set of climate experiments carried out with state-of-the art
GCMs.
* •
The application of EOS-ESTM to the case of a CO2-dominated atmosphere in
maximum greenhouse conditions (Kasting et al., 1993) yields a detailed
description of the transition to a snowball state that takes place when the
insolation decreases in the proximity of the outer edge of the HZ. Thanks to
the flexibility of our model we can explore how this transition develops in
different planetary conditions (e.g. rotation rate, radius, axis tilt, ocean
coverage), taking also into account the presence of climate bistability.
The possibility to easily adapt the input parameters to simulate a broad
spectrum of planetary and atmospheric quantities allows one to apply EOS-ESTM
to simulate a large variety of terrestrial-type exoplanets. As in the case of
the original ESTM, this flexibility can be used to explore in detail the
habitability conditions of individual exoplanets (Silva et al., 2017b) or to
perform statistical studies of exoplanetary habitability (Murante et al.,
2020). With EOS-ESTM it will be possible to extend these types of studies with
a more accurate treatment of the climate effects of land, oceans, ice and
clouds, and expanding the palette of atmospheres to non-terrestrial
compositions and the host stars to non-solar types.
The flexibility of EOS-ESTM paves the road for building up multiparameter
habitable zones, each parameter being representative a planetary property that
affects the climate. To achieve this ambitious goal it is important to assess
the consistency with respect to a hierarchy of climate models, devising a
dedicated series of experiments with the same set of initial conditions. Given
the vastness of possibilities to be tested, a collaborative effort is required
in order to establish proper protocols for a meaningful comparison of models
developed by independent research groups, such as the TRAPPIST-1 Habitable
Atmosphere Intercomparison (THAI, Fauchez et al., 2020, 2021a), and the future
larger project Climates Using Interactive Suites of Intercomparisons Nested
for Exoplanet Studies (CUISINES) NExSS242424https://nexss.info/ Working Group
(Fauchez et al., 2021b).
## Acknowledgements
The Authors wish to thank the Italian Space Agency for co-funding the Life in
Space project (ASI N. 2019-3-U.0). The research reported in this work was
supported by OGS and CINECA under HPC-TRES program award number 2022-02. We
thank the referee for his/her careful reading of the manuscript and helpful
comments.
## DATA AVAILABILITY
The data used for this article will be shared on reasonable request to the
corresponding author.
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|
# Opinion Dynamics in Financial Markets via Random Networks
Mateus F. B. Granha<EMAIL_ADDRESS>Física de Materiais, Universidade de
Pernambuco, Recife, PE 50720-001, Brazil André L. M. Vilela
<EMAIL_ADDRESS>Universidade de Pernambuco, Recife, PE 50720-001, Brazil
Center for Polymer Studies and Department of Physics, Boston University,
Boston, MA 02215, USA Chao Wang<EMAIL_ADDRESS>College of Economics
and Management, Beijing University of Technology, Beijing, 100124, China
Kenric P. Nelson<EMAIL_ADDRESS>Photrek LLC, Watertown, MA 02472, USA H.
Eugene Stanley<EMAIL_ADDRESS>Center for Polymer Studies and Department of
Physics, Boston University, Boston, MA 02215, USA
###### Abstract
We investigate the financial market dynamics by introducing a heterogeneous
agent-based opinion formation model. In this work, we organize the individuals
in a financial market by their trading strategy, namely noise traders and
fundamentalists. The opinion of a local majority compels the market exchanging
behavior of noise traders, whereas the global behavior of the market
influences the fundamentalist agents’ decisions. We introduce a noise
parameter $q$ to represent a level of anxiety and perceived uncertainty
regarding the market behavior, enabling the possibility for an adrift
financial action. We place the individuals as nodes in an Erdös-Rényi random
graph, where the links represent their social interaction. At a given time,
they assume one of two possible opinion states $\pm 1$ regarding buying or
selling an asset. The model exhibits such fundamental qualitative and
quantitative real-world market features as the distribution of logarithmic
returns with fat-tails, clustered volatility, and long-term correlation of
returns. We use Student’s t distributions to fit the histograms of logarithmic
returns, showing the gradual shift from a leptokurtic to a mesokurtic regime,
depending on the fraction of fundamentalist agents. We also compare our
results with the distribution of logarithmic returns of several real-world
financial indices.
Econophysics, Sociophysics, Monte Carlo simulation, Phase transitions, Complex
Networks
###### pacs:
89.65.Gh Economics; econophysics, financial markets, business and management,
87.23.Ge Dynamics of social systems, 05.10.Ln Monte Carlo methods, 64.60.Cn
Order-disorder transformations
††preprint: APS/123-QED
## I INTRODUCTION
For decades now, especially over the last years, financial markets and
economic systems have fascinated researchers and investors worldwide. The
effectiveness of methods and techniques from Statistical Mechanics has been a
compelling element for the comprehension of financial dynamics as a complex
system stanley2000introduction ; Stanley2002 ; Farmer2005 ; Baldovin2007 ;
Li2020 . Simultaneously, it has been responsible for developing the
interdisciplinary field of Econophysics, where agent-based models have a
significant contribution. In this framework, several models use a set of
elementary rules to represent the agent behavior and interactions. Thus,
yielding intense emergent collective phenomena bonabeau2002agent ;
macal2005tutorial ; macal2009agent ; vilela2019majority ; zubillaga2019three ;
Feng2012 ; Zhao2011 .
Oliveira de1992isotropic ; Oliveira1993 pioneered in the study of opinion
dynamics modeled by the majority-vote model (MVM) with noise, a sociological
adaptation of a magnetic spin model that presents similar critical behavior.
Lima et al., Campos et al. and Pereira et al. lima2008majority ;
campos2003small ; pereira2005majority investigated the effects of complex
networks on the MVM social dynamics, thus revealing an ordered phase over the
increasing complexity of these networks. Vilela et al. vilela2019majority
incorporated the rational and emotional behavior of agents in financial
markets introducing the global-vote model, inspired by the majority-vote
dynamics.
The MVM consists of a comprehensive approach to the study of social dynamics
with interactions embedded in regular and complex networks de1992isotropic ;
Oliveira1993 ; santos1995anisotropic ; vieira2016phase ; pereira2005majority ;
lima2008majority ; campos2003small ; vilela2009 ; vilela2018effect ;
vilela2021 . In this agent-based model with two states, the individual’s
opinion in a given time instant may assume one of the two values, $\pm 1$,
regarding some social discussion. Furthermore, an agent in the social network
assumes the opinion of the majority of its neighboring spins with probability
$(1-q)$ and the opposite opinion with probability $q$. The variable $q$ stands
for the noise parameter of the model and measures the social unrest or social
temperature of the system. The MVM exhibits a second-order phase transition
for a critical noise value $q=q_{c}\approx 0.075$ in square lattice networks
of social interactions de1992isotropic .
Complex Networks consist of a natural preference to the study of real-world
complex systems, such as climate analysis, biological neural connections,
World Wide Web, public transportation, airline networks, financial markets,
among others feldhoff2015complex ; reijneveld2007application ;
barabasi2000scale ; an2014synchronization . In this context, the random graph
networks describe the topology of a set of $N$ nodes connected with chance
$p$, thus introducing a fundamental probability distribution over graphs. The
Erdös-Rényi algorithm is one well-known method for the assembly of random
graph networks. The method connects an initial set of $N$ isolated nodes by
adding a total of $pN(N-1)/2$ random links between them, while double
connections are forbidden erdos1960evolution ; bollobas1985random .
In Figure 1, we illustrate the Erdös-Rényi process of building a random graph
network with $N=10$ and $\left<k\right>=3$. We also present the averaged
degree distribution $P(k)$ over ten networks with $N=2\times 10^{4}$ nodes and
several values of $\left<k\right>$. The lines correspond to Poisson fits for
the data with average degree $\left<k\right>=pN$, in agreement for the degree
distribution of random graphs for large networks.
Figure 1: Illustration of the Erdös-Rényi process of building random graph
networks. From (a) to (d), $N=10$ isolated nodes are connected by adding a
total of $pN(N-1)/2$ random links between them. While double connections are
forbidden, the final network has $\left<k\right>=3$. In (e) we show the degree
distribution $P(k)$ averaged over ten networks for $\left<k\right>=6,8,10$ and
$20$ with $N=2\times 10^{4}$. The lines represent Poisson fits for the data.
Modern financial markets operate as a complex system. Its dynamics depend not
only on the rational strategy of the individual but also on the emotional
circumstances that define the investor’s psyche. At first, individual
decisions might seem challenging to model; still, social agents tend to follow
a herding behavior, as they feel sheltered when the crowd is endorsing their
choices. In this way, trader social dynamics is reasonably attainable
cont2000herd ; sznajd2002simple . This bias to follow a significant group is
often embraced by less experienced agents, denominated noise traders. They are
highly susceptible to the dominant opinion and tend to overreact to fresh news
regarding buying or selling.
In contrast, other traders, denominated noise contrarian traders or
fundamentalists, follow the global minority in a given market as an investment
strategy. Thus they buy when stock prices decline and sell when the prices
increase sznajd2002simple ; lux1999scaling ; bornholdt2001expectation ;
kaizoji2002dynamics ; takaishi2005simulations . Therefore, their decision-
making process drives the asset price to its fundamental value.
Inspired by the global-vote model for financial markets vilela2019majority ,
we propose to investigate the influence of random graph networks on the time
evolution of financial markets quantity measures. This work employs an opinion
formation model to study the economic mechanics of noise traders and
contrarian individuals interacting through a random graph of social influence.
This work is organized as follows. In Section II, we describe the global-vote
opinion formation model for financial markets and present the relevant
quantities analyzed in our simulations. In Section III, we present the
numerical results obtained along with the corresponding discussions. Section
IV concludes with our final remarks.
## II THE MODEL
We represent the financial agents on the market and their interactions using a
complex network structure. We place $N$ agents on the nodes of a random graph
network, where the links represent the interaction between the agents in the
market. We map the agent’s financial decision (opinion or option) at a given
time $t$ by a spin variable, which may assume one of two values: $+1$ or $-1$,
regarding buying or selling an asset. Furthermore, to model the essential
dynamics of real-world financial markets, we randomly distribute two sets of
individuals: a fraction $f$ of fundamentalist traders, also referred to as
noise contrarians, and the remaining fraction $1-f$ of noise traders. We use
the spin variables $\lambda$ and $\alpha$ to stand for the financial option of
noise traders and contrarian agents, respectively.
We introduce a noise parameter variable $q$ to model a financial level of
anxiety and perceived uncertainty present in the market. In this study, $q$
influences contrarian and noise traders’ decisions and represents the
probability of performing an inaccurate financial action. In other words, $q$
is the chance for an agent to choose the opposite of his standard strategy
when negotiating in the financial market.
### II.1 Noise Traders
A noise trader individual $i$, with opinion $\lambda_{i}$, assumes the same
opinion of the majority of its neighbors with probability $1-q$, and the
opposite option with probability $q$. We write the opinion flipping
probability for the noise trader agent $i$ as follows
$\omega(\lambda_{i})=\frac{1}{2}\left[1-(1-2q)c_{\lambda}\lambda_{i}\textrm{sgn}(m)\right],$
(1)
where $\textrm{sgn}(x)=-1,0,+1$ for $x<0$, $x=0$ and $x>0$, respectively.
Here, $c_{\lambda}$ stands for the agent strategy, where we set
$c_{\lambda}=+1$ to model the noise traders trend to agree with the local
majority of their neighbors. The variable $m$ quantifies the local predominant
opinion, or local magnetization, defined as
$m=\sum_{\delta=1}^{k_{i}}\lambda_{i+\delta}.$ (2)
The summation is over all the $k_{i}$ neighboring agents connected to the
trader at node $i$, with opinion $\lambda_{i}$. From Eq. (1), when $q=0$ the
noise trader adopts its local predominant opinion. When we increase the market
anxiety parameter $q$, the noise trader tends to follow the opposite opinion
of its local majority.
### II.2 Noise Contrarians
A fundamentalist agent tends to follow the market’s minority opinion with
probability $1-q$, while following the majority opinion with chance $q$. We
define the prevailing option of the system as a global magnetization, which
influences noise contrarian traders’ financial option. The option of a noise
contrarian agent $j$, $\alpha_{j}$, flips with probability
$\omega(\alpha_{j})=\frac{1}{2}\left[1-(1-2q)c_{\alpha}\alpha_{j}\textrm{sgn}(M)\right],$
(3)
where $c_{\alpha}$ is defined as the fundamentalist’s strategy, and since
these individuals tend to agree with the global minority of the system,
$c_{\alpha}=-1$, for all noise contrarian agents.
The variable $M$ measures the average market opinion of the system, thus
revealing the economic order. The magnetization $M$ accounts for the financial
opinion of every agent on the market, and is evaluated as
$M=\frac{1}{(N_{\lambda}+N_{\alpha})}\left(\sum_{i=1}^{N_{\lambda}}\lambda_{i}+\sum_{j=1}^{N_{\alpha}}\alpha_{j}\right),$
(4)
where $N_{\lambda}=N(1-f)$ stands for the number of noise trader agents in the
network and $N_{\alpha}=Nf$ represents the number of noise contrarian agents.
For $M=1$, every agent on the market has an option equal to $+1$. Similarly,
$M=-1$ denotes a market configuration where every agent opinion is equal to
$-1$. $M=0$ represents the case where half of the agents have opinion $+1$ and
the other half $-1$. Additionally, intermediate values of $M$ infer the
dominant opinion of the market.
We remark that when $q=0$, a contrarian agent always follow the opposite
opinion of the global magnetization: buy (sell) when the majority sells (buy).
As $q$ increases, there is a greater probability for the contrarian agent to
follow the majority, adopting the opposite of his inherent financial market
strategy.
In the global-vote dynamics for financial markets, we recover the standard
majority-vote model on random graphs when the fraction of contrarians $f$ is
zero. In this case, we observe a order-disorder phase transition at a critical
point $q_{c}$ for each value of the average connectivity $\left<k\right>$ in
the thermodynamic limit $N\rightarrow\infty$ pereira2005majority . This $f=0$
system presents an ordered phase, with large clusters of agents that share the
same opinion for values of noise $q$ below the critical point $q_{c}$. By
increasing $q$, the same opinion clusters fade, and the magnetization (order
parameter) approaches zero.
In this work, we analyze the global-vote dynamics for financial markets on
random graphs for several values of the average connectivity $\left<k\right>$
and noise contrarian traders fraction $f\neq 0$. We set $q$ near
$q_{c}(f=0,\left<k\right>)$ to model real-world market dynamics adequately
since previous investigations suggest the strong market phase emerge when the
system is close to its critical melting point bornholdt2001expectation ;
vilela2019majority .
## III RESULTS AND DISCUSSION
We perform Monte Carlo simulations on Erdös-Rényi random graphs of size
$N=10201$ and several values of the average connectivity $\left<k\right>$. For
the network of financial interactions, we randomly place a fraction $1-f$ of
noise trader on the network nodes, and we occupy the remaining fraction $f$
with noise contrarian agents. In this way, $N=N_{\lambda}+N_{\alpha}$.
This work considers that a real-world market presents a small fraction of
noise contrarian agents. To this extent, we investigate the influence of the
noise contrarian agents for $q$ near its critical value for the $f=0$ case,
obtained in previous studies within several values of $\left<k\right>$
pereira2005majority , where our model exhibits key real-world market features
vilela2019majority . Additionally, we consider small values for the average
number of connections of an agent in a market, i.e., small values of
$\left<k\right>$, supported by the agreement between the data of real-world
markets and the results of our simulations.
The market dynamics evolve as follows. We select a randomly chosen agent and
update its opinion accordingly with the probabilities given by Eqs. (1) or (3)
for a noise trader or a noise contrarian agent, respectively. We repeat this
process $N$ times, thus defining a unity of time in one Monte Carlo step
(MCS). This way, each agent’s opinion is updated once on average on one MCS.
To discard the transient regime, we allow this dynamic to run during $10^{3}$
MCS, and we perform our analysis in the subsequent $10^{5}$ MCS.
In the financial context, we shall relate the global magnetization of the
system to the aggregate excess demand of a particular asset, for its impact on
stock prices bornholdt2001expectation ; kaizoji2002dynamics ;
takaishi2005simulations . Thus a positive demand ($M>0$) causes prices to
rise, while a negative demand ($M<0$) causes prices to fall. Moreover, markets
that fluctuate around equilibrium cont2000herd exhibit an average excess
demand that oscillates around zero.
The financial return time series represents the price variation of a given
asset over time. Positive (negative) returns relate to profit (loss) during
the period of analysis. We quantify the logarithmic return at a given time $t$
as follows
$r(t)=\textrm{log}\left[\left|M(t)\right|\right]-\textrm{log}\left[\left|M(t-1)\right|\right].$
(5)
In Figure 2, we present the logarithmic return for the closing values of the
S&P 500 index in US dollars, from Nov 01, 2012 to Dec 08, 2020. Similar to any
other traditional asset, the S&P 500 index depends mainly on market supply and
demand, a fundamental financial mechanism that yields several periods of
strong return variations. Periods of significant fluctuations of returns are
compressed in time, denoting the clustered volatility effect for the analyzed
index price. This financial phenomenon is known as volatility clustering, and
it can be elucidated by Mandelbrot’s observation that “large changes tend to
be followed by large changes - of either sign - and small changes tend to be
followed by small changes” mandelbrot1963variation .
Figure 2 also shows that the period around t = 1850 presents the most
considerable time-series return volatility. It denotes the Coronavirus
pandemic phase, where government officials halted the economic activity. The
panic and uncertainty triggered by the financial impacts of such measures led
to a expressive stock market crash vega2021 ; wei2021 .
Figure 2: Logarithmic returns for the daily closing price of S&P 500 index in
US dollars from Nov 01, 2012 to Dec 08, 2020. The high volatility observed for
the period around t = 1850 represents the Coronavirus Stock Market Crash of
2020.
In Figure 3, we analyze the influence of the average connectivity
$\left<k\right>$ and the noise contrarian fraction $f$ on the market order of
the model on random graphs. Figure 3(a) exhibits two distinct market phases: a
“strong market phase” for $f=0.20$ (dark blue), where the system is highly
volatile and the magnetization exhibits a irregular wave pattern; and a “weak
market phase” for $f=0.70$ (yellow), where magnetization values are roughly
randomly distributed vilela2019majority . This result demonstrates that the
increase of contrarians tends to stabilize the market dynamics, where the high
volatile periods are present but to a substantially lower extent.
Additionally, the increase of the agent’s average number of connections
$\left<k\right>$ also yields stochastic patterns for the market demand, but
driving a contraction on the amplitude of the fluctuations, as observed in
Figure 3(b).
Figure 3: Time series of the order parameter $M$ for two sets of parameters:
(a) $\left<k\right>=6$ for $q=0.240$, and (b) $\left<k\right>=50$ for
$q=0.411$.
Figure 4, displays the logarithmic returns of the absolute value of the order
parameter. In financial markets, we define volatility as a measure of the
variation around the average returns observed in an asset’s time series. In
this figure, we note intensive market fluctuations for $f=0.20$, embodied by
the large spikes in the plot. The result also shows clustered volatility, a
real-world market feature. We observe that the presence of contrarians tends
to stabilize the market, indicated by attenuated fluctuations of the returns.
Furthermore, when comparing Figures 4(a) and 4(b), it becomes clear that
increasing the average connectivity of the network will similarly increase the
market’s volatility and simultaneously deviate the behavior from the expected
of real-world financial markets. Figure 4 also shows that periods of high
volatility tend to be clustered, for lower values of $f$.
Figure 4: Logarithmic returns of the absolute magnetization for (a)
$\left<k\right>=6$ and $q=0.240$, (b) $\left<k\right>=50$ and $q=0.411$.
In order to quantify the effects of volatility clustering, we shall compute
the autocorrelation of absolute returns as follows
$A(\tau)=\frac{\sum_{t=\tau+1}^{T}\left[\left|r(t)\right|-|\overset{-}{r}|\right]\left[\left|r(t-\tau)\right|-|\overset{-}{r}|\right]}{\sum_{t=1}^{T}\left[\left|r(t)\right|-|\overset{-}{r}|\right]^{2}},$
(6)
where $1\leq\tau\leq 10^{5}\ \textrm{MCS}$ is the time-step difference between
observations, $T=10^{5}\ \textrm{MCS}$ is the time of simulation, $r(t)$ is
the return at a time $t$ and $\overset{-}{r}$ the average value of the return.
The function defined by Eq. (6) measures nonlinear correlations in
observations of the absolute value of log-returns as a function of the time
delay between them. Figure 5 displays the autocorrelation for several values
of $f$, and it shows that the returns present a strong correlation in time
with exponential decay kaizoji2002dynamics ; takaishi2005simulations ;
bornholdt2001expectation ; vilela2019majority .
Figure 5: Linear-log plot of the autocorrelation of absolute logarithmic
returns for (a) $\left<k\right>=6$ and $q=0.240$, and (b) $\left<k\right>=8$
and $q=0.275$. The dashed red lines represent exponential fits for the data.
Figure 6: Linear-log plot of the autocorrelation of absolute logarithmic
returns of the closing values for several financial indices. Also shown is the
autocorrelation of absolute log-returns for $\left<k\right>=6$, $q=0.240$ and
$f=0.20$ (dark blue).
To illustrate this exponential behavior, we perform an exponential fit of the
autocorrelation function $A(q,f,t)\sim\textrm{exp}(-t/t_{0})$ for $f=0.20$,
and we obtain $1/t_{0}\approx 2.5\times 10^{-7}$ and $1/t_{0}\approx 3.0\times
10^{-7}$ for Figures 5(a) and 5(b), respectively. Other values for the
fraction of noise contrarians yield similar results.
In Figure 6, we compare the results of our model with real-world financial
indices. We calculate the autocorrelation function for the daily log-returns
of the closing values of the indices: Dow Jones (DJI) observed from Jan 29,
1985 to Dec 02, 2020; Ibovespa (BVSP) observed from July 24, 1993 to Dec 02,
2020; Nikkei (N225) observed from Jan 05, 1980 to Dec 02, 2020; S&P 500 (GSPC)
observed from Jan 02, 1985 to Dec 02, 2020; Nasdaq (IXIC) observed from Oct
01, 1985 to Dec 02, 2020. We analyze each index for roughly $7\times 10^{3}$
days. We note that the returns for each investigated index display an
expressive correlation which decays exponentially in time.
Figure 7 displays the histogram of the log-returns for two sets of parameters
and several values of the noise contrarian fraction $f$ in $10^{5}$ MCS. The
real-world market systems display fat-tailed distributions as a reflection of
lower non-zero probabilities of obtaining above or below-average returns. We
observe such behavior only for lower values of the fraction of contrarians
$f$, substantiating a real-world market region for our agent-based dynamics.
To quantify the return distributions, we perform a statistical analysis of the
data. We obtain that the kurtosis $\textrm{K}(\left<k\right>,f)$ for $f=0.20$
(strong market phase) are $\textrm{K}(6,0.20)=5.85$ and
$\textrm{K}(8,0.20)=4.98$. For $f=0.70$ (weak market phase),
$\textrm{K}(6,0.70)=2.41$ and $\textrm{K}(8,0.70)=2.37$. We remark that for a
normal distribution $\textrm{K}=3$, thus, increasing the fraction of
contrarians $f$ gradually shifts the system’s behavior, from a leptokurtic
(fat-tailed) regime, to a mesokurtic (Gaussian) regime.
To qualify the distributions of Figure 7, we perform a comparative normal
quantile-quantile (Q-Q) plot. Fig. 8 displays Q-Q plots for the distribution
of log-returns using several values for the fraction of fundamentalists $f$.
The red line represents the expected results of a Gaussian distribution, and
if a particular distribution exhibits similar behavior, its data points should
lie on that reference line. For lower values of $f$, for instance $f\leq
0.40$, both Figures 8(a) and 8(b) display a non-linear behavior in the Q-Q
plot, thus indicating that the distributions of log-returns feature fat-tails,
and therefore, non-Gaussian. In such distributions, one observes a greater
chance of obtaining high return values than the expected for normal
distributions, consistent with real-world market systems
stanley2000introduction .
Figure 7: Distribution of logarithmic returns for $10^{5}$ MCS and several
values of the fraction of contrarians $f$: (a) $\left<k\right>=6$ and
$q=0.240$, and (b) $\left<k\right>=8$ and $q=0.275$. Figure 8: Normal
quantile-quantile plots of the logarithmic return distributions for (a)
$\left<k\right>=6$ and $q=0.240$, and (b) $\left<k\right>=8$ and $q=0.275$.
Here we use $10^{5}$ MCS. Figure 9: Plot of the absolute log-return
distributions $F(q,\ f,\ r)$ for several values of the fraction $f$ and
$10^{5}$ MCS . The lines for $f\leq 0.40$ correspond to Student’s t fits,
whereas the lines for $f\geq 0.5$ correspond to Gaussian fits. We use (a)
$\left<k\right>=6$ and $q=0.240$, and (b) $\left<k\right>=8$ with $q=0.275$.
Figure 9 displays the probability distribution of the absolute log-returns for
several values of $f$ in $10^{5}$ MCS with $\left<k\right>=6$ and
$\left<k\right>=8$. We find that Student’s t-distributions can depict the
strong market’s phase data ($f\leq 0.40$), while the weak market’s phase is
well fitted by Gaussian distributions vilela2019majority . Assuming that the
mean values $\mu$ of the distributions are zero, as illustrated by Figure 10,
we perform generalized Student’s t fits for the data
$S(r;\nu,\mu,\sigma)=S(r;\nu,\sigma)$, where $\nu$ is the degree of freedom
and $\sigma$ represents the scale. The function $S$ is defined as follows
$S(r;\nu,\sigma)=\frac{1}{\sqrt{\nu\sigma^{2}}\textrm{B}\left(\frac{\nu}{2},\frac{1}{2}\right)}\left(1+\frac{r^{2}}{\nu\sigma^{2}}\right)^{-\frac{\nu+1}{2}},$
(7)
where
$\textrm{B}\left(\frac{\nu}{2},\frac{1}{2}\right)=\int_{0}^{1}t^{\frac{\nu}{2}-1}(1-t)^{-\frac{1}{2}}\textrm{dt},$
(8)
is the Beta function. Values obtained for the degree $\nu$ and scale $\sigma$
are displayed in Table 1. As the fraction of contrarians increase, we find
that the absolute log-return distributions are well fitted by a Gaussian
distribution $g(r)$
$g(r)=\frac{1}{\sigma\sqrt{2\pi}}\
\textrm{e}^{-\frac{1}{2}\left(\frac{r-\mu}{\sigma}\right)^{2}},$ (9)
where $\sigma$ is the standard deviation and $\mu$ is the mean, which we shall
consider as zero once more. In Table 2, we display the Gaussian fit
information.
| | $\left<k\right>=6$, $q=0.240$ | | $\left<k\right>=8$, $q=0.275$
---|---|---|---|---
$f$ | | $\nu$ | $\sigma$ | | $\nu$ | $\sigma$
0.20 | | 5.0 $\pm$ 0.3 | 0.80 $\pm$ 0.01 | | 3.0 $\pm$ 0.1 | 0.785 $\pm$ 0.008
0.25 | | 7.4 $\pm$ 0.5 | 0.558 $\pm$ 0.006 | | 6.3 $\pm$ 0.4 | 0.621 $\pm$ 0.006
0.30 | | 12 $\pm$ 2 | 0.444 $\pm$ 0.006 | | 10 $\pm$ 1 | 0.559 $\pm$ 0.008
0.40 | | 13 $\pm$ 2 | 0.252 $\pm$ 0.005 | | 38 $\pm$ 2 | 0.319 $\pm$ 0.006
Table 1: Correlation between the scale $\sigma$ and degree $\nu$ as a function of the fraction of contrarians f, the average connectivity $\left<k\right>$ and noise $q$. | | $\left<k\right>=6$, $q=0.240$ | | $\left<k\right>=8$, $q=0.275$
---|---|---|---|---
$f$ | | $\sigma$ | | $\sigma$
0.50 | | 0.1916 $\pm$ 0.0008 | | 0.2254 $\pm$ 0.0009
0.70 | | 0.1108 $\pm$ 0.0004 | | 0.1284 $\pm$ 0.0005
Table 2: Dependence of the standard deviation $\sigma$ on the fraction f, on
the average connectivity $\left<k\right>$ and on noise parameter $q$.
Figure 10 displays the cumulative distribution of logarithmic returns $\Phi$
for the data in $10^{5}$ MCS, from which we imply that the mean for the
distributions remains zero for $\left<k\right>=6$ with $q=0.240$, and
$\left<k\right>=8$ with $q=0.275$ for several values of $f$.
Figure 10: Plot of the cumulative distribution $\Phi$ of log-returns in
$10^{5}$ MCS for (a) $\left<k\right>=6$ and $q=0.240$, and (b)
$\left<k\right>=8$ and $q=0.275$. In the inset we analyze the details of the
plot for values of $r(t)$ near zero.
Figure 11: Plot of the absolute log-return distributions $F(r)$ of the closing
values of several financial indices. Also shown is the absolute log-return
distributions for $\left<k\right>=6$, $q=0.240$, $f=0.20$. The lines
correspond to Student’s t fits.
In Figure 11, we show the distribution of the absolute log-returns of the
closing values of several financial indices to the results found in our
investigation. We analyze each index for roughly $7\times 10^{3}$ days and use
Student’s t-distributions to fit the data, represented by the lines in the
plots. In Table 3, we show the values for the degree $\nu$ and scale $\sigma$.
We note that the degree of freedom primarily determines the shape of each
distribution, and comparison shows that our model behaves as the expected for
real-world financial markets. On the other hand, the scale parameter measures
fluctuations around the mean of the distribution, and comparison displays a
disparity in the order of magnitude obtained in the fits. Analyzing Figures 2
and 4, we observe that the daily log-returns of financial markets are one
order of magnitude smaller than the log-returns obtained in our simulations,
and thus accountable for the divergence in the results. Hence, we observe that
our model represents satisfactorily real-world financial market behavior.
Index | $\nu$ | $\sigma$
---|---|---
Dow Jones | 9.5 $\pm$ 0.2 | 0.01248 $\pm$ 8 $\times 10^{-5}$
Ibovespa | 4.2 $\pm$ 0.3 | 0.0197 $\pm$ 6 $\times 10^{-4}$
Nikkei | 15.3 $\pm$ 0.6 | 0.0202 $\pm$ 2 $\times 10^{-4}$
S&P 500 | 15.6 $\pm$ 0.5 | 0.0151 $\pm$ 1 $\times 10^{-4}$
Nasdaq | 6.5 $\pm$ 0.3 | 0.0180 $\pm$ 3 $\times 10^{-4}$
Table 3: Values for the scale $\sigma$ and degree $\nu$ for several financial
indices with Student’s t fits.
We also perform an extensive investigation of the model for several values of
the average connectivity $\left<k\right>$ and different values of the noise
parameter $q$. Figure 12 displays a multi-plot table of the histograms of
logarithmic returns for several ($q,f,\left<k\right>$) triplets. Columns
correspond to values of $q$, below criticality $q<q_{c}(\left<k\right>)$
(left), at criticality $q=q_{c}(\left<k\right>)$ (center) and above
criticality with $q>q_{c}(\left<k\right>)$ (right). In this work, we use the
values of $q=q_{c}(\left<k\right>)$ obtained in previous investigations of the
MVM on random graphs, where the fraction of contrarian agents $f$ is set to
zero pereira2005majority . The values of noise above and below criticality are
taken to be $q_{c}(1\pm 10\%)$.
We observe that distributions exhibit fat-tails for lower values of $f$, and
especially for small values of $\left<k\right>$, i.e., $\left<k\right>=6$ and
$\left<k\right>=8$, and at its correspondent critical value of $q$. On the
contrary, such behavior is lost for higher values of the average connectivity
and values of $q$ that deviate from criticality. This result illustrates and
supports our choice for $\left<k\right>$ and $q_{c}$ used in this
investigation.
Figure 12: Distributions of logarithmic returns for several of the average
degree of connectivity $\left<k\right>$ in the vicinity of
$q_{c}(\left<k\right>)$ for several values of $f$: 0.20; 0.25; 0.30; 0.50;
0.70 (dark blue, purple, violet, pink, orange, yellow, respectively). Here, we
use $10^{5}$ MCS.
We conclude that the adoption of random graph networks on the global-vote
model for financial markets has demonstrated to be effective. Our model is
able to reproduce qualitatively and quantitatively real-world market features
for lower values of the average connectivity $\left<k\right>$ and near
criticality $q(\left<k\right>)=q_{c}(\left<k\right>)$.
We remark that other combination of values for $\left<k\right>$, $f$ and $q$
might yield similar results. Nevertheless, our particular choice adopts simple
key ideas: limited and small number of interacting agents, near critical
values of the noise parameter $q$ and small number of contrarian agents in the
market. Despite the model’s simplicity, it has shown its capability of
characterizing the mechanisms that drive social behavior and decision making
in economic systems.
## IV CONCLUSION AND FINAL REMARKS
This work proposes a generalization of the two-state global-vote model for
financial markets on random networks. The global-vote model suggests that any
stock market dynamics consist primarily of different agent strategies driven
by economic and social interactions. In its standard version, the stock market
consists of a heterogeneous population with two distinct kinds of investors:
noise traders, who follow the local majority of its neighbors, and noise
contrarian traders, influenced by the global minority of the system
vilela2019majority .
We aim to investigate the return’s distribution dependence on the average
connectivity $\left<k\right>$ of the random graph network. We relate
variations of the global magnetization of the system to the daily return of a
given asset. Our simulations reproduce the typical qualitative and
quantitative real-world financial time series. Thus, yielding key features as
fat-tailed distributions of returns, volatility clustering and long-term
memory volatility. We demonstrate that higher values for the average
connectivity $\left<k\right>$, or the noise contrarian fraction $f$, as well
as values of $q$ far from criticality, may eliminate similar real-world
behavior in our model.
This investigation combines two distinct fields of research - Econophysics and
Sociophysics - to better understand the mechanisms at play in the complexity
of the human decision-making process, which drives economic systems’ dynamics.
Consequently, it demonstrates advancements towards comprehending real-world
heterogeneous complex systems, such as financial markets.
Acknowledgements
The authors acknowledge financial support from Brazilian and Chinese
institutions and funding agents UPE, FACEPE (APQ-0565-1.05/14,
APQ-0707-1.05/14), CAPES, CNPq, National Natural Science Foundation of China
(72071006), the International Postdoctoral Exchange Fellowship Program
(20170016) and Beijing Social Science Foundation of China (16JDGLC005). The
Boston University Center for Polymer Studies is supported by NSF Grants
PHY-1505000, CMMI-1125290, and CHE-1213217, by DTRA Grant HDTRA1-14-1-0017,
and by DOE Contract DE-AC07-05Id14517.
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# The effect of disorder on phases across two-dimensional thermal melting
Prashanti Jami Department of Physics, Indian Institute of Science Education
and Research (IISER) Kolkata, Mohanpur - 741246, West Bengal, India Pinaki
Chaudhuri The Institute of Mathematical Sciences, Taramani, Chennai 600113,
India Chandan Dasgupta Department of Physics, Indian Institute of Science,
Bangalore 560012, India International Centre for Theoretical Sciences, Tata
Institute of Fundamental Research, Bangalore 560089, India Amit Ghosal
Department of Physics, Indian Institute of Science Education and Research
(IISER) Kolkata, Mohanpur - 741246, West Bengal, India
###### Abstract
We study melting in a two-dimensional system of classical particles with
Gaussian-core interactions in disordered environments. The pure system
validates the conventional two-step melting with a hexatic phase intervening
between the solid and the liquid. This picture is modified in the presence of
pinning impurities. A random distribution of pinning centers forces a hexatic-
like low temperature phase that transits into a liquid at a single melting
temperature $T^{\rm RP}_{\rm m}$. In contrast, pinning centers located at
randomly chosen sites of a perfect crystal anchors a solid at low temperatures
which undergoes a direct transition to the liquid at $T^{\rm CP}_{\rm m}$.
Thus, the two-step melting is lost in either cases of disorder. We discuss the
characteristics of melting depending on the nature of the impurities.
Introduction — Enhanced fluctuations make two-dimensional melting a topic of
immense research interest. Unlike their three-dimensional counterparts
undergoing “Lindemann melting” [1, 2], 2D melting is mediated by the unbinding
of topological defects. The positional order (PO) and bond-orientational order
(BOO) decouple in 2D, producing a “hexatic phase” sandwiched between the solid
and the liquid. Hexaticity, a rich concept, is realized in colloids [3, 4],
the vortex lattice in superconductors [5], in active Brownian disks [6] and
recently in van der Waals magnet [7]. The celebrated KTHNY theory [8, 9, 10,
11, 12], pictures 2D-melting as a two-step process involving successive
unbinding of dislocations and disclinations– presented schematically in Fig.
1(a). However, the relevance of the two-step 2D melting has also been debated
[13, 14, 15, 16].
Figure 1: A schematic representation of 2D-melting in: (a) KTHNY description
in a pure system, (b) A system with randomly placed impurities, and (c) With a
fraction of particles frozen at randomly chosen zero-temperature positions of
the pure system, suggesting that the clean-melting may be obscured by
impurities (See text).
Quenched disorder, inherent to real materials, can not only move around the
phase boundaries but is also capable of modifying the mechanism of melting.
For example, impurities can generate unbounded defects even at $T=0$, and
thereby mask the unbinding of thermal defect-pairs. This could strike out
solidity even at the lowest $T$, as suggested by Nelson [17] – portrayed
schematically in Fig. 1(b). In contrast, impurities which pin a given fraction
of particles on sites of the underlying perfect lattice could stabilize the
solid by anchoring it via these commensurate locations, and thereby consume
the phase space of hexaticity (see Fig. 1(c)). Role of disorder in
destabilizing the hexatic phase [18] and in enhancing long-range correlation
has also been pointed out [19]. Thus, a careful analysis of 2D-melting in
disordered media can potentially uncover new paradigms.
Experiments on colloids [20], vortex lattices [21, 22] and multi component
mixtures [23] indicate a broadened stability of the hexatic phase in the
presence of disorder consistent with recent calculations [24, 25, 26, 27, 28,
32]. Study of 2D melting is also popular in confined geometry [29, 30, 31]
mimicking disordered background. Zeng et al.[33] have argued that a solid
(“Bragg glass”) phase with power-law decay of translational correlations
cannot occur in a 2D system with impurities. Pronounced hexatic correlations
are expected to be present [34, 35] if the disorder is not strong, though
there are controversies [36, 37, 38] about the existence of a hexatic glass
phase with long- or quasi-long-range hexatic order in 2D. In contrast, phase
boundaries of the two-step melting are found to be insensitive to quenched
defects on a spherical surface [39].
In this letter, we investigate the phases across melting of a bulk 2D system
of soft-core particles, modeled via Gaussian-core interactions [40], which is
known to validate the KTHNY-melting scenario in a pure system [41]. Addressing
the role of quenched disorder in the phase behaviour this model, our key
results are summarized as follows: (i) Random-pinning (RP) destabilizes
solidity causing a single transition from a low-$T$ hexatic-like phase to a
high-$T$ liquid. Here, the low-$T$ phase undergoes a likely crossover from
hexatic-glass to hexatic-liquid. (ii) On the other hand, the commensurate-
pinning (CP) anchors solidity and engulfs hexaticity – even the high-$T$
liquid phase supports inhomogeneous pockets of crystallinity. The defect
locations correlate oppositely with pinning centers in the two models of
disorder – defects tend to bind with the pinning centers for RP-systems,
whereas they stay away from the impurities in CP-systems. Thus, in either
realization of the quenched disorder, the two-step melting is lost.
Model & method: — We introduce disorder in two different ways: (a) Random
pinning (RP), in which we freeze a given fraction ($n_{\rm imp}$) of
particles, chosen randomly in space, within a high-$T$ liquid configuration.
Here, these immobile particles act as disorder. (b) Commensurate pinning (CP),
where $n_{\rm imp}$ fraction of particles are frozen at randomly chosen
positions of an ideal triangular lattice – the ground state configuration of
the pure system. Note that CP represents correlated disorder with a long-range
positional correlation of a perfect lattice. In contrast, RP constitutes
nearly uncorrelated disorder though weak short-range correlation of a high-$T$
liquid may exist. We investigated a system of $N=4356$ particles, with $n_{\rm
imp}=3.5\%$. These results were compared with those from a pure system with
$N=4096$ particles. For these systems, we sample configurations via molecular
dynamics. [42] within the canonical ensemble, using LAMMPS [43] We consider a
simulation box having dimensions $L_{x}=\frac{2}{\sqrt{3}}L_{y}$, having
periodic boundary conditions. $L_{x}$ is adjusted to keep the density, $\rho$
of particles fixed for all our studies ($\rho=0.628$). We carried out $2\times
10^{7}$ MD steps with a time step $\delta t=0.005$. We use dimensionless
parameters: $t^{\prime}=t\sqrt{\epsilon/m\sigma^{2}}$ and
$E^{\prime}=E/\epsilon$, where $m$ is the mass of each particle. $T$ is
expressed in terms of $\Gamma^{-1}$[44], where
$\Gamma=\epsilon\exp(-\sqrt{3}/2\rho)/K_{B}T$. The physical observables are
averaged over $8$ \- $10$ independent pinning configurations for a given
$n_{\rm imp}$.
Positional and bond orientational order: — A pure 2D solid is characterized by
two kinds of ordering: (i) PO measured by $\psi_{\rm
T}=\frac{1}{N}\left\langle|\Psi_{\rm T}|\right\rangle$, where $\Psi_{\rm
T}=\sum_{i=1}^{N}\exp(i{\bf G}.{\bf r}_{i})$. $\bf{G}$ is a first shell
reciprocal-lattice vector of the underlying triangular crystal and ${\bf
r}_{i}$ is the position of particle $i$, and (ii) BOO, quantified by
$\psi_{\rm 6}=\frac{1}{N}\left\langle|\sum_{k=1}^{N}\Psi_{\rm
6}(r_{k})|\right\rangle$ where $\Psi_{\rm
6}=\frac{1}{N_{b}(k)}\sum_{l=1}^{N_{b}(k)}\exp{(i6\theta_{kl})}$. The sum is
over the $N_{b}(k)$ nearest neighbors of particle $k$ identified by a Voronoi
construction [45] and $\theta_{kl}$ is the angle that a line joining particle
$k$ and particle $l$ makes with a reference axis. KTHNY theory predicts two
critical temperatures: $\Gamma^{-1}_{\rm SH}$ and $\Gamma^{-1}_{\rm HL}$, for
the thermal depletion of quasi-long-range PO (solid to hexatic) and BOO
(hexatic to liquid) respectively, leaving a hexatic phase with quasi-long-
range BOO between the solid and isotropic liquid phases.
In Fig. 2(a,b), we plot the thermal evolution of $\psi_{\rm T}$ and $\psi_{\rm
6}$ for pure, RP and CP systems. While the pure system follows KTHNY melting
111The three phases are clearly identified by the snapshots at three
representatives $T$s in the SM, Fig. S1(a-c) with $\Gamma^{-1}_{\rm
SH}=0.0140$ and $\Gamma^{-1}_{\rm HL}=0.0162$, $\psi_{\rm T}$ in the CP-system
is found to survive to larger $T$. The RP-system shows a much weaker
$\psi_{\rm T}$ than the other two, even at the lowest $T$ and depletes very
gradually with $T$ without any threshold behavior. A threshold behavior near
$\Gamma^{-1}_{\rm HL}$ is also seen in the pinned systems in Fig. 2(b), albeit
the transitions are broader. Unlike $\psi_{\rm T}$, the $\psi_{\rm 6}$ is
comparable at low-$T$ in pure, CP- and RP-systems. We also note that
$\psi_{\rm T}$ and $\psi_{\rm 6}$ show a significant drop at the same critical
temperature in a CP-system, implying a direct transition from solid to liquid,
which we discuss further below.
In addition, the fluctuations of $\psi_{\rm T}$ and $\psi_{\rm 6}$ define
generalised susceptibilities
$\chi_{\alpha}=\frac{1}{N}\left[\langle|\Psi_{\alpha}|^{2}\rangle-\langle|\Psi_{\alpha}|\rangle^{2}\right]$,
(with $\alpha=T$ or $6$), and help to identify $\Gamma^{-1}_{\rm SH}$ and
$\Gamma^{-1}_{\rm HL}$, as shown in Fig. 2(c,d). Their behavior confirms that
the pure system shows sharp transitions. Consistent with our finding in panel
(a), $\chi_{\rm T}$ in the RP-system features only a broad and low hump,
hinting that a low-$T$ phase in such a system represents a broad crossover
between a hexatic glass [34] and a hexatic liquid 222The low-$T$ phase support
significant hexatic order $\psi_{\rm 6}$ for our model parameters, while
$\psi_{\rm T}$ and the snapshots establish the amorphous and glassy nature.
The $\Gamma^{-1}$-dependence of $\psi_{\rm 6}$ and $\psi_{\rm T}$ is
indicative of a crossover from a hexatic-glass to a hexatic-liquid, before the
RP-system transits to a liquid at $\Gamma^{-1}_{\rm RP}$. However, the
resolution of our simulation is inadequate for drawing a firm conclusion..
This is also consistent with the trajectory picture of RP system at
$\Gamma^{-1}=0.0028$ in the supplementary material (SM) [48] Fig. S1(d).
Congruous with our findings in panels (a,b), the locations of the peak of
$\chi_{\rm T}$ and $\chi_{\rm 6}$ verify that PO and BOO in the CP-system
vanish at a single $\Gamma^{-1}_{\rm CP}$. While our results from Fig. 2 seem
to support the schematic phase-diagram of Fig. 1, we emphasize that the
‘impure’ phases at low- and high-$T$ defy conventional wisdom. These include
the presence of unbound defects even at $T=0$ in RP-systems, and pockets of
crystallinity deep into the liquid phase in CP-systems, as seen from Fig.
S1(g-i) in the SM [48], and discussed below.
Figure 2: Positional and orientational ordering tendencies are shown as a
function of $\Gamma^{-1}$ for pure (green circles), RP (red triangles) and CP
(blue square) systems. (a) Decay of PO ($\psi_{\rm T}$) with $\Gamma^{-1}$
(the inset shows $\psi_{\rm T}$ for $\Gamma^{-1}\rightarrow 0$). (b) The
softening of the BOO ($\psi_{\rm 6}$). Here $\psi_{\rm
6}(\Gamma^{-1}\rightarrow 0)$ is shown in inset. Panel (c) and (d) show the
corresponding susceptibilities $\chi_{\rm T}$ and $\chi_{\rm 6}$. The location
of the peaks in $\chi_{\rm T}$ and $\chi_{\rm 6}$ identify the transitions. We
find $\Gamma^{-1}_{\rm RP}=0.0145$ from $\chi_{\rm 6}$. Interestingly, the
peak in $\chi_{\rm T}$ and $\chi_{\rm 6}$ for CP-systems appear at the same
$\Gamma^{-1}_{\rm CP}=0.0187$. Figure 3: The distribution of distances
between dislocation pairs, $P(r_{\rm dp})$ in different phases. Panels (a)-(c)
show results for pure, RP- and CP-systems, respectively, at low-$T$. $a_{0}$
is the average inter-particle distance. $P(r_{\rm dp})$ is sharply peaked at
the typical distance between dislocation pairs, though there are differences
in details. The distribution has a short tail for the pure system, a very long
tail for RP-systems, and essentially no tail for CP-systems. Panel (d)
displays $\Delta P(r_{\rm dp})$ taken in pure system between $T^{\prime}s$
just above and below $\Gamma^{-1}_{\rm SH}$. The inset shows corresponding
$P(r_{\rm dp})$. The largest zero-crossing distance of $\Delta P(r_{\rm dp})$
is taken as $r^{c}_{\rm dp}$, and is marked as an arrow in panel (a). Panels
(e) and (f) show the spatial density of unbound defects just prior to melting
into a liquid for a given realization of RP- and CP-system. The pinning
centers are marked via magenta dots. Figure 4: Evolution of defects: Panels
(a)-(c) represent the variation of defects with $\Gamma^{-1}$ (green: pure,
red: RP, and blue: CP-system). (a) The evolution of unbound dislocations (in
squares) as well as disclinations (in circles) validate KTHNY melting in pure
systems; (traces of disclinations are multiplied by $10$ for visual clarity).
(b) The presence of significant unbound dislocations in RP-systems at low
$\Gamma^{-1}$ prohibits solidity. In CP-systems, shown in panel (c), the
proliferation of unbound defects (both dislocations and disclinations)
commences at a single threshold ($\Gamma^{-1}$=0.0198), implying a direct
solid-liquid transition. Panel (d) shows the comparative number defects in RP-
systems with reference to a pure system, illustrating how the number of bound
(B) and unbound (UB) defects grow with increasing temperature.
Defects analysis: — In KTHNY theory, a 2D-solid transits to the hexatic phase
by the unbinding of paired dislocations [10, 11]. We proceed to examine the
consistency of this picture in Fig. 2. This requires an estimation of the
critical distance between two dislocations (with equal and opposite Burger
vectors) below which they are bound. We first employ the Hungarian algorithm
[49], which chooses ‘correct’ partners of dislocations by minimizing the sum
of the distances between all partners [50]. The distribution of resulting pair
distances, $P(r_{\rm dp})$, at low-$T$ is presented in Fig. 3(a-c). $P(r_{\rm
dp})$ is sharply peaked for pure systems in Fig. 3(a), with insignificant
weight at larger $r_{\rm dp}$. In contrast, its long tail at low $T$ (shown
for $\Gamma^{-1}=0.0056$ in Fig. 3(b)) for the RP-system arises from unbound
dislocations even for $T\rightarrow 0$, which destabilize a true solid.
$P(r_{\rm dp})$ in CP-systems (Fig. 3(c)) consists of the initial sharp peak,
and nearly no weight for larger $r_{\rm dp}$. A discernible tail in $P(r_{\rm
dp})$ for pure systems develops when dislocation pairs start unbinding. An
integrated distribution of $P(r_{\rm dp})$ features a threshold behavior at
this transition (see SM [48]).
To obtain the critical $r_{\rm dp}$ for the pure system, we plot in Fig. 3(d)
the difference of these distributions, $\Delta P(r_{\rm dp})$, at temperatures
just above and below $\Gamma^{-1}_{\rm SH}$, while the corresponding $P(r_{\rm
dp})$-s are shown as the inset. The total positive and negative weights of
$\Delta P(r_{\rm dp})$ cancel out, and $r^{c}_{\rm dp}\approx 2.15a_{0}$ is
identified as the last zero-crossing point. This identification is found
robust for $T$’s near $\Gamma^{-1}_{\rm SH}$. A study of distances of
disclination pairs yielded a similar critical distance between disclination
pairs. Once extracted for the pure system, these critical distances were used
for analyzing pinned systems. Subsequently, we explored the thermal evolution
of the defects and their unbinding in Fig. 4(a-c). For the lowest $T$, defects
are essentially absent in the pure system. Unbound disclinations proliferate
at $\Gamma^{-1}\approx 0.0162$, whereas dislocations unbind at
$\Gamma^{-1}\approx 0.0145$, with a hexatic phase at intervening temperatures
[41], consistent with Fig. 2. The CP-system (Fig. 4(c)) behaves like a
‘better’ solid at low-$T$ due to the absence of any free defects up to
$\Gamma^{-1}=0.0182$ beyond that unbound dislocations and disclinations start
proliferating at the same $\Gamma^{-1}_{\rm CP}$. There are significant number
of impurity-induced unbound dislocations in the RP-system for $T\rightarrow
0$, as also concluded from Fig. 2. Here, unpaired dislocations are not only
present for all $T$, they even outnumber bound dislocations at low-$T$. Fig.
4(d) addresses the role of the impurity induced free defects (at $T\rightarrow
0$) in RP-systems, on the thermal defects, whose unbinding drives the two
transitions in a pure system. Number of bound (B) and unbound (UB) defects,
with corresponding numbers subtracted for an equivalent pure system, are
examined separately in Fig. 4(d). These numbers increase sharply with $T$
until the system transits to the liquid. Thus, the impurity induced defects
help promote further thermal defects, than in pure systems for
$\Gamma^{-1}<\Gamma^{-1}_{\rm HL}$. Such a rise disappears in the liquid. In
fact, this difference in bound defects in the liquid goes down to a even lower
value than the corresponding number at $T\rightarrow 0$.
Figure 5: Correlations: Panels (a), (b) show the orientational correlation
function, $g_{6}(r)$, of RP- and CP-systems, respectively, for the values of
$\Gamma^{-1}$ as labelled in each case. In panel (a), power law decay of the
correlations is visible (for $\Gamma^{-1}<{}^{\rm RP}\Gamma^{-1}_{\rm m}$)
down to a very low $T$ in the RP-system; e.g. we find
$g_{6}(r,\Gamma^{-1}=0.0112)\sim(r/a_{0})^{-0.14}$,
$g_{6}(r,\Gamma^{-1}=0.0.014)\sim(r/a_{0})^{-0.2}$,
$g_{6}(r,\Gamma^{-1}=0.0154)\sim(r/a_{0})^{-0.64}$, while
$g_{6}(r,\Gamma^{-1}=0.0162)\sim{\rm exp}^{-0.16(r/a_{0})}$, as shown via
dashed lines. In contrast, in CP-system, as shown in (b), correlations get
modified. Even in the liquid phase, the conventional exponential decay
flattens out at large $r$, implying a “remnant crystallinity” (See text).
Correlations: — Finally, we discuss the orientational correlations as measured
by the correlation function $g_{6}(r)=\langle\Psi_{6}({\bf r}_{i})\Psi_{\rm
6}^{*}({\bf r}_{j})\rangle$ where $r=\lvert{\bf r}_{i}-{\bf r}_{j}\rvert$. The
$T$-dependence of orientational correlations in the pure system follows KTHNY
scenario [10, 11, 12] as claimed earlier [41]. The evolution of $g_{6}(r)$ for
various $T$ is shown in Fig. 5 for RP- and CP-systems. Our
$\chi^{2}$-minimization analysis [51] of the large-$r$ decay of $g_{6}(r)$ in
the RP-system (Fig. 5(a)) identified a power-law behavior for nearly the
entire low-$T$ phase. This power-law behavior continues until an exponential
decay sets in for $\Gamma^{-1}\geq 0.0162$, signaling the onset of liquidity.
Intriguingly, $g_{6}(r)$ in CP-systems shows the enhanced solidity for
$\Gamma^{-1}\leq 0.0187$, where its traces remain largely flat. Beyond the
direct melting from solid to liquid for $\Gamma^{-1}>0.0190$, $g_{6}(r)$ in
CP-systems displays a tendency of plateauing at large-$r$, though it decays at
intermediate $r$. This is a signature of ‘remnant solidity’ arising from local
crystalline pockets surrounding the impurities whose locations are
commensurate with the perfect crystal and hence anchoring crystallinity in the
vicinity (See SM [48]). This is a direct consequence of the correlated nature
of the CP-impurities.
Conclusion: — To summarize, we demonstrate that the conventional picture of
2D-melting undergoes significant changes in the presence of impurities. While
RP-disorder destabilizes solidity and CP-disorder removes the hexatic phase,
the low-$T$ phase in RP-systems is not the conventional hexatic. Similarly,
the high-$T$ phase in the CP-systems mixes remnant solidity with the liquid
phase. The inhomogeneous melting (Fig. S1 in SM [48] generates defects which
correlate differently with pinning centers: For RP-systems the defects tend to
bind with the pinning centers, whereas the defects stay away from the
impurities in CP-systems. While defects are found essential for driving the
melting, our MD configurations indicate that they often bunch up in various
shapes of macroscopic size (See videos in SM [48]). An extension of our study
to larger systems exploring possible role of grain boundaries on melting is a
promising future direction. It will also be interesting to inspect the role of
quantum fluctuations in these thermal phases. We hope that our findings will
motivate future experiments for shedding new light.
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Supplementary material for ‘The effect of disorder on phases across two-
dimensional thermal melting’
In order to support the key conclusions reported in the main manuscript, we
include below additional results in this supplementary materials (SM).
## I Model and Methods
For our study, we consider the Gaussian-core model in a 2D system with
periodic boundary conditions, within which particles act via the following
interaction. $V(r)=\epsilon exp(-r^{2}/\sigma^{2})$, with $\epsilon>0$,
$\epsilon$ and $\sigma$ are the energy and length scales, respectively. We
carry out molecular dynamics (MD) simulations of this model system, using
LAMMPS, starting with a high temperature and cooling the system down to the
desired temperature, as detailed in the main text. We use $10^{5}$ MD steps
($t=5000$) with sampling time window of $t=0.1$, which are recorded after
$10^{7}$ sweeps of equilibration runs. The desired temperatures are maintained
via the Berendsen thermostat. To ensure correct equilibration before taking
statistics, we looked into the distribution of velocities of the particles,
which assumes the form of an Maxwell-Boltzmann distribution in thermal
equilibrium as well as independence of the temporal correlation of the
observables on the time origin, i.e.
$\zeta(t_{1},t_{2})=\zeta_{(}t_{1}-t_{2})$, where $\zeta$ is any temporal
correlation defined between two time points $t_{1}$ and $t_{2}$.
Figure S1: Trajectories: The trajectories of the particles studied at
different temperatures ($\Gamma^{-1}$) for all three systems (the pinning
centers marked in red) for $\Delta t=5000$. Panels (a-c) represent the
trajctories of particles in the clean system. For lowest $\Gamma^{-1}$ (panel
a), the particles remain firmly tied to the equilibrium position of a
triangular lattice reflecting the solid phase. In the hexatic phase at
$\Gamma^{-1}=0.0145$ (panel b), as per Fig. 2 of the main text, particles tend
to move preferentially along the principal direction. In the liquid phase
(panel c), particles execute isotropic motion, covering the entire space. The
RP-system at low-$T$ (panel d) features distorted lattice lines and a
resulting irregular array of particles. Together with the results from Fig. 2
of main text, the trajectory of particles indicate that the lowest-$T$ phase
of RP-system constitute a hexatic glass. Panel (e) depicts the trajectory plot
near ${}^{\rm RP}\Gamma^{-1}_{\rm m}$ and features incipient melting which
results into motional signatures which are rather inhomogeneous in space.
Finally, the liquid phase in the RP-system in panel (f) displays particles’
motion everywhere except for a small region surrounding the impurities. The
motion in CP-system looks similar to the clean system in the solid phase
(panel g). Close to the ${}^{\rm CP}\Gamma^{-1}_{\rm m}$ in panel (h) the
dynamics shows inhomogeneity, but unlike in RP-system the mobile particles
tend to stay away from pinning centers. the concentration of pinned centers
keeps away the melting from the pinned centers (panel h). While most particles
delocalize significantly in the high-$T$ phase of the CP-system (panel i),
local patches of solid-like regions still survive, as seen from the trajectory
plot.
## II Snapshots of particle trajectories
Here we plot the trajectories of particles during their equilibrium dynamics
for the clean, RP- and CP-systems. Such motional signatures are displayed in
Fig.S1 for three representative temperatures, one at low-$T$ in the solid
phase (likely a hexatic glass in case of RP-system), at intermediate
temperature (just before the system melts into an isotropic liquid (in hexatic
phase in case of the clean system), and finally at a high temperature (deep
into the liquid state). The following points are worth mentioning:
The results for clean system in Fig. S1(a-c) are consistent with the KTHNY
picture of 2D-melting in a pure systems. Note that the particles in a hexatic
phase (Fig. S1(b)) preferentially move along the three principal directions of
the underlying triangular lattice of the solid.
The lowest temperature phase for the RP-system in Fig. S1(d) represents a
hexatic glass (or, an amorphous solid) in which the pinned impurities are
marked red, with no apparent positional order. The snapshot at
$\Gamma^{-1}=0.0140$ for RP-system in Fig.S1(e) appears inhomogeneous, where
the mobile particles close to pinned particles delocalize more. Other
particles tend to carry the motional signature of a hexatic phase in a clean
system. The snapshot in Fig.S1(f) for RP-system for $\Gamma^{-1}=0.0210$
indicates that delocalization of mobile particles is nearly engulfed the whole
space, except for small region surrounding the repulsive pinning centers.
The snapshots in the CP-system in Fig.S1(g-i) supports the notion of enhanced
solidity in the following manner: The solidity is nearly perfect at low $T$
(solidity here is stronger than in clean system) – the impurities are pinned
at commensurate locations of the underlying triangular lattice hold onto
perfect solidity. Unlike for RP-system, here the regions rich in pinning
centers anchors solidity around them, whereas, the regions relatively free of
impurity feature incipient melting. This is seen from Fig.S1(h) for
$\Gamma^{-1}=0.0182$, i.e. at a temperature just below the onset of melting.
As a result, the CP-system displays a local pockets of ‘remnant solidity’
around impurities. This remnant solidity weakens with $\Gamma^{-1}$, however,
persists up to a large temperature (Fig.S1(i) at $\Gamma^{-1}=0.0210$). Note
that the pinning centers maintain solid-like correlation at all $T$. The
remnant solidity has discernible effects on the bond-orientational
correlations $g_{\rm 6}(r)$, discussed in connection with Fig. 5(d) in the
main manuscript, and is elaborated further in the later part of this SM.
## III Identification of critical distances of defects unbinding
Figure S2: The distribution of distances between dislocation pairs: $P(r_{\rm
dp})$ shown for different phases. Panels (a-c), (d-f), and (g-i) show results
for clean, RP-, and CP-systems, respectively. $P(r_{\rm dp})$ at low-$T$ in
panels (a), (d), and (g) are sharply peaked at the typical distance of
dislocation pairs, though there are differences in details. The distribution
has a short tail for the clean system, and a very long tail for RP-systems,
and essentially no tail for CP-systems, indicating the enhanced solidity due
to commensurate pinning. The distributions for the three systems develop
smooth and healthy tails at a $T$, where defects unbinding sets in, as shown
in panels (b, e, h). Panels (c, f, i) show that they become indistinguishable
at large $T$, beyond the melting to a liquid.
In fig. S2, we report the distribution of the separation of the two
dislocations with equal and opposite Burger’s vector (here the two constitute
a bound pair of dislocations). We denote this distribution by $P(r_{\rm dp})$,
where $r_{\rm dp}$ is the separation in question. At lower $T$
($\Gamma^{-1}=0.0112$), Fig. S2(a) show that $P(r_{\rm dp})$ in a clean system
is sharply peaked at $r_{\rm dp}\approx a_{0}$, the average inter-particle
distance, with some additional weight for larger $r_{\rm dp}$. In contrast, a
weak yet very long tail in $P(r_{\rm dp})$ (Fig. S2(b)) is found for the RP-
system at low-$T$. This demonstrates the presence of unbound dislocations even
at the lowest $T$ ($\Gamma^{-1}=0.0056$), prohibiting true solidity. In
contrast, the first peak in $P(r_{\rm dp})$ for CP-systems at low-$T$ in Fig.
S2(g) is even sharper than the one in clean system (Fig. S2(a)). This reflects
an enhanced solidity in the CP-system. The nature of the $P(r_{\rm dp})$
changes with $T$, and when the dislocation pairs start unbinding, $P(r_{\rm
dp})$ begins to develops smooth tails, see Fig. S2(b, e, h). At these $T$’s
and beyond $P(r_{\rm dp})$ becomes qualitatively similar for clean, RP- and
CP-systems. The distributions become nearly identical deep into the liquid
phase (hight-$T$) as shown in Fig. S2(c, f, i).
Figure S3: Difference in $P(r_{\rm dp})$: The two panels display $\Delta
P(r_{\rm dp})$ in the pure system, taken between two pair of temperatures –
one in the solid and the other in the hexatic phase, both being close to
${}^{C}\Gamma^{-1}_{\rm SH}$. The corresponding traces of $P(r_{\rm dp})$ is
shown as the inset. The largest $r_{\rm dp}$ corresponding to a zero-crossing
is taken as $r_{dp}^{c}$.
Fig. S3 supplements the Fig. 3(d) in the main text, and illustrates the
insensitivity of $r_{dp}^{c}$ on the chosen $T$-values for its extraction, as
long as those temperatures are close to ${}^{C}\Gamma^{-1}_{\rm SH}$. We plot
in Fig. S3 the difference $\Delta P(r_{\rm dp})$ of $P(r_{\rm dp})$ for two
different pair of temperatures around the ${}^{C}\Gamma^{-1}_{\rm SH}$, while
the corresponding distributions are shown as the inset. Both plots yield
$r_{dp}^{c}=2.15a_{0}$ – the same value obtained in the main test for a yet
another pair of $T$’s.
Figure S4: integrated distribution (ID): as a function of $r_{\rm dp}$ for
different $Ts$. The attainment of unity of ID occurs sharply in the solid
phase and more gently beyond ${}^{C}\Gamma^{-1}_{\rm SH}$. The curvature of ID
is studied further to estimate the critical distance ($r_{dp}^{c}$), which is
presented as an inset.
An alternative extraction of $r_{\rm dp}^{c}$ for the pure system, employing
differently the same concept as above is the following: In Fig. S4, we plot
the integrated distribution (ID) of $P(r_{\rm dp})$ for various $T$’s. The
traces of ID shows a threshold behaviour marking the solid-to-hexatic
transition. ID comes up very steeply for the lower $T$’s corresponding the
solid phase (as inerred from the Fig. 2 of the main text), whereas the rise of
ID becomes distinctively gradual for the two higher $T$’s representing a
hexatic phase. In the inset, we plot the distribution of the change of the
slope of ID-traces in the solid phase, which attains the peak-value for
$r_{\rm dp}^{c}\approx 2.15a_{0}$.
Figure S5: distribution of distances between disclination pairs: Panels (a-c)
represent the distribution of the distances of the disclination pairs
$P({\tilde{r}}_{\rm dp})$ at different $\Gamma^{-1}$. Panel (a) is the
$P({\tilde{r}}_{\rm dp})$ corresponds to the hexatic phase, which attains a
sharp peak around the lattice spacing reflecting the only presence of tightly
bounded disclinations in the phase. When the system hits the transition $T$,
the defects unbounds. The unbinding of these defects modifies the
distribution, as shown in panels (b) in the liquid phase, and remains
identical for the higher $\Gamma^{-1}$ value (panel (c)).
We also studied the thermal evolution of the distribution of the disclination
pair (${\tilde{r}}_{\rm dp}$) and we present our results in fig .S5 for three
representative $T$’s. This distribution $P({\tilde{r}}_{\rm dp})$ features a
single peak at ${\tilde{r}}_{\rm dp}\sim a_{0}$ for up to
${}^{C}\Gamma^{-1}_{\rm HL}$, and identify a second peak for larger $T$’s at
${\tilde{r}}_{\rm dp}\approx 2a_{0}$. Thus, we conclude that
${\tilde{r}}^{c}_{\rm dp}\sim r^{c}_{\rm dp}$.
## IV Correlations:
The pair correlation is given by
$g(r)=\frac{1}{2\pi rN}\sum_{i=1}^{N}\sum_{j\neq
i=1}^{N}\langle\delta(r-\lvert\bf{r}_{i}-\bf{r}_{j}\rvert)\rangle,$ (1)
with $r=\lvert{\bf r}_{i}-{\bf r}_{j}\rvert$. We present the low-$T$ behavior
of $g(r)$ in Fig. S6(a). Its evolution in the clean (blue) and CP (green)
systems is nearly identical, consistent with the findings of Fig. 2(a) and
Fig. 2(b) in the main text. The $g(r)$ for the RP-system exhibit damped
modulations which prohibits solidity even at low temperatures. However, these
damped modulations are long-ranged than what’s expected in a liquid. Fig.
S6(b) shows $g(r)$ for large $T$ where all three systems turns into ‘liquid.’
A role reversal occurs – $g(r)$ for clean and RP-system overlap this time,
yielding a liquid-like behavior, while the long-range modulations for CP-
systems survives implying ‘remnant solidity’ as discussed already in other
context.
Figure S6: Pair correlation function: Panel (a) represents the pair
correlation (red: clean-, blue: random- and green: CP-system) for two
different $\Gamma^{-1}$ values. At the low $\Gamma^{-1}$ value, the clean
system’s behavior is comparable to the CP-system, while in the higher
$\Gamma^{-1}$, as shown in panel-b (for visual clarity, the traces of RP-
system is shown in a dotted line), it is comparable to the RP-system. Figure
S7: Orientational correlation function: of a clean system at different $Ts$.
The pure system displays the thermal evolution of $g_{6}(r)$ consistent with
KTHNY theory.
The $T$-dependence of orientational correlations in the clean system is shown
in Fig. S7 implies that $g_{6}(r)$ remains nearly independent of $\Gamma^{-1}$
for up to $\Gamma^{-1}=0.0145$. It shows a power-law decay $\sim
r^{-\eta(T)}$, in the hexatic phase for $0.0151\geq\Gamma^{-1}\geq 0.0162$
(exponents $\eta(T)$ are listed in Fig. S7). Finally, $g_{6}(r)$ features an
exponential decay in the liquid phase for $\Gamma^{-1}\geq 0.0165$. We find,
e.g., $g_{6}(r,\Gamma^{-1}=0.0140)\sim(r/a_{0})^{-0.003}$,
$g_{6}(r,\Gamma^{-1}=0.0.0157)\sim(r/a_{0})^{-0.01}$,
$g_{6}(r,\Gamma^{-1}=0.0159)\sim(r/a_{0})^{-0.23}$, while
$g_{6}(r,\Gamma^{-1}=0.0182)\sim{\rm exp}^{-0.24(r/a_{0})}$ and
$g_{6}(r,\Gamma^{-1}=0.0210)\sim{\rm exp}^{-0.49(r/a_{0})}$.
Figure S8: Orientational correlation function, The $g_{\rm 6}(r)$ is
calculated in two separate components. (i) The dotted line shows the
contribution of $g_{\rm 6}(r)$ only for the particles which lie within a
distance of $2a_{0}$ from pinned particles at the higher $Ts$ beyond the
melting temperature and (ii) the solid traces shows the $g_{\rm 6}(r)$
contributed by all other particles which are more than $2a_{0}$ distance.
In order to address ‘remnant solidity’ in the CP-system at high $T$, we plot
the two separate components of $g_{\rm 6}(r)$ in Fig. S8. Once component
estimates the contribution of $g_{\rm 6}(r)$ (dotted trace) only for the
particles which lie within a distance of $2a_{0}$ from pinned particles. The
other component measures $g_{\rm 6}(r)$ (solid trace) contributed by all other
particles. We present data of such two contributions for two values of $T$
(both in the high-$T$ liquid phase). The message from Fig. S8 is clear – the
particles close to the pinning centers at commensurate locations of a perfect
crystal anchor the crystallinity around them, which is reflected in the
corresponding $g_{\rm 6}(r)$ which remains nearly constant at large $r$ like
in a solid. The particles further away from the pinned particles, on the other
hand, expectedly behaves like a liquid. Thus, it is the particles near pinning
centers which contribute to the ‘remnant solidity.’ As such, that pinned
particles must show a perfect solid like correlation by construction!
|
# missing
Entanglement Dynamics in
Quantum Continuous–Variable States
DOCTORAL THESIS
Ankit Kumar
Indian Institute of Technology Roorkee, India
_jointly supervised by_
Prof. P. Arumugam
Department of Physics
Indian Institute of Technology Roorkee
Roorkee 247667, India
Prof. Tomasz Paterek
Institute of Theoretical Physics and Astrophysics
Faculty of Mathematics, Physics and Informatics
University of Gdańsk, Gdańsk 80-308, Poland
July, 2023
©INDIAN INSTITUTE OF TECHNOLOGY ROORKEE, 2023
ALL RIGHTS RESERVED
headsepline=true Abstract
Due to the weakness of gravitational coupling, all quantum experiments up to
date in which gravity plays a role utilized the field of the Earth. Since this
field undergoes practically undetectable back-action from quantum particles,
it effectively admits a classical description as a fixed background Newtonian
field or spacetime. This argument strongly motivates theoretical and
experimental research towards a demonstration of gravitation between two
quantum masses, as this is one of the most straightforward scenarios where
quantum features of gravity could be observed. Several proposals studied the
possibility of generating entanglement between two massive objects. Along the
same lines, with a particular focus on gravity, this thesis introduces general
tools to tackle interaction-mediated entanglement and applies them to two
particles prepared in continuous-variable states.
In order to pursue this aim systematically, this dissertation begins by
introducing methods to precisely simulate the dynamics of quantum systems
coupled by weak interactions. We improve the accuracy of the numerical
implementation of Cayley’s operator and develop a methodology to avoid
reflections from numerical infinities. We derive a condition under which a
product state from the laboratory (LAB) perspective remains a product state in
the center-of-mass (COM) frame, which reveals that only certain states are
transformed into disentangled states. Even though the primary focus is on
gravity, all the developed methods apply to arbitrary central interactions,
and considerable parts of this thesis are devoted to explicit demonstrations
of this versatility. Accordingly, the first application is to investigate the
head-on collision in the Rutherford experiment, with the projectile treated as
(realistic) localized wave packets. We observe various nonclassical effects in
the average trajectories and trace them back to the convexity properties of
the Coulomb potential with the help of Jensen’s inequality. The concluding
chapter also comments on the projectile-target entanglement.
Our next goal is to simplify the possible observation of weak gravitational
entanglement in an inevitably noisy laboratory. The basic idea is to amplify
correlations by pushing the particles toward each other, hoping that an ever-
increasing gravitational interaction will automatically lead to a higher
accumulated entanglement. A toolbox is developed that quantifies the
entanglement gain between the two particles directly in the COM frame of
reference, thereby eliminating the need for inverse transformations back to
the LAB frame. We start with the standard practice of the second-order
truncation of quantum Newtonian potential, which has long kept the
mathematical complexities at a minimum by forcefully constraining the system
into the regime of (very well-understood) Gaussian Quantum Information. While
it is known that an analytical solution exists, we utilize Ehrenfest’s theorem
to derive the covariance matrix in an exact closed form. The resultant
entanglement is insensitive to relative motion between the two particles.
The less-understood non-Gaussian regime triggered by the cubic and higher-
order potentials is considered next. We develop a hybrid analytical-numerical
scheme to faithfully estimate the entanglement gain with the help of
algorithms in Google TensorNetwork. The entanglement is found to be sensitive
to relative motion only when the system evolves into the non-Gaussian regime.
We prove that the position-momentum correlations originate from the force
gradient in relative motion. A derivation of closed forms for the non-Gaussian
entanglement gain follows with informed guesswork. In experiments, it will be
challenging to screen the system from all interactions but gravity. With this
in mind, we develop tools to quantify the entanglement with multiple central
forces acting simultaneously.
As the final application, the thesis discusses an entanglement-based test of
the Modified Newtonian Dynamics (MOND), a candidate explanation of dark matter
effects which proposes to modify Newton’s second law and/or the gravitational
force law for accelerations smaller than $\sim 10^{-10}$ m/s2. One verifies
that the masses recently cooled by the Aspelmeyer group in Vienna, when
separated by a distance of a few times their radius, are into the regime of
accelerations where MOND is relevant. Accordingly, the tools developed in this
thesis offer an opportunity to test the assumptions behind MOND through
entanglement between two nearby quantum masses. We develop an experiment where
departures from Newtonian gravity are certified by simply witnessing the
entanglement generation starting from thermal states. Publications
## Contributions in Peer-Reviewed Journals
1. 1)
Nonclassical Trajectories in Head-On Collisions
Ankit Kumar, Tanjung Krisnanda, P. Arumugam, and Tomasz Paterek
Quantum 5, 506 (2021) arXiv:2011.06470
2. 2)
Continuous-Variable Entanglement through Central Forces: Application to
Gravity between Quantum Masses
Ankit Kumar, Tanjung Krisnanda, P. Arumugam, and Tomasz Paterek
Quantum 7, 1008 (2023) arXiv:2206.12897
3. 3)
Probing Modified Gravity with Entanglement of Microspheres
Ankit Kumar, Yen-Kheng Lim, P. Arumugam, Tom Złośnik, and Tomasz Paterek
Physical Review D 109, L101501 (2024) arXiv:2306.14938
## Contributions in Peer-Reviewed Conference Proceedings
1. 1)
Closest Approach of a Quantum Projectile
Ankit Kumar, Tanjung Krisnanda, P. Arumugam, and Tomasz Paterek
Journal of Physics: Conference Series 1850, 012074 (2021) arXiv:2112.13296
## Contributions in Public Scientific Repositories
1. 1)
An Accurate Pentadiagonal Matrix Solution for the Time-Dependent Schrödinger
Equation
Ankit Kumar, and P. Arumugam
Zenodo.7275667 GitHub/vyason/Cayley-TDSE arXiv:2205.13467
###### Contents
1. Contributions in Public Scientific Repositories
|
${\mathcal{F}}_{2}(q,\bar{\rho})=\frac{1}{2}\sum_{L\in\Gamma_{8}}q^{(L+\frac{\rho}{2})^{2}}\,.$
(A.44)
The functions ${\mathcal{F}}_{1}$, ${\mathcal{F}}_{2}$, and
${\mathcal{F}}_{3}$ below, match the Mikhailov’s generating functions $F_{1}$,
$F_{2}$ and $F_{3}$ given in equations (3.3)-(3.5) in [11] up to an overall
factor $q/\eta^{24}$. The meaning of these functions will be explained
shortly.
Twisted sector
The twisted sector is much simpler. The two terms in
${\mathcal{Z}}_{\mathrm{II}}(g)$ are given in equations (A.24) and (A.26).
Combining them gives
${\mathcal{Z}}_{\mathrm{II}}(g)=\frac{1}{\bar{\eta}\eta^{17}}\sum_{n\in{\mathbb{Z}},\,\ell\in
2{\mathbb{Z}}+1}\bar{q}^{\frac{1}{2}p_{R}^{2}}\,q^{\frac{1}{2}p_{L}^{2}}\,\sum_{\rho\hskip
1.0pt\in\Gamma_{8}}q^{\frac{1}{4}\rho^{2}}\,{\mathcal{F}}_{3}\\!\left(q,(\tfrac{\rho^{2}}{2}+n)\\!\\!\\!\mod\\!2\right)\,.$
(A.45)
The momenta $p_{R}$ and $p_{L}$ are again given in (A.31). The function
${\mathcal{F}}_{3}$ reads
${\mathcal{F}}_{3}\\!\left(q,(\tfrac{\rho^{2}}{2}+n)\\!\\!\\!\\!\mod\\!2\right)=\frac{1}{2}\left[\left(\frac{\eta^{3}}{\vartheta_{4}}\right)^{\\!\\!4}-e^{i\pi(n+\frac{\rho^{2}}{2})}\left(\frac{\eta^{3}}{\vartheta_{3}}\right)^{\\!\\!4}\right]\,.$
(A.46)
The twisted states constitute class C characterized by $\rho\in\Gamma_{8}$ and
$\ell\in 2{\mathbb{Z}}+1$.
Full orbifold
Adding ${\mathcal{Z}}_{\mathrm{II}}({\mathbb{1}})$ and
${\mathcal{Z}}_{\mathrm{II}}(g)$ yields
${\mathcal{Z}}_{{\mathrm{II}}_{\text{o}}}=\frac{1}{\bar{\eta}\eta^{17}}\left\\{\sum_{\begin{subarray}{c}n\in{\mathbb{Z}}\\\
\ell\in
2{\mathbb{Z}}\end{subarray}}\bar{q}^{\frac{1}{2}p_{R}^{2}}\,q^{\frac{1}{2}p_{L}^{2}}\left[\sum_{\rho\hskip
1.0pt\in 2\Gamma_{8}}q^{\frac{1}{4}\rho^{2}}\,{\mathcal{F}}_{1}\
+\sum_{\rho\hskip
1.0pt\in\Gamma_{8}/2\Gamma_{8}}q^{\frac{1}{4}\rho^{2}}{\mathcal{F}}_{2}\right]+\sum_{\begin{subarray}{c}n\in{\mathbb{Z}}\\\
\ell\in
2{\mathbb{Z}}+1\end{subarray}}\bar{q}^{\frac{1}{2}p_{R}^{2}}\,q^{\frac{1}{2}p_{L}^{2}}\,\sum_{\rho\hskip
1.0pt\in\Gamma_{8}}q^{\frac{1}{4}\rho^{2}}\,{\mathcal{F}}_{3}\right\\}\,.$
(A.47)
The arguments of ${\mathcal{F}}_{1}$, ${\mathcal{F}}_{2}$ and
${\mathcal{F}}_{3}$ are omitted to simplify the expression. From this result
we can read off the content of orbifold states classified according to the
possible domains of $\ell$, $n$ and $\rho$. This information is summarised in
Table 8.
class | $\ell$ | $n$ | $\rho$ | sector | generating function
---|---|---|---|---|---
A | $2{\mathbb{Z}}$ | ${\mathbb{Z}}$ | $2\Gamma_{8}$ | untwisted | ${\mathcal{F}}_{1}(q,n\\!\\!\\!\mod\\!2)$
B | $2{\mathbb{Z}}$ | ${\mathbb{Z}}$ | $\Gamma_{8}/2\Gamma_{8}$ | untwisted | ${\mathcal{F}}_{2}(q,\bar{\rho})$
C | $2{\mathbb{Z}}+1$ | ${\mathbb{Z}}$ | $\Gamma_{8}$ | twisted | ${\mathcal{F}}_{3}\\!\left(q,(\tfrac{\rho^{2}}{2}+n)\\!\\!\\!\mod\\!2\right)$
Table 8: Classes of orbifold states
#### A.4.3 Reading ${\mathrm{II}}_{(1)}$ from the partition function
It turns out that some identities relating the functions ${\mathcal{F}}_{1}$,
${\mathcal{F}}_{2}$ and ${\mathcal{F}}_{3}$ are needed to show that the
orbifold states lie in the lattice ${\mathrm{II}}_{(1)}$. In [11] the
relations are proven analytically for the generating functions $F_{c}$,
$c=1,2,3$, connected to the ${\mathcal{F}}_{c}$ by
$F_{c}=\frac{q}{\eta^{24}}{\mathcal{F}}_{c}\,.$ (A.48)
The identities can be verified by comparing the $q$-expansions of the
${\mathcal{F}}_{c}$. For ${\mathcal{F}}_{1}(q,n\\!\\!\\!\mod\\!2)$ and
${\mathcal{F}}_{3}\\!\left(q,(\tfrac{\rho^{2}}{2}+n)\\!\\!\\!\mod\\!2\right)$
these expansions are easily found from their definitions. The expansion of
${\mathcal{F}}_{2}(q,\bar{\rho})$ is less direct because it depends on
$\bar{\rho}$, which is the conjugacy class of $\rho$ in
$\Gamma_{8}/2\Gamma_{8}$. As explained in [11], if $\frac{1}{2}\rho^{2}$ is
odd, then $\rho$ equals a root modulo $2\Gamma_{8}$. We denote the conjugacy
class by $\Delta_{2}$. If instead $\frac{1}{2}\rho^{2}$ is even, then $\rho$
modulo $2\Gamma_{8}$ is equal to a vector $\varrho\in\Gamma_{8}$ with
$\varrho^{2}=4$. We denote the conjugacy class by $\Delta_{4}$. From the
expansions we can check the identities
$\displaystyle{\mathcal{F}}_{1}(q,1)$
$\displaystyle={\mathcal{F}}_{2}(q,\Delta_{4})={\mathcal{F}}_{3}(q,0)=8q+64q^{2}+224q^{3}+512q^{4}+\cdots,$
(A.49a) $\displaystyle{\mathcal{F}}_{2}(q,\Delta_{2})$
$\displaystyle={\mathcal{F}}_{3}(q,1)=q^{\frac{1}{2}}(1+28q+126q^{2}+344q^{3}+\cdots)\,.$
(A.49b)
The Mikhailov’s generating functions $F_{c}$ verify the same identities
because they are equal to the ${\mathcal{F}}_{c}$ up to an overall factor.
The meaning of the generating functions can be understood by looking at the
simple lattice partition function of the 10-dimensional heterotic string in
(A.2). In this case there is a generating function $1/\eta^{16}$. Now, we know
that for each vector in the lattice there is a tower of excited states created
by acting with the oscillators of the $Y^{\hat{I}}$, $\hat{I}=1,\cdots,16$.
Moreover, the coefficients in the $q$-expansion of $1/\eta^{16}$ precisely
count the number of states at each excited level. The meaning of the
Mikhailov’s generating functions $F_{c}$ is completely analogous. The
prefactor $q/\eta^{24}$ in the relation with the ${\mathcal{F}}_{c}$ is well
justified. The power of $\eta$ corresponds to the 24 left-moving coordinates,
and the power of $q$ just offsets the normal ordering constant. In this way
their $q$-expansion will be of the form
$F(q)=\sum_{N^{\prime}}d(N^{\prime},(\ell,n,\rho))q^{N^{\prime}}\,,$ (A.50)
where now $N^{\prime}$ corresponds to the full left-moving oscillator number.
For each state $(\ell,n,\rho)$, the coefficients $d(N^{\prime},(\ell,n,\rho))$
count the number of states with oscillators acting on it and given oscillator
number $N^{\prime}$. The dependence on $(\ell,n,\rho)$ is necessary because,
as seen in Table 8, for each type of state there is an associated generating
function.
We are finally ready to state Mikhailov’s proof that the spectrum of orbifold
states can be put into correspondence with the points in the lattice
${\mathrm{II}}_{(1)}$ displayed in (A.35). This correspondence is summarised
in Table 9. For example, the points of type 1 where $(\ell,n,\rho)\in
2{\mathrm{II}}_{(1)}$ can only correlate with points of orbifold class A which
have $\rho\in 2\Gamma_{8}$, and $\ell\in 2{\mathbb{Z}}$, provided that also
$n\in 2{\mathbb{Z}}$.
${\mathrm{II}}_{(1)}$ type | $(l,n,\rho)$ | $\frac{1}{2}\rho^{2}+\ell n$ | orbifold class
---|---|---|---
1 | $2{\mathrm{II}}_{(1)}$ | $2{\mathbb{Z}}$ | $\left[{\bf A},n\in 2{\mathbb{Z}}\right]$
2 | ${\mathrm{II}}_{(1)}/2{\mathrm{II}}_{(1)}$ | $2{\mathbb{Z}}$ | $\left[{\bf A},n\in 2{\mathbb{Z}}+1\right]$, $\left[{\bf B},\frac{1}{2}\rho^{2}\in 2{\mathbb{Z}}\right]$, $\left[{\bf C},\left(\frac{1}{2}\rho^{2}+n\right)\in 2{\mathbb{Z}}\right]$
3 | ${\mathrm{II}}_{(1)}$ | $2{\mathbb{Z}}+1$ | $\left[{\bf B},\frac{1}{2}\rho^{2}\in 2{\mathbb{Z}}+1\right]$, $\left[{\bf C},\left(\frac{1}{2}\rho^{2}+n\right)\in 2{\mathbb{Z}}+1\right]$
Table 9: Points in ${\mathrm{II}}_{(1)}$ vs orbifold classes
For points of type 2 and 3 there can be more than one orbifold class as can be
understood by looking at Table 8. For these points consistency requires
precise identities among the generating functions. For example, the points of
type 3 must appear with the same generating function whether they arise in
class B with $\frac{1}{2}\rho^{2}\in 2{\mathbb{Z}}+1$, or in class C with
$(\frac{1}{2}\rho^{2}+n)\in 2{\mathbb{Z}}+1$. This means that
${\mathcal{F}}_{2}(q,\Delta_{2})$ must be equal to ${\mathcal{F}}_{3}(q,1)$,
which is precisely the identity in (A.49b). Similarly for the points of type 2
the functions ${\mathcal{F}}_{1}(q,1)$, ${\mathcal{F}}_{2}(q,\Delta_{4})$ and
${\mathcal{F}}_{3}(q,0)$ must be the same, which is true by virtue of of the
identity (A.49a). For points of type 1 they just fall in class A and occur
with generating function ${\mathcal{F}}_{1}(q,0)$.
#### A.4.4 T-duality
The previous results can be used to show that the partition function of the
$S^{1}/{\mathbb{Z}}_{2}$ orbifold is invariant under T-duality. We consider
the simpler situation with Wilson line $a=0$ in which T-duality is the action
$n\leftrightarrow\ell$, $R_{M}\to 1/R_{M}$. The relevant piece of the
partition function is the lattice contribution
${\mathcal{Z}}_{{\mathrm{II}}_{\text{o}}}$ displayed in (A.47). From the
corresponding spectrum of states summarised in Table 8 it is evident that
T-duality mixes twisted and untwisted states as remarked in [11].
To establish T-duality it is enough to show that the quantity between brackets
in (A.47) is invariant. The parts with both $n$ and $\ell$ even (odd), arising
in the untwisted (twisted) sector, are clearly invariant by themselves. The
remaining question is whether the untwisted sector terms with $n$ odd and
$\ell$ even do match the twisted sector terms with $n$ even and $\ell$ odd.
The answer is yes as follows from the equality
$\sum_{\begin{subarray}{c}n\in 2{\mathbb{Z}}+1\\\ \ell\in
2{\mathbb{Z}}\end{subarray}}\bar{q}^{\frac{1}{2}p_{R}^{2}}\,q^{\frac{1}{2}p_{L}^{2}}\left[\sum_{\rho\hskip
1.0pt\in 2\Gamma_{8}}q^{\frac{1}{4}\rho^{2}}\,{\mathcal{F}}_{1}(q,1)\
+\sum_{\rho\hskip
1.0pt\in\Gamma_{8}/2\Gamma_{8}}q^{\frac{1}{4}\rho^{2}}{\mathcal{F}}_{2}(q,\bar{\rho})\right]\\!=\\!\\!\sum_{\begin{subarray}{c}n\in
2{\mathbb{Z}}\\\ \ell\in
2{\mathbb{Z}}+1\end{subarray}}\bar{q}^{\frac{1}{2}p_{R}^{2}}\,q^{\frac{1}{2}p_{L}^{2}}\,\sum_{\rho\hskip
1.0pt\in\Gamma_{8}}q^{\frac{1}{4}\rho^{2}}\,{\mathcal{F}}_{3}\\!\left(q,\tfrac{\rho^{2}}{2}\\!\\!\\!\\!\mod\\!2\right)\,.$
(A.51)
In turn this identity can be shown using the properties
${\mathcal{F}}_{1}(q,1)={\mathcal{F}}_{3}(q,0)$,
${\mathcal{F}}_{2}(q,\Delta_{4})={\mathcal{F}}_{3}(q,0)$ and
${\mathcal{F}}_{2}(q,\Delta_{2})={\mathcal{F}}_{3}(q,1)$, given in (A.49).
## Appendix B World-sheet realisation of gauge symmetries
In this appendix we briefly discuss the Kac-Moody algebras that realize the
space-time gauge symmetries of the CHL theory in 9 dimensions and its toroidal
compactifications.
The space-time ${\mathrm{E}}_{8}\times{{\mathrm{E}}}^{\prime}_{8}$ gauge
symmetry is realized on the world-sheet by dimension (1,0) currents
$J_{1}^{a}\otimes 1$ and $1\otimes J^{b}_{2}$, $a,b=1,...,248$, that obey the
OPE
${J}_{i}^{a}(z){J}_{i}^{b}(0)\sim\frac{\tilde{k}_{i}\delta^{ab}}{z^{2}}+\frac{i}{z}f^{ab}{}_{c}{J}_{i}^{c}(0)\,,\quad
i=1,2$ (B.1)
at level $k_{i}=\frac{2\tilde{k}_{i}}{\psi_{i}^{2}}=1$, where
$\tilde{k}_{i}=1$, $\psi_{i}^{2}=2$ is the norm of the highest root and
$f^{ab}{}_{c}$ are the structure constants of the simply-laced Lie algebra of
${\mathrm{E}}_{8}$.
The Sugawara construction induces a representation of the Virasoro algebra
with central charge
$c=\sum_{i=1}^{2}c_{i}=\sum_{i=1}^{2}\frac{k_{i}\,{\rm dim}\ G_{i}}{k_{i}+{\rm
g}_{i}}\,,$ (B.2)
where ${\rm g}_{i}$ is the dual Coxeter number of the group $G_{i}$ (see Table
10). These formulae hold in general for arbitrary products of groups [35, 36].
For simply laced algebras at level one it follows that $c_{i}={\rm
rank}\,G_{i}$. In the ten dimensional heterotic string with
$G_{i}={\mathrm{E}}_{8}$, clearly $c_{i}=8$ and $c=16$.
$G$ | ${\mathrm{A}}_{n}$ | ${\mathrm{D}}_{n}$ | ${\mathrm{E}}_{6}$ | ${\mathrm{E}}_{7}$ | ${\mathrm{E}}_{8}$ | ${\mathrm{B}}_{n}$ | ${\mathrm{C}}_{n}$ | ${\mathrm{F}}_{4}$ | ${\mathrm{G}}_{2}$
---|---|---|---|---|---|---|---|---|---
${\rm g}$ | $n+1$ | $2n-2$ | 12 | 18 | 30 | $2n-1$ | $n+1$ | 9 | 4
dim $G$ | $n(n+2)$ | $n(2n-1)$ | 78 | 133 | 248 | $n(2n+1)$ | $n(2n+1)$ | 52 | 14
Table 10: Dual Coxeter number ${\rm g}$ and dimension of the gauge group $G$
The currents of the level $k=1$ untwisted affine Kac Moody algebras associated
with simple Lie algebras which are simply-laced were constructed using the
vertex operators of the massless gauge bosons of the string spectrum in [37,
38]. In the ten dimensional theory, the 248 gauge bosons of each
${\mathrm{E}}_{8}$ comprise the 8 Cartan $\alpha^{I}_{-1}|0,0\rangle$ or
$\alpha^{I+8}_{-1}|0,0\rangle,I=1,...,8$ and the 240 roots
$|p^{I},p^{I+8}\rangle=|r_{1}^{I},0\rangle$ or $|0,r_{2}^{I}\rangle$, with
$r_{1}^{I},r_{2}^{I}\in\Gamma_{8}$. Their vertex operators can be written in
terms of the free bosons $Y^{I}(z)$ and ${Y^{\prime}}^{I}(z)$, and the
corresponding currents $J_{1,2}^{a}$ have the following realisation in the
Cartan basis
$\displaystyle H^{I}_{1}(z)=i\partial Y^{I}(z)\,,\qquad\qquad E_{1}^{\pm
r_{1}}(z)=c_{r_{1}}:e^{\pm ir_{1}\cdot Y(z)}:\,,$ (B.3) $\displaystyle
H^{I}_{2}(z)=i\partial{Y^{\prime}}^{I}(z)\,,\qquad\qquad E_{2}^{\pm
r_{2}}(z)=c_{r_{2}}:e^{\pm ir_{2}\cdot Y^{\prime}(z)}:\,,$ (B.4)
where $c_{r_{1}},c_{r_{2}}$ are cocycle factors. Using the OPEs
$\partial Y^{I}(z)\partial
Y^{J}(0)=-\frac{\delta^{IJ}}{z^{2}}\,,\qquad\partial{Y^{\prime}}^{I}(z)\partial{Y^{\prime}}^{J}(0)=-\frac{\delta^{IJ}}{z^{2}}\,,$
(B.5)
the current algebra of
$\widehat{{\mathrm{E}}_{8}\times{\mathrm{E}}^{\prime}_{8}}$ is realized at
level $k_{1}=k_{2}=\frac{2\tilde{k}_{i}}{|\psi_{i}|^{2}}=1$, as can be read
from
$\displaystyle H^{I}_{i}(z)H^{J}_{j}(0)$
$\displaystyle\sim\frac{\delta_{ij}\delta^{IJ}}{z^{2}}\,,$ (B.6a)
$\displaystyle H_{i}^{I}(z)E_{j}^{\pm r_{j}}(0)$ $\displaystyle\sim\frac{\pm
r_{j}^{I}E_{j}^{\pm r_{j}}(0)\delta_{ij}}{z}\,,$ (B.6b) $\displaystyle
E_{i}^{r_{i}}(z)E_{i}^{-r_{i}}(0)$
$\displaystyle\sim\frac{1}{z^{2}}+\frac{r_{i}\cdot H_{i}(0)}{z}\,,$ (B.6c)
As we have seen, the CHL string in 9 and lower dimensions can be constructed
as a ${\mathbb{Z}}_{2}$ orbifold involving the outer automorphism that
exchanges ${\mathrm{E}}_{8}$ and ${{\mathrm{E}}}^{\prime}_{8}$. In 10
dimensions the orbifold by this exchange simply reproduces the original
theory. This can be verified computing the partition function as discussed in
Appendix A, with $g$ corresponding to the action
${\mathrm{E}}_{8}\leftrightarrow{\mathrm{E}}^{\prime}_{8}$, and using the
identities (A.49). However, if the exchange of ${\mathrm{E}}_{8}$ and
${\mathrm{E}}^{\prime}_{8}$ is accompanied by an additional $2\pi$ rotation of
the ten dimensional space-time one gets the non-supersymmetric
${\mathrm{E}}_{8}$ string [39, 40] in which some sectors of the Hilbert space
are projected out. As explained in [40], only the products ${\cal H}_{\rm
spF}\otimes{\cal H}_{\rm s}$ and ${\cal H}_{\rm spB}\otimes{\cal H}_{\rm as}$
survive, where ${\cal H}_{\rm spB}({\cal H}_{\rm spF})$ denotes the Hilbert
subspace of space-time bosons (fermions) which is symmetric (antisymmetric)
under the $2\pi$ rotation. Since the internal Hilbert space ${\cal H}_{\rm
int}$ is an irreducible representation of
$\widehat{{\mathrm{E}}_{8}\times{\mathrm{E}}^{\prime}_{8}}$, its symmetric and
antisymmetric subspaces ${\cal H}_{\rm s}$ and ${\cal H}_{\rm as}$ are not
invariant under the full ${\mathrm{E}}_{8}\times{\mathrm{E}}^{\prime}_{8}$
current algebra, but are invariant under the algebra of the diagonal currents
$\displaystyle T^{a}(z)=J_{1}^{a}(z)\otimes 1+1\otimes J^{a}_{2}(z)\,,$ (B.7)
since $T^{a}$ is invariant under the exchange of the two ${\mathrm{E}}_{8}$’s.
The diagonal $\hat{\mathrm{E}}_{8}$ is a subalgebra of
$\widehat{{\mathrm{E}}_{8}\times{\mathrm{E}}^{\prime}_{8}}$ and clearly, the
current algebra is realized at level $k_{1}+k_{2}=2$. In this case, the
central charge obtained from (B.2) is $\frac{31}{2}$, and the missing
$\frac{1}{2}$ is provided by the coset theory
$\frac{({\mathrm{E}}_{8}\times{\mathrm{E}}_{8})_{k=1}}{({\mathrm{E}}_{8})_{k=2}}$,
which is equivalent to the Ising model [40].
Let us now turn to CHL strings. As we have reviewed in the main text, the
9-dimensional theory can be described by a $S^{1}/{\mathbb{Z}}_{2}$ orbifold,
with ${\mathbb{Z}}_{2}$ action given by the exchange
${\mathrm{E}}_{8}\leftrightarrow{\mathrm{E}}^{\prime}_{8}$ together with a
translation in the compactified direction $x^{9}\rightarrow x^{9}+\pi R$. For
arbitrary values of the compactification radius $R$ and Wilson lines $a$, only
the 8 diagonal Cartan gauge bosons in the untwisted massless sector survive
the orbifold projection, and together with the KK gauge boson of the
compactified $x^{9}$, they account for the ${\mathrm{U}(1)}^{9}$ abelian
symmetry of the theory, with generators
$T^{I}_{+}=H^{I}_{1}+H^{I}_{2}=i\left(\partial
Y^{I}(z)+\partial{Y}^{{}^{\prime}I}(z)\right)\qquad{\rm and}\qquad
H^{9}=i\partial X^{9}(z)\,,$ (B.8)
where $X^{9}$ is a free boson.
For arbitrary $R$ and $a=0$ the untwisted states
$\frac{1}{\sqrt{2}}\left(|r^{I},0\rangle+|0,r^{I}\rangle\right)$ are also
massless when $r^{I}$ is a root of ${\mathrm{E}}_{8}$ and $m=n=0$. Together
with the nine Cartan above, they give rise to the rank nine gauge symmetry
${\mathrm{E}}_{8}\times{\mathrm{U}(1)}$, with ${\mathrm{E}}_{8}$ the diagonal
subgroup of the original ${\mathrm{E}}_{8}\times{\mathrm{E}}_{8}$. The
corresponding raising and lowering currents are
$T_{+}^{\pm r^{I}}=E_{1}^{\pm r^{I}}+E_{2}^{\pm r^{I}}\,.$ (B.9)
The ${\mathrm{E}}_{8}$ current algebra is realized at level 2, just as in the
10-dimensional ${\mathrm{E}}_{8}$ string theory.
Lower rank groups and higher level algebras are a hallmark of CHL strings. The
total central charge $c$ of the Kac-Moody algebra associated to the gauge
symmetry also gives useful information. In general, in $(10-d)$ dimensions
there is a bound $c\leq c_{L}^{int}$, where the internal piece is
$c_{L}^{int}=16+d$. This follows because keeping transverse degrees of freedom
the total left-moving central charge is $c_{L}=24$ and the world-sheet bosons
corresponding to the space-time coordinates contribute $(8-d)$. It is
convenient to write the bound on $c$ as
$\Delta=16+d-c\geq 0\,.$ (B.10)
We will refer to $\Delta$ as the missing central charge. A consistency
condition is that when $\Delta<1$ it must be equal to the central charge of a
unitary minimal model given by
$c_{j}=1-\frac{6}{j(j+1)},\quad j=3,4,\ldots.$ (B.11)
For instance, in the above ${\mathrm{E}}_{8}\times{\mathrm{U}(1)}$ example the
Kac-Moody central charge is $c=\frac{31}{2}+1$ and the missing
$\Delta=\frac{1}{2}$ is provided by the $j=3$ minimal model, i.e. the Ising
model which is furthermore equivalent to the coset theory
$\frac{({\mathrm{E}}_{8}\times{\mathrm{E}}_{8})_{k=1}}{({\mathrm{E}}_{8})_{k=2}}$.
Continuing with the 9-dimensional CHL string, at the particular radius
$R=\sqrt{2}$ and $a=0$, the states with $\ell=\pm 1,n=\pm 1,\rho=0$ in the
twisted sector become massless and enhance the ${\mathrm{U}(1)}$ of the KK
vector to ${\mathrm{SU}}(2)$. The vertex operators that create these states
involve the left moving currents
$\qquad H^{1}(z)=i\partial X^{1}(z)\,,\qquad E^{\pm}(z)=c_{\pm}\Lambda e^{\pm
ip_{L}\cdot X(z)}\,,$ (B.12)
where the fields $X^{\rm a}(z)=e^{\rm a}_{i}X^{i}(z)$ with ${\rm a},i=1,...,d$
have tangent space indices ${\rm a}$ and standard propagator
$\langle X^{\rm a}(z)X^{\rm b}(w)\rangle=-\delta^{\rm{ab}}ln(z-w)\,.$ (B.13)
The momentum in the tangent space is $p_{L{\rm
a}}=\hat{e}_{\rm{a}}{}^{i}p_{Li}=1$ and $\Lambda$ is a twist field with
conformal dimension $h=\frac{1}{2}$ and OPE
$\Lambda(z)\Lambda(0)=\frac{1}{z}+{\rm reg}\,,$ (B.14)
which is necessary to build spin 1 currents [41, 42]. From the OPEs
$\displaystyle H(z)H(0)$ $\displaystyle\sim$ $\displaystyle\frac{1}{z^{2}}\,,$
(B.15) $\displaystyle H(z)E^{\pm}(0)$ $\displaystyle\sim$
$\displaystyle\pm\frac{p_{L{\rm a}}E^{\pm}(0)}{z}\,,$ (B.16) $\displaystyle
E^{+}(z)E^{-}(0)$ $\displaystyle\sim$
$\displaystyle\frac{1}{z^{2}}+\frac{p_{L{\rm a}}H^{\rm a}(0)}{z}\,,$ (B.17)
we see that the affine ${\mathrm{SU}}(2)$ algebra is realized at level
$k=\frac{2\tilde{k}}{p_{L}^{2}}=2$. The central charge of the
${\mathrm{E}}_{8}\times{\mathrm{A}}_{1}$ model at level $k=2$ saturates
$c_{L}^{int}=17$, as may be verified using the data in Table 10.
The central charges of all the maximal enhancements listed in Table 3 can be
readily computed. Except for the ${\mathrm{D}}_{9}$ and
${\mathrm{E}}_{8}\times{\mathrm{A}}_{1}$ models, the internal Kac-Moody
algebras do not saturate $c_{L}^{int}=17$. In some cases, the missing central
charge $\Delta$ is provided by unitary minimal models, cf. (B.11). For
instance the ${\mathrm{E}}_{6}\times{\mathrm{A}}_{3}$ at level 2 requires
$\Delta=6/7=c_{6}$. On the other hand, the
${\mathrm{A}}_{1}\times{\mathrm{A}}_{2}\times{\mathrm{A}}_{6}$ current algebra
at level 2 leads to $\Delta=\frac{49}{30}$ which could arise combining two
minimal models with $j=4$ and $j=9$. However in the case
${\mathrm{D}}_{5}\times{\mathrm{A}}_{4}$, with $\Delta=8/7$, a candidate
world-sheet CFT is not obvious. It would be interesting to understand if there
is a realisation of the missing CFTs in terms of coset models involving the
original and the enhanced gauge groups.
In compactifications of the CHL string to 8 dimensions, the gauge group is
${\mathrm{U}(1)}^{10}$ for generic values of the background fields. To analyze
maximal enhancement at a special point in moduli space let us choose
$E_{11}=2,E_{22}=1,E_{12}=E_{21}=0$ and $a_{1}=a_{2}=0$. These moduli actually
correspond to starting with the 9-dimensional CHL model with group
${\mathrm{E}}_{8}\times{\mathrm{A}}_{1}$ at level $k=2$ discussed above, and
further compactifying on a circle of radius $R_{2}=1$. Following the analysis
in section 3, and using equations (3.7) and (3.8), we see that there are
additional untwisted states with $Z^{2}=4$, having $\rho=0$ and
$(\ell^{1},\ell^{2},n_{1},n_{2})=\pm(0,2,0,1)$. These states enhance the
$({\mathrm{E}}_{8}\times{\mathrm{A}}_{1})_{2}\times{\mathrm{U}(1)}$ gauge
symmetry to
$({\mathrm{E}}_{8}\times{\mathrm{A}}_{1})_{2}\times({\mathrm{C}}_{1})_{1}$.
Using that $p_{L}^{\rm a}=\frac{1}{\sqrt{2}}l^{i}e_{i}{}^{\rm
a}=(0,\sqrt{2})$, the vertex operators contain the currents
$H^{2}=i\partial X^{2}\,,\qquad E^{\pm}=c^{\pm}e^{\pm i\sqrt{2}X^{2}}\,,$
(B.18)
which realize the current algebra of ${\mathrm{C}}_{1}$ at level $k=1$.
We next consider an example with short and long roots. Taking $E_{11}=2$,
$E_{12}=-2$, $E_{21}=0$, $E_{22}=1$ and $a_{1}=a_{2}=0$, gives gauge symmetry
${\mathrm{E}}_{8}\times{\mathrm{C}}_{2}$. The quantum numbers
$(\ell^{1},\ell^{2},n_{1},n_{2})$ of the massless states that enhance the
${\mathrm{U}(1)}^{2}$ to ${\mathrm{C}}_{2}$ are $\pm(0,2,-2,1)$,
$\pm(2,2,0,1)$, $\pm(1,0,1,0)$ and $\pm(1,2,-1,1)$, and they all have
$\rho=0$. The vertex operators contain the currents
$\displaystyle E_{1}^{\pm}(z)$ $\displaystyle=c_{1}^{\pm}\Lambda(z)e^{\pm
iX^{1}(z)},\hskip 56.9055ptp^{\rm a}_{1L}=\alpha_{1}=(1,0)\,,$ (B.19a)
$\displaystyle E_{2}^{\pm}(z)$ $\displaystyle=c_{2}^{\pm}e^{\mp
i(X^{1}(z)-X^{2}(z))},\qquad\qquad p^{\rm a}_{2L}=\alpha_{2}=(-1,1)\,,$
(B.19b) $\displaystyle E_{3}^{\pm}(z)$
$\displaystyle=c_{3}^{\pm}\Lambda(z)e^{\pm iX^{2}(z)},\hskip 48.36958pt\quad
p^{\rm a}_{3L}=\alpha_{3}=(0,1)\,,$ (B.19c) $\displaystyle E_{4}^{\pm}(z)$
$\displaystyle=c_{4}^{\pm}e^{\pm i(X^{1}(z)+X^{2}(z))},\qquad\qquad p^{\rm
a}_{4L}=\alpha_{4}=(1,1)\,.$ (B.19d)
Together with the Cartan operators $H^{1}=i\partial X^{1}$ and
$H^{2}=i\partial X^{2}$, the current algebra of $\hat{\mathrm{C}}_{2}$ is
realized at level $k=1$ since, $\tilde{k}=1$ and the square of the highest
root $\alpha_{4}$ is 2.
It is straightforward to calculate the central charge of the Kac-Moody
algebras of the eight dimensional models listed in Table 7. As in the nine-
dimensional case, they do not saturate $\Delta=0$ in general, but again in
most cases one can find combinations of minimal models that account for the
missing contribution. A consistency check is that when $\Delta<1$ it is always
equal to the central charge of a unitary minimal model.
Finally, let us remark that vertex operators for the twisted states in the
examples (B.18) and (B.19) discussed above do not involve the fields
$Y^{I},Y^{\prime I}$ explicitly. However, when $\rho$ is non-vanishing, they
are expected to be part of the exponentials in the currents. For instance,
with Wilson line $a=\frac{1}{2}w_{6}$ and $R^{2}=\frac{3}{2}$ in nine
dimensions, the states with quantum numbers $(\ell,n,\rho)$ given by
$\pm(1,0,-w_{6})$ and $\pm(1,1,0)$ become massless and enhance the gauge group
to ${\mathrm{E}}_{7}\times{\mathrm{A}}_{2}$, with current algebra realized at
level 2. Free field representations of the affine ${\mathrm{SU}}(3)$ current
algebra at level 2 are known [35] as well as of the level 1 non-simply laced
algebras [43, 44, 45] involved in the enhanced gauge groups of the eight
dimensional theory (see also [46] for constructions of Kac Moody algebras in
terms of free fields). But not all of them can be directly related to the
vertex operators of the twisted states of the CHL theory. We postpone a
detailed analysis of the twisted vertex operators that realize these current
algebras to a future publication.
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|
# Nbias: A Natural Language Processing Framework for bias Identification in
Text
Shaina Raza<EMAIL_ADDRESS>Muskan Garg<EMAIL_ADDRESS>Deepak John Reji<EMAIL_ADDRESS>Syed Raza Bashir
<EMAIL_ADDRESS>Chen Ding<EMAIL_ADDRESS>Vector Institute
for Artificial Intelligence, Toronto, ON, Canada Artificial Intelligence &
Informatics, Mayo Clinic, Rochester, MN, USA Environmental Resources
Management, Bengaluru, Karnataka, India Toronto Metropolitan University,
Toronto, ON, Canada
###### Abstract
Bias in textual data can lead to skewed interpretations and outcomes when the
data is used. These biases could perpetuate stereotypes, discrimination, or
other forms of unfair treatment. An algorithm trained on biased data may end
up making decisions that disproportionately impact a certain group of people.
Therefore, it is crucial to detect and remove these biases to ensure the fair
and ethical use of data. To this end, we develop a comprehensive and robust
framework Nbias that consists of four main layers: data, corpus construction,
model development and an evaluation layer. The dataset is constructed by
collecting diverse data from various domains, including social media,
healthcare, and job hiring portals. As such, we applied a transformer-based
token classification model that is able to identify bias words/ phrases
through a unique named entity BIAS. In the evaluation procedure, we
incorporate a blend of quantitative and qualitative measures to gauge the
effectiveness of our models. We achieve accuracy improvements ranging from 1%
to 8% compared to baselines. We are also able to generate a robust
understanding of the model functioning. The proposed approach is applicable to
a variety of biases and contributes to the fair and ethical use of textual
data.
###### keywords:
Bias detection , Dataset , Token classification , Nbias
## 1 Introduction
The recent surge in Natural Language Processing (NLP) applications,
encompassing fields from recommendation systems to social justice and
employment screening, has sparked a critical concern - the emergence of bias
within these systems [1]. Instances of racial and gender bias have been
increasingly reported [2], indicating an urgent need for scrutiny. These
biases often originate from the training data used in NLP models, and a
majority of these large datasets harbor inherent biases . Regrettably, many
NLP practitioners lack the necessary awareness or knowledge to effectively
identify and address these biases, highlighting a significant gap in the
field.
Furthermore, there is a notable lack of discussion on data specifics - its
origin, generation, and pre-processing - in many NLP publications. Given these
circumstances, the importance of addressing biases in NLP applications cannot
be overstated. These biases, if unchecked, not only compromise the validity of
the models, but can also have unfavorable and detrimental consequences. The
objective of this research is to provide insights into the detection of bias
in NLP datasets, contributing to the development of more equitable and
unbiased Artificial Intelligence (AI) systems.
Bias in text data is a pervasive and deeply-rooted issue. The bias in data
often stems from cognitive predispositions that influences our dialogues,
views, and understanding of information [3]. This bias can be explicit which
are often seen in discriminatory language targeting certain racial or ethnic
groups [4], as in social media. Implicit bias [5], on the other hand, subtly
perpetuates prejudice through unintentional language use but is equally
harmful.
The necessity for unbiased, trustworthy text data has grown across sectors
like healthcare [6], social media [4, 7], and recruitment [8]. This data is
essential for training NLP models for various downstream tasks, like
formulating healthcare diagnoses and treatment plans, handling discriminatory
language on social media, and promoting fair recruitment practices. Figure 1
illustrates the complexities of biases in text data in various domains,
including job hiring, social media, and healthcare. These biases are primarily
conveyed through lexical choices [9] and demand sophisticated detection
methods, motivating this research. The primary aim of this study is to further
foundational research on the fairness and reliability of the textual data.
Figure 1: Visual Representation of Implicit and Explicit Biases in Textual
Data: Examples from Job Hiring, Social Media, and Healthcare.
Although NLP has advanced much, the state-of-the-art techniques [2, 10, 11]
often concentrate on bias detection in specific domains and lack
generalizability. To address this, our research offers a generalizable bias
detection method proven effective across the various domains. We present
Nbias, a comprehensive framework for detecting bias in text data. This
involves data preparation where bias-indicative terms are marked using a
transformer-based token classification method like Named Entity Recognition
(NER).
Current NER solutions can manage general [12], biomedical [13], and social
media [14] entities, but often neglect BIAS as a separate entity. To address
this, we introduce a new entity type, BIAS, to identify biased terms in text
data. In this context, bias refers to unfair and often harmful favoritism or
prejudice towards a particular group, person, or idea, which can manifest
through profanity, unjustified criticism, or discriminatory language.
A key contribution of this study is the development of the first comprehensive
framework for bias detection in text data. This framework is based on latest
language model technology and incorporates four crucial layers: data
gathering, corpus construction, model development, and rigorous evaluation.
The specific contributions of the work are as follows:
1. 1.
Development of Annotated Datasets: Acknowledging the scarcity of bias
annotations in text-based data, we designed a solution by generating multiple
annotated datasets. Our work fills a critical gap in the available resources,
thereby providing a solid foundation for future research in the realm of bias
detection.
2. 2.
Semi-Autonomous Labeling: To alleviate the time-intensive manual annotation
process, we pioneered a novel methodology termed “semi-autonomous labeling”.
This strategy provides a faster and more efficient way of annotating bias-
related terms within textual content. This innovative approach has significant
implications for improving the speed and accuracy of bias detection.
3. 3.
Unique Entity Type - BIAS: In an effort to enhance the precision of bias
identification within text, we introduced a unique entity type, BIAS. This new
entity type is specifically designed for detecting biased words and phrases
within the text data. This has the potential to dramatically improve the
process of bias identification and quantification in text-based analysis.
4. 4.
Comprehensive Evaluation Process: We subjected our proposed framework to a
thorough evaluation process, utilizing both quantitative and qualitative
analysis methods. The results confirm the reliability and efficiency of our
approach, making it compatible for its application in real-world scenarios.
This rigorous evaluation sets a benchmark for assessing the efficacy of bias
detection methodologies.
## 2 Related Work
### 2.1 Identifying Bias in NLP
One of the key challenges associated with NLP systems lies in the presence of
bias, a manifestation of unfair and systematic discrimination observed in
their outcomes [15]. Moreover, the past studies [16, 10, 11, 2, 17] have shown
the societal and cultural prejudices are deeply embedded within the training
data due to the presence of bias. As such, the biases, whether explicit or
implicit, can significantly impact the functionality of the NLP systems,
leading to skewed results and perpetuating existing societal biases. Thus, the
detection and mitigation of these biases are crucial to promoting fairness and
inclusiveness within NLP systems [7, 11].
Researchers have proposed and implemented various strategies to identify bias,
including employing statistical methods to discover patterns of bias within
the training data [2, 18]. Under this approach, specific words or phrases that
appear to be disproportionately associated with certain demographic groups,
such as genders or races, are identified. For example, certain adjectives
might be used more frequently in descriptions of women than men [2], or vice
versa. The identification and debiasing of such patterns can highlight areas
of potential bias, providing a starting point for efforts to eliminate these
biases [19].
The field of bias detection in NLP has seen a surge of innovative methods in
recent years, primarily leveraging advanced machine learning techniques. One
such study has considered the combination of a speech detection system with an
explanatory method to identify potential bias [20]. In this method, not only
is the system trained to detect instances of hate speech, but it also provides
a rationale or explanation for its classification decisions. Another area of
research that has attracted considerable attention is the investigation of
bias in event detection datasets and models [21]. Event detection tasks
involve identifying and classifying real-world events within text data. These
tasks can be susceptible to a range of bias-related issues, including data
sparsity, labeling task, and annotations.
Additionally, NLP techniques have been employed to address various aspects of
bias. For instance, in a related study [22] on gender bias and sentiment
towards political leaders in the news were quantified using word embeddings
and sentiment analysis. Another work focused on investigating ableist bias in
NLP systems, particularly at the intersection of gender, race, and disability
[23]. Similarly, a methodology was proposed to eliminate gender bias from word
embeddings [24]. Furthermore, marked attribute bias in natural language
inference was identified and analyzed, with an evaluation of existing methods
for bias mitigation [9]. These studies provide a deep understanding of the
social and cultural factors that contribute to bias identification.
Another work [25] presents bias analysis in NLP beyond demographic bias,
focusing on predicting interpersonal group relationships using fine-grained
emotions. A related study [26] evaluates gender bias in NLP research,
highlighting the lack of explicit gender theorization. In another work,
authors [27] introduce an effective bias-conflicting scoring method and
gradient alignment strategy to identify and mitigate dataset biases. Overall,
these studies underscore the importance of continuous efforts in identifying
and mitigating biases in models to ensure fairness and equity.
### 2.2 Named Entity Recognition (NER)
Named Entity Recognition (NER) is a significant task in NLP that is aimed at
identifying and classifying named entities within textual data. NER is a token
classification task that focuses on identifying and classifying named entities
such as individuals, organizations, and locations within a given text. In the
past, many traditional methods have been employed for NER, each with its
unique characteristics and benefits.
* 1.
Rule-based methods rely on predefined sets of rules to identify named entities
[28]. This method usually employs regular expressions or dictionary-based
techniques to extract entities. Although rule-based methods can be effective
for well-defined and specific contexts, their performance can decrease in the
face of variability and ambiguity in language usage.
* 2.
Supervised learning methods leverage annotated data to train a model for NER
[14, 29]. These methods use statistical models such as Support Vector Machines
(SVM), Conditional Random Fields (CRF), and others to classify the named
entities. The performance of supervised learning methods can be impressive,
given sufficient high-quality annotated data.
* 3.
Deep learning methods, which are more contemporary approaches, utilize complex
architectures like recurrent neural networks (RNNs) and transformer-based
language models to extract named entities [30, 13]. These methods have shown
promising results in NER tasks, owing to their capacity to capture intricate
language patterns and contextual information.
A recent study introduced a contrastive learning-based approach for multimodal
NER [31]. This approach leverages both textual and non-textual data to
identify and classify named entities, harnessing the complementary information
offered by different modalities to improve the model’s performance. Another
research work into event detection from social media posts, evaluating the
effectiveness of a pre-trained NER model followed by graph-based spectral
clustering [32]. The study also explored transformer-based methods to weight
the edges of the graph for event detection, further refining the detection
process. A span-based NER model eliminates the need for label dependency [32].
This approach addresses the issue of cascade label mis-classifications, a
common challenge in traditional NER models that depend on label sequences.
While our work on token classification is inspired by these studies, we
identify a notable gap in the literature: the existing seminal work does not
recognize BIAS as an entity. In this work, we detect biased expressions within
unstructured texts, designating them under the ’BIAS’ entity label.
### 2.3 Data Annotation
Data annotation is a crucial task in NLP as it involves labeling and
categorizing information to extract valuable insights from text data [33]. By
enriching text data with relevant metadata, such as part-of-speech tags, named
entity tags, and sentiment tags, data annotation provides contextual
information that is essential for subsequent analysis [34]. Quality annotated
data enhances model learning, boosting prediction accuracy. In contrast,
inadequate annotations impede learning, resulting in subpar performance.
Various methods of data annotation cater to different requirements of speed,
quality, and computational resources:
* 1.
Manual annotation is carried out by human annotators who carefully review and
label the data. This method typically yields high-quality results, given the
nuanced understanding that humans have of language. However, manual annotation
is often time-consuming and labor-intensive, and its feasibility may be
limited by the availability of qualified annotators and financial resources
[28].
* 2.
Semi-automatic annotation combines manual efforts with automated tools to
accelerate the annotation process and minimize human error. These tools can
range from rule-based systems to pre-trained machine learning models [35].
While semi-automatic annotation can improve efficiency, its accuracy may still
depend on the quality of the automated tools and the manual review process.
* 3.
Automatic annotation leverages machine learning models and algorithms to
annotate text data without human intervention [36]. Although automatic
annotation can process vast amounts of data in a relatively short time, its
accuracy may be compromised, particularly for complex or ambiguous texts.
Therefore, a common practice is to combine automatic annotation with manual
review to ensure data quality.
Various strategies have been developed to address these challenges and
optimize the annotation process. One study presents a comprehensive comparison
of different annotation tools, highlighting their strengths and limitations
[37]. Another research work proposes a method for automatically generating
high-quality labeled data for NER tasks by leveraging existing knowledge bases
[38]. A similar study has developed an annotation framework that combines
statistical machine translation and human annotation to create a parallel
corpus [39]. Other researchers have investigated methods for improving the
reliability and consistency of manual annotations, such as developing
guidelines and protocols for annotation tasks [40]or implementing quality
control mechanisms to ensure data quality.
Ultimately, the choice of annotation method and tools will depend on the
specific requirements of a project, such as the desired level of accuracy, the
available resources, and the nature of the data being annotated.To this end,
we employ a semi-automatic annotation strategy, integrating human proficiency
with semi-supervised learning methodologies.
## 3 Proposed Framework for Bias Identification in Texts
In this section, we present Nbias, an innovative framework designed to detect
biases within textual data, as illustrated in Figure 2. The Nbias framework is
structured into four distinct layers: (i) the data collection layer, (ii) the
corpus construction layer, (iii) the model development layer, and (iv) the
evaluation Layer. Each layer is designed to collaborate seamlessly with the
others, providing an effective and comprehensive approach for detecting biases
in textual content.
Figure 2: Nbias: A Natural Language Processing Framework for Bias
Identification.
### 3.1 Data Layer
The Data Layer serves as the framework’s primary interface with the data for
analysis. It handles data collection, pre-processing and data consolidation
from a variety of sources, such as social media, online articles, and
databases. This layer ensure adaptability and high performance for the entire
framework.
#### Data Gathering
Our study adopts a methodological data collection approach, incorporating
diverse sources from various domains. To analyze biases in medical narratives
and healthcare, we include data from two important clinical text databases:
MIMIC-III [41] and MACCROBAT [37]. The MIMIC-III dataset is a publicly
available database with de-identified health data from over 40,000 ICU
patients. It offers rich clinical narratives, including nursing notes,
radiology reports, and discharge summaries, enabling a deep understanding of
biases in healthcare communication. The textual data were primarily obtained
from the NOTEEVENTS table.
The MACCROBAT dataset provides valuable pediatric critical care data,
including admission notes and medical summaries. It contains 200 original
documents along with corresponding annotated versions centered around clinical
case reports.
To detect bias in news articles and social media streams, we use the BABE
(Bias Annotations By Experts) dataset [10]. This dataset includes 3700
articles and tweets, offering a comprehensive perspective on linguistic bias
in media and public opinion. It features marked statements, enabling
recognition of bias at both granular (word-level) and broader (sentence-level)
scopes, covering diverse topics.
To examine biases in employment practices, we incorporate the Job
Hiring/Recruitment dataset [42], comprising 20,000 job posts with titles,
descriptions, and associated tags from various businesses. Each advertisement
includes job details and manually assigned tags by recruiters, suggesting jobs
to potential candidates with analogous skills.
#### Data Consolidation
After gathering and pre-processing data from various sources, all datasets are
harmonized into a single consolidated dataframe. This dataframe includes the
following columns:
* 1.
Dataset: Specifies the source dataset, such as MIMIC-III, MACCROBAT, Job
Hiring, or BABE
* 2.
Text: Contains the actual textual data extracted from the respective datasets,
including clinical notes, case reports, job descriptions, or annotated
statements.
* 3.
Biased Words: Includes the words or phrases identified as biased in the text,
crucial for granular bias detection.
* 4.
Aspect of Bias: Denotes the specific type or aspect of bias present in the
text, categorized by gender, racial, or age biases, to understand the nature
of the biases detected.
* 5.
Label: Indicates whether the text is biased or non-biased, serving as the
target variable for the token classifier and for evaluation purposes.
A sample record in JSON format is shown below:
⬇
{
"Record": {
"Dataset": "MIMIC-III",
"Text": "Clinical notes of patient XYZ indicate a history of superficial
hypertension due to overly emotional personality.",
"BiasedWords": "superficial, overly emotional personality",
"AspectOfBias": "age",
"Label": "biased"
}
}
In the consolidated dataframe, each row represents a unique sample from the
original dataset, supplying information for bias detection and assessment.
Further pre-processing is conducted to prepare the data for subsequent layers
of the Nbias framework, particularly the NLP model performing token
classification.
#### Data Pre-processing
The pre-processing of textual data involves a series of sequential operations
to refine and structure the data for machine learning algorithms. This
includes tokenization, which involves breaking raw text into meaningful tokens
(words or subwords) for semantic understanding and subsequent NLP tasks; text
cleaning, which involves removing punctuation, numbers, special characters,
and converting text to lowercase to ensure uniformity and clarity; and
handling missing values, which involves identifying and appropriately managing
missing data to avoid bias and improve model performance. These pre-processing
steps convert raw text into a clean, structured format, enabling the NLP token
classification model in the subsequent layer.
### 3.2 Corpus Construction
Our group, consisting of three seasoned professionals from the disciplines of
healthcare, computer science, and journalism, was joined by two diligent
students to perform the task of detecting and labeling bias in our dataset.
Their collective role centered around the critical task of carefully
annotating bias within our dataset. This endeavor is important to ensure the
integrity and fairness of any subsequent analysis or research. The foundation
for this task was based on comprehensive guidelines that clearly delineated
the concept of bias in this context.
Bias, as per the instructions, was defined as any terminology or phraseology
that could potentially provoke prejudiced comprehension or induce
stereotyping, as mentioned in most of the literature [11, 7, 10] also. The
factors from which biases could stem were identified as gender, race,
socioeconomic status, age, or disability for this NLP work. Such biases could
inadvertently skew the dataset and, consequently, the results derived from it.
Thus, the identification and annotation of such biases are of high importance
to uphold the accuracy and reliability of our dataset. Highlighting both
explicit and implicit biases was emphasized as a critical part of our work.
#### Annotation Scheme
Under the light of these provided guidelines, our team proceeded by using a
carefully compiled list of terms and phrases, known as “bias-indicative”
lexicons. These lexicons provided a comprehensive guide to potential areas
where bias could lurk within our dataset. A portion of this list is exhibited
in Table 1 for reference. This bias-indicative lexicon served as a
navigational tool for our team to identify and mark “BIAS” entities scattered
within our textual data. These entities can be individual words or phrases
that express or imply bias. This systematic approach ensured that we could
account for most biases that exists in the data.
Table 1: Bias Dimensions and Fewer Sample Biased Words/Phrases Shown Due to Brevity Bias Dimension | Biased Words/Phrases
---|---
Gender | ‘hysterical’, ‘emotional’, ‘weak’, ‘bossy’, ‘fragile’, ‘nagging’, ‘man up’, ‘tomboy’
Race | ‘inner city’, ‘illegal alien’, ‘thug’, ‘exotic’, ‘uncivilized’, ‘model minority’, ‘white trash’
Social Status | ‘trailer park’, ‘lazy’, ‘freeloader’, ‘welfare queen’, ‘ghetto’, ‘lazy bum’, ‘filthy rich’
Age | ‘senile’, ‘slow’, ‘old-fashioned’, ‘whippersnapper’, ‘elderly’, ‘young and naive’, ‘generation gap’
Disability | ‘handicapped’, ‘crippled’, ‘invalid’, ‘sufferer’, ‘differently-abled’, ‘victim’
Religion | ‘radical’, ‘terrorist’, ‘infidel’, ‘heathen’, ‘fanatic’, ‘holy roller’
Profession | ‘greedy’, ‘dishonest’, ‘corrupt politician’, ‘crooked lawyer’, ‘greedy CEO’, ‘lazy government worker’
National | ‘unpatriotic’, ‘alien’, ‘foreigner’, ‘outsider’, ‘immigrant’, ‘nationalist’
Education | ‘uneducated’, ‘illiterate’, ‘dropout’, ‘underachiever’, ‘overachiever’, ‘smarty-pants’
Body Size | ‘fat’, ‘slob’, ‘skinny’, ‘lardass’, ‘beanpole’, ‘plus-sized’
We adopted the Inside-Outside-Beginning (IOB) annotation scheme [43] to
classify and annotate ‘BIAS’ entities. This technique categorizes tokens in
the text as the beginning (B), inside (I), or outside (O) of a bias entity.
‘B’ represents the first token of a bias entity, ‘I’ for tokens inside the
entity, and ‘O’ for tokens not part of any bias entity. This approach ensured
consistent and precise annotations, enhancing the reliability and accuracy of
our study.
#### Annotation Approach
We leveraged semi-supervised learning methodologies [35, 13, 33] to enhance
both efficiency and precision of the annotation process. The integration of
BERT (Bidirectional Encoder Representations from Transformers) , known for its
superior text comprehension abilities, substantially improved our approach.
Our annotation process initiated with the manual tagging of 20% of the
complete dataset. This critical yet time-consuming task was strategically
limited to a subset of data, ensuring a balance between accuracy and
efficiency. The “BIAS” entities were carefully annotated in compliance with
our predefined guidelines. This annotated subset then fed into our BERT model,
serving as training data for the token-classification task. Once sufficiently
trained, the model was assigned the task of predicting “BIAS” entities within
the remaining 80% of the data. The extensive dataset was effectively managed
by breaking it down into 20% increments, a process we refer to as “semi-
autonomous labelling”. Expert reviews cross-verified the “BIAS” entities
labelled by the model. This combination of semi-supervised learning with
expert validation enabled us to create an annotation process that is both
optimized and trustworthy.
#### Working Instance
To demonstrate our annotation scheme, we consider the example sentence: “The
overpriced product from the highly successful company was surprisingly
popular”. Table 2 presents the corresponding BIO format annotations for this
sentence. Assuming the term “overpriced” holds potential bias, it would be
tagged as “B” in the BIO scheme, indicating the start of a bias entity. All
other tokens not part of this bias entity would be labeled “O”. This example
shows our extensive annotation process across our dataset. This approach
allows us to quantify and comprehend biases in a consistent manner.
Table 2: Bias Annotation using BIO scheme Word | Bias Annotation
---|---
The | O
overpriced | B-BIAS
product | O
from | O
the | O
highly | B-BIAS
successful | I-BIAS
company | O
was | O
surprisingly | B-BIAS
popular | I-BIAS
. | O
#### Resolving Discrepancies
An integral part of our process was addressing discrepancies between
annotators, a common criteria in multi-person annotation tasks. We implemented
a consensus-driven approach to uphold consistency and reliability in our
annotations. Any disagreement was discussed collectively, considering each
annotator’s viewpoint and reaching a unified decision based on predefined
annotation guidelines. This process ensured collective agreement on all
annotations, minimizing potential bias or error and boosting reliability. This
consensus strategy was uniformly applied across all data sources including the
BABE, MIMIC, MACCROBAT, or Job Hiring datasets.
#### FAIR Principles
After reaching consensus on all annotations, we saved the final annotated data
in the widely-accepted CoNLL-2003 format [44]. This format represents data in
a tab-separated manner, associating each word with its part of speech tag,
chunk tag, and named entity tag. Sentences are separated by an empty line, and
each row corresponds to a token with its annotation.
The CoNLL-2003 format offers multiple benefits. It ensures compatibility with
existing NLP tools and models, facilitating future analysis and model
training. Additionally, it promotes collaboration and peer review by allowing
easy data sharing and comprehension among researchers. Lastly, it enhances the
reproducibility of our study, enabling others to use our data for model
validation and findings replication. By adhering to the FAIR principles, our
dataset is made Findable, Accountable, Interoperable, and Reusable, enhancing
the transparency, accessibility, and reliability of our research.
#### Inter-Annotator Agreement
In our research, we placed considerable emphasis on establishing rigorous
protocols to guarantee the reliability and consistency of the data
annotations. Two independent reviewers were assigned to carefully assess the
annotated data, promoting objective evaluation devoid of influence from
initial annotators. Rather than relying on subjective judgment, we quantified
their agreement through Cohen’s Kappa coefficient—a statistical measure common
in categorical data studies, accounting for potential chance agreement. Scores
over 0.6 denote “substantial” agreement and above 0.8 represent “almost
perfect” agreement. Our reviewers attained a Cohen’s Kappa score of 78%,
demonstrating high concordance on the annotations. This high score
substantiates the uniformity, consistency, and quality of our annotations.
Moreover, it demonstrates the objectivity of the assessment process,
highlighting the well-built nature of our annotated data. This, in turn,
enhances the trustworthiness of prospective findings drawn from this dataset.
### 3.3 Model Development Layer
In this layer, we leverage the BERT language model for token-classification
and adapt it for the task of NER . The choice of BERT is motivated by its
powerful capability of understanding both left and right context of a word,
and its effectiveness in recognizing and classifying multi-word phrases. These
features make it particularly well-suited for the complex task of bias
detection and annotation in our text data.
The advantage of using BERT in Nbias model development lies in its more
effective token-level bias identification. Nbias incorporates enhancements to
the standard BERT architecture, such as modifications in the attention
mechanism, loss function, and fine-tuning approaches, specifically tailored
for better capturing biases in complex text data. The subsequent section
provides a detailed explanation of the model development.
The token classifier architecture (shown in as the middle component in Figure
2) consists of a multi-layer bidirectional transformer encoder that captures
contextual information from both directions. Given an input sequence
$X=\\{x_{1},x_{2},...,x_{n}\\}$, the words are tokenized and embedded as shown
in Equation (1):
$E(X)=\\{e(x_{1}),e(x_{2}),\ldots,e(x_{n})\\}$ (1)
where $E(X)$ represents the set of embedded representations for an input
sequence $X$, $X$ consists of $n$ words $\\{x_{1},x_{2},...,x_{n}\\}$,
$e(x_{i})$ is the embedding function that maps each word $x_{i}$ from the
input sequence to a continuous vector representation. The embedded input
sequence is then passed through the transformer layers.
BERT employs self-attention mechanisms to weigh the importance of different
words in the input sequence, enabling it to better identify and understand
complex relationships between words. The self-attention $Att$ score between
word $i$ and word $j$ is computed as shown in Equation (2):
$Att(i,j)=\text{Softmax}\left(\frac{Q(e(x_{i}))\cdot
K(e(x_{j}))^{T}}{\sqrt{d_{k}}}\right)$ (2)
where $Q$, $K$ are the query and key matrices, and $d_{k}$ is the key
dimension.
Following the transformer encoder, the output after applying self-attention
and passing through the bidirectional transformer encoder is represented, as
shown in Equation (3):
$R(X)=\\{r(x_{1}),r(x_{2}),...,r(x_{n})\\}$ (3)
where $R(X)$ represents the set of contextualized representations for an input
sequence $X$. The function $r(x_{i})$ is the representation function that maps
each word $x_{i}$ from the input sequence to a continuous vector
representation after passing through the transformer encoder.
A linear layer with a softmax activation function is added for entity
classification. This layer transforms the representations generated by the
transformer encoder into a probability distribution over the possible output
classes. To simplify our annotation and prediction task, we have merged the
’B’ (Beginning) and ’I’ (Inside) tags from the standard BIO tagging scheme
into a single ’BIAS’ tag. The ’BIAS’ tag represent any part of a bias entity,
while ’O’ represents non-entity. The probability distribution is calculated as
shown in Equation (4):
$P(y|x)=\text{Softmax}(W\cdot c(x)+b)$ (4)
where $W$ is the weight matrix, $b$ is the bias vector in the softmax
function, and $P(y|x)$ is the probability distribution over the output classes
‘BIAS’ and ‘O’. The final output of the model indicates the presence of biased
words or phrases within the input sequence by labeling them as ‘BIAS’. This
simplification enables our model to recognize biased phrases more effectively,
without differentiating between their start or continuation.
We show an example of the model output on a sample from the test set in Figure
3.
Figure 3: BIAS Annotation on a Piece of Text
The pseudocode algorithm steps for the Nbias model development are given in
Algorithm 1. As seen in Algorithm 1, the Nbias model, built on BERT, tokenizes
and contextualizes input text using transformer encoders. Through self-
attention mechanisms, it weighs relationships between words and classifies
each token as biased or unbiased using a softmax-activated linear layer.
Algorithm 1 Nbias Model Development
1:Text sequence $X=\\{x_{1},x_{2},\dots,x_{n}\\}$
2:Initialize BERT with token-classification architecture
3:Tokenize input sequence $X$
4:Embed input sequence: $E(X)=\\{e(x_{1}),e(x_{2}),\dots,e(x_{n})\\}$
5:for each token in $E(X)$ do
6: Compute self-attention:
7: $Att(i,j)=\text{Softmax}\left(\frac{Q(e(x_{i}))\times
K(e(x_{j}))^{T}}{\sqrt{dk}}\right)$
8:end for
9:Pass $E(X)$ through bidirectional transformer encoder:
$R(X)=\\{r(x_{1}),r(x_{2}),\dots,r(x_{n})\\}$
10:for each token representation in $R(X)$ do
11: Compute probability distribution: $P(y|x_{i})=\text{Softmax}(W\times
c(x_{i})+b)$
12:end for
13:for each token in $X$ do
14: if probability corresponds to BIAS then
15: label as ‘BIAS’
16: else
17: label as ‘O’
18: end if
19:end for
20:return the labeled sequence
### 3.4 Evaluation Layer
The evaluation layer plays a critical role in assessing the performance of our
model. This layer encompasses both quantitative and qualitative evaluation
methods, providing a comprehensive perspective on the model’s performance.
#### Quantitative Evaluation
The quantitative evaluation is typically statistical in nature and involves
the use of various metrics to numerically measure the model’s performance.
Metrics such as F1-score, AUC-ROC and accuracy are commonly used in this
context. F1 score balances precision (the ability of the model to correctly
identify positive instances) and recall (the ability of the model to identify
all relevant instances), providing a single measure of the model’s overall
performance.
#### Qualitative Evaluations
In addition to these numerical measures, we also conduct a qualitative
evaluation. This type of evaluation is more about the quality, relevance, and
usefulness of the model’s output. It involves an expert review of a subset of
the model’s predictions to measure how well the model is performing in
practical terms. Factors such as the model’s ability to correctly identify
complex or subtle bias entities, and the interpretability of its output, are
examined in the qualitative evaluation.
In our study, we focus on qualitative evaluations, specifically assessing
model robustness and conducting perpetuation tests. Our robustness analysis
[45] explores the model’s stability under various conditions including
adversarial inputs and data variations. Perpetuation tests [46] help us
understand if the model inadvertently reinforces or introduces societal
biases. We also consider a human evaluation, to assess the model’s performance
in real-world conditions.
## 4 Experimental Setup
In this section, we detail the settings, evaluation metrics, baselines and
hyperparameters of our experimental design for replication and validation.
### 4.1 Dataset
Our study uses diverse datasets: MIMIC-III [41], MACCROBAT [37], BABE [10],
and Job Hiring [42]. After annotation (detailed in Section 3.1 and 3.2), each
dataset is split into training, validation, and test sets using an 80-10-10
ratio. The division allows for efficient model training, validation, and
testing. Modifications are made for the MACCROBAT dataset to maintain balance
despite its limited entries. Table 3 presents the detailed dataset
information.
Table 3: Dataset Details with Training (train), Development (dev), Test (test) sets and Total Samples Data Source | Domain | train | dev | test | Total
---|---|---|---|---|---
BABE | News/Social Media | 15,300 | 1,700 | 1,700 | 18,700
MIMIC (Clinical) | Healthcare | 1,800 | 200 | 200 | 2,200
MACCROBAT | Healthcare | 160 | – | 40 | 200
Job Hiring | Occupational | 16,000 | 2,000 | 2,000 | 20,000
Total | | 33,260 | 3,900 | 3,940 | 41,100
### 4.2 Hardware Settings
The experiments conducted in this study were performed on a dedicated research
server with specific hardware configurations. The server was equipped with an
Intel Xeon CPU E5-2690 v4 running at 2.60GHz, 128GB of RAM, and a powerful
NVIDIA GeForce RTX 3090 GPU. The operating system installed on the server was
Ubuntu 18.04 LTS. These hardware settings provided substantial computational
power, enabling us to efficiently execute resource-intensive tasks, such as
training complex machine learning algorithms and deep learning models.
### 4.3 Time measurements
Time measurements during the training, validation, and testing phases were
recorded for our models across the diverse datasets. Utilizing our hardware
setup, we ensured peak performance with minimized hardware-induced delays.
Specifically, the BABE dataset took 4.5 hours for training with 30 minutes
each for validation and testing. The MIMIC dataset required 2 hours of
training, and 10 minutes for both validation and testing. For the smaller
MACCROBAT dataset, training was completed in 0.5 hours, with validation and
testing taking 5 minutes each. Lastly, the Job Hiring dataset took the longest
at 5 hours for training and 40 minutes each for validation and testing.
### 4.4 Baselines
For the comparison of models for token classification model performance, we
consider a range of diverse baseline approaches. These include BiLSTM-CRF,
which combines BiLSTM and CRF [29]; BERT-CRF, a blend of BERT and CRF [47];
RoBERTa, an optimized variant of BERT [48]; BART-NER, an application of the
BART model for NER [49]; CNN-NER, a CNN-based method for capturing named
entities [50]; and TENER, an NER model that utilizes an adapted Transformer
Encoder for character and word-level features [51]. We also consider the few-
shot NER models like [52] and model-agnostic meta-learning (MAML) [53] and
zero-shot named entity typing (NET) [54] model.
The selected baselines represent a collection of different architectures such
as BiLSTM, BERT, RoBERTa, BART, CNN, and Transformer Encoder, each combined
with either the CRF or NER task. These models were chosen because they
represent the state-of-the-art and constitute a robust set of baselines for
comparing token classification model performance.
### 4.5 Hyperparameter Settings
The chosen hyperparameters for our token classifier are provided in Table 4.
Table 4: Hyperparameter Settings and Training Details Parameter/Method | Details/Value
---|---
Model | bert-base-uncased
Optimizer | Adam
Learning Rate | $1\times 10^{-2}$
Momentum | 0.5
Weight Decay | 0.01
Epochs | 5
Batch Sizes | 4, 8, 16, 32, 64
Batch Size (training) | 16
Input Sequence Length | 128 subword tokens
Dropout | Applied on input and hidden layers
Convergence Criteria | Negligible decrease in validation loss
Validation Strategy | Hold-out
Early Stopping | Implemented
Training Environment | Google Colab Pro
Hardware | NVIDIA Tesla T4 GPU
$\beta_{1}$ | 0.9
$\beta_{2}$ | 0.999
Epsilon | $1\times 10^{-8}$
Hidden Units | (Leaky) Rectified Linear Units (ReLUs)
In the comparative experiments with the baselines, the models were optimized
using a learning rate between 1e-5 and 5e-5 over several training epochs,
typically 3 to 10. The batch size varied between 16 and 64, based on memory
constraints, and the input sequence length was limited to 512 tokens. To
prevent overfitting, we used regularization techniques such as dropout and
weight decay. We generally employed the Adam or AdamW optimizer. All
hyperparameters were fine-tuned according to the specific task requirements
and dataset characteristics.
## 5 Results
### 5.1 Overall Performance
Table 5 presents a comprehensive comparison of our proposed method, Nbias,
with various baseline models in the token-classification task across three
distinct categories: Social Media Bias, Health-related, and Job Hiring. Due to
space constraints, we are only reporting the F1-scores in this overall
comparoson, which are the harmonic mean of precision and recall, as it is a
commonly used single metric that combines both precision and recall. The
F1-scores are expressed as percentages, accompanied by the standard deviation
(±) to indicate the variability in scores across five separate runs. The
highest F1-score in each category is highlighted in bold to easily identify
the best performing model.
Table 5: Comparison of Token Classification Models on Three Different Categories: Social Media Bias, Health-related, and Occupational. The performance metric used is F1-score (harmonic mean of precision and recall), expressed as a percentage, accompanied by the standard deviation (±) indicating the variability in scores across 5 runs. The best score is highlighted in bold. Model | Social Media | Health-related | Job Hiring
---|---|---|---
Rule-based [55] | 65.4 $\pm$ 1.4 | 70.2 $\pm$ 0.7 | 72.3 $\pm$ 0.9
BiLSTM-CRF [29] | 72.6 $\pm$ 1.0 | 75.8 $\pm$ 0.9 | 78.1 $\pm$ 0.8
BERT-CRF [47] | 80.7 $\pm$ 1.3 | 82.3 $\pm$ 0.7 | 83.5 $\pm$ 0.6
RoBERTa [48] | 82.8 $\pm$ 0.7 | 83.6 $\pm$ 0.9 | 80.5 $\pm$ 0.5
CNN-NER [50] | 76.2 $\pm$ 1.1 | 78.1 $\pm$ 0.0 | 73.4 $\pm$ 0.9
BART-NER [49] | 84.7 $\pm$ 0.9 | 84.2 $\pm$ 0.7 | 82.0 $\pm$ 0.8
TENER [51] | 85.7 $\pm$ 0.5 | 86.4 $\pm$ 0.6 | 85.1 $\pm$ 0.5
Few-shot NER [52] | 70.2 $\pm$ 3.4 | 73.1 $\pm$ 2.9 | 69.2 $\pm$ 1.7
NET [54] | 70.1 $\pm$ 1.4 | 72.2 $\pm$ 1.2 | 67.1 $\pm$ 1.2
MAML [53] | 62.1 $\pm$ 1.8 | 65.3 $\pm$ 1.2 | 60.5 $\pm$ 2.5
Nbias | 86.9 $\pm$ 0.2 | 89.1 $\pm$ 0.8 | 90.3 $\pm$ 0.4
The results presented in Table 5 conclusively demonstrate the performance of
the Nbias model in all tested scenarios. In the Social Media area, the Nbias
model got an F1-score of 86.9% with a small deviation of ± 0.2. In the Health
area, it performs even better with an F1-score of 89.1% and a deviation of ±
0.8, which means the scores ranged between 88.3% and 89.9%. In the Job Hiring
area, the model got an F1-score of 90.3%, with scores ranging between 89.9%
and 90.7%. These small deviations show that the model’s performance is
consistent in different tests.
Amongst the baselines models, the TENER model performs better. The BERT-CRF
and RoBERTa models, on the other hand, exhibit good performances. Both the
CNN-NER and BART-NER models also display satisfactory performance, although
they comes behind the Nbias and TENER models. Contrastingly, the Rule-based
model underperforms compared to transformer and BiLSTM based baselines. The
Few-shot NER, NET, and MAML models also showed average performance. Even
though few-shot models can work well with just a few examples, the results
show there is room for improvement. This could be achieved by creating custom
training methods or tasks that are specific to a certain area.
Overall, the Nbias model emerges as the most effective across all categories.
While other BERT-based baselines may also attempt bias identification, Nbias
outperforms them due to its custom-designed model features optimized for this
specific purpose. The performance gain could be in terms of better debiasing
results, increased fairness in predictions, or improved overall model accuracy
in scenarios where bias reduction is critical. These findings provide valuable
insights for the future development and selection of token classification
models across different domains.
#### Accuracy Analysis of Token Classification Models
Figure 4 shows how different models perform in classifying tokens over
different test sets.
Figure 4: Comparative Accuracy Scores of Token Classification Models across
Three Different Categories: Social Media, Health-related , and Job Hiring for
Bias Text Identification
As depicted in Figure 4, the Nbias model exhibits superior performance,
achieving accuracy scores of 88.4% in Social Media Bias, 90.6% in Health-
related texts, and 91.8% in Job Hiring texts. Following closely are the TENER
and BART-NER models in terms of accuracy. While other models such as RoBERTa,
BERT-CRF, BiLSTM-CRF, and CNN-NER also demonstrate commendable performance,
they fall short of the scores attained by Nbias, TENER, and BART-NER in this
experiment. Models like Few-shot NER, NET, and MAML, although not scoring the
best, exhibit promising potential. Lastly, the Rule-based model, which relies
on predefined rules rather than learning from the data, still manages to
perform above 60%.
Overall, these results underscore the enhanced capability of the latest
transformer-based models like BART and TENER to extract contextual information
from text data, as evidenced by this experiment. Moreover, it affirms that a
model carefully designed for bias detection, such as ours, can indeed yield
highly effective results.
### 5.2 Performance Analysis using ROC Curves and AUC Scores
(a) Models applied to Social Media Data.
(b) Models applied to Health-related Data.
Figure 5: ROC curves and AUC Scores for Various datasets (continued on next
page) (a) Models applied to Job Hiring Data. Figure 6: ROC Curves and AUC
Scores for Various Datasets.
In this study, we compare the performance of different models in token
classification tasks using Receiver Operating Characteristic (ROC) curves and
corresponding Area Under the Curve (AUC) scores on Social media, Health-
related, and Job Hiring data. Figures 5(a), 5(b), and 6(a) displays the AUC-
ROC curves for all the baseline models and our Nbias token classification
model.
The results presented in Figure 6 shows the superior capability of the Nbias
model, as evidenced by their better True Positive Rates at minimal False
Positive Rates. While models like Rule-based, RoBERTa, Zero-shot and few-shot
NER models demonstrated low-to-moderate performance, others such as TENER,
BiLSTM-CRF, CNN-NER, BART-NER yielded commendable results, particularly in the
early segments of their respective curves. All these models also exhibited
better performance specifically in the health and job hiring datasets.
Overall, these findings suggest that some models excel in specific domains.
This could be attributed to several factors, including but not limited to:
1. 1.
Training on analogous data points that make the model more aware of the
specific features of a domain.
2. 2.
The architecture of the model being inherently better suited for certain types
of data.
3. 3.
Hyperparameter choices that resonate better with specific data
characteristics.
4. 4.
Preprocessing and feature engineering steps that align closely with the
requirements of a domain.
Thus, choosing the optimal model for a specific domain is important for
achieving the best performance.
### 5.3 Confusion Matrix and Error Analysis
We present the results of the BIAS entity identification task for “Health-
related Bias”, “Political Bias”, and “Occupational Bias” using Nbias. The
model’s performance is evaluated based on confusion matrices and error
analysis (Table 6), providing insights into its strengths and limitations of
the model.
Table 6: Confusion Matrix and Error Analysis for BIAS Entity Identification using Nbias: The table presents the True Positives (TP), False Positives (FP), True Negatives (TN), and False Negatives (FN) for various bias types identified in the dataset, along with the Precision in percentage. The categorization of biases is based on a predefined analysis of the content and context in which they appear. Dataset | Bias types | TP | FP | TN | FN | Precision
---|---|---|---|---|---|---
Health | healthy lifestyle | 98 | 12 | 145 | 5 | 89.1%
medical advancements | 85 | 56 | 142 | 15 | 60.2%
research findings | 98 | 19 | 138 | 2 | 83.7%
Social Media | biased news source | 112 | 10 | 157 | 8 | 91.8%
political affiliation | 95 | 7 | 162 | 16 | 93.1%
political agenda | 86 | 14 | 154 | 16 | 86.0%
Occupational | gender bias in hiring | 63 | 5 | 172 | 8 | 92.7%
ethnicity bias in hiring | 49 | 4 | 173 | 11 | 92.5%
age bias in hiring | 45 | 8 | 170 | 13 | 84.2%
Health-related Bias: The Nbias exhibits strong performance in identifying
“healthy lifestyle” entities, achieving a precision of 89.1%. However, it
missed 5 instances of this entity, leading to false negatives. For “medical
advancements”, the precision is lower at 60.2%, and the model identified 56
false positives, misclassifying non-biased terms as biased. On the other hand,
the model achieved a relatively high precision of 83.7% for “research
findings” yet it missed 2 instances, resulting in false negatives. These
findings suggest that the model performs well for more explicit health-related
biases, but subtle biases and rare terms might pose challenges.
Social Media: Our Nbias demonstrates high precision in identifying “biased
news source” entities (91.8%), correctly capturing biased sources. However, it
produced a few false positives, misclassifying some non-biased sources as
biased. For “political affiliation” entities, the precision is 93.1%,
indicating a reliable performance. However, some false positives occurred,
classifying neutral statements as biased based on political association. For
“political agenda” entities, the model achieved a precision of 86.0%, although
it misclassified a few non-biased mentions as biased. These results highlight
the model’s ability to detect explicit political biases but also suggest room
for improvement in handling ambiguous language.
Occupational Bias: In the “Occupational Bias” category, the Nbias exhibits
strong precision for identifying “gender bias in hiring” entities (92.7%),
effectively capturing biased terms. However, it produced a few false
positives, misclassifying neutral statements as biased based on gender. For
“ethnicity bias in hiring” entities, the precision is 92.5%, indicating
accurate identification. Yet, a few false positives occurred, misclassifying
non-biased mentions as biased. The model achieved a precision of 84.2% for
“age bias in hiring” entities. However, some neutral statements were
misclassified as biased, revealing areas for enhancement. These findings
suggest that the model can effectively identify biased occupational entities,
but improvements are needed to reduce false positives.
Actionable Insights:
* 1.
The proposed NER model demonstrates robust precision in identifying biased
entities for all three categories with clear biases.
* 2.
Addressing false positives can enhance the model’s discrimination between
biased and non-biased entities. Fine-tuning the model to better understand
nuanced language can be beneficial.
* 3.
Augmenting the training data with diverse instances of subtle biased entities
can improve recall and help detect rare biased terms.
* 4.
Considering context-aware models, such as transformer-based models, might help
tackle challenges arising from sarcasm and subtle biases more effectively.
Overall, these results provide valuable insights into the strengths of Nbias
and areas for improvement in identifying biased entities across different
categories.
### 5.4 Ablation Study on the Nbias Model
To understand the importance of different components in the Nbias model, we
conducted an ablation study on the combined dataframe from all the data
sources. We systematically remove or replace elements/ components of the model
to observe their influence on bias detection performance. The study assessed
the following model variants:
* 1.
Nbias Full: Original model with all features intact.
* 2.
Nbias -NoAttn: Exclusion of the self-attention mechanism.
* 3.
Nbias-GloVe: GloVe embeddings replace the BERT defaults.
* 4.
Nbias -HalfBERT: A version with half the transformer layers.
* 5.
Nbias-RandInit: Trained without leveraging the pre-trained BERT weights.
Table 7 illustrates the outcomes of the ablation study for the F1-score,
precision, and recall metrics on the combined dataframe.
Table 7: Ablation Study Results for Nbias. Bold means best score Model Variant | Precision (%) | Recall (%) | F1-Score (%)
---|---|---|---
Nbias -Full | 94.8 | 95.6 | 95.2
Nbias -NoAttn | 89.5 | 91.0 | 90.1
Nbias -GloVe | 93.0 | 92.6 | 92.8
Nbias -HalfBERT | 93.7 | 93.3 | 93.5
Nbias -RandInit | 87.8 | 89.2 | 88.4
The analysis of the ablation study reveals some insightful observations. From
Table 7, it is evident that the fully featured Nbias-Full model outperforms
all other variants, with a highest F1-Score of 95.2%, highlighting the
combined effect of all its components working together. The significant
performance drop observed in the Nbias-NoAttn model, which does not
incorporate the self-attention mechanism. It shows the role that self-
attention plays in capturing contextual relationships in the text for
effective bias detection.
Additionally, the slight performance reduction in the Nbias-GloVe model, which
uses GloVe embeddings instead of the default BERT embeddings, suggests that
BERT embeddings are better suited for this specific task, possibly because
they are trained on a more diverse and comprehensive corpus. Similarly, the
negligible performance variation in the Nbias-HalfBERT model indicates that
the model can achieve almost equivalent performance with half the transformer
layers, which may be a crucial consideration in resource-constrained
environments. However, it is also worth noting that this minimal reduction
might lead to missing out on some complexities in the data that can only be
captured with a deeper network. Lastly, the reduced performance of the Nbias-
RandInit model, which does not leverage pre-trained BERT weights, highlights
the significant benefits of transfer learning and the importance of
initializing the model with pre-trained weights to achieve optimal
performance. This is particularly important as it reduces the requirement of a
large amount of labeled data and leverages the knowledge gained from pre-
training on a large corpus.
In conclusion, the Nbias model, with its full set of features, proves to be
the most effective model for bias detection.
### 5.5 Robustness Testing
Robustness testing is a type of evaluation used to assess the performance and
resilience of a system or model against various inputs or scenarios [56]. In
the context of our testing, we programmatically measure the robustness of
Nbias using three key factors: case sensitivity, semantics, contextand
spelling. In Table 8, we showcase the robustness testing on a sample of 5
examples (due to brevity reasons) from the test set.
The results of the robustness testing of the model in Table 8 are summarized
as:
* 1.
Spelling: The model partially passed the test. It was able to map “superrior”
to “superior” but failed with “un like”.
* 2.
Semantics: The model successfully mapped “females” to “women” and “exaggerate”
to “amplify”, thus passing the test.
* 3.
Case Sensitivity and Semantics: The model was case insensitive and identified
synonyms, mapping “ASIANS” to “Asians” and “more prone” to “higher
susceptibility”, passing the test.
* 4.
Semantics: The model passed another semantics test by mapping “females” to
“women” and “not fit” to “not suitable”.
* 5.
Context: The model partially passed the test by mapping “senior” to “older”
and “resist” to “resistant”, but it failed to capture the change in severity
from “severely lack” to “lack”.
Overall, the model showed strengths in identifying synonyms, being case
insensitive, and recognizing some contextual changes, but had limitations in
identifying misspellings and capturing changes in severity.
We also conducted the robustness testing on a test case of 300 samples
manually prepared by the our team of 5 members to evaluate the robustness of
our system and report the results in Figure 8.
Table 8: Robustness Test on Nbias Predictions: evaluating the model
performance in identifying misspellings, synonyms, case sensitivity,
semantics, and context. Assessment based on Pass: ✓, Partial: ⚫, Fail: ✗
Original: Conservative views are superior, unlike liberal ideas.
---
Test Case: Conservative views are superrior, un like liberal ideas.
Expected: Identifies misspellings.
Actual: Maps “superior” to “superrior”, fails with “un like”.
Test Type: Spelling.
Pass: ⚫
Original: Women tend to exaggerate pain due to their emotional nature.
Test Case: Females tend to amplify pain due to their emotional nature.
Expected: Identifies synonyms.
Actual: Correctly maps “females” to “women” and “exaggerate” to “amplify”.
Test Type: Semantics.
Pass: ✓
Original: Asians are more prone to diabetes due to lifestyles.
Test Case: ASIANS have a higher susceptibility to diabetes, due to lifestyle.
Expected: Case insensitive and identifies synonyms.
Actual: Maps “ASIANS” to “Asians” and “more prone” to “higher susceptibility”.
Test Type: Case Sensitivity, Semantics.
Pass: ✓
Original: Women are not fit for leadership roles.
Test Case: Females are not suitable for leadership roles.
Expected: Identifies synonyms.
Actual: Maps “females” to “women” and “not fit” to “not suitable”.
Test Type: Semantics.
Pass: ✓
Original: Senior employees resist change and severely lack technological
skills.
Test Case: Older employees are resistant to change and lack technological
skills.
Expected: Recognizes context and severity change.
Actual: Maps “senior” to “older” , “resist” to “resistant”, fails to capture
change in severity.
Test Type: Context.
Pass: ⚫
(a) Case Sensitivity Robustness Test.
(b) Contexts Robustness Test.
Figure 7: Robustness test (continued on next page)
(a) Semantics Robustness Test.
(b) Spellings Robustness Test.
Figure 8: Robustness Test: each plot illustrates the performance of the Nbias
model across 5 development runs in robustness tests: spelling, semantics, case
sensitivity, and context. The x-axis represents the different test instances
used in each run, while the y-axis displays the corresponding scores, referred
to as the ’Pass Count’ achieved by the model on these tests.
As observed in Figure 8, the Nbias model appears to be improving over time in
all four robustness test categories: spelling, semantics, case sensitivity,
and context. This is evident as the scores increase with each successive run,
demonstrating the model’s adaptability and improvement in its learning
approach.
In spelling, the model begins with a score of 70 and ends at 90 in the fifth
run. A similar upward trend is seen in semantics, starting from a score of 72
and concluding at 93 in the final run. The model also consistently improves in
the case sensitivity test, beginning at 80 and finishing at 95. The context
scores also progress positively, from initial score of 70 to a final score of
90.
The Nbias model shows the highest performance in case sensitivity, as it
reaches a score of 95 in the final run. It also performs well in the semantics
category, achieving a score of 93. However, the model’s performance in the
context and spelling categories is slightly lower. While these are still
strong results, there may be room for further optimization in these areas to
achieve results comparable to the case sensitivity and semantics tests.
### 5.6 Perpetuation Bias Tests for Bias Detection
To assess whether our model unintentionally perpetuates biases present in its
training data, we conducted perpetuation bias tests. These tests evaluated the
performance of our model in identifying and labeling potentially biased words
or phrases as BIAS entity.
In our testing approach, we curated a diverse list of terms and phrases
representing various social groups and contexts prone to bias. This list
included phrases like “elderly person”, “young woman”, “African immigrant”,
“gay man” and “blue-collar worker”. We inserted these phrases into neutral
sentences to evaluate the model’s perception of potential bias. Upon
processing the sentences through our model, we observed the following pattern:
The person was described as a [Phrase]
* —
Ethnicity:
* (a)
African immigrant (Flagged: 25 out of 30 times, 83%)
* (b)
Asian immigrant (Flagged: 20 out of 30 times, 67%)
* (c)
European immigrant (Flagged: 10 out of 30 times, 33%)
* —
Gender:
* (a)
young woman (Flagged: 10 out of 30 times, 33%)
* (b)
young man (Flagged: 5 out of 30 times, 17%)
* (c)
elderly man (Flagged: 5 out of 30 times, 17%)
* —
Occupation:
* (a)
blue-collar worker (Flagged: 15 out of 30 times, 50%)
* (b)
white-collar worker (Flagged: 8 out of 30 times, 27%)
* —
Age:
* (a)
elderly person (Flagged: 5 out of 30 times, 17%)
* (b)
young adult (Flagged: 3 out of 30 times, 10%)
The provided data showcases the results of a bias detection test on a language
model. Various phrases associated with different demographics (ethnicity,
gender, occupation, and age) were inserted into a neutral sentence, and the
model flagged certain phrases as “BIAS ENTITY” with varying frequencies.
Specifically, the phrases “African immigrant” and “Asian immigrant” were
flagged 83% and 67% of the time, respectively, whereas “European immigrant”
was only flagged 33% of the time. Similarly, “blue-collar worker” was flagged
50% of the time, while “white-collar worker” was flagged 27% of the time. In
contrast, phrases related to age and gender, such as “elderly person”, “young
woman”, “young man”, and “elderly man”, were flagged much less frequently,
ranging from 10% to 33%.
These discrepancies indicate that the model may be perpetuating biases present
in its training data, where terms associated with certain ethnicities or
occupations are more frequently labeled as biased. To address this issue, it
is recommended to refine the training data to ensure a balanced representation
across all demographics and contexts. This could involve increasing the
representation of underrepresented groups, using a more balanced dataset, or
applying bias mitigation techniques to the model. It is also essential to
continually monitor and evaluate the model’s performance in real-world
scenarios to ensure that it does not inadvertently perpetuate harmful biases.
### 5.7 Human Evaluation
To validate the model’s bias detection, a human evaluation was initiated.
Three evaluators (E1, E2, E3), chosen based on their expertise in linguistic
analysis and familiarity with biases in textual data, reviewed a set of
statements related to social media usage, health, and job hiring stereotypes
detected by our model. Each statement was rated on a scale from 1 to 5, with 1
indicating no perceived bias and 5 indicating a high level of bias.
The evaluation results are summarized in Table 9, which provides the
statements, the corresponding identified entities, and the scores assigned by
each evaluator. The average score for each entity is also provided, giving an
overall indication of the perceived bias.
Table 9: Evaluation of Perceived Bias in Various Statements by Three Evaluators (E1, E2, E3).: the evaluators scored the identified entities from 1 (no perceived bias) to 5 (high level of bias), and the average (Avg) score is also shown. | Score
---|---
Text | Identified Entity | E1 | E2 | E3 | Avg
Everyone using social media is just seeking attention. | seeking attention | 4 | 4 | 3 | 3.67
People who don’t exercise are lazy. | lazy | 5 | 5 | 5 | 5.0
This job role is more suitable for men. | more suitable for men | 5 | 5 | 5 | 5.0
She posts too much on social media, she must be narcissistic. | narcissistic | 4 | 3 | 4 | 3.67
Overweight people simply lack self-control. | Overweight, lack self-control | 5 | 5 | 5 | 5.0
Only people with an engineering background are best fit in this role. | best fit | 3 | 2.5 | 3 | 2.83
The results in Table 9 show that the evaluators reached a consensus on the
presence of bias in the statements, as evident from the high scores assigned
to certain identified entities. Entities such as ‘seeking attention,’ ‘lazy,’
‘more suitable for men,’ ‘narcissistic,’ and ‘overweight people, lack self-
control’ received average scores exceeding 3, indicating a significant
presence of bias in these statements. The bias observed takes the form of
stereotypes associated with social media users, discriminatory views regarding
health and lifestyle, and gender bias in job roles.
However, the last statement, which suggests that ‘only people with an
engineering background are the best fit for a role,’ received a lower bias
score compared to the others. The identified entity in this statement obtained
an average score of 2.83. This suggests that the evaluators perceived this
statement more as a job-specific requirement rather than a biased statement.
## 6 Discussion
### 6.1 Performance Analysis
The detection and identification of biases in textual data have significant
implications for ensuring fairness and ethical usage of information. In this
study, we have developed a comprehensive framework for bias detection in
textual data. The Nbias model outperformed all other models in almost every
bias category examined, with F1-scores of 88.4%, 90.6%, and 91.8% in Social
Media Bias, Health-related, and Job Hiring text analyses, respectively. The
model exhibited a strong capability in diverse token classification tasks, as
evidenced by high AUC values of 0.74, 0.90, and 0.91 across the respective
domains. The model’s high accuracy scores further shows its efficacy.
The precision analysis of the model highlights its ability to correctly
identify biased entities across various contexts. However, there remains scope
for reducing false positives. Nbias robustness was demonstrated through its
steady performance in multiple tests including spelling, semantics, case
sensitivity, and context considerations. Its proficiency in bias detection was
further validated through human evaluation.
### 6.2 Theoretical Impact
The Nbias framework offers a novel approach on text-based bias detection. Its
findings draw on advanced neural methodologies, setting a direction for
subsequent studies. The framework emphasizes the intricacies of bias in
textual content. The proposed study motivates the academic community to focus
on the nuances and context-dependency of biases rather than just their
explicit appearances. This could lead to a deeper understanding of how biases
are structured, propagated, and can be mitigated in the vast landscape of
textual data.
### 6.3 Practical Impact
Nbias’s practical use is vast and diverse. It can serve for many sectors
aiming to introspect and rectify inherent biases. Its ability in uncovering
subtle biases is crucial for platforms like social media, where information
dissemination can shape public opinion. Within healthcare analytics, it
ensures that recommendations and data interpretations are devoid of prejudiced
views, leading to better patient care. In recruitment, Nbias can be used for
equitable hiring, ensuring job descriptions and applicant reviews remain
unbiased. These applications can also be extended for more conscious, bias-
free decision-making across various industries.
### 6.4 Limitations
While our work represents a significant step forward in identifying biases in
text-based data, aiming to contribute to a more inclusive and unbiased
information landscape, it has some limitations.
Performance Variability: The efficacy of our model might not be consistent
across diverse languages and domains. Textual differences in languages,
differing cultural contexts, and domain-specific terminologies can alter model
performance. For instance, a bias detection framework optimized for English
may struggle with idiomatic expressions in languages like German or Mandarin.
Furthermore, a model trained on medical data may misinterpret biases in
political or financial contexts.
Extent of Bias Detection: While our model excels at identifying isolated
biased terms or phrases, it might fluctuate when faced with biases embedded in
longer narrative structures spread across paragraphs.
Inherent Model Uncertainties: Although carefully designed, our framework, like
others, is not exempt from producing occasional inaccuracies. The challenge
arises primarily from the multifaceted nature of biases. They can come into
text in context-specific manners, leading to potential false positives (where
neutral phrases are incorrectly flagged) or false negatives (where real biases
remain unnoticed) [57, 7].
Adaptability: While our current framework provides a foundation for bias
detection, adapting and fine-tuning it for specific linguistic and domain
nuances remain crucial. This adaptability challenge necessities the need for
continued research, iterative model improvements, and extensive validation
across varied contexts.
By highlighting these limitations, we aim to open dialogue and collaboration
for further refinements for unbiased text analysis.
### 6.5 Future Directions
Recognizing the potential of Nbias and considering the highlighted
limitations, we recommend several directions for future research to enhance
bias detection capabilities in textual data:
Incorporating Multilingual Support: Bias is not confined to any particular
language. Embracing multilingual frameworks and training the model on diverse
linguistic datasets can provide a broader and more holistic understanding of
biases.
Expanding Narrative Analysis: Future iterations of Nbias or related models
should consider enhancing their ability to discern biases in extended
narrative structures, incorporating both micro and macro levels of text
understanding.
Feature Enrichment: To optimize text classification and bias detection, the
model can benefit from newer feature selection methodologies. Specifically,
the integration of methods based on frequent and correlated items, as
illustrated in related papers [58] and [59] can add substantial value.
Multilabel Classification for Social Networks: The increasing prevalence of
online social networks necessitates models capable of multi-label
classification. Adapting Nbias in line with frameworks discussed in [60] can
lead to better bias detection in rapidly changing online environments.
Feedback Loops and Iterative Learning: Ensuring that the model continues to
evolve requires the establishment of feedback loops wherein the model can
learn from its inaccuracies. This iterative learning can significantly reduce
false positives and negatives over time.
Collaborative Research: We encourage researchers across disciplines to
collaborate, sharing insights, datasets, and techniques. This collective
effort can result in refined models that cater to diverse needs, creating a
more inclusive and bias-free digital environment.
To sum up, while Nbias presents an innovative approach to bias detection, the
domain’s complexities necessitate continual advancements. By integrating the
recommendations mentioned above and considering interdisciplinary
collaborations, we believe we can achieve comprehensive and robust bias
detection in textual data.
## 7 Conclusion
This paper presents a comprehensive framework for the detection and
identification of biases in textual data. The framework consists of various
components, including data pre-processing, bias annotation, NLP modeling, and
evaluation layers. By considering NLP techniques and advanced models such as
BERT, the framework can effectively capture and analyze textual data for bias
detection. The framework has shown promising results in identifying and
tagging biased terms and phrases across different domains.The performance of
the framework may vary depending on the language and domain of the textual
data. Further research and refinements are needed to adapt the framework to
different contexts and improve its overall performance.
CRediT authorship contribution statement
Shaina Raza: Conceptualization, Investigation, Formal analysis, Methodology,
Project administration, Software, Validation, Visualization, Writing –
original draft, Writing – review & editing. Muskan Garg: Investigation. Formal
analysis, Validation, Writing – review & editing. Deepak John Reji :
Methodology, Writing – review& editing. Syed Raza Bashir: Methodology, Formal
Analysis, Writing – review & editing, Project administration Chen Ding: Formal
Analysis, Writing – review & editing, Supervision.
Declaration of competing interest
The authors declare that they have no known competing financial interests or
personal relationships that could have appeared to influence the work reported
in this paper.
Data availability
Data will be made available on request.
Acknowledgments
Resources used in preparing this research were provided, in part, by the
Province of Ontario, the Government of Canada through CIFAR, and companies
sponsoring the Vector Institute.
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# GYOTO 2.0: a polarized relativistic ray-tracing code
N Aimar1111The three first authors in the list contributed equally to the
article., T Paumard1††footnotemark: , F H Vincent1††footnotemark: , E
Gourgoulhon2,3, and G Perrin1 1 LESIA, Observatoire de Paris, CNRS, Université
Pierre et Marie Curie, Université Paris Diderot, 5 place Jules Janssen, 92190
Meudon, France
2 Laboratoire Univers et Théories, Observatoire de Paris, CNRS, Université
PSL, Université Paris Cité, 5 place Jules Janssen, 92190 Meudon, France
3 Laboratoire de Mathématiques de Bretagne Atlantique, CNRS, Université de
Bretagne Occidentale, 6 avenue Le Gorgeu, 29238 Brest, France
<EMAIL_ADDRESS>
###### Abstract
Polarized general-relativistic radiative transfer in the vicinity of black
holes and other compact objects has become a crucial tool for probing the
properties of relativistic astrophysics plasmas. Instruments like GRAVITY, the
Event Horizon telescope, ALMA, or IXPE make it very timely to develop such
numerical frameworks. In this article, we present the polarized extension of
the public ray-tracing code Gyoto, and offer a python notebook allowing to
easily perform a first realistic computation. The code is very modular and
allows to conveniently add extensions for the specific needs of the user. It
is agnostic about the spacetime and can be used for arbitrary compact objects.
We demonstrate the validity of the code by providing tests, and show in
particular a perfect agreement with the ipole code. Our article also aims at
pedagogically introducing all the relevant formalism in a self-contained
manner.
## 1 Introduction
Generating synthetic images of black hole environments is tricky because of
relativistic effects such as aberration, Doppler beaming, gravitational
redshift and light bending. General-Relativistic Radiative Transfer (GRRT)
computation through ray-tracing method, i.e. integration of photons
trajectories (null geodesics, assuming no impact other than gravitation on the
trajectory), allows to naturally account for all relativistic effects to
generate synthetic observables of such environments that can be compared to
observational data. This technique is key to obtain meaningful constraints on
the parameters describing the emitting accretion flow or the spacetime
geometry. It can be used either on analytically described accretion flows, or
in post-processing of general relativistic magnetohydrodynamical (GRMHD)
simulations. For example, the Numerical Observatory of Violent Accreting
system (NOVAs; Varniere et al., 2018; Mignon-Risse et al., 2021) combines
GRMHD simulations of accretion flows around black holes as computed by the
code GR-AMRVAC (Casse et al., 2017), with the ray-tracing code Gyoto (Vincent
et al., 2011) to produce various synthetic observables.
The recent development of instruments now allows to measure the polarization
of light coming from the extremely close environment of black holes. The Event
Horizon Telescope (EHT) released the millimetric polarized image of M87*
(Event Horizon Telescope Collaboration et al., 2022). GRAVITY (GRAVITY
Collaboration et al., 2018, 2023) in the near infrared, and the Atacama Large
Millimeter/submillimeter Array (ALMA, Wielgus et al., 2022) have detected a
polarized signature of the radiation flares associated with the supermassive
black hole at the center of the Galaxy, Sagittarius A* (Sgr A*). Moreover, the
Imaging X-ray Polarimetry Explorer (IXPE) has obtained important constraints
on the geometry of the accretion flow of an X-ray binary by measuring its
polarized radiation (Krawczynski et al., 2022).
Thus, polarized ray-tracing codes are of particular interest to generate
synthetic observables from accretion models, be they analytic (see e.g.
Broderick et al., 2016; Gralla et al., 2018, 2019; Vincent et al., 2019;
Gralla et al., 2020; GRAVITY Collaboration et al., 2020; Nalewajko et al.,
2020; Dovčiak et al., 2022; Vos et al., 2022; Aimar et al., 2023; Cárdenas-
Avendaño and Lupsasca, 2023), or numeric (see e.g. Chan et al., 2015;
Mościbrodzka et al., 2016; Chael et al., 2018; Davelaar et al., 2018; Chael et
al., 2019; Event Horizon Telescope Collaboration et al., 2019; Anantua et al.,
2020; Dexter et al., 2020; Porth et al., 2021). In particular, polarization
signatures might allow probing the nature of spacetime close to the event
horizon of black hole candidates, making GRRT a crucial tool for constraining
general relativity (GR) in the strong-field regime (Himwich et al., 2020;
Jiménez-Rosales et al., 2021; Vincent et al., 2023).
The need for ray tracing when computing images in GR led to the development of
multiple codes. Many of them were developed for unpolarized light (Noble et
al., 2007; Dexter and Agol, 2009; Dauser et al., 2010; Vincent et al., 2011;
Pu et al., 2016; Chan et al., 2018; Bronzwaer et al., 2018; Younsi et al.,
2020). Some codes also keep track of the electric vector position angle and
polarization degree, assuming a geometrically thin equatorial accretion flow
(Dovčiak et al., 2008; Gelles et al., 2021; Cárdenas-Avendaño et al., 2023),
the latter two being specialized for highly-lensed features by implementing
adaptive ray tracing. Only a handful of codes are able of treating the most
demanding problem of integrating the full polarized radiative transfer:
grtrans (Dexter, 2016), ipole (Mościbrodzka and Gammie, 2018a), Arcmancer
(Pihajoki et al., 2018), Bhoss (Younsi et al., 2020), Raptor (Bronzwaer et
al., 2020), Blacklight (White, 2022), as well as Lemon (Xiao-lin et al., 2021)
which specializes on polarized radiative transfer with scattering. Some of
these polarized GRRT codes were recently compared by Prather et al. (2023).
Gyoto (Vincent et al., 2011) is a backwards ray-tracing code (i.e. integrating
from the observer to the source), operating in any given (analytically or
numerically computed) metric, and solving the radiative transfer equation. It
can also integrate timelike geodesics for computing e.g. stellar orbits
(Grould et al., 2017). The code is publicly available
222https://github.com/gyoto/Gyoto/blob/master/INSTALL.Gyoto.md , built to be
modular so that extensions are easy to integrate and user-friendly with XML
and python interfaces.
The goals of this paper are the following: (i) providing the new version of
Gyoto, with full polarized GRRT included, publicly available at the same
address as the older version, with the addition of a python snippet
333https://github.com/gyoto/Gyoto/blob/master/doc/examples/Gyoto_Polar_example.ipynb
to allow the interested reader to immediately be able to compute a non-trivial
setup; (ii) providing a pedagogical, in-depth presentation of the formalism of
GR polarization, as well as a detailed description of the technical
implementation. In the following discussion, we will focus on polarized
synchrotron radiation, as it is the dominant emission mechanism for our
sources of primary scientific interest (Sgr A* and M87). But the code is able
to compute polarized observables for other emission mechanism as soon as the
electric field is provided in the model. Section 2 presents the formalism of
GR polarized radiative transfer. In section 3, we present various tests that
we made to validate our code. The last section is dedicated to discussion and
conclusion.
## 2 Formalism
We will discuss the problem of polarized GRRT taking the usual point of view
of ray tracing. We thus consider a light ray (mathematically speaking, a null
geodesic) integrated backwards from a distant observer’s screen towards some
source of radiation. The problem can then be divided into three main parts
that will be discussed hereafter:
* •
the definition of a wave vector at the distant observer’s screen, tangent to
the considered null geodesic, together with a pair of spacelike vectors
forming an orthonormal basis of the observer’s screen;
* •
the backwards parallel propagation along the considered null geodesic of the
wave vector together with the screen basis, until a source of radiation is
reached;
* •
the integration of polarized radiative transfer within the source.
Before describing these three steps in detail, we will start by providing
important definitions in the next section.
### 2.1 Geometric optics, light ray, covariant and observer-specific
polarization vectors
We consider a monochromatic plane electromagnetic wave propagating in a given
spacetime. The geometrical optics approximation of Maxwell’s equations under
Lorenz gauge condition allows to describe this wave as follows.
The complex 4-potential 1-form reads
$\mathbf{\hat{A}}=\mathbf{\hat{a}}\,e^{i\Phi},$ (1)
where $\mathbf{\hat{a}}$ is the amplitude 1-form, assumed to vary much slower
than the phase $\Phi$ (this is the basic idea of the geometrical optics
approximation). The hat reminds that we are dealing with complex quantities.
The Faraday electromagnetic 2-form,
$\boldsymbol{\hat{\mathcal{F}}}=\mathbf{d\hat{A}},$ (2)
then reads in components
$\displaystyle\hat{\mathcal{F}}_{\alpha\beta}$ $\displaystyle=$
$\displaystyle\nabla_{\alpha}\hat{A}_{\beta}-\nabla_{\beta}\hat{A}_{\alpha}$
$\displaystyle=$ $\displaystyle
e^{i\Phi}\nabla_{\alpha}\hat{a}_{\beta}+\hat{a}_{\beta}\,i\,e^{i\Phi}\,\nabla_{\alpha}\Phi-\left(e^{i\Phi}\nabla_{\beta}\hat{a}_{\alpha}+\hat{a}_{\alpha}\,i\,e^{i\Phi}\,\nabla_{\beta}\Phi\right)$
$\displaystyle\approx$ $\displaystyle
i\left(\hat{a}_{\beta}\,k_{\alpha}-\hat{a}_{\alpha}\,k_{\beta}\right)\,e^{i\Phi},$
where we introduce the wave vector
$\mathbf{k}\equiv\boldsymbol{\nabla}{\Phi},$ (4)
and use the geometric optics approximation to neglect the variations of the
amplitude, i.e. the $\nabla_{\alpha}\hat{a}_{\beta}$ terms.
Plugging this into Maxwell’s equations and assuming Lorenz gauge (that is, the
divergence of $\mathbf{\hat{A}}$ should vanish) leads to the following
results:
* •
$\mathbf{k}$ is a null vector parallel propagated along itself,
$\mathbf{k}\cdot\mathbf{k}=0,\quad\boldsymbol{\nabla}_{\mathbf{k}}\,\mathbf{k}=\mathbf{0},$
(5)
so that it defines a null geodesic, which we define as a light ray;
* •
we can introduce the unit spacelike covariant polarization vector
$\mathbf{\hat{f}}\equiv\frac{\mathbf{\hat{a}}}{a},$ (6)
where $\mathbf{\hat{a}}$ is the complex vector corresponding to the amplitude
1-form introduced in Eq. 1 by metric duality (we use the same notation for
both quantities in order to simplify the notation). The scalar quantity $a$ is
the modulus of the complex vector $\mathbf{\hat{a}}$,
$a=\sqrt{\hat{a}_{\mu}\hat{a}^{\mu*}}.$ (7)
Our naming convention specifies that this vector is covariant in order to
differentiate it from the polarization vector as observed by a specific
observer, which will be our quantity of prime interest in the following. The
covariant polarization vector satisfies the two following important
properties: (i) it is perpendicular to the wave vector $\mathbf{k}$ (this is a
consequence of the Lorenz gauge choice), and (ii) it is parallel transported
along $\mathbf{k}$ in vacuum (this is a consequence of Maxwell’s equations):
$\mathbf{\hat{f}}\cdot\mathbf{k}=0,\quad\boldsymbol{\nabla}_{\mathbf{k}}\,\mathbf{\hat{f}}=\mathbf{0}.$
(8)
We can thus reexpress the Faraday tensor in terms of the polarization and wave
vectors as follows
${\hat{\mathcal{F}}_{\alpha\beta}=i\,a\,\left(\hat{f}_{\beta}\,k_{\alpha}-\hat{f}_{\alpha}\,k_{\beta}\right)\,e^{i\Phi}.}$
(9)
From this expression we deduce the important property that it is possible to
add any multiple of the wave vector $\mathbf{k}$ to the polarization vector
$\mathbf{\hat{f}}$ without altering the Faraday tensor. So the polarization
vector can be arbitrarily transformed under
$\mathbf{\hat{f}}\mapsto\mathbf{\hat{f}}+q\mathbf{k}$ (10)
for any scalar field $q$ (note that $q$ is not necessarily a constant, it is
an arbitrary scalar field).
So far, we have only used global quantities that are not defined with respect
to any particular observer. We now want to introduce the electric and magnetic
fields as observed by the distant observer, $\mathcal{O}$, the oscillations of
which define the observed electromagnetic wave. Let us denote by
$\mathbf{u_{0}}$ the 4-velocity of observer $\mathcal{O}$. By definition, the
electric linear form and the magnetic vector as measured by a generic observer
with 4-velocity $\mathbf{u}$ read 444We highlight that the electric and
magnetic fields discussed here are the electromagnetic fields describing the
monochromatic wave that reaches the observer’s screen. They should not be
confused with the electromagnetic fields that might exist at the source
location, for instance the magnetic field of the accretion flow surrounding
the black hole.
$\displaystyle\hat{E}_{\alpha}=\hat{\mathcal{F}}_{\alpha\mu}\,u^{\mu},$ (11)
$\displaystyle\hat{B}^{\alpha}=-\frac{1}{2}\epsilon^{\alpha\mu\nu}_{\>\>\>\>\>\>\>\rho}\,\hat{\mathcal{F}}_{\mu\nu}\,u^{\rho}$
where $\boldsymbol{\epsilon}$ is the Levi-Civita tensor. The electric field
vector $\mathbf{\hat{E}_{0}}$ as observed by the distant observer
$\mathcal{O}$ thus reads
$\displaystyle\hat{E}^{\rho}$ $\displaystyle=$ $\displaystyle
g^{\rho\alpha}\,\hat{E}_{\alpha}$ $\displaystyle=$ $\displaystyle
g^{\rho\alpha}\,\hat{\mathcal{F}}_{\alpha\beta}\,u^{\beta}$ $\displaystyle=$
$\displaystyle
i\,a\,e^{i\Phi}\,g^{\rho\alpha}\left(\hat{f}_{\beta}\,k_{\alpha}-\hat{f}_{\alpha}\,k_{\beta}\right)u^{\beta}$
$\displaystyle=$ $\displaystyle
i\,a\,e^{i\Phi}\,\left(\hat{f}_{\beta}\,k^{\rho}-\hat{f}^{\rho}\,k_{\beta}\right)u^{\beta}$
$\displaystyle=$ $\displaystyle
i\,a\,e^{i\Phi}\,\left((\mathbf{\hat{f}_{0}}\cdot\mathbf{u_{0}})\,k^{\rho}+\omega_{0}\hat{f}^{\rho}\right)$
where we drop the lower index $0$ for the components of the various tensors
for simplicity (all of them being evaluated at the distant observer’s
location), and we introduce
$\omega_{0}\equiv-\mathbf{k_{0}}\cdot\mathbf{u_{0}}$, where $\mathbf{k_{0}}$
is the wave vector at the distant observer’s location. This quantity
$\omega_{0}$ is the pulsation of the photon as measured by $\mathcal{O}$. All
vectors with a lower index $0$ are defined at the distant observer’s screen.
Let us decompose the vectors $\mathbf{\hat{f}_{0}}$ and $\mathbf{k_{0}}$ in
parts parallel and orthogonal to the observer’s 4-velocity:
$\displaystyle\mathbf{k_{0}}$ $\displaystyle=$
$\displaystyle\omega_{0}\,\mathbf{u_{0}}+\mathbf{K_{0}},\qquad\qquad\mathbf{K_{0}}\perp\mathbf{u_{0}},$
(13) $\displaystyle\mathbf{\hat{f}_{0}}$ $\displaystyle=$
$\displaystyle-(\mathbf{\hat{f}_{0}}\cdot\mathbf{u_{0}})\,\mathbf{u_{0}}+\mathbf{\hat{f}_{0}^{\perp}},\quad\mathbf{\hat{f}_{0}^{\perp}}\perp\mathbf{u_{0}}.$
Note that
$\mathbf{\hat{f}_{0}^{\perp}}\cdot\mathbf{\hat{f}_{0}^{\perp}}=1+\left(\mathbf{\hat{f}_{0}}\cdot\mathbf{u_{0}}\right)^{2}$
(14)
so that $\mathbf{\hat{f}_{0}^{\perp}}$ is not a unit vector in general. It is
normalized only if $\mathbf{\hat{f}_{0}}\cdot\mathbf{u_{0}}=0$, in which case
we simply have $\mathbf{\hat{f}_{0}^{\perp}}=\mathbf{\hat{f}_{0}}$. Similary,
$\mathbf{K_{0}}$ is not a unit vector and it is easy to show that
$\mathbf{K_{0}}\cdot\mathbf{K_{0}}=\omega_{0}^{2}.$ (15)
The vector $\mathbf{K_{0}}$ coincides with the incident wave vector as
measured by observer $\mathcal{O}$.
In terms of the vectors normal to $\mathbf{u_{0}}$ we immediately obtain the
final expression of the electric vector as observed by $\mathcal{O}$:
${\mathbf{\hat{E}_{0}}=i\,a\,e^{i\Phi}\left(\omega_{0}\,\mathbf{\hat{f}_{0}^{\perp}}-\frac{\mathbf{K_{0}}\cdot\mathbf{\hat{f}_{0}^{\perp}}}{\omega_{0}}\,\mathbf{K_{0}}\right).}$
(16)
This vector is clearly orthogonal to the direction of propagation
$\mathbf{K_{0}}$, and it is also orthogonal to the 4-velocity
$\mathbf{u_{0}}$, as it should for a vector living in the local rest space of
the observer.
Then, the magnetic field vector $\mathbf{\hat{B}_{0}}$ as observed by
$\mathcal{O}$ reads
$\displaystyle\hat{B}^{\rho}$ $\displaystyle=$
$\displaystyle-\frac{1}{2}\,\epsilon^{\rho\mu\nu}_{\>\>\>\>\>\>\alpha}\hat{\mathcal{F}}_{\mu\nu}u^{\alpha}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}i\,a\,e^{i\Phi}\,\epsilon^{\rho\mu\nu}_{\>\>\>\>\>\>\alpha}\left(\hat{f}_{\nu}k_{\mu}-\hat{f}_{\mu}k_{\nu}\right)u^{\alpha}$
$\displaystyle=$
$\displaystyle-i\,a\,e^{i\Phi}\,\epsilon^{\rho\mu\nu}_{\>\>\>\>\>\>\alpha}\hat{f}_{\nu}k_{\mu}u^{\alpha}$
$\displaystyle=$
$\displaystyle-i\,a\,e^{i\Phi}\,\epsilon^{\rho}_{\>\>\mu\nu\alpha}u^{\alpha}k^{\mu}\hat{f}^{\nu}$
$\displaystyle=$ $\displaystyle
i\,a\,e^{i\Phi}\,\epsilon^{\>\>\>\>\>\>\>\>\rho}_{\alpha\mu\nu}u^{\alpha}k^{\mu}\hat{f}^{\nu}$
$\displaystyle=$ $\displaystyle
i\,a\,e^{i\Phi}\,\epsilon^{\>\>\>\>\>\>\>\>\rho}_{\alpha\mu\nu}u^{\alpha}K^{\mu}\hat{f}^{\perp\nu},$
where we have used extensively the antisymmetric nature of the Levi-Civita
tensor. The last expression exactly coincides with the definition of the cross
product in the vector space orthogonal to $\mathbf{u_{0}}$ (which we label by
$\times_{\mathbf{u_{0}}}$), such that finally the magnetic field vector as
measured by $\mathcal{O}$ reads
${\mathbf{\hat{B}_{0}}=i\,a\,e^{i\Phi}\,\mathbf{K_{0}}\times_{\mathbf{u_{0}}}\mathbf{\hat{f}_{0}^{\perp}}.}$
(18)
This vector is also obviously orthogonal to the direction of propagation
$\mathbf{K_{0}}$, and to the electric vector.
We can now define the notion of polarization vector as measured by observer
$\mathcal{O}$:
$\mathbf{\hat{F}_{0}}=\mathbf{K_{0}}\times_{\mathbf{u_{0}}}\mathbf{\hat{B}_{0}},$
(19)
which is by construction normal to the direction of propagation and to the
magnetic field vector. We note that this quantity depends on the observer,
just as the electric and magnetic fields, while the covariant polarization
vector $\mathbf{\hat{f}_{0}}$, defined in Eq. 6, is a covariant quantity. They
obviously differ given that by construction $\mathbf{\hat{F}_{0}}$ is
orthogonal to the observer’s 4-velocity $\mathbf{u_{0}}$, while
$\mathbf{\hat{f}_{0}}$ is defined independently from $\mathbf{u_{0}}$. A
natural question is to investigate the relation between $\mathbf{\hat{F}_{0}}$
and $\mathbf{\hat{f}_{0}^{\perp}}$ that both live in the vector space
orthogonal to $\mathbf{u_{0}}$. These two vectors are completely independent a
priori, because $\mathbf{\hat{F}_{0}}$ is by construction orthogonal to both
$\mathbf{K_{0}}$ and $\mathbf{\hat{B}_{0}}$ (see Eq. 19), while
$\mathbf{\hat{f}_{0}^{\perp}}$ is only orthogonal to $\mathbf{\hat{B}_{0}}$
(see Eq. 18), but not to $\mathbf{K_{0}}$. Indeed, from Eq. 8 and 13, we have
$\displaystyle\mathbf{\hat{f}_{0}}\cdot\mathbf{k_{0}}=0$ (20)
$\displaystyle\Leftrightarrow$
$\displaystyle\left(\mathbf{\hat{f}_{0}}\cdot\mathbf{u_{0}}\right)\,\omega_{0}+\mathbf{\hat{f}_{0}^{\perp}}\cdot\mathbf{K_{0}}=0$
so that only if $\mathbf{\hat{f}_{0}}\cdot\mathbf{u_{0}}=0$ (which has no
reason to hold in general) is $\mathbf{\hat{f}_{0}^{\perp}}$ orthogonal to
$\mathbf{K_{0}}$. In this special case, we saw in Eq. 13 that
$\mathbf{\hat{f}_{0}^{\perp}}=\mathbf{\hat{f}_{0}}$, so that when
$\mathbf{\hat{f}_{0}}\cdot\mathbf{u_{0}}=0$, and only then,
$\mathbf{\hat{f}_{0}}$ is the unit vector along $\mathbf{\hat{F}_{0}}$.
However, we have seen in Eq. 10 that the covariant polarization vector is
defined up to a term proportional to the wavevector, the proportionality
coefficient being a scalar field. We can thus choose to work with a covariant
polarization vector $\mathbf{f^{\prime}}$ such that, at the distant observer’s
location
$\mathbf{\hat{f}^{\prime}_{0}}=\mathbf{\hat{f}_{0}}+\frac{\mathbf{\hat{f}_{0}}\cdot\mathbf{u_{0}}}{\omega_{0}}\,\mathbf{k_{0}}.$
(21)
This vector is such that
$\mathbf{\hat{f}^{\prime}_{0}}\cdot\mathbf{u_{0}}=0,$ (22)
and we saw just above that this implies that $\mathbf{\hat{f}^{\prime}_{0}}$
is then the unit vector along $\mathbf{\hat{F}_{0}}$. Thanks to the degree of
liberty in the definition of the covariant polarization vector expressed by
Eq. 10, we can thus confuse the covariant and non-covariant polarization
vectors at the observer, $\mathbf{\hat{f}_{0}}$ and $\mathbf{\hat{F}_{0}}$.
By virtue of the double vector product law we have
$\displaystyle(\mathbf{K_{0}}\times_{\mathbf{u_{0}}}\mathbf{\hat{B}_{0}})^{\alpha}$
$\displaystyle=$ $\displaystyle
i\,a\,e^{i\phi}\left[(\mathbf{K_{0}}\cdot\mathbf{\hat{f}_{0}^{\perp}})K^{\alpha}-(\mathbf{K_{0}}\cdot\mathbf{K_{0}})\hat{f}^{\perp\alpha}\right]$
$\displaystyle=$ $\displaystyle
i\,a\,e^{i\phi}\,\omega_{0}\,\left[\frac{\mathbf{K_{0}}\cdot\mathbf{\hat{f}_{0}^{\perp}}}{\omega_{0}}K^{\alpha}-\omega_{0}\hat{f}^{\perp\alpha}\right]$
so that finally
${\mathbf{\hat{F}_{0}}=(\mathbf{K_{0}}\times_{\mathbf{u_{0}}}\mathbf{\hat{B}_{0}})=-\omega_{0}\,\mathbf{\hat{E}_{0}}}.$
(24)
We thus conclude that the polarization vector as measured by $\mathcal{O}$
coincides, up to a normalization factor, with the electric field as measured
by $\mathcal{O}$. The various vectors lying in observer $\mathcal{O}$’s local
rest space (that is, the vector space orthogonal to the 4-velocity
$\mathbf{u_{0}}$) are depicted in Fig. 1. We introduce in this figure the
electric vector position angle (EVPA), defined in the local frame of the
distant observer, which is the angle between a reference direction (the local
North of the distant observer) and the polarization vector
$\mathbf{\hat{F}_{0}}$.
Figure 1: Vectors lying in the distant observer $\mathcal{O}$’s local rest
space (the vector space orthogonal to $\mathcal{O}$’s 4-velocity
$\mathbf{u_{0}}$). $\mathbf{K_{0}}$ is the wave vector projected orthogonal to
$\mathbf{u_{0}}$. The vectors $\mathbf{\bar{w}_{0}}$ and
$\mathbf{\bar{n}_{0}}$ are unit vectors pointing towards the local West and
North directions, so that
$(\mathbf{K_{0}},\mathbf{\bar{w}_{0}},\mathbf{\bar{n}_{0}})$ forms a direct
orthogonal triad. $\mathbf{\hat{E}_{0}}$ and $\mathbf{\hat{B}_{0}}$ are the
electric and magnetic vectors as measured by $\mathcal{O}$, associated with
the incident light wave, so that
$(\mathbf{K_{0}},\mathbf{\hat{E}_{0}},\mathbf{\hat{B}_{0}})$ is a direct
orthogonal triad. $\mathbf{\hat{F}_{0}}$ is the polarization vector as
measured by $\mathcal{O}$ (defined in Eq. 19), which is antiparallel to
$\mathbf{\hat{E}_{0}}$ as stated by Eq. 24. $\mathbf{\hat{f}_{0}^{\perp}}$ is
the covariant polarization vector (defined in Eq. 6) projected orthogonal to
$\mathbf{u_{0}}$. The plane orthogonal to $\mathbf{K_{0}}$ (observer’s screen
plane) is drawn in green. It contains the electric and magnetic field vectors,
and the polarization vector as measured by $\mathcal{O}$. The plane orthogonal
to $\mathbf{\hat{B}_{0}}$ is drawn in blue-green. It contains the photon’s
wave vector $\mathbf{K_{0}}$, the electric vector $\mathbf{\hat{E}_{0}}$, and
both the covariant and $\mathcal{O}$’s specific polarization vectors. The
observed Electric Vector Position Angle (EVPA0), measured East of North in the
screen’s plane, is shown in dark red.
### 2.2 Polarization basis defined at the distant observer’s screen
We take here the typical point of view of a ray-tracing problem where the
initial conditions are fixed at the far-away observer’s screen, and the
integration is performed backwards in time from the screen towards the source.
This allows to save a lot of computing time by integrating only those
geodesics that will approach the source by shooting light rays only within a
small solid angle subtending the source. The aim of this subsection is to
explicitly describe our initial conditions at the observer’s screen, which is
illustrated in Fig. 2. In order to be specific, we will consider a black hole
spacetime, but the discussion is very general and is not restricted to this
particular case.
Figure 2: Initial condition of the polarized ray-tracing problem at the
distant observer’s screen. Left: A black hole (BH) spacetime is represented
for being specific, but the figure is very general and applies to any kind of
spacetime. We consider spherical coordinates and the spacelike orthonormal
basis associated with these coordinates are labeled
$(\mathbf{\bar{e}_{r}},\mathbf{\bar{e}_{\theta}},\mathbf{\bar{e}_{\varphi}})$.
The observer’s local rest space is described by a direct orthonormal triad,
$(\mathbf{\bar{e}_{1}},\mathbf{\bar{e}_{2}},\mathbf{\bar{e}_{3}})$, where
$\mathbf{\bar{e}_{3}}=-\mathbf{\bar{e}_{r}}$, and where the screen’s plane is
contained in $\mathrm{Span}(\mathbf{\bar{e}_{1}},\mathbf{\bar{e}_{2}})$. We
consider here that the “upwards” direction of the observer’s screen coincides
with the projection of the black hole’s angular momentum on the screen, i.e.
that $\mathbf{\bar{e}_{2}}=-\mathbf{\bar{e}_{\theta}}$. The unit direction of
photon reception at the observer’s screen is $\mathbf{\bar{K}_{0}}$. The local
polarization basis at the observer’s screen,
$(\mathbf{\bar{K}_{0}},\mathbf{\bar{w}_{0}},\mathbf{\bar{n}_{0}})$, is shown,
and corresponds to the central pixel of the screen, that is, to the purely
radial incoming direction of the photon. Right: zoom on the local rest space
$(\mathbf{\bar{e}_{1}},\mathbf{\bar{e}_{2}},\mathbf{\bar{e}_{3}})$ and local
celestial sphere of the observer. For a source of radiation located at S on
the local celestial sphere, the unit direction of incidence is
$\mathbf{\bar{K}_{0}}$ (it is $-\mathbf{\bar{K}_{0}}$ on the figure because
$\mathbf{\bar{e}_{3}}$ points towards the source, while the incidence
direction is of course in the opposite direction). The vector
$\mathbf{\bar{K}_{0}}$ is here not purely along the radial direction
$\mathbf{\bar{e}_{3}}$ (in contrast with the left panel): it thus corresponds
to a pixel that is not located at the center of the screen. The corresponding
equatorial angles labeling the source, $(\alpha,\delta)$, are shown, together
with the corresponding spherical angles on the observer local sky, $(a,b)$.
For typical ray-tracing problems where the observer is far away, we have $a\ll
1$.
The observer’s screen is considered to be a pin-hole camera, with the various
pixels corresponding to different directions on sky. The local rest space of
the observer is spanned by a direct orthonormal triad
$(\mathbf{\bar{e}_{1}},\mathbf{\bar{e}_{2}},\mathbf{\bar{e}_{3}})$. Here and
in the following, a bar on top of a vector denotes a spacelike unit vector.
The vector $\mathbf{\bar{e}_{3}}$ is along the line of sight, normal to the
screen, towards the black hole. If we consider spherical coordinates centered
on the black hole (e.g. Boyer-Lindquist coordinates) and a direct orthonormal
triad
$(\mathbf{\bar{e}_{r}},\mathbf{\bar{e}_{\theta}},\mathbf{\bar{e}_{\varphi}})$
associated with these coordinates, then
$\mathbf{\bar{e}_{3}}=-\mathbf{\bar{e}_{r}}$. The screen’s plane is spanned by
$(\mathbf{\bar{e}_{1}},\mathbf{\bar{e}_{2}})$, and we consider the special
case $\mathbf{\bar{e}_{2}}=-\mathbf{\bar{e}_{\theta}}$, which boils down to
assuming that the projection of the black hole’s angular momentum on the
screen is along $\mathbf{\bar{e}_{2}}$. For a $N\times N$ pixels screen, one
pixel with indices $(i,j)$, with $i,j=1..N$, corresponds to a pair of
equatorial angles (see Fig. 2)
$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\frac{f}{N}\left(i-\frac{N+1}{2}\right),$ (25)
$\displaystyle\delta$ $\displaystyle=$
$\displaystyle\frac{f}{N}\left(j-\frac{N+1}{2}\right),$
where $f$ is the field of view of the observer. The corresponding spherical
angles (see Fig. 2) on the local sky of the observer are given by standard
spherical trigonometry relations:
$\displaystyle\cos a$ $\displaystyle=$ $\displaystyle\cos\alpha\,\cos\delta,$
(26) $\displaystyle\tan b$ $\displaystyle=$
$\displaystyle\frac{\tan\alpha}{\sin\delta}.$
The local unit direction of photon incidence then reads
$\displaystyle\mathbf{\bar{K}_{0}}$ $\displaystyle=$ $\displaystyle-\sin a\cos
b\,\boldsymbol{\bar{e}_{1}}-\sin a\sin b\,\boldsymbol{\bar{e}_{2}}-\cos
a\,\boldsymbol{\bar{e}_{3}}$ $\displaystyle=$ $\displaystyle\frac{\cos
a}{\sqrt{g_{rr}}}\,\boldsymbol{\partial_{r}}+\frac{\sin a\sin
b}{\sqrt{g_{\theta\theta}}}\,\boldsymbol{\partial_{\theta}}+\frac{\sin a\cos
b}{\sqrt{g_{\varphi\varphi}}}\,\boldsymbol{\partial_{\varphi}},$
where we have used the relations
$\mathbf{\bar{e}_{r}}=\frac{\boldsymbol{\partial_{r}}}{\sqrt{g_{rr}}},\quad\mathbf{\bar{e}_{\theta}}=\frac{\boldsymbol{\partial_{\theta}}}{\sqrt{g_{\theta\theta}}},\quad\mathbf{\bar{e}_{\varphi}}=\frac{\boldsymbol{\partial_{\varphi}}}{\sqrt{g_{\varphi\varphi}}},$
(28)
where $\boldsymbol{\partial_{i}}$ are spherical coordinate basis vectors, and
$g_{ii}=\boldsymbol{\partial_{i}}\cdot\boldsymbol{\partial_{i}}$ are the
corresponding metric coefficients. The null 4-vector tangent to the null
geodesic when incident on the observer’s screen then reads
$\mathbf{k_{0}}=\boldsymbol{\partial_{t}}+\mathbf{\bar{K}_{0}}$ (29)
where we consider that the observer’s 4-velocity is
$\mathbf{u_{0}}=\boldsymbol{\partial_{t}}$ (assuming a static observer, and
that $g_{tt}\to-1$ at the observer’s location) and where we have assumed that
$-\mathbf{k_{0}}\cdot\mathbf{u_{0}}=1$, so that the spacelike vector
$\mathbf{\bar{K}_{0}}$ in the last equation is normalized. This last
assumption means that the photon’s energy as measured by the far-away observer
is unity, which does not change anything to the problem, it simply scales the
energy, and physical values of energies can be easily retrieved when radiative
transfer calculations are performed.
Once the photon arrival direction $\mathbf{\bar{K}_{0}}$ has been defined, it
must be completed by two other vectors to form the local orthonormal
polarization basis
$(\mathbf{\bar{K}_{0}},\mathbf{\bar{w}_{0}},\mathbf{\bar{n}_{0}})$. The vector
$\mathbf{\bar{K}_{0}}$ corresponding to the central pixel of the screen
coincides with a purely radial direction of arrival,
$\mathbf{\bar{K}_{0}^{\mathrm{cen}}}=-\mathbf{\bar{e}_{3}}$ (the superscript
’cen’ refers to the central pixel of the screen), see the left panel of Fig.
2. This particular vector can be easily completed by
$\mathbf{\bar{w}_{0}^{\mathrm{cen}}}=-\mathbf{\bar{e}_{1}}$ and
$\mathbf{\bar{n}_{0}^{\mathrm{cen}}}=\mathbf{\bar{e}_{2}}$, see the left panel
of Fig. 2. In the general case of a vector $\mathbf{\bar{K}_{0}}$ defined by
the two spherical angles $(a,b)$ (see the right panel of Fig. 2), A shows that
the observer’s screen polarization basis reads
$\displaystyle\mathbf{\bar{w}_{0}}$ $\displaystyle=$
$\displaystyle\left[-\sin^{2}b\left(1-\cos a\right)-\cos
a\right]\mathbf{\bar{e}_{1}}+\sin b\cos b\left(1-\cos
a\right)\mathbf{\bar{e}_{2}}$ $\displaystyle+\cos b\sin
a\,\mathbf{\bar{e}_{3}},$ $\displaystyle\mathbf{\bar{n}_{0}}$ $\displaystyle=$
$\displaystyle-\sin b\cos b\left(1-\cos
a\right)\mathbf{\bar{e}_{1}}+\left[\cos^{2}b\left(1-\cos a\right)+\cos
a\right]\mathbf{\bar{e}_{2}}$ $\displaystyle-\sin b\sin
a\,\mathbf{\bar{e}_{3}}.$
It is straightforward to check that these vectors are unit vectors, orthogonal
to each other, and to $\mathbf{\bar{K}_{0}}$. Moreover, in a typical ray-
tracing problem where $a\ll 1$, we have as we should
$\mathbf{\bar{w}_{0}}\approx\mathbf{\bar{w}_{0}^{\mathrm{cen}}}=-\mathbf{\bar{e}_{1}}$,
and
$\mathbf{\bar{n}_{0}}\approx\mathbf{\bar{n}_{0}^{\mathrm{cen}}}=\mathbf{\bar{e}_{2}}$.
We note that the plane $(\mathbf{\bar{w}_{0}},\mathbf{\bar{n}_{0}})$ (where
the polarization angle will be defined) strictly speaking only coincides with
the screen’s plane $(\mathbf{\bar{e}_{1}},\mathbf{\bar{e}_{2}})$ for the
central pixel of the screen (with $a=0$). We will neglect this small
difference between the polarization plane and the screen’s plane, which is
perfectly valid as long as $a\ll 1$, that is, as long as the field of view is
sufficiently small.
At this point, we have fully defined our initial condition by specifying the
triad $(\mathbf{k_{0}},\mathbf{\bar{w}_{0}},\mathbf{\bar{n}_{0}})$ at the
observer’s screen. The next step is to parallel transport these vectors along
the light ray towards the source.
### 2.3 Relevant frames, parallel transport of polarization basis, EVPA
Let us start by introducing the three relevant frames for describing the
polarized GRRT problem. We will focus on synchrotron radiation, which is our
primary science interest, but most of the discussion is rather general. The
frames of interest are:
* •
the observer frame, described in detail in the previous section, defined by
the 4-velocity $\mathbf{u_{0}}$,
* •
the fluid frame, defined by the 4-velocity $\mathbf{u}$ describing the bulk
motion of the emitting fluid (for instance, Keplerian motion around a black
hole),
* •
the particle frame, which follows the helical motion of the synchrotron-
emitting electron around the magnetic field lines described in the fluid
frame.
This section will mostly deal with the fluid frame, and the link with the
particle frame is further discussed in B.
We consider a light ray, modeled by a null geodesic, joining the far-away
observer to the emitting accretion flow surrounding a black hole. We want to
parallel-transport the null 4-vector $\mathbf{k}$ tangent to the null
geodesic, backwards from the observer towards the emitter. We will also
parallel-transport the local West and North unit spacelike directions,
$\mathbf{\bar{w}}$ and $\mathbf{\bar{n}}$. Note that the index $0$ used for
these three vectors in the previous section meant that they were considered at
the screen position. We now consider their evolution along the ray and drop
the index. So we must integrate the following equations
$\displaystyle\boldsymbol{\nabla_{\mathbf{k}}}\mathbf{k}$ $\displaystyle=$
$\displaystyle\mathbf{0},$ (31)
$\displaystyle\boldsymbol{\nabla_{\mathbf{k}}}\mathbf{\bar{w}}$
$\displaystyle=$ $\displaystyle\mathbf{0},$
$\displaystyle\boldsymbol{\nabla_{\mathbf{k}}}\mathbf{\bar{n}}$
$\displaystyle=$ $\displaystyle\mathbf{0},$
with the initial conditions that
$(\mathbf{k},\mathbf{\bar{w}},\mathbf{\bar{n}})=(\mathbf{k_{0}},\mathbf{\bar{w}_{0}},\mathbf{\bar{n}_{0}})$
at the screen. Given that the parallel transport preserves the scalar product
between vectors 555This is an obvious property: let us consider two vectors
$a^{\mu}$ and $b^{\mu}$ parallel-transported along $k^{\mu}$, then
$\nabla_{\mathbf{k}}\left(\mathbf{a}\cdot\mathbf{b}\right)=k^{\mu}\nabla_{\mu}\left(g_{\alpha\beta}a^{\alpha}b^{\beta}\right)=g_{\alpha\beta}\left(a^{\alpha}k^{\mu}\nabla_{\mu}b^{\beta}+b^{\beta}k^{\mu}\nabla_{\mu}a^{\alpha}\right)=0$,
because of the parallel-transport relations $k^{\mu}\nabla_{\mu}a^{\alpha}=0$
and $k^{\mu}\nabla_{\mu}b^{\beta}=0$. We used the fact that the connexion
$\nabla$ is compatible with the metric to get $\nabla_{\mu}g_{\alpha\beta}=0$
and take the metric tensor out of the covariant derivative. , and given that
$(\mathbf{k},\mathbf{\bar{w}},\mathbf{\bar{n}})$ are mutually orthogonal at
the observer, they remain mutually orthogonal when parallel-transported at the
emitter, and $\mathbf{\bar{w}}$ and $\mathbf{\bar{n}}$ remain unit vectors.
It is useful at this point to note that, in vacuum, the EVPA is a conserved
quantity along a geodesic. Let us demonstrate this result. We have seen that,
at the distant observer’s location, we might confuse the covariant and non-
covariant polarization vectors, $\mathbf{\hat{f}_{0}}$ and
$\mathbf{\hat{F}_{0}}$. Let us consider the point along a photon’s geodesic
corresponding to the exit from the emitting source region, meaning that the
part of the geodesic located in between this point and the distant observer is
in vacuum. We hereafter call this point the exit point. We can make the exact
same reasoning at the exit point as we made at the distant observer’s
location, and conclude that we can confuse the covariant and non-covariant
polarization vectors at the exit point, $\mathbf{\hat{f}_{\mathrm{exit}}}$ and
$\mathbf{\hat{F}_{\mathrm{exit}}}$, where $\mathbf{\hat{F}_{\mathrm{exit}}}$
is the polarization vector as measured by the emitter at the exit point. This
implies more generally that $\mathbf{\hat{f}}$ and $\mathbf{\hat{F}}$ can be
confused at any point along the part of the geodesic located in vacuum. Given
that $\mathbf{\hat{f}}$ and the screen basis
$(\mathbf{\bar{w}},\mathbf{\bar{n}})$ are parallel propagated along the
geodesic in vacuum (see Eqs. 8 and 31), the angle between $\mathbf{\hat{f}}$
and the basis vectors is conserved in vacuum, hence the EVPA is conserved
along the part of the geodesic located in vacuum. This is of course no longer
valid in the source region, where the covariant polarization vector is no
longer parallel propagated (the parallel propagation of $\mathbf{\hat{f}}$ is
a consequence of Maxwell’s equations in vacuum).
We want to project the parallel-transported basis vectors
$(\mathbf{k},\mathbf{\bar{w}},\mathbf{\bar{n}})$ orthogonally to the
4-velocity $\mathbf{u}$ of the emitting fluid, that is, project them in the
local rest space of the fluid. By doing so without further precaution, we
would of course lose the mutual orthogonality between these vectors, which is
not preserved in a projection. Let us consider
$\displaystyle\mathbf{\bar{w}^{\prime}}=\mathbf{\bar{w}}-\frac{\mathbf{\bar{w}}\cdot\mathbf{u}}{\mathbf{k}\cdot\
\mathbf{u}}\,\mathbf{k},$ (32)
$\displaystyle\mathbf{\bar{n}^{\prime}}=\mathbf{\bar{n}}-\frac{\mathbf{\bar{n}}\cdot\mathbf{u}}{\mathbf{k}\cdot\
\mathbf{u}}\,\mathbf{k},$
where the denominator, $\mathbf{k}\cdot\mathbf{u}$, is minus the energy of the
photon as measured in the fluid frame, and as such, non zero, so that these
expressions are well defined. It is easy to check that these two vectors are
spacelike unit vectors, orthogonal to $\mathbf{u}$, to each other, and to
$\mathbf{\bar{K}}=\frac{\mathbf{k}+\left(\mathbf{k}\cdot\mathbf{u}\right)\,\mathbf{u}}{|\mathbf{k}\cdot\mathbf{u}|},$
(33)
the normalized projection of $\mathbf{k}$ orthogonal to $\mathbf{u}$, which
coincides with the unit direction of emission of the photon in the fluid
frame. We thus obtain a well-defined orthonormal direct triad
$(\mathbf{\bar{K}},\mathbf{\bar{w}^{\prime}},\mathbf{\bar{n}^{\prime}})$ of
the fluid frame, illustrated in Fig. 3. We note that if we consider a vector
$\mathbf{F}$ in the fluid frame, normal to $\mathbf{\bar{K}}$, then our
definition leads to
$\mathbf{F}\cdot\mathbf{\bar{n}}=\mathbf{F}\cdot\mathbf{\bar{n}^{\prime}},$
(34)
and similarly for $\mathbf{w}$, so that our definition allows to keep
unchanged the angles between such a vector $\mathbf{F}$ and the reference
directions, be they primed or unprimed. This will be important later.
Figure 3: Frames of interest for the polarized ray-tracing problem. The far-
away observer’s rest frame (orthogonal to the observer’s 4-velocity
$\mathbf{u_{0}}$) is shown in green, it is a simplified version of Fig. 1 and
shows the local incidence direction $\mathbf{\bar{K}_{0}}$ (spacelike vector)
of the observed light ray, together with the null 4-vector tangent to the
incident null geodesic $\mathbf{k_{0}}$, the local North and West spacelike
directions $(\mathbf{\bar{w}_{0}},\mathbf{\bar{n}_{0}})$, such that
$(\mathbf{\bar{K}_{0}},\mathbf{\bar{w}_{0}},\mathbf{\bar{n}_{0}})$ is a direct
orthonormal triad, and the polarization vector as measured by the far-away
observer, $\mathbf{\bar{F}_{0}}$. The observed EVPA is labelled EVPA0, lying
between the screen’s North direction and the observed polarization vector
$\mathbf{\bar{F}_{0}}$. Starting from the far-away observer, a null geodesic
is integrated backwards towards the source (red line), until it reaches the
accretion flow (black line) surrounding the black hole (black disk). The fluid
frame (orthogonal to the emitter’s 4-velocity $\mathbf{u}$) is shown in blue.
The null 4-vector tangent to the null geodesic at the emission point is called
$\mathbf{k}$. The magnetic field spacelike 4-vector as measured in the fluid
frame (thus, orthogonal to $\mathbf{u}$) is called $\mathbf{b}$. The
synchrotron-emitting electron’s trajectory is represented by the pale black
helix. The local direction of photon emission, as measured in the fluid frame,
is the spacelike unit vector $\mathbf{\bar{K}}$. The pair of vectors
$(\mathbf{\bar{n}^{\prime}},\mathbf{\bar{w}^{\prime}})$ is related to the pair
$(\mathbf{\bar{n}_{0}},\mathbf{\bar{w}_{0}})$, parallel-propagated along the
null geodesic (see text for details), such that
$(\mathbf{\bar{K}},\mathbf{\bar{w}^{\prime}},\mathbf{\bar{n}^{\prime}})$ is a
direct orthonormal triad. The unit polarization vector as measured by the
emitter is called $\mathbf{\bar{F}}$. The radiation field
$\mathbf{E_{\mathrm{rad}}}$ associated with the helical motion of the electron
is shown in pale black, and lies along $\mathbf{\bar{F}}$ for a relativistic
electron (see B for a demonstration). The unit projection of the magnetic
4-vector orthogonal to $\mathbf{\bar{K}}$ is called
$\boldsymbol{\bar{b}_{\perp}}$. Thus
$(\mathbf{\bar{K}},\boldsymbol{\bar{b}_{\perp}},\mathbf{\bar{F}})$ is also a
direct orthonormal triad, rotated with respect to
$(\mathbf{\bar{K}},\mathbf{\bar{w}^{\prime}},\mathbf{\bar{n}^{\prime}})$ by
the emission EVPA, labeled EVPAe, lying between the parallel-transported North
direction and the fluid-frame polarization vector $\mathbf{\bar{F}}$.
Let us now consider the magnetic field 4-vector $\mathbf{b}$ of the accretion
flow, as measured in the fluid frame. By construction, this vector lies in the
local rest frame of the fluid, so it is orthogonal to $\mathbf{u}$. We are
also interested in its normalized projection orthogonally to
$\mathbf{\bar{K}}$, which reads
$\boldsymbol{\bar{b}_{\perp}}=\frac{\mathbf{b}-\left(\mathbf{b}\cdot\mathbf{\bar{K}}\right)\mathbf{\bar{K}}}{||\mathbf{b}-\left(\mathbf{b}\cdot\mathbf{\bar{K}}\right)\mathbf{\bar{K}}||},$
(35)
(note that the minus sign in the numerator and denominator of the rhs,
compared to the plus sign in the numerator of the rhs of Eq. 33, comes from
the fact that $\mathbf{\bar{K}}$ is spacelike while $\mathbf{u}$ is timelike),
and in the unit polarization vector as measured in the fluid frame
$\mathbf{\bar{F}}=\frac{\mathbf{\bar{K}}\times\mathbf{b}}{||\mathbf{\bar{K}}\times\mathbf{b}||}=\mathbf{\bar{K}}\times\boldsymbol{\bar{b}_{\perp}}.$
(36)
We thus have constructed a second orthonormal direct triad of the fluid rest
space, $(\mathbf{\bar{K}},\boldsymbol{\bar{b}_{\perp}},\mathbf{\bar{F}})$. We
note that it is not obvious that the vector defined by Eq. 36 coincides with
the emission polarization vector for synchrotron radiation, that is, with the
direction of the radiation electric field emitted by an electron moving around
the $\mathbf{b}$ field lines, given that we have never discussed the emitting
electron motion so far. In B, by relating the particle frame and the fluid
frame, we demonstrate that, provided the emitting electron is relativistic,
this is indeed so.
We have thus at hand two orthonormal triads of the fluid frame, the observer-
related
$(\mathbf{\bar{K}},\mathbf{\bar{w}^{\prime}},\mathbf{\bar{n}^{\prime}})$, and
the magnetic-field-related
$(\mathbf{\bar{K}},\boldsymbol{\bar{b}_{\perp}},\mathbf{\bar{F}})$. These two
frames are rotated with respect to each other by the angle
$\chi\equiv(\mathbf{\bar{n}^{\prime}},\mathbf{\bar{F}})=(\mathbf{\bar{n}},\mathbf{\bar{F}})\equiv\mathrm{EVPA_{e}},$
(37)
the EVPA in the fluid frame, where the index e reminds that we are dealing
with an emission EVPA, as compared to the observed EVPA of Fig. 1. Both angles
are illustrated in Fig. 3. We note that the emission EVPA evolves as the light
ray evolves through the emitting fluid; the EVPA is only conserved in vacuum
as demonstrated above. So a sequence of emission EVPAs corresponds to a unique
observed EVPA. Note that the second equality in Eq. 37 is a consequence of Eq.
34. This emission EVPA will be crucial in the polarized radiative transfer
formalism that we introduce in the next section. A practical, code-friendly
expression for the emission EVPA is the following
$\mathrm{EVPA_{e}}=\frac{\pi}{2}-\mathrm{atan2}\left(\mathbf{\bar{b}_{\perp}}\
\cdot\mathbf{\bar{w}^{\prime}},\mathbf{\bar{b}_{\perp}}\cdot\mathbf{\bar{n}^{\prime}}\right).$
(38)
After having discussed the parallel transport of the vectors of interest along
the null geodesic, the last step of the polarized GRRT problem is to integrate
the polarized radiative transfer within the accretion flow surrounding the
compact object.
### 2.4 Polarized radiative transfer
#### 2.4.1 Stokes parameters
The most general monochromatic electromagnetic wave has an elliptical
polarization, in the sense that the electric field vector describing the wave
draws an ellipse during its time evolution in the plane normal to the
direction of propagation, see Fig. 4.
Figure 4: Polarization ellipse of a general monochromatic wave (in blue). The
black axes $(x,y)$ label the frame of interest where we want to formulate the
problem, while the blue axes $(x^{\prime},y^{\prime})$ are along the major and
minor axes of the ellipse and are therefore naturally adapted to the
elliptically polarized wave. The angle between the two bases is called $\chi$
(it would coincide with the notion of EVPA for a linearly polarized wave along
the $x$ axis), and the quantity $\tan\beta$ encodes the ellipse axes ratio.
The dashed black axes $(a,b)$ are tilted by $45^{\circ}$ relative to $(x,y)$
and are useful for defining the U Stokes parameter.
Let us introduce the electric field complex vector of the monochromatic wave
$\mathbf{\hat{E}}=\hat{E}_{x}\mathbf{\bar{e}_{x}}+\hat{E}_{y}\mathbf{\bar{e}_{y}},$
(39)
decomposed in an arbitrary orthonormal basis
$(\mathbf{\bar{e}_{x}},\mathbf{\bar{e}_{y}})$ of the plane orthogonal to the
direction of propagation, where $\hat{E}_{x}$ and $\hat{E}_{y}$ are the
complex components along the axes. The polarization ellipse described by the
vector $\mathbf{\hat{E}}$ can be equivalently described by two sets of
parameters. From a geometrical point of view, it is very natural to provide
the total intensity $|\mathbf{\hat{E}}|^{2}$, together with two angles $\chi$
and $\beta$. The angle $\chi$ lies between the basis
$(\mathbf{\bar{e}_{x}},\mathbf{\bar{e}_{y}})$ and the basis corresponding to
the axes of the ellipse, while $\tan\beta$ encodes the ellipse axes ratio (see
Fig. 4). However, this parametrization is not practical from a physical point
of view given that the two angles are not directly observable. So from a
physical point of view, it is more natural to consider the following set of
four Stokes parameters (Rybicki and Lightman, 1979)
$\displaystyle I$ $\displaystyle=$
$\displaystyle|\hat{E}_{x}|^{2}+|\hat{E}_{y}|^{2},$ (40) $\displaystyle Q$
$\displaystyle=$ $\displaystyle|\hat{E}_{x}|^{2}-|\hat{E}_{y}|^{2}=I\,\cos
2\chi\,\cos 2\beta,$ $\displaystyle U$ $\displaystyle=$
$\displaystyle|\hat{E}_{a}|^{2}-|\hat{E}_{b}|^{2}=I\,\sin 2\chi\,\cos 2\beta,$
$\displaystyle V$ $\displaystyle=$
$\displaystyle|\hat{E}_{r}|^{2}-|\hat{E}_{l}|^{2}=I\,\sin 2\beta,$
where $\hat{E}_{a}$ and $\hat{E}_{b}$ are the complex components of the
electric field in an orthonormal basis
$(\mathbf{\bar{e}_{a}},\mathbf{\bar{e}_{b}})$ rotated by $45^{\circ}$ compared
to $(\mathbf{\bar{e}_{x}},\mathbf{\bar{e}_{y}})$, see the dashed black axes in
Fig. 4. The quantities $\hat{E}_{r}$ and $\hat{E}_{l}$ are the complex
components of the field in an orthonormal complex basis,
$\mathbf{\bar{e}_{l,r}}=\sqrt{2}/2(\mathbf{\bar{e}_{x}}\pm
i\mathbf{\bar{e}_{y}})$. These relations clearly show that the Stokes
parameters are all sums or differences of intensities along specific
directions, and as such are directly observable and well adapted to being
evolved in a radiative transfer problem. Equations 40 show how to construct
the Stokes parameters from the geometrical angular parameters $\chi$, $\beta$
of the polarization ellipse. The reverse expression is easy to find and reads
$\displaystyle\tan 2\chi$ $\displaystyle=$ $\displaystyle\frac{U}{Q},$ (41)
$\displaystyle\sin 2\beta$ $\displaystyle=$ $\displaystyle\frac{V}{I}.$
For a circular polarization, $\beta=\pi/4$, so
$Q=0,\quad U=0,\quad V=I\quad(\mathrm{circular\>polarization}),$ (42)
while for a linear polarization, $\beta=0$, and
$Q=I\cos 2\chi,\quad U=I\sin 2\chi,\quad
V=0\quad(\mathrm{linear\>polarization}),$ (43)
and if the wave is polarized along the $x$ axis of Fig. 4, then $Q=I$ and
$U=0$, while if the wave is polarized at $45^{\circ}$ from the $x$ axis, then
$Q=0$ and $U=I$. So $Q$ and $U$ encode linear polarization along the
directions $\mathbf{\bar{e}_{x}}$ or $\mathbf{\bar{e}_{y}}$ and
$\mathbf{\bar{e}_{a}}$ or $\mathbf{\bar{e}_{b}}$, respectively, and $V$
encodes circular polarization.
Let us consider the Stokes parameters $(I,Q,U,V)$ defined in a basis
$(\mathbf{\bar{e}_{x}},\mathbf{\bar{e}_{y}})$, and the parameters
$(I^{\prime},Q^{\prime},U^{\prime},V^{\prime})$ defined in a basis
$(\mathbf{e^{\prime}_{x}},\mathbf{e^{\prime}_{y}})$, rotated by an angle
$\chi$ with respect to $(\mathbf{\bar{e}_{x}},\mathbf{\bar{e}_{y}})$, see Fig.
4. It is easy to show that the $Q$ and $U$ Stokes parameters transform
following
$\left(\begin{array}[]{c}Q\\\
U\end{array}\right)=\left(\begin{array}[]{cc}\cos 2\chi&-\sin 2\chi\\\ \sin
2\chi&\cos 2\chi\end{array}\right)\left(\begin{array}[]{c}Q^{\prime}\\\
U^{\prime}\end{array}\right),$ (44)
while $I$ and $V$ are invariant.
For a monochromatic radiation, there must exist a relation between the four
Stokes parameters, that are equivalent to the set of three parameters
$(I,\chi,\beta)$, and thus cannot be independent. This relation reads
$I^{2}=Q^{2}+U^{2}+V^{2}\quad(\mathrm{monochromatic\>/\>fully\>polarized})$
(45)
and the radiation is then said to be fully polarized. For a superimposition of
waves at different frequencies, the radiation is only partially polarized and
the resulting Stokes parameters verify
$I^{2}\geq Q^{2}+U^{2}+V^{2}\quad(\mathrm{partially\>polarized}).$ (46)
It is then useful to introduce the degree of polarization
$\mathrm{d_{p}}=\frac{\sqrt{Q^{2}+U^{2}+V^{2}}}{I},$ (47)
and the degree of linear polarization
$\mathrm{d_{lp}}=\frac{\sqrt{Q^{2}+U^{2}}}{I}.$ (48)
#### 2.4.2 Stokes parameters for synchrotron radiation, conventions
We are primarily interested in polarized synchrotron radiation given that our
main science interest is the millimeter and infrared radiation emitted by
nearby supermassive black hole environments. Let us consider a single electron
following a helical motion around the field lines of a magnetic field
$\mathbf{b}$ as measured in the fluid frame. The emitted synchrotron radiation
is elliptically polarized, with the minor axis of the polarization ellipse
aligned along the direction of the magnetic field projected orthogonally to
the direction of propagation, and major axis along the fluid-frame
polarization vector (Huang et al., 2009). The Stokes parameters are thus
naturally expressed in a basis aligned with the axes of this polarization
ellipse, that is, the $(\mathbf{\bar{F}},-\mathbf{\bar{b}_{\perp}})$ basis
(see the illustration in Fig. 5). We call this basis the synchrotron
polarization basis. This basis is rotated by the emission EVPA with respect to
the observer-related $(\mathbf{\bar{n}^{\prime}},-\mathbf{\bar{w}^{\prime}})$
basis. This last basis is called the parallel-transported polarization basis.
For integrating the radiative transfer in the observer’s frame, we will need
to take care of this rotation between the synchrotron and the parallel-
transported polarization bases. This is described in the next section.
Our sign conventions for the Stokes parameters are illustrated in Fig. 5. It
complies with the convention of the International Astronomical Union
(Transactions of the International Astronomical Union, 1973; Hamaker and
Bregman, 1996, see Fig. 1 of the second reference).
Figure 5: Geometry of the polarized synchrotron problem and Stokes parameter
illustration. All quantities depicted here are defined in the fluid frame. The
magnetic field is $\mathbf{b}$, which lies along the $\mathbf{\bar{e}_{z}}$
unit vector. The direction of emission in this frame is $\mathbf{\bar{K}}$,
which lies along the $\mathbf{\bar{e}_{\textfrak}{c}}$ unit vector. We define
the $\mathbf{\bar{e}_{\textfrak}{a}}$ unit vector as lying along the major
axis of the synchrotron polarization ellipse (shown in dashed pale blue). This
vector is defined up to an unimportant sign convention. The
$\mathbf{\bar{e}_{\textfrak}{b}}$ unit vector is such that
$(\mathbf{\bar{e}_{\textfrak}{a}},\mathbf{\bar{e}_{\textfrak}{b}},\mathbf{\bar{e}_{\textfrak}{c}})$
is a direct orthonormal triad of the fluid frame. The vector
$\mathbf{\bar{e}_{x}}$ is parallel to $\mathbf{\bar{e}_{\textfrak}{a}}$. The
$\mathbf{\bar{e}_{y}}$ unit vector is such that
$(\mathbf{\bar{e}_{x}},\mathbf{\bar{e}_{y}},\mathbf{\bar{e}_{z}})$ is a direct
orthonormal triad of the fluid frame. The angle between $\mathbf{b}$ and
$\mathbf{\bar{K}}$ is called $\theta_{B}$. The Stokes parameters Q and U are
defined in the
$(\mathbf{\bar{e}_{\textfrak}{a}}=\mathbf{F},\mathbf{\bar{e}_{\textfrak}{b}}=-\mathbf{\bar{b}_{\perp}})$
basis, that we call the synchrotron polarization basis, illustrated by the
zoom on the right of the figure. This zoom shows the synchrotron polarization
basis (in blue; subscript ’synch’) as well as the parallel-transported
polarization basis (in green; subscript ’$\parallel$trans’),
$(\mathbf{\tilde{e}_{\textfrak}{a}}=\mathbf{\bar{n}},\mathbf{\tilde{e}_{\textfrak}{b}}=-\mathbf{\bar{w}})$.
These two bases are rotated by the emission EVPA angle. The sign conventions
of the Stokes parameters are as shown in this zoom. Note that the orientation
convention used in this figure is the same as that used by e.g. Dexter (2016);
Huang and Shcherbakov (2011), which results in a positive emission coefficient
for Stokes Q. Some authors use an alternative orientation convention, taking
$\mathbf{\bar{e}_{\textfrak}{a}}$ along the minor axis of the polarization
ellipse, see Shcherbakov (2008); Marszewski et al. (2021). This simply leads
to Stokes Q being multiplied by $-1$, and to a negative emission coefficient
for Stokes Q.
#### 2.4.3 Transfer equation
Just like in the unpolarized version of Gyoto (Vincent et al., 2011), we will
integrate the radiative transfer equation in the fluid frame, and then
transform the quantities to the observer’s frame. The unpolarized radiative
transfer equation used by Gyoto reads
$\frac{\mathrm{d}I^{\mathrm{em}}_{\nu}}{\mathrm{d}s}=j_{\nu}-\alpha_{\nu}\,I^{\mathrm{em}}_{\nu},$
(49)
where $I^{\mathrm{em}}_{\nu}$ is the specific intensity (the index $\nu$ means
that we are considering an intensity per unit of frequency), $\mathrm{d}s$ is
the element of optical path length, $j_{\nu}$ and $\alpha_{\nu}$ are the
specific emission and absorption coefficients, all these quantities being
measured in the fluid frame (hence the superscript ’em’ for the intensity,
referring to the emitting fluid; we discard it for the other quantities for
simplicity). The intensity in the observer frame (superscript ’obs’) then
follows from
$I^{\mathrm{obs}}_{\nu}=I^{\mathrm{em}}_{\nu}\,\left(\frac{\nu^{\mathrm{obs}}}{\nu^{\mathrm{em}}}\right)^{3},$
(50)
which is a consequence of Liouville’s theorem (Misner et al., 1973).
The polarized radiative transfer equation is naturally written in the
synchrotron polarization basis of the fluid frame. In this basis, the transfer
equation reads
$\frac{\mathrm{d}}{\mathrm{d}s}\left(\begin{array}[]{c}I_{\nu}^{\mathrm{em;synch}}\\\
Q_{\nu}^{\mathrm{em;synch}}\\\ U_{\nu}^{\mathrm{em;synch}}\\\
V_{\nu}^{\mathrm{em;synch}}\end{array}\right)=\left(\begin{array}[]{c}j_{\nu,I}\\\
j_{\nu,Q}\\\ j_{\nu,U}\\\
j_{\nu,V}\end{array}\right)-\left(\begin{array}[]{cccc}\alpha_{\nu,I}&\alpha_{\nu,Q}&\alpha_{\nu,U}&\alpha_{\nu,V}\\\
\alpha_{\nu,Q}&\alpha_{\nu,I}&r_{\nu,V}&-r_{\nu,U}\\\
\alpha_{\nu,U}&-r_{\nu,V}&\alpha_{\nu,I}&r_{\nu,Q}\\\
\alpha_{\nu,V}&r_{\nu,U}&-r_{\nu,Q}&\alpha_{\nu,I}\end{array}\right)\left(\begin{array}[]{c}I_{\nu}^{\mathrm{em;synch}}\\\
Q_{\nu}^{\mathrm{em;synch}}\\\ U_{\nu}^{\mathrm{em;synch}}\\\
V_{\nu}^{\mathrm{em;synch}}\end{array}\right),$ (51)
where the ’em;synch’ label is here to remind that we are dealing with Stokes
parameters expressed in the synchrotron polarization basis of the emitting
fluid frame; moreover, $j_{\nu,X}$ and $\alpha_{\nu,X}$ are emission and
absorption coefficients for the Stokes parameter X, $r_{\nu,Q}$ and
$r_{\nu,U}$ are Faraday conversion parameters, and $r_{\nu,V}$ is the Faraday
rotation parameter. All these transfer coefficients are defined in the
synchrotron polarization basis of the fluid frame (we discard the superscript
for simplicity). We refer to Fig. 5 for the details of the sign conventions.
However, we rather want to integrate this equation in the parallel-transported
polarization basis, $(\mathbf{\bar{n}^{\prime}},-\mathbf{\bar{w}^{\prime}})$,
which is rotated by the emission EVPA with respect to the synchrotron
polarization basis, see Fig. 5. In the parallel-transported polarization basis
of the fluid frame, the transfer equation reads
$\displaystyle\frac{\mathrm{d}}{\mathrm{d}s}\left(\begin{array}[]{c}I_{\nu}^{\mathrm{em;\parallel
trans}}\\\ Q_{\nu}^{\mathrm{em;\parallel trans}}\\\
U_{\nu}^{\mathrm{em;\parallel trans}}\\\ V_{\nu}^{\mathrm{em;\parallel
trans}}\end{array}\right)$ $\displaystyle=$
$\displaystyle\mathbf{R}\left(\chi_{\mathrm{e}}\right)\left(\begin{array}[]{c}j_{\nu,I}\\\
j_{\nu,Q}\\\ j_{\nu,U}\\\ j_{\nu,V}\end{array}\right)$ (69)
$\displaystyle-\mathbf{R}\left(\chi_{\mathrm{e}}\right)\left(\begin{array}[]{cccc}\alpha_{\nu,I}&\alpha_{\nu,Q}&\alpha_{\nu,U}&\alpha_{\nu,V}\\\
\alpha_{\nu,Q}&\alpha_{\nu,I}&r_{\nu,V}&-r_{\nu,U}\\\
\alpha_{\nu,U}&-r_{\nu,V}&\alpha_{\nu,I}&r_{\nu,Q}\\\
\alpha_{\nu,V}&r_{\nu,U}&-r_{\nu,Q}&\alpha_{\nu,I}\end{array}\right)\mathbf{R}\left(-\chi_{\mathrm{e}}\right)\left(\begin{array}[]{c}I_{\nu}^{\mathrm{em;\parallel
trans}}\\\ Q_{\nu}^{\mathrm{em;\parallel trans}}\\\
U_{\nu}^{\mathrm{em;\parallel trans}}\\\ V_{\nu}^{\mathrm{em;\parallel
trans}}\end{array}\right),$
where the superscript ’em;$\parallel$trans’ reminds that we are dealing with
Stokes parameters defined in the parallel-transported polarization basis of
the emitting fluid frame, and
$\mathbf{R}\left(\chi_{\mathrm{e}}\right)=\left(\begin{array}[]{cccc}1&0&0&0\\\
0&\cos 2\chi_{\mathrm{e}}&-\sin 2\chi_{\mathrm{e}}&0\\\ 0&\sin
2\chi_{\mathrm{e}}&\cos 2\chi_{\mathrm{e}}&0\\\ 0&0&0&1\end{array}\right)$
(70)
is a rotation matrix describing the rotation by the angle
$\chi_{\mathrm{e}}\equiv EVPA_{\mathrm{e}}$, the emission EVPA, between the
synchrotron and the parallel-transported polarization bases. This is the exact
same transformation as that described by Eq. 44.
Solving Eq. 69 is a well-known problem that we briefly discuss in C. The
corresponding Stokes parameters in the observer frame then follow from
$X^{\mathrm{obs}}_{\nu}=X^{\mathrm{em;\parallel
trans}}_{\nu}\,\left(\frac{\nu^{\mathrm{obs}}}{\nu^{\mathrm{em}}}\right)^{3},$
(71)
similarly as in Eq. 50, where $X$ is either of the Stokes parameters.
#### 2.4.4 Polarized synchrotron coefficients
We now need to express the synchrotron coefficients in the synchrotron
polarization basis. In this basis, the transfer coefficients for the Stokes
parameter U, i.e. $j_{\nu,U}$, $\alpha_{\nu,U}$ and $r_{\nu,U}$ are zero by
definition. However, the computation of emission, absorption and rotation
synchrotron coefficients for the others Stokes parameters, from an arbitrary
distribution of electrons could be quite heavy. Indeed, even for isotropic
distributions, the computation of the emission coefficients require a double
integral (Rybicki and Lightman, 1979) and the others are even more complex
using the susceptibility tensor (see Appendix B of Marszewski et al., 2021).
Fortunately, Marszewski et al. (2021); Dexter (2016); Huang and Shcherbakov
(2011) derived fitted formulae for the emissivities, absorptivities and
rotativities for well-defined isotropic distributions of electrons : Thermal
(Maxwell-Jüttner), Power Law or Kappa (thermal core with a power law tail). We
choose to implement the formulae of Marszewski et al. (2021) in Gyoto to
compute the synchrotron coefficients as they are the only one who provides
formulae for a kappa distribution. These formulae are valid for a specific
range of parameters.
For a thermal distribution, parametrized by the dimensionless temperature
$\Theta_{e}=k_{B}T/m_{e}c^{2}$, the fits are accurate for $3<\Theta_{e}<40$
and for $\nu/\nu_{c}\gg 1$ with $\nu_{c}=eB/(2\pi m_{e}c)$ the cyclotron
frequencies (Marszewski et al., 2021).
For a Power Law distribution, parametrized by a minimum and maximum Lorentz
factor, $\gamma_{\mathrm{min}}$ and $\gamma_{\mathrm{max}}$ respectively, and
by a power law index $p$, the fits are accurate for
$\gamma_{\mathrm{min}}<10^{2}$, $1.5<p<6.5$ and, as before, for
$\nu/\nu_{c}\gg 1$ (Marszewski et al., 2021).
The Kappa distribution is characterized by two parameters $w$ (equivalent to
the dimensionless temperature) and $\kappa=p+1$. Contrary to the other
distributions where the fits are continuous functions of the parameters, the
fits for rotation coefficients for the Kappa distribution have been done for
four specific values of $\kappa=(3.5,4.0,4.5,5.0)$ and are not defined for any
other value. The fits are valid while $3<w<40$, $\nu/\nu_{c}\gg 1$ and
$X_{\kappa}\gg 10^{-1}$ where $X_{\kappa}=\nu/\nu_{c}(w\kappa)^{2}\sin\theta$
with $\theta$ the angle between the magnetic field vector and the photon
tangent vector (Marszewski et al., 2021).
For some tests in section 3, we will use the formulae from Dexter (2016),
especially for the comparison with the ray-tracing code ipole which use the
formulae of Dexter (2016) (as the code grtrans). The order of the maximum
relative error between all the fits in Marszewski et al. (2021) and the true
values is of 30%. For the typical parameters of the accretion flow of Sgr A*,
the difference between the formulae of Marszewski et al. (2021) and Dexter
(2016) is lower or equal to $10\%$.
## 3 Tests
### 3.1 Test of the parallel transport
The first test that we have to make is to check that the observer polarization
basis’ vectors, i.e. $\mathbf{\bar{n}}$ and $\mathbf{\bar{w}}$, are well
parallel transported along the null geodesics. The parallel transport equation
given by Eq. 31 is fully general and agnostic about the particular spacetime
considered. However, in the special case of the Kerr spacetime, it is well
known that the special algebraic kind of the spacetime allows the existence of
the Walker-Penrose constant (Walker and Penrose, 1970), defined as follows. If
$\mathbf{k}$ is the tangent vector to a null geodesic, and if $\mathbf{f}$ is
a vector orthogonal to $\mathbf{k}$ and parallel-transported along
$\mathbf{k}$, then the following complex quantity
$\displaystyle K_{1}-iK_{2}=$
$\displaystyle(r-ia\cos\theta)\bigg{[}(k^{t}f^{r}-k^{r}f^{t})+a\sin^{2}\theta(k^{r}f^{\varphi}-k^{\varphi}f^{r})\bigg{.}$
$\displaystyle\bigg{.}-i\sin\theta\left\\{(r^{2}+a^{2})(k^{\varphi}f^{\theta}-k^{\theta}f^{\varphi})-a(k^{t}f^{\theta}-k^{\theta}f^{t})\right\\}\bigg{]},$
here expressed in Boyer-Lindquist coordinates, is conserved along the null
geodesic. Thus, knowing the evolution of $\mathbf{k}$ along the null geodesic,
the vectors $\mathbf{\bar{n}}$ and $\mathbf{\bar{w}}$ (that obviously fulfill
the conditions on $\mathbf{f}$ in Eq. 3.1) can be immediately obtained without
further computation by using this constant. This Kerr-specific result is only
used for testing purposes in Gyoto. The code is agnostic about the spacetime
and does not use this property.
We thus check the conservation of $K_{1}$ and $K_{2}$ for specific geodesics
and obtain a conservation to within $10^{-5}$ for default integration
parameters of Gyoto. This can be improved by setting a lower tolerance value
for the integration steps, but at the cost of a longer calculation time.
### 3.2 EVPA calculation test
Parallel transport having been tested, the observer polarization basis is well
defined in the rest frame of the emitter
$(\mathbf{\bar{K}},\mathbf{\bar{w}^{\prime}},\mathbf{\bar{n}^{\prime}})$,
through Eq. (32). As said in section 2.3, the natural basis to express the
synchrotron coefficients is
$(\mathbf{\bar{K}},\boldsymbol{\bar{b}_{\perp}},\mathbf{\bar{F}})$ with
$\mathbf{\bar{K}}$ a common vector between the two bases. Thus, to express the
radiative transfer in the observer-related basis, rather than in the
synchrotron basis, we just need to apply the rotation matrix defined in Eq.
(70). The angle between these two bases corresponds to the emission EVPA (see
section 2.3).
We define a simple setup to check the computation of this crucial angle. We
consider a Page-Thorne disk (geometrically thin, optically thick; Page and
Thorne, 1974) in a Minkowski metric (to avoid GR effects), seen face-on. We
consider two magnetic field configurations $\mathbf{b}$, toroidal and radial.
Expressed in Boyer-Lindquist coordinates, the two magnetic field
configurations read, in the rest frame of the emitter
$\mathrm{\textbf{Toroidal:}}\>b^{\alpha}=\left\\{\begin{array}[]{l}b^{t}=\sqrt{\frac{g_{\varphi\varphi}}{g_{tt}}\frac{\Omega^{2}}{g_{tt}+g_{\varphi\varphi}\Omega^{2}}},\\\
b^{r}=0,\\\ b^{\theta}=0,\\\
b^{\varphi}=\sqrt{\frac{g_{tt}}{g_{\varphi\varphi}}\frac{1}{g_{tt}+g_{\varphi\varphi}\Omega^{2}}},\end{array}\right.$
(73)
where $\Omega=u^{\varphi}/u^{t}$ and $\mathbf{u}$ is the 4-velocity of the
emitting fluid assumed to be Keplerian, and
$\mathrm{\textbf{Radial: }}\>b^{\alpha}=\left\\{\begin{array}[]{l}b^{t}=0,\\\
b^{r}=\sqrt{\frac{1}{g_{rr}}},\\\ b^{\theta}=0,\\\
b^{\varphi}=0.\end{array}\right.$ (74)
In the toroidal case, for all azimuthal angles of the disk, the wave-vector
$\mathbf{\bar{K}}$ is made of two components, one almost vertical (face-on
view), and one azimuthal component resulting from special-relativistic
aberration (Vincent et al., 2023). The magnetic field $\mathbf{b}$ is in the
toroidal direction. Thus, the resulting polarization vector
$\mathbf{\bar{F}}=\mathbf{\bar{K}}\times\boldsymbol{\bar{b}_{\perp}}$ is in
the radial direction. Similarly, for the radial magnetic field, the resulting
polarization vector is in the toroidal direction.
We consider an arbitrary $I_{\nu}=1$ emission for unpolarized intensity. Gyoto
computes images for the four Stokes parameters from which we can compute the
orientation of the polarization vector, i.e. the EVPA. As we are only
interested by the computation of the EVPA for the moment, i.e. without
radiative transfer, we assume a fully linearly polarized radiation by taking
$Q_{\nu}=I_{\nu}\,\cos\left(2*EVPA\right)$ and
$U_{\nu}=I_{\nu}\,\sin\left(2*EVPA\right)$. This means that we do not take
into account any absorption nor Faraday rotation.
Figure 6: Image of total (unpolarized) intensity of a Page-Thorne disk with
Keplerian orbit in a Minkowski space-time seen face-on for two magnetic field
configurations (toroidal left, radial right). The inner radius is at
$r=3\,r_{g}$. The green lines represent the orientation of polarization
vectors.
We show in the left panel of Fig. 6 in background the unpolarized images of
the setup described above and the polarization vector with the green ticks.
The position angle of these vectors is the observed EVPA. As expected, the
polarization vectors are radial for a toroidal magnetic field and toroidal for
radial magnetic field.
We now want to make a similar test in curved spacetime using a Kerr metric. To
validate the computation of EVPA in Gyoto, we compute the polarization from a
geometrically thin ring as in Gelles et al. (2021) taking as previously
$I_{\nu}=1$, $Q_{\nu}=I_{\nu}\,\cos\left(2*EVPA\right)$ and
$U_{\nu}=I_{\nu}\,\sin\left(2*EVPA\right)$. Gelles et al. (2021) consider a
synchrotron emission that we discard here, our only interest being in testing
the EVPA. Here, we are only interested in the orientation of the polarization
vector, that is, in the EVPA, at $r_{1}=3\,r_{g}$ and $r_{2}=6\,r_{g}$ (the
inner and outter radius of the ring). Figure 7 shows the tick plots for three
magnetic field configurations : radial, toroidal and vertical with the same
setup as in Fig. 1 of Gelles et al. (2021), the fluid being assumed to be
comoving with the Zero Angular Momentum Observer frame. The vertical
configuration implemented in Gyoto reads
$\mathrm{\textbf{Vertical}}\>b^{\alpha}=\left\\{\begin{array}[]{l}b^{t}=0,\\\
b^{r}=\frac{\cos{\theta}}{\sqrt{g_{rr}}},\\\
b^{\theta}=\frac{\sin{\theta}}{\sqrt{g_{\theta\theta}}},\\\
b^{\varphi}=0.\end{array}\right.$ (75)
The results of Gyoto shown in Fig. 7 are in perfect agreement with the ones in
the Fig. 1 in Gelles et al. (2021). We note that Gelles et al. (2021) scale
the length of the ticks by the observed synchrotron intensity, while our ticks
are all of unit length: we are only interested in the EVPA. This confirms that
the calculation of the EVPA works correctly and we can now test the radiative
transfer part and compare the results with another ray-tracing code.
Figure 7: Polarized tick plots for three idealized magnetic field
configurations: radial (left), toroidal (middle), and vertical (right) from a
geometrically thin ring seen by an almost face-on observer $i=0.1^{\circ}$.
The fluid is comoving with the Zero Angular Momentum Observer frame. Each plot
shows two spins ($a=0$ and $a=-0.99$ in red and blue, respectively) as two
emission radii ($r_{1}=3\,r_{g}$ and $r_{2}=6\,r_{g}$, corresponding to the
inner and outer rings, respectively).
### 3.3 Comparison with ipole
To check that all parts of our code work correctly, we compare the results of
Gyoto with the ones of another polarized ray-tracing code : ipole
(Mościbrodzka and Gammie, 2018b). We will focus on polarized observables of a
thick disk around a Schwarzschild black hole. We take power law profiles of
the physical quantities following Vos et al. (2022). To compute the emission,
absorption and rotation coefficients for the radiative transfer, we assume a
thermal distribution of electrons (as in Vos et al., 2022), and, for this
comparison only, we use the fitting formula of Dexter (2016) as used in Ipole
(we remind that Gyoto implements the formulae from Marszewski et al., 2021).
We compared the three magnetic field configurations (toroidal, radial and
vertical) described in Vos et al. (2022) at two inclinations, close to face-on
with $i=20$° and close to edge-on with $i=80$°. We define, as in Prather et
al. (2023), the normalized mean squared error (NMSE) as
$NMSE(A,B)=\frac{\sum|A_{j}-B_{j}|^{2}}{\sum|A_{j}|^{2}}$ (76)
where $A_{j}$ and $B_{j}$ are the intensities of a particular Stokes parameter
in two images at pixel $j$. The results of Gyoto are in perfect agreement with
Ipole with a NMSE $<10^{-4}$ for all configurations and Stokes parameters
except for Stoke U in the radial cases at high inclination for which the NMSE
is around $10^{-3}$. This can be compared to the worst NMSE of $\sim 0.01$
obtained in the code comparison made in Prather et al. (2023) showing the
perfect agreement between Gyoto and Ipole.
We also performed pixel-to-pixel comparisons, not restricting our comparison
to integrated quantities like the NMSE. Fig. 8 illustrates this for 128x128
pixels images of the four Stokes parameters computed by Gyoto and their
relative difference with Ipole, in a field of view of $40\,r_{g}$, for the
three magnetic field configurations described above, at low inclination
$i=20^{\circ}$. The relative error map is very close to zero for the vast
majority of pixels, with typical values $\lesssim 0.1\%$. Higher values are
only reached on specific tracks, that correspond to the zeroes of the
corresponding Stokes parameters (as is clear by comparing with the panels
showing the maps of the corresponding Stokes parameters). It is thus not
surprising to get higher residuals there, and it does not affect the radiative
transfer, given that the Stokes parameters are anyway close to zero in these
regions. Besides these tracks corresponding to the zeroes of the Stokes
parameters, the interior of the “shadow” region (i.e. geodesics that
asymptotically approach the horizon when backward ray traced) lead to higher
error. This is due to the stop condition of the geodesic integration that
differs in the two codes. Given that this part of the image is anyway strongly
redshifted and leads to a very low flux, this has no impact on the field-of-
view integrated comparison of the NMSE.
Figure 8: The first, third and fifth lines show images from left to right of
the four Stokes parameters (I, Q, U, V) generated by Gyoto and $Q^{2}+U^{2}$
as the last column for the three magnetic field configuration at low
inclination ($i=20$°). Their relative error maps with ipole images are shown
in the second, fourth and sixth lines. The relative errors scale is linear
between $\pm 5\%$.
## 4 Conclusion
This article presents the polarized version of the ray-tracing code Gyoto.
After reviewing the formalism for polarized GRRT, our main aim is to explain
the details of our implementation, and to provide tests of our code. In
particular, we have shown that in the framework of the GRRT computation of a
geometrically thick, optically thin accretion flow, we find results in perfect
agreement with the ipole code.
Polarized GRRT is of fundamental importance for interpreting current and
future observed data, in particular that of GRAVITY, the EHT, the polarized
loops of ALMA associated with Sgr A* flares, or the data of IXPE. Properly
interpreting these data is key to better understanding the properties of
plasmas in the extreme environments of black holes, and might offer new
interesting probes of strong-field gravity.
The polarized Gyoto code is public, actively maintained, and in constant
development for offering ever more diversed setups for relativistic
astrophysics. The recent polarized version of the code is accompanied by a
python notebook, available at
https://github.com/gyoto/Gyoto/blob/master/doc/examples/Gyoto_Polar_example.ipynb,
offering a quick and user-friendly first example of the new environment.
## Appendix A Observer’s screen polarization basis
Figure 9: Local observer’s basis
$(\mathbf{\bar{e}_{1}},\mathbf{\bar{e}_{2}},\mathbf{\bar{e}_{3}})$, in green,
as represented in Fig. 2. The polarization basis corresponding to the central
pixel of the screen, that is, to a purely radial direction of incidence, is
$(\mathbf{\bar{K}_{0}^{\mathrm{cen}}},\mathbf{\bar{w}_{0}^{\mathrm{cen}}},\mathbf{\bar{n}_{0}^{\mathrm{cen}}})$,
and is aligned with the oberver’s local basis vectors. When considering a non-
central pixel, the direction of photon incidence is no longer purely radial.
The corresponding vector $\mathbf{\bar{K}_{0}}$ is obtained from
$\mathbf{\bar{K}_{0}^{\mathrm{cen}}}$ by a rotation of angle $a$ around the
vector $\mathbf{v}$ (in red) normal to the red plane that contains
$\mathbf{\bar{e}_{3}}$ and makes an angle $b$ with $\mathbf{\bar{e}_{1}}$. The
same applies for $\mathbf{\bar{w}_{0}}$ and $\mathbf{\bar{n}_{0}}$.
A general vector $\mathbf{\bar{K}_{0}}$, defined by the two spherical angles
$(a,b)$, see the right panel of Fig. 2, is obtained by rotating the vector
$\mathbf{\bar{K}_{0}^{\mathrm{cen}}}$ by an angle $a$ in a plane containing
$\mathbf{\bar{e}_{3}}$ and making an angle $b$ with the $\mathbf{\bar{e}_{1}}$
vector (see the red plane in Fig. 9). This corresponds to a rotation of angle
$a$ around the vector $\mathbf{v}=-\sin b\,\mathbf{\bar{e}_{1}}+\cos
b\,\mathbf{\bar{e}_{2}}$. The corresponding rotation matrix reads
$R(a,b)=\left(\begin{array}[]{cccc}\sin^{2}b\left(1-\cos a\right)+\cos a&-\sin
b\cos b\left(1-\cos a\right)&\cos b\sin a\\\ -\sin b\cos b\left(1-\cos
a\right)&\cos^{2}b\left(1-\cos a\right)+\cos a&\sin b\cos a\\\ -\cos b\sin
a&-\sin b\sin a&\cos a\end{array}\right).$ (77)
The general vector $\mathbf{\bar{K}_{0}}$ then reads
$\mathbf{\bar{K}_{0}}=R(a,b)\,\mathbf{\bar{K}_{0}^{\mathrm{cen}}}$, and
similarly for the two other polarization vectors. It is then easy to express
the general local unit West and North directions that complete the local
incoming photon direction given by Eq. 2.2. They read
$\displaystyle\mathbf{\bar{w}_{0}}$ $\displaystyle=$
$\displaystyle\left[-\sin^{2}b\left(1-\cos a\right)-\cos
a\right]\mathbf{\bar{e}_{1}}+\sin b\cos b\left(1-\cos
a\right)\mathbf{\bar{e}_{2}}$ $\displaystyle+\cos b\sin
a\,\mathbf{\bar{e}_{3}},$ $\displaystyle\mathbf{\bar{n}_{0}}$ $\displaystyle=$
$\displaystyle-\sin b\cos b\left(1-\cos
a\right)\mathbf{\bar{e}_{1}}+\left[\cos^{2}b\left(1-\cos a\right)+\cos
a\right]\mathbf{\bar{e}_{2}}$ $\displaystyle-\sin b\sin
a\,\mathbf{\bar{e}_{3}}.$
## Appendix B Electron gyration and polarization vector direction
Our discussion of the polarization vector in section 2.3 took place in the
fluid frame. Here, we need to discuss a third natural frame (after the
observer frame and the fluid frame) naturally associated with the GRRT
problem, namely the particle frame, the frame comoving with an individual
electron swirling around the magnetic field lines and emitting synchrotron
radiation as a consequence of this accelerated motion (see Fig. 3). The
description of this infinitesimal 666Infinitesimal as compared to the natural
scale of our problem coinciding with the gravitational radius. motion is the
topic of this appendix. Our goal is to express the radiation electric field of
an individual electron, as measured in the fluid frame. The direction of this
vector should coincide with that of the polarization vector, which was
introduced in Eq. 36 without any reference to the electron’s motion, nor to
its radiation field.
Let us consider the standard picture of synchrotron emission by a relativistic
electron illustrated in Fig. 10, using the same notation and following the
derivation of Westfold (1959). In the fluid frame, we consider a direct
orthonormal triad,
$(\boldsymbol{\mathcal{I}},\boldsymbol{\mathcal{J}},\boldsymbol{\mathcal{K}})$,
such that $\boldsymbol{\mathcal{I}}$ is antiparallel to the acceleration
vector at some initial time $t=0$ (coinciding with the proper time of the
fluid frame), $\boldsymbol{\mathcal{K}}$ is along the ambient magnetic field
$\mathbf{b}$ (measured in the fluid frame), and $\boldsymbol{\mathcal{J}}$
completes the triad. We call $\boldsymbol{\beta}$ the velocity 3-vector of the
electron in the fluid frame and $\boldsymbol{\dot{\beta}}$ the corresponding
acceleration. A synchrotron wave is emitted by the accelerated electron in the
unit direction $\mathbf{\bar{K}}$ (measured in the fluid frame). The emission
angle $\theta_{B}$ and the pitch angle $\alpha$ are illustrated in Fig. 10.
The velocity and acceleration read
$\displaystyle\boldsymbol{\beta}$ $\displaystyle=$
$\displaystyle\beta\sin\alpha\left(-\sin\omega_{B}t\,\boldsymbol{\mathcal{I}}+\cos\omega_{B}t\,\boldsymbol{\mathcal{J}}\right)+\beta\cos\alpha\,\boldsymbol{\mathcal{K}},$
(79) $\displaystyle\boldsymbol{\dot{\beta}}$ $\displaystyle=$
$\displaystyle-\omega_{B}\beta\sin\alpha\left(\cos\omega_{B}t\,\boldsymbol{\mathcal{I}}+\sin\omega_{B}t\,\boldsymbol{\mathcal{J}}\right),$
where $\beta$ is the velocity of the electron in units of the speed of light,
and $\omega_{B}$ is the cyclotron gyrofrequency. It is easy to check that at
$t=0$, the projection of the velocity vector orthogonal to $\mathbf{b}$ is
along $\boldsymbol{\mathcal{J}}$, and the acceleration vector is along
$-\boldsymbol{\mathcal{I}}$, which is the setup illustrated in Fig. 10.
The radiation field of the moving charge in the fluid frame satisfies
$\mathbf{E_{\mathrm{rad}}}\propto\mathbf{\bar{K}}\times\left[\left(\mathbf{\bar{K}}-\boldsymbol{\beta}\right)\times\boldsymbol{\dot{\beta}}\right].$
(80)
Let us consider the unit direction of emission of a synchrotron photon,
written in full generality as
$\mathbf{\bar{K}}=a\,\boldsymbol{\mathcal{I}}+b\,\boldsymbol{\mathcal{J}}+c\,\boldsymbol{\mathcal{K}},$
(81)
where $(a,b,c)$ are arbitrary real numbers such that $\mathbf{\bar{K}}$ is a
unit vector. It is easy to show that
$\mathbf{E_{\mathrm{rad}}}\cdot\boldsymbol{\mathcal{K}}\propto\beta\cos\alpha-c.$
(82)
The lhs quantity represents the projection of the radiating electric field
along the ambient magnetic field direction. It is well known that the beaming
effect leads to the radiation being confined within a narrow cone around the
pitch angle of the relativistically moving electron (see Fig. 6.5 of Rybicki
and Lightman, 1979). This exactly means that
$\beta\cos\alpha-c\approx 0,$ (83)
such that the radiating electric field is orthogonal to the ambient magnetic
field. It follows that
$\mathbf{E_{\mathrm{rad}}}\propto\mathbf{\bar{K}}\times\mathbf{b},$ (84)
so that the rhs can be used to define the direction of the synchrotron
polarization vector, as done in Eq. 36.
Figure 10: Geometry of the synchrotron-emitting electron motion in the fluid
frame. $\boldsymbol{\beta}$ is the velocity 3-vector, $\mathbf{\bar{K}}$ is
the direction of emission, $\mathbf{b}$ is the magnetic vector, all these
quantities defined in the fluid frame. The emission angle between the magnetic
field direction and the direction of emission is called $\theta_{B}$ while
$\alpha$ is the pitch angle between the magnetic field direction and the
velocity of the electron.
## Appendix C Solving the polarized radiative transfer equation
Let us start with the unpolarized radiative transfer equation in the emitter
frame
$\frac{\mathrm{d}I}{\mathrm{d}s}=-\alpha I+j$ (85)
with obvious notations. The formal solution reads
$I(\tau)=\int_{s_{0}}^{s}\mathrm{exp}\left(-(\tau-\tau^{\prime})\right)S(\tau^{\prime})\mathrm{d}\tau^{\prime}$
(86)
where $S=j/\alpha$ is the source function, $\mathrm{d}\tau=\alpha\,\delta s$
is the optical depth, $s$ is the proper length in the emitter frame and
$s_{0}$ is some initial value of $s$ where the intensity is assumed to be
zero. This equation can equivalently be written
$I(s)=\int_{s_{0}}^{s}\mathrm{exp}\left(-\alpha(s-s^{\prime})\right)j(s^{\prime})\mathrm{d}s^{\prime}$
(87)
where $\alpha$ is assumed constant over the integration interval. Considering
a small range $\delta s=s-s^{\prime}$ between some previous location
$s^{\prime}$ and some current location $s$, over which interval $j$ and
$\alpha$ can be considered constant in a realistic problem, the increment of
intensity reads
$\delta I(s)=j(s)\,\delta s\,\mathrm{exp}\left(-\alpha(s)\,\delta s\right).$
(88)
Let us now come back to the polarized version of the radiative transfer
equation (Eq. 69) which reads
$\frac{\mathrm{d}\boldsymbol{\mathcal{I}}}{\mathrm{d}s}=-\boldsymbol{\mathcal{K}}\boldsymbol{\mathcal{I}}+\boldsymbol{\mathcal{J}}$
(89)
where $\boldsymbol{\mathcal{I}}$ is the vector of Stokes parameters and
$\boldsymbol{\mathcal{K}}=\mathbf{R}\left(\chi\right)\left(\begin{array}[]{cccc}\alpha_{I}&\alpha_{Q}&\alpha_{U}&\alpha_{V}\\\
\alpha_{Q}&\alpha_{I}&r_{V}&-r_{U}\\\ \alpha_{U}&-r_{V}&\alpha_{I}&r_{Q}\\\
\alpha_{V}&r_{U}&-r_{Q}&\alpha_{I}\end{array}\right)\mathbf{R}\left(-\chi\right);\qquad\boldsymbol{\mathcal{J}}=\mathbf{R}\left(\chi\right)\left(\begin{array}[]{c}j_{I}\\\
j_{Q}\\\ j_{U}\\\ j_{V}\end{array}\right)$ (90)
with $\mathbf{R}\left(\chi\right)$ being the rotation matrix given in Eq. 70.
Its formal solution is the direct generalization of Eq. 86, provided that
$\boldsymbol{\mathcal{K}}$ is constant with $s$:
$\boldsymbol{\mathcal{I}}(s)=\int_{s_{0}}^{s}\mathrm{exp}\left(-\boldsymbol{\mathcal{K}}(s-s^{\prime})\right)\boldsymbol{\mathcal{J}}(s^{\prime})\mathrm{d}s^{\prime}.$
(91)
Let us introduce the matrix
$\mathbf{O}(s,s^{\prime})=\mathrm{exp}\left(-\boldsymbol{\mathcal{K}}(s-s^{\prime})\right).$
(92)
Over a small interval of proper length $\delta s=s-s^{\prime}$, over which the
absorption matrix and emission vector can be considered constant, the
elementary increase of Stokes parameters is
$\delta\boldsymbol{\mathcal{I}}(s)=\mathbf{O}(\delta
s)\boldsymbol{\mathcal{J}}(s)\delta s$ (93)
which is the direct generalization of Eq. 88. This is the equation used in
Gyoto to update the Stokes parameters along the light ray.
We still need to compute the exponential of matrix appearing in Eq. 92. Landi
Degl’Innocenti and Landi Degl’Innocenti (1985) have given an expression for
this matrix. It reads
$\displaystyle\mathbf{O}(\delta s)=\mathrm{exp}\left(-\alpha_{I}\delta
s\right)$ $\displaystyle\left\\{\left[\mathrm{cosh}\left(\Lambda_{1}\delta
s\right)+\cos\left(\Lambda_{2}\delta s\right)\right]\mathbf{M_{1}}/2\right.$
$\displaystyle\left.-\sin\left(\Lambda_{2}\delta
s\right)\mathbf{M_{2}}\right.$
$\displaystyle\left.-\mathrm{sinh}\left(\Lambda_{1}\delta
s\right)\mathbf{M_{3}}\right.$
$\displaystyle\left.+\left[\mathrm{cosh}\left(\Lambda_{1}\delta
s\right)-\cos\left(\Lambda_{2}\delta s\right)\right]\mathbf{M_{4}}/2\right\\}$
with
$\displaystyle\mathbf{M_{1}}$ $\displaystyle=$ $\displaystyle\mathbf{1},$ (95)
$\displaystyle\mathbf{M_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{\Theta}\left(\begin{array}[]{cccc}0&\Lambda_{2}\tilde{\alpha}_{Q}-\sigma\Lambda_{1}\tilde{r}_{Q}&\Lambda_{2}\tilde{\alpha}_{U}-\sigma\Lambda_{1}\tilde{r}_{U}&\Lambda_{2}\tilde{\alpha}_{V}-\sigma\Lambda_{1}\tilde{r}_{V}\\\
\Lambda_{2}\tilde{\alpha}_{Q}-\sigma\Lambda_{1}\tilde{r}_{Q}&0&\sigma\Lambda_{1}\tilde{\alpha}_{V}+\Lambda_{2}\tilde{r}_{V}&-\sigma\Lambda_{1}\tilde{\alpha}_{U}-\Lambda_{2}\tilde{r}_{U}\\\
\Lambda_{2}\tilde{\alpha}_{U}-\sigma\Lambda_{1}\tilde{r}_{U}&-\sigma\Lambda_{1}\tilde{\alpha}_{V}-\Lambda_{2}\tilde{r}_{V}&0&\sigma\Lambda_{1}\tilde{\alpha}_{Q}+\Lambda_{2}\tilde{r}_{Q}\\\
\Lambda_{2}\tilde{\alpha}_{V}-\sigma\Lambda_{1}\tilde{r}_{V}&\sigma\Lambda_{1}\tilde{\alpha}_{U}+\Lambda_{2}\tilde{r}_{U}&-\sigma\Lambda_{1}\tilde{\alpha}_{Q}-\Lambda_{2}\tilde{r}_{Q}&0\end{array}\right),$
(100) $\displaystyle\mathbf{M_{3}}$ $\displaystyle=$
$\displaystyle\frac{1}{\Theta}\left(\begin{array}[]{cccc}0&\Lambda_{1}\tilde{\alpha}_{Q}+\sigma\Lambda_{2}\tilde{r}_{Q}&\Lambda_{1}\tilde{\alpha}_{U}+\sigma\Lambda_{2}\tilde{r}_{U}&\Lambda_{1}\tilde{\alpha}_{V}+\sigma\Lambda_{2}\tilde{r}_{V}\\\
\Lambda_{1}\tilde{\alpha}_{Q}+\sigma\Lambda_{2}\tilde{r}_{Q}&0&-\sigma\Lambda_{2}\tilde{\alpha}_{V}+\Lambda_{1}\tilde{r}_{V}&\sigma\Lambda_{2}\tilde{\alpha}_{U}-\Lambda_{1}\tilde{r}_{U}\\\
\Lambda_{1}\tilde{\alpha}_{U}+\sigma\Lambda_{2}\tilde{r}_{U}&\sigma\Lambda_{2}\tilde{\alpha}_{V}-\Lambda_{1}\tilde{r}_{V}&0&-\sigma\Lambda_{2}\tilde{\alpha}_{Q}+\Lambda_{1}\tilde{r}_{Q}\\\
\Lambda_{1}\tilde{\alpha}_{V}+\sigma\Lambda_{2}\tilde{r}_{V}&-\sigma\Lambda_{2}\tilde{\alpha}_{U}+\Lambda_{1}\tilde{r}_{U}&\sigma\Lambda_{2}\tilde{\alpha}_{Q}-\Lambda_{1}\tilde{r}_{Q}&0\end{array}\right),$
(105) $\displaystyle\mathbf{M_{4}}$ $\displaystyle=$
$\displaystyle\frac{2}{\Theta}$ (110) $\displaystyle\times$
$\displaystyle\left(\begin{array}[]{cccc}(\tilde{\alpha}^{2}+\tilde{r}^{2})/2&\tilde{\alpha}_{V}\tilde{r}_{U}-\tilde{\alpha}_{U}\tilde{r}_{V}&\tilde{\alpha}_{Q}\tilde{r}_{V}-\tilde{\alpha}_{V}\tilde{r}_{Q}&\tilde{\alpha}_{U}\tilde{r}_{Q}-\tilde{\alpha}_{Q}\tilde{r}_{U}\\\
\tilde{\alpha}_{U}\tilde{r}_{V}-\tilde{\alpha}_{V}\tilde{r}_{U}&\tilde{\alpha}_{Q}^{2}+\tilde{r}_{Q}^{2}-(\tilde{\alpha}^{2}+\tilde{r}^{2})/2&\tilde{\alpha}_{Q}\tilde{\alpha}_{U}+\tilde{r}_{Q}\tilde{r}_{U}&\tilde{\alpha}_{V}\tilde{\alpha}_{Q}+\tilde{r}_{V}\tilde{r}_{Q}\\\
\tilde{\alpha}_{V}\tilde{r}_{Q}-\tilde{\alpha}_{Q}\tilde{r}_{V}&\tilde{\alpha}_{Q}\tilde{\alpha}_{U}+\tilde{r}_{Q}\tilde{r}_{U}&\tilde{\alpha}_{U}^{2}+\tilde{r}_{U}^{2}-(\tilde{\alpha}^{2}+\tilde{r}^{2})/2&\tilde{\alpha}_{U}\tilde{\alpha}_{V}+\tilde{r}_{U}\tilde{r}_{V}\\\
\tilde{\alpha}_{Q}\tilde{r}_{U}-\tilde{\alpha}_{U}\tilde{r}_{Q}&\tilde{\alpha}_{V}\tilde{\alpha}_{Q}+\tilde{r}_{V}\tilde{r}_{Q}&\tilde{\alpha}_{U}\tilde{\alpha}_{V}+\tilde{r}_{U}\tilde{r}_{V}&\tilde{\alpha}_{V}^{2}+\tilde{r}_{V}^{2}-(\tilde{\alpha}^{2}+\tilde{r}^{2})/2\end{array}\right)$
where
$\displaystyle\tilde{\alpha}^{2}$ $\displaystyle=$
$\displaystyle\tilde{\alpha}_{Q}^{2}+\tilde{\alpha}_{U}^{2}+\tilde{\alpha}_{V}^{2},$
(111) $\displaystyle\tilde{r}^{2}$ $\displaystyle=$
$\displaystyle\tilde{r}_{Q}^{2}+\tilde{r}_{U}^{2}+\tilde{r}_{V}^{2},$
$\displaystyle\Lambda_{1}$ $\displaystyle=$
$\displaystyle\sqrt{\sqrt{\frac{1}{4}(\tilde{\alpha}^{2}-\tilde{r}^{2})^{2}+(\tilde{\alpha}_{Q}\tilde{r}_{Q}+\tilde{\alpha}_{U}\tilde{r}_{U}+\tilde{\alpha}_{V}\tilde{r}_{V})^{2}}+\frac{1}{2}(\tilde{\alpha}^{2}-\tilde{r}^{2})},$
$\displaystyle\Lambda_{2}$ $\displaystyle=$
$\displaystyle\sqrt{\sqrt{\frac{1}{4}(\tilde{\alpha}^{2}-\tilde{r}^{2})^{2}+(\tilde{\alpha}_{Q}\tilde{r}_{Q}+\tilde{\alpha}_{U}\tilde{r}_{U}+\tilde{\alpha}_{V}\tilde{r}_{V})^{2}}-\frac{1}{2}(\tilde{\alpha}^{2}-\tilde{r}^{2})},$
$\displaystyle\Theta$ $\displaystyle=$
$\displaystyle\Lambda_{1}^{2}+\Lambda_{2}^{2},$ $\displaystyle\sigma$
$\displaystyle=$
$\displaystyle\mathrm{sign}\left(\tilde{\alpha}_{Q}\tilde{r}_{Q}+\tilde{\alpha}_{U}\tilde{r}_{U}+\tilde{\alpha}_{V}\tilde{r}_{V}\right),$
and where the tilde quantities $\tilde{\alpha}_{X}$, $\tilde{r}_{X}$ take into
account the basis rotation by an angle $\chi$ and read
$\left(\begin{array}[]{c}\tilde{\alpha}_{Q}\\\
\tilde{\alpha}_{U}\end{array}\right)=\left(\begin{array}[]{cc}\cos 2\chi&-\sin
2\chi\\\ \sin 2\chi&\cos
2\chi\end{array}\right)\left(\begin{array}[]{c}\alpha_{Q}\\\
\alpha_{U}\end{array}\right),$ (112)
and similarly for $\tilde{r}_{Q}$ and $\tilde{r}_{U}$, while
$\tilde{\alpha}_{I,V}$ and $\tilde{r}_{V}$ are the same as their counterparts
without a tilde, given that $I$ and $V$ are not affected by the rotation.
## Acknowledgements
It is a pleasure for the authors to acknowledge continuous discussions with
Maciek Wielgus on the topic of the paper. The authors gratefully acknowledge
fruitful discussions with B. Cerutti and B. Crinquand, as well as very useful
exchanges with J. Vos for the code comparison of section 3.3. E. G.
acknowledges funding by l’Agence Nationale de la Recherche, Project StronG
ANR-22-CE31-0015.
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# Dynamical Consequence of Shadows Cast to the Outer Protoplanetary Disks: I.
Two-dimensional Simulations
Zehao Su School of Physics and Astronomy, Beijing Normal University, Beijing
100875, China<EMAIL_ADDRESS>Institute for Frontier in Astronomy and
Astrophysics, Beijing Normal University, Beijing 102206, China; Xue-Ning Bai
Institute for Advanced Study, Tsinghua University, Beijing 100084, China;
<EMAIL_ADDRESS>Department of Astronomy, Tsinghua University, Beijing
100084, China
###### Abstract
There has been increasing evidence of shadows from scattered light
observations of outer protoplanetary disks (PPDs) cast from the (unresolved)
disk inner region, while in the meantime these disks present substructures of
various kinds in the submillimeter. As stellar irradiation is the primary
heating source for the outer PPDs, the presence of such shadows thus suggest
inhomogeneous heating of the outer disk in azimuth, leading to a “thermal
forcing” with dynamical consequences. We conduct a suite of idealized 2D disk
simulations of the outer disk with azimuthally-varying cooling prescription to
mimic the effect of shadows, generally assuming the shadow is static or
slowly-rotating. The linear response to such shadows is two-armed spirals with
the same pattern speed as the shadow. Towards the nonlinear regime, we find
that shadows can potentially lead to the formation of a variety of types of
substructures including rings, spirals and crescents, depending on viscosity,
cooling time, etc. We have conducted systematic and statistical
characterization of the simulation suite, and as thermal forcing from the
shadow strengthens, the dominant form of shadow-induced disk substructures
change from spirals to rings, and eventually to crescents/vortices. Our
results highlight the importance of properly modeling the dynamical impact of
inhomogeneous stellar irradiation, while call for more detailed modeling
incorporating more realistic disk physics.
Protoplanetary disks-shadows-hydrodynamics substructures
## 1 Introduction
Thanks to the advent of new observational facilities and instruments to
conduct spatially resolved observations of protoplanetary disks (PPDs), it has
now been well established that disk substructures are ubiquitous (e.g. Bae et
al., 2023). In the millimeter/sub-millimeter wavelengths, the Atacama Large
Millimeter Array (ALMA) has revealed the richness of disk substructures that
are primarily in the form of rings and gaps in addition to other features such
as spirals and crescents at different radii (e.g. ALMA Partnership et al.,
2015; Monnier et al., 2017; Avenhaus et al., 2018; Isella et al., 2018; Huang
et al., 2020; Gratton et al., 2019; Andrews, 2020). These observations reflect
the thermal emission from mm-sized dusts around the disk midplane, which are
biased tracers of the gas density profiles due to finite aerodynamic coupling
between gas and dust. In the optical and near infrared (NIR), high-contrast
imaging with extreme adaptive optics (e.g., VLT/SHERE, VLT/CRIRES, GPI) reveal
even richer and more complex features (e.g. Benisty et al., 2015; Pinilla et
al., 2015; Pohl et al., 2017; van Boekel et al., 2017; Garufi et al., 2018;
Benisty et al., 2023). Emission at optical/NIR mainly result from the
starlight scattered by micron-sized dust (better coupled to the gas) suspended
in the disk, which are better tracers of the disk surface layers. As a result,
features seen in scattered light do not necessarily have direct correspondence
to substructures recognized by ALMA (e.g. van der Marel et al., 2016; Uyama et
al., 2018; Pérez et al., 2018; Muro-Arena et al., 2018).
At least partly contributing to the complexity in features seen in scattered
light is the presence of shadows, typically defined as low-intensity regions
that are confined to specific azimuthal angles (Benisty et al., 2023). They
must be cast from the (unresolved) disk inner region, and can be mainly
classified into two types: broad extended shadows in azimuthal directions
(e.g. Muro-Arena et al., 2020) and narrow shadow lanes with only a few degrees
(e.g. Ginski et al., 2021). Considerable effort has been devoted to
understanding the origin of shadows because the morphology and temporal
variation of shadows can provide indirect information about the disk’s inner
regions. The most common case for shadow casting is the presence of a
misaligned/warped inner disk. For instance, TW Hya shows a moving shadow
pattern that could suggests a precessing inner disk (Debes et al., 2017),
shadows in HD 143006 can be reproduced using a $30^{\circ}$ misaligned inner
disk (Benisty et al., 2018), fast time variations of shadows in RX
J1604.3-2130A may come from dust very close to the central star in an
irregular misaligned inner disk (Pinilla et al., 2015), narrow shadow lanes in
SU Aur that possibly suggest misalignment caused by late-time interactions
with infalling materials (Kuffmeier et al., 2021), shadows in HD 139614 can be
explained by the combination of a misaligned inner ring and disk (Muro-Arena
et al., 2020), and the flux ratio switches sides in brightest nebula of
IRAS40302 can be achieved by applying a tilted inner disk (Villenave et al.,
2023). Even for disks with nearly aligned inner regions, subtle shadowing
effects can still be recognized (Monnier et al., 2017). In addition to the
misalignment of the inner disk regions, variations in the scale height of the
inner disk atmosphere could also be responsible for generating shadows, such
as in HD 163296 (Rich et al., 2019, 2020; Varga et al., 2021; GRAVITY
Collaboration et al., 2021).
Most effort aiming to understand disk shadows so far has focused on the
modeling the (inner) disk morphology to explain the shadow features using
radiative transfer calculations (e.g. Casassus et al., 2018; Benisty et al.,
2018; Nealon et al., 2019; Muro-Arena et al., 2020). On the other hand, we
note that as stellar irradiation is the primary source of heating in the bulk
of the (outer) PPDs, the presence of shadows must also give rise to dynamical
consequences by “thermal forcing”: the disk gas experiences (quasi-)periodic
cooling and heating as it enters and exits the shadow, which hardly settles to
thermal equilibrium and constantly exerts modest or even strong pressure
perturbations to the neighboring fluid. This effect was first explored in
Montesinos et al. (2016), who conducted 2D hydrodynamic simulations that take
into account both stellar irradiation and periodic forcing of shadows with an
opening angle of $28^{\circ}$ in the context of the transition disk HD 142527.
They found that azimuthal pressure gradients generated by shadows can trigger
$m=2$ spirals, which are enhanced by self-gravity and give rise to observable
quasi-steady spiral signals.
In this work, motivated by the diversity of shadowing features seen in
scattered light images and the case study by Montesinos et al. (2016) for the
HD 142527 disk, we aim at a systematic exploration on the dynamical
consequences of shadows cast onto outer PPDs. As an initial effort, we
restrict ourselves to vertically-integrated systems by 2D hydrodynamic
simulations. We follow the evolution of a passive, viscous gaseous disk with a
thermal relaxation prescription towards a target temperature, which is set by
stellar irradiation subject to shadowing. By exploring a large suite of
simulations varying the viscosity, cooling time, and shadow geometry, we find
that shadows can result in the formation of a wide variety of disk
substructures, and perform a statistical analysis of all the substructures
generated from our simulations.
This paper is organized as follows. We detail our simulation setup in Section
2, followed by a description representative features of shadow-driven disk
substructures in Section 3. We present statistical analysis of all our
simulations and describe the substructure-forming process from linear to non-
linear regimes in Section 4. Finally, we discuss the caveats and conclude in
Section 6.
## 2 Numerical Methods
### 2.1 Simulation Setup
We solve the vertically integrated viscous hydrodynamic equations using the
grid-based higher-order Godunov code ATHENA++ Stone et al. (2020) in
cylindrical coordinates $(r,\varphi)$. The conservative form of control
equations are:
$\frac{\partial\Sigma}{\partial t}+\nabla\cdot(\Sigma\textbf{v})=0,$ (1)
$\frac{\partial\Sigma\textbf{v}}{\partial
t}+\nabla\cdot(\Sigma\textbf{v}\textbf{v}+P\mathcal{I}+\mathcal{T}_{vis})=-\Sigma\nabla\Phi,$
(2)
$\frac{\partial E}{\partial
t}+\nabla\cdot\left[(E+P+\mathcal{T}_{vis})\textbf{v}\right]=-\Sigma\textbf{v}\cdot\nabla\Phi+\Lambda,$
(3)
where $\Sigma$ is the gas surface density in the disk, v is the gas velocity,
$P=\Sigma T$ is the vertically integrated pressure where $T$ is the disk
temperature, $\mathcal{I}$ is the identity tensor, $\Phi$ is the gravitational
potential written as $\Phi=-GM_{*}/r$ where $M_{*}$ is mass of central star,
$E$ is total energy density, and $\Lambda$ gives the cooling source terms. The
viscous stress tensor $\mathcal{T}_{visc}$ in momentum equation reads
$\mathcal{T}_{vis}=-\Sigma\nu\left(\frac{\partial v_{i}}{\partial
x_{j}}+\frac{\partial v_{j}}{\partial x_{i}}-\frac{2}{3}\frac{\partial
v_{k}}{\partial x_{k}}\delta_{ij}\right),$ (4)
with $\nu$ being the kinematic viscosity.
The total energy density $E$ is given by the combination of kinetic energy and
internal energy:
$E=\frac{1}{2}\Sigma v^{2}+\frac{P}{\gamma-1},$ (5)
where $\gamma=7/5$ is the adiabatic index for molecular gas. Note that viscous
heating is automatically included in the energy equation, although it is
generally unimportant in the outer PPDs. The gas temperature $T$ is associated
with the isothermal sound speed as $T=c_{s}^{2}$, which yields the disk scale
height $H=c_{s}/\Omega_{K}$, where $\Omega_{K}=(GM_{*}/r^{3})^{1/2}$ is the
Keplerian angular frequency. The disk aspect ratio is then given by $h=H/r$.
With this, viscosity follows the standard $\alpha$ prescription (Shakura &
Sunyaev, 1973), $\nu=\alpha c_{s}H$. It is worth noticing that viscosity
varies as the disk evolves.
Figure 1: Illustration of our shadow morphology depicted by normalized
temperature in disks with $\sigma_{\phi}=0.236$ and $0.079$ for fast
($\beta=0.001$) and slow ($\beta=1$) cooling processes. With fast cooling, the
low-temperature regions caused by shadows largely reflect the target
temperature we prescribe at the shadow location. With longer cooling
timescale, these regions deviate from the prescribed shadow positions towards
the leading side, and exhibit significantly weaker amplitude.
We choose the initial density profile to be power-law:
$\Sigma_{\rm init}=\Sigma_{0}\left(\frac{r}{r_{0}}\right)^{d}.$ (6)
where $\Sigma_{0}$ is the density at reference radius $r_{0}$, which we take
to be the radius of the inner boundary. We specify the disk initial
temperature as
$T_{\rm init}\equiv
c_{s_{0}}^{2}\left(\frac{r}{r_{0}}\right)^{p}=\left(h_{0}r_{0}\Omega_{K_{0}}\right)^{2}\left(\frac{r}{r_{0}}\right)^{p},$
(7)
where $c_{s_{0}}$, $h_{0}$, $\Omega_{K_{0}}$ are the value of isothermal sound
speed, aspect ratio and Keplerian angular velocity at reference radius
$r_{0}$.
The radial force balance leads to the initial rotation profile calculated by
$v_{\varphi}(r)=\left[\left(p+d\right)c_{s}^{2}+\frac{GM_{*}}{r}\right]^{1/2}\
.$ (8)
With viscosity, the radial velocity is set by the accretion velocity given by
$v_{r}(r)=-\frac{3}{2}\frac{\alpha c_{s}^{2}}{r\Omega_{K}}\ .$ (9)
To ensure steady-state accretion in the initial equilibrium (shadow not
included) with constant $\alpha$, the initial temperature and density profile
should satisfy $d+p=-3/2$.
Our simulations are scale-free, adopting $GM_{*}=\Sigma_{0}=r_{0}=1$ in code
units, with $h_{0}=0.1$. As a result, we have $\Omega_{K_{0}}=c_{s_{0}}=1$.
The computational domain ranges from $r_{\rm in}=1$ to $r_{\rm out}=30$ in
code units, to ensure sufficient dynamical range. We employ a logarithmic grid
in radial direction and a uniform grid in azimuthal direction with
$N_{r}\times N_{\varphi}=512\times 512$, achieving a gird resolution of 15
cells per $H$ in $r$ while keeping cell aspect ratio $\Delta r\approx
0.5r\Delta\phi$.
#### 2.1.1 Shadow prescription
In our simulations, we assume an obscuring structure present in disk inner
region, which is outside of our simulation domain (inside the inner boundary).
As an initial study, we tentatively take this obscuring structure as a
slightly misaligned inner disk (inclination angle $\sim h$). In this case,
near half of the outer disk (in azimuth) is illuminated from one hemisphere,
and the opposite side being illuminated from the other hemisphere. In the
vertically-integrated sense, the two sides of the “pie-chart” are heated
largely equally. It is the transition region, the disk can be largely blocked
by the inner disk from both hemisphere, which is mostly affected by the
shadow. The shadow then introduces a thermal forcing to the system, causing
system’s temperature to approach the target temperature $T_{\rm tar}$. For
simplicity, we prescribe this target temperature by
$T_{\rm tar}(r,\phi)=T_{\rm init}(r)\left(1-\epsilon
e^{-\frac{\phi^{2}}{2\sigma_{\phi}^{2}}}\right)\left(1-\epsilon
e^{-\frac{(\phi-\pi)^{2}}{2\sigma_{\phi}^{2}}}\right),$ (10)
where $\epsilon$ reflects the amplitude of the shadow and $\sigma_{\phi}$
characterizes the azimuthal width of the shadow. Although we have argued that
two-sided shadows are the most basic case, we still examine the one-sided case
in C for reference. Also, in most cases in this paper, except for the
simulation mentioned in Section 5.3, the shadow is static with pattern speed
($\Omega_{\rm shadow}$) being zero.
The final temperature structure depends on heating and cooling process, which
is often modeled using the $\beta$ cooling approximation (Gammie, 2001). The
cooling term is given by thermal relaxation towards the target temperature
$\Lambda=-\frac{\Sigma}{\gamma-1}\times\frac{T-T_{\rm tar}}{t_{\rm cool}}\ ,$
(11)
where the cooling timescale is specified by the dimensionless parameter
$\beta$:
$t_{\rm cool}=\beta\Omega_{K}^{-1}\ .$ (12)
It describes the disk’s thermodynamic timescale, which can range from $\sim
10^{-3}$ (approaching the isothermal limit) to at least $\sim 10$ (approaching
the adiabatic limit) in our simulations. Figure 1 shows the expected
temperature structure for four representative shadow prescriptions, with
different shadow widths ($15^{\circ}$ and $45^{\circ}$) and cooling times,
calculated by following fluid elements undergoing heating and cooling on
circular orbits. We have fixed shadow amplitude of $\epsilon=0.8$. With fast
cooling ($\beta=0.001$), we see that the shadow aligns with its expected
position, and the temperature at its center is approximately $0.2T_{0}$ as
desired. When cooling is inefficient, the observed shadow center on the
leading side from its expected location, and the lowest temperature becomes
well above $0.2T_{0}$.
It should be noted that our shadow and cooling prescriptions are highly
simplified and are not necessarily always physical (for instance, a flat disk
with $p=-1$ would not be irradiated). We emphasize that the goal of this work
is not to precisely model any particular system, but to explore the general
phenomenology in a qualitative manner.
#### 2.1.2 Boundary Conditions
We use modified outflow boundary conditions, where hydrodynamic variables are
copied from the last grid zone assuming $\Sigma\propto r^{d}$, $P\propto
r^{d+p}$, $v_{\phi}\propto r^{-1/2}$, with $v_{r}$ unchanged—except we set
$v_{r}=0$ in case of inflow. To further dampen unphysical waves, we adopt
wave-killing functions in the form described by de Val-Borro et al. (2006):
$\frac{dx}{dt}=-\frac{x-x_{0}}{\tau_{\rm damp}}R(r)$ (13)
where $x$ represents any fluid quantities (e.g. $\Sigma$, v, etc.). The
damping timescale $\tau_{\rm damp}$ is defined as $\tau_{\rm
damp}=\eta\Omega_{K}^{-1}$, where $\eta$ is the damping rate and is set to 1
for all of our simulations. The function $R(r)$ is a parabolic function
expressed as:
$R(r)=\left(\frac{r-r_{\rm damp}}{L_{\rm damp}}\right)^{2},\ {\rm
for}\left|r-r_{\rm damp}\right|<L_{\rm damp}\ ,$ (14)
where $r_{\rm damp}$ is the boundary of the damping area which we take to be
$2.08$ and $26.57$ in inner and outer part of our computation domain,
respectively, and $L_{\rm damp}$ is the length of wave killing zone.
### 2.2 Simulation Runs
In order to comprehensively investigate the dynamical effects of shadows in
PPDs, we conducted a wide parameter scan. Five main parameters are included in
our simulations: dimensionless cooling timescale $\beta$ ranging from
$10^{-3}$ to $10$, viscosity coefficient $\alpha$ ranging from $0$ to
$10^{-2}$, shadow amplitude coefficient $\epsilon=0.5$ and $\epsilon=0.8$,
shadow width $\sigma_{\phi}=0.236$ and $\sigma_{\phi}=0.079$, and the
temperature slope $p=-1$ (flat case) and $p=-0.5$ (flared case). In the
viscous simulations, they translate to density gradient $d=-0.5$ and $d=-1$ to
ensure steady state accretion. In most simulations, the shadows do not rotate,
and we fix the disk aspect ratio $h_{0}=0.1$ at $r=1$, thus $h=0.1$ is
constant in most $p=-1$ (flat) cases. All of these simulations will be
discussed in Section 4. In Sections 5.3 and 5.4, we also briefly explore
simulations with rotating shadows and vary $h_{0}$ from $0.05$ to $0.15$.
To further comment on our choice of parameters, we first note that in outer
disk conditions, we generally expect $\beta\lesssim 1$ (e.g. Lin & Youdin,
2015; Pfeil & Klahr, 2019), though the finite thermal coupling between dust
and gas may significantly enhance the effective $\beta$ (e.g. Bae et al.,
2021). In inviscid simulations, we further examine the influence of the
density profile ($d=-0.5$ and $d=-1$, which affects thermal forcing) since
this parameter is no longer free when viscosity is included (dependent on
$p$). Note that in the new paradigm of wind-driven accretion, the disk is more
laminar and the surface density profile can be more arbitrary (e.g. Bai, 2016;
Suzuki et al., 2016; Tabone et al., 2022). Although we do not incorporate
wind-driven accretion, this exploration also serves the purpose to partly
mimic “windy” disk conditions. On the choice of shadow amplitudes, note that
given the $T^{4}$ dependence, the two choices correspond to the shadowed
region receiving about $0.2^{4}\approx 0.002$ and $0.5^{4}\approx 0.06$ of the
stellar irradiation compared to the non-shadowed regions.
In all of our simulations, the total run time is chosen to be $T=20000P_{0}$,
where $P_{0}=2\pi/\Omega_{K_{0}}$ is the orbital period at the inner boundary.
This is significantly longer than the timescales for substructure formation,
which we find to be within $5000P_{0}$ for most cases. In only a few cases
(especially $\beta\sim 10,\alpha\sim 0$), even on the timescale of
$20000P_{0}$, we cannot unambiguously identify the dominant form of disk
substructure. However, we can infer their evolution trend from a statistical
point of view.
To facilitate comparison of the various simulations discussed in the following
sections, we provide a list of all our runs and their parameters in Table 1.
Our naming convention is structured as follows. We use “L” for runs in the
linear regime ($\epsilon=0.001$) and “NL” for runs in the nonlinear regime
($\epsilon=0.5,0.8$). The labels “hs”, “hm” and “hl” indicate runs with
$h_{0}=0.05$, $h_{0}=0.1$, and $h_{0}=0.15$, respectively, with $h_{0}=0.1$ as
fiducial. To specify the dominant substructure, we use “S” for spiral-
dominant, “R” for ring-dominant, and “V” for vortex-dominant. Shadow
precession speeds are denoted as “NR” for nonrotating (fiducial), “FR” for
fast rotating, “MR” for moderately rotating, and “SR” for slow rotating. For
simulations dedicated to parameter searches discussed in Section 4, we use the
label “S-h-all,” as we do not discuss individual runs for these simulations.
Table 1: Summary of All Highlighted Simulations.1
Run | $\sigma_{\phi}$ | $\epsilon$ | $\alpha$ | $\beta$ | p | $h_{0}$ | $\Omega_{\rm shadow}$ |
---|---|---|---|---|---|---|---|---
Representative runs (Section 3) |
NL-hm-S-NR | 0.236 | 0.8 | $10^{-3}$ | 10 | -1.0 | 0.1 | 0 |
NL-hm-R-NR | 0.236 | 0.5 | $10^{-4}$ | 1 | -1.0 | 0.1 | 0 |
NL-hm-V-NR | 0.236 | 0.5 | $0$ | $10^{-3}$ | -1.0 | 0.1 | 0 |
Statistical runs (Section 4) |
S-h-all2 | $(0.236,0.079)$ | $(0.5,0.8)$ | $(0,10^{-4},10^{-3},10^{-2})$ | $(10^{-3},10^{-2},10^{-1},1,10)$ | $(-1.0,-0.5)$ | 0.1 | 0 |
Linear run (Section 5.1) |
L-hm-S-NR | 0.236 | 0.001 | 0 | $10^{-3}$ | -1.0 | 0.1 | 0 |
Rotating shadow runs (Section 5.3) |
L-hm-S-FR3 | 0.236 | 0.001 | 0 | $10^{-3}$ | -1.0 | 0.1 | $\Omega_{0}$ |
L-hm-S-MR | 0.236 | 0.001 | 0 | $10^{-3}$ | -1.0 | 0.1 | $0.03\Omega_{0}$ |
L-hm-S-SR | 0.236 | 0.001 | 0 | $10^{-3}$ | -1.0 | 0.1 | $0.003\Omega_{0}$ |
Aspect ratio test runs (Section 5.4) |
NL-hs-S-NR | 0.236 | 0.8 | $10^{-3}$ | 10 | -1.0 | 0.05 | 0 |
NL-hs-R-NR | 0.236 | 0.5 | $10^{-4}$ | 1 | -1.0 | 0.05 | 0 |
NL-hs-V-NR | 0.236 | 0.5 | 0 | $10^{-3}$ | -1.0 | 0.05 | 0 |
NL-hl-S-NR | 0.236 | 0.8 | $10^{-3}$ | 10 | -1.0 | 0.15 | 0 |
NL-hl-R-NR | 0.236 | 0.5 | $10^{-4}$ | 1 | -1.0 | 0.15 | 0 |
NL-hl-V-NR | 0.236 | 0.5 | $0$ | $10^{-3}$ | -1.0 | 0.15 | 0 |
11footnotetext: Simulations mentioned in Appendix A and Appendix C are not
included.
22footnotetext: The unified name for parameter sruvey simulations, with
parameters being all combinations of those listed in this row, totaling 160
simulations.
33footnotetext: The resolution of this run is set to $N=2048$.
Note. — $\sigma_{\phi}$: shadow width parameter; $\epsilon$: shadow amplitude
parameter; $\alpha$: viscosity parameter; $\beta$: cooling rate parameter; p:
temperature slope; $h_{0}$: disk aspect ratio at inner boundary; $\Omega_{\rm
shadow}$: shadow procession angular frequency; $\Omega_{0}$: Keplerain angular
velocity at $r=1$. All runs, except for run L-hm-S-FR, use a resolution of
$N=512$.
### 2.3 Diagnostics of Substructures
As we will demonstrate, our simulations generate a variety of substructures of
all types. In this section, we provide the diagnostics we employ to identify
and characterize these substructures. To minimize the influence of the
boundaries and wave-killing, we restrict the analysis domain to be
$r\in[3,21]$.
Vortices exhibit as anti-cyclonic flows with pressure maxima at the center
that can potentially be strong dust traps. They are identified as regions with
negative vorticity, which is defined as $\nabla\times\delta\textbf{v}$ with
$\delta\textbf{v}$ being the difference between current fluid velocity and
background fluid velocity. We quantify individual vortices based on their mean
vorticity (normalized by background Keplerian angular velocity), density
contrast, spacing, and aspect ratio. In doing so, we first choose the vortex
boundary to be where the density is $10\%$ of the density at the vortex center
after subtracting background, while ensuring that the vorticity remains below
zero. This is motivated from the analytical work of Lyra & Lin (2013) while
being robust to the influence of density waves. In our simulations, vortices
are constantly generated and destroyed; only the largest vortices are chosen
(usually can survive for at least 100 local orbits). We measure the density
contrast by comparing the average density in vortex with the average density
at the same radius. The spacing of vortices is calculated by the radial
distances between neighboring vortices, which are normalized by the local
scale height at the midpoint radius between the two vortices. As vortices can
be highly time variable, all quantities are calculated and averaged over
several snapshots (see in Section 4).
In ring-forming disks, we measure the density contrast, width, spacing and
eccentricity of the rings. We identify the rings by first fitting the
background density as a power law, and consider peaks/troughs above/below the
fitted profile as ring peaks/gap centers. The boundaries of the rings are
identified as the radius at the midpoint between peak and valley densities,
with ring width being the distance between the two boundaries for each ring.
Density contrast is calculated by comparing the density between peaks and
boundaries. The final ring width is obtained by averaging the widths of all
identified rings, and each ring width is normalized to the local scale height
of the disk. Ring spacing is measured as the radial distances between the
boundaries of two neighboring rings, normalized in a way similar to that for
vortices, and averaged over several snapshots. In the above, we have treated
the rings as axsymmetric by working with 1D profiles, whereas in practice we
have found that the rings can be eccentric. For identified rings, we further
track the maximum density in 2D data and measure their eccentricity by fitting
an ellipse. Incomplete rings at the boundary of the analysis domain are
excluded from the statistics.
For spirals, we quantify their density contrast, number of spiral arms,
pattern speed and pitch angle. The density contrast is obtained by comparing
density of the spiral spine and the fitted background density at same radii.
In our simulations, we obtain the spiral phase at each radius using Fourier
decomposition and the pitch angle is obtained by fitting the phase angle with
a logarithmic function $\varphi=m(\tan\alpha_{p})^{-1}\ln r+\phi_{0}$, where
$\alpha_{p}$ is the pitch angle and $m$ is number of spiral arms. The constant
$\phi_{0}$ is further employed to measure the pattern speed of the spirals.
## 3 Representative Results
In this section, we present representative outcomes of shadow-driven
substructures at fixed disk aspect ratio $h$ before giving more comprehensive
statistical results. The three representative runs, denoted as“NL-hm-S-NR,”
“NL-hm-R-NR,” and “NL-hm-V-NR,” can be found in Table 1. We show snapshots of
major fluid quantities of interest (i.e.
$\Sigma,T,\textbf{v}_{r},\textbf{v}_{\phi},\nabla\times\textbf{v}$) from our
simulations, and discuss the results below.
### 3.1 Spirals
Figure 2: Density (a), temperature (b), radial velocity (c) and azimuthal
velocity (d) evolution in spiral-forming disk
($\sigma_{\phi}=0.236,\epsilon=0.8,\alpha=10^{-3},\beta=10,p=-1.0$). All
quantities except for $v_{r}$ are normalized by their initial values, and
$v_{r}$ is normalized by initial $v_{\phi}$. The color range for temperature
differs between the formation process of the early stage (first and second
rows) and the relative quasi-steady state (third to fifth rows). The white
solid lines represent the fitted spirals based on density contrast.
Spirals typically form in disks where the shadow is relatively weak, such as
those characterized by slow cooling or weak shadow amplitude. We choose
spirals formed in a disk with the following parameters as an example:
$\sigma_{\phi}=0.236,\epsilon=0.8,\alpha=10^{-3},\beta=10,p=-1.0$ (run NL-hm-
S-NR). As depicted in Figure 2, spirals form relatively quickly (first row in
Figure 2), typically within approximately 20 local orbits, and once formed,
they remain highly stable111The growth in density perturbations observed in
the last two rows of Figure 2 is primarily due to the combined effects of
strong viscous heating and the influence of wave damping zones.. These spirals
are clearly density waves, showing spiral patterns in all diagnostic physical
quantities. The spiral patterns are stationary (i.e., pattern speed is zero),
which is related to the fact that our shadow patterns have zero angular
velocity. Further discussions regarding the relationship between the
properties of the spirals and the other two substructures will be provided in
Section 5.2. In addition, by examining the second column of Figure 2, we see
that with inefficient cooling, the overall temperature is systematically
cooler than the initial temperature by $\sim 15\%$. The azimuthal varies
smoothly through the shadowing regions, with a maximum temperature variation
of about $4\%$.
### 3.2 Rings
Figure 3: Density (a), temperature (b), radial velocity (c) and azimuthal
velocity (d) evolution in ring-forming disk
($\sigma_{\phi}=0.236,\epsilon=0.5,\alpha=10^{-4},\beta=1,p=-1.0$). All
quantities except for $v_{r}$ are normalized by their initial values, and
$v_{r}$ is normalized by initial $v_{\phi}$. The black dashed lines represent
the fourth and fifth fitted rings in this disk.
The conditions for ring formation generally require either slow cooling or a
combination of moderate viscosity and shadow amplitude (for more detailed
information, see Section 4). In Figure 3, we adopt parameters
$\sigma_{\phi}=0.236,\epsilon=0.5,\alpha=10^{-4},\beta=1,p=-1.0$ (run NL-hm-R-
NR) to illustrate the typical formation process and properties of ring
structures.
–Formation. The formation of rings begins with the presence of two-arm spirals
following a transient period (as seen in the first to third rows of Figure 3).
The spirals appear only marginally stable, which later break apart and
reconnect to form concentric rings in surface density (as shown in the fourth
and fifth rows of Figure 3), which takes a relatively long time of $\sim 100$
local orbits. On the other hand, the spiral patterns remain in the velocity
structure even after ring formation, although they are distorted (as opposed
to the spirals discussed in Section 3.1 and could become a distorted ring
patterns in some cases).
–Evolution and main properties. Once formed, the amplitudes of the rings
continue to increase slowly, reaching a steady state over a few hundred local
orbits, where the gas density in rings are about $10\%$ higher than the
background. However, the density within one ring at quasi-steady state is
unevenly distributed, with surface density near the broken/reconnection
location being smaller, which will be further discussed in Section 5.2 and
Appendix A. The typical ring width is approximately twice the local scale
height, and the spacing is regular (about $4H$ between peaks of two
neighbouring rings) across the disk (further discussed in Section 4). We find
the rings to be eccentric (but centered on the star), with the eccentricity
measured to be $e\sim 0.12$. As can be inferred from the third and fourth
columns in Figure 3, the ratio $v_{r}/v_{\phi}$ is approximately $10^{-3}\ll
e$, suggesting that these rings do not directly correspond to the gas moving
in eccentric orbit. Also, the eccentric rings do not precess, analogous to
spiral patterns that remain stationary, thus coroborating the fact that the
rings emerge as the aftermath of spiral patterns. With moderate cooling, the
azimuthal temperature contrast reaches 15$\%$ and may cause azimuthal
brightness variations in observed rings, although we caution for our highly
simplified thermodynamic treatment (further discussed in Section 5.5).
### 3.3 Vortices and Crescents
Figure 4: Density (a), temperature (b), vorticity (c) and radial velocity (d)
evolution in vortex-forming disk
($\sigma_{\phi}=0.236,\epsilon=0.5,\alpha=0,\beta=0.001,p=-1.0$). The color
ranges for density, vorticity and radial velocity differ between the early
spiral formation stage (first and second rows) and the vortex-dominant stage
(third and fourth rows). The simulation duration increases sequentially from
top to bottom. The white frame delineates the boundary of each identified
vortex, and there are no significant structures inside the white frame in the
temperature plot (b) (compare with density (a), vorticity (c) and radial
velocity (d)) due to rapid cooling. A white star within each frame indicates
the center of the vortex. The red numbers highlight the selected vortices.
Crescents can be described as rings that exhibit an azimuthal variation in
intensity (Bae et al., 2023). Physically, the crescents discussed in this
paper are all induced by vortices, thus we use “vortices” and “crescents”
interchangeably. Figure 4 shows an example of shadow-driven formation of
vortices/crescents. This usually occurs with strong shadow amplitude and rapid
cooling, thus strong thermal forcing, and we adopt $\epsilon=0.5,\beta=0.001$
in this example (run NL-hm-V-NR).
–Formation. With rapid cooling, the disk temperature almost instantly relaxes
to the target temperature both within and outside of the shadow region,
resulting in a 50$\%$ variation in azimuthal temperature given our setup. This
leaves two symmetric low-pressure regions that form quickly at the shadow
locations. In the initial stages (first and second rows in Figure 4), it leads
to the appearance of spiral features in surface density. With strong thermal
forcing that constantly perturbing the disk, the system subsequently becomes
more chaotic (third row in Figure 4) where the velocity field undergoes
significant alterations. Although the physical process is not entirely clear,
vortex/crescent formation ensues, as identified in fourth row of Figure 4.
Selected vortices and crescents are marked by white frames in Figure 4.
–Evolution and main properties. Shadow-driven vortices are all anti-cyclonic
in nature, which can be observed either from the negative vorticity (the 3rd
column of Figure 4) or from the change in the sign of radial velocity across
the vortex center (changing from negative to positive when viewed along the
direction of rotation (counterclockwise), as shown in the 4th column of Figure
4). We observe that vortices started small and are continuously generated.
They merge to form larger ones under the influence of differential rotation
within approximately 60 local orbits, ultimately manifesting as relatively
large crescent-shaped structures. In Figure 4, vortices labeled as 4a and 4b
are undergoing a merger into one single vortex. We find that these patterns
largely corotate with the gas, as expected, and their azimuthal locations are
found to be largely random, with no preference to stay in or out of the
shadows. The disk gas remains turbulent and chaotic throughout the evolution
due to strong perturbations from thermal forcing. Velocity deviations from
local Keplerian inside the vortex region are around $0.5c_{s}$. Additionally,
the local level of turbulence, measured in terms of root mean square (rms)
velocity fluctuations averaged in azimuth, is approximately $10\%$ of the
local sound speed. The typical aspect ratio of the vortices/crescents is about
6, with their density contrast being 1.4. The normalized vorticity in this
case is 0.2. Despite of modest to strong level of turbulence, the large
vortices are relatively long-lived, with typical lifetime of at least 300
local orbits.
## 4 Statistics of Substructures
To gain deeper insights into the dynamical consequences of shadows, we
conducted a comprehensive exploration of parameter space. We performed a total
of 160 simulations (run S-h-all), encompassing a wide combination of
parameters. Most results show similarities with one of the aforementioned
three representative cases. We thus primarily summarize the outcomes in a
statistical manner.
For simulations that exhibit the formation of rings and spirals, we only
measure their properties at the end of the simulations when the system has
already reached a steady state. For simulations with vortex/crescent formation
which are generically chaotic, we select four specific snapshots, denoted as
$P_{\rm orb1}=5000P_{0}$, $P_{\rm orb2}=10000P_{0}$, $P_{\rm
orb3}=15000P_{0}$, and $P_{\rm orb4}=20000P_{0}$. The statistical values for
vorticity, density contrast, spacing, and aspect ratio of the vortices are
calculated by averaging the results at these snapshots.
The simplified statistical results are presented in Figure 5, and more
detailed ones are provided in Figures 14 and 15. It is important to emphasize
that panels shaded with red or blue lines are actually undergoing a vortex-
ring transition or a ring-spiral transition state (see discussion in Appendix
A).
Generally speaking, shadows are capable of generating different kinds of
substructures under different parameter settings. Additionally, we found that
the dominant form of shadow-driven substructures changes from spirals to rings
and eventually becomes vortices/crescents as cooling timescales and/or
viscosity decreases. Where exactly the transition occurs depends on other
parameters such as the shadow amplitude, width, and disk aspect ratio, etc.,
and these will be discussed in more detail in the following subsections.
### 4.1 Statistics for Spirals
Two-arm spirals are fundamental substructures in our simulations, dominating
in disks with cooling timescales significantly longer than the dynamical
timescale, high viscosity ($\alpha>10^{-3}$), or very weak shadow amplitude
(see Section 5). Here, we focus on discussing their density contrast, pattern
speed, and pitch angle.
–Density contrast. In general, stronger thermal forcing, higher shadow
amplitude, wider shadow width, etc. leads to stronger density contrast in the
spirals. However, as the spiral-dominated regime generally requires weak
thermal forcing, the spiral amplitudes are typically low (with upper limit
only $1\%$ higher than background density at the same radius).
–Pattern speed. Spirals found in our simulations are density wave patterns
with zero pattern speed, which also results in non-precessing rings. More
generally, the spiral pattern speed exactly matches the shadow’s pattern
speed, which will be further discussed in Section 5.3.
–Pitch angle. The pitch angle is solely affected by the disk aspect ratio.
With weak thermal forcing, we consider the dispersion relation of spiral
density waves in the linear regime under the WKB approximation (Lin & Shu,
1964)
$m^{2}(\Omega_{p}-\Omega)^{2}=k^{2}c_{s}^{2}+\kappa^{2}.$ (15)
Here, $\Omega_{p}$ represents the spiral pattern speed, $k$ is the radial wave
number, and $\kappa\approx\Omega_{K}$ is the epicyclical frequency. The spiral
pitch angle can be estimated by $\alpha_{p}=\partial r/(r\partial\phi)\approx
m/(|k|r)$. With $\Omega_{p}=0$ and $m=2$, we obtain
$\alpha_{p}\sim\frac{2}{\sqrt{3}}h=constant$ for $p=-1$ disks and
$\alpha_{p}\sim\frac{2}{\sqrt{3}}h_{0}r^{0.25}$ for $p=-0.5$ disks. Taking the
disk parameters used in our simulations (with $h_{0}=0.1$) and averaging over
radius gives $\alpha_{p}=6.6^{\circ}$ for disks with $p=-1$ and
$\alpha_{p}=12.1^{\circ}$ for disks with $p=-0.5$. These estimated values
agree well with our simulation results, which we find to be
$7.344_{-0.535}^{+0.607}$∘ and $13.202_{-1.766}^{+2.16}$∘ (see Figure 15),
respectively.
Figure 5: Statistics of shadow-driven substructures based on 80 of 160
simulations with constant disk aspect ratio ($h=0.1$). The figure is divided
into two parts by a dashed line, representing the shadow width being 45
degrees on the left and 15 degrees on the right. Within each part, the left
column corresponds to disks with $\epsilon=0.5$, while the right column
corresponds to disks with $\epsilon=0.8$. Each subfigure in the
$\beta$-$\alpha$ sections represents a specific combination of parameters.
Within each $\beta$-$\alpha$ section, there are three rows representing the
dominant structures in the disk: vortices/crescents, rings, and spirals from
top to bottom. Each type of structure is represented by a different type of
square marker, whose color represent the density contrast of the
substructures. The figure also includes red and blue line shaded areas,
indicating disks undergoing transitions from vortex-ring and ring-spiral
phases, respectively. Figure 6: The average and error of normalized ring
spacing. The black solid line represents simulations with $p=-1.0$, while the
red solid line represents simulations with $p=-0.5$. Points marked with an ’x’
do not have error bars. In this case, there are at most two selected rings
within the disk. Upward arrow represents lower limit points, where there is
only one ring within the disk. Figure 7: The average and error of normalized
vortex/crescent spacing. The black solid line represents simulations with
$p=-1.0$, while the red solid line represents simulations with $p=-0.5$.
Points marked with an ’x’ do not have error bars. In this case, there are at
most two selected vortices/crescents within the disk. Upward arrow represents
lower limit points, where there is only one selected vortex/crescent within
the disk.
### 4.2 Statistics for Rings
In our simulations, rings dominate in disks with cooling timescales comparable
to the dynamical timescale ($\beta\sim 1$) when $\alpha$ is roughly below
$10^{-3}$. For much higher viscosity, rings dominate even when the cooling
rate approaches the isothermal limit ($\beta=10^{-3}$). Typically, this value
is $\alpha=10^{-2}$ for disks with $\sigma_{\phi}=0.236$ and $\alpha=10^{-3}$
for disks with $\sigma_{\phi}=0.079$. Overall, the parameter space for the
dominance of rings is modest thermal forcing, in between the cases that form
vortices/crescents (strong forcing, see next subsection) and spirals (weak
forcing). In fact, we pose that rings can be viewed either as “reconnected
spirals” (stated in Section 3.2), or “failed vortices”, where the latter
connection arises from the finding that vortex-ring transitions often involve
crescents with very large aspect ratios, although the boundary between this
transition is not necessarily clear-cut, and will be further discussed in
Appendix A. Below, we will focus on “normal” rings (not under transition), and
will discuss the density contrast, ring radial width, ring spacing,
eccentricity, and the parameters that have strong influence on them.
–Density contrast. As shown in Figure 5 and 14, gas densities are typically
$1-20\%$ higher than the background density in ring-dominant disks, and ring
density contrast is enhanced by larger shadow amplitude and width. Density
contrast could reach very small values, such as $0.3\%$, in the ring-spiral
transition, and very large values, such as $50\%$, in the vortex-ring
transition.
–Width and spacing. The ring widths in our simulations are usually 2 times the
local scale height, regardless of shadow parameters. Similarly, for almost all
cases, the spacing between neighboring rings is approximately $4H$, as
depicted in Figure 6. There is very small deviations from the mean, indicating
a highly uniform distribution of rings within the disk.
–Eccentricity. As will be stated in Section 5, ring structures are generated
following the “reconnection” of two-armed spirals in the early stages of disk
evolution, causing the ring to become eccentric with zero pattern speed (as
shadows are stationary). More flared disk morphology results in larger spiral
pitch angles, making the spirals less tightly wound. As a result, the rings
formed in this case tend to be more eccentric. Additionally, we find that
viscosity has a strong impact on eccentricity. Typically, ring eccentricity
varies from 0.1 to 0.7 as $\alpha$ increases from $0$ to $10^{-2}$ in our
simulations (see Figure 15 for details). The angle between the ring’s major
axis and the effective shadow center (e.g. $\phi=0^{\circ},180^{\circ}$ when
$\beta=0.001$) is typically between $80^{\circ}$ and $110^{\circ}$.
### 4.3 Statistics for Vortices and Crescents
As we mentioned in Section 3.3 and better seen in Figure 5, vortices/crescents
tend to dominate in disks characterized by fast cooling processes ($\beta<1$),
low viscosity ($\alpha=0,10^{-4}$), high shadow amplitudes ($\epsilon=0.8$),
and wide shadow widths ($\sigma_{\phi}=0.236$). Such parameter settings all
point to strong thermal forcing. Below, we discuss the properties of the
shadow-driven vortices/crescents, focusing on density contrast, spacing and
aspect ratio of vortices/crescents under the influence of these parameters.
–Vorticity and density contrast. The density contrast of substructures is a
crucial factor as it directly influences their detectability. From our
explorations, the density of the crescents are typically $10-50\%$ higher than
the average density at same radius for all vortex-dominated disks. The density
contrast is generally slightly higher for stronger shadow intensity, larger
shadow width, and faster cooling, but the trend is not definitive given the
chaotic nature of the system. The normalized vorticity ranges from 0.1 to 0.6
in vortex-dominated disks, with vorticity around 0.2 in most cases,
potentially reaching up to 0.6 in the most extreme cases (large $\epsilon$ and
$\sigma_{\phi}$). No clear relationship is found between vorticity and density
contrast due to the high turbulence level, which is around $0.1c_{s}$. The
velocity deviations from local Keplerian inside the vortex region ranges from
0.4 to 1.2 $c_{s}$, indicative of strong rotation in the vortices.
–Spacing. The statistical results of the spacing of vortices/crescents are
plotted in Figure 7. In all simulations, the distance between neighboring
vortices/crescents is typically between $2H$ and $4H$. The spacing is less
uniform compared to rings, and is related to the fact that vortex-dominated
disks are usually turbulent. Note that the small error bars in a few cases are
related to very limited number of vortices/crescents (2 or 3); the lower limit
point represents the case where there is only one vortex-induced crescent in
the disk. Similar to the case shown in Figure 4, the azimuthal locations of
the vortices/crescents are largely random with no direct correlation with the
position of the shadows.
–Aspect ratio. The aspect ratio of crescents/vortices is less affected by
different parameters. Typically, in vortex-dominated disks, this value is
about 6. However, for cases close to (for example, $\sigma_{\phi}=0.079$,
$p=-1$, $\epsilon=0.5$, $\alpha=0$, $\beta=0.01$) or undergoing (for example,
$\sigma_{\phi}=0.236$, $p=-0.5$, $\epsilon=0.8$, $\alpha=10^{-4}$, $\beta=1$)
the vortex-ring transition in parameter space, the aspect ratio can be very
large (greater than 12). More detailed results are shown in Figure 15 in
Appendix B.
## 5 Discussion
Figure 8: Density (a), temperature (b), radial velocity (c) and azimuthal
velocity (d) evolution in linear evolution disk
($\sigma_{\phi}=0.236,\epsilon=0.001,\alpha=0,\beta=0.001,p=-1.0$). All
quantities except for $v_{r}$ are normalized by their initial values, and
$v_{r}$ is normalized by initial $v_{\phi}$. Figure 9: Formation processes of
substructures from linear to non-linear regime. All plots show density
contrast in disks in logarithmic scale.
In this paper, we have conducted simple numerical experiments to study the
dynamical consequence of shadows cast from the inner disk to the outer disk as
a result of thermal forcing. We have restricted ourselves to a small number of
parameters, and the discussion has been largely phenomenological. In this
section, while not going into full detail, we conduct additional studies to
help better understand the origin and trend of shadow-driven substructures,
and briefly discuss their potential implications.
### 5.1 Linear regime
Based on the analysis and discussions in the previous sections, here we
provide further analysis to gain better physical insights on the shadow-driven
substructure formation. As we observe that in all cases, substructure
formation starts from the formation of two-armed spirals under our shadow
prescriptions. This suggests that spirals are the most fundamental form of
shadow-driven substructure, and it can be instructive to look into how spirals
form and evolve under very weak thermal forcing to avoid nonlinear effects. We
thus further conducted a series of 2D inviscid hydrodynamic simulations with
varying perturbation strengths ($\epsilon=0.001$, $\epsilon=0.01$,
$\epsilon=0.1$) while keeping the cooling timescales consistent
($\beta=0.001$). Without viscosity, the simulations are in hydrostatic
equilibrium to start with before thermal forcing is introduced.
In Figure 8, we present the results from the $\epsilon=0.001$ simulation (run
L-hm-S-NR in Table 1). When the shadow is introduced, gas flows into the
shadowed region in a counterclockwise manner. The gas between the shadow
center (pressure minimum) and its rear edge, i.e., between $315^{\circ}$ and
$360^{\circ}$ in the first row of Figure 8, gets accelerated, while the gas
between the shadow center and its leading edge, i.e., between $0^{\circ}$ and
$45^{\circ}$ in the first row of Figure 8, gets decelerated. This leads to gas
piling up near the shadow center, while the neighboring gas is slightly
rarefied, which naturally launch density waves.
As the disk evolves, such density waves wind up due to differential rotation
(see second row of Figure 8). In the meantime, the periodic forcing at the
shadow location continues, keep launching new density waves, leading to
interference. After a few local orbits, the system reaches a relatively steady
pattern of two-arm spirals (see third and forth rows of Figure 8), which
remain stable over long-term. The spirals share the same pattern speed of the
shadows (in this case, zero), and the pitch angle also remains unchanged. We
note that this is very different from planet-induced spirals in that a planet
launches density waves through discrete Lindblad resonances, while as shadows
are cast over a wide range of radii, each radius can excite its own density
waves. In our case, the pattern speed of the shadow is zero, and the only
relevant resonance condition is simply given by $\Omega=\kappa/m$, where
$m=1,2,\cdots$. However, taking $m=2$, we see that with $\kappa\approx\Omega$
for Keplerian disks, no resonance condition is satisfied. In other words, the
two-armed spirals are not driven by Lindblad resonances, but are the effective
eigen-state of thermally-forced oscillations.
### 5.2 Towards the nonlinear regime
We note that even in the linear regime, the spiral patterns are distorted due
to thermal forcing. These can be most easily seen from the velocity
perturbations in the last three columns of Figure 8. They are also present in
the density perturbations where the amplitude of the spirals varies across the
shadow region. The form of the distortion can depend on system parameters,
which is found to be different in Figure 2 where cooling time is significantly
longer. We speculate that such distortions are the source of instability when
thermal forcing enters the nonlinear regime.
Based on our discussions in the previous sections, we summarize the formation
of shadow-driven substructures in Figure 9. Irrespective of whether thermal
forcing is linear or nonlinear, the initial phase of the development is
similar, involving the formation of two-armed spirals, as shown in (a)-(b).
The spirals persist under linear and weakly nonlinear thermal forcing, as seen
in the “linear branch” and “spiral branch” in (c)-(f). The properties of
spirals are similar between the linear and weakly nonlinear regimes, in terms
of pitch angle and pattern speed.
When thermal forcing becomes slightly stronger, the spiral arms undergo a
relatively quiescent transformation by “reconnecting” into eccentric rings
(see Figure 9(g), (h)). The eccentricity of these rings is largely set by the
pitch angles of the original two-arm spirals stage and disk viscosity.
However, when the thermal forcing becomes too strong, the spirals break in a
highly chaotic manner (see Figure 9(i), (j)), leading to the formation of more
localized vortices/crescents.
Figure 10: Normalized density perturbation for simulations with
$\Omega_{\text{shadow}}=1\Omega_{0}$ (10), $0.03\Omega_{0}$ (10), and
$0.003\Omega_{0}$ (10) at the final steady state under the parameters setting
as $\sigma_{\phi}=0.236,\epsilon=0.001,\alpha=0,\beta=0.001,p=-1.0$. The radii
of corotation resonances (CR), inner Lindblad resonances (ILR), and outer
Lindblad resonances (OLR) are shown as yellow, green, and purple dashed lines
respectively.
### 5.3 Rotating shadows
In this paper, we have only discussed the situation when the shadow’s pattern
speed is zero. However, if the misaligned inner disk precesses around the
central star, the shadow cast from the inner region would have a pattern
speed, which then changes the resonance condition discussed in Secton 5.1. To
extend our study to more general conditions, we have conducted additional
simulations with rotating shadows in the linear regime, with three different
shadow pattern speeds, $\Omega_{\text{shadow}}=1\Omega_{0}$ (run L-hm-S-FR),
$0.03\Omega_{0}$ (run L-hm-S-MR), and $0.003\Omega_{0}$ (run L-hm-S-SR). Here,
$\Omega_{0}$ is Keplerain angular velocity at $r=1$. The detailed parameter
settings can be found in Table 1. The density structure from these simulations
in the final states are shown in Figures 10. We measure the pattern speed of
the spirals $\Omega_{p}$ in these situations, and we confirm that in all three
cases, the spirals all have $\Omega_{p}=\Omega_{\text{shadow}}$.
Given the pattern speed, the radii of corotation resonances (CR), inner
Lindblad resonances (ILR), and outer Lindblad resonances (OLR) can be
calculated by $\Omega_{p}=\Omega$, $\Omega_{p}=\Omega\pm\kappa/m$ (with
$m=2$), and shown as yellow, green, and purple dashed lines in Figures 10.
With the WKB dispersion relation 15, the permitted regions for density wave
propagation are outside the Lindblad resonances. In the fast-rotating case
$\Omega_{p}=\Omega_{0}$, density waves are permitted beyond the OLR, and the
spirals are tightly wound towards outer radii with pitch angle $\alpha_{p}\sim
h(\Omega/\Omega_{p})\sim h(r/r_{0})^{-3/2}$. Even with a resolution of
$N=2048$ in Figure 10, it is still insufficient to resolve the spirals across
the entire disk, weakening the spirals at the outer disk by numerical
dissipation. With intermediate $\Omega_{\rm shadow}=0.03\Omega_{0}$, the ILR
and OLR are located at $r=6.5$ and $r=13.5$, respectively. Clearly, there are
well-defined spirals outside the Lindblad resonances, which break inside the
Lindblad resonances. In the slow-rotating case with $\Omega_{\rm
shadow}=0.003\Omega_{0}$, even the ILR is beyond the computational domain, and
the results are largely identical to the stationary case described in Section
5.1.
Given the discussion above, we expect the results presented in this paper
largely applies to regions inside the ILR in slowly-precessing shadows.
Although not the focus of this paper, it is worth noting the significance of
moderately rotating shadows, where the corotation radius lies within the disk
region. Our findings are morphologically similar to those of Montesinos &
Cuello (2018), who demonstrated that the morphology of shadow-driven spirals
notably resembles the planetary wakes caused by embedded planets in the disc
using radiative transfer. For better comparison with planet-induced spirals,
more detailed investigation with more realistic physics (especially dust and
radiative processes) is necessary for the slow-rotating case, especially in
regions between the ILR and OLR.
### 5.4 Dependence on disk aspect ratio
Figure 11: Normalized density perturbation for runs with different disk
aspect ratios. The parameters are identical to those in the representative
runs (NL-hm runs) in Section 3, except for the disk aspect ratio. Detailed
parameter settings are provided in Table 1. (a): snapshot with $P_{\rm
orb}=20000P_{0}$ for run NL-hs-S-NR
($h_{0}=0.05,\sigma_{\phi}=0.236,\epsilon=0.8,\alpha=10^{-3},\beta=10,p=-1.0$).
(b): snapshot with $P_{\rm orb}=20000P_{0}$ for run NL-hs-R-NR
($h_{0}=0.05,\sigma_{\phi}=0.236,\epsilon=0.5,\alpha=10^{-4},\beta=1,p=-1.0$).
(c): snapshot with $P_{\rm orb}=20000P_{0}$ for run NL-hs-V-NR
($h_{0}=0.05,\sigma_{\phi}=0.236,\epsilon=0.5,\alpha=0,\beta=0.001,p=-1.0$).
(d): snapshot with $P_{\rm orb}=3000P_{0}$ for run NL-hl-S-NR
($h_{0}=0.15,\sigma_{\phi}=0.236,\epsilon=0.8,\alpha=10^{-3},\beta=10,p=-1.0$).
(e): snapshot with $P_{\rm orb}=10000P_{0}$ for run NL-hl-R-NR
($h_{0}=0.15,\sigma_{\phi}=0.236,\epsilon=0.5,\alpha=10^{-4},\beta=1,p=-1.0$).
(f): snapshot with $P_{\rm orb}=1500P_{0}$ for run NL-hl-V-NR
($h_{0}=0.15,\sigma_{\phi}=0.236,\epsilon=0.5,\alpha=0,\beta=0.001,p=-1.0$).
In the preceding discussion, we observed that shadow-driven substructures are
closely tied to thermal forcing, which is influenced not only by the cooling
process but also by disk temperature. Additionally, detailed characteristics
of substructures, such as pitch angle or eccentricity, are affected by the
disk aspect ratio h. Therefore, it is natural to further investigate the
influence of $h_{0}$. We conducted additional simulations with $h_{0}$ ranging
from 0.03 to 0.15, focusing on $h_{0}=0.05$ and $h_{0}=0.15$. These
simulations, denoted as NL-hs-S-NR, NL-hs-R-NR, NL-hs-V-NR and NL-hl-S-NR, NL-
hl-R-NR, NL-hl-V-NR respectively, maintained the same parameters as the
representative runs discussed in Section 3 except for $h_{0}$ (see Table 1).
We note that here “S”, “R”, and “V” do not necessarily indicate dominant form
of substructures but rather serve to guide the reader that these runs only
vary $h_{0}$ compared to representative runs.
In the NL-hs run series ($h_{0}=0.05$), with lower target temperature, we see
that the NL-hs-S-NR (Figure 11) and NL-hs-R-NR (Figure 11) runs maintain
spirals and rings as the primary substructure, respectively. We see the
spirals are more tightly wound and the rings spacing remains uniform except
for being smaller. The changes are exactly in proportion to $h_{0}$, and the
general properties of the rings and spirals are otherwise identical to those
discussed in the NL-hm runs. For the NL-hs-V-NR run, while the vortices are
clearly the dominant, many of the overdensities close a full circle, and we
identify this run as in the vortex-ring transition state.
In the NL-hl run series ($h_{0}=0.15$), with higher target temperature, we see
that all three NL-hl runs retain their spiral, ring and crescent/vortex as the
dominant substructure, respectively. Similarly, the spirals are more open, the
rings are more eccentric, and the vortices are larger and more widely spaced,
as expected.
Overall, we find that varying $h$ slightly alters the boundary where different
forms of substructures dominate, while the general properties for individual
substructures largely remain consistent with what we have found in the
fiducial simulations with $h_{0}=0.1$.
### 5.5 Observational implications
Given the diverse dynamical consequence of shadowing, such disks is expected
to exhibit a variety signatures that are potentially observable. However, it
should be noted that our work serves as a general exploration without detailed
modeling, including radiation transport, dust dynamics, shadow precession
rates (e.g., Pinilla et al., 2015; Stolker et al., 2016; Wolff et al., 2016;
Debes et al., 2017), and realistic shadow morphologies may differ from our
prescription (e.g., Muro-Arena et al., 2020; Debes et al., 2017).
Additionally, as there are a variety of other mechanisms that can drive
substructures (e.g., see reviews by Andrews, 2020; Bae et al., 2023; Benisty
et al., 2023), such as planet-disk interactions, icelines etc. Our shadowed
disk simulations implicitly assumed a smooth disk to start with, and it is
conceivable that the final outcome is set by the interplay between the
existing substructures and shadowing. Besides such dynamical interplay,
substructures themselves can self-shadow (e.g., Zhang et al., 2021), which can
further complicate the situation. Therefore, a systematic observational
comparison with specific sources is beyond the scope of this work. Below, we
mainly discuss general aspects of potential observational implications.
–Spirals. Spirals generated from shadows may not be easily detectable in the
submm continuum or in kinematics, but may be observable in scatter light.
Nearly all spiral-dominant disks correspond to weak thermal forcing, resulting
in only about $0.1\%$ higher gas density than the background. This not only
makes pressure variations across the spirals small that is difficult for dust
trapping, and only sufficiently small particles with a stopping time shorter
than the spiral crossing time (typically requiring the Stokes number much less
than 0.1) can potentially be trapped by the spiral (e.g. Sturm et al., 2020;
Speedie et al., 2022). With the weak spirals, the gas velocity is found to
show very small deviationsfrom Keplerian ($\sim 0.1\%$ $v_{k}$, as opposed to
$\gtrsim 0.5\%$ $v_{k}$ for typical ALMA observations (Pinte et al., 2023).),
making it difficult for kinematic detection. On the other hand, such spirals
may be detectable in scattered light, as suggested by Montesinos et al. (2016)
for the HD 142527 disk, thanks to azimuthal variation of disk scale heights
across the spirals, though three-dimensional simulations are needed for proper
characterization.
–Rings. For full disks, our simulations predict the presence of multiple gas
rings that are uniformly spaced and weakly eccentric. The relatively high
density contrast in our simulations suggests that these rings likely
concentrate dust, making them readily observable in sub-mm wavelengths. While
the resulting dust rings formed are also likely uniformly spaced, whether they
can be eccentric remains uncertain (as the eccentric gas ring is a pattern and
does not reflect real motion), requiring simulations incorporating dust
dynamics. From all simulations, we find that the azimuthal temperature
contrast in the ring-dominant disks are typically greater than $8\%$ and can
reach up to $50\%$ as they approach to vortex-ring transition in disks with
high viscosity and rapid cooling. Such azimuthal temperature variations should
result in azimuthal brightness variations in the mm continuum image, which
however has not been revealed in in real shadowed disks with rings (e.g., HD
143006). This suggests that thermal forcing by shadows in these systems are
likely not as strong as given in our prescriptions, but we caution that
without detailed modeling of shadow morphology, radiation transport and dust
dynamics, we cannot make specific predictions for individual systems. On the
other hand, we comment that both the weakly eccentric ring pattern and low-
level of azimuthal temperature variation, if present, may affect the
interpretation of azimuthal asymmetries seen in multi-ring systems (e.g. Doi &
Kataoka, 2021; Liu et al., 2022). Finally, we note that detection by kinematic
signatures, with velocity disturbances being $\sim 1\%$ of the Keplerian
velocity, is possible but challenging since they are close to ALMA’s detection
limits.
–Crescents. Vortices generate significant velocity perturbations and are
favored sites for dust trapping. Given the adopted turbulent viscosity
parameter $\alpha_{t}\gtrsim 10^{-2}$ in most vortex-dominated simulations,
dust with Stokes number $St>\alpha_{t}\sim 10^{-2}$ is expected to concentrate
inside vortices overcoming turbulent diffusion (Birnstiel et al., 2013), and
can be readily observable in sub-millimeter wavelengths (Zhu et al., 2014).
Previous studies have found that detecting kinematic signatures of vortices
can be possible but challenging (Huang et al., 2018), despite the relatively
large vorticity (typically around $0.2$) and significant velocity deviations
from local Keplerian ($\delta v$ up to $1.2c_{s}$) inside vortex region . It
is expected that sources with modest inclination favors detection but requires
long integration time with ALMA (more than 10h) to achieve the necessary
signal-to-noise ratio.
## 6 Summary and Future Prospects
In this work, we have systematically studied the dynamical consequence of
thermal forcing by shadows cast to the outer protoplanetary disks. With a
large survey of parameters, we have identified a diverse forms of
substructures generated by shadows and studied their trends under different
thermodynamic and viscosity prescriptions. Our results apply in regimes where
the shadow is static or slowly-rotating (prograde), so that the corotation
radius is further than regions of interest. The main findings of our studies
are as follows.
1\. Two-arm spirals with identical pattern speed as the shadow are fundamental
substructures generated by weak thermal forcing ($\epsilon<=0.5$,
$\sigma_{\phi}=0.079$, $\beta>1$) or high viscosity ($\alpha>10^{-3}$). They
represent linear response to thermal forcing, and their pitch angle well
agrees with standard density waves. Both the density contrast (0.1-1$\%$
higher than background) and velocity disturbance up to 0.5$\%$ $v_{K}$) are
small and scale with the strength of thermal forcing.
2\. Disks with moderate thermal forcing are dominated by ring-like
substructures. In this regime (parameter space in between crescent/vortex and
spiral-dominant disks), the gas density contrast reaches 1-20$\%$ above the
background. The rings are uniformly spaced ($\Delta r/H\sim 4H$) and exhibit
pattern eccentricities on the order of $h/r$ or higher which rotates at the
same rate of the shadow.
3\. Crescents/vortices dominate disks under strong thermal forcing
($\epsilon>0.5$, $\beta\lesssim 0.1$, $\sigma_{\phi}=0.236$) and low viscosity
($\alpha<=10^{-4}$). In this case, the density contrast is typically 10-50$\%$
higher than the average density at the same radius. The vortices in our
simulations exhibit relatively large vorticity (ranging from 0.1 to 0.6,
typically around 0.2) and significant velocity deviations from local Keplerian
inside the vortex region (ranging from 0.4 to 1.2 $c_{s}$). Due to the chaotic
nature (local turbulence level is 0.1 $c_{s}$) of the vortex-dominant disk,
these structures are not uniformly spaced, with $\Delta r/H$ between 2 and 4.
4\. Thermodynamics and viscosity significantly influence the formation of
shadow-driven disk substructures. The dominant substructure transitions from
spirals to rings and eventually to vortices as cooling timescales and/or
viscosity decrease.
5\. Owing to the simplicity of our problem setup, it is premature to
definitely assess the observability of such shadow-driven substructures. We
anticipate that the azimuthal brightness contrast in the sub-mm continuum to
offer important constraints on the strength of the thermal forcing, while
detecting in-plane kinematic signatures is likely challenging.
Through our suite of physically-motivated while highly simplified simulations,
we highlight the importance on the dynamical impact of shadows or more
generally, inhomogeneous stellar irradiation, on the gas dynamics of PPDs
through thermal forcing. Given the fact that shadows are often observed in
scattered light images of disks, our results call for proper consideration and
incorporation of such effects for adequate modeling of such systems.
Our simulations can be considered as a starting point to understand the
dynamical effects of shadows on PPDs, yet real systems are likely much more
complex. This leaves several aspects to be considered and tested in the
future. Proper characterizing disk thermodynamics is a pre-requisite to
accurately model thermal forcing from shadows, which requires better modeling
of the shadow geometry, together with self-consistent radiation transport.
Such modeling under typical disk parameters (that are likely nearly optically-
thin) will likely reduce the azimuthal temperature contrast due to in-plane
radiation transport. Incorporation of dust dynamics is essential to obtain
dust response to the shadow-driven substructures. Such simulations are
expected to link the results with specific sources, as we are aware of efforts
underway (Ziampras et al., in preparation). We have also assumed the shadows
are cast to a full disk, whereas shadows are also observed in transition
disks, and it is also pertinent to account for the interplay other physical
mechanisms that cause disk substructures, with additional effect of self-
shadowing.
Finally, all existing studies of shadow-driven disk dynamics are conducted in
2D in the disk plane, whereas the shadow-driven thermal forcing is also
expected to also drive oscillations in the vertical direction (Liu & Bai, in
preparation). Future studies should incorporate 3D effects, which is essential
to further assess the fidelity of 2D simulation results, and make more
realistic observational predictions and comparisons.
Acknowledgements We thank Yanqin Wu and Shangjia Zhang for useful
discussions, Pinghui Huang for helpful instructions on problem setup, and
Alexandros Ziampras for constructive exchanges. This work is supported by
National Science Foundation of China under grant No. 12233004, 12325304. We
also acknowledge the China Center of Advanced Science and Technology for
hosting the Protoplanetary Disk and Planet Formation Summer School in 2022
when this work was initiated. Numerical simulations are conducted in the Orion
and Sirius clusters at Department of Astronomy, Tsinghua University and
TianHe-1 (A) at National Supercomputer Center in Tianjin, China.
## Appendix A Transition State
The vortex-ring transition represents the parameter regime where both the
features of vortices/crescents and rings can be observed in the disk. Four
examples of vortex-ring transition are illustrated in Figure 12. They are
recognized as vortex-ring transitions generally based on two reasons: rings
and vortices/crescents are simultaneously present in the disk (Figure 12), or
the basic morphology appears as rings but with significant asymmetry (Figure
12, 12, 12). In Figures 14 and 15, the left side of vortex-ring transition
cases depicts vortex-dominated disks, while the right side illustrates ring-
dominated disks. Further decreases in $\beta$ or $\alpha$ lead to the disk
being completely dominated by vortices/crescents.
The ring-spiral transition represents the parameter regime where both the
features of rings and spirals can be identified in the disk. Four examples of
ring-spiral transitions are shown in Figure 13. They either exhibit regularly
broken rings (Figure 13 and 13) or clearly display both rings and spirals
within the same disks (Figure 13 and 13). These transition regions lie between
ring-dominated disks and spiral-dominated disks in Figure 14 and 15. The disk
becomes dominated by spirals as $\beta$ or $\alpha$ increases.
From the transition states shown in Figure 12 and 13, we can verify that rings
exhibit characteristics of both vortices/crescents and spirals, as discussed
in Section 4.2. Slightly excessive thermal forcing, relative to ring-dominant
disks, can hamper reconnection (Figure 12) mentioned in Section 5.2, leading
the disk into a vortex-ring transition state with strongly asymmetric rings
(Figure 12) or crescents with large aspect ratios (Figure 12). Conversely,
with weak thermal forcing, the breaking of two-armed spirals is partial
(Figure 13), placing the disk into a ring-spiral transition state.
Figure 12: The demonstration of vortex-ring transition situation. The
parameters of these 4 plots are
$\sigma_{\phi}=0.236,p=-0.5,\epsilon=0.5,\alpha=0.0001,\beta=1$;
$\sigma_{\phi}=0.236,p=-0.5,\epsilon=0.8,\alpha=0.01,\beta=0.001$;
$\sigma_{\phi}=0.236,p=-1.0,\epsilon=0.8,\alpha=0.01,\beta=0.001$;
$\sigma_{\phi}=0.079,p=-0.5,\epsilon=0.8,\alpha=0,\beta=0.1$ respectively. All
these cases can be found in Figure 14 and Figure 15 as vortex or ring blocks
covered by red hatch lines.
Figure 13: The demonstration of ring-spiral transition situation. The
parameters of these 4 plots are
$\sigma_{\phi}=0.236,p=-0.5,\epsilon=0.8,\alpha=0,\beta=10$;
$\sigma_{\phi}=0.079,p=-1.0,\epsilon=0.8,\alpha=0.001,\beta=1$;
$\sigma_{\phi}=0.079,p=-0.5,\epsilon=0.5,\alpha=0,\beta=10$;
$\sigma_{\phi}=0.079,p=-0.5,\epsilon=0.5,\alpha=0.001,\beta=1$ respectively.
All these cases can be found in Figure 14 and Figure 15 as ring or spiral
blocks covered by blue hatch lines.
## Appendix B Simulation statistics
The detailed statistical plot of vorticity and density contrast (Figure 14),
along with other parameters (Figure 15) of substructures, is presented here.
These two figures share the same structures. Each of these figures is divided
into two sections by a dashed line, representing shadow ranges of 45 degrees
($\sigma_{\phi}=0.236$) and 15 degrees ($\sigma_{\phi}=0.079$), respectively.
In the left column, disks with a temperature slope of $-1$ are shown, while
the right column represents disks with a temperature slope of $-0.5$. Each
row, from top to bottom, corresponds to shadow amplitudes of $\epsilon=0.5$
and $\epsilon=0.8$. The $x$-axis of the subfigures represents $\beta$, while
the $y$-axis represents $\alpha$. Within each $\beta$-$\alpha$ section, there
are three rows indicating the dominant structures in the disk:
vortices/crescents, rings, and spirals, each represented by different types of
squares, colored by the relevant physical properties as indicated in the color
bars. The figure also includes red and blue line shaded areas, indicating
disks undergoing transitions from vortex-ring and ring-spiral phases,
respectively. We note that the inviscid ($\alpha=0$) simulations maintain the
same temperature gradient with a slope of $-1$ (indicating that the $p$ value
shown in the title of each subfigure only applies to viscid runs) and vary the
density gradient of $d=-0.5$ and $d=-1$ in the left and right columns,
respectively, which help us exclude the influence of density gradient.
Figure 14: Statistics of substructures’ vorticity and density ratio for 160
simulations. The structure of the figure is described in Appendix B and is
similar to Figure 5. Figure 15: Statistics of vortices’ aspect ratio, rings’
eccentricity and spirals’ pitch angles for 160 simulations. The structure of
the figure is same as Fig14.
## Appendix C One-sided shadow test
In this Appendix, we briefly examine how the morphology and form of
substructures can be affected by the morphology of the shadow region. As an
experiment, we performed simulations with only the right side of the shadow
shown in Figure 1 present, and the target temperature is taken as
$T_{\rm tar}(r,\phi)=T_{\rm init}(r)\left(1-\epsilon
e^{-\frac{\phi^{2}}{2\sigma_{\phi}^{2}}}\right).$ (C1)
The remaining parameters for the disk and shadow are the same as those in the
representative simulations (NL-hm runs). For detailed parameter settings for
the NL-hm runs, please refer to Table 1. It can be seen from Figure 16 that
the types of dominant substructures have not changed compare with NL-hm runs.
The dominant spiral now has $m=1$, and the rings become asymmetric (with
$m=1$, as opposed to eccentric with $m=2$), while crescents are generated as
usual. These outcomes similarly follow the formation process described in
Section 5.2. These simulations illustrate that besides a morphological change
from $m=2$ to $m=1$, the general trends of shadow-driven substructures are not
sensitive to shadow prescriptions.
Figure 16: Normalized density perturbation for one-sided shadow. Left panel:
Spiral forming disk with parameter taken to be same as run NL-hm-S-NR (Figure
2). Middle panel: Ring forming disk with parameter taken to be same as run NL-
hm-R-NR (Figure 3). Right panel: Vortex forming disk with parameter taken to
be same as run NL-hm-V-NR (Figure 4).
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# Onset of metallic transition in molecular liquid hydrogen
Jianqing Guo School of Physics, Peking University, Beijing 100871, People’s
Republic of China International Center for Quantum Materials, Peking
University, Beijing 100871, People’s Republic of China Bingqing Cheng The
Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg,
Austria Limei Xu School of Physics, Peking University, Beijing 100871,
People’s Republic of China International Center for Quantum Materials, Peking
University, Beijing 100871, People’s Republic of China Interdisciplinary
Institute of Light-Element Quantum Materials and Research Center for Light-
Element Advanced Materials, Peking University, Beijing, China. Enge Wang
School of Physics, Peking University, Beijing 100871, People’s Republic of
China International Center for Quantum Materials, Peking University, Beijing
100871, People’s Republic of China Interdisciplinary Institute of Light-
Element Quantum Materials and Research Center for Light-Element Advanced
Materials, Peking University, Beijing, China. Songshan Lake Materials Lab,
Institute of Physics, Chinese Academy of Sciences, Guangdong, China. School
of Physics, Liaoning University, Shenyang, China Ji Chen<EMAIL_ADDRESS>School of Physics, Peking University, Beijing 100871, People’s Republic of
China Interdisciplinary Institute of Light-Element Quantum Materials and
Research Center for Light-Element Advanced Materials, Peking University,
Beijing, China. Collaborative Innovation Center of Quantum Matter, Beijing
100871, People’s Republic of China
###### Abstract
Liquid-liquid phase transition of hydrogen is at the center of hydrogen phase
diagram as a promising route towards emergent properties such as the Wigner-
Huntington metallization, superconductivity, and superfluidity. Here we report
a study on the liquid-liquid phase transition of hydrogen using the state-of-
the-art diffusion quantum Monte Carlo and density functional theory
calculations. Our results suggest that the metallization process happens at
lower pressures and temperatures compared to the structural phase transition
of molecular to atomic hydrogen. The consequence is that metallized molecular
hydrogen is stable at a wide range of pressures and temperatures. Our study
breaks the conventional assumption that metallization coinciding with
dissociation of hydrogen molecule, and the molecular metallic hydrogen liquid
phase is likely to become the frontier of studying hydrogen phase transitions.
Hydrogen, the simplest and lightest element in nature, is famous for a complex
phase diagram and a rich variety of phase transitions. Among the many phases
of hydrogen, the liquid-liquid phase transition (LLPT) is of particular
interest ever since metallic hydrogen was predicted in 1935, and LLPT is
considered as a path to metallic hydrogen McMahon _et al._ (2012); Nellis
(2021). Experimentally, the LLPT of hydrogen has been achieved under dynamic
and static compression in the fluid phase, and the evidence of metallization
was mostly provided by the change in optical reflectivity Dzyabura _et al._
(2013); Zaghoo _et al._ (2016, 2018); Jiang _et al._ (2020); Weir _et al._
(1996); Knudson _et al._ (2015); Celliers _et al._ (2018). Nevertheless,
challenges and controversies still exist in experiments especially at low
temperatures, including the uncertainty in determining the extreme pressure
and temperature conditions, the difficulties in structure characterizations,
and the lack of accurate transport measurements of conductivity McMahon _et
al._ (2012); Fang _et al._ (2019); Silvera and Dias (2021).
Theoretically, LLPT of hydrogen is often described as a transition from the
insulating molecular phase to the metallic atomic phase, supported by
computational simulations Scandolo (2003); Holst _et al._ (2008); Morales
_et al._ (2010); Lorenzen _et al._ (2010); Morales _et al._ (2013); Geng
_et al._ (2019); Hinz _et al._ (2020); Tian _et al._ (2020); van de Bund
_et al._ (2021), indicating or assuming that the molecular-atomic transition
(MAT) and the insulating-metallic transition (IMT) occur simultaneously. In
recent years, new attention has been paid to the supercritical behavior and
the location of liquid-liquid critical point in the computational community,
involving studies using density functional theory (DFT), machine learning
potentials and quantum Monte Carlo (QMC) Li _et al._ (2015); Gorelov _et
al._ (2020); Cheng _et al._ (2020); Karasiev _et al._ (2021); Cheng _et
al._ (2021); Tirelli _et al._ (2021). Based on these studies, it is
understood that below the critical point, if the temperature is above the
melting line, LLPT is one phase transition where $\text{H}_{2}$ molecule
dissociates and metallizes simultaneously; but above the critical point, the
dissociation and metallization happen in a smooth and continuous manner. In
literature, both the molecular-atomic transition and the insulating-metallic
transition have been analysed in detail using the state-of-the-art
computational methods (see e.g. Refs. Pierleoni _et al._ , 2018; Gorelov _et
al._ , 2020). However, to the best of our knowledge, there is no theoretical
guarantee or experimental evidence that the metallization of molecular
hydrogen should exactly coincide with the dissociation, regardless if LLPT is
first order or not. On the contrary, experimental studies performed in the
solid regime of dense hydrogen has suggested that metallization can happen
without a molecular-atomic transition, motivating a re-examination of the
nature of LLPT Eremets _et al._ (2019).
In this paper, we study the molecular-atomic and insulating-metallic phase
transition of liquid hydrogen, to answer the basic question of whether they
coincide with each other or one occurs before another. To provide microscopic
insights, we propose the molecular fraction of liquid hydrogen as the key
order parameter to compare the two kinds of phase transitions on the same
foot. We use the state-of-the-art diffusion quantum Monte Carlo (DMC) method
to benchmark DFT functionals and find that vdw-DF can accurately describe the
structural phase transition of dense liquid hydrogen. We also study the
electronic phase transition of liquid hydrogen by calculating the fundamental
gap as a function of molecular fraction using DMC and DFT methods. We find
that the insulating-metallic transition begins when around 20% hydrogen
molecules dissociate into hydrogen atoms, which requires much lower
dissociation than the half molecular-atomic phase transition. By mapping such
results onto the pressure-temperature phase diagram, we estimate that the
pressure of IMT is tens of GPa lower than structural MAT.
Figure 1: Molecular and atomic liquid hydrogen from AIMD simulations. (a-b)
Snapshots of typical molecular liquid hydrogen (x = 0.96) and half-molecular-
half-atomic liquid hydrogen (x = 0.48) from the vdW-DF simulations. The
criterion for drawing the H-H bond is 0.9 Å. (c) Molecular fraction as a
function of temperature using different DFT functionals. The molecular
fraction order parameter x is defined in detail in the main text and the
Supplementary Material. Qualitatively the molecular faction x describes the
portion of $\text{H}_{2}$ molecules in liquid hydrogen. (d) HH pair
correlation functions $\text{G}_{\text{HH}}(\text{r})$ at different
temperatures calculated from vdw-DF AIMD simulations. Simulations were carried
out at the density of $\text{r}_{\text{s}}$ = 1.50 Bohr in the NVT ensemble
with 128 hydrogen atoms.
DFT-based ab initio molecular dynamics (AIMD) simulations were carried out
using the Quantum Espresso (QE) package Giannozzi _et al._ (2009, 2017) with
the Perdew-Burke-Ernzerhof (PBE) Perdew _et al._ (1996), BLYP-D2 Grimme
(2006), the vdW-DF Dion _et al._ (2004), the vdW-DF2 Klimeš _et al._ (2009)
and the SCAN Sun _et al._ (2015) functionals. The interactions between the
valence electrons were treated with Hamann-Schlüter-Chiang-Vanderbilt
pseudopotentials Hamann _et al._ (1979); Vanderbilt (1985). Each simulation
was performed in the canonical (NVT) ensemble, and the temperature was
controlled by a Nosé-Hoover thermostat Martyna _et al._ (1992). DFT total
energy calculations with the PBE, the vdW-DF, the BLYP-D2, the SCAN, the HSE06
Heyd _et al._ (2003) functionals were performed with both the QE package and
the VASP code Kresse and Furthmüller (1996). DMC calculations were performed
using the CASINO package Needs _et al._ (2020), with the size-consistent DMC
algorithm ZSGMA Zen _et al._ (2016). The recently developed energy consistent
correlated electron pseudopotentials (eCEPPs) Trail and Needs (2017) were
used. Further computational details can be found in the Supplemental Material
si .
To investigate the molecular-atomic phase transition we use an order
parameter, namely the molecular fraction. As suggested in previous studies, a
molecular fraction of 0.5 is a good criterion to characterize the structural
molecular-atomic phase transition Tamblyn and Bonev (2010); Geng _et al._
(2019); Cheng _et al._ (2020). Specifically, we follow the definition of
Cheng et al.Cheng _et al._ (2020), where the molecular fraction x was defined
as the fraction of atoms with one neighbour within a smooth cutoff function
starts from 0.8 Å and decays to 0 at 1.1 Å (see Supplementary Material II for
more details and further validation using an agnostic classification scheme si
). There are other definitions of molecular fraction Tamblyn and Bonev (2010);
Geng _et al._ (2019) and other order parameters, e.g. density/volume, heat
capacity, compressibility, and diffusion coefficient, that can be used to
describe the molecular-atomic transition on the temperature pressure phase
diagram Li _et al._ (2015); Cheng _et al._ (2020); Karasiev _et al._
(2021). A detailed discussion on the thermodynamic properties involved in the
molecular-atomic LLPT is beyond the scope of this study. In this study we
focus on the molecular fraction, which provides the clearest microscopic
insight to the molecular-atomic transition. Fig. 1 (a) and (b) show snapshot
of two structures with a high molecular fraction of (x=0.96) and a low
molecular fraction (x=0.48), respectively.
In Fig. 1 (c) we plot the molecular fraction as a function of temperature by
performing systematic AIMD simulations using different DFT exchange
correlation functionals at the density of $\text{r}_{\text{s}}$ = 1.50 Bohr
with a system size of 128 H atoms. Tests on e.g. size effects and density
dependence are further presented and discussed in Fig. S2 and S3 si . The
simulations confirm that the molecular fraction can be used to monitor the
molecular-atomic LLPT as a function of temperature. The structural transition
is further illustrated by the pair correlation functions in Fig. 1 (d) and
Fig. S1 si , where the first peak is suppressed when the molecular fraction
decreases to $\sim$0.5. However, different functionals show quite different
transition temperatures at a constant density, which are consistent with
shifts of phase boundaries on a pressure-temperature phase diagram observed in
previous studies. Among the various functionals considered, we find that PBE
predict the lowest phase transition temperature and vdW-DF2 has the highest
phase transition temperature. The other three functionals, namely SCAN, vdW-
DF, and BLYP-D2, show behaviors that are in between the cases of PBE and vdW-
DF2. Overall, these computational observations of the qualitative difference
among different DFT functionals agree with the comparisons reported in
literature Clay _et al._ (2014); Geng _et al._ (2019).
Figure 2: Benchmark of DFT against DMC. 14 structures covering a wide range of
molecular fraction x were selected for the benchmark. The structures are
collected from vdW-DF AIMD and machine learning potential simulations of 128
hydrogen atoms with a density of rs=1.5 Bohr at different temperatures (see
Supplementary Material II for more details si ). The relative binding energy
is calculated as
$\text{E}_{\text{b}}^{\text{DFT}}-\text{E}_{\text{b}}^{\text{QMC}}$, where
$\text{E}_{\text{b}}=(\text{E}_{\text{liquid}}-\text{N}_{\text{H}}\times\text{E}_{\text{H}})/\text{N}_{\text{H}}$
is the binding energy.
The different results from AIMD simulations suggest that it is necessary to
further evaluate the accuracy of DFT functionals for liquid hydrogen.
Therefore, we calculate the binding energies of a set of structures via DMC
and use them as the benchmark to evaluate the performance of DFT functionals,
a strategy that has been adopted successfully for solid hydrogen Chen _et
al._ (2014); Clay _et al._ (2014); Drummond _et al._ (2015). In Fig. 2, we
plot the relative binding energies of liquid hydrogen structures as a function
of the molecular fraction with respect to the DMC calculations
($\text{E}_{\text{b}}^{\text{DFT}}-\text{E}_{\text{b}}^{\text{QMC}}$). The
binding energy is defined as
$\text{E}_{\text{b}}=(\text{E}_{\text{liquid}}-\text{N}_{\text{H}}\times\text{E}_{\text{H}})/\text{N}_{\text{H}}$,
where $\text{E}_{\text{liquid}}$ is the total energy computed using DFT or
DMC, and $\text{E}_{\text{H}}$ is the energy of a single H atom. As one can
see, vdw-DF and BLYP+D2 describe the interactions in liquid hydrogen most
accurately among all the functionals. From the atomic side to the molecular
side, the relative binding energy differs by less than 10 meV per atom, and
the minor slope of the curve also benefits the prediction of phase boundary
between the molecular phase and the atomic phase. We also find that PBE and
SCAN tend to overestimate the stability of liquid hydrogen, and a large
positive slope means an severe overestimate of the relative stability of the
atomic phase. In contrast, vdw-DF2 underestimates the stability of liquid
hydrogen and the underestimation of the stability of the atomic phase is
stronger. Based on the benchmark, we conclude that vdw-DF and BLYP+D2 describe
the structural phase transition of liquid hydrogen most accurately, whereas
PBE and SCAN tend to facilitate the molecular-atomic transition and vdw-DF2
hinders the hydrogen molecules from dissociating into atoms. The benchmark
conclusions are in line with the AIMD results: the vdw-DF and BLYP+D2 results
coincide with each other within a small variation; PBE and SCAN predict lower
phase transition temperatures while vdw-DF2 has a higher phase transition
temperature. Note that the DFT methods that we used are different in both the
nature of the underlying exchange correlation functionals and the van der
Waals correction scheme. The importance of van der Waals interactions in dense
hydrogen have been mentioned in some studies, however further analyses show
that the differences in describing the molecular and atomic liquid hydrogen
are mainly dominated by the underlying exchange correlation functional while
the van der Waals interactions do not play a significant role (Fig. S4 si ).
Figure 3: Fundamental energy gap for liquid hydrogen as a function of
molecular fraction x calculated using different methods. Fundamental gap is
defined as
$\Delta=\text{E}(\text{N}+1)+\text{E}(\text{N}-1)-2\text{E}(\text{N})$, where
$\text{N}=128$ is the number of electrons, which equals the number of hydrogen
atoms in the system. Lines with different color show fundamental gap computed
using different methods. The structures are selected from those used in Fig.
2, and are detailed in the Supplementary Material. AMH, MMH and MIH indicates
regimes of atomic metallic hydrogen, molecular metallic hydrogen and molecular
insulating hydrogen, respectively. The background is colored with the
molecular fraction.
We now consider the electronic insulating-metallic phase transition of liquid
hydrogen. The fundamental gap of liquid hydrogen as a function of molecular
fraction calculated by DMC is shown in Fig. 3, computed according to
$\Delta=\text{E}(\text{N}+1)+\text{E}(\text{N}-1)-2\text{E}(\text{N})$. We
find that the fundamental gap $\Delta$ closes at molecular fraction $x=0.8$
(red dashed line), which is far from the half molecular-atomic phase
transition boundary $x=0.5$ (green dashed line). A fraction of $x=0.8$ means
liquid hydrogen becomes conductive when about 20% hydrogen molecules
dissociate into atomic hydrogen. Note that the structures used in computing
the fundamental gap are collected from simulations covering a wide range of
temperatures and pressures si . Therefore, despite that hydrogen metallizes
easier under higher pressures, the molecular fraction can be a robust order
parameter to characterize the metallization of hydrogen.
We can also compute the fundamental gap using DFT methods. As show in Fig. 3,
DFT calculations generally underestimate the fundamental gap compared with
DMC. Specifically, HSE06 and vdW-DF underestimates the gap by up to 30%,
whereas PBE may underestimate by more than 50%. However, an interesting fact
is that the molecular fraction where the fundamental gap closes is independent
of the calculation methods. All DFT functionals predict consistently a
threshold of molecular fraction $x=0.8$ for gap closing, therefore we can
directly calculate the fundamental gap using more economic methods to test the
size effects in fundamental gap calculations. So we select more configurations
from vdw-DF AIMD simulations with 512 hydrogen atoms at different densities
($\text{r}_{\text{s}}$ = 1.35 Bohr,$\text{r}_{\text{s}}$ = 1.50 Bohr and
$\text{r}_{\text{s}}$ = 1.65 Bohr respectively) and calculate their
fundamental gaps using PBE, as shown in Fig. S5 si . We find that the
insulating-metallic phase transition boundary is always around molecular
fraction $x=0.8$.
Figure 4: LLPT on the pressure-temperature phase diagram of hydrogen.
Insulating-metallic transition and molecular-atomic transition boundaries
determined in this work are shown as dark red and green solid lines,
respectively. NVT AIMD simulations of 512 hydrogen atoms were performed with
the vdW-DF functional at different temperatures and densities from which the
molecular fraction and pressure were calculated. The two transition boundaries
were estimated by the temperature and the pressure that reproduce the
molecular fraction x=0.8 (dark red line) and x=0.5 (green line), respectively.
Futher details can be found in Fig. S6 of the Supplemental Material si . MIH,
MMH and AMH indicates the molcular-insulating hydrogen, molecular-metallic
hydrogen and the atomic-metallic hydrogen in the liquid regime. The light red
line indicates the melting line adapted from Ref. Nellis, 2021. Symbols with
dashed lines (dark red) show several experimental measurements of insulating-
metallic phase transition. The dark blue dashed line shows the MAT phase
boundary obtained by Geng et al. using the vdW-DF functional Geng _et al._
(2019). The light blue squares are IMT points calculated by Gorelov et al.
Gorelov _et al._ (2020) using quantum Monte Carlo. Recent experimental
reports of the insulating-metallic transition boundary by Celliers et al.
Celliers _et al._ (2018), Knudson et al. Knudson _et al._ (2015) and Zaghoo
et al. Zaghoo _et al._ (2016); Zaghoo and Silvera (2017) are presented by the
dark red points.
A clear prediction from the above results is that dense liquid hydrogen can be
metallized before the molecular-atomic phase transition, which implies that
experiments can find metallic hydrogen at lower temperatures/pressures than
atomic hydrogen. To provide a more quantitative prediction, we draw the
insulating-metallic ($x=0.8$) and molecular-atomic ($x=0.5$) phase transition
boundaries in the pressure-temperature phase diagram, using AIMD simulations
with the benchmarked vdW-DF functional (See Supplementary Material I and Fig.
S6 for more details si ). As shown by the pink shaded area in Fig. 4, at a
constant temperature the insulating-metallic transition pressure is shifted
downward 30-60 GPa from the molecular-atomic transition pressure. At a
constant pressure, the insulating-metallic transition temperature is lower
than the molecular-atomic transition temperature by a range of 300 to 1000 K.
The two phase boundaries mean that liquid hydrogen can be divided into three
phases, namely the molecular insulating hydrogen (MIH), the molecular metallic
hydrogen (MMH), and the atomic metallic hydrogen (AMH), as labelled in Fig. 3
and Fig. 4. Molecular semi-metallic phase has been observed in the low
temperature solid regime of dense hydrogenEremets _et al._ (2019), and
partially dissociated molecular phases of solid hydrogen have also been
discussed experimentally and theoretically Dalladay-Simpson _et al._ (2016);
Monserrat _et al._ (2018). Here our prediction suggests that there is a wide
range of molecular metallic hydrogen in the liquid regime, where we can
further explore the intriguing electronic transitions of hydrogen. We note
that a different definition of molecular fraction was used in Ref. Geng _et
al._ , 2019 to determine the MAT boundary, and our estimates agree quite well
with theirs, suggesting that the conclusions reached are insensitive to the
definition of the molecular fraction.
In experiments of liquid hydrogen, most of the measures are monitoring the
metallization instead of the structural phase transition. Therefore, the
separation of the two phase transition boundaries allows us to make further
quantitative comparison with experimental data, some of which are also plotted
in Fig. 4. Apart from the original results of Knudson et al., which was later
re-interpreted by Celliers et al.Celliers _et al._ (2018), our new prediction
of the insulating-metallic transition boundary are in better agreements with
experimental data Zaghoo and Silvera (2017); Celliers _et al._ (2018).
Vibrational measurements using e.g. Raman and Infrared spectroscopy have been
performed to identify structural transitions of dense solid hydrogen Howie
_et al._ (2015), and further extension of such techniques are desired to
determine the molecular-atomic LLPT. On the contrary, theoretical studies of
LLPT have mostly focused on structural phase transitions, and there is space
for further developments of theoretical methods, e.g. employing other
electronic structure calculations. Nuclear quantum effects can also lead to a
shift of both boundaries by a small amount Morales _et al._ (2013); van de
Bund _et al._ (2021), but it remains to be investigated whether nuclear
quantum effects on MAT and IMT are effectively the same, and how sensitive are
nuclear quantum effects to the choice of DFT functional.
To conclude, we have performed DMC calculations and AIMD simulations of liquid
hydrogen at a wide range of pressures and temperatures. Our main finding is
that the LLPT of hydrogen is in fact two separate transitions that do not
coincide with each other. Specifically, the insulting-metallic transition
occurs before the molecular-atomic phase transition as the temperature and
pressure increases, leading to a regime of molecular metallic hydrogen with a
width of 30-60 GPa and 300-1000 K. Our results provide an important addition
to the current understanding of the phase diagram of hydrogen. Last but not
the least, the onset of metallic transition in molecular liquid hydrogen is
encouraging for the experimental chasing of the holy grail of metallic
hydrogen and the seeking of high temperature superconductor in dense hydrides.
###### Acknowledgements.
The authors thank X.-Z. Li and A. Zen for helpful discussions. This work was
supported by the Strategic Priority Research Program of Chinese Academy of
Sciences under Grant No. XDB33000000, and the National Natural Science
Foundation of China under Grant No. 11974024, 92165101, 11935002. We are
grateful for computational resources provided by TianHe-1A, the High
Performance Computing Platform of Peking University, Shanghai Supercomputer
Center, and the Platform for Data Driven Computational Materials Discovery of
the Songshan Lake Materials Lab.
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|
# Wideband Audio Waveform Evaluation Networks: Efficient, Accurate Estimation
of Speech Qualities
Andrew Catellier, and Stephen Voran A. Catellier and S. Voran are with the
Institute for Telecommunication Sciences, Boulder, CO, 80305 USA e-mail:
<EMAIL_ADDRESS>
###### Abstract
Wideband Audio Waveform Evaluation Networks (WAWEnets) are convolutional
neural networks that operate directly on wideband audio waveforms in order to
produce evaluations of those waveforms. In the present work these evaluations
give qualities of telecommunications speech (e.g., noisiness, intelligibility,
overall speech quality). WAWEnets are no-reference networks because they do
not require “reference” (original or undistorted) versions of the waveforms
they evaluate. Our initial WAWEnet publication introduced four WAWEnets and
each emulated the output of an established full-reference speech quality or
intelligibility estimation algorithm.
We have updated the WAWEnet architecture to be more efficient and effective.
Here we present a single WAWEnet that closely tracks seven different quality
and intelligibility values. We create a second network that additionally
tracks four subjective speech quality dimensions. We offer a third network
that focuses on just subjective quality scores and achieves very high levels
of agreement. This work has leveraged 334 hours of speech in 13 languages,
over two million full-reference target values and over 93,000 subjective mean
opinion scores.
We also interpret the operation of WAWEnets and identify the key to their
operation using the language of signal processing: ReLUs strategically move
spectral information from non-DC components into the DC component. The DC
values of 96 output signals define a vector in a 96-D latent space and this
vector is then mapped to a quality or intelligibility value for the input
waveform.
###### Index Terms:
convolutional neural network, no-reference objective estimator, speech
intelligibility, speech quality, subjective testing, wideband speech.
## I Introduction
Wired and wireless telecommunications options continue to proliferate and
evolve. Speech quality and speech intelligibility can vary dramatically
between systems and devices, and further variation is driven by changes in
acoustic environment, network loading, and radio conditions. Efficient real-
time measurements of received speech quality and intelligibly are invaluable.
Researchers have been developing such measurement tools for decades but
telecommunications systems, devices, and use cases continue to evolve. This
can cause entirely new classes of speech impairments and existing measurement
tools may then fail to give meaningful results. This scenario motivates the
development of new tools that do give meaningful results in the current
environment.
Measurement tools fall into two major classifications—full-reference (FR)
tools require reference (transmitted) and impaired (received) speech and are
practical in off-line applications. No-reference (NR) tools operate on
impaired speech alone and are much more practical for real-time, in-service
monitoring. Some examples of FR tools developed over the years can be found in
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. The most effective FR tools apply
psycho-acoustic transformations to reference and impaired speech and then
compare these internal representations in ways that mimic key attributes of
human judgement.
NR tools eliminate the need for time alignment and the issue of comparisons,
but they require an embodied model for how speech should sound, independent of
what speech was sent. This is a significant challenge, but success allows for
practical real-time, in-service monitoring of speech quality or
intelligibility. Thus these tools have also been named “single-ended,” “in-
service,” “non-intrusive,” or “output only.” Some examples of early NR tools
are found in [14, 15, 16].
### I-A Existing Machine Learning Approaches
As machine learning (ML) has become more powerful and accessible, numerous
research groups have sought to apply ML to develop NR tools [17, 18, 19, 20,
21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39,
40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50]. Some of these NR tools produce
estimates of subjective test scores that report speech or sound quality mean
opinion score (MOS) [17, 18, 19, 24, 25, 26, 27, 30, 35, 38, 40, 41, 46, 47],
naturalness [28, 34, 36], listening effort [23], noise intrusiveness [47], and
speech intelligibility [20, 32]. The non-intrusive speech quality assessment
model called NISQA [50] produces estimates of subjective speech quality as
well as four constituent dimensions: noisiness, coloration, discontinuity, and
loudness. Other NR tools produce estimates of objective values including FR
speech quality values [22, 29, 31, 42, 48], FR speech intelligibility values
[29, 31, 42, 49], speech transmission index [21], codec bit-rate [43], and
detection of specific impairments, artifacts, or noise types [33, 37, 39, 49].
Some of these tools perform a single task and others perform multiple tasks.
The works cited here cover a wide variety of ML architectures. They address
application areas that include room acoustics (noise and reverberation),
speech enhancers, telecommunications systems, and hearing aids. Each addresses
one or more of narrowband (NB) (nominally 300 Hz–3.5 kHz), wideband (WB)
(nominally 50 Hz–7 kHz), super-wideband (SWB) (nominally 50 Hz–16 kHz), or
fullband (FB) (nominally 20 Hz–20 kHz) speech, except for [33, 43] which
address music. Recent work shows that NR-tools can measure the speech quality
at a system input in spite of the fact that such tools can only access the
system output. [44].
It is natural to consider agreement with subjective test results as the
ultimate goal for NR tools. For ML-based tools this requires large datasets of
subjective test results (most commonly speech quality MOS values) for training
and validation. Sufficient datasets are rare and expensive to generate through
laboratory testing, so crowd-sourced tests are becoming common. Joint training
or transfer learning can leverage objective FR quality values [25, 30, 47] or
impairment categories [46] alongside MOS values to maximize the benefit of
those MOS values. Semi-supervised learning [40] is also an effective way to
compensate for scarce MOS values.
It is important to recognize that the MOS scale is relative, not absolute, and
the use of the scale in any given subjective test will depend on the
conditions included in that test. For example, a given condition might be be
rated 4.0 when it appears in a test with lower quality conditions. But that
same condition might be rated 3.0 when it appears in a test with higher
quality conditions. Per-subject corrections are explored in [47]. A method to
learn per-test correction factors is given in [45] and this approach may be
viewed as the ML version of the linear algebra solution previously given in
[51].
Key considerations when using ML to develop NR tools are the total amount of
data available, how that data is used to both train and test a model, and the
homogeneity or diversity of the training and testing data. The number of
talkers, languages, and sources of impairment are also potentially important
factors, depending on the application. Because the speech quality measurement
community has not yet settled on standardized datasets, some published work is
backed by extensive data and rigorous testing while other work could be
described as an initial proof-of-concept for some specific architecture or
application area, backed by a domain-specific dataset. For this reason three
separate datasets with three different application areas are considered in
this work.
### I-B The WAWEnet Machine Learning Approach
The most common ML approach applies ML to a set of features that form a
spectral description, often one that uses a perceptual frequency scale (e.g.
mel, gammatone, or third-octave bands) and that may use the log of the
spectral magnitudes since the log function can very roughly approximate the
human perception of loudness. Alternately, ML has been applied to features
such as mel-frequency cepstral coefficients, delta MFCCs, perceptual linear
prediction coefficients, pitch, voice activity, frame energy, and features
used in the legacy NR tools [15, 16].
One distinction of WAWEnets is that they apply ML directly to speech waveforms
instead of applying it to features extracted from speech waveforms. Extracting
features from waveforms is a data compression step—it reduces the number of
inputs per unit time that the ML algorithm must process. Good features
identify the most useful information in the speech waveform so that the ML
algorithm may more easily and efficiently map that information to the desired
target value.
But in general, feature extraction is not lossless data compression. The most
popular features apply the magnitude function to complex-valued time-to-
frequency transform results. This clearly removes information, and that
information cannot be restored. Feature extraction makes assumptions about
what information is important and what can be discarded. While many feature-
based approaches have been successful in this problem space, WAWEnets
demonstrate that using waveforms directly is both practical and effective.
Thanks to the convolutional neural network architecture, WAWEnets are not
excessively large, yet they can process waveform samples directly, thus
gaining access to all the information those samples carry and effectively
generating the best features for the task internally, rather than having
potentially sub-optimal features mandated externally.
Before our initial WAWEnet publication [31] we found just one other NR
research effort that considered speech waveforms for input to ML. In [21] ML
is applied to a spectrogram representation of audio in order to estimate
speech transmission index. But this spectrogram representation is also
learned—waveforms are processed with 128 filters, each with length 128, and
those filters are “initialized with a Fourier basis (sine waves at different
frequencies).” So in effect, ML is actually applied to the audio waveform, but
with a strong suggestion that the first ML step should be to calculate a
spectral representation. This spectral representation does not use the
magnitude function and thus it has the potential to preserve all of the
information in the audio waveform.
We are aware of three additional waveform-oriented research efforts that have
emerged since [31] was published. In [40] ML is applied to $\mu$-law
compressed speech in order to estimate speech quality MOS. But the value of
$\mu$ is learned (it is initialized to 256 as in G.711 [52]) so in effect, ML
is actually applied to the speech waveform, but with the strong suggestion
that the first ML step should be compression. No quantization is added, so
this approach can preserve all of the information in the speech waveform.
In [42] the MOS estimation task is learned in two different ways. Speech
waveforms are first processed by either a learnable 1-D convolution layer, or
by the “conventional STFT,” and then supplied to the main ML architecture.
Authors report that compared to the STFT, “the learnable 1-D convolution layer
leads to slight improvements for all targets in nearly all criteria.” This
result suggests that allowing ML to operate on waveforms is preferable to
providing it a spectral representation. And finally, in [53] a WAWEnet-style
architecture is applied to waveforms to generate quality information to guide
dialog remixing.
In [31] we introduced four WAWEnets. Three of these emulated FR speech quality
tools POLQA [10], WB-PESQ [8], and PEMO [7]. The fourth WAWEnet emulated the
FR speech intelligibility tool STOI [54].
Since that time we have updated the WAWEnet architecture to be more efficient
and effective and have leveraged large amounts of data to train three very
effective WAWEnets. Section II describes the new architecture. In Section III
we describe how we used 252 hours of speech from 13 languages to train and
test individual WAWEnets to very closely emulate seven FR tools (four speech-
quality and three speech-intelligibility). For comparison purposes we provide
results from FR versions of these WAWEnets as well. Next we present a single
WAWEnet that emulates all seven FR tools at once.
Section IV introduces 30 hours of subjectively-rated speech and we train
another WAWEnet to produce values for four subjective rating dimensions as
well as the seven FR quality and intelligibility values. In Section V we train
and evaluate a third WAWEnet using 52 hours of speech with MOS scores. This
WAWEnet tracks MOS scores quite closely, confirming that the single
architecture can be trained to produce very good results for both FR objective
targets and subjective targets.
Using ML to develop NR tools for evaluating speech and audio is presently a
rich and active field, with many successful results already published. Our
work is novel because we adopt a relatively simple convolutional architecture,
we apply it directly to speech waveforms, we successfully train to closely
emulate eleven different targets, and we use an unprecedented quantity and
diversity of speech data. Finally Section VII uses the language of signal
processing to explain how WAWEnets process speech signals to produce
evaluations of themselves. To our knowledge, interpretations of this sort have
never before been published.
Figure 1: Diagram depicting the overall shape of our WAWEnet model. Each
rectangle represents one section and the enclosed text describes the shape of
the section’s input vector. Starting with 48,000 speech samples, the model
coalesces information in the time domain until a 96-dimension feature vector
remains.
## II WAWEnet Architecture
Similar to the model we described in [55], WAWEnets accept 3 seconds of speech
sampled at 16,000 smp/s as input. The architecture is composed of sections
that use one-dimensional, 96 channel convolutional layers to filter the signal
and pooling layers which effectively downsample the signal.
This model differs from the model described in [55] in a few key ways. We
added additional convolutional sections that downsample all the way to one
“sample” per channel for each of the 96 channels, thus yielding a 96-dimension
feature vector. A dense layer then maps the feature vector to an estimate.
This model therefore only uses convolutional layers and downsampling to
coalesce temporal information, whereas the previous model also used a dense
layer to coalesce information across a temporal and a feature dimension. All
PReLU activations have been replaced with ReLU, and average pooling is used
throughout. The first section downsamples the input signal by 4, thereby
significantly reducing memory and computation requirements.
In this work we allow the dense layer in $S_{14}$ to generate $N_{T}\geq 1$
estimates. This model is visualized in Fig. 1 and is fully described by Tables
I and II. With $N_{T}=1$, the new formulation has a total of 335,905
parameters, a roughly 50% increase over the 225,025 parameters of our previous
model. For brevity, we refer to specific WAWEnet implementations using a
subscript that can indicate the type and number ($N_{T}$) of estimates
produced. For example, $\text{WAWEnet}_{11}$ is a WAWEnet with one input
channel and 11 estimates. In addition, $\text{WAWEnet}_{\text{S}1}$ estimates
one subjective target and $\text{WAWEnet}_{\text{O}1}$ estimates one objective
FR target.
We also made a separate WAWEnet configuration that allows $S_{1}$ to accept
two 48,000 sample vectors. We used this configuration to create an FR WAWEnet
that uses reference and impaired speech ($\text{WAWEnet}_{1}^{\text{FR}}$)
and, for comparison purposes, a WAWEnet that uses two identical copies of the
impaired speech ($\text{WAWEnet}_{1}^{\text{2}}$).
TABLE I: WAWEnet architecture: sections $S_{1}$–$S_{13}$ are composed of one of the two section types listed in Table II. Number of input and output samples per channel are given by $l_{in}$ and $l_{out}$, effective sample rate by $\hat{f_{s}}$, and effective sample spacing by $s_{l}$. The dense layer $S_{14}$ maps $f_{n}=96$ scalar outputs from $S_{13}$ to the $N_{T}$ estimates. $S$ | section type | $\hat{f_{s}}$ (Hz) | $l_{in}$ | $s_{l}$ (ms) | $l_{out}$
---|---|---|---|---|---
$S_{1}$ | Conv A-4 | 16,000 | 48,000 | 0.0625 | 12,000
$S_{2}$ | Conv A-2 | 4,000 | 12,000 | 0.25 | 6,000
$S_{3}$ | Conv A-2 | 2,000 | 6,000 | 0.5 | 3,000
$S_{4}$ | Conv A-4 | 1,000 | 3,000 | 1 | 750
$S_{5}$ | Conv A-2 | 250 | 750 | 4 | 375
$S_{6}$ | P Conv A-2 | 125 | 376 | 8 | 188
$S_{7}$ | Conv A-2 | 62.50 | 188 | 16 | 94
$S_{8}$ | Conv A-2 | 31.25 | 94 | 32 | 47
$S_{9}$ | P Conv A-2 | 15.63 | 48 | 64 | 24
$S_{10}$ | Conv A-2 | 7.81 | 24 | 128 | 12
$S_{11}$ | Conv A-2 | 3.91 | 12 | 256 | 6
$S_{12}$ | Conv A-2 | 1.95 | 6 | 512 | 3
$S_{13}$ | Conv A-3 | 0.98 | 3 | 1024 | 1
$S_{14}$ | Dense | — | $f_{n}$ | — | $N_{T}$
TABLE II: WAWEnet section types. Each section contains a 1-D convolution layer C-$f_{n}$-$f_{l}$ with $f_{n}$$=96$ filters per channel and $f_{n}$ channels, filter length $f_{l}=3$, stride of 1, and zero padding $\lfloor\frac{f_{l}}{2}\rfloor=1$. Padding layer $\text{Pad}(a,b)$ prepends $a$ zeros and appends $b$ zeros to the input vector. ReLU indicates a ReLU activation. $k$ denotes average pooling layer kernel size. name | Conv A-$k$ | P Conv A-$k$
---|---|---
| | $\text{Pad}(0,1)$
| C-$f_{n}$-$f_{l}$ | C-$f_{n}$-$f_{l}$
| BatchNorm | BatchNorm
| ReLU | ReLU
layers | AvgPool-$k$ | AvgPool-$k$
## III WAWEnet for Seven Objective FR Targets
### III-A Data
The Institute for Telecommunication Sciences (ITS) dataset is formed from
high-quality WB speech recordings. We carefully selected these from an array
of sources including [56, 57, 58, 59, 60, 61, 62, 63, 64, 65] and also from
recordings made in our lab. We extracted three-second “reference segments”
from these recordings such that each reference segment has a speech activity
factor (SAF) of 50% or greater (determined by the P.56 Speech Voltmeter found
in [66]) and any segment has at most 50% (1.5 sec) content in common with any
other segment. We then normalized each reference segment to have an active
speech level of $26\pm 0.2$ dB below the overload point, again using the P.56
Speech Voltmeter.
This process provided 84 hours of speech in the form of 100,681 reference
segments, representing 13 languages and 1230 different talkers. Spanish,
Mandarin, and North American English each account for 29% of the segments.
Korean contributes 6% of the segments, African-accented French 3%, Japanese
2%, while Italian, French, German, Portuguese, Hindi, British English,
Finnish, Dutch, and Polish combine to contribute the remaining 2%.
We strategically split the reference segments into training, testing, and
validation groups, as well as an unseen group. We generated the unseen group
by reserving 10% of the talkers in each language, to the extent possible. The
resulting 127 talkers associated with the unseen group do not contribute any
segments to the remaining data and are only used for evaluation. The unseen
dataset contains 10,391 segments (9 hours) of speech. We split the remaining
segments into training (50%), testing (40%), and validation groups (10%) with
approximate sizes of 38, 30, and 7 hours respectively.
We processed each reference segment with software to simulate the impairments
found in a wide range of current telecommunications environments. These
impairments cover three classes: background noise and suppression, speech
codecs, and packet loss and concealment. Background noises include coffee
shop, party, and street noise at SNRs between 5 and 25 dB. Noise suppression
ranged from mild (minimal artifacts but significant noise) to very aggressive
(significant artifacts and distortion). We applied 6 WB codec algorithms: EVS,
AMR-WB, Opus, G.711.1, G.722.1, and G.722. We selected bit-rates ranging from
8 to 64 kbps for a total of 49 WB codec modes. We used 13 different NB codecs
including EVS, AMR, Opus, G.711, G.729, G.723.1, G.726, MELP and others. Bit-
rates range from 1.2 to 64 kbps for a total of 40 NB codec modes. We applied
both independent and bursty packet losses at rates ranging from 5 to 40%
followed by packet loss concealment (PLC). Finally, we normalized each
impaired segment to have an active speech level of $26\pm 0.2$ dB below the
overload point.
Each reference segment was impaired in three different ways: a randomly
selected NB noise or codec impairment, a randomly selected WB noise or codec
impairment, and a randomly selected NB or WB impairment that combined a codec
with noise or PLC or both. This produced roughly 302,000 segments of impaired
speech. This is 252 hours (114 training, 90 testing, 21 validation, and 27
unseen). A high-level summary of the impairment distribution is given in Table
III. In each row 50% of the speech is NB and 50% is WB. A total of 321
distinct conditions are present in the ITS dataset.
TABLE III: Distribution of impairments in ITS dataset. Hours | Noise & | Speech Codecs | Packet Loss &
---|---|---|---
| Suppression | | Concealment
$84$ | ✓ | |
$84$ | | ✓ |
$28$ | ✓ | ✓ |
$28$ | | ✓ | ✓
$28$ | ✓ | ✓ | ✓
Each impaired segment in the dataset was then labeled with values from seven
established FR estimators: WB-PESQ [8], POLQA [10], ViSQOL (compliance version
c310) [12], and PEMO (software available via [67]) estimate the quality of the
impaired segment, while STOI [54], ESTOI [9], and SIIBGauss [68] give
estimates of the speech intelligibility. Each of these seven FR tools compared
every impaired segment with the corresponding WB reference segment. We then
trained WAWEnets to estimate these FR values using only the impaired segments.
### III-B Training
To prepare each segment for the training process, we converted the speech data
to float and normalized it to the range [-1, 1]. We used affine
transformations to map values from PESQ ([1.02, 4.64]), POLQA ([1, 4.75]),
PEMO ([0, 1]), ViSQOL ([1, 5]), STOI ([0.45, 1]), ESTOI ([0.23, 1]), and
SIIBGauss ([0, 750]) to [-1, 1] before use as targets. As in [29], we
performed inverse phase augmentation (IPA) to augment all datasets in order to
train WAWEnet to learn invariance to waveform phase inversion. This
augmentation increased the amount of data available to just over 500 hours of
total speech data.
When training our model from scratch on the ITS dataset we used one set of
initial weights for each training process. This set of weights was initialized
using the Kaiming-Normal initialization method [69]. In the cases where
$N_{T}>1$, the weights in the last layer were duplicated $N_{T}$ times,
resulting in a shape of $N_{T}\times 96$.
We seeded all random number generators such that batch order and batch
contents were consistent for every training run. WAWEnets were trained using
mini-batches that were as large as GPU memory would allow; in this case, 60
segments per batch. We used the Adam optimizer [70] with $10^{-4}$ learning
rate, and $L_{2}$ regularization parameter set to $10^{-5}$. When the network
had trained for an entire epoch we evaluated the validation set and logged the
epoch RMSE (root mean-squared error) loss $E_{l}$ and per-segment correlation
between the target and the WAWEnet output, $\rho_{seg}$. In the case that
$E_{l}$ on the validation set had not decreased by at least $10^{-4}$ for 5
epochs, we multiplied the learning rate by $10^{-1}$. The network was trained
for 30 epochs. Training $\text{WAWEnet}_{\text{O}1}$ on one NVIDIA GTX 1070
took about 14 hours.111Certain products are mentioned in this paper to
describe the experiment design. The mention of such entities should not be
construed as any endorsement, approval, recommendation, prediction of success,
or that they are in any way superior to or more noteworthy than similar
entities not mentioned.
WAWEnets are NR tools, but for completeness of the research effort, we also
created some FR versions. An NR WAWEnet processes 48,000 samples
($3\text{s}\times 16,000$ smp/s). The FR version
($\text{WAWEnet}_{1}^{\text{FR}}$) processes reference speech and the
corresponding impaired speech. Thus the first convolutional layer of
$\text{WAWEnet}_{1}^{\text{FR}}$ has more parameters (input size $2\
\text{channels}\times 48,000\ \text{samples}\times 96\ \text{filters}$)
compared to the NR WAWEnet (input size $1\ \text{channel}\times 48,000\
\text{samples}\times 96\ \text{filters}$). During initialization, all weights
in the $S_{1}$ were duplicated resulting in a shape of $2\times 48,000\times
96$. In order to make a fair comparison, we also created a dual input NR
version called $\text{WAWEnet}_{1}^{\text{2}}$.
$\text{WAWEnet}_{1}^{\text{2}}$ and $\text{WAWEnet}_{1}^{\text{FR}}$ have
identical architecture and number of parameters.
$\text{WAWEnet}_{1}^{\text{2}}$ processes two identical copies of the impaired
speech. Results for all three versions are presented and compared below.
### III-C Individual network for each of seven FR targets
We trained individual WAWEnets ($\text{WAWEnet}_{\text{O}1}$) for each of the
seven FR targets. Table IV gives the resulting per-segment Pearson
correlations and Table V shows the corresponding per-segment RMS errors. To
allow for direct comparisons, these errors are normalized and shown as a
percentage of the full scale for each target.
The network outputs are highly correlated to the FR target values across a
vast amount of data that spans a wide range of impairment types, talkers and
languages, in spite of the fact that the networks have no access to the
reference speech for comparison purposes. In effect, the networks embody very
effective generalized models for speech quality (or intelligibility). These
generalized models are invariant to speech content, talker, and language and
this allows them to operate without comparison to a reference speech signal.
The “Dual-NR” column of Table IV shows that the additional weights in
$\text{WAWEnet}_{1}^{\text{2}}$’s $S_{1}$ have minimal effect on correlation.
And the “FR” column ($\text{WAWEnet}_{1}^{\text{FR}}$) shows that access to
reference speech (and necessarily increasing the network size) does produce
some additional benefit, as expected. Table V tells the same story in the RMSE
domain. The networks produce impressive estimation error values that range
from 5 to 9% of full scale. These values barely change when network size is
increased, but are further reduced when reference speech is provided.
TABLE IV: Pearson correlations between predictions from three WAWEnets with $N_{T}=1$ and seven objective FR targets, unseen portion of ITS dataset, individual network for each target, correlations calculated per-segment. Target | NR | Dual-NR | FR
---|---|---|---
| $\text{WAWEnet}_{\text{O}1}$ | $\text{WAWEnet}_{1}^{\text{2}}$ | $\text{WAWEnet}_{1}^{\text{FR}}$
WB-PESQ | 0.94 | 0.95 | 0.98
POLQA | 0.93 | 0.93 | 0.97
ViSQOL | 0.95 | 0.95 | 0.98
PEMO | 0.95 | 0.95 | 0.97
STOI | 0.91 | 0.90 | 0.98
ESTOI | 0.95 | 0.95 | 0.99
SIIBGauss | 0.96 | 0.97 | 0.99
TABLE V: Normalized RMSE between predictions from three WAWEnets with $N_{T}=1$ and seven objective FR targets, unseen portion of ITS dataset, individual network for each target, errors calculated per-segment, values are percent of full scale. Target | NR | Dual-NR | FR
---|---|---|---
| $\text{WAWEnet}_{\text{O}1}$ | $\text{WAWEnet}_{1}^{\text{2}}$ | $\text{WAWEnet}_{1}^{\text{FR}}$
WB-PESQ | 9.2 | 9.1 | 4.9
POLQA | 9.4 | 9.4 | 6.6
ViSQOL | 7.2 | 7.0 | 4.9
PEMO | 6.1 | 6.2 | 5.1
STOI | 5.1 | 5.2 | 2.4
ESTOI | 6.1 | 6.1 | 3.2
SIIBGauss | 4.9 | 4.6 | 2.2
### III-D Single network for all seven FR targets
The outstanding success of individual WAWEnets for each target suggests that
the WAWEnet architecture has plenty of capacity for this class of problems.
This then suggests the possibility of a single network that maps speech
signals to points in a single latent space such that those points can then be
mapped to all four quality and all three intelligibility targets. Speech
quality indicates how pleasing a speech signal is to the ear and speech
intelligibility measures the information transferred by the speech. These are
certainly different quantities, but they are also related, and this bodes well
for the possibility of a single network.
We trained a single WAWEnet to produce four, five, six, and all seven target
values and found that this co-training is both possible and beneficial. Seven
networks are replaced with a single network of nearly identical size without
compromise in performance. In fact, co-training with seven targets appears to
regularize the problem and results in improved performance. Table VI provides
correlation and RMSE values for the single network called
$\text{WAWEnet}_{7}$. The single network correlation values are better than
those for the individual networks (see Table IV) in four cases and they are
matched in the other three cases. The single network RMSE values are better
than those for the individual networks (see Table V) in six cases and they are
matched in the seventh case. Fig. 2 contains two-dimensional histograms that
show the joint distribution of $\text{WAWEnet}_{7}$ per-segment estimates and
actual target values for four of the seven targets.
TABLE VI: Pearson correlation, RMSE, and normalized RMSE between seven estimates from a single network ($\text{WAWEnet}_{7}$) and seven objective FR targets, unseen portion of ITS dataset, correlation and error calculated per-segment, error shown as actual values and percent of full scale. | Correlation | RMSE | RMSE (%)
---|---|---|---
WB-PESQ | 0.95 | 0.31 | 7.8
POLQA | 0.94 | 0.33 | 8.3
ViSQOL | 0.95 | 0.28 | 6.9
PEMO | 0.96 | 0.06 | 6.1
STOI | 0.92 | 0.05 | 4.7
ESTOI | 0.95 | 0.06 | 5.8
SIIBGauss | 0.96 | 35.1 | 4.7
(a) WB-PESQ (b) POLQA (c) ViSQOL (d) STOI
Figure 2: Two-dimensional histogram showing joint distribution of
$\text{WAWEnet}_{7}$ per-segment estimates and actual target values with per-
segment Pearson correlation and unnormalized RMSE for four of the seven
targets on the unseen portion of the ITS dataset. Number of segments per bin
is given by the scale at the right.
## IV WAWEnet for Seven Objective FR Targets and Four Subjective Targets
Having successfully developed a single WAWEnet that emulates seven FR tools,
we asked if that network might be further trained to also emulate subjective
scores.
### IV-A Data
We added to our collection the dataset described in [50] and generously
provided by the Quality and Usability Lab of the Technische Universität
Berlin. We designate this the TUB dataset. It contains a variety of speech
sources, simulated impairments (added background noise, selected codecs,
packet loss, bandpass filtering, and clipping) and live impairments
(background noise, landline-to-mobile calls, and VoIP calls) Additional
details are given in [50]. We successfully computed seven FR target values for
14,220 of the TUB speech files.222The “TEST_LIVETALK” database had no
reference files so FR targets could not be calculated. FR estimators
occasionally fail to produce valid results so this reduces the usable number
of files as well. Each of these files also has crowd-sourced subjective
ratings of overall speech quality, noisiness, coloration, discontinuity, and
loudness.
We used the subjective ratings “overall speech quality,” “noisiness,”
“coloration,” and “discontinuity” (labeled as MOS, NOI, COL, and DIS,
respectively) as targets for WAWEnet training. The “loudness” rating is not a
practical target for WAWEnets because WAWEnets use normalized input speech
levels. This removes variation in overall signal level which is a dominating
factor behind the “loudness” ratings. This normalization could be removed if
training for “loudness” is desired.
We divided the dataset in two ways. The first was according to the labeling
that was provided with it. We used the 10,903 files (77%) from “TRAIN_LIVE”
and “TRAIN_SIM” for training, the 642 (4%) files from “TEST_FOR,” “TEST_NSC,”
and “TEST_P501” for testing, and the 2,675 files (19%) from “VAL_LIVE” and
“VAL_SIM” for validation. Note that “TEST_NSC” contains German language speech
and the remainder of the dataset is English language speech. The second way we
divided this dataset was through random sampling: 50% (7,110 files) were used
for training, 40% (5,688 files) for testing, and 10% (1,422 files) for
validation. The results presented below are based on the testing portion in
both cases.
File lengths range from 4.5 s to 14.6 s with a mean of 8.8 s and a median of
9.0 s. WAWEnets work on three-second segments where the SAF is at least 50
percent. For each file we find all such distinct segments in the file—98% of
the files produce two segments, 48% produce three and 10% produce four. The
result is approximately 28,200 training segments, 6,800 validation segments,
and 1400 testing segments—a total of 30 hours of speech. Using G.191 tools
[66], we converted the data from 48,000 smp/s to 16,000 smp/s and normalized
each segment to $26\pm 0.2$ dB below the overload point.
Each subjective rating is based on an entire file. For training we replicate
that file rating to create an identical target for each segment extracted from
that file. If the file shows little temporal variation in the rated attribute,
then this target replication incurs only a minor approximation. But if there
is major variation (e.g. localized packet loss or non-stationary background
noise) then replication of targets can be a significant approximation and
source of error. For testing, the correlations and RMSE values compare each
per-segment WAWEnet output with the subjective rating of the corresponding
file.
Signal bandwidths create an additional approximation in this work. The
subjective ratings in the TUB dataset are based on FB speech signals but
WAWEnets are WB and only analyze the lower 8 kHz of these signals.
(a) MOS (b) DIS (c) WB-PESQ (d) STOI
Figure 3: Two-dimensional histogram showing joint distribution of
$\text{WAWEnet}_{11}$ per-segment estimates and actual target values with per-
segment Pearson correlation and unnormalized RMSE for four of the eleven
targets on the test portion of the TUB dataset when using the 50/40/10 split.
Number of segments per bin is given by the scale at the right.
### IV-B Training and Results
Starting with the weights from $\text{WAWEnet}_{7}$ we allowed the optimizer
to update the weights in each section. This strategy improved overall
performance compared to the random initialization strategy used in Section
III-B. We used affine transformations to map all subjective scores from [1, 5]
to [-1, 1]. Besides those exceptions, we followed the training process
described in Section III-B.
We trained a WAWEnet to emulate the four subjective ratings:
$\text{WAWEnet}_{\text{S}4}$. The per-segment correlation and normalized RMSE
values are shown in Table VII. We also trained WAWEnets to emulate the four
subjective ratings and three, four, five, six, or all seven of the FR values.
Table VII also shows the result for a single WAWEnet that emulates four
subjective and seven objective FR targets: $\text{WAWEnet}_{11}$. Fig. 3
contains two-dimensional histograms that show the joint distribution of
$\text{WAWEnet}_{11}$ per-segment estimates and actual target values for four
of the eleven targets using the 50/40/10 split.
Table VII makes clear that co-training with the objective FR targets improves
correlations and RMS errors for the four subjective targets. It appears that
the extra constraints regularize the problem and lead to a better solution.
Considering the test data prescribed by the TUB dataset, Table VII shows
dramatic correlation drops and RMS error increases for the FR targets. But the
correlation drop is smaller when evaluated on the random split. Table VI and
Table VII are based on different and dissimilar datasets so the comparison is
not exact.
Estimators are often judged by per-condition statistics. Target values are
averaged for all results from each condition (e.g., codec mode or noise
environment) and the same is done for the estimates. These per-condition
averages are then compared by correlation or error statistics. This averaging
reflects a common and relevant estimation situation—it removes variation due
to talkers, utterances, and other factors so we can draw clear conclusions
about the systems we are testing. Table VIII provides per-condition results
analogous to those in Table VII. Here we see that $\text{WAWEnet}_{11}$ gives
per-condition correlations of 0.85 or better for four subjective and three
objective FR targets.
TABLE VII: Per-segment Pearson correlation ($\rho_{seg}$) and normalized RMSE between single WAWEnet trained for 4 ($\text{WAWEnet}_{\text{S}4}$) or 11 ($\text{WAWEnet}_{11}$) targets.Testing portion of TUB dataset. Error values are percent of full scale. | 77/4/19 Split | 50/40/10 Split
---|---|---
| $\rho_{seg}$ | RMSE (%) | $\rho_{seg}$ | RMSE (%)
$N_{T}$ | 4 | 11 | 4 | 11 | 4 | 11 | 4 | 11
MOS | 0.80 | 0.82 | 16 | 15 | 0.84 | 0.85 | 15 | 14
NOI | 0.80 | 0.82 | 14 | 13 | 0.80 | 0.82 | 14 | 14
COL | 0.79 | 0.81 | 13 | 12 | 0.78 | 0.80 | 15 | 14
DIS | 0.75 | 0.78 | 17 | 16 | 0.71 | 0.73 | 17 | 16
WB-PESQ | - | 0.79 | - | 19 | - | 0.89 | - | 15
POLQA | - | 0.80 | - | 17 | - | 0.88 | - | 14
ViSQOL3 | - | 0.83 | - | 14 | - | 0.91 | - | 12
PEMO | - | 0.58 | - | 17 | - | 0.85 | - | 8
STOI | - | 0.52 | - | 16 | - | 0.85 | - | 8
ESTOI | - | 0.58 | - | 22 | - | 0.88 | - | 10
SIIBGauss | - | 0.75 | - | 16 | - | 0.82 | - | 19
TABLE VIII: Per-condition Pearson correlation ($\rho$) and normalized RMSE between single WAWEnet trained for 4 ($\text{WAWEnet}_{\text{S}4}$) or 11 ($\text{WAWEnet}_{11}$) targets. Testing portion of TUB dataset. Error values are percent of full scale. | 77/4/19 Split | 50/40/10 Split
---|---|---
| $\rho$ | RMSE (%) | $\rho$ | RMSE (%)
$N_{T}$ | 4 | 11 | 4 | 11 | 4 | 11 | 4 | 11
MOS | 0.91 | 0.91 | 12 | 12 | 0.96 | 0.96 | 9 | 9
NOI | 0.91 | 0.91 | 10 | 9 | 0.97 | 0.97 | 9 | 9
COL | 0.91 | 0.91 | 9 | 9 | 0.97 | 0.97 | 7 | 8
DIS | 0.90 | 0.90 | 12 | 12 | 0.97 | 0.96 | 10 | 10
WB-PESQ | - | 0.85 | - | 15 | - | 0.94 | - | 10
POLQA | - | 0.91 | - | 10 | - | 0.96 | - | 10
ViSQOL3 | - | 0.89 | - | 12 | - | 0.97 | - | 8
PEMO | - | 0.66 | - | 18 | - | 0.93 | - | 13
STOI | - | 0.66 | - | 19 | - | 0.95 | - | 11
ESTOI | - | 0.60 | - | 25 | - | 0.91 | - | 15
SIIBGauss | - | 0.80 | - | 13 | - | 0.88 | - | 10
## V WAWEnet for Subjective Scores Only
We also trained a WAWEnet to closely emulate just subjective speech quality
scores.
### V-A Data
We are very grateful that the Audio, Speech, and Information Retrieval Group
at Indiana University Bloomington provided us with the dataset described in
[71]. We call this the IUB dataset and it is better suited to WAWEnets than
the TUB dataset is. The IUB dataset is WB, so WAWEnets need not approximate FB
subjective scores from WB signals as was the case with the TUB dataset. In
addition, the IUB file lengths range from 2.0 to 7.8 s with a mean of 3.8 s
and a median of 3.7 s thus providing a much closer match to the WAWEnet three-
second window than was possible with the TUB data.
The IUB dataset includes high-quality speech from close-talking microphones
and lower quality speech from more distant microphones. The distant
microphones necessarily capture more natural and artificial environmental
noise (SNRs reported in the range -10 to +11 dB) and natural reverberation
(speech-to-reverberation ratios reported in the range -5 to +4 dB)[71]. In
addition, some recordings were subjected to 3.4 kHz low-pass filtering in
order to create anchor conditions for subjective testing. Subjective testing
was crowd-sourced. Scores were collected, filtered, and normalized [71] to
produce speech quality scores on a scale of zero to ten. The result is 36,000
speech files, each with a scaled subjective speech quality MOS value.
The IUB dataset contains 35,428 files that are six seconds or shorter. From
each of these files we selected all disjoint three-second segments with SAF at
or above 50%. Any file shorter than three seconds was zero-padded to create a
single three-second segment. By this process 27,004 files produced two
segments each and the remaining 8,424 files produced one segment each. This
gives over 62,000 segments or 52 hours of speech data. Each segment was
assigned the scaled MOS value of the file that it came from. Using only files
six seconds or shorter means that a three-second segment contains at least
half of the file. This minimizes error associated with assigning per-file MOS
values to individual segments when speech quality is not constant throughout
the file. Using G.191 tools [66], we normalized each file to $26\pm 0.2$ dB
below the overload point.
### V-B Training and Results
We used affine transformations to map all subjective scores from [0, 10] to
[-1, 1]. Starting with the weights from $\text{WAWEnet}_{7}$ we allowed the
optimizer to update the weights in each section. However, with this dataset
equivalent results are achieved using the initialization method described in
III-B. We randomly selected 50% of the segments for training, 10% for
validation and 40% for testing. The rest of the training process was similar
to the process described in Section III-B except we allowed training for 60
epochs in this case. The results that follow are based on the approximately
25,000 testing segments, which comprise about 21 hours of speech.
$\text{WAWEnet}_{\text{S}1}$ achieves a per-segment correlation to MOS of 0.96
and normalized RMSE of 6 % of full scale. Fig. 4 shows this correlation
graphically. Six other recently proposed NR tools are applied to the IUB
dataset in [42] and the resulting Pearson correlations to MOS range from 0.93
to 0.96. The $\text{WAWEnet}_{\text{S}1}$ MOS correlation of 0.965 matches the
best of these. Mean absolute errors (MAEs) given in [42] range from 0.40 to
0.50 and the $\text{WAWEnet}_{\text{S}1}$ MAE is 0.41, which places
$\text{WAWEnet}_{\text{S}1}$ near the best previous result. Note that this is
not a complete comparison because the best tools in [42] produce MOS and three
additional estimates, while $\text{WAWEnet}_{\text{S}1}$ produces just MOS.
But there is no question that this WAWEnet architecture performs on par with
these other top-performers on this dataset.
Figure 4: Two-dimensional histogram showing joint distribution of
$\text{WAWEnet}_{\text{S}1}$ per-segment estimates and actual target values
with per-segment Pearson correlation and unnormalized RMSE for scaled MOS on
the test portion of the IUB dataset. Number of segments per bin is given by
the scale at the right.
## VI Discussion and Extensions
We have shown that the NR WAWEnet architecture can evaluate wideband speech
signals in a manner consistent with a variety of objective and subject
evaluations without using any reference speech signal. $\text{WAWEnet}_{7}$
can simultaneously very closely emulate seven diverse objective targets.
$\text{WAWEnet}_{\text{S}1}$ closely emulates MOS values. And simultaneous
emulation of seven objective targets and four subjective targets with
$\text{WAWEnet}_{11}$ appears to improve performance on subjective targets
compared to emulating solely subjective targets.
The weights present in $\text{WAWEnet}_{7}$ appear to be relevant to both the
TUB and the IUB datasets. Training on the TUB dataset with the 50/40/10 split
with a random initialization and 11 targets resulted in MOS, NOI, COL, and DIS
per-segment correlations of 0.80, 0.80, 0.75, and 0.64 respectively—lower than
the correlations achieved when starting from $\text{WAWEnet}_{7}$. When
training on the IUB dataset, the weights present in $\text{WAWEnet}_{7}$ don’t
improve estimation performance, but the training process converged to the best
performance roughly 5 epochs sooner. With both the TUB and the IUB datasets,
solely allowing the weights in $\text{WAWEnet}_{7}$ $S_{14}$ to be trained
resulted in worse performance than allowing the weights in all sections to be
trained.
It is possible to extend the WAWEnet architecture to wider bandwidths (higher
sample rates) by adding one or more sections to the input of the network. For
example, adding a section with a $k=3$ would allow WAWEnets to process 3
seconds of speech with a sample rate 48 kHz.
Likewise, there are several strategies suitable for extending WAWEnets to
process longer signals. One strategy would be to insert a section with a $k=2$
between $S_{13}$ and $S_{14}$. This would allow WAWEnets to generate estimates
for six-second signals. Another strategy would be to use either the 96-D
feature vector or target estimates as an input to a recurrent neural network
of some kind, e.g. long short-term memory (LSTM), or gated recurrent unit
(GRU), thus allowing WAWEnets to process arbitrary-length speech. With
sufficient data either strategy might learn to properly account for the
various principles at work when speech quality varies. These principles
include (see [72]): long-term ratings are lower bounded by minima and upper-
bounded by averages, larger and more frequent quality variations reduce
quality, and recency. Alternately, longer signals can be accommodated by a
sliding three-second WAWEnet processing window followed by averaging or more
sophisticated processing of the multiple results produced.
## VII Signal Processing Interpretation
We have established that the WAWEnet architecture offers an efficient and
effective tool for evaluating wideband speech waveforms. It does this by
mapping a wideband speech waveform to a 96-dimensional vector in a latent
space. Different projections in that space produce scalar values that can
track different objective or subjective values related to speech quality or
intelligibility. We also seek to understand, to the extent possible, _how_ the
WAWEnet architecture maps waveforms to this space. As is often the case with
algorithms developed through machine learning, a fully-satisfying
interpretation is elusive. But we can describe the signal processing that a
WAWEnet applies and give a high-level description of how this signal
processing converts a waveform into an evaluation of that waveform.
### VII-A Functions
To map waveforms to the latent space a WAWEnet uses 13 sections and each
section consists of four layers. In the language of ML, these four layers are
convolution, batch normalization, ReLU, and average pooling. Table IX shows
how the four layers map to six signal-processing (SP) functions. Linear time-
invariant (LTI) systems are often relatively amenable to analysis. Table IX
shows that WAWEnets are linear except for the bias and half-wave rectification
(HWR). Of course these non-linear functions are what enable WAWEnets (and many
other ML based algorithms) to accomplish the assigned tasks and also prevent
the network from simplifying into a trivial and ineffective one. Time-
invariance is satisfied by all functions except the sub-sampling (which shows
time-invariance only for time shifts that are integral multiples of the output
sampling period).
TABLE IX: WAWEnet ML layers expanded into signal processing functions ML Layer | SP Function | Equation | Linear | Time-Invar
---|---|---|---|---
Conv. | FIR Filtering | $y_{k}=\sum\limits_{i=0}^{2}h_{i}x_{k-i}$ | ✓ | ✓
Batch Norm. | Gain | $y_{k}=ax_{k}$ | ✓ | ✓
Bias | $y_{k}=x_{k}+b$ | | ✓
ReLU | Half-Wave Rect. | $y_{k}=\text{max}(x_{k},0)$ | | ✓
Avg. Pool | FIR Filtering | $y_{k}=\frac{1}{m}\sum\limits_{i=0}^{m-1}x_{k-i}$ | ✓ | ✓
Sub-sampling | $y_{k}=x_{mk}$ | ✓ |
$S_{1}$ has one input channel and 96 output channels. $S_{1}$ begins by
splitting the input audio signal into 96 identical copies which then feed into
the 96 channels. These 96 channels are processed in parallel and independent
of each other. Each channel starts with FIR (finite impulse response)
filtering, followed by the application of gain and bias, then HWR, and finally
a low-pass FIR filter and sub-sampling with a factor of four.
$S_{2}$ through $S_{13}$ have 96 input channels and 96 output channels. The
filtering layer of these sections can be described as a full matrix of
filtering. That is, $96^{2}=9216$ filters are used to produce 96 filtered
versions of each input channel. Then each of the 96 output channels is formed
by summing one filtered version of each input channel (96 signals in each
sum). The remaining layers of $S_{2}$ through $S_{13}$ are the same as those
in $S_{1}$, although the sub-sampling may use a factor of two, three, or four.
In each of these layers the 96 channels are processed in parallel and
independent of each other. $S_{6}$ and $S_{9}$ start with zero padding
(appending one zero at the end of the signal) to allow sub-sampling by a
factor of two at the end of the section.
### VII-B Per-Function Operation
A time-domain description of the operations says that WAWEnets replace samples
with weighted local averages, add up signals from 96 different channels, scale
and shift all samples in a channel uniformly, replace negative samples with
zero, and replace blocks of samples with their average value. A frequency-
domain description provides more insight, so we next describe how these
functions change the spectra of the signals as they move through the network.
We describe separately changes to the DC component of the signal spectra and
changes to the other components. This distinction is key to the description of
the overall operation that follows.
The second-order FIR filters have three unconstrained real coefficients,
resulting in either a pair of complex-conjugate zeros, or two zeros on the
real line. The result is gentle spectral peaks and gentle or deep spectral
nulls, depending on the location of the zeros relative to the unit circle.
Forty percent of these filters are lowpass, 29% are highpass, 19% bandstop,
and 12% bandpass. These proportions change by less than 1% between
$\text{WAWEnet}_{7}$, $\text{WAWEnet}_{11}$, and $\text{WAWEnet}_{\text{S}1}$.
This commonality is consistent with the fact that both $\text{WAWEnet}_{11}$
and $\text{WAWEnet}_{\text{S}1}$ use the weights from $\text{WAWEnet}_{7}$ as
their initial state during the training process.
The gain function simply scales all values of the spectrum (DC and non-DC) by
a single value. The bias function adjusts only the DC value of the spectrum
and leaves the rest unchanged. The effect of HWR is controlled by the bias. If
the bias forces all samples to be negative, HWR removes the signal. If the
bias forces all samples to be positive, HWR does nothing. If the bias is such
that the signal is bipolar, the most common and prominent spectral effect is
the creation of new spectral components, thus increasing the spectral density.
An example is given in the middle panel of Fig. 5. The effect of HWR on the DC
value of the signal depends strongly on the signal.
Figure 5: Top panel shows spectra of two tones (345 Hz, shown in light blue,
and 6789 Hz, shown in gold), middle panel shows the result of half-wave
rectification with zero bias, bottom panel shows the additional effect of
average pooling (pooling factor is 2). Colors emphasize non-linearity: light
blue shows spectral components that are produced by the 345 Hz tone alone,
gold shows components produced by the 6789 Hz tone alone, blue shows
components that only appear when both tones are present. A significant DC
component will be produced in any of these three cases and it is shown in
black.
The FIR filtering that precedes the subsampling is length $m$ with all
coefficients equal to $m^{-1}$ ($m=$ 2, 3, or 4). These are low-pass filters
and each one has a perfect null at any frequency that would alias to DC so the
subsequent subsampling cannot change the DC value of the spectrum. (This is
consistent with the fact that averaging cannot change the DC value of a
signal.) The subsampling will remove all spectral content above the new
Nyquist frequency and can produce aliasing at any non-DC frequency below the
new Nyquist frequency. The aliasing is significant because these FIR low-pass
filters are very short and have responses that are far from the near brick-
wall responses needed to achieve alias-free sub-sampling. For example, when
$m=$2 spectral components just above the new Nyquist frequency are aliased to
those just below the new Nyquist frequency with only 3 dB of attenuation.
Aliasing involves addition of complex values, so aliasing may reduce or
increase the original spectral magnitudes depending on the relative phases of
the two addends. The bottom panel of Fig. 5 shows an example.
Note that in conventional sample rate reduction, a filter calculates one
output sample for each input sample, then subsampling retains every $m^{th}$
sample. The avgpool function integrates filtering and subsampling so that only
every $m^{th}$ sample is calculated. Because the filtering is FIR and the
filter length matches the downsampling factor, this unconventional approach
produces the same results as the conventional approach.
### VII-C Overall Operation
We can view the end-to-end mapping from audio signals to the 96-D latent space
as 96 individual (but coupled) signal processors. The job of these processors
is to shorten the signals and to strategically shape and move relevant
spectral information to DC. In $S_{13}$ the length-3 filtering and 3-to-1 sub-
sampling (the avgpool layer) serve to extract the DC value of a signal that
has 3 samples. The ensemble of the 96 DC values defines a vector in the latent
space which is then mapped to a final output value by $S_{14}$.
The particulars of shaping spectral information and moving it to DC are shown
graphically for one section in Fig. 6. In each section non-DC spectral values
of the signals are modified by FIR filtering, gain, HWR, the pooling FIR
filter, and sub-sampling. The DC spectral values of the signals are modified
by just four of the six functions. In the FIR filtering, gain, and bias
functions the modification of the DC value is determined solely by the
processor. These modifications are independent of the signal itself. But in
the HWR function the modification of the DC spectral component is driven by
the non-DC spectral components. The HWR is the stage where relevant non-DC
spectral information is strategically moved to DC.
Figure 6: In each section four functions change DC component and five change
non-DC components. Non-DC components affect the DC component _only_ in the
half-wave rectification function. Sections sequentially move non-DC
information to DC.
For example, consider a speech signal and noise signal each with a DC value of
zero. When added, the noisy speech signal still has a DC value of zero. But
after HWR, the original and noisy speech signal can have very different DC
values and this DC value can thus serve to indicate that noise was present in
the speech signal. This is a very simple example but by using many sections of
intricate spectral shaping and folding, this processor can also accurately
assess a broad array of much more nuanced perturbations to speech signals.
Fig. 6 emphasizes the fact that every function modifies either the DC
component or the non-DC component, or both, and that the HWR is the sole
function where non-DC information influences the DC value. The input audio
signal will typically have a DC value near zero, but as the processing
continues, the spectral shaping and the flow of information to DC at HWRs
results in DC values that describe important characteristics of the original
audio signal.
Fig. 7 gives a visual example of DC values in the first 13 sections of
$\text{WAWEnet}_{\text{S}1}$ (1 input signal plus the outputs of 6 SP
functions $\times$ 13 sections gives 79 rows) for each of the 96 channels.
Input speech is shown at the top of the figure and DC values there are all
zero as expected. As signals move through the network (down the figure) DC
values build in various channels for part or all of a section. No continuous
downward “flow” of DC appears because channels are fully connected with each
other in every section and channel numbers are arbitrary. The most dramatic
distribution of DC values is in the second half of the network (lower half of
the figure), and this distribution then moderates to form the output.
Figure 7: Example DC values (dark purple for large negative, bright yellow for
large positive) in the first 13 sections (shown as labeled macro-rows) and all
96 channels (shown as columns) of $\text{WAWEnet}_{\text{S}1}$.The six rows
within each section show the DC output value of six signal processing
functions (FIR, gain, bias, HWR, FIR, sub-sampling, in that order) in that
section. The DC value of the input signal is replicated 96 times above
$S_{1}$.
In effect, the wideband input speech signal passes through 96 parallel coupled
signal processors (composed of $S_{1}$–$S_{13}$) and the 96 DC values of the
96 output signals form a vector that is then mapped by $S_{14}$ to an estimate
of some quality of that speech signal. Fig. 8 shows examples of these 96-D
$\text{WAWEnet}_{7}$ outputs for 30 different conditions (1,320 segments
averaged for each condition). These examples show how different dimensions
respond to different attributes while they work together to produce an
estimate of speech quality or intelligibility.
Figure 8: 96 outputs of $S_{13}$ for 30 conditions. From left to right, six
increasing bit-rates for each of four codecs, then three noise suppression
thresholds for each of two SNRs. Example observations: channel 32 responds
strongly to WB vs NB and also to noise suppression thresholds, channel 17
responds to codec bit-rates and to noise levels.
## VIII Conclusion
WAWEnets are no-reference wideband audio waveform evaluation networks that
process waveforms directly into evaluations of those waveforms. We have
trained and evaluated multiple WAWEnets and this work is based on 334 hours of
speech in 13 different languages produced by well over 1000 different talkers.
We have demonstrated that a $\text{WAWEnet}_{7}$ can produce speech quality
and intelligibility estimates that agree (correlations of 0.92 to 0.96) with
values from seven established FR objective estimators. $\text{WAWEnet}_{11}$
demonstrates that this architecture is useful for estimating both objective
and subjective speech qualities at the same time. $\text{WAWEnet}_{\text{S}1}$
agrees with the subjective MOS values of the IUB dataset with per-segment
correlation 0.965, commensurate with top results from other current
approaches. We expect that WAWEnets will generate accurate predictions of
speech qualities for applications where speech impairments resemble those
represented in our wide-ranging training dataset. In addition, we expect they
are useful for narrower applications (for example, measuring reverb present in
a room) if trained to do so. WAWEnets’ small size make them approachable for
applications where computing power is at a premium.
WAWEnets are distinct from the vast majority of alternatives because they
apply ML directly to speech waveforms instead of applying it to features
extracted from speech waveforms. This gives WAWEnets access to all the
information in the waveforms and allows them to effectively generate the best
features for the task internally, rather than having potentially sub-optimal
features mandated externally. Our work shows that this is indeed a viable
approach.
We have proposed several strategies to extend WAWEnets to wider bandwidths and
longer signals as data becomes available.
WAWEnets are composed of six common signal processing functions and, as
expected, the non-linear functions (bias and HWR) are critical. These
functions move spectral information (shaped by FIR filtering and gains) to DC
so that the DC values of the final, very short, signals provide a 96-D
description of the original, much longer signals. We encourage further
experimentation and development with WAWEnets and that is enabled by the
software repository at https://github.com/NTIA/WEnets.
## Acknowledgments
Sincere thanks to Gabriel Mittag, Sebastian Möller, and the members of the
Quality and Usability Lab of the Technische Universität Berlin for creating
and sharing the TUB dataset; to Zhuohuang Zhang, Donald Williamson, and the
members of the Audio, Speech, and Information Retrieval Group at Indiana
University for creating and providing the IUB dataset; and to Michael Chinen,
Andrew Hines, and Jan Skoglund for support with ViSQOL 3.0.
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Andrew Catellier (Senior Member, IEEE) attended the University of Wyoming
where he earned a B.S. in Electrical Engineering in 2006 and an M.S. in
Electrical Engineering in 2007. Since then, Andrew conducted traditional and
crowd-sourced subjective experiments for audio and video signals and explored
objective speech quality assessment using ML techniques at the Institute for
Telecommunication Sciences (ITS) in Boulder, Colorado. He worked to facilitate
crop yield measurements, forecasts, and classification via remote sensing,
computer vision, and ML at GeoVisual Analytics from 2017 until 2022.
Currently, Andrew uses computer vision and ML to validate identification
documents and identity at Nametag, Inc.
---
Stephen Voran (Senior Member, IEEE) received the B.A. degree in Mathematics
from Carleton College, Northfield, MN, in 1985, and the M.S. degree in
Electrical Engineering from the University of Colorado, Boulder, in 1989.
Since 1990 he has been with the Institute for Telecommunication Sciences,
Boulder, Colorado and has been contributing to signal-processing based
advances in objective assessment of telecommunications speech quality and
intelligibility, as well as subjective audio testing, speech coding, and audio
quality enhancement.
---
|
Entangling transmons with low-frequency protected superconducting qubits
Andrea Maiani, Morten Kjaergaard, and Constantin Schrade
Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
Novel qubits with intrinsic noise protection constitute a promising route for improving the coherence of quantum
information in superconducting circuits. However, many protected superconducting qubits exhibit relatively low transition frequencies, which could make their integration with conventional transmon circuits challenging. In this work, we propose and study a scheme for entangling a
tunable transmon with a Cooper-pair parity-protected qubit, a paradigmatic example of a low-frequency protected qubit that stores quantum information in opposite Cooper-pair parity states on a superconducting island. By tuning the external flux on the transmon, we show that non-computational states can mediate a two-qubit entangling gate that preserves the Cooper-pair parity independent of the detailed pulse sequence. Interestingly, the entangling gate bears similarities to a controlled-phase gate in conventional transmon devices. Hence, our results suggest that standard high-precision gate calibration protocols could be repurposed for operating hybrid qubit devices.
Superconducting transmon qubits [1] are a highly promising platform for noisy intermediate-scale quantum (NISQ) devices [2] and error-corrected quantum computers [3, 4, 5, 6] with applications ranging from quantum simulations [7, 8, 9, 10] to the first experiments on quantum advantage [11, 12]. Among the most attractive features of the transmon circuit are its reproducibility, insensitivity to charge noise-induced dephasing, and coherence times that have seen steady improvements over the past decade [13]. Interestingly though, despite notable advances in prolonging the coherence of transmons, the transmon circuit does not exhibit intrinsic protection to qubit relaxation errors. It is thus an important question how transmon devices can be further optimized with complementary qubit technologies to accelerate the path to fault-tolerant quantum computation.
Motivated by the challenge of exploring complementary qubit modalities, several alternative qubit encodings have been proposed [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32] and experimentally studied [33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45].
A particular class of such novel qubit encodings are Cooper-pair parity-protected qubits (PPQ) [30, 31, 32, 33, 34], which rely on a special Josephson element that only permits tunneling of pairs of Cooper-pairs. Similar to the transmon qubit, the two nearly-degenerate ground states of the PPQ have a nearly flat charge dispersion, which makes them insensitive to charge-noise induced dephasing. Similar to the fluxonium qubit [39, 40, 41, 42, 43, 44], the two qubit states also have disjoint support, since they carry opposite Cooper-pair parity. This disjoint support ensures that, if the qubit-environment coupling conserves the Cooper-pair parity, relaxation errors between the computational states are prevented.
While considerable efforts have been devoted to the development of a gate set for protected superconducting qubits [16, 46, 47, 48, 49, 50] for their use as an independent quantum computing platform, a different approach is to integrate protected qubits as memory elements in a conventional transmon-based quantum computing architecture.
In such a scheme, the qubit state would be stored on the protected qubit during idle times and transferred to the transmon qubits for fast, high-fidelity operations, using the full machinery of well-established high-fidelity transmon operation. However, many protected qubits exhibit relatively low qubit transition frequencies, which could make this integration with transmon devices challenging. This motivates the question of efficiently generating entanglement between protected superconducting qubits and transmon qubits.
In this work, we propose and study a capacitive coupling scheme for entangling a tunable transmon with a PPQ, a paradigmatic example of a protected superconducting qubit featuring a nearly-flat charge dispersion and qubit states with disjoint support.
By tuning the external flux on the transmon, we show that non-computational states can mediate an entangling gate that
preserves the Cooper-pair parity irrespective of the detailed pulse sequence. Besides opening the way to coherent state transfer, the proposed entangling gate also bears similarities with a controlled-phase gate in conventional capacitively coupled transmon qubits. Consequently, our results suggest standard high-precision two-qubit calibration protocols could be repurposed for the operation of hybrid qubit devices.
§ SETUP
As depicted in Fig. <ref>(a), we consider a direct capacitively coupling between a frequency-tunable transmon qubit and
a PPQ, realized by a capacitively-shunted $\cos(2\phi)$ Josephson element for the tunneling of pairs of Cooper-pairs. The individual Hamiltonians of the transmon, $H_{t}$,
and of the PPQ, $H_{p}$, are given by,
\begin{equation}
\begin{split}
H_{t} &= 4 E^{}_{C,t} (n_{t} - n^{}_{g,t})^2 - E^{}_{J,t}\, \text{cos}(\phi_{t}) \,, \\
H_{p} &= 4 E^{}_{C,p} (n_{p} - n_{g,p})^2 - E^{}_{J,p} \, \text{cos}(2\phi_{p})\,.
\end{split}
\label{Eq1}
\end{equation}
Here, $(n_{t},\phi_{t})$ and $(n_{p},\phi_{p})$ denote Cooper-pair charge and phase degrees of freedom of the transmon and PPQ. Moreover, $E_{J,t}$ is the transmon Josephson energy and $E_{J,p}$ is the two-Cooper-pair tunneling amplitude of the PPQ. The charging energies of the two qubit circuits are $E^{}_{C,t} =e^{2}/2C_{t}$ and $E^{}_{C,p} =e^{2}/2C_{p}$ with the shunt capacitances $C_{t}$ and $C_{p}$.
Both Hamiltonians in Eq. (<ref>) can be diagonalized exactly by rewriting the eigenvalue problems as Mathieu equations. For the transmon [1], the energy splitting between the ground and first-excited state, which form the qubit basis $\left|0_{t}\right\rangle$ and $\left|1_{t}\right\rangle$, is $\omega_{t}=\sqrt{8E_{J,t}E_{C,t}}+\delta\omega_{t}$ with $\delta \omega_t\propto \text{exp}(-\sqrt{8E_{J,t}/E_{C,t}}) \cos(2\pi n_{g,t})$ for $E_{J,t}\gg E_{C,t}$, see the left panel of Fig. <ref>(b). For the PPQ [31], the qubit basis is given by the two lowest-energy states with even and odd Cooper-pair parity, $\left|0_{p}\right\rangle$ and $\left|1_{p}\right\rangle$. These states have an exponentially suppressed energy splitting, $\omega_{p}\propto \text{exp}(-\sqrt{2E_{J,p}/E_{C,p}}) |\cos(\pi n_{g,p})|$ for $E_{J,p}\gg E_{C,p}$, see the right panel of Fig. <ref>(b). Unlike the transmon, the PPQ is thus a low-frequency qubit with the computational states exhibiting an exact degeneracy if $\text{cos}(\pi n_{g,p})=0$ and a near-degeneracy otherwise. However, like the transmon, the energy splitting of the PPQ is insensitive to variations in $n_{g,p}$ if $E_{J,p}\gg E_{C,p}$, which ensures insensitivity to charge noise dephasing.
To explain the protection of the PPQ against parity-preserving relaxation errors, we consider the wavefunctions of the computational basis. In phase space, these wavefunctions are symmetric/anti-symmetric combinations of states that are localized in the $0$- and $\pi$-valleys of the Josephson potential, see Fig. <ref>(c). In charge space, the same wavefunctions are superpositions of states with even/odd Cooper-pair number. Due to this disjoint support of the charge space wavefunctions, $\left\langle 0_{p} | \mathcal{O} | 1_{p}\right\rangle = 0$ for any operator $\mathcal{O}$ that preserves the Cooper-pair parity, which is the condition for protection against parity-preserving relaxation errors [30].
Hybrid qubit setup.
(a) A frequency-tunable transmon (red) coupled to a PPQ (blue) realized by a capacitively-shunted tunneling element for pairs of Cooper-pairs (Josephson junction symbol with double lines). The two qubits are coupled via a coupling capacitance $C_c$.
(b) Zoom-in on the charge dispersion relation for the transmon (left panel) and the PPQ (right panel). In the gray area, the ground (first excited) state carries odd (even) Cooper-pair parity. In the white area, the order is inverted.
(c) Wavefunctions of the decoupled system $(C_c=0)$. The system parameters are $(E_{J,t}, E_{J,p},E_{C,t},E_{C,p})
= 2\pi (10,3,0.25,0.25)\,\text{GHz}$.
Low-energy spectrum of the hybrid qubit setup and $\textsf{CZ}_\phi$ gate.
(a) Low-energy spectrum of the hybrid qubit setup for $n_{g, p}=0$ and $(E_{J,t}, E_{J,p},E_{C,t},E_{C,p},E_{C,c})
= 2\pi (12,2.7,0.2,0.15,0.025)\,\text{GHz}$ as a function of the external flux $\Phi_{\text{ext}}$ of the tunable transmon.
The $\textsf{CZ}_\phi$ gate is realized by a rapid excursion from $\Phi_{\text{ext}}=0$ to the vicinity of the
$\left|1_{t},0_{p}\right\rangle\leftrightarrow\left|0_{t},3_{p}\right\rangle$ anti-crossing at $\Phi_{\text{ext}}=\Phi^{*}_{\text{ext}}$. The bottom panel shows the coupling strengths, $g^{yz}$ and $g^{zz}$, upon approaching the anti-crossing.
(b) Same as (a) but for $n_{g, p}=0.5$. Each shown energy level is now exactly two-fold degenerate.
Having introduced the two decoupled qubit circuits with the associated computational subspace $\mathcal{P}_0=
\{
\left|1_{t},1_{p}\right\rangle,
\left|1_{t},0_{p}\right\rangle,
\left|0_{t},1_{p}\right\rangle,
\left|0_{t},0_{p}\right\rangle
\}
$, we proceed by coupling the qubits via a standard capacitive coupling (see also Fig. <ref>a) corresponding to a coupling Hamiltonian given by
\begin{equation}
H_{c} = 4 E_{C,c} (n_{p} - n_{g,p})(n_{t} - n_{g,t}).
\label{Eq2}
\end{equation}
Here, $E_{C,c} = e^2 C_{c}/(C_{p}C_{t})$ with the coupling capacitance $C_{c}$. In summary, the full Hamiltonian of our setup is $H_{}=H_{p}+H_{t}+H_{c}$. In the next section, we will derive the effective qubit interaction due to this direct capacitive coupling.
§ EFFECTIVE HAMILTONIAN
To motivate the derivation of the effective qubit interaction, we first recall the case of two capacitively coupled transmon qubits, $t1$ and $t2$, which are both `high-frequency' qubits. In this example, the capacitive coupling mediates a $\sigma_{t1}^{y}\sigma_{t2}^{y}$ interaction when projected onto the computational subspace and a $\sigma_{t1}^{z}\sigma_{t2}^{z}$ interaction due to the mixing of computational and non-computational states [51]. In our setup, which involves the coupling of a `high-frequency' transmon qubit and a `low-frequency' PPQ, we will show that the couplings to non-computational states will play an even more essential role. To anticipate this result, we note that the coupling Hamiltonian of Eq. (<ref>) at $n_{g,p}=0$ vanishes exactly when projected onto the computational subspace, $\langle s_{t}, s'_{p}|H_{c}|s''_{t},s'''_{p}\rangle=0$ for any two states $|s_{t},s'_{p}\rangle, |s''_{t},s'''_{p}\rangle$ in $\mathcal{P}_0$ since $\langle 0_{t}|n_{t}|0_{t}\rangle=\langle 1_{t}|n_{t}|1_{t}\rangle=0$ and $\langle 0_{p}|n_{p}|1_{p}\rangle=0$. A direct coupling within the computational subspace is thus fully absent at $n_{g,p}=0$ and any qubit interaction, if present, is necessarily mediated by virtual transitions through non-computational states.
§.§ Special case: $\boldsymbol{n_{g,p}}=0$
To identify the origin of such virtual transitions, we initially compare two special cases with the offset charge on the PPQ set to either $n_{g,p}=0$ or $n_{g,p}=0.5$. Starting with the $n_{g,p}=0$ case, we show the low-energy spectrum as a function of external magnetic flux $\Phi^{\text{ext}}_{t}$ of the tunable transmon in Fig. <ref>(a). The spectrum comprises not only the four qubit levels of $\mathcal{P}_0$ but also two additional levels corresponding to the $\left|0_{t},2_{p}\right\rangle$ and $\left|0_{t},3_{p}\right\rangle$ state of the uncoupled system.
Interestingly, the non-computational states exhibit two anti-crossings with the computational states, $\left|1_{t},1_{p}\right\rangle\leftrightarrow\left|0_{t},2_{p}\right\rangle$ and $\left|1_{t},0_{p}\right\rangle\leftrightarrow\left|0_{t},3_{p}\right\rangle$, at certain values of external flux. These anti-crossings arise because the respective couplings preserve the Cooper-pair parity on the PPQ. On the other hand, anti-crossing between $\left|1_{t},0_{p}\right\rangle\leftrightarrow\left|0_{t},2_{p}\right\rangle$ and $\left|1_{t},1_{p}\right\rangle\leftrightarrow\left|0_{t},3_{p}\right\rangle$ are absent from the spectum in Fig. <ref>(a), as such couplings violate the Cooper-pair parity conservation on the PPQ. We will now show that in the vicinity of the two anti-crossings, virtual transitions in-and-out of the computational subspace are enhanced and, consequently, can induce a sizable effective qubit interaction between the transmon and the PPQ.
For computing the effective qubit interaction at $n_{g,p}=0$, we initially project our setup Hamiltonian $H$ onto the four qubit states of $\mathcal{P}_0$ and on the additional $\left|0_{t},2_{p}\right\rangle$ and $\left|0_{t},3_{p}\right\rangle$ states. This yields the following low-energy Hamiltonian,
\begin{equation}
\begin{pNiceArray}{cccc|cc}
\omega_{11} & 0 & 0 & 0 &\lambda' & 0 \\
0 & \omega_{10} & 0 & 0 & 0 & -\lambda'' \\
0 & 0 & \omega_{01} & 0 & 0 & 0 \\
0 & 0 & 0 & \omega_{00} & 0& 0 \\
\hline
\lambda' & 0 & 0 & 0 & \omega_{02} & 0 \\
0 & -\lambda'' & 0 & 0 & 0 & \omega_{03}\\
\end{pNiceArray}
\label{lowenergy1}
\end{equation}
Here, the frequency of the the $\left|s_{t},s'_{p}\right\rangle$ state in the uncoupled system is denoted by $\omega_{ss'}=\omega_{t,s}(\Phi^\mathrm{ext}_t)+\omega_{p,s'}$. Moreover, the coupling matrix elements are given by $\lambda'=\left\langle 1_{t},1_{p}|H_{c}|0_{t},2_{p}\right\rangle$ and $\lambda''=\left\langle 1_{t},0_{p}|H_{c}|0_{t},3_{p}\right\rangle$, where we picked a wavefunction gauge for which ($\lambda',\lambda''$) are real-valued. We point out that the low-energy Hamiltonian of Eq. (<ref>) is different from the one of capacitively-coupled transmons [51], because the conservation of Cooper-pair parity prohibits a coupling of the $\left|0_{t},1_{p}\right\rangle$ to the $\left|1_{t},0_{p}\right\rangle$ state. Also, for two coupled transmons only the highest energy computational state exhibits crossing with non-computational states. In our case, the two computational states, $\left|1_{t},0_{p}\right\rangle$ and $\left|1_{t},1_{p}\right\rangle$, both cross with non-computational states, albeit at different values of external flux.
Next, we integrate out the non-computational states to second order in $\lambda'$ and $\lambda''$ by a Schrieffer-Wolff transformation [52]. Provided that $\lambda'^{2}\ll |\omega_{02}-\omega_{11}|$ and $\lambda''^{2}\ll |\omega_{03}-\omega_{10}|$, we find that the effective qubit Hamiltonian reads [53],
\begin{align}
\left(\omega_{t}+\frac{g^{zz}_{+}}{2}\right)
\frac{
\sigma^{z}_{t}
\left(\omega_{p}+\frac{g^{zz}_{-}}{2}\right)
\frac{
\sigma^{z}_{p}
\sigma^{z}_{t}
\frac{
\sigma^{z}_{p}
\\
\label{Heffng0}
\end{align}
where $\omega_{p/t}=\omega_{p/t,1}-\omega_{p/t,0}$ denote the bare qubit frequencies. The key insight from Eq. (<ref>) is that the interaction between the two qubits is of $\sigma^{z}_{p}\sigma^{z}_{t}$ type. As anticipated, this interaction arises from a two-step perturbative sequence involving virtual transitions in-and-out of the
$\left|0_{t},2_{p}\right\rangle$ and $\left|0_{t},3_{p}\right\rangle$ state. For example, in a perturbative sequence close to the $\left|1_{t},1_{p}\right\rangle\leftrightarrow\left|0_{t},2_{p}\right\rangle$ anti-crossing, the system exhibits a first virtual transition from the computational state $\left|1_{t},1_{p}\right\rangle$ to the non-computational state $\left|0_{t},2_{p}\right\rangle$ and, subsequently, a second virtual transition back to $\left|1_{t},1_{p}\right\rangle$. Such a sequence preserves the state of the transmon, which explains why the interaction is $\propto\sigma^{z}_{t}$. The dependence of the interaction on $\sigma^{z}_{p}$ arises because the coupling Hamiltonian of Eq. (<ref>) preserves the Cooper-pair parity.
§.§ Special case: $\boldsymbol{n_{g,p}}=0.5$
Having derived the qubit interaction at $n_{g,p}=0$, we want to compare the results of Eq. (<ref>) with the $n_{g,p}=0.5$ case. We therefore plot the low-energy spectrum at $n_{g,p}=0.5$ in Fig. <ref>(b). Unlike in the previous case, we find that each depicted energy level exhibits an exact two-fold degeneracy, corresponding to opposite Cooper-pair parity sectors. This finding is consistent with our results of Fig. <ref>(b), where we pointed out
that the levels on the uncoupled PPQ are exactly degenerate at $n_{g,p}=0.5$. In particular, the anti-crossings $\left|1_{t},1_{p}\right\rangle\leftrightarrow\left|0_{t},2_{p}\right\rangle$ and $\left|1_{t},0_{p}\right\rangle\leftrightarrow\left|0_{t},3_{p}\right\rangle$ occur now at the same value of external flux and overlap exactly. Couplings between $\left|1_{t},0_{p}\right\rangle\leftrightarrow\left|0_{t},2_{p}\right\rangle$ and $\left|1_{t},1_{p}\right\rangle\leftrightarrow\left|0_{t},3_{p}\right\rangle$ remain absent (they are forbidden since the states belong to a different parity sector). We will now show that this new scenario at $n_{g,p}=0.5$ will lead to a different effective qubit Hamiltonian compared to Eq. (<ref>).
We begin again by projecting the setup Hamiltonian $H$ onto the qubit subspace $\mathcal{P}_{0}$ and onto the states $\left|0_{t},2_{p}\right\rangle$ and $\left|0_{t},3_{p}\right\rangle$. The resulting low-energy Hamiltonian reads,
\begin{equation}
\begin{pNiceArray}{cccc|cc}
\omega_{11} & 0 & -i\eta & 0 & \lambda & 0 \\
0 & \omega_{10} & 0 & i\eta & 0 & -\lambda \\
i\eta & 0 & \omega_{01} & 0 & 0 & 0 \\
0 & -i\eta & 0 & \omega_{00} & 0 & 0 \\
\hline
\lambda & 0 & 0 & 0 & \omega_{02} & 0 \\
0& -\lambda & 0 & 0& 0 & \omega_{03}
\end{pNiceArray}.
\end{equation}
Here, we have $\lambda=\left\langle 1_{t},1_{p}|H_{c}|0_{t},2_{p}\right\rangle=-\left\langle 1_{t},0_{p}|H_{c}|0_{t},3_{p}\right\rangle$ and $\eta=i\left\langle 1_{t},1_{p}|H_{c}|0_{t},1_{p}\right\rangle=-i\left\langle 1_{t},0_{p}|H_{c}|0_{t},0_{p}\right\rangle$ in a wavefunction gauge for which $(\lambda,\eta)$ are real-valued. By inserting the coupling Hamiltonian in the expressions for the matrix elements, we note that $\eta\propto\left\langle s_{p}|n_{p}|s_{p}\right\rangle-n_{g,p}$. In the previous case when $n_{g,p}=0$, we had $\left\langle s_{p}|n_{p}|s_{p}\right\rangle=0$ and, consequently, $\eta$ vanished. In the present case when $n_{g,p}=0.5$ and $E_{J,p}\gtrsim E_{C,p}$, we have $\left\langle s_{p}|n_{p}|s_{p}\right\rangle\neq n_{g,p}$ so that $\eta$ is finite yet gets successively smaller upon increasing $E_{J,p}$.
In particular, when $E_{J,p}\gg E_{C,p}$, we have $\left\langle s_{p}|n_{p}|s_{p}\right\rangle\rightarrow n_{g,p}$ so that the contribution of $\eta$ to the low-energy Hamiltonian is negligible.
Lastly, we note that due to the degeneracy of the PPQ levels, $\omega_{p,0}=\omega_{p,1}$ and $\omega_{p,2}=\omega_{p,3}$. Hence, the frequencies of the hybrid setup satisfy $\omega_{11}=\omega_{10}$, $\omega_{01}=\omega_{00}$, and $\omega_{02}=\omega_{03}$.
We now proceed by integrating out the effects of the non-computational states, $\left|0_{t},2_{p}\right\rangle$ and $\left|0_{t},3_{p}\right\rangle$, with a Schrieffer-Wolff transformation. The resulting effective Hamiltonian is of the form,
\begin{equation}
\begin{split}
\left(\omega_{t}+\frac{g^{zz}_{+}}{2}\right)
\frac{
\sigma^{z}_{t}
\end{split}
\end{equation}
Contrasting this result with Eq. (<ref>), we note that both terms $\propto\sigma^{z}_{p}$ and $\propto\sigma^{z}_{t}\sigma^{z}_{p}$ have vanished because $\omega_{p}=0$ and $g^{zz}_{-}=0$. As a result, the effective qubit interaction is not of $\sigma^{z}_{t}\sigma^{z}_{p}$ but rather of $\sigma^{y}_{t}\sigma^{z}_{p}$ type. The physical origin of the interaction at $n_{g,p} = 0.5$ is different from the $n_{g,p} = 0$ case, since it arises directly from the finite matrix elements of the charge operators of the parity-protected qubit and the transmon qubits. The non-computational states induce only a renormalization of the transmon frequency through the $g^{zz}_{+}$ contribution in the coefficient of $\sigma^{z}_{t}$.
§.§ General case
So far, we have seen that the capacitive coupling between the transmon and the parity-protected qubit induces a qubit interaction that is substantially different for $n_{g,p}=0$ and $n_{g,p}=0.5$. In a last step, we want to interpolate between those two representative cases. This interpolation is achieved by studying the dependence on the offset charge $n_{g,p}$ of the various matrix elements. Since the procedure for obtaining the effective interaction is otherwise identical to the special cases, we only note that for generic values of $n_{g,p}$ the effective Hamiltonian acquires both a $\sigma^{z}_{t}\sigma^{z}_{p}$ and a $\sigma^{y}_{t}\sigma^{z}_{p}$ interaction term [53],
\begin{align}
\left(\omega_{t}+\frac{g^{zz}_{+}}{2}\right)
\frac{
\sigma^{z}_{t}
\left(\omega_{p}+\frac{g^{zz}_{-}}{2}\right)
\frac{
\sigma^{z}_{p}
\sigma^{z}_{t}
\nonumber
\\
\sigma^{z}_{t}
\frac{
\sigma^{z}_{p}
\end{align}
While $g^{zz}_{\pm}$ are defined as in Eq. (<ref>), the definition of $g^{yz}$ is now slightly generalized to $g^{yz}=(\eta'+\eta'')/2$ with
$\eta'=i\left\langle 1_{t},1_{p}|H_{c}|0_{t},1_{p}\right\rangle$ and $\eta''=-i\left\langle 1_{t},0_{p}|H_{c}|0_{t},0_{p}\right\rangle$. The transition from
a pure $\sigma^{z}_{t}\sigma^{z}_{p}$ at $n_{g,p}=0$ to a pure $\sigma^{y}_{t}\sigma^{z}_{p}$ at $n_{g,p}=0.5$ is gradual. As for the the dependence on the transmon offset charge, we remark that in the deep-transmon regime, $E_{J,t}\gg E_{C,t}$, the qubit interaction is almost independent of $n_{g,t}$.
So far, we have derived the effective qubit interaction and have demonstrated that it depends on the anti-crossings with the non-computational states, $\left|0_{t},2_{p}\right\rangle$ and $\left|0_{t},3_{p}\right\rangle$. To realize the respective anti-crossings,
we note that it is essential that,
\begin{equation}
\omega_{02}<\omega_{10}.
\label{condition}
\end{equation}
The transmon energy levels are approximated by $\omega_{t,n}\approx\sqrt{8E_{J,t}E_{C,t}}(n+1/2)-E_{J,t}$ while the PPQ energy levels by
$\frac{\omega_{p,2} + \omega_{p,3}}{2} \approx2\sqrt{8E_{J,p}E_{C,p}}-4 E_{J,p}$. Neglecting
the anharmonicity corrections on both qubits, we find that the necessary condition in Eq. (<ref>) simplifies to $2\sqrt{E_{J,p}E_{C,p}}<\sqrt{E_{J,t}E_{C,t}}$. This condition is satisfied for the parameters chosen in Fig. <ref>.
§ QUANTUM GATES
We will now use the effective Hamiltonian for the hybrid PPQ/transmon setup to implement a controlled-phase gate ($\textsf{CZ}_\phi$), which will preserve the Cooper-pair parity irrespective of the detailed pulse sequence. In addition, we will also discuss a complete set of single-qubit gates realized by controllably driving the system in-and-out of protection [37]. In combination with the $\textsf{CZ}_\phi$ gate, these single-qubit gates will permit the coherent state transfer, a operation, between the transmon and PPQ.
§.§ $\textsf{CZ}_\phi$ gate
For deriving the $\textsf{CZ}_\phi$ gate protocol, we initially move to the frame that rotates with the bare qubit frequencies, $\tilde{H}^{(n_{g,p})}_{\text{eff}}=U^{\dag}(t)H^{(n_{g,p})}_{\text{eff}}U(t)-iU^{\dag}(t)\dot{U}(t)$ with
\omega_{p}\sigma^{z}_{p})t/2}$. Within this rotating frame, the effective Hamiltonian reads,
\begin{equation}
\begin{split}
\tilde{H}^{(n_{g,p})}_{\text{eff}}&=
\frac{g^{zz}_{+}}{2}
\frac{
\sigma^{z}_{t}
\frac{g^{zz}_{-}}{2}
\frac{
\sigma^{z}_{p}
\sigma^{z}_{t}
\frac{
\sigma^{z}_{p}
\\
\sigma^{+}_{t}\sigma^{z}_{p}
\sigma^{+}_{t})+\text{H.c.}],
\label{rotframeHamiltonian}
\end{split}
\end{equation}
where we introduced $\sigma^{\pm}_{}=(\sigma^{x}_{}\pm i\sigma^{y}_{})/2$. We note that the terms $\propto g^{yz}\sigma^{\pm}_{t}\sigma^{z}_{p}$ and $\propto g^{y}\sigma^{\pm}_{t}$ vanish if $n_{g,p}=0$. In this situation, the free evolution of the effective Hamiltonian can implement a $\textsf{CZ}^{10}_\phi$ gate. For executing this $\textsf{CZ}^{10}_\phi$ gate, we carry out a rapid excursion from $\Phi^{\text{ext}}_t\approx 0$ to a flux $\Phi^{\text{ext}}_t\approx \Phi^{\text{ext}}_{*}$ close to the anti-crossing $\left|1_{t},0_{p}\right\rangle\leftrightarrow\left|0_{t},3_{p}\right\rangle$. We then let the system evolve freely for a time $t_{*}=\phi$. This free evolution gives rise to a rotation in the space of $\left|1_{t},0_{p}\right\rangle$ and $\left|0_{t},3_{p}\right\rangle$.
After the time $t_{*}$, the $\left|1_{t},0_{p}\right\rangle$ state will have acquired a finite phase factor and we rapidly return to the idle configuration at $\Phi^{\text{ext}}_t\approx 0$. Because $g^{zz}_{-}\approx-g^{zz}_{+}$ near the anti-crossing, the result of this rapid excursion is a $\textsf{CZ}^{10}_{\phi}$ gate of the form,
\begin{equation}
\begin{split}
\textsf{CZ}^{10}_\phi
|0_{t}\rangle\langle 0_{t}| \otimes \textsf{I}_{p}
|1_{t}\rangle\langle 1_{t}| \otimes \textsf{P}_{p}
\\
\textsf{P}_{p}&=e^{-i\phi}|0_{p}\rangle\langle 0_{p}|+|1_{p}\rangle\langle 1_{p}|
\end{split}
\end{equation}
Unlike for the case of two capacitively coupled transmons $t1$ and $t2$, we remark that the phase factor is not acquired by the $\left|1_{t1},1_{t2}\right\rangle$ state but by the $\left|1_{t},0_{p}\right\rangle$ state. Also, as announced at the beginning of this section, we highlight that the Cooper-pair parity is preserved for the full duration of the $\textsf{CZ}^{10}_\phi$ gate.
In the protocol for the $\textsf{CZ}^{10}_{\phi}$ gate, we have assumed that the offset charge on the PPQ is gate-tuned to $n_{g,p}=0$. Such a tuning is beneficial as it maximizes the coefficient of the $\sigma^{z}_{t}\sigma^{z}_{p}$ terms, thereby allowing for improved gate speed. Furthermore, the tuning should always be achievable because higher levels of the PPQ are strongly offset charge sensitive and can be used for adjusting $n_{g,p}$. However, the fine-tuning to $n_{g,p}=0$ is not essential for the gate protocol. To see this, we note that the terms $\propto g^{yz}\sigma^{\pm}_{t}\sigma^{z}_{p}$ and $\propto g^{y}\sigma^{\pm}_{t}$ in Eq. (<ref>), which appear when $n_{g,p}$ is detuned from zero, share a fast-oscillating prefactor $\propto e^{i\omega_{t}t}$. This fast-oscillating prefactor suggests that such terms are average to zero when invoking a `rotating-wave approximation'. For making this argument more precise, we have integrated out the fast-oscillating terms to second order in $g^{y}$ and $g^{yz}$ with a time-dependent Schrieffer-Wolff transformation [54, 55]. The resulting modified effective Hamiltonian reads [53],
\begin{align}
\tilde{H}^{(n_{g,p})}_{\text{eff}}(t)&\approx
\left(
\frac{g^{zz}_{+}}{2}
\frac{4[\tilde{g}^{y}(t)^{2}+\tilde{g}^{yz}(t)^{2}]}{\omega_{t}}
\right)
\frac{
\sigma^{z}_{t}
\frac{g^{zz}_{-}}{2}
\frac{
\sigma^{z}_{p}
\\
\left(
\frac{16\tilde{g}^{y}(t)\tilde{g}^{yz}(t)}{\omega_{t}}
\right)
\,\frac{
\sigma^{z}_{t}
\frac{
\sigma^{z}_{p}
\label{rotframeHamiltonian2}
\end{align}
with $\tilde{g}(t)=g\sin(\omega_{t}t/2)$.
Provided that $g^{y}\ll \omega_{t}$ and $g^{yz}\ll \omega_{t}$, we see that the correction terms to the effective Hamiltonian are indeed negligibly small. For the realistic parameters chosen in Fig. <ref>, we have $g^{y} / (2 \pi) =\SI{345}{\kilo\Hz}$ and $g^{yz}/ (2\pi) =\SI{3.88}{\mega\Hz}$ if $n_{g,p}=0.1$.
Single-qubit gates.
(a) A generalized PPQ circuit with three circuits elements; a $\cos(2\phi_{p})$ element (blue), a $\cos(\phi_p)$ element (red), and a $\sin(\phi_p)$ element (green). No magnetic flux is threading the gray areas of the circuit. (b) Dependence of the matrix element $h^{x}$ (resulting from the $\cos(\phi_p)$ element) and $h^{y}$ (resulting from the $\sin(\phi_p)$ element) as a function of $n_{g,p}$. The system parameters are $(E_{J, p}, E_{C, p}) = 2\pi (2.7, 0.18)\,\text{GHz}$ and $(\varepsilon_x/E_{J, p},\varepsilon_y/E_{J, p})=(0.04,0.2)$. (c) The $\cos(\phi_p)$ element element induces rotations around the $x$-axis of the Bloch sphere (red arrow). The $\sin(\phi_p)$ element induces rotations around the $y$-axis of the Bloch sphere. The wavefunctions of the PPQ in the $z$-basis ($|0_{p}\rangle$,$|1_{p}\rangle$) and in the $x$-basis ($|+_{p}\rangle$,$|-_{p}\rangle$) are shown schematically.
§.§ Single-qubit gates
Having introduced the $\textsf{CZ}_\phi$ gate, we now discuss the implementation of single-qubit gates on the PPQ. For implementing these single-qubits gates, we consider the generalized circuit for a PPQ depicted in Fig. <ref>(a). The circuit comprises not only a $\cos(2\phi_{p})$ element for the tunneling of pairs of Cooper-pairs, but also a $\cos(\phi_{p})$ and $\sin(\phi_{p})$ element that describe the tunneling of single Cooper-pairs.
The Hamiltonians for these additional circuit elements are given by,
\begin{equation}
\begin{split}
\\
\label{single-qubit-perturbations}
\end{split}
\end{equation}
While both additional circuit elements permit single Cooper-pair tunnelings and temporarily lift the qubit protection, they are typically tuned by different control parameters, depending on the experimental implementation of the PPQ [33, 34]. For example, if the PPQ is realized in a nanowire Josephson interferometer, then sinusoidal term arises if the interferometer junctions are tuned out of balance by local gate electrodes. In contrast, the cosinusoidal term arise when the interferometer magnetic flux is biased away from half flux quantum [32, 33].
We now project the Hamiltonians $H_{p}+H^{x}_{}$ and $H_{p}+H^{y}_{}$ onto the computational subspace of the PPQ. The resulting qubit Hamiltonians read,
\begin{align}
\delta\omega_{p}\cos(\pi n_{g,p})\,\sigma^{z}_{p}\,/2
\delta h^{x}\,\sigma^{x}_{p},\nonumber
\\
\delta\omega_{p}\cos(\pi n_{g,p})\,\sigma^{z}_{p}\,/2
\delta h^{y}\sin(\pi n_{g,p})\,\sigma^{y}_{p}.
\label{singlequbitham}
\end{align}
From this result, we see that the $\cos(\phi_{p})$ and $\sin(\phi_{p})$ elements induce rotations around the $x$- and $y$-axis of the Bloch sphere.
The respective matrix elements are given by $\delta h^{x}=\left\langle 0_{p}| H^{x}_{}|1_{p}\right\rangle$ and $\delta h^{y}\sin(\pi n_{g,p})=\left\langle 0_{p}|H^{y}_{}|1_{p}\right\rangle$. The dependence of these matrix elements on the offset charge $n_{g,p}$ is shown in Fig. <ref>(b). Since we can reach any point on the Bloch sphere through a combined rotation around the $x$- and $y$-axis, we conclude that the free time evolution of the Hamiltonians in Eq. (<ref>) can implement a complete set of single-qubit gates. However, we also emphasize that these single-qubit gates break the Cooper-pair parity conservation so that the PPQ is prone to relaxation errors during the operation time of the single-qubit gates.
§.§ $\textsf{CNOT}$ and $\textsf{SWAP}$ gate
We now combine the proposed method for single-qubit gates with the $\textsf{CZ}^{10}_\phi$ (with $\phi = \pi$) gate to realize a $\textsf{CNOT}_{tp}$ gate with the transmon as control and the PPQ as target by the gate sequence,
\begin{equation}
\textsf{CNOT}_{tp} =
\begin{quantikz}[column sep=0.2cm, row sep = 0.75cm]
& \ctrl{1} & \qw \\
& \targ{} & \qw
\end{quantikz}
=\begin{quantikz}[column sep=0.2cm, row sep = 0.35cm]
\qw & \qw &\gate[wires=2]{\textsf{CZ}^{10}} & \qw & \qw\\
\qw & \gate{\textsf{Y}_{\frac{\pi}{2}}} & & \gate{\textsf{Y}_{-\frac{\pi}{2}}} & \qw
\end{quantikz}
\end{equation}
A $\textsf{CNOT}_{pt}$ gate that uses the PPQ as control and the transmon as target is similarly given by $\textsf{CNOT}_{pt}=\textsf{H}_{t}\cdot \textsf{H}_{p} \cdot \textsf{CNOT}_{tp}\cdot \textsf{H}_{t}\cdot \textsf{H}_{p}$ with the Hadamards, $\textsf{H}_{t/p}=(\sigma^{x}_{t/p}+\sigma^{z}_{t/p})/\sqrt{2}$. Most notably, the $\textsf{CNOT}_{tp}$ and $\textsf{CNOT}_{pt}$ gate can now be combined to realize a $\textsf{SWAP}=\textsf{CNOT}_{tp}\cdot \textsf{CNOT}_{pt}\cdot \textsf{CNOT}_{tp}$ operation. The operation enables the coherent transfer of quantum information between the transmon and the PPQ. Interestingly, this coherent state transfer also gives a novel read-out method for the PPQ by swapping the quantum information onto the transmon and performing the read-out on the latter.
§ POSSIBLE ERRORS
In the previous sections, we have focused on deriving a scheme for a $\textsf{CZ}_\phi$ gate within our hybrid qubit setup. For our scheme, we have assumed that the Cooper-pair parity on the PPQ is conserved during the gate operation time. An interesting question is if the gate protocol modifies if errors due to unintentional single Cooper-pair tunneling terms, as given by Eq. (<ref>), are present on the PPQ?
§.§ $\boldsymbol{\sin(\phi_{p})}$ errors
To address this question, we consider the PPQ at its $n_{g,p}=0$ operation point for optimal gate-speed. We initially consider an error term, $H^{y}_{}
-\varepsilon^{y}\,\sin(\phi_{p})$, with an amplitude $\varepsilon^{y}$ that is small compared to the remaining energy scales of the setup. This $\sin(\phi_p)$ error arises in a PPQ realized by a nanowire Josephson interferometer if the two interferometers junctions are not in balance [32].
Due to the error term, we find that the low-energy Hamiltonian of Eq. (<ref>) changes to,
\begin{equation}
\rightarrow
\begin{pNiceArray}{cccc|cc}
\omega_{11} & 0 & 0 & 0 &\lambda' & 0 \\
0 & \omega_{10} & 0 & 0 & 0 & -\lambda'' \\
0 & 0 & \omega_{01} & 0 & 0 & \kappa \\
0 & 0 & 0 & \omega_{00} & \kappa & 0 \\
\hline
\lambda' & 0 & 0 & \kappa & \omega_{02} & 0 \\
0 & -\lambda'' & \kappa & 0 & 0 & \omega_{03}
\end{pNiceArray}
\end{equation}
Here, we introduced the real-valued matrix element $\kappa=\left\langle 0_{t},1_{p}|H^{y}|0_{t},3_{p}\right\rangle=\left\langle 0_{t},0_{p}|H^{y}|0_{t},2_{p}\right\rangle$. Moreover, in accordance with Eq. (<ref>), couplings of states with opposite Cooper-pair parity within the qubit subspace $\mathcal{P}_0$ are found to be absent at $n_{g,p}=0$, .
Next, we integrate out the non-computational states, $\left|0_{t},2_{p}\right\rangle$ and $\left|0_{t},3_{p}\right\rangle$, with a Schrieffer-Wolff transformation
and move to the rotating frame of the bare qubit frequencies. The effective rotating frame Hamiltonian of Eq. (<ref>) then modifies to,
\begin{align}
\frac{g^{zz}_{+}}{2}
\frac{
\sigma^{z}_{t}
\frac{g^{zz}_{-}}{2}
\frac{
\sigma^{z}_{p}
\sigma^{z}_{t}
\frac{
\sigma^{z}_{p}
\\
\sigma^{+}_{t}\sigma^{+}_{p}
\sigma^{+}_{t}\sigma^{-}_{p}+\text{H.c.}),\nonumber
\end{align}
with the coefficients,
\begin{align}
g^{++}&=\frac{\kappa\lambda'}{2(\omega_{11}-\omega_{02})}, \quad g^{+-}=\frac{\kappa\lambda''}{2(\omega_{03}-\omega_{10})}.
\end{align}
It is now instructive to compare this result to the case of two capactively coupled transmons, $t1$ and $t2$, near the operation point of the $\textsf{iSWAP}$ gate [51].
In the latter case, the effective Hamiltonian comprises similar terms, $\propto \sigma^{+}_{t1}\sigma^{-}_{t2}$ and $\propto \sigma^{+}_{t1}\sigma^{+}_{t2}$, that are
`rotating' with a factor $e^{i(\omega_{t1}-\omega_{t2})t}$ and `counter-rotating' with a factor $e^{i(\omega_{t1}+\omega_{t2})t}$, respectively. For $\omega_{t1}\approx\omega_{t2}$, the `counter-rotating' terms, which are fast-oscillating, average to zero within a `rotating-wave approximation'. Only the `rotating' terms, which oscillate slowly, are thus retained in the effective qubit Hamiltonian. In our case, the situation is very different. Because $\omega_{t}\gg\omega_{p}$, both factors, $e^{i(\omega_{t}+\omega_{p})t}$ and $e^{i(\omega_{t}-\omega_{p})t}$, are fast-oscillating. Within a `rotating-wave approximation', we thus expect that both error terms average to zero.
To formalize this `rotating-wave approximation' argument, we integrate out the fast-oscillating terms with a time-dependent Schrieffer-Wolff transformation. To second order in $g^{++}$ and $g^{+-}$, we find that [53],
\begin{equation}
\begin{split}
\tilde{H}^{(n_{g,p})}_{\text{eff}}
\left(
\frac{g^{zz}_{+}}{2}
\frac{2[\tilde{g}^{xx}(t)-\tilde{g}^{yy}(t)]^{2}}{\omega_{t}}
\right)
\frac{
\sigma^{z}_{t}
\\
\left(
\frac{g^{zz}_{-}}{2}
\frac{2[\tilde{g}^{xx}(t)-\tilde{g}^{yy}(t)]^{2}}{\omega_{t}}
\right)
\frac{
\sigma^{z}_{p}
\\
\left(
\frac{4[\tilde{g}^{xx}(t)+\tilde{g}^{yy}(t)]^{2}}{\omega_{t}}
\right)
\frac{
\sigma^{z}_{t}
\frac{
\sigma^{z}_{p}
\label{rotframeHamiltonian3}
\end{split}
\end{equation}
with $\tilde{g}(t)=g\sin(\omega_{t}t/2)$. From this expression for the effective rotating-frame Hamiltonian, we conclude that the mitigation of the effects of $\sin(\phi_{p})$ errors requires us to operate the setup in the regime when $g^{xx}\ll \omega_{t}$ and $g^{yy}\ll \omega_{t}$.
§.§ $\boldsymbol{\cos(\phi_{p})}$ errors
It is now interesting to compare our results for $\sin(\phi_p)$ errors with $\cos(\phi_p)$ errors that are described by an error term $H^{x}_{}
-\varepsilon^x\,\cos(\phi_{p})$ in the Hamiltonian. Such an error term can arise in an implementation of the PPQ with a nanowire Josephson interferometer if
the external flux that threading the interferometer loop is detuned from half flux quantum [32]. In this situation, the low-energy Hamiltonian of Eq. (<ref>) modifies to,
\begin{equation}
\rightarrow
\begin{pNiceArray}{cccc|cc}
\omega_{11} & \delta h^{x} & 0 & 0 &\lambda' & 0 \\
\delta h^{x} & \omega_{10} & 0 & 0 & 0 & -\lambda'' \\
0 & 0 & \omega_{01} & \delta h^{x} & 0 & 0 \\
0 & 0 & \delta h^{x} & \omega_{00} & 0 & 0 \\
\hline
\lambda' & 0 & 0 & 0 & \omega_{02} & \chi \\
0 & -\lambda'' &0 & 0 & \chi & \omega_{03}
\end{pNiceArray}
\end{equation}
with the matrix element $\chi=\left\langle 0_{t},2_{p}|H^{y}|0_{t},3_{p}\right\rangle$. Importantly, we see that the $\cos(\phi_p)$ errors do not lead to off-diagonal terms
that couple the matrix blocks representing the qubit subspace $\mathcal{P}_0$ and the non-computational subspace $\{|0_{t},2_{p}\rangle,|0_{t},3_{p}\rangle\}$. Consequently, we note that the $\cos(\phi_p)$ errors primarily induces mixing of opposite-parity states on the PPQ as described by $H^{x}_{\text{eff}}$ in Eq. (<ref>).
A cosine error on the PPQ.
Low-energy spectrum of the hybrid qubit setup for $n_{g, p}=0$ and $(E_{J,t}, E_{J,p},E_{C,t},E_{C,p},E_{C,c})
= 2\pi (12,2.7,0.2,0.15,0.025)\,\text{GHz}$ as a function of the external flux $\Phi_{\text{ext}}$ of the tunable transmon in the presence of a cosine error, $H^{x} =- \varepsilon^x\,\cos(\phi_{p})$ with $\varepsilon^x=0.05 E_{J, p}$. The error term introduce additional anti-crossings between states of opposite Cooper-pair parity.
In summary, we have found that the nature of $\sin(\phi_p)$ errors and $\cos(\phi_p)$ errors is different in our hybrid qubit. While the $\sin(\phi_p)$ errors lead primarily to
additional two-qubit interactions that become less relevant in the limit when $g^{xx}\ll \omega_{t}$ and $g^{yy}\ll \omega_{t}$, the $\cos(\phi_p)$ errors lead primarily to additional single-qubit terms. Finding strategies of mitigating such flux errors, for example by concatenating multiple imperfect PPQs [35, 32], is an important open challenge of the field.
§ CONCLUSION
To conclude, we have proposed a coupling scheme for entangling a parity-protected superconducting qubit
with a conventional transmon qubit and discussed coherent state transfer as an application. While our scheme could open the way for using PPQs as quantum memories in a transmon architecture, it could also allow for a comparison of coherence times of the two qubits types within the same device.
§ ACKNOWLEDGEMENTS
We gratefully acknowledge discussions
with K. Flensberg and A. Gyenis.
This work was supported by the Danish National Research Foundation, the Danish Council for Independent Research Natural Sciences.
MK gratefully acknowledges support from the Villum Foundation (grant
37467) through a Villum Young Investigator grant. We acknowledge support from the Microsoft Corporation.
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Supplemental Material to `Entangling transmons with low-frequency protected superconducting qubits'
Andrea Maiani, Morten Kjaergaard, and Constantin Schrade
Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
In the Supplemental Material, we provide details on the derivations of the effective Hamiltonians presented in the main text. We also provide more details on the energy level structure of the hybrid qubit setup.
§ TIME-INDEPENDENT EFFECTIVE HAMILTONIANS
In this section of the Supplemental Material, we give details on the derivation of the time-independent effective Hamiltonians for the coupled qubit setup as presented in
Eq. (7) of the main text.
As a starting point, we project the setup Hamiltonian $H$ onto $\{\left|0_{t},0_{p}\right\rangle,\left|1_{t},0_{p}\right\rangle,\left|0_{t},1_{p}\right\rangle,\left|1_{t},1_{p}\right\rangle,\left|0_{t},2_{p}\right\rangle,\left|0_{t},3_{p}\right\rangle \}$, which correspond to the relevant low-energy states of the uncoupled Hamiltonian $H_{0}$. The resulting projected Hamiltonian is given by $H^{(n_{g,p})}_{\text{low}}=H^{(0)}_{\text{low}}+H^{(1)}_{\text{low}}+H^{(2)}_{\text{low}}$ with,
\begin{equation}
\begin{pNiceArray}{cccc|cc}
\omega_{11} & 0 & 0 & 0 & 0 & 0 \\
0 & \omega_{10} & 0 & 0 & 0 & 0 \\
0 & 0 & \omega_{01}& 0 & 0& 0 \\
0 & 0 & 0 & \omega_{00} & 0 & 0 \\
\hline
0 & 0 & 0 & 0 & \omega_{02} & 0 \\
0 & 0 & 0 & 0 & 0 & \omega_{03}
\end{pNiceArray}
\quad
\begin{pNiceArray}{cccc|cc}
0 & 0 & -i\eta' & 0 & 0 & 0 \\
0 & 0 & 0 & i\eta'' & 0 & 0 \\
i\eta' & 0 &0& 0 & 0& 0 \\
0 & -i\eta'' & 0 & 0& 0 & 0 \\
\hline
0 & 0 & 0 & 0 &0 & 0 \\
0 & 0 & 0 & 0 & 0 & 0
\end{pNiceArray}
\quad
\begin{pNiceArray}{cccc|cc}
0 & 0 & 0 & 0 & \lambda' & 0 \\
0 & 0 & 0 & 0 & 0 & -\lambda'' \\
0 & 0 & 0& 0 & 0& 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
\hline
\lambda' & 0 & 0 &0 &0 & 0 \\
0 & -\lambda'' &0 & 0 & 0 & 0
\end{pNiceArray}.
\end{equation}
Here we have chosen a gauge of the wavefunctions in the uncoupled system such that $\langle 2_{p}|n_{p}|1_{p}\rangle$, $\langle 3_{p}|n_{p}|0_{p}\rangle$, and $\langle 0_{t}|n_{t}|1_{t}\rangle$ are purely imaginary-valued. As a result of this gauge choice, the following quantities are real-valued,
\begin{equation}
\begin{split}
\lambda'&=\left\langle 1_{t},1_{p}|H_{c}|0_{t},2_{p}\right\rangle,
\\
-\lambda''&=\left\langle 1_{t},0_{p}|H_{c}|0_{t},3_{p}\right\rangle,
\\
-i\eta'&=\left\langle 1_{t},1_{p}|H_{c}|0_{t},1_{p}\right\rangle,
\\
i\eta''&=\left\langle 1_{t},0_{p}|H_{c}|0_{t},0_{p}\right\rangle.
\end{split}
\end{equation}
Next, we integrate out the effects of the non-computational states $\{\left|0_{t},2_{p}\right\rangle,\left|0_{t},3_{p}\right\rangle \}$ by means of a Schrieffer-Wolff transformation [52, 57]. As the generator of the Schrieffer-Wolff transformation, we use
\begin{equation}
\begin{pNiceArray}{cccc|cc}
0 & 0 & 0 & 0 & -\Omega'/\lambda' & 0 \\
0 & 0 & 0 & 0 & 0 & \Omega''/\lambda'' \\
0 & 0 & 0& 0 & 0& 0 \\
0 & 0 & 0 & 0 & 0 & 0 \\
\hline
\Omega'/\lambda' & 0 & 0 & 0 &0 & 0 \\
0 & - \Omega''/\lambda''& 0 & 0 & 0 & 0
\end{pNiceArray}
\quad
\text{with}
\quad
\Omega'=\frac{\lambda'^{2}}{\omega_{11}-\omega_{02}}\quad\text{and} \quad
\Omega''=\frac{\lambda''^{2}}{\omega_{10}-\omega_{03}}.
\end{equation}
We note that the generator satisfies $[H^{(0)}_{\text{low}},S]=-H^{(2)}_{\text{low}}$. To second order in $\lambda'$ and $\lambda''$, the Schrieffer-Wolff generator produces an effective Hamiltonian $H^{(n_{g,p})}_{\text{eff}}=H^{(0)}_{\text{low}}+H^{(1)}_{\text{low}}+[H^{(2)}_{\text{low}},S]/2$. Evaluating this expression yields,
\begin{equation}
\begin{split}
\left(\omega_{t}+\frac{g^{zz}_{+}}{2}\right)
\frac{
\sigma^{z}_{t}
\left(\omega_{p}+\frac{g^{zz}_{-}}{2}\right)
\frac{
\sigma^{z}_{p}
\sigma^{z}_{t}
\frac{
\sigma^{z}_{p}
\quad \text{with} \quad
\\
\end{split}
\end{equation}
This corresponds to the effective Hamiltonian presented in Eq. (7) of the main text.
§ EFFECTIVE HAMILTONIANS FOR THE TIME-EVOLUTION
In this section of the Supplemental Material, we give details on the derivation of the effective Hamiltonian in Eq. (10) of the main text that approximates the time-evolution of our hybrid qubit system.
§.§ Time-dependent Schrieffer-Wolff transformation
As a starting point, we provide a brief review of the time-dependent version of the Schrieffer-Wolff transformation.
Given some time-dependent effective Hamiltonian $H(t)$, the time-dependent Schrieffer-Wolff transformation generates an effective Hamiltonian $H_{\text{eff}}(t)$ via a unitary transformation with a time-dependent generator $S(t)$ with $S(t)=-S(t)^{\dagger}$.
By using the Baker-Campbell-Haussdorf formula, we can formulate the action of the time-dependent Schrieffer-Wolff unitary transformation on the Hamiltonian $H(t)$ as,
\begin{equation}
\begin{split}
\\
\left[
\right]
\frac{1}{2}
\left[
\left[
\right],
\right]
\dots
\\
\left[
\right]
\frac{1}{2}
\left[
\left[
\right],
\right]
\frac{i}{2}
\left[
\dot{S}(t),
\right]
\dots
\end{split}
\end{equation}
Next, we choose the time-dependent Hamiltonian to be of the specific form,
\begin{equation}
\begin{split}
H(t)&=H^{(0)}+\xi H^{(2)}(t).
\end{split}
\end{equation}
Here, $H^{(0)}$ is a time-independent unperturbed Hamiltonian and $ H^{(2)}(t)$ is a time-dependent perturbation. The parameter $\xi$ is an aid to count the order in perturbation theory and can be set to $\xi=1$ at the end of the derivation.
Besides specifying the form of the time-dependent Hamiltonian, we also require that the generator of the time-dependent Schrieffer-Wolff transformation satisfies the following differential equation,
\begin{equation}
\begin{split}
\xi H^{(2)}(t) + \left[ H^{(0)}, S(t) \right] - i \dot{S}(t) =0.
\end{split}
\end{equation}
Using these two conditions on the time-independent Hamiltonian and the generator $S(t)$, we find that the expression for the effective Hamiltonian $H_{\text{eff}}(t)$ can be simplified to,
\begin{equation}
\begin{split}
\left[
\right]
\frac{1}{2}
\left[
\left[
\right],
\right]
\frac{i}{2}
\left[
\dot{S}(t),
\right]
\dots
\\
\frac{1}{2}
\left[
\xi H^{(2)}(t)
\right]
\frac{1}{2}
\left[
\left[
\xi H^{(2)}(t),S(t)
\right],
\right]
\dots
\end{split}
\end{equation}
We now proceed by assuming that the generator $S(t)$ can expanded in a perturbative series,
\begin{equation}
\begin{split}
S(t)&=\xi S_{1}(t)+\xi^{2} S_{2}(t) + \cdots
\end{split}
\end{equation}
Inserting this series into the expression for the effective Hamiltonian $H_{\text{eff}}$ and only retaining terms up to order $\xi^{2}$, we find that,
\begin{equation}
\begin{split}
\frac{\xi^{2}}{2}
\left[
\right]
\mathcal{O}(\xi^{3})
\end{split}
\end{equation}
Finally, we set $\xi=1$ and arrive at the following form of the effective Hamiltonian,
\begin{equation}
\begin{split}
\frac{1}{2}
\left[
\right]
\end{split}
\end{equation}
§.§ Rotating frame for the hybrid qubit setup
We now want to apply the time-dependent Schrieffer-Wolff transformation to our hybrid qubit setup. For that purpose, it is helpful to initially move to a rotating reference frame, which is achieved by separating the full qubit entangling Hamiltonian into,
\begin{equation}
\begin{split}
\quad
\text{with}
\quad
\left(\omega_{t}+\frac{g^{zz}_{+}}{2}\right)
\frac{
\sigma^{z}_{t}
\left(\omega_{p}+\frac{g^{zz}_{-}}{2}\right)
\frac{
\sigma^{z}_{p}
\sigma^{z}_{t}
\frac{
\sigma^{z}_{p}
} ,
\quad
\end{split}
\end{equation}
and applying the following time-dependent unitary transformation,
\begin{equation}
\omega_{p}\sigma^{z}_{p})t/2}.
\end{equation}
This transformation yields the qubit entangling Hamiltonian in the frame that rotates at the bare qubit frequencies with components,
\begin{equation}
\begin{split}
\tilde{H}^{(0)}&=U^{\dag}(t)H^{(0)}U(t)-iU^{\dag}(t)\dot{U}(t)=\frac{g^{zz}_{+}}{2}
\frac{
\sigma^{z}_{t}
\frac{g^{zz}_{-}}{2}
\frac{
\sigma^{z}_{p}
\sigma^{z}_{t}
\frac{
\sigma^{z}_{p}
} ,\\
\tilde{H}^{(2)}(t)&=U^{\dag}(t)H^{(2)}U(t)-iU^{\dag}(t)\dot{U}(t)=
g^{yz}\cos(\omega_{t}t)\,\sigma^{y}_{t}\sigma^{z}_{p} .
\end{split}
\end{equation}
The rotating frame Hamiltonian can also be explicitly written in matrix form as,
\begin{equation}
\begin{split}
\tilde{H}^{(0)}&=
\frac{1}{4}
\begin{pmatrix}
g^{zz}_{+}+2g^{zz}_{-} & 0 &0 & 0 \\
0 & g^{zz}_{+}-2g^{zz}_{-} & 0 &0 \\
0 & 0 & -g^{zz}_{+}& 0 \\
0 & 0 & 0 & -g^{zz}_{+}
\end{pmatrix}, \\
\tilde{H}^{(2)}(t)&=
\begin{pmatrix}
0 & 0 &-i (g^{y}+g^{yz}) e^{i\omega_{t}t} & 0 \\
0 & 0 & 0 &-i (g^{y}-g^{yz}) e^{i\omega_{t}t} \\
i (g^{y}+g^{yz})e^{-i\omega_{t}t} & 0 &0& 0 \\
0 & i (g^{y}-g^{yz}) e^{-i\omega_{t}t} & 0 &0
\end{pmatrix}.
\end{split}
\end{equation}
§.§ Time-dependent Schrieffer-Wolff transformation for hybrid qubit setup
We now want to perform a time-dependent Schrieffer-Wolff transformation that eliminates the fast-oscillating terms, $\propto e^{\pm i\omega_{t}t}$, in $\tilde{H}^{(2)}(t)$
to second order in $g^{y}$ and $g^{yz}$. We, therefore, introduce the following Schrieffer-Wolff generator,
\begin{equation}
\begin{pmatrix}
0 & 0 &f_{1}(t)& 0 \\
0 & 0 & 0 &f_{2}(t) \\
-f_{1}(t)^{*} & 0 &0& 0 \\
0 &-f_{2}(t)^{*} & 0 &0
\end{pmatrix}
\end{equation}
with the functions,
\begin{equation}
}{g^{zz}_{-}+g^{zz}_{+}+2\omega_{t}}, \quad
\end{equation}
This generator satisfies,
\begin{equation}
S_{1}(t)=-S_{1}(t)^{\dagger}, \quad
\tilde{H}^{(2)}(t) + \left[ \tilde{H}^{(0)}, S_{1}(t) \right] - i \dot{S}_{1}(t) =0,
\quad
\text{and} \quad
\end{equation}
Moreover, the generator allows us to compute the effective correction term to $\tilde{H}^{(0)}(t)$ to second order in $g^{y}$ and $g^{yz}$,
\begin{equation}
\begin{split}
\frac{1}{2}
\left[
\tilde{H}^{(2)}(t),S_{1}(t)
\right]
\begin{pmatrix}
h_{1}(t)& 0 &0 & 0 \\
0 & h_{2}(t) & 0 &0 \\
0 & 0 & -h_{1}(t) & 0 \\
0 & 0 & 0 & -h_{2}(t)
\end{pmatrix},
\end{split}
\end{equation}
with the functions,
\begin{equation}
\begin{split}
&h_{1}(t) = \frac{4(g^{y}+g^{yz})^{2}\sin([g^{zz}_{-}+g^{zz}_{+}+2\omega_{t}]t/4)^{2}}{g^{zz}_{-}+g^{zz}_{+}+2\omega_{t}},
\quad
h_{2}(t) = -\frac{4(g^{y}-g^{yz})^{2}\sin([g^{zz}_{-}-g^{zz}_{+}-2\omega_{t}]t/4)^{2}}{g^{zz}_{-}-g^{zz}_{+}-2\omega_{t}}
\end{split}
\end{equation}
Provided that $\omega_{t}\gg g^{zz}_{\pm}$, we neglect the terms in the denominators and sine functions that are $\propto g^{zz}_{\pm}$. When then add the correction term to
$\tilde{H}^{(0)}(t)$, which yields the full effective Hamiltonian,
\begin{equation}
\begin{split}
\tilde{H}^{(n_{g,p})}_{\text{eff}}(t)&\approx
\left(
\frac{g^{zz}_{+}}{2}
\frac{4[(g^{y})^{2}+(g^{yz})^{2}]\sin(\omega_{t}t/2)^{2}}{\omega_{t}}
\right)
\frac{
\sigma^{z}_{t}
\frac{g^{zz}_{-}}{2}
\frac{
\sigma^{z}_{p}
\left(
\frac{16g^{y}g^{yz}\sin(\omega_{t}t/2)^{2}}{\omega_{t}}
\right)
\,\frac{
\sigma^{z}_{t}
\frac{
\sigma^{z}_{p}
\end{split}
\end{equation}
This concludes our derivation of the effective Hamiltonian for the time-evolution of our hybrid qubit setup.
§ MORE DETAILS ON THE POSSIBLE ERRORS
In this section of the Supplemental Material, we provide details on the derivation of the effective Hamiltonians presented in Eq. (15) and Eq. (16) of the main text. These Hamiltonians account for the presence of $\sin(\phi_p)$ error terms in our hybrid qubit setup.
First, we note that the derivations for the effective Hamiltonians with the $\sin(\phi_p)$ error terms are very similar to the derivations for the effective Hamiltonians without $\sin(\phi_p)$ the error terms. Since the latter derivations have been discussed in great detail in the previous sections of the Supplemental Material, we will focus only on the main modifications.
§.§ Time-independent effective Hamiltonian
For deriving the time-independent effective Hamiltonian of Eq. (15), we note that the low-energy Hamiltonian at $n_{g,p}=0$ is given by
$H^{(n_{g,p}=0)}_{\text{low}}=H^{(0)}_{\text{low}}+H^{(2)}_{\text{low}}$ with,
\begin{equation}
\begin{pNiceArray}{cccc|cc}
\omega_{11} & 0 & 0 & 0 & 0 & 0 \\
0 & \omega_{10} & 0 & 0 & 0 & 0 \\
0 & 0 & \omega_{01}& 0 & 0& 0 \\
0 & 0 & 0 & \omega_{00} & 0 & 0 \\
\hline
0 & 0 & 0 & 0 & \omega_{02} & 0 \\
0 & 0 & 0 & 0 & 0 & \omega_{03}
\end{pNiceArray}
\quad
\begin{pNiceArray}{cccc|cc}
0 & 0 & 0 & 0 & \lambda' & 0 \\
0 & 0 & 0 & 0 & 0 & -\lambda'' \\
0 & 0 & 0& 0 & 0& \kappa \\
0 & 0 & 0 & 0 & \kappa & 0 \\
\hline
\lambda' & 0 & 0 &\kappa &0 & 0 \\
0 & -\lambda'' &\kappa & 0 & 0 & 0
\end{pNiceArray}.
\end{equation}
Here, the matrix element $\kappa=\left\langle 0_{t},1_{p}|H^{y}|0_{t},3_{p}\right\rangle=\left\langle 0_{t},0_{p}|H^{y}|0_{t},2_{p}\right\rangle$ is real-valued (in the same gauge as the one used in the first section of the Supplemental Material) and accounts for the presence of the $\sin(\phi_p)$ error terms.
Next, we write down the generator of the Schrieffer-Wolff transformation,
\begin{equation}
\begin{pmatrix}
0 & 0 & 0 & 0 & -\Omega'/\lambda' & 0 \\
0 & 0 & 0 & 0 & 0 & \Omega''/\lambda'' \\
0 & 0 & 0& 0 &0& \Gamma' \\
0 & 0 & 0 & 0 & \Gamma'' & 0 \\
\Omega'/\lambda' & 0 & 0 & -\Gamma'' &0 & 0 \\
0 & - \Omega''/\lambda''& -\Gamma' & 0 & 0 & 0
\end{pmatrix}
\quad
\text{with}
\quad
\Gamma'=\frac{\kappa}{\omega_{03}-\omega_{01}}\quad\text{and} \quad
\Gamma''=\frac{\kappa}{\omega_{02}-\omega_{00}},
\end{equation}
The generator satisfies $[H^{(0)}_{\text{low}},S]=-H^{(2)}_{\text{low}}$ and yields the effective Hamiltonian, $H^{(n_{g,p}=0)}_{\text{eff}}=H^{(0)}_{\text{low}}+[H^{(2)}_{\text{low}},S]/2$. Projected onto the qubit subspace $\mathcal{P}_0$, the effective Hamiltonian evaluates to,
\begin{equation}
\begin{split}
\left(\omega_{t}+\frac{g^{zz}_{+}}{2}\right)
\frac{
\sigma^{z}_{t}
\left(\omega_{p}+\frac{g^{zz}_{-}}{2}\right)
\frac{
\sigma^{z}_{p}
\sigma^{z}_{t}
\frac{
\sigma^{z}_{p}
\end{split}
\end{equation}
with the coefficients,
\begin{equation}
\frac{\kappa}{4}
\left(
\frac{\lambda'}{\omega_{11}-\omega_{02}}
\frac{\lambda''}{\omega_{03}-\omega_{10}}
\right)
\quad
\left(
\frac{\lambda'}{\omega_{02}-\omega_{11}}
\frac{\lambda''}{\omega_{03}-\omega_{10}}
\right).
\end{equation}
Here, we have dropped terms $\propto 1/(\omega_{01}-\omega_{03})$ and $\propto 1/(\omega_{00}-\omega_{02})$ due to the large separation of the respective energy levels.
§.§ Time-dependent effective Hamiltonian
For deriving the time-dependent effective Hamiltonian of Eq. (16), we transform the effective Hamiltonian, $H^{(n_{g,p}=0)}_{\text{eff}}$, to the frame that rotates with the bare qubit frequencies. The rotating-frame Hamiltonian is of the form $\tilde{H}^{(0)}+\tilde{H}^{(2)}(t)$ with the two contributions,
\begin{equation}
\begin{split}
\tilde{H}^{(0)}&=
\frac{1}{4}
\begin{pmatrix}
g^{zz}_{+}+2g^{zz}_{-} & 0 &0 & 0 \\
0 & g^{zz}_{+}-2g^{zz}_{-} & 0 &0 \\
0 & 0 & -g^{zz}_{+}& 0 \\
0 & 0 & 0 & -g^{zz}_{+}
\end{pmatrix}, \\
\tilde{H}^{(2)}(t)&=
\begin{pmatrix}
0 & 0 &0 & (g^{xx}-g^{yy}) e^{i(\omega_{t}+\omega_{p})t} \\
0 & 0 & (g^{xx}+g^{yy}) e^{i(\omega_{t}-\omega_{p})t} &0 \\
0& (g^{xx}+g^{yy}) e^{-i(\omega_{t}-\omega_{p})t} &0& 0 \\
(g^{xx}-g^{yy}) e^{-i(\omega_{t}+\omega_{p})t} &0& 0 &0
\end{pmatrix}.
\end{split}
\end{equation}
We now introduce the generator of the time-dependent Schrieffer-Wolff transformation,
\begin{equation}
\begin{pmatrix}
0 & 0 &0& f_{1}(t) \\
0 & 0 & f_{2}(t) &0 \\
0 & -f_{2}(t)^{*} &0& 0 \\
-f_{1}(t)^{*} &0 & 0 &0
\end{pmatrix},
\end{equation}
with the functions,
\begin{equation}
\quad
\end{equation}
This generator satisfies,
\begin{equation}
S_{1}(t)=-S_{1}(t)^{\dagger}, \quad
\tilde{H}^{(2)}(t) + \left[ \tilde{H}^{(0)}, S_{1}(t) \right] - i \dot{S}_{1}(t) =0,
\quad
\text{and} \quad
\end{equation}
The generator yields the effective Hamiltonian,
\tilde{H}^{(0)}+
\tilde{H}^{(2)}(t),S_{1}(t)
]/2$, which evaluates to,
\begin{equation}
\begin{split}
\tilde{H}^{(n_{g,p}=0)}_{\text{eff}}(t)&\approx
\left(\frac{g^{zz}_{+}}{2}
\frac{2(g^{xx}-g^{yy})^{2}\sin(\omega_{t}t/2)^{2}}{\omega_{t}}
\right)
\frac{
\sigma^{z}_{t}
\left(\frac{g^{zz}_{-}}{2}
\frac{2(g^{xx}-g^{yy})^{2}\sin(\omega_{t}t/2)^{2}}{\omega_{t}}
\right)
\frac{
\sigma^{z}_{p}
\\
\left(
\frac{4(g^{xx}+g^{yy})^{2}\sin(\omega_{t}t/2)^{2}}{\omega_{t}}
\right)
\,\frac{
\sigma^{z}_{t}
\frac{
\sigma^{z}_{p}
\end{split}
\end{equation}
This concludes our derivation of the effective Hamiltonians presented in Eq. (15) and Eq. (16) of the main text.
§ NUMERICAL SCHRIEFFER-WOLFF TRANSFORMATION
In this section of the Supplemental Material, we provide more details on the theory and application of the Schrieffer-Wolff transformation for the coupled qubits problem. The numerical Schrieffer-Wolff method is used in the text to derive the 6-levels effective Hamiltonians integrating out the effect of high-energy levels.
Let us consider the Hamiltonian of the coupled system $H = H_0 + H_c$ with $H_0 = H_t \otimes \mathbb{I} + \mathbb{I} \otimes H_p$ being the decoupled system Hamiltonian and $H_c$ the capacitive coupling Hamiltonian. The Hilbert space of the system can be decomposed as $\mathcal{H} = \mathcal{P}_{0} \oplus \mathcal{Q}_0 = \mathcal{P} \oplus \mathcal{Q}$ where $\mathcal{P}_0$ and $\mathcal{P}$ are the low energy Hilbert spaces of the decoupled and coupled system. The computational space of the system is identified by $\mathcal{P}_0$. For this reason, we are interested in finding a unitary $U \in \mathrm{End}(\mathcal{H})$ that maps the low-energy subspace of the interacting Hamiltonian $\mathcal{P}$ to the one of the uncoupled one $\mathcal{P}_{0}$. In other words, defining $P$ and $P_0$ the orthogonal projectors on the low-energy susbspaces
\begin{align}
P = \sum_{i}^{d} \ket{\psi_i}\bra{\psi_i} \\
P_0 = \sum_{i}^{d} \ket{\psi_i^0}\bra{\psi_i^0}
\end{align}
where $\ket{\psi_i}$ and $\ket{\psi_i^0}$ are, respectively, the coupled and decoupled system eigenstates and $d=4$, we are seeking a unitary $U$ that satisfies
\begin{equation}
U P U^\dag = P_0 \qquad \Rightarrow \qquad U P = P_0 U \,.
\end{equation}
This is achieved by the Schrieffer-Wolff transformation [56].
One way to see this is as a direct rotation that can be written as the square root of the product of the two reflections
\begin{equation}
U = \sqrt{\mathcal{M}_{\mathcal{P}_0}\mathcal{M}_{\mathcal{P}}} = \sqrt{(P_0 - Q_0)(P - Q)} = \sqrt{(2 P_0 - \mathbb{I})(2 P - \mathbb{I})}
\end{equation}
where $Q$ and $Q_0$ are the orthogonal projectors to the high-energy subspaces and $\mathcal{M}_j$ are the reflections upon the lower energy subspaces.
The low energy Hamiltonian is then:
\begin{equation}
H_\mathrm{eff} = P_0 U H U^\dag P_0 = U P H P U^\dag
\end{equation}
An efficient way to tackle the problem numerically is the procedure developed in [58, 59] that we will now discuss. Since the final objective is the effect of the Schrieffer-Wolff transformation only in the low-energy sector, we can focus on the following operator product
\begin{equation}
P_0 U = U P = \sum_{ij} A_{ij} \ket{\psi_i^0}\bra{\psi_j} = A
\end{equation}
where $A \in \mathrm{Hom}\qty(\mathcal{P}_0, \mathcal{P})$ is a rank $d$ operator. For later use, we introduce also the rank $d$ operator $B \in \mathrm{Hom}\qty(\mathcal{P}_0, \mathcal{P})$ defined as $B = P_0 P$. Since both the operators belongs to $\mathrm{Hom}\qty(\mathcal{P}_0, \mathcal{P})$, $P_0$ acts as a left identity and $P$ is a right identity. Moreover, $A$ and $B$ are related, indeed
\begin{equation}
(U P)^2 = (P_0 A P)^2 = P_0 A P P_0 A P = P_0 A B^\dag A P = A B^\dag A
\end{equation}
at the same time
\begin{equation}
( U P)^2 = P_0 U^2 P = P_0 (P_0 - Q_0)(P - Q) P = P_0 P = B
\end{equation}
and therefore $A B^\dag A = B$. Using singular values decomposition, we can decompose $B = W \Sigma V^\dag$ where $W$ and $V$ are unitaries and $\Sigma$ is a diagonal matrix. The equation $A B^\dag A = B$ is then solved by $A = W V^\dag$.
In a practical implementation, by starting with the Hamiltonian $H_0$ and $H$ in whatever basis, we can calculate the incomplete low-energy $d$-dimensional orthogonal eigenbasis $V_0$ and $V$ ($n \times d$ matrices) with eigenvalue matrices $W_0$ and $W$ ($d \times d$ diagonal matrices) by using the Lanczos algorithm. With these, we can calculate $B = V_0^\dag V$, that is a dimension $d$ matrix and perform SVD to calculate the unitary $A$. The effective low-energy Hamiltonian in the computational basis is then $H_\mathrm{eff} = A W A^\dag$. The drawback of this method is that we lose the information on the dressed states that is encoded in the matrix $U$.
§.§ Smooth gauge for parametric sweeps
The purpose of the Schrieffer-Wolff transformation is to derive an effective Hamiltonian $H_\mathrm{eff}$ written in the basis of the unperturbed system $H_0$. Since $H_\mathrm{eff}$ is not an observable of the system, the effective Hamiltonian derived is not unique but it depends on the choice of the gauge for the unperturbed eigenstates $\qty{\ket{\psi_i^0}}$. This means that it is crucial to fix a smooth gauge when sweeping over a parameter that appear also in the unperturbed Hamiltonian $H_0$. Notice that, when numerical diagonalization is employed, control over the global phase of the eigenvectors is not guaranteed. Therefore a smooth gauging algorithm needs to be applied after the diagonalization.
In the case addressed in this paper, we are free to fix the gauge of the two qubits independently since they are decoupled in $H_0$. In the case the charge offsets $n_{g,t}$ and $n_{g,p}$ are both zero, it is possible to fix the gauge consistently by imposing that the wavefunction is real at a reference point. A convenient choice is represented by picking $\phi=0$ for $\ket{0_p}$, $\ket{1_p}$ and $\ket{0_t}$ and $\phi = \pi/4$ for $\ket{1_t}$.
To keep a smooth gauge fixing during the sweep of the offset charge, we use a smooth gauge fixing procedure. We first discretize the $n_{g, i}$ axes in a set of $N$ points in the interval $\qty[0, k]$ where $k$ is $1$ for the transmon and $2$ for the parity qubit. We will assume an homogeneous discretization with inter-site distance $\Delta n = k / (N-1)$ for simplicity. First, an arbitrary gauge fixing, like the one described in the previous paragraph, is applied for the wavefunction at point $i=0$. Next, for each wavefunction we impose that the overlap integral with the previous point is real. In other words, we calculate the fixing phase $\beta_i$ as
\begin{equation}
\beta_{n, i} = \sum_{j=0}^{i-1} \Im \ln \bra{\psi_{n, j}^0}\ket{\psi_{n, j+1}^0}
\end{equation}
and then the wavefunctions are updated as $\ket{\tilde{\psi}^0_{n,i}} = e^{-i \beta_{n,i}} \ket{\tilde{\psi}^0_{n,i}}$. This is possible because we have assumed that the index $n$ identifies corresponding eigenstates at different indexes $i$. In other words, if the index $n$ orders the states by energy we are assuming that there are no level crossings in the charge-Brillouin zone. This is not true for the parity qubit, but in this case we have labeled by $n=0,1$ the even and odd lowest states that can be identified, for example, by comparing the amplitude at $\phi=0$.
A more general method is available for the case when it is not possible to easily identify corresponding eigenstates at different values of the parameters. In that case, the application of an additional unitary point by point is necessary.
§ ADDITIONAL RESULTS ON THE ENERGY LEVELS
In this section of the Supplemental Material, we provide additional details on the energy spectrum of the two qubit system and design principles of an hybrid qubit.
We recall that, for the transmon, it is possible to approximate the eigenvalues distribution by expanding the transmon Hamiltonian to fourth order. In this way, the Hamiltonian is mapped to a quantum Duffing oscillator [1]. This gives for the transmon the following approximate spectrum:
\begin{equation}
E_{t, m} = - E_{J, t} + \sqrt{8 E_{C, t} E_{J, t}}\qty(m + 1/2) - \frac{E_{C, t}}{12}\qty(6 m^2 + 6 m + 3)\,.
\end{equation}
A similar approach can be used to model the parity-protected qubit in the transmon regime by treating it as a double Duffing oscillator. We can expand the potential in the two wells as
\begin{equation}
V(\phi) = E_{J, p} \cos(2 \phi - \pi/4) = -E_{J, p} + 4 E_{j, p} \frac{(\phi-\phi_l)^2}{2} - 16 E_{j, p} \frac{(\phi-\phi_l)^4}{24} + o(\phi^5)
\end{equation}
with $\phi_L = \pm \pi/4$. Next we introduce a hopping amplitude between the two wells. For semplicity, we consider allowed only hoppings between the same energy levels in the left and right well, i.e. $t_{mm'} = t_m \delta_{m m'}$. Therefore, the approximate Hamiltonian is
\begin{equation}
H = \sum_{l=L, R} \qty[\omega_p a^\dag_l a_l - E_{j, p} - \frac{E_{C, p}}{3} (a_l + a_l^\dag)^4] + \sum_{m} \qty[ t_m \qty(1 + e^{i 2 \pi n_{g,p}})( a^\dag_L)^m(a_R)^m]
\end{equation}
where $\omega_p = 2 \sqrt{8 E_{j, p} E_{C, p}}$.
The eigenvalues of the Hamiltonian can be calculated by first order perturbation theory using the number basis $\qty{\ket{m_L m_R}}$.
The double Duffing oscillator spectrum is composed by pairs of states located around the mean value
\begin{equation}
\mu_{n, n+1} = \frac{E_{n} + E_{n+1}}{2} \simeq - E_{J, p} + 2 \sqrt{8 E_{C, p} E_{J, p}}\qty(\frac{n}{2} + 1/2) - 4 \frac{E_{C, p}}{12}\qty(6 \qty(\frac{n}{2})^2 + 6 \frac{n}{2} + 3)\,,
\end{equation}
for even $n$, with a splitting $\delta_{n, n+1} = E_{n+1}-E_{n} \simeq \frac{t_n}{2} \cos(\pi n_{g, p})$. Each state belong to either the even or odd parity sector. In the regime $-0.5< n_{g, p} < 0.5$ the order of the states is even, odd, odd, even, …, while in the regime $0.5< n_{g, p} <1.5$ is odd, even, even, odd, …. In the PPQ the $E_{J, p} / E_{C, p}$ ratio has a twofold role. On the one hand higher ratios reduce the splitting between pair of states (i.e, $\omega_{p, 1}$ and $\delta\omega_{p, 23} = \omega_{p, 2} - \omega_{p,3}$) on the other hand increase the separation between the pairs of states (i.e. $\mu_{p, 23}$).
Low energy spectrum of: (a) a transmon with $E_{C, t} = (2\pi) \SI{0.2}{\giga\Hz}$ and (b) a parity-protected qubit with $E_{C, p} = (2\pi)\SI{0.18}{\giga\Hz}$ as function of the Josephson energy and zero offset charge. Panel (c): low energy spectrum of a parity protected qubit as a function of the offset charge with parameters $(E_{J, p}, E_{C, p}) = (2 \pi) (\SI{2.7}{\giga\Hz}, \SI{0.18}{\giga\Hz}) $
At the optimal point $n_{g, p}=0$, states $\vert 0_t, 2_p\rangle$ belongs to the odd sector while $\vert 0_t, 3_p\rangle$ belongs to the even sector. This pair of states show a splitting $\propto \abs{\cos (\pi n_{g, p})}$ and the mean is located approximately at energy $\mu_{p, 23} = \qty(\omega_{p, 2} + \omega_{p,3})/2 \simeq 2 \sqrt{8 E_{C, p} E_{J, p}} - 4 E_{C, p}$. Depending on the parameters of the system, the pair of states can be placed above the $\vert 1_t, 0_p\rangle$ $\vert 1_t, 1_p\rangle$ pair (when $\mu_{p, 23} \gtrsim \omega_{t, 1}$) or below (when $\mu_{p, 23} \lesssim \omega_{t, 1}$). In the presence of sizable splitting of the pair, the situation in which one of the two states lies below and one above is also possible. To obtain a controllable coupling, it is needed that at least one of the two excited states lays below the pair of computational states. For this reason, the choice of the parameters is crucial. For this reason, the approximate condition
\begin{equation}
\sqrt{8 E_{C, t} E_{J, t}} - E_{C, t} > 2 \sqrt{8 E_{C, p} E_{J, p}} - 4 E_{C, p}
\end{equation}
has to be satisfied.
[56]
Bravyi, S., DiVincenzo, D. P., and Loss, D. Schrieffer-Wolff transformation for quantum many-body systems.
Ann. Phys. 326, 2793 (2011).
[57]
M. J. Rancic, S. Hoffman, C. Schrade, J. Klinovaja, D. Loss, Physical Review B 99 (16), 165306 (2019).
[58]
G. Consani and P. A. Warburton, Effective Hamiltonians for interacting superconducting qubits: local basis reduction and the Schrieffer-Wolff transformation, New J. Phys. 22, 053040 (2020).
[59]
M. Hita-Pérez, G. Jaumà, M. Pino, J. J. García-Ripoll, Three-Josephson Junctions Flux Qubit Couplings,
arXiv:2111.05373 [quant-ph].
|
# Proximity and Remoteness in Graphs: a survey
Mustapha Aouchichea and Bilal Ahmad Ratherb,111Corresponding author
aPolytechnique Montreal, Montreal, QC, Canada
<EMAIL_ADDRESS>
bDepartment of Mathematical Sciences, College of Science,
United Arab Emirate University, Al Ain 15551, Abu Dhabi, UAE
<EMAIL_ADDRESS>
###### Abstract
The proximity $\displaystyle\pi=\pi(G)$ of a connected graph $\displaystyle G$
is the minimum, over all vertices, of the average distance from a vertex to
all others. Similarly, the maximum is called the remoteness and denoted by
$\displaystyle\rho=\rho(G)$. The concepts of proximity and remoteness, first
defined in 2006, attracted the attention of several researchers in Graph
Theory. Their investigation led to a considerable number of publications. In
this paper, we present a survey of the research work.
Keywords: Distance; Transmission; Proximity; Remoteness; Extremal graphs
AMS subject classification: 05C12, 05C35, 15A18.
## 1 Introduction
Models involving paths, distances and location on graphs are much studied in
operations research and mathematics. Models from operations research (see e.g.
[11, 25]) usually use weighted graphs to describe some well-defined class of
problems or some specific applications. Models from mathematics most often
consider unweighted graphs and relations between graph invariants, i.e.,
numerical quantities whose values do not depend on the labelling of edges or
vertices. In this paper, we present a survey of two graph invariants:
proximity and remoteness, defined as the minimum and maximum of the average
distance from a vertex to all others.
Let $\displaystyle G=(V,E)$ denote a simple and connected graph, with vertex
set $\displaystyle V$ and edge set $\displaystyle E$, containing
$\displaystyle n=|V|$ vertices and $\displaystyle m=|E|$ edges. All the graphs
considered in the present paper are finite, simple and connected.
For a vertex $\displaystyle u\in V$, the set of its neighbors in
$\displaystyle G$ is denoted as
$\displaystyle N_{u}=\\{v\in V:uv\in E\\}.$
The degree of $\displaystyle u$ is the number of its neighbors, i.e.,
$\displaystyle d(u)=d_{u}=|N_{u}|$. The maximum degree $\displaystyle\Delta$
and the minimum degree $\displaystyle\delta$ of $\displaystyle G$ are defined
as
$\displaystyle\Delta=\max_{u\in V}d(u)\qquad\mbox{ and
}\qquad\delta=\min_{u\in V}d(u).$
A graph $\displaystyle G$ is said to be $\displaystyle d$-regular, or regular
of degree $\displaystyle d$, if all of its vertices have degree $\displaystyle
d$.
The distance between two vertices $\displaystyle u$ and $\displaystyle v$ in
$\displaystyle G$, denoted by $\displaystyle d(u,v)$, is the length of a
shortest path between $\displaystyle u$ and $\displaystyle v$. The average
distance between all pairs of vertices in $\displaystyle G$ is denoted by
$\displaystyle\overline{l}$. The eccentricity $\displaystyle e(v)$ of a vertex
$\displaystyle v$ in $\displaystyle G$ is the largest distance from
$\displaystyle v$ to another vertex of $\displaystyle G$. The minimum
eccentricity in $\displaystyle G$, denoted by $\displaystyle r$, is the radius
of $\displaystyle G$. The maximum eccentricity of $\displaystyle G$, denoted
by $\displaystyle D$, is the diameter of $\displaystyle G$. The average
eccentricity of $\displaystyle G$ is denoted $\displaystyle ecc$. That is
$\displaystyle r=\min_{v\in V}e(v),\qquad D=\max_{v\in V}e(v) \qquad\mbox{ and
}\qquad ecc=\frac{1}{n}\sum_{v\in V}e(v).$
The Wiener index $\displaystyle W=W(G)$, introduced in [55], of a connected
graph $\displaystyle G$ is defined to be the sum of all distances in
$\displaystyle G$, i.e.,
$\displaystyle W(G)=\frac{1}{2}\sum_{u,v\in V}d(u,v).$
The transmission $\displaystyle t(v)$ of a vertex $\displaystyle v$ is defined
to be the sum of the distances from $\displaystyle v$ to all other vertices in
$\displaystyle G$, i.e.,
$\displaystyle t(v)=\sum_{u\in V}d(u,v).$
A connected graph $\displaystyle G=(V,E)$ is said to be $\displaystyle
k$–transmission regular if $\displaystyle t(v)=k$ for every vertex
$\displaystyle v\in V$. The transmission regular graphs are exactly the
distance–balanced graphs introduced in [28]. They are also called self–median
graphs in [14]. It is clear that any vertex–transitive graph (a graph
$\displaystyle G$ in which for every two vertices $\displaystyle u$ and
$\displaystyle v$, there exist an automorphism $\displaystyle f$ on
$\displaystyle G$ such that $\displaystyle f(u)=f(v)$) is a transmission
regular graph. The converse is not true in general. Indeed, the graph on
$\displaystyle 9$ vertices illustrated in Figure 1 is $\displaystyle
14$–transmission regular graph but not degree regular and therefore not
vertex–transitive. Actually, the graph in Figure 1 is the smallest
transmission regular but not degree regular (see e.g. [3, 32]). For more
examples of transmission regular but not degree regular graphs see [3, 6, 32,
34, 54].
Figure 1: The transmission regular but not degree regular graph with the
smallest order
The proximity $\displaystyle\pi=\pi(G)$ of $\displaystyle G$ is the minimum
average distance from a vertex of $\displaystyle G$ to all others. Similarly,
the remoteness $\displaystyle\rho=\rho(G)$ of $\displaystyle G$ is the maximum
average distance from a vertex to all others. The two last concepts were
recently introduced in [1, 2]. They are close to the concept of transmission
$\displaystyle t(v)$ of a vertex $\displaystyle v$. That is, if we denote
$\displaystyle\tilde{t}(v)$ the average distance from a vertex $\displaystyle
v$ to all other vertices in $\displaystyle G$, we have
$\displaystyle\pi=\min_{v\in V}\tilde{t}(v)=\min_{v\in V}\frac{t(v)}{n-1}
\qquad\mbox{ and }\qquad\rho=\max_{v\in V}\tilde{t}(v)=\max_{v\in
V}\frac{t(v)}{n-1}.$
The transmission of a vertex is also known as the distance of a vertex [24]
and the minimum distance (transmission) of a vertex is studied in [46]. A
notion very close to the average distance from a vertex is the vertex
deviation introduced by Zelinka [57] as
$\displaystyle m_{1}(v)=\frac{1}{n}\sum_{u\in V}d(u,v)=\frac{t(v)}{n}.$
The vector composed of the vertex transmissions in a graph was first
introduced by Harary [29] in 1959, under the name the status of a graph, as a
measure of the ”weights” of individuals in social networks. The same vector
was called the distance degree sequence by Bloom, Kennedy and Quintas [12]. It
was used to tackle the problem of graph isomorphism. Randić [47] conjectured
that two graphs are isomorphic if and only if they have the same distance
degree sequence. The conjecture was refuted by several authors such as Slater
[51], Buckley and Harary [13], and Entringer, Jackson and Snyder [24]. The
transmission of a graph was also introduced by Sabidussi [48] in 1966 as a
measure of centrality in social networks. The notion of centrality is widely
used in different branches of sciences (see for example [36] and the
references therein) such as transportation–network theory, communication
network theory, electrical circuits theory, psychology, sociology, geography,
game theory and computer science. Notions closely related to that of the
distance from a vertex are those of a center and a centroid introduced by
Jordan [35] in 1869. For mathematical properties of these two concepts see the
survey, as well as the references therein, [52]. In 1964, Hakimi [27] used for
the first time the sum of distances in solving facility location problems. In
fact, Hakimi [27] considered two problems, subsequently considered in many
works: the first problem was to determine a vertex $\displaystyle u\in V$ so
as to minimize $\displaystyle max_{v\in V}\\{d(u,v):u\in V\\}$, i.e., the
center of a graph; and the second problem is to determine a vertex
$\displaystyle u\in V$ so as to minimize the sum of distances from
$\displaystyle u$, i.e., the centroid. Interpretations of these problems can
be found, for instance, in [26]. In view of the interest of the transmission
vector in different domains of sciences, it is natural to study the properties
of its extremal values themselves, and among the set of graph parameters. The
study of proximity and remoteness, since closely related to respectively the
minimum and maximum values of the transmissions, appears to be convenient,
specially, with other metric invariants, such as the diameter, radius, average
eccentricity and average distance. Indeed, it follows from the definitions
that
$\displaystyle\pi\leq r\leq ecc\leq D,\qquad\pi\leq\overline{l}\leq\rho\leq
D\qquad\mbox{ and }\qquad\overline{l}=\frac{1}{n(n-1)}\sum_{v\in V}t(v).$
In addition to these inequalities, several related ones can be found in the
graph theory literature. Recall that a subset $\displaystyle S$ of vertices of
$\displaystyle G$ is said to be independent if its vertices are pairwise
nonadjacent. The maximum cardinality of such a subset is called the
independence number of $\displaystyle G$ and is denoted by
$\displaystyle\alpha=\alpha(G)$. Then $\displaystyle\overline{l}\leq\alpha$
[15] and $\displaystyle r\leq\alpha$ [23]. Recall, also, that a matching in a
graph is a set of disjoint edges. The maximum possible cardinality of a
matching in a graph $\displaystyle G$ is called the matching number of
$\displaystyle G$ and denoted by $\displaystyle\mu=\mu(G)$. The inequality
$\displaystyle r\leq\mu$ can be found in [23]. It was proved in [4] that
$\displaystyle\rho\leq ecc$. All these inequalities are illustrated in Figure
2, where vertices correspond to invariants $\displaystyle a,b,\ldots$, and
directed arcs to the relations $\displaystyle a\leq b$. Observe that all
relations mentioned are tight as all of them but $\displaystyle r\leq\mu$
become equalities for the complete graph $\displaystyle K_{n}$ (all invariants
but $\displaystyle\mu$ being equal to 1) and $\displaystyle r=\mu=1$ for the
star $\displaystyle S_{n}$.
Figure 2: Relations between the invariants.
Since their introduction in [1, 2], the proximity and the remoteness attracted
the attention of several authors.
## 2 Proximity and Remoteness
As for any graph invariant, the first questions asked about the proximity and
the remoteness are: “what are their minimum and maximum values for given order
$\displaystyle n$?” and “which extremal graphs are associated with these
extremal values for a given order $\displaystyle n$?” Both questions are
answered in the following proposition proved in [4].
###### Proposition 2.1 ([4]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n\geq 3$ vertices
with proximity $\displaystyle\pi$ and remoteness $\displaystyle\rho$. Then
$\displaystyle 1\leq\pi\leq\left\\{\begin{array}[]{ccl}\frac{n+1}{4}&&\mbox{
if $\displaystyle n$ is odd}\\\ \\\ \frac{n+1}{4}+\frac{1}{4(n-1)}&&\mbox{ if
$\displaystyle n$ is even}\end{array}\right.$
and
$\displaystyle 1\leq\rho\leq\frac{n}{2}.$
The lower bound on $\displaystyle\pi$ is reached if and only if $\displaystyle
G$ contains a dominating vertex; the upper bound on $\displaystyle\pi$ is
attained if and only if $\displaystyle G$ is either the cycle $\displaystyle
C_{n}$ or the path $\displaystyle P_{n}$; the lower bound on
$\displaystyle\rho$ is reached if and only if $\displaystyle G$ is the
complete graph $\displaystyle K_{n}$; the upper bound on $\displaystyle\rho$
is attained if and only if $\displaystyle G$ is the path $\displaystyle
P_{n}$.
Since the proximity $\displaystyle\pi$ and the remoteness $\displaystyle\rho$
are respectively the minimum and the maximum of the same function on a graph,
it is natural to ask about how large can the difference
$\displaystyle\rho-\pi$ be, or in other words, how large can the spread of the
average distances from vertices be. This problem was solved in [4].
###### Proposition 2.2 ([4]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n\geq 3$ vertices
with remoteness $\displaystyle\rho$ and proximity $\displaystyle\pi$. Then
$\displaystyle\rho-\pi\leq\left\\{\begin{array}[]{lll}\frac{n-1}{4}&&\mbox{if
$\displaystyle n$ is odd, }\\\ \frac{n-1}{4}-\frac{1}{4n-4}&&\mbox{if
$\displaystyle n$ is even. }\end{array}\right.$
Equality holds if and only if $\displaystyle G$ is a graph obtained from a
path $\displaystyle P_{\left\lceil\frac{n}{2}\right\rceil}$ and any connected
graph $\displaystyle H$ on $\displaystyle\left\lfloor n/2\right\rfloor+1$
vertices by a coalescence of an endpoint of the path with any vertex of
$\displaystyle H$.
The problem of find bounds on $\displaystyle\pi$ and $\displaystyle\rho$ for a
graph $\displaystyle G$ with given order $\displaystyle n$ and diameter
$\displaystyle D$ was considered in [4]. Actually, the best possible lower
bound on $\displaystyle\pi(G)$ and the best possible upper bound on
$\displaystyle\rho(G)$ were established. These involve a particular class of
graphs, next described. A double-tailed comet $\displaystyle DTC_{n,p,q}$ (see
Figure 3 for an example, i.e., $\displaystyle DTC_{15,4,4}$), with
$\displaystyle n\geq p+q+1$, $\displaystyle p\geq 2$ and $\displaystyle q\geq
2$ is the tree obtained from a path $\displaystyle u_{0}u_{1}\cdots
u_{p}u_{p+1}\cdots u_{p+q}$ by attaching $\displaystyle n-p-q-1$ pendant edges
to $\displaystyle u_{p}$. It is said to be balanced if $\displaystyle p=q$. It
is easy to see that the diameter of $\displaystyle DTC_{n,p,q}$ is
$\displaystyle D=p+q$ and its maximum degree is
$\displaystyle\Delta=d(u_{p})=n-p-q+1$. Assuming, without loss of generality,
that $\displaystyle p\geq q$, we have
$\displaystyle\displaystyle\pi(DTC_{n,p,q})$ $\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{p(p+1)}{2(n-1)}+\frac{q(q+1)}{2(n-1)}-\frac{p+q}{n-1}+1$
$\displaystyle\displaystyle=$
$\displaystyle\displaystyle\frac{p(p-1)}{2(n-1)}+\frac{q(q-1)}{2(n-1)}+1$
and
$\displaystyle\rho(DTC_{n,p,q})=\frac{(p+q)(p+q+1)}{2(n-1)}-\frac{p^{2}+pq}{n-1}+p.$
Figure 3: The double-tailed comet $\displaystyle DTC_{15,4,4}$.
###### Proposition 2.3 ([4]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n$ vertices with
diameter $\displaystyle D$. Then
$\displaystyle\pi(G)\geq\frac{p(p-1)}{2(n-1)}+\frac{q(q-1)}{2(n-1)}+1$
where $\displaystyle p=\left\lceil D/2\right\rceil$ and $\displaystyle
q=\left\lfloor D/2\right\rfloor$. The bound is best possible as shown by the
double-tailed comet $\displaystyle DTC_{n,p,q}$.
A comet $\displaystyle CO_{n,\Delta}$ is the tree obtained from a star
$\displaystyle S_{\Delta+1}$ and a path $\displaystyle P_{n-\Delta}$ by
coalescence of an endpoint of $\displaystyle P_{n-\Delta}$ with a pendant
vertex of $\displaystyle S_{\Delta+1}$. Easy computations lead to the
following expressions for the diameter and the remoteness of a comet:
$\displaystyle D(CO_{n,\Delta})=n-\Delta+1$
and
$\displaystyle\rho(CO_{n,\Delta})=\frac{(n-\Delta+1)(n+\Delta-2)}{2(n-1)}.$
A kite $\displaystyle KI_{n,\omega}$ is the connected graph obtained from a
clique $\displaystyle K_{\omega}$ and a path $\displaystyle P_{n-\omega}$ by
adding an edge between an endpoint of the path and a vertex from the clique. A
pseudo-kite $\displaystyle PKI_{n,p}$ is any connected graph which is a
spanning subgraph of the kite $\displaystyle KI_{n,p}$ and that contains the
comet $\displaystyle CO_{n,p}$ as a spanning tree. Note that $\displaystyle
CO_{n,p}$, $\displaystyle KI_{n,p}$ and $\displaystyle PKI_{n,p}$ have the
same proximity and the same remoteness.
The first relationship between the pseudo-kites and the notion of remoteness
is given in the following proposition.
###### Proposition 2.4 ([4]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n\geq 3$ vertices
with diameter $\displaystyle D$. Then
$\displaystyle\rho(G)\leq\rho(PKI_{n,n-D+1})$
with equality if and only if $\displaystyle G$ is a pseudo-kite $\displaystyle
PKI_{n,n-D+1}$.
Let $\displaystyle G$ be a graph and $\displaystyle\overline{G}$ its
complement. If $\displaystyle I$ is an invariant of $\displaystyle G$, we
denote by $\displaystyle\overline{I}$ the same invariant but in
$\displaystyle\overline{G}$. Nordhaus-Gaddum relations for the graph invariant
$\displaystyle I$ are inequalities of the following form
$\displaystyle L_{1}(n)\leq I+\overline{I}\leq U_{1}(n)\qquad\mbox{ and
}\qquad L_{2}(n)\leq I\cdot\overline{I}\leq U_{2}(n),$
where $\displaystyle L_{1}(n)$ and $\displaystyle L_{2}(n)$ are lower bounding
functions of the order $\displaystyle n$, and $\displaystyle U_{1}(n)$ and
$\displaystyle U_{2}(n)$ upper bounding functions of the order $\displaystyle
n$. Note that sometimes, in addition to the order $\displaystyle n$, other
graph invariants are used in the bounds. This type of relations is so called
after Nordhaus and Gaddum [41] who were the first authors to give such
relations, namely
$2\sqrt{n}\leq\chi+\overline{\chi}\leq n+1\qquad\mbox{ and }\qquad
n\leq\chi\cdot\overline{\chi}\leq\left(\frac{n+1}{2}\right)^{2},$ (1)
where $\displaystyle\chi$ is the chromatic number of a graph. Since then many
graph theorists have been interested in finding such relations for various
graph invariants. See [7] for a survey of Nordhaus–Gaddum type results. For
proximity and remoteness, Nordhaus–Gaddum inequalities were proved in [5].
###### Theorem 2.5 ([5]).
For any connected graph $\displaystyle G$ on $\displaystyle n\geq 5$ vertices
for which $\displaystyle\overline{G}$ is connected
$\displaystyle\frac{2n}{n-1}\leq\pi+\overline{\pi}\leq\left\\{\begin{array}[]{ccl}\frac{n+1}{4}+\frac{n+1}{n-1}&&\mbox{
if $\displaystyle n$ is odd, }\\\
\frac{n}{4}+\frac{n}{4(n-1)}+\frac{n+1}{n-1}&&\mbox{ if $\displaystyle n$ is
even. }\end{array}\right.$
The lower bound is attained if and only if
$\displaystyle\Delta(G)=\Delta(\overline{G})=n-2$. The upper bound is attained
if and only if either $\displaystyle G$ or $\displaystyle\overline{G}$ is the
cycle $\displaystyle C_{n}$.
###### Theorem 2.6 ([5]).
For any connected graph $\displaystyle G$ on $\displaystyle n\geq 5$ vertices
for which $\displaystyle\overline{G}$ is connected
$\displaystyle\frac{n^{2}}{(n-1)^{2}}\leq\pi\cdot\overline{\pi}\leq\left\\{\begin{array}[]{ccl}\frac{(n+1)^{2}}{4(n-1)}&&\mbox{
if $\displaystyle n$ is odd, }\\\
\frac{n(n+1)}{4(n-1)}+\frac{n(n+1)}{4(n-1)^{2}}&&\mbox{ if $\displaystyle n$
is even. }\end{array}\right.$
The lower bound is attained if and only if
$\displaystyle\Delta(G)=\Delta(\overline{G})=n-2$. The upper bound is attained
if and only if either $\displaystyle G$ or $\displaystyle\overline{G}$ is the
cycle $\displaystyle C_{n}$.
Recall that a comet $\displaystyle Co_{n,\Delta}$ is obtained from a star
$\displaystyle S_{\Delta+1}$ by appending a path $\displaystyle
P_{n-\Delta-1}$ to one of its pending vertices. Moreover, a path-complete
graph $\displaystyle PK_{n,m}$ on $\displaystyle n$ vertices and
$\displaystyle m$ edges is the graph obtained from a path $\displaystyle
P_{k}$, $\displaystyle k\geq 1$, and a clique $\displaystyle K_{n-k}$ by
adding at least one edge between one endpoint of the path and the vertices of
$\displaystyle K_{n-k}$, where $\displaystyle(n-k)(n-k-1)/2+k\leq
m\leq(n-k+1)(n-k)/2+k-1$. One can verify that there is exactly one path-
complete graph $\displaystyle PK_{n,m}$ for all $\displaystyle n$ and
$\displaystyle m$ such that $\displaystyle 1\leq n-1\leq m\leq n(n-1)/2$.
###### Theorem 2.7 ([5]).
For any connected graph $\displaystyle G$ on $\displaystyle n\geq 6$ vertices
for which $\displaystyle\overline{G}$ is connected
$\displaystyle 3\leq\rho+\overline{\rho}\leq\frac{n+2}{2}+\frac{2}{n-1}.$
The lower bound is attained if and only if $\displaystyle n\geq 8$,
$\displaystyle G$ is regular and $\displaystyle D=\overline{D}=2$. The upper
bound is attained if and only if $\displaystyle G$ or
$\displaystyle\overline{G}$ is the path $\displaystyle P_{n}$, the comet
$\displaystyle Co_{n,3}$ or the path-complete graph $\displaystyle PK_{n,n}$
when $\displaystyle n\geq 7$, and if and only if $\displaystyle G$ or
$\displaystyle\overline{G}$ is the path $\displaystyle P_{6}$, the comet
$\displaystyle Co_{6,3}$, the path-complete graph $\displaystyle PK_{6,6}$ or
one of the graphs in Fig. 4.
Figure 4: Graphs with $\displaystyle D=3$ that maximize
$\displaystyle\rho+\overline{\rho}$ for $\displaystyle n=6$.
###### Theorem 2.8 ([5]).
For any connected graph $\displaystyle G$ on $\displaystyle n\geq 7$ vertices
for which $\displaystyle\overline{G}$ is connected
$\displaystyle\rho\cdot\overline{\rho}\leq\left\\{\begin{array}[]{lll}\frac{16n+20}{27}+\frac{8}{9(n-1)}+\frac{4}{27(n-1)^{2}}&
\mbox{ if }&n=\,\,0\,\,(mod\,\,3),\\\ \frac{16n+20}{27}+\frac{2}{3(n-1)}&
\mbox{ if }&n=\,\,1\,\,(mod\,\,3),\\\
\frac{16n+20}{27}+\frac{8}{9(n-1)}+\frac{5}{27(n-1)^{2}}& \mbox{ if
}&n=\,\,2\,\,(mod\,\,3).\\\ \end{array}\right.$
The upper bound is the best possible as shown by the comets $\displaystyle
Co_{n,\left\lceil\frac{n}{3}\right\rceil+1}$, and $\displaystyle
Co_{n,\left\lceil\frac{n}{3}\right\rceil}$ if $\displaystyle
n=\,\,1\,\,(mod\,\,3)$.
Note that the bound provided in Theorem 2.8 is not valid for $\displaystyle
n=4,5,6$ (if $\displaystyle n\leq 3$, then $\displaystyle G$ and
$\displaystyle\overline{G}$ cannot be connected simultaneously). For the path
$\displaystyle P_{4}$ (resp. the comets $\displaystyle Co_{5,3}$ and
$\displaystyle Co_{6,4}$), $\displaystyle\rho\cdot\overline{\rho}=4$ (resp.
9/2 and 24/5) while the corresponding value provided by Theorem 2.8 is
$\displaystyle 10/3$ (resp. 63/16 and 112/25).
Czabarka et al. [16] obtained results on proximatity in triangulations and
quadrangulations, that is graphs with maximal planer graphs and graphs with
maximal bipartite planer graphs, respectiely.
###### Theorem 2.9 ([16]).
Let $\displaystyle G$ be a planar graph of order $\displaystyle n$ and
$\displaystyle v$ a central vertex of $\displaystyle G$.
* (i)
If $\displaystyle G$ is a triangulation, then
$\displaystyle\pi\leq\frac{n+19}{12}+\frac{25}{3(n-1)}.$
* (ii)
If $\displaystyle G$ is a $\displaystyle 4$-connected triangulation, then
$\displaystyle\pi\leq\frac{n+35}{16}+\frac{91}{4(n-1)}.$
* (iii)
If $\displaystyle G$ is a $\displaystyle 5$-connected triangulation, then
$\displaystyle\pi\leq\frac{n+57}{20}+\frac{393}{10(n-1)}.$
* (iv)
If $\displaystyle G$ is a quadrangulation, then
$\displaystyle\pi\leq\frac{n+11}{8}+\frac{9}{2(n-1)}.$
* (v)
If $\displaystyle G$ is a $\displaystyle 3$-connected quadrangulation, then
$\displaystyle\pi\leq\frac{n+25}{12}+\frac{169}{12(n-1)}.$
Pei et al. [45] gave following intresting results relating proximity, average
eccentricity and domination, results earlier conjuctured in [1]. The
domination number $\displaystyle\gamma(G)$ is the minimum cardinality of a
dominating set of G.
Recall that $\displaystyle\gamma(G)\leq\left\lfloor\frac{n}{2}\right\rfloor.$
In fact, the graphs with domination number
$\displaystyle\left\lfloor\frac{n}{2}\right\rfloor$ have been determined in
the following result.
Figure 5: Graphs in family $\displaystyle\mathcal{A}$
Figure 6: Graphs in family $\displaystyle\mathcal{B}$
###### Lemma 2.10 ([42]).
A connected graph $\displaystyle G$ of order $\displaystyle n$ satisfies
$\displaystyle\gamma(G)=\left\lfloor\frac{n}{2}\right\rfloor$ if and only if
$\displaystyle G\in\bigcup_{i=1}^{6}\mathcal{G}_{i}$, where
$\displaystyle\mathcal{G}_{i},~{}i=1,\dots,6,$ is the set defined in the
following. Let $\displaystyle H$ be any graph with vertex set
$\displaystyle\\{v_{1},\dots,v_{k}\\}$. Denote by $\displaystyle f(H)$ the
graph obtained from $\displaystyle H$ by adding new vertices $\displaystyle
u_{1},\dots,u_{k}$ and the edges $\displaystyle v_{i}u_{i},~{}i=1,\dots,k.$
Define
$\displaystyle\mathcal{G}_{1}=\\{C_{4}\\}\cup\\{G|G=f(H)~{}\text{for some
connected graph}~{}H\\}.$
Let $\displaystyle\mathcal{F}=\mathcal{A}\cup\mathcal{B}$ and
$\displaystyle\mathcal{G}_{2}=\mathcal{F}-\\{C_{4}\\},$ where
$\displaystyle\mathcal{A}=\\{C_{4},G(7,i)|i=1,\dots,6\\}$ and
$\displaystyle\mathcal{B}=\\{K3,G(5,i)|i=1,\dots,4\\},$ as shown in Fig. 5 and
Fig. 6, respectively. For any graph $\displaystyle H$, let
$\displaystyle\phi(H)$ be the set of connected graphs, each of which can be
formed from $\displaystyle f(H)$ by adding a new vertex $\displaystyle x$ and
edges joining $\displaystyle x$ to one or more vertices of $\displaystyle H$.
Then define $\displaystyle\mathcal{G}_{3}=\\{G|G=\phi(H)~{}\text{for some
graph}~{}H\\}.$ Let $\displaystyle G\in\mathcal{G}_{3}$ and $\displaystyle y$
be a vertex of a copy of $\displaystyle C_{4}$. Denote by
$\displaystyle\theta(G)$ the graph obtained by joining $\displaystyle G$ to
$\displaystyle C_{4}$ with the single edge $\displaystyle xy,$ where
$\displaystyle x$ is the new vertex added in forming G. Then define
$\displaystyle\mathcal{G}_{4}=\\{G|G=\theta(H)~{}\text{for some graph
H}~{}\in\mathcal{G}_{3}\\}.$
Let $\displaystyle u,v,w$ be the vertex sequence of a path $\displaystyle
P_{3}.$ For any graph $\displaystyle H,$ let $\displaystyle\mathcal{P}(H)$ be
the set of connected graphs which may be formed from $\displaystyle f(H)$ by
joining each of $\displaystyle u$ and $\displaystyle w$ to one or more
vertices of M Then define
$\displaystyle\mathcal{G}_{5}=\\{G|G=\mathcal{P}(H)~{}\text{for some
graph}~{}H\\}.$
For a graph $\displaystyle X\in\mathcal{B}$, let $\displaystyle U\subset V(X)$
be a set of vertices such that no fewer than $\displaystyle\gamma(X)$ vertices
of $\displaystyle X$ dominate $\displaystyle V(X)\setminus U.$ Let
$\displaystyle\mathcal{R}(H,X)$ be the set of connected graphs which may be
formed from $\displaystyle f(H)$ by joining each vertex of $\displaystyle U$
to one or more vertices of $\displaystyle H$ for some set $\displaystyle U$ as
defined above and any graph $\displaystyle H.$ Then define
$\displaystyle\mathcal{G}_{6}=\\{G|G\in\mathcal{R}(H,X)~{}\text{for
some}~{}X\in\mathcal{B}~{}\text{and some}~{}H\\}.$
###### Lemma 2.11 ([45]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n$ with
$\displaystyle\gamma(G)\geq 2,\Delta(G)\leq n-2$ and
$\displaystyle\delta(G)=1.$ Then
$\displaystyle ecc\geq 2+\frac{\gamma(G)}{n}.$
###### Lemma 2.12 ([45]).
Let $\displaystyle G$ be a connected graph with $\displaystyle\Delta(G)\leq
n-2$ and $\displaystyle\delta(G)=2.$ Then $\displaystyle\gamma(G)=2$ or
$\displaystyle ecc\geq 2+\frac{2}{n}.$
###### Lemma 2.13 ([45]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n\geq 2$.
If $\displaystyle\Delta(G)=n-1,$ then $\displaystyle\frac{\gamma(G)}{ecc}\leq
1$ with equality if and only if $\displaystyle G\cong K_{n}.$
###### Lemma 2.14 ([45]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n$ vertices,
where $\displaystyle 2\leq n\leq 5.$ Then
$\displaystyle\frac{\gamma(G)}{ecc}\leq 1,$
with equality if and only if $\displaystyle G\in\\{P_{2},C_{3},K_{4},C_{4}\\}$
when $\displaystyle n\leq 4$, and $\displaystyle
G\in{K_{5},G(5,1),G(5,2),G(5,3),G(5,4)}\subseteq\mathcal{B}$ when
$\displaystyle n=5,$ see Fig. 6
###### Theorem 2.15 ([45]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n\geq 2.$
Then
$\displaystyle\frac{\gamma(G)}{ecc}\begin{cases}1&\text{if}~{}n\leq 5,\\\
\frac{n.\left\lfloor\frac{n}{2}\right\rfloor}{\left\lfloor\frac{5n}{2}\right\rfloor}&\text{if}~{}n\geq
6,\end{cases}$
with equality if and only if $\displaystyle
G\in\\{P_{2},C_{3},K_{4},C_{4},K_{5},G(5,1),G(5,2),G(5,3),G(5,4)\\}$ when
$\displaystyle n\leq 5$, or $\displaystyle
G\in\\{K_{\left\lceil\frac{n}{2}\right\rceil,\left\lfloor\frac{n}{2}\right\rfloor},G(7,1),G(7,2),G^{\prime},G^{\prime\prime}\\}$
when $\displaystyle n\geq 6,$ where $\displaystyle
G^{\prime},G^{\prime\prime}$ are defined in [44].
The first observation in [45] is that for a vertex $\displaystyle v$ in a
conected graph $\displaystyle G$, the proximity is
$\displaystyle\pi(v)\geq\frac{2n-2-d(v)}{n-1},$
with equality if and only if $\displaystyle e(v)\leq 2.$
we define a subset of $\displaystyle\mathcal{G}$. Namely,
$\displaystyle\mathcal{G}^{\ast}=\cup_{i\in\\{1,2,3,5,6\\}}\mathcal{G}_{i}^{\ast},$
where $\displaystyle\mathcal{G}_{i}^{\ast}$ is shown as follows,
$\displaystyle i=1,2,3,5,6.$ We continue with the meaning of the notations
$\displaystyle x,u,v,w$ in the definition of $\displaystyle\mathcal{G}_{3}$
and $\displaystyle\mathcal{G}_{5}$, where $\displaystyle x$ is the vertex
added in forming graph $\displaystyle G\in\mathcal{G}_{3},u,v,w$ is the vertex
sequence of a path $\displaystyle P_{3}$ mentioned in the constructionof
$\displaystyle G_{5}.$ Let $\displaystyle H^{\ast}$ be any connected graph
with vertex set $\displaystyle\\{v_{1},\dots,v_{k}\\}$ and
$\displaystyle\triangle(H^{\ast})=k-1.$ Assume that $\displaystyle
d_{H^{\ast}}(v_{i_{0}})=\triangle(H^{\ast})$ for some $\displaystyle
i_{0}\in\\{1,\dots,k\\},$ that is
$\displaystyle{v_{i_{0}}v_{j}|j=1,\dots,i_{0}-1,i_{0}+1,\dot{,}k}\subseteq
E(H^{\ast}).$ Let
$\displaystyle\displaystyle\mathcal{G}_{1}^{\ast}$
$\displaystyle\displaystyle=\\{G|G=f(H^{\ast})~{}\text{for some connected
graph}H^{\ast}\\},$ $\displaystyle\displaystyle\mathcal{G}_{2}^{\ast}$
$\displaystyle\displaystyle=\\{G(5,2),G(5,3),G(5,4),G(7,2),G(7,5),G(7,6)\\},$
$\displaystyle\displaystyle\mathcal{G}_{3}^{\ast}$
$\displaystyle\displaystyle=\\{G|G=\phi(H^{\ast})~{}\text{and}~{}v_{i_{0}}x\in
E(G)\\},$ $\displaystyle\displaystyle\mathcal{G}_{5}^{\ast}$
$\displaystyle\displaystyle=\\{G|G=\mathcal{P}(H^{\ast}),v_{i_{0}}u\in
E(G)~{}\text{and}~{}v_{i_{0}}w\in E(G)\\}$ and
$\displaystyle\displaystyle\mathcal{G}_{6}^{\ast}$
$\displaystyle\displaystyle=\mathcal{G}_{6}^{1}\cup\mathcal{G}_{6}^{1}\cup\mathcal{G}_{6}^{(3,2)}\cup\mathcal{G}_{6}^{(3,3)}\cup\mathcal{G}_{6}^{(3,4)},$
where $\displaystyle\displaystyle\mathcal{G}_{6}^{1}$
$\displaystyle\displaystyle=\\{G|G=\mathcal{R}(K3,H^{\ast})~{}\text{with}~{}|U|=2,~{}\text{and}~{}v_{i_{0}}s\in
E(G)~{}\text{for each vertex}~{}s\in U\\}.$
$\displaystyle\displaystyle\mathcal{G}_{6}^{2}$
$\displaystyle\displaystyle=\\{G|G\in\mathcal{G}_{6}~{}\text{with}~{}X=K_{3},~{}\text{and}~{}sv_{i}\in
E(G)~{}\text{for some vertex}~{}s\in U~{}\text{and}~{}i=1,\dots,k\\}.$
For $\displaystyle i=2,3,4,$
$\displaystyle\mathcal{G}_{6}^{(3,i)}=\\{G|G\in\mathcal{G}_{6}~{}\text{with}~{}X=G(5,i),~{}\text{and}~{}sv_{j}\in
E(G)~{}\text{for}~{}j=1,\dots,k,~{}\text{some vertex}s\in
U~{}\text{with}~{}d_{G}(5,i)(s)=3\\}.$
###### Lemma 2.16 ([45]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n\geq 5$
with $\displaystyle\gamma(G)=\left\lfloor\frac{n}{2}\right\rfloor$. Then
$\displaystyle\pi\geq\begin{cases}\frac{3}{2}-\frac{1}{2(n-1)},&n~{}\text{is
even}\\\ \frac{3}{2}-\frac{1}{n-1},&n~{}\text{is odd}\end{cases}$
with equality if and only if $\displaystyle G\in\mathcal{G}^{\ast}.$
###### Theorem 2.17 ([45]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n\geq 2$.
Then
$\displaystyle\gamma(G)-\pi\leq\begin{cases}\frac{n-3}{2}+\frac{1}{2(n-1)},&\text{if
$\displaystyle n$ is even}\\\ \frac{n-4}{2}-\frac{1}{n-1},&\text{if
$\displaystyle n$ is odd}\end{cases}$
with equality if and only if $\displaystyle
G\in\mathcal{G}^{\ast}\cup\\{P_{2},P_{3},C_{3},P_{4},C_{4}\\}.$
Recently Pie [43] obtained the following results for the domination number and
the remoteness of a graph.
###### Lemma 2.18 ([43]).
Let $\displaystyle G$ be a connected graph with $\displaystyle n\leq 6$
vertices with $\displaystyle\gamma(G)=\left\lfloor\frac{n}{2}\right\rfloor-1.$
Then
$\displaystyle\gamma(G)-\rho(G)<\begin{cases}\frac{n-5}{2}+\frac{3}{2n-2},&n~{}\text{is
even}\\\ \frac{n-6}{2}+\frac{2}{n-1},&n~{}\text{is odd and}~{}n\geq 9\\\
\frac{n-3}{4},&n=7.\end{cases}$
###### Lemma 2.19 ([43]).
Suppose that $\displaystyle G$ be a connected graph of order $\displaystyle n$
with $\displaystyle 1\leq\gamma(G)=\left\lfloor\frac{n}{2}\right\rfloor-2.$
Then
$\displaystyle\gamma(G)-\rho(G)<\begin{cases}\frac{n-5}{2}+\frac{3}{2n-2},&n~{}\text{is
even}\\\ \frac{n-6}{2}+\frac{2}{n-1},&n~{}\text{is odd and}~{}n\geq 9\\\
\frac{n-3}{4},&n~{}\text{is odd and}~{}n\leq 7.\end{cases}$
###### Lemma 2.20 ([43]).
If $\displaystyle G$ is a connected graph with order $\displaystyle n(\geq 2)$
and $\displaystyle\gamma(G)=\left\lfloor\frac{n}{2}\right\rfloor,$ then
$\displaystyle\gamma(G)-\rho(G)\leq\begin{cases}\frac{2}{3},&n=4\\\
\frac{n-5}{2}+\frac{3}{2n-2},&n~{}\text{is even and}~{}n\neq 4\\\
\frac{n-3}{4},&n~{}\text{is odd and}~{}n\leq 7\\\
\frac{n-6}{2}+\frac{2}{n-1},&n~{}\text{is odd and}~{}n\geq 9,\\\ \end{cases}$
with equality if and only if $\displaystyle
G\in\\{C_{4},K_{\frac{n}{2},\frac{n}{2}}~{}|~{}n~{}\text{is even and}~{}n\neq
4\\}\cup(\mathcal{G}_{2}-\\{G^{5}_{7})\\})\cup\\{K_{\lceil\frac{n}{2}\rceil,\lfloor\frac{n}{2}\rfloor},G^{\prime},G^{\prime\prime}$
$\displaystyle|~{}n~{}\text{is odd and}~{}n\geq 9\\},$ where $\displaystyle
G^{\prime},G^{\prime\prime}$ are defined in [44].
###### Lemma 2.21 ([43]).
If $\displaystyle G$ is a connected graph with $\displaystyle n\leq 7$
vertices and $\displaystyle\gamma(G)=\left\lfloor\frac{n}{2}\right\rfloor-1,$
then $\displaystyle n\geq 4$ and
$\displaystyle\gamma(G)-\rho(G)\leq\begin{cases}\frac{4}{5},&n=6\\\ 0&4\leq
n\leq 5\end{cases}$
with equality if and only if G is $\displaystyle 4$-regular when
$\displaystyle n=6$, and $\displaystyle G\cong K_{n}$ when $\displaystyle
4\leq n\leq 5.$
Dankelmann, Jonck and Mafunda [20], obtained boubds for $\displaystyle\pi$ and
$\displaystyle\rho$ in triangle-free and $\displaystyle C_{4}$-free graphs in
terms of order and minimum degree.
Given positive integers $\displaystyle n,\delta$ with $\displaystyle\delta\geq
3.$ Let $\displaystyle A=(1,1,\delta-1,\delta-1,1,1,\delta-1,\delta-1,\dots)$
be the infinite sequence repeating the
$\displaystyle(1,1,\delta-1,\delta-1)$-pattern indefinitely. Define the finite
sequence $\displaystyle X_{n,\delta}$ by
$\displaystyle
X_{n,\delta}=(1,\delta,\delta-1,a_{1},a_{2},\dots,a_{l(A,n-4\delta)},\delta,r_{n,\delta}),$
where $\displaystyle r_{n,\delta}=n-3\delta-\sum_{i=1}^{l(A,n-4\delta)}a_{i}.$
The sequential sum of graphs $\displaystyle G=G_{1}+G_{2}+\dots+G_{n}$ to be
the sequential join such that the vertex set $\displaystyle V(G)=V(G_{1})\cup
V(G_{2})\cup\dots\cup V(G_{n})$ and the edge set $\displaystyle
E(G)=E(G_{1})\cup E(G_{2})\cup\dots\cup E(G_{n})\bigcup_{i=1}^{n-1}\\{uv|u\in
V(G_{i}),v\in V(G_{i+1})\\}.$
For a finite sequence $\displaystyle X=(x_{0},x_{1},\dots,x_{d})$ of positive
integers we define the graph $\displaystyle G(X)$ by
$\displaystyle
G(X)=\overline{K}_{x_{0}}+\overline{K}_{x_{1}}+\dots+\overline{K}_{x_{d}}$
###### Theorem 2.22 ([20]).
Let $\displaystyle G$ be a connected, triangle free graph of order
$\displaystyle n$ and minimum degree $\displaystyle\delta$, where
$\displaystyle\delta\geq 3$ and $\displaystyle n\geq 6\delta.$ Then
$\displaystyle\rho\leq\rho(G(X_{n,\delta})).$
###### Corollary 2.23 ([20]).
If $\displaystyle G$ is a connected triangle-free graph of order
$\displaystyle n$ and minimum degree $\displaystyle\delta\geq 3$, then
$\displaystyle\rho\leq
2\left\lceil\frac{n-3\delta}{2\delta}\right\rceil+2-\frac{\delta}{n-1},$
and this bound is sharp.
###### Theorem 2.24 ([20]).
If $\displaystyle G$ is a connected, triangle-free graph of order
$\displaystyle n$ and minimum degree $\displaystyle\delta\geq 3$, then
$\displaystyle\pi\leq\frac{n}{2\delta}+2-\frac{5}{2\delta}-\frac{21\delta^{2}-8\delta-3}{2\delta(n-1)}.$
###### Theorem 2.25 ([20]).
If $\displaystyle G$ is a connected, $\displaystyle C_{4}$-free graph of order
$\displaystyle n$ and minimum degree $\displaystyle\delta\geq 3$, then
$\displaystyle\rho\leq\frac{5}{2}\left\lfloor\frac{n}{\delta^{2}-2\left\lfloor\frac{\delta}{2}\right\rfloor+1}\right\rfloor+2.$
The next theorem shows that for many values of $\displaystyle\delta$ the bound
on remoteness is close to being best possible in the sense that the ratio of
the coefficients of $\displaystyle n$ in the bound and approach $\displaystyle
1$ as $\displaystyle\delta$ gets large.
###### Theorem 2.26 ([20]).
Let $\displaystyle\delta\geq 3$ be an integer such that
$\displaystyle\delta=q-1$ for some prime power $\displaystyle q$. Then there
exists an infinite number of $\displaystyle C_{4}$-free graphs $\displaystyle
G$ of minimum degree at least $\displaystyle\delta$ with
$\displaystyle\displaystyle\rho$
$\displaystyle\displaystyle=\frac{5}{2}\frac{n}{\delta^{2}+3\delta+2}+O(1),$
$\displaystyle\displaystyle\pi$
$\displaystyle\displaystyle=\frac{5}{4}\frac{n}{\delta^{2}+3\delta+2}+O(1),$
where $\displaystyle n$ is the order of $\displaystyle G.$
###### Theorem 2.27 ([20]).
If $\displaystyle G$ is a connected $\displaystyle C_{4}$-free graph of order
$\displaystyle n$ and minimum degree $\displaystyle\delta\geq 3$, then
$\displaystyle\pi\leq\frac{5}{4}\left\lfloor\frac{n}{\delta^{2}-2\left\lfloor\frac{\delta}{2}\right\rfloor+1}\right\rfloor+\frac{147}{32}.$
Dankelmann and Mafunda [21] gave results about difference between
$\displaystyle\pi$ and distance parameters in triangle-free and $\displaystyle
C_{4}$-free graphs.
###### Theorem 2.28 ([21]).
If $\displaystyle G$ is a connected, triangle-free graph of order
$\displaystyle n\geq 7$ and minimum degree $\displaystyle\delta\geq 3$, then
$\displaystyle\rho-\pi\leq\frac{n+1}{2\delta}+4.$
###### Theorem 2.29 ([21]).
If $\displaystyle G$ is a connected, $\displaystyle C_{4}$-free graph of order
$\displaystyle n\geq 6$ and minimum degree $\displaystyle\delta\geq 3$, then
$\displaystyle\rho-\pi\leq\frac{5(n+1)}{4\left(\delta^{2}-2\left\lfloor\frac{\delta}{2}\right\rfloor+1\right)}+\frac{101}{20}.$
###### Theorem 2.30 ([22]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n,$ minimum
degree $\displaystyle\delta$ and maximum degree $\displaystyle\triangle$.
If $\displaystyle\triangle>\frac{n}{2}-1,$ then
$\displaystyle\pi\leq\frac{3(n-\triangle)^{2}}{2(n-1)(\delta+1)}+\frac{13}{2}.$
If $\displaystyle\triangle\leq\frac{n}{2}-1,$ then
$\displaystyle\pi\leq\frac{3(n^{2}-2\triangle^{2})}{4(n-1)(\delta+1)}+\frac{35}{4}.$
###### Theorem 2.31 ([22]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n$, minimum
degree $\displaystyle\delta$ and maximum degree $\displaystyle\triangle$. Then
there exists a spanning tree $\displaystyle T$ of $\displaystyle G$ with
$\displaystyle\rho(T)\leq\frac{3(n^{2}-\triangle^{2})}{2(n-1)(\delta+1)}+7.$
Since $\displaystyle\rho(G)\leq\rho(T)$ for every spanning tree of a conncetd
graph $\displaystyle G,$ the following result follows from Theorem 2.31.
###### Corollary 2.32 ([22]).
If $\displaystyle G$ is a connected graph of order $\displaystyle n$, minimum
degree $\displaystyle\delta$ and maximum degree $\displaystyle\triangle$, then
$\displaystyle\rho\leq\frac{3(n^{2}-\triangle^{2})}{2(n-1)(\delta+1)}+7.$
A thesis about the proximity and the remoteness of graphs written by Mallu in
2022 can be seen [38], jointly supervised by Dankelmann and Mafunda.
## 3 Proximity and Remoteness Compared to Metric Invariants
Aouchiche and Hansen in 2011 compared proximity and remoteness with the metric
invariants of a graph [4].
###### Proposition 3.1 ([4]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n\geq 3$ vertices
with diameter $\displaystyle D$ and proximity $\displaystyle\pi$. Then
$\displaystyle D-\pi\leq\left\\{\begin{array}[]{ll}\frac{3n-5}{4}&\quad\mbox{
if $\displaystyle n$ is odd, }\\\ \frac{3n-5}{4}-\frac{1}{4n-4}&\quad\mbox{ if
$\displaystyle n$ is even, }\end{array}\right.$
with equality if and only if $\displaystyle G$ is a path $\displaystyle
P_{n}$.
###### Proposition 3.2 ([4]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n\geq 3$ vertices
with remoteness $\displaystyle\rho$ and diameter $\displaystyle D$. Then
$\displaystyle D-\rho\leq\frac{n-2}{2},$
with equality if and only if $\displaystyle G$ is the path $\displaystyle
P_{n}$.
###### Proposition 3.3 ([4]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n\geq 3$ vertices
with radius $\displaystyle r$ and proximity $\displaystyle\pi$. Then
$\displaystyle
r-\pi\leq\left\\{\begin{array}[]{ll}\frac{n-1}{4}-\frac{1}{4(n-1)}&\mbox{if
$\displaystyle n$ is even, }\\\ \frac{n-1}{4}-\frac{1}{n-1}&\mbox{ if
$\displaystyle n$ is odd. }\end{array}\right.$
The bound is best possible as shown by the graph composed of a cycle with an
additional edge forming a triangle or two additional crossed edges on four
successive vertices of the cycle if $\displaystyle n$ is odd, and by the path
$\displaystyle P_{n}$ or the cycle $\displaystyle C_{n}$ if $\displaystyle n$
is even.
###### Proposition 3.4 ([4]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n\geq 3$ vertices
with remoteness $\displaystyle\rho$ and radius $\displaystyle r$. Then
$\displaystyle\rho-r\leq\left\\{\begin{array}[]{lll}\frac{n+1}{8}+\frac{1}{8(n-1)}&&\mbox{
if }n=0\,\,(mod\,\, 4),\\\ \frac{n+1}{8}&&\mbox{ if }n=1\,\,(mod\,\, 4),\\\
\frac{n+1}{8}-\frac{3}{8(n-1)}&&\mbox{ if }n=2\,\,(mod\,\, 4),\\\
\frac{n+1}{8}&&\mbox{ if }n=3\,\,(mod\,\, 4).\\\ \end{array}\right.$
The bound is attained if and only if $\displaystyle G$ is a pseudo-kite
$\displaystyle PKI_{n,n-2r^{*}+1}$ where
$\displaystyle r^{*}=\left\\{\begin{array}[]{lll}\frac{n}{4}&&\mbox{ if
}n=0\,\,(mod\,\, 4),\\\ \frac{n-1}{4}&&\mbox{ if }n =1\,\,(mod\,\, 4),\\\
\frac{n-2}{4}\mbox{ or }\frac{n+2}{4}&&\mbox{ if }n=2\,\,(mod\,\, 4),\\\
\frac{n+1}{4}&&\mbox{ if }n=3\,\,(mod\,\, 4).\\\ \end{array}\right.$
###### Proposition 3.5 ([4]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n\geq 3$ vertices
with remoteness $\displaystyle\rho$ and average eccentricity $\displaystyle
ecc$. Then
$\displaystyle\rho\leq ecc$
with equality in both cases if and only if $\displaystyle G$ is a complete
graph $\displaystyle K_{n}$.
Ma, Wu and Zhang [40] gave the following lemmas and proved a conjecture that
consists of an upper bound on $\displaystyle ecc-\pi$ together with the
corresponding extremal graphs.
A vertex $\displaystyle v\in V$ is called a centroidal vertex if
$\displaystyle\pi(v)=\pi(G)$, and the set of all centroidal vertices is the
centroid (sometimes known as median or barycenter) of $\displaystyle G.$
For an edge $\displaystyle e\in E(G)$ and a vertex $\displaystyle u\in V(G)$,
we denote by $\displaystyle(G-e)_{u}$ the component of $\displaystyle G-e$
containing $\displaystyle u$, and let $\displaystyle V_{u}(e)=V((G-e)_{u})$
and $\displaystyle n_{u}(e)=|V_{u}(e)|$. Clearly, for any edge $\displaystyle
e=uv\in E(G),~{}n_{u}(e)+n_{v}(e)=|V(G)|$. The following lemmas are given in
[40].
###### Lemma 3.6 ([40]).
Let $\displaystyle G$ be a tree of order $\displaystyle n$. For any edge
$\displaystyle e=uv\in E(G),$
$\displaystyle \pi(u)+\frac{1}{n-1}n_{u}(e)=\pi(v)+\frac{1}{n-1}n_{v}(e).$
###### Lemma 3.7 ([40]).
The following holds for a tree $\displaystyle G$ of order $\displaystyle n$
* (i)
If $\displaystyle x$ is a centroidal vertex of $\displaystyle G$, then
$\displaystyle ecc(x)\leq\left\lfloor\frac{n}{2}\right\rfloor,$
with equality if and only if there exists a path of length
$\displaystyle\left\lfloor\frac{n}{2}\right\rfloor$ in $\displaystyle G$,which
joins $\displaystyle x$ to a pendent vertex of $\displaystyle G$ with the
property that the degree of every internal vertex of it is equal to two in
$\displaystyle G$.
* (ii)
If there is a path $\displaystyle P$ of length
$\displaystyle\left\lfloor\frac{n}{2}\right\rfloor$ in $\displaystyle G$,
which joins a vertex $\displaystyle y$ and a pendent vertex of $\displaystyle
G$ with the property that the degree of every internal vertex of it is equal
to two in $\displaystyle G$, then $\displaystyle y$ is a centroidal vertex of
$\displaystyle G$.
###### Lemma 3.8 ([40]).
Let G be a tree of order $\displaystyle n\geq 3$. Let $\displaystyle
v_{0}v_{1}\dots v_{d}$ be a longest path in $\displaystyle G$. Set
$\displaystyle V_{0}=\\{v\in V(G)|d(v,v_{0})\geq
d(v,v_{d})\\},~{}V_{d}=\\{v\in V(G)|d(v,v_{d})\geq d(v,v_{0})\\}$. Without
loss of generality, let $\displaystyle|V_{0}|\geq|V_{d}|$. If $\displaystyle
G$ is not a path, then the following holds:
* (1)
there is a pendent vertex $\displaystyle v^{\ast}\in V_{0}$ distinct from
$\displaystyle v_{d}$,
* (2)
$\displaystyle ecc(G)-ecc(G^{\prime})\geq\frac{1}{2}$, where $\displaystyle
G^{\prime}$ is the tree obtained from $\displaystyle G$ by deleting the edge
incident with $\displaystyle v^{\ast}$ and joining $\displaystyle v^{\ast}$
and $\displaystyle v_{0}$,
* (3)
$\displaystyle\pi(G)-\pi(G^{\prime})<\frac{1}{2}$, where $\displaystyle
G^{\prime}$ is the tree as defined in (2).
Lemmas 3.6, 3.7 and 3.8 help in establishing the following result and verifies
a conjecture of [4].
###### Theorem 3.9 ([40]).
For a connected graph $\displaystyle G$ on $\displaystyle n\geq 3$ vertices,
$\displaystyle
ecc-\pi\leq\left\\{\begin{array}[]{lll}\frac{(3n+1)(n-1)}{4n}-\frac{n+1}{4}&&\mbox{
if $\displaystyle n$ is odd,}\\\ \frac{n-1}{2}-\frac{n}{4(n-1)}&&\mbox{ if
$\displaystyle n$ is even,}\end{array}\right.$
with equality if and only if $\displaystyle G$ is the path $\displaystyle
P_{n}$.
Sedlar [50] studied three AutoGraphiX conjectures involving proximity and
remoteness. She solved the conjecture by using the following graph
transformation for trees. Let $\displaystyle G$ be a tree. Then graph
transformtaion transforms $\displaystyle G$ into $\displaystyle G^{\prime}$,
where $\displaystyle G^{\prime}$ is either
* 1)
Path $\displaystyle P_{n}$,
* 2)
a tree consisting of four paths if equal length with a common end point,
* 2)
a tree consisting of three paths of almost equal length with a common end
point.
Next, the following results prove that among those graphs the difference
$\displaystyle\overline{\ell}-\pi$ is maximum for the last.
###### Lemma 3.10 ([50]).
The difference $\displaystyle\overline{\ell}-\pi$ is greater for a tree
$\displaystyle G$ on $\displaystyle n\geq 4 $ vertices consisting of three
paths of almost equal length with a common end point than for path
$\displaystyle P_{n}$.
###### Lemma 3.11 ([50]).
The difference $\displaystyle\overline{\ell}-\pi$ is greater for the tree
$\displaystyle G$ on $\displaystyle n\geq 9$ vertices, where $\displaystyle
n=1$mod$\displaystyle(4)$, consisting of three paths of almost equal length
with a common end point than for the tree $\displaystyle G^{\prime}$ on
$\displaystyle n$ vertices consisting of four paths of equal length, where
$\displaystyle G^{\prime}$ is tree obtained from $\displaystyle G$ by some
transformation.
###### Lemma 3.12 ([50]).
Let $\displaystyle G$ be a tree on $\displaystyle n\geq 6$ vertices with at
least four leafs (a vertex of degree $\displaystyle 1$ in a tree).Then there
is a tree $\displaystyle G^{\prime}$ on $\displaystyle n$ vertices with three
leafs for which the difference $\displaystyle\overline{\ell}-\pi$ is greater
or equal than for $\displaystyle G$.
###### Lemma 3.13 ([50]).
Among trees with three leafs, the difference
$\displaystyle\overline{\ell}-\pi$ is maximal for the tree $\displaystyle G$
on $\displaystyle n$ vertices consisting of three paths of almost equal length
with a common end vertex.
Lemmas 3.10, 3.11, 3.12 and 3.13 can be summarized into the following theorem.
###### Theorem 3.14 ([50]).
Among all trees on $\displaystyle n\geq 4~{}(n,\neq 5)$ vertices with average
distance $\displaystyle\overline{\ell}$ and proximity $\displaystyle\pi$, the
difference $\displaystyle\overline{\ell}-\pi$ is maximal for a tree
$\displaystyle G$ composed of three paths of almost equal lengths with a
common end vertex.
From Theorem 3.14, the following result follows and settles a conjecture
involving proximity and remoteness.
###### Theorem 3.15 ([50]).
Among all connected graphs $\displaystyle G$ on $\displaystyle n\geq 3$
vertices with average distance $\displaystyle\overline{\ell}$ and proximity
$\displaystyle\pi$, the difference $\displaystyle\overline{\ell}-\pi$ is
maximum for a graph $\displaystyle G$ composed of three paths of almost equal
lengths with a common end point.
Sedlar [50] also proved partial results related to another conjecture: while
the conjecture is stated for all connected graphs, the results are proved for
trees only.
###### Lemma 3.16 ([50]).
Let $\displaystyle G$ be a tree on $\displaystyle n$ vertices with diameter
$\displaystyle D$ and let $\displaystyle P=v_{0}v_{1}\dots v_{D}$ be a
diametric path in $\displaystyle G$. If there is $\displaystyle
j\leq\frac{D}{2}$ such that the degree of $\displaystyle v_{k}$ is at most
$\displaystyle 2$ for $\displaystyle k\geq j+1$, then the difference
$\displaystyle ecc-\rho$ is greater or equal for the path $\displaystyle
P_{n}$ than for $\displaystyle G.$
###### Theorem 3.17 ([50]).
Among all trees on $\displaystyle n\geq 3$ vertices, the difference
$\displaystyle ecc-\rho$ is maximum for the path $\displaystyle P_{n}$.
The following sequence of results are given by Sedlar [50], which relates
remotness $\displaystyle\rho$ wity radius $\displaystyle r.$
###### Lemma 3.18 ([50]).
Let $\displaystyle G$ be a tree on $\displaystyle n$ vertices. There is a
caterpillar tree $\displaystyle G^{\prime}$ on $\displaystyle n$ vertices for
which the difference $\displaystyle\rho-r$ is less or equal than for
$\displaystyle G$.
###### Lemma 3.19 ([50]).
Let $\displaystyle G\neq P_{n}$ be a caterpillar tree on $\displaystyle n$
vertices with diameter $\displaystyle D$,remoteness $\displaystyle\rho$ and
only one centroidal vertex. Let $\displaystyle P=v_{0}v_{1}\dots v_{D}$ be the
diametric path in $\displaystyle G$ such that $\displaystyle v_{j}\in P$ is
the only centroidal vertex in $\displaystyle G$ and every of the vertices
$\displaystyle v_{j+1},\dots,v_{D}$ is of the degree at most $\displaystyle
2$. Then there is a caterpillar tree $\displaystyle G^{\prime}$ on
$\displaystyle n$ vertices of the diameter $\displaystyle D+1$ and the
remoteness at most $\displaystyle\rho+\frac{1}{2}.$
###### Lemma 3.20 ([50]).
Let $\displaystyle G\neq P_{n}$ be a caterpillar tree on $\displaystyle n$
vertices with diameter $\displaystyle D$, remoteness and exactly two
centroidal vertices. Let $\displaystyle P=v_{0}v_{1}\dots v_{D}$ be a
diametric path in $\displaystyle G$ such that $\displaystyle v_{j},v_{j+1}\in
P$ are centroidal vertices and every of the vertices $\displaystyle
v_{j+1},\dots,v_{D}$ is of degree at most $\displaystyle 2$. Then there is a
caterpillar tree $\displaystyle G^{\prime}$ on $\displaystyle n$ vertices of
the diameter $\displaystyle D+1$ and the remoteness at most
$\displaystyle\rho+\frac{1}{2}.$
###### Lemma 3.21 ([50]).
Let $\displaystyle G\neq P_{n}$ be a caterpillar tree on $\displaystyle n$
vertices with diameter $\displaystyle D$, remoteness $\displaystyle\rho$ and
exactly two centroidal vertices of different degrees. Let $\displaystyle
P=v_{0}v_{1}\dots v_{D}$ be a diametric path in $\displaystyle G$ such that
$\displaystyle v_{j},v_{j+1}\in P$ are centroidal vertices and every of the
vertices $\displaystyle v_{0},\dots,v_{j-1},v_{j+2},\dots,v_{D}$ is of degree
at most $\displaystyle 2$. Then there is a caterpillar tree $\displaystyle
G^{\prime}$ on $\displaystyle n$ vertices of the diameter $\displaystyle D+1$
and the remoteness at most $\displaystyle\rho+\frac{1}{2} $.
###### Lemma 3.22 ([50]).
Let $\displaystyle G\neq P_{n}$ be a caterpillar tree on $\displaystyle n$
vertices with diameter $\displaystyle D$, remoteness $\displaystyle\rho$ and
exactly two centroidal vertices of equal degrees. Let $\displaystyle
P=v_{0}v_{1}\dots v_{D}$ be a diametric path in $\displaystyle G$ such that
$\displaystyle v_{j},v_{j+1}\in P$ are centroidal vertices and every of the
vertices $\displaystyle v_{0},\dots,v_{j-1},v_{j+2},\dots,v_{D}$ is of degree
at most $\displaystyle 2$. Then the difference $\displaystyle\rho-r$ is less
or equal for path $\displaystyle P_{n}$ than for $\displaystyle G.$
###### Lemma 3.23 ([50]).
Let $\displaystyle G$ be a caterpillar tree on $\displaystyle n$ vertices. If
$\displaystyle n$ is odd, then the difference $\displaystyle\rho-r$ is less or
equal for path $\displaystyle P_{n}$ then for $\displaystyle G$. If
$\displaystyle n$ is even, then the difference $\displaystyle\rho-r$ is less
or equal for path $\displaystyle P_{n}-1$ with a leaf appended to a central
vertex than for G.
Lemmas 3.18, 3.19, 3.20, 3.21, 3.22 and 3.23 can be summarized in the
following result, which gives minimal trees for $\displaystyle\rho-r.$
###### Theorem 3.24 ([50]).
Let $\displaystyle G$ be a tree on $\displaystyle n$ vertices. If
$\displaystyle n$ is odd, then the difference $\displaystyle\rho-r$ is less or
equal for path $\displaystyle P_{n}$ then for $\displaystyle G$. If
$\displaystyle n$ is even, then the difference $\displaystyle\rho-r$ is less
or equal for path $\displaystyle P_{n-1}$ with a leaf appended to a central
vertex than for $\displaystyle G$.
Let $\displaystyle G$ be a connected graph. A vertex $\displaystyle u\in V(G)$
is called a peripheral vertex if $\displaystyle\sigma(u)=\rho(G)$. For a
vertex $\displaystyle u\in V(G),$ let $\displaystyle V_{i}(u)=\\{v\in
V(G)|d(u,v)=i\\}$ and $\displaystyle n_{i}(u)=|V_{i}(u)|$ for each
$\displaystyle i\in\\{1,2,\dots,d\\},$where $\displaystyle d=e_{G}(u).$ In
what follows, $\displaystyle V_{i}(u)$ is simply denoted by $\displaystyle
V_{i}$ for a peripheral vertex $\displaystyle u$ of $\displaystyle G.$ Wu and
Zhang [56] proved some lemmas and two theorems, first conjectured in [4].
###### Lemma 3.25 ([56]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n\geq 3.$
Let $\displaystyle u$ be a peripheral vertex of $\displaystyle G$ and let
$\displaystyle d=e_{G}(u).$ Let $\displaystyle G^{\prime}$ be the graph
obtained from $\displaystyle G$ by joining each pair of all nonadjacent
vertices $\displaystyle x,y$ of $\displaystyle G,$ where $\displaystyle x,y\in
V_{j}\cup V_{j+1}$ for some $\displaystyle j\in\\{1,2,\dots,d-1\\}$. We have
$\displaystyle\rho(G^{\prime})-\overline{\ell}(G^{\prime})\geq\rho(G)-\overline{\ell}(G),$
with equality if and only if $\displaystyle G^{\prime}=G.$
###### Lemma 3.26 ([56]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n\geq 3$.
Let $\displaystyle u$ be a peripheral vertex of $\displaystyle G$ and
$\displaystyle e_{G}(v)=d.$ Assume that $\displaystyle G[V_{j}\cup V_{j+1}]$
is a clique for each $\displaystyle j\in\\{0,1,\dots,d-1\\}$. Let
$\displaystyle G^{\prime}$ be the graph with $\displaystyle
V(G^{\prime})=V(G)$ and $\displaystyle E(G^{\prime})=E(G)\cup[xy:x\in
V_{d},y\in V_{d}.$ If $\displaystyle
d>\left\lfloor\frac{n+1}{2}\right\rfloor,$ then
$\displaystyle\rho(G^{\prime})-\overline{\ell}(G^{\prime})\leq\rho(G)-\overline{\ell}(G),$
with equality if and only if $\displaystyle n$ is even and $\displaystyle
d=\frac{n}{2}+11.$
###### Lemma 3.27 ([56]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n\geq 3$.
Let $\displaystyle u$ be a peripheral vertex of $\displaystyle G$ and
$\displaystyle e_{G}(v)=d.$ Assume that $\displaystyle G[V_{j}\cup V_{j+1}]$
is a clique for each $\displaystyle j\in\\{0,1,\dots,d-1\\}$. Let
$\displaystyle i$ be the smallest integer in $\displaystyle\\{1,2,\dots,d\\}$
such that $\displaystyle n_{i}(u)\geq 2.$ Let $\displaystyle
V_{i-1}(u)=\\{u_{i-1}\\}$ and $\displaystyle v$ a vertex in $\displaystyle
V_{i}(u)$. Denote by $\displaystyle G^{\prime}$ the graph with $\displaystyle
V(G^{\prime})=V(G)$ and $\displaystyle
E(G^{\prime})=E(G)\setminus(\\{u_{i-1}y:y\in V_{i}\setminus\\{v\\}\\}\cup A),$
where $\displaystyle A=\\{vx:x\in V_{i+1}\\},$ if $\displaystyle i\leq d-1$,
and $\displaystyle A=\emptyset$ otherwise. If $\displaystyle
d<\left\lfloor\frac{n+1}{2}\right\rfloor$, then
$\displaystyle\rho(G^{\prime})-\overline{\ell}(G^{\prime})>\rho(G)-\overline{\ell}(G).$
###### Lemma 3.28 ([56]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n\geq 3.$
Let $\displaystyle u$ be a peripheral vertex of $\displaystyle G$ and
$\displaystyle e_{G}(v)=d.$ Assume that $\displaystyle G[V_{j}\cup V_{j+1}]$
is a clique for each $\displaystyle j\in\\{0,1,\dots,d-1\\}$ and that
$\displaystyle n_{i}(u)\geq 2$ for some $\displaystyle
i\in\\{1,2,\dots,d-1\\}$. Further, assume that $\displaystyle i$ is the
minimum subject to the above condition. Let $\displaystyle v$ be a vertex in
$\displaystyle V_{i}(u)$ and $\displaystyle V_{i-1}=\\{u_{i-1}\\}$. Let
$\displaystyle G^{\prime}$ be the graph with $\displaystyle
V(G^{\prime})=V(G)$ and $\displaystyle E(G^{\prime})=E(G)\cup
A\setminus\\{vu_{i-1}\\},$ where $\displaystyle A=\\{vy:y\in V_{i+2}\\}$ if
$\displaystyle i\leq d-2$, and $\displaystyle A=\emptyset$ otherwise. If
$\displaystyle d=\left\lfloor\frac{n+1}{2}\right\rfloor,$ then
$\displaystyle\rho(G^{\prime})-\overline{\ell}(G^{\prime})>\rho(G)-\overline{\ell}(G).$
A Soltés or path-complete graph [53] is the graph obtained from a clique and a
path by adding at least one edge between an endpoint of the path and the
clique. The Soltés graphs are known to maximize the average distance
$\displaystyle\overline{\ell}$ when the number of vertices and of edges are
fixed [53].
Lemmas 3.25, 3.26, 3.27 and 3.28 lead the following result.
###### Theorem 3.29 ([56]).
Among all connected graphs $\displaystyle G$ on $\displaystyle n\geq 3$
vertices with average distance $\displaystyle\overline{\ell}$ and remoteness
$\displaystyle\rho$, the Soltés graphs with diameter
$\displaystyle\lfloor(n+1)/2\rfloor$ maximize the difference
$\displaystyle\rho-\overline{\ell}$.
Theorem 3.29 can be equivalently states as
###### Theorem 3.30 ([56]).
Among all connected graphs $\displaystyle G$ on $\displaystyle n\geq 3$
vertices with average distance $\displaystyle\overline{\ell}$ and remoteness
$\displaystyle\rho$, the maximum value of $\displaystyle\rho-\overline{\ell}$
is attained by the Soltés graphs with diameter $\displaystyle D$, where
$\displaystyle\begin{cases}D=\frac{n+1}{2},&\text{if $\displaystyle n$ is
odd}\\\ D\in\\{\frac{n}{2},\frac{n}{2}+1\\}&\text{if $\displaystyle n$ is
even}.\end{cases}$
Wu and Zhang [56] proved the following lemma, which helped them in proving
Theorem 3.32, first conjectured in [4].
###### Corollary 3.31 ([56]).
Let $\displaystyle G$ be connected graph with order $\displaystyle n\geq 5$.
If $\displaystyle n$ is odd and $\displaystyle r=\frac{n-1}{2}$. Then
$\displaystyle\rho\geq\dfrac{n+1}{4}$, with equality if and only if
$\displaystyle G$ is the cycle $\displaystyle C_{n}$ or the graph composed of
the cycle $\displaystyle C_{n}$ together with two crossed edges on four
successive vertices of the cycle.
###### Theorem 3.32 ([56]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n\geq 3$ vertices
with remoteness $\displaystyle\rho$ and radius $\displaystyle r$. Then
$\displaystyle\rho-r\geq\left\\{\begin{array}[]{lll}\frac{3-n}{4}&&\mbox{ if
$\displaystyle n$ is odd }\\\ \frac{n^{2}}{4n-4}-\frac{n}{2}&&\mbox{ if
$\displaystyle n$ is even.}\end{array}\right.$
The inequality is best possible as shown by the cycle $\displaystyle C_{n}$ if
$\displaystyle n$ is even and by the graph composed of the cycle
$\displaystyle C_{n}$ together with two crossed edges on four successive
vertices of the cycle.
Figure 7: Graphs $\displaystyle C_{n}^{1},C_{n}^{2}$ and $\displaystyle
C_{n}^{1}$ (n is odd)
Hau, Chen and Das [31] obtained the folllowing result, earlier conjurectured
in [4].
###### Theorem 3.33 ([31]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n\geq 3$ vertices
with remoteness $\displaystyle\rho$ and radius $\displaystyle r$. If
$\displaystyle n$ is odd, then $\displaystyle\rho-r\geq\frac{3-n}{4}$ with
equality if and only if $\displaystyle G\cong Cn$ or $\displaystyle
C_{n}^{i},~{}i=1,2,3$ (see Fig. 7), and if $\displaystyle n$ is even, then
$\displaystyle\rho-r\geq\frac{2n-n^{2}}{4(n-1)}$ with equality if and only if
$\displaystyle G\cong Cn.$
Dankelmann related proximity and remoteness with minimum degree and gave
several intresting results [18].
###### Theorem 3.34 ([18]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n$ and
minimum degree $\displaystyle\delta$, where $\displaystyle\delta\geq 2$. Then
there exists a spanning tree $\displaystyle T$ of $\displaystyle G$ with
$\displaystyle\pi(T)\leq\frac{3n}{4(\delta+1)}+3$
and
$\displaystyle\rho(T)\leq\frac{3n}{2(\delta+1)}+\frac{7}{2}.$
The following is an immediate consequence of Theorem 3.34
###### Corollary 3.35 ([18]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n$ and
minimum degree $\displaystyle\delta$, where $\displaystyle\delta\geq 2$. Then
$\displaystyle\pi(T)\leq\frac{3n}{4(\delta+1)}+3$
and
$\displaystyle\rho(T)\leq\frac{3n}{2(\delta+1)}+\frac{7}{2}.$
Next, we have result given upper bound for $\displaystyle\rho-\pi$ in terms of
order $\displaystyle n$ and minimum degree $\displaystyle\delta.$
###### Theorem 3.36 ([18]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n$ and
minimum degree $\displaystyle\delta$, where $\displaystyle\delta\geq 2$. Then
$\displaystyle\rho-\pi\leq\frac{3}{4(\delta+1)}+3.$
Dankelmann [19] obtained some new bounds on proximity and remoteness.
###### Theorem 3.37 ([19]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n$ and
minimum degree $\displaystyle\delta$, where $\displaystyle n\geq 20$ and
$\displaystyle\delta\geq 2$. Then
$\displaystyle D-\pi\leq\frac{9n}{4(\delta+1)}+\frac{3\delta}{4}.$
Next result gives sharp bound for remoteness in terms of diameter and order od
graph
###### Proposition 3.38 ([19]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n$ and
diameter $\displaystyle D$. Then
$\displaystyle\rho\geq\frac{nd}{2(n-1)},$
and this bound is sharp for all $\displaystyle n$ and $\displaystyle D$ with
$\displaystyle n\geq D+1\geq 33$ for which $\displaystyle nD$ is even.
###### Corollary 3.39 ([19]).
Let $\displaystyle G$ be a connected graph of diameter $\displaystyle D.$ Then
$\displaystyle\rho>\frac{D}{2},$
and the coefficient $\displaystyle\frac{1}{2}$ is best possible.
###### Theorem 3.40 ([19]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n$ and
minimum degree $\displaystyle\delta$, where
$\displaystyle\delta<\frac{n}{4}-1.$ Then
$\displaystyle r-\pi\leq\frac{3n}{4(\delta+1)}+\frac{8\delta+5}{4(\delta+1)},$
and this bound is best possible, apart from an additive constant.
###### Corollary 3.41 ([19]).
Let $\displaystyle G$ be a connected graph. Then
$\displaystyle\rho>\frac{r}{2}.$
The girth $\displaystyle g$ of a graph $\displaystyle G$ is the length of its
smallest cycle.
A lollipop $\displaystyle L_{n,g}$ is the graph obtained from a cycle
$\displaystyle C_{g}$ and a path $\displaystyle P_{n-g}$ by adding an edge
between an endpoint of $\displaystyle P_{n-g}$ and a vertex of the cycle
$\displaystyle C_{g}$. The lollipop $\displaystyle L_{11,7}$ is illustrated in
Figure 9. For a lollipop $\displaystyle L_{n,g}$, we have
$\displaystyle\rho(L_{n,g})=\left\\{\begin{array}[]{ccc}\frac{n}{2}-\frac{g(g-2)}{4(n-1)}&&\mbox{
if $\displaystyle g$ is even }\\\ \frac{n}{2}-\frac{g(g-2)+1}{4(n-1)}&&\mbox{
if $\displaystyle g$ is odd. }\end{array}\right.$
A turnip $\displaystyle T_{n,g}$, with $\displaystyle n\geq g\geq 3$, is the
graph obtained from a cycle $\displaystyle C_{g}$ by attaching $\displaystyle
n-p$ pending edges to one vertex from the cycle. The turnip $\displaystyle
T_{10,5}$ is illustarated in Figure 9. If $\displaystyle g=n$, the turnip
$\displaystyle T_{n,g}=T_{n,n}$ is the cycle $\displaystyle C_{n}$. For a
turnip $\displaystyle T_{n,g}$, we have
$\displaystyle\pi(T_{n,g})=\left\\{\begin{array}[]{ccl}\frac{g^{2}-4g+4n-1}{4(n-1)}&&\mbox{
if $\displaystyle g$ is odd}\\\ \frac{g^{2}-4g+4n}{4(n-1)}&&\mbox{ if
$\displaystyle g$ is even.}\end{array}\right.$
Figure 8: The lollipop $\displaystyle L_{11,7}$.
Figure 9: The turnip $\displaystyle T_{10,5}$.
###### Lemma 3.42 ([10]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n\geq 3$ vertices
with girth $\displaystyle g$.
$\displaystyle(i)$ If $\displaystyle g=3$, then $\displaystyle\pi\geq 1$ with
equality if and only if $\displaystyle G$ contains a dominating vertex.
$\displaystyle(ii)$ If $\displaystyle g=4$, then $\displaystyle\pi\geq
n/(n-1)$ with equality if and only if $\displaystyle G$ contains the turnip
$\displaystyle T_{n,4}$ as a spanning subgraph and is a spanning subgraph of
the complete bipartite graph $\displaystyle K_{n-2,2}$.
$\displaystyle(iii)$ If $\displaystyle g\geq 5$, then
$\displaystyle\pi\geq\pi(T_{n,g})$ with equality if and only if $\displaystyle
G$ is the turnip $\displaystyle T_{n,g}$.
###### Theorem 3.43 ([10]).
For any connected graph $\displaystyle G$ on $\displaystyle n\geq 3$ vertices
with a finite girth $\displaystyle g$ and proximity $\displaystyle\pi$, we
have
$\left.\begin{array}[]{lr}\mbox{if $\displaystyle n$ is odd,
}&\frac{1-3n}{4}\\\ \mbox{if $\displaystyle n$ is even,
}&\frac{4n-3n^{2}}{4n-4}\end{array}\right\\}\leq\pi-g\leq\left\\{\begin{array}[]{ll}\frac{n-11}{4}-\frac{1}{n-1}&\mbox{if
$\displaystyle n$ is odd, }\\\ \frac{n-11}{4}-\frac{3}{4(n-1)}&\mbox{if
$\displaystyle n$ is even; }\end{array}\right.$ (2)
$4\leq\pi+g\leq\left\\{\begin{array}[]{lr}\frac{5n+1}{4}&\mbox{if
$\displaystyle n$ is odd, }\\\ \frac{5n^{2}-4n}{4(n-1)}&\mbox{if
$\displaystyle n$ is even; }\end{array}\right.$ (3)
$\frac{1}{2\left\lfloor\sqrt{n}\right\rfloor+1}+\frac{\left\lfloor\sqrt{n}\right\rfloor(\left\lfloor\sqrt{n}\right\rfloor-1)}{(2\left\lfloor\sqrt{n}\right\rfloor+1)(n-1)}\leq\frac{\pi}{g}\leq\left\\{\begin{array}[]{ll}\frac{n^{2}-4}{12n-12}&\mbox{if
$\displaystyle n$ is even, }\\\ \frac{n+1}{12}-\frac{1}{3n-3}&\mbox{if
$\displaystyle n$ is odd; }\end{array}\right.$ (4) $3\leq\pi\cdot
g\leq\left\\{\begin{array}[]{lr}\frac{n^{2}+n}{4}&\mbox{if $\displaystyle n$
is odd, }\\\ \frac{n^{3}}{4(n-1)}&\mbox{if $\displaystyle n$ is even;
}\end{array}\right.$ (5)
The lower bound in (2) and the upper bounds in (3) and (5) are reached if and
only if $\displaystyle G$ is the cycle $\displaystyle C_{n}$. The upper bounds
in (2) and (4) are reached if and only if $\displaystyle G$ is the lollipop
$\displaystyle L_{n,3}$. The lower bounds in (3) and (5) are reached if and
only if $\displaystyle G$ contains a dominating vertex. The lower bound in (4)
is reached if and only if $\displaystyle G$ is the turnip $\displaystyle
T_{n,s}$, where $\displaystyle s=2\left\lfloor\sqrt{n}\right\rfloor+1$ when
$\displaystyle\sqrt{n}$ is not an integer, and if and only if $\displaystyle
G$ is any one of the turnips $\displaystyle T_{n,2\sqrt{n}-1}$, $\displaystyle
T_{n,2\sqrt{n}}$ or $\displaystyle T_{n,2\sqrt{n}+1}$ when
$\displaystyle\sqrt{n}$ is an integer.
###### Lemma 3.44 ([10]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n\geq 4$ vertices
with girth $\displaystyle g\leq n-1$ and remoteness $\displaystyle\rho$. Then
$\displaystyle\rho\leq\rho(L_{n,g})$
with equality if and only if $\displaystyle G$ is the lollipop $\displaystyle
L_{n,g}$.
###### Theorem 3.45 ([10]).
For any connected graph $\displaystyle G$ on $\displaystyle n\geq 3$ vertices
with remoteness $\displaystyle\rho$ and girth $\displaystyle g$, we have
$\left.\begin{array}[]{lr}\mbox{if $\displaystyle n$ is even,
}&\frac{4n-3n^{2}}{4n-4}\\\ \mbox{if $\displaystyle n$ is odd,
}&\frac{1-3n}{4}\end{array}\right\\}\leq\rho-g\leq\frac{(n+1)(n-2)}{2n-2}-3;$
(6) $4\leq\rho+g\leq\left\\{\begin{array}[]{ll}\frac{5n^{2}-4n}{4n-4}&\mbox{if
$\displaystyle n$ is even, }\\\ \frac{5n+1}{4}&\mbox{if $\displaystyle n$ is
odd; }\end{array}\right.$ (7) $\frac{\rho}{g}\leq\frac{(n+1)(n-2)}{6n-6};$ (8)
$3\leq\rho\cdot g\leq\rho(L_{n,g^{*}})\cdot g^{*}$ (9)
where $\displaystyle g^{*}$ is the girth for which
$\displaystyle\rho(L_{n,g_{i}})\cdot g_{i}$, $\displaystyle i=1,\ldots,4$, is
maximum with $\displaystyle
g_{1}=\left\lfloor\frac{2+\sqrt{6n^{2}-6n+4}}{3}\right\rfloor$, $\displaystyle
g_{2}=\left\lceil\frac{2+\sqrt{6n^{2}-6n+4}}{3}\right\rceil$, $\displaystyle
g_{3}=\left\lfloor\frac{2+\sqrt{6n^{2}-6n+7}}{3}\right\rfloor$ and
$\displaystyle g_{4}=\left\lceil\frac{2+\sqrt{6n^{2}-6n+7}}{3}\right\rceil$.
The lower bound in (6) and the upper bound in (7) are reached if and only if
$\displaystyle G$ is the cycle $\displaystyle C_{n}$. The upper bounds in (6)
and (8) are reached if and only if $\displaystyle G$ is the lollipop
$\displaystyle L_{n,3}$. The lower bounds in (7) and (9) are reached if and
only if $\displaystyle G$ is the complete graph $\displaystyle K_{n}$. The
upper bound in (9) is reached if and only if $\displaystyle G$ is the lollipop
$\displaystyle L_{n,g^{*}}$.
The lower bound on $\displaystyle\rho/g$, first conjectured using AGX [1], was
proved in [30].
###### Theorem 3.46 ([30]).
For any connected graph $\displaystyle G$ on $\displaystyle n\geq 3$ vertices
with remoteness $\displaystyle\rho$ and girth $\displaystyle g$,
$\displaystyle\frac{\rho}{g}\geq\left\\{\begin{array}[]{ll}\frac{n}{4n-4}&\mbox{if
$\displaystyle n$ is even, }\\\ \frac{n+1}{4n}&\mbox{if $\displaystyle n$ is
odd. }\end{array}\right.$
with equality if and only if $\displaystyle G$ is a cycle $\displaystyle
C_{n}$.
###### Theorem 3.47 ([30]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n\geq 2$ vertices
with proximity $\displaystyle\pi$ and average distance
$\displaystyle\overline{\ell}$. Then
$\displaystyle\frac{\pi}{\overline{\ell}}\geq\frac{n}{2(n-1)},$
with equality if and only if $\displaystyle G$ is isomorphic to the star
$\displaystyle S_{n}$.
The following is an immediate consequence of Theorem 3.47.
###### Corollary 3.48 ([30]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n\geq 2$ vertices
with proximity $\displaystyle\pi$, average degree $\displaystyle\overline{d}$
and average distance $\displaystyle\overline{\ell}$. Then
$\displaystyle\pi.\overline{d}\geq\overline{\ell},$
with equality if and only if $\displaystyle G$ is isomorphic to the star
$\displaystyle S_{n}$.
Dankelmann and Mafunda [21] gave results relating $\displaystyle\pi$ with
diameter and radius of triangle-free and $\displaystyle C_{4}$-free graphs.
###### Theorem 3.49 ([21]).
If $\displaystyle G$ is a connected, triangle-free graph of order
$\displaystyle n\geq 8$, minimum degree $\displaystyle\delta\geq 3$ and
diameter $\displaystyle D$, then
$\displaystyle\pi\geq\frac{\delta(D-4)(D-1)}{8(n-1)}.$
###### Corollary 3.50 ([21]).
If $\displaystyle G$ is a connected, triangle-free graph of order
$\displaystyle n\geq 8$ and minimum degree $\displaystyle\delta\geq 3$, then
$\displaystyle D-\pi\leq\frac{3(n-1)}{2\delta}+\frac{5}{2}.$
This bound is sharp apart from an additive constant.
###### Theorem 3.51 ([21]).
If $\displaystyle G$ is a connected, triangle-free graph of order
$\displaystyle n\geq 6$, minimum degree $\displaystyle\delta\geq 3$ and radius
$\displaystyle r\geq 1$, then
$\displaystyle\pi\geq\frac{\delta}{2(n-1)}\left(r^{2}-7r+\frac{47}{8}\right).$
###### Corollary 3.52 ([21]).
If $\displaystyle G$ is a connected, triangle-free graph of order
$\displaystyle n\geq 6$ and minimum degree $\displaystyle\delta\geq 3$, then
$\displaystyle r-\pi\leq\frac{n-1}{2\delta}+\frac{11}{2}.$
This bound is sharp apart from an additive constant.
###### Theorem 3.53 ([21]).
If $\displaystyle G$ is a connected, $\displaystyle C_{4}$-free graph of order
$\displaystyle n\geq 16$ and minimum degree $\displaystyle\delta\geq 3$, then
$\displaystyle\pi\geq\frac{\delta^{2}-2\left\lfloor\frac{\delta}{2}\right\rfloor+1}{5(n-1)}\left(r^{2}-8r+\frac{127}{8}\right).$
###### Corollary 3.54 ([21]).
If $\displaystyle G$ is a connected, $\displaystyle C_{4}$-free graph of order
$\displaystyle n\geq 16$ and minimum degree $\displaystyle\delta\geq 3$, then
$\displaystyle
r-\pi\leq\frac{5(n-1)}{4\left(\delta^{2}-2\left\lfloor\frac{\delta}{2}\right\rfloor+1\right)}+4.$
Czabarka et al. [17] gave the upper bounds on $\displaystyle\pi$ in
triangulations and quadrangulations.
The plane graph $\displaystyle G$ is a triangulation (resp. quadrangulation)
if every face is a triangle (resp. 4-cycle).
###### Proposition 3.55 ([17]).
* (a)
Let $\displaystyle G$ be a $\displaystyle 5$-connected triangulation of order
$\displaystyle n$. Then
$\displaystyle\rho\leq\frac{n+4}{10}+\epsilon_{n},$
where $\displaystyle\epsilon_{n}=-\frac{3}{5(n-1)}$ if $\displaystyle n\equiv
0$ (mod $\displaystyle 5$), $\displaystyle\epsilon_{n}=-\dfrac{1}{n-1}$ if
$\displaystyle n\equiv 1$ (mod $\displaystyle 5$),
$\displaystyle\epsilon_{n}=\dfrac{2}{5(n-1)}$ if $\displaystyle n\equiv 2$
(mod $\displaystyle 5$), and $\displaystyle\epsilon_{n}=-\frac{2}{5(n-1)}$ if
$\displaystyle n\equiv 3,4$ (mod $\displaystyle 5$).
* (b)
If $\displaystyle G$ is a $\displaystyle 3$-connected quadrangulation of order
$\displaystyle n$, then
$\displaystyle\rho\leq\frac{n+2}{6}+\epsilon_{n},$
where $\displaystyle\epsilon_{n}=-\frac{5}{3(n-1)}$ if $\displaystyle n\equiv
0$ (mod $\displaystyle 3$), $\displaystyle\epsilon_{n}=-\frac{1}{n-1}$ if
$\displaystyle n\equiv 1$ (mod $\displaystyle 3$), and
$\displaystyle\epsilon_{n}=\dfrac{1}{3(n-1)}$ if $\displaystyle n\equiv 2$(mod
$\displaystyle 3$).
## 4 Proximity and Remoteness Compared to other Invariants
Aouchiche and Hasen [4] related remoteness with the independence number and
presented the following results.
###### Proposition 4.1 ([4]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n\geq 8$ vertices
with remoteness $\displaystyle\rho$ and independence number
$\displaystyle\alpha$. Then
$\displaystyle\rho-\alpha\geq 3-n-\frac{1}{n-1}$
and
$\displaystyle\rho-\alpha\leq\left\\{\begin{array}[]{lcl}\frac{n-3}{8}-\frac{3}{8(n-1)}&&\mbox{
if } n=0\,\,(mod\,\,4),\\\ \frac{n-3}{8}&&\mbox{ if } n=1\,\,(mod\,\,2),\\\
\frac{n-3}{8}+\frac{1}{8(n-1)}&&\mbox{ if }
n=2\,\,(mod\,\,4).\end{array}\right.$
The lower bound is attained if and only if $\displaystyle G$ is the star
$\displaystyle S_{n}$. The bound is best possible as shown by the kites
$\displaystyle KI_{n,n_{0}}$ where $\displaystyle n_{0}=(n+n\,\,(mod\,\,
4))/2$.
###### Proposition 4.2 ([4]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n\geq 3$ with
remoteness $\displaystyle\rho$ and matching number $\displaystyle\mu$. Then
$\displaystyle\rho-\mu\leq\left\\{\begin{array}[]{lll}\frac{n+1}{8}+\frac{1}{8(n-1)}&&\mbox{
if }n=0\,\,(mod\,\, 4),\\\ \frac{n+1}{8}&&\mbox{ if }n=1\,\,(mod\,\, 4),\\\
\frac{n+1}{8}-\frac{3}{8(n-1)}&&\mbox{ if }n=2\,\,(mod\,\, 4),\\\
\frac{n+1}{8}&&\mbox{ if }n=3\,\,(mod\,\, 4),\\\ \end{array}\right.$
with equality if and only if $\displaystyle G$ is the comet $\displaystyle
CO_{n,n-D^{*}+1}$ where $\displaystyle
D^{*}=2\left\lfloor\frac{n+2}{4}\right\rfloor$.
The (vertex) connectivity $\displaystyle\nu$ of a connected graph
$\displaystyle G$ is the minimum number of vertices whose removal disconnects
$\displaystyle G$ or reduces it to a single vertex. The algebraic connectivity
$\displaystyle a$ of a graph $\displaystyle G$ is the second smallest
eigenvalue of its Laplacian $\displaystyle L=Diag-A$, where $\displaystyle
Diag$ is the diagonal square matrix indexed by the vertices of $\displaystyle
G$ whose diagonal entries are the degrees in $\displaystyle G$, and
$\displaystyle A$ is the adjacency matrix of $\displaystyle G$.
###### Theorem 4.3 ([49]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n\geq 2$ vertices
with vertex connectivity $\displaystyle\nu$, algebraic connectivity
$\displaystyle a$ and remoteness $\displaystyle\rho$. Then
$\displaystyle\nu\cdot\rho\leq n-1$
with equality if and only if $\displaystyle G$ is the complete graph
$\displaystyle K_{n}$; and
$\displaystyle a\cdot\rho\leq n$
with equality if and only if $\displaystyle G$ is the complete graph
$\displaystyle K_{n}$. Moreover, if $\displaystyle G$ is not complete, then
$\displaystyle a\cdot\rho\leq n-1-\frac{1}{n-1}$
with equality if and only if $\displaystyle G\cong K_{n}-M$, where
$\displaystyle M$ is any non empty set of disjoint edges.
###### Theorem 4.4 ([30]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n\geq 3$ vertices
with remoteness $\displaystyle\rho$ and maximum degree
$\displaystyle\triangle$. If
$\displaystyle\triangle\geq\left\lceil\frac{n}{4}\right\rceil+1,$ then
$\displaystyle\rho+\triangle\geq\begin{cases}\frac{n+9}{4}&\text{if
$\displaystyle n$ is odd},\\\ 2+\frac{n^{2}}{4n-4}&\text{if $\displaystyle n$
is even},\end{cases}$
with equality if and only if $\displaystyle G\cong C_{3}$ or $\displaystyle
G\cong C_{4}.$
###### Theorem 4.5 ([30]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n$ vertices,
$\displaystyle m$ edges with proximity $\displaystyle\pi$ and average degree
$\displaystyle\overline{d}$. If $\displaystyle m\leq\frac{2(n-1)^{2}}{n},$
then
$\displaystyle\pi.\overline{d}\leq n-1.$
###### Theorem 4.6 ([30]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n\geq 3$ vertices
with clique number $\displaystyle\omega$ and remoteness $\displaystyle\rho$.
Then
$\displaystyle\rho\leq\frac{n^{2}-\omega^{2}-n+3\omega-2}{2(n-1)},$
with equality if and only if $\displaystyle G\cong Ki_{n,\omega}.$
###### Theorem 4.7 ([30]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n\geq 3$ vertices
with clique number $\displaystyle\omega$, remoteness $\displaystyle\rho$ and
proximity $\displaystyle\pi$. Then
$\displaystyle\rho+\pi\leq\begin{cases}\frac{3n^{2}-2\omega^{2}-2n+6\omega-5}{4(n-1)}&\text{if
$\displaystyle n$ is odd},\\\
\frac{3n^{2}-2\omega^{2}-2n+6\omega-4}{4(n-1)}&\text{if $\displaystyle n$ is
odd},\end{cases}$
with equality if and only if $\displaystyle G\cong P_{n}.$
The next result gives an upper bound on $\displaystyle\rho-\pi$ in terms of
$\displaystyle n$ and $\displaystyle\omega.$
###### Theorem 4.8 ([30]).
Let $\displaystyle G$ be a connected graph on $\displaystyle n\geq 3$ vertices
with clique number $\displaystyle\omega$, remoteness $\displaystyle\rho$ and
proximity $\displaystyle\pi$. Then
$\displaystyle\rho-\pi\leq\frac{(n-\omega)(n+\omega-3)}{2(n-1)},$
with equality if and only if $\displaystyle G\cong Ki_{n,n-1}.$
###### Theorem 4.9 ([30]).
Let $\displaystyle G$ be a connected bipartite graph with each partite set of
cardinality $\displaystyle n\geq 2$. If the average distance
$\displaystyle\overline{\ell}\leq\frac{3}{2}+\frac{n-2}{2n(2n-1)},$
then $\displaystyle G$ is Hamiltonion.
###### Corollary 4.10 ([30]).
Let $\displaystyle G$ be a connected bipartite graph with each partite set of
cardinality $\displaystyle n\geq 2$. If
$\displaystyle\rho\leq\frac{3}{2}+\frac{n-2}{2n(2n-1)},$
then $\displaystyle G$ is Hamiltonion.
The distance eigenvalues of a connected graph, denoted by
$\displaystyle\partial_{1},\partial_{2},\ldots,\partial_{n}$, are those of its
distance matrix, and are indexed such that
$\displaystyle\partial_{1}\geq\partial_{2}\geq\ldots\geq\partial_{n}$. For a
survey on distance spectra of graphs see [8]. Next, we have results relating
proximity $\displaystyle\pi$ and remoteness $\displaystyle\rho$ with the
distance eigenvalues of graphs.
###### Theorem 4.11 ([9]).
Let $\displaystyle G$ be a graph on $\displaystyle n\geq 4$ vertices with
largest distance eigenvalue $\displaystyle\partial_{1}$, proximity
$\displaystyle\pi$ and remoteness $\displaystyle\rho$. Then
$\displaystyle\pi\leq\overline{\ell}\leq\frac{\partial_{1}}{n-1}\leq\rho$
with equalities if and only if $\displaystyle G$ is a transmission regular
graph.
###### Corollary 4.12 ([9]).
Let $\displaystyle G$ be a graph on $\displaystyle n\geq 2$ vertices with
largest distance eigenvalue $\displaystyle\partial_{1}$ and proximity
$\displaystyle\pi$. Then
$\displaystyle\partial_{1}-\pi\geq n-2$
with equalities if and only if $\displaystyle G$ is the complete graph
$\displaystyle K_{n}$.
###### Corollary 4.13 ([9]).
Let $\displaystyle G$ be a graph on $\displaystyle n\geq 4$ vertices with
second largest distance eigenvalue $\displaystyle\partial_{2}$ and remoteness
$\displaystyle\rho$. Then
$\displaystyle\rho+\partial_{2}\geq 0$
with equality if and only if $\displaystyle G$ is the complete graph
$\displaystyle K_{n}$.
The bound in the above corollary is best possible among the bounds of the form
$\displaystyle\rho+\partial_{k}\geq 0$, with a fixed integer $\displaystyle
k$, over the class of all connected graphs. Indeed, if we consider the
complete bipartite graphs $\displaystyle K_{\left\lfloor
n/2\right\rfloor,\left\lceil n/2\right\rceil}$, on $\displaystyle n\geq 3$, by
direct calculation we get
$\displaystyle\rho(K_{\left\lfloor n/2\right\rfloor,\left\lceil
n/2\right\rceil})+\partial_{3}(K_{\left\lfloor n/2\right\rfloor,\left\lceil
n/2\right\rceil})=\left\\{\begin{array}[]{lcl}-\frac{1}{2}&&\mbox{ if
$\displaystyle n$ is odd,}\\\ \\\ -\frac{1}{2}-\frac{1}{2(n-1)}&&\mbox{ if
$\displaystyle n$ is even,}\\\ \end{array}\right.$
which is negative for $\displaystyle n\geq 3$.
###### Proposition 4.14 ([9]).
Let $\displaystyle T$ be a tree on $\displaystyle n\geq 4$ vertices with
remoteness $\displaystyle\rho$, diameter $\displaystyle D$ and distance
spectrum
$\displaystyle\partial_{1}\geq\partial_{2}\geq\cdots\geq\partial_{n}$. Then
$\displaystyle\rho+\partial_{\left\lfloor\frac{D}{2}\right\rfloor}>0.$
###### Corollary 4.15 ([9]).
Let $\displaystyle G$ be a graph on $\displaystyle n\geq 4$ vertices with
second largest distance eigenvalue $\displaystyle\partial_{2}$ and proximity
$\displaystyle\pi$. Then
$\displaystyle\pi+\partial_{n}\leq 0$
with equality if and only if $\displaystyle G$ is the complete graph
$\displaystyle K_{n}$.
###### Theorem 4.16 ([9]).
Let $\displaystyle G$ be a graph on $\displaystyle n\geq 4$ vertices with
largest distance eigenvalue $\displaystyle\partial_{1}$ and remoteness
$\displaystyle\rho$. Then
$\displaystyle\partial_{1}-\rho\geq n-2$
with equalities if and only if $\displaystyle G$ is the complete graph
$\displaystyle K_{n}$.
###### Proposition 4.17 ([9]).
Let $\displaystyle G$ be a graph on $\displaystyle n\geq 4$ vertices with
least distance eigenvalue $\displaystyle\partial_{n}$ and remoteness
$\displaystyle\rho$. Then
$\displaystyle\partial_{n}+\rho\leq 0$
with equallity if and only if $\displaystyle G$ is $\displaystyle K_{n}$.
###### Theorem 4.18 ([9]).
Let $\displaystyle G$ be a graph on $\displaystyle n\geq 4$ vertices with
second largest distance eigenvalue $\displaystyle\partial_{2}$ and proximity
$\displaystyle\pi$. Then
$\displaystyle\pi+\partial_{2}\geq 0$
with equality if and only if $\displaystyle G$ is the complete graph
$\displaystyle K_{n}$.
###### Proposition 4.19 ([9]).
Let $\displaystyle T$ be a graph on $\displaystyle n\geq 4$ vertices with
proximity $\displaystyle\pi$, diameter $\displaystyle D$ and distance spectrum
$\displaystyle\partial_{1}\geq\partial_{2}\geq\cdots\geq\partial_{n}$. Then
$\displaystyle\pi+\partial_{\left\lfloor\frac{D}{2}\right\rfloor}>0.$
Lin, Das and Wu [37] proved two theorems, earlier as conjectures in [9].
Following lemmas and results were given.
###### Lemma 4.20 ([37]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n$ with
diameter $\displaystyle D$ and remoteness $\displaystyle\rho$. Then
$\displaystyle\rho\geq\frac{D}{2}.$
###### Theorem 4.21 ([37]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n\geq 4$
with diameter $\displaystyle D$, remoteness $\displaystyle\rho$ and distance
eigenvalues $\displaystyle\partial_{1}\geq\dots\geq\partial_{n}.$ Then we have
the following statements.
* (i)
If $\displaystyle D=2,$ then
$\displaystyle\rho+\partial_{3}\geq\frac{\left\lceil\frac{n}{2}\right\rceil-2}{n-1}-1,$
with equality holding if and only if $\displaystyle G\cong K_{n_{1},n_{2}}.$
* (ii)
If $\displaystyle D\geq 3$, then
$\displaystyle\rho+\partial_{3}>\frac{d}{2}-1.2.$
###### Theorem 4.22 ([37]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n\geq 4$
with diameter $\displaystyle D$, remoteness $\displaystyle\rho$ and distance
spectrum $\displaystyle\partial_{1}\geq\dots\geq\partial_{n}.$ Then
$\displaystyle\rho+\partial_{\left\lfloor\frac{7D}{8}\right\rfloor}>0.$
Besides above results, more results regarding remoteness and distance
eigenvalues were given in [37].
Denote by $\displaystyle H_{n-D}(n>D)$ a graph of order $\displaystyle n-D$
such that $\displaystyle V(H_{n-d})=V(\overline{K}_{n-d})$ and $\displaystyle
E(H_{n-D})\supseteq E(\overline{K}_{n-D}),$ where
$\displaystyle\overline{K}_{n-D}$ is a null graph of order $\displaystyle
n-D.$ Let $\displaystyle H_{n,D}$ be a graph of order $\displaystyle n$ with
diameter $\displaystyle D$ obtained by joining $\displaystyle n-D$ edges
between one end of the path $\displaystyle P_{D}$ with each vertex of
$\displaystyle H_{n-D}.$
###### Lemma 4.23 ([37]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n$ with
diameter $\displaystyle D$ and remoteness $\displaystyle\rho$. Then
$\displaystyle\rho\leq D-\frac{D^{2}-D}{2(n-1)},$
with equality holding if and only if $\displaystyle G\cong H_{n,D}.$
###### Theorem 4.24 ([37]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n$ with
diameter $\displaystyle D$ and remoteness $\displaystyle\rho$. Then
$\displaystyle\rho+\partial_{n}\leq-\frac{D^{2}-D}{2(n-1)},$
with equality holding if and only if $\displaystyle G\cong K_{n}.$
###### Lemma 4.25 ([37]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n$ with
diameter $\displaystyle D\geq 3.$ Then
$\displaystyle\partial_{1}>n-2+D.$
###### Theorem 4.26 ([37]).
Let $\displaystyle G\ncong K_{n}$ be a connected graph of order $\displaystyle
n$ and remoteness $\displaystyle\rho$. Then
$\displaystyle\partial_{1}-\rho\geq\frac{n-1+\sqrt{(n-1)^{2}+8}}{2}-\frac{n}{n-1},$
with equality holding if and only if $\displaystyle G\cong K_{n}-e,$ where
$\displaystyle e$ is an edge of $\displaystyle G.$
Jia and Song [33] obtained various results related to remoteness and distance,
distance (signless) Laplacian eigenvalues of graphs.
###### Theorem 4.27 ([33]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n\geq 4$
with remoteness $\displaystyle\rho$. Thenn
$\displaystyle n\leq\rho+\partial_{1}\leq\rho(P_{n})+\partial_{1}(P_{n}),$
with the left equality holding if and only if $\displaystyle G\cong K_{n}$ and
the right equality holding if and onlyif $\displaystyle G\cong P_{n}.$
###### Theorem 4.28 ([33]).
Let $\displaystyle G\ncong K_{n}$ be a connected graph of order $\displaystyle
n\geq 4$ with remoteness $\displaystyle\rho$. Then
$\displaystyle\rho+\partial_{1}\geq\frac{n}{n-1}+\frac{n-1+\sqrt{(n-1)^{2}+8}}{2},$
with equality holding if and only if $\displaystyle G\cong K_{n}-e.$
###### Theorem 4.29 ([33]).
Let $\displaystyle G\ncong(K_{n},K_{n}-e)$ be a connected graph of order
$\displaystyle n\geq 4$ with remoteness $\displaystyle\rho$. Then
$\displaystyle\rho+\partial_{1}\geq\frac{n}{n-1}+\frac{n-1+\sqrt{(n-1)^{2}+16}}{2},$
with equality holding if and only if $\displaystyle G\cong K_{n}-2e,$ where
$\displaystyle 2e$ are two matching edges.
###### Theorem 4.30 ([33]).
Let $\displaystyle G$ be a complete bipartite graph of order $\displaystyle
n\geq 4$ with remoteness $\displaystyle\rho$. Then
$\displaystyle\rho+\partial_{2}\geq n-\frac{1}{n-1}-\sqrt{n^{2}-3n+3},$
with equality holding if and only if $\displaystyle G$ is star.
For connected graphs Jia and Song [33] proposed the following conjecture.
###### Conjecture 1 ([33]).
Let $\displaystyle G\ncong(K_{n},K_{n}-e)$ be a connected graph of order
$\displaystyle n\geq 4$ with remoteness $\displaystyle\rho$. Then
$\displaystyle\rho+\partial_{2}\geq\frac{n}{n-1}+\frac{n-1-\sqrt{(n-1)^{2}+8}}{2},$
with equality holding if and only if $\displaystyle G\cong K_{n}-2e,$ where
$\displaystyle 2e$ are two matching edges.
###### Theorem 4.31 ([33]).
Let $\displaystyle G\ncong K_{n}$ be a connected graph of order $\displaystyle
n\geq 4$ with remoteness $\displaystyle\rho$. Then
$\displaystyle\rho-\partial_{n}\geq 2,$
with equality holding if and only if $\displaystyle G\cong K_{n}.$
###### Theorem 4.32 ([33]).
Let $\displaystyle G\ncong K_{n}$ be a connected graph of order $\displaystyle
n\geq 4$ with remoteness $\displaystyle\rho$. Then
$\displaystyle\rho-\partial_{n}\geq 3+\frac{1}{n-1},$
with equality holding if and only if $\displaystyle G\cong K_{n}-me,$ where
$\displaystyle me$ denotes $\displaystyle m$ matching edges.
The distance Laplacian eigenvalues of a connected graph, denoted by
$\displaystyle\partial_{1}^{L},\partial_{2}^{L},\ldots,\partial_{n}^{L}$, are
those of its distance Laplacian matrix, and are indexed such that
$\displaystyle\partial_{1}^{L}\geq\partial_{2}^{L}\geq\ldots\geq\partial_{n}^{L}$.
###### Theorem 4.33 ([33]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n\geq 4$
with remoteness $\displaystyle\rho$. Then
$\displaystyle
n+1\leq\rho+\partial_{1}^{L}\leq\rho(P_{n})+\partial_{1}^{L}(P_{n}),$
with the left equality holding if and only if $\displaystyle G\cong K_{n}$ and
the right equality holding if and only if $\displaystyle G\cong P_{n}.$
###### Theorem 4.34 ([33]).
Let $\displaystyle G\ncong K_{n}$ be a connected graph of order $\displaystyle
n\geq 4$ with remoteness $\displaystyle\rho$. Then
$\displaystyle\rho+\partial_{1}^{L}\geq n+\frac{1}{n-1}+3,$
with equality holding if and only if $\displaystyle G\cong K_{n}-me,$ where
$\displaystyle me$ denotes $\displaystyle m$ matchings.
###### Theorem 4.35 ([33]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n\geq 4$
with remoteness $\displaystyle\rho$. Then
$\displaystyle\partial_{1}^{L}-\rho\geq n-1,$
with equality holding if and only if $\displaystyle G\cong K_{n}.$
###### Theorem 4.36 ([33]).
Let $\displaystyle G\ncong K_{n}$ be a connected graph of order $\displaystyle
n\geq 5$ with remoteness $\displaystyle\rho$. Then
$\displaystyle\partial_{1}^{L}-\rho\geq n+1-\frac{1}{n-1},$
with equality holding if and only if $\displaystyle G\cong K_{n}-me,$ where
$\displaystyle 1\leq m\leq\left\lfloor\frac{n}{2}\right\rfloor.$
The distance signless Laplacian eigenvalues of a connected graph, denoted by
$\displaystyle\partial_{1}^{Q},\partial_{2}^{Q},\ldots,\partial_{n}^{Q}$, are
those of its distance signless Laplacian matrix, and are indexed such that
$\displaystyle\partial_{1}^{Q}\geq\partial_{2}^{Q}\geq\ldots\geq\partial_{n}^{Q}$.
###### Theorem 4.37 ([33]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n\geq 4$
with remoteness $\displaystyle\rho$. Then
$\displaystyle 2n\leq 2\rho+\partial_{1}^{Q}\leq
2\rho(P_{n})+\partial_{1}^{Q}(P_{n}),$
with the left equality holding if and only if $\displaystyle G\cong K_{n}$ and
the right equality holding if and only if $\displaystyle G\cong P_{n}.$
###### Theorem 4.38 ([33]).
Let $\displaystyle G\ncong K_{n}$ be a connected graph of order $\displaystyle
n\geq 4$ with remoteness $\displaystyle\rho$. Then
$\displaystyle
2\rho+\partial_{1}^{Q}\geq\frac{3n-2+\sqrt{(n-2)^{2}+16}}{2}+\frac{2n}{n-1},$
with equality holding if and only if $\displaystyle G\cong K_{n}-e.$
###### Theorem 4.39 ([33]).
Let $\displaystyle G\ncong(K_{n},K_{n}-e)$ be a connected graph of order
$\displaystyle n\geq 4$ with remoteness $\displaystyle\rho$. Then
$\displaystyle
2\rho+\partial_{1}^{Q}\geq\frac{3n-2+\sqrt{(n-2)^{2}+32}}{2}+\frac{2n}{n-1},$
with equality holding if and only if $\displaystyle G\cong K_{n}-2e,$ where
$\displaystyle 2e$ are two matching edges.
###### Theorem 4.40 ([33]).
Let $\displaystyle G$ be a connected graph of order $\displaystyle n\geq 4$
with remoteness $\displaystyle\rho.$ Then
$\displaystyle\partial_{1}^{Q}-2\rho\geq 2n-4,$
with equality holding if and only if $\displaystyle G\cong K_{n}.$
###### Theorem 4.41 ([33]).
Let $\displaystyle G\ncong K_{n}$ be a connected graph of order $\displaystyle
n\geq 4$ with remoteness $\displaystyle\rho$. Then
$\displaystyle\partial_{1}^{Q}-2\rho\geq\frac{3n-2+\sqrt{(n-2)^{2}+16}}{2}-\frac{2n}{n-1},$
with equality holding if and only if $\displaystyle G\cong K_{n}-e$.
Mojallal and Hansen [39] obtained relation between proximity and the third
largest distance eigenvalue of a graph, some results earlier conjuctured in
[9].
Figure 10: The tree $\displaystyle T_{1}$ and the graph $\displaystyle G_{1}$
###### Lemma 4.42 ([39]).
Let $\displaystyle T_{1}$ be the tree of order $\displaystyle n$ given in Fig
10. Then
$\displaystyle\pi(T_{1})+\partial_{3}(T_{1})>0.$
###### Lemma 4.43 ([39]).
Let $\displaystyle G_{1}$ be the graph of order $\displaystyle n$ given in Fig
10. Then
$\displaystyle\pi((G_{1}))+\partial_{3}(G_{1})>0.$
###### Lemma 4.44 ([39]).
Let $\displaystyle G$ be a graph of order $\displaystyle n$ with the diameter
$\displaystyle D=3$ and let $\displaystyle i_{1}$ and $\displaystyle i_{4}$ be
two vertices of $\displaystyle G$ with distance $\displaystyle 3$. If
$\displaystyle d_{G}(i_{1})\geq 2$ and $\displaystyle d_{G}(i_{4})\geq 2.$
Then
$\displaystyle\pi+\partial_{3}>0.$
###### Lemma 4.45 ([39]).
Let $\displaystyle G$ be a graph of order $\displaystyle n$ with the diameter
$\displaystyle D=3$ and let $\displaystyle i_{1}$ and $\displaystyle i_{4}$ be
two vertices of $\displaystyle G$ with distance $\displaystyle 3$. If
$\displaystyle d_{G}(i_{1})=1$ or $\displaystyle d_{G}(i_{4})=1.$ Then
$\displaystyle\pi+\partial_{3}>0.$
###### Theorem 4.46 ([39]).
Let $\displaystyle G$ be a graph with the diameter $\displaystyle D\geq 3$,
proximity $\displaystyle\pi(G)$ and third largest distance eigenvalue
$\displaystyle\partial_{3}.$ Then
$\displaystyle\pi+\partial_{3}>0.$
###### Corollary 4.47 ([39]).
Let $\displaystyle G$ be a graph with the diameter $\displaystyle D\geq 3,$
and the third largest distance eigenvalue $\displaystyle\partial_{3}.$ Then
$\displaystyle\displaystyle(i)~{}D+\partial_{3}>0\quad(ii)~{}\rho+\partial_{3}>0\quad(iii)~{}\overline{\ell}+\partial_{3}>0\quad(iv)~{}ecc+\partial_{3}>0\quad(v)~{}r+\partial_{3}>0.$
## 5 Conclusion
The survey collects all the results related to proximity $\displaystyle\pi$
and remoteness $\displaystyle\rho$ published in different journals from 2006
to September 2023.
## Data Availability
There is no data associated with this article.
## Conflict of interest
The authors declare that they have no competing interests.
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|
(see also Fig. 5). Firstly, the number of bound states at a given mass is
shown on the left of Fig. 6. We have argued that production of a bound state
with a given $S$ and $L>0$ will generically lead to production (via one or
more decays) of an $L=0$ bound state with the same $S$, so it unnecessary to
compute the rate to produce SM particles directly from those $L>0$ states. We
confirm this numerically on the right of Fig. 6: however measured, the
branching fraction is always greater than 90%. For instance, at 100 TeV, the
deepest $Q=0$ $L+S$ odd $p$-wave bound state decays to the deepest $s$-wave
state with 99.6% probability and to the second deepest $s$-wave state with a
likelihood of 0.4%. (The next most dominant transition is to the third deepest
$s$-wave bound state, and only occurs with a probability of 0.006%.)
We also discussed decays of the $S=1,\,L=0$ bound states, noting that they
generate only a power suppressed contribution to the photon endpoint spectrum.
Nevertheless, the contribution from the $S=1$ states can be comparable to that
from the $S=0$ states, after accounting for both their (enhanced) production
rate and (suppressed) endpoint spectrum from annihilation. However, in
practice this means that the contributions to the endpoint spectrum from both
the $S=0$ and $S=1$ bound states are suppressed compared to direct
annihilation (either due to power suppression or because of the small
formation rate).
For the aforementioned reasons, we focus on the $L=S=0$ bound states to
estimate the size of the (generally subdominant) bound-state contribution.
Doing so implies we can reuse our SCET results from the direct annihilation
case, with appropriate modification of the wavefunction factors. This choice
does mean that the total bound-state contribution to the endpoint photon
spectrum could increase by an $\mathcal{O}(1)$ factor in an improved
calculation (once the $S=1$ states are included). Because the bound-state
contribution to the endpoint spectrum is suppressed, this typically
corresponds to a percent-level theoretical uncertainty in the overall endpoint
spectrum, with larger uncertainties at specific mass points where the bound
state formation cross section is enhanced relative to direct annihilation. The
contribution to continuum photons (not near the endpoint) from bound state
formation can be markedly larger, compared to the effect on the endpoint
spectrum, as there is no power suppression in the continuum contribution from
annihilation of the $S=1$ bound states. We will discuss each of these
contributions to the total spectrum in the next section.
## 5 The Combined Photon Spectrum and Numerical Results
At this stage we have all ingredients required to determine the quintuplet
annihilation spectrum, including both direct annihilation and the contribution
of bound states. In this section we collect our results to determine the full
energy distribution of photons the quintuplet generates at the thermal mass of
13.6 TeV, but also for a wider range of masses. We will estimate the impact of
several uncertainties on our results, such as the residual theoretical
uncertainty on the NLL computations, but also from astrophysical uncertainties
such as the distribution of $v$ values and on the DM density in the inner
Galaxy. Finally, we will put these results together to estimate the
sensitivity of existing and upcoming IACTs to quintuplet DM.
### 5.1 Predictions for the spectrum and rate of photon production
A central goal of this work is to accurately determine the distribution of
photons that emerge when two SU(2) quintuplets annihilate. This spectrum forms
the signal template for telescopes searching for high energy photons, and
therefore is a central theoretical input. To achieve this, throughout we have
computed differential cross sections $d\sigma/dz$, both for the direct
annihilation in Eq. (31), and also for the bound state contribution by
combining the results of the previous sections with Eq. (1). For indirect
detection, observables are sensitive to $d\langle\sigma v\rangle/dz$. To begin
with we will assume the DM states are incident with a fixed $v=10^{-3}$,
revisiting the validity of this approximation in the next subsection. In order
to extract the shape of the photon distribution from the differential cross
section, it is common in indirect detection to introduce a photon spectrum
$dN/dE$, and our convention for doing so is the following,151515Further
discussion of the connection between spectra used in indirect detection and
the corresponding field theoretic quantities can be found in, for instance,
Ref. Bauer:2020jay .
$\displaystyle\frac{d\langle\sigma v\rangle}{dE}=\langle\sigma v\rangle_{\rm
line}\times\frac{dN}{dE}.$ (84)
This choice follows Ref. Baumgart:2017nsr , and implies that our spectrum is
normalized with respect to the _line cross section_ , $\langle\sigma
v\rangle_{\rm line}\equiv\langle\sigma
v\rangle_{\gamma\gamma}+\tfrac{1}{2}\langle\sigma v\rangle_{\gamma Z}$, which
is defined as the rate to produce two photons at exactly $E=M_{\chi}$. By
construction, $dN/dE$ will contain a contribution of exactly
$2\delta(E-M_{\chi})$ for the line, but it will also contain contributions
from endpoint photons, bound state decays, and continuum photons arising
primarily from the unstable particles the direct annihilations or bound state
decays can produce. For each of these latter components, however, their
additions to $dN/dE$ will be weighted by a branching fraction $\langle\sigma
v\rangle_{i}/\langle\sigma v\rangle_{\rm line}$, with $\langle\sigma
v\rangle_{i}$ the cross section for that particular contribution. The
rationale for anchoring our calculations to the line cross section is that
$\chi\chi\to\gamma\gamma$, which has a spectrum of exactly
$dN/dE=2\delta(E-M_{\chi})$, is a common experimental target, and therefore
there are a wide number of existing constraints on $\langle\sigma
v\rangle_{\rm line}$ which we can then directly compare with. Further
discussion of this point can be found in Ref. Baumgart:2017nsr .
Figure 7: The cross section for line photons, breaking down the contributions
from the direct annihilation and bound states. While generically the direct
annihilation dominates, for isolated masses near Sommerfeld peaks the bound
state contribution can be the leading one. For all masses lower than those
shown bound states are strictly subdominant.
The spectra of line and endpoint photons produced by decay of the bound states
is computed using the methods of Sec. 4.161616In this work we do not track the
much lower-energy photons radiated in bound state formation and transitions;
these signals have been studied for the quintuplet in Refs. Mitridate:2017izz
; Mahbubani:2020knq . For our bound state formation and transition
calculations, we include only states with $L=0,1,2$. Capture into $L=3$ and
higher states requires at least $L=2$ for the initial state, and we expect the
contributions from components of the initial state with high $L$ to be
suppressed at low velocities, by a factor that is parametrically
$(M_{\chi}v/m_{\scriptscriptstyle W})^{2L}$ (although for sufficiently high
masses, $M_{\chi}\gtrsim 100$ TeV, this suppression is lifted). It is also
worth noting that for essentially all the parameter space most relevant for
experimental searches with H.E.S.S, at $M_{\chi}<20$ TeV, we find that no
$L=3+$ bound states exist in the spectrum. We independently expect capture
into states with high principal quantum number $n$ (which is required for high
$L$) to be suppressed, as (1) the finite range of the potential means only a
limited number of states are bound at all, so unlike in unbroken gauge
theories there is no infinite tower of high-$n$ states, (2) capture into
weakly-bound states is suppressed by a phase space factor, and (3) analytic
approximations (App. F) suggest that we can expect the leading contribution to
the capture cross section to be exponentially suppressed for large $n$. In
practice, our numerical calculation expresses the bound states as a linear
combination of 30 basis states for each combination of $L$, $Q$ and
$(-1)^{L+S}$, allowing us to access up to 30 distinct bound states indexed by
different values of $n$ (although as we approach this upper bound we expect
the spectrum to become less accurate), and we include all these states in our
calculation. We have checked at sample mass points that our binding energies
and cross sections for capture into lower-$n$ states are not significantly
affected by doubling the number of basis states. For the reasons given above,
we generally expect the error due to the omission of higher-$n$ states to be
small.
Before showing the full distributions of line and endpoint photons, we can
already consider one measure of the importance of bound states to the
resulting photon signal: their contribution to $\langle\sigma
v\rangle_{\text{line}}$. This is shown in Fig. 7, where we separate the
contribution to the line from the direct annihilation to that of processes
involving an intermediate bound state. At this stage we only include bound-
state contributions that produce line photons, with energy essentially at
$M_{\chi}$. The figure makes clear a point already estimated earlier: direct
annihilation generally dominates the production of line photons at $E\simeq
M_{\chi}$ by $1-2$ orders of magnitude. However, the bound-state contribution
can be significant and even dominate at isolated mass points, for instance as
at $M_{\chi}=68.1~{}{\rm TeV}$, and therefore a reliable prediction at
arbitrary masses must include this contribution.
Figure 8: The quintuplet annihilation spectrum for two masses, the thermal
mass of 13.6 TeV (left), and a mass where bound state contributions are
appreciable, 68.1 TeV (right). For each mass we show results convolved with
the H.E.S.S. energy resolution (top) and unsmeared (below). The full spectrum
is broken into five individual components: the line, endpoint, bound state
line and endpoint, the direct annihilation continuum, and bound state
contribution to the continuum. Details of each are provided in the text.
Moving beyond the line, in Fig. 8 we show the full spectrum, broken down by
various contributions, for two masses: the thermal mass of
$M_{\chi}=13.6~{}{\rm TeV}$, and a mass where bound state contributions are
significant, $68.1~{}{\rm TeV}$. For each mass we show two versions of the
spectrum. In the lower panels, we show the unsmeared spectra, which is the
distribution of photons that emerge from the annihilations. (Note in this case
the line contribution is simply $dN/dE=2\delta(E-M_{\chi})$ and so represented
by a vertical line.) In the upper panels, we have convolved the raw spectra
with a finite experimental energy resolution in order to model what would
actually be seen at a realistic instrument. For this we take the energy
resolution of the H.E.S.S. telescope, determined from Ref. HESS:2013rld . In
detail, we fix the relative width $\Delta E/E$ to 0.17 and 0.11 for
$E=500~{}{\rm GeV}$ and $E=10~{}{\rm TeV}$, respectively, and then vary
logarithmically between these endpoints, freezing the ratio either side. From
this we compute $(dN/dE)_{\rm smeared}$ as $dN/dE$ convolved with a Gaussian
of width equal to the energy resolution.
In terms of these two notions of the spectra, Fig. 8 shows five contributions
to the photons distributions for the two masses. The first three of these are:
1. the direct annihilation line; 2. direct annihilation endpoint; and 3. the
bound state contribution to the line and endpoint. Again we see clearly a
point noted for the wino in Refs. Baumgart:2017nsr ; Baumgart:2018yed : the
endpoint contribution makes a considerable modification to the observed number
of photons with $E\sim M_{\chi}$, with the peak smeared spectra enhanced by
1.9 and 3.1 for $M_{\chi}=13.6$ and 68.1 TeV. The bound state contribution is
more modest: it is effectively negligible at the thermal mass, and a factor of
1.7 enhancement at 68.1 TeV, which again is a mass with an anomalously large
bound state contribution to the hard photon spectrum.
Beyond these three, we also show the contribution of two continuum sources,
both of which can generate lower energy photons. The first of these is the
continuum emission arising from direct annihilation. This results from tree
level annihilation of the quintuplets into $W$ or $Z$ bosons. The latter of
these arises from $\gamma Z$ and $ZZ$ final states, and is a simple
reweighting of the line cross section as
$\frac{\langle\sigma v\rangle_{ZZ}+\tfrac{1}{2}\langle\sigma
v\rangle_{Z\gamma}}{\langle\sigma
v\rangle_{\gamma\gamma}+\tfrac{1}{2}\langle\sigma v\rangle_{\gamma
Z}}=\frac{c_{\scriptscriptstyle W}^{2}}{s_{\scriptscriptstyle W}^{2}}.$ (85)
For the $W^{+}W^{-}$ final state, the tree level annihilation rate, with the
Sommerfeld effect included can be computed as,
$\displaystyle\langle\sigma v\rangle_{\scriptscriptstyle
WW}=\frac{\pi\alpha_{\scriptscriptstyle W}^{2}}{M_{\chi}^{2}}$
$\displaystyle\Big{[}18|s_{00}|^{2}+25|s_{0\pm}|^{2}+4|s_{0\pm\pm}|^{2}$ (86)
$\displaystyle+30\sqrt{2}{\rm Re}(s_{00}s_{0\pm}^{*})+12\sqrt{2}{\rm
Re}(s_{00}s_{0\pm\pm}^{*})+20{\rm Re}(s_{0\pm}s_{0\pm\pm}^{*})\Big{]}.$
As discussed for the case of the wino in Ref. Hryczuk:2011vi , higher order
corrections to this should not be appreciable, and so we do not include them.
We can then add the $W^{+}W^{-}$ final state to $dN/dE$ with weighting
$\langle\sigma v\rangle_{\scriptscriptstyle WW}/\langle\sigma v\rangle_{\rm
line}$ along with the $Z$ contribution. In each case, to determine the
spectrum of photons that result from these electroweak bosons we use PPPC4DMID
Cirelli:2010xx with the electroweak corrections turned off, to avoid any
double counting of the endpoint contributions we computed. As seen in Fig. 8,
these contributions are important for $E_{\gamma}\ll M_{\chi}$.
The final contribution to the spectrum we consider is continuum photons
arising from the bound state decays. This contribution is not the main focus
of this work, but to get an estimate of its size and spectrum, we assume the
most common SM decay products are light quarks (equally weighted between
flavors), and employ the corresponding gamma-ray spectrum from PPPC4DMID. The
motivation for this choice is that the bound states will decay through their
couplings to (on-shell or off-shell) $W$ and $Z$ bosons, with the exact
channel depending on their $L$ and $S$ quantum numbers, and the gauge bosons
in turn decay dominantly to quarks due to their large associated degrees of
freedom. We weight the continuum spectrum by the ratio between the bound state
capture cross section and $\langle\sigma v\rangle_{\textrm{line}}$, similar to
how we weight the $Z$ and $W$ continuum components above. At the thermal mass,
this ratio is roughly 31, and the visible contribution can be seen in Fig. 8.
The contribution is dominated by the $Q=0$ $p\to s$ capture cross section
which sits near a Sommerfeld peak in this capture rate at 13.6 TeV (cf. Fig.
4). To highlight this, at the edges of the uncertainty band on the thermal
mass, 12.8 and 14.4 TeV, the equivalent ratio is reduced significantly, to
0.15 and 0.70, respectively. At 68.1 TeV the ratio is larger still – just over
471 – and is dominated by the $Q=0$ and $Q=1$ $p\to s$ capture rates.
Figure 9: The full quintuplet spectrum for three different masses,
$M_{\chi}=14,\,16,$ and $18~{}{\rm TeV}$. As shown, the spectrum can change
significantly as a function of mass, a feature which does not arise for the
wino or higgsino.
Figure 8 highlights the various contributions to the spectrum, but does not
capture the variation of the spectrum as a function of mass. The variation can
be considerable, as shown in Fig. 9. From the definition, the line
contribution to this spectrum is fixed at $2\delta(E-M_{\chi})$ independent of
mass. What is not fixed, however, is the endpoint and continuum contributions,
which can vary significantly even for small changes in mass. (The bound state
contributions are not significant for the masses shown.) As shown in Ref.
Montanari:2022buj , such rapid variations can lead to sharp features in the
instrumental sensitivity to $\langle\sigma v\rangle_{\rm line}$, as the shape
of the DM signal being searched for varies rapidly with mass. These effects do
not occur for the wino or higgsino, where the spectra varies relatively
smoothly with mass (see Ref. Montanari:2022buj ).
The origin of this behavior seems to be interference between the different
Sommerfeld factors, associated with the distinct mass eigenstates for the
final annihilation: $\chi^{0}\chi^{0}$, $\chi^{+}\chi^{-}$, and
$\chi^{++}\chi^{--}$. These states have different branching ratios into the
various SM final states, and the positions of the resonance peaks can differ
between the interfering Sommerfeld factors. As we vary the mass, we can move
rapidly from the sharp turn off of one Sommerfeld peak to the sharp turn on of
another, and in doing so transition rapidly in the strength of the associated
endpoint and continuum contributions, as seen in Fig. 9.
One might ask why this behavior was not seen for the wino, which also has
multiple Sommerfeld factors that can interfere with each other. For the line
cross section, one might suspect that the issue is that only the
$\chi^{+}\chi^{-}$ state can annihilate to photons at tree-level; however, we
also do not see sharp quintuplet-like features in annihilation of wino DM to W
bosons, which is allowed at tree-level from both the $\chi^{0}\chi^{0}$ and
$\chi^{+}\chi^{-}$ states. More insight can be gained by working in the basis
of eigenstates of the potential at small $r$, rather than mass eigenstates;
this corresponds to the basis of potential eigenstates in the limit of
unbroken SU(2), as discussed in App. F. In the limit of unbroken SU(2), the
relevant potential for the quintuplet (coupling states with total charge zero
and even $L+S$) has two eigenstates that experience attractive interactions
and one eigenstate that experiences repulsive interactions. We expect that the
linear combination of Sommerfeld factors corresponding to the repulsed
eigenstate at small $r$ should be suppressed, as the SU(2) symmetry is
restored in the small-$r$ regime. We have confirmed numerically that this
suppression is quite pronounced, typically several orders of magnitude at the
velocities we consider. We would also expect a difference in the linear
combination of Sommerfeld factors corresponding to the two attracted
eigenstates, with the larger-magnitude eigenvalue yielding a larger Sommerfeld
enhancement, but this difference is much less dramatic, and so the two
attracted eigenstates still experience meaningful interference.
We thus attribute the sharp features observed in the quintuplet case to this
interference between the different attracted eigenstates (in the small-$r$
potential-dominated regime); its absence in the wino case is presumably
related to the fact that the wino has only one attracted eigenstate (e.g. Ref.
Asadi:2016ybp ). We thus expect this behavior (sharp variations in the
spectrum with mass) to be ubiquitous for larger SU(2) representations.
### 5.2 Uncertainty associated with the velocity distribution of dark matter
The complete initial-state wavefunctions naturally depend on the relative
velocity of the incoming DM particles, which in the discussion so far we have
simply set as $v=10^{-3}$. In this subsection we explore the systematic
uncertainties associated with our modeling of the velocity. We first discuss
the detailed dependence of the cross sections on the relative DM velocity, and
then explore the effects on our spectra of averaging over different plausible
velocity distributions. The effects of the long-range potential on the
wavefunction saturate when $v\lesssim m_{\scriptscriptstyle W}/M_{\chi}$,
which is true for halo velocities ($v\sim 10^{-3}$) for $M_{\chi}\lesssim
80~{}{\rm TeV}$; consequently, except near resonances or for very heavy DM, we
do not expect the Sommerfeld enhancement from the weak interactions to depend
sensitively on the velocity distribution. However, the bound-state formation
rate from $L>0$ partial-wave components of the initial state will have a non-
trivial velocity dependence even below this threshold. Furthermore, for the
quintuplet the thermal mass is only a factor of few below the saturation
threshold, and in systems with higher velocities than the Sun’s neighborhood –
such as galaxy clusters – both the direct annihilation cross section and the
bound-state formation rates are expected to depend sensitively on the
velocity.
Figure 10: The dependence of the cross sections on $v$ for four different
fixed masses. We show the case of direct endpoint annihilation (left, the
analogue of Fig. 11), and bound state capture (right, as in Fig. 12).
In Fig. 10 we show how annihilation and capture vary as a function of velocity
at four different masses. We observe that for a mass of $M_{\chi}=14$ TeV,
noticeable velocity dependence is present at $v\gtrsim 4\times 10^{-3}$. As we
discuss in depth in App. F, the oscillatory behavior observed at high
velocities can be understood in the limit of unbroken SU(2). This behavior
originates from interference between the different eigenvalues of the
potential. At low velocities, by contrast, SU(2) breaking effects are expected
to suppress the oscillations. For higher DM masses, where the velocity
dependence is relevant even for $v\lesssim 10^{-3}$, our previous annihilation
cross section plots should be taken as an illustrative estimate. A full
calculation at high mass would involve integrating the formulae given in this
paper over the true velocity distribution in the region of interest. The
oscillatory behavior of the cross section at high velocities means that
assuming a single velocity could in principle lead to large errors in this
case.
We now estimate the effect of averaging over the velocity distribution. The
characteristic scale of the DM velocity dispersion should be comparable to the
circular velocity of the visible matter, which in the vicinity of the Sun has
been measured to be $v_{\text{circ}}\simeq 240$ km/s 2016MNRAS.463.2623H .
Since the Milky Way’s rotation curve is roughly flat at the Sun’s location, we
expect the velocity dispersion to be of a similar order over much of the
Galaxy. However, close to the Galactic Center the DM velocity is not well-
known. In DM-only simulations the velocity dispersion falls as one approaches
the Galactic Center (e.g. Ref. Navarro:2008kc ) but simulations including
baryons have demonstrated the opposite behavior (e.g. Refs.
2010MNRAS.406..922T ; Board:2021bwj ; 2022MNRAS.513…55M ). Even at the Sun’s
location, the full DM velocity distribution is not well-understood: the
distribution is often treated as Maxwellian up to some escape velocity,
although this is only a crude approximation (e.g. Ref. Herzog-
Arbeitman:2017fte ). The escape velocity is determined to be $\sim$500 km/s at
the location of the solar system Necib:2021vxr ; kh2021 ; mfcEtAl2018
.171717Ref. mfcEtAl2018 finds that the escape velocity increases slightly
toward the Galactic Center. However, they only present results in to a radius
of around 5 kpc, where it is closer to 650 km/s. The precise value of this
cutoff is numerically unimportant for this work though, due to the exponential
suppression in the distribution (cf. Eq. (87)). In practice, we truncate the
particle velocity at 500 km/s, but the numerical difference between this and
2400 km/s is at most a part per million in the annihilation rate. Within the
Maxwellian approximation, the distribution is specified by that escape
velocity and the velocity dispersion, with the latter having a greater effect
on the annihilation rate.
For the Milky Way, we use the velocity dispersion values obtained in Ref.
Boddy:2018ike for a variety of NFW-profiles. In particular, we take the
slowest and fastest velocities for locations interior to the solar system.
This gives a range $v_{\rm disp}\in[130,330]$ km/s. As a function of $v_{\rm
disp}$, the magnitude of the relative WIMP velocity is drawn from the
following 1D probability distribution,
$f(v)=\sqrt{\frac{27}{4\pi}}\,\frac{v^{2}}{v_{\rm disp}^{3}}e^{-3v^{2}/4v_{\rm
disp}^{2}}.$ (87)
Here $v_{\rm disp}$ is the RMS velocity for a single DM particle, which is
equal to the three-dimensional velocity dispersion $\sigma_{v,3d}$ defined in
Ref. Boddy:2018ike by
$v_{\rm disp}^{2}=\sigma_{v,3d}^{2}=\frac{\int dv\,v^{4}f(r,v)}{\int
dv\,v^{2}f(r,v)}.$ (88)
Here $f(r,v)$ is the speed distribution for a single DM particle at a distance
$r$ from the Galactic Center.
Figure 11: An estimate of the range of uncertainty in our results associated
with the DM velocity distribution for the dominant direct annihilation. We use
two Maxwell distributions with $v_{\rm disp}$ – cf. Eq. (87) – at the extremal
values found by Ref. Boddy:2018ike . We divide their resulting $\langle\sigma
v\rangle$ by that of the simplified case of all quintuplets annihilating with
$v=10^{-3}$.
In Fig. 11, we plot the leading contribution to endpoint photon production,
direct annihilation from an $s$-wave initial state, for two different velocity
distributions, normalized to the simple assumption of all quintuplets having a
fixed $v=10^{-3}$. Except on resonances, the uncertainty is typically
negligible. We also see that off resonance, and particularly at lower masses,
the simple fixed-velocity assumption is a good approximation of either more
realistic model. The reason for this is simply that we are in the saturation
regime, as seen in Fig. 10. Therefore, we conclude that in general the fixed
velocity assumption is a good one at low masses, although at higher masses one
is generically underestimating the actual cross section, sometimes by more
than an order of magnitude. Accordingly, for an actual experimental analysis,
completeness would require an appropriate weighting of the cross section
according to the specific region of interest studied.
Figure 12: Capture rate by emission of a dipole photon from $p$-wave initial
state to $s$-wave quintuplet bound state for $Q=0$ states. The three choices
of velocity mirror those in Fig. 11. We see two different values of
$v_{\text{disp}}$ plotted, plus a $v_{\text{fixed}}=10^{-3}$ whose capture
rate we divide by.
For bound state capture, the off-resonance uncertainties are generally larger
than for direct annihilation, as anticipated. This is demonstrated in Fig. 12,
where we show $p$-to-$s$ capture, the leading single-photon dipole transition.
As we see by comparing the rates with those in Fig. 7 though, capture is
generally far subdominant to direct annihilation. In this channel, however,
the simple assumption of all DM having $v=10^{-3}$ generally provides a result
in the middle of the band given by the remaining two options. Even where it
does not, we see that it still provides a good approximation to the more
realistic velocity profiles. Accordingly, just as with direct annihilation, we
will take this value as a representative approximation of this subleading
contribution to endpoint photons when forecasting experimental sensitivity.
### 5.3 Estimating the experimental sensitivity to quintuplet DM
Finally we turn to an estimate of the experimental sensitivity to the
quintuplet DM hypothesis using the spectra we have computed. Using our
definition of the spectrum in Eq. (84), the average DM-generated flux an
instrument observes from a region of interest (ROI) of solid angle
$\Omega_{\rm ROI}$ is,
$\frac{d\Phi}{dE}=\frac{\langle\sigma v\rangle_{\rm line}}{8\pi
M_{\chi}^{2}}\left(\frac{dN}{dE}\right)_{\rm
smeared}\left(\frac{1}{\Omega_{\rm ROI}}\int
ds\,d\Omega\,\rho_{\chi}^{2}\right)\\!.$ (89)
As defined, the flux has units of $[{\rm counts}/{\rm cm}^{2}/{\rm s}/{\rm
TeV}/{\rm sr}]$ (for a detailed discussion of the units, see Ref.
Lisanti:2017qoz ). The final term in parentheses here is often referred to as
the $J$-factor, and is an integral over the DM density squared in the region
being observed. If the DM density $\rho_{\chi}$ was known exactly, then for a
model like the thermal quintuplet the flux is fully determined, as we have
computed both the cross section and spectrum (up to residual uncertainties
from higher order terms in the theory prediction, and velocity distribution as
discussed in the last subsection).
To test the quintuplet hypothesis, we need to compare the flux in Eq. (89) to
experimental measurements. For this, we will estimate the sensitivity of
H.E.S.S. to the endpoint photon signal using the “mock analysis” method
described in Ref. Baumgart:2017nsr . The approach in that work was to make use
of the publicly available H.E.S.S. data in Ref. HESS:2013rld , where a search
for $\chi\chi\to\gamma\gamma$ was performed using 112 hours of Galactic Center
observations taken by the instrument between $2004-2008$. This is a small
fraction of the observations H.E.S.S. has taken towards the galactic center.
As emphasized in Ref. Montanari:2022buj , the collaboration has already
collected roughly 800 hours of data in the region, and continues to collect
roughly 150 hours each year. In that sense, the dataset we consider represents
a small fraction of what is available. A further limitation of our approach is
that the analysis in Ref. HESS:2013rld was a search purely for line photons,
and therefore adopted a flexible background model that absorbed all smooth
components. The analysis is therefore unsuitable to consider continuum
contributions, which can play an important role in these sorts of analyses, as
emphasized in, for instance, Ref. Rinchiuso:2020skh . Our rationale for
adopting this mock analysis, however, is that Ref. HESS:2013rld provided
enough information that the full analysis they undertook can be performed
reconstructed, as was shown in Ref. Baumgart:2017nsr . Later on we will
provide a rough estimate of how sensitive more recent and upcoming analyses
could be.
Figure 13: (Left) The estimated sensitivity of H.E.S.S. I using 112 hours of
galactic center (GC) data to the quintuplet. Assuming an Einasto profile, we
show sensitivity to $\langle\sigma v\rangle_{\rm line}$ as defined in Eq.
(84), which can then be compared to the equivalent theoretical prediction.
Across the entire mass range considered here, H.E.S.S. would be able to
exclude the quintuplet assuming an Einasto profile. (Right) If the DM profile
is cored as in Eq. (91), the core sizes that would be required to make a non-
observation consistent with our quintuplet predictions. At the thermal mass,
13.6 TeV, a core of 1 kpc is required, whereas at upper edge of the thermal
band, 14.4 TeV, the results would be consistent with a 0.5 kpc core.
For the mock analysis, we fit the data provided in Ref. HESS:2013rld to a
combination of the flux in Eq. (89) and a parametric background model adopted
by the experiment—full details are provided in Ref. Baumgart:2017nsr . If we
have a prediction for $\rho_{\chi}$, this approach combined with our
prediction for the spectrum can be used to obtain an estimated sensitivity for
$\langle\sigma v\rangle_{\rm line}$. For this we adopt the Einasto profile
1965TrAlm…5…87E used by the H.E.S.S. analysis (based on Ref. Pieri:2009je ),
$\rho_{\rm
Einasto}(r)\propto\exp\left[-\frac{2}{\alpha}\left(\left(\frac{r}{r_{s}}\right)^{\alpha}-1\right)\right]\\!,$
(90)
with $\alpha=0.17$, $r_{s}=20~{}{\rm kpc}$, and the normalization fixed to
$0.39~{}{\rm GeV/cm}^{3}$ at the solar radius, $r=8.5~{}{\rm kpc}$. The
resulting estimated constraint is shown in the left of Fig. 13.181818One can
find an early attempt to make projected $\gamma$-ray constraints from the
H.E.S.S. galactic center data in Ref. Cirelli:2015bda . Their Fig. 7 is
analogous to our Fig. 13. The earlier paper did not include LL (or NLL)
resummation, nor the NLO corrections to the electroweak potential. For similar
Einasto parameters, their bounds on the line cross section are about an order
of magnitude weaker than ours (however, note that our predicted line cross
section is also smaller). We emphasize, that even though this is a constraint
on $\langle\sigma v\rangle$ it is not based solely on the line prediction—the
results are based on the estimated detectability of the entire spectrum
resulting from line, endpoint, and bound state photons. For the mass range
considered, the bound state contribution is negligible (cf. Fig. 7), however
the endpoint is not: at 13.6 TeV, it enhances the sensitivity by a factor of
1.9. As mentioned above, by default, the results do not include either
continuum contribution considered in Fig. 8. The motivation for this is the
particular background model adopted in Ref. HESS:2013rld was designed solely
to search for a narrow line feature, and there can be considerable degeneracy
with the continuum emission (see the discussion in Ref. Baumgart:2017nsr ).
Nevertheless, we have tested adding the continuum emission from direct
annihilation to $W$ and $Z$ final states, and found for most masses it has no
impact on the estimated limits, although there is a slight fluctuation around
the thermal mass, which increases the sensitivity by $\simeq$$20\%$. We also
note that there are several locations, such as just below $M_{\chi}=3~{}{\rm
TeV}$ and just above the thermal mass where there is a larger theoretical
error. This results from the sharp variations in the spectra observed in Fig.
9. In fact, the sensitivity to these features is reduced by the insensitivity
of the background model used in this work to smooth features. These results
can also be compared to Ref. Montanari:2022buj , where an alternate H.E.S.S.
analysis is performed using the spectra from this work, and much sharper
variations are observed in their sensitivity. (We note that the results of
that work made use of the LO Sommerfeld potential calculations, not the NLO
results we have adopted here.)
The results of the mock analysis suggest that for the central value of the
thermal mass, even 112 hours of H.E.S.S. data can exclude the thermal
prediction by a factor of 10. Nevertheless, this varies considerably across
the uncertainty band on the thermal mass: at 12.8 TeV, the exclusion factor is
55, whereas it is only 4 at 14.4 TeV. The sharp variation is a result of the
thermal window sitting near a Sommerfeld resonance, as shown in Fig. 13. We
emphasize once more that although we use the NLO potential in our
computations, the thermal mass range was computed with the LO potential, and
given the sensitivity of our findings to the exact mass, computing the thermal
mass at NLO will be important for narrowing the fate of the thermal
quintuplet. If we relax the thermal cosmology assumption and consider a
broader range of masses, we see the quintuplet is excluded across the full
$0.5-20~{}{\rm TeV}$ mass range. Yet this statement is contingent on the form
of $\rho_{\chi}$ adopted, about which there is considerable uncertainty. In
particular, the density profile may flatten toward the inner Galaxy. As the
annihilation signal is sensitive to $\rho_{\chi}^{2}$ flattening the profile
has a marked impact on the flux, making this one of the dominant uncertainties
in $\rho_{\chi}$ for our purposes. We parameterize a possible flattening of
the profile by replacing the Einasto density profile with a constant value for
Galactocentric distances $r<r_{c}$, where we will refer to $r_{c}$ as the
“core size”:191919To be explicit, we fix the normalization of the Einasto
profile in Eq. (90) to the solar radius before we impose the core restriction
of Eq. (91). This implies that for a core size larger than the solar radius,
the profile will predict less than 0.39 GeV/cm3 at our location. Of course,
the more significant concern is that such a large core is not consistent with
observations, as discussed in the text, and should solely be viewed as a proxy
for how much the $J$-factor needs to be reduced.
$\rho(r)=\begin{cases}\rho_{\rm Einasto}(r)&r>r_{c},\\\ \rho_{\rm
Einasto}(r_{c})&r<r_{c}.\end{cases}$ (91)
We can then ask what choice of $r_{c}$ would raise the estimated constraint on
$\langle\sigma v\rangle_{\text{line}}$ above the theoretical prediction. This
is plotted in the right-hand panel of Fig. 13, both employing our full
endpoint spectrum and in the case where we (incorrectly) use only the line
cross section in setting the bounds. To provide an estimate for what core
sizes are consistent with data, we note that simulations of Milky Way like
galaxies can generate ${\cal O}(1~{}{\rm kpc})$ cores Chan:2015tna , however,
measurements of stars in the Bulge seem to disfavor $r_{c}\gtrsim 2~{}{\rm
kpc}$ 2015MNRAS.448..713P ; Hooper:2016ggc . At the lower end of the thermal
mass range, 12.8 TeV, the thermal Quintuplet would already be in tension with
this, requiring a 2.8 kpc core. At the central (13.6 TeV) and upper (14.4 TeV)
end, however, the required core size is 1.0 and 0.5 kpc, respectively, and
therefore not obviously in tension with our mock analysis. A more recent study
claims evidence for a few kpc core that could potentially saturate the earlier
limits Ou:2023adg . If confirmed, this could suppress the $J$-factor by nearly
an order of magnitude. That would challenge the indirect-detection community
to set aggressive limits, but as we estimate, even cores of this size are in
reach of CTA and possibly H.E.S.S (cf. Fig. 14).
As already mentioned, the mock analysis we consider makes use of only a very
small amount of existing data, and with that we forecast a sensitivity to
$\langle\sigma v\rangle_{\rm line}\simeq 8.5\times 10^{-27}~{}{\rm cm^{3}/s}$
at the central thermal mass. (The error on this value due to uncertainty on
the spectrum from our NLL calculation is less than 1%, far smaller than the
variation across the thermal mass range, which is closer to 10%.) Using 500
hours of H.E.S.S. data and an identical Einasto profile, Ref.
Montanari:2022buj forecast a sensitivity at the thermal mass of
$\simeq$$9.3\times 10^{-28}~{}{\rm cm^{3}/s}$, almost a factor of ten better
than used here. This is significantly more than the naive $\sqrt{5}$ the
additional data would suggest, which can be primarily attributed to the fact
that work used H.E.S.S. II observations, whereas we adopt the sensitivity from
H.E.S.S. I, combined with a different analysis used in that work. With such a
sensitivity, we would require a core size slightly larger than $3.5~{}{\rm
kpc}$ to save the thermal quintuplet, which would be in tension with
observations. Nevertheless, repeating the process at 14.4 TeV, the required
core size would only be $1.6~{}{\rm kpc}$, and therefore not yet clearly
excluded. We can also give a crude estimate for the sensitivity the upcoming
CTA could have for the quintuplet. Although no dedicated forecast for the
quintuplet has been performed using the spectra in our work, we can estimate
the improved sensitivity as follows. Reference Baumgart:2018yed performed an
identical mock analysis to ours for the NLL wino spectrum, estimating
sensitivity at $M_{\chi}=13.6~{}{\rm TeV}$ of $\langle\sigma v\rangle_{\rm
line}\simeq 8\times 10^{-27}~{}{\rm cm^{3}/s}$, slightly stronger than the
sensitivity to the quintuplet. Using the identical NLL spectrum, Ref.
Rinchiuso:2020skh then estimated that with 500 hours of data, CTA could reach
$\simeq 1\times 10^{-28}~{}{\rm cm^{3}/s}$, a factor of eighty improvement.
Assuming the same improvement for the quintuplet, CTA would be sensitive to
$\langle\sigma v\rangle_{\rm line}\simeq 1.1\times 10^{-28}~{}{\rm cm^{3}/s}$,
excluding the thermal value by a factor of eight hundred. To not have seen the
thermal quintuplet, we would need to core $\rho_{\chi}$ out to almost
$8.6~{}{\rm kpc}$ – beyond the solar radius – which is simply inconsistent
with observations. Even at the upper end of the mass range, a core of
$6.4~{}{\rm kpc}$ would be required. In this sense, CTA would provide the
definitive word on the whether the thermal quintuplet is the DM of our
Universe.
Figure 14: An estimate for the required core size of the Einasto profile,
given no preference for a signal is seen in our H.E.S.S. mock analysis, and if
no signal emerges in an analysis of the full H.E.S.S. dataset, or at CTA. The
central values correspond to 13.6 TeV, the central thermal mass, whereas the
upper and lower error bars correspond to 12.8 and 14.4 TeV, the edges of the
thermal mass window. The dashed corresponds to the rough core size constraint
from Ref. Hooper:2016ggc . Our results suggest that H.E.S.S. can already
considerably test the quintuplet, with the final word likely being left to
CTA.
These results are summarized in Fig. 14, where the point shows the core size
required for the central thermal mass, 13.6 TeV, whereas the upper and lower
error bars corresponds to the lower and upper ends of the thermal mass range.
We also show the core size disfavored by the analysis in Ref. Hooper:2016ggc .
The figure summarizes the conclusion reached above: CTA will seemingly have
the final word of the thermal quintuplet, however if a full analysis of
H.E.S.S. data sees no sign of the signal, the model would already begin to be
disfavored. There are two important caveats to this conclusion. The first is
that our findings are based off an extrapolation from a mock analysis of
H.E.S.S. I data, and are no substitution for a full analysis or projection
using the present and forecast H.E.S.S. and CTA instrumental responses. To
give one example of what could change, an analysis that accounts for the
continuum emission could be able to even more strongly test the quintuplet.
Secondly, the range of masses we have considered is the thermal mass window of
$13.6\pm 0.8~{}{\rm TeV}$ that was determined using the LO potential, whereas
the remainder of our calculations use the NLO results, as emphasized several
times already. Updating the thermal mass using the NLO potentials will be
important. To give a sense for the impact this could have, repeating the
analysis in Fig. 14 for 13.6 TeV using the LO potential in our calculations,
the core size would change from 1.0, 3.5, and 8.5 kpc, to 0.7, 2.5, and 7.1
kpc.
## 6 Conclusions
For all the vastness of the DM parameter space, a thermal WIMP has remained a
constant focus for decades. Minimal DM is an exemplar of thermal DM, and
through indirect detection, many of the associated models are on the verge of
being detected or firmly excluded, as we have shown for the quintuplet in the
present work. Either way, these are important times in the search for DM.
With this in mind, the present work has computed the quintuplet annihilation
spectrum to NLL accuracy, and established the formalism to straightforwardly
extend this to higher odd SU(2) representations. We plot the spectrum along
with projected limits from a simple extension of the H.E.S.S. I analysis in
Fig. 13. In doing so, we have demonstrated the power of the EFT of Heavy DM,
and also extended this formalism to include the contribution from the rich set
of bound states the model contains. While the bound states can make a
significant contribution to the continuum photon emission, their impact on the
number of photons with $E_{\gamma}\sim M_{\chi}$ is minimal, except at
isolated masses. As seen in earlier studies of the wino, the same cannot be
said for endpoint photons from direct annihilation, which again provide an
${\cal O}(1)$ correction to the line signal seen in IACTs.
Taken together, we estimated that with our spectra, H.E.S.S. should almost be
able to probe the entire allowed range for the quintuplet once uncertainties
on the DM density in the inner galaxy are accounted for. Performing this
analysis using the existing data, and the soon-to-be-collected data with CTA
will be critical (cf. Fig. 14). The use of background models which enhance
sensitivity to smooth features such as the continuum as well as the full
contribution from the endpoint-photon spectrum can provide an additional piece
of experimental leverage beyond conventional line searches.
On the theory side, the thermal abundance should be recomputed using NLO
potentials, as the sensitivity to the quintuplet depends strongly on what end
of the predicted thermal mass range one sits. Finally, it will be interesting
to extend the techniques in this work to additional representations, such as a
$\mathbf{7}$ of SU(2), where we expect key features of the quintuplet such as
the strong variation in the spectrum as a function of mass, to appear and be
even more pronounced.
###### Acknowledgements.
The work we have presented benefited from useful discussions with Tobias
Binder, Marco Cirelli, Tongyan Lin, Alessandro Montanari, Emmanuel Moulin,
Nikhil Raghuram, and Diego Redigolo. MB is supported by the DOE (HEP) Award
DE-SC0019470. VV is supported by startup funds from the University of South
Dakota. TRS’ work is supported by the Simons Foundation (Grant Number 929255,
T.R.S), by the National Science Foundation under Cooperative Agreement
PHY-2019786 (The NSF AI Institute for Artificial Intelligence and Fundamental
Interactions), and by the U.S. Department of Energy, Office of Science, Office
of High Energy Physics of U.S. Department of Energy under grant Contract
Number DE-SC0012567. TRS thanks the Aspen Center for Physics, which is
supported by National Science Foundation grant PHY-2210452, for hospitality
during the completion of this work.
## Appendix
## Appendix A Quintuplet Dark Matter: A Brief Review
Here we provide a brief review of quintuplet DM, also referred to as 5-plet
electroweak DM. In particular, we firstly outline the relevant group theory
necessary to specify the interactions used for the calculations in the main
text. We will then review phenomenological aspects of the model beyond
indirect detection.
### A.1 Interactions
Quintuplet DM consists of adding to the SM five Majorana fermions that
transform together in the $\mathbf{5}$ representation of SU(2), and as a
singlet under the remaining SM forces. Above the electroweak symmetry breaking
scale, we collect the five fields into a multiplet
$\chi=(\chi^{1},\,\ldots,\,\chi^{5})^{T}$, in terms of which the DM Lagrangian
takes the following form
$\displaystyle\mathcal{L}_{\scriptscriptstyle\textrm{DM}}=\frac{1}{2}\bar{\chi}(i\not{D}-M_{\chi})\chi=\frac{1}{2}\bar{\chi}([i\not{\partial}-M_{\chi}]\mathbb{1}+g_{\scriptscriptstyle
W}T_{\mathbf{5}}^{a}\not{W}^{a})\chi.$ (92)
In the final expression, the first two contributions represent the kinetic
terms for each of the fields, which are diagonal amongst the multiplets as
indicated by $\mathbb{1}$. The final term describes the interaction between
the additional fields and the SM electroweak bosons. Importantly, we emphasize
that the interaction strength is specified by the SM SU(2) gauge coupling,
$g_{\scriptscriptstyle W}$, and is not a free parameter. Instead, $M_{\chi}$
is the unique free parameter in the theory. If we assume a conventional
thermal origin for the quintuplet, then even the mass can be fixed by the
observed relic density to $M_{\chi}=13.6\pm 0.8~{}{\rm TeV}$ Mitridate:2017izz
; Bottaro:2021snn .202020As emphasized above, this value was computed with the
LO electroweak potential. Redoing the analysis with the NLO potential would
reduce an important theoretical uncertainty. The 13.6 TeV quintuplet is
therefore a zero parameter DM model.
After electroweak symmetry breaking, the five Majorana fermions rearrange
themselves into three mass eigenstates: a Majorana neutral fermion $\chi^{0}$,
and two charged Dirac fermions $\chi^{+}$ and $\chi^{++}$. At leading order,
each of these states maintains a mass of $M_{\chi}$. However, radiative
corrections to the charged states break this degeneracy, raising the masses of
the charged fermions, singling out $\chi^{0}$ as the lightest state and the DM
candidate. These corrections have been computed, and in detail
$\delta_{0}=M_{\chi^{+}}-M_{\chi^{0}}\simeq 164~{}{\rm MeV}$, and
$\delta_{+}=M_{\chi^{++}}-M_{\chi^{+}}=3\delta_{0}$ Cirelli:2005uq ;
Ibe:2012sx . For most aspects of our calculations these mass splittings will
be irrelevant and we will take $\delta_{0}\simeq\delta_{+}\simeq 0$. However,
we do include them in the electroweak potential used to compute Sommerfeld
factors, and scattering- & bound-state wavefunctions. This is done by adding
$2\delta_{0}$ to the diagonal term in the potential matrix correspondingto the
$\chi^{+}\chi^{-}$ component of any state, $8\delta_{0}$ to the diagonal term
corresponding to the $\chi^{++}\chi^{--}$ component, and similar, appropriate
shifts for diagonal elements in the potential matrix corresponding to
components of $Q\neq 0$ states (the shift is given by the difference between
the rest mass of the state constituents and $\chi^{0}\chi^{0}$).
We now turn to the interaction term in Eq. (92),
$\tfrac{1}{2}g_{\scriptscriptstyle
W}\bar{\chi}T_{\mathbf{5}}^{a}\not{W}^{a}\chi$. Here $a=1,2,3$ indexes the
electroweak gauge bosons, which transform together in an adjoint of SU(2). In
the broken theory, these are mapped to the charge and mass eigenstates in the
usual way,
$\displaystyle
W^{1}_{\mu}=\frac{1}{\sqrt{2}}(W^{+}_{\mu}+W^{-}_{\mu}),\;\;W^{2}_{\mu}=\frac{i}{\sqrt{2}}(W^{+}_{\mu}-W^{-}_{\mu}),\;\;W^{3}_{\mu}=s_{\scriptscriptstyle
W}A_{\mu}+c_{\scriptscriptstyle W}Z_{\mu}.$ (93)
The only part of Eq. (92) that remains undetermined is $T_{\mathbf{5}}^{a}$,
the three generators of SU(2) in the quintuplet representation. A convenient
basis in which to specify $T_{\mathbf{5}}^{a}$ is the basis of charged states
discussed above, where the DM can be cleanly identified. We can determine the
charged states through their couplings to the bosons in Eq. (93). In
particular, as $A_{\mu}$ couples to charge, we can read off the charges of the
states as soon as we know $T^{3}$. It will also be convenient to introduce
$T^{\pm}=\frac{1}{\sqrt{2}}(T^{1}\pm iT^{2}),$ (94)
in terms of which $T^{a}W^{a}=T^{+}W^{+}+W^{-}T^{-}+W^{3}T^{3}$, independent
of representation.
Before evaluating the generators for the quintuplet, let us review how the
argument proceeds for the simpler case of the wino—a triplet or $\mathbf{3}$
of SU(2). In this case, it is conventional to exploit the fact that the
generators are given by the structure constants of SU(2), so that
$\bar{\chi}T_{\mathbf{3}}^{a}\gamma^{\mu}\chi=\bar{\chi}_{b}(T_{\mathbf{3}}^{a})_{bc}\gamma^{\mu}\chi_{c}=-i\epsilon_{abc}\bar{\chi}_{b}\gamma^{\mu}\chi_{c}.$
(95)
This approach depended on the fact we already knew a representation of the
generators for the adjoint, and so does not simply generalise to larger
representations. However, we can derive a representation more systematically
as follows. Recall that one representation of $\mathbf{n}$ in SU(2) is an
$n-1$ index symmetric tensor, with each index transforming in the fundamental.
In this representation we denote the adjoint as $\chi^{ij}$, and the
quintuplet as $\chi^{ijkl}$, where the indices take values 1 and 2. Beginning
with the wino, $\chi^{ij}$ has three unique components, which we can embed
into a vector as212121The $\sqrt{2}$ ensures that $\chi$ as it appears in
$\bar{\chi}\not{\partial}\chi/2=\bar{\chi}^{ij}\not{\partial}\chi^{ij}/2$ is
canonically normalized. An identical argument explains the coefficients in Eq.
(100).
$\chi=\begin{pmatrix}\chi^{1}\\\ \chi^{2}\\\
\chi^{3}\end{pmatrix}=\begin{pmatrix}\chi^{11}\\\ \sqrt{2}\chi^{12}\\\
\chi^{22}\end{pmatrix}\\!.$ (96)
We can use this representation to explicitly construct $T^{a}_{\mathbf{3}}$ as
follows. The key is that we know exactly how the generators act on $\chi^{ij}$
as each index transforms in the fundamental. Accordingly,
$T^{a}(\chi^{ij})=[(T^{a}_{F})^{i}_{k}\delta^{j}_{l}+\delta^{i}_{k}(T^{a}_{F})^{j}_{n}]\chi^{kl},$
(97)
where $T^{a}_{F}=\sigma^{a}/2$, with $\sigma^{a}$ the Pauli matrices. Consider
a generic infinitesimal transformation, $U=1+iu$, with $u=u_{a}T^{a}$. If we
take $u^{a}=(0,0,\kappa)$ with $\kappa\ll 1$, then the components of $\chi$
transform as
$\displaystyle\delta\chi^{1}=i\kappa\chi^{1},\;\;\delta\chi^{2}=0,\;\;\delta\chi^{3}=i\kappa\chi^{3}\\!.$
(98)
From this, we can read off the action of an infinitesimal $\mathbf{U}$ on
$\chi$, and hence infer that we must have
$T^{3}_{\mathbf{3}}={\rm diag}(+1,\,0,\,-1).$ (99)
We can now identify $\chi^{1}$, $\chi^{2}$, and $\chi^{3}$ as having charges
$+1$, $0$, and $-1$, yielding the expected spectrum in the broken
phase.222222The representation in the charge basis is not unique. For
instance, the transformation $\chi^{1,2}\to e^{\pm i\phi}\chi^{1,2}$ leaves
the charge assignments unchanged, but will introduce a phase into the off-
diagonal $W^{\pm}$ couplings. The same will be true for the off-diagonal
quintuplet couplings. The remaining components of $T^{a}_{\mathbf{3}}$ can be
derived identically.
This approach readily generalizes to the quintuplet. The five unique
components of $\chi^{ijkl}$ can be embedded into a vector as follows,
$\chi=\begin{pmatrix}\chi^{1}\\\ \chi^{2}\\\ \chi^{3}\\\ \chi^{4}\\\
\chi^{5}\end{pmatrix}=\begin{pmatrix}\chi^{1111}\\\ 2\chi^{1112}\\\
\sqrt{6}\chi^{1122}\\\ 2\chi^{1222}\\\
\chi^{2222}\end{pmatrix}=\begin{pmatrix}\chi^{++}\\\ \chi^{+}\\\ \chi^{0}\\\
\chi^{-}\\\ \chi^{--}\end{pmatrix}\\!.$ (100)
To justify the charge assignments, we repeat the above analysis to find
$T^{3}_{\mathbf{5}}={\rm diag}(+2,\,+1,\,0,\,-1,\,-2).$ (101)
Further, we can compute
$\displaystyle T^{+}_{\mathbf{5}}=\begin{pmatrix}0&\sqrt{2}&0&0&0\\\
0&0&\sqrt{3}&0&0\\\ 0&0&0&\sqrt{3}&0\\\ 0&0&0&0&\sqrt{2}\\\
0&0&0&0&0\end{pmatrix}\\!,$ (102)
and $T^{-}_{\mathbf{5}}=(T^{+}_{\mathbf{5}})^{T}$. In the main body we will
exclusively work in the charged basis of Eq. (100), using the form of
$T_{\mathbf{5}}^{a}$ as above whenever necessary.
### A.2 Phenomenology
We end this section with a brief description of quintuplet phenomenology
beyond indirect detection. As already noted, the case where $M_{\chi}\simeq
13.6$ TeV is particularly appealing, as this is the mass singled out from a
conventional cosmology via a WIMP-miracle-style argument. The interactions of
the quintuplets, reviewed in the previous subsection, are sufficient to keep
it in thermal equilibrium in the early Universe. As the Universe cools,
eventually the quintuplet undergoes a conventional freeze-out. When this
occurs dictates the final abundance, and matching to the observed DM density
fixes the single parameter of the theory, $M_{\chi}$. For larger (smaller)
values of $M_{\chi}$ than 13.6 TeV, one naively over (under) produces the
observed DM density. Nevertheless, it is entirely possible that there are
effects in the early Universe that modify this simple picture. The presence of
additional beyond-the-SM states that either decay to the SM or directly to the
quintuplet can dilute or increase its abundance, respectively, making a wider
mass range viable. Further, for $M_{\chi}\leq 13.6$ TeV, even with no
additional states, the quintuplet would represent a well motivated (and
predictable) fraction of DM. For these reasons, in the main body we considered
a wide range of quintuplet masses, although we emphasize once more that the
scenario where $M_{\chi}\simeq 13.6$ TeV is compelling.
As with any DM candidate, one can also consider searching for the quintuplet
with either direct detection or at a collider. For direct detection, as the
quintuplet carries no U(1) hypercharge, it does not couple to the $Z$ boson at
tree level. Nevertheless, couplings to SM nucleons can arise at loop level.
The spin-independent cross-section is $\sigma\simeq(1.0\pm 0.3)\times
10^{-46}~{}{\rm cm}^{2}$ Bottaro:2021snn (see also Ref. Hisano:2015rsa ;
Chen:2023bwg ), beyond the reach of current searches at 13.6 TeV, and even
next generation instruments such as LZ Mount:2017qzi . Nevertheless, the
cross-section is a factor of $\sim 4$ above the neutrino floor, and is in
reach of generation-3 instruments such as DARWIN DARWIN:2016hyl .
Detection at colliders is similarly challenging, but also potentially within
reach of future instruments. Existing LHC searches reach masses of $\sim
270~{}\text{GeV}$; even the high-luminosity dataset will only reach to $\sim
520~{}\text{GeV}$ Ostdiek:2015aga . Even future hadron colliders are unlikely
to reach the thermal mass. A future 100 TeV hadron collider will just be able
to reach the thermal masses for canonical neutralinos such as the higgsino and
wino Cirelli:2014dsa ; Low:2014cba . Given these two candidates have
significantly lower thermal masses of 1 and 2.9 TeV respectively Hisano:2006nn
; Cirelli:2007xd ; Hryczuk:2010zi ; Beneke:2016ync , the prospects of probing
the 13.6 TeV quintuplet appear discouraging. Future muon colliders operating
at lower center-of-mass energies could reach the quintuplet, although they
would still need to obtain $\sqrt{s}\simeq 35~{}{\rm TeV}$ Bottaro:2021snn
(see also Refs. Han:2020uak ; Han:2022ubw ). Taken together, in the short term
indirect detection remains the most likely avenue for probing quintuplet DM.
## Appendix B Operators for higher-$L$ Bound State annihilation
As argued in Sec. 4.1, the higher-$L$ bound states preferentially decay to the
deeper bound states with lower $L$ instead of directly annihilating to SM
particles. Nevertheless, for completeness here we provide the complete set of
relevant operators up to ${\cal O}(v)$.
We consider the structure of the operators which support up to $p$-wave bound
states. Thus we need to keep operators (at the amplitude level) suppressed by
at most one power of the DM 3-momentum; the $\mathcal{O}(v^{0})$ operators
will support both the direct annihilation as well as $s$-wave bound state
annihilation. By matching to the full tree level amplitude, we can obtain the
structure that supports $S=0,\,L=1$ states,232323Throughout this appendix we
will work with 4-component DM fields and not reduce it to 2 components as was
done for the $L=S=0$ operator in the main text.
$\mathcal{O}_{1}=\mathbf{v}_{\chi}\cdot\mathbf{n}\left(\bar{\chi}\Big{[}T^{a},T^{b}\Big{]}\gamma^{0}\gamma^{5}\chi\right)i\epsilon^{ijk}(n-\bar{n})^{k}{\cal
B}_{\perp n}^{i,a}{\cal B}_{\perp\bar{n}}^{j,b},$ (103)
where it is understood that the subscript $\chi$ on $\mathbf{v}_{\chi}$
indicates that the velocity vector is the velocity of the state
created/annihilated by the $\chi$ field (as opposed to the $\bar{\chi}$
field). As is evident, this operator supports a bound state with $L=1$ and
$S=0$. Likewise we can write down the next set of operators,
$\displaystyle\mathcal{O}_{2}$
$\displaystyle=\mathbf{v}_{\chi}\cdot\mathbf{n}\left(\bar{\chi}\Big{\\{}T^{a},T^{b}\Big{\\}}\gamma^{i}\chi\right)(n-\bar{n})^{i}{\cal
B}_{\perp n}^{\mu,a}{\cal B}^{b}_{\perp\bar{n}\mu},$ (104)
$\displaystyle\mathcal{O}_{3}$ $\displaystyle=\mathbf{v}_{\chi}\cdot{\cal
B}^{a}_{\perp n}{\cal
B}^{b}_{\perp\bar{n}\mu}\left(\bar{\chi}T^{b}T^{a}\gamma^{\mu\perp}\chi\right)\\!,$
$\displaystyle\mathcal{O}_{4}$ $\displaystyle=\mathbf{v}_{\chi}\cdot{\cal
B}^{b}_{\perp\bar{n}}{\cal B}^{a}_{\perp
n\mu}\left(\bar{\chi}T^{a}T^{b}\gamma^{\mu\perp}\chi\right)\\!.$
From their Dirac structure, it is clear that these operators support $L=1,S=1$
bound states. There is also another operator which supports an $L+S$ odd bound
state, specifically the $L=1$, $S=0$ bound state which arises out of a
correction to an ultra-soft gauge boson emission off the heavy DM particles.
The details of this operator are involved and hence are separately given
further below.242424A complete anaysis and resummation of this channel at NLL
would require us to do a full two loop computation in order to recover the
anomalous dimension as well matching to stage II of the EFT. We leave this
work for the future.
We note that decays of SU(2)-singlet bound states with $L+S$ odd into two
(transverse) gauge bosons are constrained by charge conjugation invariance and
parity. This is because the underlying Lagrangian in Eq. (92) which couples
the electroweak gauge sector and a Majorana quintuplet fermion is $C$ and $P$
invariant. A fermion-antifermion bound state has $C$ eigenvalue $(-1)^{L+S}$
and $P$ eigenvalue $(-1)^{L+1}$. A $\gamma\gamma$, $\gamma Z$, or $ZZ$ final
state has $C$ eigenvalue $+1$, thereby forbidding an $L+S$ odd bound state
decay into them by $C$ alone. Decays to the $W^{+}W^{-}$ final state are
allowed regardles of the $L,\,S$ quantum numbers of the bound state.252525One
might think the Landau-Yang theorem also forbids decay from $L+S$-odd initial
states into two bosons at all orders, but the application of the theorem to
non-Abelian theories is subtle; there is a generalized Landau-Yang theorem
that holds so long as the decay products are in a color-singlet state
Beenakker:2015mra , but this is not the case for the decay of $L+S$-odd
states. In Ref. Asadi:2016ybp , one can see explicitly that for wino-onium,
decays to $W^{+}W^{-}$ are allowed for all combinations of initial-state $L$
and $S$. In all cases, decays to longitudinal gauge bosons are potentially
allowed, as well.
Finally, let us return to the additional operator which is sub-leading in
velocity that we can write down by looking at the emission of another gauge
boson which usually contributes to the $Y_{v}$ Wilson line. We do not get any
such correction from the $Y_{n}$ or the $Y_{\bar{n}}$ Wilson line since we do
not wish to consider sub-leading terms in the SCET power counting parameter.
We start again with our ${\cal O}(v^{0})$ operator at the amplitude level
before we dress it with soft Wilson lines
$\displaystyle\bar{\chi}\\{T_{\chi}^{a},T_{\chi}^{b}\\}\Gamma\chi{\cal
B}_{\perp n}^{a}{\cal B}^{b}_{\perp\bar{n}},$ (105)
where $\Gamma$ is an arbitrary Lorentz structure. Consider the emission of an
SCET ultra-soft gauge boson of momentum k off the initial $\chi$ or
$\bar{\chi}$ particle. First looking at the $\chi$ particle,
$\displaystyle
u_{e}(\tilde{p})+ig\frac{i(\not{p}-\not{k}+M_{\chi})}{(p-k)^{2}-M_{\chi}^{2}+i\epsilon}\gamma^{\mu}\epsilon_{\mu}^{a}(T_{\chi}^{a})_{ce}u_{e}(p)$
(106)
$\tilde{p}$, p is the momentum of the $\chi$ particle. Let us now expand this
result to $\mathcal{O}(v)$ (except inside the spinor). We see that apart from
the usual Wilson line contribution, we also get two other relevant terms,
$\displaystyle u_{e}(\tilde{p})-g_{\scriptscriptstyle
W}(T_{\chi}^{a})_{ce}\left(-\frac{\epsilon_{0}^{a}}{k_{0}}+\frac{\mathbf{v}\cdot\mathbf{\epsilon}^{a}(k)}{k_{0}}-\frac{\epsilon_{0}^{a}(k)\mathbf{v}\cdot\mathbf{k}}{k_{0}^{2}}\right)u_{e}(M_{\chi}).$
(107)
If we sum the infinite series of gauge bosons with one of the propagators
expanded out to $\mathcal{O}(v)$, we then have a structure
$\displaystyle\chi\rightarrow Y_{v}\mathbf{v}\cdot\mathbf{B_{s}}\chi$ (108)
where we have defined the following Hermitian operator,
$\displaystyle\mathbf{B}_{s}=\frac{Y^{\dagger}_{v}\left(\mathbf{\mathcal{P}}-g_{\scriptscriptstyle
W}\mathbf{A}_{s}\right)Y_{v}}{v\cdot\mathcal{P}},$ (109)
where $\mathcal{P}$ is the momentum label operator. If this term is included
in a larger expression, the $v\cdot\mathcal{P}$ factor in the denominator only
acts on the terms in the numerator while the one in the numerators acts only
on the $Y_{v}$ Wilson line to the right. We can now combine this with the
emissions off $\bar{\chi}$ to give us an effective operator, now dressed with
soft Wilson lines
$\displaystyle\bar{\chi}\Big{\\{}Y_{v}^{\dagger}\\{T^{a},T^{b}\\}\Gamma
Y_{v},\mathbf{v}\cdot\mathbf{B}_{s}\Big{\\}}\chi{\cal B}_{\perp
n}^{a^{\prime}}{\cal
B}^{b^{\prime}}_{\perp\bar{n}}Y_{n}^{aa^{\prime}}Y_{\bar{n}}^{bb^{\prime}}.$
(110)
We can once again use our Wilson line identity to write this in terms of our
usual soft function
$\displaystyle\bar{\chi}\Big{\\{}\\{T^{a},T^{b}\\}\Gamma,\mathbf{v}\cdot\mathbf{B}_{s}\Big{\\}}\chi{\cal
B}_{\perp n}^{c}{\cal B}^{d}_{\perp\bar{n}}Y^{abcd}.$ (111)
We can then formally define a new object $\mathbfcal{B}_{s}$ to explicitly
separate out all the soft fields from the heavy DM fields
$\displaystyle\mathbfcal{B}_{s}^{a}T^{a}=\mathbf{B}_{s}.$ (112)
Then we see that
$\displaystyle\mathbfcal{B}_{s}^{a}=\mathrm{Tr}[\mathbf{B}_{s}T^{a}],$ (113)
where we have used $\mathrm{Tr}[T^{a}T^{b}]=\delta^{ab}$. Our operator now
becomes
$\displaystyle\bar{\chi}\Big{\\{}\\{T^{a},T^{b}\\}\Gamma,T^{e}\Big{\\}}\chi\,{\cal
B}_{\perp n}^{c}{\cal
B}^{d}_{\perp\bar{n}}Y^{abcd}\mathbf{v}\cdot\mathbfcal{B}_{s}^{e}.$ (114)
In summary, we have an additional soft function which is explicitly suppressed
in velocity. The Dirac structure for this operator is $\gamma^{0}\gamma^{5}$
so that this is the only operator that supports an L+S odd bound state.
However, it is clear that the soft operator only begins at one loop and hence
we are always forced into at least a 3 gauge boson final state.
At the amplitude squared level, we simply have a square of this operator and
there will no interference terms with other bound state operators since all
other operators support an L+S even bound state. Let us consider the soft
operator,
$\displaystyle S^{abea^{\prime}b^{\prime}e^{\prime}}=\langle
0|(Y^{a^{\prime}b^{\prime}3d}\mathbf{v}\cdot\mathbfcal{B}_{s}^{e^{\prime}})^{\dagger}\mathcal{M}|X_{s}\rangle\langle
X_{s}|Y^{ab3d}\mathbf{v}\cdot\mathbfcal{B}_{s}^{e}|0\rangle,$ (115)
where $\mathcal{M}$ is the measurement performed on the soft operator and
$X_{s}$ are the soft modes. This soft operator now has 6 free indices which
must be contracted into the DM wavefunction factor. Since the soft final state
is completely inclusive, we can simplify the operator as
$\displaystyle S^{abea^{\prime}b^{\prime}e^{\prime}}=|\mathbf{v}|^{2}\langle
0|(Y^{a^{\prime}b^{\prime}3d}\mathbfcal{B}_{s}^{e^{\prime}i})^{\dagger}\mathcal{M}|X_{s}\rangle\langle
X_{s}|Y^{ab3d}\mathbfcal{B}_{s}^{ei}|0\rangle.$ (116)
We can now move the $|\mathbf{v}|^{2}$ factor to the DM wavefunction and
instead redefine a soft operator
$\displaystyle S^{abea^{\prime}b^{\prime}e^{\prime}}=\langle
0|(Y^{a^{\prime}b^{\prime}3d}\mathbfcal{B}_{s}^{e^{\prime}i})^{\dagger}\mathcal{M}|X_{s}\rangle\langle
X_{s}|Y^{ab3d}\mathbfcal{B}_{s}^{ei}|0\rangle.$ (117)
The only term that contributes at one loop is the $\mathbfcal{B}$ since it is
0 at ${\cal O}(\alpha_{\scriptscriptstyle W}^{0})$. So we can set all other
Wilson lines to their tree level values. Explicitly,
$\displaystyle
Y^{ab3d}\big{|}_{\textrm{tree}}=(Y_{v}^{fa}Y_{n}^{f3})(Y_{v}^{gb}Y_{\bar{n}}^{gd})\big{|}_{\textrm{tree}}=\delta^{a3}\delta^{bd}.$
(118)
Thus, our operator at one loop becomes
$\displaystyle S^{abea^{\prime}b^{\prime}e^{\prime}}_{1\mathchar
45\relax\mathrm{loop}}=\delta^{a3}\delta^{a^{\prime}3}\delta^{bb^{\prime}}\langle
0|(\mathbfcal{B}_{s}^{e^{\prime},i})^{\dagger}\mathcal{M}|X_{s}\rangle\langle
X_{s}|\mathbfcal{B}_{s}^{e,i}|0\rangle.$ (119)
We will only calculate this operator to one loop to elucidate its properties.
The one-loop integrand up to an overall factor take the form,
$\displaystyle I=$ $\displaystyle 2\delta^{ee^{\prime}}g_{\scriptscriptstyle
W}^{2}\int\frac{d^{d}k}{(2\pi)^{(d-1)}}\frac{\delta^{+}(k^{2}-M_{\chi}^{2})\delta(q^{+}-k^{+})}{k_{0}^{2}}$
(120) $\displaystyle-$ $\displaystyle\delta^{ee^{\prime}}g_{\scriptscriptstyle
W}^{2}\int\frac{d^{d}k}{(2\pi)^{(d-1)}}\frac{\delta^{+}(k^{2}-M_{\chi}^{2})\delta(q^{+}-k^{+})k^{2}}{k_{0}^{4}}.$
where $q^{+}$ is the contribution to the final photon momentum from the soft
function. The second term is proportional to $M_{\chi}^{2}$ and gives a power
correction in $M_{\chi}^{2}/(q^{+})^{2}$ and hence can be ignored. The first
term does not give a UV divergence, but will contribute a log which will be
relevant for stage 2 of the EFT. In detail,
$2\delta^{ee^{\prime}}g_{\scriptscriptstyle
W}^{2}\int\frac{d^{d}k}{(2\pi)^{(d-1)}}\frac{\delta^{+}(k^{2}-M_{\chi}^{2})\delta(q^{+}-k^{+})}{k_{0}^{2}}=2\delta^{ee^{\prime}}\frac{\alpha_{\scriptscriptstyle
W}}{\pi}\frac{q^{+}}{(q^{+})^{2}+M_{\chi}^{2}}.$ (121)
Going to Laplace space and expanding out in the limit $M_{\chi}\rightarrow 0$,
we have
$\displaystyle I=-2\delta^{ee^{\prime}}\frac{\alpha_{\scriptscriptstyle
W}}{\pi}\ln(M_{\chi}se^{\gamma_{E}}).$ (122)
This result has an IR divergence. At first glance, this may not be that
surprising since all our soft operators in the direct-channel annihilation
also were IR divergent. In those cases, we could trace back the IR divergence
to the violation of the KLN theorem due to the semi-inclusive nature of the
final state. In this case however, the interesting point is that, even when
the final state is completely inclusive, i.e., we do not constrain the final
state to be just a photon, our soft function is still IR divergent. Here we
can trace this to the exclusive nature of the initial state where we demand
that our operator support an $L=1$, $S=0$ state. This forces the emission of
the 3rd gauge boson which has no virtual counterpart, leading to an IR
divergence. This is very similar to the IR divergence that appears in the
computation of PDFs in QCD.
## Appendix C Unstable Particle Effective Theory
In this section, we justify our use of Eq. (61) for computing the decay rate
of bound states. Let us now look at the effective theory for resonances
systematically. In the literature, this is referred to as unstable particle
effective theory (for a review see Ref. Beneke:2015vfa ).
To begin with, if we have an intermediate resonance state we expect a
propagator in our amplitude of the form
$\displaystyle D=\frac{i}{p^{2}-M_{*}^{2}},$ (123)
where $M_{*}$ is a complex pole. If we write $M_{*}=M+i\Gamma/2$, then we have
the result
$\displaystyle D=\frac{i}{p^{2}-M^{2}-i\Gamma M+\Gamma^{2}/4}.$ (124)
We wish to work in a regime of narrow width, i.e. $p^{2}-M^{2}\sim\Gamma M\ll
M^{2}$, so that the propagator is well approximated by
$\displaystyle D\simeq\frac{i}{p^{2}-M^{2}-i\Gamma M},$ (125)
and we want to develop an effective theory with an expansion in the small
parameter $\lambda=\Gamma/M$. For the case of inclusive decays of our DM bound
state, $\Gamma M_{\chi}\sim\alpha_{\scriptscriptstyle W}^{5}M_{\chi}^{2}$; we
are interested in the inclusive decay rate, so there are no large logarithms
and the perturbative cross section begins at $\alpha_{\scriptscriptstyle
W}^{2}$, and due to the nontrivial wavefunction, we have an additional factor
of at least $\alpha_{\scriptscriptstyle W}^{3}$. The effective coupling thus
scales as $\lambda\sim\alpha_{\scriptscriptstyle W}^{5}\ll 1$.
The hard scale in this process is just the resonance mass, which in our case
is simply $\sim$$M_{\chi}$. We can now treat this as an HQET theory writing
$p=M_{\chi}v+k$, where $v$ is the four-velocity of the resonance with
$v^{2}=1$ and $k$ is the residual momentum. Given the scaling above of
$p^{2}-M^{2}\sim\Gamma M$, we can immediately see that $k\sim\Gamma$, so that
we have a soft mode $k^{\mu}\sim(\Gamma,\Gamma,\Gamma)\equiv
M_{\chi}(\alpha_{\scriptscriptstyle W}^{5},\alpha_{\scriptscriptstyle
W}^{5},\alpha_{\scriptscriptstyle W}^{5})$. Obviously, the question is how
does this scale relate to the mass scale $m_{\scriptscriptstyle W}\sim
vM_{\chi}$ that we already have. Now the $\alpha_{\scriptscriptstyle W}$ in
the decay rate is evaluated at the scale $M_{\chi}$ while the
$\alpha_{\scriptscriptstyle W}$ in the HQET scaling is at the scale
$m_{\scriptscriptstyle W}$; however, we can see by the equations relating
$\alpha_{\scriptscriptstyle W}$ at the two scales that they are parametrically
of the same order. If that is the case, then our mode has $k^{\mu}\sim
M_{\chi}(\alpha_{\scriptscriptstyle W}^{5},\alpha_{\scriptscriptstyle
W}^{5},\alpha_{\scriptscriptstyle W}^{5})$ with $k^{2}\ll
m_{\scriptscriptstyle W}^{2}$ and hence can only be populated by a massless
mode such as the photon.
Given the fact that in our case $\lambda\sim\alpha_{\scriptscriptstyle
W}^{5}$, any corrections of the order $\lambda^{1}$ are minute and will be
sub-dominant in any error band at the accuracy we are aiming for. So we can
safely work at leading order in $\lambda$. Following Beneke:2015vfa , it is
clear that at this order the only term that exists is the propagator with the
1PI self-energy corrections. Any communication between the production and
decay states via radiative emissions of our mode $k^{\mu}$ only occurs at
$\mathcal{O}(\lambda)$ and hence is severely suppressed. Additionally, since
$\Gamma\sim\alpha_{W}^{5}M_{\chi}$, but the splitting between bound states,
$\Delta E_{n}\sim\alpha_{W}^{2}M_{\chi}$, any interference between bound
states is also subleading. Therefore, we can safely ignore any radiative
corrections by this mode. This suppression is a manifestation of the length
separation in space-time between the process of production and decay.
Now let us look at a cross-section for production of a resonance and its
subsequent decay. We assume that we are in a regime $p^{2}-M^{2}\sim\Gamma M$,
where $p$ is the momentum of the intermediate resonance state and $M$ is the
real part of the pole. Let $N$ be our initial scattering state that will
create the resonance, and we focus on the cross section to produce an observed
final state $f$ and an ultrasoft photon $\gamma_{\textrm{us}}$ indicating that
a bound state was formed. In detail, the differential cross section is
$\displaystyle\frac{d\sigma}{dz}=\frac{1}{\mathcal{N}}\int
d\Pi_{\gamma}d\Pi_{f}|{\cal M}(N\rightarrow
f+\gamma_{\textrm{us}})|^{2}\delta^{(4)}(p_{\gamma}+p_{f}-p_{N})\mathcal{M}_{z}(f),$
(126)
where ${\cal M}_{z}$ is the measurement (in this case the photon energy)
function on the final state particles, and ${\cal N}$ is a normalizing
kinematic factor. Since the photon emitted during the bound state formation is
an ultrasoft photon, there is no measurement on it (via a multipole expansion
of the measurement function) and its phase space is integrated over fully. The
amplitude squared will contain the squared propagator for the resonance,
$\displaystyle J=\frac{1}{(p^{2}-M^{2})^{2}+\Gamma^{2}M^{2}}.$ (127)
Now, the key point is that if we are not interested in the details of the
variation of the cross section near resonance, and we are sufficiently
inclusive over $p^{2}$ around the resonance (by a value $\gg\Gamma M$), then
we can make the following substitution
$\displaystyle\lim_{\Gamma/M\rightarrow 0}J\rightarrow\frac{\pi}{\Gamma
M}\delta(p^{2}-M^{2}).$ (128)
This substitution is true only in the distribution sense, i.e. under the
integral which at least encompasses the region of the size of the width about
the resonance. This is the narrow width approximation. This substitution then
puts the intermediate resonance on-shell. The cross section can then be
written as
$\displaystyle\frac{d\sigma}{dz}$
$\displaystyle=\frac{1}{\mathcal{N}}\frac{\pi}{\Gamma M}\int
d\Pi_{\gamma}d\Pi_{f}|{\cal M}(N\rightarrow
B(p)+\gamma_{\textrm{us}})|^{2}|{\cal M}(B(p)\rightarrow f)|^{2}$ (129)
$\displaystyle\times\delta^{(4)}(p_{\gamma}+p_{f}-p_{N})\delta(p^{2}-M^{2})\mathcal{M}_{z}(f).$
Here $B(p)$ represents a bound state with momentum $p$, and again this result
is true at leading order in $\Gamma/M$, which forbids any communication
between the production and decay states. If we then insert a factor of unity,
$1=\int d^{4}p\,\delta^{(4)}(p-p_{f})$, the result can be rearranged to yield,
$\displaystyle\frac{d\sigma}{dz}=$
$\displaystyle\frac{1}{\mathcal{N}}\frac{\pi}{M}\Big{[}\int
d\Pi_{\gamma}d\Pi_{R}|{\cal M}(N\rightarrow
B(p)+\gamma_{\textrm{us}})|^{2}\delta^{(4)}(p_{\gamma}+p-p_{N})\Big{]}$ (130)
$\displaystyle\times$ $\displaystyle\frac{1}{\Gamma}\Big{[}\int d\Pi_{f}|{\cal
M}(B(p)\rightarrow f)|^{2}\delta^{(4)}(p-p_{f})\mathcal{M}_{z}(f)\Big{]}$
$\displaystyle=$ $\displaystyle\sigma(N\rightarrow
B+\gamma_{\textrm{us}})\frac{1}{\Gamma}\frac{d\Gamma_{B\rightarrow f}}{dz},$
which is simply the product of the production cross section and the
differential branching ratio. The above separation holds where the process
proceeds solely through the long lived bound state, but in practice the result
should be summed over all bound states in the spectrum, as well as the direct
annihilation case where no bound states are formed. Applied to our specific
case we arrive at Eq. (61).
## Appendix D Proof of the Wilson Line Identity
In this section, we prove the identity involving soft Wilson lines used in Eq.
(10) that eventually leads to a universal factorization of the IR physics in
terms of soft and jet functions independent of the representation. The
property that we wish to show is
$\displaystyle S_{v}T^{a}S_{v}^{\dagger}=T^{a^{\prime}}S_{v}^{a^{\prime}a},$
(131)
where $T$ is a generator in an arbitrary representation (we used $\mathbf{5}$
for the quintuplet), and on the left hand side we have two Wilson lines in the
same representation, whereas on the right it is in the adjoint. (In the main
text we used $Y_{v}$ for the latter, we keep all as $S$ here for notational
convenience.) In the main text we actually used
$S_{v}^{\dagger}T^{a}S_{v}=S_{v}^{aa^{\prime}}T^{a^{\prime}}$, although this
follows from the above by applying various inverses. In position space, the
Wilson lines are defined as
$\displaystyle S_{v}(x)$ $\displaystyle=Pe^{ig\int_{-\infty}^{x}ds\,v\cdot
A_{s}(vs)}=Pe^{ig\int_{-\infty}^{\bar{v}\cdot x}d\bar{v}\cdot y\,v\cdot
A_{s}(\bar{v}\cdot y)},$ (132) $\displaystyle S_{v}^{\dagger}(x)$
$\displaystyle=\bar{P}e^{-ig\int_{-\infty}^{x}ds\,v\cdot A_{s}(vs)}.$
where $v\cdot A_{s}=v\cdot A_{s}^{a}T^{a}$, with $T^{a}$ in the appropriate
representation for the Wilson line, whilst $P$ is path ordering and $\bar{P}$
indicates anti-path ordering. The variable $s$ parametrizes the path along the
light cone direction $n$ from $x$ to $-\infty$. The statement in Eq. (131) is
a generalization of the identity applied for QCD Becher:2014oda for other
group representations. The soft Wilson line obeys the equation
$\displaystyle\frac{d}{d\bar{v}\cdot x}S_{v}(x)$ $\displaystyle=ig\,v\cdot
A_{s}(\bar{v}\cdot x)S_{v}(x),$ (133) $\displaystyle\frac{d}{d\bar{v}\cdot
x}S^{\dagger}_{v}(x)$ $\displaystyle=-S_{v}^{\dagger}(x)ig\,v\cdot
A_{s}(\bar{v}\cdot x).$
Coming back to our question, let us define
$U^{a}(x)=S_{v}(x)T^{a}S_{v}^{\dagger}(x)$. We can then immediately see
$\displaystyle\frac{d}{d\bar{v}\cdot x}U^{a}(x)=\left[igv\cdot
A_{s}(\bar{v}\cdot x),U^{a}(x)\right]\\!.$ (134)
We will solve this equation by recursion, order by order in the coupling $g$
to build up the full solution. The tree level result is simply
$U^{a(0)}(x)=T^{a}=T^{a^{\prime}}\delta^{a^{\prime}a}$. At the next order,
$\displaystyle U^{a(1)}(x)$ $\displaystyle=\int_{-\infty}^{\bar{v}\cdot
x}d\bar{v}\cdot y\left[ign\cdot A_{s}(\bar{v}\cdot y),T^{a}\right]$ (135)
$\displaystyle=ig\int_{-\infty}^{\bar{v}\cdot x}d\bar{v}\cdot y\,v\cdot
A_{s}^{b}(\bar{v}\cdot y)if^{baa^{\prime}}T^{a^{\prime}}$
$\displaystyle=T^{a^{\prime}}ig\int_{-\infty}^{\bar{v}\cdot x}d\bar{v}\cdot
y\,v\cdot(A_{s}(\bar{v}\cdot y))^{a^{\prime}a}.$
At ${\cal O}(g^{2})$,
$\displaystyle U^{a(2)}(x)$ $\displaystyle=\int_{-\infty}^{\bar{v}\cdot
x}d\bar{v}\cdot y_{1}\int_{-\infty}^{\bar{v}\cdot y_{1}}d\bar{v}\cdot
y_{2}\left[igv\cdot A_{s}(\bar{v}\cdot y_{1}),\left[igv\cdot
A_{s}(\bar{v}\cdot y_{2}),T^{a}\right]\right]$ (136)
$\displaystyle=\frac{1}{2}\int_{-\infty}^{\bar{v}\cdot x}d\bar{v}\cdot
y_{1}\int_{-\infty}^{\bar{v}\cdot x}d\bar{v}\cdot
y_{2}P\Big{\\{}\left[igv\cdot A_{s}(\bar{v}\cdot y_{1}),\left[igv\cdot
A_{s}(\bar{v}\cdot y_{2}),T^{a}\right]\right]\Big{\\}}$
$\displaystyle=T^{a^{\prime}}\frac{(ig)^{2}}{2}\int_{-\infty}^{\bar{v}\cdot
x}d\bar{v}\cdot y_{1}\int_{-\infty}^{\bar{v}\cdot x}d\bar{v}\cdot
y_{2}P\Big{\\{}\left(v\cdot A_{s}(\bar{v}\cdot y_{1})v\cdot A_{s}(\bar{v}\cdot
y_{2})\right)^{a^{\prime}a}\Big{\\}}.$
From here, we can see that the $n^{th}$ term will be
$\displaystyle U^{a(n)}(x)$
$\displaystyle=\frac{1}{n!}\int_{-\infty}^{\bar{v}\cdot x}d\bar{v}\cdot
y_{1}\int_{-\infty}^{\bar{v}\cdot x}d\bar{v}\cdot
y_{2}\ldots\int_{-\infty}^{\bar{v}\cdot x}d\bar{v}\cdot y_{n}$ (137)
$\displaystyle\times P\Big{\\{}\left[igv\cdot A_{s}(\bar{v}\cdot
y_{1}),\left[igv\cdot A_{s}(\bar{v}\cdot y_{2}),\left[\ldots\left[igv\cdot
A_{s}(\bar{v}\cdot y_{n}),T^{a}\right]\right]\right]\right]\Big{\\}}$
$\displaystyle=T^{a^{\prime}}\frac{(ig)^{n}}{n!}\int_{-\infty}^{\bar{v}\cdot
x}d\bar{v}\cdot y_{1}\int_{-\infty}^{\bar{v}\cdot x}d\bar{v}\cdot
y_{2}\ldots\int_{-\infty}^{\bar{v}\cdot x}d\bar{v}\cdot y_{n}$
$\displaystyle\times P\Big{\\{}\left(v\cdot A_{s}(\bar{v}\cdot y_{1})v\cdot
A_{s}(\bar{v}\cdot y_{2})\ldots v\cdot A_{s}(\bar{v}\cdot
y_{n})\right)^{a^{\prime}a}\Big{\\}}.$
Summing to all orders then proves our result.
## Appendix E Subtle Signs in the Bound-state Formation and Decay
Calculations
In the bulk of the paper, we have freely used results from Ref. Harz:2018csl
that are written in terms of two-body states of the form $|ij\rangle$.
However, in general our 2-body states for non-identical particles will in fact
be combinations of the form
$\frac{1}{\sqrt{2}}(|ij\rangle+(-1)^{L+S}|ji\rangle)$. (This convention choice
is also discussed in the context of Sommerfeld enhancement in Ref.
Beneke:2014gja , where these two approaches are labeled “method-1” and
“method-2”; we largely adopt “method-2”, where we treat $|ij\rangle$ and
$|ji\rangle$ as components of a single state, rather than tracking them
separately.) The factor of $(-1)^{L+S}$ arises from a factor of $(-1)^{S+1}$
from the behavior of the spin configuration under particle exchange, a factor
of $(-1)^{L}$ from the parity of the spatial wavefunction, and a factor of
$(-1)$ from the exchange of two fermions.
This means that when considering a transition of the form
$|ij\rangle\rightarrow|i^{\prime}j^{\prime}\rangle$, we also need to include
transitions between the $|ji\rangle$ and $|j^{\prime}i^{\prime}\rangle$
states. In many cases this does not make a difference and it is adequate to
represent states purely by one component $|ij\rangle$. For example, if the
$|ij\rangle\rightarrow|i^{\prime}j^{\prime}\rangle$ and
$|ji\rangle\rightarrow|j^{\prime}i^{\prime}\rangle$ processes have equal
rates, but $|ij\rangle\rightarrow|j^{\prime}i^{\prime}\rangle$ and
$|ji\rangle\rightarrow|i^{\prime}j^{\prime}\rangle$ are forbidden (for
example, this occurs if $|ij\rangle=|0,++\rangle$), then the combined rate is
the same as what one would obtain from purely considering the
$|ij\rangle\rightarrow|i^{\prime}j^{\prime}\rangle$ process. However, this
behavior is not universal.
As an example of a case where this matters, consider the transition between
$Q=1$ bound state components $|\\!+0\rangle\rightarrow|\\!+0\rangle$. Writing
out the individual components of these states (labeled as $23$ and $32$,
following the notation of Sec. 3),262626One might be tempted to write the 23
state as $|\\!+0\rangle$ and 32 as $|0+\rangle$. However, we reserve
$|\\!+0\rangle$ for the full quantum state such that
$|\\!+0\rangle=(|2\,3\rangle+(-1)^{L+S}|3\,2\rangle)/\sqrt{2}$. the full
matrix element should be:
$\displaystyle\mathcal{M}=\frac{1}{2}$
$\displaystyle\left(\mathcal{M}_{22,33}+(-1)^{(L+S)_{i}}\mathcal{M}_{32,23}+(-1)^{(L+S)_{f}}\mathcal{M}_{23,32}\right.$
(138)
$\displaystyle\left.+(-1)^{(L+S)_{i}+(L+S)_{f}}\mathcal{M}_{33,22}\right)\\!.$
Now we can write $(L+S)_{f}=(L+S)_{i}+1\,\textrm{(mod 2)}$ for dipole
transitions, and consequently:
$\mathcal{M}=\frac{1}{2}\left(\mathcal{M}_{22,33}-\mathcal{M}_{33,22}+(-1)^{(L+S)_{i}}\left(\mathcal{M}_{32,23}-\mathcal{M}_{23,32}\right)\right)\\!.$
(139)
Now as calculated in Sec. 3, if $\psi_{i}$ and $\psi_{f}$ denote the initial-
and final-state wavefunctions, we have:
$\displaystyle\mathcal{M}^{3}_{22,33}$
$\displaystyle=i\sqrt{2^{6}\pi\alpha_{\text{rad}}M_{\chi}}\left[(T^{3})_{22}-(T^{3})_{33}\right]\int
d^{3}\mathbf{r}\,\psi_{f}^{*}\nabla\psi_{i}$ (140)
$\displaystyle=i\sqrt{2^{6}\pi\alpha_{\text{rad}}M_{\chi}}\int
d^{3}\mathbf{r}\,\psi_{f}^{*}\nabla\psi_{i},{}$
$\displaystyle\mathcal{M}^{3}_{33,22}$
$\displaystyle=i\sqrt{2^{6}\pi\alpha_{\text{rad}}M_{\chi}}\left[(T^{3})_{33}-(T^{3})_{22}\right]\int
d^{3}\mathbf{r}\,\psi_{f}^{*}\nabla\psi_{i}{}$
$\displaystyle=i\sqrt{2^{6}\pi\alpha_{\text{rad}}M_{\chi}}\int
d^{3}\mathbf{r}\,(-\psi_{f}^{*}\nabla\psi_{i}),{}$
$\displaystyle\mathcal{M}^{3}_{23,32}$
$\displaystyle=i\sqrt{2^{6}\pi\alpha_{\text{rad}}M_{\chi}}\left\\{\left[-i((T^{1})_{32}(T^{2})_{23}-(T^{2})_{32}(T^{1})_{23}\right]M_{\chi}\alpha_{\text{NA}}\int
d^{3}\mathbf{r}\,\hat{\mathbf{r}}\psi_{f}^{*}\psi_{i}\right\\}{}$
$\displaystyle=i\sqrt{2^{6}\pi\alpha_{\text{rad}}M_{\chi}}\left\\{-3M_{\chi}\alpha_{\text{NA}}\int
d^{3}\mathbf{r}\,\hat{\mathbf{r}}\psi_{f}^{*}\psi_{i}\right\\}\\!,{}$
$\displaystyle\mathcal{M}^{3}_{32,23}$
$\displaystyle=i\sqrt{2^{6}\pi\alpha_{\text{rad}}M_{\chi}}\left\\{\left[-i((T^{1})_{23}(T^{2})_{32}-(T^{2})_{23}(T^{1})_{32}\right]M_{\chi}\alpha_{\text{NA}}\int
d^{3}\mathbf{r}\,\hat{\mathbf{r}}\psi_{f}^{*}\psi_{i}\right\\}{}$
$\displaystyle=i\sqrt{2^{6}\pi\alpha_{\text{rad}}M_{\chi}}\left\\{3M_{\chi}\alpha_{\text{NA}}\int
d^{3}\mathbf{r}\,\hat{\mathbf{r}}\psi_{f}^{*}\psi_{i}\right\\}\\!,{}$
recalling that the “3” superscript is for $\gamma,\,Z$ emission, depending on
the value of $\alpha_{\text{rad}}$, the coupling of the emitted boson to the
charged particle that radiated it. We also recall that the terms with
$\alpha_{\text{NA}}$ correspond to emission off the virtual particles sourcing
the potential. The $\alpha_{\text{NA}}$ factor is thus the coupling of the
virtual line to the WIMPs. For the case of capture by a radiated $\gamma$,
$\alpha_{\text{rad}}=\alpha_{\text{em}}$ and $\alpha_{\text{NA}}=\alpha_{W}$.
Thus, overall we have $\mathcal{M}_{32,23}=-\mathcal{M}_{23,32}$ and
$\mathcal{M}_{33,22}=-\mathcal{M}_{22,33}$, and consequently:
$\displaystyle\mathcal{M}$
$\displaystyle=\mathcal{M}_{22,33}+(-1)^{(L+S)_{i}}\mathcal{M}_{32,23}{}$
$\displaystyle=i\sqrt{2^{6}\pi\alpha_{\text{rad}}M_{\chi}}\left[\int
d^{3}\mathbf{r}\,\psi_{f}^{*}\nabla\psi_{i}+(-1)^{(L+S)_{i}}3M_{\chi}\alpha_{\text{NA}}\int
d^{3}\mathbf{r}\,\hat{\mathbf{r}}\psi_{f}^{*}\psi_{i}\right]\\!.$ (141)
The $(-1)^{(L+S)_{i}}$ factor is obtained by treating the different components
correctly, and is required to ensure the correct behavior of the matrix
element under time reversal.
## Appendix F Analytic Approximate Results for Annihilation and Bound-state
Formation
In this final appendix we provide analytic estimates for the annihilation and
bound state formation rate of DM. We consider the quintuplet case of interest
first, followed by providing equivalent results for a general representation.
### F.1 Results for the quintuplet
In the limit of unbroken SU(2), the wavefunctions and their integrals, and
hence the bound-state capture rate, can be computed analytically in the low-
velocity limit. In this regime the Sommerfeld enhancement can also be computed
analytically. These results can also be applied (approximately, and with
caveats we will discuss below) to the case where SU(2) is broken but the DM
mass is very heavy relative to the symmetry breaking scale. These calculations
can be useful both as a cross-check on our numerical results, and to develop
intuition for which channels are likely to dominate the overall annihilation
signal. As such, we present the details of these analytic calculations below,
beginning with the DM in the quintuplet representation.
#### F.1.1 Capture and annihilation rates
As an opening example, let us estimate the spin-averaged capture rate into the
spin-triplet ground state via photon emission. The total cross section for
this process is given by (see App. C of Ref. Asadi:2016ybp ),272727This
expression as written includes capture only from the components of the
incoming state that experience an attractive potential; we expect the
contribution from repulsed incoming states to be suppressed, due to their
small overlap with the bound states.
$\displaystyle\sigma v$
$\displaystyle=\frac{3}{2}\times\frac{2^{8}\pi\alpha\,k}{3}\left|\sum_{i}({\bf
I}\cdot\eta_{i})\alpha_{\scriptscriptstyle W}(\alpha_{\scriptscriptstyle
W}\lambda_{f}M_{\chi}/2)^{-3/2}e^{-2\lambda_{i}/\lambda_{f}}e^{\pi\alpha_{\scaleto{W}{3.5pt}}\lambda_{i}/(2v)}\Gamma(1-i\alpha_{\scriptscriptstyle
W}\lambda_{i}/v)\right.{}$
$\displaystyle\left.\times\eta_{f}^{\dagger}\left[\lambda_{i}\hat{C}_{1}+\hat{C}_{2}\lambda_{i}/\lambda_{f}\right]\eta_{i}\right|^{2}\\!,$
(142)
The notation we employ follows Ref. Asadi:2016ybp ; $\eta_{i}$ and $\eta_{f}$
are potential eigenvectors which for the quintuplet are given by (in our
basis) $\eta_{f}=\\{-2,1,0\\}/\sqrt{5}$, $i=1,2$ with
$\eta_{1}=\\{\sqrt{2},-\sqrt{2},1\\}/\sqrt{5}$,
$\eta_{2}=\\{-2,-1,\sqrt{2}\\}/\sqrt{7}$, with corresponding attractive
eigenvalues $\lambda_{f}=5$, $\lambda_{1}=6$, $\lambda_{2}=3$. The $I$ vector
describes the fraction of the incoming plane wave in each state; we will
choose $I=\\{0,0,1\\}$, as the state asymptotes to two noninteracting, neutral
DM particles. The energy of the outgoing photon is $k$, which at low
velocities can be approximated as the binding energy of the ground state,
$\lambda_{f}^{2}\alpha_{\scriptscriptstyle W}^{2}M_{\chi}/4$. Lastly, the
$\hat{C}_{1}$ and $\hat{C}_{2}$ matrices describe the couplings between the
different components of the initial and final states; for capture via photon
(or $Z$) emission, they take the form:
$\hat{C}_{1}=\begin{pmatrix}2&0&0\\\ 0&1&0\\\
0&0&0\end{pmatrix}\\!,\quad\hat{C}_{2}=\begin{pmatrix}0&2&0\\\
-2&0&3\sqrt{2}\\\ 0&-3\sqrt{2}&0\end{pmatrix}\\!.$ (143)
In the wino and positronium cases studied in Ref. Asadi:2016ybp , there was
only one eigenstate that experienced an attractive initial-state potential,
and so the sum in Eq. (142) was trivial. There is a simple expression for
$|e^{\pi\alpha_{\scaleto{W}{3.5pt}}\lambda_{i}/(2v)}\Gamma(1-i\alpha_{\scriptscriptstyle
W}\lambda_{i}/v)|^{2}\simeq 2\pi\alpha_{\scriptscriptstyle W}\lambda_{i}/v$ in
the limit of small $v$ (for positive $\lambda_{i}$), which scales purely as
$1/v$, and consequently in those cases (wino and positronium) $\sigma v$ had a
simple $1/v$ scaling at low relative velocities. However, in the quintuplet
case, we see there are multiple terms in the sum that can interfere with each
other, and so even in the limit of unbroken SU(2), we expect there to be a
non-trivial velocity dependence in the capture cross section.
Similarly, the Sommerfeld factors can be read off from the components of the
scattering-state wavefunction at the origin, and in the unbroken limit this
wavefunction has the form
$\sum_{i}(I\cdot\eta_{i})\eta_{i}\,\phi(\lambda_{i}\alpha_{\scriptscriptstyle
W},r)$, where $\phi(\alpha,r)$ is the solution to the scalar Schrodinger
equation with an attractive Coulomb potential with coupling $\alpha$. In
principle this sum runs over both positive and negative eigenvalues of the
potential (corresponding to both attracted and repulsed eigenstates), but for
low velocities we expect the contribution of the eigenstates experiencing a
repulsive interaction to be very small. Nonetheless, where (as in the
quintuplet case) there are two eigenstates experiencing an attractive
potential, the value of the wavefunction at the origin (and hence the
Sommerfeld factors) will experience a non-trivial interference between the two
contributions. This can give rise to a velocity dependence differing from the
simpler case where there is only one attractive eigenstate. Similar
interference effects can be seen in the form of rapid changes in spectrum with
respect to $M_{\chi}$ in the case of broken SU(2) symmetry, where the
interference occurs between the various Sommerfeld factors with resonances at
different positions (as discussed in the context of Fig. 9). In contrast, the
manifestation of the eigenstate interference identified here persists in the
SU(2)-symmetric limit and does not require any resonance structure, only
differing (velocity-dependent) phases between the interfering contributions.
However, as noted in Ref. Schutz:2014nka ; Asadi:2016ybp , at low velocities
the system is often in an “adiabatic” regime where the incoming particle
wavefunction evolves such that at short distances it has complete overlap with
the eigenvector with the largest-magnitude attractive eigenvalue. The
criterion for this behavior is roughly $v\lesssim\delta/m_{\scriptscriptstyle
W}$, where $\delta$ is the mass splitting between the states; for $\delta=164$
MeV, we expect this behavior to hold roughly for $v\lesssim 2\times 10^{-3}$,
i.e. for Milky-Way-scale velocities and lower. Note that this criterion is
independent of the DM mass, so even when the DM is very heavy and the ratio
$m_{\scriptscriptstyle W}/M_{\chi}$ is small, the effect of SU(2) breaking can
still be seen in the presence of this adiabatic regime. In this case, the
interference will be suppressed for both bound-state formation and Sommerfeld
enhancement, with only the $i=1$, $\lambda_{i}=6$ term contributing
significantly, and with the coefficient $I\cdot\eta_{i}$ replaced with
$\delta_{i1}$.
This is an important simplifying approximation within its regime of validity.
Note that the presence of this regime relies on SU(2) being broken, and also
on low velocity ($v\lesssim\delta/m_{\scriptscriptstyle W}$); it will not
appear if an unbroken symmetry ensures the degeneracy of the mass eigenstates,
and it will also not generally be relevant in the early universe (e.g. for
relic density calculations) where velocities are much higher. However, it is
well-suited to the case of indirect detection in the Milky Way halo.
For example, within this approximation, we obtain the spin-averaged capture
rate to the ground state as:
$\sigma v\simeq\frac{2^{8}}{5}\frac{\pi\alpha\alpha_{\scriptscriptstyle
W}}{M_{\chi}^{2}}\times\frac{3^{3}\cdot
2^{9}}{5^{2}}e^{-24/5}\frac{\pi\alpha_{\scriptscriptstyle
W}}{v}=\frac{2^{17}\cdot
3^{3}}{5^{3}}e^{-24/5}\frac{\pi^{2}\alpha\alpha_{\scriptscriptstyle
W}^{2}}{M_{\chi}^{2}v}\simeq\frac{233\pi^{2}}{v}\frac{\alpha\alpha_{\scriptscriptstyle
W}^{2}}{M_{\chi}^{2}}.$ (144)
Here we have employed the low-velocity approximation
$|e^{\pi\alpha_{\scaleto{W}{3.5pt}}\lambda_{i}/(2v)}\Gamma(1-i\alpha_{\scriptscriptstyle
W}\lambda_{i}/v)|^{2}\simeq 2\pi\alpha_{\scriptscriptstyle W}\lambda_{i}/v$.
In the same regime, where the initial state rotates into the most-attracted
eigenstate, the $s$-wave direct annihilation cross section to gauge bosons can
be computed as,
$\sigma v\simeq\frac{720\pi^{2}\alpha_{\scriptscriptstyle
W}^{3}}{M_{\chi}^{2}v}.$ (145)
In this unbroken limit, the effective branching ratio to the line (i.e. to
$\gamma\gamma$ \+ half the branching ratio to $\gamma Z$) should be given by
$(s_{\scriptscriptstyle W}^{4}+s_{\scriptscriptstyle
W}^{2}c_{\scriptscriptstyle W}^{2})/3=s_{\scriptscriptstyle W}^{2}/3$, so the
line cross section should be:
$(\sigma v)_{\text{line}}\simeq\frac{240\pi^{2}\alpha_{\scriptscriptstyle
W}^{2}\alpha}{M_{\chi}^{2}v}.$ (146)
As for the bound-state formation, at higher velocities ($v\gtrsim 2\times
10^{-3}$), we expect to see the onset of interference between the
contributions from the two attracted eigenstates, resulting in a non-monotonic
dependence of the cross section on velocity even in the $s$-wave case. This
behavior, and its onset at roughly Milky Way-scale velocities, can be observed
in Fig. 10. (Note that the velocity dependence can also be suppressed if the
DM mass is small enough that the Sommerfeld enhancement is fully saturated,
such that the velocity dependence of the individual Sommerfeld factors is very
different from the case of unbroken SU(2) symmetry.)
This would suggest the cross-sections for capture (to the spin-triplet ground
state) and for annihilation producing a line should be very similar, at least
when this adiabatic approximation holds (numerical calculations indicate the
non-adiabatic cross section for bound-state capture, as estimated in Eq. 142,
can range between larger than the adiabatic result by a factor of $\sim$$2$
and smaller by a factor of $\sim$$5$, as $v$ is varied). However, for $v\ll
m_{\scriptscriptstyle W}/M_{\chi}$, we have the SU(2)-breaking effect that
$p$-wave processes should be parametrically suppressed by a factor of order
$(vM_{\chi}/m_{\scriptscriptstyle W})^{2}$, which at a 13.6 TeV mass can
suppress the $p$-wave capture cross section by $\sim$$2$ orders of magnitude.
We can also study the cross-section for spin-singlet $s\rightarrow p$ capture
to an $n=2$, $l=1$ state. In this case we have
$k=(25/16)\alpha_{\scriptscriptstyle W}^{2}M_{\chi}$, and
$\displaystyle\sigma v$ $\displaystyle=\frac{2^{13}\pi\alpha
k}{3^{3}}\frac{1}{\alpha_{\scriptscriptstyle
W}}M_{\chi}^{-3}\frac{1}{\lambda_{f}^{3}}\left|\sum_{i}({\bf
I}\cdot\eta_{i})e^{-4\lambda_{i}/\lambda_{f}}e^{\pi\alpha_{\scaleto{W}{3.5pt}}\lambda_{i}/(2v)}\Gamma(1-i\alpha_{\scriptscriptstyle
W}\lambda_{i}/v)\eta_{f}^{\dagger}\right.$ (147)
$\displaystyle\left.\times\left[\lambda_{i}\left(\frac{4\lambda_{i}}{\lambda_{f}}-3\right)\hat{C}_{1}+\hat{C}_{2}\left(3-12\frac{\lambda_{i}}{\lambda_{f}}+8\frac{\lambda_{i}^{2}}{\lambda_{f}^{2}}\right)\right]\eta_{i}\right|^{2}$
$\displaystyle=\frac{2^{9}\pi\alpha\alpha_{\scriptscriptstyle W}}{5\cdot
3^{3}M_{\chi}^{2}}\Bigg{|}\frac{3}{25}\sqrt{\frac{2}{5}}e^{-24/5}e^{3\pi\alpha_{\scaleto{W}{3.5pt}}/(2v)}\left(37e^{12/5}\Gamma(1-3i\alpha_{\scriptscriptstyle
W}/v)\right.$ $\displaystyle\hskip
184.9429pt\left.+89e^{3\pi\alpha_{\scaleto{W}{3.5pt}}/(2v)}\Gamma(1-6i\alpha_{\scriptscriptstyle
W}/v)\right)\Bigg{|}^{2},$
or in the adiabatic regime,
$\displaystyle\sigma v$
$\displaystyle\rightarrow\frac{2^{9}\pi\alpha\alpha_{\scriptscriptstyle
W}}{5\cdot
3^{3}M_{\chi}^{2}}\left|\frac{267}{25}\sqrt{2}e^{-24/5}e^{3\pi\alpha_{\scaleto{W}{3.5pt}}/(2v)}\Gamma(1-6i\alpha_{\scriptscriptstyle
W}/v)\right|^{2}$ (148) $\displaystyle=\frac{2^{12}\cdot
89^{2}\pi^{2}}{5^{5}\,v}e^{-48/5}\frac{\alpha\alpha_{\scriptscriptstyle
W}^{2}}{M_{\chi}^{2}}$
$\displaystyle\simeq\frac{0.70\pi^{2}}{v}\frac{\alpha\alpha_{\scriptscriptstyle
W}^{2}}{M_{\chi}^{2}}.$
We see that the scale for this cross-section is naturally two orders of
magnitude smaller than the previous ones, which arises from the various
numerical prefactors, primarily the factor of $e^{-48/5}$ compared to
$e^{-24/5}$ for the capture to the $n=1$ state, corresponding to a suppression
factor of $8\times 10^{-3}$. If these exponential terms were removed, the
other prefactors would differ by less than a factor of 3. (Numerical
calculations indicate the non-adiabatic cross section is larger than the
adiabatic one in this case, by factors between 1 and 3.6.) This is suggestive
that only the capture to the ground-state is likely to be comparable to direct
annihilation, and the capture from the $p$-wave initial-state component
suffers from a $v^{2}$ suppression once $v$ drops below $m_{\scriptscriptstyle
W}/M_{\chi}$, which renders it subdominant at the $\mathcal{O}(1\%)$ level for
our 13.6 TeV benchmark point.
This suppression for higher-$n$ capture also suggests that the contribution to
the endpoint hard photon spectrum from bound state formation and decay will be
suppressed, as these contributions are dominated by capture into states with
odd $L$ (and thus $n>1$) and $S=0$ that decay to $L=S=0$ states before
annihilating (see Sec. 4 for a more in-depth discussion). Capture to the
ground-state via emission of a dipole gauge boson changes $L$ by 1, thus
requiring an initial $L=1$ state (which must then have $S=1$ if it contains
identical DM particles), and $S=1$ states do not produce a leading-power
contribution to the endpoint spectrum when they decay.
For the $Q=1$ sector, let us again consider capture from the spin-triplet
$p$-wave incoming wave to the spin-triplet $s$-wave state. In the unbroken
limit the potential matrix for the final state takes the form:
$V=\begin{pmatrix}-2&\sqrt{6}\\\ \sqrt{6}&-3\end{pmatrix}\\!.$ (149)
Here the first row refers to the $++-$ state and the second to the $+\,0$
state. The transition matrices are now,
$\hat{C}_{1}=\begin{pmatrix}-\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&0\\\
0&-\frac{\sqrt{3}}{2}&\sqrt{\frac{3}{2}}\end{pmatrix}\\!,\quad\hat{C}_{2}=\begin{pmatrix}2\sqrt{2}&\sqrt{2}&0\\\
0&\sqrt{3}&0\end{pmatrix}\\!.$ (150)
In this case we must also replace the $\alpha$ prefactor in the cross section
with $\alpha_{\scriptscriptstyle W}$. The attractive eigenvalue for the final
state has $\lambda_{f}=5$, $\eta_{f}=\\{-\sqrt{2},\sqrt{3}\\}/\sqrt{5}$.
Again, the binding energy of the ground state is $k=\alpha_{\scriptscriptstyle
W}^{2}\lambda_{f}^{2}M_{\chi}/4=(25/4)\alpha_{\scriptscriptstyle
W}^{2}M_{\chi}$ (and since in the unbroken limit $m_{\scriptscriptstyle W}=0$,
we do not need to include a kinematic suppression for the $W$ mass). Then we
obtain for the unbroken limit:
$\displaystyle\sigma v=\frac{2^{7}3^{2}\pi\alpha_{\scriptscriptstyle
W}^{2}}{5^{4}M_{\chi}^{2}}$
$\displaystyle\left|16e^{-12/5}e^{6\pi\alpha_{\scaleto{W}{3.5pt}}/(2v)}\Gamma(1-6i\alpha_{\scriptscriptstyle
W}/v)\right.$ (151)
$\displaystyle\left.+7e^{-6/5}e^{3\pi\alpha_{\scaleto{W}{3.5pt}}/(2v)}\Gamma(1-3i\alpha_{\scriptscriptstyle
W}/v)\right|^{2}\\!,$
When we assume the adiabatic regime, the result reduces to,
$\sigma v=\frac{2^{17}\cdot
3^{3}\pi^{2}}{5^{3}}e^{-24/5}\frac{\alpha_{\scriptscriptstyle
W}^{3}}{vM_{\chi}^{2}}\simeq 233\frac{\pi^{2}\alpha_{\scriptscriptstyle
W}^{3}}{vM_{\chi}^{2}}.$ (152)
This is for capture to the $Q=+1$ state; there is an equal rate for capture to
the $Q=-1$ state. Note the $\alpha_{\scriptscriptstyle W}$ prefactor (rather
than $\alpha$); including formation of the $Q=0$ state through $Z$ emission
(as well as photon emission) would similarly promote that capture rate to have
a prefactor of $\alpha_{\scriptscriptstyle W}$ rather than $\alpha$, for an
overall capture rate (summing across all three channels) of $\sim
700\pi^{2}\alpha_{\scriptscriptstyle W}^{3}/(M_{\chi}^{2}v)$, similar to the
full $s$-wave direct annihilation rate. The primary difference between this
$p\rightarrow s$ capture rate and the inclusive direct annihilation rate will
arise from velocity suppression of the $p\rightarrow s$ capture cross section
in the broken-SU(2) case (with this suppression being lifted in the truly
unbroken limit).
#### F.1.2 Presence of metastable bound states
The unbroken-SU(2) limit is also helpful for studying the question of whether
there could be $L>0$ states in the spectrum whose decays to more deeply bound
states are highly suppressed, leading them to decay through annihilation to SM
particles with a substantial branching ratio. States which are degenerate in
the unbroken limit are likely to remain close in energy as we reduce
$M_{\chi}$, and therefore decays between them will be suppressed (although if
this is decisive in determining whether a state is metastable, a more careful
analysis will be required). The $L+S$-even potential for the quintuplet has
two attractive eigenvalues, $Z=6$ and $Z=3$, whereas the $L+S$-odd potential
has a single attractive eigenvalue, $Z=5$. Thus for spin-singlet states
($S=0$) we expect $L$-even bound states with energies
$E_{n}/\alpha_{\scriptscriptstyle W}^{2}M_{\chi}=-9/n^{2},-2.25/n^{2}$, and
$L$-odd bound states with energies $E_{n}/\alpha_{\scriptscriptstyle
W}^{2}M_{\chi}=-6.25/n^{2}$. For spin-triplet states ($S=1$) we expect $L$-odd
states with energies $E_{n}/\alpha_{\scriptscriptstyle
W}^{2}M_{\chi}=-9/n^{2},-2.25/n^{2}$, and $L$-even states with energies
$E_{n}/\alpha_{\scriptscriptstyle W}^{2}M_{\chi}=-6.25/n^{2}$.
We first consider the case of $L$-odd states. Given a spin-singlet $L$-odd
state with $n>L>0$ (binding energy $6.25/n^{2}$), there should always be a
more deeply bound state with $n^{\prime}=n$, $L^{\prime}=L-1$ (binding energy
$9/n^{2}$), which is accessible through a dipole transition. So in the spin-
singlet case and Coulomb limit there should be no metastable states with
$L>0$. For the spin-triplet, the case is slightly more complicated, as given
an $L$-odd state with $n>L>0$ (binding energy $9/n^{2}$ or $2.25/n^{2}$), the
accompanying state with $n^{\prime}=n$, $L^{\prime}=L-1$ has binding energy
$6.25/n^{2}$. We see that the $L$-odd states with binding energies
$2.25/n^{2}$ will always have an accompanying more-deeply-bound $L$-even spin-
triplet state, to which they can decay, but this is not necessarily true for
the states with binding energies $9/n^{2}$. A state with $L^{\prime}=L-1$ will
be available if $9/n^{2}<6.25/m^{2}$ for some $m$ with $L\leq m<n$, i.e. if
$m<n\sqrt{6.25/9}=n/1.2$ is consistent with $m\geq L$. This will be true for
$n>1.2L$, so the dangerous range is states with $L<n\leq 1.2L$. In order for
this range to include an integer, we must have $L>5$. For example, consider
the spin-triplet state with $L=7$ and $n=8$, with dimensionless binding energy
$9/8^{2}\simeq 0.14$ in the Coulombic limit. The lowest-lying $L=6$ spin-
triplet state that is accessible via $\Delta L=1$, $\Delta S=0$ transitions
has $n=7$, and consequently binding energy $6.25/7^{2}=0.13$ in the Coulombic
limit; thus the $L=7$ state cannot decay through such a transition. For the
$L=5$ case, in the Coulombic limit the states are degenerate, and so we would
need to perform a more careful calculation.
If we now consider the case of even-$L$ states, the situation is reversed
between the spin-singlet and the spin-triplet; in the spin-triplet case we
expect the even-$L$ states will always be able to decay to their accompanying,
more deeply bound state with $L^{\prime}=L-1$, with the exception of the $L=0$
case where no such state exists (we will consider the $L=0$ case below). In
the spin-singlet case, the same argument as previously tells us that for
$L>0$, a state with $L^{\prime}=L-1$ will be available except in the case
where $L<n\leq 1.2L$, which in the case of even $L$ is potentially relevant
for $L\geq 6$.
Therefore, based on the Coulombic limit, we would predict that the only
possible (meta)stable states with $L>0$ are $L+S$-even states (spin-triplet
with $L$ odd or spin-singlet with $L$ even) with $L\geq 5$, $L<n\leq 1.2L$,
and the eigenstate structure corresponding to the $Z=6$ eigenvalue. These
high-$L$ states may not even be bound for masses of interest to us, and in any
case the capture rate into them is likely to be very small.
### F.2 Results for general representations
Let us now consider the more general situation where the DM is the lightest
component of an SU(2) multiplet in a real representation of odd dimension $N$.
Larger representations require higher DM masses to obtain the correct relic
density (e.g. Ref. Bottaro:2021snn ), and hence the unbroken-SU(2)
approximation is likely to be better at their thermal masses. Recall, however,
that the condition to be in the adiabatic regime is mass-independent,
$v\lesssim\delta/m_{\scriptscriptstyle W}$, so while this is a feature of the
broken SU(2) symmetry, we expect it to be retained at sufficiently low
velocities even for very heavy DM.
In the unbroken-SU(2) limit we can use the results of Ref. Mitridate:2017izz
for general representations. They proceed by decomposing the two-particle
state into eigenstates of isospin $I$ ($I=1,3,\cdots,2N-1$); this corresponds
to identifying the eigenstates of the potential in our language. They find the
eigenvalue associated with the state with isospin $I$ is
$\lambda=(2N^{2}-1-I^{2})/8$ (where positive eigenvalues correspond to
attracted states, as per our previous convention) and so the most-attracted
channel is the singlet, where $\lambda=(N^{2}-1)/4$. The isospin singlet
corresponds to an $L+S$-even state with total charge $Q=0$, and for the
quintuplet $\lambda=6$, as discussed above. In general, states with
$I<\sqrt{2N^{2}-1}$ can support bound states; for the quintuplet this means we
have $I=1,3,5$ bound states.
In the adiabatic regime, SU(2)-breaking effects cause the lowest-energy state
at large distances (i.e. the $L+S$-even, $Q=0$ state of two identical DM
particles) to smoothly transition into the isospin-singlet state at short
distances. Thus in this regime, we expect both the Sommerfeld-enhanced direct
annihilation and the bound state capture rates to be consistent with an
initial isospin-singlet state. Since the dominant bound-state capture process
(dipole emission of a gauge boson) changes isospin by 2, the final state must
then have $I=3$, i.e. it is a SU(2) adjoint (this state is $L+S$-odd and the
three components have $Q=0,\pm 1$).
Consequently, within the adiabatic approximation, we only need to concern
ourselves with (Sommerfeld enhanced) direct annihilation from the isospin-
singlet state, and capture from an isospin-singlet initial state to an
isospin-triplet final state, with the relevant eigenvalues being
$\lambda_{i}=(N^{2}-1)/4$ and $\lambda_{f}=(N^{2}-5)/4$. (Transitions amongst
bound states can involve higher-$I$ states; in particular $I=5$ for the
quintuplet, with $\lambda=3$.)
We will thus focus in this appendix on singlet-to-adjoint transitions. We
reiterate that this approximation is not appropriate if the gauge symmetry is
actually unbroken or at the high velocities associated with thermal freezeout
in the early universe (as studied in e.g. Ref. Binder:2023ckj ), where other
transitions can also contribute significantly and may dominate. The quality of
this approximation—i.e. the degree to which the incoming state retains non-
singlet components at small $r$, which could contribute significantly to the
capture rate—is an interesting question, but we ignore it here, as our main
purpose is simply to develop some simple intuition for the importance of
bound-state effects for the gamma-ray endpoint signal. The corresponding
approximation for the quintuplet appears to do a reasonable job of estimating
the relative size of bound-state capture and annihilation, as we see in Fig.
4.
For these singlet-to-adjoint transitions, we can write the group theory
coefficients for bound state formation from Ref. Mitridate:2017izz in the
simplified form:
$\displaystyle C^{a1b}_{\mathcal{J}}$
$\displaystyle=\frac{1}{\sqrt{T_{R}d_{R}}}\text{Tr}(T^{b}T^{a}),$ (153)
$\displaystyle C^{a1b}_{\mathcal{\tau}}$
$\displaystyle=i\frac{1}{\sqrt{T_{R}d_{R}}}\text{Tr}(T^{b}T^{c}T^{d})f^{acd}=-\frac{1}{\sqrt{T_{R}d_{R}}}\text{Tr}(T^{b}T^{c}T^{d})(T^{a}_{\text{adj}})^{cd}.$
We can now use $\text{tr}(T^{a}T^{b})=T_{R}\delta^{ab}$, and also
$\displaystyle\text{Tr}(T^{b}T^{c}T^{d})(T^{a}_{\text{adj}})^{cd}$
$\displaystyle=\frac{1}{2}T_{R}T_{\text{adj}}\delta^{ab}.$ (154)
Thus, finally we obtain:
$\displaystyle C^{a1b}_{\mathcal{J}}$
$\displaystyle=\sqrt{\frac{T_{R}}{d_{R}}}\delta^{ab},\quad
C^{a1b}_{\mathcal{\tau}}=-\frac{1}{2}\sqrt{\frac{T_{R}}{d_{R}}}T_{\text{adj}}\delta^{ab},$
(155)
where we will show how the $C^{a1b}_{\mathcal{J}}$ and
$C^{a1b}_{\mathcal{\tau}}$ coefficients enter the bound state capture rate in
Secs. F.2.2 and F.2.3.
Now if $R$ is the representation of size $N$, for SU(2) we have
$T_{R}=N(N^{2}-1)/12$, and so $T_{\text{adj}}=2$, while $d_{R}=N$ (and in
particular $d_{\text{adj}}=3$). Thus for SU(2) we obtain the coefficients:
$\displaystyle C^{a1b}_{\mathcal{J}}$
$\displaystyle=\sqrt{\frac{N^{2}-1}{12}}\delta^{ab},\quad
C^{a1b}_{\mathcal{\tau}}=-C^{a1b}_{\mathcal{J}}.$ (156)
Let us also note that we can now extend the argument given in App. F.1.2 to
general representations. A bound state of isospin $I$ will generally have open
decay channels to states with lower isospin (by 2 units) which are hence more
deeply bound due to the larger eigenvalue $\lambda$. The exception is $I=1$
states, which must decay to $I=3$ states which are more shallowly bound for
the same principal quantum number. For this reason, excited $I=1$ states can
be metastable if they have sufficiently large $L$ that all the $I=3$ states
differing by only one unit in $L$ are more shallowly bound. This can occur for
a general representation if
$L<n\leq\frac{N^{2}-1}{N^{2}-5}L=\left(1+\frac{4}{N^{2}-5}\right)L$; this
range will contain an integer if $L>(N^{2}-5)/4$. Thus the threshold $L$ at
which this effect can occur increases as the representation size goes up.
#### F.2.1 Direct annihilation
In this case, if we can evaluate the tree-level cross section for annihilation
from an isospin-singlet initial state to any desired SM final state, we can
account for the Sommerfeld enhancement by simply multiplying the tree-level
cross section by $S=2\pi\alpha_{\scriptscriptstyle W}\lambda_{i}/v$, in the
low-velocity limit. This cross section is given for Majorana fermion DM by
Ref. Mitridate:2017izz as:
$\displaystyle(\sigma v)_{\text{tree,I=1}}$
$\displaystyle=\frac{\pi\alpha_{\scriptscriptstyle
W}^{2}}{M_{\chi}^{2}}\frac{T_{R}^{2}d_{\text{adj}}}{d_{R}}{}$
$\displaystyle=\frac{\pi\alpha_{\scriptscriptstyle W}^{2}}{2^{4}\times 3\times
M_{\chi}^{2}}N(N^{2}-1)^{2}.$ (157)
Multiplying by the Sommerfeld factor gives:
$\displaystyle(\sigma v)_{\text{I=1}}$
$\displaystyle=\frac{\pi^{2}\alpha_{\scriptscriptstyle W}^{3}}{2^{5}\times
3\times
M_{\chi}^{2}v}N(N^{2}-1)^{3}\rightarrow\frac{\pi^{2}\alpha_{\scriptscriptstyle
W}^{3}}{M_{\chi}^{2}v}\frac{N^{7}}{96},$ (158)
where in the final step we have assumed $N\gg 1$.
Checking, for the wino and quintuplet this yields:
$\displaystyle(\sigma v)_{\text{I=1}}$
$\displaystyle=\frac{\pi^{2}\alpha_{\scriptscriptstyle
W}^{3}}{M_{\chi}^{2}v}\begin{cases}16,&N=3,\\\ 720,&N=5.\end{cases}$ (159)
For the quintuplet this agrees with our calculation above. This also agrees
with the wino result from Ref. Asadi:2016ybp , accounting for our assumption
that the adiabatic condition holds.
#### F.2.2 Capture to the ground state
At small velocities, from Ref. Mitridate:2017izz we can read off the low-
velocity bound state capture cross section to the $n=1$ state as:
$\displaystyle(\sigma
v)^{n=1,l=0}_{\text{bsf}}=\frac{\pi\alpha_{\scriptscriptstyle
W}^{2}}{M_{\chi}^{2}}\frac{2S+1}{g_{\chi}^{2}}\frac{2^{11}\pi}{3}\sum_{ab}|C^{a1b}_{\mathcal{J}}+(1/\lambda_{f})C^{a1b}_{\mathcal{\tau}}|^{2}\frac{\lambda_{i}^{3}\alpha_{\scriptscriptstyle
W}}{\lambda_{f}v}e^{-4\lambda_{i}/\lambda_{f}}.$ (160)
This expression involves an average over initial states, with degrees of
freedom denoted by $g_{\chi}$; since we are interested in the case where
$100\%$ of the DM captures from the singlet state, we only need to average
over spin degrees of freedom, so $g_{\chi}=2$. Our initial state must be
$L+S$-even and thus for capture to an $L=0$ final state, it must have $S=1$.
Thus we obtain:
$\displaystyle(\sigma v)^{n=1,l=0}_{\text{bsf}}$
$\displaystyle=\frac{\pi\alpha_{\scriptscriptstyle
W}^{2}}{M_{\chi}^{2}}\frac{3}{4}\frac{2^{11}\pi}{3}d_{\text{adj}}\left(\frac{N^{2}-1}{12}\right)|1-(1/\lambda_{f})|^{2}\frac{\lambda_{i}^{3}\alpha_{\scriptscriptstyle
W}}{\lambda_{f}v}e^{-4\lambda_{i}/\lambda_{f}}{}$
$\displaystyle=\frac{8\pi^{2}\alpha_{\scriptscriptstyle
W}^{3}}{M_{\chi}^{2}v}\frac{(N^{2}-9)^{2}(N^{2}-1)^{4}}{(N^{2}-5)^{3}}e^{-4(N^{2}-1)/(N^{2}-5)}.$
(161)
where $d_{\text{adj}}=3$ counts the number of generators and arises from
$\sum_{ab}|\delta^{ab}|^{2}=d_{\text{adj}}$.
In the limit of $N\gg 3$, which is helpful for comparison against direct
annihilation, we obtain the simplified result:
$\displaystyle(\sigma v)^{n=1,l=0}_{\text{bsf}}$
$\displaystyle\rightarrow\frac{\pi^{2}\alpha_{\scriptscriptstyle
W}^{3}}{M_{\chi}^{2}v}8N^{6}e^{-4}\simeq\frac{\pi^{2}\alpha_{\scriptscriptstyle
W}^{3}}{M_{\chi}^{2}v}\frac{N^{6}}{6.8},\quad N\gg 3.$ (162)
(Note in the quintuplet case this approximate value is larger than the truth
by about a factor of 3; it will be a better approximation for larger $N$.)
#### F.2.3 Capture to the $n=2$ states
Now let us consider capture to the $n=2$, $L=1$ states, as their subsequent
decays and annihilations can give rise to endpoint photons, unlike capture
directly to the $n=1$ state. In this case the final state must have $S=0$ (so
the initial state has $L+S$ even). From Ref. Mitridate:2017izz in the low-
velocity limit the $s$-wave (1st line) and $d$-wave (2nd line) contributions
are:
$\displaystyle(\sigma v)^{n=2,l=1}_{\text{bsf}}$
$\displaystyle=\frac{\pi\alpha_{\scriptscriptstyle
W}^{2}}{M_{\chi}^{2}}\left(\frac{2S+1}{g_{\chi}^{2}}\right)\frac{2^{12}\pi\lambda_{i}}{9\lambda_{f}^{5}}\frac{\alpha_{\scriptscriptstyle
W}}{v}\left[\sum_{ab}|C^{a1b}_{\mathcal{J}}(\lambda_{f}\lambda_{i}(3\lambda_{f}-4\lambda_{i})+C^{a1b}_{\mathcal{\tau}}(-3\lambda_{f}^{2}+12\lambda_{f}\lambda_{i}-8\lambda_{i}^{2})|^{2}\right.{}$
$\displaystyle\left.+\sum_{ab}2^{5}\lambda_{i}^{4}|C^{a1b}_{\mathcal{J}}\lambda_{f}+2C^{a1b}_{\mathcal{\tau}}|^{2}\right]e^{-8\lambda_{i}/\lambda_{f}}{}$
$\displaystyle=\frac{\pi^{2}\alpha_{\scriptscriptstyle
W}^{3}}{M_{\chi}^{2}v}\frac{2^{8}\lambda_{i}}{9\lambda_{f}^{5}}(N^{2}-1)\left[|(\lambda_{f}\lambda_{i}(3\lambda_{f}-4\lambda_{i})-(-3\lambda_{f}^{2}+12\lambda_{f}\lambda_{i}-8\lambda_{i}^{2})|^{2}\right.{}$
$\displaystyle\left.+2^{5}\lambda_{i}^{4}|\lambda_{f}-2|^{2}\right]e^{-8\lambda_{i}/\lambda_{f}}{}$
$\displaystyle=\frac{\pi^{2}\alpha_{\scriptscriptstyle
W}^{3}}{M_{\chi}^{2}v}\frac{2^{4}(N^{2}-1)^{2}}{9(N^{2}-5)^{5}}\left[(N^{6}+9N^{4}-165N^{2}-37)^{2}\right.{}$
$\displaystyle\left.+2^{5}(N^{2}-1)^{4}(N^{2}-13)^{2}\right]e^{-8(N^{2}-1)/(N^{2}-5)}.$
(163)
The first line here agrees with the $s\rightarrow p$ quintuplet result
computed in Eq. (148), once we multiply that result (which was for photon-
mediated capture into a specific $n=2$ $L=1$ bound state) by a factor of 3 to
account for $W$\- and $Z$-mediated capture and a second factor of 3 to account
for the $m=0,\pm 1$ states.
In the limit of large $N$, this expression has the scaling:
$\displaystyle(\sigma v)^{n=2,l=1}_{\text{bsf}}$
$\displaystyle\rightarrow\frac{\pi^{2}\alpha_{\scriptscriptstyle
W}^{3}}{M_{\chi}^{2}v}\frac{2^{4}N^{6}}{9}\left[1+2^{5}\right]e^{-8}\simeq\frac{\pi^{2}\alpha_{\scriptscriptstyle
W}^{3}}{M_{\chi}^{2}v}\frac{N^{6}}{1700}\left[1+32\right],\quad N\gg 5$ (164)
So we see that compared to direct annihilation, in the large-$N$ limit we
expect the various contributions to scale as:
* •
$p\rightarrow s$ capture to the $n=1,L=0,S=1$ state (contributions to endpoint
photons are power-suppressed): direct annihilation rate $\times 14/N$, in
addition to (when SU(2) is broken) any kinematic suppression of $W/Z$ emission
(by a factor as small as $\alpha/(3\alpha_{\scriptscriptstyle W})$) or
velocity suppression due to the $p$-wave initial state (parametrically
$\mathcal{O}(M_{\chi}^{2}v^{2}/m_{\scriptscriptstyle W}^{2})$ for $v\lesssim
m_{\scriptscriptstyle W}/M_{\chi}$),282828As mentioned above, the large-$N$
approximation overestimates this ratio for the quintuplet by about a factor of
3, and so the rates are actually very comparable.
* •
$s\rightarrow p$ capture to the $n=2,L=1$ states collectively: direct
annihilation rate $\times 0.06/N$, in addition to any kinematic suppression of
$W/Z$ emission (up to a factor of $3\alpha_{\scriptscriptstyle W}/\alpha$),
* •
$d\rightarrow p$ capture to the $n=2,L=1$ states collectively: direct
annihilation rate $\times 1.8/N$, in addition to any kinematic suppression of
$W/Z$ emission (up to a factor of $3\alpha_{\scriptscriptstyle W}/\alpha$) or
velocity suppression due to the $d$-wave initial state (parametrically
$\mathcal{O}(M_{\chi}^{4}v^{4}/m_{\scriptscriptstyle W}^{4})$ for $v\lesssim
m_{\scriptscriptstyle W}/M_{\chi}$).
We see that the only contribution that is not suppressed at low velocities and
that gives rise to leading-power contributions to the endpoint photon spectra
(via its decay and subsequent annihilation) is generically expected to have a
cross section 2 or more orders of magnitude below direct annihilation.
Consequently, it is quite plausible for bound state formation to be a large or
even dominant contribution to the inclusive annihilation rate when the
velocity suppression for higher partial waves is mild or absent and the gauge
bosons are massless (as in the case of freezeout), while simultaneously having
a generically small effect on the endpoint spectrum for indirect detection,
especially at low velocities ($v\ll m_{\scriptscriptstyle W}/M_{\chi}$) or
where the $n=2$ states are too loosely bound to allow $W$\- or $Z$-mediated
capture.
One might ask about the contribution from capture into states with $n>2$.
While in the unbroken limit bound states with large $L$ and $n$ may play a
large role in the capture rate (e.g. Binder:2023ckj ), for the parameter space
we have considered in this paper, the number of bound states is always
truncated by the non-zero SU(2) breaking scale, preventing large enhancements
from the proliferation of high-$n$ states. Furthermore, we expect velocity
suppressions (of order $(M_{\chi}v/m_{\scriptscriptstyle W})^{2L}$) for all
capture rates with initial $L>0$. Finally, within our adiabatic approximation
both initial and final states always experience attractive interactions, with
couplings that obey $\lambda_{i}/\lambda_{f}>1$, leading to increasingly
strong exponential suppression for large-$n$ states and avoiding the
potentially unitarity-violating region of parameter space identified in Ref.
Binder:2023ckj .
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|
# Retention of Water in Terrestrial Magma Oceans and Carbon-rich Early
Atmospheres
Dan J. Bower Center for Space and Habitability, University of Bern
Gesellschaftsstrasse 6
3012 Bern, Switzerland Kaustubh Hakim Center for Space and Habitability,
University of Bern
Gesellschaftsstrasse 6
3012 Bern, Switzerland Paolo A. Sossi Institute of Geochemistry and
Petrology, Department of Earth Sciences, ETH Zurich
Clausiusstrasse 25
8092 Zurich, Switzerland Patrick Sanan Institute of Geophysics, ETH Zurich
Sonneggstrasse 5
8092 Zurich, Switzerland
(Received July 14, 2021; Revised October 12, 2021; Accepted March 21, 2022)
###### Abstract
Massive steam and CO2 atmospheres have been proposed for magma ocean
outgassing of Earth and terrestrial planets. Yet formation of such atmospheres
depends on volatile exchange with the molten interior, governed by volatile
solubilities and redox reactions. We determine the evolution of magma
ocean–atmosphere systems for a range of oxygen fugacities, C/H ratios and
hydrogen budgets that include redox reactions for hydrogen (H2–H2O), carbon
(CO–CO2), methane (CH4), and solubility laws for H2O and CO2. We find that
small initial budgets of hydrogen, high C/H ratios, and oxidizing conditions,
suppress outgassing of hydrogen until the late stage of magma ocean
crystallization. Hence early atmospheres in equilibrium with magma oceans are
dominantly carbon-rich, and specifically CO-rich except at the most oxidizing
conditions. The high solubility of H2O limits its outgassing to melt fractions
below $\sim$30%, the fraction at which the mantle transitions from vigorous to
sluggish convection with melt percolation. Sluggish melt percolation could
enable a surface lid to form, trapping water in the interior and thereby
maintaining a carbon-rich atmosphere (equilibrium crystallization).
Alternatively, efficient crystal settling could maintain a molten surface,
promoting a transition to a water-rich atmosphere (fractional
crystallization). However, additional processes, including melt trapping and H
dissolution in crystallizing minerals, further conspire to limit the extent of
H outgassing, even for fractional crystallization. Hence, much of the water
delivered to planets during their accretion can be safely harbored in their
interiors during the magma ocean stage, particularly at oxidizing conditions.
Planetary interior — Planetary atmospheres — Planetary structure — Super
Earths — Extrasolar rocky planets — Exoplanet atmospheric composition —
Exoplanet evolution
††journal: The Planetary Science Journal††software: Simulating Planetary
Interior Dynamics with Extreme Rheology (SPIDER) (Bower et al., 2018, 2019) is
open source and hosted at https://github.com/djbower/spider. SPIDER Version
0.2.1 was used in this study and is also available on Zenodo (Bower et al.,
2021).
## 1 Introduction
The origin and distribution of light volatile elements such as C, H, and O, in
the major planetary reservoirs (core, mantle, and atmosphere), regulates how
terrestrial planet atmospheres form and evolve (Gaillard et al., 2021, and
references therein). During planetary accretion and contemporaneous iron-core
formation—both of which promote the formation of a global magma ocean—the
siderophile nature of both C (Corgne et al., 2008) and H (Tagawa et al., 2021)
strongly partitions them into the core. Nevertheless, the mantle retains some
fraction of C at core-forming conditions, in addition to replenishment of both
C and H after core formation (Hirschmann, 2016); hence, these volatiles
subsequently participate in short- and long-term geochemical cycling (e.g.,
Dasgupta, 2013). Here we examine how the inventories of C and H exchange
between the interior and atmosphere as the magma ocean evolves after core
formation.
The magma ocean outgasses volatiles as it cools from molten to mostly solid
silicate rock to form a secondary atmosphere. Therefore, chemical and physical
processes operating in the magma ocean, both at its surface and at depth,
exert strong controls on the size and composition of such early atmospheres.
Early atmospheres establish the initial conditions for the subsequent long-
term evolution of planetary interiors and their atmospheres through
interaction with the surface environment via geochemical cycles. Therefore,
magma ocean atmospheres provide the connection between the volatile budgets
emplaced during planet formation and the establishment of long-term climate
states.
The shock degassing of substantial H2O from hydrated minerals (Lange & Ahrens,
1982) to simulate impacts during planetary accretion motivated investigation
of the blanketing effect of a steam atmosphere above the molten early Earth
(Abe & Matsui, 1985). Additional justification for the widespread presence of
H2O and CO2 is provided by experiments and accompanying equilibrium models for
the devolatilization of ordinary and carbonaceous chondrites, which are shown
to produce oxidized species in abundance (Gooding & Muenow, 1977; Schaefer &
Fegley, 2010; Thompson et al., 2021). Consequently, coupled magma
ocean–atmosphere models have focused almost exclusively on the role of
oxidized species (H2O and CO2) in controlling the lifetime of magma oceans and
growth of early atmospheres (e.g., Lebrun et al., 2013; Hier-Majumder &
Hirschmann, 2017; Salvador et al., 2017; Nikolaou et al., 2019; Barth et al.,
2021).
More reduced materials (such as enstatite chondrites that are thought to have
contributed to Earth’s accretion on isotopic grounds) may instead produce
H2-rich or CO-rich gases (Schaefer & Fegley, 2010). This has partly inspired
calculations of the chemistry and cooling timescales of reducing atmospheres
(Zahnle et al., 2020; Lichtenberg et al., 2021). However, chondrites may not
be appropriate analogs for the materials that degas to produce early
atmospheres around terrestrial planets (Sossi, 2021). This is because mass
transfer is much more efficient when materials are liquid rather than solid,
allowing for the more rapid replenishment of surface material via convection
and thus facilitating atmosphere formation. This has further led coupled
models to understand the astrophysical observables of magma oceans to
interpret the spectroscopic data from next-generation telescopes (Hamano et
al., 2015; Bonati et al., 2019; Bower et al., 2019; Katyal et al., 2020).
The redox state (more precisely, the oxygen fugacity, $f{\rm O_{2}}$) of the
mantle depends on pressure, temperature, and composition and dictates the
speciation of outgassing products. It controls the relative fugacities of
reduced and oxidized species, which is equivalent to partial pressure at the
low total pressures of terrestrial atmospheres. Core formation on Earth
established $f{\rm O_{2}}$ around two log10 units below the iron-wüstite (IW)
buffer at equilibrium at depth, based on the Fe content of the mantle and core
(e.g., Rubie et al., 2015). However, the volume change of the reaction between
ferric and ferrous iron means that silicate melt transported to the surface
defines higher $f{\rm O_{2}}$ than that set by core formation (Armstrong et
al., 2019; Hirschmann, 2012). The surface $f{\rm O_{2}}$, rather than the
$f{\rm O_{2}}$ at depth, imposes the redox state at the magma ocean–atmosphere
interface and thereby controls outgassing chemistry (Sossi et al., 2020).
Evolutionary models of magma oceans are usually derived from geochemical or
geodynamic considerations. Chemical models focus on tracking the compositional
evolution of cumulate mineral assemblages that form as the magma ocean cools
(e.g., Elkins-Tanton, 2008). However, they ignore dynamics during the
crystallization process and often assume fractional crystallization. In
(stepwise) fractional crystallization, cumulates that form owing to cooling
are isolated from the molten magma ocean, and the evolving composition of the
melt is determined from mass balance. By contrast, dynamic models track
thermal energy transport but at the expense of reduced compositional
complexity. However, dynamic formulations that include a local representation
of melt–solid separation (Abe, 1993; Bower et al., 2018) can replicate
fractional crystallization. They can also replicate equilibrium
crystallization, in which melt and solid freeze together before any
significant relative motion has occurred.
Equilibrium chemistry calculations propose a $\sim$100 bar CO-rich atmosphere
for an Earth-like magma ocean around 2200 K (Sossi et al., 2020), although it
remains unclear how the atmospheric size and speciation change during the
cooling and crystallization of the magma ocean. In particular, it is unknown
whether all early atmospheres are expected to be CO-rich and whether they can
transition to atmospheres that are instead dominated by hydrogen species, such
as H2, H2O, and CH4—with implications for habitability and the formation of
surface water oceans. Furthermore, it remains to be determined how the
transition is influenced by planetary conditions (e.g., $f{\rm O_{2}}$, C/H
ratio, H budget), which is pertinent given the diversity of formation
environments for terrestrial planets outside the solar system.
To this end, we combine mass balance and equilibrium chemistry in a self-
consistent and time-evolved model to probe magma ocean–atmosphere exchange.
Unlike previous models, our model accounts for the CO2/CO, H2O/H2, and CH4/CO2
ratio set by the oxygen fugacity, C/H ratio and temperature of the magma
ocean, as well as their relative solubilities. We explore a range of initial
endowments of the hydrogen and carbon volatile inventory and quantify the
relationship between these variables and the partial pressures of the degassed
species. Finally, atmospheric escape is also modeled to determine the extent
to which the preferential loss of hydrogen can influence the subsequent
outgassing of dissolved volatiles.
## 2 Interior–Atmosphere Coupling
### 2.1 Overview
The SPIDER code (Bower et al., 2018, 2019) solves for the coupled evolution of
the silicate mantle and atmosphere, where the mantle can be molten, solid, or
a mixture (partial melt). It considers interior energy transport by
convection, conduction, and relative motion of melt and solid through mixing
and separation and is a true 1D model in the sense that energy fluxes are
determined locally (Abe, 1993; Bower et al., 2018). Cooling of the interior is
regulated by radiative transfer in the atmosphere, and the atmosphere itself
grows through outgassing of the interior as the mantle solidifies owing to
cooling. We consider a two-phase system of single composition (MgSiO3) for
simplicity, where the thermophysical properties of solid and melt are
determined from Mosenfelder et al. (2009) and Wolf & Bower (2018),
respectively. The melting curves adhere to peridotite melting data in the
upper mantle and measurements on chondritic material in the lower mantle
(Andrault et al., 2011); they are not perturbed according to the mantle
volatile content. The atmosphere is treated as gray with no scattering, and
the two-stream approximation is applied to solve for radiative transfer (Abe &
Matsui, 1985); Radiation limits are not considered. Opacities of the gas
species are provided in Appendix A.
We modified the SPIDER code to additionally account for mass exchange between
volatiles according to equilibrium chemistry (Section 2.3), as well as
atmospheric escape (Section 2.4). The mathematical description of the volatile
mass balance and its adherence to equilibrium chemistry is given in detail in
Appendix C. For each chemical reaction, we introduce a term into the volatile
mass balance to account for the exchange of mass necessary to retain chemical
equilibrium between participating gas species as dictated by oxygen fugacity
and temperature. Similarly, another term in the mass balance equation tracks
volatile loss due to atmospheric escape. Hence, chemical equilibrium and
escape are self-consistently determined as part of the same system of
equations that are integrated to describe the coupled interior–atmosphere
evolution.
Our parameter choices are similar to those in Bower et al. (2018, 2019), so we
only present pertinent details here. We consider a planet with Earth
dimensions and a magma ocean that crystallizes from the bottom up, an approach
that is justified owing to the steeper melting curves compared to the mantle
adiabat. Convection and melt–solid mixing are determined by an eddy
diffusivity that is based on a constant mixing length, and internal heat
sources are not considered because their influence is negligible over the
lifetime of a magma ocean. Viscosity controls the dynamic regime of the mantle
and varies from $10^{2}$ Pa s for pure silicate melt to $10^{22}$ Pa s for
pure solid, increasingly abruptly at a melt fraction of 40% (Costa et al.,
2009; Bower et al., 2018). Other controls on viscosity, such as temperature
and composition, are second order and therefore neglected. The crystal size,
which controls the efficiency of melt–solid separation, is set at either 1 mm
or 5 cm (Appendix B) to simulate equilibrium or fractional crystallization,
respectively.
We track the reservoir evolution of five volatile species during the magma
ocean stage: H2, H2O, CO, CO2, and CH4. For convenience, we refer to H2O and
H2 as hydrogen species and CO, CO2, and CH4 as carbon species. The dissolved
content of H2O and CO2 in the magma ocean is determined by their solubility,
and reduced species are set to zero solubility (Section 2.2). Redox reactions
impose H2–H2O, CO$-$CO2, and CO2–H2–CH4 equilibrium (Section 2.3).
Partitioning of volatiles into solids (crystals) is not considered because its
effect is small compared to the partitioning of volatiles between the melt and
atmosphere. Therefore, the amount of volatiles stored in the mantle during
solidification is a minimum estimate. We simulate cooling of a magma ocean
from an initial surface temperature of 2700 K until it reaches 1650 K. For
surface temperatures cooler than 1650 K the assumption of equilibrium
dissolution begins to break down. This is because a rheological transition
caused by increasing crystal fraction results in mass transfer that is too
sluggish to maintain equilibrium with the atmosphere; for conciseness we
subsequently refer to this event as ”surface lid formation.”
### 2.2 Solubility
A solubility law relates the dissolved volatile content of a particular
species in melt $X$ to its fugacity $f$. Pressures at the magma
ocean–atmosphere interface of an Earth-sized planet are expected to be around
several hundred bars, which motivates our subsequent choice of solubility
laws. In the following, the dissolved content of a given volatile in the magma
ocean is set solely by its fugacity at the interface and the relevant
solubility law. The volatile species are considered to be homogeneously
distributed in the magma ocean.
#### 2.2.1 Solubility of H2O
We employ a general expression for solubility with a power-law dependence:
$X=\alpha f^{1/\beta},$ (1)
where $\alpha$ and $\beta$ are constants specific to a particular equilibrium
between a gas species and its dissolved component. In Equation 1, $\alpha$ is
an empirically determined coefficient that encompasses the temperature,
pressure, and composition of the liquid in which the solubility is being
determined, while $\beta$ reflects the stoichiometric relationship between the
mole fraction of the dissolved species and the fugacity of the gas species.
Henry’s law is recovered when $\beta=1$, which indicates no chemical reaction
(hence no change in speciation) when the gas dissolves in the solvent.
Throughout this work we assume gas ideality such that fugacity and partial
pressure are equivalent, and hence symbols for fugacity $f$ and partial
pressure $p$ are used interchangeably. In Equation 1, the relevant partial
pressure (fugacity) is determined at the interface of the magma ocean and
atmosphere.
At low H2O fugacities, water dissolves in silicate melts as hydroxyl (OH-)
group, so $\beta_{\rm H_{2}O}=2$ describes the solubility of H2O in melt
(e.g., McMillan, 1994). However, this relationship breaks down at high H2O
fugacities because H2O replaces OH- as the predominant melt species (above
$\sim 2000$ bars, Berndt et al., 2002; Stolper, 1982). Therefore, a single
$\beta_{\rm H_{2}O}$ determined by simply best-fitting data across large
$f$H2O ranges yields spurious values of $\beta_{\rm H_{2}O}$ because different
reactions govern solubility at different $f$H2O.
Table 1 summarizes recent experimental constraints on H2O solubility for
compositions, pressures, and temperatures relevant to the magma
ocean–atmosphere interface of a terrestrial planet, i.e. a few hundred bars
and greater than about 1500 K. The table is ordered based on the silica
content of the sample, increasing from less than 45 wt% SiO2 to more than 50
wt% SiO2. Experiments on evolved compositions (e.g. rhyolitic/granitic melts)
are not included because they are not representative of compositions during
the magma ocean stage: these are instead summarized in Iacono-Marziano et al.
(2012).
Peridotite is a rock composed predominantly of olivine, which is the major
constituent of Earth’s mantle and likely exoplanetary mantles (Putirka &
Rarick, 2019). The solubility of water in liquid peridotite (red line, Figure
1) is determined at high temperature and low $f$H2O, so its solubility law is
most appropriate for a terrestrial magma ocean (Sossi et al., 2022). For the
relevant pressure–temperature conditions of the magma ocean surface, the mole
fraction of H2O dissolved depends largely on variations in $f$H2O, while the
effect of temperature is a subordinate (Hamilton & Oxtoby, 1986), if not
poorly constrained, variable. As such, temperature does not explicitly appear
as a functional dependence in Equation 1, and $\alpha$ is taken to be
constant.
#### 2.2.2 Solubility of CO2
There are no experimental constraints on the solubility of CO2 in peridotite,
so instead we use a solubility law formulated for basalt as a proxy. At low
CO2 fugacities, carbon dioxide dissolves in silicate melts as either molecular
CO2 or the CO${}_{3}^{2-}$ group (Stolper & Holloway, 1988). Because the
stability of the carbonate ion increases according to melt basicity (Mg2+,
Ca2+ cations, Holloway & Blank (1994)), the Mg-rich nature of planetary
mantles ensures that it occurs exclusively as CO${}_{3}^{2-}$ in such liquids.
Determining the mole fraction of dissolved carbonate for pressures up to 815
bars $f$CO2, Dixon et al. (1995) constrained the solubility of CO2 in mid-
ocean ridge basalt (MORB) melt:
$X^{m}_{\rm CO_{2}}=(3.8\times 10^{-7})f{\rm CO_{2}}\exp\left(-23\ \frac{f{\rm
CO_{2}}-1}{10RT}\right),$ (2)
where $X^{m}_{\rm CO_{2}}$ is the mole fraction of CO2 in the melt, $f{\rm
CO_{2}}$ is fugacity in bars, and $T$ temperature and $R$ gas constant in SI
units. The exponential function on the right-hand side is known as the
Poynting factor, which captures the influence of pressure and temperature on
CO2 fugacity. The abundance of CO2 (ppmw) in melt is then
$X_{\rm CO_{2}}=10^{4}\left(\frac{4400X^{m}_{\rm CO_{2}}}{36.594-44X^{m}_{\rm
CO_{2}}}\right).$ (3)
This solubility law compares favorably to experimental results at 10 kbar (Pan
et al., 1991). At water concentrations less than about 3 wt% there is a
negligible influence of the presence of dissolved H2O on CO2 solubility (Dixon
et al., 1995). Above about 6 wt%, water can enhance the solubility of CO2 by
around a factor of two (Iacono-Marziano et al., 2012), although CO2 remains
insoluble compared to H2O. In our model, the water concentration only
increases beyond 3 wt% at the end of the magma ocean stage for the most
oxidizing condition and largest initial hydrogen budget; otherwise, the water
concentration is significantly less. Hence, we ignore the minor influence of
both carbon and hydrogen on each other’s solubility.
#### 2.2.3 Solubility of H2, CO, and CH4
We assume that the solubilities of H2, CO, and CH4 are negligible compared to
H2O and CO2, respectively, and therefore set $\alpha_{\rm H_{2}}=\alpha_{\rm
CO}=\alpha_{\rm CH_{4}}=0$ (Equation 1). This is motivated by high-pressure
experiments that find that the equilibrium constants for the dissolution of H2
gas in basaltic melts result in solubilities more than two orders of magnitude
lower than that for H2O (Li et al., 2015b; Hirschmann et al., 2012).
Similarly, the mole fraction of reduced carbon dissolved in silicate melts is
typically one to two orders of magnitude lower than for CO2 (Yoshioka et al.,
2019).
Methane dissolved in silicate melts has been identified by spectroscopy in
quenched glasses (Mysen et al., 2011; Ardia et al., 2013; Wetzel et al., 2013;
Armstrong et al., 2015; Dalou et al., 2019). These studies together
demonstrate that the fraction of dissolved methane increases in melts
equilibrated under increasingly reducing, high-pressure and H-rich conditions.
To quantify the solubility of methane in the melt at a given methane fugacity,
Ardia et al. (2013) performed experiments with Fe-free basaltic melts at
pressures between 0.7 and 3 GPa. The equilibrium constant of the reaction
CH4(l) = CH4(g) results in an order of magnitude less dissolved C for a given
$f$CH4 compared to an equivalent $f$CO2. Because estimated CH4 contents are
lower than several hundred ppmw even at $f$CH4 of 1000 bars (Ardia et al.,
2013), its solubility is neglected. It is noteworthy that solubility data for
CH4 in silicate melts at low total pressures relevant to magma oceans ($<$0.7
GPa) are lacking, preventing an accurate assessment of its solubility
behavior.
Table 1: H2O Solubilities Constrained by Experimental Studies
${\alpha}^{\ddagger}$ | $\beta$ | Composition | Temp. (K) | Pres. (bars) | $f{\rm O_{2}}$ ($\Delta$IW) | Reference
---|---|---|---|---|---|---
534 | 2.0 | Peridotite | 2173 | 1.013 | –1.4 to +6.5 | Sossi et al. (2022)
683 | 2.0 | Lunar Glass | 1623 | 1.013 | –3.0 to +4.8 | Newcombe et al. (2017)
727 | 2.0 | Anorthite-Diopside | 1623 | 1.013 | –3.0 to +4.8 | Newcombe et al. (2017)
965 | 2.0 | Basalt (MORB) | 1473 | 176-717 | +3.9 to +13.2 | Dixon et al. (1995)
1007 | 2.0 | Basalt (MORB) | 1473 | 503-2021 | +3.5 and +7.9 | Berndt et al. (2002)
215 | 1/0.7 | Basalt | 1373 | 1034-6067 | unbuffered | Wilson & Head III (1981)†
Note. — † Used in several modeling studies, e.g. Lebrun et al. (2013),
Salvador et al. (2017), Nikolaou et al. (2019), and Bower et al. (2019). ‡
$\alpha$ has units of ppmw/bar1/β and where relevant is determined by refits
to existing data by constraining $\beta=2$ and no solubility at zero fugacity.
We only consider experimental studies of basaltic or more mafic melts (i.e.,
Mg and Fe rich) at relatively low $f{\rm H_{2}O}$ ($\lesssim$2000 bars).
Figure 1: Experimentally determined solubility data of H2O for several melt
compositions and temperatures, for fugacities from 0 to 200 bars (Table 1).
Dotted lines denote $\beta=2.0$, and $\alpha$ is reported at the end of the
lines (Equation 1).
### 2.3 Redox Reactions
When at equilibrium, the magma ocean and atmosphere must be considered as a
single thermodynamic unit. This is because, based on the scaling of convective
velocity (Solomatov, 2000), the magma ocean mixing timescale is a few weeks,
which is considerably shorter than the cooling timescale of around 1 Myr.
Accordingly, $f{\rm O_{2}}$ defined by the magma ocean at its surface is
equivalent to the oxygen partial pressure (or fugacity) in the atmosphere.
Therefore, the fugacity ratio of gaseous redox couples (H2/H2O and CO/CO2), as
well as CO2–H2–CH4, satisfy chemical equilibrium at the $T$ and $f{\rm O_{2}}$
imposed by the magma ocean at its surface. We consider three reactions, where
all species exist in the gas phase, albeit partially dissolved in the melt
according to their respective solubilities (Sect. 2.2). For the hydrogen redox
couple,
$\displaystyle\rm{H}_{2}+0.5\rm{O}_{2}$ $\displaystyle=\rm{H}_{2}O,$ (4)
$\displaystyle\log_{10}{\rm K_{eq}}$ $\displaystyle=\frac{13152}{T}-3.039,$
for the carbon redox couple,
$\displaystyle\rm{CO}+0.5\rm{O}_{2}$ $\displaystyle=\rm{CO}_{2},$ (5)
$\displaystyle\log_{10}{\rm K_{eq}}$ $\displaystyle=\frac{14468}{T}-4.348,$
and for CO2–H2–CH4,
$\displaystyle\rm{CO}_{2}+2\rm{H}_{2}$ $\displaystyle=\rm{CH}_{4}+\rm{O}_{2},$
(6) $\displaystyle\log_{10}{\rm K_{eq}}$
$\displaystyle=\frac{-16276}{T}-5.4738.$
The equilibrium constants $\rm{K_{eq}}$ for the redox couples are determined
from the Gibbs free energy of reaction using data from the JANAF database fit
for temperatures between 1500 and 3000 K (Chase, 1998). The equilibrium
constant for the CO2–H2–CH4 reaction is taken from the IVTANTHERMO database
fit for temperatures between 300 and 2000 K (Schaefer & Fegley, 2017). The
most abundant redox-sensitive element in Earth’s mantle is iron, so the amount
of oxygen that is free to participate in reactions will be approximately
regulated by Gibbs free energy changes along the IW buffer during the magma
ocean stage. Due to the presence of melt and the difficulty of determining the
thermodynamic properties of a nonstoichiometric phase (wüstite), it is
preferable to consider equilibrium of solid metallic Fe with liquid FeO (the
IW buffer), where the Gibbs free energy of reaction constrains $f{\rm{O_{2}}}$
(O’Neill & Eggins, 2002):
$\displaystyle\rm{Fe}+0.5\rm{O}_{2}=\rm{FeO},$ (7)
$\displaystyle\log_{10}{f{\rm O_{2},IW}}=\frac{-244118+115.559T-8.474T\ln
T}{0.5\ln(10)RT}.$
It is convenient to define oxygen fugacity in log10 units relative to the IW
buffer ($\Delta{\rm IW}$):
$\Delta{\rm IW}=\log_{10}f{\rm O_{2}}-\log_{10}f{\rm O_{2},IW}.$ (8)
In our nominal models we set $\Delta{\rm IW}=0.5$ based on a recent
determination of the inferred oxygen fugacity of Earth’s magma ocean at its
surface using the Fe3+/Fe2+ ratio from a global compilation of peridotites
(Sossi et al., 2020). Hence, by fixing $f{\rm O_{2}}$ relative to the IW
buffer, H2O/H2 (Equation 4) and CO2/CO (Equation 5) are constrained as a
function of temperature, and by extension CH4/CO2 (Equation 6). Therefore,
while solubility controls the abundance of species in the atmosphere versus
the interior, redox reactions dictate the relative abundance of oxidized to
reduced species in the atmosphere. The thermodynamic coupling between the
atmosphere and the interior is set by the surface temperature at the magma
ocean–atmosphere interface. Appendices C.1 and C.3 describe how reactions are
self-consistently incorporated into the time stepper that evolves the magma
ocean–atmosphere system.
### 2.4 Atmospheric Escape
Atmospheric escape from terrestrial planets is challenging to parameterize
given the complex interaction of thermal and nonthermal processes, as well as
uncertainty regarding the history and details of the stellar environment and
planetary magnetic field (e.g., Gronoff et al., 2020). In the context of this
study, we concern ourselves with the impact of H2 loss from the atmosphere on
the evolution of the magma ocean–atmosphere system. We focus on hydrogen since
it is the lightest atmospheric component and therefore the most prone to
escape. Escape of H2 may be buffered by outgassing from the interior,
resulting in efficient depletion of hydrogen from the magma ocean. We apply a
model that obeys an upper limit for H2 loss based on the diffusion rate of H2
through a hydrostatic carbon-dominated atmosphere (Zahnle et al., 2019). Below
this limit, the escape flux follows the energy-limit that considers the
stellar energy input necessary for a volatile to escape Earth’s gravitational
field. Therefore, the H2 escape flux is (Zahnle et al., 2019)
$\phi_{\rm H_{2}}\approx\frac{\gamma 10^{16}\rm{[VMR]}_{\rm
H_{2}}S}{N_{A}\sqrt{1+0.006S^{2}}}\qquad\text{moles}\ {\rm m}^{-2}\ {\rm
s}^{-1},$ (9)
where $N_{A}$ is the Avogadro constant and ${\rm[VMR]}_{\rm H_{2}}$ is the
volume mixing ratio of H2 that provides coupling between the evolving
atmospheric speciation and H2 escape. Here $\gamma$ is a scaling factor (unity
by default), and $S$ is the normalized solar irradiation according to Earth’s
present-day value:
$S(t)=F_{\rm XUV}/F_{\rm XUV\odot},$ (10)
where XUV combines the influence of X-ray, extreme ultraviolet, and far
ultraviolet radiation. We adopt an upper estimate of $S=40$ at 4.0 Ga (Tu et
al., 2015). For ${\rm[VMR]}_{\rm H_{2}}$ of 30% and $\gamma=1$, the H2 mass-
loss rate is 2.1$\times 10^{5}$ kg s-1 (Equation C4). Both the escape
prefactor $\gamma$ (constant for a given case) and ${\rm[VMR]}_{\rm H_{2}}$
(time dependent) scale the mass-loss rate linearly.
## 3 Results
### 3.1 Outgassing of Hydrogen (One Earth Ocean)
Figure 2: Coupled interior and atmosphere evolution during magma ocean
outgassing at $\Delta$IW=0.5 with a hydrogen budget of one Earth ocean (no
carbon), where E1 shows equilibrium crystallization and F1 shows fractional
crystallization. For E1 and F1: (a, d) Interior temperature with gray showing
the mixed phase region where melt and solid coexist; (b, e) interior melt
fraction. (c) Atmospheric partial pressure of H2 and H2O for E1 and F1, where
the pressures of F1 (black lines) initially follow E1 (gray lines) before
departing at late time. Colored vertical ticks and lines show times in panels
(a), (b), (d), and (e), and labels show when a surface lid forms for E1 and
F1. (f) Interior depletion of melt (‘Melt’) and hydrogen (‘Hydr.’) for E1 and
F1.
We first present magma ocean outgassing models (Cases E1 and F1) where the
total hydrogen (H) budget is equivalent to the mass of H in Earth’s present-
day water ocean (hereinafter one ocean) (Figure 2). This budget is equivalent
to a fully outgassed atmosphere of 270 bars for a single species atmosphere
consisting of H2O. These models allow us to describe H2 and H2O outgassing
before subsequently introducing further complexity in the form of carbon
species, redox variation, and atmospheric escape. The total budget of H is
partitioned between a reduced (H2) and an oxidized (H2O) species according to
the $f{\rm O_{2}}$ buffer set at $\Delta$IW=0.5, with an initial surface
temperature of 2700 K. We recall that in our model H2 has zero solubility in
the melt and so can only reside in the atmosphere. The only parameter
difference between Case E1 (hereinafter E1) and Case F1 (hereinafter F1) is
the crystal size, where $a=$ 1 mm for E1 and $a=$ 5 cm for F1. Both of these
crystal sizes are reasonable for a crystallizing magma ocean and demonstrate
the end-member scenarios of equilibrium and fractional crystallization that
arise depending on the efficiency of melt–solid separation, which scales
according to the crystal size squared (Appendix B).
Both E1 and F1 chart a similar thermal evolution when the melt fraction is
high. Cooling proceeds quickly since most H remains in the magma ocean in the
form of H2O; this keeps the atmospheric pressure low ($<$ 1 bar). The
atmospheric composition (mole fraction of H2 compared to H2O) is determined by
chemical equilibrium, where $f{\rm O_{2}}$ decreases with decreasing surface
temperature (Equation 4) but parallels the IW buffer. Vigorous convection
keeps crystals mostly in suspension assuming efficient re-entrainment of
crystals (Solomatov et al., 1993; Solomatov & Stevenson, 1993a) and the
interior is approximately adiabatic. Due to the curvature of the melting
curves and the two-phase adiabat, the deepest mantle has the lowest melt
fraction and hence reaches the ‘rheological transition’ first. The rheological
transition is defined by an abrupt increase in viscosity around 30–40% melt
fraction, where the formation of an interconnected solid matrix begins to
dictate the convective timescale. The ”rheological front” is the interface
between melt-dominated dynamics above (lower pressure) and solid creep below
(higher pressure), and it moves upward through the mantle as cooling proceeds.
Hence, convection in the deep mantle becomes sluggish, akin to the present-day
solid mantle, acting as a brake on deep mantle cooling (Andrault et al.,
2016). Gravitational separation of melt and solid becomes a dominant driver of
energy transport at the rheological transition (Abe, 1993). Hence, the E1 and
F1 models exhibit different interior and outgassing evolution because of the
timescale of melt–solid separation (percolation) compared to net cooling rate.
E1 represents equilibrium crystallization, where the timescale for melt–solid
separation is longer than the timescale for the advancement of the rheological
front (Tonks & Melosh, 1990; Solomatov & Stevenson, 1993b). The rheological
front rapidly advances through the mantle, causing the thermal profile to
collapse on top of the rheological transition (Figure 2b). When it reaches the
surface, it initiates lid formation since convection has become so sluggish
that heat transport by convection to the surface cannot prevent top-down
cooling. Lid formation brings about the end of interior–atmosphere dissolution
equilibrium (‘E1 lid’ at 4.3 kyr; Figure 2). At this time, the mantle is
around 30% molten since it remains pinned at the rheological transition
because it has not had time to undergo any significant differentiation through
melt–solid separation prior to the formation of a surface lid. Therefore, the
mantle contains sufficient melt to trap the majority of H as dissolved H2O
beneath the surface lid (Figure 2f). Hence, the subsequent interaction of
these sequestered volatiles with the surface and atmosphere will be regulated
by geological processes operating over long timescales (millions to billions
of years), rather than by dissolution equilibrium with a comparatively short-
lived magma ocean.
By contrast, for F1 the melt percolation velocity keeps pace with the upward
velocity of the rheological front. This enables the complete solidification of
deep mantle layers below the rheological front by crystal settling, as well as
stalling the upward progression of the rheological front. This occurs because
efficient upward draining of melt keeps the upper regions molten at the
expense of cooling and fully crystallizing the lowermost mantle (Figure 2e).
Therefore, dissolution equilibrium with the atmosphere is maintained while
crystals form and are displaced to depth. Hence, F1 represents fractional
crystallization, where formed solids deep in the mantle can be considered
chemically isolated from the molten reservoir above. Below the rheological
transition, efficient melt–solid separation causes the thermal profile to
abruptly drop to the solidus, where now subsolidus cooling is greatly
restricted by the viscous transport timescale and partly by the core that
buffers the cooling of the mantle (Figure 2d). This ultimately enables a
substantial reduction of the melt reservoir ($98\%$ crystallized) at a surface
temperature of 1650 K compared to E1, where the crystal fraction only reaches
70% (Figure 2f). Due to the high solubility of H2O in melt, for F1 the
additional $\approx 30\%$ crystallization compared to E1 further depletes H
from the interior by $65\%$ before a surface lid forms. We note that
fractional crystallization could also occur for E1 after the surface lid has
formed, but in this case the melt reservoir is separated from the atmosphere
by a surface lid, and therefore outgassing is not driven by dissolution
equilibrium.
The atmospheric pressures of H2O and H2 increase in unison during cooling
(Figure 2c) because the relative abundance of these species is controlled by
chemical equilibrium at $\Delta$IW=0.5 (Equation 4). Since H2O is highly
soluble in silicate liquid, early in the lifetime of the magma ocean (prior to
1 kyr while melt fractions are high) the atmospheric mass remains mostly
constant and the partial pressures are controlled by the temperature
dependence of the $f{\rm O_{2}}$ buffer and equilibrium constant (Figure 2c,
Equation 4). After 1 kyr, the extent to which H outgasses is proportional to
the degree of crystallinity (i.e., continued cooling and crystallization)
according to the lever rule. Despite the mantle reaching 70% crystal fraction
for E1 before a surface lid forms, the atmospheric pressures of H2O and H2
only reach 4 and 2.4 bars, respectively (E1 lid, Figure 2c). For F1, the
atmospheric pressures of H2 and H2O (and hence their depletion in the magma
ocean) track E1 until around 2 kyr (gray lines, Figure 2c,f). At this time,
efficient melt–solid separation enables growth of a more substantial H2O (80
bars) and H2 (50 bars) atmosphere prior to surface lid formation (”F1 lid”,
Figure 2c). It takes around 4.3 and 26 kyr for a surface lid to form for E1
and F1, respectively, which compares favorably to a timescale of 10 kyr
determined by a model with a more advanced treatment of radiative transfer
(Lichtenberg et al., 2021). When a surface lid forms, the atmospheric pressure
of E1 and F1 differs by more than an order of magnitude.
### 3.2 Outgassing of Hydrogen (1–10 Earth Oceans)
Planet formation models predict substantial delivery of water to the inner
solar system (Raymond & Izidoro, 2017; Raymond et al., 2007), which can
account for the 1–10 oceans of water currently stored in Earth (Lécuyer et
al., 1998). Recent geochemical estimates propose that around 2.5 oceans are
currently stored in the mantle (Marty, 2012; Hirschmann, 2018). Therefore, we
now present cases the same as E1 and F1 but with an initial H budget up to 10
Earth oceans. Since the main features of the coupled evolution of the
interior–atmosphere are described for E1 and F1 (Section 3.1), only
differences relative to these fiducial cases are now presented. Figure 3
reveals the interior depletion of H as a percentage of the H inventory, where
the circles with a solid line and containing a 1 are E1 (Figure 3a) and F1
(Figure 3b).
For both equilibrium and fractional crystallization, as the H inventory is
increased from 1 to 10 oceans, the magma ocean lifetime increases by around
two orders of magnitude, reaching a maximum duration slightly less than 1 Myr.
This is because the optical depth of the atmosphere scales with pressure, and
a larger inventory results in a larger outgassed atmosphere and hence a higher
atmospheric pressure. Therefore, due to the thicker atmosphere, the magma
ocean takes longer to cool before the rheological front reaches the surface
and initiates lid formation. In addition, the interior depletion of H
increases for larger total inventories as a direct consequence of power-law
solubility where the exponent $\beta=2$. This is because, during cooling, the
atmospheric reservoir of H grows with the water mole fraction to the second
power (see Appendix C.6 for analysis). Hence, for H outgassing, both the
pressure of the atmosphere and relative depletion of the interior increase
with the total inventory at a given melt fraction. For a volatile species that
obeys a linear solubility relation ($\beta=1$), its relative depletion at a
given melt fraction is independent of its total inventory (Appendix C.6).
Water is expected to transition to such a linear solubility above around 10
ocean masses on Earth owing to the prevalence of dissolved H2O.
Figure 3: Depletion of hydrogen (H) from the interior during magma ocean
outgassing at $\Delta$IW=0.5 for H budgets from 1 to 10 Earth oceans: (a)
equilibrium crystallization, and (b) fractional crystallization. Lines show
constant melt fraction, and circles show the H budget. Magma ocean stage ends
at around 30% melt for equilibrium crystallization and 2% melt for fractional
crystallization. Cases E1 and F1 (Figure 2) correspond to 1 H ocean for
equilibrium and fractional crystallization, respectively. Dashed lines in
panel (a) with a ‘W’ suffix denote models that only include H as H2O, i.e. a
completely oxidized mantle.
For equilibrium crystallization, lid formation occurs soon after 30% melt is
reached (Figure 3a). For all equilibrium cases, melt–solid separation is slow
compared to the cooling timescale, so when a lid forms, the melt fraction is
only marginally reduced for cases with a long compared to a short cooling
timescale. At 30% melt, a maximum of around 23% of the total H budget is
outgassed for 10 oceans, although this decreases to about 4% for 1 ocean
(Figure 3a). For fractional crystallization, the melt fraction decreases below
30% until lid formation occurs around 2% melt. Compared to equilibrium
crystallization, this extra depletion of melt drives outgassing of hydrogen to
around 80% of its total budget (Figure 3b). Regardless of total inventory,
depletion must tend to 100% when the melt fraction reaches zero because we
assume that no volatiles are retained in solids. This explains the flattening
of the depletion curve for 2% melt fraction compared to, for example, 10% melt
fraction.
The magma ocean lifetimes for $\Delta{\rm IW}=0.5$ cases (solid lines, Figure
3a) are comparable to cases that are fully oxidized where H can only exist as
water (W cases, dashed lines, Figure 3a). This is because for all cases H2O
dominates the opacity of the atmosphere. However, H depletion of the
$\Delta$IW=0.5 cases and that of the fully-oxidized (W) cases differ by up to
about 11%. This is because at $\Delta$IW=0.5 outgassing of H2O must be
accompanied by commensurate growth of H2 in the atmosphere to maintain
chemical equilibrium. Since the H budget is conserved, increasing H2 in the
atmosphere necessitates a decrease in the H2O reservoir dissolved in the magma
ocean. In short, H2O solubility determines H2O outgassing, but oxygen fugacity
dictates the additional depletion of interior hydrogen to maintain equilibrium
between H2 and H2O in the atmosphere.
### 3.3 Outgassing of Hydrogen and Carbon
Figure 4: Atmospheric composition during magma ocean outgassing at
$\Delta$IW=0.5. Columns from left to right show hydrogen (H) budgets of 1, 3,
and 10 oceans, respectively. Rows from top to bottom show C/H by weight of 0,
0.1, 1, and 5. Magma ocean lifetime corresponds to the termination of the
colored region (at $\approx$ 2% melt for fractional crystallization, between
$10^{4}$ and $10^{7}$ yr). White dotted lines correspond to 30% melt, which
indicates both the duration and atmospheric composition if instead the mantle
underwent equilibrium crystallization. Initial partial pressures of volatile
species ($p_{\rm init}$) are for a completely molten magma ocean and $T_{\rm
surf}=2700$ K. Final partial pressures ($p_{\rm final}$) are for complete
solidification and outgassing of all volatile species and $T_{\rm surf}=1400$
K.
In contrast to the high solubility of water in silicate melt, carbon dioxide
is relatively insoluble (e.g., Dixon et al., 1995). Hence, due to the high
abundance of C in rocky planets, the presence of gaseous CO, CO2, and CH4 may
regulate the lifetime of a magma ocean and thereby the timescale over which H
outgasses. Carbon also suppresses the outgassing of H through its influence on
the mean molar mass of the atmosphere (Bower et al., 2019). Therefore, we now
present fractional crystallization cases that additionally include C species
(CO2, CO, and CH4), with their relative abundances in the gas phase determined
by redox reactions. The initial carbon-to-hydrogen ratio (C/H by weight) is
varied from 0.1 to 5, which is a range compatible with C/H ratios in
chondritic meteorites and in the bulk silicate Earth (1.1 and 1.4 by mass,
respectively (Hirschmann, 2016)).
We independently verified the results of our model by tabulating the final
total pressure and composition (in terms of moles of H, C, and O) of select
outgassed atmospheres calculated using our model (Appendix E). These were used
as input parameters for the Equilib module of FactSage 8.0 (Bale et al.,
2009), which calculates the equilibrium partial pressures of gases and stable
condensed phases using a Gibbs free energy minimizer with a database of more
than 40 gas species in the H-C-O system. For the sake of comparison, we
assumed ideal gas behavior. Provided that graphite is not predicted to
precipitate (Section 4.4) the agreement with our models, which utilize a
comparatively simple chemical network and five gas species, is commendable;
the partial pressures differ by at most a few percent (Table E).
Figure 4 compares the influence of the C/H ratio on magma ocean outgassing for
an H budget of 1, 3, and 10 oceans. Cases F1 (Figure 4a), F3 (Figure 4b), and
F10 (Figure 4c) serve as carbon-free reference cases. We first focus attention
on increasing C/H for an H budget of one ocean (left column, Figure 4). Even
for an addition of only 10% C by weight (i.e., C/H=0.1), the atmosphere is
dominated by C species (almost 90% volume mixing ratio) for a fully molten
mantle. This is due to the low solubility of CO2 compared to H2O, coupled with
equilibrium chemistry that establishes the partial pressure of CO around a
factor of seven greater than the partial pressure of CO2. During magma ocean
cooling, the CO pressure in the atmosphere steadily decreases. This occurs as
a result of the compounding effects of the equilibrium constant of Equation 5,
which results in a decrease of $f{\rm CO}/f{\rm CO_{2}}$ as the surface
temperature decreases, and the continued outgassing of CO2 as the melt
fraction decreases. The latter also drives an initial increase in CO2
pressure, although the CO2 pressure decreases later once H begins to outgas in
earnest. Hence, even though the ever-decreasing melt fraction drives
outgassing of both H and C throughout the magma ocean evolution, the partial
pressure of volatile species can either mostly decrease (CO), mostly increase
(H2, H2O), or increase and then decrease (CO2) (Figure 5).
Figure 5: Atmospheric pressures during magma ocean outgassing for different
C/H and $f{\rm O_{2}}$. Columns from left to right show hydrogen (H) budgets
of 1, 3, and 10 oceans, respectively. Rows from top to bottom show $f{\rm
O_{2}}$ of $\Delta$IW=$-2$, $\Delta$IW=0.5, and $\Delta$IW=4. Colors denote
volatiles (red: CO; green: CO2; orange: H2; blue: H2O; purple: CH4, where
solid lines show C/H=1 and dashed lines show C/H=5.
For an H budget of one ocean, the initial pressure of H2 and H2O is 0.5 and
0.4 bars, respectively, regardless of C/H. However, at higher initial C/H
ratios, the H species constitute a smaller fraction (volume mixing ratio) of
the atmosphere owing to the relative insolubility of C-bearing species
compared to H2O. For C/H=5, hydrogen species constitute less than 1% of the
atmosphere during the early evolution of the magma ocean ($>$30% melt). For
this case, the mixing ratio of H species increases by a factor of 230 as the
magma ocean cools from fully molten to fully solid (Figure 4j). In contrast,
for C/H=0.1 the mixing ratio of H species only increases by a factor of about
nine (Figure 4d). Even though the total inventory of H is fixed, the final
outgassed pressures of H2 and H2O depend on C/H, increasing from 62 and 133
bars (C/H=0) to 102 and 221 bars (C/H=5.0), respectively. White dotted lines
in Figure 4 denote a melt fraction of 30%, which corresponds to when a surface
lid forms for equilibrium crystallization. Hence, for equilibrium
crystallization, outgassed atmospheres are usually more dominated by C species
versus H species. For C/H=0.1, the mixing ratio of H species increases by
about a factor of two from 30% melt to 2% melt (Figure 4d). In contrast, for
C/H=5, H species increase from about 1% to more than 50% over the same range
of melt fraction (Figure 4j).
Figure 6: Depletion of hydrogen (H) versus melt fraction at $\Delta$IW=0.5 for
H budgets of 1, 3, and 10 oceans and different C/H by weight. For each shaded
region, the upper bound is C/H=0, lower bound is C/H=5, and black line is
C/H=1. Symbols indicate times of C/H=1 cases near 20% melt and 5% melt.
For an inventory of three oceans (middle column, Figure 4), a larger fraction
of the atmosphere consists of H species during the evolution compared to an
inventory of one ocean. This is a direct consequence of the power-law
solubility of H2O and the higher amount of H (Appendix C.6). When no melt
remains—as for the end of fractional crystallization—the entire initial
inventory of H and C sets the atmospheric composition where equilibrium
chemistry continues to dictate the partitioning between reducing and oxidizing
species. Therefore, for fixed C/H at the end of fractional crystallisation,
the volume mixing ratios are also set. If CH4 is insignificant, the final
partial pressures ($p_{\rm final}$) scale according to the initial inventory;
for example, the final pressures for C/H=0.1 are a factor of three larger for
a three-ocean inventory compared to a one-ocean inventory (compare $p_{\rm
final}$ in Figure 4d,e). For large H inventories, the volume mixing ratio of H
species changes less during outgassing because initially more H (compared to
C) resides in the atmosphere owing to the power-law solubility of H2O. The
case with a three-ocean budget and C/H=1 (Figure 4h, 5e) is similar to the
bulk silicate Earth calculated in Sossi et al. (2020). At high temperatures, a
CO-dominated atmosphere forms with $\sim$200-bar C-bearing species and
$\sim$10-bar H-bearing species in which fH2 and fH2O are subequal, while fCO
$>$ fCO2.
Figure 7: Depletion of hydrogen (H) versus time at $\Delta$IW=0.5 for
different C/H, where colored lines show constant melt fraction. Dashed lines
show C/H=0 for H budgets from 1 to 10 Earth oceans; these are the solid lines
in Figure 3b. Departing from the dashed lines, circles show 3 oceans and
C/H=X, where X is 0, 1, or 5. Similarly, rectangles show 10 oceans and C/H=X,
where X is 0, 1, or 5.
Figure 6 summarizes the interior depletion of H for 1–10 oceans and C/H by
weight varying from zero to five. To first order, depletion at a given melt
fraction is dictated by the total inventory of H due to the power-law
solubility of H2O. For 10% melt, this gives rise to around a 40% increase in
depletion as the H inventory increases from 1 to 10 oceans. A secondary
influence on depletion is C/H, where higher C/H suppresses the outgassing and
hence depletion of H. This effect is not connected to the solubility of H2O,
but rather is due to the dominant presence of C species in the atmosphere that
increases the mean molar mass of the atmosphere relative to H species
(Appendix C.1). Figure 7 reveals the influence of C/H on the depletion during
magma ocean outgassing and can be compared with Figure 3. For a fixed H
inventory, increasing C/H prolongs the time taken to reach a given melt
fraction since the optical depth of the atmosphere increases. Increasing C/H
increases the volume mixing ratio of C species in the atmosphere, and this
alone would actually decrease the opacity of the atmosphere since CO has the
lowest opacity of all the considered volatiles (Appendix A). However,
increasing C/H also increases the total surface pressure substantially because
CO2 is relatively insoluble compared to H2O. Hence, the optical depth
increases in response to the larger surface pressure, and this increase
predominates over the change in speciation; the net outcome is prolonged
cooling for larger C/H. Figure 7 further demonstrates the trend of Figure 6,
in which increasing C/H suppresses the outgassing and hence interior depletion
of H at a given melt fraction.
### 3.4 Oxygen Fugacity
Figure 8: Atmospheric composition during magma ocean outgassing for C/H=1 and
different $f{\rm O_{2}}$. Columns from left to right show hydrogen (H) budgets
of 1, 3, and 10 oceans, respectively. Rows from top to bottom show $f{\rm
O_{2}}$ of $\Delta$IW=$-2$, $\Delta$IW=0.5, $\Delta$IW=2, and $\Delta$IW=4.
Magma ocean lifetime corresponds to the termination of the colored region (at
$\approx$ 2% melt for fractional crystallization, between $10^{4}$ and
$10^{7}$ yr). White dotted lines correspond to 30% melt, which indicates both
the duration and atmospheric composition if instead the mantle underwent
equilibrium crystallization. Initial volatile partial pressures ($p_{\rm
init}$) are for a completely molten magma ocean and $T_{\rm surf}=2700$ K.
Final partial pressures ($p_{\rm final}$) are for complete solidification and
outgassing of all volatiles and $T_{\rm surf}=1400$ K. Figure 9: Depletion of
hydrogen (H) vs. melt fraction for H budgets of 1 and 10 oceans and different
$f{\rm O_{2}}$. For each shaded region, the upper bound is $\Delta$IW=$-2$,
lower bound is $\Delta$IW=4, and black line (same as Figure 6) is
$\Delta$IW=0.5. Symbols indicate times of $\Delta$IW=0.5 cases near 20% and 5%
melt. Figure 10: Depletion of hydrogen (H) vs. time for different $f{\rm
O_{2}}$, where colored lines show constant melt fraction. Dashed lines show
$\Delta$IW=0.5 and C/H=0 for H budgets from 1 to 10 Earth oceans; these are
the solid lines in Figure 3b. Departing from the dashed lines, circles show 3
oceans and C/H=1 for $\Delta$IW=X, where X is $-2$, 0.5, or 4. Similarly,
rectangles show 10 oceans and C/H=1 for $\Delta$IW=X, where X is $-2$, 0.5, or
4.
Iron-core formation established the deep Earth’s redox state at
$\Delta$IW=$-2$ whereas the present-day upper mantle redox is $\Delta$IW=4.
This indicates that mantle redox can vary with both time (e.g., Scaillet &
Gaillard, 2011) and pressure (Armstrong et al., 2019), potentially giving rise
to atmospheres that are more reduced or oxidized than our nominal cases at
$\Delta$IW=0.5 (e.g., Hirschmann, 2012). Hence, we supplement our calculations
at $\Delta$IW=0.5 by additionally considering outgassing scenarios at
$\Delta$IW=$-2$, $\Delta$IW=2, and $\Delta$IW=4. Figure 8 summarizes the
results for C/H=1, and detailed figures for all C/H cases at each $f{\rm
O_{2}}$ are presented in Appendix D. Increasing the oxygen fugacity from
$\Delta$IW=0.5 to $\Delta$IW=4 progressively increases the ratio of oxidized
to reduced species in the atmosphere at a given temperature. Since oxidized
species of C and H have a higher molar mass than their reduced counterparts,
this also increases the mean molar mass of the atmosphere which influences
partial pressures through mass balance (e.g., Equation C1). Cooling times are
generally extended for oxidized versus reduced atmospheres, due to the higher
surface pressure and the intrinsic higher opacity of oxidized species
(Appendix A). However, for $\Delta$IW=$-2$ and C/H=5, production of CH4
increases the cooling time to be comparable to an oxidized atmosphere
dominated by CO2 and H2O.
Oxidized interiors mitigate the interior depletion of H relative to reducing
conditions (e.g., compare $\Delta$IW=$-2$ and $\Delta$IW=4 in Figure 9 and
10). At 30% melt, H depletion is around a factor of 10 larger for
$\Delta$IW=$-2$ compared to $\Delta$IW=4. At 2% melt, H depletion at
$\Delta$IW=$-2$ is approximately 20% larger than $\Delta$IW=4; we recall that
depletion for all cases must reach 100% when no melt remains. Depletion of H
is largest for $\Delta$IW=$-2$ because it exists mainly as the less soluble H2
and CH4 in the gas, so only a small quantity of H is dissolved in the melt
reservoir as H2O. Whereas increasing C/H mitigates H depletion and extends
cooling times, decreasing $f{\rm O_{2}}$ greatly enhances H depletion (i.e.,
outgassing) and generally reduces cooling times. For C/H=1, early atmospheres
are almost always dominated by carbon species, as either CO (reduced cases) or
CO2 (oxidized cases). For 10 H oceans and $\Delta$IW=$-2$ the early atmosphere
is instead dominated by H2. Even for the most oxidized scenario when
$\Delta$IW=4, H2O is never the dominant species in the atmosphere until the
melt fraction drops below about 20% or even less. Hence, magma oceans that
undergo fractional crystallization are more likely to produce oxidized and
H2O-rich atmospheres versus equilibrium crystallization which produces reduced
and often CO-rich atmospheres.
### 3.5 Atmospheric Escape of H During Outgassing
Figure 11: Comparison of atmospheric composition and interior depletion at
$\Delta$IW=0.5 for (1) C/H=0.1, three H oceans, $\gamma=1$ (2) C/H=1, three H
oceans, $\gamma=1$ (3) C/H=0.1, one H ocean, $\gamma=1$ and (4) ”Large escape”
with C/H=0.1, three H oceans, and an escape prefactor $\gamma=1000$ (Equation
9). (a, b) H2 partial pressure and volume mixing ratio; (d, e) CO partial
pressure and volume mixing ratio; (c, f) interior depletion of H and C,
respectively.
During the magma ocean stage, escape of H2—the lightest atmospheric
component—could impact the reservoir evolution of volatiles, so we explore
this possibility in Figure 11. The reference case has C/H=0.1, three H oceans,
and unity escape prefactor ($\gamma=1$, Equation 9). However, an equivalent
case with no escape is visually indistinguishable from $\gamma=1$. This
immediately demonstrates that H2 escape due to irradiation of a
carbon–hydrogen atmosphere does not appreciably alter the volatile reservoirs
during the magma ocean stage for an Earth-sized body at 1 AU, largely because
of the magma ocean’s short lifetime ($\sim$105 yr). Even for a larger H budget
of 10 oceans where cooling is relatively prolonged, $\gamma\lesssim 100$ has
no appreciable impact on the evolution of the volatile reservoirs. Therefore,
we show a case in which escape is increased by setting $\gamma=1000$ (”large
escape”) to investigate an extreme end-member escape scenario, which may
arise, for example, as a result of tidal effects. For comparison,
$\gamma=1000$ gives a mass-loss rate around $2.1\times 10^{8}$ kg s-1, which
is within the range of rates determined for volatile stripping of the proto-
Moon by Earth (Charnoz et al., 2021) or for a Mars-sized body near 1 AU
undergoing EUV-driven thermal escape (Benedikt et al., 2020). For large escape
the inventory of hydrogen decreases by almost 50% over the lifetime of the
magma ocean of around $10^{5}$ yr. By comparison, the magma ocean lifetime for
an equivalent case without escape is around $1.5\times 10^{5}$ yr.
Early on (equivalently times at which melt $>$40%), the H2 partial pressure of
all cases with three H oceans follows a comparable trajectory. At later times
(equivalently $<$40% melt), the cases with three oceans and $\gamma=1$
continue to track a similar trend, although the final H2 partial pressure
depends on the initial C/H (e.g., Figure 4e, h). For high escape rate,
however, the growth of H2 in the atmosphere is mitigated by escape such that
the H2 partial pressure is reduced compared to the cases with an unchanging H
inventory of three oceans. Nevertheless, equilibrium chemistry continues to
dictate $f{\rm H_{2}}/f{\rm H_{2}O}$ so only the magnitudes of the partial
pressures of H species are modulated by escape. The buffering of the
atmosphere by the magma ocean presupposes that the loss of H2 is not
sufficient to influence the redox state of the magma ocean itself. In
practice, the preferential loss of H2 promotes oxidation of the residual
mantle (e.g., Olson & Sharp, 2019). The precise amount depends on the
abundance of H, but the net effect is to self-arrest loss of H as H2. This is
because as escape proceeds, necessarily more H2 is converted to H2O, given
that O-bearing species are heavier and thus escape less readily than H2. Since
for large escape almost 50% of the initial three-ocean inventory is lost,
$f{\rm H_{2}}$ is steered toward the final pressure (at 0% melt) of the one-
ocean case, which has the same C/H=0.1. For fixed C/H=0.1, the volume mixing
ratio of H2 is the same at 0% melt regardless of the final H inventory.
However, for large escape, H is lost relative to C, and hence the H2 volume
mixing ratio at the end of the magma ocean stage is slightly reduced compared
to cases with unchanging C/H=0.1.
Atmospheric escape modulates two controls on outgassing during magma ocean
cooling that we have previously investigated: (1) escape decreases the
inventory of H and hence modulates the partitioning of H between the interior
and atmosphere when melt is present according to the power-law solubility of
H2O, and (2) escape increases C/H, resulting in a final atmosphere richer in C
compared to H. For example, the loss of H2 for large escape increases the
volume mixing ratio of CO at later time compared to C/H=0.1. This abides by
our previous models that show a larger contribution of carbon species to the
atmosphere for a larger C/H. For large escape, C/H by weight increases from
0.1 initially to 0.2 at the end of the magma ocean stage. For minimal escape,
both decreasing the H inventory and/or increasing C/H suppresses the interior
depletion of H at a given melt fraction, relative to the total inventory
(Figure 6). However, for large escape, the depletion of H is enhanced relative
to other cases (Figure 11). This demonstrates that the direct loss of H due to
escape is a stronger control on its depletion in the magma ocean than the
secondary effects on its solubility (through H inventory) and the suppression
of H outgassing due to a heavier, carbon-rich atmosphere. In short, H2 escape
driven by irradiation given best estimates for the XUV flux of the young Sun
has an insignificant impact on volatile reservoir evolution of an Earth-sized
body at 1 AU during the magma ocean stage. However, more extreme escape
scenarios can reduce the partial pressure of H species in the atmosphere and
drive faster depletion of the interior H reservoir.
## 4 Discussion
### 4.1 Atmospheric Composition and Evolution
Figure 12: Summary of atmospheric composition and hydrogen depletion of the
interior during magma ocean cooling at 100%, 30%, and 0% melt, where the
surface temperature is given below the melt fraction. Transition from carbon-
dominated to hydrogen-dominated atmospheres depends on surface lid formation
and the style of magma ocean crystallization, which may prevent or hinder
interior outgassing below 30% melt fraction.
For a redox state inferred for the early Earth, the earliest outgassed
atmospheres of terrestrial planets are dominantly CO-rich by volume mixing
ratio when the mantle is mostly molten (Figure 12). This is borne out of the
equilibrium chemistry of CO–CO2 and the low solubility of C species compared
to H species. With C present, H species can in principle constitute more than
50% of the atmosphere if the H budget is greater than about 6 Earth oceans
owing to the power-law solubility of H2O. However, this is only possible if
C/H by weight is low—around an order of magnitude less than for the bulk
silicate Earth. As C/H increases to unity (Earth-like) and beyond, the
atmosphere is CO-rich regardless of the H budget. Hence, only for C/H$\lesssim
0.1$ is an H-rich atmosphere dominant during the early cooling phase, when
most of the mantle is molten. Throughout the magma ocean stage, H2 and H2O
have roughly constant mixing ratios owing to the weak temperature dependence
of the equilibrium constant (Equation 4), while $f{\rm H_{2}}/f{\rm H_{2}O}$
is approximately unity at $\Delta$IW=0.5 at magmatic temperatures.
For C/H$\gtrsim$1, H species can only begin outgassing in earnest once the
melt fraction drops below about 30%, regardless of the redox state. However,
this is contingent on the surface remaining molten to ensure equilibrium
between the interior and atmosphere; this is less likely if the magma ocean
undergoes equilibrium crystallization where crystals and melt freeze together
and a surface lid forms quickly. In comparison, fractional
crystallization—driven by melt–solid separation—can more likely maintain a
molten surface while the deeper mantle crystallizes and exsolves volatiles.
Hence, the transition from a C-rich to H-rich atmosphere depends on the style
of crystallization, and crucially the behavior of the mantle as it transitions
from mostly molten to mostly solid. A large steam-dominated atmosphere can
only form for a relatively oxidized mantle toward the end of mantle
crystallization, although it may persist for a longer duration than the early
carbon-rich atmosphere in the absence of an efficient loss mechanism. An
atmosphere that contains initially more H species by volume mixing ratio
(C/H$\lesssim 0.1$ and particularly for H inventories $>$ 6 oceans) undergoes
less modification during outgassing to reach its final volatile mixing ratios
when the mantle has fully crystallized.
The chemical boundary layer at the surface of the magma ocean facilitates
interior–atmosphere equilibrium, where the equilibrium timescale is rapid
relative to a crystallization timescale of around a million years (Pahlevan et
al., 2019). For low $f{\rm O_{2}}$ or small atmospheres, the crystallization
timescale decreases by no more than two orders of magnitude ($\sim 10^{4}$ yr)
and this decrease would be further mitigated by a greater H2 greenhouse
forcing (Lichtenberg et al., 2021). Other mechanisms to facilitate
equilibrium, such as bubble formation due to volatile supersaturation (Elkins-
Tanton, 2008), would also reduce the equilibrium timescale. Nevertheless, if
crystallization proceeds too rapidly compared to the equilibrium timescale,
then disequilibrium chemistry may arise between the interior and atmosphere.
### 4.2 Hydrogen in the Interior
For otherwise highly soluble volatiles, such as H2O, to exist in the
atmosphere, the magma ocean must have crystallized below 30% melt fraction
before a surface lid could form and persist at the surface. Otherwise,
thermodynamic communication between the interior and atmosphere is broken, and
dissolved volatiles remain trapped in the mantle. Below 30% melt fraction,
continued outgassing of dissolved volatiles requires that significant
solidification occurs through melt percolation and solid compaction before the
rheological front reaches the surface or a quench crust forms (fractional
crystallization). In an end-member scenario, this can enable near-complete
outgassing of all volatiles, which has received the most attention in the
literature. At a given melt fraction, we find that a larger fraction of H can
be retained in the interior for smaller total inventories of H, larger C/H by
weight, and more oxidized interiors. Although C does not influence H
solubility directly, it impacts H retention since it suppresses H outgassing
through its influence on the mean molar mass of the atmosphere (Bower et al.,
2019).
Equilibrium crystallization results in a substantial reservoir of melt and
hence volatiles trapped in the interior when the surface forms a lid. This
later possibility has received less attention in the context of volatile
evolution during the magma ocean stage, but it provides a mechanism to safely
harbor a significant quantity of volatiles in the mantle to protect against
loss from atmospheric escape and impacts. Hence, the high solubility of H2O in
silicate melt may be a crucial property of this life-supporting molecule that
enables it to survive during the violent early years of a terrestrial planet’s
life.
Volatile retention would be further enhanced if a melt–crystal density
crossover enabled the formation of a voluminous basal magma ocean (Labrosse et
al., 2007; Caracas et al., 2019) or if melt is captured as the rheological
front advances through the mantle (Hier-Majumder & Hirschmann, 2017). Trapped
melt, and hence soluble volatiles, could then be sequestered in the deep
mantle by Rayleigh-Taylor instability (Maurice et al., 2017; Miyazaki &
Korenaga, 2021). Moreover, we assume no solubility of H in crystallizing
minerals, whereas experimental data indicate significant quantities of water
may be stored in ringwoodite (Fei & Katsura, 2020). In short, there are
additional processes not included in our models that further conspire against
the complete outgassing of soluble volatiles, namely, water, during the magma
ocean stage. Furthermore, estimates of the volatile concentration in the bulk
silicate Earth are incompatible with complete outgassing (e.g., Hier-Majumder
& Hirschmann, 2017).
Therefore, complete outgassing of soluble volatiles—frequently an outcome of
magma ocean models—only occurs with fractional crystallization if the
aforementioned processes are absent or inefficient. Furthermore, it requires
that a surface lid or quench crust, if present, does not hinder volatile
outgassing as fractional crystallization proceeds. Hence, at the end of the
magma ocean stage it is reasonable to expect that a trapped reservoir of H2O
in the interior will interact with the surface environment and atmosphere.
This could occur owing to post-magma ocean cumulate overturn or owing to
processes operating over geological timescales (millions to billions of
years). The detailed chemical and physical processes governing the formation
and sustenance of cumulate layers and a surface lid in a dynamic magma ocean
require further investigation; external influences such as projectile
bombardment could also stifle lid formation (e.g., Perera et al., 2018).
For an Earth-like planet orbiting a Sun-like star at 1 AU during its magma
ocean stage, we find that atmospheric escape of H2 due to irradiation (energy
and diffusion limited) does not significantly impact the evolution of volatile
reservoirs, in agreement with Hamano et al. (2013) and Katyal et al. (2020).
This is because escape rates are sufficiently low and the magma ocean duration
is sufficiently short (at most a few million years). In future work, a
feedback to probe is that loss of H2 increases $f{\rm O_{2}}$ slightly,
causing more H2 to convert to H2O and thereby partly self-arresting the loss
process. Furthermore, we have not considered photodissociation of H2O, but
this could interplay with the included geochemical reactions that depend on
mantle redox.
### 4.3 Crystallisation Style
Based on energetic considerations of convection versus gravitational settling,
Solomatov & Stevenson (1993a) propose a critical crystal size of 1 cm above
which fractional crystallization is inevitable. Furthermore, they show that
the critical crystal size is weakly dependent on crystal fraction and
therefore depth. Hence, their results are compatible with our fractional
crystallization models with a constant crystal size of 5 cm. In detail,
gravitational settling in our models becomes dominant at the rheological
transition owing to the substantial decrease of turbulence that enabled
crystals to remain mostly in suspension. It is reasonable to assume that
crystals are suspended at high melt fraction (Tonks & Melosh, 1990), although
they could begin to settle and thereby initiate fractional crystallization
before the rheological transition is obtained (Patoc̆ka et al., 2020),
assuming inefficient re-entrainment (Solomatov et al., 1993) or a high
planetary rotation rate (Maas & Hansen, 2019). Settling also depends on
mineral buoyancy, and if the minerals float rather than sink, this could
hinder interior–atmosphere communication. For example, preferential
partitioning of Fe into melt during crystallization continuously changes the
level of neutral buoyancy between crystals and melt (Caracas et al., 2019). In
contrast to fractional crystallization, equilibrium crystallization prevents
differentiation of the mantle and occurs for a smaller crystal size of 1 mm
(Solomatov & Stevenson, 1993b).
Most previous dynamic models of magma ocean cooling do not consider melt–solid
separation (Lebrun et al., 2013; Hamano et al., 2013; Schaefer et al., 2016;
Nikolaou et al., 2019); rather, the interior is assumed to always be
adiabatic. Yet some of these previous models report agreement with the
solidification timescale derived from geochemical models that explicitly
consider fractional crystallization. The previous models implicitly assume
that although the deep mantle reaches the rheological transition first, it
continues cooling as efficiently by convection as the uppermost mantle; this
allows the melt fraction to continue decreasing at a similar rate. However,
this is unlikely given the rheological behavior of a melt–solid aggregate
(e.g., Costa et al., 2009), which at the rheological transition predicts a
large reduction in the convective velocity and rapid melt–solid separation
(Abe, 1993). Hence, somewhat coincidentally, previous models decrease the melt
reservoir and outgas volatiles at a similar rate to geochemical fractional
crystallization models, even though the key ingredient to justify fractional
crystallisation (i.e., melt–solid separation) is not included.
### 4.4 Graphite, Diamond, and Water Precipitation
Our model assumes that all volatile species dissolved at the surface of the
magma ocean continue to remain so, irrespective of pressure (depth of the
magma ocean) and temperature (cooling of the magma ocean). Phase
transformations involving these volatile elements are likely to occur as
pressure and temperature change (e.g., Hirschmann, 2012). However,
experimental data for the speciation and partitioning of H-, C-, and O-bearing
volatile species rarely exceed 7 GPa (Section 2.2), which precludes a
holistic, bottom-up model of magma ocean crystallization. Nevertheless, we now
systematically investigate possible phase transformations of H and C to
identify scenarios under which the pressures of outgassed atmospheres could
diverge from those calculated by our model.
Reducing conditions may induce graphite or diamond precipitation (Hirschmann,
2012; Takahashi et al., 2013; Keppler & Golabek, 2019). The precipitation of
graphite occurs if the CO fugacity of the atmosphere exceeds that defined by
the graphite–carbon monoxide (CCO) buffer:
$\displaystyle\rm{C(s)}+0.5\rm{O_{2}(g)}$ $\displaystyle=\rm{CO(g)},$ (11)
$\displaystyle\log_{10}{\rm K_{eq}}$ $\displaystyle=\frac{6254}{T}+4.334,$
where $\mathrm{K_{eq}}$ as a function of temperature is provided by the JANAF
database (Chase, 1998). Equation 11 indicates that $\log_{10}f$CO in
equilibrium with graphite decreases proportional to $0.5\log_{10}f$O2 and
increases with temperature. Thus, graphite precipitation is favored under more
reducing conditions and lower temperatures (Figure 13). For our cases at
$f\rm{O_{2}}$s of $\Delta$IW=0.5 or more oxidized, graphite precipitation
would not occur because $f$CO never exceeds that defined by the CCO buffer.
This suggests that it is unlikely that a terrestrial magma ocean was saturated
in graphite at high melt fractions ($>30$%). However, it may occur for most
atmospheres as the surface cools below the temperature at which a surface lid
forms around 1650 K (Sossi et al., 2020). At this stage, however, the
atmosphere and mantle are not necessarily in equilibrium and the atmosphere
can evolve as a near-closed system.
For a magma ocean more reduced than IW, the modeled cases at $f\rm{O_{2}}$ of
$\Delta$IW=$-2$ and C/H=5 show that graphite precipitation would occur in a
cooling atmosphere, even accounting for the production of CH4 that buffers
$f$CO (Figure 13). Specifically, higher initial C budgets result in higher
$f$CO, where, for C/H=5, the scenario with 10 oceans crosses the CCO buffer at
$\sim$2300 K, while it does so at $\sim$2000 K for five oceans. As such, the
quoted fugacities of carbon-bearing species for these two scenarios are upper
limits. Cases at C/H=1 and $\Delta$IW=$-2$ do not result in graphite
saturation at any temperature, irrespective of the number of oceans, owing to
the lower $f$CO.
Figure 13: The fugacity of CO in equilibrium with graphite (Equation 11)
assuming ideal gas behavior as a function of temperature at $\Delta$IW=$-2$
(dashed black line) and $\Delta$IW=0.5 (dashed gray line). All cases at
$\Delta$IW=$-2$ and C/H=5 (colored lines) intercept the CCO buffer at
$\Delta$IW=$-2$, indicating graphite precipitation. Only cases that intercept
the CCO buffer are plotted.
We presume that the magma ocean is well mixed, such that it remains
isochemical throughout its depth. However, the saturation of graphite in
equilibrium with silicate melt depends on pressure and temperature (Dasgupta
et al., 2013; Stanley et al., 2014; Chi et al., 2014). As such, chemical
gradients could arise from its precipitation during magma ocean cooling and/or
at depth followed by its isolation from the melt (Hirschmann, 2012). To assess
the potential effect of graphite/diamond precipitation on the calculated
fugacities of carbon-bearing species, we compare the C contents dissolved in
the magma ocean predicted by our models with the graphite/diamond
precipitation curve. Above the IW buffer, C dissolves predominantly as
CO${}_{3}^{2-}$ in mafic and ultramafic melts (Armstrong et al., 2015; Duncan
et al., 2017). Holloway et al. (1992) devised an expression to determine
CO${}_{3}^{2-}$ solubility at graphite saturation. To calculate whether
graphite is expected to precipitate from the silicate melt in our simulations,
we adopt the calibration of Duncan et al. (2017) extrapolated to peridotite
compositions. At constant pressure (1 GPa) and for the temperature range of
cooling considered in our models (1650 to 2500 K), little or no C
precipitation occurs in the magma ocean under any of the scenarios modeled at
IW+0.5 and above. At 1650 K and 1 GPa, the model of Duncan et al. (2017)
predicts that $\sim$350 ppmw CO2 is dissolved in a peridotite melt or 100 ppmw
for a komatiitic melt (expected for an evolved magma ocean) at graphite
saturation, compared to $\sim$300 ppmw in the melt in the most extreme, C/H=5,
10-ocean case.
Below the IW buffer, CO${}_{3}^{2-}$ is no longer the prevailing melt species
of dissolved carbon (instead, carbon likely occurs as some CO-bearing
molecule, Wetzel et al., 2013; Dalou et al., 2019); therefore other
expressions for the prediction of C solubility at graphite saturation need to
be considered. Here we adopt the calibration of Yoshioka et al. (2019) that
links CO dissolved in the melt to its fugacity, $f\rm{CO}$. We fix $f\rm{CO}$
as a function of pressure in equilibrium with graphite at IW$-2$ using
thermodynamic data and the equation of state of Jakobsson & Oskarsson (1994).
At 2 GPa, this yields an $f\rm{CO}$ of $\sim$8.3 GPa, which, using the
expression of Yoshioka et al. (2019), leads to 500 ppmw of dissolved CO. These
estimates far exceed those derived by solely considering CO${}_{3}^{2-}$ as
the dissolved species ($\sim$2 ppmw, Duncan et al., 2017). Given that
dissolved C contents reach 20 ppmw at most in our simulations at IW$-2$, this
implies that graphite and diamond should not precipitate at any stage in the
magma ocean, even under reducing conditions.
Another important phase change can occur in cooling atmospheres, namely, that
between liquid water and steam:
${\rm H_{2}O(l)}={\rm H_{2}O(g)}.$ (12)
Its equilibrium constant as a function of temperature indicates that, at 1400
K, H2O condenses at about 600-bar $f$H2O. This phase transformation is
neglected in our simulations primarily because these pressures and
temperatures exceed the critical point of water, which is 650 K and 220 bars.
Hence at 600-bar $f$H2O the atmosphere is a supercritical fluid rather than an
ideal gas as modeled herein. Moreover, the mixing of CO2 and H2O is nonideal,
such that departures from ideal gas behavior are expected in these solutions
(Duan et al., 1992; Frost & Wood, 1997).
### 4.5 Meteoritic Degassing
Theoretical calculations (Schaefer & Fegley, 2010) and outgassing experiments
(Thompson et al., 2021) predict the composition of gases evolved from the
degassing of meteoritic materials, some of which may be representative of
planetary building blocks. By examining equilibrium between gas and condensed
phases, thermodynamic models find that H2 and CO gases form from reduced
chondrites (such as EH) at high temperature (Schaefer & Fegley, 2010). Only
for the most oxidizing materials (CI chondritic composition) is H2O predicted
to be a major constituent of the gas mixture. Since water is only an important
gaseous species during degassing of one particular class of chondrite, this
calls into question the applicability of steam atmospheres to the evolution of
terrestrial planets at large.
Although instructive, directly correlating gases produced from meteorites to
the composition of planetary atmospheres is complicated for several reasons.
Meteorites are imperfect analogs of the material that went on to form the
terrestrial planets (Sossi, 2021), particularly as regards volatile elements
due to the ubiquitous alteration processes occurring on their parent body
(Brearley, 2006), as well as thermal metamorphism and the likelihood that
planetesimals are already differentiated when they accrete. More importantly,
the likely presence of a deep magma ocean during and immediately after
accretion would have modified the budgets of elements available to degas at
the planetary surface (e.g., Grewal et al., 2020; Dalou et al., 2019). As
shown in our work, equilibrium reactions between silicate melt and volatiles
in the gas phase influence the partitioning of gas species between the
atmosphere and magma ocean according to their solubility. Furthermore, some
volatiles (including H and C) may be permanently sequestered into an iron core
owing to their siderophile nature (e.g., Tagawa et al., 2021).
Therefore, connecting gas mixtures to atmospheric composition necessitates an
assumption that the starting bulk composition—usually derived from
chondrites–adequately captures elemental abundances in the atmosphere during
and after the magma ocean phase. This assumes that volatiles did not
experience significant partitioning due to magma ocean and core formation
processes and that atmosphere formation proceeds by the breakdown of solid
mineral assemblages (i.e., degassing occurs subsolidus). Solid bodies cannot
efficiently replenish surface material from deeper regions, whereas for molten
bodies the entire melt mass can potentially communicate with the atmosphere,
subject to the interior–atmosphere equilibration timescale (e.g., Pahlevan et
al., 2019). Alternatively, partitioning can be permitted if ongoing impact-
induced outgassing of chondritic materials after a surface lid or quench crust
forms (e.g. in a late-veneer scenario or during the late heavy bombardment) is
sufficient to displace or dilute any previously equilibrated atmosphere
(Zahnle et al., 2020).
Extraction of volatiles to solids in the core or mantle locks them away from
outgassing to the atmosphere over a short timescale, and perhaps indefinitely.
By contrast, volatiles partitioned in melt are readily available to outgas
during mantle crystallization as long as the interior and atmosphere remain in
thermodynamic communication (e.g., for fractional crystallization). Accounting
for thermodynamic equilibrium with silicate melt, we demonstrate that the
earliest atmosphere evolves in both mass and composition during magma ocean
cooling. An atmosphere can transition from dominantly CO-rich (”reducing”) to
H2O-rich (”oxidizing”) owing to solubility and redox reactions as the magma
ocean cools (Figure 12). Therefore, the solubility of volatiles in silicate
melt prevents drawing a simple connection between the composition of planetary
building blocks and the earliest atmospheres of terrestrial planets.
## 5 Summary and Conclusions
The investigation of catastrophic outgassing of CO2 and H2O from terrestrial
magma oceans—as well as their influence on the nature of early atmospheres
around rocky planets—was originally motivated by the degassing of hydrated
minerals and oxidized chondritic materials. However, for redox conditions
appropriate for the early Earth, we find that CO is usually the dominant
atmospheric species during the early molten stage, where the melt fraction is
greater than 30% (Figure. 12). This is due to the relatively high solubility
of hydrogen species (H2O) compared to carbon species (CO2), as well as redox
reactions that govern CO–CO2, H2–H2O, and CO2–H2–CH4 equilibria. Only when C/H
by weight is small (C/H$<$0.1) and the H budget large ($>$6 oceans) can H
species constitute more than 50% of the atmosphere, with H2 and H2O in
approximately equal abundance (Figure 4). For more oxidizing conditions, early
atmospheres are dominated by CO2 with H2O a minor species (Figure 8).
For the late molten stage ($<30\%$ melt), more H is retained (relative to the
total budget) for smaller H budgets, larger C/H, and more oxidizing conditions
(Figure 12). At this stage, continued outgassing driven by dissolution
equilibrium requires that the surface remains molten and a persistent lid does
not form; this requirement can be satisfied if a magma ocean undergoes
fractional crystallization. However, for magma oceans that freeze more quickly
by equilibrium crystallization, formation of a surface lid around 30% melt can
prevent subsequent outgassing of highly soluble volatiles such as H2O. This
would leave the atmosphere as carbon/CO dominated by stifling the formation of
a steam atmosphere. Methane only forms for reduced conditions
($\Delta$IW$\lesssim$0.5) when pressures are sufficiently high or temperatures
sufficiently low; otherwise, CH4 is absent. Graphite precipitation from the
atmosphere during magma ocean cooling is expected only for very high C budgets
(higher than those anticipated for Earth) and for $\Delta$IW$\lesssim$0.5.
Nevertheless, for all cases, graphite in the atmosphere may precipitate with
continued cooling.
Complete outgassing of volatiles during magma ocean solidification can arise
as a result of fractional crystallisation. However, by additionally
considering equilibrium crystallization and other processes, we expect that a
substantial reservoir of H could remain in planetary mantles and outgas over
geological timescales, potentially impacting the depth of surface oceans and
the ability to desiccate planets. In this case, hydrogen species would only
play a minor role in determining atmospheric opacity or modulating atmospheric
escape during the early magma ocean; rather, the behavior of carbon species is
dominant. The style of magma ocean crystallization—fractional or
equilibrium—therefore controls the composition and mass of early atmospheres
and the efficiency of volatile delivery to the planetary atmosphere and
surface. Ultimately, the high solubility of H2O in magma oceans may enable its
safe storage during the tumultuous phase of planet formation.
## 6 Data Availability
All of the data generated as part of this study can be obtained by contacting
Dan J. Bower.
DJB acknowledges Swiss National Science Foundation (SNSF) Ambizione Grant
173992. KH is supported by the European Research Council via Consolidator
Grant ERC-2017-CoG-771620-EXOKLEIN. PAS acknowledges Swiss National Science
Foundation (SNSF) Ambizione Grant 180025. PS acknowledges financial support
from the Swiss University Conference and the Swiss Council of Federal
Institutes of Technology through the Platform for Advanced Scientific
Computing (PASC) program. This research was partly inspired by discussions and
interactions within the framework of the National Center for Competence in
Research (NCCR) PlanetS supported by the SNSF. The calculations were performed
on UBELIX (http://www.id.unibe.ch/hpc), the HPC cluster at the University of
Bern. We thank S. Grimm and D. Kitzmann for computing Rosseland mean
opacities, and A. Wolf, M. Tian, and T. Lichtenberg for discussions and
feedback on the manuscript. Three anonymous referees provided comments that
further enhanced the manuscript.
## Appendix A Opacities
We computed Rosseland mean opacities at 1 bar between 1700 and 2700 K for H2O,
H2, CO2, CO, and CH4 using Helios-k (Grimm et al., 2021; Grimm & Heng, 2015).
The opacity (Rosseland mean mass absorption coefficient) for H2O using the
full BT2 list (Barber et al., 2006) is $1.5\times 10^{-2}$ m2 kg-1 at 1700 K
and $5.0\times 10^{-4}$ m2 kg-1 at 2700 K; the opacity at 1800 K compares
favorably with $10^{-2}$ m2 kg-1 which has been used extensively in previous
magma ocean studies. For simplicity we also adopt a constant value (i.e.,
independent of temperature) of $10^{-2}$ m2 kg-1. The opacity of H2 is
determined by collision-induced absorption, and using HITRAN (Karman et al.,
2019; Abel et al., 2011) it is essentially constant ($5\times 10^{-5}$ m2
kg-1) from 2700 to 1700 K.
For CO2, HITEMP (Rothman et al., 2010) provided the line list, and we
determined an opacity of $1.5\times 10^{-4}$ m2 kg-1 at 1700 K and $3\times
10^{-5}$ m2 kg-1 at 2700 K. Hence the CO2 opacity at 1700 K is compatible with
$10^{-4}$ m2 kg-1, which is the lowest opacity considered for CO2 in previous
magma ocean models. To be consistent with our selection of the H2O opacity
around 1800 K, we similarly adopt a constant CO2 opacity of $10^{-4}$ m2 kg-1.
For CO we determined an opacity of $1.2\times 10^{-5}$ m2 kg-1 at 1700 K and
$3\times 10^{-6}$ m2 kg-1 at 2700 K using HITEMP (Li et al., 2015a). Again, we
selected an opacity of $10^{-5}$ m2 kg-1 based on the value at 1800 K. We
computed a CH4 opacity of $10^{-2}$ m2 kg-1 at 1800 K based on the line lists
of Yurchenko et al. (2013); Yurchenko & Tennyson (2014). The opacities of
species at 1 bar are reference opacities that are used to compute a pressure-
dependent opacity (Equation A22, Abe & Matsui, 1985).
## Appendix B Melt–Solid Separation
The fluxes to describe energy transport in a planetary mantle (which is
molten, partially molten, or solid) are described in detail in Bower et al.
(2018); Abe (1995). However, we recap the physics underpinning melt–solid
separation owing to its importance for determining whether a mantle undergoes
fractional or equilibrium crystallization. At high melt fraction, the settling
or flotation of crystals within a magma ocean is determined by Stokes’s law:
$u_{m}-u_{s}=\frac{2a^{2}g(\rho_{m}-\rho_{s})}{9\eta_{m}},$ (B1)
where $u$ is velocity (positive radially outward), $a$ crystal size, $g$
gravity (negative), $\rho$ density, $\eta$ viscosity, and subscripts $m$ and
$s$ denote melt and solid, respectively. Interaction amongst crystals is not
considered, and therefore free settling or flotation of crystals is an upper
limit on the efficiency of melt–solid separation at high melt fraction. At low
melt fraction, separation occurs via Darcy flow:
$u_{m}-u_{s}=\frac{k_{p}g(\rho_{m}-\rho_{s})}{\phi_{p}\eta_{m}},$ (B2)
where $k_{p}$ is permeability and $\phi_{p}$ porosity (synonymous with the
volume fraction of melt). The Rumpf–Gupte permeability law is appropriate at
intermediate porosity (Rumpf & Gupte, 1971),
$k_{rg}=\frac{a^{2}\phi_{p}^{5.5}}{1.4},$ (B3)
and the Blake–Kozeny–Carman permeability law at low porosity,
$k_{bkc}=\frac{a^{2}\phi_{p}^{3}}{1000(1-\phi_{p})^{2}}.$ (B4)
The constants in these permeability laws are constrained by experiments (see
McKenzie, 1984; Abe, 1995, for discussion). The flow laws can be represented
by a single description (Abe, 1995):
$u_{m}-u_{s}=\frac{a^{2}g(\rho_{m}-\rho_{s})}{\eta_{m}}f,$ (B5) $\displaystyle
f=$ $\displaystyle\dfrac{2}{9}$ $\text{for }0.77\leq\phi_{p}$, (B6a)
$\displaystyle f=$ $\displaystyle\dfrac{k_{rg}}{a^{2}\phi_{p}}$ $\text{for
}0.08<\phi_{p}<0.77$, (B6b) $\displaystyle f=$
$\displaystyle\dfrac{k_{bkc}}{a^{2}\phi_{p}}$ $\text{for }\phi_{p}\leq 0.08$,
(B6c)
where the range of applicability of each flow law ensures that $f$ is a
continuous function of porosity. Through consideration of the local
barycentric velocity, the mass flux of melt is
$J_{m}=\rho\phi(1-\phi)(u_{m}-u_{s}),$ (B7)
where $\phi$ is melt fraction (the mass fraction of melt). Equation B7
naturally satisfies the requirement that $J_{m}=0$ for $\phi=0$ and $\phi=1$.
The energy flux of melt–solid separation is therefore
$F_{\rm grav}=J_{m}T_{\rm fus}\Delta S_{\rm fus},$ (B8)
where $T_{\rm fus}$ is the temperature at 50% melt fraction and $\Delta S_{\rm
fus}$ is the entropy of fusion, both of which are functions of pressure
according to the melting curves. Equation B5 reveals the common factors
appearing in all of the flow laws and emphasizes the importance of crystal
size owing to its squared dependence: larger crystals enhance melt–solid
separation. Since we consider a single-component mantle (MgSiO3), everywhere
$\rho_{m}-\rho_{s}<0$ and hence crystals always sink and melt always drains
upward to the surface.
## Appendix C Magma ocean and atmosphere equilibrium
### C.1 Volatile mass balance and evolution
We extended Bower et al. (2019) to express the mass balance for a volatile $v$
in a chemically reactive environment with atmospheric escape:
$\displaystyle X_{v}k_{v}M^{s}+X_{v}M^{l}+4\pi
R_{p}^{2}\left(\frac{\mu_{v}}{\bar{\mu}}\right)\frac{p_{v}}{|g|}+m_{v}^{e}+\sum_{w=1}^{r}m_{v}^{w}$
$\displaystyle=X_{v}^{0}M^{m},$ (C1)
where:
* •
$X_{v}$ is the volatile abundance in the pure melt;
* •
$k_{v}$ is the distribution coefficient between solid and melt;
* •
$M^{s}$ is the mantle mass of solid;
* •
$M^{l}$ is the mantle mass of melt;
* •
$M^{m}=M^{s}+M^{l}$ is the total mantle mass, which is constant;
* •
$R_{p}$ is the planetary surface radius;
* •
$g$ is the surface gravity;
* •
$\mu_{v}$ is the molar mass of the volatile;
* •
$\bar{\mu}$ is the mean molar mass of the atmosphere;
* •
$p_{v}$ is the surface partial pressure of the volatile; and
* •
$X_{v}^{0}$ is the initial total volatile abundance relative to the total
mantle mass.
A solubility law relates the volatile concentration in the melt ($X_{v}$) to
the partial pressure of the volatile in the atmosphere ($p_{v}$). The first
three terms on the left-hand side of Equation C1 represent the mass of the
volatile stored in the solid mantle, the molten mantle, and the atmosphere.
The final two terms on the left-hand side are fictitious reservoirs used to
account for mass loss due to escape ($m_{v}^{e}$; Appendix C.2), and mass
transfer due to participation of this volatile in reaction $w$ ($m_{v}^{w}$,
Appendix C.3). The time derivative of Equation C1 derives the evolution
equation (cf., Equation A4, Bower et al., 2019):
$\displaystyle\frac{dX_{v}}{dt}\left(k_{v}M^{m}+(1-k_{v})M^{l}\right)+X_{v}(1-k_{v})\frac{dM^{l}}{dt}$
$\displaystyle+\frac{4\pi
R_{p}^{2}\mu_{v}}{|g|}\left[\frac{1}{\bar{\mu}}\frac{dp_{v}}{dt}-\frac{p_{v}}{\bar{\mu}^{2}}\sum_{w=1}^{n}\frac{\mu_{w}}{P_{s}}\left(\frac{dp_{w}}{dt}-\frac{p_{w}}{P_{s}}\frac{dP_{s}}{dt}\right)\right]$
$\displaystyle+\frac{dm_{v}^{e}}{dt}+\sum_{w=1}^{r}\frac{dm_{v}^{w}}{dt}=0,$
(C2)
where $P_{s}$ is the total surface pressure, which is equal to the summation
of the partial pressures of all volatiles according to Dalton’s law. The
derivative of $X_{v}$ with respect to $t$ can be determined from the
solubility law relevant for volatile $v$, which for power-law solubility
(Equation 1) is
$\frac{dX_{v}}{dt}=\frac{dX_{v}}{dp_{v}}\frac{dp_{v}}{dt}=\left(\frac{\alpha_{v}{p_{v}}^{1/\beta_{v}-1}}{\beta_{v}}\right)\frac{dp_{v}}{dt}.$
(C3)
Similarly, $X_{v}$ can be eliminated from Equation C2 using the solubility
law. This results in a system of nonlinear equations that can be solved
simultaneously to determine $dp_{v}/dt$ for each volatile $v$. Solving for
$dp_{v}/dt$ rather than $dX_{v}/dt$ enables us to treat the case of zero
solubility. Oxygen fugacity $f{\rm O_{2}}$ (Equation 7) depends on surface
temperature, so oxygen is not explicitly tracked by mass balance as with the
other volatiles.
### C.2 Mass loss due to atmospheric escape
The escape flux of H2 (Equation 9) is used to determine a mass-loss rate of H2
(e.g., Katyal et al., 2020):
$\frac{dm_{H_{2}}^{e}}{dt}=4\pi R_{p}^{2}\phi_{\rm H_{2}}\mu_{\rm H_{2}}.$
(C4)
The loss rate depends on the time-dependent volume mixing flux of H2 through
$\phi_{\rm H_{2}}$. In turn, the volume mixing ratio of H2 is computed from
the partial pressure of H2 and the total surface pressure. In the present
work, the mass-loss rate for volatiles other than H2 is set to zero.
### C.3 Mass exchange due to chemical reactions
During the magma ocean stage, the silicate melt and overlying atmosphere can
accommodate mass exchange via chemical reactions. We consider a general
reaction with two reactants ($A$, $B$) and two products ($C$, $D$):
$aA+bB\rightleftharpoons cC+dD,$ (C5)
where $a$, $b$, $c$, and $d$ are stoichiometric coefficients. The reaction
quotient $Q$ is a function of time $t$:
$Q(t)=\frac{\\{C\\}^{c}\\{D\\}^{d}}{\\{A\\}^{a}\\{B\\}^{b}}\doteq\frac{Q_{p}(t)}{Q_{r}(t)},$
(C6)
where curly brackets $\\{\\}$ denote the activity of the species and the
activity of a gas is its fugacity divided by a reference pressure. At chemical
equilibrium, the reaction quotient is equal to the equilibrium constant $K$,
which is a function of temperature. An appropriate temperature is that of the
magma ocean–atmosphere interface; hence,
$\frac{\\{C\\}^{c}\\{D\\}^{d}}{\\{A\\}^{a}\\{B\\}^{b}}-K(T)=0.$ (C7)
The reaction in Equation C5 can also be expressed in terms of a mass balance
of reactants and products:
$m_{A}+m_{B}\rightleftharpoons m_{C}+m_{D}.$ (C8)
This is because the fractional change of moles corresponds to the fractional
change of mass, so correlating terms in Equation. C5 and C8:
$m_{B}=\left(\frac{b\mu_{B}}{a\mu_{A}}\right)m_{A},\quad
m_{C}=\left(\frac{c\mu_{C}}{a\mu_{A}}\right)m_{A},\quad{\rm etc.}$ (C9)
where $\mu$ is the molar mass. It is convenient in Equation C9 to express the
mass of the reactants and products using the mass of the first reactant
($m_{A}$) multiplied by a constant factor that depends only on molar masses
and reaction stoichiometry; mathematically this choice is arbitrary, and
another mass could equally be used. Hence, each chemical reaction introduces a
single unknown ”reaction mass” ($m_{A}$) and one equation (Equation C7) to the
complete system of equations that we solve. The functional dependencies of all
quantities in Equation C7 are known, since we assume ideal gas behavior to
connect gas activities to volatile partial pressures and the functional
dependence of the equilibrium constant on temperature $T$ is documented in
thermochemical tables (Chase, 1998).
Evolution equations of Equations C7 and C9 are necessary to time-step the
model. The time derivatives of the reactant and product masses are trivial to
compute since stoichiometric coefficients and molar masses are constant. For
example (c.f., Equation C9):
$\frac{dm_{B}}{dt}=\left(\frac{b\mu_{B}}{a\mu_{A}}\right)\frac{dm_{A}}{dt},$
(C10)
and the time derivative of the reactant masses ($m_{C}$, $m_{D}$) can be
similarly computed. The time derivative of the equilibrium condition (Equation
C7) is:
$\frac{1}{Q_{r}}\frac{dQ_{p}}{dt}-\frac{Q_{p}}{Q_{r}^{2}}\frac{dQ_{r}}{dt}-\frac{dK}{dT}\frac{dT}{dt}=0,$
(C11)
where the reaction quotient numerator $Q_{p}=\\{C\\}^{c}\\{D\\}^{d}$ and
denominator $Q_{r}=\\{A\\}^{a}\\{B\\}^{b}$ depend on the volatile partial
pressures and reaction stoichiometry. The equilibrium constant $K$ depends
only on temperature $T$ (Equations 4 and 5) and therefore $dK/dT$ is
straightforward to evaluate. The code also solves for $dT/dt$ in the mantle at
each staggered mesh point, although only the change of surface temperature is
required since this temperature defines the interface of the magma ocean and
atmosphere.
Hence, to include reactions in the time stepper, Equation C11 constitutes an
extra equation per reaction that must be satisfied, where the extra unknowns
we solve for are the change of reaction mass for each reaction ($dm_{A}/dt$).
Equation C10 relates the change of reaction mass for reactant $B$ and products
$C$ and $D$ to the solution quantity ($dm_{A}/dt$). For each volatile, we can
thus evaluate the total rate of change of its mass due to all reactions that
it participates in; this is the final term on the left-hand side of Equation
C2.
### C.4 Initial volatile abundances
For the initial condition, we solve for the abundance of each volatile in the
melt using a set of coupled equations consisting of all the volatile mass
balances (Equation C1) and $r$ chemical reactions (Equation C7). To do this,
estimates of the initial total volatile abundances $X_{v}^{\rm 0,est}$ are
prescribed, and the initial atmospheric escape masses are set to zero
($m_{v}^{e}=0$). We solve for the partial pressure of each volatile in the
atmosphere and the reaction masses, the latter of which are likely to be
nonzero. Hence, it is convenient—although formally not necessary—to offset the
estimates of the initial total volatile abundances ($X_{v}^{\rm 0,est}$) using
the solved reaction masses to enforce that the initial total volatile
abundances ($X_{v}^{0}$) adhere to chemical equilibrium constraints with zero
reaction mass:
$X_{v}^{0}=X_{v}^{\rm 0,est}-\frac{1}{M^{m}}\sum_{w}^{r}m_{v}^{w}.$ (C12)
Hence, the reaction masses are zero by definition for the initial condition
and can increase or decrease during the time integration.
### C.5 Alternative using thermodynamic components
The system of volatiles we consider can be represented by two components: C
and H2, where we recall that oxygen is not explicitly tracked. Hence, an
alternative approach to solving the mass balance for each volatile is to
instead solve for the total abundance of C and H2 moles. Conservation laws for
C and H2 abundances can be derived by converting the relevant volatile mass
balances to C or H2 moles and then summing. Without chemical reactions,
conserving species mass or abundance is the same because there is no exchange
of material between species. With reactions, either one can continue to track
species mass by accounting for mass exchange by reactions (as we choose to
do), or alternatively one can instead track the abundance of components.
Considering species mass allows one to retain the familiar mass formulation
derived without reactions, but at the expense of more equations (e.g., five
volatile species plus three reactions). By contrast, considering components
reduces the number of equations (e.g., two components plus three reactions),
but requires a departure from the traditional species mass balance approach.
Ultimately, both approaches are valid and equivalent and will likely find
utility in different applications.
### C.6 Simplified mass balance and interior depletion
The partitioning of volatiles between a fully molten magma ocean and
atmosphere can be analyzed using a reduced form of the volatile mass balance
(Equation C1). This aids the interpretation of the results—notably
demonstrating their universality—as well as providing analytical expressions
and scaling relations. We consider the mass balance for a single volatile that
is only partitioned between the melt and atmosphere. However, we retain the
mean molar mass of the atmosphere $\bar{\mu}$ to provide insight when the
atmosphere is dominated by the presence of another (or mixture) of volatiles
that are less soluble than the volatile under consideration. Following
Equation C1:
$X_{v}M^{l}+4\pi
R_{p}^{2}\left(\frac{\mu_{v}}{\bar{\mu}}\right)\frac{p_{v}}{|g|}=X_{v}^{0}M^{m}.$
(C13)
For power-law solubility (Equation 1),
$\frac{\phi_{g}}{C}X_{v}+{X_{v}}^{\beta_{v}}=\frac{X_{v}^{0}}{C},$ (C14)
where $\phi_{g}=M^{l}/M^{m}$ is the global melt fraction and
$C=\frac{4\pi R_{p}^{2}\mu_{v}{\alpha_{v}}^{-\beta_{v}}}{M^{m}\bar{\mu}|g|}.$
(C15)
For a single species ($\bar{\mu}=\mu_{v}$), an Earth-sized mantle, and values
of $\alpha_{v}$ and $\beta_{v}$ from the peridotite solubility law (Table 1),
the coefficient $C\approx 17/4$.
#### C.6.1 Volatile depletion for $\beta_{v}=2$
For the peridotite solubility law (as well as other laws), $\beta_{v}=2$ and
Equation C14 is a quadratic equation that can be solved for the volatile
abundance in the melt:
$X_{v}=\frac{-\phi_{g}}{2C}+\frac{1}{2}\sqrt{\frac{\phi_{g}^{2}}{C^{2}}+\frac{4X_{v}^{0}}{C}}.$
(C16)
Hence, comparing to Equation C14, the scaled volatile mass in the liquid is
$m_{v}^{l}=\frac{-\phi_{g}^{2}}{2C^{2}}+\frac{\phi_{g}}{2C}\sqrt{\frac{\phi_{g}^{2}}{C^{2}}+\frac{4X_{v}^{0}}{C}},$
(C17)
the scaled volatile mass in the atmosphere is
$m_{v}^{a}=\frac{X_{v}^{0}}{C}+\frac{\phi_{g}^{2}}{2C^{2}}-\frac{\phi_{g}}{2C}\sqrt{\frac{\phi_{g}^{2}}{C^{2}}+\frac{4X_{v}^{0}}{C}},$
(C18)
and the scaled total volatile mass is
$m_{v}^{t}=\frac{X_{v}^{0}}{C}.$ (C19)
Therefore, the volatile depletion $\mathcal{D}$ expressed as a fraction
relative to the total volatile mass is
$\mathcal{D}=1-\frac{m_{v}^{l}}{m_{v}^{t}}=\frac{m_{v}^{a}}{m_{v}^{t}}=1+\frac{\phi_{g}^{2}}{2CX_{v}^{0}}-\frac{\phi_{g}}{2X_{v}^{0}}\sqrt{\frac{\phi_{g}^{2}}{C^{2}}+\frac{4X_{v}^{0}}{C}}.$
(C20)
Taking the derivative of Equation C20 with respect to the initial volatile
abundance $X_{v}^{0}$, we obtain
$\frac{\partial\mathcal{D}}{\partial
X_{v}^{0}}=\frac{\phi_{g}\left(-\phi_{g}\sqrt{\phi_{g}^{2}+4CX_{v}^{0}}+\phi_{g}^{2}+2CX_{v}^{0}\right)}{2(X_{v}^{0})^{2}C\sqrt{\phi_{g}^{2}+4X_{v}^{0}C}}.$
(C21)
All variables must be positive to be physically meaningful. The denominator is
clearly always positive and the numerator is also positive since the following
condition is always satisfied:
$\phi_{g}^{2}+2CX_{v}^{0}>\phi_{g}\sqrt{\phi_{g}^{2}+4CX_{v}^{0}}\,.$ (C22)
Therefore,
$\frac{\mathcal{\partial D}}{\partial X_{v}^{0}}>0\,.$ (C23)
Hence, at a given melt fraction $\phi_{g}$, the relative depletion of the
interior increases with the initial volatile abundance $X_{v}^{0}$ (Figure 3).
#### C.6.2 Volatile depletion for $\beta_{v}=1$
For a volatile that obeys Henrian solubility behavior ($\beta_{v}=1$), similar
to Appendix C.6.1, we can write:
$X_{v}=\frac{X_{v}^{0}}{C+\phi_{g}},\quad
m_{v}^{l}=\frac{\phi_{g}X_{v}^{0}}{C(C+\phi_{g})},\quad
m_{v}^{a}=\frac{X_{v}^{0}}{C+\phi_{g}}.$ (C24)
Hence, the interior depletion $\mathcal{D}$ is
$\mathcal{D}=\frac{C}{C+\phi_{g}},\qquad\frac{\partial\mathcal{D}}{\partial
X_{v}^{0}}=0.$ (C25)
Notably, interior depletion is independent of the initial volatile abundance
$X_{v}^{0}$ when $\beta_{v}=1$.
## Appendix D Oxygen fugacity and atmosphere speciation
Figure 4 shows the atmospheric composition at an oxygen fugacity of
$\Delta$IW=0.5. The following figures show the same at $\Delta$IW=$-2$ (Figure
14), $\Delta$IW=2 (Figure 15), and $\Delta$IW=4 (Figure 16).
Figure 14: Atmospheric composition during magma ocean outgassing at $f{\rm
O_{2}}$ of $\Delta$IW=$-2$. See Figure 4 caption. Figure 15: Atmospheric
composition during magma ocean outgassing at $f{\rm O_{2}}$ of $\Delta$IW=2.
See Figure 4 caption. Figure 16: Atmospheric composition during magma ocean
outgassing at $f{\rm O_{2}}$ of $\Delta$IW=4. See Figure 4 caption.
## Appendix E Comparison with FactSage 8.0
We independently verified the results of our model by comparison to FactSage
calculations, assuming ideal gas behavior (Table E).
Table E: Comparison of Select Outgassed Atmospheres Calculated Using Our Model
Compared to FactSage Calculations
Case | Our model (basr) | FactSage (bars)
---|---|---
H Oceans | C/H | $f\rm O_{2}$ ($\Delta$IW) | CO | CO2 | H2 | H2O | CH4 | CO | CO2 | H2 | H2O | CH4
3 | 1 | -2.0 | 7.1 | 0.4 | 186 | 23 | 42 | 7.2 | 0.4 | 187.5 | 22.4 | 41
3 | 1 | +0.5 | 51 | 48 | 213 | 460 | 22 | 48.9 | 49.7 | 216.1 | 458.6 | 20.7
1 | 0.1 | +2.0 | 0.7 | 3.7 | 20 | 240 | 0 | 0.6 | 3.7 | 20.2 | 239.9 | 0.00043
1 | 5 | +4.0 | 7 | 368 | 3.7 | 444 | 0 | 6.4 | 367.6 | 3.7 | 445 | 0
Note. — This comparison reveals that the partial pressure of a given volatile
differs by at most a few percent.
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# Bounds on Dao numbers and applications to regular local rings
Antonino Ficarra Department of Mathematics and Computer Sciences, Physics and
Earth Sciences, University of Messina, Viale Ferdinando Stagno d’Alcontres 31,
98166 Messina, Italy<EMAIL_ADDRESS>, Cleto B. Miranda-Neto
Departamento de Matemática, Universidade Federal da Paraíba, 58051-900 João
Pessoa, PB, Brazil<EMAIL_ADDRESS>and Douglas S. Queiroz Departamento
de Matemática, Universidade Federal da Paraíba, 58051-900 João Pessoa, PB,
Brazil<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.
The so-called Dao numbers are a sort of measure of the asymptotic behaviour of
full properties of certain product ideals in a Noetherian local ring $R$ with
infinite residue field and positive depth. In this paper, we answer a question
of H. Dao on how to bound such numbers. The auxiliary tools range from
Castelnuovo-Mumford regularity of appropriate graded structures to reduction
numbers of the maximal ideal. In particular, we substantially improve previous
results (and answer questions) by the authors. As an application, we provide
new characterizations of when $R$ is regular; for instance, we show that this
holds if and only if the maximal ideal of $R$ can be generated by a
$d$-sequence (in the sense of Huneke) if and only if the third Dao number of
any (minimal) reduction of the maximal ideal vanishes.
###### Key words and phrases:
Full ideal, Castelnuovo-Mumford regularity, Rees algebra, reduction number,
regular local ring
###### 2020 Mathematics Subject Classification:
Primary: 13C05, 13C13; Secondary: 13A30, 13H99
## 1\. Motivation: Dao’s problem on the fullness of certain ideals
Throughout this note, by a ring we mean a commutative, Noetherian, unital
ring. Let $R$ be either a local ring with residue field $K$ and maximal ideal
$\mathfrak{m}$, or a standard graded algebra over a field $K$ having a unique
graded maximal ideal $\mathfrak{m}$. We will assume throughout that $K$ is
infinite and $\mathrm{depth}\ R>0$ (i.e., $\mathfrak{m}$ contains a non-
zerodivisor), and in addition $I\subset R$ stands for an ideal which we assume
to be homogeneous whenever $R$ is graded.
In this paper we focus on the properties of $\mathfrak{m}$-fullness, fullness,
and weak $\mathfrak{m}$-fullness (to be recalled in the next section) of
certain ideals. More precisely, we are interested in the so-called Dao numbers
of the given ideal $I$, i.e., three non-negative integers
${\mathfrak{d}}_{i}(I)$, $i=1,2,3$, defined below, which in some sense provide
a measure for the asymptotic behaviour of the full properties of certain
product ideals involving $I$.
###### Definition 1.1.
The Dao numbers of $I$ are defined as:
$\displaystyle{\mathfrak{d}}_{1}(I)$ $\displaystyle=$ $\displaystyle{\rm
min}\\{t\geq 0\mid\mbox{$I\mathfrak{m}^{k}$ is $\mathfrak{m}$-full for all
$k\geq t$\\}};$ $\displaystyle{\mathfrak{d}}_{2}(I)$ $\displaystyle=$
$\displaystyle{\rm min}\\{t\geq 0\mid\mbox{$I\mathfrak{m}^{k}$ is full for all
$k\geq t$\\}};$ $\displaystyle{\mathfrak{d}}_{3}(I)$ $\displaystyle=$
$\displaystyle{\rm min}\\{t\geq 0\mid\mbox{$I\mathfrak{m}^{k}$ is weakly
$\mathfrak{m}$-full for all $k\geq t$\\}}.$
It is worth observing that, if $(R,\mathfrak{m},K)$ and $I$ are as above, then
as shown in [17, Proposition 2.2] the basic relations among the Dao numbers of
$I$ are
${\mathfrak{d}}_{2}(I)\leq{\mathfrak{d}}_{1}(I)={\mathfrak{d}}_{3}(I).$
Our motivation is the following problem suggested by H. Dao, which was first
addressed in [17] and then in [6].
###### Question 1.2.
([4, Question 4.5]) Can we find good lower and upper bounds for the
${\mathfrak{d}}_{i}(I)$’s?
In the case where $I$ is a reduction of $\mathfrak{m}$, a lower bound for
${\mathfrak{d}}_{3}(I)$ is given by ${\rm r}_{I}(\mathfrak{m})$, the reduction
number of $\mathfrak{m}$ with respect to $I$, which follows immediately from
[17, Theorem 3.4] (the question as to whether ${\rm
r}_{I}(\mathfrak{m})\leq{\mathfrak{d}}_{2}(I)$ remains unanswered). Otherwise,
for a general $I$ (not necessarily a reduction of $\mathfrak{m}$), a more
elaborated lower bound was established in [6, Proposition 1.8].
So, in the present paper, our main goal is to answer the upper bound part of
Dao’s question, by using two fundamental numerical invariants in commutative
algebra: the Castelnuovo-Mumford regularity (of appropriate graded structures)
and, again, the reduction number. The connection between these numbers and
Dao’s problem was first exploited in [17] and later developed even further
(concerning specifically the Castelnuovo-Mumford regularity) in [6]. The
present work establishes, in fact, generalizations and substantial
improvements of the results given in these two papers. Additionally, we shall
finally derive, as an application of some of our results, new
characterizations of regular local rings.
## 2\. Auxiliary notions and basic properties
In this section, we invoke some basic concepts and facts which we shall freely
use in this note (without explicit mention).
### 2.1. Full properties of ideals
Let $K$ be an infinite field and $(R,\mathfrak{m})$ be either a local ring
with residue field $K$ or a standard graded $K$-algebra having a unique
homogeneous maximal ideal $\mathfrak{m}$. Assume $\mathrm{depth}\ R>0$, and
let $I\subset R$ stand for an ideal (homogeneous whenever $R$ is graded).
###### Definition 2.1.
The following notions are central in this paper:
1. (a)
$I$ is $\mathfrak{m}$-full if $I\mathfrak{m}:x=I$ for some element
$x\in\mathfrak{m}\setminus{\mathfrak{m}}^{2}$;
2. (b)
$I$ is full if $I:x=I:\mathfrak{m}$ for some element
$x\in\mathfrak{m}\setminus{\mathfrak{m}}^{2}$;
3. (c)
$I$ is weakly $\mathfrak{m}$-full if $I\mathfrak{m}:\mathfrak{m}=I$.
For completeness, we recall a few interesting properties. It is clear that
$\mathfrak{m}$-full ideals are weakly $\mathfrak{m}$-full. If $I$ is
$\mathfrak{m}$-primary, then $I$ is weakly $\mathfrak{m}$-full if and only if
$I$ is basically full in the sense of [8]. Moreover, $\mathfrak{m}$-full
ideals satisfy the so-called Rees property, and if $R$ is a normal domain then
any integrally closed ideal is $\mathfrak{m}$-full; see [24] (also [7]).
### 2.2. Castelnuovo-Mumford regularity
Let $S=\bigoplus_{m\geq 0}S_{m}$ be a finitely generated standard graded
algebra over a ring $S_{0}$. As usual, by standard we mean that $S$ is
generated by $S_{1}$ as an $S_{0}$-algebra. We write $S_{+}=\bigoplus_{m\geq
1}S_{m}$ for the ideal generated by all elements of $S$ of positive degree.
For a graded $S$-module $A=\bigoplus_{m\in{\mathbb{Z}}}A_{m}$ satisfying
$A_{m}=0$ for all $m\gg 0$, we let $a(A)=\textrm{max}\\{m\in{\mathbb{Z}}\ |\
A_{m}\neq 0\\}$ if $A\neq 0$, and $a(A)=-\infty$ if $A=0$.
Now, for a finitely generated graded $S$-module $N\neq 0$ and an integer
$j\geq 0$, we take $A=H_{S_{+}}^{j}(N)$ and use the notation
$a_{j}(N)\,:=\,a(H_{S_{+}}^{j}(N)),$
where $H_{S_{+}}^{j}(-)$ stands for the $j$-th local cohomology functor with
respect to the ideal $S_{+}$. It is known that $H_{S_{+}}^{j}(N)$ is a graded
module with $H_{S_{+}}^{j}(N)_{n}=0$ for all $n\gg 0$ (see, e.g., [1,
Proposition 15.1.5(ii)]). Thus, $a_{j}(N)<+\infty$.
###### Definition 2.2.
Maintain the above setting and notations. The Castelnuovo-Mumford regularity
of $N$ is defined as
$\mathrm{reg}_{S}N\,:=\,\mathrm{max}\\{a_{j}(N)+j\,\mid\,j\geq 0\\}.$
It is well-known that $\mathrm{reg}\,N$ governs the complexity of the graded
structure of $N$ and is relevant in commutative algebra and algebraic
geometry, for example in the study of degrees of syzygies over polynomial
rings (see, e.g., [1, Chapter 15]).
###### Remark 2.3.
A classical instance of interest is when $S=\mathcal{R}(J)$, the Rees algebra
of an ideal $J$ in a ring $R$ (to be recalled in the next subsection), which
is known to be a finitely generated standard graded $R$-algebra. In
particular, we can consider the case where $R$ is local and $J=\mathfrak{m}$,
the maximal ideal of $R$.
We recall below a few basic rules about this invariant. For details, we refer
to [2, p. 277, (a), (c) and (d)], [5, Corollary 20.19] and [10, Lemma 3.1].
* (i)
As usual, given an integer $j$, we denote by $N(j)$ the module $N$ with
degrees shifted by $j$, that is, $N(j)_{i}=N_{i+j}$ for all $i$. Then,
$\mathrm{reg}_{S}N(j)=\mathrm{reg}_{S}N-j.$
* (ii)
Let $0\rightarrow M\rightarrow N\rightarrow P\rightarrow 0$ be a short exact
sequence of finitely generated graded $S$-modules. Then:
* $\bullet$
$\mathrm{reg}_{S}N\leq\mathrm{max}\\{\mathrm{reg}_{S}M,\mathrm{reg}_{S}P\\},$
with equality if $\mathrm{reg}_{S}P\neq\mathrm{reg}_{S}M-1$ or $M_{k}=0$ for
$k\gg 0$.
* $\bullet$
$\mathrm{reg}_{S}M\leq\mathrm{max}\\{\mathrm{reg}_{S}N,\mathrm{reg}_{S}P+1\\},$
with equality if $\mathrm{reg}_{S}N\neq\mathrm{reg}_{S}P.$
* $\bullet$
$\mathrm{reg}_{S}P\leq\mathrm{max}\\{\mathrm{reg}_{S}N,\mathrm{reg}_{S}M-1\\},$
with equality if $\mathrm{reg}_{S}N\neq\mathrm{reg}_{S}M.$
* (iii)
If $N_{j}=0$ for all $j\gg 0$, then $\mathrm{reg}_{S}N=a(N)$.
### 2.3. Rees algebra and Dao module
Here, we consider some useful graded structures. Let $R$ be a ring and $J$ an
ideal of $R$.
###### Definition 2.4.
The Rees algebra of $J$ is the graded ring
$\mathcal{R}(J)=\bigoplus_{k\geq 0}J^{k}=R\oplus J\oplus J^{2}\oplus\cdots$
###### Notation 2.5.
Given an $R$-module $M$, it is customary to write
$\mathcal{R}(J,M)=\bigoplus_{k\geq 0}J^{k}M$ for the Rees module of $J$
relative to $M$. In this paper, if $I$ is another $R$-ideal, we will be
particularly interested in
$\mathcal{R}(J,I)=\bigoplus_{k\geq 0}IJ^{k}.$
Note $\mathcal{R}(J,I)=I\mathcal{R}(J)$, the extension of $I$ to the ring
$\mathcal{R}(J)$, which is therefore a finitely generated ideal of
$\mathcal{R}(J)$.
Now let $(R,\mathfrak{m})$ be as in Section 1. Its associated graded ring is
defined by $\mathrm{gr}_{\mathfrak{m}}(R)=\bigoplus_{k\geq
0}\mathfrak{m}^{k}/\mathfrak{m}^{k+1}$. In [6], the following graded structure
is introduced for a given ideal $I$ of $R$.
###### Definition 2.6.
The Dao module of $I$ is given by
$\mathfrak{D}_{\mathfrak{m}}(I)=\bigoplus_{k\geq
0}\frac{I\mathfrak{m}^{k+1}:\mathfrak{m}}{I\mathfrak{m}^{k}},$
which is a graded $\mathcal{R}(\mathfrak{m})$-module.
###### Remark 2.7.
The $k$th component of the Dao module vanishes if and only if the ideal
$I{\mathfrak{m}}^{k}$ is weakly $\mathfrak{m}$-full. Since
$\mathfrak{D}_{\mathfrak{m}}(I)_{k}=0\quad\mbox{for\, all}\quad
k\geq{\mathfrak{d}}_{3}(I),$
it follows that $\mathfrak{D}_{\mathfrak{m}}(I)$ has finite length and
therefore is a finitely generated graded $\mathcal{R}(\mathfrak{m})$-module,
satisfying
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathfrak{D}_{\mathfrak{m}}(I)={\mathfrak{d}}_{3}(I)-1$
whenever ${\mathfrak{d}}_{3}(I)\geq 1$ (e.g., if $I$ is not weakly
$\mathfrak{m}$-full).
### 2.4. Ratliff-Rush operation
Let $I$ be an ideal of a ring $R$.
###### Definition 2.8.
The Ratliff-Rush closure $\widetilde{I}$ of the ideal $I$ is given by
$\widetilde{I}=\bigcup_{m\geq 1}\,I^{m+1}:I^{m}.$
This is an ideal of $R$ containing $I$ which in fact refines the integral
closure of $I$, so that $\widetilde{I}=I$ whenever $I$ is integrally closed.
For details, see [18].
Now suppose $I$ contains a regular element, i.e., a non-zerodivisor on $R$.
Then it is well-known that $\widetilde{I}$ is the largest ideal that shares
with $I$ the same sufficiently high powers; hence,
$\widetilde{I^{m}}=I^{m}\quad\mbox{for\, all}\quad m\gg 0.$
This enables us to consider the following helpful number (inspired by [19,
Proposition 4.2]).
###### Notation 2.9.
If $I$ contains a regular element, we set
$s(I)=\mathrm{min}\,\\{n\geq 1\,\mid\,\widetilde{I^{i}}=I^{i}\ \mathrm{for\
all}\ i\geq n\\}.$
###### Remark 2.10.
Let us invoke a couple of useful properties. First, according to [14, Lemma
2.2] we can write
$\widetilde{I^{k+1}}:I=\widetilde{I^{k}}\quad\mbox{for \,all}\quad k\geq 0.$
Moreover, if $\mathrm{gr}_{I}(R)=\bigoplus_{k\geq 0}I^{k}/I^{k+1}$ denotes the
associated graded ring of $I$, then by [19, Remark 1.6] we get that
$\widetilde{I^{k}}=I^{k}$ for all $k\geq 0$ (i.e., $s(I)=1$) if and only if
$\mathrm{depth}\,\mathrm{gr}_{I}(R)>0.$
### 2.5. Reduction number
One last auxiliary notion is in order.
###### Definition 2.11.
Let $J$ be a proper ideal of a ring $R$. An ideal $I\subset J$ is said to be a
reduction of $J$ if $IJ^{r}=J^{r+1}$ for some integer $r\geq 0$. Such a
reduction $I$ is minimal if it is minimal with respect to inclusion. If $I$ is
a reduction of $J$, we define the reduction number of $J$ with respect to $I$
as the number
${\rm r}_{I}(J)=\mathrm{min}\,\\{m\in\mathbb{N}\,\mid\,IJ^{m}=J^{m+1}\\},$
and the reduction number of $J$ as
${\rm r}(J)=\mathrm{min}\,\\{{\rm r}_{I}(J)\,\mid\,\mbox{$I$ is a minimal
reduction of $J$}\\}.$
Of special interest in this paper will be the case where $(R,\mathfrak{m})$ is
a local ring and $J=\mathfrak{m}$.
## 3\. Upper bounds on Dao numbers via Castelnuovo-Mumford regularity
Before establishing the main results of the section, we fix a piece of
notation. For a graded $R$-module $M=\bigoplus_{k\geq 0}M_{k}$ and an integer
$\ell\geq 0$, we can consider the truncation
$M_{\geq\ell}=\bigoplus_{k\geq\ell}M_{k}$. Specifically, with notation as in
(2.5) and in case $(R,\mathfrak{m},K)$ is a local ring or a standard graded
$K$-algebra, we will be interested in the truncation
$\mathcal{R}(\mathfrak{m},I)_{\geq 1}=\bigoplus_{k\geq 1}I{\mathfrak{m}}^{k}.$
Here we are interested in tackling the upper bound part of Question 1.2 in
terms of the Castelnuovo-Mumford regularity of appropriate graded structures.
The first result in this direction, in case $R$ is local and $I$ is a
reduction of $\mathfrak{m}$, was proved in [17] and can be stated as follows.
###### Theorem 3.1.
$($[17, Theorem 3.10]$)$ Let $(R,\mathfrak{m})$ be a local ring with infinite
residue field and ${\rm depth}\,R>0$, and let $I$ be a reduction of
$\mathfrak{m}$. Then,
${\mathfrak{d}}_{3}(I)\leq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m}).$
Later, in [6], the following general answer to Question 1.2 was provided.
###### Theorem 3.2.
$($[6, Theorem 1.1]$)$ Let $(R,\mathfrak{m},K)$ be either a local ring or a
standard graded $K$-algebra, with $K$ infinite and $\mathrm{depth}\,R>0$. Let
$I\subset R$ be an ideal $($homogeneous if $R$ is graded$)$. Then,
${\mathfrak{d}}_{3}(I)\leq\mathrm{max}\\{\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m},I),\
\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m},I)_{\geq
1}:_{\mathcal{R}(R)}\mathfrak{m}\\}.$
Here, we prove the following result, which establishes [6, Conjecture 0.1] in
its full generality.
###### Theorem 3.3.
Let $R$ and $I$ be as in Theorem 3.2. Then,
${\mathfrak{d}}_{3}(I)\leq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m},I).$
Proof. We may suppose $\mathfrak{d}_{3}(I)>0$, otherwise there is nothing to
prove. Let
$\mathfrak{M}=\mathcal{R}(\mathfrak{m})_{+}=\bigoplus_{k>0}\mathfrak{m}^{k}$
be the homogeneous maximal ideal of $\mathcal{R}(\mathfrak{m})$. It was shown
in [6, proof of Theorem 1.3] that
(1) $\mathrm{Soc}(\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I))\ =\
0:_{\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I)}\mathfrak{M}\ =\
\bigoplus_{k\geq
0}\frac{(I\mathfrak{m}^{k+1}:\mathfrak{m})\cap\mathfrak{m}^{k}}{I\mathfrak{m}^{k}}.$
Now, since
$\frac{(I\mathfrak{m}^{k+1}:\mathfrak{m})\cap\mathfrak{m}^{k}}{I\mathfrak{m}^{k}}\
\subset\
\frac{I\mathfrak{m}^{k+1}:\mathfrak{m}}{I\mathfrak{m}^{k}}\quad\mbox{for\,
all}\quad k\geq 0,$
we have
(2) $\mathrm{Soc}(\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I))\
\subset\ \mathfrak{D}_{\mathfrak{m}}(I)$
as a graded $\mathcal{R}(\mathfrak{m})$-submodule of
$\mathfrak{D}_{\mathfrak{m}}(I)$. Next, let
$t=\max\\{k\mid\mathfrak{D}_{\mathfrak{m}}(I)_{k}\neq 0\\}$. Then,
$t=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathfrak{D}_{\mathfrak{m}}(I)=\mathfrak{d}_{3}(I)-1$.
Because $\mathfrak{M}\cdot\mathfrak{D}_{\mathfrak{m}}(I)_{t}=0$, it follows
that
$\mathfrak{D}_{\mathfrak{m}}(I)_{t}\ \subset\
[0:_{\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I)}\mathfrak{M}]_{t}\
=\ \mathrm{Soc}(\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I))_{t}$
and note that the opposite inclusion always holds by equation (2). Therefore,
$\mathrm{Soc}(\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I))_{t}=\mathfrak{D}_{\mathfrak{m}}(I)_{t}\neq
0$
and consequently
$\displaystyle\mathfrak{d}_{3}(I)-1\ =\ t\ $ $\displaystyle\leq\
\max\\{j\mid\mathrm{Soc}(\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I))_{j}\neq
0\\}$ $\displaystyle\leq\
\textrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I)$
$\displaystyle=\
\textrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)-1.$
The assertion follows. $\square$
Naturally, we can ask when the inclusion (2) is actually an equality.
###### Proposition 3.4.
Let $R$ and $I$ be as in Theorem 3.2. If $R$ is a standard graded $K$-algebra
or $\mathrm{depth}\,\mathrm{gr}_{\mathfrak{m}}(R)>0$, then
$\mathrm{Soc}(\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I))=\mathfrak{D}_{\mathfrak{m}}(I)$.
Proof. In view of equation (1), it suffices to show that
$I\mathfrak{m}^{k+1}:\mathfrak{m}\ \subset\ \mathfrak{m}^{k}\quad\mbox{for\,
all}\quad k\geq 0.$
First, if $R$ is a standard graded $K$-algebra, we can represent it as a
quotient $S/J$ where $S=K[x_{1},\dots,x_{n}]$ is a standard graded polynomial
ring and $J\subset S$ is a homogeneous ideal. Now let $f\in
I\mathfrak{m}^{k+1}:\mathfrak{m}$ be an homogeneous element. Then
$(x_{1}+J)f\in I\mathfrak{m}^{k+1}\subset\mathfrak{m}^{k+1}$
and so $\deg f+1\geq k+1$. Thus $f\in\mathfrak{m}^{k}$ and consequently
$I\mathfrak{m}^{k+1}:\mathfrak{m}\subset\mathfrak{m}^{k}$ for all $k\geq 0$.
Otherwise, if $\mathrm{depth}\,\mathrm{gr}_{\mathfrak{m}}(R)>0$ then, as we
know, $\widetilde{\mathfrak{m}^{k}}=\mathfrak{m}^{k}$ for all $k\geq 0$.
Consequently,
$I\mathfrak{m}^{k+1}:\mathfrak{m}\subset\mathfrak{m}^{k+1}:\mathfrak{m}=\widetilde{\mathfrak{m}^{k+1}}:\mathfrak{m}=\widetilde{\mathfrak{m}^{k}}=\mathfrak{m}^{k}.$
$\square$
We conclude this section with some other consequences of Theorem 3.3.
###### Corollary 3.5.
Let $R$ and $I$ be as in Theorem 3.2. If $\mathfrak{d}_{3}(I)>0$, then
$\mathfrak{d}_{3}(I)$ is equal to the highest socle degree of
$\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I)$ increased by one, that
is,
$\mathfrak{d}_{3}(I)\ =\
\max\\{j\mid\mathrm{Soc}(\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I))_{j}\neq
0\\}+1.$
Proof. Set
$h=\max\\{j\mid\mathrm{Soc}(\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I))_{j}\neq
0\\}$. In the proof of Theorem 3.3 we have seen that $\mathfrak{d}_{3}(I)\leq
h+1$. On the other hand, since $\mathfrak{d}_{3}(I)>0$, we can write
$\mathfrak{d}_{3}(I)=\max\\{j\mid\mathfrak{D}_{\mathfrak{m}}(I)_{j}\neq
0\\}+1$. By equation (2) we have an inclusion
$\mathrm{Soc}(\mathcal{R}(\mathfrak{m})/\mathcal{R}(\mathfrak{m},I))\subset\mathfrak{D}_{\mathfrak{m}}(I)$,
so that
$\mathfrak{d}_{3}(I)\ =\ \max\\{j\mid\mathfrak{D}_{\mathfrak{m}}(I)_{j}\neq
0\\}+1\ \geq\ h+1,$
thus completing the proof. $\square$
Next, we estimate the Castelnuovo-Mumford regularity of some natural blowup
algebras that are related to the Dao module of $I$.
###### Definition 3.6.
The associated graded module of $\mathfrak{m}$ relative to $I$ is the
$\mathcal{R}(\mathfrak{m})$-module
$\mathrm{gr}_{\mathfrak{m}}(I)=\mathcal{R}(\mathfrak{m},I)/\mathfrak{m}\mathcal{R}(\mathfrak{m},I)=\bigoplus_{k\geq
0}I\mathfrak{m}^{k}/I\mathfrak{m}^{k+1}.$
Note that, for each $k\geq 0$, we have the following commutative diagram with
exact rows and columns:
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I\mathfrak{m}^{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I\mathfrak{m}^{k+1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I\mathfrak{m}^{k}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{I\mathfrak{m}^{k+1}:\mathfrak{m}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cfrac{I\mathfrak{m}^{k+1}:\mathfrak{m}}{I\mathfrak{m}^{k}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cfrac{I\mathfrak{m}^{k}}{I\mathfrak{m}^{k+1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cfrac{I\mathfrak{m}^{k+1}:\mathfrak{m}}{I\mathfrak{m}^{k+1}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\cfrac{I\mathfrak{m}^{k+1}:\mathfrak{m}}{I\mathfrak{m}^{k}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$
###### Notation 3.7.
For simplicity, we set
$\mathcal{P}_{\mathfrak{m}}(I)=\mathcal{R}(\mathfrak{m},I)_{\geq
1}:_{\mathcal{R}(R)}\mathfrak{m}\quad\mbox{and}\quad\mathcal{Q}_{\mathfrak{m}}(I)=(\mathcal{R}(\mathfrak{m},I)_{\geq
1}:_{\mathcal{R}(R)}\mathfrak{m})/\mathcal{R}(\mathfrak{m},I)_{\geq 1}.$
Taking the direct sum in the above diagram, we obtain the following
commutative diagram of finitely generated graded
$\mathcal{R}(\mathfrak{m})$-modules with exact rows and columns:
$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{R}(\mathfrak{m},I)_{\geq
1}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{R}(\mathfrak{m},I)_{\geq
1}(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{R}(\mathfrak{m},I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{P}_{\mathfrak{m}}(I)(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{D}_{\mathfrak{m}}(I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathrm{gr}_{\mathfrak{m}}(I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathcal{Q}_{\mathfrak{m}}(I)(1)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\mathfrak{D}_{\mathfrak{m}}(I)\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$$\textstyle{0}$$\textstyle{0}$
The lemma below is a special case of [25, Corollary 3].
###### Lemma 3.8.
There is an equality
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathrm{gr}_{\mathfrak{m}}(I)$.
Now we prove:
###### Corollary 3.9.
Let $R$ and $I$ be as in Theorem 3.2. Then,
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{P}_{\mathfrak{m}}(I),\,\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{Q}_{\mathfrak{m}}(I)\
\leq\ \mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)+1.$
In particular,
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{P}_{\mathfrak{m}}(I)=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{Q}_{\mathfrak{m}}(I)=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)+1$
if either $\mathfrak{d}_{3}(I)=0$ or
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)>\mathfrak{d}_{3}(I)>0$.
Proof. There are short exact sequences of finitely generated graded
$\mathcal{R}(\mathfrak{m})$-modules
$0\rightarrow\mathcal{R}(\mathfrak{m},I)\rightarrow\mathcal{P}_{\mathfrak{m}}(I)(1)\rightarrow\mathfrak{D}_{\mathfrak{m}}(I)\rightarrow
0,$
$0\rightarrow\mathrm{gr}_{\mathfrak{m}}(I)\rightarrow\mathcal{Q}_{\mathfrak{m}}(I)(1)\rightarrow\mathfrak{D}_{\mathfrak{m}}(I)\rightarrow
0.$
Recall that Lemma 3.8 gives
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathrm{gr}_{\mathfrak{m}}(I)$.
First, assume that $\mathfrak{d}_{3}(I)=0$. Then,
$\mathfrak{D}_{\mathfrak{m}}(I)=0$. In this case,
$\mathcal{P}_{\mathfrak{m}}(I)(1)\cong\mathcal{R}(\mathfrak{m},I)$ and
$\mathcal{Q}_{\mathfrak{m}}(I)(1)\cong\mathrm{gr}_{\mathfrak{m}}(I)$, so that
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{P}_{\mathfrak{m}}(I)-1=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathrm{gr}_{\mathfrak{m}}(I)=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{Q}_{\mathfrak{m}}(I)-1.$
Now, suppose $\mathfrak{d}_{3}(I)>0$. In particular
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathfrak{D}_{\mathfrak{m}}(I)=\mathfrak{d}_{3}(I)-1$.
On the other hand, Theorem 3.3 yields
$\mathfrak{d}_{3}(I)\leq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathrm{gr}_{\mathfrak{m}}(I)$
and thus the above exact sequence implies
$\displaystyle\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{P}_{\mathfrak{m}}(I)(1)\
$ $\displaystyle\leq\
\max\\{\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I),\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathfrak{D}_{\mathfrak{m}}(I)\\}$
$\displaystyle=\
\max\\{\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I),\mathfrak{d}_{3}(I)-1\\}$
$\displaystyle=\
\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I).$
Similarly,
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{Q}_{\mathfrak{m}}(I)(1)\leq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)$.
Hence,
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{P}_{\mathfrak{m}}(I),\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{Q}_{\mathfrak{m}}(I)\leq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)+1.$
Finally, if we suppose that
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)>\mathfrak{d}_{3}(I)>0$,
we derive
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)-1\neq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathfrak{D}_{\mathfrak{m}}(I),$
and the equalities
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{P}_{\mathfrak{m}}(I)=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{Q}_{\mathfrak{m}}(I)=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m},I)+1$
follow. $\square$
Next, we strengthen [6, Proposition 1.6] by finding the relation between the
regularity of an ideal as in (2.5) and that of the ambient Rees algebra
$\mathcal{R}(J)$.
###### Lemma 3.10.
Let $(R,\mathfrak{m},K)$ be either a local ring or a standard graded
$K$-algebra. Let $I,J$ be ideals of $R$. Then,
$\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J)\leq\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J,I),$
with equality if $I$ is a reduction of $J$.
Proof. Since $\mathcal{R}(J,I)$ is a homogeneous ideal of $\mathcal{R}(J)$,
the short exact sequence
$0\rightarrow\mathcal{R}(J,I)\rightarrow\mathcal{R}(J)\rightarrow\frac{\mathcal{R}(J)}{\mathcal{R}(J,I)}\rightarrow
0$
yields
$\displaystyle\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J)$ $\displaystyle\leq$
$\displaystyle\mathrm{max}\Big{\\{}\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J,I),\mathrm{reg}_{\mathcal{R}(J)}\frac{\mathcal{R}(J)}{\mathcal{R}(J,I)}\Big{\\}}$
$\displaystyle=$
$\displaystyle\mathrm{max}\Big{\\{}\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J,I),\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J,I)-1\Big{\\}}$
$\displaystyle=$ $\displaystyle\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J,I).$
For the equality part, the proof of
$\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J)\geq\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J,I)$
is exactly the same as the one given in [6, Proposition 1.6] for the case
$J=\mathfrak{m}$. $\square$
###### Remark 3.11.
Notice that, as a direct consequence of Theorem 3.3 and Lemma 3.10, we
rediscover Theorem 3.1.
In our view, and in connection to Lemma 3.10, it is worth asking the
following.
###### Question 3.12.
Let $(R,\mathfrak{m},K)$ be either a local ring or a standard graded
$K$-algebra. Let $I\subset J$ be ideals of $R$. When does the condition
$\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J)=\mathrm{reg}_{\mathcal{R}(J)}\mathcal{R}(J,I)$
force $I$ to be a reduction of $J$?
Inspired by Theorem 3.1 and Theorem 3.3, we might wonder whether the
comparison
${\mathfrak{d}}_{3}(I)\leq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m})$
holds in general. However, in the case where $I$ is not a reduction of
$\mathfrak{m}$, the relationship with the regularity of
$\mathcal{R}(\mathfrak{m})$ may become rather wild, as we can see in the
examples below.
###### Example 3.13.
Let $(R,\mathfrak{m})$ be a local ring with infinite residue field and ${\rm
depth}\,R>0$. By [20, Proposition 1.5], we have
$\mathfrak{m}^{n+1}:\mathfrak{m}=\mathfrak{m}^{n}$ for all
$n\geq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m})$.
Now set $I={\mathfrak{m}}^{k}$ for any given $k\geq 2$ (note $I$ cannot be a
reduction of $\mathfrak{m}$). We can write
(3)
$I\mathfrak{m}^{n}=\mathfrak{m}^{n+k}=\mathfrak{m}^{n+k+1}:\mathfrak{m}=I\mathfrak{m}^{n+1}:\mathfrak{m}$
for all
$n\geq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m})$ and
therefore
${\mathfrak{d}}_{3}(I)\leq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m})$.
Furthermore notice that, by using (3) whenever
$n\geq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m})-k\geq
0$, we obtain
${\mathfrak{d}}_{3}(I)\leq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m})-k<\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m}).$
Finally, if
$k\geq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m})$
then it is easy to see that ${\mathfrak{d}}_{3}(I)=0$. In particular, if
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m})\leq 2$
then ${\mathfrak{d}}_{3}(I)=0$.
###### Example 3.14.
Let $R$ be as in [17, Example 4.3]. As observed there,
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m})=8$.
Consider the ideal $I={\mathfrak{m}}^{2}$. By Example 3.13 above, we can write
${\mathfrak{d}}_{3}(I)\leq\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m})-2=6,$
whereas, on the other hand, a computation shows $I{\mathfrak{m}}^{5}\neq
I{\mathfrak{m}}^{6}:\mathfrak{m}$. Therefore, we must have
${\mathfrak{d}}_{3}(I)=6$.
We are also able to illustrate the opposite situation, as follows.
###### Example 3.15.
Let $I=(x^{a},y^{a})\subset R=k[\\![x,y]\\!]$, where $k$ is a field and $a\geq
2$. Clearly, $I$ is not a reduction of $\mathfrak{m}=(x,y)$. From [4, Example
4.1] we can write
${\mathfrak{d}}_{3}(I)=a-1>0=\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\,\mathcal{R}(\mathfrak{m}),$
where the last equality holds because $R$ is a regular local ring (see Theorem
5.3 for a more general statement).
## 4\. Approach (and a conjecture) via reduction numbers
We begin recalling the following result in dimension one.
###### Proposition 4.1.
$($[17, Corollary 3.7]$)$ If $(R,\mathfrak{m})$ is a one-dimensional Cohen-
Macaulay local ring with infinite residue field, then
$\mathfrak{d}_{3}(I)={\rm r}(\mathfrak{m})$
for any minimal reduction $I$ of $\mathfrak{m}$.
It is worth pointing out that this proposition is no longer true if the
reduction $I$ is not required to be minimal, as exemplified in [17, Example
4.2].
In higher dimension, there is the conjecture below.
###### Conjecture 4.2.
$($[17, Conjecture 3.8]$)$ If $(R,\mathfrak{m})$ is a Cohen-Macaulay local
ring with infinite residue field and ${\rm dim}\,R\geq 2$, then
$\mathfrak{d}_{3}(I)={\rm r}_{I}(\mathfrak{m})$
for any minimal reduction $I$ of $\mathfrak{m}$.
Our main purpose in this section is to provide partial answers to this
conjecture. A crucial tool in this investigation is given by the following
result.
###### Theorem 4.3.
$($[17, Theorem 3.4]$)$ Let $(R,\mathfrak{m})$ be a local ring with infinite
residue field and ${\rm depth}\,R>0$, and let $I$ be a reduction of
$\mathfrak{m}$. Then
$\mathfrak{d}_{3}(I)=\mathrm{max}\\{{\rm
r}_{I}(\mathfrak{m}),s(\mathfrak{m})-1\\}.$
Now we can illustrate Conjecture 4.2 in the case ${\rm dim}\,R=2$.
###### Example 4.4.
Let $(R,\mathfrak{m})$ be the local ring of a rational triple point as in [17,
Example 4.1], where we highlighted that $s(\mathfrak{m})=1$. Then, by Theorem
4.3, we obtain $\mathfrak{d}_{3}(I)={\rm r}_{I}(\mathfrak{m})$ for any
reduction (in particular, minimal reduction) $I$ of $\mathfrak{m}$.
The above example actually shows that Conjecture 4.2 is true in the case
$s(\mathfrak{m})=1$, or equivalently, ${\rm depth}\,{\rm
gr}_{\mathfrak{m}}(R)>0$ (see also Proposition 4.9 below). Furthermore, it is
clear that $\widetilde{\mathfrak{m}}=\mathfrak{m}$, which forces
$s(\mathfrak{m})\neq 2$. Therefore, in tackling the conjecture we can suppose
$s(\mathfrak{m})\geq 3$.
###### Theorem 4.5.
Conjecture 4.2 holds true in case $s:=s(\mathfrak{m})\geq 3$ and
$I\widetilde{\mathfrak{m}^{s-2}}=I\mathfrak{m}^{s-2}.$
Proof. By virtue of Theorem 4.3, it suffices to show that ${\rm
r}_{I}(\mathfrak{m})\geq s-1$. Suppose, by way of contradiction, that
${\rm r}_{I}(\mathfrak{m})\leq s-2.$
Since $I$ is a minimal reduction of $\mathfrak{m}$, we can apply [14,
Proposition 2.4 and Theorem 2.10] to obtain
$I\widetilde{\mathfrak{m}^{k}}=\widetilde{\mathfrak{m}^{k+1}}$ for all
$k\geq{\rm r}_{I}(\mathfrak{m})$. Consequently,
$\widetilde{\mathfrak{m}^{s-1}}=I\widetilde{\mathfrak{m}^{s-2}}=I\mathfrak{m}^{s-2}=\mathfrak{m}^{s-1},$
which contradicts the definition of $s$. $\square$
Since $\mathfrak{m}=\widetilde{\mathfrak{m}}$ is always true, the case
$s(\mathfrak{m})=3$ is a straightforward consequence of Theorem 4.5.
###### Corollary 4.6.
Conjecture 4.2 is true if $s(\mathfrak{m})=3$.
###### Example 4.7.
Let $K$ be an infinite field and
$R=\frac{K[\\![x_{1},x_{2},x_{3},x_{4},x_{5},x_{6},x_{7}]\\!]}{(x_{1}^{2},x_{1}x_{2},x_{1}x_{3},x_{1}x_{4},x_{2}x_{3},x_{2}x_{4},x_{3}x_{4},x_{2}^{3}-x_{1}x_{5},x_{3}^{3}-x_{1}x_{6},x_{4}^{3}-x_{1}x_{7})},$
which is a three-dimensional Cohen-Macaulay local ring. Notice that
$x_{1}\in\widetilde{\mathfrak{m}^{2}}\setminus\mathfrak{m}^{2}$
and $\widetilde{\mathfrak{m}^{n}}=\mathfrak{m}^{n}$ for all $n\geq 3$. Thus,
$s(\mathfrak{m})=3$. By Corollary 4.6, we obtain $\mathfrak{d}_{3}(I)={\rm
r}_{I}(\mathfrak{m})$ for any minimal reduction $I$ of $\mathfrak{m}$.
When $s:=s(\mathfrak{m})\geq 4$, it may be difficult to determine whether
$I\widetilde{\mathfrak{m}^{s-2}}=I\mathfrak{m}^{s-2}$. However, by employing a
different argument, we are able to establish Conjecture 4.2 in the case
$s(\mathfrak{m})=4$.
###### Proposition 4.8.
Conjecture 4.2 is true if $s(\mathfrak{m})=4$.
Proof. Suppose ${\rm r}_{I}(\mathfrak{m})\leq s(\mathfrak{m})-2=2$. Then, by
[3, Theorem 3.6], the ring $\mathrm{gr}_{\mathfrak{m}}(R)$ must be Cohen-
Macaulay. Hence ${\rm depth}\,\mathrm{gr}_{\mathfrak{m}}(R)={\rm
dim}\,\mathrm{gr}_{\mathfrak{m}}(R)={\rm dim}\,R>0$. But this implies
$s(\mathfrak{m})=1$, which is a contradiction. Therefore,
${\rm r}_{I}(\mathfrak{m})\geq s(\mathfrak{m})-1$
and the result follows from Theorem 4.3. $\square$
We close the section with yet another affirmative case of the conjecture.
###### Proposition 4.9.
Conjecture 4.2 is true if $R$ has minimal multiplicity.
Proof. For such $R$, the ring ${\rm gr}_{\mathfrak{m}}(R)$ is Cohen-Macaulay
(see [22, Theorem 2]), hence its depth is equal to ${\rm dim}\,R\geq 2>0$. In
this case, as we already know, the conjecture holds true. $\square$
## 5\. Application: Regular local rings
In this section, as applications of some of our results, we provide new
characterizations of regular local rings, and describe a potential approach to
the long-standing Zariski-Lipman conjecture about derivations.
### 5.1. Characterizations of regular local rings
Our result in this part is Theorem 5.3 below. We observe that the implication
(a) $\Rightarrow$ (d) recovers [17, Corollary 3.11]. Moreover, a crucial fact
here (which, as far as we know, is new) is given by (b) $\Rightarrow$ (a),
i.e., $(R,\mathfrak{m})$ must be regular if $\mathfrak{m}$ is generated by a
$d$-sequence (see [11], [12] for the definition of this type of sequence and
its properties), which in particular solves the problem suggested in [17,
Remark 3.12]. Finally, the equivalence between assertions (a) and (e) reveals
the curious role played by ${\mathfrak{d}}_{3}(I)$ in regard to the theory of
regular local rings, which can be re-expressed by means of equivalence to the
structural assertion (f).
For the proof, the following two interesting facts will be useful.
###### Lemma 5.1.
([23, Corollary 5.2]) Let $(R,\mathfrak{m})$ be a local ring with infinite
residue field. Then, $\mathfrak{m}$ is generated by a $d$-sequence if and only
if $\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m})=0$.
###### Lemma 5.2.
([15, Theorem 1.1]) Let $(R,\mathfrak{m})$ be a local ring with ${\rm
depth}\,R>0$. If $Q$ is a parameter ideal of $R$ such that $Q^{n}$ is
$\mathfrak{m}$-full for some $n\geq 1$, then $R$ is regular.
###### Theorem 5.3.
Let $(R,\mathfrak{m},K)$ be either a local ring or a standard graded
$K$-algebra, with $K$ infinite and $\mathrm{depth}\,R>0$. The following
assertions are equivalent:
* (a)
$R$ is regular;
* (b)
$\mathfrak{m}$ is generated by a $d$-sequence;
* (c)
$s(\mathfrak{m})=1$ and ${\rm r}_{I}(\mathfrak{m})=0$, for any (minimal)
reduction $I$ of $\mathfrak{m}$;
* (d)
${\mathfrak{d}}_{1}(I)={\mathfrak{d}}_{2}(I)={\mathfrak{d}}_{3}(I)=0$, for any
(minimal) reduction $I$ of $\mathfrak{m}$;
* (e)
${\mathfrak{d}}_{3}(I)=0$, for any (minimal) reduction $I$ of $\mathfrak{m}$;
* (f)
$\mathcal{R}(\mathfrak{m},I)=(\mathcal{R}(\mathfrak{m},I)_{\geq
1}:_{\mathcal{R}(R)}\mathfrak{m})(1)$, for any (minimal) reduction $I$ of
$\mathfrak{m}$.
Proof. (a) $\Rightarrow$ (b) If $R$ is regular, then $\mathfrak{m}$ is
generated by a regular sequence, which is therefore a $d$-sequence.
(b) $\Rightarrow$ (c) According to [20, Theorem 2.1(ii)], we have ${\rm
max}\\{\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m}),1\\}\geq
s(\mathfrak{m})$. Hence, using Lemma 5.1, we obtain $s(\mathfrak{m})=1$. In
order to deal with ${\rm r}_{I}(\mathfrak{m})$, consider first the case where
the reduction $I$ is minimal. Applying [20, Theorem 1.3] and [21, p. 12], we
derive
$\mathrm{reg}_{\mathcal{R}(\mathfrak{m})}\mathcal{R}(\mathfrak{m})\geq{\rm
r}_{I}(\mathfrak{m}),$
which gives ${\rm r}_{I}(\mathfrak{m})=0$. Now, if $I$ is not minimal, then it
necessarily contains a minimal reduction $J$ of $\mathfrak{m}$, so that ${\rm
r}_{I}(\mathfrak{m})\leq{\rm r}_{J}(\mathfrak{m})=0$.
(c) $\Rightarrow$ (d) This follows readily from the fact (proved in [17,
Theorem 3.4]) that
${\mathfrak{d}}_{3}(I)\leq\mathrm{max}\\{{\rm
r}_{I}(\mathfrak{m}),s(\mathfrak{m})-1\\}.$
Now, by [17, Proposition 2.2], the vanishing of ${\mathfrak{d}}_{3}(I)$ forces
that of ${\mathfrak{d}}_{1}(I)$ and ${\mathfrak{d}}_{2}(I)$.
(d) $\Rightarrow$ (e) This is obvious.
(e) $\Rightarrow$ (a) In the present setting, $\mathfrak{m}$ admits a
reduction $Q$ which is a parameter ideal (see, e.g., [13, Exercise 8.11(ii)]).
As ${\mathfrak{d}}_{3}(Q)={\mathfrak{d}}_{1}(Q)$, we obtain
${\mathfrak{d}}_{1}(Q)=0$
and consequently $Q$ is $\mathfrak{m}$-full. We are now in a position to apply
Lemma 5.2 with $n=1$ to conclude that $R$ is regular.
It remains to prove that assertions (a) and (f) are equivalent. To this end,
simply note that (f) holds if and only if
$(\mathcal{R}(\mathfrak{m},I))_{k}=(\mathcal{R}(\mathfrak{m},I)_{k\geq
1}:_{\mathcal{R}(R)}\mathfrak{m})(1)_{k}$ for all $k\geq 0$ and any (minimal)
reduction $I$ of $\mathfrak{m}$, which means
$I\mathfrak{m}^{k}=I\mathfrak{m}^{k+1}:\mathfrak{m}\quad\mbox{for\, all}\quad
k\geq 0.$
This is, by definition, tantamount to ${\mathfrak{d}}_{3}(I)=0$, which as we
have shown above is equivalent to $R$ being regular. $\square$
### 5.2. Potential approach to a classical conjecture
For a field $K$ and a $K$-algebra $R$, we write as usual ${\rm Der}_{K}(R)$
for the module of $K$-derivations of $R$, i.e., the additive maps $D\colon
R\to R$ that vanish on $K$ and satisfy Leibniz rule: $D(\alpha\beta)=\alpha
D(\beta)+\beta D(\alpha)$ for all $\alpha,\beta\in R$. Now assume that $R$ is
a positive-dimensional local ring which is either
(4) ${K}[x_{1},\ldots,x_{m}]_{\mathfrak{q}}/I\ \ \ ({\mathfrak{q}}\in{\rm
Spec}\,{K}[x_{1},\ldots,x_{m}])\quad\mbox{or}\quad{K}[\\![x_{1},\ldots,x_{m}]\\!]/I,$
with $I$ a proper radical ideal and $x_{1},\ldots,x_{m}$ indeterminates over a
field ${K}$ of characteristic zero. In this setting, there is the following
long-held classical problem.
###### Conjecture 5.4.
(Zariski-Lipman) Let $R$ be as in (4). If ${\rm Der}_{K}(R)$ is free, then $R$
is regular.
This problem has an interesting long history, featuring in particular a strong
geometric counterpart, and remains open in some cases. For details and
references, see [9, Section 2] (in particular, see [9, Theorem 2.3] for a
simple proof of the Zariski-Lipman conjecture in the graded case).
Additionally, [16, Section 4] shows a relation between this conjecture and the
$\mathfrak{m}$-full property of ideals in the (open) two-dimensional local
case, which makes it somewhat natural to expect further connections.
We point out that, under the hypotheses of the conjecture, $R$ must be a
(normal) domain, and we can write
${\rm Der}_{K}(R)=\bigoplus_{1\leq j\leq t}RD_{j}\leavevmode\nobreak\
\leavevmode\nobreak\ \cong R^{t},\quad t={\rm dim}\,R,$
for some free basis $\\{D_{j}\\}$ consisting of precisely $t$ derivations.
Conversely, if ${\rm Der}_{K}(R)$ admits a free basis
$\\{D_{1},\ldots,D_{s}\\}$, then necessarily $s=t$.
###### Remark 5.5.
Let $(R,\mathfrak{m})$ be as in (4). If, as above, the $R$-module ${\rm
Der}_{K}(R)$ is free, then our guess is that, for each minimal reduction $I$
of $\mathfrak{m}$, there exist $\alpha_{j}^{(I)},\beta_{j}^{(I)}\in R$,
$j=1,\ldots,t$, such that the element given by
${\ell}^{(I)}=\sum_{j=1}^{t}\beta_{j}^{(I)}D_{j}(\alpha_{j}^{(I)})$
satisfies ${\ell}^{(I)}\in\mathfrak{m}\setminus{\mathfrak{m}}^{2}$ and
$I{\mathfrak{m}}^{k+1}\colon({\ell}^{(I)})=I{\mathfrak{m}}^{k}$ for all $k\geq
0$. This would confirm the validity of Conjecture 5.4, because such conditions
yield $I{\mathfrak{m}}^{k}$ to be $\mathfrak{m}$-full for all $k\geq 0$, which
means $\mathfrak{d}_{1}(I)=0$ and hence, as we already know,
$\mathfrak{d}_{3}(I)=0$. Now, Theorem 5.3 ensures that $R$ is regular.
Acknowledgements. The second-named author was partially supported by CNPq
(grants 406377/2021-9 and 313357/2023-4).
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|
# Conformal invariance and multifractality at Anderson transitions in
arbitrary dimensions
Jaychandran Padayasi Ilya A. Gruzberg<EMAIL_ADDRESS>The Ohio State
University, Department of Physics, Columbus OH, 43210, USA
###### Abstract
Electronic wave functions at Anderson transitions exhibit multifractal scaling
characterized by a continuum of generalized multifractal exponents
$\Delta_{\gamma}$ with vector indices $\gamma=(q_{1},\ldots,q_{n})$. In a
field theory description of the transitions, there are corresponding
multifractal operators $\mathcal{O}_{\gamma}$ with scaling dimensions
$\Delta_{\gamma}$. Assuming conformal invariance and using the conformal
bootstrap framework, we derive a constraint that implies that the generalized
multifractal spectrum $\Delta_{\gamma}$ must be quadratic in all $q_{i}$ in
any dimension $d>2$. As several numerical studies have shown deviations from
parabolicity, we argue that conformal invariance is likely absent at Anderson
transitions in dimensions $d>2$.
††preprint: APS/123-QED
Introduction. Continuous Anderson transitions (ATs) between metals and
insulators, as well as transitions between topologically different localized
phases are a major focal point in the study of disordered systems [1]. Despite
major breakthroughs in the scaling theory [2], symmetry classification [3],
critical properties at ATs have eluded a rigorous theoretical treatment
because of the strongly-coupled nature of the critical points.
A remarkable property of ATs is the multifractality of critical wave
functions, or the local density of states $\nu(r)$: their moments scale (in
any spatial dimension $d$) with the system size $L$ as
$\overline{\nu^{q}(r)}\sim L^{-\Delta_{q}}$, where the exponents $\Delta_{q}$,
called the multifractal (MF) spectrum, depend on the continuous (complex)
parameter $q$ in a non-linear way. (The bar denotes disorder averaging.) Also,
there are more general combinations $P_{\gamma}$ of critical wave functions
[4, 5, 6, 7] labeled by vectors $\gamma=(q_{1},\ldots,q_{n})$ of complex
numbers $q_{i}$, with scaling dimensions $\Delta_{\gamma}$ 111Notice that what
we denote as $\Delta_{q}$ and $\Delta_{\gamma}$ here is usually denoted by
$x_{q}$ and $x_{\gamma}$ in other papers on Anderson multifractality..
Field theories of Anderson localization are non-linear sigma models whose
target spaces are cosets $\mathcal{G}/\mathcal{K}$ of certain Lie supergroups
[9, 10, 11, 1, 12]. In these models, the generalized MF observables
$P_{\gamma}$ are represented by gradientless composite operators
$\mathcal{O}_{\gamma}$. A key fact is that $\mathcal{O}_{\gamma}$ can be
constructed as highest-weight vectors under the action of the Lie superalgebra
of $\mathcal{G}$ with weights $\gamma$. Then the $\mathcal{G}$-symmetry of the
target space (not broken at the critical point) strongly restricts the fusion
of MF operators and implies certain symmetries of the generalized
($\Delta_{\gamma}$) and simple ($\Delta_{q}$) MF spectra [13, 4].
Specifically, the MF operators $\mathcal{O}_{\gamma}$ satisfy 1) _Abelian
fusion_ (addition of highest weights)
$\displaystyle\mathcal{O}_{\gamma_{1}}\times\mathcal{O}_{\gamma_{2}}\sim\mathcal{O}_{\gamma_{1}+\gamma_{2}}+\ldots,$
(1)
where the ellipses denote derivatives of
$\mathcal{O}_{\gamma_{1}+\gamma_{2}}$; 2) _Weyl symmetry_ of the MF spectra
$\Delta_{\gamma}=\Delta_{w\gamma}$, $w\in W$, where $W$ is the Weyl group that
acts in the space of weights $\gamma$ and is generated by the sign inversions
of ${\tilde{q}}_{j}\equiv q_{j}+c_{j}/2$ and interchanges of ${\tilde{q}}_{i}$
and ${\tilde{q}}_{j}$:
$\displaystyle q_{i}$ $\displaystyle\rightarrow-c_{i}-q_{i},$ $\displaystyle
q_{i}\rightarrow q_{j}+(c_{j}-c_{i})/2.$ (2)
The parameters $c_{i}$ are the coefficients of the half-sum of the positive
roots $\rho_{b}=\sum_{j=1}^{n}c_{j}e_{j}$ in a standard basis $e_{j}$. They
are known for all families of symmetric superspaces [14, 13, 4, 5, 6, 7, 15].
A consequence of Eq. (2) is the existence of the operator
$\mathcal{O}_{-\rho_{b}}=\mathcal{O}_{(-c_{1},-c_{2},\ldots,-c_{n})}$ with
vanishing scaling dimension $\Delta_{-\rho_{b}}=0$. The simple MF spectrum
$\Delta_{q}$ corresponds to $\gamma=(q,0,0,\ldots,0)$. In this case one
denotes $c_{1}=-q_{*}$, and the Weyl symmetry reduces to
$\Delta_{q}=\Delta_{q_{*}-q}$. The Weyl symmetry is fully supported by
numerical and analytical results for various symmetry classes and dimensions
$d\geq 2$ [16, 17, 14, 18, 19, 20, 21, 22, 23, 24, 25, 5, 6, 7, 15].
As for other critical points, the scale invariance at ATs may be enhanced to
conformal invariance (though this is not guaranteed [26, 27]), in which case
critical properties are described by a conformal field theory (CFT). In this
letter we derive a general constraint on the MF spectra in an _arbitrary_
dimension $d$ and for _any_ symmetry class, that results from the properties
(1)–(2) and the assumption of presence of conformal symmetry at ATs. Our main
result is that in this situation the MF spectra have exactly parabolic form:
$\displaystyle\Delta_{\gamma}$ $\displaystyle=-b\sum_{i}q_{i}(q_{i}+c_{i}),$
$\displaystyle\Delta_{q}$ $\displaystyle=bq(q_{*}-q),$ (3)
where the overall prefactor $b$ cannot be determined from symmetry
considerations alone.
Our result is very general, and should apply to other situations where
multifractality and conformal invariance may both be present, such as critical
points in disordered systems [28, 29], turbulence [30, 31, 32], and other
systems.
Multifractality and CFT in $d=2$. Two-dimensional (2D) CFTs possess the
infinite-dimensional Virasoro symmetry. In this setting, the ellipses in Eq.
(1) represent Virasoro descendants, and lead to a single Virasoro block in a
4-point function of MF operators, and a Vafa-Lewellen [33, 34] constraint on
the MF spectra. The unique solution of this constraint subject to the
symmetries (2) is the parabolic spectrum (3) [35, 36, 5]. The MF operators
appear as vertex operators in a 2D Coulomb gas theory (Gaussian free field
with a background charge).
Both the simple and the generalized parabolicity (3) were tested analytically
and numerically, and were found to be violated at 2D ATs in various symmetry
classes [21, 22, 37, 5, 6, 7, 15]. This has led to the understanding that
conformal invariance might be lost at these critical points. Similarly,
numerical studies of multifractality in $d=3,4,5$ have found strong deviations
from parabolicity [25, 38, 39, 40] but there has not been any prediction in
$d>2$ from a CFT perspective. Our paper provides such a prediction.
The conformal bootstrap program [41] has brought the study of higher-
dimensional CFTs into the limelight with extensive work on both analytical and
numerical fronts. The bootstrap philosophy attempts to solve the _crossing
symmetry_ conditions coming from associativity of the operator product
expansion (OPE) of the CFT operators, with minimal inputs from global symmetry
and expected fusion rules for the operators. Crossing symmetry relates three
possible ways (or channels) of reducing a 4-point function
$\langle\prod_{i=1}^{4}O_{i}(x_{i})\rangle$ to 2-point functions by replacing
pairs of operators with their OPEs (Fig. 1). The $s$-channel fusion ($1\to 2$,
$3\to 4$) and the $t$-channel fusion ($1\to 4$, $2\to 3$) result in expansions
of the 4-point function that have overlapping regions of convergence and give
the crossing equation
$\displaystyle\sum_{O_{s}}\lambda_{12}^{O_{s}}\lambda_{34}^{O_{s}}W_{O_{s}}=\sum_{O_{t}}\lambda_{14}^{O_{t}}\lambda_{23}^{O_{t}}W_{O_{t}}.$
(4)
$\displaystyle\sum_{O_{s}}$ $O_{1}$$O_{2}$$O_{4}$$O_{3}$$O_{s}$ =
$\displaystyle\sum_{O_{t}}$ $O_{1}$$O_{2}$$O_{4}$$O_{3}$$O_{t}$
Figure 1: Schematic representation of the $s$-$t$ crossing equation.
Here the factors $W_{O}$ are fully determined by conformal symmetry, while the
CFT data $\\{\Delta_{i},\lambda_{ijk}\\}$ consisting of the spectrum of
scaling dimensions and the OPE coefficients are to be found. Solutions of the
crossing equations determine consistent CFTs fully defined by the CFT data,
circumventing the need for a Lagrangian description.
The $s$ and $t$ channels are obtained from each other by interchanges of
indices of the operators (and their points of insertion): $s\leftrightarrow
t\equiv 1\leftrightarrow 3$. Accordingly, starting with a function
$f^{(s)}\equiv f(x_{1},x_{2};x_{3},x_{4})$ of four ordered arguments we
obtain, by permuting $1\leftrightarrow 3$, another function $f^{(t)}\equiv
f(x_{3},x_{2};x_{1},x_{4})$. Using this notation, we can write the 4-point
function as a product of a conformally-covariant kinematic factor
$\mathbb{K}_{4}^{(c)}$ and a $G$-function (different in each channel)
$\displaystyle\Big{\langle}\prod_{i=1}^{4}O_{i}(x_{i})\Big{\rangle}=\mathbb{K}_{4}^{(s)}G^{(s)}=\mathbb{K}_{4}^{(t)}G^{(t)}.$
(5)
The $G$-functions depend on the cross ratios
$\displaystyle u$
$\displaystyle=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}},$
$\displaystyle v$
$\displaystyle=\frac{x_{14}^{2}x_{23}^{2}}{x_{13}^{2}x_{24}^{2}},$ where
$\displaystyle x_{ij}$ $\displaystyle=x_{i}-x_{j},$ (6)
and operator labels. The cross ratios get transformed upon crossing so that
$\displaystyle G^{(s)}$ $\displaystyle=G_{12,34}(u,v),$ $\displaystyle
G^{(t)}$ $\displaystyle=G_{32,14}(v,u).$ (7)
In terms of the $G$-functions the $s$-$t$ crossing equation is
$\displaystyle G^{(s)}u^{-\frac{\Delta_{1}+\Delta_{2}}{2}}$
$\displaystyle=G^{(t)}v^{-\frac{\Delta_{2}+\Delta_{3}}{2}}.$ (8)
Much of the bootstrap formalism is geared towards solving Eq. (8) self-
consistently for unitary CFTs. Since any putative CFT for MF correlators
contains infinitely many relevant operators and, thus, is non-unitary, we
resort to novel, unorthodox methods that focus on the $G$-function and various
physical inputs (similar in spirit to the “inverse bootstrap” method in [42]).
We start by studying the Coulomb gas theories in the language of modern
conformal bootstrap and use them as signposts to generalize the notion of
Abelian fusion to higher dimensions. Then we show that the generalized Abelian
fusion and crossing symmetry together yield a constraint on the spectrum of
scaling dimensions in any $d$ that is analogous to the Vafa-Lewellen
constraints [33, 34] known in CFT in $d=2$. Finally, additional physical
assumptions specific to MF observables allow us to solve the constraint,
leading to a quadratic dependence of the MF spectrum $\Delta_{\gamma}$ on
$q_{i}$ in any dimension $d$.
Coulomb gas theories with global conformal blocks. In $d=2$, the Coulomb gas
theories arise out of breaking the U(1) symmetry of the free boson $\phi$ by
including a background charge $Q$ in the action [43, 44, 45]. We will be
interested in the correlators of the vertex operators $V_{\alpha}\sim
e^{d\alpha\phi}$. A Coulomb gas CFT can be defined [44, 45] in any dimension
$d\in\mathbb{N}$ by considering an action with a possibly non-local kinetic
term $\propto\phi(-\square)^{\frac{d}{2}}\phi$. Such CFTs also arise as limits
of generalized free fields, where the scaling dimension of the free boson is
tuned to $\Delta_{\phi}=0$. In this limit $\langle\phi\phi\rangle$ is
logarithmic in any dimension which allows us to study vertex operators.
Following the conventions in [44, 45], the spectrum and the multi-point
functions in such higher-dimensional Coulomb gas theories are
$\displaystyle\Delta_{\alpha}$ $\displaystyle=d\alpha(Q-\alpha),$
$\displaystyle\Big{\langle}\prod V_{\alpha_{i}}(x_{i})\Big{\rangle}$
$\displaystyle=\prod_{i<j}|x_{ij}|^{-2d\alpha_{i}\alpha_{j}},$ (9)
where the correlators satisfy the charge neutrality $\sum{\alpha_{i}}=Q$. Our
first goal is to derive the OPE of vertex operators in terms of primaries of
the global conformal group by studying the conformal block expansion.
Consider a four-point function of vertex operators which can be written in the
form (5) with the $G$-function
$\displaystyle G^{(s)}_{\text{CG}}$
$\displaystyle=u^{\frac{d}{2}(\alpha_{3}+\alpha_{4})(\alpha_{1}+\alpha_{2})}v^{-d\alpha_{2}\alpha_{3}}$
$\displaystyle=u^{\frac{1}{2}\Delta_{\alpha_{1}+\alpha_{2}}}v^{\frac{1}{2}(\Delta_{\alpha_{2}+\alpha_{3}}-\Delta_{\alpha_{2}}-\Delta_{\alpha_{3}})}.$
(10)
This function is explicitly crossing symmetric [satisfies Eq. (8)] and has a
convergent conformal block expansion [41] in the $s$-channel in any
dimensionality $d$,
$\displaystyle
G^{(s)}_{\text{CG}}=\sum_{O}\lambda_{12}^{O}\lambda_{34}^{O}\,g_{\Delta_{O},l_{O}}(u,v).$
(11)
The conformal blocks $g_{\Delta_{O},l_{O}}$ are known analytically in even
dimensions. (Their dependence on the dimensions $\Delta_{1}$–$\Delta_{4}$ of
the external operators will not be important for us.) The blocks are often
written in conformal frame coordinates $(z,\bar{z})$ related to the cross
ratios $(u,v)$ by
$\displaystyle u$ $\displaystyle=z\bar{z},$ $\displaystyle v$
$\displaystyle=(1-z)(1-\bar{z}).$ (12)
In the $s$-channel limit, $x_{12}\approx x_{34}\ll x_{13}\approx x_{24}\approx
x_{23}\approx x_{14}$, and thus $u\rightarrow 0$, $v\rightarrow 1$, see Eq
(6). This corresponds to $z,\bar{z}\rightarrow 0$, and the $G$-function
(Conformal invariance and multifractality at Anderson transitions in arbitrary
dimensions) has the form
$\displaystyle G^{(s)}$
$\displaystyle=(z\bar{z})^{\frac{\Delta^{(s)}}{2}}f(z,\bar{z}),$ (13)
where $f(z,\bar{z})$ is a Taylor series symmetric in $(z,\bar{z})$, and
$\Delta^{(s)}=\Delta_{\alpha_{1}+\alpha_{2}}$. In the Supplementary material
[46] we use the leading behavior of the conformal blocks in the $s$-channel
[47] to show that any $G$-function of the form (13) admits the conformal block
expansion
$\displaystyle G^{(s)}=\sum_{n,l\geq
0}\mu^{(n,l)}g_{\Delta^{(s)}+2n+l,l}(z,\bar{z})$ (14)
in arbitrary dimensions $d\geq 2$. Conversely, any $G$-function that can be
expanded as in Eq. (14) can also be written in the form of Eq. (13).
Let us write global primaries in any dimension as $[\Delta,l]$ specifying
their scaling dimension $\Delta$ and spin $l$. Then we say that the expansion
(14) contains just one twist family [42] consisting of the leading primary
$[\Delta^{(s)},0]$ and subleading operators $[\Delta^{(s)}+2n+l,l]$ which are
constructed from its derivatives. The superscript of the product of the OPE
coefficients $\mu^{(n,l)}\equiv\lambda_{1,2}^{(n,l)}\lambda_{3,4}^{(n,l)}$
identifies the operator $[\Delta^{(s)}+2n+l,l]$ in the twist family.
Expanding the Coulomb gas $G$-function (Conformal invariance and
multifractality at Anderson transitions in arbitrary dimensions) in global
conformal blocks gives the OPE of $V_{\alpha_{1}}\times V_{\alpha_{2}}$ as
[46]
$\displaystyle[\Delta_{\alpha_{1}},0]\times[\Delta_{\alpha_{2}},0]\sim\sum_{n,l\geq
0}\lambda^{(n,l)}_{1,2}[\Delta_{\alpha_{1}+\alpha_{2}}+2n+l,l],$ (15)
where $n,l$ are non-negative integers, and the $(n,l)=(0,1)$ term is absent in
the OPE. For two identical operators ($\alpha_{1}=\alpha_{2}$), their OPE is
completely specified by the conformal block expansion since we can extract the
squared OPE coefficients (see [46] for explicit expressions in the $d=2$ and
$d=4$ cases).
Generalized Abelian fusion. In the strict sense, Abelian fusion (1) cannot
hold in CFTs in $d>2$, since an OPE written with finitely many global
conformal primaries cannot satisfy crossing [48, 49, 50, 51]. Thus, we need to
generalize the notion of Abelian fusion to $d>2$. Global conformal block
expansions of Coulomb gas correlators exhibit certain features that we adopt
as the _definition_ of Abelian fusion in $d>2$:
* •
All primary multifractal operators of the CFT can be grouped into twist
families.
* •
The OPE of any two leading multifractal primaries contains only one twist
family
$\displaystyle[\Delta_{1},0]\times[\Delta_{2},0]\sim\sum_{n,l\geq
0}\lambda_{1,2}^{(n,l)}[\Delta+2n+l,l].$ (16)
This definition is more general than Eq. (15), where the three operators
involved, $V_{\alpha_{1}}$, $V_{\alpha_{2}}$, an $V_{\alpha_{1}+\alpha_{2}}$,
are labeled by continuous parameters that add upon fusion.
The generalized Abelian fusion (16) and the related conformal block expansion
(14) constrain a general scalar 4-point $G$-function to have the form (13).
Since $z\bar{z}=u$ and $z+\bar{z}=u+1-v$ form a basis in the ring of symmetric
functions of $z$ and $\bar{z}$, Eq. (13) can also be written as
$G^{(s)}=u^{\frac{\Delta^{(s)}}{2}}\sum_{n\geq 0}f^{(s)}_{n}(v)u^{n}.$ (17)
All powers of $u$ in this expansion differ from the leading power
$\Delta^{(s)}/2$ by non-negative integers $n$, consistent with the even
integer gaps between the spin-zero conformal blocks in Eq. (14). The functions
$f_{n}(v)$ are arbitrary so far, and quantities with the superscript $(s)$
depend on the external dimensions in a channel-covariant manner. At this
point, one can make a further simplification by assuming that the functions
$f_{r}(v)$ can be represented as (possibly infinite) sums of power laws in
$v$, i.e.
$G^{(s)}=u^{\frac{\Delta^{(s)}}{2}}\sum_{n,m\geq
0}C_{nm}v^{\sigma^{(s)}_{m}}u^{n}$ (18)
where $\sigma_{m}$ are unrelated real numbers and the coefficients $C_{nm}$ do
not depend on the cross-ratios $(u,v)$ or the external dimensions. The Coulomb
gas theories are of this form with a single term, see Eq. (Conformal
invariance and multifractality at Anderson transitions in arbitrary
dimensions). The generalized free field correlators [50] with
$\Delta_{\phi}\geq 0$ are similarly composed of sums of power-laws in $u$ and
$v$ although they do not satisfy Abelian fusion (integer gaps in powers of
$u$). The above ansatz made an appearance for the case of a correlator of
identical operators in Ref. [42] which discussed the idea of building
crossing-symmetric $G$-functions. The issue of its generality was raised and
left unanswered there. Similarly, the authors of Ref. [51] use a version where
the coefficients $C_{nm}$ are functions of $\log u,\log v$ (the logarithms
come from anomalous dimensions of the subleading operators in the twist
family). As our definition of Abelian fusion, Eq. (16), exactly fixes the
dimensions of all subleading operators, the logarithms are unnecessary in our
treatment.
Constraints on the $G$-function from crossing. Now we explore constraints on
the Abelian $G$-function (18) imposed by crossing symmetry. Substituting this
function into the crossing equation (8), we obtain
$\displaystyle
u^{\frac{\Delta^{(s)}}{2}-\frac{\Delta_{1}+\Delta_{2}}{2}}\sum_{n,m\geq
0}C_{nm}v^{\sigma^{(s)}_{m}}u^{n}$
$\displaystyle=v^{\frac{\Delta^{(t)}}{2}-\frac{\Delta_{2}+\Delta_{3}}{2}}\sum_{n,m\geq
0}C_{nm}u^{\sigma^{(t)}_{m}}v^{n}.$ (19)
This equation enforces a structure on the $G$-function that is understandable
in terms of crossing symmetric building blocks [42], such that any truncation
up to $(N,M)$ of the double sum in Eq. (18) is also crossing symmetric. In
this case all terms in Eq. (18) fall into two groups: single terms that are
crossing-symmetric by themselves, and pairs of terms that get exchanged under
crossing. The result is the $G$-function
$\displaystyle G^{(s)}(u,v)$
$\displaystyle=u^{\frac{\Delta^{(s)}}{2}}v^{\frac{\Delta^{(t)}}{2}-\frac{\Delta_{2}+\Delta_{3}}{2}}\bigg{(}\sum_{k\geq
0}S_{k}(uv)^{k}$ $\displaystyle\quad+\sum_{j\geq 1,k\geq
0}D_{jk}(uv)^{k}(u^{j}+v^{j})\bigg{)}.$ (20)
where we use $S_{k}$ and $D_{jk}$ to represent the coefficients of the
crossing-symmetric terms and pairs respectively. Thus, we adopt Eq. (Conformal
invariance and multifractality at Anderson transitions in arbitrary
dimensions) as the generic form of the Abelian $G$-function (18) that also
satisfies $s$-$t$ crossing.
Excluding the spin-1 operator. Focusing on the last part of the puzzle, we
expand the $G$-function (Conformal invariance and multifractality at Anderson
transitions in arbitrary dimensions) in conformal blocks in arbitrary
dimensions to first few orders in $z$ and $\bar{z}$. By construction, the
first block that appears in the expansion is $[\Delta^{(s)},0]$. The product
of the OPE coefficients of the leading block is read off as $\mu^{(0,0)}\equiv
S_{0}$. The coefficient $\mu^{(0,1)}$ of the spin-1 block $[\Delta^{(s)}+1,1]$
can be obtained by matching the coefficients of the series for the order
$\sim(z\bar{z})^{\Delta^{(s)}/2}(z+\bar{z})$ as
$\displaystyle\frac{\mu^{(0,1)}}{S_{0}}$
$\displaystyle=\frac{\Delta_{2}+\Delta_{3}-\Delta^{(t)}}{2}-\frac{(\Delta^{(s)}-\Delta_{12})(\Delta^{(s)}+\Delta_{34})}{4\Delta^{(s)}}$
$\displaystyle+\frac{\Delta_{2}+\Delta_{3}-\Delta^{(t)}}{2}\sum_{j\geq
1}\frac{D_{j0}}{S_{0}}-\sum_{j\geq 1}\frac{jD_{j0}}{S_{0}}.$ (21)
Both sums over $j$ must converge for the OPE coefficient to be well-defined,
and thus we introduce
$\displaystyle\sum_{j\geq 1}\frac{D_{j0}}{S_{0}}$
$\displaystyle\equiv\mathcal{P},$ $\displaystyle\sum_{j\geq
1}\frac{jD_{j0}}{S_{0}}$ $\displaystyle\equiv\mathcal{Q}.$ (22)
In the context of MF correlators, we identify
$\Delta^{(s)}\equiv\Delta_{\gamma_{1}+\gamma_{2}}$,
$\Delta^{(t)}\equiv\Delta_{\gamma_{3}+\gamma_{2}}$. The neutrality condition
$\sum_{i}\gamma_{i}=-\rho_{b}$ fixes
$\Delta_{4}=\Delta_{-\rho_{b}-\gamma_{1}-\gamma_{2}-\gamma_{3}}=\Delta_{\gamma_{1}+\gamma_{2}+\gamma_{3}}$.
This spin-1 operator cannot appear in any OPE of two identical scalar
operators on general grounds. Indeed, $O(x_{1})O(x_{2})$ is even with respect
to the interchange $x_{1}\leftrightarrow x_{2}$ but a spin-1 operator must
appear in the OPE as $\sim(x_{1}-x_{2})\cdot\partial
O_{\Delta^{(s)}}((x_{1}+x_{2})/2)$ which is odd. Exploiting this fact, we set
$O_{2}\equiv O_{1}$ in which case $\mu^{(0,1)}=0$, and Eq. (Conformal
invariance and multifractality at Anderson transitions in arbitrary
dimensions) becomes a constraint on $\Delta$’s:
$\displaystyle\Delta^{(s)}+\Delta_{3}-\Delta_{4}+4\mathcal{Q}=2(\Delta_{1}+\Delta_{3}-\Delta^{(t)})(1+\mathcal{P}).$
(23)
Furthermore, the continuity of MF spectra allows us to choose
$\gamma_{1}=\gamma_{2}=\epsilon e_{i}$, where $e_{i}=(0,\ldots,1,\ldots,0)$
(unit in the $i$-th place), with $\epsilon\ll 1$, and $\gamma_{3}=\gamma$ in
Eq. (23), which gives
$\displaystyle\Delta_{2\epsilon
e_{i}}+\Delta_{\gamma}-\Delta_{\gamma+2\epsilon e_{i}}+4\mathcal{Q}$
$\displaystyle\quad=2(\Delta_{\epsilon
e_{i}}+\Delta_{\gamma}-\Delta_{\gamma+\epsilon e_{i}})(1+\mathcal{P}).$ (24)
Now we expand in orders of $\epsilon$. At order $\epsilon^{0}$ we get
$4\mathcal{Q}=(1+2\mathcal{P})\Delta_{0}=0$, because $\Delta_{0}=0$ by
definition. At order $\epsilon$ we get
$\mathcal{P}\partial_{q_{i}}(\Delta_{0}-\Delta_{\gamma})=0$ which implies
either $\mathcal{P}=0$ or
$\partial_{q_{i}}\Delta_{0}=\partial_{q_{i}}\Delta_{\gamma}$. The second
condition [together with Eq. (2)] yields the trivial spectrum
$\Delta_{\gamma}=\text{const}=0$. Hence, we consider the case $\mathcal{P}=0$.
Then at order $\epsilon^{2}$ we get
$\partial_{q_{i}}^{2}(\Delta_{0}-\Delta_{\gamma})=0$, which implies that
$\Delta_{\gamma}$ is a quadratic polynomial in all $q_{i}$. Any such
polynomial that vanishes at $\gamma=0$ and satisfies the symmetry properties
(2) is exactly of the form given in Eq. (3) [5]. Thus, we arrive at our main
result:
> _The only MF spectrum $\Delta_{\gamma}$ which satisfies generalized Abelian
> fusion and crossing symmetry has the form given in Eq. (3)._
Going back to Eq. (23), we substitute $\mathcal{P}=\mathcal{Q}=0$ and the
quadratic solution for $\Delta_{\gamma}$ to find that the constraint
$2\Delta_{\gamma_{1}}+\Delta_{\gamma_{3}}-2\Delta_{\gamma_{1}+\gamma_{3}}-\Delta_{2\gamma_{1}}+\Delta_{2\gamma_{1}+\gamma_{3}}=0$
(25)
correctly picks out Abelian CFTs in $d\geq 2$, and thus is the appropriate
generalization of the 2D Vafa-Lewellen constraint with a single exchanged
Virasoro primary.
Summary and Outlook. Using conformal invariance, we have shown that any
Abelian CFT in $d>2$ must be intimately related to the Coulomb gas theory, and
have a quadratic spectrum. To summarize, the set of assumptions used to prove
the exact parabolicity of the MF spectrum in $d\geq 2$ is the following:
1. 1.
The MF operators $\mathcal{O}_{\gamma}$ are global primaries in some CFT, so
that their correlators solve the crossing equations.
2. 2.
The fusion of two MF operators is of the general form (16) (generalized
Abelian fusion).
3. 3.
The $G$-functions of the charge-neutral 4-point correlators of MF operators
admit an expansion of the form
$G^{(s)}=\sum_{n,m}C_{nm}u^{\Delta_{\gamma_{1}+\gamma_{2}}/2+n}v^{\sigma_{m}}$.
The second and third assumptions are our strongest, and are fundamentally
related to each other. As in the case of weakly perturbed CFTs [51], it
remains to be seen if the generalized Abelian CFT defined here could be
perturbed so that the derivative operators gain anomalous dimensions.
Finally, let us discuss the implication of the first assumption, that of
conformal invariance. As we summarised earlier, the perturbative analytical
results in $d=2+\epsilon$ and numerical simulations in $d=3,4,5$ have shown
that the MF spectra for generic Anderson transitions are in fact, not
parabolic. In the light of our result, it follows that conformal symmetry is
likely lost at Anderson transitions. The only alternative is that the
symmetries of the sigma models that were used to deduce Abelian fusion and
Weyl symmetry are lost at the critical point. We believe this to be a very
unlikely scenario that would contradict the vast body of literature on
Anderson transitions. Thus, we identify Anderson transitions as examples of
systems where scale invariance does not imply conformal invariance.
Going forward, we believe that it is very important to understand the
relationship between scale and conformal invariance, especially for non-
unitary theories. Anderson transitions may provide a fertile ground for such
understanding. Moreover, it would be interesting to translate our analysis to
other physical systems exhibiting multifractal scaling. Finally, it is
exciting to consider possible generalizations of rational CFTs to $d>2$. This
line of inquiry may provide important answers in the much bigger project of
tracking the evolution of CFTs with $d$.
Acknowledgements. We thank Dalimil Mazáč, Marco Meineri, Alexander Mirlin,
Yaron Oz, Lorenzo Quintavalle, Sylvain Ribault, and Bernardo Zan for useful
discussions. This research was supported by Grant No. 2020193 from the United
States-Israel Binational Science Foundation (BSF).
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## Supplementary material
### .1 Subleading behavior of global conformal blocks
Start with the Casimir equation solved by the spin-0 conformal blocks
$\mathcal{D}g_{\Delta,0}^{\Delta_{12},\Delta_{34}}(z,\bar{z})=C_{\Delta,0}g_{\Delta,0}^{\Delta_{12},\Delta_{34}}(z,\bar{z})$
(26)
where $\mathcal{D}$ is the Casimir operator of $SO(d+1,1)$:
$\mathcal{D}=\mathcal{D}_{z}+\mathcal{D}_{\bar{z}}+2(d-2)\frac{z\bar{z}}{z-\bar{z}}\left[(1-z)\partial_{z}-(1-\bar{z})\partial_{\bar{z}}\right],$
(27)
$\mathcal{D}_{z}=2z^{2}(1-z)\partial_{z}^{2}-(2+\Delta_{34}-\Delta_{12})z^{2}\partial_{z}+\frac{\Delta_{12}\Delta_{34}}{2}z$
(28)
and $C_{\Delta,0}\equiv\Delta(\Delta-d)$ is the eigenvalue of the Casimir
operator. Based on the leading form of $g_{\Delta,0}$ and the symmetricity
with respect to $z$ and $\bar{z}$, we use the following power series as an
ansatz for the solution:
$g_{\Delta,0}(z,\bar{z})=(z\bar{z})^{\alpha}\sum_{r,s=0}^{\infty}\frac{\kappa_{rs}}{1+\delta_{rs}}(z^{r}\bar{z}^{s}+z^{s}\bar{z}^{r}).$
(29)
For the subleading behavior of the block, we need to compute the coefficient
$\kappa_{10}$. Substituting the ansatz into the Casimir equation and
extracting the terms at leading order gives
$C_{\Delta,0}=\Delta(\Delta-d)=4\alpha(\alpha-1)-2(d-2)\alpha$ (30)
which is solved by $\alpha=\Delta/2$ as expected. $\kappa_{00}$ is a
normalization that we can set to unity. Now considering the subleading order
terms, we have
$\displaystyle
2\kappa_{10}\bigg{(}(\alpha+1)_{2}-(\alpha)_{2}-(d-2)\alpha\bigg{)}-2(\alpha)_{2}-\alpha(2+\Delta_{34}-\Delta_{12})+\frac{\Delta_{12}\Delta_{34}}{2}=\kappa_{10}\Delta(\Delta-d).$
(31)
$\kappa_{10}$ is read off from the equation as
$\kappa_{10}=\frac{(\Delta-\Delta_{12})(\Delta+\Delta_{34})}{4\Delta}$ (32)
with no dependence on dimension $d$.
### .2 Global conformal block decomposition of Coulomb gas-like correlator
Consider a $G$-function which admits a series expansion at
$z,\bar{z}\rightarrow 0$ of the form
$G(z,\bar{z})=(z\bar{z})^{p}\sum_{i,j=0}^{\infty}\mathscr{C}_{ij}z^{i}\bar{z}^{j}$
(33)
with $\mathscr{C}_{ij}=\mathscr{C}_{ji}$. The Coulomb gas correlator
(Conformal invariance and multifractality at Anderson transitions in arbitrary
dimensions) is a particular example, with
$2p=\Delta_{\alpha_{1}+\alpha_{2}}\hskip 36.135pt\text{and}\hskip
36.135pt\mathscr{C}_{ij}=\binom{i+d\alpha_{2}\alpha_{3}-1}{i}\binom{j+d\alpha_{2}\alpha_{3}-1}{j}.$
(34)
Let us show that the given $G(z,\bar{z})$ can be expanded in terms of global
conformal blocks in any dimension $d$ with a single double trace family
$[2p+2n+l,l].$
In other words, we show that
$G(z,\bar{z})=\sum_{n,l\geq 0}\mu^{(n,l)}g_{2p+2n+l,l}(z,\bar{z})$ (35)
where $\mu^{(n,l)}$ is a product of two OPE coefficients. The proof will be
based upon the idea that the expansion Eq. 35 can be uniquely fixed order-by-
order in powers of $z$ and $\bar{z}$.
As noted in the main text, The leading order behavior of the global conformal
blocks in any dimensions is
$g_{\Delta,l}\sim\mathcal{N}_{d,l}(z\bar{z})^{\Delta/2}\text{Geg}_{l}^{d/2-1}\left(\frac{z+\bar{z}}{2\sqrt{z\bar{z}}}\right)$
(36)
where we work with the normalization
$\mathcal{N}_{d,l}=\frac{l!}{\left(d/2-1\right)_{l}}.$ (37)
Focusing on the double-trace blocks $g_{2p+2n+l,l}$, we know further that the
subleading terms should be expandable as a power series
$g_{\Delta+2n+l,l}(z,\bar{z})=(z\bar{z})^{\Delta/2}\left(\mathcal{N}_{d,l}(z\bar{z})^{l/2+n}\text{Geg}_{l}^{d/2-1}\left(\frac{z+\bar{z}}{2\sqrt{z\bar{z}}}\right)+\sum_{r+s>(n+l/2)}\frac{\kappa^{(n,l)}_{rs}}{1+\delta_{rs}}z^{r}\bar{z}^{s}\right)$
(38)
where $r,s$ are integers and $\kappa_{rs}=\kappa_{sr}$ [52].
The Gegenbauer polynomial for $l=0$ is just 1, so for the leading degree, the
block $g_{2p,0}\sim(z\bar{z})^{p}$ has the correct power-law to match
$G(z,\bar{z})=\mathscr{C}_{00}(z\bar{z})^{p}+\ldots$.
Next, consider degree 1 terms in the series,
$(z\bar{z})^{p}(\mathscr{C}_{10}(z+\bar{z})+\ldots)$. On the conformal blocks
side, the subleading contribution from the leading block at this degree can be
balanced by adding a new term $[2p+1,1]$ with the correct power-law behavior,
$g_{2p+1,1}\sim(z\bar{z})^{p+\frac{1}{2}}\left(\frac{z+\bar{z}}{2\sqrt{z\bar{z}}}\right).$
(39)
The OPE coefficient is read off by equating the two series upto this order,
$\mu^{(1,1)}=\mathscr{C}_{10}-\mu^{(0,0)}\frac{\partial}{\partial
z}\left.\frac{g_{2p,0}(z,\bar{z})}{(z\bar{z})^{p}}\right|_{z=\bar{z}=0}=\mathscr{C}_{10}-\mathscr{C}_{00}\frac{(2p-\Delta_{12})(2p+\Delta_{34})}{8p}$
(40)
where we have used the subleading coefficient of the spin-zero block derived
in .1. Now, consider an arbitrary term in the $G$-function expansion
$\sim\mathscr{C}_{ij}z^{i}\bar{z}^{j}$. There is always a conformal block in
the double-trace family (in this case $g_{2p+i+j,i-j}$) with the correct
leading behavior to fix the series at this order. Looking closely at Eq. 36,
we have (with $j=n$ and $i-j=l$)
$\displaystyle g_{2p+2n+l,l}$
$\displaystyle\sim(z\bar{z})^{p+n+l/2}\frac{l!}{(d/2-1)_{l}}\text{Geg}_{l}^{d/2-1}\left(\frac{z+\bar{z}}{2\sqrt{z\bar{z}}}\right)$
(41)
$\displaystyle=(z\bar{z})^{p+n+l/2}\frac{l!}{(d/2-1)_{l}}\sum_{k=0}^{\lfloor\frac{l}{2}\rfloor}\frac{\left(d/2-1\right)_{l-k}}{k!(l-2k)!}(-1)^{k}\left(\frac{z+\bar{z}}{\sqrt{z\bar{z}}}\right)^{l-2k}$
(42)
$\displaystyle=(z\bar{z})^{p}\sum_{k=0}^{\lfloor\frac{l}{2}\rfloor}\frac{(-1)^{k}(l-2k)_{2k}}{k!(d/2+l-k-1)_{k}}(z\bar{z})^{k+n}(z+\bar{z})^{l-2k}.$
(43)
Setting $k=0$ in Eq. (43) yields a term proportional to
$(z\bar{z})^{p}(z^{i}\bar{z}^{j})$ with coefficient 1 as desired. There are
additional terms even at the leading order in the conformal block, but all of
them are of the same degree in $z,\bar{z}$: $i+j$. Therefore, for each degree,
the terms in the series of the same degree must be fixed in descending order
in $l$, so that the $l=0$ or $l=1$ term is fixed last. For example, in the
quadratic degree, there are two terms in the G-function series,
$\mathscr{C}_{20}(z\bar{z})^{p}(z^{2}+\bar{z}^{2})$ and
$\mathscr{C}_{11}(z\bar{z})^{p}(z\bar{z})$. Of the two corresponding blocks
$g_{2p+2,2}$ and $g_{2p+2,0}$,
$g_{2p+2,2}\sim(z\bar{z})^{p}(z^{2}+\bar{z}^{2}+z\bar{z})$ contributes to the
$z\bar{z}$ term but the block $g_{2p+2,0}\sim(z\bar{z})^{p+1}$ does not feed
back into the former.
In summary, the recursive process can be codified into a formula for
calculating $\mu^{(n,l)}$ given $C_{ij}$ as
$\mu^{(j,i-j)}=\mathscr{C}_{ij}-\frac{1}{i!j!}\frac{\partial^{i}}{\partial
z^{i}}\frac{\partial^{i}}{\partial\bar{z}^{j}}\left(\frac{1}{(z\bar{z})^{p}}\left[\sum_{2\alpha<(i+j)}\sum_{\beta=0}^{i+j-1-2\alpha}\mu^{(\alpha,\beta)}g_{2p+2\alpha+\beta,\beta}+\sum_{\alpha=0}^{j-1}\mu^{(\alpha,i+j-2\alpha)}g_{(2p+i+j,i+j-2\alpha)}\right]\right)_{z=\bar{z}=0}.$
(44)
Using the form of double-trace blocks in Eq. 38, this becomes
$\mu^{(j,i-j)}=\mathscr{C}_{ij}-\left[\sum_{2\alpha<(i+j)}\sum_{\beta=0}^{i+j-1-2\alpha}\mu^{(\alpha,\beta)}\kappa^{(\alpha,\beta)}_{ij}+\sum_{\alpha=0}^{j-1}\mu^{(\alpha,i+j-2\alpha)}\mathscr{K}_{\alpha}\right].$
(45)
where the numbers $\mathscr{K}_{\alpha}$ are given by
$\mathscr{K}_{\alpha}=\sum_{k=j-\alpha}^{\lfloor\frac{i+j}{2}-\alpha\rfloor}\binom{i+j-2\alpha-2k}{i-\alpha-k}\frac{(-1)^{k}(i+j-2\alpha-2k)_{2k}}{k!(d/2+i+j-2\alpha-k-1)_{k}}.$
(46)
Of course, to calculate $\mu^{(j,i-j)}$ more explicitly one requires the
coefficients $\kappa_{ij}$ in the definition of the conformal blocks, which is
not available in generic dimensions. But we still have the result
> Any G-function of the form $G(z,\bar{z})=(z\bar{z})^{p}f(z,\bar{z})$ where
> $f(z,\bar{z})$ has a convergent Taylor series expansion at $z=\bar{z}=0$ can
> be expanded in conformal blocks from a single twist family $[2p+2n+l,l]$.
### .3 Coulomb gas OPE coefficients
Following the procedure outlined in .2, we performed the global conformal
block expansion of the Coulomb Gas $G$-function in $d=2$ and $d=4$. In this
section we present some interesting observations and the first few
coefficients in the OPE of identical scalars.
Consider the Coulomb Gas $G$-function reproduced here for convenience,
$G(u,v)=u^{\frac{d}{2}(\alpha_{3}+\alpha_{4})}v^{-d\alpha_{2}\alpha_{3}}.$
(47)
One sees that it can be rewritten as a series expansion of the kind discussed
in .2,
$G(z,\bar{z})=(z\bar{z})^{\frac{\Delta_{\alpha_{1}+\alpha_{2}}}{2}}\left(\sum_{i,j=0}^{\infty}\binom{i+d\alpha_{2}\alpha_{3}-1}{i}\binom{j+d\alpha_{2}\alpha_{3}-1}{j}z^{i}\bar{z}^{j}\right)$
(48)
where we have used
$\Delta_{\alpha_{1}+\alpha_{2}}=d(\alpha_{1}+\alpha_{2})(\alpha_{3}+\alpha_{4})$
and charge neutrality. At leading order in any $d$, the coefficient
$\mathscr{C}_{00}=1$ in the series above. Thus the leading block in the
expansion is $[\Delta_{\alpha_{1}+\alpha_{2}},0]$ with coefficient
$\mu^{(0,0)}=1$. At the next degree, the only block to be considered is
$[\Delta_{\alpha_{1}+\alpha_{2}}+1,1]$, with the product OPE coefficient given
by Eq. 40, which turns out to be
$\mu^{(0,1)}=\binom{d\alpha_{2}\alpha_{3}}{1}-\frac{(\Delta_{\alpha_{1}+\alpha_{2}}-\Delta_{12})(\Delta_{\alpha_{1}+\alpha_{2}}+\Delta_{34})}{4\Delta_{\alpha_{1}+\alpha_{2}}}.$
(49)
As the discussion pans out in the key argument of the main text, we notice
upon simplification that generically, for all the Coulomb Gas theories,
$\mu^{(0,1)}=0.$ (50)
Let us stress that for $d=2$, it is expected that this OPE coefficient
vanishes in the global block expansion. It is a manifestation of the result
that in a Verma module, there are no quasiprimaries at level 1. But as we have
shown, this fact explicitly generalizes to higher dimensions for Coulomb Gas
theories.
Computing the expansion to higher degrees in $d=2$, we observe that the twist-
two $(n=1)$ family of operators $[\Delta_{\alpha_{1}+\alpha_{2}}+l+2,l]$ is
missing from the OPE (their OPE coefficients vanish for any $G$-function
independent of $\alpha_{i}$). Additionally, as we expect, in the OPE of
identical operators, the coefficients of odd-spin operators vanish. For
example,
$\mu^{(0,3)}=\frac{2\alpha_{1}\alpha_{2}\alpha_{3}\alpha_{4}(\alpha_{1}-\alpha_{2})(\alpha_{3}-\alpha_{4})}{3(1+(\alpha_{1}+\alpha_{2})(\alpha_{3}+\alpha_{4}))(2+(\alpha_{1}+\alpha_{2})(\alpha_{3}+\alpha_{4}))}$
(51)
and we see that $\mu^{(0,3)}=0$ if $\alpha_{1}=\alpha_{2}$ or
$\alpha_{3}=\alpha_{4}$. The non-zero coefficients in the OPE of two identical
scalars are tabulated in Table 1. (For the case of non-identical scalars, see
[53]).
$(n,l)$ | $(\lambda^{(n,l)})^{2}$
---|---
$(0,0)$ | 1
$(0,2)$ | $\dfrac{2\alpha^{4}}{8\alpha^{2}+1}$
$(0,4)$ | $\dfrac{\alpha^{4}(2\alpha^{2}+1)^{2}}{2(64\alpha^{4}+64\alpha^{2}+15)}$
$(2,0)$ | $\dfrac{4\alpha^{8}}{(8\alpha^{2}+1)^{2}}$
$(0,6)$ | $\dfrac{\alpha^{4}(2\alpha^{4}+3\alpha^{2}+1)^{2}}{3(8\alpha^{2}+5)(8\alpha^{2}+7)(8\alpha^{2}+9)}$
$(2,2)$ | $\dfrac{\alpha^{8}(2\alpha^{2}+1)^{2}}{(8\alpha^{2}+1)(8\alpha^{2}+3)(8\alpha^{2}+5)}$
$(0,8)$ | $\dfrac{\alpha^{4}(4\alpha^{6}+12\alpha^{4}+11\alpha^{2}+3)^{2}}{24(8\alpha^{2}+7)(8\alpha^{2}+9)(8\alpha^{2}+11)(8\alpha^{2}+13)}$
$(2,4)$ | $\dfrac{2\alpha^{8}(2\alpha^{4}+3\alpha^{2}+1)^{2}}{3(8\alpha^{2}+1)(8\alpha^{2}+5)(8\alpha^{2}+7)(8\alpha^{2}+9)}$
$(4,0)$ | $\dfrac{\alpha^{8}(2\alpha^{2}+1)^{4}}{4(64\alpha^{4}+64\alpha^{2}+15)^{2}}$
Table 1: All non-zero OPE coefficients up to degree 8 $(n=8)$ in $d=2$ for the
coulomb gas correlator $\langle
V_{\alpha}V_{\alpha}V_{\alpha}V_{\alpha}\rangle$ in the $s$-channel with
$\alpha\equiv Q/4$ to satsify charge neutrality.
Similarly, for $d=4$, we obtain the global conformal block expansion of the
$G$-function. We notice that the odd spin OPE coefficients still vanish, the
twist-two family is no longer missing in the OPE. We also see that not all OPE
coefficient squares are positive definite, owing to the non-unitary nature of
the theory. The coefficients are tabulated in Table 2 .
$(n,l)$ | $(\lambda^{(n,l)})^{2}$
---|---
$(0,0)$ | 1
$(0,2)$ | $\dfrac{8\alpha^{4}}{16\alpha^{2}+1}$
$(1,0)$ | $\dfrac{8\alpha^{4}}{1-16\alpha^{2}}$
$(0,4)$ | $\dfrac{2\alpha^{4}(4\alpha^{2}+1)^{2}}{(16\alpha^{2}+3)(16\alpha^{2}+5)}$
$(2,2)$ | $-\dfrac{2\alpha^{4}(4\alpha^{2}+1)^{2}}{3+64(4\alpha^{4}+\alpha^{2})}$
$(2,0)$ | $\dfrac{64\alpha^{8}}{256\alpha^{4}-1}$
$(0,6)$ | $\dfrac{4\alpha^{4}(8\alpha^{4}+6\alpha^{2}+1)^{2}}{3(16\alpha^{2}+5)(16\alpha^{2}+7)(16\alpha^{2}+9)}$
$(1,4)$ | $-\dfrac{4\alpha^{4}(8\alpha^{4}+6\alpha^{2}+1)^{2}}{3(16\alpha^{2}+3)(16\alpha^{2}+5)(16\alpha^{2}+7)}$
$(2,2)$ | $\dfrac{16\alpha^{8}(4\alpha^{2}+1)^{2}}{(16\alpha^{2}-1)(16\alpha^{2}+3)(16\alpha^{2}+5)}$
$(3,0)$ | $-\dfrac{16\alpha^{8}(4\alpha^{2}+1)^{2}}{(16\alpha^{2}+1)^{2}(16\alpha^{2}+3)}$
Table 2: All non-zero OPE coefficients up to degree 6 in $d=4$ for the coulomb
gas correlator $\langle V_{\alpha}V_{\alpha}V_{\alpha}V_{\alpha}\rangle$ in
the $s$-channel with $\alpha\equiv Q/4$ to satsify charge neutrality.
|
# Yang-Gaudin model: A paradigm of many-body physics
Xi-Wen Guan<EMAIL_ADDRESS>State Key Laboratory of Magnetic Resonance
and Atomic and Molecular Physics, Wuhan Institute of Physics and Mathematics,
APM, Chinese Academy of Sciences, Wuhan 430071, China Department of
Theoretical Physics, Research School of Physics and Engineering, Australian
National University, Canberra ACT 0200, Australia Hai-Qing Lin
<EMAIL_ADDRESS>Beijing Computational Science Research Center, Beijing
100193, China Department of Physics, Beijing Normal University, Beijing,
100875, China
###### Abstract
Using Bethe’s hypothesis, C N Yang exactly solved the one-dimensional (1D)
delta-function interacting spin-1/2 Fermi gas with an arbitrary spin-imbalance
in 1967. At that time, using a different method, M Gaudin solved the problem
of interacting fermions in a spin-balanced case. Later, the 1D delta-function
interacting fermion problem was named as the Yang-Gaudin model. It has been in
general agreed that a key discovery of C N Yang’s work was the cubic matrix
equation for the solvability conditions. This equation was later independently
found by R J Baxter for commuting transfer matrices of 2D exactly solvable
vertex models. The equation has since been referred to Yang-Baxter equation,
being the master equation to integrability. The Yang-Baxter equation has been
used to solve a wide range of 1D many-body problems in physics, such as 1D
Hubbard model, $SU(N)$ Fermi gases, Kondo impurity problem and strongly
correlated electronic systems etc. In this paper, we will briefly discuss
recent developments of the Yang-Gaudin model on several breakthroughs of many-
body phenomena, ranging from the universal thermodynamics to the Luttigner
liquid, the spin charge separation, the Fulde-Ferrell-Larkin-Ovchinnikov
(FFLO)-like pairing state and the quantum criticality. These developments
demonstrate that the Yang-Gaudin model has laid out a profound legacy of the
Yang-Baxter equation.
###### pacs:
03.75.Ss, 03.75.Hh, 02.30.IK, 05.30.Fk
The Bethe ansatz (BA), i.e. a particular form of wavefunction, was first
introduced in 1931 by Hans Bethe Bethe:1931 as a way to obtain the
eigenspectrum of the one-dimensional (1D) spin-1/2 Heisenberg chain. In
Bethe’s method, $N!$ plane waves are $N$-fold products of individual
exponential phase factors $e^{\mathrm{i}k_{i}x_{j}}$, where the $N$ distinct
wave numbers, $\left\\{k_{i}\right\\}$, are permuted among the $N$ distinct
coordinates, $x_{j}$. Each of the $N!$ plane waves has an amplitude
coefficient in each of regions, i.e. superposition of all possible plane waves
of $N$ particles. However, only more than few decades later, physicists, L
Hulthén Hulthen , R. Orbach Orbach , L R Walker Walker , R B Griffith
Griffith , J des Cloizeaux and Pearson Cloizeau , and few others, studied the
Bethe’s method and the Heisenberg chain in terms of Bethe’s solution. In the
mid-60’s, C N Yang and C P Yang YY-1 ; YY-2 ; YY-3 presented a systematic
study of the BA equations for the Heisenberg spin chain throughout the full
range of anisotropic parameter $\Delta$ with a presence of magnetic field.
While they coined Bethe’s method as Bethe’s hypothesis.
In 60’s, Bethe’s hypothesis had been proved to be invaluable for the field of
exactly solvable models in statistical mechanics. In 1963, using Bethe’s
hypothesis, E Lieb and W Liniger Lieb-Liniger solved the 1D Bose gas with a
delta-function interaction. In 1967, C N Yang Yang solved the 1D delta-
interacting Fermi gas with a discovery of the necessary condition for the
Bethe ansatz solvability, which is now known as the Yang-Baxter equation, i.e.
the factorization condition–the scattering matrix of a quantum many-body
system can be factorized into a product of many two-body scattering matrices.
In the same year, M Gaudin also rigorously derived the BA equations for the
spin-$1/2$ Fermi gas with a spin balance Gaudin . In 1972, R J Baxter Baxter
independently showed that such a factorization relation also occurred as the
conditions for commuting transfer matrices in 2D vertex models in statistical
mechanics. The Bethe ansatz approach has since found success in the realm of
condensed matter physics, such as the 1D Hubbard model Lieb-Wu , SU(N)
interacting Fermi gases Sutherland:1968 , Kondo impurity problems Andrei:1983
, BCS pairing models Dukelsky:2004 , strongly correlated electron systems
1D-Hubbard ; Korepin ; Sutherland-book ; Takahashi-b ; Wang-book and spin
chain and ladder compounds Batchelor:2007 , quantum gases of cold atoms
Cazalilla:2011 ; yangyou ; Guan:2013 ; Batchelor:2016 ; Mistakidis:2022 , and
also many other problems in physics and mathematics.
The next significant progress was made by C N Yang and C P Yang Yang-Yang in
1969 on the finite temperature problem for the Lieb-Liniger Bose gas. They
showed that the thermodynamics of the model can be determined from the
minimisation conditions of the Gibbs free energy subject to the Bethe ansatz
equations. Later Takahashi showed Takahashi:1971 ; Takahashi:1972 that the
Yang-Yang method was an elegant way to analytically obtain thermodynamics of
integrable models, e.g. 1D Heisenberg spin chains, Hubbard model, etc. Recent
developments on the exactly solvable models in ultracold atoms have shown
Guan:2013 ; Zhao ; Guan2007 ; Guan:2013PRL ; Yang:2017 ; He:2020 ;
PhysRevB.101.035149 that Yang-Yang method provides an elegant way to study
universal thermodynamics, Luttinger liquid, spin charge separation, transport
properties and critical phenomena for a wide range of low-dimensional quantum
many-body systems. In this short review, we shall discuss how the exact
solution of the Yang-Gaudin model provides a rigorous understanding of such
fundamental many-body physics in terms of the legacy of Yang-Baxter equation
and Yang-Yang thermodynamics.
In this paper, we will present Bethe ansatz solution of the Yang-Guadin model
in Section I and will discuss the ground state, spin charge separation,
universal thermodynamics and quantum criticality for the model with a
repulsive interaction in Section II. In section III, we will briefly review
novel quantum phases of pairing and depairing, the Fulde-Ferrell-Larkin-
Ovchinnikov (FFLO) pairing correlation, multicomponent Luttinger liquids and
dimensionless ratios for the Yang-Gaudin model with an attractive interaction.
A brief conclusion and a short discussion on the future study of the Yang-
Gaudin model will be presented in Section IV.
## I I. The Yang-Gaudin Model
The Hamiltonian for the 1D contact interacting fermion problem Yang ; Gaudin
$H=-\frac{\hbar^{2}}{2m}\sum_{i=1}^{N}\frac{\partial^{2}}{\partial
x_{i}^{2}}+g_{1D}\sum_{1\leq i<j\leq N}\delta(x_{i}-x_{j})$ (1)
describes $N$ fermions of the same mass $m$ with two internal spin states
confined to a 1D system of length $L$ interacting via a $\delta$-function
potential. In this Hamiltonian, we denote the numbers of fermions in the two
hyperfine levels $|\uparrow\rangle$ and $|\downarrow\rangle$ as $N_{\uparrow}$
and $N_{\downarrow}$, respectively. While we denoted the total number of
fermions and the magnetization as $N=N_{\uparrow}+N_{\downarrow}$ and
$M^{z}=(N_{\uparrow}-N_{\downarrow})/2$. The coupling constant $g_{1D}$ can be
expressed in terms of the interaction strength $g_{1D}=\hbar^{2}c/m$ with
$c=-2/a_{1D}$, where $a_{1D}$ is the effective 1D scattering length
Olshanii_PRL_1998 . In the following discussion, we let $2m=\hbar=1$ for our
convenience. We define a dimensionless interaction strength $\gamma=c/n$ for
our later physical analysis. Here the linear density is defined by $n=N/L$.
For a repulsive interaction, $c>0$ and for an attractive interaction, $c<0$.
Using Bethe’s hypothesis, C N Yang solved the model (1) with the following
many-body wave function
$\psi=\sum_{P}A_{\sigma_{1}\ldots\sigma_{N}}(P|Q)\exp\textrm{i}(k_{P1}x_{Q1}+\ldots+k_{PN}x_{QN})$
(2)
for the domain $0<x_{Q1}<x_{Q2}<\ldots<x_{QN}<L$. Where $\\{k_{i}\\}$ denote a
set of unequal wave numbers and $\sigma_{i}$ with $i=1,\ldots,N$ indicate the
spin coordinates. Both $P$ and $Q$ are permutations of indices
$\\{1,2,\ldots,N\\}$, i.e. $P=\left\\{P_{1},\ldots,P_{N}\right\\}$ and
$Q=\left\\{Q_{1},\ldots,Q_{N}\right\\}$. The sum runs over all $N!$
permutations $P$ and the coefficients of the exponentials are column vectors
with each of the $N!$ components representing a permutation $Q$. To determine
the wave function associated with the irreducible representations of the
permutation group $S_{N}$ and the irreducible representation of the Young
tableau for different up- and down-spin fermions, one first needs to consider
the boundary conditions of the wave function, i.e. the continuity of the wave
function and discontiuity of its derivative. It follows that the two adjacent
coefficients must meet the two-body scattering relation
$A_{\sigma_{1}\ldots\sigma_{N}}(P_{i}P_{j}|Q_{i}Q_{j})=Y_{ij}(k_{Pj}-k_{Pi})A_{\sigma^{\prime}_{1}\ldots\sigma^{\prime}_{N}}(P_{j}P_{i}|Q_{i}Q_{j}),$
(3)
where the two-body scattering matrix $Y_{ij}$ is given by
$Y_{ij}(k_{Pj}-k_{Pi})=\left[\frac{\textrm{i}(k_{Pj}-k_{Pi})T_{ij}+cI}{\textrm{i}(k_{Pj}-k_{Pi})-c}\right]^{\sigma^{\prime}_{1}\ldots\sigma^{\prime}_{N}}_{\sigma_{1}\ldots\sigma_{N}}$
(4)
with the operator $T_{ij}=-P_{ij}$, here $P_{ij}$ is the permutation operator.
The key discovery of C N Yang’s work is that the two-body scattering matrix
acting on three linear tensor spaces $V_{1}\otimes V_{2}\otimes V_{3}$
$Y_{ij}(u)=\frac{\textrm{i}uT_{ij}+cI}{\textrm{i}u-c}$ (5)
satisfies the following cubic equation
$\displaystyle Y_{12}(k_{2}-k_{1})Y_{23}(k_{3}-k_{1})Y_{12}(k_{3}-k_{2})$
$\displaystyle=Y_{23}(k_{3}-k_{2})Y_{12}(k_{3}-k_{1})Y_{23}(k_{2}-k_{1}),$ (6)
which has been known as the Yang-Baxter equation. This equation guarantees no
diffraction in the three-particle scattering process, i.e.
$(k^{\prime}_{1},k^{\prime}_{2},k^{\prime}_{3})=(k_{1},k_{2},k_{3})$. The
Yang-Baxter equation in a graphical representation presents a kind of
topological invariance for interchanging the three operators according to the
two paths, see Fig. 1. This equation was immediately seen as a necessary
condition to the quantum integrability.
$\textstyle{Y_{213}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{231}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{123}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{321}}$$\textstyle{Y_{132}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{Y_{312}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$
Figure 1: The graphical representation of the Yang-Baxter equation Eq. (6).
In general, the scattering matrix $Y_{ab}(u)$ satisfies the following
relations
$\displaystyle Y_{ab}(u)Y_{cd}(v)$ $\displaystyle=$ $\displaystyle
Y_{cd}(v)Y_{ab}(u),$ $\displaystyle Y_{ab}(u)Y_{bc}(u+v)Y_{ab}(v)$
$\displaystyle=$ $\displaystyle Y_{bc}(v)Y_{ab}(u+v)Y_{bc}(u),$ $\displaystyle
Y_{ab}(u)Y_{ba}(-u)$ $\displaystyle=$ $\displaystyle 1.$
Let us further introduce the operator $R_{ij}=T_{ij}Y_{ij}$ which acts on the
space $\prod_{n=1}^{N}\otimes V_{n}$. Then we define the following monodromy
matrix:
$\mathcal{T}_{N}(u)=L_{N}(k_{N}-u)\ldots L_{2}(k_{2}-u)L_{1}(k_{1}-u)$ (7)
where $L_{i}(k_{i}-u)\equiv R_{i,a}(k_{i}-u)$. By taking a trance of the
transfer matrix over the auxiliary space $a$, we have
$\tau(u)=\textrm{Tr}_{a}(\mathcal{T}_{N}(u))$, then we have
$\tau(u)|_{u=k_{i}}=\mathfrak{R}_{i}(k_{i}),$ (8)
where
$\displaystyle\mathfrak{R}_{i}(k_{i})$ $\displaystyle=$ $\displaystyle
R_{i+1,i}(k_{i+1}-k_{i})\ldots R_{N,i}(k_{N}-k_{i})R_{1,i}(k_{1}-k_{i})\ldots$
(9) $\displaystyle\times R_{i-1,i}(k_{i-1}-k_{i}).$
The above notations were borrowed from the quantum inverse scattering method
which had been developed in the 1980’s by Faddeev and others Korepin .
Solving the eigenvalue problem of $N$ interacting particles (1) in a periodic
box of length $L$, one needs to apply the periodic boundary condition
$\psi(x_{1},\ldots,x_{i},\ldots,x_{N})=\psi(x_{1},\ldots,x_{i}+L,\ldots,x_{N}).$
(10)
This is equivalent to solving the following eigenvalue problem
$\mathfrak{R}_{i}(k_{i})A_{E}(P|Q)=\exp(\textrm{i}k_{i}L)A_{E}(P|Q)$ (11)
with the transfer matrix $\mathfrak{R}_{i}(k_{i})$ given by (9). By some
algebraic manipulations, the eigenvalue of the transfer matrix
$\tau(u)=\textrm{Tr}_{a}(\mathcal{T}_{N}(u))$ can be straightforwardly
obtained. Then one can obtain the following C N Yang’s Bethe ansatz equations
(BAE) Yang for the Fermi gas
$\displaystyle{\rm
e}^{ik_{j}L}=\prod_{\alpha=1}^{M}\frac{k_{j}-\lambda_{\alpha}+ic/2}{k_{j}-\lambda_{\alpha}-ic/2},$
(12)
$\displaystyle\prod_{j=1}^{N}\frac{\lambda_{\alpha}-k_{j}+ic/2}{\lambda_{\alpha}-k_{j}-ic/2}=-\prod_{\beta=1}^{M}\frac{\lambda_{\alpha}-\lambda_{\beta}+ic}{\lambda_{\alpha}-\lambda_{\beta}-ic},$
(13)
with $j=1,2,\cdots,N$ and $\alpha=1,2,\cdots,M$. Here $M$ is the number of
atoms with down-spins. The energy eigenspectrum is given in terms of the
quasimomenta $\left\\{k_{i}\right\\}$ of the fermions via
$E=\frac{\hbar^{2}}{2m}\sum_{j=1}^{N}k_{j}^{2}$. All quasimomenta
$\left\\{k_{i}\right\\}$ are distinct and uniquely determine the wave function
of model Eq.(2).
The fundamental physics of the model (1) can be in principle obtained by
solving the transcendental Bethe ansatz equations (12) and (13). However, the
difficulty with this interacting fermion problem lies in finding all the
solutions to these Bethe ansatz equations. In the following discussion, we
will briefly review recent breakthroughs in the study of the Bethe ansatz
solutions of the Yang-Gaudin model (1).
## II II. Spin Charge Separation and Spin Incoherent Liquid for the Repulsive
Fermi Gas
Finding the solution of the Bethe ansatz equations (12) and (13) is
cumbersome. In the thermodynamic limit, i.e., $L,N\to\infty$, $N/L$ is finite,
the above Bethe ansatz equations can be written as the generalized Fredholm
equations
$\displaystyle{\rho}(k)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}+\int_{-B_{2}}^{B_{2}}a_{1}(k-\lambda){\sigma_{1}}(\lambda)d\lambda,$
(14) $\displaystyle{\sigma_{1}}(\lambda)$ $\displaystyle=$
$\displaystyle\int_{-B_{1}}^{B_{1}}a_{1}(\lambda-k){\rho}(k)dk$ (15)
$\displaystyle-\int_{-B_{2}}^{B_{2}}a_{2}(\lambda-\lambda^{\prime}){\sigma_{1}}(\lambda^{\prime})d\lambda^{\prime}.$
The associated integration boundaries $B_{1}$, $B_{2}$ are determined by the
relations
$\displaystyle n:$ $\displaystyle\equiv$ $\displaystyle
N/L=\int_{-B_{1}}^{B_{1}}{\rho}(k)dk,$ $\displaystyle n_{\downarrow}:$
$\displaystyle\equiv$ $\displaystyle
N_{\downarrow}/L=\int_{-B_{2}}^{B_{2}}{\sigma_{1}}(k)dk,$ (16)
where $n$ denotes the linear density while $n_{\downarrow}$ is the density of
spin-down Fermions. In the above equations we introduced the quasimomentum
distribution function $\rho(k)$ and distribution function of the spin rapidity
$\sigma_{1}(\lambda)$ for the ground state. The boundary $B_{1}$ characterizes
the Fermi point in the quasimomentum space whereas the boundary $B_{2}$
characterizes the spin rapidity distribution interval with respect to the
polarization. They can be obtained by solving the equations given in (16). In
the above equations, we denoted the kernel function by
$a_{\ell}(x)=\frac{1}{2\pi}\frac{\ell c}{(\ell c/2)^{2}+x^{2}}.$ (17)
The ground state energy per unit length is given by
$E=\int_{-B_{1}}^{B_{1}}k^{2}{\rho}(k)dk.$ (18)
The ground state energy for weak and strong interactions can be calculated
directly from the integral forms of the Bethe ansatz equations (14) and (15),
namely
$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{1}{12}n^{3}\pi^{2}+\frac{1}{2}n^{2}c+O(c^{2}),\,\,{\rm
for}\,c\ll 1,$ (19) $\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{n^{3}\pi^{2}}{3}\left[1-\frac{4\ln 2}{\gamma}+\frac{12(\ln
2)^{2}}{\gamma^{2}}-\frac{32(\ln 2)^{3}}{\gamma^{3}}\right.$ (20)
$\displaystyle\left.+\frac{8\pi^{2}\zeta(3)}{5\gamma^{3}}\right]+O(c^{-4})\,\,{\rm
for}\,c\gg 1.$
This is a good approximation for the spin-balanced Fermi gas with weakly and
strongly repulsive interactions. More detailed study of the solutions of the
BA equations was presented in Guan:PRA12 .
In general, for a repulsive interaction, the Bethe ansatz equations (12) and
(13) only admit real roots in the charge degree of freedom $k_{j}$, whereas in
the spin sector, the spin string states are given by
$\displaystyle\lambda_{\alpha}^{n,j}=\lambda_{\alpha}^{n}+\frac{ic}{2}(n+1-2j),\quad
j=1,2,\cdots,n,$ (21)
which are called the length-$n$ spin strings. Such a spin structure comprises
a rich magnetism like what the 1D Heisenberg spin chain has. Accordingly, the
Bethe ansatz equations (12) and (13) with string hypothesis (21) reduce to the
following two sets of BA equations associated with the quantum number
$\\{I_{j}\\}$ and $\\{J_{\alpha}^{n}\\}$
$\displaystyle k_{j}L=2\pi
I_{j}-\sum_{n=1}^{\infty}\sum_{\alpha=1}^{M_{n}}\theta\left(\frac{2(k_{j}-\lambda_{\alpha}^{n})}{nc}\right),$
(22)
$\displaystyle\sum_{j=1}^{N}\theta\left(\frac{2(k_{j}-\lambda_{\alpha}^{n})}{nc}\right)=2\pi
J_{\alpha}^{n}$
$\displaystyle+\sum_{m=1}^{\infty}\sum_{\beta=1}^{M_{m}}\Theta_{mn}\left(\frac{2(\lambda_{\alpha}^{n}-\lambda_{\beta}^{m})}{c}\right),\
$ (23)
where $j=1,2,\cdots,N$, $\alpha=1,2,\cdots,M_{n},\;n\geq 1$ and $M_{n}$ is the
number of length-$n$ string, $\theta(x)=2\tan^{-1}(x)$, and the function
$\Theta_{mn}(x)$ is defined by
$\displaystyle\Theta_{mn}(x)=\left\\{\begin{array}[]{rcl}&&\theta\left(\frac{x}{|n-m|}\right)+2\theta\left(\frac{x}{|n-m|+2}\right)+\cdots\\\
&&+2\theta\left(\frac{x}{n+m-2}\right)+\theta\left(\frac{x}{m+n}\right),\,\,\text{for}\,\,n\neq
m,\\\
&&2\theta\left(\frac{x}{2}\right)+2\theta\left(\frac{x}{4}\right)+\cdots\\\
&&+2\theta\left(\frac{x}{2n-2}\right)+\theta\left(\frac{x}{2n}\right),\quad\text{for}\quad
n=m.\end{array}\right.$ (28)
The quantum number $I_{j}$ for charge degree of freedom takes distinct
integers (or half-odd integers) for even (odd) $\sum_{\alpha}M_{\alpha}$,
explicitly
$I_{j}\in\sum_{n=1}^{\infty}\frac{M_{n}}{2}+\mathbb{Z}.$ (29)
The spin quantum number $J_{\alpha}^{n}$ are distinct integers (half-odd
integers) for odd (even) $N-M_{m}$, which satisfy
$\displaystyle J_{\alpha}^{n}\in\frac{N-M_{n}}{2}+\frac{1}{2}+\mathbb{Z},$
$\displaystyle|J_{\alpha}^{n}|\leq
J_{+}^{n}=\frac{N}{2}-\sum_{m=1}^{n}mM_{m}-n\sum_{m=n+1}^{\infty}M_{m}+\frac{M_{n}}{2}-\frac{1}{2},$
$\displaystyle
J_{\alpha}^{n}=-J_{+}^{n},-J_{+}^{n}+1,-J_{+}^{n}+2,\cdots,J_{+}^{n}-1,J_{+}^{n}.$
The total momentum of the system is given by
$K=\sum_{j=1}^{N}k_{j}=\frac{2\pi}{L}\left(\sum_{j=1}^{N}I_{j}+\sum_{\alpha=1}^{M_{n}}\sum_{n=1}^{\infty}J_{\alpha}^{n}\right).$
(30)
Figure 2: Exact low energy excitation spectra for charge (yellow green) and
spin (dark brown) at $\gamma=5.03$ with the Fermi surface $k_{F}=n\pi$,
density $n=N/L=3\times 10^{6}$. The yellow green spectrum shows the particle-
hole continuum excitation. The dark brown spectrum shows the notable two-
spinon excitation spectra. In long wave limit, i.e. $\Delta K\ll 1$, the spin
and charge spectra show two independent linear dispersion with velocities
$v_{s}$ and $v_{c}$, respectively. From He et. al. He:2020 .
It is significantly found He:2020 from the above Bethe ansatz equations (12)
and (13) that the excitations in charge sector display a particle-hole
continuum spectrum, see Fig. 2. It shows a linear dispersion structure for
arbitrary strongly interacting fermions in a long wave limit
$\omega(q)=v_{c}|q|\pm\frac{\hbar q^{2}}{2m^{*}}+\cdots,$ (31)
where the charge velocity and the effective mass are given by the following
expressions
$\displaystyle v_{c}$ $\displaystyle=$
$\displaystyle\frac{\varepsilon_{c}^{\prime}(k_{0})}{2\pi\rho_{c}(k_{0})},$
(32) $\displaystyle\frac{1}{2m^{*}}$ $\displaystyle=$
$\displaystyle\frac{\varepsilon_{c}^{\prime\prime}(k_{0})}{2(2\pi\rho_{c}(k_{0})^{2}}-\frac{\pi\rho_{c}^{\prime}(k_{0})\varepsilon_{c}^{\prime}(k_{0})}{(2\pi\rho_{c}(k_{0}))^{3}},$
(33)
respectively. For a strong coupling limit, the charge velocity and the
effective mass are given by He:2020
$\displaystyle v_{c}\approx 2\pi n_{c}\left(1-\frac{4\ln
2}{\gamma}\right),\quad m^{*}=m\left(1+\frac{4\ln 2}{\gamma}\right).$ (34)
The low-energy spin flipping excitation in the spin sector is also displayed
in Fig. 2. It shows a typical two-spinon excitation spectrum in spin
elementary excitations for the Yang-Gaudin model. The total excited momentum
is given by
$\Delta K_{\rm spinon}=n\pi-2\pi\sum_{j=1}^{2}\int_{0}^{\lambda_{j}^{\rm
h}}\rho_{s}^{0}(\lambda){\rm d}\lambda,$ (35)
while the energy of two-spinon excitation is given by
$\displaystyle\Delta E_{\rm
spinon}=-\sum_{j=1}^{2}\phi_{s}^{0}(\lambda_{j}^{\rm h}),$ (36)
that present a microscopic origin of the two deconfined spinons with
spin-$1/2$, showing a fractional excitation. This result confirms an
antiferromagnetic ordering in the Fermi gas with internal degrees of freedom.
However, when the interaction increases, the spin excitation band becomes
lower, and even it vanishes in the limit $\gamma\to\infty$.
Moreover, Fig. 2 remarkably displays the origin of the separated excitations
in spin and charge sectors. In fact, in one dimension such low-lying
excitations with different sound velocities form two collective motions of
bosons, i.e. the so-called spin-charge separation – fermions dissolve into
spinons and holons. The spin-charge separation phenomenon is the hallmark of
one-dimensional physics and has not been unambiguously confirmed by
experiments, either in solids Kim:1996 ; auslaender2005spin ; Kim:2006 ;
Jompol:2009 or ultracold atoms Hulet:2018 ; Vijayan:2020 . For theoretical
understanding such unique 1D many-body phenomenon, it acquires low temperature
thermodynamics and dynamical correlation functions for such excitations in
spin and charge degrees of freedom. Such novel physics spin-charge separation
has been recently observed with ultracold atoms in Senaratne:2021
On the other hand, the finite temperature problem for the Lieb-Liniger Bose
gas was solved by C. N. Yang and C. P. Yang in 1969 Yang-Yang . Extension to
the Yang-Gaudin model in terms of the Bethe ansatz equations (22) and (23) was
done some time ago by Lai Lai:1971 ; Lai:1973 and M Takahashi Takahashi:1971
, namely, the so-called thermodynamic Bethe ansatz (TBA) equations for the
Yang-Gaudin model are given by
$\displaystyle\varepsilon(k)$ $\displaystyle=$ $\displaystyle
k^{2}-\mu-\frac{H}{2}-T\sum_{n=1}^{\infty}a_{n}*{\rm ln}[1+{\rm
e}^{-\phi_{n}(\lambda)/T}],\quad$ (37) $\displaystyle\phi_{n}(\lambda)$
$\displaystyle=$ $\displaystyle nH-Ta_{n}*\ln[1+{\rm e}^{-\varepsilon(k)/T}]$
(38) $\displaystyle+$ $\displaystyle T\sum_{m=1}^{\infty}T_{mn}*{\rm
ln}[1+{\rm e}^{-\phi_{m}(\lambda)/T}]$
with $n=1,\ldots,\infty$. Here $*$ denotes the convolution, $\varepsilon(k)$
and $\phi_{n}(\lambda)$ are the dressed energies for the charge and the
length-$n$ spin strings, respectively, with $k$’s and $\lambda$’s being the
rapidities; and the function $T_{mn}(x)=d\Theta_{mn}(x)/dx$ is given in Refs.
Takahashi:1971 . The pressure is given by
$p=\frac{T}{2\pi}\int_{-\infty}^{\infty}{\rm ln}[1+{\rm
e}^{-\varepsilon(k)/T}]{\rm d}k,$ (39)
from which all the thermal and magnetic quantities can be derived according to
the standard statistical relations.
Figure 3: (color online) Contour plot of the dimensionless specific heat
$\tilde{c}_{V}=c_{v}/T$, showing the phase diagram in the
$\tilde{T}$-$\tilde{H}$ plane. Here dimensionless chemical potentials
$\tilde{\mu}=2.5$, $\tilde{H}_{c}=2.9145$. The black dashed lines denote the
peak positions of specific heat, whereas the white-dot-dashed line shows the
boundary of the linear $T$ dependence of specific heat. $COR1$ denotes the
crossover regions between QC and the TLL, giving a spin incoherent liquid.
From He et. al. He:2020 .
At low temperatures, the TBA equations (37) and (38) are extremely hard to be
solved either numerically or analytically. However, one can solve these
coupled integral equations for certain physical regimes, for examples
$k_{B}T\ll\mu,H$ or $c^{2}\gg E_{F}$, etc. For a fixed chemical potential, the
contour plot of the specific heat in temperature-magnetic-field plane
naturally reads off different critical regions near the quantum phase
transition from the spin charge separated Tomonaga-Luttinger liquid (TLL)
phase to a fully-polarized phase (FP), see Fig. 3. The quantum critical region
fans out from the critical point, forming a critical cone.
For $T\ll E_{F}$, we can safely neglect the contributions from the high
strings and just retain the leading length-1 string in the TBA equations (37)
and (38). Throughout the TLL phase with $H<H_{c}$, where $H_{c}$ is the
critical field for a fixed chemical potential
$H_{c}=\left(\frac{c^{2}}{2\pi}+2\pi n^{2}\right)\tan^{-1}\left(\frac{2\pi
n}{c}\right)-cn.$ (40)
The pressure can, in general, be given by
$p-p_{0}=\frac{\pi T^{2}}{6}\left(\frac{1}{v_{c}}+\frac{1}{v_{s}}\right),$
(41)
where $p_{0}=\int_{-k_{0}}^{k_{0}}\varepsilon(k)dk$ is the pressure at $T=0$
and the charge and spin velocities are given by
$v_{c}=\frac{t_{c}}{2\pi\rho(k_{0})}\,,\;\;\;\;v_{s}=\frac{t_{s}}{2\pi\sigma_{1}(\lambda_{0})}\,,$
(42)
respectively, with $\rho$ and $\sigma_{1}$ being the distribution functions at
Fermi points $k_{0}$ and $\lambda_{0}$ for the charge and the spin sectors,
respectively. While $t_{c}=\frac{{\rm d}\varepsilon(k)}{{\rm
d}k}\bigg{|}_{k=k_{0}}$ and $t_{s}=\frac{{\rm d}\phi_{1}(\lambda)}{{\rm
d}\lambda}\bigg{|}_{\lambda=\lambda_{0}},$ are the respective linear slopes of
the dispersion at the Fermi points. Although the proof of the form (41) is
rather cumbersome, it gives a simple universal low temperature form of spin-
charge separation theory Affleck1986 . The velocities can be numerically
obtained for arbitrary interaction strength. For the strongly interacting
regime, the charge and spin excitation velocities are given by
$v_{c}\approx 2\pi n\left(1-\frac{4\ln 2}{\gamma}\right),\qquad
v_{s}\approx\frac{2\pi^{3}n}{3\gamma}\left(1-\frac{6\ln 2}{\gamma}\right),$
respectively. For the external field approaching the saturation field $H_{c}$,
the charge and spin velocities can be derived from the relations (42). The
leading terms in the velocities are then found to be
$\displaystyle v_{c}$ $\displaystyle=$ $\displaystyle 2\pi
n\left(1-\frac{12}{\pi\gamma}\sqrt{1-\frac{H}{H_{c}}}\right),\,\,v_{s}=\frac{H_{c}}{n}\sqrt{1-\frac{H}{H_{c}}}.$
In the TLL regime, the dispersion relations for Yang-Gaudin model are
approximately linear. Conformal field theory predicts that the energy per unit
length has a universal finite size scaling form $E=E_{0}+\Delta/L^{2}$ where
$E_{0}$ is the ground state energy per unit length for the infinite system and
$\Delta$ is a universal term. In this scenario, Cardy Cardy1986 showed that
the two-point correlation function between primary fields can be directly
derived from conformal mapping using transfer matrix techniques and expressed
the conformal dimensions in terms of finite-size corrections to the energy
spectrum. For example, at zero temperature and zero magnetic field, by using
conformal field theory Belavin1984 ; Blote1986 , the asymptotic of single
particle correlation function can be given explicitly
$\displaystyle
G_{\uparrow}(x,t)\sim\langle\psi_{\uparrow}^{\dagger}(x,t)\psi_{\uparrow}(0,0)\rangle$
$\displaystyle\approx\frac{A_{\uparrow,1}e^{-\mathrm{i}k_{F\uparrow}x}}{(x-\mathrm{i}v_{c}t)^{2\Delta_{c}^{+}}(x+\mathrm{i}v_{c}t)^{2\Delta_{c}^{-}}(x-\mathrm{i}v_{s}t)^{2\Delta_{s}^{+}}}+h.c.$
(43)
with the conformal dimensions
$\displaystyle 2\Delta_{c}^{+}$ $\displaystyle=$
$\displaystyle\frac{9}{16}-\frac{3\ln
2}{4\gamma}+\frac{3}{2\pi^{2}}\left(\frac{H}{H_{c}}\right)+\frac{3}{2\gamma}\left(\frac{H}{H_{c}}\right)$
(44) $\displaystyle-\frac{\ln 2}{\pi^{2}\gamma}\left(\frac{H}{H_{c}}\right),$
$\displaystyle 2\Delta_{c}^{-}$ $\displaystyle=$
$\displaystyle\frac{1}{16}-\frac{3\ln
2}{4\gamma}-\frac{1}{2\pi^{2}}\left(\frac{H}{H_{c}}\right)+\frac{1}{2\gamma}\left(\frac{H}{H_{c}}\right)$
(45) $\displaystyle+\frac{3\ln 2}{\pi^{2}\gamma}\left(\frac{H}{H_{c}}\right),$
$\displaystyle 2\Delta_{s}^{+}$ $\displaystyle=$
$\displaystyle\frac{1}{2}-\frac{2}{\pi^{2}}\left(\frac{H}{H_{c}}\right)-\frac{1}{\gamma}\left(\frac{H}{H_{c}}\right)+\frac{4\ln
2}{\pi^{2}\gamma}\left(\frac{H}{H_{c}}\right).$ (46)
We see clearly that the correlation function decays as some power of distance
governed by the critical exponent.
On the other hand, universal scaling behaviour can be derived in the vicinity
of the critical point at low temperatures. Near the critical point, the spin
dressed energy $\phi_{1}(\lambda)$ in (38) only has a small negative part,
which makes a major contribution to the spin dressed energy near the critical
point at low temperatures. Therefore, we can expand the integration kernel
$a_{n}(k-\lambda)$ in terms of the functions of small variables $\lambda$.
After a tedious calculation, we obtain the pressure of the Yang-Gaudin model
(1) near the phase transition from the TLL phase to FP phase
$\displaystyle p$ $\displaystyle\approx$ $\displaystyle
p_{0}-\frac{\arctan\left(\frac{2}{c}k_{0}\right)T^{3/2}}{\pi^{3/2}\sqrt{(a+c_{2})}}\operatorname{Li}_{\frac{3}{2}}\left(-{\rm
e}^{\frac{s_{0}\Delta H-c_{1}}{T}}\right)$ (47)
$\displaystyle+\frac{T^{5/2}}{4\pi^{3/2}(a+c_{2})^{3/2}}\frac{ck_{0}}{\left(c^{2}/4+k_{T}^{2}\right)^{2}}\operatorname{Li}_{\frac{5}{2}}\left(-{\rm
e}^{\frac{s_{0}\Delta H-c_{1}}{T}}\right),$
where the regular part $p_{0}$ of the pressure is given by
$\displaystyle p_{0}=\frac{\pi
T^{2}}{6v_{c}}+\frac{2}{3\pi}\left(\mu_{c}+\frac{H}{2}\right)^{3/2}=p_{0}^{\rm
Liquid}+p_{0}^{\rm BG}.$ (48)
The above expression $p_{0}^{\rm
BG}=\frac{2}{3\pi}\left(\mu_{c}+\frac{H}{2}\right)^{3/2}$ can be regarded as
the background part of charge, whereas $p_{0}^{\rm Liquid}=\frac{\pi
T^{2}}{6v_{c}}$ denotes the Luttinger liquid contribution from charge degrees
of freedom. However, the Luttinger liquid in the spin sector dissolves into
the free fermion quantum criticality in the quantum critical (QC) region, see
Fig. 3. It is worth noting that in the QC region the pressure (47) presents a
universal scaling form of the equation of states
$\displaystyle p=p_{0}^{\rm Liquid}+p_{0}^{\rm
BG}+T^{\frac{1}{z}+1}\mathcal{G}\left(\frac{s_{0}\Delta H}{T^{1/\nu
z}}\right),$ (49)
from which one reads off the dynamical critical exponent $z=2$ and correlation
length critical exponent $\nu=1/2$. Consequently, the scaling functions of all
thermodynamic quantities can be derived based on this exact expression of the
equation of states. The two crossover temperatures of the QC region are given
by $T_{l}^{*}=\alpha_{1}|H-H_{c}|^{\nu z}$ and
$T_{r}^{*}=\alpha_{2}|H-H_{c}|$, here $\alpha_{1,2}=s_{0}/y_{1,2}$ with
$y_{1}=-1.5629$, $y_{2}=3.6205$ are numerical constants. In the QC region, all
thermodynamic quantities can be cast into universal scaling forms.
In Fig. 3, we also identify a crossover region COR1 in the region $E_{\rm
spin}\ll k_{B}T\ll E_{F}$. Here $E_{\rm spin}$ and $E_{F}$ are the energy of
spin sector and Fermi energy, respectively. In this region, the interplay
between the spin and the charge degrees of freedom leads to a large deviation
from the linear dispersion in the spin sector, leading to a large disruption
of the spin charge coherent TLLs. Consequently, the spin-spin correlation
function exponentially decays, while the charge-charge correlation still
remains power law decay with distance. Breakdown of the conformal field theory
gives rise to an asymptotic of single particle correlation function
$\displaystyle G_{\uparrow}(x,t)$ $\displaystyle\approx$
$\displaystyle\frac{A_{\uparrow,1}e^{-\mathrm{i}k_{F\uparrow}x}}{(x-\mathrm{i}v_{c}t)^{2\Delta_{c}^{+}}(x+\mathrm{i}v_{c}t)^{2\Delta_{c}^{-}}}$
(50) $\displaystyle\times\frac{(\pi T/v_{s})^{2\Delta^{+}_{s}}}{e^{\frac{2\pi
T}{v_{s}}x{\Delta_{s}^{+}}}}+h.c.$
The conformal dimensions can be given by Eqs. (44)-(46). In this regime, the
temperature is low enough in comparison with the Fermi energy, whereas it is
high enough in comparison with the spin excitation energy. Therefore the spin
velocity $v_{s}\to 0$ in this crossover region COR1, which is a reminiscence
of the spin incoherent liquid Fiete:2007 ; Cheianov:2004 .
The novel phase diagram of Fig. 3 reveals a key concept that the low-lying
excitations near the Fermi points dissolve into two separate collective
motions of charge and spin, i.e., TLLs of charge and spin. Evidence for the
spin-charge separation was reported in solid state materials. but none of
those existing experiments in this study seemed to provide a conclusive
observation of the spin-charge separated Luttinger liquids. In a recent
experiment with ultracold atoms presented a confirmative observation of this
phenomenon through the spin and charge dynamic structure factors
Senaratne:2021 .
## III III. FFLO Pairing Correlation and Universal Thermodynamics for the
Attractive Fermi Gas
For the attractive regime, i.e. $c<0$, the root patterns of the BA equations
(12) and (13) are significantly different from that of the model for $c>0$.
For an attractive interaction, the quasimomenta $\left\\{k_{i}\right\\}$ of
the fermions with different spins form two-body bound states Takahashi-a ; Gu-
Yang , i.e., $k_{i}=\lambda_{i}\pm\mathrm{i}\frac{1}{2}c$, accompanied by the
real spin parameter $\lambda_{i}$. Here $i=1,\ldots,N_{\downarrow}$. The
excess fermions have real quasimomenta $\left\\{k_{j}\right\\}$ with
$j=1,\ldots,N-2N_{\downarrow}$. Thus the Bethe ansatz equations for the ground
state are transformed into the Fredholm equations according to the densities
of the pairs $\rho_{2}(k)$ and density of single fermionic atoms
$\rho_{1}(k)$. They satisfy the following Fredholm equations Yang ;
Takahashi-a
$\displaystyle\rho_{1}(k)$ $\displaystyle=$
$\displaystyle\frac{1}{2\pi}+\int_{-Q_{2}}^{Q_{2}}a_{1}(k-\lambda)\rho_{2}(\lambda)d\lambda,$
(51) $\displaystyle\rho_{2}(\lambda)$ $\displaystyle=$
$\displaystyle\frac{2}{2\pi}+\int_{-Q_{1}}^{Q_{1}}a_{1}(\lambda-k^{\prime})\rho_{1}(k^{\prime})dk^{\prime}$
(52)
$\displaystyle+\int_{-Q_{2}}^{Q_{2}}a_{2}(\lambda-\lambda^{\prime})\rho_{2}(\lambda^{\prime})d\lambda^{\prime}$
with the integration boundaries $Q_{1}$ and $Q_{2}$, which are the Fermi
points of the single particles and pairs, respectively. Here $Q_{1}$ and
$Q_{2}$ can be determined by the conditions
$\displaystyle n$ $\displaystyle\equiv:$
$\displaystyle\frac{N}{L}=2\int_{-Q_{2}}^{Q_{2}}\rho_{2}(k)dk+\int^{Q_{1}}_{-Q_{1}}\rho_{1}(k)dk,$
$\displaystyle n_{\downarrow}$ $\displaystyle\equiv:$
$\displaystyle\frac{N_{\downarrow}}{L}=\int_{-Q_{2}}^{Q_{2}}\rho_{2}(k)dk.$
(53)
The ground state energy per length is given by
$E=\int_{-Q_{2}}^{Q_{2}}\left(2k^{2}+E_{B}\right)\rho_{2}(k)dk+\int_{-Q_{1}}^{Q_{1}}k^{2}\rho_{1}(k)dk.$
(54)
In the above equation, the binding energy is given by $E_{B}=c^{2}/2$.
For an attractive interaction, there exist two Fermi seas in charge degree of
freedom at the ground state. So that the Fermi points $Q_{1}$ and $Q_{2}$ are
finite. This allows us to asymptotically solve the BA equations (51) and (52).
For weak and strong attractions with an arbitrary polarization
$P=(n_{\uparrow}-n_{\downarrow})/(n_{\uparrow}+n_{\downarrow})$, the ground
state energy are respectively given by Guan:PRA12
$\displaystyle E$ $\displaystyle=$
$\displaystyle\frac{1}{3}n_{\uparrow}^{3}\pi^{2}+\frac{1}{3}n_{\downarrow}^{3}\pi^{2}-2|c|n_{\uparrow}n_{\downarrow}+O(c^{2}),$
(55) $\displaystyle E$ $\displaystyle\approx$
$\displaystyle\frac{\hbar^{2}n^{3}}{2m}\left\\{-\frac{(1-P)\gamma^{2}}{4}+\frac{\pi^{2}(1-3P+3P^{2}+15P^{3})}{48}\right.$
(56)
$\displaystyle\left.+\frac{\pi^{2}(1-P)(1+P-5P^{2}+67P^{3})}{48|\gamma|}\right.$
$\displaystyle\left.+\frac{\pi^{2}(1-P)^{2}(1+5P+3P^{2}+247P^{3})}{64\gamma^{2}}\right.$
$\displaystyle\left.-\frac{\pi^{2}(1-P)}{1440|\gamma|^{3}}\left[-15+31125{P}^{4}+1861{\pi}^{2}{P}^{5}\right.\right.$
$\displaystyle\left.\left.-15765P^{5}-659{\pi}^{2}{P}^{4}+346{\pi}^{2}{P}^{3}-14{\pi}^{2}{P}^{2}\right.\right.$
$\displaystyle\left.\left.+\pi^{2}P+{\pi}^{2}-105P-150P^{2}-15090P^{3}\right]\right\\},$
respectively. This result was also obtained from dressed energy equations
Guan2007 ; Wadati . We notice that the energy (55) continuously connects to
the repulsive ground state energy (19) at $c\to 0$. But this does not mean
that the energy analytically connects because of the divergence in the small
region $c\to\mathrm{i}0$, see a discussion Guan:FP ; Takahashi:1970-m .
At finite temperatures, except the two-body bound states
$k_{j}=\lambda_{j}\pm\mathrm{i}\frac{1}{2}c$ in charge sector, the spin
quasimomenta of the BA equations (12) and (13) form complex strings
$\lambda_{\alpha,j}^{n}=\lambda^{n}_{\alpha}+\mathrm{i}\frac{1}{2}(n+1-2j)c$
with $j=1,\ldots,n$ Takahashi:1971 ; Guan2007 . Here $\alpha=1,\ldots,N_{n}$
is the number of strings. The equilibrium states are determined by the
minimization condition of the Gibbs free energy, which gives rise to a set of
coupled nonlinear integral equations. In terms of the dressed energies
$\epsilon_{2}(k):=T\ln(\rho_{2}^{h}(k)/\rho_{2}(k))$ and
$\epsilon_{1}(k):=T\ln(\rho_{1}^{h}(k)/\rho_{1}(k))$ for paired and unpaired
fermions Takahashi:1971 ; Guan2007 ; Schlottmann:1993 , the thermodynamic
Bethe ansatz equations for the 1D attractive Fermi gas are given by
$\displaystyle\epsilon_{2}(k)$ $\displaystyle=$ $\displaystyle
2(k^{2}-\mu-\frac{1}{4}{c^{2}})+Ta_{2}*\ln(1+\mathrm{e}^{-\epsilon_{2}(k)/T})$
$\displaystyle+\,Ta_{1}*\ln(1+\mathrm{e}^{-\epsilon_{1}(k)/{T}}),$
$\displaystyle\epsilon_{1}(k)$ $\displaystyle=$ $\displaystyle
k^{2}-\mu-\frac{1}{2}{H}+Ta_{1}*\ln(1+\mathrm{e}^{-\epsilon_{2}(k)/{T}})$
$\displaystyle-T\sum_{n=1}^{\infty}a_{n}*\ln(1+\eta_{n}^{-1}(k)),$
$\displaystyle\ln\eta_{n}(\lambda)$ $\displaystyle=$
$\displaystyle\frac{nH}{T}+a_{n}*\ln(1+\mathrm{e}^{-\epsilon_{1}(\lambda)/{T}})$
(57) $\displaystyle+\sum_{n=1}^{\infty}T_{mn}*\ln(1+\eta^{-1}_{n}(\lambda)),$
where $n=1,\ldots,\infty$ and the function
$\eta_{n}(\lambda):=\xi^{h}(\lambda)/\xi(\lambda)$ is the ratio of the string
densities. While the function $T_{mn}(\lambda)$ is given in Takahashi:1971 ;
Guan2007 . The Gibbs free energy per unit length, i.e. the pressure, is given
by
$\displaystyle p$ $\displaystyle=$
$\displaystyle\frac{T}{\pi}\int_{-\infty}^{\infty}dk\ln(1+\mathrm{e}^{-\epsilon_{2}(k)/{T}})$
(58)
$\displaystyle+\,\frac{T}{2\pi}\int_{-\infty}^{\infty}dk\ln(1+\mathrm{e}^{-\epsilon_{1}(k)/{T}}).$
This serves as the equation of states, from which all the thermal and magnetic
quantities can be derived according to the standard statistical mechanic
relations.
From the TBA equations (57), at low temperatures, the pressure of the
attractive Fermi gas with a strong interaction can be written as a sum of two
components: $p=p_{1}+p_{2}$, where $p_{1}$ is the pressure for unpaired
fermions and $p_{2}$ for pairs, given explicitly in Guan:2011-QC
$p_{1}=F^{1}_{\frac{3}{2}}\left[1+\frac{p_{2}}{4|c|^{3}}\right],\,p_{2}=F^{2}_{\frac{3}{2}}\left[1+\frac{4p_{1}}{|c|^{3}}+\frac{p_{2}}{4|c|^{3}}\right],$
(59)
respectively. Here $F^{r}_{a}=-\sqrt{{r}/{4\pi}}T^{a}{\rm
Li}_{a}[-\exp({A_{r}/T})]$ ($r=1,2$) with
$A_{1}=\mu_{1}-\frac{2p_{2}}{|c|}+\frac{1}{4|c|^{3}}F^{2}_{\frac{5}{2}}+Te^{-\frac{H}{T}}e^{-\frac{J}{T}}I_{0}\left(\frac{J}{T}\right)$
and
$A_{2}=2\mu_{2}-\frac{4p_{1}}{|c|}-\frac{p_{2}}{|c|}+\frac{8}{|c|^{3}}F^{1}_{\frac{5}{2}}+\frac{1}{4|c|^{3}}F^{2}_{\frac{5}{2}}$.
In the above equations $\mathrm{Li}_{n}(x)=\sum_{k=1}^{\infty}{x^{k}}/{k^{n}}$
is the polylogarithm function, $J=2p_{1}/|c|$ and
$I_{0}(x)=\sum_{k=0}^{\infty}\left(x/2\right)^{2k}/{(k!)^{2}}$. This is a
remarkable result of the low temperature thermodynamics of the Yang-Gaudin
model Guan:2011-QC . Based on this result, the exact results of quantum
criticality of the attractive Yang-Gaudin model can be found in Guan:2011-QC ;
Yin:2011 ; Chen:2014 ; He-WB:2016 .
Figure 4: (Color online) Contour plot of the susceptibility Wilson ratio
$R_{W}^{\chi}$ (60) of the attractive Fermi gas for dimensionless interaction
$|\gamma|=10$. In this figure the reduced temperature $t=T/\varepsilon_{b}$
was used. $\varepsilon_{b}$ denotes the binding energy. The susceptibility
Wilson ratio is temperature independent in the FFLO-like phase The dashed
lines indicate the crossover temperature $T^{*}=\alpha|H-H_{c}|$ separating
the relativistic TLL liquid from the free fermion quantum criticality.
$R_{W}=0$ for both the TLL of pairs (PP) and the TLL of excess fermions (F).
From Guan et. al., Guan:2013PRL .
A significant feature of the Yang-Gaudin model with an attractive interaction
is the existence of the FFLO pairing state Larkin1965 ; Fulde1964 ; Yang2001 .
The FFLO physics in Yang-Gaudin model has received extensive study in theory
Orso ; HuiHu ; Feiguin2007 ; Rizzi2008 ; Zhao2008 ; Lee2011 ; Schlottmann2012
and experiment Liao ; Revelle:2016 . In 2007, three groups Guan2007 ; Orso ;
HuiHu predicted the phase diagram of the attractive Fermi gas where the novel
FFLO-like pairing phase sits in a large parameter space. The phase diagram can
be mapped out from specific heat or dimensionless ratios, such as Grüneisen
parameter Peng:2019 ; Yu:2020 , Wilson ratios Guan:2013PRL ; Yu:2016 . The
dimensionless susceptibility $\chi$ and compressibility $\kappa$ Wilson ratios
are given by
$\displaystyle R_{W}^{\chi}=\frac{4}{3}\left(\frac{\pi
k_{B}}{g\mu_{B}}\right)^{2}\frac{\chi}{c_{V}/T},\quad
R_{W}^{\kappa}=\frac{\pi^{2}}{3}\frac{\kappa}{c_{V}/T}$ (60)
that map out the phase diagram of the model in low temperature limit
Guan:2013PRL ; Yu:2016 .
It is very interesting to note Guan:2013PRL ; Yu:2016 that additivity rules
of thermodynamical properties within subsystems of the FFLO-like phase are a
reminiscence of the rules for multi-resistor networks in series and parallel.
Such simple additivity rules indicate a novel and useful characteristic of
multi-component TLLs regardless of microscopic details of the systems. For the
$SU(2)$ Fermi gas (1) with an attraction, we have the relations
$H/2=\left({\mu}_{1}-{\mu}_{2}\right)+E_{B}/2$ and $n=2n_{2}+n_{1}$. Thus we
may define the susceptibilities of subsystems in the FFLO state
$\chi_{1}=(\mu_{\rm B}{g}_{\mathrm{Lande}})^{2}\left({\partial
n_{1}}/{\partial\mu_{1}}\right)_{n}$, $\chi_{2}=2(\mu_{\rm
B}{g}_{\mathrm{Lande}})^{2}\left({\partial
n_{2}}/{\partial\mu_{2}}\right)_{n}$, and the compressibility
$\kappa_{1}=\left({\partial n_{1}}/{\partial\mu_{1}}\right)_{H}$ and
$\kappa_{2}=2\left({\partial n_{2}}/{\partial\mu_{2}}\right)_{H}$ in the grand
canonical ensemble. Here $\mu_{B}$ is the Bohr magneton and
$g_{\mathrm{Lande}}$ is the Lande factor. Consequently, the compressibility
and susceptibility satisfy the following additivity rules
$\displaystyle\kappa$ $\displaystyle=$ $\displaystyle\kappa_{1}+\kappa_{2},$
(61) $\displaystyle\frac{1}{\chi}$ $\displaystyle=$
$\displaystyle\frac{1}{\chi_{1}}+\frac{1}{\chi_{2}}.$ (62)
The compressibilities and susceptibilities are given explicitly in units of
$\hbar^{2}/(2m)$
$\displaystyle\kappa_{\mathrm{2}}^{-1}$ $\displaystyle\approx$
$\displaystyle\frac{\pi^{2}n_{2}}{4}\left(1+\frac{6n_{1}}{|c|}+\frac{4n_{2}}{|c|}+\frac{n_{2}^{2}}{2|c|n_{1}}+\frac{24n_{1}^{2}}{c^{2}}\right.$
(63)
$\displaystyle\left.+\frac{24n_{1}n_{2}}{c^{2}}+\frac{17n_{2}^{2}}{c^{2}}-\frac{2n_{2}^{3}}{c^{2}n_{1}}+\frac{n_{2}^{4}}{4c^{2}n_{1}^{2}}\right),$
$\displaystyle\kappa_{\mathrm{1}}^{-1}$ $\displaystyle\approx$ $\displaystyle
2\pi^{2}n_{1}\left(1+\frac{12n_{2}}{|c|}+\frac{16n_{1}^{2}}{|c|n_{2}}+\frac{96n_{2}^{2}}{c^{2}}+\frac{384n_{1}^{2}}{c^{2}}\right.$
(64)
$\displaystyle\left.-\frac{8n_{1}n_{2}}{c^{2}}-\frac{96n_{1}^{3}}{c^{2}n_{2}}+\frac{256n_{1}^{4}}{c^{2}n_{2}}\right)$
and
$\displaystyle\chi_{1}^{-1}$ $\displaystyle=$
$\displaystyle\pi^{2}n_{2}\left[1+\frac{4}{|c|}(n-3n_{2})+\frac{3}{c^{2}}(4n^{2}\right.$
(65) $\displaystyle\left.-24nn_{2}+30n_{2}^{2})\right],$
$\displaystyle\chi_{2}^{-1}$ $\displaystyle=$ $\displaystyle
8\pi^{2}n_{1}\left[1+\frac{4}{|c|}(n-2n_{1})+\frac{4}{c^{2}}(2n^{2}\right.$
(66) $\displaystyle\left.+10n_{1}^{2}-12nn_{1})\right].$
Moreover, the interaction effect is revealed from the sound velocities
$v_{1}$, $v_{2}$ of the excess fermions and bound pairs, for example, for
strong attraction
$\displaystyle v_{1}$ $\displaystyle\approx$
$\displaystyle\frac{\hbar}{2m}2\pi
n_{1}\left(1+8n_{2}/|c|+48n_{2}^{2}/c^{2}\right),$ $\displaystyle v_{2}$
$\displaystyle\approx$ $\displaystyle\frac{\hbar}{2m}\pi
n_{2}\left(1+2A/|c|+3A^{2}/c^{2}\right),$ (67)
with $A=2n_{1}+n_{2}$. We similarly find that the specific heat satisfies
$c_{V}=c_{V,1}+c_{V,2},$ (68)
where $c_{V,r}=\frac{\pi k_{B}^{2}T}{3\hbar}\frac{1}{v_{r}}$ with $r=1,\,2$.
It is remarkably observed that such additivity rules of compressibility (61),
susceptibility (62) and the rescaled specific heat $c_{V}/T$ with (68) do not
depend on the temperature in the FFLO-like state below a certain temperature.
Such additivity rules are a characteristic of multi-component TLLs Yu:2016 .
This nature presents a universal feature of thermodynamics for a two-component
TLLs of pairs and single fermions in 1D.
Moreover, it was proved Yu:2016 that the susceptibility and compressibility
Wilson ratios
$\displaystyle R^{\chi}_{\mathrm{W},r^{\prime}}$ $\displaystyle=$
$\displaystyle\left(\sum_{r=1}^{2}\frac{D_{r}^{\chi}}{r^{2}}\right)^{-1}\left(\sum_{r=1}^{2}\frac{1}{v_{r}}\right)^{-1},$
(69) $\displaystyle R^{\kappa}_{\mathrm{W}}$ $\displaystyle=$
$\displaystyle\left(\sum_{r=1}^{2}\frac{r^{2}}{D_{r}^{\kappa}}\right)\left(\sum_{r=1}^{2}\frac{1}{v_{r}}\right)^{-1}.$
(70)
are dimensionless and uniquely determined by the sound velocities and
stiffnesses. In the above expressions, the individual stiffnesses
$D^{\kappa}_{r}$ and $D^{\chi}_{r}$ can be respectively given in terms of
$\kappa_{r}$ and $\chi_{r}$ via
$\displaystyle D_{r}^{\kappa}$ $\displaystyle=$
$\displaystyle\frac{r^{2}}{\pi\hbar}\frac{1}{\kappa_{r}},\qquad
D_{r}^{\chi}=\frac{r^{2}}{\pi\hbar}\frac{1}{\chi_{r}}.$ (71)
In this context the Wilson ratios of the Yang-Gaudin model with polarization
elegantly determine the TLL nature of the FFLO-like state, see Fig. 4. It also
indicates a universal feature of the quantum criticality of the attractive
Fermi gas.
Figure 5: Contour plot of the negative Grüneisen parameter (72), i.e.
$-\Gamma$, mapping out the full phase diagram of the Yang-Gaudin model with an
attractive interaction in $h-\mu$ plane, see the main text. Here we set the
dimensionless temperature $t=0.001$. From Yu, et. al. Yu:2020 .
On the other hand, Grüneisen parameter Gruneisen_AdP_1908 , which was
introduced in the beginning of 20th century in the study of the effect of
volume change of a crystal lattice on its vibrational frequencies, has been
extensively studied for the exploration of the caloric effect of solids and
phase transitions associated with volume change. Recently, one of the authors
of this paper and coworkers Peng:2019 ; Yu:2020 studied interaction- and
chemical potential-driven caloric effects and Grüneisen parameter in ultracold
atomic gases in 1D. By using the Maxwell’s relations , the Grüneisen parameter
of quantum gases in grand canonical ensemble is given by Yu:2020
$\Gamma=V\frac{\left.\frac{\mathrm{d}p}{\mathrm{d}T}\right|_{V,N}}{\left.\frac{\mathrm{d}E}{\mathrm{d}T}\right|_{V,N}}=\frac{1}{T}\frac{\frac{\partial^{2}p}{\partial\mu^{2}}\frac{\partial
p}{\partial T}-\frac{\partial^{2}p}{\partial\mu\partial T}\frac{\partial
p}{\partial\mu}}{\frac{\partial^{2}p}{\partial\mu^{2}}\frac{\partial^{2}p}{\partial
T^{2}}-\left(\frac{\partial^{2}p}{\partial\mu\partial T}\right)^{2}},$ (72)
that remarkably maps out the phase diagram of the Yang-Gaudin model with an
attractive interaction, see Fig. 5. This is intimately related to the
expansionary caloric effect
$\displaystyle\left.\frac{\partial T}{\partial V}\right|_{S,N,H}$
$\displaystyle=$ $\displaystyle\frac{T}{V}\Gamma.$ (73)
Similarly one can find
$\displaystyle\left.\frac{\partial T}{\partial H}\right|_{S,N,V}$
$\displaystyle=$ $\displaystyle\frac{T}{H}\Gamma_{\text{mag}},$ (74)
$\displaystyle\left.\frac{\partial T}{\partial c}\right|_{S,N,V,H}$
$\displaystyle=$ $\displaystyle\frac{T}{c}\Gamma_{\text{int}}$ (75)
that establish important relations between magnetocaloric/interaction-driven
caloric effect and the Grüneisen parameter, respectively. In the above
equations, the magnetic and interaction Grüneisen parameters in grand
canonical ensemble are given by
$\displaystyle\Gamma_{\text{mag}}$ $\displaystyle=$
$\displaystyle-\frac{H}{T}\frac{\frac{\partial^{2}p}{\partial\mu^{2}}\frac{\partial^{2}p}{\partial
H\partial T}-\frac{\partial^{2}p}{\partial\mu\partial
H}\frac{\partial^{2}p}{\partial\mu\partial
T}}{\frac{\partial^{2}p}{\partial\mu^{2}}\frac{\partial^{2}p}{\partial
T^{2}}-(\frac{\partial^{2}p}{\partial\mu\partial T})^{2}},$ (76)
$\displaystyle\Gamma_{\text{int}}$ $\displaystyle=$
$\displaystyle-\frac{\frac{\partial^{2}p}{\partial\mu^{2}}\frac{\partial^{2}p}{\partial
c\partial T}-\frac{\partial^{2}p}{\partial\mu\partial
c}\frac{\partial^{2}p}{\partial\mu\partial
T}}{\frac{\partial^{2}p}{\partial\mu^{2}}\frac{\partial^{2}p}{\partial
T^{2}}-(\frac{\partial^{2}p}{\partial\mu\partial T})^{2}}\frac{c}{T},$ (77)
respectively. It is interesting to note that like the adiabatic
demagnetization cooling in solid, the interaction ramp-up and -down in quantum
gases provides a promising protocol of quantum refrigeration.
The Yang-Gaudin model (1) with an attractive interaction exhibits three phases
of quantum states: a fully paired phase with polarization $P=0$, a partially-
polarized FFLO-like phase with $0<P<1$ as well as a fully-polarized phase with
$P=1$ at zero temperature, see Fig. 5. The Grüneisen parameter has a sudden
enhancement near the quantum phase transition that gives a universal divergent
scaling $\Gamma\sim t^{-1/2}$. The key features of this phase diagram were
experimentally confirmed using finite temperature density profiles of trapped
fermionic 6Li atoms Liao .
In the FFLO-like phase, at zero temperature, the leading order of the long
distance asymptotics for the pair correlation function
$G_{p}(x,t)=\langle\psi_{\uparrow}^{\dagger}(x,t)\psi_{\downarrow}^{\dagger}(x,t)\psi_{\uparrow}(0,0)\psi_{\downarrow}(0,0)\rangle$
oscillates with wave number $\pi(n_{\uparrow}-n_{\downarrow})\equiv\pi n_{f}P$
where $n_{f}=n_{\uparrow}+n_{\downarrow}$. The proof can be straightforward by
using the conformal field theory Lee2011 , i.e.
$\displaystyle G_{p}(x,t)$ $\displaystyle\approx$
$\displaystyle\frac{A_{p,1}\cos\left(\pi(n_{\uparrow}-n_{\downarrow})x\right)}{|x+\mathrm{i}v_{u}t|^{\theta_{1}}|x+\mathrm{i}v_{b}t|^{\theta_{2}}},$
where $\theta_{1}\approx\frac{1}{2}$ and
$\theta_{2}\approx\frac{1}{2}-\frac{(1-P)}{2|\gamma|}$ for a strong
attraction. The oscillations in $G_{p}(x,t)$ are caused by an imbalance in the
densities of spin-up and spin-down fermions, i.e.
$n_{\uparrow}-n_{\downarrow}$ which is a mismatch in Fermi surfaces between
both species. The spatial modulation is characteristic of the FFLO state. The
backscattering among the Fermi points of bound pairs and unpaired fermions
results in a 1D analog of the FFLO-like state and displays a microscopic
origin of the FFLO pairing correlation. These results are consistent with the
Larkin-Ovchinikov phase Larkin1965 and the wave numbers coincide with the
ones discovered through the density matrix renormalization group method
Feiguin2007 ; Tezuka2008 ; Rizzi2008 , quantum Monte Carlo method Batrouni2008
, the mean field approach Liu2008 ; Zhao2008 and bosonization technique
Yang2001 .
## IV IV. Conclusion and Outlook
We have presented a brief review of recent developments of the Yang-Gaudin
model from integrability perspectives. It turns out that the Bethe ansatz
solution of the model provides a rigorous understanding of many-body phenomena
ranging from fractional excitations to spin charge separation, FFLO pairing
state and universal thermodynamics and quantum criticality as well. The legacy
of Yang-Baxter equation significantly contributes to developments of
analytical methods for cold atoms, spin liquids and condensed matter physics,
particularly with regard to the fundamental many-body physics to be gained
from exactly solvable quantum many-body systems. In this short review, we have
also discussed a number of the authors’ contributions in the study of the
Yang-Gaudin model, which have led to direct applications in recent
breakthrough experiments on low-dimensional many-body physics of ultracold
atoms Liao ; Yang:2017 ; Hulet:2018 ; Revelle:2016 ; Senaratne:2021 . An
outlook for future research on the Yang-Gaudin model includes:
(a) The observation of spin-charge separation phenomenon is a promising
research in many-body physics. People, in several important papers Kim:1996 ;
auslaender2005spin ; Kim:2006 ; Jompol:2009 ; Hulet:2018 ; Vijayan:2020 ,
experimentally probed spin-charge separation with evidence. Recent new
experiment Senaratne:2021 has provided a conclusive observation of the spin-
charge separated Luttinger liquids. This experimental confirmation of this 1D
unique phenomenon should include: 1) identification of the separate collective
excitation spectra of charge and spin, 2) confirmation of the spin and charge
dynamical response correlation functions; 3) determination of the independent
spin and charge sound velocities and their Luttinger parameters. This opens to
further study of spin coherent and incoherent Luttinger liquids in quantum
gases with higher spin symmetries. Applications of such unique 1D behaviour in
quantum metrology and quantum information will be highly expected.
(b) The wave function of the Yang-Gaudin model is largely unexplored due to
the complexity of the Bethe’s superpositions of $N!$ plane waves. So far there
has been a little understanding of correlation functions and dynamical
response functions for ground state and excited states of this model. Such a
lack of study prevent an access to quantum entanglement behaviour and
metrological useful information for quantum technology. It was recently shown
Hauke:2016 that the dynamical response function can be used to measure
multipartite entanglement in quantum spin systems. This opens a promising
opportunity to further explore realistic applications of fractional
excitations, spin liquids and impurities in quantum metrology.
(c) Cooling fermions in ultracold atomic experiments remain elusive. To
achieve this goal, it is essential to understand caloric effects induced by
magnetic field, trapping potential and dynamical interaction in quantum gases.
Therefor in this research there exist many open questions regarding adiabatic
processes and heat exchanges between the system and baths. The Yang-Gaudin
model has rich phases of quantum matter which hold a promise for studying
quantum transport, hydrodynamics, quantum heat engine and quantum
refrigeration by driving external trapping potentials and interactions.
Acknowledgement This article particularly delicates to the centenary of
Professor C. N. Yang’s birthday. XWG is very grateful to professor C N Yang
for his mentoring, constant help and encouragements since the first time I met
him in 2010. He also acknowledges Institute for Advanced study, Tsinghua
University and Beijing Computational Science Research Center for their kind
hospitality. X.-W. G. is supported by NSFC Key Grant No.12134015 and NSFC
Grant No.11874393. HQL thanks professor C N Yang for his endless support and
great contributions to the physics department at the Chinese University of
Hong Kong.
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# DeepScaleTool : A Tool for the Accurate Estimation of Technology Scaling in
the Deep-Submicron Era
Satyabrata Sarangi and Bevan Baas Department of Electrical and Computer
Engineering
University of California, Davis
Email: {ssarangi<EMAIL_ADDRESS>
###### Abstract
The estimation of classical CMOS “constant-field” or “Dennard” scaling methods
that define scaling factors for various dimensional and electrical parameters
have become less accurate in the deep-submicron regime, which drives the need
for better estimation approaches especially in the educational and research
domains. We present _DeepScaleTool,_ a tool for the accurate estimation of
deep-submicron technology scaling by modeling and curve fitting published data
by a leading commercial fabrication company for silicon fabrication technology
generations from 130 nm to 7 nm for the key parameters of area, delay, and
energy. Compared to 10 nm–7 nm scaling data published by a leading foundry,
the DeepScaleTool achieves an error of 1.7% in area, 2.5% in delay, and 5% in
power. This compares favorably with another leading academic estimation method
that achieves an error of 24% in area, 9.1% in delay, and 24.9% in power.
## I Introduction
Moore’s law [1] has been pivotal in the advancement of the semiconductor
industry for decades, which lays out a projection of doubling of the
transistors on an IC every two years. Similarly, Dennard scaling [2, 3]
pioneered the progress by showing scaling across physical dimensions,
substrate doping, and supply voltage, which in turn results in lower area,
delay, and power dissipation for MOSFETs. The resulting changes due to scaling
are depicted in terms of an entity called scaling factor $K$, which is defined
as the ratio of two technology nodes. These scaling factors are discussed in
various textbooks [4, 5] and the literature [6, 7], and are shown in Table I.
The key points from the traditional scaling factors shown in Table I are the
following: transistor physical dimensions shrink down by the scaling factor
$K$, which in turn scales down the transistor area by a factor of $K^{2}$.
Similarly, the speed of the transistor increases by a factor of $K$ as the
delay reduces by $1{/}K$.
The traditional scaling factors were accurate until the advent of the deep-
submicron era. As the transistors get smaller in the deep-submicron regime,
due to short channel effects, effect of leakage current and thermal runaway,
and process variation, the traditional scaling estimations are no longer
accurate [7, 8]. Moreover, resulting performance gain over recent technology
generations is minimal unlike the predictions by traditional scaling
estimations.
The inaccuracy in the traditional scaling factors can also be depicted from
the real silicon data from various foundries across technology fabrication
generations. Notably, Holt [9] discusses transistor scaling and its effect on
transistor area, gate delay, switching energy, and energy delay product in the
deep-submicron regime based on Intel’s data. Bohr and Young [6] describe
Intel’s scaling trends for area, transistor performance, and cost per
transistor over the past decade. A method to propose accurate scaling
predictions has been demonstrated using data from PTM [10], ITRS [11], and
simulated measurements of Fan Out 4, or FO4 circuit [7]. However, accuracy
achieved in this method differs significantly from both TSMC [12] based
silicon technology scaling data and estimations presented in this article,
which is discussed in Section IV.
TABLE I: MOSFET device parameters and traditional scaling factors
Parameter | Scaling factor
---|---
Device dimension $(W,L,tox)$ | $1{/}K$
Doping concentration $Na$ | $K$
Voltage $V$ | $1{/}K$
Current $I$ | $1{/}K$
Capacitance $\epsilon{A{/}t}$ | $1{/}K$
Delay time $VC{/}I$ | $1{/}K$
Power dissipation $VI$ | $1{/}{K^{2}}$
Power density $VI{/}A$ | $1$
The prediction of accurate scaling factors is also important for a fair
comparison of design performance and other metrics across different technology
fabrication nodes. Although the ITRS and PTM based simulation and modeling
approach looks viable for predicting scaling factors in the deep-submicron
regime, supply voltage information is not always publicly released, which is
accounted for modeling the delay and power scaling equations in the article
[7]. Moreover, such estimation method doesn’t necessarily align with the
actual silicon technology scaling trends, which are discussed in Section IV.
In academia, popular textbooks [4, 5] that cover digital VLSI design and
scaling of CMOS transistors describe traditional scaling factors and reasons
associated with the discontinuity in traditional scaling trends. However, an
accurate estimation of scaling factors across technology fabrication nodes and
correlation between traditional scaling factors and scaling factors resulting
from actual silicon are usually not covered. Therefore, we propose a scaling
tool whose modeling is based on the industrial technology scaling trends and
polynomial based curve fitting approach for an easy and accurate estimation of
scaling factors in the deep-submicron era.
The major contributions of our work are as follows:
* •
We demonstrate DeepScaleTool, a spreadsheet-based tool for accurate estimation
of scaling factors for area, delay, and energy from 130 nm to 7 nm for
educational and research purposes.
* •
We show and analyze the percentage errors in between classical and estimated
scaling factors from real silicon data in the deep-submicron range.
* •
We also illustrate examples of scaling factors estimation using DeepScaleTool
and compare the results both with PTM and ITRS based modeling [7] and scaling
data from TSMC [12].
The remainder of the paper is organized as follows. Section II describes the
published silicon technology scaling trends [6, 9], method of scaling data
modeling adopted in this work, and the tool framework. Section III describes
the usage of DeepScaleTool and examples of scaling factors computations using
the tool. Section IV discusses comparison of traditional scaling factors vs.
accurately estimated scaling factors and differences in scaling estimation
methods. Section V concludes the paper.
## II Transistor Scaling Trends, Data Modeling, and DeepScaleTool Framework
Figure 1: Procedure of Data Collection and Framework
Figure 1 provides an overview of the steps leading to the design of
DeepScaleTool. We analyze published transistor scaling trends [6, 9], curve
fit scaling data that are available for certain technology nodes using second-
order polynomial based models, and then extrapolate scaling data for the rest
of the technology nodes. Finally, we design and update the spreadsheet-based
framework using modeled scaling factors for a combination of starting and
target technology nodes.
### II-A Transistor Scaling Trends
The following two notable transistor scaling trends that are based on Intel’s
silicon results have been analyzed for our work. Holt [9] discusses
generational technology benefits over reduction in gate delay, switching
energy, and energy delay product. All the presented scaling data in the
article span around 65 nm to 10 nm technology nodes and are relative to 65 nm.
Moreover, the normalized transistor area across 130 nm to 14 nm technology
nodes have been demonstrated. The scaling trends shown in the article over
technology fabrication generations infer the following for
circuits—acceleration in transistor density, higher performance, and lower
power.
Similarly, Bohr and Young [6] discuss scaling logic circuit area from 45 nm to
10 nm. The key takeaway from the presented circuit area scaling data is that
with the advent of newer technology nodes and transistor level innovations
more aggressive scaling is possible than the traditional scaling estimation.
For example, both 14 nm and 10 nm technology nodes achieve 0.37 times logic
area scaling than the previous generation. This article also presents the
trends in improved transistor performance, active power, and performance per
watt metrics. However, due to unavailability of proper axis labeling the
corresponding trends have not been considered for data modeling purposes in
this work.
### II-B Data Extraction and Modeling
The g3data [13] tool has been used to extract the digitized data from the
plots with technology scaling trends shown in the articles [6, 9]. The plots
with proper axis labeling have been considered for data extraction. To obtain
scaling trends from 130 nm to 7 nm for key circuit parameters like area,
delay, and energy, available scaling data across technology fabrication
generations have been extrapolated to obtain the scaling data of the
corresponding parameter at the missing technology generations. The polynomial
based extrapolation models that are used to curve fit for various circuit
parameters yield a coefficient of determination or $R^{2}$ value of equal or
greater than 0.99. The small differences in area scaling factors that are
obtained after modeling the scaling trends in both articles [6, 9], have been
offset by taking the average of the corresponding two scaling factors for any
given starting and target technology generations.
### II-C DeepScaleTool Framework
Instead of providing big tables consisting of scaling factors across
technology fabrication nodes for each circuit parameter, we present a
spreadsheet-based framework for the automated generation of scaling factors
for various circuit parameters. The framework is designed using visual basic
for applications (VBA) programming language. The scaling factor values fields
in the spreadsheet are programmed for any of the supported current and target
technology nodes. The DeepScaleTool is available as an open source tool and it
can be accessed at https://sourceforge.net/projects/deepscaletool/ [14].
Figure 2 depicts a screenshot of the tool. The tool can be updated easily for
future nodes with the availability of future scaling trends.
## III Usage of DeepScaleTool and Scaling Factor Computation Examples
### III-A DeepScaleTool Usage
Figure 2: Screenshot of the DeepScaleTool page showing the values given for
current node and target node, generated scaling factors, and user
instructions.
The usage of the DeepScaleTool is simple, which requires three-fold steps as
shown in Figure 2. Currently, the tool supports scaling factors for the
following fabrication nodes in units of nm: 130, 90, 65, 45, 40, 32, 28, 22,
14, 10, and 7. The user inputs one of those values for the current node and
target node. Next, the user can press the corresponding button for any
performance metric or parameter to display the scaling factor. Finally, the
value of any metric at a target node can be found based on the value of that
metric at the current node and resulting scaling factor using the following
equation, where $x$ and $y$ denote target node and current node respectively,
$Value_{x}=~{}{Value_{y}~{}{/}~{}Scalingfactor}$ (1)
### III-B Examples of Scaling Factor Computation
The following examples are shown to illustrate the scaling factor computation
procedure. To scale a circuit from 130 nm to 45 nm, area scaling factor is 8.3
per the current version of the tool as shown in Figure 2. If the circuit
occupies an area of 100 um2 in 130 nm node, the resulting area in 45 nm node
using equation (1) will be 100 / 8.3 = 12.05 um2. Similarly, to scale a
circuit from 45 nm to 32 nm, the tool displays a power scaling factor of
1.238. If the circuit dissipates 100 mW in 45 nm node, the resulting power
dissipation in 32 nm node using equation (1) will be 100 / 1.238 = 80.775 mW.
The scaling computations for delay, energy, energy delay product, and
throughput can be performed by generating the corresponding scaling factors
from the tool and applying equation (1) like the above examples. The scaling
factors for derived metrics like throughput/area and power density can also be
found out using the scaling factors for the corresponding primary metrics that
are generated from the tool.
## IV Comparison of Scaling Factors Estimation Methods and Accuracy with
Traditional Scaling
Figure 3: Scaling trends for area, delay, power from 130 nm to 7 nm based on
the modeling presented in this work.
### IV-A Scaling Trends and Accuracy Analysis with Traditional Scaling
Figure 3 shows the scaling trends for transistor area, delay, energy,
throughput, and power based on the modeling presented in this work. Among all
the considered parameters transistor area achieves a remarkable scaling over
the years, which is better than the traditional scaling trends. The delay and
throughput achieves minimal scaling with the recent technology nodes.
Figure 4: Traditional scaling factors vs. modeled scaling factors presented in
this work for the deep-submicron regime.
Figure 4 shows the percentage error of value of scaling factors modeled for
each parameter with respect to traditional scaling factors. The percentage
errors shown for each technology node are relative to the values at 130 nm.
Transistor area shows the least variation among all parameters as transistor
area still scales by the traditional estimation or even better with the advent
of transistor innovations like high-dielectric metal gates, FinFET, and 3D
FinFET structures [6]. However, delay and power dissipation trends vary
significantly as compared to traditional scaling improvements of $1{/}K$ and
$1{/}K^{2}$.
Due to the effect of leakage current threshold voltage scaling doesn’t happen
aggressively and thus the same effect on scaling for supply voltage as well.
Therefore, there is a limit on energy efficiency scaling than the traditional
scaling factor of $1{/}K^{3}$. Similarly, transistor gate delay scaling is
primarily limited by the slower interconnect scaling. The poor trends in gate
delay and power dissipation scaling shown in Figure 4 affect the energy
efficiency scaling and thereby a larger percentage of error is observed for
scaling energy. Moreover, as we advance for the recent technology nodes we
observe a greater percentage of error in relative scaling values with respect
to 130 nm due to the cumulative effect of the error margins across technology
nodes.
### IV-B Comparison of Scaling Factor Estimation Methods
Figure 5: Comparison of area scaling factors between our work and Stillmaker [7] for starting nodes from 130 nm to 14 nm with a target node of 14 nm, where in our work shows better correlation to the reference data by Holt [9]. TABLE II: Comparison of area, delay, and power scaling (% reduction) factors for 10 nm to 7 nm scaling. The reference scaling data [12] corresponds to TSMC’s technology scaling. Source | Area | Delay | Power
---|---|---|---
Cai [12] | 30 – 35 | 10 | 35
Stillmaker [7] | 59 | 19.1 | 9.1
This work | 36.7 | 7.5 | 30
Table II shows the variation in area, delay, and power percentage scaling from
10 nm to 7 nm. The scaling factor values modeled in this work which are based
on Intel’s silicon technology scaling trends [6, 9] achieve better correlation
with TSMC based scaling data [12] than the ITRS and PTM based modeling
approach [7] in terms of area, delay, and power. The delay and power scaling
data for the article [7] have been calculated using the given corresponding
coefficients, supply voltage per ITRS data, and modeling expressions. The
errors in between modeled data presented in this work and TSMC’s scaling data
are 1%, 2.5%, and 5% for area, delay, and power respectively, which states the
reliability of DeepScaleTool across major foundries. However, area, delay, and
power scaling factors presented in the article [7] differ from TSMC based data
by 24–29%, 9.1%, and 24.9% respectively, which is significant given the
percentage scaling occurring for those parameters with the said technology
nodes.
Moreover, area scaling factors presented in the article [7] varies
significantly when compared to modeling based on silicon data as presented by
DeepScaleTool as shown in Figure 5. For example, 130 nm to 7 nm technology
scaling brings down the area by a factor of 110 per the article [7], while the
current version of DeepScaleTool suggests the corresponding area scaling
factor value of 754.55. The later scaling factor seems more accurate since
area scales down by a factor of approximately 303 in general over eight
generations from 130 nm to 7 nm. Moreover, the aggressive scaling than the
normal rate of 0.49 makes the overall factor shoot up to 754.55. Therefore,
DeepScaleTool caters to provide more accurate estimation of scaling factors
for various design parameters irrespective of major foundries and it avoids
the prediction inaccuracy from ITRS and PTM models.
## V Conclusion
We present DeepScaleTool, a tool designed to provide accurate estimation of
scaling factors using published silicon trends and polynomial based curve-
fitting method. The scaling factors presented in this work shows that the
traditional scaling factors go obsolete in deep-submicron era. Although the
primary data sets considered for this work belong to Intel’s published
technology scaling trends, the proposed tool achieves good correlation to the
TSMC based scaling trends as well. Moreover, we show that published silicon
data based modeling and estimation is more accurate than the simulation based
modeling and data per ITRS and PTM, which is the state-of-the-art in
estimating scaling factors in the deep-submicron regime. In conclusion,
DeepScaleTool provides an easy platform to obtain reliable scaling factors for
various design parameters in the deep-submicron era, understand the
discrepancies with traditional scaling factors, and also helps in performing
fair comparisons of circuits performance over different technology nodes.
## References
* [1] G. E. Moore, “Cramming more components onto integrated circuits,” _Proceedings of the IEEE_ , vol. 86, no. 1, pp. 82–85, 1998.
* [2] R. H. Dennard, F. H. Gaensslen, H. Yu, V. L. Rideout, E. Bassous, and A. R. LeBlanc, “Design of ion-implanted mosfet’s with very small physical dimensions,” _IEEE Journal of Solid-State Circuits_ , vol. 9, no. 5, pp. 256–268, 1974.
* [3] G. Baccarani, M. R. Wordeman, and R. H. Dennard, “Generalized scaling theory and its application to a 1/4 micrometer MOSFET design,” _IEEE Transactions on Electron Devices_ , vol. 31, no. 4, pp. 452–462, 1984.
* [4] J. M. Rabaey, A. P. Chandrakasan, and B. Nikolić, _Digital integrated circuits: a design perspective_. Pearson Education, 2003, vol. 7.
* [5] J. Uyemura, _Introduction to VLSI Circuits and Systems_ , 1st ed. Hoboken, NJ: John Wiley ‘—&’ Sons, Inc., 2002.
* [6] M. T. Bohr and I. A. Young, “CMOS scaling trends and beyond,” _IEEE Micro_ , vol. 37, no. 6, pp. 20–29, 2017.
* [7] A. Stillmaker and B. Baas, “Scaling equations for the accurate prediction of CMOS device performance from 180 nm to 7 nm,” _Integration, the VLSI Journal_ , vol. 58, pp. 74–81, 2017, http://vcl.ece.ucdavis.edu/pubs/2017.02.VLSIintegration.TechScale/.
* [8] K. J. Kuhn, “CMOS transistor scaling past 32 nm and implications on variation,” in _2010 IEEE/SEMI Advanced Semiconductor Manufacturing Conference (ASMC)_ , 2010, pp. 241–246.
* [9] W. M. Holt, “1.1 moore’s law: A path going forward,” in _2016 IEEE International Solid-State Circuits Conference (ISSCC)_ , 2016, pp. 8–13.
* [10] (2015, Oct.) Predictive technology model. [Online]. Available: http://ptm.asu.edu/
* [11] (2015, Oct.) International technology roadmap for semiconductors. [Online]. Available: http://www.itrs.net/
* [12] M. Cai _et al._ , “7nm mobile soc and 5g platform technology and design co-development for ppa and manufacturability,” in _2019 Symposium on VLSI Technology_ , 2019, pp. T104–T105.
* [13] (2019, Oct.) g3data, a tool for extracting data from scanned graphs. [Online]. Available: https://github.com/pn2200/g3data
* [14] S. Sarangi and B. Baas. (2021, Feb.) Deepscaletool. [Online]. Available: https://sourceforge.net/projects/deepscaletool/
|
# Topological phase transitions in strongly correlated systems: application to
Co3Sn2S2
V. Yu. Irkhin<EMAIL_ADDRESS>Yu. N. Skryabin M. N. Mikheev
Institute of Metal Physics, 620108 Ekaterinburg, Russia
###### Abstract
The topological transition in the strongly correlated half-metallic
ferromagnetic compound Co3Sn2S2 from Weyl semimetal (including chiral massless
fermions) to a non-magnetic state is treated. This transition goes with a
change in topological invariant, and is accompanied by a non-topological
transition from saturated ferromagnetic to paramagnetic state, the minority
Fermi surface being transformed from ghost (hidden) to real. A corresponding
description is given in terms of slave fermion representation for the
effective narrow-band Hubbard model. The system Co3Sn2S2 provides a bright
example of coexistence of non-trivial topology and strong low-dimensional
ferromagnetism. A comparison is performed with other compounds where
frustrations result in formation of a correlated paramagnetic state.
## 1 Introduction
Recently, the layered kagome lattice compound Co3Sn2S2 has been a subject of
numerous experimental [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13] and
theoretical [14, 15] investigations. In particular, giant topological
anomalous Hall effect was observed in this system [3, 6, 8]. Its electronic
structure contains Weyl points, Fermi arcs and nodal rings (Fig. 1), which
play an important role in the anomalous Hall effect [3, 8, 14].
Single-crystal experimental data on the Co3InxSn2-xS2 kagome system [18] show
that these systems have an almost two-dimensional itinerant magnetism and a
chiral spin state; in addition, a strongly correlated state with a high
electronic heat capacity is formed. The important role of correlations is
confirmed by a considerable enhancement of $\gamma T$-linear specific heat
even in the ferromagnetic phase [19, 18], especially at approaching the
magnetic-nonmagnetic critical point somewhat below $x=1$. At $x=1$, $\gamma$
vanishes, but strongly increases with further increasing $x$ [18].
The ferromagnetism in Co3Sn2S2 breaks time-reversal $T$-symmetry and is
necessary for the existence of topological Weyl points. The strongly
correlated metallic state in this system emerges as a result of magnetic
fluctuations. Above $T_{C}$, intrinsic magnetic field disappears, the Weyl
points annihilate and the Dirac points acquire a gap. This restores
$T$-symmetry and eliminates the topological behavior.
Figure 1: Schematic energy spectra for the Dirac semimetal (a), Weyl semimetal
(b), and nodal line semimetal (c), according to Refs.[16, 17].
A similar, but quantum transition occurs with disappearance of ferromagnetism
in the Co3InxSn2-xS2 system at the hole doping [6, 14]. The doping shifts the
Weyl nodes away from the Fermi level. For small doping, the nodal rings are
located around the Fermi energy, and for $x\sim 0.2$, the nodal lines
surrounding the L point in the Brillouin zone cross the Fermi surfaces (Fig.
2). With further increasing $x$, the nodal lines are split into two rings as
with the annihilation of Weyl points in the presence of the spin-orbit
coupling (the role of the latter is discussed in detail in Ref. [12]). For
$x>0.6$, the nodal lines are located far from the Fermi level, resulting in
the small Berry curvature on the whole Fermi surfaces [14]. At $x=1$ the
system becomes insulating [6]. According to [18], this anomalous nonmetallic
state may originate from the Fermi energy tuning through a Dirac point.
Figure 2: Schematic Fermi surfaces (solid green lines) for Co3Sn2S2 in the
$k_{x}-k_{z}$ plane at $k_{y}=0$ according to Ref.[14]. The thin black solid
line shows the nodal lines in the absence of spin-orbit coupling for (a)
$x=0.2$ and (b) $x=0.4$. The upper and lower triangles on the nodal lines in
(a) stand for the Weyl points with topological charges $+1$ and $-1$ in the
presence of spin-orbit coupling.
In the present work we treat the model picture of correlated half-metallic
ferromagnetism in Co3Sn2S2 and provide a description of these transitions
within the topological classification [20, 21, 22].
## 2 Half-metallic ferromagnetism and Weyl points in the effective Hubbard
model
The half-metallic ferromagnetism of Co3Sn2S2 occurs in the partially filled Co
3$d_{x^{2}-y^{2}}$ band which crosses the Fermi level. The itinerant magnetism
cannot be fully described by a simple one-electron approach including band
splitting. In particular, the persistence of local moments from in the Curie
Weiss susceptibility suggests a many-body correlated behavior [15]. The
associated moment of 1$\mu_{B}$ is spread over three Co atoms, in agreement
with the 0.33$\mu_{B}$ per Co magnetic moment from first-principle
calculations and the experimental moment which is slightly less than
1$\mu_{B}$/f.u. The configuration with five electrons and one hole results in
a spin-1/2 ferromagnetic state. This enables one to formulate a local Hubbard
model for the three Co atom cluster [15].
According to [15], across the magnetic transition, Co3Sn2S2 evolves from a
Mott ferromagnet to a correlated metallic state. In fact, the “Mott
ferromagnet” is a half-metallic ferromagnetic state, so that we have a partial
Mott transition in the minority spin subband. The physical reason of half-
metallic ferromagnetism in a wide doping region is the presence of Van Hove
singularity at the Fermi level, which is connected with quasi-two-dimensional
kagome structure (see Fig.2 in Ref. [14]). First-principle calculations [14]
and experimental data [6, 18] demonstrate that the doping in Co3InxSn2-xS2
system preserves half-metallic ferromagnetism with a linearly decreasing
magnetic moment.
The picture of half-metallic ferromagnetism can be qualitatively described by
the simplest narrow-band Hubbard model with large on-site repulsion $U$. In
this model, doubly occupied states (doubles) are absent owing to the Hubbard
splitting, but states with both spin projections are still present. Thus the
situation is different from the Stoner model where spin splitting becomes
infinitely large. The corresponding Hamiltonian reads
$\mathcal{H}_{\mathrm{H}}=\sum_{ij\sigma}t_{ij}\tilde{c}^{\dagger}_{i\sigma}\tilde{c}_{j\sigma}$
(1)
where
$\tilde{c}^{\dagger}_{i\sigma}=X_{i}(0,\sigma)=|i0\rangle\langle i\sigma|$ (2)
are the Hubbard projection operators describing motion of holes in the
correlated state on the background of magnetic moments. We can use the slave
fermion representation
$X_{i}(0,\sigma)=f_{i}^{\dagger}b_{i\sigma},\,X_{i}(+,-)=b^{\dagger}_{i\uparrow}b_{i\downarrow},$
(3)
where $f_{i}$ are fermions and $b_{i\sigma}$ are Schwinger boson operators
(cf. [23, 24]), so that
$\sum_{\sigma}b^{\dagger}_{i\sigma}b_{i\sigma}+f_{i}^{\dagger}f_{i}=1.$ (4)
In the saturated ferromagnetic state the $b_{i\uparrow}$ boson is condensed,
$\langle b_{i\uparrow}\rangle\simeq 1$, and $b_{i\downarrow}$ becomes magnon
annihilation operator.
Consider the hole Green’s functions for a saturated ferromagnet in the
representation (3).
$G_{\mathbf{k}\sigma}(E)=\langle\langle
X_{\mathbf{k}}(\sigma,0)|X_{\mathbf{-k}}(0,\sigma)\rangle\rangle_{E}.$ (5)
The spin-up (majority) states propagate freely on the background of strong
ferromagnetic ordering,
$G_{\mathbf{k}\uparrow}(E)=(E-t_{\mathbf{k}})^{-1}$ (6)
with $t_{\mathbf{k}}$ the bare band spectrum. For our system, these possess an
exotic spectrum of chiral Weyl fermions in the internal field owing to
ferromagnetic ordering of local moments. This just leads to unusual
topological properties.
The spin down (minority) Green’s function in the leading approximation is
obtained as a convolution of the Green’s functions for free fermions and
bosons, so that
$G_{\mathbf{k}\downarrow}(E)=\sum_{\mathbf{q}}\frac{N_{B}(\omega_{\mathbf{q}})+f(t_{\mathbf{k}+\mathbf{q}})}{E-t_{\mathbf{k}-\mathbf{q}}+\omega_{\mathbf{q}}}$
(7)
where $N_{B}(\omega)$ and $f(E)$ are the Bose and Fermi functions,
$\omega_{\mathbf{q}}$ is the magnon spectrum. Similar results for a Hubbard
ferromagnet were obtained earlier in the many-electron representation of
X-operators [25, 26], the analogy with Anderson’s spinons [27] being
discussed.
The Green’s function (7) has a purely non-quasiparticle nature. Because of
very weak $\mathbf{k}$-dependence of the corresponding distribution function
the non-quasiparticle (incoherent) states do not carry current. As demonstrate
more exact calculations [26], at small doping the spin down Green’s function
(3) has no poles above the Fermi level of holes $E_{F}$, so that the above
conclusions are not changed. However, with increasing doping, it can acquire a
spin-polaron pole above $E_{F}$, and the half-metallic ferromagnetism is
destroyed (Fig.3).
Figure 3: Schematic picture of the density of states in the narrow-band
Hubbard model illustrating the doping-induced Lifshitz transition, the data
are from Ref. [29]. Dashed line is the bare (and also spin-up) density of
states, solid and dot-dashed lines are the spin-down densities of states for
the half-metallic and usual ferromagnetic states.
Despite such unusual properties, the minority states cannot be fully
neglected. In fact, their number is equal to the number of majority states
$n_{0}$ owing to the sum rule
$\sum_{\mathbf{k}}\langle
X_{\mathbf{-k}}(0,\sigma)X_{\mathbf{k}}(\sigma,0)\rangle=\langle
X_{i}(0,0)\rangle=n_{0}$ (8)
for both projections $\sigma$ (which is satisfied in the approximation (7)),
so that the current carriers (Hubbard’s holes) are in a sense spinless (see
Fig.3: the corresponding areas below the Fermi level determining the
occupations numbers are nearly equal). The minority states in the ground state
are absent at the Fermi surface and occur there at finite temperatures owing
to thermal magnon excitations. The physics does not qualitatively change in
the case of finite Hubbard $U$, since the doubles are automatically eliminated
in the saturated half-metallic state [26].
The description of the transition to the half-metallic state (vanishing of the
quasiparticle residue for one spin projection) is similar to that of the Mott
transition in the paramagnetic Hubbard model [28]. Here, the Fermi surface
becomes ghost (hidden) after the metal-insulator transition in the insulating
(Mott) phase, and the fractionalization of electron states occurs, including
spin-charge separation into a neutral fermion (spinon) and charged boson
(holon). Our situation can be described as a partial Mott transition in the
minority spin subband, cf. the discussion of the orbital-selective Mott
transition [30].
The electron states in a strongly correlated system need not to have purely
quasiparticle nature. They can be described by both poles and branch cuts of
the Green’s function. For minority states, the quasiparticle residue $Z$ is
completely suppressed, which means formation of the energy gap and takes
place, e.g., for the Mott transition [20]. The violation of the standard
Fermi-liquid picture can be described in terms of the formation of the
Luttinger surface which is the surface of zeros of the electron Green’s
function.
The Lifshitz transitions with vanishing quasiparticle poles can be viewed as
quantum phase transitions with a change of the topology of the Fermi surface,
but without symmetry breaking. Indeed, the Fermi surface itself is the
singularity in the Green’s function, which is characterized by topological
invariant $N_{1}$ and topologically protected: it is the vortex line in the
frequency-momentum space [20, 21]. In the gapped phase, usual Fermi surface
does not exist, but its topology is preserved if we take into account the
Luttinger contribution. Then the Luttinger theorem (the conservation of the
volume enclosed by the Fermi surface) is still valid [20]. Thus the Fermi
surface becomes ghost (hidden) in the Mott phase for both spin projections and
in our half-metallic situation for minority states, since the Fermi level lies
in the corresponding gap.
On the contrary, the transition with disappearance of the Weyl points is
essentially topological: topological invariants are changed. In the Weyl
semimetal phase, the Weyl points have topological charges $N_{3}=+1$ and $-1$
and annihilate in the critical Dirac semimetal. Further on, in the normal
paramagnetic state the topology owing to the Berry curvature vanishes. Thus
the conservation law for the topological charge [20] is fulfilled. In the
insulator case, we have a transition from topological to normal insulator with
restoring time-reversal symmetry. A still more complicated situation occurs in
the case of Chern insulators with a change of the Chern number [31, 7, 32].
At the hole doping in the Co3InxSn2-xS2 system, suppression of ferromagnetism
is also accompanied with decreasing the Berry curvature, and in the
paramagnetic phase both the Weyl points and nodal rings vanish [14].
## 3 Discussion
As stated in Ref. [15], the topological properties and strong correlations in
Co3Sn2S2 are intricately linked, so that one cannot be adequately considered
without the other. The scanning tunnelling microscopy data in Co3Sn2S2
demonstrated a pronounced peak at the Fermi level, which was identified as
arising from the kinetically frustrated kagome flat band [9]. The occurrence
of an electronic band connecting the two Weyl cones and flattened by
electronic correlations was demonstrated in Ref. [10], the Coulomb-interaction
strength being estimated as $U\sim 4$ eV.
The electron correlations in pyrochlore iridates $R_{2}$Ir2O7 were discussed
by first-principle calculations in Ref. [33]. Depending on the strength of
correlations $U$, a Mott insulating phase with a magnetic structure or Weyl
semimetal phase with Weyl points at the Fermi energy were treated. A variety
of phases ranging from normal metal at small $U$ to Weyl semimetal at
intermediate $U\sim$ 1.5 eV and Mott insulating phase at $U$ above 2 eV with
non-collinear magnetic all-in/all-out ordering were predicted. The giant
anomalous Hall effect is observed in Nd2Mo2O7, a pyrochlore ferromagnet with
geometrically frustrated lattice structure, is mostly due to the spin
chirality and the associated Berry phase originating from the Mo spin tilting
[34]. However, the situation in pyrochlores with strong spin-orbit coupling
differs from half-metallic ferromagnetism.
We have demonstrated that in the half-metallic ferromagnetic state Hubbard
correlations do not result in narrowing of bare bands for majority states, but
in the paramagnetic state the situation changes: we come to the regime of
narrow correlated bands for both spin projections. These may be characterized
either by strongly renormalized quasiparticle residue, or even by a non-Fermi-
liquid (e.g., marginal Fermi-liquid [28]) behavior. Besides absence of
$T$-breaking internal magnetic field in the paramagnetic phase, this can be
important for vanishing of topological effects.
Since magnetic frustration effects in the kagome lattice should play a role,
the magnetic structure of Co3Sn2S2 is to be discussed in more detail.
According to [13], at finite temperatures this includes the out-of-plane
ferromagnetism, in-plane antiferromagnetism, and hidden phases. A metastable
magnetic phase exists in some interval $T_{com}<T<T_{C}$. The corresponding
values of transition temperatures are $T_{C}=182$ K, $T_{N}=177$ K, and
$T_{com}=150$ K. The out-of-plane magnetization of Co3Sn2S2-xSex demonstrates
a first-order phase transition, which may again indicate strong half-metallic
magnetism. This may also important for a combined description of the non-
topological ferromagnetic and topological transitions. A possibility of a
chiral spin state is discussed in [18].
From the $\mu$SR measurements, the ground ferromagnetic state with moment
along c axis exists in Co3Sn2S2 below $T^{*}_{C}$ = 90 K, and a coexistence of
the ferromagnetic order and an in-plane 120∘ antiferromagnetic order was
proposed at $T>T^{*}_{C}$ [11] (see, however, Ref.[35]). The antiferromagnetic
volume fraction grows with increasing temperature and dominates around 170 K,
before it disappears at $T_{C2}=172$ K. Above $T_{C2}$, the sample has the
small volume fraction with the out-of-plane ferromagnetic order, the rest of
the volume being occupied by the paramagnetic state up to $T_{C1}=177$ K. The
temperature-dependent magnetic fraction shows a rather sharp transition
between the paramagnetic and magnetic states with the coexistence of their
regions in the interval $T_{C2}<T<T_{C1}$.
Thus, although frustrations in Co3Sn2S2 seem to be important, they turn out to
be insufficient for formation of a paramagnetic spin-liquid-like state. The
situation seems to be different for the correlated kagome systems YCr6Ge6 [36]
and Na2/3CoO2 [37] without magnetic ordering.
Besides pyrochlores, comparison can be made with other three-dimensional
systems. Two topological Weyl cones with band crossing points were identified
around the X point for the Heusler alloy Co2MnGe, which can induce large
anomalous Hall effect owing to the Berry flux in the half-metallic ferromagnet
structure [38].
To conclude, Co3Sn2S2 provides a bright example of coexistence of non-trivial
topology and half-metallic ferromagnetism in a quasi-two-dimensional system.
Both these factors are important for non-usual phase transitions and anomalies
of electronic properties, including giant anomalous Hall effect.
The authors are grateful to A. V. Zarubin for the help in preparing the
manuscript. The research was carried out within the state assignment of FASO
of Russia (theme “Flux” No AAAA-A18-118020190112-8 and theme “Quantum” No.
AAAA-A18-118020190095-4).
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|
# Joint Deep Multi-Graph Matching and 3D Geometry Learning
from Inhomogeneous 2D Image Collections
Zhenzhang Ye, Tarun Yenamandra, Florian Bernard, Daniel Cremers
Technical University of Munich
{zz.ye, tarun.yenamandra, f.bernard<EMAIL_ADDRESS>
###### Abstract
Graph matching aims to establish correspondences between vertices of graphs
such that both the node and edge attributes agree. Various learning-based
methods were recently proposed for finding correspondences between image key
points based on deep graph matching formulations. While these approaches
mainly focus on learning node and edge attributes, they completely ignore the
3D geometry of the underlying 3D objects depicted in the 2D images. We fill
this gap by proposing a trainable framework that takes advantage of graph
neural networks for learning a deformable 3D geometry model from inhomogeneous
image collections, i.e. a set of images that depict different instances of
objects from the same category. Experimentally we demonstrate that our method
outperforms recent learning-based approaches for graph matching considering
both accuracy and cycle-consistency error, while we in addition obtain the
underlying 3D geometry of the objects depicted in the 2D images.
Previous approach [55] Our proposed approach Sparse non-rigid geometry 3D
reconstruction
Figure 1: We consider a deep graph matching approach for bringing 2D image key
points into correspondence. Left: Existing deep graph matching methods
completely ignore the underlying 3D geometry of the 3D objects depicted in the
2D images. In addition, they lead to cycle errors, as shown by the red line.
Middle: Our method obtains the underlying 3D geometry from a collection of
inhomogeneous 2D images (indicated by the coloured points and the bike sketch
in the centre), while at the same time guaranteeing cycle consistency. Right:
To model nonlinear 3D object deformations, we infer coarse 3D geometry and in
addition use a 3D deformation module to refine the underlying 3D geometry
based on the 2D image key point observations.
## 1 Introduction
Graph matching is a widely studied problem in computer vision, graphics and
machine learning due its universal nature and the broad range of applications.
Intuitively, the objective of graph matching is to establish correspondences
between the nodes of two given weighted graphs, so that the weights of
corresponding edges agree as well as possible. Diverse visual computing tasks
fit into the generic graph matching framework. In this work we focus in
particular on the task of matching 2D key points defined in images, which has
a high relevance for 3D reconstruction, tracking, deformation model learning,
localisation, and many more. In this case, a graph is constructed for each
image by using the key points as graph nodes, and by connecting neighbouring
key points with edges, according to some suitable neighbourhood criterion. The
edges typically contain information about geometric relations, such as the
Euclidean distance between nodes in the simplest case.
Image key point matching was traditionally addressed based on finding nearest
neighbours between feature descriptors such as SIFT [33], SURF [3] or learned
features [23]. A downside to this approach is that the geometric relation
between the key points are completely ignored, which is in particular
problematic if there are repetitive structures that lead to similar feature
descriptors. Instead, we can use a graph matching formulation to establish
correspondences between key points while taking into account geometric
relations between points. Yet, the sequential nature of first computing
features and then bringing them into correspondence may lead to sub-optimal
results, since both tasks are solved independently from each other – despite
their mutual dependence. More recently, several deep learning-based graph
matching methods have been proposed that learn task-specific optimal features
while simultaneously solving graph matching in an end-to-end manner [58, 51,
55, 41]. While such deep graph matching approaches lead to state-of-the-art
results in terms of the matching accuracy, they have profound disadvantages,
particularly in the context of 2D key point matching in image collections. On
the one hand, most existing approaches only consider the matching of pairs of
images, rather than the entire collection. This has the negative side-effect
that so-obtained matchings are generally not cycle-consistent, i.e. matching
image A via B to C may be different than matching image A directly to C. To
circumvent this, there are approaches that use a post-processing procedure
[52] to establish cycle consistency based on permutation synchronisation [6,
37]. Yet, they do not directly obtain cycle-consistent matchings but rather
achieve it based on post-processing. On the other hand, and perhaps more
importantly, approaches that use graph matching for 2D image key point
matching have the strong disadvantage that the underlying 3D structure of the
objects whose 2D projections are depicted in the images is not adequately
considered. In particular, the spatial relations in the 2D image plane are
highly dependent on the 3D geometric structure of the object, as well as on
the camera parameters. Hence, learning graph features directly based on the
image appearance and/or 2D image coordinates is sub-optimal, at best, since
the neural network implicitly needs to learn the difficult task of reasoning
about the underlying 3D structure.
In this work we address these issues by proposing a deep multi-graph matching
approach that simultaneously learns the 3D structure of objects. We summarise
our main contributions as follows:
* •
For the first time we propose a solution for jointly considering deep multi-
graph matching and inferring sparse 3D geometry from inhomogeneous 2D image
collections, see Fig. Joint Deep Multi-Graph Matching and 3D Geometry Learning
from Inhomogeneous 2D Image Collections.
* •
To effectively deal with the inhomogeneity of the image collection, in which
different instances of objects of the same category are present (e.g.
different types of bikes as shown in Fig. Joint Deep Multi-Graph Matching and
3D Geometry Learning from Inhomogeneous 2D Image Collections), we introduce a
novel deformable 3D model that we directly learn from the image collection
based on a graph neural network.
* •
Rather than performing pairwise image-to-image matching, we consider an image-
to-deformable-3D-model matching formulation that by construction guarantees
cycle consistency.
* •
Our approach substantially outperforms the previous state of the art in
learning-based graph matching approaches considering accuracy and cycle error
simultaneously.
## 2 Related Work
In the following we summarise the works that we consider most relevant to our
approach. For a more detailed background on image key point matching we refer
interested readers to the recent survey paper [34].
Feature-Based Matching. Feature descriptors extracted from images at key point
locations, e.g. based on SIFT [33], SURF [3], or deep neural networks [23],
are often used for image matching. In order to bring extracted features into
correspondence, commonly a nearest neighbour strategy [4] or a linear
assignment problem (LAP) [13] formulation are used. However, these methods
suffer from the problem that geometric relation between the key points in the
images are not taken into account.
Graph Matching and Geometric Consistency. Geometric relations can be taken
into account by modelling feature matching as graph matching problem. Here,
the image key points represent the graph nodes, and the edges in the graph
encode geometric relations between key points (e.g. spatial distances).
Mathematically, graph matching can be phrased in terms of a quadratic
assignment problem [24, 38, 30, 13]. There are many existing works for
addressing the graph matching problem in visual computing, including [17, 61,
45, 18, 5, 45]. A drawback of these approaches is that they mostly rely on
handcrafted graph attributes and/or respective graph matching cost functions
based on affinity scores. In [59], a learned-based approach that directly
obtains affinity scores from data was introduced. More recently, the
differentiation of the power iteration method has been considered in a deep
graph matching approach [58]. A more general blackbox differentiation approach
was introduced in [41]. Various other deep learning approaches have been
proposed for graph matching [28, 20], and some approaches also address image
key point matching [51, 60, 55]. In this case, optimal graph features are
directly learned from the image appearance and/or 2D image coordinates, while
simultaneously solving graph matching in an end-to-end manner. Although these
methods consider geometric consistency, they are tailored towards matching a
pair of graphs and thus lead to cycle-inconsistent matchings when pairwise
matchings of more than two graphs are computed.
Synchronisation and Multi-Matching. Cycle-consistency is often obtained as a
post-processing step after obtaining pairwise matchings. Methods that
establish cycle consistency in the set of pairwise matchings are commonly
referred to as permutation synchronisation methods [37, 62, 35, 6, 8]. There
are also methods for directly obtaining cycle-consistent multi-matchings [49,
50, 7]. Recently, permutation synchronisation has also been considered in a
deep graph matching framework, where a separate permutation synchronisation
module is utilised to generalise a two-graph matching approach to the matching
of multiple graphs [52]. However, when applying such multi-matching approaches
to image key point matching they have the significant shortcoming that they
ignore the underlying 3D geometry of the 2D points. This makes it extremely
difficult to establish correct matchings across images, which after all depict
2D projections of 3D objects in different poses, possibly even under starkly
varying perspective projections.
3D Reconstruction. Obtaining 3D geometric information from 2D data is a widely
studied problem known as 3D reconstruction. When relying on a monocular input
only, 3D reconstruction is generally an ill-posed problem. To circumvent this,
statistical 3D models [9, 32, 39] are commonly used as priors in such
reconstruction tasks. Reconstruction from a single image or video using a
deformable 3D prior has for example been achieved by fitting a 3D morphable
model of a specific object class such as humans bodies, faces, or cars, and
then finding the parameters of the model that best explain the image [47, 10,
53]. However, the availability of a suitable 3D prior is a rather strong
assumption.
Video-based methods are able to utilise temporal information and temporal
consistency constraints to reconstruct dynamic scenes. For the specific object
class of faces, recently a method was proposed that is able to simultaneously
learn a 3D face prior model while performing 3D reconstruction [46]. There are
also some methods that do not assume explicit 3D prior models, but they
exploit different geometric and photometric properties of 3D shapes to
reconstruct 3D information from images [21, 48, 56].
An alternative to address the ill-posedness of single-view reconstruction is
to consider multiple views. Traditional static scene reconstruction methods
use a collection of images from one or many cameras to match the images,
obtain camera parameters, and reconstruct an object [43, 42, 1].
More recent methods for multi-view reconstruction assume camera parameters and
use self-supervised learning based on a neural renderer to reconstruct static
and dynamic objects with novel 3D representations [36, 40, 31, 44]. A downside
of multi-view reconstruction methods is that they require many different
images of the same object, which is often hard to obtain and unavailable in
existing datasets.
Contrary to existing approaches, we simultaneously solve deep multi-graph
matching and infer sparse 3D geometry fromi nhomogeneous 2D image collections.
In particular, our approach obtains cycle-consistent multi-matchings and does
not rely on a hand-crafted template or any other prior 3D model.
## 3 Problem Formulation & Preliminaries
Symbol | Meaning
---|---
$N$ | number of graphs/images
$m_{j}$ | number of vertices in $j$-th graph $\mathcal{G}_{j}$
$d$ | number of vertices in universe graph $\mathcal{U}$
$X_{jk}$ | pairwise matching between $\mathcal{G}_{j}$ and $\mathcal{G}_{k}$
$X_{j}$ | matching between $\mathcal{G}_{j}$ and $\mathcal{U}$
$U$ | homogeneous coordinates of universe points
$\mathcal{V}_{j}$ | homogeneous coordinates of pixels in $j$-th image
$j,k,l$ | indices used to refer to images and graphs
$i$ | index used to refer to the $i$-th point
Table 1: Overview of our notation.
In this section we summarise how to achieve cycle-consistency for multiple
graph matching by utilising the notion of universe points. In order to
explicitly construct such universe points, we formalise it as a sparse
reconstruction of the 3D key points from multiple 2D images.
### 3.1 Multi-Matching and Cycle Consistency
Given is the set $\\{\mathcal{G}_{j}\\}_{j=1}^{N}$ of $N$ undirected graphs,
where each graph $\mathcal{G}_{j}=(\mathcal{V}_{j},\mathcal{E}_{j})$ comprises
of a total of $m_{j}$ nodes $\mathcal{V}_{j}=\\{v_{1},\dots,v_{m_{j}}\\}$ and
a total of $n_{j}$ edges $\mathcal{E}_{j}=\\{e_{1},\dots,e_{n_{j}}\\}$ that
connect pairs of nodes in $\mathcal{V}_{j}$. We assume that each node
represents an image key point, and that the node $v_{i}\in\mathbb{R}^{2}$ is
identified with the respective 2D image coordinates. The pairwise graph
matching problem is to find a node correspondence
$X_{jk}\in\mathbb{P}_{m_{j}m_{k}}$ between $\mathcal{G}_{j}$ and
$\mathcal{G}_{k}$. Here, $\mathbb{P}_{m_{j}m_{k}}$ is the set of
$(m_{j}{\times}m_{k})$-dimensional partial permutation matrices.
Let $\mathcal{X}=\\{X_{jk}\in\mathbb{P}_{m_{j}m_{k}}\\}_{j,k=1}^{N}$ be the
set of pairwise matching between all graphs in
$\\{\mathcal{G}_{j}\\}_{j=1}^{N}$. The set $\mathcal{X}$ is said to be cycle-
consistent if for all $j,k,l\in\\{1,\dots,N\\}$, the following three
properties hold [22, 49, 6]:
1. 1.
$X_{jj}=\mathbb{I}_{m_{j}}$, where $\mathbb{I}_{m_{j}}$ denotes the identity
matrix of size $m_{j}$.
2. 2.
$X_{jk}=X_{kj}^{T}$.
3. 3.
$X_{jk}X_{kl}\preceq X_{jl}$ (element-wise).
When solving multi-graph matchings in terms of pairwise matching, cycle
consistency is desirable since it is an intrinsic property of the (typically
unknown) ground truth matching. Rather then explicitly imposing the above
three constraints, it is possible to achieve cycle consistency by representing
the pairwise matching in terms of a universe graph [22, 49, 6]:
###### Lemma 1
The set $\mathcal{X}$ of pairwise matchings is cycle-consistent, if there
exists a collection
$\\{X_{j}\in\mathbb{P}_{m_{j}d}:X_{j}\mathbf{1}_{d}=\mathbf{1}_{m_{j}}\\}_{j=1}^{N}$
such that for each $X_{jk}\in\mathcal{X}$ it holds that
$X_{jk}=X_{j}X_{k}^{T}$.
Here, the $X_{j}$ can be considered as the pairwise matching between the graph
$\mathcal{G}_{j}$ and a universe graph $\mathcal{U}=(\mathcal{V},\mathcal{E})$
with similar definition to $\mathcal{G}_{j}$, where
$\mathcal{V}=\\{u_{1},\dots,u_{d}\\}$ denote the universe points and
$\mathcal{E}=\\{e_{1},\dots,e_{n}\\}$ the universe edges. Intuitively, the
universe graph can be interpreted as assigning each point in $\mathcal{G}_{j}$
to exactly one of the $d$ universe points in $\mathcal{U}$. Therefore, rather
than modelling the cubic number of cycle consistency constraints on
$\\{\mathcal{G}_{j}\\}_{j=1}^{N}$ explicitly, we use an object-to-universe
matching formulation based on the $\\{X_{j}\\}_{j=1}^{N}$.
### 3.2 3D Reconstruction
Though the idea of the universe graph is a crucial ingredient for
synchronisation approaches [37, 22, 6], the universe graph is never explicitly
instantiated in these methods. That is because it is merely used as an
abstract entity that must exist in order to ensure cycle consistency in multi-
matchings. Considering that the graphs in this work come from image
collections, we assume that the nodes $u_{i}\in\mathbb{R}^{3}$ of the universe
graph represent 3D points, which will allow us to address their explicit
instantiation based on multiple-view geometry.
We denote the homogeneous coordinate representation of the universe point
$u_{i}\in\mathbb{R}^{3}$ (represented in world coordinates) as
$U_{i}=(u_{i},1)\in\mathbb{R}^{4}$. Its projection onto the $j$-th image
plane, denoted by $\mathcal{V}_{ij}=(v_{ij},1)\in\mathbb{R}^{3}$, is given by
$\mathcal{V}_{ij}=\lambda_{ij}K_{j}\underbrace{\left(\begin{matrix}1&0&0&0\\\
0&1&0&0\\\
0&0&1&0\end{matrix}\right)}_{\Pi_{0}}\underbrace{\left(\begin{matrix}R_{j}&T_{j}\\\
0&1\end{matrix}\right)}_{g_{j}}U_{i}.$ (1)
Here, $g_{j}$ represents the world-to-camera space rigid-body transformation
comprising of the rotation $R_{j}\in\mathbb{R}^{3\times 3}$ and the
translation $T_{j}\in\mathbb{R}^{3}$, $\Pi_{0}$ is the canonical projection
matrix, $K_{j}\in\mathbb{R}^{3\times 3}$ is the intrinsic camera matrix, and
$\lambda_{ij}\in\mathbb{R}$ is the scale parameter. For brevity, we define the
general projection matrix $\Pi_{j}=K_{j}\Pi_{0}g_{j}$. Let
$U\in\mathbb{R}^{4\times d}$ be the stacked universe points in homogeneous
coordinates, let $\mathcal{V}_{j}\in\mathbb{R}^{3\times d}$ be the respective
projection onto the $j$-th image plane $j$, and let
$\Lambda_{j}=\operatorname{diag}(\lambda_{1j},\ldots,\lambda_{dj})\in\mathbb{R}^{d\times
d}$ be the diagonal matrix with the $\lambda_{ij}$ on its diagonal. The matrix
formulation of Eq. (1) is
$\mathcal{V}_{j}=\Pi_{j}U\Lambda_{j}.$ (2)
Once we have a collection of $N$ images of different objects from the same
category (not necessarily the same object instance, e.g. two images of
different bicycles), reconstructing the universe points $U$ can be phrased as
finding the minimiser of the least-squares problem
$\displaystyle\operatorname{arg\,min}_{U}\sum_{j=1}^{N}||\Pi_{j}U\Lambda_{j}-\mathcal{V}_{j}||_{F}^{2}.$
(3)
Note that in practice the variables $U,\\{\Lambda_{j}\\}$ and $\\{\Pi_{j}\\}$
are generally unknown, so that without further constraints this is an under-
constrained problem. In the next section, we will elaborate on how we approach
this.
Figure 2: Overview of our algorithm. Given an image with features, we predict
the corresponding features in 3D as a combination of universe 3D points and
deformations. The universe 3D points are learned during training for a given
class of objects, while the deformations are predicted per image. We create
edges for among the 2D and 3D features and find a matching between the two
graphs using a graph matching network. Since the matchings are between
universe points and images, our algorithm is cycle consistent.
## 4 Proposed Method
Our learning framework consists of four main components. The first two
components have the purpose to obtain 3D universe points, along with a
deformation of these 3D points representing the underlying 3D structure of the
2D key points in the $j$-th image. The purpose of the other two components is
to predict the matching between the 2D points of $\mathcal{G}_{j}$ and the 3D
points of $\mathcal{U}$. Thus, rather than learning pairwise matchings between
$\mathcal{G}_{j}$ and $\mathcal{G}_{k}$, we utilise an object-to-universe
matching formulation. Therefore, the underlying 3D structure and cycle-
consistent multi-matchings are both attained by our method. The whole pipeline
is illustrated in Fig. 2 and comprises the following four main components:
1. 1.
Learnable 3D Universe Points: the 2D key points
$\\{\mathcal{V}_{j}\\}_{j=1}^{N}$ of all images in the collection are used to
reconstruct the 3D universe points $U$ by incorporating a reconstruction loss
that approximates Eq. (3).
2. 2.
Deformation Module: the retrieved universe points $U$ are static and therefore
they cannot accurately model the geometric variability present in different
instances of an object from the same category (e.g. different bicycles). To
address this, the universe points are non-linearly deformed by the deformation
module that takes the 2D points and the (learned) 3D universe points as input.
3. 3.
Assignment Graph Generation: by connecting the 2D points and universe points,
respectively, the 2D graph and the 3D universe graph are constructed. The
assignment graph is then constructed as the product of these two graphs.
4. 4.
Graph Matching Network: a graph matching network performs graph convolutions
on the assignment graph, and eventually performs a binary node classification
on the assignment graph representing the matching between the 2D graph and the
universe graph.
### 4.1 Learnable 3D Universe Points
As discussed in Sec. 3.2, the universe points can be retrieved by minimising
(3). This problem, however, is generally under-determined, since
$U,\\{\Lambda_{j}\\}$ and $\\{\Pi_{j}\\}$ in (3) are generally unknown in most
practical settings. Additionally, although all objects share a similar 3D
geometry, the nonlinear deformations between different instances are
disregarded in (3). Thus, instead of an exact solution we settle for an
approximation that we later refine in our pipeline. To this end, we assume a
weak perspective projection model, i.e. all universe points are assumed to
have the same distance from the camera. With this condition, the diagonal of
$\Lambda_{j}$ is constant and can absorbed into $\Pi_{j}$. This leads to the
least-squares problem
$\operatorname{arg\,min}_{U}\sum_{j=1}^{N}||\Pi_{j}U-\mathcal{V}_{j}||_{F}^{2},$
(4)
which can be solved in an end-to-end manner during neural network training
based on commonly used ‘backpropagable’ pseudo-inverse implementations. The
variable $\Pi_{j}$ can be written as $\Pi_{j}=\mathcal{V}_{j}U^{+}$ where
$U^{+}$ is the right pseudo-inverse that satisfies $UU^{+}=\mathbb{I}_{4}$.
Therefore, we introduce the reconstruction loss
$\mathcal{L}_{\text{rec}}=\frac{1}{N}\sum_{j=1}^{N}||\mathcal{V}_{j}U^{+}U-\mathcal{V}_{j}||^{2}_{F}.$
(5)
### 4.2 Deformation Module
The universe points retrieved in the previous step are static, meaning that
they can only reflect the rough geometric information of the underlying 3D
object, but that they are unable to accurately represent finer-scale
variations between different instances of the same object category.
Thus, we introduce the deformation module to learn an additional nonlinear
deformation.
This modules takes the universe points $U$ and the 2D points $\mathcal{V}_{j}$
as input. As shown in the bottom left of Fig. 2, $\mathcal{V}_{j}$ is passed
to a 2D Point encoder. The encoder first performs a nonlinear feature
transform of all input points based on multi-layer perceptron (MLP), and then
performs a max pooling to get a global feature representing the input object.
As can be seen in the top left in Fig. 2, an MLP is utilised to perform a
nonlinear feature transform for each of the 3D points in $U$. Each 3D point
feature is then concatenated with the same global feature from the 2D Point
Encoder. The concatenated per 3D point features are fed into an MLP to compute
the motion of each point. The output is a set of per-point offsets
$S\in\mathbb{R}^{3\times d}$ that are added to $U$ to generate the deformed 3D
universe points. The computation of the per-point offsets is summarised as
$\displaystyle S_{j}$
$\displaystyle=\text{MLP}\left(\text{MLP}(U)\circ\text{Encoder}(\mathcal{V}_{j})\right),$
(6)
where $\circ$ represents the concatenation operation.
Similarly as for the reconstruction loss, the projection of the deformed
universe points onto the image plane should be close to the observed 2D
points. Additionally, since the static 3D universe points should reflect the
rough geometry of the underlying 3D object, the offset $S_{j}$ should be
small. Therefore, we introduce the deformed reconstruction loss and the offset
regulariser as
$\displaystyle\mathcal{L}_{\text{def}}$
$\displaystyle=\frac{1}{N}\sum_{j=1}^{N}||\mathcal{V}_{j}(U{+}S_{j})^{+}(U{+}S_{j})-\mathcal{V}_{j}||^{2}_{F},\text{
and}$ (7) $\displaystyle\mathcal{L}_{\text{off}}$
$\displaystyle=||S_{j}||^{2}_{F}.$ (8)
### 4.3 Assignment Graph Generation
To obtain graphs from the 2D points and the deformed 3D universe points,
respectively, we apply the Delaunay algorithm [11] to generate edges, see Fig.
2.
Moreover, we define the attribute of each edge as the concatenation of the
respective coordinates of the respective connecting points.
Once the 3D universe graph $\mathcal{U}$ and the 2D graph $\mathcal{G}_{j}$
are generated, we construct the assignment graph $\mathcal{G}^{A}_{j}$ as the
product graph of $\mathcal{U}$ and $\mathcal{G}_{j}$ following [25]. To be
more specific, the nodes in $\mathcal{G}^{A}_{j}$ are defined as the product
of the two node sets $\mathcal{V}_{j}$ and $\mathcal{V}$ of $\mathcal{G}_{j}$
and $\mathcal{U}$, respectively, i.e.
$\mathcal{V}^{A}_{j}=\\{v_{jk}:v_{jk}=(v_{j},u_{k})\in\mathcal{V}_{j}\times\mathcal{V}\\}$.
The edges in $\mathcal{G}^{A}_{j}$ are built between nodes $v_{jk}$,
$v_{mn}\in\mathcal{V}^{A}_{j}$ if and only if there is an edge between $v_{j}$
and $v_{m}$ in $\mathcal{E}_{j}$, as well as between $u_{k}$ and $u_{n}$ in
$\mathcal{E}$. The attribute of each node and edge in $\mathcal{G}^{A}_{j}$ is
again the concatenation of the attribute of corresponding nodes and edges in
$\mathcal{G}_{i}$ and $\mathcal{U}$, respectively.
### 4.4 Graph Matching Network
The graph matching problem is converted to a binary classification problem on
the assignment graph $\mathcal{G}^{A}$. For example, an assignment graph is
shown on the top right of Fig. 2. Classifying nodes $\\{1c,2b,3a\\}$ as
positive equals to matching point $1$ to $c$, $2$ to $b$ and $3$ to $a$, where
numeric nodes correspond to the 2D graph, and alphabetic nodes correspond the
3D universe graph.
The assignment graph is then passed into the graph matching network [55],
which encodes the attributes of the assignment graph into a latent
representation. This is achieved by alternatingly applying edge convolutions
and node convolutions. The edge convolution assembles the attributes of the
connected nodes, while the node convolution aggregates the information from
its adjacent edges and updates the attributes of each node. The overall
architecture is based on the graph network from [2].
Car (Willow Dataset) | Duck (Willow Dataset)
---|---
| | | | | | |
Bicycle (Pascal VOC Dataset) | Cow (Pascal VOC Dataset)
| | | | | | |
Figure 3: Qualitative results of the proposed method on the Willow Dataset and
Pascal VOC Dataset. It can be seen that our method achieves accurate results
for non-deformable objects of different types (car, bike), and that it
achieves still reasonable results for different instances of articulated
objects (duck, cow).
### 4.5 Loss Function
Similarly as existing deep graph matching approaches, we train our network in
a supervised way based on the ground-truth matching matrix $X_{j}^{\text{gt}}$
between $\mathcal{G}_{j}$ and $\mathcal{U}$. To this end, we adopt the mean
squared loss to compute a matching loss
$\mathcal{L}_{\text{match}}=\frac{1}{N}\sum_{j=1}^{N}||X_{j}^{\text{gt}}-X_{j}||^{2}.$
(9)
Furthermore, one-to-one matching is a reasonable prior for graph matching,
which was already adopted in previous work [54, 55]. To include a one-to-one
matching soft-constraint, we first convert the predicted permutation matrix
$X_{j}$ to a binary node label matrix $Y_{j}\in\\{0,1\\}^{m_{j}d\times 2}$
that we define as
$Y_{j}=\begin{pmatrix}1{-}\text{vec}(X_{j}),\text{vec}(X_{j})\end{pmatrix}.$
(10)
Here, $\text{vec}(X_{j})$ is the vectorisation of $X_{j}$. We can compute the
corresponding index vector $y_{j}\in\\{0,1\\}^{m_{j}d}$ defined as
$(y_{j})_{i}=\operatorname{arg\,max}_{k\in\\{1,2\\}}(Y_{j})_{ik}.$ (11)
By leveraging the auxiliary matrix $B\in\\{0,1\\}^{(m_{j}+d)\times m_{j}d}$
and the ground-truth permutation matrix $X_{j}^{\text{gt}}$ [54], the one-to-
one matching regularisation is
$\mathcal{L}_{\text{reg}}=||B(y-\text{vec}(X_{j}^{\text{gt}}))||^{2}.$ (12)
The total loss that we minimise during training is
$\displaystyle\mathcal{L}=$
$\displaystyle~{}\omega_{\text{m}}\mathcal{L}_{\text{match}}+\omega_{\text{d}}\mathcal{L}_{\text{def}}+\omega_{\text{r}}\mathcal{L}_{\text{rec}}+\omega_{\text{o}}\mathcal{L}_{\text{off}}+\omega_{\text{reg}}\mathcal{L}_{\text{reg}}.$
(13)
### 4.6 Training
We train a single network that is able to handle multiple object categories at
the same time. To this end, we learn separate 3D universe points for each
category, and in addition we introduce a separate learnable linear operator
for each category that is applied to the global feature obtained by the 2D
Point Encoder. The linear operator has the purpose to transform the generic
global feature to a category-specific representation, and also helps in
resolving ambiguities between categories with objects that are somewhat
similar (e.g. cat and dog).
In practice, we apply a warm start for learning the universe points
$\mathcal{U}$, which are randomly initialised for each category. To this end,
at the beginning of the training, $\omega_{\text{r}}$ is set to 1 and all
other weights are 0. After a certain number of iterations, 4k in our setting,
we replace the reconstruction loss $\mathcal{L}_{\text{rec}}$ by the deformed
reconstruction loss $\mathcal{L}_{\text{def}}$ by setting $\omega_{r}=0$ ,
$\omega_{\text{m}}=1.0$, $\omega_{\text{d}}=0.5$, $\omega_{\text{o}}=0.05$ and
$\omega_{\text{reg}}=0.1$ (in all our experiments). The batch size is 16 and
the number of iterations after warm start is 150k. The learning rate is
$0.008$ and scheduled to decrease exponentially by 0.98 after each 3k
iterations.
## 5 Experiments
In the following, we evaluate our method in various settings. We compare our
method to different state-of-the-art methods on two datasets, and we evaluate
our deformation module based on a dataset of 3D objects. Additional results
can be found in the supplementary material.
### 5.1 Comparisons to State of the Art
For the comparison experiments, we follow the testing protocol that was used
in CSGM [55]. In the following, we summarise the experimental setting for each
dataset and discuss our results. Parts of the matching results are visualised
in Fig. 3
#### 5.1.1 Willow Dataset
We simultaneously train our model for all categories of the Willow dataset
[14]. The dataset consists of images from $5$ classes, car, duck, face,
motorbike, and wine bottle. It is compiled from Caltech-256 and Pascal VOC
2007 datasets, and it consists of more than $40$ images per class with $10$
manually labelled distinctive features each.
To compare our method against existing methods, we use the same training/test
split as in CSGM [55]. For training, $20$ images are randomly chosen from each
class and the rest are used for testing. For non-learning based methods, the
affinity matrix is constructed using the appearance similarity of SIFT
descriptors [33] as done in [54]. Delaunay triangulation is applied to build
graph edges. Each edge $(k,l)\in\mathcal{E}_{j}$ requires two features
$w_{kl}$ and $\theta_{kl}$, where $w_{kl}$ is the pairwise distance between
the connected nodes $v_{k}$ and $v_{l}$, and $\theta_{kl}$ is the absolute
angle between the edge and the horizontal line with
$0\leq\theta_{kl}\leq\pi/2$. The edge affinity between edges $(k,l)$ in
$\mathcal{G}_{1}$ and $(a,b)$ in $\mathcal{G}_{2}$ is computed as
$e_{(k,a),(l,b)}=\text{exp}(-(|w_{kl}-w_{ab}|+|\theta_{kl}-\theta_{ab}|)/2)$.
In our method, we use the 2D key point coordinates as attributes of nodes in
$\mathcal{G}_{i}$, while the nodes in $\mathcal{U}$ have the 3D coordinates of
the (learned) universe points as attributes. The attributes of edges are the
concatenation of the coordinates of connected nodes. The attribute of nodes
and edges in the assignment graph is described in Sec. 4.3.
| car | duck | face | motor. | wine. | avg. acc. | cyc.-cons.
---|---|---|---|---|---|---|---
IPFP [26] | 74.8 | 60.6 | 98.9 | 84.0 | 79.0 | 79.5 | ✗
RRWM [15] | 86.3 | 75.5 | 100 | 94.9 | 94.3 | 90.2 | ✗
PSM [19] | 88.0 | 76.8 | 100 | 96.4 | 97.0 | 91.6 | ✗
GNCCP [29] | 86.4 | 77.4 | 100 | 95.6 | 95.7 | 91.0 | ✗
ABPF [54] | 88.4 | 80.1 | 100 | 96.2 | 96.7 | 92.3 | ✗
HARG [14] | 71.9 | 72.2 | 93.9 | 71.4 | 86.1 | 79.1 | ✗
GMN [58] | 74.3 | 82.8 | 99.3 | 71.4 | 76.7 | 80.9 | ✗
PCA [51] | 84.0 | 93.5 | 100 | 76.7 | 96.9 | 90.2 | ✗
CSGM [55] | 91.2 | 86.2 | 100 | 99.4 | 97.9 | 94.9 | ✗
Ours | 98.8 | 90.3 | 99.9 | 99.8 | 100 | 97.8 | ✓
Table 2: Matching accuracy on the Willow dataset. Our method beats the
previous state of the art and achieves the best average accuracy.
Table 2 shows the accuracy of our method compared to other methods on the
Willow dataset. It can be seen that on an average, our method achieves an
accuracy of $97.8\%$, outperforming other methods by at least $2.9\%$, while
also being able to reconstruct the 3D structure of objects, see Fig. Joint
Deep Multi-Graph Matching and 3D Geometry Learning from Inhomogeneous 2D
Image Collections. In the car category, our method outperforms the others
noticeably. Additionally, although there is non-rigid motion in the duck
category that is cause by articulation, our method can still achieve a
reasonable accuracy. Further, our method is the only one that can guarantee
cycle-consistent matchings thanks to the object-to-universe matching
formulation.
#### 5.1.2 Pascal VOC Keypoints Dataset
The Pascal VOC Keypoints dataset [12] contains $20$ categories of objects with
labelled key point annotations. The number of key points varies from 6 to 23
for each category. Following [55], we use $7020$ images for training and
$1682$ for testing. The accuracy computations are the same as described in
Sec. 5.1.1.
| aero | bike | bird | boat | bottle | bus | car | cat | chair | cow | table | dog | horse | mbike | person | plant | sheep | sofa | train | tv | avg. acc. | cyc.-cons.
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
GMN [58] | 31.9 | 47.2 | 51.9 | 40.8 | 68.7 | 72.2 | 53.6 | 52.8 | 34.6 | 48.6 | 72.3 | 47.7 | 54.8 | 51.0 | 38.6 | 75.1 | 49.5 | 45.0 | 83.0 | 86.3 | 55.3 | ✗
PCA [51] | 40.9 | 55.0 | 65.8 | 47.9 | 76.9 | 77.9 | 63.5 | 67.4 | 33.7 | 65.5 | 63.6 | 61.3 | 68.9 | 62.8 | 44.9 | 77.5 | 67.4 | 57.5 | 86.7 | 90.9 | 63.8 | ✗
CSGM [55] | 46.9 | 58.0 | 63.6 | 69.9 | 87.8 | 79.8 | 71.8 | 60.3 | 44.8 | 64.3 | 79.4 | 57.5 | 64.4 | 57.6 | 52.4 | 96.1 | 62.9 | 65.8 | 94.4 | 92.0 | 68.5 | ✗
Ours | 41.1 | 64.7 | 64.9 | 64.6 | 94.5 | 79.3 | 69.9 | 56.6 | 66.0 | 68.6 | 69.5 | 56.8 | 53.6 | 67.2 | 57.3 | 98.3 | 54.2 | 58.8 | 86.3 | 69.7 | 67.1 | ✓
Table 3: Results on Pascal VOC Keypoints dataset [12]. Note that in terms of
accuracy we achieve comparable results to the previous state of the art method
CSGM [55], while we are the only one that additionally achieves cycle
consistency.
We randomly sample from the training data to train our model. In Table 3 it
can be seen that in terms of matching accuracy our method is almost on par
with the previous state of the art [55]. Moreover, if a key point point is not
visible in both images that are considered for obtaining the pairwise
matching, the other methods discard this point and merely aim to find
correspondences between key points common in both images. This procedure is
not necessary for our method because the universe graph contains all possible
key points in one category. Additionally, unlike any earlier methods, our
method achieves cycle-consistent matchings, and also infers 3D geometry. We
emphasise that accuracy alone is not a fully descriptive measure of
performance, since the ground truth matchings must be cycle-consistent.
Figure 4: Illustration of 3D universe points obtained from 2D projections of
D3DFACs heads. Left: the coarse 3D universe points (error to ground truth:
0.3388). Right: the 3D universe points after refinment via our deformation
module (error to ground truth: 0.0905).
### 5.2 3D Geometry and Deformation Evaluation
As our network learns 3D universe points along with a nonlinear deformation
model, we evaluate our 3D deformation model by training it on samples from the
3D head dataset D3DFACs [16, 27]. We use a similar pre-processing pipeline as
in i3DMM [57] to obtain $8$ facial landmarks on each head in the dataset. We
refer interested readers to [57] for more details. Please note that the
landmarks are all in correspondence, as we use a template-registered dataset
[27]. For our evaluation, we compute 2D projections of the 3D landmarks using
a pinhole camera model and randomly sampled rotations and translations.
The goal of this experiment is to show that reconstructed 3D universe points
are plausible, and that our deformation module is indeed useful for model
nonlinear deformations. As shown in Fig. Joint Deep Multi-Graph Matching and
3D Geometry Learning from Inhomogeneous 2D Image Collections (right) and Fig.
4, the learned 3D universe points (before applying the deformation module) can
only reflect rough 3D geometry. Subsequently, the deformation module is able
to refine the 3D universe points. The average error between the ground truth
3D points and the obtained 3D universe points are 0.356 before the deformation
module, and 0.148 after the deformation module, confirming the merits of the
deformation module. Although the reconstructed 3D geometry is not perfect,
considering that we are inferring 3D geometry from multiple images that depict
different instances of an object, we believe the obtained results clearly
confirm the utility of our approach.
## 6 Conclusion
We have introduced a novel graph neural network approach that jointly learns
multi-graph matching and 3D geometry from an inhomogeneous image collection.
For the first time, we achieve several favourable properties simultaneously:
Our matchings are guaranteed to be cycle-consistent, which is an important
property since the (unknown) ground truth matchings are cycle-consistent. Our
approach does not rely on the availability of an initial 3D geometry model, so
that we can train it on virtually any object category, as opposed to object-
specific approaches that are for example tailored towards faces only [47].
Instead, during training we learn a (sparse) deformable 3D geometric model
directly from 2D image data. Moreover, our methods merely requires multiple
images of _different object instances_ of the same category. This is in
contrast to typical multi-view reconstruction approaches that require multiple
images of the _same object_ from different views. We believe that the joint
consideration of deep graph matching and 3D geometry inference will open up
interesting new research directions and that our approach may serve as
inspiration for follow-up works on matching problems, 3D reconstruction, 3D
deformation model learning, and many more.
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## Appendix A Ablation Study
To confirm the importance of the individual components of our approach we
conducted an ablation study. To this end we evaluate the accuracy on the
Pascal VOC dataset in cases where we omit individual terms of the loss
function, omit the warm start for learning the universe points $\mathcal{U}$,
and omit deformation module, see Table 4.
When we omit the one-to-one matching regulariser by setting
$\omega_{\text{reg}}$ to 0, the matching accuracy is depressed substantially.
When we do not conduct a warm start for finding initial universe points, the
matching accuracy deteriorates. Similarly, the matching accuracy lowers
without the use of our deformation module. Further, the offset regularisation
and the deformed reconstruction loss can refine the universe points for each
object, which brings a better matching accuracy as shown in the last two
experiments. Overall, the accuracy is highest when using all components
together.
## Appendix B Cycle Consistency
We further provide quantitative evaluations of the cycle consistency on the
Pascal VOC dataset, as shown in Table 5. We quantify in terms of the cycle
consistency score, which is computed as follows:
1. 1.
Given three graphs $\\{\mathcal{G}_{j}\\}$, $\\{\mathcal{G}_{k}\\}$ and
$\\{\mathcal{G}_{l}\\}$, we use the trained network to predict $X_{jk}$,
$X_{jl}$ and $X_{kl}$.
2. 2.
We compute the composed pairwise matching between $\\{\mathcal{G}_{k}\\}$ and
$\\{\mathcal{G}_{l}\\}$ by $X^{\prime}_{kl}=X_{jk}^{T}X_{jl}$.
3. 3.
We denote the number of points that $X^{\prime}_{kl}$ equals to $X_{kl}$ as
$m_{\text{cycle}}$ and the number of points in $X_{kl}$ as $m_{kl}$. The cycle
consistency score is then computed as
$\text{cycle consistency score}=100\times\frac{m_{\text{cycle}}}{m_{kl}}\%.$
(14)
Note that in this case, we only consider the common points that are observed
in $\\{\mathcal{G}_{j}\\}$, $\\{\mathcal{G}_{k}\\}$ and
$\\{\mathcal{G}_{l}\\}$.
Loss | Avg. acc.
---|---
$\omega_{\text{reg}}=0$ | 58.11
w/o warm start | 58.49
w/o deformation module | 60.33
$\omega_{\text{o}}=0$ | 64.19
$\omega_{\text{d}}=0$ | 64.81
Ours | 67.1
Table 4: Matching accuracy on the Pascal VOC dataset with the variants on
regularisation terms or training strategies. PCA [51]CSGM [55]Ours$0$ %$20$
%$40$ %$60$ %$80$ %$100$ %matching accuracycycle consistency score Figure 5:
The average matching accuracy and cycle consistency score of three learning-
based methods on Pascal VOC dataset.
| aero | bike | bird | boat | bottle | bus | car | cat | chair | cow | table | dog | horse | mbike | person | plant | sheep | sofa | train | tv | Avg.
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
PCA [51] | 40.92 | 15.48 | 44.91 | 45.30 | 14.55 | 41.83 | 55.97 | 42.97 | 35.99 | 44.30 | 41.59 | 49.10 | 43.68 | 33.33 | 35.04 | 24.67 | 53.93 | 45.87 | 44.00 | 29.39 | 39.19
CSGM [55] | 49.08 | 51.50 | 60.13 | 67.84 | 81.13 | 80.36 | 67.40 | 57.10 | 51.26 | 61.42 | 56.16 | 55.28 | 61.61 | 54.17 | 54.57 | 96.84 | 60.71 | 58.30 | 96.6 | 93.60 | 65.75
Ours | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100 | 100
Table 5: Cycle consistency scores (in percent) of three learning-based methods
on the Pascal VOC Keypoints dataset. Our method is the only one that
guarantees cycle consistency for all categories.
In Fig. 5, we show the average matching accuracy and cycle consistency score
of our method and compare it with PCA [51] and CSGM [55]. It is clear that our
method can achieve comparable accuracy and the best cycle consistency at the
same time.
## Appendix C More Deformation Results
We provide more qualitative results for our deformation module, see Fig. 6. As
shown in the figure, the deformation module is able to refine the 3D universe
points. Although 3D reconstructions are not perfect, we can observe that they
represent the overall 3D structure well, and are thus valuable for matching
respective key points.
$\mathcal{G}$ | Right View | Front View | Left View | $\mathcal{G}$ | Right View | Front View | Left View
---|---|---|---|---|---|---|---
Ground Truth | | | | Universe Points | | |
Case 1 | | | | Case 2 | | |
Case 3 | | | | Case 4 | | |
Case 5 | | | | Case 6 | | |
Case 7 | | | | Case 8 | | |
Figure 6: Qualitative results of deformation module. The top-left part shows
the ground truth points on a reference shape, and the top-right part shows the
universe points before the deformation module is applied. The remaining parts
show individual cases, where it can be seen that the deformation module
adequately deforms the universe points (top right), and that it is able to
approximate the overall 3D geometry of the face well.
|
# T-STAR: Truthful Style Transfer using AMR Graph as Intermediate
Representation
Anubhav Jangra Preksha Nema* Aravindan Raghuveer
Google Research, India
<EMAIL_ADDRESS>
denotes equal contribution
###### Abstract
Unavailability of parallel corpora for training text style transfer (TST)
models is a very challenging yet common scenario. Also, TST models implicitly
need to preserve the content while transforming a source sentence into the
target style. To tackle these problems, an intermediate representation is
often constructed that is devoid of style while still preserving the meaning
of the source sentence. In this work, we study the usefulness of Abstract
Meaning Representation (AMR) graph as the intermediate style agnostic
representation. We posit that semantic notations like AMR are a natural choice
for an intermediate representation. Hence, we propose T-STAR: a model
comprising of two components, text-to-AMR encoder and a AMR-to-text decoder.
We propose several modeling improvements to enhance the style agnosticity of
the generated AMR. To the best of our knowledge, T-STAR is the first work that
uses AMR as an intermediate representation for TST. With thorough experimental
evaluation we show T-STAR significantly outperforms state of the art
techniques by achieving on an average $15.2$% higher content preservation with
negligible loss ($\sim$3%) in style accuracy. Through detailed human
evaluation with $90,000$ ratings, we also show that T-STAR has upto $50$%
lesser hallucinations compared to state of the art TST models.
## 1 Introduction
A well accepted definition of style refers to the manner (via linguistic
elements like word choices, syntactic structures, metaphors) in which
semantics of a sentence are expressed McDonald and Pustejovsky (1985); Jin et
al. (2020). Text Style Transfer (TST) is the task of rephrasing an input
sentence to contain specific stylistic properties without altering the meaning
of the sentence Prabhumoye et al. (2018). We refer the reader to Jin et al.
(2020) for a detailed survey of approaches towards TST problem formulation,
metrics and models. In the practical scenario, that we consider in this paper,
where a large corpus of parallel data is not available Niu and Bansal (2018);
Ma et al. (2020); Wu et al. (2020), two family of approaches have been
proposed in the literature Jin et al. (2020). 1\. Disentanglement: Content and
Style are disentangled in a latent space and only the style information is
varied to transform the sentence. 2\. Prototype Editing: Style bearing words
in the source sentence are replaced with those corresponding to the target
style. The sentence may be further re-arranged for fluency and naturalness.
Both the above approaches have drawbacks in the way the style agnostic
intermediate representation is constructed, described as follows. First, in
the disentangling approaches, it is not easy to verify the efficacy of
separation between style and content. Recent approaches Subramanian et al.
(2018); Samanta et al. (2021) even state that as content and style are so
subtly entangled in text, it is difficult to disentangle both of them in a
latent space. Consequently, this affects the model’s interpretability in that
it is hard to attribute an effect we see in the output to the latent
intermediate vector or the style vector. Second, with prototype editing
approaches for lingustic styles (such as author, formality), where the content
and style are tightly coupled it is not feasible to segregate style and
content carrying words (word examples: cometh, thou). Furthermore, style-
marker detection is a non-trivial NLP task and needs to be addressed for every
new style that is added to the system causing scalabilty concerns Jin et al.
(2020).
Figure 1: Different syntactic variations, leads to the same AMR as they all
are similar in meaning.
In this paper, we propose T-STAR (Truthful Style Transfer using AMR Graph as
Intermediate Representation) that uses a symbolic semantic graph notation
called Abstract Meaning Representation (AMR) as the style agnostic
intermediate stage. AMR Banarescu et al. (2013) is designed to capture
semantics of a given sentence in all entirety while abstracting away the
syntactic variations, inflections and function words. In other words, two
sentences with the same meaning but written in very different styles will have
a very similar AMR if not exactly the same (See Figure 1).
This addresses the shortcomings with Disentanglement and Prototype Editing
approaches. First, AMR being a representation with well-defined semantics, we
can inspect, interpret and measure quality of the intermediate representation
and the provenance of knowledge transfer between source and target sentence.
Second, AMR being a well known standard has high quality, robust reference
implementations, especially for head languages (e.g., English). Our
contributions are as follows:
1. We propose a novel TST approach with AMRs, an interpretable, symbolic intermediate representation, to achieve better content preservation. To this end, we enhance AMR parsing techniques to better suit the TST task. To the best of our knowledge, we are the first work to use AMR representations for the style transfer task (Sections 3, 4).
2. Through novel experimentation, we show that an AMR, as a style agnostic intermediate representation, has better content preservation and less style information of the given source sentence compared to competitive baselines.(Sections 5, 6)
3. On multiple datasets we show T-STAR is able to beat competitive baselines by producing sentences with significantly higher content preservation with similar style transfer scores. (Section 7)
4. With thorough human evaluations spanning 90,000 ratings we show T-STAR has $\sim 70$% better content preservation compared to state of art baseline with $50$% lesser hallucinations. (Section 8)
(a) MRPC
(b) QQP
(c) STS
Figure 2: AMR comparison for MRPC, QQP, and STS datasets
## 2 Related Work
### 2.1 Unsupervised Text Style Transfer
TST systems broadly use two family of approaches: disentanglement Shen et al.
(2017) and prototype-editing Jin et al. (2020). Prior works Hu et al. (2017);
John et al. (2019); Fu et al. (2017); Singh and Palod (2018); Logeswaran et
al. (2018) disentangle the content and style information in latent space using
style-based classifier or adversarial learning. Prototype-editing based
approaches Li et al. (2018); Madaan et al. (2020); Sudhakar et al. (2019) are
used to gain better controllability and interpretability.
Recently, some works propose to jointly optimize for content and style
information, to overcome the limitations of explicitly disentangling the style
and content information. Yamshchikov et al. (2019b) illustrates that
architectures with higher quality of information decomposition perform better
on style transfer tasks. Subramanian et al. (2018) argues that it is often
easy to fool style discriminators without explicitly disentangling content and
style information, which may lead to low content preservation Xu et al.
(2018). Instead, Subramanian et al. (2018); Logeswaran et al. (2018) use back-
translation to optimize for content preservation. Luo et al. (2019); Liu et
al. (2020) use reinforcement learning framework with explicit rewards designed
for content preservation. Wang et al. (2019) pushes the entangled latent
representation in the targeted-style direction using style discriminators.
Samanta et al. (2019) uses normalizing flow to infuse the content and style
information back before passing it to the decoder.
Cheng et al. (2020) proposes a context aware text style transfer framework
using two separate encoders for input sentence and the context. Similar to our
work, Krishna et al. (2020) also generates an interpretable intermediate
representation. The authors first paraphrase the given source to first convert
it to a destylized version before passing it to the targeted style-specific
decoder. Complementary to our work, Shi et al. (2021) uses syntactic graphs
(dependency tree) as an intermediate representation for attribute transfer to
retain the linguistic structure. On the other hand, we focus on retaining the
semantics using AMR graphs as intermediate representation and modifying the
linguistic structure (authorship style).
### 2.2 Text $\Longleftrightarrow$ AMR
In order to improve the parsing performance for AMRs, neural models are
receving increasing attention. Neural AMR parsers can be divided into
following categories: i) sequence-to-sequence based AMR-parsers Xu et al.
(2020a), ii) sequence-to-graph based AMR parsers Zhang et al. (2019), where
the graph is incrementally built by spanning one node at a time. A more
detailed survey on related works can be found in Appendix A.
## 3 AMR as an Intermediate Representation
Abstract Meaning Representations (AMRs) is a semantic formalism construct that
abstracts away the syntactic information and only preserves the semantic
meaning, in a rooted, directed and acyclic graph. In Figure 1, we present
certain syntactic variations (changing the voice, and tense) for a sentence
without altering meaning. All variations result in the same AMR graph. The
nodes in the AMR graphs (“produce-01”, “they”, “before”, etc.) are concepts
(entities/predicates) that are canonicalized and mapped to semantic role
annotations present in Propbank framesets111https://propbank.github.io/. The
edges (“ARG0”, “time”, “duration”, etc.) are then relations between these
concepts. In other words, AMRs aim at decoupling “what to say” from “how to
say” in an interpretrable way. We posit that this could be beneficial for text
style transfer, where the goal is to alter the “how to say” aspect while
preserving “what to say”.
Recently, semantic meaning representation property of AMRs has been shown to
be useful in other generation tasks. In abstractive summarization, Liao et al.
(2018) uses AMR as an intermediate representation to first obtain a summary
AMR graph from document and then generate a summary from it. Hardy and Vlachos
(2018) use AMRs to cover for the lack of explicit semantic modeling in
sequence-to-sequence models. In machine translation, Song et al. (2019)
adopted AMRs to enforce meaning preservation while translating from English to
German. For Paraphrase Generation, Cai et al. (2021) found that using AMRs as
intermediate representation reduces the semantic drift. Moreover,
incorporating symbolic representations as an intermediate representation
provides a way to effectively understand the reasons behind a model’s
shortcomings. We utilize this advantage to analyse the weaknesses of T-STAR in
Section 8.2.
In order to demonstrate the semantic meaning preservation property of AMRs, we
design an experiment using three publicly available paraphrase benchmarks,
i.e., MRPC Dolan and Brockett (2005),
QQP222https://www.quora.com/profile/Ricky-Riche-2/First-Quora-Dataset-Release-
Question-Pairs, and STS. MRPC and QQP are sentence pair datasets with each
pair labeled yes if they are paraphrases of each other no otherwise. STS
dataset assigns a scores from 0 (not similar) to 5 (exactly similar) to a
sentence pair. We hypothesize that if AMRs are indeed semantic meaning
preserving, two sentences with similar meaning should have highly similar
AMRs. To measure the similarity between two AMRs, we use the Smatch score Cai
and Knight (2013) that calculates the number of overlapping triplets between
two AMRs. We use an off-the-shelf AMR parser
333https://amrlib.readthedocs.io/en/latest/ to generate AMR given a sentence.
We plot the distribution of the Smatch scores for MRPC, QQP and STS datasets
in Figure 2. For MRPC, we infer that the Smatch scores for paraphrases are
significantly higher than the Smatch scores for non-paraphrases. Similarly,
for QQP, quartile distribution of Smatch scores for paraphrases is higher in
comparison to non-paraphrases. For STS dataset, we observe a gradual increase
in quartile distribution of Smatch scores as we move towards more similar
sentences.
The experiments above corroborate the claim that AMRs can preserve meaning
under lexical variations like paraphrasing, tense and voice changes. Recent
research discussed earlier, have successfully used this property to show task
improvements. Building on the above qualitative, quantitative and prior
research evidence, we further explore the applicability of AMR for the TST
task.
Vanilla T-STAR Encoder | T-STAR Encoder
---|---
Sentence: To make us feel existence, and to shew
(a/and :op1(m/make-02 :ARG1(f/feel-01 :ARG0(w/we) :ARG1(e/exist-01 :ARG1 w))) :op2 (s / shew-01 :ARG0 w)) | (h/have-purpose-91 :ARG2(a/and :op1(m/make-02 :ARG1(f/feel-01 :ARG0(w/we) :ARG1(e/exist-01 :ARG1 w))) :op2(s/show-01 :ARG0 w :ARG1 e)))
Sentence: But trust not this; too easy Youth, beware!
(m/multi-sentence :snt1(c/contrast-01 :ARG2(t/trust-01 :polarity - :mode imperative :ARG0(y/you) :ARG1(t2/this))) :snt2(b/beware-01 :mode imperative :ARG0(y2/youth :ARG1-of(e/easy-05 :ARG2-of(h/have-degree-91 :ARG1 y2 :ARG3 (t3/too)))))) | (c/contrast-01 :ARG2 (a/and :op1 (t/trust-02 :polarity - :mode imperative :ARG0 (y/you :mod (y2/youth)) :ARG1 (t2/this)) :op2 (h/have-degree-91 :ARG1 t2 :ARG2 (e/easy-05 :ARG1 t2) :ARG3 (t3/too))))
Table 1: Comparison between AMRs from vanilla T-STAR Encoder and T-STAR
Encoder. T-STAR Encoder generates better AMRs for stylized sentences.
Dimension | Metric | Description | Metric used in related works
---|---|---|---
AMR similarity | SMATCH$\uparrow$ | SMATCH Cai and Knight (2013) measures degree of overlap between AMR graphs of _S,T_. The score is computed based on triplet (edge) overlap by finding a variable node mapping that maximizes the count of matching triplets. | Used extensively to measure similarity between two AMRs, across AMR-parsing literature.
Lexical Diveristy | Self-BLEU $\downarrow$ | BLEU-4 Papineni et al. (2002) between _S,T_ | Hu et al. (2017); Li et al. (2018); Logeswaran et al. (2018); Xu et al. (2018)
Content Preservation (C.P.) | WMD $\downarrow$ | Word Mover Distance Kusner et al. (2015) measures dissimilarity between _S,T_ as the minimum distance between their embedded words. | Yamshchikov et al. (2020) states that WMD correlates best with human-evaluations on semantic similarity
SIM $\uparrow$ | SIM Wieting et al. (2019a) uses an embedding model proposed in Wieting et al. (2019b) to measure semantic similarity | Krishna et al. (2020); Luo et al. (2019)
Style Transfer (S.T.) | Style Accuracy $\uparrow$ | Score of 4-way and 2-way fine-tuned RoBERTa-Large Liu et al. (2019) model for styles in CDS and Author-Imitation datasets respectively | Krishna et al. (2020)*,Hu et al. (2017); Fu et al. (2017); Madaan et al. (2020)*
C.P. & S.T. | Weighted Style Accuracy $\uparrow$ | Style Accuracy weighed by their corresponding semantic similarity scores averaged across all test instances. | Krishna et al. (2020)*,Li et al. (2018)*
Table 2: Evaluation Metrics: Source and Target sentences denoted as _S_ ,_T_
respectively. We use all the dimensions except AMR similarity to measure the
performance of TST. Note that Weighted Style Accuracy is the only metric that
encompasses two crucial dimension : C.P. and S.T. in one metric. AMR
similarity is used to select best performing TSTAR-Encoder. * represents the
works that use slight variations of the mentioned metrics. Note that across
all metrics we compute an average scores across all test instances. Figure 3:
An overview of T-STAR model architecture. It consists of two modules: T-STAR
Encoder, that transforms a given sentence $s_{i}$ in style $i$ to its AMR
representation $A_{i}$. To convert the sentence to style $j$, $A_{i}$ is
passed to T-STAR Decoder specific to style $j$.
## 4 Proposed Solution
Our proposed model T-STAR consists of two modules (refer Figure 3). First,
T-STAR Encoder generates an AMR given source sentence in style $i$. Second,
T-STAR Decoder generates a sentence in style $j$ with similar meaning as
preserved in the generated intermediate AMR. We take T5-Base Raffel et al.
(2020) pre-trained model as our basic seq2seq architecture for both the
modules. In order to use AMR as a sequence in T5, we borrow the choice of DFS
Traversal from Bevilacqua et al. (2021), that thoroughly study the effect of
various traversals on AMR parsing.
Figure 4: Generic sentences from AMR 3.0 corpus are stylized using a TST
model. The corresponding AMR and stylized sentences are mapped together as
silver training dataset to finetune T-STAR Encoder.
### 4.1 T-STAR Encoder
We train our simplest encoder, called the vanilla T-STAR Encoder, by fine-
tuning T5 base with the open source AMR 3.0 dataset Knight et al. (2021). The
AMR 3.0 dataset consists of roughly 59,000 generic English sentences (denoted
as $s_{i}$ for the $i^{th}$ sentence) and their corresponding AMRs (denoted as
$A_{i}$). In a qualitative analysis, we observe that the vanilla T-STAR
encoder under performs in two significant ways as illustrated in Table 1.
First, style bearing words (such as “shew”) become concepts in the AMRs
(Sentence-1 in Table 1) as opposed to being canonicalized to their respective
propbank role annotations. Second, the meaning of the stylized sentences get
incorrectly encoded in the AMRs, as shown in the second example in Table 1. To
overcome this, we propose a style-agnostic fine-tuning strategy as follows.
Style-agnostic Fine Tuning: We hypothesize that vanilla Text to AMR encoder is
unable to effectively transform stylized sentences to AMRs because it has only
been trained on generic English sentences. Therefore, we propose a data-
augmentation strategy (refer to Figure 4), where we use an off the shelf style
transfer model, e.g., STRAP Krishna et al. (2020), to stylize a generic
English sentence $s_{i}$ in style $p$ ($\hat{s}^{p}_{i}$). While converting
$s_{i}$ to $\hat{s}^{p}_{i}$, we alter the style of original sentence, keeping
the meaning intact. For a high quality synthetic dataset, we filter out
samples with low semantic similarity between $s_{i}$ and $\hat{s}^{p}_{i}$. We
provide a detailed empirical analysis on this filtering strategy in Appendix
C. Since the meaning is preserved, we can now map $\hat{s}^{p}_{i}$ to the
same AMR $A_{i}$. We then fine-tune our T-STAR Encoder on
$\bar{\mathbf{S}}=\mathbf{S}\cup\mathbf{\hat{S}}$ , where
$\mathbf{\hat{S}}=\\{\hat{s}^{p}_{i},A_{i}\\}_{N}\forall{p\in P}$ and $P$ is
the total number of styles in the dataset.
1
Input: parallel corpora ($\mathbf{S}$), mono-lingual corpus
($\mathbf{R}^{p},\forall p\in P$)
2
3$\mathbf{\hat{S}}=\\{\hat{s}^{p}_{i},A_{i}\\}_{N}\forall{p\in P}$ using an
existing TST model, e.g., STRAP
4$X:=\mathbf{S}\cup\mathbf{\hat{S}}$
5while _Convergence_ do
6 fine-tune TSTAR-Encoder $f_{amr}(.)$ on $X$
7 $\mathbf{\hat{S}}:=\\{\\}$
8 for _$p\in P$_ do
9 Use $f_{amr}(.)$ to create
$\mathbf{\hat{R}}^{p}:=\\{r^{p}_{i},\tilde{A}^{p}_{i}\\}_{M}$
10 fine-tune TSTAR-Decoder $f_{p}(.)$ for style $p$
11 Use TSTAR-Decoder to get
$\mathbf{\hat{S}^{p}}=\\{\hat{s}^{p}_{i},A_{i}\\}_{N}$
12 $\mathbf{\hat{S}}=\mathbf{\hat{S}}\cup\mathbf{\hat{S}^{p}}$
13 $X:=\mathbf{S}\cup\mathbf{\hat{S}}$
Algorithm 1 Iterative T-STAR
### 4.2 T-STAR Decoders
Due to the unavailability of parallel style corpora, we are provided $P$ mono-
style corpora $\mathbf{R}^{p}=\\{r^{p}_{i}\\}_{M_{p}}$ written in style $p$,
where $P$ refers to the total number of styles and $r^{p}_{i}$ refers to the
$i^{th}$ sentence of the $p$ style dataset which has $M_{p}$ sentences.
Training a style specific decoder ($f_{p}(.)$) to generate a sentence in style
$p$ from an AMR consists of two steps. First, we use our fine-tuned T-STAR
Encoder (Section 4.1) to generate AMRs $\hat{A}^{p}_{i}$ for every sentence
$\mathbf{r}^{p}_{i}$ for every style corpora. Second, we fine tune a T5 base
model to recover the original sentence $\mathbf{r}^{p}_{i}$ given the AMR
$\hat{A}^{p}_{i}$, obtaining style-specific decoders. In other words, we fine
tune using $M_{p}$ pairs of ( $\mathbf{r}^{p}_{i}$, $\hat{A}^{p}_{i}$)
constructed from the first step. Note that, we experimented with a data
augmentation technique for the decoders as well, however it did not lead to an
improvement in the style transfer performance (refer to Appendix E).
Once style specific decoders have been trained for every style in $P$, we can
use the T-STAR Encoder in tandem with the T-STAR Decoders to convert between
arbitrary style combinations as in Krishna et al. (2020).
### 4.3 Iterative T-STAR
The performance of our modules, T-STAR Encoder and T-STAR Decoders depends on
the quality of the synthetic datasets ($\mathbf{\hat{S}}$,
$\mathbf{\hat{R}^{p}}$) generated by their complementary modules. We adopt the
iterative back-translation technique used in unsupervised machine translation
Hoang et al. (2018). Iteration proceeds in rounds of training an encoder and
decoder from mono-style data. In every round, we aim to iteratively improve
the quality of the encoder and decoder modules by generating increasingly
better synthetic data from the previous round. We briefly describe this
process in Algorithm 1.
## 5 Experimental Setup
In this section we briefly describe the various T-STAR variations that are
analyzed in the subsequent sections, baselines, and the implementation
details. The models are validated against the metrics summarized in Table 2.
### 5.1 T-STAR variations
Vanilla T-STAR: The T-STAR Encoder used in this version, is only trained on
AMR 3.0 dataset, and not finetuned for stylized sentences.
T-STAR: We train the encoder and decoders using Algorithm 1 for only one
iteration.
Iterative T-STAR: We follow two iterations of Algorithm 1 to obtain better
quality synthetic dataset.
### 5.2 Baselines
UNMT: Subramanian et al. (2018) models style transfer as unsupervised machine
translation task. DLSM He et al. (2020) is a deep generative model that
unifies back-translation and adversarial loss. RLPrompt Deng et al. (2022a)
uses a discrete prompt optimization approach using reinforcement learning. It
is adopted in a zero-shot setting where we use Distil-BERT Sanh et al. (2019)
and run the optimization for 1k steps.
STRAP Krishna et al. (2020) first normalizes the style information by
paraphrasing the source text into generic English sentence, which is then
passed through a style-specific GPT-2 based model to generate the styled
output.
### 5.3 Datasets
We evaluate performance of T-STAR on two English datasets that capture
Linguistic Styles Jin et al. (2020). First, Shakespeare Author Imitation
Dataset Xu et al. (2012) consists of $18$K pairs of sentences written in two
styles. Original Shakespeare’s plays have been written in Early Modern
English, a significantly different style. Second, Corpus of Diverse Styles
(CDS) Krishna et al. (2020). This dataset consists of non-parallel sentences
written in 11 different styles. We will present our results on a subset of
four styles : Bible, Poetry, Shakespeare and Switchboard which consists of
$34.8$K, $29.8$K, $27.5$K, $148.8$K instances respectively (CDS uses MIT
License).
### 5.4 Implementation Details
We use pre-trained T5-Base model architecture for both the encoder and
decoder. Following iterative back translation literature Kumari et al. (2021),
we run Iterative T-STAR for two iterations. The AMRs are preprocessed and
postprocessed based on the method mentioned in Appendix B. The best modules
are selected based on the performance on validation set. Finer details about
the model architecture and hyperparameters can be found in Appendix B.
Style | Model | Content Preservation | S. R. $\uparrow$
---|---|---|---
WMD $\downarrow$ | SIM $\uparrow$
Bible | STRAP | 0.200 | 0.715 | 0.979
T-STAR | 0.170 | 0.792 | 0.979
Poetry | STRAP | 0.290 | 0.664 | 0.969
T-STAR | 0.215 | 0.760 | 0.965
Shakespeare | STRAP | 0.328 | 0.610 | 0.971
T-STAR | 0.222 | 0.754 | 0.972
Switchboard | STRAP | 0.222 | 0.751 | 0.999
T-STAR | 0.163 | 0.848 | 1.0
Table 3: Reconstructing the original sentences using the intermediate semantic
representation outperforms the baseline with a significant margin with respect
to content preservation. T-STAR is on par with style retention (S.R.) with the
baseline model.
## 6 Robustness of AMRs for Text Style Transfer
An ideal style agnostic intermediate representation for TST should a) encode
complete semantic information and b) minimal style information. We
quantitatively measure AMR’s efficacy in these two dimensions. We compare
T-STAR with STRAP Krishna et al. (2020) for it uses a human readable
intermediate representation as well.
### 6.1 Semantic Containment of AMR
If semantics of input sentence is completely preserved in the intermediate
representation, we should be able to reconstruct the input sentence from it.
To evaluate the robustness of AMRs across all styles for content preservation,
we first generate intermediate AMR given the sentence in style $p$ using our
encoder and then reconstruct the same sentence using our decoder $f_{p}(.)$ in
style $p$. We study how close the generated sentence (from STRAP, T-STAR) is
with respect to the original sentence. We can infer from Table 3 that AMRs as
an intermediate representation performs significantly better on content
preservation as compared to STRAP across all the styles. Specifically, it
gives an average of $0.10$ and $0.06$ absolute improvement on SIM and WMD
scores across the four styles respectively. Also, we get comparable
performance on retaining the original style. We present an ablation study on
AMR parser in Appendix C, and propose a new unsupervised Text-AMR evaluation
metric to measure the content preservation of AMR parsing along with its
results on the CDS dataset in Appendix D.
Style | Original | Paraphrase | AMR
---|---|---|---
Vanilla-T-STAR | T-STAR
Bible | 97.72 | 79.97 | 74.64 | 71.90
Switchboard | 99.47 | 89.79 | 87.74 | 64.66
Poetry | 83.17 | 81.04 | 49.05 | 61.61
Shakespeare | 70.07 | 78.07 | 88.8 | 80.73
Average | 87.61 | 82.22 | 75.06 | 69.73
Table 4: Accuracy of style classifiers trained on original, paraphrase and AMR
inputs. AMR has least performance denoting it encodes least style information.
### 6.2 Style Agnosticity of AMR
A style classifier $C$ assigns a style class label to an input sentence $S$.
If $S$ does not encode style information, it should result in poor classifier
performance. We use this observation to design an experiment to evaluate the
style-agnosticity of AMRs as follows. We train three versions of 4-way style
classifier using original sentences, paraphrased sentences (as used in STRAP)
and AMRs as the input sequences to the classifier. Two models are used to
generate AMRs: Vanilla T-STAR Encoder and T-STAR Encoder. For all versions of
the style classifier, we mask content bearing words like entities, numbers,
common nouns and AMR tags in the input sequences.
The accuracy of the 4 classifiers is shown in Table 4. First, even Vanilla
T-STAR Encoder has lower classifier accuracy for three out of four styles in
comparison to original and paraphrases sentences. Second, with the T-STAR
Encoder, we observe further reduction in classifier’s performance, and obtain
an average absolute drop of 15.19% and 7.1% as compared to paraphrase and
Vanilla T-STAR Encoder respectively.
We illustrate some examples in Table 1, where the T-STAR Encoder generates
better AMRs as compared to the Vanilla T-STAR Encoder. In the first
example,the T-STAR Encoder, unlike the vanilla T-STAR Encoder, is able to map
style-specific word “shew” to the valid concept “show” and also associate
“existence” to it. In the second example, the T-STAR Encoder does not split
the AMR into two sentences while parsing. Additionally, it is able to make the
association between “you” and “youth”. Through the above quantitative and
qualitative analysis we demonstrate that the T-STAR Encoder generates AMRs
which are robust in preserving meaning of the source sentence while
predominantly losing style information.
## 7 Performance on Style Transfer Tasks
In this section, we compare our model performance T-STAR and Iterative-T-STAR
to the baselines across two datasets: Shakespeare Imitation Dataset and CDS
dataset.
Model | Content Preserv. | Lex. Div. | S.T. | WSAcc.
---|---|---|---|---
SIM $\uparrow$ | WMD $\downarrow$ | Self-BLEU $\downarrow$
Original to Modern Style
UNMT | 0.461 | 0.318 | 0.118 | 0.586 | 0.256
DLSM | 0.447 | 0.369 | 0.079 | 0.192 | 0.115
RLPrompt | 0.508 | 0.387 | 0.164 | 0.354 | 0.292
STRAP | 0.647 | 0.337 | 0.118 | 0.886 | 0.552
Vanilla-TSTAR | 0.848 | 0.182 | 0.269 | 0.601 | 0.497
TSTAR | 0.754 | 0.257 | 0.175 | 0.754 | 0.554
Iterative-TSTAR | 0.799 | 0.227 | 0.209 | 0.715 | 0.556
Modern to Original Style
UNMT | 0.373 | 0.375 | 0.057 | 0.414 | 0.158
DLSM | 0.421 | 0.373 | 0.086 | 0.391 | 0.174
RLPrompt | 0.550 | 0.348 | 0.261 | 0.547 | 0.203
STRAP | 0.656 | 0.332 | 0.139 | 0.681 | 0.433
Vanilla TSTAR | 0.897 | 0.14 | 0.379 | 0.47 | 0.418
TSTAR | 0.842 | 0.185 | 0.324 | 0.490 | 0.402
Iterative-TSTAR | 0.853 | 0.181 | 0.329 | 0.540 | 0.446
Table 5: T-STAR models comparison with baseline models on Shakespeare Author
Imitation Dataset for both the directions. Iterative T-STAR outperforms all
the baselines on Weighted Style Accuracy.
Direction | Model | L.D. | Cont. Preserv. | S.R. $\downarrow$ | S.T. $\uparrow$ | WSAcc $\uparrow$
---|---|---|---|---|---|---
S-BLEU $\downarrow$ | WMD $\downarrow$ | SIM $\uparrow$
poetry $\rightarrow$ bible | STRAP | 0.067 | 0.314 | 0.571 | 0.335 | 0.548 | 0.289
Van-TSTAR | 0.170 | 0.183 | 0.812 | 0.430 | 0.289 | 0.226
TSTAR | 0.084 | 0.268 | 0.670 | 0.202 | 0.618 | 0.391
Itr-TSTAR | 0.106 | 0.241 | 0.721 | 0.231 | 0.566 | 0.383
shak. $\rightarrow$ bible | STRAP | 0.073 | 0.343 | 0.535 | 0.291 | 0.677 | 0.346
Van-TSTAR | 0.212 | 0.194 | 0.817 | 0.540 | 0.422 | 0.337
TSTAR | 0.123 | 0.286 | 0.656 | 0.277 | 0.705 | 0.437
Itr-TSTAR | 0.149 | 0.260 | 0.712 | 0.271 | 0.701 | 0.477
switch. $\rightarrow$ bible | STRAP | 0.042 | 0.323 | 0.476 | 0.062 | 0.670 | 0.307
Van-TSTAR | 0.128 | 0.196 | 0.729 | 0.085 | 0.456 | 0.330
TSTAR | 0.758 | 0.260 | 0.605 | 0.007 | 0.725 | 0.419
Itr-TSTAR | 0.097 | 0.241 | 0.637 | 0.006 | 0.745 | 0.456
poetry $\rightarrow$ shak. | STRAP | 0.067 | 0.327 | 0.571 | 0.159 | 0.810 | 0.450
Van-TSTAR | 0.207 | 0.166 | 0.821 | 0.398 | 0.576 | 0.460
TSTAR | 0.131 | 0.244 | 0.717 | 0.241 | 0.733 | 0.509
Itr-TSTAR | 0.164 | 0.207 | 0.768 | 0.278 | 0.704 | 0.522
switch. $\rightarrow$ shak. | STRAP | 0.045 | 0.328 | 0.489 | 0.012 | 0.956 | 0.461
Van-TSTAR | 0.152 | 0.187 | 0.740 | 0.034 | 0.948 | 0.696
TSTAR | 0.089 | 0.250 | 0.651 | 0.009 | 0.971 | 0.628
Itr-TSTAR | 0.122 | 0.22 | 0.696 | 0.010 | 0.973 | 0.675
bible $\rightarrow$ shak. | STRAP | 0.110 | 0.242 | 0.634 | 0.231 | 0.764 | 0.465
Van-TSTAR | 0.214 | 0.151 | 0.842 | 0.377 | 0.613 | 0.508
TSTAR | 0.100 | 0.227 | 0.713 | 0.309 | 0.676 | 0.472
Itr-TSTAR | 0.126 | 0.210 | 0.743 | 0.299 | 0.683 | 0.499
bible $\rightarrow$ switch. | STRAP | 0.088 | 0.238 | 0.625 | 0.080 | 0.918 | 0.565
Van-TSTAR | 0.171 | 0.162 | 0.810 | 0.186 | 0.813 | 0.649
TSTAR | 0.084 | 0.227 | 0.713 | 0.082 | 0.916 | 0.646
Itr-TSTAR | 0.101 | 0.213 | 0.73 | 0.083 | 0.911 | 0.658
shak. $\rightarrow$ switch. | STRAP | 0.039 | 0.344 | 0.534 | 0.000 | 0.998 | 0.533
Van-TSTAR | 0.138 | 0.212 | 0.767 | 0.002 | 0.988 | 0.755
TSTAR | 0.089 | 0.283 | 0.661 | 0.000 | 0.998 | 0.660
| Itr-TSTAR | 0.110 | 0.249 | 0.719 | 0.002 | 0.991 | 0.712
poetry $\rightarrow$ switch. | STRAP | 0.054 | 0.312 | 0.616 | 0.007 | 0.993 | 0.610
Van-TSTAR | 0.164 | 0.184 | 0.825 | 0.028 | 0.972 | 0.801
TSTAR | 0.113 | 0.250 | 0.740 | 0.015 | 0.985 | 0.728
Itr-TSTAR | 0.141 | 0.213 | 0.791 | 0.019 | 0.981 | 0.774
bible $\rightarrow$ poetry | STRAP | 0.060 | 0.275 | 0.633 | 0.346 | 0.621 | 0.374
Van-TSTAR | 0.094 | 0.195 | 0.777 | 0.533 | 0.417 | 0.315
TSTAR | 0.054 | 0.253 | 0.684 | 0.323 | 0.597 | 0.394
Itr-TSTAR | 0.064 | 0.237 | 0.700 | 0.348 | 0.559 | 0.375
shak. $\rightarrow$ poetry | STRAP | 0.071 | 0.353 | 0.550 | 0.160 | 0.818 | 0.449
Van-TSTAR | 0.155 | 0.219 | 0.753 | 0.228 | 0.748 | 0.551
TSTAR | 0.108 | 0.286 | 0.641 | 0.166 | 0.815 | 0.512
Itr-TSTAR | 0.128 | 0.258 | 0.693 | 0.190 | 0.794 | 0.538
switch. $\rightarrow$ poetry | STRAP | 0.037 | 0.331 | 0.524 | 0.094 | 0.844 | 0.432
Van-TSTAR | 0.101 | 0.215 | 0.715 | 0.261 | 0.486 | 0.334
TSTAR | 0.064 | 0.268 | 0.631 | 0.106 | 0.701 | 0.427
Itr-TSTAR | 0.084 | 0.239 | 0.673 | 0.158 | 0.603 | 0.388
All Styles Avg. | STRAP | 0.063 | 0.311 | 0.563 | 0.148 | 0.801 | 0.440
Van-TSTAR | 0.159 | 0.189 | 0.784 | 0.259 | 0.644 | 0.497
TSTAR | 0.093 | 0.256 | 0.674 | 0.145 | 0.787 | 0.519
Itr-TSTAR | 0.116 | 0.232 | 0.715 | 0.158 | 0.768 | 0.538
Table 6: Performance comparison of T-STAR models with STRAP on 12 different
directions, across four styles. T-STAR and Iterative T-STAR beats STRAP for 11
directions out of 12. S.R. - Style Retention, S.T. - Style Transfer, WSAcc -
Weighted Style Accuracy.
### 7.1 Performance Analysis on Shakespeare Imitation Dataset
In Table 5, we present the results of T-STAR and Iterative-T-STAR in
comparison with the baselines for the Shakespeare Author Imitation dataset for
both the directions :original to modern style and vice versa. The Weighted
Style Accuracy is the primary metric as it is shown to effectively combine
both style accuracy and content preservation Krishna et al. (2020).
We observe that our model, T-STAR slightly performs better than STRAP model
for original to modern style, but has lower performance for modern to original
style. However Iterative T-STAR, outperforms all the baselines for both the
directions on Weighted Style Accuracy. We observe that STRAP has very high
style accuracy that it achieves by compromising on Content Preservation.
Through human evaluation (Section 8) we see STRAP employs significantly higher
hallucinations to achieve style transfer. Vanilla T-STAR on the other hand
achieves high content preservation via significant copying from source as seen
by the substantially high self-BLEU score. Iterative T-STAR finds the middle
ground of achieving style transfer while not compromising on content
preservation. We also found that the length of generated sentence is similar
to the input sentence, with on average one word difference. In the subsequent
sections, we compare T-STAR to only the best performing baseline, STRAP.
### 7.2 Performance Analysis on CDS Dataset
In Table 6, we compare the performance of T-STAR and Iterative T-STAR against
STRAP and Vanilla T-STAR, across all 12 directions for {Poetry, Shakespeare,
Switchboard, Bible} styles. We make the following observations: First, both
our models T-STAR and Iterative T-STAR outperform the state-of-the-art
baseline, STRAP, on 11 out of 12 directions, with an average absolute
improvement of 7.7% and 9.7% respectively. Second, Vanilla T-STAR is observed
to be a stronger baseline than STRAP, as it beats STRAP on 8 out of 12
directions. Third, when we compare iterative T-STAR against vanilla T-STAR, we
beat it in 7 out of 12 directions, where the average absolute improvement
across these 7 directions is 8%, as compare to the average absolute loss in 5
out of 12 styles is 2%. Fourth, Iterative T-STAR model outperforms T-STAR
model on 9 out of 12 directions, which shows that the improvement in synthetic
dataset quality, is boosting the performance of the downstream task.
Consistent with the findings of the previous experiment, we see Iterative
T-STAR is able to find the middle ground of transferring style without
compromising on content preservation.
Direction | Input Sentence | STRAP | Vanilla T-STAR | T-STAR | Iterative T-STAR
---|---|---|---|---|---
Poetry$\rightarrow$Bible | “Dear Lord! It has a fiendish look”– | But they laughed, and said, Oh my God! | Thou lookest fiendish, O LORD | Behold, it beholdeth to be fiendish, saith the LORD. | Dear Lord, it looketh fiendish unto thee.
Bible$\rightarrow$Poetry | And he said unto another, Follow me. | And bade the other follow me; | And another: “O follow me!” said he, | And follow me! and I will be be ye gone, | And thou, my love, will follow me.
Shake.$\rightarrow$Switch. | Dead art thou, dead! | did you get a uh a uh you have a dead art | you’re dead i say you are dead | oh you’d die | you’re dead yeah
Shake.$\rightarrow$Switch. | Thus with a kiss I die. | so i’m i’m dying to get a kiss | so i die with a kiss | i die with a kiss | so i die with a kiss
Table 7: Example of generated stylized sentences for STRAP, Vanilla T-STAR,
T-STAR and Iterative T-STAR models for the given input sentence.
#### 7.2.1 Qualitative Analysis
In Table 7, we enumerate few examples with generated stylized sentences using
STRAP and T-STAR variations. We can infer that, although STRAP performs well
in transforming the sentence to the given style, it alters the meaning (row
1,3, and 4). On the other hand, vanilla T-STAR does not always transform the
style (row 1 and 2). However, with T-STAR and Iterative T-STAR are able to
transform the sentences while keeping the style intact.
We further quantify these observations through an extensive set of human
annotations as described below.
## 8 Human Evaluations
Model | $>$ STRAP | $<$ STRAP | = STRAP
---|---|---|---
T-STAR | 70.8% | 26.4% | 2.8%
Itr. T-STAR | 77.5% | 20.5% | 2.0%
Table 8: Comparison of T-STAR and Iterative T-STAR models against STRAP for
content preservation.
Automatic metrics are insufficient to thoroughly understand subjective quality
measures like content preservation. Therefore, we conduct an extensive case
study with human evaluations. Our analysis is two folds, first we compare
STRAP with our models T-STAR and Iterative T-STAR on meaning preservation.
Second, we further understand the various categories of meaning loss failures.
For both the human evaluation tasks below, the criteria for choosing
annotators were i) proficient in English with a minimum education of Diploma.
ii) The annotators have to first qualify on two simpler questions, else they
are not allowed to continue on the task. Each instance is annotated by three
taskers, and the final annotation is a majority vote.
### 8.1 Comparison on Meaning Preservation
In order to study the faithfulness of the T-STAR models, we do a side-by-side
human evaluation. In this task, a source sentence is shown with 2 stylized
target sentence (one from T-STAR and another from STRAP). We present the
annotators with three options to judge content preservation with respect to
source sentence: option on left better than one on right, right better than
left and both equal.
We extensively compare the two models across all 12 directions for four
styles. For each direction we randomly sample 500 instances. Each instance is
rated by 3 annotators leading to a total of 18,000 ratings. We summarize our
findings in Table 8. Both T-STAR ad Iterative T-STAR significantly outperform
STRAP in terms of being better at content preservation (The $>$ STRAP column
in Figure 8). Further Iterative T-STAR has 7% higher meaning preservation
compared to T-STAR. In addition to the quantitative content preservation
metrics discussed in Section 7, this analysis gathers additional qualitative
evidence towards AMRs as an effective intermediate representation for content
preserving style transfer. The complete statistics per direction are available
in Appendix F.
Model | Type of Error
---|---
None$\uparrow$ | Hal.$\downarrow$ | Sem. Drift.$\downarrow$ | Incomp.$\downarrow$
STRAP | 10.48% | 39.38% | 35.38% | 14.75%
T-STAR | 19.36% | 24.46% | 33.71% | 22.45%
Itr. T-STAR | 24.83% | 22.6% | 36.98% | 15.58%
Table 9: Aggregate Error Analysis for error types: Hallucinations, Semantic
Drift and Incomplete for 6,000 samples. With Iterative T-STAR, the number of
samples with no errors and less hallucinations increase significantly.
### 8.2 Error Analysis
In the next study, we further aim to study the nature of meaning loss errors
made by style tranfer models. We categorize these errors into three categories
i) Hallucinations: new information not present in the source sentence is added
to target ii) Semantic Drift: the target sentence has a different meaning to
source sentence iii) Incomplete: some important content information is missed
in the target. The taskers also have the option to select “No Error” if the
meaning is preserved in the generated target. As in the previous experiment,
we collect 18,000 ratings and the results are summarized in Table 9.
We observe that our models T-STAR and Iterative T-STAR consistently beat STRAP
in the “No Error” category. Furthermore, the amount of hallucinations
significantly drops to 24.46% and further 22.6% with Iterative T-STAR across
all styles from 39.3% for STRAP. Reduction in hallucination can be clearly
seen as a benefit of encoding critical information in the source sentence
using a semantic parse representation like AMR. As a sign of improving the AMR
parsing quality, we see that iterative T-STAR further reduce the Incomplete to
15.5%. For further details refer to Appendix F.
#### 8.2.1 Usefulness of an interpretable intermediate representation
With intermediate AMRs being interpretable, it is possible to broadly
understand if such errors are emerging from either encoder or decoder module.
To intuitvely understand the reason for high number in Incomplete and Semantic
Drift errors, we qualitatively analyzed some instances along with the
generated intermediate AMRs. We have listed down these examples in Appendix F.
We observed that for Incomplete errors, the generated AMRs were not encoding
complete information, and thus this error percolated from the T-STAR encoder.
For the majority of the instances, either some entities were missing, and if
the clause was separated using “:,;”, it was not parsed in the intermediate
graph. Semantic Drift errors indicate shortcomings in both the modules, for
some instances the encoder is not abstracting out the meaning efficiently and
for others the decoder is not able to generate sentences with the meaning
encoded in the intermediate AMR.
## 9 Conclusion
We explored the use of AMR graphs as an intermediate representation for the
TST task. We see that the performance of the proposed method T-STAR surpasses
state of the art techniques in content preservation with comparable style
accuracy. Through qualitative analysis we show that obtaining very high style
accuracy scores without altering meaning is indeed a challenging problem.
## 10 Limitations
Some of the limitations for T-STAR based models are the following. First,
although our proposed models are performing better in the joint objective of
content preservation and style transfer, but they are not able to outperform
vanilla T-STAR (overall best performing model for CP) and STRAP (overall best
performing model for ST). This is a promising future direction, to keep
boosting the performance on both the directions without comprising on the
other dimension. Second, we are not incorporating graph structure in our
models, and thus there could be some information loss while interpreting and
generating the AMRs. Third, based on our error analysis, although our T-STAR
encoder is able to generate better AMRs for stylized sentences as compared to
vanilla T-STAR model, we are generating significant incomplete AMRs that are
missing out on important entities and relations to preserve meaning of source
sentence. Fourth, similar to prior research to generate synthetic dataset,
initial iteration of our model are dependant on an existing off the shelf TST
model, however the quality of the generated AMRs improves significantly using
the described data augmentation strategy. Fifth, our work is dependant on a
robust AMR parsing approach, which makes it challenging to adopt our approach
for other languages. However, with the recent advancements in multilingual AMR
parsing, it will be feasible in upcoming future works.
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## Appendix A Extended Related Works
### A.1 TST Metrics
Table 2 summarizes different metrics that have been used to measure content
preservation and style transfer efficacy. Yamshchikov et al. (2020) presents a
comprehensive analysis and categorization of several such metrics with respect
to human evaluations. Tikhonov et al. (2019) also points out some flaws in
traditional evaluation techniques and insists on using human written
reformulations for better evaluation444Due to the difficulty of our task, we
instead restrict to human evaluations instead of obtaining human gold-standard
benchmarks..
### A.2 Text to AMR
Recent works Bevilacqua et al. (2021); Cai and Lam (2020); Zhou et al. (2020)
for text-to-AMR task have pushed the SOTA, that makes it feasible to
automatically construct AMR given a sentence. As a consequence, semantic-
preserving NLG tasks, such as Neural Machine Translation Song et al. (2019);
Xu et al. (2020b), Abstractive Summarization Takase et al. (2016), and
Question Decomposition for multi-hop question answering Deng et al. (2022b)
use AMRs as intermediate representations. However, AMRs have not been explored
for style transfer tasks before our work.
The increase in AMRs being adopted for several seq2seq tasks is due to the
boost in the quality of AMR parsers. Earlier works, relied on statistical
approaches Peng et al. (2017); Flanigan et al. (2014, 2016) to generate AMRs
for a given piece of text. With the emergence of deep learning, various AMR
parsers are being proposed, which can be divided into following categories: i)
sequence-to-sequence based AMR-parsers Xu et al. (2020a), ii) sequence-to-
graph based AMR parsers Zhang et al. (2019), where the graph is incrementally
built by spanning one node at a time. More recently, several works have
adopted pretrained models for AMR parsers, and have observed a boost in
performance. Bai et al. (2022) uses BART model and posit the AMR parsing task
as a seq2seq task, and generates a traversal of the AMR parser as the output.
Bai et al. (2022) incorporates a pretraining strategy to better encode graph
information in the BART architecture. Xu et al. (2020a) uses sentence encoding
generated from BERT model. In this work, we adopt the pretrained technique
based AMR parser, to generate high quality AMRs for the given stylized
sentences.
Although off-the-shelf AMR parsers work well for some problems Fan and Gardent
(2020), they often need to be modified to be useful in the downstream tasks.
For instance, Deng et al. (2022b) proposed graph segmentation strategy to
perform question decomposition on a multi-hop query.
Xia et al. (2021) and Du and Flanigan (2021) illustrated that using silver
data augmentation can help improve in the task of AMR parsing. In this work,
we also illustrate the benefit of using silver data towards improving the
style agnosticity of AMR graphs as an intermediate representation.
### A.3 AMR to Text
Similar to text-to-AMR models, AMR-to-text frameworks can also be categorised
into two types - i ) sequence-to-sequence generation frameworks , ii) graph-
encoder based frameworks Song et al. (2018); Wang et al. (2021, 2020). Bai et
al. (2020) propose a decoder that back-predicts projected AMR graphs to better
preserve the input meaning than standard decoders. Bai et al. (2022) argues
that PLMs are pretrained on textual data, making is sub-optimal for modeling
structural knowledge, and hence propose self-supervised graph-based training
objectives to improve the quality of AMR-to-text generation.
## Appendix B Implementation Details
Offensive language We used the “List of Dirty, Naughty, Obscene or Otherwise
Bad Words”555https://github.com/LDNOOBW/List-of-Dirty-Naughty-Obscene-and-
Otherwise-Bad-Words to validate that the source and the generated target text
do not contain any offensive text.
Model Architecture: We use a standard t5-base encoder-decoder model as
described in Raffel et al. (2020). The pre-trained
HuggingFace666https://github.com/huggingface/transformers T5 transformer is
used for both text-to-AMR and AMR-to-text parts of the proposed architecture.
The model is pre-trained on the Colossal Clean Crawled Corpus (C4)
777https://www.tensorflow.org/datasets/catalog/c4 comprising of $\sim$750 GBs
worth of text articles. The model comprises of 220 billion parameters, and is
pre-trained for $2^{1}9$ steps before fine-tuning. For pre-training, AdaFactor
optimizer Shazeer and Stern (2018) is used with “inverse square root” learning
rate schedule.
AMR Graph Construction: We use the SPRING model Bevilacqua et al. (2021) to
generate AMR graphs from source style text. We use
amrlib888https://github.com/bjascob/amrlib package to generate the AMR graphs.
This implementation uses T5-base Raffel et al. (2020) as its underlying model,
as opposed to the BART model Lewis et al. (2019) in the SPRING architecture.
It is trained on AMR 3.0 (LDC2020T02) dataset Knight et al. (2021) that
consists of 59K manually created sentence-AMR pairs. The model is trained for
8 epochs using a learning rate of $10^{-4}$. The source and target sequence
lengths are restricted to 100 and 512 tokens respectively. Note that t5-based
SPRING model achieves an SMATCH score of $83.5$, which guarantees the quality
of obtained AMR representations $z_{i}$.
AMR-Based Style Transfer: We use the T5wtense (T5 with tense) architecture
from the amrlib package. The T5wtense architecture encodes part-of-speech
(POS) tags to the concepts in the AMR graph, which helps the generation model
to predict the tense of the output sentence since AMR graphs do not retain any
tense information from their corresponding sentence. This model outperforms
the standard T5-based model by 10 BLEU points on the AMR 3.0 (LDC2020T02)
dataset Knight et al. (2021). To keep the training steps comparable for the
subsets of the CDS dataset Krishna et al. (2020), we train this t5-base model
for $20$ epochs for the Bible, Romantic Poetry, and Shakespeare datasets, and
$5$ epochs for the Switchboard dataset. The model was trained for $20$ epochs
for the Shakespeare Author Imitation Dataset Xu et al. (2012) as well. We used
a learning rate of $10^{-4}$ for both datasets, and restricted source and
target sequence lengths to 512 and 90 throughout, respectively. Everything
else was kept same as the amrlib implementation to keep the results
consistent.
STRAP baseline: We train the model keeping the same hyperparameter
configuration as reported in Krishna et al. (2020). We train each style-
specific decoder for 3 epochs with learning rate of $5\times 10^{-5}$ with
Adam optimizer Kingma and Ba (2014). During inference we set the p-value for
nucleus sampling Holtzman et al. (2019) to 0.7 to have an appropriate balance
between content preservation and style accuracy scores.
Style classifiers: Similar to Krishna et al. (2020), we fine-tune a RoBERTa-
large model Liu et al. (2019) using the official implementation in the fairseq
package999https://github.com/pytorch/fairseq to train the style classifiers
mentioned in Table 3. For all classifier variants, learning rate of $10^{-5}$
and a mini-batch size of $32$ was used. The models were trained for 10 epochs
using Adam optimizer Kingma and Ba (2014). We also masked out named entities,
nouns and numbers from the input before training the classifier. We used spacy
package101010https://spacy.io/ to obtain the named entities and POS tags. To
obtain the named-entities and POS for AMR graphs, information extracted from
original sentences was used.
Train-Validation-Test splits: The data splits were kept the same as the
baseline model, STRAP Krishna et al. (2020) and can be found in Table 10.
Dataset | Train split | Validation split | Test Split
---|---|---|---
CDS | Bible | 31,404 | 1,714 | 1,714
Switchboard | 145,823 | 1,487 | 1,488
Poetry | 26,880 | 1,464 | 1,470
Shakespeare | 24,852 | 1,313 | 1,293
SAID | Original | 18,395 | 1,128 | 1,462
Modern | 18,395 | 1,218 | 1,462
Table 10: Size of train, validation and test sets for CDS dataset Krishna et
al. (2020) and Shakespeare Author Imitation Dataset (SAID) Xu et al. (2012).
Computational time and device setup: All experiments were done on a 16 GB
NVIDIA V100 GPU system with 120 GB n1-standard-32 Intel Broadwell CPU. It took
$\sim$16 hrs to train the AMR-to-Text models for Shakespeare Author Imitation
dataset Xu et al. (2012) and $\sim$25 hrs for the CDS dataset.
Evaluation Metrics: We used the gensim111111https://radimrehurek.com/gensim/
package to compute the Word Mover Distance (WMD) Kusner et al. (2015),
nltk121212https://www.nltk.org/ package to compute the BLEU scores Papineni et
al. (2002), smatch package131313https://github.com/snowblink14/smatch to
compute the SMATCH score Cai and Knight (2013) and implementation by Krishna
et al. Krishna et al.
(2020)141414https://github.com/martiansideofthemoon/style-transfer-paraphrase
for the cosine similarity using SIM embeddings Wieting et al. (2019a).
License of the packages used: The following packages use the MIT
License151515https://opensource.org/licenses/MIT \- amrlib, spacy, fairseq,
smatch, and STRAP. The following packages use the Apache License
2.0161616https://www.apache.org/licenses/LICENSE-2.0 \- nltk, and
huggingface’s transformer. The following packages use the GNU LGPL license
3.0171717https://www.gnu.org/licenses/lgpl-3.0.en.html \- gensim.
## Appendix C T-STAR-Encoder Ablation Study
The performance of our T-STAR-Encoder heavily depends on the quality of
synthetic dataset generated while stylizing the sentences present in AMR 3.0
dataset. Therefore, we conducted thorough empirical analysis to identify a
filtering strategy to boost the performance of the encoder.
To obtain the initial set of stylized sentences, we use the state-of-the-art
model available to transform generic English sentences to relevant styles
{Poetry, Shakespeare, Bible, Switchboard}, i.e., seq2seq inverse paraphrase
module Krishna et al. (2020).
We then fine-tune our T-STAR-Encoder on synthetic datasets obtained from
different filtering strategies, and compare the performan on test split of AMR
3.0, to ensure that the quality of the generated AMRs do not drop
significantly. We present our findings in Table 11. We observe that using the
whole set of generated samples, leads to a significant drop in the performance
(row-1).
Therefore, we filter out the augmented stylized sentences with SIM similarity
score Wieting et al. (2019a) below the threshold $\delta$. This filtering
strategy was able to significantly improve over the T-STAR-Encoder
performance, giving competitive results to the non-augmented Vanilla T-STAR-
Encoder. We select the best performing $\delta$ based on the performance on
the test-split of AMR 3.0. Note that we have used the best performing
threshold, $\delta=0.7$
Datasets | Precition | Recall | F-Score
---|---|---|---
Vanilla T-STAR Encoder | 0.829 | 0.794 | 0.811
T-STAR-Encoder | 0.671 | 0.390 | 0.493
T-STAR Encoder-Flt ($\delta$=0.5) | 0.830 | 0.790 | 0.810
T-STAR Encoder-Flt ($\delta$=0.6) | 0.807 | 0.753 | 0.779
T-STAR Encoder-Flt ($\delta$=0.7) | 0.836 | 0.798 | 0.816
T-STAR Encoder-Flt ($\delta$=0.8) | 0.828 | 0.793 | 0.810
Iterative T-STAR-Encoder($\delta$=0.7) | 0.829 | 0.794 | 0.811
Table 11: SMATCH scores on various versions of T-STAR-Encoder on AMR 3.0
dataset’s test split.
## Appendix D Unsupervised Evaluation of AMR parsing
Since we use AMR graphs as the intermediate representation for the TST task,
it is important to validate the generation quality of generated AMRs in terms
of content preservation with respect to the input sentence. However, there
does not exist an unsupervised metric to evaluate content overlap between an
AMR graph and a sentence. Hence we propose to use a slight variation of the
Word Mover Distance Kusner et al. (2015) for this purpose. We choose WMD over
other content preservation metrics for the following reasons -
* •
Yamshchikov et al. (2019a) illustrate the efficacy of WMD to evaluate text
style transfer over other metrics based on correlation with human evaluations.
Which means that it is more robust to the domain difference in the input and
the output sentence, making it an ideal candidate for text-AMR similarity
measurement.
* •
Syntactic metrics like BLEU would not be able to compute the content overlap
between an AMR graph and a sentence because word representation in an AMR
graph discards noun forms and tense information, and some verb tokens are
mapped to a different PropBank verb. These modifications along with the
disparity in sequential-graphical representation makes syntactic metrics
infeasible for the task.
* •
Semantic representations like SIM are fragile to the input sequence order, and
affected by non-content bearing words as well. However, WMD adopts on a bag-
of-words paradigm, making it more suitable for the task.
We propose the following two variants of WMD -
* •
WMD Overall \- In this variant, we aimed to keep the content bearing tokens
from the sentence and AMR graphs. For sentence, we removed the stopwords
(after doing a detailed corruption study on sentences, refer to Table 13),
while for AMR Graphs we removed AMR notation specific tokens (like “:op?”,
“ARG?”), punctuations (like “(”, ‘” ’), assigned variables and propbank code
for verbs (eg. changing “s / say-01” to “say”).
* •
WMD Verb Overall \- In this variant we specifically want to compute the
similarity of verbs in the parsed AMR graphs and input sentence. For this, use
nltk POS tagging tool to extract out verbs from the input sentence, and
directly extract out propbank based verbs from the AMR graph.
Refer to Table 12 for an example of the preprocessing strategy adopted.
Input Sentence -
---
Malaysian vice-prime minister Anwar ended a visit to China this
afternoon , and left Shanghai for Tokyo.
Extracted content from Input Sentence -
malaysian vice-prime minister anwar ended visit china afternoon ,
left shanghai tokyo.
Extracted Verbs from Input Sentence -
ended left
Input AMR-
(a2 / and :op1 (e2 / end-01 :ARG0 (p / person
:name (n / name :op1 ”Anwar”) :ARG0-of (h / have-org-role- 91
:ARG1 (c7 / country :name (n3 / name
:op1 ”Malaysia”)) :ARG2 (m / minister
:mod (p2 / prime) :mod (v /
vice)))) :ARG1 (v2 / visit-01 :ARG0 p
:ARG1 (c6 / country :name (n2 / name :op1
”China”))) :time (d / date-entity :dayperiod (a3 / afternoon)
:mod (t / today))) :op2 (l / leave-11
:ARG0 p :ARG1 (c8 / city :name (n4 / name
:op1 ”Shanghai”)) :ARG2 (c9 / city
:name (n5 / name :op1 ”Tokyo”))))
Extracted sequence-
and end person name Anwar have-org-role country name
Malaysia minister prime vice visit country name China
afternoon today leave city name Shanghai city name Tokyo
Corresponding Verb extraction (AMR) -
end visit leave
Table 12: Illustrative example of the preprocessing done in proposed text-AMR
unsupervised WMD Overall and WMD Verb Overall metrics.
Model | WMD Mean / Std dev. | SIM Mean / Std dev.
---|---|---
Original | 0.0 / 0.0 | 1.0 / 0.0
Stop Words | 0.1049 / 0.0761 | 0.9373 / 0.0734
Lowercase | 0.0754 / 0.1319 | 0.9946 / 0.0321
Stop Words + Lowercase | 0.1663 / 0.1347 | 0.9373 / 0.0734
POS | 0.1081 / 0.0731 | 0.8680 / 0.1929
Stop Words + Lowercase + POS | 0.1966 / 0.1438 | 0.8492 / 0.1957
Synonym Replacement | 0.1864 / 0.1266 | 0.6946 / 0.1832
Synonym + Stop Words | 0.2228 / 0.1335 | 0.6545 / 0.1979
Synonym + Lowercase | 0.2133 / 0.1511 | 0.6946 / 0.1832
Synonym + Stop Words + Lowercase | 0.2476 / 0.1534 | 0.6545 / 0.1979
Synonym + POS | 0.2320 / 0.1306 | 0.5948 / 0.2209
Synonym + Stop Words + Lowercase + POS | 0.2686 / 0.1525 | 0.5817 / 0.2243
Table 13: WMD scores on various corruption strategies against original
sentence on test set of AMR 3.0 dataset. POS refers to removing tags other
than nouns, verbs and adjectives from the sentence.
We also study the effect on text-AMR WMD score juxtaposed to text-text WMD
scores on varying degree of similarity between the compared sequences. For
this we use the diverse paraphraser trained by Krishna et al. (2020) on the
test set of AMR 3.0 dataset, and generate paraphrases with varying nucleus
sampling p-values Holtzman et al. (2019) from 0.0 to 1.0 with step size of
0.1. We notice that the WMD scores for text-text WMD (blue line in Fig. 5) and
text-AMR WMD (red line in Fig. 5) are similar to each other throughout.
Figure 5: WMD Scores on AMR 3.0 dataset for sentence-paraphrase pairs (blue)
and amr-paraphrase pairs (red). The shaded region denotes standard deviations
from the mean.
We present the WMD Overall scores and WMD Verb Overall scores for the CDS
dataset in Table 14; validating the content retention in parsed AMR graphs for
different strategies. We notice that Iterative T-STAR Encoder outperforms the
T-STAR Encoder-Flt ($\delta$=0.7) and T-STAR Encoder (unfiltered) baselines in
both WMD overall and Verb WMD overall. Even though comparable, it is still
however lesser than the Vanilla T-STAR Encoder numbers. However, we believe we
can credit that to more style information retention in Vanilla T-STAR Encoder
(refer to Section 6.2), leading to poorer performance in downstream text style
transfer task.
Models | Styles | WMD Overall | Verb WMD Overall
---|---|---|---
Vanilla T-STAR Encoder | Bible | 0.243 | 0.409
Poetry | 0.272 | 0.532
Shakespeare | 0.322 | 0.484
Switchboard | 0.292 | 0.467
T-STAR Encoder (Unfiltered) | Bible | 0.339 | 0.511
Poetry | 0.357 | 0.599
Shakespeare | 0.403 | 0.552
Switchboard | 0.341 | 0.537
T-STAR Encoder Flt ($\delta$=0.7) | Bible | 0.290 | 0.461
Poetry | 0.322 | 0.568
Shakespeare | 0.365 | 0.522
Switchboard | 0.323 | 0.512
Iterative T-STAR Encoder Flt ($\delta$=0.7) | Bible | 0.281 | 0.439
Poetry | 0.300 | 0.550
Shakespeare | 0.344 | 0.500
Switchboard | 0.301 | 0.491
Table 14: WMD Overall and WMD Verb Overall scores for evaluation of
unsupervised content preservation of different models across different styles
in the CDS dataset Krishna et al. (2020).
## Appendix E Data Augmentation for T-STAR Decoder
We hypothesize that the T-STAR decoder performances will improve if the
underlying model, is better at generating text given an AMR graph. To this
end, we create synthetic dataset using sentences from Wikipedia corpus. We
sample 10 million sentences from it and generate the corresponding AMRs using
our vanilla T-STAR model. We further filter out the samples for which WMD
Overall, mentioned in Appendix D and keep samples with a WMD score below 0.15,
which results in 2.3M instances.
We first fine-tune the T5-Base model for AMR to Text task on this filter
dataset. We obtain a BLEU score of 49.13 on Gold AMR test set. Note that this
performance is very close to the state-of-the art result 49.2 BLEU for this
task Bai et al. (2022). Table 15 lists down the different filtering strategy
and dataset sizes we experimented to identify the best strategy to improve the
performance.
Dataset | BLEU Score
---|---
GoldAMR (Baseline) | 44.36
Wikipedia 1M (Unfiltered) | 48.14
Wikipedia 120K- Filter($<$0.15) | 44.56
Wikipedia 1M -Filter ($<$0.15) | 48.58
Wikipedia 2.3M - Filter ($<$0.15) | 49.14
Table 15: Fine-tuning model for AMR-To-Text generation task using data
augmentation technique using Wikipedia Sentences
We then compare the performance of the best performing model on the style
transfer task again STRAP. We observe that this model is not beating the STRAP
performance across various style directions. Therefore, we conclude that a
vanilla fine-tuning of model for AMR to text task, does not necessarily boost
the performance in the downstream tasks.
## Appendix F Error Analysis
### F.1 Comparison on Meaning Preservation
We present the results across all the 12 directions for content preservation
comparitive analysis in Table 16 and 17 respectively. We can observe that for
every direction our models are consistently better in content preservation
with respect to STRAP model.
Direction | TSTAR >STRAP | STRAP <TSTAR | TSTAR=STRAP
---|---|---|---
bible $\rightarrow$ poetry | 60.8% | 37.6% | 1.6%
bible $\rightarrow$ shak. | 66.6% | 32.2% | 1.2%
bible$\rightarrow$switch. | 67% | 30.4% | 2.6%
poetry$\rightarrow$bible | 68.6% | 29.4% | 2%
poetry $\rightarrow$ shak. | 78.6% | 19.6% | 1.8%
poetry$\rightarrow$switch. | 69.4% | 27.4% | 3.2%
shak.$\rightarrow$bible | 73% | 24.8% | 2.2%
shak.$\rightarrow$poetry | 66.2% | 27.4% | 3.2%
shak.$\rightarrow$switch | 73.6% | 24% | 2.4%
switch.$\rightarrow$bible | 73% | 24% | 3%
switch.$\rightarrow$poetry | 73.6% | 23.4% | 3%
switch.$\rightarrow$shak. | 79.4% | 18.6% | 4%
Table 16: Comparitive analysis of TSTAR and STRAP model to understand which
model generates more meaning preserving outputs
Direction | TSTAR=STRAP | TSTAR $<$ STRAP | TSTAR $>$ STRAP
---|---|---|---
bible$\rightarrow$switch. | 1.8 | 31.6 | 66.6
poetry$\rightarrow$switch. | 2 | 15.6 | 82.4
shak.$\rightarrow$switch. | 4 | 20.6 | 75.4
switch.$\rightarrow$bible | 1.4 | 19.4 | 79.2
switch.$\rightarrow$poetry | 2.2 | 16.2 | 81.6
switch.$\rightarrow$shak. | 0.8 | 13 | 86.2
bible$\rightarrow$poetry | 1.8 | 27.8 | 70.4
bible$\rightarrow$shak. | 0.8 | 28.8 | 70.4
poetry$\rightarrow$bible | 2 | 20.2 | 77.8
poetry$\rightarrow$shak. | 2.8 | 12.2 | 85
shak.$\rightarrow$bible | 2.4 | 19.6 | 78
Table 17: Comparison on content preservation using human evaluations on
Iterative T-STAR and STRAP across all 12 direction
| STRAP | TSTAR | Itr-TSTAR | STRAP | TSTAR | Itr-TSTAR | STRAP | TSTAR | Itr-TSTAR | STRAP | TSTAR | Itr-TSTAR
---|---|---|---|---|---|---|---|---|---|---|---|---
Direction | No Error $\uparrow$ | Hallucination $\downarrow$ | Incomplete $\downarrow$ | Semantic Drift $\downarrow$
bible$\rightarrow$poetry | 42 | 43 | 62 | 98 | 68 | 70 | 239 | 270 | 228 | 121 | 119 | 140
bible$\rightarrow$shakespeare | 45 | 72 | 93 | 224 | 76 | 87 | 55 | 190 | 149 | 176 | 162 | 171
bible$\rightarrow$switchboard | 54 | 98 | 106 | 203 | 65 | 87 | 59 | 189 | 132 | 184 | 148 | 175
poetry$\rightarrow$bible | 46 | 85 | 110 | 257 | 232 | 175 | 22 | 27 | 15 | 175 | 156 | 200
poetry$\rightarrow$shakespeare | 43 | 134 | 166 | 224 | 89 | 79 | 50 | 53 | 47 | 183 | 224 | 208
poetry$\rightarrow$switchboard | 80 | 142 | 160 | 171 | 63 | 39 | 63 | 82 | 65 | 186 | 213 | 236
shakespeare$\rightarrow$bible | 44 | 69 | 120 | 253 | 203 | 186 | 19 | 43 | 14 | 184 | 185 | 180
shakespeare$\rightarrow$poetry | 44 | 55 | 85 | 187 | 190 | 161 | 77 | 89 | 69 | 192 | 166 | 187
shakespeare$\rightarrow$switchboard | 70 | 132 | 161 | 151 | 69 | 62 | 42 | 98 | 47 | 237 | 201 | 230
switchboard$\rightarrow$bible | 50 | 85 | 129 | 241 | 213 | 221 | 42 | 54 | 14 | 167 | 148 | 135
switchboard$\rightarrow$poetry | 49 | 98 | 114 | 165 | 113 | 97 | 140 | 165 | 123 | 146 | 124 | 166
switchboard$\rightarrow$shakespeare | 62 | 149 | 186 | 189 | 87 | 92 | 77 | 87 | 31 | 172 | 177 | 191
Table 18: Type of Error Analysis across three models STRAP, TSTAR, Iterative-
TSTAR across all the four styles.
### F.2 Error Analysis per direction
In this section, we present the error analysis for each direction in Table 18.
We observe that T-STAR and Iterative T-STAR models are consistently better on
No-Error and Hallucinations across all the directions. Moreover, we observe
that for 7 out of 12 direction, Iterative T-STAR model is better than STRAP
for Incompleteness error. Note that our T-STAR model was under-performing,
however another iteration of model improvements increases that number
significantly. Note that all the models are giving high error in Semantic
Drift, and improving the model for this type of errors can be explored in
future works.
S.No | Input Stylized Sentence | Output Stylized Sentence | TSTAR-Encoder AMR
---|---|---|---
1. | And she answered and said unto him, Yes, Lord: yet the dogs under the table eat of the children’s crumbs. | Yet the dogs under the table Eat children’s crumbs. | | (e / eat-01
---
:ARG0 (d / dog
:location (u / under
:op1 (t / table)))
:ARG1 (c / crumb
:poss (c2 / child))
:mod (s / still))
2. | And the priest said unto them, Go in peace: before the LORD is your way wherein ye go. | Before the Lord go in peace on high; | | (g / go-02
---
:mode imperative
:ARG0 (y / you)
:manner (p / peace)
:time (b / before
:op1 (p2 / person
:name (n / name
:op1 ”Lord”))))
3. | And there were made on them, on the doors of the temple, cherubims and palm trees, like as were made upon the walls; and there were thick planks upon the face of the porch without. | The cherubims and palm-trees are at the temple doors, And the thick planks on the porch face. | | (a / and
---
:op1 (b / be-located-at-91
:ARG1 (a2 / and
:op1 (c / cherubim)
:op2 (t / tree
:mod (p / palm)))
:ARG2 (d / door
:part-of (t2 / temple)))
:op2 (b2 / be-located-at-91
:ARG1 (p2 / plank
:ARG1-of (t3 / thick-03))
:ARG2 (f / face
:part-of (p3 / porch))))
4. | and uh they’ve had catalytic con- you know catalytic converters on those on uh i think they’re required in California and Oregon and and Washington | In Caledonia, Oregon, and Washington catalytic conversion required. | | (r / require-01
---
:ARG1 (t / thing
:ARG0-of (c / convert-01
:mod (c2 / catalytic)))
:location (a / and
:op1 (s / state
:name (n / name
:op1 ”California”))
:op2 (s2 / state
:name (n2 / name
:op1 ”Oregon”))
:op3 (s3 / state
:name (n3 / name
:op1 ”Washington”))))
5. | That crowns a lofty clift, which proudly towers | crowns the lofty towers | | (c / crown-01
---
:ARG0 (ii / it)
:ARG1 (t / tower
:mod (l / lofty)))
6. | And the rough goat is the king of Grecia: and the great horn that is between his eyes is the first king. | The rough goat o’er the king of Greece | | (h / have-org-role-91
---
:ARG0 (g / goat
:ARG1-of (r / rough-04))
:ARG1 (c / country
:name (n / name
:op1 ”Greece”))
:ARG2 (k / king))
Table 19: Samples from Human Evaluations where the T-STAR output was marked
incomplete. We observe that entities and relations missing in the sentence,
were not present in their corresponding intermediate AMRs.
Input Stylized Sentence | Output Stylized Sentence | Input generated AMR | Potential Error
---|---|---|---
Ay, mine own fortune in my misery. | my fortune is my misery | | (m / misery
---
:poss (ii / i)
:domain (f / fortune
:poss ii))
T-STAR Decoder
yeah it makes you wonder um had the jury been a different group of people | It maketh you to wonder at the diversities of the jury. | | (m / make-02
---
:ARG0 (ii / it)
:ARG1 (w / wonder-01
:ARG0 (y / you)
:ARG1 (d / differ-02
:ARG1 (j / jury))))
T-STAR Decoder
Tis not Maria’s whispering call; | It is not whispered by Maria. | | (w / whisper-01
---
:polarity -
:ARG0 (p / person
:name (n / name
:op1 ”Maria”))
:ARG1 (ii / it))
T-STAR Encoder
Yea,” quoth he, “Dost thou fall upon thy face? | Quoth he, ”Dost thou fall thy face, | | (q / quote-01
---
:ARG1 (f / fall-01
:ARG1 (y / you
:part (f2 / face))
:ARG4 f2
:polarity (a / amr-unknown))
:ARG2 (h / he))
T-STAR Encoder
Around my grave they’ll wither. | They will wither about my grave. | | (w / wither-01
---
:ARG1 (t / they)
:location (a / around
:op1 (g / grave
:poss (ii / i))))
T-STAR Decoder
Justice is sworn ’gainst tears, and hers would crave | uh Justice has sworn to cried and cried | | (s / swear-01
---
:ARG0 (p / person
:name (n / name
:op1 ”Justice”))
:ARG1 (a / and
:op1 (c / cry-02
:ARG0 p)
:op2 (c2 / cry-02
:ARG0 p)))
T-STAR Encoder
Table 20: Various Instances that had Semantic Drift as a type of error. We
manually analyze the same, and hypothesize that the potential error in the
listed module.
### F.3 Qualitative Analysis for Incompleteness and Semantic Drift
As we are using interpretable intermediate representations, it is easily
possible to understand the intuition behind these errors, and broadly
understand which modules (encoder or decoder) needs to be improved further.
Therefore, we study few instances and analyze the generated Intermediate AMRs
to understand the reason for high number in Semantic Drift and Incomplete
errors. We list down some intstances in Table 20 and Table 19
Across the various instances that we analyzed, we observe that the generated
AMRs were not encoding the complete information themselves. For instance,
either missing some entities (example 1, 4, 5 and 6 in Table 19), if the
clauses were separated using ”:”, ”;”, only one of those were parsed in the
intermediate graph (example 2 and 3 in Table 19. For semantic drift, we
observed that the errors was arising due to the shortcomings in both the
modules, i.e., it was leading to meaning change if the Encoder didn’t generate
an efficient AMR graph, or if the decoder was not able to interpret AMR
correctly. We have listed the modules that could be the potential reason for
the error in the last column in Table 20.
It is important to note that, the source of errors is very easy to identify
now because we are using robust, interpretable and symbolic representation as
pivot to transfer from style A to style B. We have also provided a case study
of performance of various baselines and proposed model in Table 7 on the CDS
dataset.
|
# On Fully Nonlinear Loewner-Nirenberg Problem of Ricci curvature
Zhenan Sui Institute for Advanced Study in Mathematics of HIT, Harbin
Institute of Technology, Harbin, China<EMAIL_ADDRESS>
###### Abstract.
We prove the existence of a smooth complete conformal metric with constant kth
elementary symmetric function of negative Ricci curvature under certain
condition on general domain in Euclidean space. We then formulate this problem
for more general equations.
###### 2010 Mathematics Subject Classification:
Primary 53C21; Secondary 35J60
## 1\. Introduction
In this paper, we discuss the fully nonlinear version of Loewner-Nirenberg
problem: let $\Omega\subsetneq\mathbb{R}^{n}$ be a domain in Euclidean space
with $n\geq 3$ and $g$ the Euclidean metric. Assume that $\partial\Omega$
consists of a finite number of disjoint, non-self-intersecting, smooth,
compact surfaces. We want to find a smooth complete metric
$g_{u}=u^{\frac{4}{n-2}}g$ with $u>0$, which satisfies
(1.1)
$\sigma_{k}\Big{(}-g_{u}^{-1}\text{Ric}_{g_{u}}\Big{)}:=\sigma_{k}\Big{(}\lambda\big{(}-g_{u}^{-1}\text{Ric}_{g_{u}}\big{)}\Big{)}=1\quad\text{
in }\Omega,$
where $\text{Ric}_{g}$ is the Ricci tensor of $g$,
$\lambda(A)=(\lambda_{1},\ldots,\lambda_{n})$ are the eigenvalues of the
matrix $A$, and
$\sigma_{k}(\lambda)=\sum\limits_{1\leq i_{1}<\cdots<i_{k}\leq
n}\lambda_{i_{1}}\cdots\lambda_{i_{k}}$
is the $k$th elementary symmetric function defined on Garding’s cone
$\Gamma_{k}=\\{\lambda\in\mathbb{R}^{n}:\sigma_{j}(\lambda)>0,\,j=1,\ldots,k\\}.$
Under conformal deformation of metric $g_{u}=u^{\frac{4}{n-2}}g$, the Ricci
tensor $\text{Ric}_{g_{u}}$ and $\text{Ric}_{g}$ are related by the formula
$\frac{n-2}{2}\mbox{Ric}_{g_{u}}=\frac{n-2}{2}\mbox{Ric}_{g}-(n-2)\frac{\nabla^{2}u}{u}-\bigg{(}\frac{\Delta
u}{u}+\frac{|\nabla u|^{2}}{u^{2}}\bigg{)}g+\frac{n}{u^{2}}du\otimes du,$
where $\nabla{u}$, $\nabla^{2}u$ and $\Delta u$ are the gradient, Hessian and
Laplace-Beltrami operator of $u$ with respect to the background metric $g$
respectively. We note that $\mbox{Ric}_{g}\equiv 0$ if $g$ is Euclidean.
Consequently, the geometric problem (1.1) is equivalent to the second order
partial differential equation
(1.2)
$\sigma_{k}^{\frac{1}{k}}\big{(}W[u]\big{)}=\frac{n-2}{2}u^{\frac{n+2}{n-2}}\quad\text{
in }\Omega$
along with the boundary condition
(1.3) $u=\infty\quad\text{ on }\partial\Omega,$
where
(1.4) $W[u]=g^{-1}\bigg{(}(n-2)\nabla^{2}u-\frac{n}{u}du\otimes
du+\Big{(}\Delta u+\frac{|\nabla
u|^{2}}{u}\Big{)}g-\frac{n-2}{2}u\text{Ric}_{g}\bigg{)}.$
We shall call a $C^{2}$ function $u$ $k$-admissible (or admissible if there is
no ambiguity) in $\Omega$ if
$\lambda\big{(}W[u]\big{)}(x)\in\Gamma_{k}$
for any $x\in\Omega$. We note that equation (1.2) is elliptic if $u$ is
admissible. For convenience, we call equation (1.2) the conformal $k$-Ricci
curvature equation. Our goal in this paper is to seek smooth positive
admissible solutions to (1.2).
When $k=1$, (1.2)-(1.3) reduces to
(1.5) $\left\\{\begin{aligned} \frac{4(n-1)}{n-2}\Delta
u-S_{g}u=&u^{\frac{n+2}{n-2}}\quad\text{ in }\,\,\Omega,\\\
u=&\infty\quad\text{ on }\,\,\partial\Omega,\end{aligned}\right.$
where $S_{g}$ is the scalar curvature with respect to $g$. In this case, the
above problem is known as Loewner-Nirenberg problem. If $\Gamma$ is a smooth
compact submanifold of $\mathbb{R}^{n}$ of codimension $m$ such that
$\partial\Omega\setminus\Gamma$ is also compact, Loewner and Nirenberg [13]
proved the existence of a smooth conformally flat metric with constant
negative scalar curvature in $\Omega$ satisfying
$u(x)\rightarrow\infty\quad\text{ as }x\rightarrow\Gamma$
if $m<\frac{n}{2}+1$. They also gave the nonexistence result if
$m>\frac{n}{2}+1$ and conjectured the nonexistence for the borderline case
$m=\frac{n}{2}+1$, which was later proved by Aviles and Véron [2, 15].
When $k>1$, equation (1.2) becomes fully nonlinear. Following [5], we call the
above problem fully nonlinear Loewner-Nirenberg problem. If $\text{Ric}_{g}$
in (1.1) is replaced by the Schouten tensor $A_{g}$:
$A_{g}=\frac{1}{n-2}\Big{(}Ric_{g}-\frac{S_{g}}{2(n-1)}g\Big{)},$
González, Li and Luc [5] proved the existence of a Lipschitz continuous
complete conformally flat metric $g_{u}=u^{\frac{4}{n-2}}g$ satisfying
$\sigma_{k}\Big{(}-g_{u}^{-1}A_{g_{u}}\Big{)}=1$
if the vector
$u_{m}=\big{(}\underbrace{1,\ldots,1}_{n-m+1},\underbrace{-1,\ldots,-1}_{m-1}\big{)}\in\Gamma_{k}.$
They also gave the nonexistence result if $u_{m}\notin\overline{\Gamma}_{k}$.
The borderline case
(1.6) $u_{m}\in\partial\Gamma_{k}$
remains open.
Motivated by the above literatures, we investigate fully nonlinear Loewner-
Nirenberg problem of Ricci curvature. Different from the Schouten tensor case,
we are able to obtain smooth solutions, thanks to the interior second order
estimates of Guan [6] and Evans-Krylov theory [3, 11].
Our first result is an existence result when $\partial\Omega$ is composed of
hypersurfaces.
###### Theorem 1.1.
Let $\Omega\subsetneq\mathbb{R}^{n}$ be a domain whose boundary
$\partial\Omega$ consists of finitely many disjoint non-self-intersecting
smooth compact hypersurfaces. Then there exists a smooth complete metric
$g_{u}=u^{\frac{4}{n-2}}g$ satisfying (1.1), or equivalently, there exists a
smooth positive admissible solution $u$ to (1.2)-(1.3). Moreover, the
conformal factor $u$ has the following growth rate
(1.7)
${\rho}^{\frac{n}{2}-1}(x)u(x)\rightarrow\big{(}(n-1)(C_{n}^{k})^{\frac{1}{k}}\big{)}^{\frac{n-2}{4}}\quad\text{
as }x\rightarrow\partial\Omega,$
where $\rho(x)$ is the distance of $x$ to $\partial\Omega$. If $\Omega$ is
bounded or $k=1$, the solution $u$ is unique.
We remark that on smooth compact Riemannian manifold with boundary, the growth
rate (1.7) was discovered by Gursky, Streets and Warren [10]. In Theorem 2.8,
we shall give a simplified proof for general smooth manifolds with compact
boundaries.
Our main result on the solvability of fully nonlinear Loewner-Nirenberg
problem of Ricci curvature is as follows.
###### Theorem 1.2.
Let $\Omega\subsetneq\mathbb{R}^{n}$ be a domain. Suppose that the boundary
$\partial\Omega$ consists of finitely many disjoint non-self-intersecting
smooth compact surfaces, and the maximal solution $u_{\Omega}$ of (1.2)
satisfies $u_{\Omega}\not\equiv 0$. If each component of $\partial\Omega$ has
codimension $m$ satisfying
(1.8)
$v_{m}=\big{(}\underbrace{n-m,\ldots,n-m}_{n-m+1},\underbrace{2-m,\ldots,2-m}_{m-1}\big{)}\in\Gamma_{k},$
then there exists a smooth complete metric $g_{u}=u^{\frac{4}{n-2}}g$
satisfying (1.1), or equivalently, there exists a smooth positive admissible
solution to (1.2)-(1.3). If some component has codimension $m$ satisfying
$v_{m}\notin\overline{\Gamma}_{k}$, then there does not exist a complete
conformal metric satisfying (1.1), or equivalently, equation (1.2)-(1.3) is
not solvable.
The borderline case $v_{m}\in\partial\Gamma_{k}$ is still open. The difficulty
lies in the lack of a suitable upper barrier. We observe that when $k=1$, the
borderline case (1.6) agrees with (1.8), which was solved in [2, 15] by some
integral estimates. We wish to reinvestigate the borderline case in the
future.
Motivated by [9], the second goal in this paper is to investigate the
existence of complete conformal metrics subject to the following more general
equation
(1.9)
$\sigma_{k}\Big{(}-g_{u}^{-1}\text{Ric}_{g_{u}}\Big{)}+\alpha(x)\sigma_{k-1}\Big{(}-g_{u}^{-1}\text{Ric}_{g_{u}}\Big{)}=\alpha_{0}(x),$
where $\alpha_{0}(x)>0$ and $\alpha(x)$ are real valued smooth functions on
$\overline{\Omega}$. Under conformal deformation of metric
$g_{u}=u^{\frac{4}{n-2}}g$, (1.9) is equivalent to
(1.10)
$\displaystyle\sigma_{k}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}+\alpha(x)\sigma_{k-1}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}=\alpha_{0}(x).$
As proved in [9], equation (1.10) is elliptic when $u$ is $(k-1)$-admissible.
In what follows, we may assume that $k\geq 2$ in (1.10).
Our first result associated to (1.10) shows that similar to (1.2), there
exists smooth positive $(k-1)$-admissible solutions to Dirichlet problems.
###### Theorem 1.3.
Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ whose boundary is
composed of finitely many disjoint non-self-intersecting smooth compact
hypersurfaces. For any smooth positive function $\varphi$ defined on
$\partial\Omega$, there exists a smooth positive $(k-1)$-admissible solution
$u$ to equation (1.10) which satisfies the boundary condition
(1.11) $u=\varphi\quad\text{ on }\partial\Omega.$
The proof of Theorem 1.3 relies on the establishment of $C^{2}$ a priori
estimates (see Theorem 4.2, 4.4, 4.5) and the existence of a subsolution. We
notice that the method for $C^{2}$ estimates is similar to Guan [6] but may be
more complicated due to some extra terms. In addition, the estimates in
section 4 depend explicitly on $\inf\alpha_{0}$. Also, we give a new proof of
the second order boundary estimate. For deriving preliminary estimates and
conducting continuity process, we construct a smooth subsolution to the
Dirichlet problem (1.10)–(1.11), synthesizing the ideas of Guan [6] and Guan
[8].
Our next results associated to (1.10) are on the formulation of fully
nonlinear Loewner-Nirenberg type problem. When the boundary $\partial\Omega$
is composed of hypersurfaces, we have the following result.
###### Theorem 1.4.
Let $\Omega$ be a domain in $\mathbb{R}^{n}$ with nonempty smooth compact
boundary which is composed of closed hypersurfaces. Then there exists a smooth
complete metric $g_{u}=u^{\frac{4}{n-2}}g$ satisfying (1.9), or equivalently,
there exists a smooth positive $(k-1)$-admissible solution $u$ to equation
(1.10)–(1.3), provided that
$\inf\limits_{\Omega}\alpha_{0}(x)>0,\quad\inf\limits_{\Omega}\alpha(x)>-\infty,\quad\sup\limits_{\Omega}\alpha_{0}(x)<\infty,\quad\sup\limits_{\Omega}\alpha(x)<\infty.$
The proof of Theorem 1.4 relies further on the interior second order estimates
for equation (1.10), which are derived in Theorem 4.1, 4.3. Also, we need to
find a global upper barrier and a global lower barrier to conduct the diagonal
process. The latter is given by Proposition 5.4 and the former is provided by
Loewner and Nirenberg [13].
When $\Omega$ is a general domain in $\mathbb{R}^{n}$ with smooth compact
boundary, in order to define the maximal solution (see Definition 5.6), we
have to assume that $\alpha(x)\leq 0$ in $\Omega$ so that we can apply the
maximum principle Theorem 5.3. We obtain the following result.
###### Theorem 1.5.
Let $\Omega\subsetneq\mathbb{R}^{n}$ be a domain with finitely many disjoint
non-self-intersecting smooth compact surfaces as the boundary. Suppose that
$\alpha(x)\leq 0\text{ in
}\Omega,\quad\inf\limits_{\Omega}\alpha_{0}(x)>0,\quad\inf\limits_{\Omega}\alpha(x)>-\infty,\quad\sup\limits_{\Omega}\alpha_{0}(x)<\infty.$
In addition, suppose that the maximal solution $u_{\Omega}$ of (1.10)
satisfies
$u_{\Omega}\not\equiv 0\quad\text{ in }\Omega.$
If each component of $\partial\Omega$ has codimension $m$ satisfying
$v_{m}\in\Gamma_{k}$, then there exists a smooth complete metric
$g_{u}=u^{\frac{4}{n-2}}g$ satisfying (1.9), or equivalently, there exists a
smooth positive $k$-admissible solution to (1.10)–(1.3). If some component has
codimension $m$ satisfying $v_{m}\notin\overline{\Gamma}_{k}$, then there does
not exist a complete conformal metric satisfying (1.9), or equivalently,
equation (1.10)–(1.3) is not solvable.
This paper is organized as follows. We prove Theorem 1.1 in section 2 and
Theorem 1.2 in section 3. Section 4 is devoted to $C^{2}$ a priori estimates
and interior $C^{2}$ estimates for equation (1.10). Then we apply continuity
method and degree theory to prove Theorem 1.3. In section 5, we discuss the
fully nonlinear Loewner-Nirenberg problem associated to equation (1.10) and
give the proof of Theorem 1.4 and Theorem 1.5.
## 2\. Solutions on Euclidean domains with smooth compact boundary consisting
of closed hypersurfaces
In this paper, we shall mainly discuss admissible solutions to equation
(1.2)-(1.3) in Euclidean domains which have smooth compact boundaries. Within
this section, we focus on the case when these boundaries are composed of
closed hypersurfaces.
### 2.1. Preliminaries
###### Definition 2.1.
A function $0<\underline{u}\in C^{2}(\Omega)$ is a subsolution of (1.2) in
$\Omega$ if
$\lambda\big{(}W[\underline{u}]\big{)}\in\Gamma_{k}\text{ and
}\sigma_{k}^{\frac{1}{k}}\big{(}W[\underline{u}]\big{)}\geq\frac{n-2}{2}\underline{u}^{\frac{n+2}{n-2}}\quad\text{
in }\Omega.$
A function $0<\overline{u}\in C^{2}(\Omega)$ is a supersolution of (1.2) in
$\Omega$ if
$\text{either }\lambda\big{(}W[\overline{u}]\big{)}\notin\Gamma_{k}\text{ or
}\sigma_{k}^{\frac{1}{k}}\big{(}W[\overline{u}]\big{)}\leq\frac{n-2}{2}\overline{u}^{\frac{n+2}{n-2}}\quad\text{
in }\Omega.$
In view of the form of equation (1.2), we first give a general property on
supersolutions and subsolutions.
###### Proposition 2.2.
If $u$ is a positive subsolution of (1.2), so does $cu$ for any constant
$0<c<1$. If $u$ is a positive supersolution of (1.2), so does $cu$ for any
constant $c>1$; if in addition $\lambda\big{(}W[u]\big{)}\notin\Gamma_{k}$
everywhere in $\Omega$, then $cu$ is a supersolution of (1.2) for any $c>0$.
###### Proof.
The conclusion follows from the fact that $W[cu]=cW[u]$ and
$\sigma_{k}^{\frac{1}{k}}(W[cu])=\sigma_{k}^{\frac{1}{k}}(cW[u])=c\sigma_{k}^{\frac{1}{k}}(W[u])\quad\text{if
}\lambda\big{(}W[u]\big{)}\in\Gamma_{k}.$
∎
We will also need the following maximum principle. The proof is similar to
[10].
###### Theorem 2.3.
Let $M$ be a smooth compact manifold with boundary. Suppose that $u$ and $v$
are $C^{2}$ positive subsolution and supersolution of (1.2) respectively on
$M$. If $u\leq v$ on $\partial M$, then $u\leq v$ on $M$.
###### Proof.
Suppose that $u>v$ somewhere in the interior of $M$. Let $C$ be the maximum of
$\frac{u}{v}$ on $M$, which is attained at $x_{0}$ in the interior of $M$.
Since $C>1$, by Theorem 2.2 we know that $w=\frac{u}{C}$ is a strict
subsolution, that is,
$\sigma_{k}^{\frac{1}{k}}(W[w])>\frac{n-2}{2}w^{\frac{n+2}{n-2}}.$
On the other hand, since $w(x_{0})=v(x_{0})$ while $w\leq v$ near $x_{0}$,
thus at $x_{0}$ we have
$\nabla w(x_{0})=\nabla
v(x_{0}),\quad\nabla^{2}w(x_{0})\leq\nabla^{2}v(x_{0})$
and consequently $W[w](x_{0})\leq W[v](x_{0})$. It follows that
$\sigma_{k}^{\frac{1}{k}}(W[v])(x_{0})\geq\sigma_{k}^{\frac{1}{k}}(W[w])(x_{0})>\frac{n-2}{2}w^{\frac{n+2}{n-2}}(x_{0})=\frac{n-2}{2}v^{\frac{n+2}{n-2}}(x_{0}),$
contradicting with the fact that $v$ is a supersolution. ∎
Next, we provide some special solutions on balls and exterior of balls in
$\mathbb{R}^{n}$ of equation (1.2), which will be used frequently as
supersolutions or subsolutions. In Euclidean coordinates, $W[u]$ can be
expressed as
(2.1) $W_{ij}[u]=(n-2)u_{ij}-n\frac{u_{i}u_{j}}{u}+\Big{(}\Delta
u+\frac{|\nabla u|^{2}}{u}\Big{)}\delta_{ij}.$
Let
$B_{R}(x_{0})=\big{\\{}x\in\mathbb{R}^{n}\big{|}|x-x_{0}|<R\big{\\}}.$
We can verify the following proposition.
###### Proposition 2.4.
For any fixed $s>0$,
$u(x)=\Big{(}4(n-1)(C_{n}^{k})^{\frac{1}{k}}s^{2}\Big{)}^{\frac{n-2}{4}}\Big{(}s^{2}-{|x-x_{0}|}^{2}\Big{)}^{1-\frac{n}{2}}\quad\text{
in }B_{s}(x_{0}),$
$v(x)=\Big{(}4(n-1)(C_{n}^{k})^{\frac{1}{k}}s^{2}\Big{)}^{\frac{n-2}{4}}\Big{(}{|x-x_{0}|}^{2}-s^{2}\Big{)}^{1-\frac{n}{2}}\quad\text{
in }\mathbb{R}^{n}\setminus\overline{B_{s}(x_{0})}$
are admissible solutions of (1.2) which approach to $\infty$ on $\partial
B_{s}(x_{0})$.
###### Proof.
We first prove that $u(x)$ is an admissible solution of (1.2). For
convenience, denote
$c:=\Big{(}4(n-1)(C_{n}^{k})^{\frac{1}{k}}s^{2}\Big{)}^{\frac{n-2}{4}},\quad
x=(x^{1},\ldots,x^{n}),\quad x_{0}=(x_{0}^{1},\ldots,x_{0}^{n}).$
Direct calculation shows that
$W_{ij}[u]=2(n-1)(n-2)c\Big{(}s^{2}-|x-x_{0}|^{2}\Big{)}^{-\frac{n}{2}-1}s^{2}\delta_{ij}.$
Hence
$\sigma_{k}^{\frac{1}{k}}\big{(}W[u]\big{)}=2(n-1)(n-2)c\Big{(}s^{2}-|x-x_{0}|^{2}\Big{)}^{-\frac{n}{2}-1}s^{2}\big{(}C_{n}^{k}\big{)}^{\frac{1}{k}},$
which agrees with the right hand side
$\frac{n-2}{2}u^{\frac{n+2}{n-2}}=\frac{n-2}{2}c^{\frac{n+2}{n-2}}\Big{(}s^{2}-|x-x_{0}|^{2}\Big{)}^{-\frac{n}{2}-1}.$
$v(x)$ can be verified similarly. ∎
The following result is a direct consequence of Proposition 2.4 and Theorem
2.3.
###### Corollary 2.5.
For any positive admissible solution $u$ of (1.2) in $B_{R}(x_{0})$, we have
(2.2)
$u(x)\leq\Big{(}4(n-1)(C_{n}^{k})^{\frac{1}{k}}R^{2}\Big{)}^{\frac{n-2}{4}}\Big{(}R^{2}-{|x-x_{0}|}^{2}\Big{)}^{1-\frac{n}{2}}\quad\text{
in }B_{R}(x_{0}).$
In particular, we have
(2.3)
$u(x_{0})\leq\Big{(}4(n-1)(C_{n}^{k})^{\frac{1}{k}}\Big{)}^{\frac{n-2}{4}}R^{1-\frac{n}{2}}.$
As a consequence of (2.3), we immediately obtain
###### Corollary 2.6.
There does not exist a positive admissible solution to (1.2) on
$\mathbb{R}^{n}$.
###### Proof.
If $u$ is a positive admissible solution to (1.2) on $\mathbb{R}^{n}$, then
$u$ is obviously a positive admissible solution on $B_{R}(0)$ for any $R>0$.
By (2.3) we have
$u(0)\leq\Big{(}4(n-1)(C_{n}^{k})^{\frac{1}{k}}\Big{)}^{\frac{n-2}{4}}R^{1-\frac{n}{2}}.$
Letting $R\rightarrow\infty$ yields
$u(0)\leq 0,$
which contradicts with the fact that $u(0)>0$. ∎
We also find the decay property at $\infty$ of any positive admissible
solution of (1.2).
###### Corollary 2.7.
Let $\Omega$ be an unbounded domain in $\mathbb{R}^{n}$ with smooth compact
boundary. If $u$ is a smooth positive admissible solution of (1.2), then
(2.4) $u(x)\rightarrow 0\quad\text{ as }\,\,|x|\rightarrow\infty.$
###### Proof.
Applying (2.3) in a ball centered at $x$ of radius $|x|/2$ for $|x|$ large, we
have
$u(x)\leq\Big{(}4(n-1)\big{(}C_{n}^{k}\big{)}^{\frac{1}{k}}\Big{)}^{\frac{n-2}{4}}\Big{(}{\frac{|x|}{2}}\Big{)}^{1-\frac{n}{2}},$
which implies (2.4). ∎
### 2.2. Growth rate near codimension $1$ boundary
In this section, we assume that the boundary of $\Omega$ is composed of smooth
compact hypersurfaces. For positive admissible solution of (1.2) which
approaches $\infty$ at $\partial\Omega$, we characterize the growth rate.
###### Theorem 2.8.
Let $(M,g)$ be a smooth manifold with boundary. Assume that $\partial M$ is
compact. If $u$ is a smooth positive admissible solution of (1.2) on $M$ which
approaches to $\infty$ at $\partial M$, then
(2.5)
${\rho}^{\frac{n}{2}-1}(x)u(x)\rightarrow\big{(}(n-1)(C_{n}^{k})^{\frac{1}{k}}\big{)}^{\frac{n-2}{4}}\quad\text{
as }x\rightarrow\partial M,$
where $\rho(x)$ is the distance from $x$ to $\partial M$.
###### Proof.
We first construct a positive admissible subsolution in a neighborhood of
$\partial M$. Consider
$\underline{u}=c_{0}(\rho+\epsilon)^{1-\frac{n}{2}}e^{w},$
where $\epsilon$ is a small positive constant,
$c_{0}=\big{(}(n-1)(C_{n}^{k})^{\frac{1}{k}}\big{)}^{\frac{n-2}{4}},$
and
$w=w(\rho)=\frac{1}{\rho+\delta}-\frac{1}{\delta}$
with $\delta$ a small positive constant to be chosen later.
For any point $x\in M\setminus\partial M$ at which $\rho$ is smooth, we choose
a local orthonormal frame $e_{1},\ldots,e_{n}$ around $x$ such that
$e_{1}=\frac{\partial}{\partial\rho}$ (recall that $\nabla\rho$ =
$\frac{\partial}{\partial\rho}$ and $|\nabla\rho|=1$ at $x$). Elementary
calculation shows that at $x$,
$\nabla\underline{u}=c_{0}e^{w}(\rho+\epsilon)^{-\frac{n}{2}}\Big{(}1-\frac{n}{2}+(\rho+\epsilon)w^{\prime}\Big{)}\nabla\rho,$
$\displaystyle\nabla^{2}\underline{u}=$ $\displaystyle
c_{0}e^{w}(\rho+\epsilon)^{-\frac{n}{2}-1}\Bigg{(}\bigg{(}\Big{(}1-\frac{n}{2}\Big{)}(\rho+\epsilon)+(\rho+\epsilon)^{2}w^{\prime}\bigg{)}\nabla^{2}\rho$
$\displaystyle+\bigg{(}\frac{n}{2}\Big{(}\frac{n}{2}-1\Big{)}+(\rho+\epsilon)^{2}w^{\prime\prime}-(n-2)(\rho+\epsilon)w^{\prime}+(\rho+\epsilon)^{2}w^{\prime
2}\bigg{)}\nabla\rho\otimes\nabla\rho\Bigg{)},$
and consequently,
$\displaystyle W[\underline{u}]=$ $\displaystyle
c_{0}e^{w}(\rho+\epsilon)^{-\frac{n}{2}-1}\Bigg{(}\Big{(}(n-2)(\rho+\epsilon)^{2}w^{\prime}-\frac{1}{2}(n-2)^{2}(\rho+\epsilon)\Big{)}\nabla^{2}\rho$
$\displaystyle+\Big{(}(n-2)(\rho+\epsilon)^{2}w^{\prime\prime}+2(n-2)(\rho+\epsilon)w^{\prime}-2(\rho+\epsilon)^{2}w^{\prime
2}\Big{)}\nabla\rho\otimes\nabla\rho$
$\displaystyle+\bigg{(}\Big{(}(\rho+\epsilon)^{2}w^{\prime}-\frac{1}{2}(n-2)(\rho+\epsilon)\Big{)}\Delta\rho$
$\displaystyle+(\rho+\epsilon)^{2}w^{\prime\prime}-2(n-2)(\rho+\epsilon)w^{\prime}+2(\rho+\epsilon)^{2}w^{\prime
2}+\frac{1}{2}(n-1)(n-2)\bigg{)}I$
$\displaystyle-\frac{n-2}{2}(\rho+\epsilon)^{2}\text{Ric}_{g}\Bigg{)}.$
For $\rho_{0}>0$, denote
$M_{\rho_{0}}=\big{\\{}x\in M\big{|}\rho(x)<\rho_{0}\big{\\}}.$
We may choose $\rho_{0}$ sufficiently small such that on $M_{\rho_{0}}$,
$u>c_{0}$ and
(2.6) $\Delta\rho I+(n-2)\nabla^{2}\rho\leq\lambda I,\quad\text{Ric}_{g}\leq
C_{0}I,$
where $I$ is the unit matrix and $\lambda$, $C_{0}$ are positive constants
depending only on $g=I$. Denoting
$\displaystyle\Psi=$
$\displaystyle(n-2)(\rho+\epsilon)^{2}w^{\prime\prime}\nabla\rho\otimes\nabla\rho+(n-2)(\rho+\epsilon)^{2}w^{\prime}\nabla^{2}\rho+(\rho+\epsilon)^{2}w^{\prime\prime}I$
$\displaystyle+2(\rho+\epsilon)^{2}w^{\prime
2}(I-\nabla\rho\otimes\nabla\rho)+(\rho+\epsilon)^{2}w^{\prime}\Delta\rho
I-\frac{n-2}{2}(\rho+\epsilon)^{2}\text{Ric}_{g},$ $\displaystyle\Phi=$
$\displaystyle-\frac{1}{2}(n-2)(\rho+\epsilon)\Delta\rho
I-\frac{1}{2}(n-2)^{2}(\rho+\epsilon)\nabla^{2}\rho$
$\displaystyle+2(n-2)(\rho+\epsilon)w^{\prime}(\nabla\rho\otimes\nabla\rho-I)+\frac{1}{2}(n-1)(n-2)I,$
then
$W[\underline{u}]=c_{0}e^{w}(\rho+\epsilon)^{-\frac{n}{2}-1}(\Psi+\Phi).$
Note that
$w^{\prime}=\frac{-1}{(\rho+\delta)^{2}}<0,\quad
w^{\prime\prime}=\frac{2}{(\rho+\delta)^{3}}>0.$
In view of (2.6) we obtain
$\Psi\geq(\rho+\epsilon)^{2}\Big{(}w^{\prime\prime}+\lambda
w^{\prime}-\frac{n-2}{2}C_{0}\Big{)}I.$
We note that
$w^{\prime\prime}+\lambda
w^{\prime}-\frac{n-2}{2}C_{0}=\frac{2}{(\rho+\delta)^{3}}-\frac{\lambda}{(\rho+\delta)^{2}}-\frac{n-2}{2}C_{0}\geq
0$
for $\rho_{0}+\delta$ sufficiently small depending only on $n$, $\lambda$ and
$C_{0}$. Hence we conclude that $\Psi\geq 0$.
Also, by (2.6) and for $\rho_{0}+\delta\leq\sqrt{\frac{2}{\lambda}}$, we have
$\displaystyle\Phi\geq$ $\displaystyle-\frac{n-2}{2}(\rho+\epsilon)\lambda
I+\frac{2(n-2)(\rho+\epsilon)}{(\rho+\delta)^{2}}(I-\nabla\rho\otimes\nabla\rho)+\frac{1}{2}(n-1)(n-2)I$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}(n-1)(n-2)\left(\begin{array}[]{cccc}1-\frac{\lambda(\rho+\epsilon)}{n-1}&&&\\\
&1+\frac{\lambda(\rho+\epsilon)}{n-1}&&\\\ &&\ddots&\\\
&&&1+\frac{\lambda(\rho+\epsilon)}{n-1}\\\ \end{array}\right).$
Letting $\epsilon<\rho_{0}$ and $\rho_{0}<\frac{n-1}{2\lambda}$, we have
$\Phi>0$ and
$\displaystyle\sigma_{k}^{\frac{1}{k}}(W[\underline{u}])\geq$ $\displaystyle
c_{0}e^{w}(\rho+\epsilon)^{-\frac{n}{2}-1}\sigma_{k}^{\frac{1}{k}}(\Phi)$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}(n-1)(n-2)c_{0}e^{w}(\rho+\epsilon)^{-\frac{n}{2}-1}(C_{n}^{k})^{\frac{1}{k}}\cdot$
$\displaystyle\Big{(}1+\frac{\lambda(\rho+\epsilon)}{n-1}\Big{)}^{\frac{k-1}{k}}\bigg{(}1+\Big{(}1-\frac{2k}{n}\Big{)}\frac{\lambda(\rho+\epsilon)}{n-1}\bigg{)}^{\frac{1}{k}}.$
Elementary calculation shows that
$\Big{(}1+\frac{\lambda(\rho+\epsilon)}{n-1}\Big{)}^{k-1}\bigg{(}1+\Big{(}1-\frac{2k}{n}\Big{)}\frac{\lambda(\rho+\epsilon)}{n-1}\bigg{)}\geq
1$
when $k\leq\frac{n}{2}$, or when $k>\frac{n}{2}$ and
$\rho_{0}\leq\frac{(n-1)(n-2)k}{2(2k-n)(k-1)\lambda}$. Therefore,
$\sigma_{k}^{\frac{1}{k}}(W[\underline{u}])\geq\frac{1}{2}(n-1)(n-2)c_{0}(C_{n}^{k})^{\frac{1}{k}}e^{w}(\rho+\epsilon)^{-\frac{n}{2}-1}.$
On the other hand,
$\frac{n-2}{2}\underline{u}^{\frac{n+2}{n-2}}=\frac{n-2}{2}{c_{0}}^{\frac{n+2}{n-2}}(\rho+\epsilon)^{-\frac{n}{2}-1}e^{\frac{n+2}{n-2}w}.$
Note that $w\leq 0$. Hence $\underline{u}$ is a subsolution of (1.2).
Observe that
$\underline{u}=c_{0}\epsilon^{1-\frac{n}{2}}\leq u=\infty\quad\text{ on
}\partial M,$
and on $\rho=\rho_{0}$,
$\underline{u}\leq
c_{0}{\rho_{0}}^{1-\frac{n}{2}}e^{(\rho_{0}+\delta)^{-1}-{\delta}^{-1}}\leq
c_{0}{\rho_{0}}^{1-\frac{n}{2}}e^{-\delta^{-1}/2}\leq c_{0}\leq u$
for $\delta$ sufficiently small depending in addition on $\rho_{0}$. By the
maximum principle,
$\underline{u}\leq u\quad\text{ on }M_{\rho_{0}}.$
Letting $\epsilon\rightarrow 0^{+}$ and then $x\rightarrow\partial M$ we
obtain
$\liminf\limits_{x\rightarrow\partial M}\rho^{\frac{n}{2}-1}u\geq c_{0}.$
We next prove
(2.7) $\limsup\limits_{x\rightarrow\partial M}\rho^{\frac{n}{2}-1}u\leq
c_{0}.$
For this, we construct a supersolution in
$M_{\rho_{1}}\setminus\overline{M_{\epsilon}}$ for any $0<\epsilon<\rho_{1}$:
$\overline{u}=c_{0}(\rho-\epsilon)^{1-\frac{n}{2}}e^{v},$
where
$v=v(\rho)=\frac{1}{2}\Big{(}\ln(\delta+\rho)-\ln\delta\Big{)}$
with $\rho_{1}$, $\delta$ small positive constants to be chosen later.
From the above calculation, it is easy to obtain
$\displaystyle\sigma_{1}\big{(}W[\overline{u}]\big{)}=$ $\displaystyle
c_{0}e^{v}(\rho-\epsilon)^{-\frac{n}{2}-1}\bigg{(}\Big{(}2(n-1)(\rho-\epsilon)^{2}v^{\prime}-(n-2)(n-1)(\rho-\epsilon)\Big{)}\Delta\rho$
$\displaystyle+2(n-1)(\rho-\epsilon)^{2}v^{\prime\prime}-2(n-2)(n-1)(\rho-\epsilon)v^{\prime}+2(n-1)(\rho-\epsilon)^{2}v^{\prime
2}$
$\displaystyle+\frac{n(n-1)(n-2)}{2}-\frac{n-2}{2}(\rho-\epsilon)^{2}S_{g}\bigg{)},$
where $S_{g}$ is the scalar curvature on $M$. We may choose $\rho_{1}$
sufficiently small such that on $M_{\rho_{1}}$,
$|\Delta\rho|\leq C_{1},\quad S_{g}\geq-C_{1},$
where $C_{1}$ is a positive constant depending only on $g$. Note that
$v^{\prime}=\frac{1}{2(\delta+\rho)},\quad
v^{\prime\prime}=-\frac{1}{2(\delta+\rho)^{2}}.$
It follows that
$\displaystyle
2(n-1)v^{\prime}\Delta\rho+2(n-1)v^{\prime\prime}+2(n-1)v^{\prime
2}-\frac{n-2}{2}S_{g}$ $\displaystyle\leq$
$\displaystyle\frac{(n-1)C_{1}}{\delta+\rho}-\frac{n-1}{2(\delta+\rho)^{2}}+\frac{n-2}{2}C_{1}\leq
0,$
and
$\displaystyle-(n-2)(n-1)\Delta\rho-2(n-2)(n-1)v^{\prime}\leq(n-2)(n-1)\Big{(}C_{1}-\frac{1}{\delta+\rho}\Big{)}\leq
0$
for $\rho_{1}+\delta$ sufficiently small depending only on $n$ and $C_{1}$.
Consequently,
$\sigma_{1}\big{(}W[\overline{u}]\big{)}\leq
c_{0}e^{v}(\rho-\epsilon)^{-\frac{n}{2}-1}\frac{n(n-1)(n-2)}{2}.$
Then applying Newton-Maclaurin inequality we arrive at
$\sigma_{k}^{\frac{1}{k}}\big{(}W[\overline{u}]\big{)}\leq\frac{(C_{n}^{k})^{\frac{1}{k}}}{n}\sigma_{1}\big{(}W[\overline{u}]\big{)}\leq\frac{n-2}{2}{\overline{u}}^{\frac{n+2}{n-2}}\quad\text{
in }M_{\rho_{1}}\setminus\overline{M_{\epsilon}}.$
Also, we note that on $\rho=\epsilon$,
$\overline{u}=\infty>u,$
and on $\rho=\rho_{1}$,
$\overline{u}\geq
c_{0}{\rho_{1}}^{1-\frac{n}{2}}\sqrt{\frac{\delta+\rho_{1}}{\delta}}\geq u$
if $\delta$ is chosen sufficiently small depending in addition on $\rho_{1}$
and $u$. By the maximum principle, we have
$\overline{u}\geq u\quad\text{ on }M_{\rho_{1}}\setminus M_{\epsilon}.$
Letting $\epsilon\rightarrow 0^{+}$ and then $x\rightarrow\partial M$ we
obtain (2.7). ∎
###### Remark 2.9.
In [10], a global subsolution
$\underline{u}=c_{0}(\rho+\epsilon)^{1-\frac{n}{2}}e^{A\big{(}(\rho+\delta)^{-p}-{\delta}^{-p}\big{)}}$
is constructed on compact manifold with boundary which yields the lower bound
of the growth rate. We simplify the proof by defining the subsolution only in
a neighborhood of the boundary $\partial M$. For the upper bound, we also
adopt a new supersolution near $\partial M$. For the special case when $M$ is
a domain in $\mathbb{R}^{n}$, we may use González-Li-Nguyen’s method (see
Lemma 3.4 in [5]) to give an easier proof.
### 2.3. Existence of smooth solutions
We now cite an important result of Loewner and Nirenberg [13] when the
boundary of the domain is composed of hypersurfaces for the scalar case
($k=1$), which will be used as an upper barrier later.
###### Lemma 2.10.
Let $\Omega$ be a domain in $\mathbb{R}^{n}$ having smooth compact
hypersurfaces as boundary. There exists a unique positive solution $u\in
C^{\infty}(\Omega)$ of (1.5). In addition, if $\rho(x)$ denotes the distance
of $x$ to the boundary of $\Omega$, then
${\rho}^{\frac{n}{2}-1}(x)u(x)\rightarrow\big{(}n(n-1)\big{)}^{\frac{n-2}{4}}\quad\text{
as }x\rightarrow\partial\Omega.$
If $\Omega$ is unbounded, the solution $u(x)$ has the property that
$|x|^{n-2}u(x)\rightarrow c\quad\text{ as }|x|\rightarrow\infty,$
where $c$ is a positive constant depending only on $n$ and $\Omega$.
We can now state the main theorem in this section.
###### Theorem 2.11.
Let $\Omega$ be a domain in $\mathbb{R}^{n}$ with nonempty smooth compact
boundary which are composed of closed hypersurfaces. Then there exists a
positive admissible solution $u\in C^{\infty}(\Omega)$ to equation (1.2) which
tends to $\infty$ on $\partial\Omega$. If $\Omega$ is bounded or $k=1$, the
solution is unique.
###### Proof.
If $\Omega$ is a bounded domain, by Corollary 1.3 in [6], there exists a
positive admissible solution $u\in C^{\infty}(\Omega)$ to equation (1.2) with
infinite boundary value.
When $\Omega$ is unbounded, we may assume without loss of generality that
$0\notin\overline{\Omega}$. Let $B_{s}(0)$ be a fixed ball such that
$\overline{\Omega}\subset\mathbb{R}^{n}\setminus\overline{B_{s}(0)}$. By
Proposition 2.4, there exists a smooth solution
$\underline{u}=C_{1}(|x|^{2}-s^{2})^{1-\frac{n}{2}},\quad
C_{1}=\big{(}4(n-1)(C_{n}^{k})^{\frac{1}{k}}s^{2}\big{)}^{\frac{n-2}{4}}$
to equation (1.2) in $\mathbb{R}^{n}\setminus\overline{B_{s}(0)}$.
For any $R>\max\limits_{\partial\Omega}\underline{u}$ large enough such that
$\partial\Omega\subset B_{R}(0)$, by Corollary 5.4 in [6], there exists a
smooth positive admissible solution $u_{R}$ to the Dirichlet problem
(2.8) $\left\\{\begin{aligned}
&\sigma_{k}^{\frac{1}{k}}(W[u])=\frac{n-2}{2}{u}^{\frac{n+2}{n-2}}\quad\text{
in }B_{R}(0)\cap\Omega,\\\ &u=R\quad\text{ on }\partial\Omega,\\\
&u=\underline{u}\quad\text{ on }\partial B_{R}(0).\end{aligned}\right.$
By Lemma 2.10, we are able to find a positive solution $\bar{u}\in
C^{\infty}(\Omega)$ of (1.5) which tends to $\infty$ at $\partial\Omega$ with
the growth rate
$\rho^{\frac{n}{2}-1}(x)\overline{u}(x)\rightarrow\big{(}n(n-1)\big{)}^{\frac{n-2}{4}}\quad\text{
as }x\rightarrow\partial\Omega,$
and decays to $0$ at $\infty$ with the decay rate
$|x|^{n-2}\overline{u}(x)\rightarrow c\quad\text{ as }|x|\rightarrow\infty,$
where $c$ is the positive constant in Lemma 2.10.
Comparing the decay rate as $|x|\rightarrow\infty$, we have on $\partial
B_{R}(0)$ with $R$ sufficiently large,
$u_{R}(x)=\underline{u}(x)\leq\frac{2C_{1}}{c}\overline{u}(x).$
Meanwhile, we have
$u_{R}(x)=R<\infty=\frac{2C_{1}}{c}\overline{u}(x)\quad\text{ on
}\partial\Omega.$
By the maximum principle, we obtain
$u_{R}(x)\leq\max\Big{\\{}\frac{2C_{1}}{c},1\Big{\\}}\overline{u}(x)\quad\text{
on }B_{R}(0)\cap\Omega.$
In proving the above inequality, we have applied the Newton-Maclaurin
inequality as well as Proposition 2.2. On the other hand, by the maximum
principle we have
$u_{R}(x)\geq\underline{u}(x)\quad\text{ on }B_{R}(0)\cap\Omega.$
Now we can apply the interior estimates of Guan (see Theorems 2.1 and 3.1 in
[6]), followed by the Evans-Krylov interior estimates [3, 11] and a standard
diagonal process to obtain a smooth positive admissible solution $u$ of (1.2)
on $\Omega$ which tends to $\infty$ at $\partial\Omega$.
When $\Omega$ is bounded, we shall prove the uniqueness. Suppose $v$ is
another positive admissible solution satisfying (1.2)-(1.3). By Theorem 2.8,
$u$ and $v$ have the same growth rate as $x\rightarrow\partial\Omega$. Hence
for any $\epsilon>0$, on $\\{\rho=c\\}$ where $c>0$ is a sufficiently small
positive constant, we have
$u\leq(1+\epsilon)v.$
By Proposition 2.2 and the maximum principle,
$u\leq(1+\epsilon)v\quad\text{ in }\,\,\\{x\in\Omega|\rho(x)>c\\}.$
Letting $c\rightarrow 0$, we have
$u\leq(1+\epsilon)v\quad\text{ in }\,\,\Omega.$
Then let $\epsilon\rightarrow 0$. It follows that
$u\leq v\quad\text{ in }\,\,\Omega.$
Similarly, we can show that $v\leq u$ in $\Omega$. Hence we proved the
uniqueness. ∎
## 3\. Maximal solution on general domain in $\mathbb{R}^{n}$
For any domain $\Omega\subset\mathbb{R}^{n}$ with smooth compact boundary
$\partial\Omega$, we hope to associate a smooth positive admissible solution
$u_{\Omega}$ of (1.2) which is maximal, in the sense that it is greater than
or equal to any smooth positive admissible solution of (1.2) in $\Omega$. We
will then investigate when $u_{\Omega}$ tends to $\infty$ on $\partial\Omega$.
### 3.1. Construction of the maximal solution
Let $\Omega_{(1)}\Subset\Omega_{(2)}\Subset\ldots$ be an increasing sequence
of bounded subdomains of $\Omega$ with smooth compact boundaries
$\partial\Omega_{(j)}$ which are closed hypersurfaces such that
$\Omega=\cup\Omega_{(j)}$. By Theorem 2.11, we can find a unique smooth
positive admissible solution $u_{(j)}$ of (1.2) in $\Omega_{(j)}$ which tends
to $\infty$ on the boundary $\partial\Omega_{(j)}$. By the maximum principle,
we see that $\\{u_{(j)}\\}$ is a monotone decreasing sequence of positive
functions. It follows that $u_{(j)}$ converges to a nonnegative function
$u_{\Omega}$ in $\Omega$.
We have the following dichotomy, which is similar to [5].
###### Lemma 3.1.
We have either $u_{\Omega}>0$ in $\Omega$ or $u_{\Omega}\equiv 0$ in $\Omega$.
###### Proof.
For the sake of completeness, we provide the proof. Suppose that
$u_{\Omega}\not\equiv 0$. Then we must have $u_{\Omega}>c$ on some
$B(x_{0},r_{0})\subset\Omega$ for some constant $c>0$. We may choose
$0<r_{1}<r_{0}$ such that the solution
$v=\big{(}4(n-1)(C_{n}^{k})^{\frac{1}{k}}r_{1}^{2}\big{)}^{\frac{n-2}{4}}\big{(}{|x-x_{0}|}^{2}-r_{1}^{2}\big{)}^{1-\frac{n}{2}}\quad\text{
in }\mathbb{R}^{n}\setminus\overline{B_{r_{1}}(x_{0})}$
(see Proposition 2.4) satisfies
$\big{(}4(n-1)(C_{n}^{k})^{\frac{1}{k}}r_{1}^{2}\big{)}^{\frac{n-2}{4}}\big{(}r_{0}^{2}-r_{1}^{2}\big{)}^{1-\frac{n}{2}}=c.$
Let $\Omega_{(j)}$ and $u_{(j)}$ be as above. By the maximum principle, we
know that $u_{(j)}\geq v$ in $\Omega_{(j)}\setminus B(x_{0},r_{0})$ for any
$j$. Hence $u_{\Omega}\geq v>0$ in $\Omega\setminus B(x_{0},r_{0})$. We thus
have $u_{\Omega}>0$ in $\Omega$. ∎
###### Remark 3.2.
Several remarks are as follows.
1. (1)
In view of Lemma 3.1 and the interior regularity of Guan [6] and Evans-Krylov
[3, 11], we know that $u_{\Omega}$ is smooth.
2. (2)
If there exists a positive admissible subsolution $v$ of (1.2) in $\Omega$
(for example, when $\mathbb{R}^{n}\setminus\overline{\Omega}\neq\emptyset$),
then $u_{\Omega}>0$ in $\Omega$.
3. (3)
When $u_{\Omega}>0$ in $\Omega$, $u_{\Omega}$ is the so-called maximal
solution of equation (1.2) in $\Omega$. In fact, if $w$ is any smooth positive
admissible solution of (1.2) in $\Omega$, then $u_{(j)}\geq w$ on
$\Omega_{(j)}$ by the maximum principle. Hence $u_{\Omega}\geq w$ in $\Omega$.
4. (4)
When $\Omega$ is bounded and $\partial\Omega$ is composed of closed
hypersurfaces, $u_{\Omega}$ coincides with the solution given by Theorem 2.11.
As in [13], we call a compact subset $\Gamma\subset\partial\Omega$ regular, if
(3.1) $u_{\Omega}(x)\rightarrow\infty\quad\text{ as }x\rightarrow\Gamma.$
The rest of this section discusses the validity of (3.1).
### 3.2. Regularity of a portion on the boundary
We consider a portion $\Gamma\subset\partial\Omega$ which is a smooth compact
non-self-intersecting surface of codimension $m$. We also assume that
$\partial\Omega\setminus\Gamma$ is smooth compact. As in [13, 5], we first
give a necessary and sufficient condition for (3.1).
###### Theorem 3.3.
Let $\Omega$ be a domain in $\mathbb{R}^{n}$ and $\Gamma$ be a compact subset
of $\partial\Omega$ such that $\partial\Omega\setminus\Gamma$ is also compact.
Suppose that $u_{\Omega}\not\equiv 0$. Then $u_{\Omega}(x)\rightarrow\infty$
as $x\rightarrow\Gamma$ if and only if there exists an open neighborhood $U$
of $\Gamma$ and a $C^{2}$ positive admissible subsolution $\phi(x)$ of (1.2)
defined in $\Omega\cap U$ which tends to $\infty$ as $x\rightarrow\Gamma$.
###### Proof.
Necessity is obvious. For sufficiency, without loss of generality we may
assume that $\overline{U}$ is compact,
$\overline{U}\cap(\partial\Omega\setminus\Gamma)=\emptyset$ and $0<\phi\in
C^{2}(\overline{U}\cap\Omega)$. For $j$ sufficiently large, we have
$\partial(U\cap\Omega_{(j)})=\partial U\cup(U\cap\partial\Omega_{(j)}).$
Since $u_{\Omega}$ is positive in $\Omega$ and $\partial U\subset\Omega$ is
compact, we have $\inf\limits_{\partial U}u_{\Omega}:=m>0$. Denote
$M:=\sup\limits_{\partial U}\phi(x)>0$ and $A:=\max\\{\frac{M}{m},1\\}$. We
note that $u_{(j)}\geq u_{\Omega}\geq m\geq\frac{\phi}{A}$ on $\partial U$,
and $u_{(j)}=\infty>\frac{\phi}{A}$ on $U\cap\partial\Omega_{(j)}$. By
Proposition 2.2, $\frac{\phi}{A}$ is again a positive admissible subsolution
of (1.2). By the maximum principle, we arrive at $u_{(j)}\geq\frac{\phi}{A}$
in $U\cap\Omega_{(j)}$. Consequently, we obtain $u_{\Omega}\geq\frac{\phi}{A}$
in $U\cap\Omega$. Hence we proved the sufficiency. ∎
Now we aim at constructing $\phi(x)$ in a neighborhood of $\Gamma$ as in
Theorem 3.3. Let $\rho(x)$ denote the distance of $x$ to $\Gamma$. Define
$\Gamma_{\rho_{0}}=\\{x\in\Omega|\rho(x)<\rho_{0}\\}.$
For $\rho_{0}$ sufficiently small, $\rho$ is a smooth function in
$\Gamma_{\rho_{0}}\setminus\Gamma$. Define on
$\Gamma_{\rho_{0}}\setminus\Gamma$
$\phi(x)=c\rho^{1-\frac{n}{2}}(x),$
where $c>0$ is a constant to be determined later. Recall that
$|\nabla\rho|=1$. Direct calculation shows that
$\phi_{i}=c\Big{(}1-\frac{n}{2}\Big{)}{\rho}^{-\frac{n}{2}}\rho_{i},$
$\phi_{ij}=c\Big{(}\frac{n}{2}-1\Big{)}\frac{n}{2}{\rho}^{-\frac{n}{2}-1}\rho_{i}\rho_{j}+c\Big{(}1-\frac{n}{2}\Big{)}{\rho}^{-\frac{n}{2}}\rho_{ij},$
$|\nabla\phi|^{2}=c^{2}\Big{(}1-\frac{n}{2}\Big{)}^{2}{\rho}^{-n},$
$\Delta\phi=c\Big{(}\frac{n}{2}-1\Big{)}\frac{n}{2}{\rho}^{-\frac{n}{2}-1}+c\Big{(}1-\frac{n}{2}\Big{)}{\rho}^{-\frac{n}{2}}\Delta\rho.$
Thus, we have
$W_{ij}[\phi]=\frac{1}{2}(n-2)c{\rho}^{-\frac{n}{2}-1}\Big{(}(n-1-\rho\Delta\rho)\delta_{ij}-(n-2)\rho\rho_{ij}\Big{)}.$
As shown in [5], as $\rho\rightarrow 0$,
$\lambda(\rho\nabla^{2}\rho)\rightarrow\big{(}\underbrace{0,\ldots,0}_{n-m+1},\underbrace{1,\ldots,1}_{m-1}\big{)}.$
Consequently, as $\rho\rightarrow 0$,
$\lambda\big{(}{\rho}^{\frac{n}{2}+1}W_{ij}[\phi]\big{)}\rightarrow\frac{1}{2}(n-2)c\big{(}\underbrace{n-m,\ldots,n-m}_{n-m+1},\underbrace{2-m,\ldots,2-m}_{m-1}\big{)}.$
If we assume that
$v_{m}=\big{(}\underbrace{n-m,\ldots,n-m}_{n-m+1},\underbrace{2-m,\ldots,2-m}_{m-1}\big{)}\in\Gamma_{k},$
then we have as $\rho\rightarrow 0$,
${\rho}^{\frac{n}{2}+1}\sigma_{k}^{\frac{1}{k}}\big{(}W_{ij}[\phi]\big{)}\rightarrow\frac{1}{2}(n-2)c\sigma_{k}^{\frac{1}{k}}(v_{m}).$
On the other hand,
$\frac{n-2}{2}\phi^{\frac{n+2}{n-2}}=\frac{n-2}{2}c^{\frac{n+2}{n-2}}\rho^{-\frac{n}{2}-1}.$
If we choose
$0<c<\Big{(}\sigma_{k}^{\frac{1}{k}}(v_{m})\Big{)}^{\frac{n-2}{4}},$
then $\phi$ is the one as in Theorem 3.3, which implies that
$u_{\Omega}\rightarrow\infty$ as $x\rightarrow\Gamma$ if $u_{\Omega}\not\equiv
0$.
### 3.3. The case when $v_{m}\notin\overline{\Gamma}_{k}$
We adopt the same function as [5]
$\psi=\psi_{c,d}=(c\rho^{-\alpha}+d)^{\beta},$
where $\alpha$, $\beta$, $c$ and $d$ are positive constants. Recall that
$\rho(x)$ is the distance of $x$ to $\Gamma$. By direct calculation, we have
$\displaystyle\psi_{i}=$ $\displaystyle-\alpha\beta
c(c\rho^{-\alpha}+d)^{\beta-1}\rho^{-\alpha-1}\rho_{i},$
$\displaystyle\psi_{ij}=$
$\displaystyle\alpha^{2}\beta(\beta-1)c^{2}(c\rho^{-\alpha}+d)^{\beta-2}\rho^{-2\alpha-2}\rho_{i}\rho_{j}$
$\displaystyle+\alpha(\alpha+1)\beta
c(c\rho^{-\alpha}+d)^{\beta-1}\rho^{-\alpha-2}\rho_{i}\rho_{j}-\alpha\beta
c(c\rho^{-\alpha}+d)^{\beta-1}\rho^{-\alpha-1}\rho_{ij},$
$\displaystyle|\nabla\psi|^{2}=$
$\displaystyle\alpha^{2}\beta^{2}c^{2}(c\rho^{-\alpha}+d)^{2\beta-2}\rho^{-2\alpha-2},$
$\displaystyle\Delta\psi=$
$\displaystyle\alpha^{2}\beta(\beta-1)c^{2}(c\rho^{-\alpha}+d)^{\beta-2}\rho^{-2\alpha-2}+\alpha(\alpha+1)\beta
c(c\rho^{-\alpha}+d)^{\beta-1}\rho^{-\alpha-2}$ $\displaystyle-\alpha\beta
c(c\rho^{-\alpha}+d)^{\beta-1}\rho^{-\alpha-1}\Delta\rho.$
It follows that
$\displaystyle\psi^{-\frac{n+2}{n-2}}W_{ij}[\psi]=$ $\displaystyle\alpha\beta
c\rho^{-\alpha-2}(c\rho^{-\alpha}+d)^{-\frac{4\beta}{n-2}-1}\bigg{(}-(n-2)\rho\rho_{ij}$
$\displaystyle+\Big{(}\alpha(-n-2\beta+2)\frac{c\rho^{-\alpha}}{c\rho^{-\alpha}+d}+(n-2)(\alpha+1)\Big{)}\rho_{i}\rho_{j}$
$\displaystyle+\Big{(}\alpha(2\beta-1)\frac{c\rho^{-\alpha}}{c\rho^{-\alpha}+d}+\alpha+1-\rho\Delta\rho\Big{)}\delta_{ij}\bigg{)}.$
Choosing an appropriate coordinate system as in [5] such that as
$\rho\rightarrow 0$,
$\rho\nabla^{2}\rho=\left(\begin{array}[]{ccc}\mathcal{O}(\rho)_{(n-m)\times(n-m)}&&\\\
&0&\\\ &&I_{(m-1)\times(m-1)}\\\ \end{array}\right)$
and
$\nabla\rho\otimes\nabla\rho=\left(\begin{array}[]{ccc}0_{(n-m)\times(n-m)}&&\\\
&1&\\\ &&0_{(m-1)\times(m-1)}\\\ \end{array}\right),$
we thus obtain
(3.2) $\frac{1}{\alpha\beta
c}\rho^{\alpha+2}(c\rho^{-\alpha}+d)^{\frac{4\beta}{n-2}+1}\psi^{-\frac{n+2}{n-2}}W_{ij}[\psi]=A_{m}+B_{\alpha\beta}(\zeta)+\mathcal{O}(\rho),$
where $\zeta=\frac{c\rho^{-\alpha}}{c\rho^{-\alpha}+d}$,
$A_{m}=\left(\begin{array}[]{ccc}(n-m)I&&\\\ &n-m&\\\ &&(2-m)I\\\
\end{array}\right),$
and
$\tiny{B_{\alpha\beta}(\zeta)=\left(\begin{array}[]{ccc}\big{(}2\alpha\beta\zeta+\alpha(1-\zeta)+2-n\big{)}I&&\\\
&(n-1)\alpha(1-\zeta)&\\\
&&\big{(}2\alpha\beta\zeta+\alpha(1-\zeta)+2-n\big{)}I\\\
\end{array}\right).}$
Since $v_{m}\notin\overline{\Gamma}_{k}$, we may choose $\alpha$ sufficiently
small, and then an appropriate $\beta$ such that $\alpha\beta$ is slightly
larger than $\frac{n}{2}-1$ in order to make
$\lambda(A_{m}+B_{\alpha\beta}(\zeta))\notin\overline{\Gamma}_{k}$
for all $0<\zeta<1$.
Then we can choose a sufficiently small $0<\rho_{0}<1$ such that on
$0<\rho<\rho_{0}$ and for all $0<\zeta<1$, the righthand side of (3.2)
satisfies
$\lambda\big{(}A_{m}+B_{\alpha\beta}(\zeta)+\mathcal{O}(\rho)\big{)}\notin\overline{\Gamma}_{k}.$
It follows that $\lambda\big{(}W_{ij}[\psi]\big{)}\notin\overline{\Gamma}_{k}$
on $\\{0<\rho<\rho_{0}\\}$ for all $c>0$ and $d>0$, which means that $\psi$ is
a supersolution of (1.2).
Now we compare $u_{\Omega}$ and $\psi$. Choosing $d$ sufficiently large
depending on $\rho_{0}$ such that on $\rho=\rho_{0}$ we have $\psi\geq
d^{\beta}\geq u_{\Omega}$. By comparing $u_{\Omega}$ with the first function
in Proposition 2.4 in the ball $B_{\rho(x)}(x)$ we can deduce that
$u_{\Omega}(x)\leq C(n,k)\rho^{1-\frac{n}{2}}\leq
c^{\beta}\rho^{-\alpha\beta}<\psi$
in $0<\rho\leq\delta$, where $\delta$ is a sufficiently small positive
constant depending on $c$. By the maximum principle we have
$u_{\Omega}\leq\psi$ in $\delta<\rho<\rho_{0}$. Letting $\delta\rightarrow 0$,
we arrive at $u_{\Omega}\leq\psi$ in $0<\rho<\rho_{0}$. Next letting
$c\rightarrow 0$ we deduce that $u_{\Omega}\leq d^{\beta}$ in
$0<\rho<\rho_{0}$.
## 4\. More general equations
By change of variable $u=e^{\frac{n-2}{2}v}$, equation (1.10) is equivalent to
(4.1)
$\sigma_{k}\bigg{(}(n-2)e^{-2v}\mathcal{W}[v]\bigg{)}+\alpha(x)\sigma_{k-1}\bigg{(}(n-2)e^{-2v}\mathcal{W}[v]\bigg{)}=\alpha_{0}(x),$
where
$\mathcal{W}[v]=g^{-1}\bigg{(}\nabla^{2}v-dv\otimes dv+\Big{(}\frac{\Delta
v}{n-2}+|\nabla v|^{2}\Big{)}g-\frac{\text{Ric}_{g}}{n-2}\bigg{)}.$
Now we derive a priori estimates for $(k-1)$-admissible solutions to equation
(4.1) on a smooth compact manifold with boundary, that is, on
$\overline{M}:=M\cup\partial M$. We write (4.1) in the following form
(4.2)
$F\big{(}\mathcal{W}[v]\big{)}:=\frac{\sigma_{k}\big{(}\mathcal{W}[v]\big{)}}{\sigma_{k-1}\big{(}\mathcal{W}[v]\big{)}}-\frac{\alpha_{0}(x)e^{2kv}}{(n-2)^{k}}\frac{1}{\sigma_{k-1}\big{(}\mathcal{W}[v]\big{)}}=-\frac{\alpha(x)}{n-2}e^{2v}.$
As shown in [9], for $(k-1)$-admissible $v$, namely
$\lambda\big{(}\mathcal{W}[v]\big{)}\in\Gamma_{k-1}$, equation (4.2) is
elliptic and the operator $F\big{(}\mathcal{W}[v]\big{)}$ is concave with
respect to $\\{\mathcal{W}_{ij}\\}$.
### 4.1. Gradient estimates
Let $v\in C^{3}(M)\cap C^{1}(\overline{M})$ be a $(k-1)$-admissible solution
of (4.1). We adopt the same test function as [6]: $\Phi=\zeta(x)we^{\eta(v)}$,
where $w=\frac{1}{2}|\nabla v|^{2}$, $\zeta$ and $\eta$ are functions to be
chosen later. Assume that $\Phi$ attains its maximum at an interior point
$x_{0}\in M$. Choose a local orthonormal frame $e_{1},\ldots,e_{n}$ about
$x_{0}$. Then, at $x_{0}$, we have
(4.3)
$\frac{\nabla_{i}\zeta}{\zeta}+\frac{\nabla_{i}w}{w}+\eta^{\prime}\nabla_{i}v=0,$
(4.4)
$F^{ij}\Big{(}\frac{\nabla_{ij}\zeta}{\zeta}-\frac{\nabla_{i}\zeta\nabla_{j}\zeta}{\zeta^{2}}+\frac{\nabla_{ij}w}{w}-\frac{\nabla_{i}w\nabla_{j}w}{w^{2}}+\eta^{\prime}\nabla_{ij}v+\eta^{\prime\prime}\nabla_{i}v\nabla_{j}v\Big{)}\leq
0,$
where $F^{ij}=\frac{\partial
F}{\partial\mathcal{W}_{ij}}\big{(}\mathcal{W}[v]\big{)}$.
By direct calculation,
$\nabla_{i}w=\nabla_{im}v\nabla_{m}v,$
$\nabla_{ij}w=\nabla_{ijm}v\nabla_{m}v+\nabla_{im}v\nabla_{jm}v.$
By (4.3), we have
(4.5) $F^{ij}\frac{\nabla_{i}w\nabla_{j}w}{w^{2}}\leq
3F^{ij}\frac{\nabla_{i}\zeta\nabla_{j}\zeta}{\zeta^{2}}+\frac{3\eta^{\prime
2}}{2}F^{ij}\nabla_{i}v\nabla_{j}v.$
Hence (4.4) becomes
(4.6) $\displaystyle
F^{ij}\Big{(}\frac{\nabla_{ij}\zeta}{\zeta}-\frac{5\nabla_{i}\zeta\nabla_{j}\zeta}{2\zeta^{2}}\Big{)}+\frac{1}{w}\Big{(}\delta_{lm}-\frac{\nabla_{l}v\nabla_{m}v}{2w}\Big{)}F^{ij}\nabla_{il}v\nabla_{jm}v$
$\displaystyle+\frac{F^{ij}\nabla_{ijm}v\nabla_{m}v}{w}+\eta^{\prime}F^{ij}\nabla_{ij}v+\Big{(}\eta^{\prime\prime}-\frac{3}{4}\eta^{\prime
2}\Big{)}F^{ij}\nabla_{i}v\nabla_{j}v\leq 0.$
Choose a smooth function $\zeta$ such that
$0\leq\zeta\leq 1,\quad|\nabla\zeta|\leq
b_{0}\sqrt{\zeta},\quad|\nabla^{2}\zeta|\leq b_{0}.$
Then (4.6) reduces to
(4.7)
$\frac{F^{ij}\nabla_{ijm}v\nabla_{m}v}{w}+\eta^{\prime}F^{ij}\nabla_{ij}v+\Big{(}\eta^{\prime\prime}-\frac{3}{4}\eta^{\prime
2}\Big{)}F^{ij}\nabla_{i}v\nabla_{j}v\leq\frac{C}{\zeta}\sum F^{ii}.$
Similarly, we have
(4.8) $\displaystyle\frac{\Delta\nabla_{m}v\nabla_{m}v}{w}+\eta^{\prime}\Delta
v+\Big{(}\eta^{\prime\prime}-\frac{3}{4}\eta^{\prime 2}\Big{)}|\nabla
v|^{2}\leq\frac{C}{\zeta}.$
Differentiating (4.2) yields
(4.9)
$F^{ij}\nabla_{l}\mathcal{W}_{ij}-\frac{\nabla_{l}\alpha_{0}e^{2kv}+\alpha_{0}e^{2kv}2k\nabla_{l}v}{(n-2)^{k}}\frac{1}{\sigma_{k-1}\big{(}\mathcal{W}[v]\big{)}}=-\frac{\nabla_{l}\alpha
e^{2v}+\alpha e^{2v}2\nabla_{l}v}{n-2},$
where
$\nabla_{l}\mathcal{W}_{ij}[v]=\nabla_{lij}v-\nabla_{li}v\nabla_{j}v-\nabla_{i}v\nabla_{lj}v+\Big{(}\frac{\nabla_{l}\Delta
v}{n-2}+2\nabla_{l}w\Big{)}\delta_{ij}-\nabla_{l}\mathcal{R}_{ij}.$
Since
$\nabla_{ijl}v=\nabla_{lij}v+R_{lij}^{m}\nabla_{m}v,$
we have
$\nabla_{l}\Delta v=\Delta\nabla_{l}v-\sum\limits_{m}R_{lmm}^{s}\nabla_{s}v.$
Assume that $|\nabla v|\geq 1$. Combining (4.7), (4.8) and (4.9) we obtain
(4.10) $\displaystyle\eta^{\prime}F^{ij}\nabla_{ij}v+\eta^{\prime}\frac{\Delta
v}{n-2}\sum F^{ii}$
$\displaystyle+\Big{(}\eta^{\prime\prime}-\frac{3}{4}\eta^{\prime
2}\Big{)}F^{ij}\nabla_{i}v\nabla_{j}v+\Big{(}\eta^{\prime\prime}-\frac{3}{4}\eta^{\prime
2}\Big{)}\frac{|\nabla v|^{2}}{n-2}\sum F^{ii}$
$\displaystyle+\frac{e^{2kv}\nabla_{l}\alpha_{0}\nabla_{l}v}{(n-2)^{k}\sigma_{k-1}w}+\frac{4k\alpha_{0}e^{2kv}}{(n-2)^{k}\sigma_{k-1}}-\frac{e^{2v}\nabla_{l}\alpha\nabla_{l}v}{(n-2)w}-\frac{4\alpha
e^{2v}}{n-2}$ $\displaystyle\leq$ $\displaystyle\frac{C}{\zeta}\sum
F^{ii}-\frac{F^{ij}\nabla_{li}v\nabla_{j}v\nabla_{l}v}{w}-\frac{F^{ij}\nabla_{i}v\nabla_{lj}v\nabla_{l}v}{w}+\frac{2\nabla_{l}w\nabla_{l}v}{w}\sum
F^{ii}.$
By (4.3),
(4.11)
$\displaystyle-\frac{F^{ij}\nabla_{li}v\nabla_{j}v\nabla_{l}v}{w}-\frac{F^{ij}\nabla_{i}v\nabla_{lj}v\nabla_{l}v}{w}+\frac{2\nabla_{l}w\nabla_{l}v}{w}\sum
F^{ii}$ $\displaystyle=$ $\displaystyle
F^{ij}\Big{(}\frac{\nabla_{i}\zeta}{\zeta}+\eta^{\prime}\nabla_{i}v\Big{)}\nabla_{j}v+F^{ij}\nabla_{i}v\Big{(}\frac{\nabla_{j}\zeta}{\zeta}+\eta^{\prime}\nabla_{j}v\Big{)}$
$\displaystyle-2\nabla_{l}v\Big{(}\frac{\nabla_{l}\zeta}{\zeta}+\eta^{\prime}\nabla_{l}v\Big{)}\sum
F^{ii}$ $\displaystyle\leq$ $\displaystyle\frac{C}{\sqrt{\zeta}}|\nabla v|\sum
F^{ii}+2\eta^{\prime}F^{ij}\nabla_{i}v\nabla_{j}v-2\eta^{\prime}|\nabla
v|^{2}\sum F^{ii}.$
Also, we have
$F^{ij}\mathcal{W}_{ij}=\frac{\sigma_{k}}{\sigma_{k-1}}+(k-1)\frac{\alpha_{0}e^{2kv}}{(n-2)^{k}\sigma_{k-1}},$
and
$F^{ij}\mathcal{W}_{ij}=F^{ij}\nabla_{ij}v-F^{ij}\nabla_{i}v\nabla_{j}v+\frac{\Delta
v}{n-2}\sum F^{ii}+|\nabla v|^{2}\sum F^{ii}-F^{ij}\mathcal{R}_{ij}.$
Therefore,
(4.12) $\displaystyle\eta^{\prime}F^{ij}\nabla_{ij}v+\eta^{\prime}\frac{\Delta
v}{n-2}\sum F^{ii}$ $\displaystyle\geq$
$\displaystyle\eta^{\prime}\frac{\sigma_{k}}{\sigma_{k-1}}+(k-1)\frac{\alpha_{0}e^{2kv}}{(n-2)^{k}\sigma_{k-1}}\eta^{\prime}$
$\displaystyle+\eta^{\prime}F^{ij}\nabla_{i}v\nabla_{j}v-\eta^{\prime}|\nabla
v|^{2}\sum F^{ii}-C|\eta^{\prime}|\sum F^{ii}.$
Taking (4.11) and (4.12) into (4.10), and in view of (4.2), we obtain
(4.13)
$\displaystyle\frac{e^{2kv}}{(n-2)^{k}\sigma_{k-1}}\Big{(}k\eta^{\prime}\alpha_{0}+4k\alpha_{0}+\frac{\nabla_{l}\alpha_{0}\nabla_{l}v}{w}\Big{)}$
$\displaystyle+\Big{(}\eta^{\prime\prime}-\frac{3}{4}\eta^{\prime
2}-\eta^{\prime}\Big{)}F^{ij}\nabla_{i}v\nabla_{j}v+\Big{(}\eta^{\prime\prime}-\frac{3}{4}\eta^{\prime
2}+(n-2)\eta^{\prime}\Big{)}\frac{|\nabla v|^{2}}{n-2}\sum F^{ii}$
$\displaystyle\leq$ $\displaystyle C\Big{(}\frac{1}{\zeta}+\frac{|\nabla
v|}{\sqrt{\zeta}}+|\eta^{\prime}|\Big{)}\sum
F^{ii}+\Big{(}\eta^{\prime}\alpha+4\alpha+\frac{\nabla_{l}\alpha\nabla_{l}v}{w}\Big{)}\frac{e^{2v}}{n-2}.$
Choose
$\eta(v)=\Big{(}\frac{3}{2}+v-\inf_{\\{\zeta>0\\}}v\Big{)}^{-N},$
where $N\geq 1$ is sufficiently large such that
$\displaystyle\eta^{\prime\prime}-\frac{3}{4}\eta^{\prime 2}-\eta^{\prime}$
$\displaystyle\geq$ $\displaystyle\eta^{\prime\prime}-\frac{3}{4}\eta^{\prime
2}+(n-2)\eta^{\prime}$ $\displaystyle\geq$
$\displaystyle\frac{N^{2}}{2}\Big{(}\frac{3}{2}+v-\inf_{\\{\zeta>0\\}}v\Big{)}^{-N-2}.$
By Newton-Maclaurin inequality
$\sigma_{k}\sigma_{k-2}\leq\frac{(n-k+1)(k-1)}{(n-k+2)k}\sigma_{k-1}^{2},$
we know that
$\displaystyle\sum F^{ii}=$
$\displaystyle(n-k+1)-\frac{(n-k+2)\sigma_{k}\sigma_{k-2}}{\sigma_{k-1}^{2}}+\frac{(n-k+2)\alpha_{0}e^{2kv}\sigma_{k-2}}{(n-2)^{k}\sigma_{k-1}^{2}}$
$\displaystyle\geq$ $\displaystyle\frac{n-k+1}{k}.$
Therefore,
$\bigg{(}\eta^{\prime}\alpha+4\alpha+\frac{\nabla_{l}\alpha\nabla_{l}v}{w}\bigg{)}\frac{e^{2v}}{n-2}\leq
C\bigg{(}|\eta^{\prime}|+1+\frac{1}{|\nabla v|}\bigg{)}\sum F^{ii}.$
Also, we know that
$k\eta^{\prime}\alpha_{0}+4k\alpha_{0}+\frac{\nabla_{l}\alpha_{0}\nabla_{l}v}{w}\geq\alpha_{0}\bigg{(}-kN\Big{(}\frac{2}{3}\Big{)}^{N+1}+4k-\frac{C}{\alpha_{0}|\nabla
v|}\bigg{)}>0$
for $N$ sufficiently large and $|\nabla v|\geq\frac{C}{2k\inf\alpha_{0}}$.
Hence, (4.13) reduces to
$|\nabla v|^{2}\leq C\Big{(}\frac{1}{\zeta}+\frac{|\nabla
v|}{\sqrt{\zeta}}\Big{)}.$
Consequently,
(4.14) $\sqrt{\zeta}|\nabla v|\leq C.$
Taking $\zeta$ to be a standard cutoff function in a geodesic ball $B_{r}$ of
radius $r>0$ satisfying
$|\nabla\zeta|\leq\frac{C}{r},$
by (4.14) we obtain
###### Theorem 4.1.
If $v\in C^{3}(B_{r})$ is a $(k-1)$-admissible solution of (4.1) in a geodesic
ball $B_{r}\subset M$ of radius $r>0$, then
$\sup\limits_{B_{\frac{r}{2}}}|\nabla v|\leq C,$
where $C$ depends on $r^{-1}$, $n$, $k$, $\|v\|_{C^{0}(B_{r})}$,
$\|\alpha\|_{C^{1}(B_{r})}$, $\|\alpha_{0}\|_{C^{1}{(B_{r})}}$ and
$\inf\limits_{B_{r}}\alpha_{0}$.
Let $\zeta\equiv 1$. By (4.14) we have
###### Theorem 4.2.
Let $v\in C^{3}(M)\cap C^{1}(\overline{M})$ be a $(k-1)$-admissible solution
of (4.1) in $\overline{M}$. Then
$\max\limits_{\overline{M}}|\nabla v|\leq C,$
where $C$ depends on $n$, $k$, $\|v\|_{C^{0}(\overline{M})}$,
$\max\limits_{\partial M}|\nabla v|$, $\|\alpha\|_{C^{1}(M)}$,
$\|\alpha_{0}\|_{C^{1}{(M)}}$ and $\inf\limits_{M}\alpha_{0}$.
### 4.2. Interior and global estimates for second derivatives
Let $v\in C^{4}(M)\cap C^{2}(\overline{M})$ be a $(k-1)$-admissible solution
of (4.1). Consider the test function $\Psi(x)=\zeta
e^{\eta(w)}\nabla_{\xi\xi}v$, where $\xi$ is a unit tangent vector to
$\overline{M}$ at $x$, $w=\frac{1}{2}|\nabla v|^{2}$, $\zeta$ and $\eta$ are
functions to be chosen later, with $\zeta$ smooth and satisfying
$0\leq\zeta\leq 1,\quad|\nabla\zeta|\leq b_{0},\quad|\nabla^{2}\zeta|\leq
b_{0}.$
Assume that $\Psi$ is attained at some interior point $x_{0}\in M$ and for
some unit vector $\xi\in T_{x_{0}}\overline{M}$. Choose a smooth local
orthonormal frame $e_{1},\ldots,e_{n}$ about $x_{0}$ such that
$e_{1}(x_{0})=\xi$ and $\big{\\{}\mathcal{W}_{ij}[v](x_{0})\big{\\}}$ is
diagonal. Assume that at $x_{0}$, $\nabla_{11}v\geq 1$. Since
$\tilde{\Psi}:=\zeta e^{\eta(w)}\nabla_{11}v$ locally attains a maximum at
$x_{0}$, thus at $x_{0}$,
(4.15)
$\frac{\nabla_{i11}v}{\nabla_{11}v}+\nabla_{i}\eta+\frac{\nabla_{i}\zeta}{\zeta}=0,$
and
(4.16)
$F^{ii}\bigg{(}\frac{\nabla_{ii11}v}{\nabla_{11}v}-\Big{(}\frac{\nabla_{i11}v}{\nabla_{11}v}\Big{)}^{2}+\nabla_{ii}\eta+\frac{\nabla_{ii}\zeta}{\zeta}-\Big{(}\frac{\nabla_{i}\zeta}{\zeta}\Big{)}^{2}\bigg{)}\leq
0.$
By (4.15) and Cauchy-Schwartz inequality,
$\bigg{(}\frac{\nabla_{i11}v}{\nabla_{11}v}\bigg{)}^{2}\leq(1+\epsilon)(\nabla_{i}\eta)^{2}+\Big{(}1+\frac{1}{\epsilon}\Big{)}\Big{(}\frac{\nabla_{i}\zeta}{\zeta}\Big{)}^{2},\quad\epsilon>0.$
Therefore, (4.16) reduces to
(4.17)
$F^{ii}\bigg{(}\frac{\nabla_{ii11}v}{\nabla_{11}v}+\nabla_{ii}\eta-(1+\epsilon)\big{(}\nabla_{i}\eta\big{)}^{2}\bigg{)}\leq\frac{C}{\zeta^{2}}\sum
F^{ii}.$
Similarly,
(4.18)
$\frac{\Delta\nabla_{11}v}{\nabla_{11}v}+\Delta\eta-(1+\epsilon)|\nabla\eta|^{2}\leq\frac{C}{\zeta^{2}}.$
Besides, we notice that
(4.19) $\displaystyle F^{ii}\nabla_{11}\bigg{(}\nabla_{ii}v+\frac{\Delta
v}{n-2}\bigg{)}=F^{ii}\nabla_{11}\Big{(}\mathcal{W}_{ii}+\big{(}\nabla_{i}v\big{)}^{2}-|\nabla
v|^{2}+\mathcal{R}_{ii}\Big{)}$ $\displaystyle\geq$ $\displaystyle
F^{ii}\nabla_{11}\mathcal{W}_{ii}+2F^{ii}\nabla_{i}v\nabla_{11i}v+2F^{ii}\big{(}\nabla_{1i}v\big{)}^{2}$
$\displaystyle-2\sum\limits_{k}\nabla_{11k}v\nabla_{k}v\sum\limits_{i}F^{ii}-2\sum\limits_{k}\big{(}\nabla_{1k}v\big{)}^{2}\sum\limits_{i}F^{ii}-C\sum
F^{ii}.$
By (4.15) and the formula for interchanging order of covariant derivatives,
$2F^{ii}\nabla_{i}v\nabla_{11i}v-2\sum\limits_{k}\nabla_{11k}v\nabla_{k}v\sum\limits_{i}F^{ii}\geq-C\bigg{(}|\nabla\eta|+\frac{1}{\zeta}\bigg{)}\nabla_{11}v\sum
F^{ii}.$
Applying the formula for interchanging order of covariant derivatives, (4.19)
reduces to
(4.20) $\displaystyle F^{ii}\nabla_{ii11}v+\frac{\Delta\nabla_{11}v}{n-2}\sum
F^{ii}$ $\displaystyle\geq$ $\displaystyle
F^{ii}\nabla_{11}\mathcal{W}_{ii}+2F^{ii}\big{(}\nabla_{1i}v\big{)}^{2}-2\sum\limits_{k}\big{(}\nabla_{1k}v\big{)}^{2}\sum\limits_{i}F^{ii}$
$\displaystyle-C\bigg{(}|\nabla\eta|\nabla_{11}v+\frac{\nabla_{11}v}{\zeta}+\sum\limits_{j,l}|\nabla_{jl}v|\bigg{)}\sum
F^{ii}.$
Differentiating equation (4.2) twice and by the concavity of
$\frac{\sigma_{k}}{\sigma_{k-1}}$,
(4.21) $\displaystyle
F^{ii}\nabla_{11}\mathcal{W}_{ii}+\frac{\alpha_{0}e^{2kv}}{(n-2)^{k}\sigma_{k-1}}\bigg{(}\frac{\sigma_{k-1}^{ij,rs}}{\sigma_{k-1}}-\frac{2\sigma_{k-1}^{ij}\sigma_{k-1}^{rs}}{\sigma_{k-1}^{2}}\bigg{)}\nabla_{1}\mathcal{W}_{ij}\nabla_{1}\mathcal{W}_{rs}$
$\displaystyle+\frac{\big{(}2\nabla_{1}\alpha_{0}e^{2kv}+4k\alpha_{0}e^{2kv}\nabla_{1}v\big{)}\sigma_{k-1}^{ii}\nabla_{1}\mathcal{W}_{ii}}{(n-2)^{k}\sigma_{k-1}^{2}}$
$\displaystyle-\frac{\nabla_{1}\alpha_{0}e^{2kv}4k\nabla_{1}v+\alpha_{0}e^{2kv}4k^{2}\big{(}\nabla_{1}v\big{)}^{2}+\alpha_{0}e^{2kv}2k\nabla_{11}v+\nabla_{11}\alpha_{0}e^{2kv}}{(n-2)^{k}\sigma_{k-1}}$
$\displaystyle\geq$ $\displaystyle-\frac{\nabla_{11}\alpha
e^{2v}+4\nabla_{1}\alpha e^{2v}\nabla_{1}v+4\alpha
e^{2v}\big{(}\nabla_{1}v\big{)}^{2}+2\alpha e^{2v}\nabla_{11}v}{n-2}.$
By the concavity of $\sigma_{k-1}^{\frac{1}{k-1}}$,
$\bigg{(}\frac{\sigma_{k-1}^{ij,rs}}{\sigma_{k-1}}+\Big{(}\frac{1}{k-1}-1\Big{)}\frac{\sigma_{k-1}^{ij}\sigma_{k-1}^{rs}}{\sigma_{k-1}^{2}}\bigg{)}\nabla_{1}\mathcal{W}_{ij}\nabla_{1}\mathcal{W}_{rs}\leq
0.$
Consequently,
$\displaystyle\frac{\alpha_{0}e^{2kv}}{(n-2)^{k}\sigma_{k-1}}\bigg{(}\frac{\sigma_{k-1}^{ij,rs}}{\sigma_{k-1}}-\frac{2\sigma_{k-1}^{ij}\sigma_{k-1}^{rs}}{\sigma_{k-1}^{2}}\bigg{)}\nabla_{1}\mathcal{W}_{ij}\nabla_{1}\mathcal{W}_{rs}$
$\displaystyle+\frac{\big{(}2\nabla_{1}\alpha_{0}e^{2kv}+4k\alpha_{0}e^{2kv}\nabla_{1}v\big{)}\sigma_{k-1}^{ii}\nabla_{1}\mathcal{W}_{ii}}{(n-2)^{k}\sigma_{k-1}^{2}}$
$\displaystyle\leq$
$\displaystyle-\frac{\alpha_{0}e^{2kv}}{(n-2)^{k}\sigma_{k-1}}\frac{k}{k-1}\frac{\sigma_{k-1}^{ij}\sigma_{k-1}^{rs}}{\sigma_{k-1}^{2}}\nabla_{1}\mathcal{W}_{ij}\nabla_{1}\mathcal{W}_{rs}$
$\displaystyle+\frac{\big{(}2\nabla_{1}\alpha_{0}e^{2kv}+4k\alpha_{0}e^{2kv}\nabla_{1}v\big{)}\sigma_{k-1}^{ii}\nabla_{1}\mathcal{W}_{ii}}{(n-2)^{k}\sigma_{k-1}^{2}}$
$\displaystyle\leq$
$\displaystyle\frac{(k-1)e^{2kv}\big{(}\nabla_{1}\alpha_{0}+2k\alpha_{0}\nabla_{1}v\big{)}^{2}}{k(n-2)^{k}\alpha_{0}\sigma_{k-1}}.$
By (4.2) and Newton-Maclaurin inequality,
$\displaystyle\frac{\alpha_{0}}{\sigma_{k-1}}=\frac{(n-2)^{k}}{e^{2kv}}\bigg{(}\frac{\sigma_{k}}{\sigma_{k-1}}+\frac{\alpha
e^{2v}}{n-2}\bigg{)}\leq C\Big{(}\sigma_{k-1}^{\frac{1}{k-1}}+1\Big{)}\leq
C(\sigma_{1}+1)\leq C\nabla_{11}v.$
Also,
$|\nabla_{ij}v|\leq C\nabla_{11}v.$
Therefore, (4.21) reduces to
$F^{ii}\nabla_{11}\mathcal{W}_{ii}\geq-C\nabla_{11}^{2}v.$
Combining this inequality with (4.20), (4.17) and (4.18) yields,
(4.22) $\displaystyle
F^{ii}\bigg{(}\nabla_{ii}\eta-(1+\epsilon)\big{(}\nabla_{i}\eta\big{)}^{2}\bigg{)}+\frac{1}{n-2}\bigg{(}\Delta\eta-(1+\epsilon)|\nabla\eta|^{2}\bigg{)}\sum
F^{ii}$ $\displaystyle\leq$ $\displaystyle
C\bigg{(}|\nabla\eta|+\nabla_{11}v+\frac{1}{\zeta^{2}}\bigg{)}\sum F^{ii}.$
Let
$\eta(w)=\bigg{(}1-\frac{3w}{4M}\bigg{)}^{-1/2},$
where
$w=\frac{|\nabla v|^{2}}{2},\quad M=\sup\limits_{\\{\zeta>0\\}}w.$
Choosing $\epsilon=\frac{1}{2}$, we have
$\eta^{\prime\prime}-(1+\epsilon)\eta^{\prime
2}=\frac{9}{64M^{2}}\bigg{(}1-\frac{3w}{4M}\bigg{)}^{-\frac{5}{2}}\Bigg{(}3-(1+\epsilon)\bigg{(}1-\frac{3w}{4M}\bigg{)}^{-\frac{1}{2}}\Bigg{)}\geq
0.$
Therefore,
$\displaystyle
F^{ii}\bigg{(}\nabla_{ii}\eta-(1+\epsilon)\big{(}\nabla_{i}\eta\big{)}^{2}\bigg{)}+\frac{1}{n-2}\bigg{(}\Delta\eta-(1+\epsilon)|\nabla\eta|^{2}\bigg{)}\sum
F^{ii}$ $\displaystyle\geq$
$\displaystyle\eta^{\prime}F^{ii}\big{(}\nabla_{iil}v\nabla_{l}v+\big{(}\nabla_{il}v\big{)}^{2}\big{)}+\frac{1}{n-2}\eta^{\prime}\bigg{(}\Delta\nabla_{l}v\nabla_{l}v+\sum\limits_{jl}\big{(}\nabla_{jl}v\big{)}^{2}\bigg{)}\sum
F^{ii}.$
By (4.9) and interchanging order of covariant derivatives,
$F^{ii}\nabla_{iil}v\nabla_{l}v+\frac{1}{n-2}\Delta\nabla_{l}v\nabla_{l}v\sum
F^{ii}\geq-C\nabla_{11}v\sum F^{ii}.$
We note that
$\frac{3}{8M}\leq\eta^{\prime}\leq\frac{3}{M},\quad|\nabla\eta|\leq
C\nabla_{11}v.$
Hence (4.22) reduces to
$\big{(}\nabla_{11}v\big{)}^{2}\leq
C\bigg{(}\nabla_{11}v+\frac{1}{\zeta^{2}}\bigg{)}.$
Consequently,
(4.23) $\zeta\nabla_{11}v\leq C.$
Finally, choosing $\zeta$ to be an appropriate cutoff function with support in
$B_{r}$, we obtain the second order interior estimate for equation (4.1).
###### Theorem 4.3.
Let $v\in C^{4}(B_{r})$ be a $(k-1)$-admissible solution of (4.1) in a
geodesic ball $B_{r}\subset M$ of radius $r>0$. Then
$\sup\limits_{B_{\frac{r}{2}}}|\nabla^{2}v|\leq C,$
where $C$ depends on $r^{-1}$, $n$, $k$, $\|v\|_{C^{1}(B_{r})}$,
$\|\alpha\|_{C^{2}(B_{r})}$, $\|\alpha_{0}\|_{C^{2}{(B_{r})}}$ and
$\inf\limits_{B_{r}}\alpha_{0}$.
If we choose $\zeta\equiv 1$ in (4.23), we obtain the global second order
estimate.
###### Theorem 4.4.
Let $v\in C^{4}(M)\cap C^{2}(\overline{M})$ be a $(k-1)$-admissible solution
of (4.1). Then
$\max\limits_{\overline{M}}|\nabla^{2}v|\leq C,$
where $C$ depends on $n$, $k$, $\|v\|_{C^{1}(\overline{M})}$,
$\max\limits_{\partial M}|\nabla^{2}v|$, $\|\alpha\|_{C^{2}(M)}$,
$\|\alpha_{0}\|_{C^{2}{(M)}}$ and $\inf\limits_{M}\alpha_{0}$.
### 4.3. Boundary estimates for second derivatives
###### Theorem 4.5.
Let $v\in C^{3}(\overline{M})$ be a $(k-1)$-admissible solution of (4.1)
satisfying
$v=\varphi\quad\text{ on }\partial M,$
where $\varphi\in C^{\infty}(\overline{M})$. Then we have the estimate
$|\nabla^{2}v|\leq C\quad\text{ on }\partial M,$
where $C>0$ is a constant depending on $n$, $k$,
$\|v\|_{C^{1}(\overline{M})}$, $\|\varphi\|_{C^{3}(\overline{M})}$,
$\|\alpha\|_{C^{1}(\overline{M})}$, $\|\alpha_{0}\|_{C^{1}(\overline{M})}$ and
$\min\limits_{\overline{M}}\alpha_{0}$.
###### Proof.
Let $x_{0}$ be an arbitrary point on $\partial M$. Choose a smooth local
orthonormal frame $e_{1},\ldots,e_{n}$ around $x_{0}$ such that $e_{n}$ is the
interior unit normal vector field to $\partial M$ along $\partial M$. We
obtain the pure tangential second derivative bound immediately
(4.24) $\big{|}\nabla_{st}v(x_{0})\big{|}\leq C,\quad\forall\,s,t<n.$
For the tangential-normal second derivative estimate
(4.25) $\big{|}\nabla_{sn}v(x_{0})\big{|}\leq C,\quad\forall\,s<n,$
we use the function
$\Theta=\beta\bigg{(}\rho-\frac{N}{2}\rho^{2}\bigg{)},$
where $\beta$ and $N$ are positive constants, and $\rho(x)$ is the distance
from $x$ to $\partial M$. Set
$M_{\delta}=\\{x\in M\,|\,\rho(x)<\delta\\}.$
Choose $\delta>0$ sufficiently small such that $\rho(x)$ is smooth in
$M_{\delta}$, on which, we have
$|\nabla\rho|=1,\quad|\nabla^{2}\rho|\leq C.$
We consider the linearized operator $\mathcal{L}$ which is locally defined by
$\mathcal{L}\eta=F^{ij}\big{(}\nabla_{ij}\eta-2\nabla_{i}v\nabla_{j}\eta\big{)}+\bigg{(}\frac{\Delta\eta}{n-2}+2\nabla_{m}v\nabla_{m}\eta\bigg{)}\sum
F^{ii}.$
It can be seen that
$\displaystyle\mathcal{L}\Theta\leq\beta\bigg{(}C(1+N\delta)-\frac{N}{n-2}\bigg{)}\sum
F^{ii}.$
Choosing
$N\geq(n-2)(2C+1),\quad\text{ and }\quad\delta\leq\frac{1}{N},$
we have
(4.26) $\mathcal{L}\Theta\leq-\beta\sum F^{ii}\quad\text{ in }M_{\delta}$
and
(4.27) $\Theta\geq\frac{\beta}{2}\rho\quad\text{ in }M_{\delta}.$
We note that
$\nabla_{ij}(\nabla_{k}v)=\nabla_{kij}v+\Gamma_{ik}^{l}\nabla_{jl}v+\Gamma_{jk}^{l}\nabla_{il}v+\nabla_{k}\Gamma_{ij}^{l}\nabla_{l}v.$
Therefore, for $t<n$,
$\displaystyle\mathcal{L}\big{(}\nabla_{t}(v-\varphi)\big{)}$ $\displaystyle=$
$\displaystyle
F^{ij}\Big{(}\nabla_{tij}(v-\varphi)+\Gamma_{it}^{l}\nabla_{jl}(v-\varphi)+\Gamma_{jt}^{l}\nabla_{il}(v-\varphi)+\nabla_{t}\Gamma_{ij}^{l}\nabla_{l}(v-\varphi)\Big{)}$
$\displaystyle-2F^{ij}\nabla_{i}v\Big{(}\nabla_{jt}(v-\varphi)+\Gamma_{jt}^{l}\nabla_{l}(v-\varphi)\Big{)}$
$\displaystyle+\Bigg{(}\frac{\nabla_{t}\Delta(v-\varphi)+2\sum_{j}\Gamma_{jt}^{l}\nabla_{jl}(v-\varphi)+\sum_{j}\nabla_{t}\Gamma_{jj}^{l}\nabla_{l}(v-\varphi)}{n-2}$
$\displaystyle+2\nabla_{m}v\Big{(}\nabla_{mt}(v-\varphi)+\Gamma_{mt}^{l}\nabla_{l}(v-\varphi)\Big{)}\Bigg{)}\sum
F^{ii}.$
By (4.9), we have
$\displaystyle
F^{ij}\Big{(}\nabla_{tij}v-2\nabla_{i}v\nabla_{tj}v\Big{)}+\Bigg{(}\frac{\nabla_{t}\Delta
v}{n-2}+\nabla_{t}|\nabla v|^{2}\Bigg{)}\sum F^{ii}$ $\displaystyle\leq$
$\displaystyle C\sum F^{ii}+\frac{C\alpha_{0}}{\sigma_{k-1}}\leq C\sum
F^{ii}+C\big{(}\sigma_{1}+1\big{)}\leq C\sum F^{ii}+C\sigma_{1}.$
Consequently, for $t<n$,
(4.28) $\displaystyle\mathcal{L}\big{(}\nabla_{t}(v-\varphi)\big{)}$
$\displaystyle\leq$ $\displaystyle C\sum
F^{ii}+C\sigma_{1}+C\sqrt{\sum_{jl}(\nabla_{jl}v)^{2}}\sum F^{ii}$
$\displaystyle\leq$ $\displaystyle
C\Bigg{(}1+\sqrt{\sum_{jl}(\nabla_{jl}v)^{2}}\Bigg{)}\sum F^{ii}.$
Similarly, we may verify that
(4.29) $\displaystyle\mathcal{L}\big{(}-\nabla_{t}(v-\varphi)\big{)}\leq
C\Bigg{(}1+\sqrt{\sum_{jl}(\nabla_{jl}v)^{2}}\Bigg{)}\sum F^{ii}.$
We also have
(4.30)
$\displaystyle\mathcal{L}\Bigg{(}\sum\limits_{s<n}\big{|}\nabla_{s}(v-\varphi)\big{|}^{2}\Bigg{)}=\sum\limits_{s<n}2\nabla_{s}(v-\varphi)\mathcal{L}\big{(}\nabla_{s}(v-\varphi)\big{)}$
$\displaystyle+\sum\limits_{s<n}2F^{ij}\nabla_{i}\big{(}\nabla_{s}(v-\varphi)\big{)}\nabla_{j}\big{(}\nabla_{s}(v-\varphi)\big{)}+\sum\limits_{s<n}\frac{2\Big{|}\nabla\big{(}\nabla_{s}(v-\varphi)\big{)}\Big{|}^{2}}{n-2}\sum
F^{ii}$ $\displaystyle\geq$
$\displaystyle-C\Bigg{(}1+\sqrt{\sum_{jl}(\nabla_{jl}v)^{2}}\Bigg{)}\sum
F^{ii}+0+\frac{\sum\limits_{l}\sum\limits_{s<n}(\nabla_{ls}v)^{2}}{n-2}\sum
F^{ii}-C\sum F^{ii}.$
We observe that if $\nabla_{nn}v\leq 0$, by the fact that
$0\leq\sigma_{1}\big{(}\mathcal{W}[v]\big{)}\leq\frac{2(n-1)}{n-2}\Delta v+C,$
we have
$0\geq\nabla_{nn}v\geq-\sum\limits_{s<n}\nabla_{ss}v-C.$
If $\nabla_{nn}v\geq 0$, by the concavity of $F$ with respect to
$\\{\mathcal{W}_{ij}\\}$, that is,
$\sum\limits_{ij}F^{ij}\big{(}\delta_{ij}-\mathcal{W}_{ij}\big{)}\geq
F\big{(}I\big{)}-F\big{(}\mathcal{W}[v]\big{)}\geq-C,$
we have
$\displaystyle\sum F^{ii}+C\geq\sum\limits_{ij}F^{ij}\mathcal{W}_{ij}$
$\displaystyle\geq$
$\displaystyle\sum\limits_{ij}F^{ij}\bigg{(}\nabla_{ij}v+\frac{\Delta
v}{n-2}\delta_{ij}\bigg{)}-C\sum F^{ii}$ $\displaystyle=$
$\displaystyle\sum\limits_{i}\sum\limits_{s<n}F^{is}\nabla_{is}v+\sum\limits_{s<n}F^{sn}\nabla_{sn}v+F^{nn}\nabla_{nn}v+\frac{\Delta
v}{n-2}\sum F^{ii}-C\sum F^{ii}$ $\displaystyle\geq$
$\displaystyle-C\sqrt{\sum_{l}\sum_{s<n}(\nabla_{ls}v)^{2}}\sum
F^{ii}+\frac{\Delta v}{n-2}\sum F^{ii}-C\sum F^{ii},$
which implies that
$0\leq\nabla_{nn}v\leq C+C\sqrt{\sum_{l}\sum_{s<n}(\nabla_{ls}v)^{2}}.$
By the above observation, (4.26), (4.28), (4.29) and (4.30), we have
$\displaystyle\mathcal{L}\Bigg{(}\Theta-\sum\limits_{s<n}\big{|}\nabla_{s}(v-\varphi)\big{|}^{2}\pm\nabla_{t}(v-\varphi)\Bigg{)}$
$\displaystyle\leq$ $\displaystyle-\beta\sum
F^{ii}+C\sqrt{\sum\limits_{l}\sum\limits_{s<n}(\nabla_{ls}v)^{2}}\sum
F^{ii}-\frac{\sum\limits_{l}\sum\limits_{s<n}(\nabla_{ls}v)^{2}}{n-2}\sum
F^{ii}+C\sum F^{ii}.$
If
$\sqrt{\sum\limits_{l}\sum\limits_{s<n}(\nabla_{ls}v)^{2}}\leq C(n-2),$
then we obtain
$\mathcal{L}\Bigg{(}\Theta-\sum\limits_{s<n}\big{|}\nabla_{s}(v-\varphi)\big{|}^{2}\pm\nabla_{t}(v-\varphi)\Bigg{)}\leq-\beta\sum
F^{ii}+C^{2}(n-2)\sum F^{ii}+C\sum F^{ii}.$
If
$\sqrt{\sum\limits_{l}\sum\limits_{s<n}(\nabla_{ls}v)^{2}}\geq C(n-2),$
then we obtain
$\mathcal{L}\Bigg{(}\Theta-\sum\limits_{s<n}\big{|}\nabla_{s}(v-\varphi)\big{|}^{2}\pm\nabla_{t}(v-\varphi)\Bigg{)}\leq-\beta\sum
F^{ii}+C\sum F^{ii}.$
Thus, choosing $\beta$ sufficiently large, we arrive at
$\mathcal{L}\Bigg{(}\Theta-\sum\limits_{s<n}\big{|}\nabla_{s}(v-\varphi)\big{|}^{2}\pm\nabla_{t}(v-\varphi)\Bigg{)}\leq
0.$
Also,
$\Theta-\sum\limits_{s<n}\big{|}\nabla_{s}(v-\varphi)\big{|}^{2}\pm\nabla_{t}(v-\varphi)=0\quad\text{
on }\partial M,$
and by (4.27), we may choose $\beta$ further large depending on $\delta$ such
that
$\Theta-\sum\limits_{s<n}\big{|}\nabla_{s}(v-\varphi)\big{|}^{2}\pm\nabla_{t}(v-\varphi)\geq\frac{\beta}{2}\delta-C\geq
0\quad\text{ on }\\{x\in M\,|\,\rho(x)=\delta\\}.$
By the maximum principle,
$\Theta-\sum\limits_{s<n}\big{|}\nabla_{s}(v-\varphi)\big{|}^{2}\pm\nabla_{t}(v-\varphi)\geq
0\quad\text{ in }M_{\delta}.$
Hence we obtain (4.25).
Next, we derive the double normal second derivative estimate
(4.31) $\nabla_{nn}v(x_{0})\leq C.$
By (4.1), that is,
$\sigma_{k}\big{(}\mathcal{W}[v]\big{)}+\frac{\alpha
e^{2v}}{n-2}\sigma_{k-1}\big{(}\mathcal{W}[v]\big{)}=\frac{\alpha_{0}e^{2kv}}{(n-2)^{k}},$
if $\nabla_{nn}v(x_{0})$ is sufficiently large, in view of (4.24) and (4.25),
we have at $x_{0}$,
$\sigma_{k}\bigg{(}\frac{\Delta
v}{n-2}\delta_{ij}-C\delta_{ij}\bigg{)}-C\sigma_{k-1}\bigg{(}\frac{\Delta
v}{n-2}\delta_{ij}+\nabla_{nn}v\delta_{ij}+C\delta_{ij}\bigg{)}\leq C,$
which further implies that
$\bigg{(}\frac{1}{n-2}\nabla_{nn}v-C\bigg{)}^{k}\sigma_{k}(I)-C\bigg{(}\frac{n-1}{n-2}\nabla_{nn}v+C\bigg{)}^{k-1}\sigma_{k-1}(I)\leq
C.$
Hence we obtain (4.31).
∎
###### Remark 4.6.
When deriving the estimates, we can also directly obtain
$\frac{1}{\sigma_{k-1}\big{(}\mathcal{W}[v]\big{)}}\leq C$
by (4.1). In fact, we have
$\displaystyle\inf\alpha_{0}\leq$
$\displaystyle\alpha_{0}(x)=\sigma_{k}\bigg{(}(n-2)e^{-2v}\mathcal{W}[v]\bigg{)}+\alpha(x)\sigma_{k-1}\bigg{(}(n-2)e^{-2v}\mathcal{W}[v]\bigg{)}$
$\displaystyle\leq$
$\displaystyle\sigma_{k-1}^{\frac{k}{k-1}}\bigg{(}(n-2)e^{-2v}\mathcal{W}[v]\bigg{)}+\sup\alpha\sigma_{k-1}\bigg{(}(n-2)e^{-2v}\mathcal{W}[v]\bigg{)}.$
Next, we shall use continuity method and degree theory to prove the existence
of a smooth $(k-1)$-admissible solution to the Dirichlet problem
(4.32) $\left\\{\begin{aligned}
F\big{(}\mathcal{W}[v]\big{)}=&-\frac{\alpha(x)}{n-2}e^{2v}\quad&\text{ in
}\Omega,\\\ v=&\varphi\quad&\text{ on }\partial\Omega,\end{aligned}\right.$
where $\Omega$ is a bounded domain in $\mathbb{R}^{n}$ with smooth compact
boundary which are composed of closed hypersurfaces.
### 4.4. Existence of subsolutions
In this subsection, we construct a subsolution synthesizing the ideas of Guan
[6] and Guan [8]. We note that there exist sufficiently large $R>r>0$ such
that $\Omega\subset B_{r}(0)$, and
$w(x)=-\ln\big{(}R^{2}-|x|^{2}\big{)}-C$
satisfies
$\mathcal{W}[w]\geq\nabla^{2}w+\frac{\Delta
w}{n-2}g\geq\frac{4R^{2}}{(n-2)\big{(}R^{2}-|x|^{2}\big{)}^{2}}g.$
It follows that
$(n-2)e^{-2w}\mathcal{W}[w]\geq e^{2C}4R^{2}g>e^{2C}2R^{2}g.$
Choosing $C>0$ sufficiently large such that
(4.33)
$\displaystyle\frac{\sigma_{k}}{\sigma_{k-1}}\bigg{(}(n-2)e^{-2w}\mathcal{W}[w]\bigg{)}-\frac{\alpha_{0}(x)}{\sigma_{k-1}\bigg{(}(n-2)e^{-2w}\mathcal{W}[w]\bigg{)}}$
$\displaystyle\geq$
$\displaystyle\frac{\sigma_{k}}{\sigma_{k-1}}\bigg{(}e^{2C}2R^{2}g\bigg{)}-\frac{\alpha_{0}(x)}{\sigma_{k-1}\bigg{(}e^{2C}2R^{2}g\bigg{)}}>-\alpha(x)\quad\text{
on }\overline{\Omega}$
and
$w\leq-\ln\big{(}R^{2}-r^{2}\big{)}-C<\varphi\quad\text{ on }\partial\Omega.$
Also, we consider
$\eta=2\ln\delta-\ln(\rho+\delta^{2})+\varphi.$
For $\delta>0$ sufficiently small,
$\mathcal{W}[\eta]\geq\nabla^{2}\eta+\frac{\Delta\eta}{n-2}g\geq\frac{1}{2(n-2)(\rho+\delta^{2})^{2}}g\quad\text{
on }\\{0\leq\rho\leq\delta\\}.$
Consequently,
$(n-2)e^{-2\eta}\mathcal{W}[\eta]\geq\frac{e^{-2\varphi}}{2\delta^{4}}g>\frac{e^{-2\varphi}}{4\delta^{4}}g\quad\text{
on }\\{0\leq\rho\leq\delta\\}.$
Choosing $\delta>0$ further small such that
(4.34)
$\displaystyle\frac{\sigma_{k}}{\sigma_{k-1}}\bigg{(}(n-2)e^{-2\eta}\mathcal{W}[\eta]\bigg{)}-\frac{\alpha_{0}(x)}{\sigma_{k-1}\bigg{(}(n-2)e^{-2\eta}\mathcal{W}[\eta]\bigg{)}}$
$\displaystyle\geq$
$\displaystyle\frac{\sigma_{k}}{\sigma_{k-1}}\bigg{(}\frac{e^{-2\varphi}}{4\delta^{4}}g\bigg{)}-\frac{\alpha_{0}(x)}{\sigma_{k-1}\bigg{(}\frac{e^{-2\varphi}}{4\delta^{4}}g\bigg{)}}>-\alpha(x)\quad\text{
on }\\{0\leq\rho\leq\delta\\},$
and
$\eta<2\ln\delta-\ln\delta+\varphi<w\quad\text{ on }\rho=\delta.$
Now we need a lemma from Guan [8].
###### Lemma 4.7.
For all $\epsilon>0$, there is an even function $h(t)\in
C^{\infty}(\mathbb{R})$ such that
1. (1)
$h(t)\geq|t|$ for all $t\in\mathbb{R}$, and $h(t)=|t|$ for all
$|t|\geq\epsilon$;
2. (2)
$\big{|}h^{\prime}(t)\big{|}\leq 1$ and $h^{\prime\prime}(t)\geq 0$ for all
$t\in\mathbb{R}$, and $h^{\prime}(t)\geq 0$ for all $t\geq 0$.
Define
$\underline{v}=\left\\{\begin{aligned}
&\frac{1}{2}(\eta+w)+\frac{1}{2}h(\eta-w)\quad\text{ on
}\\{0\leq\rho\leq\delta\\}\\\ &w\quad\text{ on
}\\{\rho\geq\delta\\}\end{aligned}\right..$
Direct calculation shows that on $\\{0\leq\rho\leq\delta\\}$,
$\nabla_{i}\underline{v}=\frac{1}{2}\Big{(}\nabla_{i}\eta+\nabla_{i}w\Big{)}+\frac{1}{2}h^{\prime}(\eta-w)\Big{(}\nabla_{i}\eta-\nabla_{i}w\Big{)},$
$\displaystyle\nabla_{ij}\underline{v}=$
$\displaystyle\frac{1}{2}\Big{(}\nabla_{ij}\eta+\nabla_{ij}w\Big{)}+\frac{1}{2}h^{\prime}\Big{(}\nabla_{ij}\eta-\nabla_{ij}w\Big{)}$
$\displaystyle+\frac{1}{2}h^{\prime\prime}\Big{(}\nabla_{i}\eta-\nabla_{i}w\Big{)}\Big{(}\nabla_{j}\eta-\nabla_{j}w\Big{)}.$
It follows that within $\\{0\leq\rho\leq\delta\\}$,
$\displaystyle\mathcal{W}[\underline{v}]\geq$
$\displaystyle\nabla^{2}\underline{v}+\frac{\Delta\underline{v}}{n-2}g$
$\displaystyle\geq$
$\displaystyle\frac{1}{2}\Big{(}\nabla^{2}\eta+\nabla^{2}w\Big{)}+\frac{1}{2}h^{\prime}\Big{(}\nabla^{2}\eta-\nabla^{2}w\Big{)}$
$\displaystyle+\frac{1}{2(n-2)}\Big{(}\Delta\eta+\Delta
w\Big{)}g+\frac{1}{2(n-2)}h^{\prime}\Big{(}\Delta\eta-\Delta w\Big{)}g$
$\displaystyle=$
$\displaystyle\frac{1+h^{\prime}}{2}\bigg{(}\nabla^{2}\eta+\frac{\Delta\eta}{n-2}g\bigg{)}+\frac{1-h^{\prime}}{2}\bigg{(}\nabla^{2}w+\frac{\Delta
w}{n-2}g\bigg{)}$ $\displaystyle\geq$
$\displaystyle\frac{1+h^{\prime}}{2}\frac{1}{2(n-2)(\rho+\delta^{2})^{2}}g+\frac{1-h^{\prime}}{2}\frac{4R^{2}}{(n-2)\big{(}R^{2}-|x|^{2}\big{)}^{2}}g.$
Since on $\\{0\leq\rho\leq\delta\\}$,
$\underline{v}\leq\sup\\{\eta,w\\}+\frac{\epsilon}{2},$
we have
$\displaystyle(n-2)e^{-2\underline{v}}\mathcal{W}[\underline{v}]$
$\displaystyle\geq$
$\displaystyle(n-2)e^{-2\underline{v}}\frac{1+h^{\prime}}{2}\frac{1}{2(n-2)(\rho+\delta^{2})^{2}}g+(n-2)e^{-2\underline{v}}\frac{1-h^{\prime}}{2}\frac{4R^{2}}{(n-2)\big{(}R^{2}-|x|^{2}\big{)}^{2}}g$
$\displaystyle\geq$ $\displaystyle\left\\{\begin{aligned}
(n-2)e^{-2\eta-\epsilon}\frac{1}{2}\frac{1}{2(n-2)(\rho+\delta^{2})^{2}}g\quad\text{
on }\\{0\leq\rho\leq\delta\\}\cap\\{\eta\geq w\\}\\\
(n-2)e^{-2w-\epsilon}\frac{1}{2}\frac{4R^{2}}{(n-2)\big{(}R^{2}-|x|^{2}\big{)}^{2}}g\quad\text{
on }\\{0\leq\rho\leq\delta\\}\cap\\{\eta\leq w\\}\end{aligned}\right.$
$\displaystyle\geq$ $\displaystyle\left\\{\begin{aligned}
\frac{1}{4\delta^{4}}e^{-2\varphi-\epsilon}g\quad\text{ on
}\\{0\leq\rho\leq\delta\\}\cap\\{\eta\geq w\\}\\\
e^{2C-\epsilon}2R^{2}g\quad\text{ on }\\{0\leq\rho\leq\delta\\}\cap\\{\eta\leq
w\\}\end{aligned}\right..$
In view of (4.33) and (4.34), we may choose $\epsilon>0$ sufficiently small
such that
$\frac{\sigma_{k}}{\sigma_{k-1}}\bigg{(}(n-2)e^{-2\underline{v}}\mathcal{W}[\underline{v}]\bigg{)}-\frac{\alpha_{0}(x)}{\sigma_{k-1}\bigg{(}(n-2)e^{-2\underline{v}}\mathcal{W}[\underline{v}]\bigg{)}}>-\alpha(x)\quad\text{
on }\overline{\Omega}.$
### 4.5. Preliminary estimates
For a $(k-1)$-admissible function $v\in C^{2}(\overline{\Omega})$ with
$v\geq\underline{v}\text{ in }\overline{\Omega}\quad\text{ and
}v=\varphi\text{ on }\partial\Omega,$
we change $v$ back into $u$ by $v=\frac{2}{n-2}\ln u$ to see that
$\sigma_{1}\big{(}W[u]\big{)}=2(n-1)\Delta u\geq 0.$
Let $h$ be the solution to
$\left\\{\begin{aligned} \Delta h=&0\quad&\text{ in }\Omega,\\\
h=&e^{\frac{n-2}{2}\varphi}\quad&\text{ on
}\partial\Omega.\end{aligned}\right.$
By the maximum principle, $u\leq h$ in $\Omega$ and hence
$v\leq\overline{v}:=\frac{2}{n-2}\ln h\quad\text{ in }\Omega.$
Then we have
$\nabla_{\nu}\underline{v}\leq\nabla_{\nu}v\leq\nabla_{\nu}\overline{v}\quad\text{
on }\partial\Omega,$
where $\nu$ is the interior unit normal to $\partial\Omega$.
### 4.6. Existence of solutions
Denote
$\displaystyle G[v]:=$ $\displaystyle G(\nabla^{2}v,\nabla
v,v)=F\big{(}\mathcal{W}[v]\big{)},$ $\displaystyle G^{ij}[v]:=$
$\displaystyle G^{ij}(\nabla^{2}v,\nabla v,v)=\frac{\partial
G}{\partial\nabla_{ij}v},$ $\displaystyle G^{i}[v]:=$ $\displaystyle
G^{i}(\nabla^{2}v,\nabla v,v)=\frac{\partial G}{\partial\nabla_{i}v},$
$\displaystyle G_{v}[v]:=$ $\displaystyle G_{v}(\nabla^{2}v,\nabla
v,v)=\frac{\partial G}{\partial v}.$
Let $C_{0}$ be a positive constant such that
(4.35)
$G[\underline{v}]=G(\nabla^{2}\underline{v},\nabla\underline{v},\underline{v})>-C_{0}\quad\text{
in }\overline{\Omega}.$
For $t\in[0,1]$, we consider the following two equations (similar construction
of the equations can be found in Su [14]).
(4.36) $\left\\{\begin{aligned} G[v]=&(1-t)G[\underline{v}]-tC_{0}\quad&\text{
in }\Omega,\\\ v=&\varphi\quad&\text{ on }\partial\Omega.\end{aligned}\right.$
(4.37) $\left\\{\begin{aligned}
G[v]=&-(1-t)C_{0}-t\frac{\alpha(x)}{n-2}e^{2v}\quad&\text{ in }\Omega,\\\
v=&\varphi\quad&\text{ on }\partial\Omega.\end{aligned}\right.$
###### Remark 4.8.
For $x\in\overline{\Omega}$ and a $C^{2}$ function $v$ which is
$(k-1)$-admissible near $x$, we have
$G_{v}(x)=-\frac{\alpha_{0}(x)e^{2kv}2k}{(n-2)^{k}}\frac{1}{\sigma_{k-1}\big{(}\mathcal{W}[v]\big{)}}<0$
if $\alpha_{0}(x)>0$.
###### Lemma 4.9.
For $t\in[0,1]$, let $\underline{V}$ and $v$ be any $(k-1)$-admissible
subsolution and solution of (4.36). Then $v\geq\underline{V}$ in $\Omega$. In
particular, (4.36) has at most one $(k-1)$-admissible solution.
###### Proof.
Suppose that $\underline{V}-v$ achieves a positive maximum at
$x_{0}\in\Omega$, at which we have
$\underline{V}(x_{0})>v(x_{0}),\quad\nabla\underline{V}(x_{0})=\nabla
v(x_{0}),\quad\nabla^{2}\underline{V}(x_{0})\leq\nabla^{2}v(x_{0}).$
It follows that $\mathcal{W}[\underline{V}](x_{0})\leq\mathcal{W}[v](x_{0})$,
and therefore
$\displaystyle F\big{(}\mathcal{W}[\underline{V}]\big{)}(x_{0})=$
$\displaystyle\frac{\sigma_{k}}{\sigma_{k-1}}\big{(}\mathcal{W}[\underline{V}]\big{)}(x_{0})-\frac{\alpha_{0}(x_{0})e^{2k\underline{V}(x_{0})}}{(n-2)^{k}}\frac{1}{\sigma_{k-1}\big{(}\mathcal{W}[\underline{V}]\big{)}(x_{0})}$
$\displaystyle\leq$
$\displaystyle\frac{\sigma_{k}}{\sigma_{k-1}}\big{(}\mathcal{W}[v]\big{)}(x_{0})-\frac{\alpha_{0}(x_{0})e^{2k\underline{V}(x_{0})}}{(n-2)^{k}}\frac{1}{\sigma_{k-1}\big{(}\mathcal{W}[v]\big{)}(x_{0})}$
$\displaystyle<$
$\displaystyle\frac{\sigma_{k}}{\sigma_{k-1}}\big{(}\mathcal{W}[v]\big{)}(x_{0})-\frac{\alpha_{0}(x_{0})e^{2kv(x_{0})}}{(n-2)^{k}}\frac{1}{\sigma_{k-1}\big{(}\mathcal{W}[v]\big{)}(x_{0})}$
$\displaystyle=$ $\displaystyle F\big{(}\mathcal{W}[v]\big{)}(x_{0}).$
But
$F\big{(}\mathcal{W}[\underline{V}]\big{)}(x_{0})\geq(1-t)G[\underline{v}](x_{0})-tC_{0}=F\big{(}\mathcal{W}[v]\big{)}(x_{0}),$
which is a contradiction. ∎
###### Theorem 4.10.
For $t\in[0,1]$, (4.36) has a unique $(k-1)$-admissible solution
$v\geq\underline{v}$.
###### Proof.
By Lemma 4.9, we immediately obtain uniqueness. For existence, we use standard
continuity method. By assumption (4.35), $\underline{v}$ is a subsolution of
(4.36). We note that the $C^{2}$ estimate for $(k-1)$-admissible solution
$v\geq\underline{v}$ of (4.36) implies uniform ellipticity of this equation
and hence gives $C^{2,\alpha}$ estimate by Evans-Krylov theory [3, 11]
(4.38) $\|v\|_{C^{2,\alpha}(\overline{\Omega})}\leq C,$
where $C$ is independent of $t$. Denote
$C_{0}^{2,\alpha}(\overline{\Omega})=\\{w\in
C^{2,\alpha}(\overline{\Omega})\,|\,w=0\text{ on }\partial\Omega\\},$
and consider the open subset of $C_{0}^{2,\alpha}(\overline{\Omega})$
$\mathcal{U}=\\{w\in
C_{0}^{2,\alpha}(\overline{\Omega})\,|\,\underline{v}+w\text{ is
}(k-1)\text{-admissible in }\overline{\Omega}\\}.$
Define $\mathcal{L}:\mathcal{U}\times[0,1]\rightarrow
C^{\alpha}(\overline{\Omega})$,
$\mathcal{L}(w,t)=G[\underline{v}+w]-(1-t)G[\underline{v}]+tC_{0},$
and set
$\mathcal{S}=\\{t\in[0,1]\,|\,\mathcal{L}(w,t)=0\text{ has a solution }w\text{
in }\mathcal{U}\\}.$
$\mathcal{S}\neq\emptyset$ since $\mathcal{L}(0,0)=0$.
$\mathcal{S}$ is open in $[0,1]$. In fact, for any $t_{0}\in\mathcal{S}$,
there exists $w_{0}\in\mathcal{U}$ such that $\mathcal{L}(w_{0},t_{0})=0$.
Note that the Fréchet derivative of $\mathcal{L}$ with respect to $w$ at
$(w_{0},t_{0})$ is a linear elliptic operator from
$C^{2,\alpha}_{0}(\overline{\Omega})$ to $C^{\alpha}(\overline{\Omega})$,
$\mathcal{L}_{w}\big{|}_{(w_{0},t_{0})}(h)=G^{ij}[\underline{v}+w_{0}]\nabla_{ij}h+G^{i}[\underline{v}+w_{0}]\nabla_{i}h+G_{v}[\underline{v}+w_{0}]h.$
Remark 4.8 implies $\mathcal{L}_{w}\big{|}_{(w_{0},t_{0})}$ is invertible.
Thus a neighborhood of $t_{0}$ is also contained in $\mathcal{S}$ by implicit
function theorem.
$\mathcal{S}$ is closed in $[0,1]$. In fact, let $t_{i}$ be a sequence in
$\mathcal{S}$ converging to $t_{0}\in[0,1]$ and $w_{i}\in\mathcal{U}$ be the
unique solution to $\mathcal{L}(w_{i},t_{i})=0$. Lemma 4.9 implies $w_{i}\geq
0$. By (4.38), $v_{i}:=\underline{v}+w_{i}$ is a bounded sequence in
$C^{2,\alpha}(\overline{\Omega})$, which possesses a subsequence converging to
a $(k-1)$-admissible solution $v_{0}$ of (4.36). Since
$w_{0}=v_{0}-\underline{v}\in\mathcal{U}$ and $\mathcal{L}(w_{0},t_{0})=0$, we
know that $t_{0}\in\mathcal{S}$. ∎
Next we may assume that $\underline{v}$ is not a solution of (4.32).
###### Lemma 4.11.
If $v\geq\underline{v}$ is a $(k-1)$-admissible solution of (4.37), then
$v>\underline{v}$ in $\Omega$ and $\nabla_{\nu}(v-\underline{v})>0$ on
$\partial\Omega$.
###### Proof.
We note that
$G[\underline{v}]-G[v]\geq-t\frac{\alpha(x)}{n-2}(e^{2\underline{v}}-e^{2v}),$
and
$\displaystyle
G[\underline{v}]-G[v]=F\big{(}\mathcal{W}[\underline{v}]\big{)}-F\big{(}\mathcal{W}[v]\big{)}$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\frac{d}{ds}\Bigg{(}\frac{\sigma_{k}}{\sigma_{k-1}}\Big{(}(1-s)\mathcal{W}[v]+s\mathcal{W}[\underline{v}]\Big{)}-\frac{\alpha_{0}(x)e^{2k((1-s)v+s\underline{v})}}{(n-2)^{k}\sigma_{k-1}\Big{(}(1-s)\mathcal{W}[v]+s\mathcal{W}[\underline{v}]\Big{)}}\Bigg{)}ds$
$\displaystyle=$
$\displaystyle\underbrace{\int_{0}^{1}\Bigg{(}\Big{(}\frac{\sigma_{k}}{\sigma_{k-1}}\Big{)}^{ij}+\frac{\alpha_{0}(x)e^{2k((1-s)v+s\underline{v})}\sigma_{k-1}^{ij}}{(n-2)^{k}\sigma_{k-1}^{2}}\Bigg{)}\Big{(}(1-s)\mathcal{W}[v]+s\mathcal{W}[\underline{v}]\Big{)}ds}_{\text{denoted
by }a_{ij}}$
$\displaystyle\cdot\Big{(}\mathcal{W}_{ij}[\underline{v}]-\mathcal{W}_{ij}[v]\Big{)}-\underbrace{\int_{0}^{1}\frac{\alpha_{0}(x)e^{2k((1-s)v+s\underline{v})}2k}{(n-2)^{k}\sigma_{k-1}\Big{(}(1-s)\mathcal{W}[v]+s\mathcal{W}[\underline{v}]\Big{)}}ds}_{\text{denoted
by }c}\cdot(\underline{v}-v)$ $\displaystyle=$ $\displaystyle
a_{ij}\bigg{(}\nabla_{ij}(\underline{v}-v)+\frac{\Delta(\underline{v}-v)}{n-2}\delta_{ij}\bigg{)}-a_{ij}\nabla_{i}\underline{v}\nabla_{j}(\underline{v}-v)-a_{ij}\nabla_{j}v\nabla_{i}(\underline{v}-v)$
$\displaystyle+\nabla(\underline{v}+v)\cdot\nabla(\underline{v}-v)\sum
a_{ii}-c(\underline{v}-v).$
Applying the Maximum Principle and Lemma H (see p. 212 of [4]) we proved the
lemma. ∎
###### Theorem 4.12.
For any $t\in[0,1]$, there is a $(k-1)$-admissible solution
$v\geq\underline{v}$ to Dirichlet problem (4.37).
###### Proof.
We obtain by classical Schauder theory the $C^{4,\alpha}$ estimate
(4.39) $\|v\|_{C^{4,\alpha}(\overline{\Omega})}<C_{4}.$
Also, we have
(4.40)
$\text{dist}\big{(}\lambda(\mathcal{W}[v]),\partial\Gamma_{k-1}\big{)}>c_{2}>0\quad\text{
in }\overline{\Omega},$
where $C_{4}$ and $c_{2}$ are independent of $t$. Denote
$C_{0}^{4,\alpha}(\overline{\Omega})=\big{\\{}w\in
C^{4,\alpha}(\overline{\Omega})\,|\,w=0\text{ on }\partial\Omega\big{\\}}$
and the open bounded subset of $C_{0}^{4,\alpha}(\overline{\Omega})$
$\mathcal{O}=\Bigg{\\{}w\in
C_{0}^{4,\alpha}(\overline{\Omega})\left|\footnotesize\begin{aligned}
&w>0\text{ in }\Omega,\nabla_{\nu}w>0\text{ on
}\partial\Omega,\|w{\|}_{C^{4,\alpha}(\overline{\Omega})}<C_{4}+\|\underline{v}\|_{C^{4,\alpha}(\overline{\Omega})},\\\
&\underline{v}+w\text{ is }(k-1)\text{-admissible in
}\overline{\Omega},\text{dist}\Big{(}\lambda\big{(}\mathcal{W}[\underline{v}+w]\big{)},\partial\Gamma_{k-1}\Big{)}>c_{2}\text{
in }\overline{\Omega}\end{aligned}\right.\Bigg{\\}}.$
Define a map $\mathcal{M}_{t}(w):\mathcal{O}\times[0,1]\rightarrow
C^{2,\alpha}(\overline{\Omega})$,
$\mathcal{M}_{t}(w)=G[\underline{v}+w]+(1-t)C_{0}+t\frac{\alpha(x)}{n-2}e^{2(\underline{v}+w)}.$
By Theorem 4.10 and Lemma 4.9, there is a unique $(k-1)$-admissible solution
$v^{0}$ of (4.36) at $t=1$, which is also the unique $(k-1)$-admissible
solution of (4.37) at $t=0$. By Lemma 4.9, $w^{0}=v^{0}-\underline{v}\geq 0$
in $\Omega$. Consequently, $w^{0}>0$ in $\Omega$ and $\nabla_{\nu}w^{0}>0$ on
$\partial\Omega$ by Lemma 4.11. We also note that $\underline{v}+w^{0}$
satisfies (4.39) and (4.40). Thus, $w^{0}\in\mathcal{O}$. By Lemma 4.11,
(4.39) and (4.40), $\mathcal{M}_{t}(w)=0$ has no solution on
$\partial\mathcal{O}$ for any $t\in[0,1]$. Since $\mathcal{M}_{t}$ is
uniformly elliptic on $\mathcal{O}$ independent of $t$, the degree of
$\mathcal{M}_{t}$ on $\mathcal{O}$ at $0$ can be defined independent of $t$.
This degree is nonzero at $t=0$. In fact, since $\mathcal{M}_{0}(w)=0$ has a
unique solution $w^{0}\in\mathcal{O}$. The Fréchet derivative of
$\mathcal{M}_{0}$ with respect to $w$ at $w^{0}$ is a linear elliptic operator
from $C^{4,\alpha}_{0}(\overline{\Omega})$ to
$C^{2,\alpha}(\overline{\Omega})$,
$\mathcal{M}_{0,w}|_{w^{0}}(h)=G^{ij}[v^{0}]\nabla_{ij}h+G^{i}[v^{0}]\nabla_{i}h+G_{v}[v^{0}]h.$
By Remark 4.8, $G_{v}[v^{0}]<0$ in $\overline{\Omega}$. Thus,
$\mathcal{M}_{0,w}|_{w^{0}}$ is invertible. The degree theory in [12] implies
that the degree at $t=0$ is nonzero, which implies that (4.37) has at least
one $(k-1)$-admissible solution $v\geq\underline{v}$ for any $t\in[0,1]$. ∎
## 5\. Fully nonlinear Loewner-Nirenberg problem for general equations
In this section, we discuss fully nonlinear Loewner-Nirenberg problem related
to general equation (1.10).
### 5.1. The case of smooth compact boundary consisting of closed
hypersurfaces
We first give the definition of supersolutions and subsolutions to equation
(1.10).
###### Definition 5.1.
A function $0<\underline{u}\in C^{2}(\Omega)$ is a subsolution of (1.10) in
$\Omega$ if
$\displaystyle\lambda\big{(}W[\underline{u}]\big{)}\in\Gamma_{k-1}\text{ and
}$
$\displaystyle\sigma_{k}\bigg{(}\frac{2}{n-2}\underline{u}^{-\frac{n+2}{n-2}}W[\underline{u}]\bigg{)}$
$\displaystyle+\alpha(x)\sigma_{k-1}\bigg{(}\frac{2}{n-2}\underline{u}^{-\frac{n+2}{n-2}}W[\underline{u}]\bigg{)}\geq\alpha_{0}(x)\quad\text{
in }\Omega.$
A function $0<\overline{u}\in C^{2}(\Omega)$ is a supersolution of (1.10) in
$\Omega$ if
$\displaystyle\text{either
}\lambda\big{(}W[\overline{u}]\big{)}\notin\Gamma_{k-1}\text{ or }$
$\displaystyle\sigma_{k}\bigg{(}\frac{2}{n-2}\overline{u}^{-\frac{n+2}{n-2}}W[\overline{u}]\bigg{)}$
$\displaystyle+\alpha(x)\sigma_{k-1}\bigg{(}\frac{2}{n-2}\overline{u}^{-\frac{n+2}{n-2}}W[\overline{u}]\bigg{)}\leq\alpha_{0}(x)\quad\text{
in }\Omega.$
###### Proposition 5.2.
If $\underline{u}$ is a subsolution of (1.10) satisfying
(5.1)
$\lambda\big{(}W[\underline{u}]\big{)}\in\overline{\Gamma}_{k}\quad\text{ in
}\Omega,$
so does $c\underline{u}$ for any constant $0<c<1$. If $\overline{u}$ is a
supersolution of (1.10) satisfying
$\lambda\big{(}W[\overline{u}]\big{)}\notin\Gamma_{k-1}\quad\text{everywhere
in }\Omega,$
then $c\overline{u}$ is a supersolution of (1.10) for any $c>0$.
###### Proof.
When $\lambda\big{(}W[\underline{u}]\big{)}\in\overline{\Gamma}_{k}$,
$\displaystyle\frac{\sigma_{k}}{\sigma_{k-1}}\bigg{(}\frac{2}{n-2}c^{-\frac{4}{n-2}}\underline{u}^{-\frac{n+2}{n-2}}W[\underline{u}]\bigg{)}-\frac{\alpha_{0}(x)}{\sigma_{k-1}\bigg{(}\frac{2}{n-2}c^{-\frac{4}{n-2}}\underline{u}^{-\frac{n+2}{n-2}}W[\underline{u}]\bigg{)}}$
$\displaystyle\geq$
$\displaystyle\frac{\sigma_{k}}{\sigma_{k-1}}\bigg{(}\frac{2}{n-2}\underline{u}^{-\frac{n+2}{n-2}}W[\underline{u}]\bigg{)}-\frac{\alpha_{0}(x)}{\sigma_{k-1}\bigg{(}\frac{2}{n-2}\underline{u}^{-\frac{n+2}{n-2}}W[\underline{u}]\bigg{)}}\geq-\alpha(x).$
Hence we proved the first statement. The second statement is straightforward
by definition. ∎
Then we observe the following maximum principle of (1.10).
###### Theorem 5.3.
Let $M$ be a smooth compact manifold with boundary. Suppose that
$\underline{u}$ is a subsolution of
$\displaystyle\sigma_{k}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}+\alpha(x)\sigma_{k-1}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}=\alpha_{0}(x)$
satisfying
$\lambda\big{(}W[\underline{u}]\big{)}\in\Gamma_{k}\quad\text{ in }\Omega,$
and $\overline{u}$ is a supersolution of
$\displaystyle\sigma_{k}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}+\beta(x)\sigma_{k-1}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}=\beta_{0}(x),$
with $\alpha(x)\leq\beta(x)$ and $\alpha_{0}(x)\geq\beta_{0}(x)$ on $M$. If
$\underline{u}\leq\overline{u}$ on $\partial M$, then
$\underline{u}\leq\overline{u}$ on $M$.
###### Proof.
Suppose that $\underline{u}>\overline{u}$ somewhere in the interior of $M$.
Let $C$ be the maximum of $\frac{\underline{u}}{\overline{u}}$ on $M$, which
is attained at $x_{0}$ in the interior of $M$. Since $C>1$, by Proposition
5.2, we know that $\underline{u}_{1}:=\frac{\underline{u}}{C}$ satisfies
$\frac{\sigma_{k}}{\sigma_{k-1}}\bigg{(}\frac{2}{n-2}\underline{u}_{1}^{-\frac{n+2}{n-2}}W[\underline{u}_{1}]\bigg{)}-\frac{\alpha_{0}(x)}{\sigma_{k-1}\bigg{(}\frac{2}{n-2}\underline{u}_{1}^{-\frac{n+2}{n-2}}W[\underline{u}_{1}]\bigg{)}}>-\alpha(x).$
On the other hand, since $\underline{u}_{1}(x_{0})=\overline{u}(x_{0})$ while
$\underline{u}_{1}\leq\overline{u}$ near $x_{0}$, thus at $x_{0}$
$\nabla\underline{u}_{1}(x_{0})=\nabla\overline{u}(x_{0}),\quad\nabla^{2}\underline{u}_{1}(x_{0})\leq\nabla^{2}\overline{u}(x_{0}).$
It follows that $W[\underline{u}_{1}](x_{0})\leq W[\overline{u}](x_{0})$,
which further implies
$\displaystyle-\alpha(x_{0})<\frac{\sigma_{k}}{\sigma_{k-1}}\bigg{(}\frac{2}{n-2}\underline{u}_{1}^{-\frac{n+2}{n-2}}W[\underline{u}_{1}]\bigg{)}(x_{0})-\frac{\alpha_{0}(x_{0})}{\sigma_{k-1}\bigg{(}\frac{2}{n-2}\underline{u}_{1}^{-\frac{n+2}{n-2}}W[\underline{u}_{1}]\bigg{)}(x_{0})}$
$\displaystyle\leq$
$\displaystyle\frac{\sigma_{k}}{\sigma_{k-1}}\bigg{(}\frac{2}{n-2}\overline{u}^{-\frac{n+2}{n-2}}W[\overline{u}]\bigg{)}(x_{0})-\frac{\beta_{0}(x_{0})}{\sigma_{k-1}\bigg{(}\frac{2}{n-2}\overline{u}^{-\frac{n+2}{n-2}}W[\overline{u}]\bigg{)}(x_{0})}\leq-\beta(x_{0}),$
which is a contradiction. ∎
Similar to Proposition 2.4, we observe the following fact.
###### Proposition 5.4.
For any real number $\alpha$, any positive constant $\alpha_{0}$, any fixed
$s>0$, there exists a positive constant $c$ such that
$u(x)=c\Big{(}s^{2}-{|x-x_{0}|}^{2}\Big{)}^{1-\frac{n}{2}}\quad\text{ in
}B_{s}(x_{0}),$
and
$v(x)=c\Big{(}{|x-x_{0}|}^{2}-s^{2}\Big{)}^{1-\frac{n}{2}}\quad\text{ in
}\mathbb{R}^{n}\setminus\overline{B_{s}(x_{0})},$
are admissible solutions of
$\displaystyle\sigma_{k}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}+\alpha\sigma_{k-1}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}=\alpha_{0},$
which approach to $\infty$ on $\partial B_{s}(x_{0})$.
###### Proof.
Consider
$u(x)=c\Big{(}s^{2}-{|x-x_{0}|}^{2}\Big{)}^{1-\frac{n}{2}}.$
We have
$W_{ij}[u]=2(n-1)(n-2)c\Big{(}s^{2}-|x-x_{0}|^{2}\Big{)}^{-\frac{n}{2}-1}s^{2}\delta_{ij},$
and therefore
$\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W_{ij}[u]=4(n-1)c^{-\frac{4}{n-2}}s^{2}\delta_{ij}.$
Hence
$\displaystyle\sigma_{k}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}+\alpha\sigma_{k-1}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}$
$\displaystyle=$
$\displaystyle\bigg{(}4(n-1)c^{-\frac{4}{n-2}}s^{2}\bigg{)}^{k}\sigma_{k}\big{(}I\big{)}+\alpha\bigg{(}4(n-1)c^{-\frac{4}{n-2}}s^{2}\bigg{)}^{k-1}\sigma_{k-1}\big{(}I\big{)}.$
Since $k\geq 2$, by intermediate value theorem, there exists a positive
constant $c$ such that the above quantity equals $\alpha_{0}$. ∎
###### Remark 5.5.
The relation between $c$ and $s$ in $u(x)$ and $v(x)$ in Proposition 5.4 can
be obtained once $\alpha$ and $\alpha_{0}$ are fixed. In fact, let
$\tilde{\mu}$ be a positive root of
$\mu^{k}\sigma_{k}\big{(}I\big{)}+\alpha\mu^{k-1}\sigma_{k-1}\big{(}I\big{)}=\alpha_{0}.$
Then choose
$4(n-1)c^{-\frac{4}{n-2}}s^{2}=\tilde{\mu}$
to obtain
$c=\bigg{(}\frac{4(n-1)}{\tilde{\mu}}\bigg{)}^{\frac{n-2}{4}}s^{\frac{n-2}{2}}.$
Proof of Theorem 1.4 If $\Omega$ is a bounded domain, for any positive integer
$m$, we consider the Dirichlet problem
(5.2) $\left\\{\begin{aligned}
F\big{(}\mathcal{W}[v]\big{)}=&-\frac{\alpha(x)}{n-2}e^{2v}\quad&\text{ in
}\Omega,\\\ v=&m\quad&\text{ on }\partial\Omega.\end{aligned}\right.$
By Theorem 4.12, there exists a smooth $(k-1)$-admissible solution $v_{m}$ to
(5.2). Let $\mu_{0}$ be the smallest positive root of the equation
$\mu^{k}+\overline{\alpha}\mu^{k-1}=\underline{\alpha}_{0},$
where
$\overline{\alpha}=\max\Big{\\{}\sup\limits_{\Omega}\alpha(x),0.01\Big{\\}}>0,\quad\underline{\alpha}_{0}=\inf\limits_{\Omega}\alpha_{0}(x)>0.$
By Loewner and Nirenberg [13], there exists a positive solution $\overline{u}$
to
(5.3) $\left\\{\begin{aligned}
\sigma_{1}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}=&\mu_{0}\quad\text{
in }\Omega,\\\ u=&\infty\quad\text{ on }\partial\Omega.\end{aligned}\right.$
Since $u_{m}=e^{\frac{n-2}{2}v_{m}}$ satisfies
$\displaystyle\underline{\alpha}_{0}\leq\alpha_{0}(x)=$
$\displaystyle\sigma_{k}\bigg{(}\frac{2}{n-2}u_{m}^{-\frac{n+2}{n-2}}W[u_{m}]\bigg{)}+\alpha(x)\sigma_{k-1}\bigg{(}\frac{2}{n-2}u_{m}^{-\frac{n+2}{n-2}}W[u_{m}]\bigg{)}$
$\displaystyle\leq$
$\displaystyle\sigma_{1}^{k}\bigg{(}\frac{2}{n-2}u_{m}^{-\frac{n+2}{n-2}}W[u_{m}]\bigg{)}+\overline{\alpha}\sigma_{1}^{k-1}\bigg{(}\frac{2}{n-2}u_{m}^{-\frac{n+2}{n-2}}W[u_{m}]\bigg{)},$
we have
$\sigma_{1}\bigg{(}\frac{2}{n-2}u_{m}^{-\frac{n+2}{n-2}}W[u_{m}]\bigg{)}\geq\mu_{0}.$
By the maximum principle Theorem 2.3, we have
$u_{m}\leq\overline{u}\quad\text{ in }\Omega\quad\text{ for any }m.$
Denote
$\underline{\alpha}=\inf\limits_{\Omega}\alpha(x),\quad\overline{\alpha}_{0}=\sup\limits_{\Omega}\alpha_{0}(x).$
Let $t>0$ be sufficiently large such that $\overline{\Omega}\subset B_{t}(0)$.
Consider the equation
(5.4)
$\sigma_{k}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}+\underline{\alpha}\sigma_{k-1}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}=\overline{\alpha}_{0}.$
For any fixed $s>t>0$, by Proposition 5.4 and Remark 5.5, there exists a
smooth solution
$\underline{u}=\bigg{(}\frac{4(n-1)}{\tilde{\mu}}\bigg{)}^{\frac{n-2}{4}}s^{\frac{n-2}{2}}\Big{(}s^{2}-|x|^{2}\Big{)}^{1-\frac{n}{2}}$
to equation (5.4) in $B_{s}(0)$, which is a subsolution of (1.10). We may
choose $s$ sufficiently large so that
$\displaystyle\underline{u}\leq\bigg{(}\frac{4(n-1)}{\tilde{\mu}}\bigg{)}^{\frac{n-2}{4}}s^{\frac{n-2}{2}}\Big{(}s^{2}-t^{2}\Big{)}^{1-\frac{n}{2}}\leq
e^{\frac{n-2}{2}}\leq e^{\frac{(n-2)m}{2}}=u_{m}\quad\text{ on
}\partial\Omega.$
By Theorem 5.3 we know that
$u_{m}\geq\underline{u}\quad\text{ in }\Omega\quad\text{ for any }m.$
By interior $C^{2}$ estimate established in last section and Evans-Krylov
theory [3, 11], we obtain a smooth positive $(k-1)$-admissible solution $u$ to
equation (1.10) which tends to $\infty$ on $\partial\Omega$.
When $\Omega$ is unbounded, we may assume without loss of generality that
$0\notin\overline{\Omega}$. Let $B_{s}(0)$ be a fixed ball such that
$\overline{\Omega}\subset\mathbb{R}^{n}\setminus\overline{B_{s}(0)}$. By
Proposition 5.4, there exists a smooth solution
$\underline{u}=c_{0}\Big{(}|x|^{2}-s^{2}\Big{)}^{1-\frac{n}{2}}\quad\text{ for
some positive constant }c_{0}$
to equation (5.4) in $\mathbb{R}^{n}\setminus\overline{B_{s}(0)}$, which is a
subsolution of (1.10).
For any $R>\max\limits_{\partial\Omega}\underline{u}$ large enough such that
$\partial\Omega\subset B_{R}(0)$, by Theorem 4.12, there exists a smooth
positive $(k-1)$-admissible solution $u_{R}$ to the Dirichlet problem
$\left\\{\begin{aligned}
&\sigma_{k}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}+\alpha\sigma_{k-1}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}=\alpha_{0}\quad\text{
in }B_{R}(0)\cap\Omega,\\\ &u=R\quad\text{ on }\partial\Omega,\\\
&u=\underline{u}\quad\text{ on }\partial B_{R}(0).\end{aligned}\right.$
By Theorem 5.3, we know that
$u_{R}\geq\underline{u}\quad\text{ in }B_{R}(0)\cap\Omega.$
By Lemma 2.10, we are able to find a positive solution $\overline{u}\in
C^{\infty}(\Omega)$ of (5.3) which decays to $0$ at $\infty$ with the decay
rate
$|x|^{n-2}\overline{u}(x)\rightarrow c\quad\text{ as }|x|\rightarrow\infty$
for some positive constant $c$. Comparing the decay rate as
$|x|\rightarrow\infty$, we have on $\partial B_{R}(0)$ with $R$ sufficiently
large,
$u_{R}(x)=\underline{u}(x)\leq\frac{2c_{0}}{c}\overline{u}(x).$
Meanwhile, we have
$u_{R}(x)=R<\infty=\frac{2c_{0}}{c}\overline{u}(x)\quad\text{ on
}\partial\Omega.$
By Proposition 2.2 and Theorem 2.3, we obtain
$u_{R}(x)\leq\max\Big{\\{}\frac{2c_{0}}{c},1\Big{\\}}\overline{u}(x)\quad\text{
in }B_{R}(0)\cap\Omega.$
Applying the interior estimates in section 4, Evans-Krylov interior estimates
[3, 11] and a standard diagonal process, we obtain a smooth positive
$(k-1)$-admissible solution $u$ of (1.10) in $\Omega$ which tends to $\infty$
on $\partial\Omega$. ∎
### 5.2. Maximal solution on general domain in $\mathbb{R}^{n}$ when
$\alpha\leq 0$
We first observe the fact that when $\alpha(x)\leq 0$, any $(k-1)$-admissible
solution to equation (1.10) must be $k$-admissible. Then we are able to apply
the maximum principle Theorem 5.3. As in section 3, we can define the maximal
solution of (1.10).
Assume throughout this subsection that $\Omega\subset\mathbb{R}^{n}$ is a
domain with smooth compact boundary $\partial\Omega$. Moreover, we assume that
$\alpha(x)\leq 0\text{ in
}\Omega,\quad\underline{\alpha}_{0}=\inf\limits_{\Omega}\alpha_{0}(x)>0,\quad\underline{\alpha}=\inf\limits_{\Omega}\alpha(x)>-\infty,\quad\overline{\alpha}_{0}=\sup\limits_{\Omega}\alpha_{0}(x)<\infty.$
###### Definition 5.6.
A smooth positive $k$-admissible solution $u_{\Omega}$ of (1.10) is said to be
maximal in $\Omega$, if it is greater than or equal to any smooth positive
$k$-admissible solution of (1.10) in $\Omega$.
Let $\Omega_{(1)}\Subset\Omega_{(2)}\Subset\ldots$ be an increasing sequence
of bounded subdomains of $\Omega$ with smooth compact boundaries
$\partial\Omega_{(j)}$ which are closed hypersurfaces such that
$\Omega=\cup\Omega_{(j)}$. By Theorem 1.4, we can find a smooth positive
$k$-admissible solution $u_{(j)}$ of (1.10) in $\Omega_{(j)}$ which tends to
$\infty$ on $\partial\Omega_{(j)}$. By Theorem 5.3, we see that
$\\{u_{(j)}\\}$ is a monotone decreasing sequence of positive functions. It
follows that $u_{(j)}$ converges to a nonnegative function $u_{\Omega}$ in
$\Omega$.
###### Lemma 5.7.
Either $u_{\Omega}>0$ in $\Omega$ or $u_{\Omega}\equiv 0$ in $\Omega$.
###### Proof.
Suppose that $u_{\Omega}\not\equiv 0$. Then we must have $u_{\Omega}>c$ on
some $B(x_{0},r_{0})\subset\Omega$ for some constant $c>0$. By Proposition 5.4
and Remark 5.5, we may choose $0<r_{1}<r_{0}$ such that
$v=\bigg{(}\frac{4(n-1)}{\tilde{\mu}}\bigg{)}^{\frac{n-2}{4}}r_{1}^{\frac{n-2}{2}}\Big{(}{|x-x_{0}|}^{2}-r_{1}^{2}\Big{)}^{1-\frac{n}{2}}\quad\text{
in }\mathbb{R}^{n}\setminus\overline{B_{r_{1}}(x_{0})}$
is a solution of
$\sigma_{k}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}+\underline{\alpha}\sigma_{k-1}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}=\overline{\alpha}_{0}$
satisfying
$\bigg{(}\frac{4(n-1)}{\tilde{\mu}}\bigg{)}^{\frac{n-2}{4}}r_{1}^{\frac{n-2}{2}}\Big{(}r_{0}^{2}-r_{1}^{2}\Big{)}^{1-\frac{n}{2}}=c.$
By Theorem 5.3, we know that $u_{(j)}\geq v$ in $\Omega_{(j)}\setminus
B(x_{0},r_{0})$ for any $j$. Hence $u_{\Omega}\geq v>0$ in $\Omega\setminus
B(x_{0},r_{0})$. We thus have $u_{\Omega}>0$ in $\Omega$. ∎
###### Remark 5.8.
1. (1)
In view of Lemma 5.7, the interior regularity in section 4, and Evans-Krylov
theory [3, 11], we know that $u_{\Omega}$ is smooth.
2. (2)
If there exists a positive $k$-admissible subsolution of (1.10) in $\Omega$
(for example, when $\mathbb{R}^{n}\setminus\overline{\Omega}\neq\emptyset$),
then $u_{\Omega}>0$ in $\Omega$.
3. (3)
When $u_{\Omega}>0$ in $\Omega$, $u_{\Omega}$ is the maximal solution of
equation (1.10) in $\Omega$. In fact, if $w$ is any smooth positive
$k$-admissible solution of (1.10) in $\Omega$, then $u_{(j)}\geq w$ in
$\Omega_{(j)}$ by Theorem 5.3. Hence $u_{\Omega}\geq w$ in $\Omega$.
###### Definition 5.9.
We call a compact subset $\Gamma\subset\partial\Omega$ regular, if
$u_{\Omega}(x)\rightarrow\infty\quad\text{ as }x\rightarrow\Gamma.$
Now we consider a portion $\Gamma\subset\partial\Omega$ which is a smooth
compact non-self-intersecting surface of codimension $m$. Meanwhile, we assume
that $\partial\Omega\setminus\Gamma$ is smooth compact.
###### Theorem 5.10.
Let $\Omega$ be a domain in $\mathbb{R}^{n}$ and $\Gamma$ be a compact subset
of $\partial\Omega$ such that $\partial\Omega\setminus\Gamma$ is also compact.
Suppose that $u_{\Omega}\not\equiv 0$. Then $\Gamma$ is regular if there
exists an open neighborhood $U$ of $\Gamma$ and a $C^{2}$ positive
$k$-admissible subsolution $\phi(x)$ of (1.10) defined in $\Omega\cap U$ which
tends to $\infty$ as $x\rightarrow\Gamma$.
###### Proof.
Without loss of generality we may assume that $\overline{U}$ is compact,
$\overline{U}\cap(\partial\Omega\setminus\Gamma)=\emptyset\text{ and
}0<\phi\in C^{2}(\overline{U}\cap\Omega).$
For $j$ sufficiently large, we have
$\partial(U\cap\Omega_{(j)})=\partial U\cup(U\cap\partial\Omega_{(j)}).$
Since $u_{\Omega}$ is positive in $\Omega$ and $\partial U\subset\Omega$ is
compact, we have $\inf\limits_{\partial U}u_{\Omega}:=m>0$. Denote
$M:=\sup\limits_{\partial U}\phi(x)>0$ and $A:=\max\\{\frac{M}{m},1\\}$. We
note that $u_{(j)}\geq u_{\Omega}\geq m\geq\frac{\phi}{A}$ on $\partial U$,
and $u_{(j)}=\infty>\frac{\phi}{A}$ on $U\cap\partial\Omega_{(j)}$. By
Proposition 5.2, $\frac{\phi}{A}$ is again a positive $k$-admissible
subsolution of (1.10). By Theorem 5.3, we arrive at
$u_{(j)}\geq\frac{\phi}{A}$ in $U\cap\Omega_{(j)}$. Consequently, we obtain
$u_{\Omega}\geq\frac{\phi}{A}$ in $U\cap\Omega$. ∎
In what follows we assume that $u_{\Omega}\not\equiv 0$. Similar to section 3,
let $\rho(x)$ be the distance of $x$ to $\Gamma$, and
$\Gamma_{\rho_{0}}=\big{\\{}x\in\Omega\,\big{|}\,\rho(x)<\rho_{0}\big{\\}}.$
For $\rho_{0}$ sufficiently small, define on
$\Gamma_{\rho_{0}}\setminus\Gamma$
$\phi(x)=c\rho^{1-\frac{n}{2}}(x),$
where $c>0$ is a constant to be determined later.
Denote
$v_{m}=\big{(}\underbrace{n-m,\ldots,n-m}_{n-m+1},\underbrace{2-m,\ldots,2-m}_{m-1}\big{)}.$
As $\rho\rightarrow 0$,
$\lambda\bigg{(}\frac{2}{n-2}\phi^{-\frac{n+2}{n-2}}W[\phi]\bigg{)}\rightarrow
c^{-\frac{4}{n-2}}v_{m}.$
If we assume that $v_{m}\in\Gamma_{k}$, then there exists $c>0$ sufficiently
small such that
$\displaystyle\sigma_{k}\Big{(}c^{-\frac{4}{n-2}}v_{m}\Big{)}+\underline{\alpha}\sigma_{k-1}\Big{(}c^{-\frac{4}{n-2}}v_{m}\Big{)}>\overline{\alpha}_{0}.$
As $\rho_{0}>0$ sufficiently small, within $\\{0<\rho<\rho_{0}\\}$,
$\sigma_{k}\bigg{(}\frac{2}{n-2}{\phi}^{-\frac{n+2}{n-2}}W[\phi]\bigg{)}+\underline{\alpha}\sigma_{k-1}\bigg{(}\frac{2}{n-2}{\phi}^{-\frac{n+2}{n-2}}W[\phi]\bigg{)}>\overline{\alpha}_{0}.$
Therefore, $\Gamma$ is regular by Theorem 5.10.
If $v_{m}\in\mathbb{R}^{n}\setminus\overline{\Gamma}_{k}$, as in section 3, we
consider
$\psi=\psi_{c,d}=(c\rho^{-a}+d)^{b},$
where $a$, $b$, $c$ and $d$ are positive constants. Direct calculation shows
that
(5.5)
$\frac{1}{abc}\rho^{a+2}(c\rho^{-a}+d)^{\frac{4b}{n-2}+1}\psi^{-\frac{n+2}{n-2}}W[\psi]=A_{m}+B_{ab}(\zeta)+\mathcal{O}(\rho),$
where $\zeta$, $A_{m}$ and $B_{ab}(\zeta)$ are as in (3.2). Choose $a$
sufficiently small, and then choose an appropriate $b$ such that $ab$ is
slightly larger than $\frac{n}{2}-1$ such that
$\lambda\big{(}A_{m}+B_{ab}(\zeta)\big{)}\notin\overline{\Gamma}_{k}\quad\text{for
all }0<\zeta<1.$
Then choose $0<\rho_{0}<1$ further small such that within $0<\rho<\rho_{0}$
and for all $0<\zeta<1$,
$\lambda\big{(}A_{m}+B_{ab}(\zeta)+\mathcal{O}(\rho)\big{)}\notin\overline{\Gamma}_{k},$
which implies that
$\lambda\big{(}W[\psi]\big{)}\notin\overline{\Gamma}_{k}\text{ in
}\\{0<\rho<\rho_{0}\\}\text{ for all }c>0\text{ and }d>0.$
Therefore, we have either $\lambda\big{(}W[\psi]\big{)}\notin\Gamma_{k-1}$,
or, if $\lambda\big{(}W[\psi]\big{)}\in\Gamma_{k-1}$, then
$\sigma_{k}\bigg{(}\frac{2}{n-2}\psi^{-\frac{n+2}{n-2}}W[\psi]\bigg{)}+\alpha\sigma_{k-1}\bigg{(}\frac{2}{n-2}\psi^{-\frac{n+2}{n-2}}W[\psi]\bigg{)}\leq
0<\alpha_{0}.$
That is, $\psi$ is a supersolution of (1.10). Now, choosing $d$ sufficiently
large depending on $\rho_{0}$ such that on $\rho=\rho_{0}$ we have $\psi\geq
d^{b}\geq u_{\Omega}$. By Proposition 5.4, Remark 5.5 and Theorem 5.3, the
solution of the equation
$\sigma_{k}\bigg{(}\frac{2}{n-2}u^{-\frac{n+2}{n-2}}W[u]\bigg{)}=\underline{\alpha}_{0}$
in the ball $B_{\rho(x)}(x)$ can serve as an upper bound for $u_{\Omega}$.
Thus, we can deduce that
$u_{\Omega}\leq\bigg{(}\frac{4(n-1)}{\tilde{\mu}}\bigg{)}^{\frac{n-2}{4}}\rho^{1-\frac{n}{2}}\leq
c^{b}\rho^{-ab}<\psi$
within $0<\rho\leq\delta$, where $\delta>0$ is a sufficiently small constant
depending on $c$. By Theorem 5.3 we have $u_{\Omega}\leq\psi$ in
$\delta<\rho<\rho_{0}$. Letting $\delta\rightarrow 0$, we arrive at
$u_{\Omega}\leq\psi$ in $0<\rho<\rho_{0}$. Next letting $c\rightarrow 0$ we
deduce that $u_{\Omega}\leq d^{b}$ in $0<\rho<\rho_{0}$. Hence $\Gamma$ is not
regular according to Definition 5.9.
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|
Kinetic equations and hierarchies of evolution
equations of quantum systems
V.I. Gerasimenko∗111E-mail<EMAIL_ADDRESS>
∗Institute of Mathematics of the NAS of Ukraine,
3, Tereshchenkivs’ka Str.,
01601, Kyiv-4, Ukraine
Abstract. The article provides an overview of some advances in the
mathematical understanding of the nature of the kinetic equations of quantum
systems of many particles. The fundamental equations of modern mathematical
physics are studied, in particular, the hierarchies of evolution equations of
quantum systems and their asymptotic behavior described by kinetic nonlinear
equations.
Key words: kinetic equation, von Neumann hierarchy, BBGKY hierarchy, density
operator, correlation operator.
2010 Mathematics Subject Classification: 35Q20; 47J35.
math
###### Contents
1. 1 A brief chronology of 150 years of the theory of kinetic equations
2. 2 Hierarchies of evolution equations of quantum systems
1. 2.1 Preliminaries: on evolution equations of quantum systems of many particles
2. 2.2 Dynamics of correlations of particle states
3. 2.3 The BBGKY hierarchies of evolution equations
3. 3 On quantum kinetic equations
1. 3.1 The origin of the kinetic evolution of a state
2. 3.2 The generalized quantum kinetic equation
3. 3.3 Scaling properties of state evolution
4. 4 The kinetic evolution of observables
1. 4.1 The hierarchy of kinetic equations for observables
2. 4.2 The propagation of the initial chaos
3. 4.3 Kinetic equations with initial correlations
5. 5 Outlook
## 1 A brief chronology of 150 years of the theory of kinetic equations
Mathematically the collective properties of many-particle systems in kinetic
theory are described by kinetic equations, namely, nonlinear equations that
describe the evolution of the state of a system of many particles by means of
the evolution of the state of a typical particle. The generator of such
evolution equations consists of a term that describes the free-evolution
(motion by inertia) of a typical particle of the system and a nonlinear term
that simulates the self-interaction of a typical particle (collision
integral). The mean values of observables of the many-particle system are
determined by a solution of the kinetic equations. A well-known historical
example of a kinetic equation is the Boltzmann equation, which describes the
process of collision of particles in rarefied gases. Quantum kinetic equations
are the corresponding generalizations of the kinetic equations of classical
particle systems.
From a physical point of view, kinetic equations describe a certain stage of
the process of transition (relaxation) from a non-equilibrium state to the
state of thermodynamic equilibrium of the system of many particles. Indeed, in
the general case, in the process of relaxation, an arbitrary non-equilibrium
state of the particle system attracts to a state that can be fully described
in terms of a one-particle probability distribution function (for quantum
systems, it is a one-particle statistical operator, the kernel of which is
known as a one-particle density matrix), that is governed by the suitable
kinetic equation depending on the interaction potential of the particles. At
the next stage of relaxation, such a state of the particle system tends to a
state of local equilibrium, which is described by the Maxwell distribution,
characterized by hydrodynamic parameters that depend on a position in space.
The evolution of hydrodynamic parameters in turn is governed by the equations
of a continuous medium (hydrodynamics or diffusion equations).
One hundred and fifty-year history of kinetic equations is derived from the
works of J. C. Maxwell (1860, 1867) [54],[55] and L. E. Boltzmann (1872) [9].
In 1872 Ludwig Boltzmann published the paper [9] where the evolution equation
for a one-particle distribution function was formulated and was showed that
the Maxwell distribution describes only the equilibrium state of a gas. He
proved the so-called H-theorem (on the increase in entropy) about a property
of the solution of this equation, which explained the irreversibility of
macroscopic dynamics. Thus, from this time began a period of development of
the theory of kinetic equations, which was based on phenomenological models of
kinetic phenomena.
Later in order to generalize the Boltzmann equation to dense gases or liquids
for a system of many hard spheres, the Enskog kinetic equation was formulated
(D. Enskog, 1922) [14]. For a particle (Brownian) in a system of many
particles (an environment) was introduced the Fokker–Planck equation (A. D.
Fokker, 1914, M. Planck, 1917) [17],[63] and the Smoluchowski equation as its
partial case (M. von Smoluchowski, 1906) [70]. In this period of development
of kinetic theory were also formulated: the Leontovich equation for stochastic
dynamics of a system of many particles (M. A. Leontovich, 1935) [53], the
Landau equation (L. D. Landau, 1936) [50], the Vlasov equation (A. A. Vlasov,
1938) [68] and the Lenard–Balescu equation (A. Lenard, R. Balescu, R. L.
Guernsey, 1960) [1],[45],[52] for systems of many charged particles (ionized
gases, plasma).
With the beginning of the development of quantum theory the Uehling–Uhlenbeck
kinetic equation was formulated as a quantum analog of the Boltzmann equation
(L. W. Nordheim, 1928; E. A. Uehling, G. E. Uhlenbeck, 1933) [56],[67] and the
quantum Bogolyubov equation (M. M. Bogolyubov, K. P. Gurov, 1947) [7]. In the
mean-field approximation for pure states, the Hartree equation (D. R. Hartree,
1928) [46] or the Hartree–Fock equation for systems of fermions and bosons (V.
A. Fock, 1930) [16] were derived, and for mixed states, the quantum Vlasov
kinetic equation (A. A. Vlasov, 1947) [69]. For quantum many-particle systems
in condensed states, it was suggested the Bogolyubov kinetic equation (M. M.
Bogolyubov, 1947) [5] and later the Gross–Pitaevskii equation (E. P. Gross, L.
D. Pitaevskii, 1961) [44],[62].
From the second half of the 1940s, a new stage in the development of kinetic
theory began, namely, the creation of a formalized theory of kinetic
phenomena. In 1945 in Kyiv at the Institute of Mathematics in the famous
monograph [8], M. M. Bogolyubov formulated a consistent approach to deriving
kinetic equations from the dynamics of systems of many particles, namely, from
fundamental equations described the evolution of all possible states of many-
particle systems, they are now known as the BBGKY hierarchy
(Bogolyubov–Born–Green–Kirkwood–Yvon) [8],[10],[49],[71]. Using the methods of
perturbation theory, an approach was developed to construct a generalization
of the Boltzmann equation, known as the kinetic equation of Bogolyubov, as
well as the Vlasov and Landau kinetic equations. Thanks to this work, the
irreversibility mechanism of the evolution of systems of many particles in the
macroscopic scale, the dynamics of which are described reversible in time
equations of motion in a microscopic scale, became clear. Initially, the
mathematical theory of the BBGKY hierarchy was developed in works
[32],[57],[59] and see also in [12],[23],[58].
Later H. Grad (H. Grad, 1958) [43] formulated an approach to the derivation of
kinetic equations as the evolution equations, which describe the corresponding
scaling asymptotic of a solution of the BBGKY hierarchy. In recent decades,
this approach has been used as an accepted method for the rigorous derivation
of kinetic equations of complex systems of various nature. In general, the
problem of the rigorous derivation of kinetic equations from the dynamics of
systems of many particles remains an open problem of kinetic theory.
At the present stage of development of kinetic theory, the most advanced is
the mathematical theory of the nonlinear Boltzmann equation, which takes its
origin from the works of H. Poincaré (1906) [64], who drew the attention of
mathematicians to the need to substantiate kinetic theory and D. Hilbert
(1912) [48], who established the connection between a solution of the
Boltzmann equation and the hydrodynamics equations of (D. Hilbert’s sixth
problem was formulated at the International Congress of Mathematics in 1900
[47]), as well as the work of T. Carleman (1957) [11] on mathematical analysis
of the spatially homogeneous Boltzmann equation.
The mathematical theory of nonlinear kinetic equations began to develop
intensively in the early 80s of the XX century. One of the achievements of
this period was the rigorous derivation of the Boltzmann equation from the
dynamics of an infinite number of hard spheres in the Boltzmann–Grad limit
[51],[60]. Rigorous results of the theory of kinetic equations and their
justification at the end of the XX century were summarized in monographs
[12],[13],[66].
Over the last decade mathematical results on the derivation of the Boltzmann
kinetic equation for classical systems of particles with short-range
interaction potential have been summarized in the monograph [18] and new
methods have been developed for deriving the Boltzmann equation [20] and the
Enskog equation [30] from the collisional dynamics [31] (for details see the
references in the above works).
In the last two decades significant progress has also been observed in
deriving quantum kinetic equations in the scaling limits of the BBCKY
hierarchy solution constructed by methods of the perturbation theory [3],
[42], [58], in particular, of the quantum Boltzmann equation [2] in a weak
coupling limit, of the nonlinear Schrödinger equation [4], [61] and the
Gross–Pitaevskii equation [15] in the mean-field limit.
Below the fundamental equations that describe the nature of substance are
studies, namely the hierarchies of evolution equations of quantum systems of
many particles, and non-perturbative methods for constructing their solutions.
Based on this, two new approaches to the description of the kinetic evolution
of quantum systems [23] are considered. One of them is consists in the
description of the evolution of quantum systems in the mean-field scaling
limit of the hierarchy of evolution equations for the observables [19] and an
approach based on a non-Markovian generalization of quantum kinetic equations
[38].
## 2 Hierarchies of evolution equations of quantum systems
As is well-known, quantum systems are described by concepts such as
observables and states. The average value functional of the observables
(mathematical expectation) determines the duality of the observables and the
state. Consequently, there are two approaches to describing the evolution of a
quantum system of a finite number of particles, namely, in terms of
observables whose evolution is governed by the Heisenberg equation, or in
terms of a state whose evolution is governed by the von Neumann equation
(quantum Liouville equation) for the density operator (statistical operator)
which kernel is known as the density matrix, respectively.
An equivalent approach to describing the evolution of systems of many
particles, both of a finite and an infinite number of particles, is to
describe the evolution in terms of a sequence of reduced operators of
observables are governing by the dual BBGKY hierarchy
(Bogolyubov–Born–Green–Kirkwood–Yvon) [23],[28], or in terms of the state
describing by a sequence of the reduced density operators governed by the
BBGKY hierarchy [8].
An alternative method for describing the evolution of the state of a quantum
system of finitely many particles is to describe the state in terms of the
operators that are determined by cluster expansions of the density operator.
Such operators are interpreted as correlations of particle states, and their
evolution is governed by the von Neumann hierarchy for a sequence of
correlation operators [26],[34],[37].
### 2.1 Preliminaries: on evolution equations of quantum systems of many
particles
For generality, we will consider below quantum systems in space
$\mathbb{R}^{\nu},\,\nu\geq 1$of a non-fixed number of identical spinless
particles, i.e. of an arbitrary but finite average number of particles, which
obeying the Maxwell–Boltzmann statistics. We will use units where
$h={2\pi\hbar}=1$ is a Planck constant, $m=1$ is particle mass.
We denote the $n$-particle Hilbert space by
$\mathcal{H}_{n}=\mathcal{H}^{\otimes n}$ and $\mathcal{H}^{\otimes
0}=\mathbb{C}$. We denote the Fock space over the space $\mathcal{H}$ by
$\mathcal{F}_{\mathcal{H}}={\bigoplus\limits}_{n=0}^{\infty}\mathcal{H}_{n}$.
The self-adjoint operator $f_{n}$ defined on the space
$\mathcal{H}_{n}=\mathcal{H}^{\otimes n}$ further we will also denote by the
symbol $f_{n}(1,\ldots,n)$. Let $\mathfrak{L}(\mathcal{H}_{n})$ be a space of
bounded operators $g_{n}\equiv
g_{n}(1,\ldots,n)\in\mathfrak{L}(\mathcal{H}_{n})$ with operator norm
$\|.\|_{\mathfrak{L}(\mathcal{H}_{n})}$. Accordingly, let the space
$\mathfrak{L}^{1}(\mathcal{H}_{n})$ be the space of trace class operators
$f_{n}\equiv f_{n}(1,\ldots,n)\in\mathfrak{L}^{1}(\mathcal{H}_{n})$ with the
norm:
$\|f_{n}\|_{\mathfrak{L}^{1}(\mathcal{H}_{n})}=\mathrm{Tr}_{1,\ldots,n}|f_{n}(1,\ldots,n)|,$
where the symbol $\mathrm{Tr}_{1,\ldots,n}$ denotes partial traces of the
operator $f_{n}$. The subspace of finite sequences of degenerate operators
with infinitely differentiated kernels with compact supports is denoted by
$\mathfrak{L}^{1}_{0}(\mathcal{F}_{\mathcal{H}})$.
For a quantum system of a non-fixed number of particles, the observables are
described by sequences $A=(A_{0},A_{1}(1),\ldots,A_{n}(1,\ldots,n),\ldots)$ of
the self-adjoint operators $A_{n}\in\mathfrak{L}(\mathcal{H}_{n})$. The
evolution of the observables
$A(t)=(A_{0},A_{1}(t,1),\ldots,A_{n}(t,1,\ldots,n),\ldots)$, where
$t\in\mathbb{R}$, is determined by the Cauchy problem for a sequence of the
Heisenberg equations:
$\displaystyle\frac{\partial}{\partial t}A(t)=\mathcal{N}A(t),$ (1)
$\displaystyle A(t)|_{t=0}=A(0),$ (2)
where $A(0)=(A_{0},A_{1}^{0}(1),\ldots,A_{n}^{0}(1,\ldots,n),\ldots)$ is the
initial observable, the generator
$\mathcal{N}=\oplus_{n=0}^{\infty}\mathcal{N}_{n}$ is defined by the formula
$\displaystyle\mathcal{N}_{n}g_{n}\doteq-i\,(g_{n}H_{n}-H_{n}g_{n}),$ (3)
and the self-adjoint operator
$H_{n}={\sum\limits}_{j=1}^{n}K(j)+\epsilon{\sum\limits}_{j_{1}<j_{2}=1}^{n}\Phi(j_{1},j_{2})$
is the Hamilton operator of a system of $n$ particles, that is, the operator
$K(j)$ is the kinetic energy operator of $j$ particle, $\Phi$ is the bounded
operator of the pair interaction potential, $\epsilon>0$ is the scaling
parameter.
If $A(0)\in\mathfrak{L}(\mathcal{F}_{\mathcal{H}})$, then for $t\in\mathbb{R}$
in the space $\mathfrak{L}(\mathcal{F}_{\mathcal{H}})$ the unique solution of
the Cauchy problem (1),(2) is represented by a one-parameter family of
mappings $\mathcal{G}(t)=\oplus_{n=0}^{\infty}\mathcal{G}_{n}(t)$:
$\displaystyle A(t)=\mathcal{G}(t)A(0),$
where the mapping $\mathcal{G}_{n}(t)$ is defined by the formula
$\displaystyle\mathbb{R}^{1}\ni t\mapsto\mathcal{G}_{n}(t)g_{n}\doteq
e^{itH_{n}}g_{n}e^{-itH_{n}}.$ (4)
In the space $\mathfrak{L}(\mathcal{H}_{n})$ one-parameter mapping (4) forms a
$\ast$-weakly continuous group of operators, the infinitesimal generator of
which coincides with operator (3) on the domain of its definition.
The average value (mathematical expectation) of the observable $A(t)$ at time
$t\in\mathbb{R}$ is determined by a continuous linear functional, which is
represented by such an expansion in a series [23]:
$\displaystyle\langle
A\rangle(t)=(I,D(0))^{-1}\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{1,\ldots,n}\,A_{n}(t)\,D_{n}^{0},$
(5)
where the sequence $D(0)=(I,D_{1}^{0},\ldots,D_{n}^{0},\ldots)$ of positive
self-adjoint operators $D_{n}^{0}\in\mathfrak{L}^{1}(\mathcal{H}_{n})$ is the
sequence of density operators by which they are described all possible states
of the quantum system of a non-fixed number of particles at the initial
instant, the coefficient
$(I,D(0))={\sum\limits}_{n=0}^{\infty}\frac{1}{n!}\mathrm{Tr}_{1,\ldots,n}D_{n}^{0}$
is the normalizing factor. Functional (5), which determines the duality of
observables and a state, exists for
$D_{n}^{0}\in\mathfrak{L}^{1}(\mathcal{H}_{n})$ and
$A_{n}(t)\in\mathfrak{L}(\mathcal{H}_{n})$.
For the normalizing factor, the following equality is valid
$\displaystyle(I,D(0))=(I,\mathcal{G}^{\ast}(t)D(0)).$ (6)
On the space $\mathfrak{L}^{1}(\mathcal{F}_{\mathcal{H}})$ the one-parameter
mapping $\mathcal{G}^{\ast}(t)=\oplus_{n=0}^{\infty}\mathcal{G}^{\ast}_{n}(t)$
adjoint to the group (4) is defined by
$\displaystyle\mathbb{R}^{1}\ni t\mapsto\mathcal{G}^{\ast}_{n}(t)f_{n}\doteq
e^{-itH_{n}}f_{n}e^{itH_{n}},$ (7)
and forms a strongly continuous isometric group of operators, which preserves
positivity and self-adjointness of operators.
Due to equality (6) for functional (5) the following representation is valid:
$\displaystyle\hskip-34.1433pt(A(t),D(0))=(I,D(0))^{-1}\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{1,\ldots,n}\,\mathcal{G}_{n}(t)A_{n}^{0}\,D_{n}^{0}=$
$\displaystyle(I,\mathcal{G}^{\ast}(t)D(0))^{-1}\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{1,\ldots,n}\,A_{n}^{0}\,\mathcal{G}^{\ast}_{n}(t)D_{n}^{0}=$
$\displaystyle(I,D(t))^{-1}(A(0),D(t)),$
that is, the evolution of quantum systems of many particles in an equivalent
way can be described as the evolution of the state. Indeed, the evolution of
all possible states, i.e. the sequence
$D(t)=(1,D_{1}(t),\ldots,D_{n}(t),\ldots)\in\mathfrak{L}^{1}(\mathcal{F}_{\mathcal{H}})$
of the density operators $D_{n}(t),\,n\geq 1$, is described by the Cauchy
problem for a sequence of the von Neumann equations (quantum Liouville
equations):
$\displaystyle\frac{\partial}{\partial t}D(t)=\mathcal{N}^{\ast}D(t),$ (8)
$\displaystyle D(t)|_{t=0}=D(0),$ (9)
where the generator
$\mathcal{N}^{\ast}=\oplus^{\infty}_{n=0}\mathcal{N}^{\ast}_{n}$ of the von
Neumann equation (8) is adjoint operator to generator (3) of the Heisenberg
equation (1) and is defined by the formula
$\displaystyle\mathcal{N}^{\ast}_{n}f_{n}\doteq-i\,(H_{n}f_{n}-f_{n}H_{n}).$
(10)
Operator (10) has such structure:
$\mathcal{N}^{\ast}_{n}=\sum_{j=1}^{n}\mathcal{N}^{\ast}(j)+\epsilon\sum_{j_{1}<j_{2}=1}^{n}\mathcal{N}^{\ast}_{\mathrm{int}}(j_{1},j_{2})$,
where the operator $\mathcal{N}^{\ast}(j)$ is the generator of the von Neumann
equation of noninteracting particles and the operator
$\mathcal{N}^{\ast}_{\mathrm{int}}$ is defined is defined by the operator of
the particle interaction:
$\mathcal{N}^{\ast}_{\mathrm{int}}(j_{1},j_{2})f_{n}\doteq-i\,(\Phi(j_{1},j_{2})f_{n}-f_{n}\Phi(j_{1},j_{2}))$.
Thus, the unique solution of the Cauchy problem (8),(9) is representing by the
group of operators (4)
$\displaystyle D(t)=\mathcal{G}^{\ast}(t)D(0).$
Note that the density operator is represented by a convex linear combination
of projectors of the first rank. Density operator, which is a projector of the
first rank $D_{n}(t)=P_{\psi_{n}}(t),\,\psi_{n}\in\mathcal{H}_{n},$ is known
as the pure state, and the arbitrary state is interpreted as a mixed state. As
a consequence of the validity for the projector of such equality:
$P_{\psi_{n}}(t)=P_{\psi_{n}(t)}$, where $\psi_{n}(t)=e^{-itH_{n}}\psi_{n}$,
the evolution of the pure state can also be described by the Cauchy problem
for a sequence of the Schrödinger equations:
$\displaystyle i\frac{\partial}{\partial t}\psi_{n}(t)=H_{n}\psi_{n}(t),$
$\displaystyle\psi(t)_{n}\big{|}_{t=0}=\psi_{n}^{0},\quad n\geq 1,$
where the operator $H_{n}$ is the Hamiltonian of the system of $n$ particles.
### 2.2 Dynamics of correlations of particle states
An alternative approach to describing the state of a quantum system of a
finite average number of particles is to describe the state using cumulants of
density operators, which are interpreted as correlations of the states of
clusters of particles [27],[34],[37].
We introduce sequence $g(t)=(I,g_{1}(t,1),\ldots,g_{s}(t,1,\ldots,s),\ldots)$
of correlation operators by means of cluster expansions of the density
operators $D(t)=(I,D_{1}(t,1),\ldots,D_{n}(t,1,\ldots,n),\ldots)$:
$\displaystyle\hskip-34.1433ptD_{n}(t,1,\ldots,n)=g_{n}(t,1,\ldots,n)+\sum\limits_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{P}:(1,\ldots,n)=\bigcup_{i}X_{i},\\\
|\mathrm{P}|>1\end{array}$}}\prod_{X_{i}\subset\mathrm{P}}g_{|X_{i}|}(t,X_{i}),\quad
n\geq 1,$ (13)
where
${\sum\limits}_{\mathrm{P}:(1,\ldots,n)=\bigcup_{i}X_{i},\,|\mathrm{P}|>1}$ is
the sum over all possible partitions $\mathrm{P}$ of the set of indices
$(1,\ldots,n)$ on $|\mathrm{P}|>1$ non-empty subsets
$X_{i}\subset(1,\ldots,n)$, which are not mutually intersecting.
The solutions of recurrent relations (13) are determined by the following
expansions:
$\displaystyle\hskip-42.67912ptg_{s}(t,1,\ldots,s)=D_{s}(t,1,\ldots,s)+$ (14)
$\displaystyle\hskip-34.1433pt\sum\limits_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{P}:(1,\ldots,s)=\bigcup_{i}X_{i},\\\
|\mathrm{P}|>1\end{array}$}}(-1)^{|\mathrm{P}|-1}(|\mathrm{P}|-1)!\,\prod_{X_{i}\subset\mathrm{P}}D_{|X_{i}|}(t,X_{i}),\quad
s\geq 1.$ (17)
The structure of expansions (14) is such that correlation operators can be
interpreted as cumulants (semi-invariants) of density operators (2.1).
If $g_{s}^{0}\in\mathfrak{L}^{1}(\mathcal{H}_{s}),\,s\geq 1$, then for
$t\in\mathbb{R}$ sequence of correlation operators (14) is the unique solution
of the Cauchy problem of the von Neumann hierarchy [34],[37]:
$\displaystyle\hskip-42.67912pt\frac{\partial}{\partial
t}g_{s}(t,1,\ldots,s)=\mathcal{N}^{\ast}_{s}g_{s}(t,1,\ldots,s)+$ (18)
$\displaystyle\hskip-34.1433pt\sum\limits_{\mathrm{P}:\,(1,\ldots,s)=X_{1}\bigcup
X_{2}}\,\sum\limits_{i_{1}\in X_{1}}\sum\limits_{i_{2}\in
X_{2}}\epsilon\,\mathcal{N}_{\mathrm{int}}^{\ast}(i_{1},i_{2})g_{|X_{1}|}(t,X_{1})g_{|X_{2}|}(t,X_{2}),$
$\displaystyle\hskip-42.67912ptg_{s}(t,1,\ldots,s)\big{|}_{t=0}=g_{s}^{0}(1,\ldots,s),\quad
s\geq 1,$ (19)
where the symbol ${\sum\limits}_{\mathrm{P}:\,(1,\ldots,s)=X_{1}\bigcup
X_{2}}$ means the sum over all possible partitions $\mathrm{P}$ of the set
$(1,\ldots,s)$ on two nonempty subsets $X_{1}$ and $X_{2}$, which are not
mutually intersect, and the operators $\mathcal{N}^{\ast}_{s}$,
$\mathcal{N}_{\mathrm{int}}^{\ast}(i_{1},i_{2})$ are defined by formulas (10).
We emphasize that the von Neumann hierarchy (18) is a set of recurrent
evolution equations.
If the initial state is described by a sequence of correlation operators
$g(0)=(I,g_{1}^{0}(1),\ldots,$
$g_{n}^{0}(1,\ldots,n),\ldots)\in\oplus_{n=0}^{\infty}\mathfrak{L}^{1}(\mathcal{H}_{n})$,
then the evolution of all possible states of a quantum system of many
particles, i.e., the sequence
$g(t)=(I,g_{1}(t,1),\ldots,g_{s}(t,1,\ldots,s),\ldots)$ of correlation
operators $g_{s}(t),\,s\geq 1$, is determined by such a group of nonlinear
operators [34]:
$\displaystyle\hskip-42.67912ptg_{s}(t,1,\ldots,s)=\mathcal{G}(t;1,\ldots,s\mid
g(0))\doteq$ (20)
$\displaystyle\hskip-28.45274pt\sum\limits_{\mathrm{P}:\,(1,\ldots,s)=\bigcup_{j}X_{j}}\mathfrak{A}_{|\mathrm{P}|}(t,\\{X_{1}\\},\ldots,\\{X_{|\mathrm{P}|}\\})\prod_{X_{j}\subset\mathrm{P}}g_{|X_{j}|}^{0}(X_{j}),\quad
s\geq 1,$
where $\sum_{\mathrm{P}:\,(1,\ldots,s)=\bigcup_{j}X_{j}}$ is the sum over all
possible partitions $\mathrm{P}$ of the set of indices $(1,\ldots,n)$ on
$|\mathrm{P}|>1$ non-empty subsets $X_{j}$, which are not mutually
intersecting, the set $(\\{X_{1}\\},\ldots,\\{X_{|\mathrm{P}|}\\})$ consists
of elements that are subsets $X_{j}\subset(1,\ldots,s)$, i.e.,
$|(\\{X_{1}\\},\ldots,\\{X_{|\mathrm{P}|}\\})|=|\mathrm{P}|$. The generating
operator $\mathfrak{A}_{|\mathrm{P}|}(t)$ of expansion (20) is the
$|\mathrm{P}|$-th order cumulant of groups of operators (7), which is
determined by such expansion
$\displaystyle\hskip-42.67912pt\mathfrak{A}_{|\mathrm{P}|}(t,\\{X_{1}\\},\ldots,\\{X_{|\mathrm{P}|}\\})\doteq$
(21)
$\displaystyle\hskip-14.22636pt\sum\limits_{\mathrm{P}^{{}^{\prime}}:\,(\\{X_{1}\\},\ldots,\\{X_{|\mathrm{P}|}\\})=\bigcup_{k}Z_{k}}(-1)^{|\mathrm{P}^{{}^{\prime}}|-1}({|\mathrm{P}^{{}^{\prime}}|-1})!\prod\limits_{Z_{k}\subset\mathrm{P}^{{}^{\prime}}}\mathcal{G}^{\ast}_{|\theta(Z_{k})|}(t,\theta(Z_{k})),$
where $\theta$ is the declusterization mapping:
$\theta(\\{X_{1}\\},\ldots,\\{X_{|\mathrm{P}|}\\})\doteq(1,\ldots,s)$.
In the absence of correlations between the particles at the initial instant
(the initial state satisfies the chaos condition [3],[66]), that is, of the
sequence of initial correlation operators
$g^{(c)}(0)=(0,g_{1}^{0}(1),0,\ldots,0,\ldots)$ (in the case of the
Maxwell–Boltzmann statistics in terms of density operators this condition
means that
$D^{(c)}(0)=(I,D_{1}^{0}(1),\ldots,\prod^{n}_{i=1}D_{1}^{0}(i),\ldots)$),
expansions (20) take the form:
$\displaystyle
g_{s}(t,1,\ldots,s)=\mathfrak{A}_{s}(t,1,\ldots,s)\,\prod\limits_{i=1}^{s}g_{1}^{0}(i),\quad
s\geq 1,$
where the $s$-th order cumulant $\mathfrak{A}_{s}(t)$ of the groups of
operators (7) is defined by the following expansion
$\displaystyle\hskip-34.1433pt\mathfrak{A}_{s}(t,1,\ldots,s)=\sum\limits_{\mathrm{P}:\,(1,\ldots,s)=\bigcup_{i}X_{i}}(-1)^{|\mathrm{P}|-1}({|\mathrm{P}|-1})!\prod\limits_{X_{i}\subset\mathrm{P}}\mathcal{G}^{\ast}_{|X_{i}|}(t,X_{i}),$
(22)
and where it was used notations accepted in formula (20).
Thus, the cumulant origin of correlation operators (14) induces a cumulant
structure of one-parameter mapping (20).
We emphasize that the dynamics of correlations, i.e. the hierarchy of
fundamental equations (18) which describes the evolution of state correlations
can be used as a basis for describing the evolution of states both a system of
a finite and infinite number of particles instead of the von Neumann equations
(8) for density operators.
### 2.3 The BBGKY hierarchies of evolution equations
To describe the evolution of quantum systems of both finite and infinite
numbers of particles, another approach to the description of observables and a
state is used, which equivalent to the approach formulated above in the case
of systems of a finite average number of particles. [6],[12].
Indeed, the mean value functional of observables (5) can be else represented
in the following form
$\displaystyle\hskip-34.1433pt\langle
A\rangle(t)=(I,D(0))^{-1}\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{1,\ldots,n}\,A_{n}(t)\,D_{n}^{0}=$
(23)
$\displaystyle\sum\limits_{s=0}^{\infty}\frac{1}{s!}\,\mathrm{Tr}_{1,\ldots,s}\,B_{s}(t,1,\ldots,s)\,F_{s}^{0}(1,\ldots,s),$
where to describe the observables and the state introduced sequences of the
reduced ($s$-particle) operators of observables
$B(t)=(B_{0},B_{1}(t,1),\ldots,B_{s}(t,1,\ldots,s),\ldots)$ and the reduced
($s$-particle) density operators
$F(0)=(I,F_{1}^{0}(1),\ldots,F_{s}^{0}(1,\ldots,s),\ldots)$, respectively
[6],[66]. Therefore, the reduced observables are defined in terms of the
observables by such expansions [28]:
$\displaystyle\hskip-34.1433ptB_{s}(t,1,\ldots,s)=\sum_{n=0}^{s}\,\frac{(-1)^{n}}{n!}\sum_{j_{1}\neq\ldots\neq
j_{n}=1}^{s}A_{s-n}(t,(1,\ldots,s)\setminus(j_{1},\ldots,j_{n})),\quad s\geq
1,$ (24)
and reduced density operators are defined by density operators as follows
[12]:
$\displaystyle\hskip-34.1433ptF_{s}^{0}(1,\ldots,s)=(I,D)^{-1}\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{s+1,\ldots,s+n}\,D_{s+n}^{0}(1,\ldots,s+n),\quad
s\geq 1.$ (25)
We emphasize that the possibility of describing observables and a state with
the help of corresponding reduced operators naturally arises as a result of
dividing the series in the expression (5) by the series of the normalization
factor, i.e. in consequence of redefining the representation of mean value
functional (23).
If at the initial moment an observable is determined by the sequence of
reduced observables
$B(0)=(B_{0},B_{1}^{0}(1),\ldots,B_{s}^{0}(1,\ldots,s),\ldots)\in\mathfrak{L}(\mathcal{F}_{\mathcal{H}})$,
then for arbitrary $t\in\mathbb{R}$ the sequence
$B(t)=(B_{0},B_{1}(t,1),\ldots,B_{s}(t,1,\ldots,s),\ldots)$ of reduced
observables (24) satisfies the Cauchy problem of the quantum dual BBGKY
hierarchy [21],[23],[19]:
$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}B_{s}(t,1,\ldots,s)=\big{(}\sum\limits_{j=1}^{s}\mathcal{N}(j)+\epsilon\hskip-5.69054pt\sum\limits_{j_{1}<j_{2}=1}^{s}\mathcal{N}_{\mathrm{int}}(j_{1},j_{2})\big{)}B_{s}(t,1,\ldots,s)+$
(26) $\displaystyle\hskip-14.22636pt+\epsilon\,\sum_{j_{1}\neq
j_{2}=1}^{s}\mathcal{N}_{\mathrm{int}}(j_{1},j_{2})B_{s-1}(t,1,\ldots,j_{1}-1,j_{1}+1,\ldots,s),$
$\displaystyle\hskip-34.1433ptB_{s}(t,1,\ldots,s)_{\mid
t=0}=B_{s}^{0}(1,\ldots,s),\quad s\geq 1,$ (27)
where the notation accepted in formula (3) was used.
We remark that the hierarchy of equations (26) has the structure of recurrent
evolution equations, for example,
$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}B_{1}(t,1)=\mathcal{N}(1)B_{1}(t,1),$
$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}B_{2}(t,1,2)=\big{(}\sum\limits_{j=1}^{2}\mathcal{N}(j)+\epsilon\,\mathcal{N}_{\mathrm{int}}(1,2)\big{)}B_{2}(t,1,2)+\epsilon\,\mathcal{N}_{\mathrm{int}}(1,2)\big{(}B_{1}(t,1)+B_{1}(t,2)\big{)}.$
The solution of the Cauchy problem (26),(27) is represented by the following
expansions [28],[19]:
$\displaystyle\hskip-34.1433ptB_{s}(t,1,\ldots,s)=\sum_{n=0}^{s}\,\frac{1}{n!}\sum_{j_{1}\neq\ldots\neq
j_{n}=1}^{s}\mathfrak{A}_{1+n}\big{(}t,\\{(1,\ldots,s)\setminus(j_{1},\ldots,j_{n})\\},$
(28)
$\displaystyle(j_{1},\ldots,j_{n})\big{)}\,B_{s-n}^{0}(1,\ldots,j_{1}-1,j_{1}+1,\ldots,j_{n}-1,j_{n}+1,\ldots,s),\quad
s\geq 1,$
where the generating operator of this expansion is the $(1+n)$-th order
cumulant of the groups of operators (4):
$\displaystyle\hskip-34.1433pt\mathfrak{A}_{1+n}(t,\\{(1,\ldots,s)\setminus(j_{1},\ldots,j_{n})\\},(j_{1},\ldots,j_{n}))\doteq$
$\displaystyle\sum\limits_{\mathrm{P}:\,(\\{(1,\ldots,s)\setminus(j_{1},\ldots,j_{n})\\},(j_{1},\ldots,j_{n}))={\bigcup}_{i}X_{i}}(-1)^{\mathrm{|P|}-1}({\mathrm{|P|}-1})!\prod_{X_{i}\subset\mathrm{P}}\mathcal{G}_{|\theta(X_{i})|}(t,\theta(X_{i})),\quad
n\geq 0,$
and used notation similar to the formula (21).
We note that expansion (28) for the solution can be represented as an
iteration series (perturbation theory series) of recurrent evolution equations
(26) in the result of the application of analogues of the Duhamel equation to
the generating operators, i.e. to the cumulants of groups of operators (4).
According to the definition of additive type observables (24) correspond to
the one-component sequences $B^{(1)}(0)=(0,b_{1}^{0}(1),0,\ldots)$ of reduced
observables, and the sequences $B^{(k)}(0)=(0,\ldots,$
$0,b_{k}^{0}(1,\ldots,k),0,\ldots)$ correspond to $k$-ary (non-additive) type
observables. Then for a certain type of observables, the structure of
expansion (28) takes the appropriate form
$\displaystyle
B_{s}^{(1)}(t,1,\ldots,s)=\mathfrak{A}_{s}(t,1,\ldots,s)\sum_{j=1}^{s}b_{1}^{0}(j),\quad
s\geq 1,$ (29)
where the generating operator $\mathfrak{A}_{s}(t)$ is the $s$-th order
cumulant of the groups of operators (4), and, if $s\geq k$,
$\displaystyle\hskip-34.1433ptB_{s}^{(k)}(t,1,\ldots,s)=\frac{1}{(s-k)!}\sum_{j_{1}\neq\ldots\neq
j_{s-k}=1}^{s}\mathfrak{A}_{1+s-k}\big{(}t,\\{(1,\ldots,s)\setminus(j_{1},\ldots,j_{s-k})\\},j_{1},\ldots,$
(30) $\displaystyle
j_{s-k}\big{)}\,b_{k}^{0}(1,\ldots,j_{1}-1,j_{1}+1,\ldots,j_{s}-k-1,j_{s}-k+1,\ldots,s),$
and, if $1\leq s<k$, we have: $B_{s}^{(k)}(t)=0$.
Traditionally, the evolution of many-particle systems is described within the
framework of the evolution of the state governed by the BBGKY hierarchy for
reduced density operators [3],[6],[12],[66].
Indeed, for functional (23) the following representation is hold
$\displaystyle(B(t),F(0))=(B(0),F(t)),$
that is, the evolution of quantum systems of many particles in an equivalent
way can be described as the evolution of the state using reduced density
operators (25).
If $F(0)\in\mathfrak{L}^{1}(\mathcal{F}_{\mathcal{H}})$, the for arbitrary
$t\in\mathbb{R}$ the sequence $F(t)=(I,F_{1}(t,1),\ldots,F_{s}(t,$
$1,\ldots,s),\ldots)$ of reduced density operators (25) satisfies the Cauchy
problem of the quantum BBGKY hierarchy [6],[12]:
$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}F_{s}(t,1,\ldots,s)=\mathcal{N}^{\ast}_{s}F_{s}(t,1,\ldots,s)+\epsilon\,\sum\limits_{j=1}^{s}\mathrm{Tr}_{s+1}\mathcal{N}^{\ast}_{\mathrm{int}}(j,s+1)F_{s+1}(t,1,\ldots,s,s+1),$
(31)
$\displaystyle\hskip-34.1433ptF_{s}(t,1,\ldots,s)\mid_{t=0}=F_{s}^{0}(1,\ldots,s),\quad
s\geq 1,$ (32)
where the notation from formula (10) is used.
The solution of the Cauchy problem (31),(32) is represented by the following
series [35],[36]:
$\displaystyle\hskip-34.1433ptF_{s}(t,1,\ldots,s)=$ (33)
$\displaystyle\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{s+1,\ldots,{s+n}}\,\mathfrak{A}_{1+n}(t,\\{1,\ldots,s\\},s+1,\ldots,{s+n})F_{s+n}^{0}(1,\ldots,{s+n}),\quad
s\geq 1,$
where the generating operator of this series
$\displaystyle\hskip-42.67912pt\mathfrak{A}_{1+n}(t,\\{1,\ldots,s\\},s+1,\ldots,{s+n})=$
(34)
$\displaystyle\sum\limits_{\mathrm{P}\,:(\\{1,\ldots,s\\},s+1,\ldots,{s+n})={\bigcup\limits}_{i}X_{i}}(-1)^{|\mathrm{P}|-1}(|\mathrm{P}|-1)!\prod_{X_{i}\subset\mathrm{P}}\mathcal{G}^{\ast}_{|\theta(X_{i})|}(t,\theta(X_{i}))$
is the $(1+n)$-th order cumulant of the groups of operators (7). In expansion
(34) symbol ${\sum\limits}_{\mathrm{P}}$ means the sum over all possible
partitions $\mathrm{P}$ of the set $(\\{1,\ldots,s\\},s+1,\ldots,{s+n})$ on
$|\mathrm{P}|$ nonampty subsets
$X_{i}\subset(\\{1,\ldots,s\\},s+1,\ldots,{s+n})$, which do not mutually
intersect, and the notations introduced in (20) are used.
We remark that one of the methods for constructing solutions (33) and (28) is
based on the application of cluster expansions [23],[36] to groups of
operators (7) and (4), which are the generating operators of the series (25)
for the reduced density operators and the expansions (24) for the reduced
observables ones, respectively.
We note that a common form of representation of the solution of the Cauchy
problem of the BBGKY hierarchy is its representation as a perturbation theory
series (iteration series of the BBGKY hierarchy) [6],[57] (see also [3] [12]
and therein references):
$\displaystyle\hskip-34.1433ptF_{s}(t,1,\ldots,s)=$ (35)
$\displaystyle\sum\limits_{n=0}^{\infty}\,\int\limits_{0}^{t}dt_{1}\ldots\int\limits_{0}^{t_{n-1}}dt_{n}\mathrm{Tr}_{s+1,\ldots,s+n}\mathcal{G}^{\ast}_{s}(t-t_{1})\sum\limits_{j_{1}=1}^{s}\mathcal{N}^{\ast}_{\mathrm{int}}(j_{1},s+1))\mathcal{G}^{\ast}_{s+1}(t_{1}-t_{2})\ldots$
$\displaystyle\mathcal{G}^{\ast}_{s+n-1}(t_{n-1}-t_{n})\sum\limits_{j_{n}=1}^{s+n-1}\mathcal{N}^{\ast}_{\mathrm{int}}(j_{n},s+n))\mathcal{G}^{\ast}_{s+n}(t_{n})F_{s+n}^{0}(1,\ldots,s+n),\quad
s\geq 1,$
where notations from the expression (10) are used. The representation of the
solution by the series (33) is equivalent to series (35) due to the validity
under the appropriate conditions for the initial data and the interaction
potential of particles of analogs of the Duhamel equation for generating
operators (34) of series (33) that is, for cumulants of groups of operators
(4).
Thus, there are two approaches to describing the evolution of quantum systems
of many particles, namely, in terms of observables whose evolution is governed
by the dual BBGKY hierarchy (26), or in terms of a state whose evolution is
governed by the BBGKY hierarchy (31). For systems of finitely many particles,
these hierarchies of evolution equations are equivalent to the Heisenberg
equation (1) and the von Neumann equation (8), respectively.
In paper [23], these hierarchies of evolution equations are generalized for
many-particle systems with multiparticle interaction potentials.
An alternative approach, as noted above, to the description of the evolution
of the state of quantum systems, consisting of both finitely and infinitely
many particles, can be formulated by means of operators defined by cluster
expansions of the reduced density operators, namely:
$\displaystyle\hskip-42.67912ptG_{s}(t,1,\ldots,s)=\sum\limits_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{P}:(1,\ldots,s)=\bigcup_{i}X_{i}\end{array}$}}(-1)^{|\mathrm{P}|-1}(|\mathrm{P}|-1)!\,\prod_{X_{i}\subset\mathrm{P}}F_{|X_{i}|}(t,X_{i}),\quad
s\geq 1,$ (37)
where the notation from formula (14) was used. Such cumulants of reduced
density operators are interpreted as reduced correlations of the state [6].
On a microscopic scale, the macroscopic characteristics of the fluctuations of
the observables are directly determined by the reduced correlation operators,
for example, the functional of dispersion of the additive type observables,
that is, of such sequences
$A^{(1)}=(0,a_{1}(1),\ldots,\sum_{i=1}^{n}a_{1}(i),\ldots)$, is represented by
the following formula
$\displaystyle\hskip-34.1433pt\langle(A^{(1)}-\langle
A^{(1)}\rangle)^{2}\rangle(t)=\mathrm{Tr}_{1}\,(a_{1}^{2}(1)-\langle
A^{(1)}\rangle^{2}(t))G_{1}(t,1)+\mathrm{Tr}_{1,2}\,a_{1}(1)a_{1}(2)G_{2}(t,1,2),$
where the mean value functional of observables of an additive type we denoted
by $\langle A^{(1)}\rangle(t)=\mathrm{Tr}_{1}\,a_{1}(1)G_{1}(t,1)$.
If $G(0)\in\in\mathfrak{L}^{1}(\mathcal{F}_{\mathcal{H}})$, then for arbitrary
$t\in\mathbb{R}$ the sequence of reduced correlation operators (37) satisfies
the Cauchy problem for the hierarchy of nonlinear evolution equations (quantun
nonlinear BBGKY hierarchy) [6]:
$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}G_{s}(t,1,\ldots,s)=\mathcal{N}^{\ast}_{s}G_{s}(t,1,\ldots,s)+$ (38)
$\displaystyle\sum\limits_{\mathrm{P}:\,(1,\ldots,s)=X_{1}\bigcup
X_{2}}\,\sum\limits_{i_{1}\in X_{1}}\sum\limits_{i_{2}\in
X_{2}}\epsilon\,\mathcal{N}_{\mathrm{int}}^{\ast}(i_{1},i_{2})G_{|X_{1}|}(t,X_{1})G_{|X_{2}|}(t,X_{2}))+$
$\displaystyle\mathrm{Tr}_{s+1}\sum_{i\in(1,\ldots,s)}\epsilon\,\mathcal{N}^{\ast}_{\mathrm{int}}(i,s+1)\big{(}G_{s+1}(t,1,\ldots,s+1)+$
$\displaystyle\sum_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{P}:(1,\ldots,s+1)=X_{1}\bigcup
X_{2},\\\ i\in X_{1};s+1\in
X_{2}\end{array}$}}G_{|X_{1}|}(t,X_{1})G_{|X_{2}|}(t,X_{2})\big{)},$ (41)
$\displaystyle\hskip-34.1433ptG_{s}(t,1,\ldots,s)\big{|}_{t=0}=G_{s}^{0}(,1,\ldots,s),\quad
s\geq 1,$ (42)
where the notations for the hierarchy of equations (18) is used.
In the case of the initial state in the absence of correlations between the
particles, i.e. the state described by the sequence
$G^{(c)}=(0,G_{1}^{0},0,\ldots,0,\ldots)$, the solution of the Cauchy problem
(38),(42) is represented by the following series:
$\displaystyle\hskip-42.67912ptG_{s}(t,1,\ldots,s)=\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{s+1,\ldots,s+n}\,\mathfrak{A}_{s+n}(t;1,\ldots,s+n)\prod_{i=1}^{s+n}G_{1}^{0}(i),\quad
s\geq 1,$ (43)
where the generating operator $\mathfrak{A}_{s+n}(t)$ is the $(s+n)$-th order
cumulant (22) of groups of operators (7). We note that for the specified
initial state, the expressions for the reduced density operators (33) and the
reduced correlation operators (43) differ only in the order of the generating
operator for the corresponding terms of the series representing these
operators.
For an arbitrary initial state (42) a solution of the Cauchy problem for the
nonlinear BBGKY hierarchy (38) was constructed in the paper [25] (see also
[27]).
Note that the description of the evolution of the state of quantum systems of
many particles by means of both reduced density operators and reduced
correlation operators can be based on an approach founded on the dynamics of
correlations, which is governed by the von Neumann hierarchy (18) for
correlation operators. Within the framework of this approach, the reduced
density operators are defined by the following series:
$\displaystyle\hskip-34.1433ptF_{s}(t,1,\ldots,s)\doteq\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{s+1,\ldots,s+n}\,\,g_{1+n}(t,\\{1,\ldots,s\\},s+1,\ldots,s+n),\quad
s\geq 1,$ (44)
where the correlation operators $g_{1+n}(t),\,n\geq 0,$ of particle cluster
and particles are represented by the following expansions:
$\displaystyle\hskip-34.1433ptg_{1+n}(t,\\{1,\ldots,s\\},s+1,\ldots,s+n)=$
(45)
$\displaystyle\sum\limits_{\mathrm{P}:\,(\\{1,\ldots,s\\},\,s+1,\ldots,s+n)=\bigcup_{i}X_{i}}\mathfrak{A}_{|\mathrm{P}|}\big{(}t,\\{\theta(X_{1})\\},\ldots,\\{\theta(X_{|\mathrm{P}|})\\}\big{)}\prod_{X_{i}\subset\mathrm{P}}g_{|X_{i}|}^{0}(X_{i}),\quad
n\geq 0,$
in which the notations from formula (20) are used. The reduced correlation
operators are defined by the corresponding series:
$\displaystyle\hskip-22.76219ptG_{s}(t,1,\ldots,s)\doteq\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{s+1,\ldots,s+n}\,\,g_{s+n}(t,1,\ldots,s+n),\quad
s\geq 1,$ (46)
where the correlation operators $g_{s+n}(t),\,n\geq 0,$ are represented by
expansions (20).
Thus, as a result of definitions (44) and (46), we establish that the cumulant
structure of generating operators of expansions for correlation operators
(20), or more general expansions (45), induces a cumulant structure of series
for reduced density operators (33) and reduced correlation operators (43),
i.e. in fact, the evolution of a system of infinitely many particles is
generated by the dynamics of correlations.
Note also that in the article [29] the hierarchies of evolution equations
(18),(26),(31),(38) are studied as evolution equations in functional
derivatives.
## 3 On quantum kinetic equations
This section describes an approach to describing the evolution of a state
using the state of a typical particle of a quantum system of many particles,
or, in other words, discusses the origin of describing the evolution of a
state by means of quantum kinetic equations [G09],[33],[38].
### 3.1 The origin of the kinetic evolution of a state
We consider a system of many particles that satisfy the Maxwell-Boltzmann
statistics, in the absence of correlations between particles at the initial
time, i.e. the initial state of which is determined by a one-particle density
operator, namely, such a sequence of reduced density operators
$F^{(c)}=(0,F_{1}^{0}(1),\ldots,\prod^{n}_{i=1}F_{1}^{0}(i),\ldots)$. We
emphasize that the formulated assumption about the initial state is inherent
in the kinetic theory of systems of many particles [3],[23],[43].
Due to the fact that the initial state is determined by a one-particle density
operator, for the mean value functional of observables (23) the following
representation is valid
$\displaystyle\big{(}B(t),F^{(c)}\big{)}=\big{(}B(0),F(t\mid
F_{1}(t))\big{)},$
that is, the evolution of all possible states is described by a sequence
$F(t\mid F_{1}(t))=\big{(}I,F_{1}(t),F_{2}(t\mid F_{1}(t)),\ldots,F_{s}(t\mid
F_{1}(t)),\ldots\big{)}$ of reduced functionals of the state
$F_{s}(t,1,\ldots,s\mid F_{1}(t)),\,s\geq 2,$ which are represented by the
series:
$\displaystyle\hskip-34.1433ptF_{s}\bigl{(}t,1,\ldots,s\mid F_{1}(t)\bigr{)}=$
(47)
$\displaystyle\sum_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{s+1,\ldots,s+n}\,\mathfrak{V}_{1+n}\bigl{(}t,\\{1,\ldots,s\\},s+1,\ldots,s+n\bigr{)}\prod_{i=1}^{s+n}F_{1}(t,i),\quad
s\geq 2,$
with respect of a one-particle density operator
$\displaystyle\hskip-34.1433ptF_{1}(t,1)=\sum\limits_{n=0}^{\infty}\frac{1}{n!}\,\mathrm{Tr}_{2,\ldots,{1+n}}\,\mathfrak{A}_{1+n}(t,1,\ldots,n+1)\prod_{i=1}^{n+1}F_{1}^{0}(i),$
(48)
which describes the evolution of the state of a typical particle of a quantum
system of many particles. The generating operators of the series (48) are the
cumulants (34) of the corresponding order of the groups of operators (7).
The reduced functionals of the state $F_{s}(t\mid F_{1}(t)),\,s\geq 2$,
describe all possible correlations that created in the process of evolution of
the quantum system of many particles in terms of the state of a typical
particle. The $(1+n)$-th order generating operator of the series (47) is
determined by the following expansion [38]
$\displaystyle\hskip-25.60747pt\mathfrak{V}_{1+n}\bigl{(}t,\\{1,\ldots,s\\},s+1,\ldots,s+n\bigr{)}=n!\,\sum_{k=0}^{n}\,(-1)^{k}\,\sum_{n_{1}=1}^{n}\ldots$
(49) $\displaystyle\hskip-19.91692pt\sum_{n_{k}=1}^{n-n_{1}-\ldots-
n_{k-1}}\frac{1}{(n-n_{1}-\ldots-
n_{k})!}\,\widehat{\mathfrak{A}}_{1+n-n_{1}-\ldots-
n_{k}}(t,\\{1,\ldots,s\\},s+1,\ldots,s+n-$
$\displaystyle\hskip-19.91692ptn_{1}-\ldots-
n_{k})\prod_{j=1}^{k}\,\sum\limits_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{D}_{j}:Z_{j}=\bigcup_{l_{j}}X_{l_{j}},\\\
|\mathrm{D}_{j}|\leq s+n-n_{1}-\dots-
n_{j}\end{array}$}}\frac{1}{|\mathrm{D}_{j}|!}\sum_{i_{1}\neq\ldots\neq
i_{|\mathrm{D}_{j}|}=1}^{s+n-n_{1}-\ldots-
n_{j}}\,\prod_{X_{l_{j}}\subset\mathrm{D}_{j}}\,\frac{1}{|X_{l_{j}}|!}\,\widehat{\mathfrak{A}}_{1+|X_{l_{j}}|}(t,i_{l_{j}},X_{l_{j}}),$
(52)
where the symbol $\sum_{\mathrm{D}_{j}:Z_{j}=\bigcup_{l_{j}}X_{l_{j}}}$ means
the sum over all possible dissections of the linearly ordered set
$Z_{j}\equiv(s+n-n_{1}-\ldots-n_{j}+1,\ldots,s+n-n_{1}-\ldots-n_{j-1})$ on no
more than $s+n-n_{1}-\ldots-n_{j}$ linearly ordered subsets and the generating
operator of this expansion $\widehat{\mathfrak{A}}_{1+n}(t)$ is the $(1+n)$-th
order cumulant (34) of the groups of scattering operators
$\displaystyle\widehat{\mathcal{G}}_{n}(t)\doteq\mathcal{G}_{n}^{\ast}(t,1,\ldots,n)\prod_{i=1}^{n}(\mathcal{G}_{1}^{\ast}(t,i))^{-1},\quad
n\geq 1,$
We give the simplest examples of generating operators of expansion (49):
$\displaystyle\hskip-22.76219pt\mathfrak{V}_{1}(t,\\{1,\ldots,s\\})=\widehat{\mathfrak{A}}_{1}(t,\\{1,\ldots,s\\}),$
$\displaystyle\hskip-22.76219pt\mathfrak{V}_{1+1}(t,\\{1,\ldots,s\\},s+1)=\widehat{\mathfrak{A}}_{2}(t,\\{1,\ldots,s\\},s+1)-\widehat{\mathfrak{A}}_{1}(t,\\{1,\ldots,s\\})\sum_{i=1}^{s}\widehat{\mathfrak{A}}_{2}(t,i,s+1).$
Thus, according to the definition of (44), the cumulant structure of
generating operators of expansions for correlation operators (45) induces a
generalized cumulant structure of generating operators for series of reduced
functionals of the state (47).
Noticing that for the initial state specified by the one-particle density
operator $F^{(c)}$, the Cauchy problem (38),(42) is not correctly well-
defined, because the initial data are not are independent for each unknown
reduced density operator of the BBGKY hierarchy. As a consequence, such a
Cauchy problem can be reformulated as a new Cauchy problem for a one-particle
density operator with an independent initial condition and a sequence of
explicitly defined functionals of the solution of the Cauchy problem of the
evolution equation for a one-particle density operator (kinetic equation). In
this case, the method of constructing reduced state functionals (47) is based
on the application of the species of cluster expansions, the so-called kinetic
cluster expansions [38], to generating operators (34) of series representing
reduced density operators (33).
### 3.2 The generalized quantum kinetic equation
If $F_{1}^{0}\in\mathfrak{L}^{1}(\mathcal{H})$, then for an arbitrary
$t\in\mathbb{R}$ a one-particle density operator (48) satisfies the Cauchy
problem for the generalized quantum kinetic equation [38]:
$\displaystyle\hskip-22.76219pt\frac{\partial}{\partial
t}F_{1}(t,1)=\mathcal{N}^{\ast}(1)F_{1}(t,1)+\epsilon\,\mathrm{Tr}_{2}\,\mathcal{N}_{\mathrm{int}}^{\ast}(1,2)F_{2}\bigl{(}t,1,2\mid
F_{1}(t)\bigr{)},$ (53)
$\displaystyle\hskip-22.76219ptF_{1}(t,1)\big{|}_{t=0}=F_{1}^{0}(1),$ (54)
where the collision integral is determined by the two-particle functional of
the state (47) and the notation is used (10).
In [38], the existence theorem of the Cauchy problem (53), (54) in the space
of the trace class operators is proved.
Thus, for the initial state in the absence of correlations between particles,
i.e. of the state specified by the one-particle density operator, the
evolution of all possible states of the quantum system of many particles can
be described without any approximations by means of a one-particle density
operator (53) and of a sequence of functionals (47) of this operator (48).
### 3.3 Scaling properties of state evolution
The generally accepted philosophy of describing evolution by kinetic equations
is as follows [8],[43]. If the initial state is determined by the state of a
typical particle of the system, i.e. at the initial moment there are no
correlations between particles (chaos condition), then in a certain scaling
approximation [3],[15],[23],[42] the evolution of the state of a system of
many particles can be effectively described within the state of a typical
particle, i.e. by a one-particle density operator, which is governed by the
corresponding nonlinear kinetic equation.
Further, the scaling asymptotic behavior of the reduced functionals of the
state $F(t\mid F_{1}(t))$ in the specific case of the mean-field limit is
considered [40].
Let there exists the mean-field limit of the initial one-particle density
operator in the following sense
$\displaystyle\lim\limits_{\epsilon\rightarrow
0}\big{\|}\epsilon\,F_{1}^{0}-f_{1}^{0}\big{\|}_{\mathfrak{L}^{1}(\mathcal{H})}=0,$
(55)
where $\epsilon>0$ is a scaling parameter.
Since for an arbitrary finite time interval for an asymptotically perturbed
first-order cumulant of groups of operators (7), i.e. for a strongly
continuous group (7), such equality holds
$\displaystyle\lim\limits_{\epsilon\rightarrow
0}\Big{\|}\mathcal{G}^{\ast}_{s}(t,1,\ldots,s)f_{s}-\prod\limits_{j=1}^{s}\mathcal{G}^{\ast}_{1}(t,j)f_{s}\Big{\|}_{\mathfrak{L}^{1}(\mathcal{H}_{s})}=0.$
then for the $(1+n)$-th order cumulant of asymptotically perturbed groups of
operators (7) the equality valid
$\displaystyle\hskip-34.1433pt\lim\limits_{\epsilon\rightarrow
0}\Big{\|}\frac{1}{\epsilon^{n}}\,\frac{1}{n!}\mathfrak{A}_{1+n}(t,\\{1,\ldots,s\\},s+1,\ldots,s+n)f_{s+n}-$
(56)
$\displaystyle\hskip-5.69054pt\int\limits_{0}^{t}dt_{1}\ldots\int\limits_{0}^{t_{n-1}}dt_{n}\prod\limits_{j=1}^{s}\mathcal{G}_{1}^{\ast}(t-t_{1},j)\sum\limits_{i_{1}=1}^{s}\mathcal{N}^{\ast}_{\mathrm{int}}(i_{1},s+1)\prod\limits_{j_{1}=1}^{s+1}\mathcal{G}_{1}^{\ast}(t_{1}-t_{2},j_{1})\ldots$
$\displaystyle\hskip-5.69054pt\prod\limits_{j_{n-1}=1}^{s+n-1}\mathcal{G}^{\ast}_{1}(t_{n-1}-t_{n},j_{n-1})\sum\limits_{i_{n}=1}^{s+n-1}\mathcal{N}^{\ast}_{\mathrm{int}}(i_{n},s+n)\prod\limits_{j_{n}=1}^{s+n}\mathcal{G}^{\ast}_{1}(t_{n},j_{n})f_{s+n}\Big{\|}_{\mathfrak{L}^{1}(\mathcal{H}_{s+n})}=0,\quad
n\geq 1.$
The last equations are a consequence of the validity for the bounded
interaction potential of analogues of the Duhamel equation for cumulants (34)
of the groups of operators (7).
According to equalities (56) for a one-particle density operator (48) the
following limit theorem holds.
If condition (55) is satisfied, then for series (48) the equality is valid
$\displaystyle\lim\limits_{\epsilon\rightarrow 0}\big{\|}\epsilon
F_{1}(t)-f_{1}(t)\big{\|}_{\mathfrak{L}^{1}(\mathcal{H})}=0,$
where for an arbitrary finite time interval the limiting one-particle density
operator $f_{1}(t)$ is determined by convergent series over the norm of space
$\mathfrak{L}^{1}(\mathcal{H})$
$\displaystyle\hskip-34.1433ptf_{1}(t,1)=\sum\limits_{n=0}^{\infty}\,\int\limits_{0}^{t}dt_{1}\ldots\int\limits_{0}^{t_{n-1}}dt_{n}\,\mathrm{Tr}_{2,\ldots,n+1}\mathcal{G}^{\ast}_{1}(t-t_{1},1)\mathcal{N}^{\ast}_{\mathrm{int}}(1,2)\prod\limits_{j_{1}=1}^{2}\mathcal{G}^{\ast}_{1}(t_{1}-t_{2},j_{1})\ldots$
(57)
$\displaystyle\prod\limits_{i_{n}=1}^{n}\mathcal{G}^{\ast}_{1}(t_{n}-t_{n},i_{n})\sum\limits_{k_{n}=1}^{n}\mathcal{N}^{\ast}_{\mathrm{int}}(k_{n},n+1)\prod\limits_{j_{n}=1}^{n+1}\mathcal{G}^{\ast}_{1}(t_{n},j_{n})\prod\limits_{i=1}^{n+1}f_{1}^{0}(i).$
For bounded interaction potentials, the series (57)the series converges in the
norm of space $\mathfrak{L}^{1}(\mathcal{H})$ provided that:
$t<t_{0}\equiv(2\,\|\Phi\|_{\mathfrak{L}(\mathcal{H}_{2})}\|f_{1}^{0}\|_{\mathfrak{L}^{1}(\mathcal{H})})^{-1}$.
For the initial state $f_{1}^{0}\in\mathfrak{L}^{1}(\mathcal{H})$ limiting
operator (57) satisfies the Cauchy problem for the quantum Vlasov kinetic
equation
$\displaystyle\hskip-14.22636pt\frac{\partial}{\partial
t}f_{1}(t,1)=\mathcal{N}^{\ast}_{1}(1)f_{1}(t,1)+\mathrm{Tr}_{2}\,\mathcal{N}^{\ast}_{\mathrm{int}}(1,2)f_{1}(t,1)f_{1}(t,2),$
(58) $\displaystyle\hskip-14.22636ptf_{1}(t)|_{t=0}=f_{1}^{0}.$ (59)
We note that for pure states, i.e. $f_{1}(t)=|\psi(t)\rangle\langle\psi(t)|$
or in terms of the kernel of this operator
$f_{1}(t,q,q^{\prime})=\psi(t,q)\psi^{\ast}(t,q^{\prime})$, the quantum Vlasov
kinetic (58) is reduced to the Hartree equation
$\displaystyle i\frac{\partial}{\partial
t}\psi(t,q)=-\frac{1}{2}\Delta_{q}\psi(t,q)+\int
dq^{\prime}\Phi(q-q^{\prime})|\psi(q^{\prime})|^{2}\psi(t,q).$
In particular, if the kernel of the interaction potential $\Phi(q)=\delta(q)$
is a Dirac measure, then the Hartree kinetic equation turns into the nonlinear
Schrödinger equation with cubic nonlinearity
$\displaystyle i\frac{\partial}{\partial
t}\psi(t,q)=-\frac{1}{2}\Delta_{q}\psi(t,q)+|\psi(t,q)|^{2}\psi(t,q).$
According to property (56) for cumulants of asymptotically perturbed groups of
operators for generating operators (49) the equalities true:
$\displaystyle\hskip-14.22636pt\lim\limits_{\epsilon\rightarrow
0}\Big{\|}\big{(}\mathfrak{V}_{1}(t,\\{1,\ldots,s\\})-I\big{)}f_{s}\Big{\|}_{\mathfrak{L}^{1}(\mathcal{H}_{s})}=0,$
and for $n\geq 1$
$\displaystyle\hskip-14.22636pt\lim\limits_{\epsilon\rightarrow
0}\Big{\|}\frac{1}{\epsilon^{n}}\,\mathfrak{V}_{1+n}(t,\\{1,\ldots,s\\},s+1,\ldots,s+n)f_{s+n}\Big{\|}_{\mathfrak{L}^{1}(\mathcal{H}_{s+n})}=0,$
respectively.
Due to such equalities and convergence in the mean-field limit of the solution
of the Cauchy problem of the generalized quantum kinetic equation (53),(54) to
the solution of the Cauchy problem of the quantum Vlasov kinetic equation
(58),(59) for reduced functionals of the state (47) the following equalities
are valid:
$\displaystyle\lim\limits_{\epsilon\rightarrow
0}\Big{\|}\epsilon^{s}F_{s}\big{(}t,1,\ldots,s\mid
F_{1}(t)\big{)}-\prod\limits_{j=1}^{s}f_{1}(t,j)\Big{\|}_{\mathfrak{L}^{1}(\mathcal{H}_{s})}=0,\quad
s\geq 2,$
where the limit operator $f_{1}(t)$ is determined by series (57), which
represents the solution of the quantum Vlasov equation (58).
The last statement describes the process of propagation of the initial chaos
in the mean-field limit, i.e., if at the initial moment there are no
correlations in the system, then in the process of evolution correlations are
not created in this approximation.
Note that the traditional approach to the problem of the propagation of
initial chaos is based on the construction of the asymptotic behavior of a
solution of the quantum BBGKY hierarchy for reduced density operators within
the perturbation theory [3],[42].
## 4 The kinetic evolution of observables
In this section, we consider the scaling asymptotic behavior of a solution of
the Cauchy problem (28) for the dual BBGKY hierarchy (26),(27) in the case of
the mean-field limit [19], or, in other words, we consider the foundations of
the description of the kinetic evolution of quantum systems of many particles
within the framework of observables.
Note that one of the advantages of such an approach to the description of
kinetic evolution is the possibility to describe the propagation of initial
correlations in scaling limits.
### 4.1 The hierarchy of kinetic equations for observables
Suppose that at the initial moment of time there exists the mean-field limit
of reduced observables (27) in the sense of $\ast$-weak convergence of the
space of bounded operators $\mathfrak{L}(\mathcal{H}_{s})$
$\displaystyle\mathrm{w^{\ast}-}\lim\limits_{\epsilon\rightarrow
0}\big{(}\epsilon^{-s}B_{s}^{\epsilon,0}-b_{s}^{0}\big{)}=0,$ (60)
where $\epsilon>0$ is a scaling parameter.
Then the following limit theorem holds for reduced observables (28), which are
the solution of the dual BBGKY hierarchy (26).
If the condition (60) is satisfied, then for an arbitrary finite time interval
there is a mean-field limit for the sequence of reduced observables (28) in
the same sense [24],[19]
$\displaystyle\mathrm{w^{\ast}-}\lim\limits_{\epsilon\rightarrow
0}\big{(}\epsilon^{-s}B_{s}(t)-b_{s}(t)\big{)}=0,$ (61)
where the reduced observables $b_{s}(t),\,s\geq 1$, are determined by the
following expansions:
$\displaystyle\hskip-34.1433ptb_{s}(t,1,\ldots,s)=\sum\limits_{n=0}^{s-1}\,\int\limits_{0}^{t}dt_{1}\ldots\int\limits_{0}^{t_{n-1}}dt_{n}\,\mathcal{G}_{s}^{0}(t-t_{1})\sum\limits_{i_{1}\neq
j_{1}=1}^{s}\mathcal{N}_{\mathrm{int}}(i_{1},j_{1})\mathcal{G}_{s-1}^{0}(t_{1}-t_{2})\ldots$
(62)
$\displaystyle\hskip-34.1433pt\mathcal{G}_{s-n+1}^{0}(t_{n-1}-t_{n})\sum\limits^{s}_{\mbox{\scriptsize$\begin{array}[]{c}i_{n}\neq
j_{n}=1,\\\
i_{n},j_{n}\neq(j_{1},\ldots,j_{n-1})\end{array}$}}\mathcal{N}_{\mathrm{int}}(i_{n},j_{n})\mathcal{G}_{s-n}^{0}(t_{n})b_{s-n}^{0}((1,\ldots,s)\setminus({j_{1}},\ldots,{j_{n}})),$
(65) $\displaystyle\hskip-34.1433pts\geq 1,$
and for the group of operators of non-interacting particles, the notation was
used
$\displaystyle\mathcal{G}_{s-n+1}^{0}(t_{n-1}-t_{n})\equiv\prod\limits_{j\in(1,\ldots,s)\setminus(j_{1},\ldots,j_{n-1})}\mathcal{G}_{1}(t_{n-1}-t_{n},j).$
For a certain type of observables, the structure of expansion (62) takes a
particular form, for example, in the case of the $k$-ary type of observables
we have:
$\displaystyle\hskip-34.1433ptb_{s}^{(k)}(t,1,\ldots,s)=\int\limits_{0}^{t}dt_{1}\ldots\int\limits_{0}^{t_{s-k-1}}dt_{s-k}\,\mathcal{G}_{s}^{0}(t-t_{1})\sum\limits_{i_{1}\neq
j_{1}=1}^{s}\mathcal{N}_{\mathrm{int}}(i_{1},j_{1})\mathcal{G}_{s-1}^{0}(t_{1}-t_{2})\ldots$
(66)
$\displaystyle\mathcal{G}_{s-n+1}^{0}(t_{s-k-1}-t_{s-k})\sum\limits^{s}_{\mbox{\scriptsize$\begin{array}[]{c}i_{s-k}\neq
j_{s-k}=1,\\\
i_{s-k},j_{s-k}\neq(j_{1},\ldots,j_{s-k-1})\end{array}$}}\mathcal{N}_{\mathrm{int}}(i_{s-k},j_{s-k})\mathcal{G}_{s-n}^{0}(t_{s-k})\times$
(69) $\displaystyle
b_{k}^{0}((1,\ldots,s)\setminus({j_{1}},\ldots,{j_{s-k}})),\quad 1\leq s\leq
k.$
If $b^{0}\in\mathfrak{L}(\mathcal{F}_{\mathcal{H}})$, then the sequence
$b(t)=(b_{0},b_{1}(t),\ldots,b_{s}(t),\ldots)$ of limit observables (62)
satisfies the Cauchy problem of the dual Vlasov hierarchy:
$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}b_{s}(t,1,\ldots,s)=\sum\limits_{j=1}^{s}\mathcal{N}(j)\,b_{s}(t,1,\ldots,s)+\sum_{j_{1}\neq
j_{2}=1}^{s}\mathcal{N}_{\mathrm{int}}(j_{1},j_{2})\,b_{s-1}(t,(1,\ldots,s)\setminus(j_{1})),$
(70)
$\displaystyle\hskip-34.1433ptb_{s}(t,1,\ldots,s)\mid_{t=0}=b_{s}^{0}(1,\ldots,s),\quad
s\geq 1.$ (71)
We give some examples of the dual Vlasov hierarchy (70) in terms of kernels of
the operators for the reduced observables:
$\displaystyle\hskip-22.76219pti\,\frac{\partial}{\partial
t}b_{1}(t,q_{1};q^{\prime}_{1})=-\frac{1}{2}(-\Delta_{q_{1}}+\Delta_{q^{\prime}_{1}})b_{1}(t,q_{1};q^{\prime}_{1}),$
$\displaystyle\hskip-22.76219pti\,\frac{\partial}{\partial
t}b_{2}(t,q_{1},q_{2};q^{\prime}_{1},q^{\prime}_{2})=\big{(}-\frac{1}{2}\sum\limits_{i=1}^{2}(-\Delta_{q_{i}}+\Delta_{q^{\prime}_{i}})+$
$\displaystyle\hskip
28.45274pt(\Phi(q^{\prime}_{1}-q^{\prime}_{2})-\Phi(q_{1}-q_{2}))\big{)}b_{2}(t,q_{1},q_{2};q^{\prime}_{1},q^{\prime}_{2}))+$
$\displaystyle\hskip
28.45274pt\big{(}\Phi(q^{\prime}_{1}-q^{\prime}_{2})-\Phi(q_{1}-q_{2})\big{)}\big{(}b_{1}(t,q_{1};q^{\prime}_{1})+b_{1}(t,q_{2};q^{\prime}_{2})\big{)}.$
We note that the sequence of evolution equations (70) has the structure of
recursive equations.
Thus, in the mean-field limit, the collective behavior (kinetic evolution) of
quantum systems of many particles is described in terms of the sequence of
limiting reduced observables (62), in particular, by sequence (66) whose
evolution is governed by the Cauchy problem of the dual Vlasov hierarchy
(70),(71).
### 4.2 The propagation of the initial chaos
Let us consider the relationship of collective behavior within the mean-field
approximation of a quantum system of many particles, which is described by the
dual Vlasov hierarchy (70) for the limiting reduced observables, and by the
Vlasov kinetic equation (58) for the state of a typical particle of a system.
Let the initial state satisfy a chaos condition, i.e. at the initial instant
there are no correlations between particles (the statistically independent
particles), namely, in the case of the Maxwell–Boltzmann statistics by such a
sequence of limiting reduced density operators
$\displaystyle
f^{(c)}\equiv\big{(}I,f_{1}^{0}(1),\ldots,\prod_{i=1}^{s}f_{1}^{0}(i),\ldots\big{)}.$
(72)
As mentioned above, this assumption regarding the initial state is
characteristic of the kinetic description of the gas, because in this case the
state is completely determined using a one-particle density operator.
For mean value functional (23) of additive type limit observables (66) and
initial state (72) on a finite time interval, the equality holds
$\displaystyle\hskip-22.76219pt\big{(}b^{(1)}(t),f^{(c)}\big{)}=\sum\limits_{s=0}^{\infty}\frac{1}{s!}\,\mathrm{Tr}_{1,\ldots,s}\,b_{s}^{(1)}(t,1,\ldots,s)\prod\limits_{i=1}^{s}f_{1}^{0}(i)=$
$\displaystyle\hskip 19.91692pt\mathrm{Tr}_{1}\,b_{1}^{0}(1)f_{1}(t,1),$
where a one-particle density operator is represented by such a series
$\displaystyle\hskip-34.1433ptf_{1}(t,1)=\sum\limits_{n=0}^{\infty}\int\limits_{0}^{t}dt_{1}\ldots\int\limits_{0}^{t_{n-1}}dt_{n}\,\mathrm{Tr}_{2,\ldots,n+1}\prod\limits_{i_{1}=1}^{1}\mathcal{G}^{\ast}_{1}(t-t_{1},i_{1})\mathcal{N}^{\ast}_{\mathrm{int}}(1,2)\prod\limits_{j_{1}=1}^{2}\mathcal{G}^{\ast}_{1}(t_{1}-t_{2},j_{1})\ldots$
(73)
$\displaystyle\prod\limits_{i_{n}=1}^{n}\mathcal{G}^{\ast}_{1}(t_{n}-t_{n},i_{n})\sum\limits_{k_{n}=1}^{n}\mathcal{N}^{\ast}_{\mathrm{int}}(k_{n},n+1)\prod\limits_{j_{n}=1}^{n+1}\mathcal{G}^{\ast}_{1}(t_{n},j_{n})\prod\limits_{i=1}^{n+1}f_{1}^{0}(i).$
The one-particle density operator (73) satisfies the Cauchy problem of the
quantum Vlasov kinetic equation (58),(59).
Thus, the hierarchy of evolutionary equations (70) for limit reduced
observables of additive type and initial state (72) describes the evolution of
quantum many-particle systems in an equivalent way compared to the Vlasov
equation (58).
Accordingly, for the mean value functional of the limit observables of
nonadditive type (66) and the initial state (72) on a finite time interval,
equations valid:
$\displaystyle\hskip-19.91692pt\big{(}b^{(k)}(t),f^{(c)}\big{)}=\sum\limits_{s=0}^{\infty}\frac{1}{s!}\,\mathrm{Tr}_{1,\ldots,s}\,b_{s}^{(k)}(t,1,\ldots,s)\prod\limits_{i=1}^{s}f_{1}^{0}(i)=$
(74)
$\displaystyle\frac{1}{k!}\mathrm{Tr}_{1,\ldots,k}\,b_{k}^{0}(1,\ldots,k)\prod\limits_{i=1}^{k}f_{1}(t,i),\quad
k\geq 2,$
where the one-particle density operator is represented by series (73), i.e.
satisfies the Cauchy problem for the quantum Vlasov kinetic equation
(58),(59).
Equality (74) describes the process of the propagation of initial chaos (72)
in the mean-field limit by means of solution (73) of the quantum Vlasov
kinetic eqation (58), namely, for an arbitrary finite time interval, the state
is represented by such a sequence of limit reduced operators:
$\displaystyle f_{k}(t,1,\ldots,k)=\prod\limits_{i=1}^{k}f_{1}(t,i),\quad
k\geq 2,$ (75)
that is, if at the initial moment of time in the system of particles there are
no correlations, then in this approximation in the process of evolution
correlations of particle states are not create.
### 4.3 Kinetic equations with initial correlations
We note that the above approach to the derivation of kinetic equations allows
us to formulate kinetic equations in the case of more general initial states,
which describe not only the gases of quantum particles (72) but also systems
of many particles in condensed states.
Further, we consider the initial states of quantum systems of many particles,
which are determined by a one-particle density operator and correlation
operators (Maxwell - Boltzmann statistics) [24]
$\displaystyle\hskip-34.1433ptf^{(cc)}=\big{(}I,f_{1}^{0}(1),g_{2}^{0}(1,2)\prod_{i=1}^{2}f_{1}^{0}(i),\ldots,g_{n}^{0}(1,\ldots,n)\prod_{i=1}^{n}f_{1}^{0}(i),\ldots\big{)},$
(76)
where the correlations of the initial states of the particles are determined
by the operators $g_{n}^{0}(1,\ldots,n)\equiv
g_{n}^{0}\in\mathfrak{L}^{1}_{0}(\mathcal{H}_{n}),\,n\geq 2$. We emphasize
that this assumption (76) with respect to the initial state is typical for the
kinetic description of systems of many particles in condensed states, which
are characterized by correlations, for example, such as in particle flows
[6],[65].
Then, using the method of derivation of kinetic equations based on the
hierarchy of kinetic equations for the observables, for an arbitrary finite
time interval we establish that the state is described by the sequence
$f(t)=\big{(}I,f_{1}(t),\ldots,f_{n}(t,1,\ldots,n),\ldots\big{)}$ of limiting
reduced density operators where a one-particle density operator is represented
by such a series
$\displaystyle\hskip-34.1433ptf_{1}(t,1)=\sum\limits_{n=0}^{\infty}\int\limits_{0}^{t}dt_{1}\ldots\int\limits_{0}^{t_{n-1}}dt_{n}\,\mathrm{Tr}_{2,\ldots,n+1}\prod\limits_{i_{1}=1}^{1}\mathcal{G}^{\ast}_{1}(t-t_{1},i_{1})\mathcal{N}^{\ast}_{\mathrm{int}}(1,2)\prod\limits_{j_{1}=1}^{2}\mathcal{G}^{\ast}_{1}(t_{1}-t_{2},j_{1})\ldots$
(77)
$\displaystyle\prod\limits_{i_{n}=1}^{n}\mathcal{G}^{\ast}_{1}(t_{n}-t_{n},i_{n})\sum\limits_{k_{n}=1}^{n}\mathcal{N}^{\ast}_{\mathrm{int}}(k_{n},n+1)\prod\limits_{j_{n}=1}^{n+1}\mathcal{G}^{\ast}_{1}(t_{n},j_{n})g_{n+1}^{0}(1,\ldots,n+1)\prod\limits_{i=1}^{n+1}f_{1}^{0}(i),$
and the limiting density operators $f_{k}(t,1,\ldots,k),\,k\geq 2,$ are
determined by the following expressions [41]:
$\displaystyle\hskip-34.1433ptf_{k}(t,1,\ldots,k)=\prod_{i_{1}=1}^{k}\mathcal{G}^{\ast}_{1}(t,i_{1})g_{k}^{0}(1,\ldots,k)\prod_{i_{2}=1}^{k}(\mathcal{G}_{1}^{\ast})^{-1}(t,i_{2})\prod\limits_{j=1}^{k}f_{1}(t,j),\quad
k\geq 2.$ (78)
Indeed, for the mean value functionals of the limiting observables (66) in the
case of the initial state (76) for a finite time interval, the equalities are
valid:
$\displaystyle\hskip-19.91692pt\big{(}b^{(k)}(t),f^{(cc)}\big{)}=\frac{1}{k!}\mathrm{Tr}_{1,\ldots,k}\,b_{k}^{0}(1,\ldots,k)f_{k}(t,1,\ldots,k),\quad
k\geq 1,$
where the sequence $f(t)$ is determined by expansions (77) and (78).
In [41], a similar result was obtained using a generalized quantum kinetic
equation with initial correlations.
In the case of initial states specified by a sequence of limiting correlation
operators (76), the process of propagation of initial correlations is
described by the following sequence of correlation operators (37):
$\displaystyle\hskip-34.1433ptg_{n}(t,1,\ldots,n)=\prod_{i_{1}=1}^{n}\mathcal{G}_{1}^{\ast}(t,i_{1})\sum\limits_{\mbox{\scriptsize$\begin{array}[]{c}\mathrm{P}:(1,\ldots,n)=\bigcup_{i}X_{i}\end{array}$}}(-1)^{|\mathrm{P}|-1}(|\mathrm{P}|-1)!\,\prod_{X_{i}\subset\mathrm{P}}g_{|X_{i}|}^{0}(X_{i})\times$
$\displaystyle\prod_{i_{2}=1}^{n}(\mathcal{G}_{1}^{\ast})^{-1}(t,i_{2})\prod\limits_{j=1}^{n}f_{1}(t,j),\quad
n\geq 2.$
Limiting one-particle density operator (77) is governed by the quantum Vlasov
kinetic equation with initial correlations [41]:
$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}f_{1}(t,1)=\mathcal{N}^{\ast}(1)f_{1}(t,1)+$ (80)
$\displaystyle\mathrm{Tr}_{2}\,\mathcal{N}^{\ast}_{\mathrm{int}}(1,2)\prod_{i_{1}=1}^{2}\mathcal{G}^{\ast}_{1}(t,i_{1})g_{2}^{0}(1,2)\prod_{i_{2}=1}^{2}(\mathcal{G}^{\ast}_{1})^{-1}(t,i_{2})f_{1}(t,1)f_{1}(t,2),$
$\displaystyle\hskip-34.1433ptf_{1}(t)|_{t=0}=f_{1}^{0},$ (81)
where the notations (10) is used and the inverse group of operators to group
(4) is denoted by $(\mathcal{G}^{\ast}_{1})^{-1}(t)$.
We note that the kinetic equation (80) is a non-Markov kinetic equation. For
pure states, equation (80) reduces to the Hartree kinetic equation with
initial correlations. For the initial states of the system of statistically
independent particles, the kinetic equation (80) coincides with the quantum
Vlasov equation (58), and reduced density operators (78) describe the process
of propagation of the initial chaos (75).
We remark also that in the case of arbitrary initial states in a mean-field
limit, a sequence of reduced density operators, as well as sequence (77),
(78), is the solution of the Cauchy problem of the Vlasov hierarchy:
$\displaystyle\hskip-34.1433pt\frac{\partial}{\partial
t}f_{s}(t,1,\ldots,s)=\sum\limits_{i=1}^{s}\mathcal{N}^{\ast}(i)f_{s}(t,1,\ldots,s)+\sum\limits_{i=1}^{s}\mathrm{Tr}_{s+1}\mathcal{N}^{\ast}_{\mathrm{int}}(i,s+1)f_{s+1}(t,1,\ldots,s+1),$
$\displaystyle\hskip-34.1433ptf_{s}(t)_{|t=0}=f_{s}^{0},\quad s\geq 1.$
Thus, in the case of initial states given by a one-particle density operator
and correlation operators (76), the dual Vlasov hierarchy (70) for the
additive type reduced observables describes the evolution of quantum systems
of many particles, just as non-Markov quantum Vlasov kinetic equation with
initial correlations (80). In the case of reduced nonadditive-type
observables, the dual Vlasov hierarchy (70) describes in an equivalent way in
the sense of equality (78) the process of propagation of initial correlations
in terms of reduced density operators (78). In other words, the alternative
method of describing the evolution of states of quantum systems of many
particles in the mean-field approximation is based on the non-Markovian Vlasov
kinetic equation with initial correlations (80).
## 5 Outlook
The possible approaches to the description of the evolution of quantum systems
of many particles are considered above, namely, in terms of the observables
ones, the evolution of which is described by the dual BBGKY hierarchy (26), or
in terms of the state whose evolution is described by the BBGKY hierarchy
(31). For systems of a finite number of particles, these hierarchies of
evolution equations are equivalent to the Heisenberg equation (1) and the von
Neumann equation (8), respectively. In particular, possible methods of
describing the evolution of the state by means of the hierarchy of fundamental
equations (18), which describes the evolution of state correlations, are
considered.
It was established that the concept of cumulant (22) of groups of operators
forms the basis of expansions for solutions of fundamental evolution
equations, which describe the evolution of quantum systems of many particles,
namely: in the case of the groups of operators (4) for the dual BBGKY
hierarchy for reduced observables; in the case of groups of operators (7) for
the von Neumann hierarchy (18) for correlation operators, for the BBGKY
hierarchy (31) for reduced density operators and for the BBGKY hierarchy of
nonlinear equations (38) for reduced correlation operators, respectively, as
well as for the basis of the kinetic description of infinitely many particles
systems (47),(53).
We emphasize that the structure of expansions for correlation operators (45),
in which the generating operators are of the appropriate order cumulants (21)
of the groups of operators (7), induces the cumulant structure of expansions
in a series for reduced density operators (33), reduced correlation operators
(46) and reduced functionals of the state (47). Thus, the dynamics of systems
of infinitely many particles are generated by the dynamics of correlations.
The article also was considered two new approaches to describing the kinetic
evolution of quantum systems of many particles. One of them consists in the
description of the kinetic evolution of the quantum system of particles by
means of the reduced observables in mean-field scaling limit (70). Another
approach is based on the non-Markovian generalization of quantum kinetic
equations (53).
One of the advantages of the considered approaches is connected with the
possibility of construction of kinetic equations in scaling approximations
taking into account correlations of particles at the initial instant which
characterize the condensed states of systems of many particles. Another
advantage is related to the problem of the rigorous derivation of kinetic
equations of the non-Markovian type based on the dynamics of quantum many-
particle systems, which allow us to describe the effects of memory in
nanostructures.
We remark that the approach to the derivation of the quantum Vlasov kinetic
equation from the dynamics of many-particle systems, which is based on the
generalized quantum kinetic equation (53), also allows the construction of
higher-order corrections to the approximation of mean-field for quantum
kinetic equations.
The above results can be extended to systems of many bosons or fermions
[34],[39].
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|
# DIALOGUE STRATEGY ADAPTATION TO NEW ACTION SETS USING MULTI-DIMENSIONAL
MODELLING
###### Abstract
A major bottleneck for building statistical spoken dialogue systems for new
domains and applications is the need for large amounts of training data. To
address this problem, we adopt the multi-dimensional approach to dialogue
management and evaluate its potential for transfer learning. Specifically, we
exploit pre-trained task-independent policies to speed up training for an
extended task-specific action set, in which the single summary action for
requesting a slot is replaced by multiple slot-specific request actions.
Policy optimisation and evaluation experiments using an agenda-based user
simulator show that with limited training data, much better performance levels
can be achieved when using the proposed multi-dimensional adaptation method.
We confirm this improvement in a crowd-sourced human user evaluation of our
spoken dialogue system, comparing partially trained policies. The multi-
dimensional system (with adaptation on limited training data in the target
scenario) outperforms the one-dimensional baseline (without adaptation on the
same amount of training data) by 7% perceived success rate.
Index Terms— dialogue systems, policy optimisation, reinforcement learning,
transfer learning
## 1 Introduction
One of the main challenges in spoken dialogue system development is their
scalability to new domains and applications. A statistical spoken dialogue
system can be built efficiently for a large slot-filling domain when
sufficient annotated data in that domain is available. However, as domains are
expanded with new slots, and new applications emerge that require new, task-
specific system actions, in-domain annotated datasets are typically hard to
get by. Therefore, numerous efforts have been made to use transfer learning
techniques [1, 2] to efficiently develop dialogue systems for new domains with
limited or no data in the target domain. Most of the efforts in statistical
dialogue management and action selection in particular focus on adaptation to
newly introduced slots and values, with the underlying task unchanged [3, 4,
5].
The multi-dimensional approach to dialogue modelling [6] offers the potential
to exploit its principled separation of domain- and application-independent
aspects of dialogue to adapt to new domains as well as new applications. In
[7], a multi-dimensional statistical dialogue manager was presented and it was
demonstrated that policy optimisation for a target domain could be improved by
re-using the policies for the application-independent dimensions (Social
Obligations Management and Auto-feedback; see Section 2) that were pre-trained
in a source domain. However, the addressed use-case was limited to adapting
across very similar domains, where the set of task-specific actions was the
same and only the set of slots and values changed.
In this paper, we adopt the multi-dimensional approach to dialogue management
[8, 9, 7], but rather than adapting to new slots and values, we focus on
adapting to a different policy action set, which is typically required when
developing a dialogue system for a new application. In addition, we present
two extensions of the model: 1) whereas [7] restricted their model to allow
only single dialogue acts being generated (despite the multi-agent design), we
allow combinations of auto-feedback and task acts that support implicit
confirmation, and 2) the agent that evaluates the generated dialogue act
candidates from the different dimensions (see Section 2) is trained jointly
with the other agents, rather than hand-coded.
Using the improved multi-dimensional design, we present policy optimisation
and evaluation experiments using our simulated user and error model,
demonstrating significant improvements in performance on limited training data
when using the proposed adaptation method (Section 3). Furthermore, a human
user evaluation was carried out to confirm this result in more realistic
conditions; the experimental design, spoken dialogue system implementation,
and evaluation results are described in Section 4. Section 5 contains a more
detailed discussion of related work in the area of transfer learning for
dialogue management. The paper is wrapped up in Section 6 with conclusions and
directions for future work.
Task | Auto-Feedback | SOM | All | Example utterance
---|---|---|---|---
offer | | | offer | How about the Rice Boat?
offer | impl-confirm | | offer+impl-confirm | The Rice Boat is a nice Indian restaurant
answer | | | answer | The address of the Rice Boat is …
request | | | request | What price range did you have in mind?
request | impl-confirm | | req+impl-confirm | Okay, Indian food. What price range did you have in mind?
| expl-confirm | | expl-confirm | You want Indian food, is that right?
| auto_negative | | auto_negative | I did not quite catch that, could you please rephrase?
| | accept_thanking | accept_thanking | You’re welcome
| | return_goodbye | return_goodbye | Have a nice day
none | none | none | |
Table 1: The dialogue act agent action sets. Note that impl-confirm is
essentially an inform act stating with a slot-value pair a preference that the
system believes the user has, whereas expl-confirm is essentially a
propositional question asking the user whether they have a specific
preference, expressed by a slot-value pair.
## 2 Multi-dimensional dialogue management
In conventional reinforcement learning-based statistical dialogue systems
[10], the dialogue policy selects an action from a single set of possible
actions in each turn. In contrast to such ‘one-dimensional’ systems, multi-
dimensional dialogue systems employ multiple dialogue act agents, each
dedicated to a different aspect of the dialogue process, using a policy that
selects actions from its own specialised action set. We propose and evaluate a
model with three dimensions and corresponding agents: Auto-feedback, Social
Obligation Management (SOM), and Task. The Auto-feedback agent has actions for
giving feedback to the user about processing their utterances, for example
indicate non-understanding when the speech recogniser does not return any
results (“I did not quite get that, could you please repeat?”) or provide
articulate feedback in the form of an explicit/implicit confirmation when the
system is unsure about the user’s input (e.g., “Expensive, you said?”), the
SOM agent deals with greeting, thanking, apologising, and other social
actions, and the Task agent focuses on the underlying task or activity
(information navigation, tutoring, negotiation, etcetera). Additional agents
to support other dimensions could be introduced as well (e.g. for turn-taking
and time-management [11]), but the three agents currently included are
considered to be the minimum requirement for a task-oriented multi-dimensional
dialogue system. An additional Evaluation agent is used for determining which
of the generated dialogue act candidates are forwarded as final dialogue acts
to the natural language generation component [12]. Figure 1 gives an outline
of the multi-agent action selection component.
Fig. 1: Multi-agent action selection component, showing the Task, Auto-
feedback, and SOM agents generating candidate dialogue acts, of which two are
selected by the Evaluation agent.
Besides reflecting the multi-dimensional nature of dialogue [6], this design
opens up opportunities for efficient adaptation of dialogue managers to new
tasks and domains. The Auto-feedback and SOM dimensions are considered to be
domain and task independent, and therefore policies for these dimensions may
be transferred, leaving only the Task policy to be trained from scratch in the
target domain/application.
We introduce a new design of the multi-agent action selection model, in which
the Evaluation agent is designed to allow two possible combinations of two
dialogue acts. Both of these combinations include an implicit confirmation
from the Auto-feedback dimension, combined with either an offer (e.g., “Prezzo
is a popular Italian restaurant.”), or a request (e.g., “You are looking for
an Italian restaurant in which area?”) from the Task dimension. Table 1 lists
all allowed action combinations, organised along the three supported
dimensions, as well as in a single action set for a one-dimensional baseline
system. In future versions, additional combinations can be considered, for
example negative feedback combined with an apology (e.g., “I’m sorry, I did
not quite get that”).
## 3 Policy optimisation and adaptation experiments
The potential for transfer learning in a multi-dimensional dialogue manager
has been demonstrated previously for the use-case of adapting from a hotel
search application to a restaurant search application [7]. Here, we introduce
a use-case where we stay within the restaurant domain, but adapt to a new
action set. Specifically, the action set of the Task agent is extended by
replacing the summary action request for requesting a slot by separate request
actions for each slot, e.g. request-pricerange. Whereas in the source scenario
the policy may select the request summary action, after which heuristics
determine which slot is requested, in the target scenario the policy may
select a request action for a particular slot, for example the area.
Therefore, in the target scenario with extended action set, the system learns
automatically which slot to request, rather than relying on heuristics in the
source scenario.
In the DSTC-2 restaurant search domain [13], this means that the request
summary action is replaced by the three slot-specific request actions,
corresponding to the slots food, area, and pricerange. Going back to the
original action sets outlined in Table 1, the action set of the Task agent
therefore grows from 4 (offer, request, answer, none) to 6 actions (offer,
request-food, request-area, request-pricerange, answer, none), whereas the
action set of the one-dimensional baseline (column ‘All’) grows from 9 to 13
actions, since the extension applies to both request and req+impl-confirm.
Hence, we first train a multi-dimensional system for the source scenario, in
which the Task agent uses a summary action for requesting slots (mdim-src).
This yields 4 trained policies, corresponding to the 3 conversational
dimensions plus the evaluation agent. Out of these, the policies for Auto-
feedback and SOM are subsequently re-used and adapted when we train a system
for the target scenario, in which the Task agent uses the extended action set
(mdim-ada). As baselines, we also train a multi-dimensional system for the
target scenario without re-using the pre-trained Auto-feedback and SOM
policies (multi-dim) and a one-dimensional system for the target scenario in
which all action combinations from the multi-dimensional system are taken as
single actions into one action set (one-dim).
### 3.1 Training details
All action selection models are trained in online interaction with an agenda-
based user simulator [14]. In each case, all 4 policies are trained
simultaneously using Monte Carlo Control reinforcement learning with linear
value function approximation [15]. Each policy selects actions from its own
action set, based on the approximated values given the current state. The
values predict the long-term cumulative reward when taking an action in a
given state and following the policy in subsequent turns, where the state is
represented by a set of features extracted from the full dialogue state.
During training, the policies use Boltzmann exploration, i.e., actions are
sampled from a softmax distribution applied to the estimated values. The
temperature hyper-parameter of the softmax is decayed linearly, gradually
reducing the level of exploration until the policy only selects actions with
the highest estimated value. The weights of the linear value function are
updated after every dialogue/episode, based on a shared reward signal. The
policies coordinate their actions only indirectly through the shared rewards,
i.e., each policy operates independently without any direct communication with
other policies.
The rewards are calculated at each turn, combining rewards obtained from the
simulated user with internal rewards. The user gives a reward of +100 upon
task completion: in the restaurant search domain, this is when the system has
recommended a restaurant matching the user’s preferences and has provided all
requested information about this restaurant. Such ‘user goals’ are randomly
initialised from the domain ontology at the start of each dialogue, and fed to
the simulated user. In addition, the user issues a penalty when the system
fails to respond to a thanking action or inserts a social act when it is not
called for (-5 for each occurrence). This is to force the system to learn
basic reactive social behaviour, though we are aware that this might be
experienced as repetitive and unnatural. In future work, we will consider
learning more sophisticated social patterns. Internally, a penalty of -1 is
applied for each turn (to encourage shorter dialogues) and a penalty of -25
when a ‘processing problem’ is encountered and the system does not signal this
to the user with a feedback act. A processing problem is recorded in the
dialogue state when speech recognition or natural language understanding
fails, i.e., returns no results. During training, such processing problems are
simulated randomly in 5% of the user turns, discarding the original simulated
user act.
The Evaluation agent from [7] was implemented via a set of rules, based on
definitions from the dialogue act annotation standard [11]. Here, we implement
it as another reinforcement learning agent, which takes as input the candidate
dialogue act for each dimension and selects which dialogue act combination
will be passed on to the natural language generation component. Given that
there are 3 supported dimensions, the actions correspond to the 8 possible
combinations of dimensions that can be selected. Using an action masking
mechanism, we ensure that the Evaluation agent only allows single dialogue
acts or a combination of impl-confirm (Auto-feedback dimension) with offer or
with any of the three request actions (Task dimension).
### 3.2 Results
The policy optimisation results in simulation for the target scenario are
shown in Figure 2. The learning curves (here shown in terms of success rates)
quite clearly show that the performance levels are much higher in the early
stages of training when using our proposed multi-dimensional adaptation method
(mdim-ada, in red) than when training a multi-dimensional action selection
model from scratch (multi-dim, in green). In the first 20k training dialogues
we also observe much higher success rates compared to the one-dimensional
baseline (one-dim, in blue).
Fig. 2: Learning curves for the target scenario in terms of success rate for
the three training methods. Note that the results are averaged over 10
training runs in each setting; in each training run, success rates over a
sliding window of 100 dialogues are recorded.
After training all policies, we ran evaluations of the fully trained policies
over 3000 simulated dialogues each, at 25% semantic error rate. The first
three rows in Table 2 correspond to the systems with the extended action set
(the target scenario) that were also shown in Figure 2. These systems all
score similar average rewards, although the adapted system (mdim-ada) gets a
slightly higher success rate, but with slightly longer dialogues. Since the
system with the original action set (the source scenario) covers the same
range of full system dialogue acts (through its summary actions and heuristics
for mapping them to complete dialogue acts), it has the same functionality as
the extended system, and can therefore be included in the evaluation for
comparison. As this system (mdim-src) achieves similar scores, automatically
learning request actions for each slot did not result in improved performance
levels in this particular scenario. Whether improvements can be achieved in
this way, however, depends on the nature of the domain ontology and database,
and on the behaviour of the users.
System | ASet | ADA | NLU | Succ | Len | Rew
---|---|---|---|---|---|---
one-dim | ext | no | sim | 96.0 | 9.28 | 86.06
multi-dim | ext | no | sim | 96.7 | 9.49 | 85.49
mdim-ada | ext | yes | sim | 98.3 | 9.71 | 87.92
mdim-src | sum | no | sim | 97.2 | 9.57 | 87.15
one-dim-asu | ext | no | asu | 99.5 | 7.69 | 91.83
mdim-ada-asu | ext | yes | asu | 88.9 | 7.51 | 81.41
Table 2: Evaluation of fully trained policies over 3000 simulated dialogues. ASet: Action Set (summary/extended), ADA: adapted (yes/no), NLU: Natural Language Understanding (asu/simulated), Succ: Success Rate, Len: Average Length, Rew: Average Reward System | Success rate | Average length | Average reward | Num. dial’s
---|---|---|---|---
one-dim | 96.0 | 96.0 | 99.5 | 9.28 | 9.28 | 7.69 | 86.06 | 86.06 | 91.83 |
mdim-ada | 98.3 | 98.3 | 88.9 | 9.71 | 9.71 | 7.51 | 87.92 | 87.92 | 81.41 | 100k
one-dim | 59.1 | 61.7 | 64.9 | 8.02 | 11.64 | 10.15 | 49.05 | 45.78 | 52.13 | 18k
mdim-ada | 73.5 | 73.8 | 73.3 | 8.95 | 10.01 | 7.71 | 62.18 | 60.77 | 64.25 |
one-dim | 48.9 | 57.5 | 60.3 | 7.55 | 11.98 | 10.63 | 37.31 | 38.05 | 45.82 | 17k
mdim-ada | 77.1 | 79.1 | 78.5 | 9.49 | 10.70 | 7.75 | 65.55 | 64.99 | 68.72 |
one-dim | 45.3 | 48.5 | 54.1 | 7.30 | 11.60 | 10.16 | 34.07 | 30.04 | 39.77 | 16k
mdim-ada | 83.5 | 83.5 | 83.2 | 9.32 | 10.36 | 8.13 | 73.04 | 71.18 | 74.21 |
one-dim | 30.6 | 33.4 | 37.5 | 5.61 | 11.94 | 10.70 | 19.86 | 13.95 | 21.48 | 15k
mdim-ada | 64.5 | 64.3 | 62.3 | 9.26 | 10.14 | 8.08 | 52.67 | 50.54 | 52.44 |
| no-expl | expl | expl+asu | no-expl | expl | expl+asu | no-expl | expl | expl+asu |
Table 3: Evaluation of partially trained policies over 3000 simulated
dialogues, reporting success rate, average dialogue length, and average reward
in three settings.
The results in the first 4 rows have been obtained using a rule-based state
tracker that takes (simulated) user dialogue act hypotheses as input, as is
the case during policy optimisation. For the human user evaluation, we use the
Action State Update (ASU) dialogue state tracker [16] that takes user
utterances as input (see also Section 4.1). We have therefore also evaluated
the one-dim and mdim-ada policies with this tracker in the loop, and included
a hybrid retrieval/template based natural language generation component that
maps simulated user dialogue acts to natural language utterances that can be
fed to the ASU component111Training the policies together with the ASU tracker
is currently not practical, because its BERT model is too slow for running the
required volume of training dialogues.. For the one-dimensional system, this
results in higher scores (one-dim-asu), whereas for the adapted multi-
dimensional system, we get considerably lower scores (mdim-ada-asu). In the
latter case, we note however that this is mainly due to one of the 10 policy
combinations performing very badly: the policy repeatedly responds with
negative feedback, after which the simulated user loses its patience and hangs
up, resulting in 0% success rate and -5.02 average reward. In contrast, the
other policies all achieve 98% success rate.
In order to decide which partially trained policies to include in the human
user evaluation, we have run simulated evaluations for the policies that were
obtained after 15k, 16k, 17k, and 18k training dialogues. This pre-selection
is based on the learning curves in Figure 2 and in particular the area where
the success rate of the one-dimensional system is starting to build up, but is
still strongly outperformed by the adapted multi-dimensional system. The
results in Table 3 report success rates, average dialogue lengths (in terms of
user turns), and average rewards over 3000 simulated dialogues at 25% error
rate. Each evaluation is carried out in three different settings, indicated in
the bottom row of the table: 1) using the rule-based state tracker and no
policy exploration, i.e. the policies always select the action with the
highest estimated value (no-expl); 2) using the rule based state tracker and
policy exploration at the level determined by the temperature setting at the
corresponding stage of training (expl); and 3) using the ASU state tracker and
policy exploration (expl+asu).
At a relatively early training stage, the policy is still exploring the state-
action space, and has not experienced many successful dialogues yet.
Therefore, the reported performance levels can differ between the no-expl and
expl settings. Overall, the results in Table 3 show lower average rewards in
the expl setting, but higher success rates. Furthermore, the performance
levels are much higher when evaluating with the ASU tracker in the loop. The
likely reason for this is that no semantic error model was used in this
setting, i.e., the true user dialogue acts from the simulator were used to
generate the input user utterances for the tracker. Hence, the language
understanding performance is probably much better in this setting, and
therefore the action selection performance as well.
Based on these results, we selected the one- and multi-dimensional models that
were trained on 17k dialogues for the human evaluation, as well as the fully-
trained multi-dimensional system. The three selected variants are highlighted
in grey in Table 3 and also pointed out in Figure 2.
System | Num. | Average | Q1 [%] | Q2 [%] | Q3 [1-6] | Q4 [1-6] | Q5 [1-6] | Q6 [1-6]
---|---|---|---|---|---|---|---|---
| Dialogues | Length | Found Venue | DialSuccess | Understand | Recognise | SysResponse | Naturalness
one-dim-17k | 203 | 7.69 (4.34) | 65.52 (3.34) | 63.05 (3.40) | 3.34 (1.68) | 3.46 (1.66) | 3.21 (1.72) | 3.01 (1.77)
mdim-ada-17k | 201 | 6.52 (3.24) | 73.00 (3.15) | 70.00 (3.25) | 3.43 (1.58) | 3.56 (1.55) | 3.48 (1.59) | 3.35 (1.67)
mdim-ada-100k | 199 | 5.88 (2.38) | 78.89 (2.90) | 79.90 (2.85) | 3.76 (1.56) | 3.86 (1.50) | 3.93 (1.46) | 3.71 (1.62)
Table 4: Human user evaluation results, where N is the number of dialogues,
AvgLen is the average number of turns per dialogue, and Q1-6 are the scores
obtained from the questionnaire. The scores for Q1-2 are percentages, and the
standard deviation for each score is indicated in brackets.
## 4 Human User Evaluation
Using the agenda-based user simulator and error model, we have shown that
significant performance gains with limited training data can be achieved with
the proposed adaptation method. However, it remains to be seen to what extent
these results are representative for the scenario of real users interacting
with the system [17, 18]. Ideally, the policy optimisation experiments should
be carried out in online interaction with human users, which has been
attempted with some success [19, 20]. However, this requires moving to more
sample efficient optimisation methods and addressing many other technical
challenges, which we will leave for future work. Instead, we have run a human
user evaluation in which we compared three system variants, and in particular
two sets of partially trained policies, corresponding to the adapted multi-
dimensional system and the non-adapted one-dimensional baseline, both trained
on 17k dialogues only.
Users were recruited through the Amazon Mechanical Turk (AMT) crowd-sourcing
platform, where they were given a task description (e.g., “You are looking for
a moderately priced French restaurant. Make sure you get the phone number and
address.”) and a link to the web-based interface that we have created. To
enable spoken dialogue on the web interface, we use the Google Web Speech API
for both ASR (for recognising the user’s speech) and TTS (for synthesizing the
system’s speech); our dialogue system server receives recognised user
utterances and responds with system utterances, both in text form.
After finishing their conversation with the system, the user can ‘hang up’ by
pressing a button and receive a token which they can use to proceed on the AMT
page, where they are given a questionnaire to fill out and submit, upon which
they complete the task. In the questionnaire, the subject is asked to state
their opinion on 6 statements about the conversation, in the form of either a
binary Yes or No (Q1 and Q2), or on a 6 point Likert scale (Q3 to Q6), ranging
from ‘Strongly disagree’ to ‘Strongly agree’.
* Q1:
The system recommended a restaurant that matched my constraints. (Yes/No)
* Q2:
I got all the information I was looking for. (Yes/No)
* Q3:
The system understood what I was saying. (6 point Likert)
* Q4:
The system recognised my speech well. (6 point Likert)
* Q5:
The system’s responses were appropriate. (6 point Likert)
* Q6:
The conversation felt natural. (6 point Likert)
### 4.1 Spoken dialogue system implementation
Our dialogue system takes as input the user utterance text, in the form of the
ASR top hypothesis obtained from the web interface. A domain-independent
dialogue act tagger is used to recognise social acts like goodbye and thanks,
after which the utterance is passed to the Action State Update model for
dialogue state tracking [16]. The dialogue state contains beliefs about the
user goal (in terms of slot-value pairs and their confidence score), which
slots are believed to be requested, which database items are or have been
discussed, whether any processing problems have occurred (i.e. speech
recognition or language understanding failed), and the dialogue act tags of
previous user utterances. Based on the dialogue state, one or more system
dialogue acts are selected using one of the trained action selection models;
the three models compared in the human evaluation are highlighted in Table 3.
A set of templates is used to generate a natural language system response from
the selected dialogue acts. The response is synthesised on the web interface.
In order to obtain results that are representative for each system variant,
all 10 policies that were trained for each variant have been evaluated and the
average results reported. For each dialogue, one policy is selected from the
pool of 10 using a round-robin system.
### 4.2 Results
The evaluation results are shown in Table 4, including the (objective) average
dialogue length and the (subjective) scores from the questionnaire. As
expected, the fully trained system mdim-ada-100k gets the best scores. More
importantly, across all metrics the partially trained mdim-ada-17k gets better
scores than the partially trained baseline one-dim-17k. Especially in terms of
average dialogue length and the perceived partial and full task completion
rates (Q1 resp. Q2) the difference between one-dim-17k and mdim-ada-17k is
substantial, as was predicted by the experiments with the user simulator. The
scores for perceived understanding (Q3) and speech recognition (Q4) could
somewhat explain these differences (rather than attributing them to the used
policies), but their variance is quite large. When assessing how appropriate
the system responses appear to users (Q5), the difference between one-
dimensional and multi-dimensional adaptation is larger.
Compared to the simulated evaluation results in Table 3, the perceived success
rates (Q2) are much lower in the case of the fully trained system (79.9% vs.
88.9%) and the multi-dimensional partially trained system (70% vs. 78.5%),
whereas they are slightly higher in the case of the partially trained one-
dimensional system (63.05% vs. 60.03%).
In terms of average dialogue length, the human evaluation dialogues turn out
to be shorter overall. This suggests that the human subjects might have given
up more quickly when the system failed to recommend a restaurant that met
their constraints, i.e. the unsuccessful dialogues were shorter. In contrast,
the simulator keeps trying to complete the goal until the maximum number of
user turns is reached (set to 30 turns in the configuration) or the system
response act is repeated too many times (set to 3 times in the configuration).
## 5 Related work
In the area of transfer learning methods for dialogue management, most
approaches have focused on cross-domain adaptation or multi-domain
optimisation for slot-filling dialogue, where each domain is defined in terms
of slots and their possible values.
In [3], Domain-Independent Parameterisation (DIP) of dialogue state and action
representations is introduced to enable transfer across domains. DIP seeks to
train a dialogue policy that abstracts away from the specific slots and values
in a particular domain, so it is applicable to the conversational search task
in other domains as well. The effectiveness of their method was demonstrated
using an agenda-based user simulator when adapting from restaurant search to
laptop search. Furthermore, they carried out a human user evaluation, showing
that their best transferred DIP policy performed at the same level as a non-
transferred in-domain policy.
A Bayesian approach to dialogue management is described in [5], which uses
Gaussian Process (GP) reinforcement learning, exploiting model priors of a
generic dialogue policy for fast domain adaptation. They also discuss a
Bayesian committee machine approach, in which each domain is handled by a
separate GP policy, but when only limited data is available for a specific
domain, its policy may rely on the output of the other policies. This approach
has been further extended into a multi-agent learning setting, further
improving performance levels during training with human users.
The Multi-Agent Dialogue Policy (MADP) is proposed in [4], which consists of
several slot-specific agents and a slot-independent agent. Adopting the Deep
Q-Network (DQN) reinforcement learning framework [21], the parameters of the
slot-independent agent and the shared parameters of the slot-specific agents
that have been learned for the source domain can be transferred to a new
target domain. So in this case, the multi-agent design is based on the slots
in the domain definition. The benefit of this method has only been
demonstrated in simulated experiments, and is focused on adapting to new slots
within the same restaurant/tourist information task. A more recent incarnation
of this approach is the AgentGraph framework [22], which employs Graph Neural
Networks. This method was evaluated using the PyDial benchmark, demonstrating
successful transfer between restaurant search and laptop shopping tasks.
Where the agents in [5] are associated with domains, and the agents in [4, 22]
are associated with slots (except for the generic slot-independent agent), the
agents in the multi-dimensional approach to dialogue management are associated
with dimensions [23]. Although this approach is very different in nature, it
is not necessarily incompatible with these other multi-agent approaches. For
example, it could benefit from the MADP approach by dividing up the Auto-
feedback agent into sub-agents for each slot in the domain definition.
The multi-dimensional design presented in this paper is an extension of the
previous models described in [9, 7]. First, the system supports the generation
of multiple dialogue acts in a single response; second, the Evaluation agent
that was extended for this purpose is trained jointly with the other agents.
Furthermore, the adaptation use-case in previous work was limited to domain
adaptation only, where the slots and values changed between source and target
scenario, but not the action sets. Finally, the human evaluation described in
[7] included fully trained policies only, showing that a multi-dimensional
system could be trained to a performance level equivalent to a one-dimensional
baseline. In this paper, we have presented a human user evaluation to confirm
the performance boost of partially adapted policies seen on simulated data
when using the multi-dimensional adaptation method.
## 6 Conclusion
We have presented a multi-dimensional approach to dialogue management, in
which system response actions are selected through a combination of 4 dialogue
policies, 2 of which are task-independent, and are therefore suitable for re-
use when moving to a new application. In simulated policy optimisation and
evaluation experiments we have shown that by re-using and adapting these task-
independent policies, significant performance gains can be achieved in the
early stages of training. As a first task adaptation use-case, we have looked
at extending the task-specific action set with multiple slot-specific request
actions, replacing the original summary request action that relied on
heuristics to determine the slot to be requested. To confirm the adaptation
results obtained with an agenda-based user simulator, we have carried out a
crowd-sourced human user evaluation. When trained on the same limited amount
of training data, the proposed multi-dimensional adaptation strategy achieved
significantly better results than the one-dimensional baseline (trained from
scratch without any adaptation) on all subjective metrics, including task
success and appropriateness of system responses, as well as on dialogue
length.
In future work, we will consider more challenging use cases for applying the
multi-dimensional adaptation method. We will extend the role of the social
agent to improve the naturalness of the interactions, explore methods for
combining different types of agents (e.g., rule-based agents, or agents
trained with supervised learning), and investigate the relation between multi-
dimensional dialogue management and natural language generation. Finally, to
enable online learning with human users and therefore evaluate the proposed
adaptation method more directly, we will explore more efficient reinforcement
learning algorithms.
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|
# Anisotropic Magnetized Asteroseismic Waves
B. Tripathi Department of Physics, University of Wisconsin–Madison, Madison,
Wisconsin 53706, USA<EMAIL_ADDRESS>Dhrubaditya Mitra Nordita, KTH Royal
Institute of Technology and Stockholm University, Hannes Alfvéns väg 12, 114
19 Stockholm, Sweden<EMAIL_ADDRESS>
###### Abstract
We solve for waves in a polytropic, stratified medium with a spatially varying
background magnetic field that points along a horizontal $x$-direction, and
with gravity that is directed along the vertical $z$-direction. Force balance
determines the magnitude of the background magnetic field, $B_{\rm 0}^{2}\sim
z^{n+1}$, where $n$ is the polytropic index. Using numerical and asymptotic
methods, we deduce an accurate and explicit dispersion relation for fast
pressure-driven waves: $\Omega^{2}\sim
K\left(2m+n\right)\left[1+(1/\mbox{M}_{\rm
A})^{2}(4-2\gamma+\cos^{2}\theta-3\cos^{4}\theta)/4\right]$. Here, $\Omega$ is
the frequency, $K$ the wavenumber, $\theta$ the angle the wave-vector makes
with the background magnetic field, $\mbox{M}_{\rm A}$ the Alfvénic Mach
number, and $m$ an integer representing the eigen state. Applications of our
result are in magnetoseismology and nonlinear asteroseismology.
††journal: ApJ
## 1 Introduction
The strengths of magnetic fields buried below the surface of stars are not
known, though they are vital for improved understanding of the magnetic
behaviour of stars. This challenge has impeded progress in the development of
stellar magnetism and the evolution of magnetized stellar interiors. To
estimate the magnetic field strengths, linear asteroseismology has stood as a
promising candidate (Aerts et al., 2010), although nonlinear asteroseismology
may also be able to provide insights. It is undisputed that the
asteroseismology, both linear and nonlinear, benefits from reliable and simple
expressions of anisotropic magnetic effect on linear waves. Specifically, the
development of nonlinear asteroseismology (Guo, 2020; Van Beeck et al., 2021,
2023) requires linear dispersion relations to evaluate mode resonances; the
linear asteroseismology directly employs dispersion relations to solve the
inverse problem of detecting subsurface magnetic fields using observed stellar
surface oscillations. Such magnetoseismology has witnessed fitful progress.
For example, observational studies report travel-time perturbations of
acoustic waves to be a critical signature of strong magnetic fields in the
stellar interior (Schunker et al., 2005; Ilonidis et al., 2011). Numerical
simulations of asteroseismic waves also suggest the possibility of detecting
deep-seated fields, before they emerge on the surface (Singh et al., 2014,
2015, 2016, 2020; Das et al., 2020; Das, 2022). To bolster such suggestive
findings, a thorough understanding of the impact of magnetic fields on
asteroseismic waves is essential (Nye & Thomas, 1976; Adam, 1977; Thomas,
1983; Campos, 1983; Cally, 2007; Campos & Marta, 2015; Tripathi & Mitra,
2022).
Waves in an unmagnetized polytropic atmosphere were exactly solved
analytically by Lamb (1911) who derived the relation
$\frac{\Omega^{2}}{2K}-\frac{(n+1)(n+1-\gamma
n)K}{2\gamma^{2}\Omega^{2}}=m+\frac{n}{2},$ (1)
where $\Omega$ is the frequency, $K$ the wavenumber, $n$ the polytropic index,
$\gamma$ the adiabatic index, and $m$ the eigen-state index, with
$m=0,1,2,...\,$. This advancement led to a series of newer and significant
understanding of hydrodynamic waves. Under fast-wave approximation,
$\Omega^{2}/K\gg 1$, the leading-order dispersion relation becomes
$\Omega^{2}\sim K\left(2m+n\right)\/.$ (2)
A similar closed-form expression for waves in a magnetized polytrope, as of
yet, is unknown. Despite several progresses (e.g., Gough & Thompson, 1990;
Bogdan & Cally, 1997; Cally & Bogdan, 1997), the problem, to this day, remains
unsolved. Gough & Thompson (1990) treat a global problem (in spherical
coordinates); there, the computation of eigenfrequenices requires numerical
evaluation of complicated integral expressions [e.g., Eqs. (4.11)–(4.13) in
Gough & Thompson (1990)]. A closed-form analytical formula is not available.
The limitations of purely numerical approach and lack of a closed-form
expression were succinctly expressed by Bogdan & Cally (1997):
> “Ideally, we would wish to proceed by writing down an equation analogous to
> Lamb’s formula for the magnetized polytrope. Unfortunately, this approach is
> not feasible and for the most part one must instead be content with a
> numerically derived visual comparison of how the allowed oscillation
> frequencies depend upon the choice of the horizontal wavenumber $k$.”
Analytic dispersion relations are also essential for developing wave
turbulence theory in the presence of both gravity and magnetic fields. In wave
turbulence theory, calculations of mode resonances require simple, explicit
dispersion relations that accurately capture the magnetic effect on observed
linear waves. We note that the Lamb’s dispersion relation (2),
$\Omega\sim\sqrt{K}$, is similar to that of surface gravity waves in oceans
(Hasselmann, 1962). However, there is a critical difference: the surface
gravity waves do not couple via three-wave resonance, thus requiring a weaker
four-wave coupling (Nazarenko & Lukaschuk, 2016). The Lamb waves, on the other
hand, can couple via three-wave resonance, because there are infinitely many
such waves (eigen states) at a given wave number, unlike only one pair of
surface gravity waves at a given wave number. Thus, the infinitely many Lamb
waves at a given wavenumber have distinct wave frequencies, which allow the
sum of three frequencies at three wavenumbers to become null. While a wave
turbulence theory for surface gravity waves has been well established, it is
yet to be developed for the asteroseismic waves, whose dispersion relation in
fully analytic form is a basic requirement for such a theory. The value that
an analytic dispersion relation offers in resonant-coupling theory cannot be
overstated when magnetic fields make the wave dispersion relation anisotropic
and complicated. Motivated by these reasons, we seek here an accurate and
simple formulae for the effect of magnetic fields on the Lamb waves.
Introducing magnetic fields, aligned orthogonal to a vertical gravity (Fig.
1), we find, as did Bogdan & Cally (1997), that the linearized
magnetohydrodynamic (MHD) equations are too cumbersome to obtain a closed-form
expression for the dispersion relation, even with the fast-wave approximation,
$\Omega^{2}/K\gg 1$. Here, we overcome this difficulty by using both numerical
simulations and extensive use of Mathematica, followed by a variant of
Jeffereys–Wentzel–Kramers–Brillouin (JWKB) approximation to deduce
$\Omega^{2}\sim
K\left(2m+n\right)\left[1+\frac{{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}(4-2\gamma+\cos^{2}\theta-3\cos^{4}\theta)}{4}\right],$
(3)
in the limit $\Omega^{2}/K\gg 1$, with $m=0,1,2,...\,$. The parameter
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ is the inverse of the Alfvénic Mach
number and $\theta$ is the angle the wave vector makes with the background
magnetic field. Equation (3) is the principal result of this paper.
This paper is organized in the following way. In § 2, we describe our model
and present the linearized compressible MHD equations. Such equations are then
numerically solved in § 3. To obtain analytical understanding of the numerical
results, the linearized equations are reduced to a wave equation in § 4. The
normal-form wave equation is then perturbatively solved using a variant of the
JWKB theory we devise; analytical understanding is gained in § 5. With
astrophysical implications and utility of our results, we conclude in § 6.
## 2 System setup and linearized perturbation equations
Figure 1: An inhomogeneous magnetic field $\bm{B}_{0}(z)$, oriented orthogonal
to a constant vertical gravity $\bm{g}$, is considered where the wave is
allowed to propagate in an arbitrary direction, shown with $\bm{K}$, making an
angle $\theta$ with $\bm{B}_{0}$. The gradient in the colormap of the box
schematically represents increasing functions with depth $z$ of the magnetic
field, fluid density, pressure, and sound speed.
To study waves in a magnetized, stratified medium, we consider the ideal
compressible MHD equations (Chandrasekhar, 1961)
$\displaystyle\partial_{t}\rho+{\bm{\nabla}}\cdot\left(\rho{\bm{U}}\right)$
$\displaystyle=0\/,$ (4a)
$\displaystyle\rho\left[\partial_{t}{\bm{U}}+({\bm{U}}\cdot\bm{\nabla}){\bm{U}}\right]$
$\displaystyle=-\bm{\nabla}P+\rho\bm{g}+\bm{J}\times\bm{B}\/,$ (4b)
$\displaystyle\partial_{t}\bm{B}$
$\displaystyle={\bm{\nabla}}\times\left({\bm{U}}\times\bm{B}\right),$ (4c)
$\displaystyle\frac{Dp}{Dt}$ $\displaystyle=c^{2}\frac{D\rho}{Dt},$ (4d)
where $\rho$, ${\bm{U}}$, $P$, and $\bm{B}$ represent the density, the
velocity, the pressure, and the magnetic field, respectively. The current
density is ${\bm{J}}={\bm{\nabla}}\times\bm{B}/\mu_{{\rm 0}}$, where
$\mu_{{\rm 0}}$ represents the magnetic permeability of the vacuum. The
magnetic field is additionally constrained to be divergence-less
${\bm{\nabla}}\cdot\bm{B}=0.$ (5)
We consider a local Cartesian domain where the equilibrium density $\rho_{\rm
0}$ and pressure $P_{\rm 0}$ satisfy the polytropic relation
$P_{\rm 0}\sim\rho_{\rm 0}^{1+1/n},$ (6)
where $n$ is the polytropic index of the gaseous atmosphere. In an
unmagnetized atmosphere, the force balance gives $\partial_{z}P_{\rm
0}(z)=\rho_{\rm 0}(z)g$, where $\bm{g}=g\hat{\bm{e}}_{z}$, with $g$ as the
constant acceleration due to gravity along the vertical $z$-axis, as depicted
in Fig. 1. The force balance requires
$\rho_{\rm 0}(z)\sim z^{n};\quad P_{\rm 0}(z)\sim z^{n+1}.$ (7)
### 2.1 Background magnetic field and sound speed
We introduce an inhomogeneous background magnetic field, $\bm{B}=B_{\rm
0}(z)\hat{\bm{e}}_{x}$. The force-balance relation with the Lorentz force,
$({\bm{\nabla}}\times\bm{B}/\mu_{{\rm 0}})\times\bm{B}$, then becomes
$\partial_{z}P_{\rm 0}=\rho_{\rm 0}g-\frac{1}{2\mu_{{\rm
0}}}\partial_{z}B_{\rm 0}^{2}.$ (8)
The solution
$B_{0}^{2}(z)\sim z^{n+1}\/$ (9)
also satisfies Eq. (7). The plasma $\beta$ is then a constant throughout the
domain
$\beta\equiv\frac{P_{\mathrm{gas}}}{P_{\mathrm{magnetic}}}=\frac{2\mu_{{\rm
0}}P_{\rm 0}}{B_{\rm 0}^{2}}.$ (10)
The sound speed ($c$) may now be deduced from Eq. (8), using $P_{\rm
0}=c^{2}\rho_{\rm 0}/\gamma$, where $\gamma$ is the adiabatic index of the gas
$\partial_{z}c^{2}=-\frac{nc^{2}}{z}+\frac{\gamma g}{1+\beta^{-1}}.$ (11)
The solution to Eq. (11) is $c^{2}\mathrm{=}c_{\rm 0}^{2}z$, where $c_{\rm
0}^{2}$ is a constant given by
$c_{\rm 0}^{2}=\frac{\gamma g}{(n+1)(1+\beta^{-1})}\/.$ (12)
It is useful to express $\beta^{-1}$ in terms of the Alfvénic Mach number
$\mbox{M}_{\rm A}$, which is the ratio of the Alfvén speed to the sound speed,
$\beta^{-1}=\frac{B_{\rm 0}^{2}}{\mu_{0}\rho_{0}}\cdot\frac{\rho_{0}}{\gamma
P_{\rm 0}}\cdot\frac{\gamma}{2}=\frac{\gamma}{2\mbox{M}_{\rm A}^{2}}\/,$ (13)
using which the sound speed becomes
$c^{2}=\frac{\gamma gz}{(n+1)(1+\gamma\mbox{M}_{\rm A}^{-2}/2)}.$ (14)
### 2.2 Linearized perturbation equations
We now linearize the MHD equations around the background profiles $[\rho_{\rm
0},\bm{0},B_{\rm 0},P_{\rm 0}]$, introduced in Eqs. (7) and (9). Such a
linearization yields evolution equations for infinitismal perturbations
$[\widetilde{\rho},\widetilde{{\bm{u}}},\widetilde{{\bm{b}}},\widetilde{p}]$
as
$\displaystyle\partial_{t}\widetilde{\rho}$
$\displaystyle=-(\widetilde{\bm{u}}\cdot\bm{\nabla})\rho_{\rm 0}-\rho_{\rm
0}(\bm{\nabla}\cdot\widetilde{\bm{u}})\/,$ (15a)
$\displaystyle\partial_{t}\widetilde{u}_{x}$
$\displaystyle=-\frac{\partial_{x}\widetilde{p}}{\rho_{0}}+\frac{\widetilde{b}_{z}\partial_{z}B_{\rm
0}}{\mu_{{\rm 0}}\rho_{\rm 0}}\/,$ (15b)
$\displaystyle\partial_{t}\widetilde{u}_{y}$
$\displaystyle=-\frac{\partial_{y}\widetilde{p}}{\rho_{0}}+\frac{B_{\rm
0}(\partial_{x}\widetilde{b}_{y}-\partial_{y}\widetilde{b}_{x})}{\mu_{{\rm
0}}\rho_{\rm 0}}\/,$ (15c) $\displaystyle\partial_{t}\widetilde{u}_{z}$
$\displaystyle=-\frac{\partial_{z}\widetilde{p}}{\rho_{\rm
0}}+\frac{\widetilde{\rho}g}{\rho_{\rm 0}}+\frac{B_{\rm
0}(\partial_{x}\widetilde{b}_{z}-\partial_{z}\widetilde{b}_{x})-\widetilde{b}_{x}\partial_{z}B_{\rm
0}}{\mu_{{\rm 0}}\rho_{\rm 0}}\/,$ (15d)
$\displaystyle\partial_{t}\widetilde{b}_{x}$ $\displaystyle=-B_{\rm
0}(\partial_{y}\widetilde{u}_{y}+\partial_{z}\widetilde{u}_{z})-\widetilde{u}_{z}\partial_{z}B_{\rm
0}\/,$ (15e) $\displaystyle\partial_{t}\widetilde{b}_{y}$
$\displaystyle=-B_{\rm 0}\partial_{x}\widetilde{u}_{y}\/,$ (15f)
$\displaystyle\partial_{t}\widetilde{b}_{z}$ $\displaystyle=-B_{\rm
0}\partial_{x}\widetilde{u}_{z}\/,$ (15g)
$\displaystyle\partial_{t}\widetilde{p}$
$\displaystyle=-(\widetilde{\bm{u}}\cdot\bm{\nabla})P_{\rm
0}-c^{2}\rho_{0}(\bm{\nabla}\cdot\widetilde{\bm{u}})\/.$ (15h)
We shall use
$\widetilde{\chi}=\bm{\nabla}\cdot\widetilde{{\bm{u}}}$ (16)
in the rest of the article, where helpful.
Equations (15a)–(15h) can be simplified to derive a set of fewer (closed)
equations
$\displaystyle\partial_{t}^{2}\widetilde{u}_{x}$
$\displaystyle=c^{2}\partial_{x}\widetilde{\chi}+g\partial_{x}\widetilde{u}_{z}\/,$
(17a) $\displaystyle\partial_{t}^{2}\widetilde{u}_{y}$
$\displaystyle=c^{2}\partial_{y}\widetilde{\chi}+g\partial_{y}\widetilde{u}_{z}+\frac{c^{2}}{\mbox{M}_{\rm
A}^{2}}\left(\partial_{xx}\widetilde{u}_{y}+\partial_{y}\widetilde{\chi}-\partial_{xy}\widetilde{u}_{x}\right)\/,$
(17b) $\displaystyle\partial_{t}^{2}\widetilde{u}_{z}$
$\displaystyle=c^{2}\partial_{z}\widetilde{\chi}-g\partial_{x}\widetilde{u}_{x}\left[1+\frac{\gamma}{\mbox{M}_{\rm
A}^{2}(1+\gamma\mbox{M}_{\rm
A}^{-2}/2)}\right]-g\partial_{y}\widetilde{u}_{y}$ $\displaystyle\hskip
5.69046pt+\frac{\gamma(1+\mbox{M}_{\rm
A}^{-2})g\widetilde{\chi}}{(1+\gamma\mbox{M}_{\rm
A}^{-2}/2)}+\frac{c^{2}}{\mbox{M}_{\rm
A}^{2}}\left(\partial_{xx}\widetilde{u}_{z}+\partial_{z}\widetilde{\chi}-\partial_{xz}\widetilde{u}_{x}\right)\/.$
(17c)
We note that the appearance of $\mbox{M}_{\rm A}^{2}$ in Eq. (17) in certain
terms may seem non-trivial at first sight; however, upon inspection, we
understand them as terms emerging from the effect of the Lorentz force on the
background states, via terms like $\partial_{z}c^{2}(z)$ and
$(\partial_{z}P_{\rm 0})/\rho_{n}$, while processing Eqs. (15a)–(15h).
Admittedly, Eqs. (17a)–(17) are somewhat challenging to proceed with clarity.
Thus, we now non-dimensionalize the equations to make them reasonably
transparent.
### 2.3 Non-dimensionalized linear equations
We define $L$ as the characteristic length scale over which the sound speed
varies (and, for that matter, pressure, density, and temperature also vary)
appreciably. Then we find that the characteristic sound speed in an
unmagnetized polytrope is, using Eq. (14), $c_{L}=\sqrt{\gamma gL/(n+1)}$.
Using $L$ and $L/c_{L}$ as the dimensional length and time units, we non-
dimensionalize all variables henceforth, starting with the sound speed
$C^{2}=\frac{c^{2}}{c_{L}^{2}}=\frac{Z}{1+\gamma\mbox{M}_{\rm A}^{-2}/2},$
(18)
where the lowercase dimensional variables ($c$ and $z$) are cast as uppercase
non-dimensional variables ($C$ and $Z$). Thus we replace all the dimensional
variables in Eqs. (17a)–(17) using
$\displaystyle(x,y,z)$ $\displaystyle=(LX,LY,LZ),$ (19a) $\displaystyle c^{2}$
$\displaystyle=c_{L}^{2}\frac{Z}{1+\gamma\mbox{M}_{\rm A}^{-2}/2},$ (19b)
$\displaystyle\partial_{t}$ $\displaystyle\equiv\frac{c_{L}}{L}\partial_{T},$
(19c)
where the uppercase characters $X,Y,$ and $T$ represent non-dimensional
variables.
We analyze perturbations by Fourier-transforming in the $(x,y)$-plane, viz.,
$\widetilde{u}_{x}(Z)=\int
dK_{X}dK_{Y}d\Omega\,\hat{u}_{x}\exp\left[i(K_{X}X+K_{Y}Y+\Omega T)\right],$
(20)
where the uppercase characters represent non-dimensional quantities, e.g.,
$\bm{K}\equiv(K_{X},K_{Y})$ is the non-dimensional wavevector in the
$(x,y)$-plane, and $\Omega$ is the nondimensional frequency. Fourier analyzing
Eqs. (17a)–(17) and representing $\mbox{M}_{\rm A}^{-1}$ by
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ henceforth, we write
$\displaystyle\left[\frac{-\Omega^{2}(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)}{iK_{X}Z}\right]\hat{u}_{x}+\left[\frac{-(n+1)(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)}{\gamma
Z}\right]\hat{u}_{z}=\hat{\chi}\/,$ (21a)
$\displaystyle\left[-i{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}K_{X}\right]\hat{u}_{x}+\left[\frac{-\Omega^{2}(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)}{iK_{Y}Z}+\frac{{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}K_{X}^{2}}{iK_{Y}}\right]\hat{u}_{y}$
$\displaystyle\hskip
56.9055pt+\left[\frac{-(n+1)(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)}{\gamma
Z}\right]\hat{u}_{z}=\hat{\chi}(1+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2})\/,$
(21b)
$\displaystyle\left[iK_{X}\left\\{\frac{1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2}{\gamma}+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}\left(1+\frac{Z\partial_{Z}}{n+1}\right)\right\\}\right]\hat{u}_{x}$
$\displaystyle+\left[\frac{iK_{Y}(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)}{\gamma}\right]\hat{u}_{y}+\left[\frac{-\Omega^{2}(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}ZK_{X}^{2}}{n+1}\right]\hat{u}_{z}$
$\displaystyle\hskip
99.58464pt=\left(\hat{\chi}+\frac{Z\partial_{Z}\hat{\chi}}{n+1}\right)(1+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2})\/.$
(21c)
## 3 Exact numerical solution
We obtain fully converged numerical solution to Eqs. (21a)–(21) by employing
the spectral code “Dedalus” (Burns et al., 2020). Referring to the Dedalus
methods paper (Burns et al., 2020) for more details, we briefly outline the
numerical procedures employed in Dedalus for eigenvalue problems. At each
horizontal wavenumber $(K_{X},K_{Y})$, the state variables—the three
components of velocity—are expanded in the Chebyshev polynomials along the
inhomogeneous $z$-axis. Because of the background inhomogeneity, different
Chebyshev coefficients couple, creating a dense linear operator.
Sparsification is provided by a change of basis from the Chebyshev polynomials
of the first kind to those of the second kind. To impose boundary conditions
and keep the matrix sparse, Dirichlet preconditioning is applied. Efficient
solution of the resulting matrices is then found by passing the matrices
$\mathcal{L}$ and $\mathcal{M}$ in the eigenvalue ($\sigma$) problem,
$\mathcal{L}\mathcal{X}=\sigma\mathcal{M}\mathcal{X}$, to the “scipy” linear
algebra packages. For a given spectral resolution along the $z$-axis, we solve
for all the eigenvalues of the matrices. Such a non-targeted, general solution
produces eigenmodes of all the linear MHD waves, including the pressure-driven
and gravity-driven modes. We specialize in the high-frequency modes to assess
the effect of magnetic fields on such pressure-driven modes. We identify the
pressure-driven modes by comparing their eigenfrequencies with those predicted
by Lamb’s solution.
Figure 2: $\Omega^{2}(K_{X},K_{Y})$ for hydrodynamic and magnetized
polytropes. The two plotted surfaces are visually indifferentiable because the
difference ($\Delta\Omega^{2}$) between them is much smaller than
$\Omega^{2}$; see Fig. 3. The parameters used are $\epsilon=0.1$, $m=20$,
$n=2.5$, and $\gamma=5/3$. For variations in these parameters, the surface
plot of $\Omega^{2}(K_{X},K_{Y})$ remains the same qualitatively.
For the boundary condition, at the lower boundary $z=L_{z}$, we require
$\widetilde{u}_{z}=0$. At the upper boundary, $z=0$, where the atmosphere
ceases, we enforce, following Lamb (1911),
$\displaystyle\frac{Dp}{Dt}$ $\displaystyle=0,$ (22)
which implies
$c^{2}\frac{D\rho}{Dt}=-c^{2}\rho_{\rm
0}\widetilde{\chi}=0=z^{n+1}\widetilde{\chi}.$ (23)
We first validate that the solver successfully reproduces Lamb’s exact
analytical dispersion relation in the absence of the magnetic field. We then
compute the eigenvalues of the magnetized polytrope, which we present in Fig.
2. Although the two surface plots of $\Omega^{2}(K_{X},K_{Y})$ are displayed
in Fig. 2—one for the hydrodynamic and other for the magnetized polytrope, the
plots are visually indistinguishable. The difference between the two surface
plots is shown in Fig. 3. For $K_{Y}=0$, $\Delta\Omega^{2}$ is negative, and
for $K_{X}=0$, $\Delta\Omega^{2}$ is positive and relatively large. We also
note a minor decrease in positive value of $\Delta\Omega^{2}$ in going from
$K_{X}\approx 0$ to $K_{X}=0$.
It turns out that $\Delta\Omega^{2}$ is related to the hydrodynamic squared-
frequency $\Omega_{\mathrm{hydro}}^{2}$; and
$\Delta\Omega^{2}/\Omega_{\mathrm{hydro}}^{2}$ is almost entirely independent
of the wavevector magnitude (Fig. 4). Only angular dependence is observed.
The extremely low wavenumbers in Fig. 4 correspond to very large scale waves
that cannot be captured in a finite box in numerical calculation. To capture
lower wavenumbers, we significantly extended domain size along the vertical
$z$-axis, which allowed us to obtain fully converged numerical results for
other wavenumbers.
Figure 3: $\Delta\Omega^{2}=\Omega^{2}-\Omega_{\mathrm{hydro}}^{2}$ is plotted
for a magnetized polytrope, with $\epsilon=0.1$, $m=20$, $n=2.5$, and
$\gamma=5/3$. This surface plot shows the difference between the two surface
plots in Fig. 2. Figure 4: Relative difference of squared-frequency depends on
the wavevector propagation angle (spanned by the black circle), but the
relative difference is insensitive to the wavevector magnitude. The parameters
chosen are $\epsilon=0.1$, $m=20$, $n=2.5$, and $\gamma=5/3$.
## 4 Reduction to wave equation
To analytically determine the dispersion relation, we solve the set of
equations (21a)–(21) perturbatively in the limit of a weak magnetic field,
i.e., ${\mathbb{\color[rgb]{0,0,0}\epsilon}}\ll 1$. By setting
${\mathbb{\color[rgb]{0,0,0}\epsilon}}=0$, we recover the Lamb’s equations for
the unmagnetized polytrope (Lamb, 1911). The Lamb’s equations reduce to a
second-order differential equation for $\hat{\chi}$. With the same goal, we
proceed in the following manner. We rewrite Eqs. (21a)–(21) as
$\begin{bmatrix}\begin{array}[]{c c c}M_{11}&M_{12}&M_{13}\\\
M_{21}&M_{22}&M_{23}\\\ M_{31}&M_{32}&M_{33}\\\
\end{array}\end{bmatrix}\begin{bmatrix}\begin{array}[]{c}\hat{u}_{x}\\\
\hat{u}_{y}\\\ \hat{u}_{z}\\\
\end{array}\end{bmatrix}=\begin{bmatrix}\begin{array}[]{c}h_{x}(\hat{\chi})\\\
h_{y}(\hat{\chi})\\\ h_{z}(\hat{\chi},\partial_{Z}\hat{\chi})\\\
\end{array}\end{bmatrix},\\\ $ (24)
where the matrix elements $M_{ij}$ are independent of $Z$-derivatives, and are
functions of
$K_{X},K_{Y},\Omega,n,\gamma,{\mathbb{\color[rgb]{0,0,0}\epsilon}},$ and $Z$
only (for their complete expressions, see Appendix A). In arriving at Eq.
(24), we have replaced $\partial_{Z}\hat{u}_{x}$ in the first term on the
left-hand side of Eq. (21) with its exact expression obtained by
differentiating Eq. (21a) with respect to $Z$. We then substitute
$\partial_{Z}$ operation in $\partial_{Z}\hat{u}_{z}$ by writing it as
$\hat{\chi}-iK_{X}\hat{u}_{x}-iK_{Y}\hat{u}_{y}$. Such a process removes
$\partial_{Z}$ operation from the matrix $M$. The functions $h_{\nu}$ are
linear in $\hat{\chi}$ and $\partial_{Z}\hat{\chi}$.
Straightforward inversion of the matrix $M$ expresses all components of the
velocity in terms of $\hat{\chi}$ and $\partial_{Z}\hat{\chi}$:
$\hat{u}_{\nu}=f_{\nu}(Z)\hat{\chi}+g_{\nu}(Z)\partial_{Z}\hat{\chi},$ (25)
where $\nu$ can be either $x,y,$ or $z$. We recognize that $f_{\nu}$ and
$g_{\nu}$ are functions of $Z$, but do not involve $\partial_{Z}$. The three
velocity components of Eq. (25) can now be subsumed into a single second-order
differential equation for $\hat{\chi}$:
$\partial_{Z}^{2}\hat{\chi}+P(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}})\partial_{Z}\hat{\chi}+R(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}})\hat{\chi}=0,$
(26)
which can be recast into the normal form of the second-order differential
equation by changing variable as
$\hat{\chi}(Z)=\hat{\psi}(Z)\exp{\left[-\frac{1}{2}\int_{Z}dZ\,P(Z)\right]},$
(27)
which reduces Eq. (26) to the wave equation
$\partial_{Z}^{2}\hat{\psi}+\Gamma^{2}(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}})\hat{\psi}=0.$
(28)
The procedure outlined above appears straightforward. However, the analytical
manipulations in arriving at Eq. (28)—a magnetized version of Lamb’s
equation—require laborious and careful calculations, as
$\Gamma^{2}(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}})$ alone conceals an
expression of exhaustive length—tens of pages of this article. In the absence
of the magnetic field, the expression for
$\Gamma^{2}(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}}=0)=\Gamma_{\rm 0}^{2}(Z)$
is beautifully short, $\Gamma_{\rm 0}(Z)=K^{2}(Z-\alpha)(\beta-Z)/Z^{2}$,
where $\alpha$ and $\beta$ are the two turning points—two zeros of
$\Gamma_{\rm 0}(Z)$.
## 5 Perturbative solution for anisotropic magnetic effect
The presence of a weak magnetic field may be considered as a perturbation to
Lamb’s two-turning point eigenvalue problem. Hence, the magnetic field changes
both the locations of the turning points and the form of the potential
$\Gamma(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}})$. In general, we write the
JWKB quantization condition (see e.g., Bender & Orszag, 1978; Tripathi, 2022)
as
$\frac{1}{\pi}\int_{Z_{1}({\mathbb{\color[rgb]{0,0,0}\epsilon}})}^{Z_{2}({\mathbb{\color[rgb]{0,0,0}\epsilon}})}\Gamma(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}})\,dZ\sim\left(m+\frac{1}{2}\right);\,\,\,m=0,1,2,...,$
(29)
where $m$ refers to the eigen-state index,
$\Gamma(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}})$ is the magnetically-modified
wavenumber, and $Z_{1}({\mathbb{\color[rgb]{0,0,0}\epsilon}})$ and
$Z_{2}({\mathbb{\color[rgb]{0,0,0}\epsilon}})$ are the magnetically-shifted
turning points.
### 5.1 Perturbative calculations
First, we expand the wavenumber
$\Gamma(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}})$ as
$\Gamma(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}})=\Gamma_{\rm
0}(Z)+{\mathbb{\color[rgb]{0,0,0}\epsilon}}\Gamma_{1}(Z)+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}\Gamma_{2}(Z)+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3}\Gamma_{3}(Z)+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{4})}.$
(30)
It may be noted that the leading-order effect of the Lorentz force on the
wavenumber appears only at the second order
$\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2})}$ in the expansion.
So, $\Gamma_{1}(Z)$ above is zero. The expressions for $\Gamma_{0}(Z)$ and
$\Gamma_{2}(Z)$ are
$\displaystyle\Gamma_{0}(Z)$
$\displaystyle=\frac{iK\sqrt{(Z-\alpha)(Z-\beta)}}{Z},$ (31a)
$\displaystyle\Gamma_{2}(Z)$
$\displaystyle=\frac{i(b_{2}Z^{2}+b_{1}Z+b_{0})}{\sqrt{(Z-\alpha)(Z-\beta)}},$
(31b)
where we have used $K_{X}=K\cos\theta$ and $K_{Y}=K\sin\theta$. The parameters
$\alpha$ and $\beta$ satisfy the following properties
$\displaystyle\alpha\beta$ $\displaystyle=\frac{n(n+2)}{4K^{2}},$ (32a)
$\displaystyle\alpha+\beta$
$\displaystyle=\left[\frac{\Omega^{2}}{K}-\frac{(n+1)(n+1-n\gamma)}{\gamma^{2}}\frac{K}{\Omega^{2}}\right]\frac{1}{K}.$
(32b)
The lengthy expressions of $b_{0}$, $b_{1}$, and $b_{2}$ are presented in
Appendix B. We note that these parameters depend on $\Omega,K,\theta,n,$ and
$\gamma$ only.
Now we expand the turning points
$Z_{1}({\mathbb{\color[rgb]{0,0,0}\epsilon}})$ and
$Z_{2}({\mathbb{\color[rgb]{0,0,0}\epsilon}})$ around the turning points of
the unmagnetized polytrope, $Z_{1}^{(0)}=\alpha$ and $Z_{2}^{(0)}=\beta$, as
$Z_{j}({\mathbb{\color[rgb]{0,0,0}\epsilon}})=Z_{j}^{(0)}+{\mathbb{\color[rgb]{0,0,0}\epsilon}}Z_{j}^{(1)}+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}Z_{j}^{(2)}+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3})},$
(33)
where $j=1$ and $j=2$ refer to the left and the right turning points,
respectively (i.e., $\alpha<\beta$). We note that, in Eq. (33), the correction
term at the first order in ${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ is zero,
i.e., $Z_{j}^{(1)}=0$. We find this result by substituting the expression for
$Z_{j}({\mathbb{\color[rgb]{0,0,0}\epsilon}})$ from Eq. (33) in
$\Gamma(Z,{\mathbb{\color[rgb]{0,0,0}\epsilon}})=0$, and by employing Eq.
(30). Solving the resulting equation order-by-order in
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ produces $Z_{j}^{(1)}=0$. We note,
however, that, at the second order in ${\mathbb{\color[rgb]{0,0,0}\epsilon}}$
(which is where the effect of the Lorentz force comes in action),
$Z_{j}^{(2)}$ becomes non-zero. The expressions for $Z_{j}^{(2)}$ are given in
Appendix C.
Because the term $Z_{j}^{(2)}$ appears at the second order in
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$, it may be tempting to assume that the
term contributes to second order in ${\mathbb{\color[rgb]{0,0,0}\epsilon}}$
itself in the JWKB integral in Eq. (29). This, however, is not the case. The
term contributes to third and higher orders in
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ as we show next. Expanding Eq. (29),
$\displaystyle\frac{1}{\pi}\int_{Z_{1}^{(0)}+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}Z_{1}^{(2)}+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3})}}^{Z_{2}^{(0)}+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}Z_{2}^{(2)}+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3})}}$
$\displaystyle\left[\Gamma_{0}(Z)+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}\Gamma_{2}(Z)+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3})}\right]\,dZ$
(34) $\displaystyle\sim\left(m+\frac{1}{2}\right);\,\,\,m=0,1,2,...\/.$
We now integrate $\Gamma_{\rm 0}(Z)$ as
$\displaystyle\int_{Z_{1}^{(0)}+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}Z_{1}^{(2)}+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3})}}^{Z_{2}^{(0)}+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}Z_{2}^{(2)}+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3})}}\Gamma_{\rm
0}(Z)\,dZ$ (35)
$\displaystyle=\left(\int_{\alpha+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}Z_{1}^{(2)}+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3})}}^{\alpha}+\int_{\alpha}^{\beta}+\int_{\beta}^{\beta+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}Z_{2}^{(2)}+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3})}}\right)\Gamma_{\rm
0}(Z)\,dZ$
$\displaystyle=-\frac{2{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3}K(\beta-\alpha)^{1/2}\left[Z_{1}^{(2)}\right]^{3/2}}{3\alpha}+\int_{\alpha}^{\beta}\Gamma_{\rm
0}(Z)\,dZ$ $\displaystyle\hskip
56.9055pt+\frac{2{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3}K(\beta-\alpha)^{1/2}\left[-Z_{2}^{(2)}\right]^{3/2}}{3\beta}+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{4})}\/,$
where we notice terms with ${\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3}$ arising
from the second-order shifts in the turning points,
${\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}Z_{1}^{(2)}$ and
${\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}Z_{2}^{(2)}$. The additional power
of ${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ emerges from the integrand
$\Gamma_{\rm 0}(Z)$, which has a term $\sqrt{Z-Z_{j}^{(0)}}$ that when
expanded around $Z_{j}^{(0)}$ in powers of
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ contributes an
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ to the integral.
In Eq. (35) the term $\int_{\alpha}^{\beta}\Gamma_{0}(Z)\,dZ$ is the integral
that one finds in Lamb’s calculations:
$\displaystyle\int_{\alpha}^{\beta}\Gamma_{0}(Z)\,dZ\equiv$ $\displaystyle
I_{\mathrm{Lamb}}=\frac{(n+1)}{2}\left[\frac{\Omega^{2}}{K(n+1)}+\frac{(\gamma
n-n-1)K}{\gamma^{2}\Omega^{2}}-1\right]+1.$
Thus, we replace the integral $\int_{\alpha}^{\beta}\Gamma_{0}(Z)\,dZ$,
appearing in Eq. (35), with $I_{\mathrm{Lamb}}$ from Eq. (5.1) to obtain
$\displaystyle I_{\mathrm{Lamb}}$
$\displaystyle+\frac{{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}}{\pi}\int_{\alpha}^{\beta}\Gamma_{2}(Z)dZ+\mathcal{O}{({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{3})}$
(36) $\displaystyle\hskip
28.45274pt\sim\left(m+\frac{1}{2}\right);\,\,\,m=0,1,2,...,$
which is accurate up to second order in
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$. We substitute $\Gamma_{2}(Z)$ from
Eq. (31b) and perform the integral in Eq. (36) to arrive at
$\displaystyle I_{\mathrm{Lamb}}-{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}$
$\displaystyle\left[b_{0}+b_{1}\left(\frac{\alpha+\beta}{2}\right)+b_{2}\left(\frac{3\alpha^{2}+2\alpha\beta+3\beta^{2}}{8}\right)\right]$
(37) $\displaystyle\hskip
28.45274pt\sim\left(m+\frac{1}{2}\right);\,\,\,m=0,1,2,...\,.$
Employing fast-wave approximation, we expand each term on the left-hand side
of Eq. (37) in powers of $\delta=K/\Omega^{2}$ as
$\displaystyle I_{\mathrm{Lamb}}$
$\displaystyle\approx\frac{1}{\delta}\left[\frac{1}{2}+\frac{\delta(1-n)}{2}+\mathcal{O}(\delta^{2})\right],$
(38a) $\displaystyle b_{0}$
$\displaystyle\approx\frac{1}{\delta}\left[\frac{2-\gamma}{4}+\mathcal{O}(\delta^{2})\right],$
(38b) $\displaystyle b_{1}\left(\frac{\alpha+\beta}{2}\right)$
$\displaystyle\approx\frac{1}{\delta}\left[\frac{\cos^{2}\theta}{4}+\mathcal{O}(\delta^{2})\right],$
(38c) $\displaystyle
b_{2}\left(\frac{3\alpha^{2}+2\alpha\beta+3\beta^{2}}{8}\right)$
$\displaystyle\approx\frac{1}{\delta}\left[\frac{-3\cos^{4}\theta}{8}+\mathcal{O}(\delta^{2})\right].$
(38d)
Thus we obtain a simplified dispersion relation from Eq. (37):
$\displaystyle\frac{\Omega^{2}}{K}\left[\frac{1}{2}-\frac{{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}}{4}\left\\{2-\gamma+\cos^{2}\theta-\frac{3\cos^{4}\theta}{2}\right\\}\right]$
(39) $\displaystyle\hskip
56.9055pt\sim\left(m+\frac{n}{2}\right);\,\,\,m=0,1,2,...\,.$
It turns out that Eq. (39) is not in excellent agreement with our numerical
results, but replacing $\cos^{2}\theta$ with $(\cos^{2}\theta)/2$ gives
excellent agreement.
Informed in this way, we write the final dispersion relation
$\boxed{\begin{aligned}
&\frac{\Omega^{2}}{K}\left[\frac{1}{2}-\frac{{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}}{4}\left\\{2-\gamma+\frac{\cos^{2}\theta}{2}-\frac{3\cos^{4}\theta}{2}\right\\}\right]\\\
&\hskip
56.9055pt\sim\left(m+\frac{n}{2}\right);\,\,\,m=0,1,2,...\,.\end{aligned}}$
(40)
### 5.2 Comparison between theory and numerics
Figure 5: Theory and numerics: Our theory predicts that
$g(\theta)=\Delta\Omega^{2}/({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}\Omega_{\rm
hydro}^{2})+\gamma/2$ depends only on $\theta$ as given in Eq. (41). The
function $g(\theta)$ is independent of the wavenumber $K$, Alfvénic Mach
number $\epsilon^{-1}$, the polytropic index $n$, the adiabatic index
$\gamma$, and the eigen-state index $m$. Here, we plot $g(\theta)$ obtained
from our numerical solutions for several different values of $m$, ranging from
$10$ to $35$, and for several different values of $K$ ($0.6\leq|K|\leq 2$),
and for two values of ${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ and two values
of $\gamma$ (different symbols). All data points collapse on the same
universal curve. The function $g(\theta)$ from our asymptotic theory, Eq.
(41), is also plotted here. The asymptotic curve is indistinguishable from the
exact numerical solutions.
Using the Lamb’s relation, $\Omega_{\rm hydro}^{2}\sim K(2m+n)$, we rewrite
Eq. (40) as
$\displaystyle g(\theta)$
$\displaystyle\equiv\gamma/2+\Delta\Omega^{2}/({\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}\Omega_{\rm
hydro}^{2})$ (41)
$\displaystyle=1+\frac{1}{4}\left(\cos^{2}\theta-3\cos^{4}\theta\right),$
where $\Delta\Omega^{2}\equiv\Omega^{2}-\Omega_{\rm hydro}^{2}$. This is a
remarkable result. The right-hand-side of Eq. (41) is independent of every
other possible parameter other than $\theta$ ! In Fig. 5, we plot the function
$g(\theta)$ from our numerical determination of the eigenfrequencies for
several different values of $m$ and $K$, and two different values of
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$, and two different values of $\gamma$.
These numerical values are shown as different symbols. All the different
curves fall on top of each other, creating a universal curve. We also plot our
asymptotic expression, Eq. (40), which agrees very well with the numerical
results. This demonstrates that our theory is in excellent agreement with
numerics even for ${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ as large as
${\mathbb{\color[rgb]{0,0,0}\epsilon}}=0.1$.
It is surprising that, in our attempt to capture the anisotropy brought in by
the magnetic field, despite a myriad of unwieldy expressions encountered on
the way, an expression as simple as Eq. (40), is obtained. This simple
expression is also highly accurate, as demonstrated in Fig. 5. The success of
our asymptotic theory is somewhat unexpected, given that the analytical
solution is accurate only up to the leading order
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ and $K/\Omega^{2}$.
More accurate solutions can be obtained by using the full expressions of
$b_{0},b_{1},$ and $b_{2}$ from Appendix B in Eqs. (38d)–(38b), and keeping
higher-order terms in ${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ in Eq. (34).
Finally, a remark on the effect of magnetic fields on the slow gravity-driven
waves is in order. For the slow waves, the first term on the left-hand side of
Eq. (1) may be neglected, which implies that such waves in unmagnetized medium
become unstable when $\gamma<(1+1/n)$. This is consistent with the energy
principle of Newcomb (1961). When a magnetic field is present, by applying the
energy principle of Newcomb, we find, for $K_{X}\neq 0$,111For $K_{X}=0$, the
criterion for the gravity waves to become unstable is slightly modified:
$\gamma<(1+1/n)(1+1/\beta)-2/\beta$. the instability threshold on $\gamma$ for
the gravity waves is lifted to $\gamma<(1+1/n)(1+1/\beta)$. When the magnetic
field is very strong ($\beta\ll 1$), the regular perturbation series in powers
of ${\mathbb{\color[rgb]{0,0,0}\epsilon}}\,(\propto 1/\sqrt{\beta})$ is
possibly inadequate for unstable gravity-driven waves. Such considerations are
clearly beyond the scope of the present paper.
## 6 Conclusions
Here we derive, for pressure-driven waves, an accurate and simple analytical
formula that captures the effects of magnetic field and five other parameters:
adiabatic index, polytropic index, eigenmode state index, wavenumbers, and the
angle between the wavevector and the magnetic field
[$\theta\mathrm{=}\cos^{-1}(\hat{\mathbf{K}}\cdot\hat{\bm{B}}_{0})$]. Such a
six-parameter-dependent formula is distilled using a perturbative solution to
magnetized version of Lamb’s hydrodynamic polytropic waves. Our explicit
analytical formula overcomes the limitation of previously attempted formulae
for the magnetized polytrope that were presented in general integral forms;
such a formulation, for instance, that of Gough & Thompson [1990; e.g., Eq.
(4.11)] and Bogdan & Cally (1997), requires numerical evaluation of the
eigenfrequencies, and thus leaves out the critical step of obtaining an
analytical understanding and expression. We achieve such here guided by our
numerical solutions and at the cost of extensive use of Mathematica for our
perturbative analyses.
We emphasize that the closed-form expression we obtain is not solely by use of
perturbative calculation. At the last step, we needed guidance from our
numerical solutions. A possible explanation for inadequacy of the perturbative
calculation alone can be the ordering scheme in the perturbation series, which
in this paper has two small parameters—one that corresponds to the WKB short-
wavelength limit approximation and the other that is the actual small
parameter, inverse of the Alfvénic Mach number. This implies that future work
should develop better perturbation theory to take this inadequacy into account
and to generalize our methodology to other complex wave problems.
The simplicity and accuracy of our formula are encouraging to employ the
formula to help solve the inverse problem of magnetoseismology. Our formula
provides an explicit, analytical dependence of the observed surface
oscillation frequency with the orientation and strength of the deep-seated and
subsurface magnetic field. Such an information can be crucial to predict the
surface-emergence location and strength of sunspots. Precursors for such may
be detected by analyzing the anisotropic magnetic effect in stellar ring
diagrams—the rings of constant frequencies over the wavenumber plane.
Nonlinear asteroseismology can also directly benefit from our analytical work,
as weak turbulence theory of asteroseismic waves inevitably requires accurate
and simple expressions of linear wave frequencies in resonant triad
interactions. The effect of the magnetic field on such is completely unknown.
However, observations now exist that suggest resonant mode interactions are
possible and can be a critical element of strongly pulsating stars (Guo,
2020). Nonlinear mode coupling of linear eigenmodes (Tripathi et al., 2023a,
b) may also need to be analyzed, in addition to mode resonances. Future
planned research will directly take advantage of the formula derived here to
assess the role of the magnetic field and other parameters in asteroseismic
wave turbulence, which is awaiting to soon enter adulthood from its infancy.
## Acknowledgments
We are pleased to thank Professor Ellen G. Zweibel for helpful discussions and
for her contribution in applying the energy principle of Newcomb (1961) to our
system. The inception of this paper took place at Nordic Institute for
Theoretical Physics (NORDITA), Sweden, while B.T. was visiting on a research
fellowship generously made available by NORDITA.
## Data Availability
Mathematica notebook—built to perform lengthy analytical manipulations—and
Dedalus script—used for numerical solution in this paper—are available at
https://github.com/BindeshTripathi/polytrope.
## Appendix A
The matrix elements $M_{ij}$ and $h_{\mu}$ that appear in Eq. (24) are
$\displaystyle M_{11}$
$\displaystyle=-\Omega^{2}(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2),$
(A1a) $\displaystyle M_{12}$ $\displaystyle=0,$ (A1b) $\displaystyle M_{13}$
$\displaystyle=\frac{-iK_{X}(n+1)(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)}{\gamma},$
(A1c) $\displaystyle M_{21}$
$\displaystyle={\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}K_{X}K_{Y}Z,$ (A1d)
$\displaystyle M_{22}$
$\displaystyle=-\Omega^{2}(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}K_{X}^{2}Z,$
(A1e) $\displaystyle M_{23}$
$\displaystyle=\frac{-iK_{Y}(n+1)(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)}{\gamma},$
(A1f) $\displaystyle M_{31}$
$\displaystyle=\frac{iK_{X}}{\gamma}\left[1+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}\left(\frac{3\gamma}{2}+\frac{K_{X}^{2}Z}{\Omega^{2}}\right)\right],$
(A1g) $\displaystyle M_{32}$
$\displaystyle=\frac{iK_{Y}}{\gamma}\left[1+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}\left(\frac{\gamma}{2}+\frac{K_{X}^{2}Z}{\Omega^{2}}\right)\right],$
(A1h) $\displaystyle M_{33}$
$\displaystyle=\frac{-\Omega^{2}(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}K_{X}^{2}Z}{n+1},$
(A1i) $\displaystyle h_{x}$ $\displaystyle=iK_{X}Z\hat{\chi},$ (A1j)
$\displaystyle h_{y}$
$\displaystyle=iK_{Y}Z\hat{\chi}(1+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}),$
(A1k) $\displaystyle h_{z}$
$\displaystyle=\hat{\chi}+\frac{Z\partial_{Z}\hat{\chi}}{n+1}+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}\hat{\chi}\left[1+\frac{K_{X}^{2}Z\left\\{\gamma-(n+1)(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)\right\\}}{(n+1)\gamma\Omega^{2}(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)}\right]+{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}\frac{Z\partial_{Z}\hat{\chi}}{n+1}\left[1+\frac{K_{X}^{2}Z}{\Omega^{2}(1+\gamma{\mathbb{\color[rgb]{0,0,0}\epsilon}}^{2}/2)}\right].$
(A1l)
## Appendix B
The parameters introduced in Eq. (31b), while writing the expression for
$\Gamma_{2}(Z)$, appear below:
$\displaystyle b_{2}$ $\displaystyle=\frac{-K^{3}\cos^{4}\theta}{\Omega^{2}},$
(A2a) $\displaystyle b_{1}$
$\displaystyle=\frac{-K\left[-\Omega^{8}\gamma^{4}+\Omega^{4}\gamma^{2}K^{2}(n+1)\left\\{(n+1+\gamma-\gamma
n)\cos(2\theta)-n-1-3\gamma\right\\}+2K^{4}(n+1)^{4}-K^{4}(n+1)^{3}(n+1-\gamma
n)\cos(2\theta)\right]}{2\Omega^{4}\gamma^{2}\mathrm{sec}^{2}\theta\,\left[\Omega^{4}\gamma^{2}-K^{2}(n+1)^{2}\right]},$
(A2b) $\displaystyle b_{0}$
$\displaystyle=\frac{\Bigg{(}\begin{gathered}4\Omega^{8}\gamma^{4}(2-\gamma)-2\Omega^{4}\gamma^{2}K^{2}\left\\{2(n+1)^{2}(4-3\gamma)+n(2n+3)\gamma^{2}\right\\}+K^{4}(n+1)^{2}\left\\{n\gamma^{2}(2n+3)-8(n+1)^{2}(\gamma-1)\right\\}\\\
+2K^{2}\gamma\cos(2\theta)\left[K^{2}(n+1)^{2}\left\\{n(\gamma-4+2n(\gamma-1))-2\right\\}+\Omega^{4}\gamma^{2}\left\\{2(n+1)^{2}-\gamma
n(2n+3)\right\\}\right]+n\gamma^{2}K^{4}(n+1)^{2}\cos(4\theta)\end{gathered}\Bigg{)}}{16\Omega^{2}\gamma^{2}K\left[\Omega^{4}\gamma^{2}-K^{2}(n+1)^{2}\right]}$
(A2e)
## Appendix C
Due to the magnetic field, the locations of the turning points, $Z_{1}$ and
$Z_{2}$, shift—which to the second order in
${\mathbb{\color[rgb]{0,0,0}\epsilon}}$ in Eq. (33) are given by $Z_{1}^{(2)}$
and $Z_{2}^{(2)}$:
$\displaystyle Z_{1}^{(2)}$
$\displaystyle=\frac{\alpha(b_{2}\alpha^{2}+b_{1}\alpha+b_{0})}{K(\beta-\alpha)},$
(A3a) $\displaystyle Z_{2}^{(2)}$
$\displaystyle=\frac{-\beta(b_{2}\beta^{2}+b_{1}\beta+b_{0})}{K(\beta-\alpha)}.$
(A3b)
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|
# Conformal partial waves in momentum space
Marc Gillioz
(_SISSA, via Bonomea 265, 34136 Trieste, Italy_)
###### Abstract
The decomposition of 4-point correlation functions into conformal partial
waves is a central tool in the study of conformal field theory. We compute
these partial waves for scalar operators in Minkowski momentum space, and find
a closed-form result valid in arbitrary space-time dimension $d\geq 3$
(including non-integer $d$). Each conformal partial wave is expressed as a sum
over ordinary spin partial waves, and the coefficients of this sum factorize
into a product of vertex functions that only depend on the conformal data of
the incoming, respectively outgoing operators. As a simple example, we apply
this conformal partial wave decomposition to the scalar box integral in $d=4$
dimensions.
###### Contents
1. 1 Introduction
2. 2 Spin eigenstates and completeness relation
1. 2.1 Kinematics of the 4-point function
2. 2.2 Polarization tensors
3. 2.3 Spin eigenstates and normalization
4. 2.4 Completeness relation
3. 3 Vertex functions
1. 3.1 Conformal Ward identities
2. 3.2 Highest-spin functions in various orderings
3. 3.3 Lower spin functions
4. 3.4 Special kinematic limits
4. 4 Example: the scalar box integral in 4 dimensions
1. 4.1 CFT optical theorem
2. 4.2 IR divergences
3. 4.3 Conformal partial wave expansion
5. 5 Conclusions
## 1 Introduction
Conformal field theory (CFT) can be defined non-perturbatively from its 2- and
3-point correlation functions, thanks to a convergent operator product
expansion (OPE) that reduces the computation of higher-point functions to
these elementary building blocks. In particular, the decomposition of 4-point
functions into conformal partial waves is at the heart of the modern conformal
bootstrap program. The bootstrap exploits the self-consistency of the OPE to
put constraints on the _dynamics_ of a theory encoded in the CFT data [1, 2,
3, 4]. The problem of computing the conformal partial waves, i.e. determining
the _kinematics_ underlying 4-point correlators, has been solved long ago [5,
6]. Closed-form results exist for scalar operators in dimensions $d=2$ and
$4$, and various techniques for an efficient computation of the conformal
partial waves in arbitrary dimensions have been developed, both for scalars
[7, 8, 9, 10, 11] and for operators that carry spin [12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29].
In this work, we address a different but related problem: we compute the
conformal partial waves in the momentum-space representation of correlation
functions in Minkowski space, as opposed to the ordinary Euclidean position-
space approach. We focus on correlation functions involving 4 scalar primary
operators $\phi_{i}$, each carrying momentum $p_{i}$, in $d\geq 3$ dimensions,
of the form
$\left\langle
0\right|[\phi_{1}(p_{1})\phi_{2}(p_{2})][\phi_{3}(p_{3})\phi_{4}(p_{4})]\left|0\right\rangle\equiv(2\pi)^{d}\delta^{d}(p_{1}+p_{2}+p_{3}+p_{4})\,G(p_{1},p_{2},p_{3}),$
(1.1)
where the square brackets around pairs of operators on the left-hand side can
mean any of the following: no particular ordering (Wightman function), a
commutator (possibly retarded or advanced), a time-ordered product or its
hermitian conjugate. The conformal wave expansion for $G$ takes the form
$G(p_{1},p_{2},p_{3})=s^{(\Delta_{1}+\Delta_{2}+\Delta_{3}+\Delta_{4}-3d)/2}\sum_{\mathcal{O}}\lambda_{12\mathcal{O}}\lambda_{\mathcal{O}34}G_{\Delta,\ell}(p_{1},p_{2},p_{3})$
(1.2)
where the sum is over primary operators $\mathcal{O}$ with scaling dimension
$\Delta$ and spin $\ell$, $\lambda_{12\mathcal{O}}$ and
$\lambda_{\mathcal{O}34}$ are OPE coefficients encoding the dynamical
information about the theory, and $s$ is the center-of-mass energy
$s=(p_{1}+p_{2})^{2}=(p_{3}+p_{4})^{2}>0,$ (1.3)
which is strictly positive by the spectral condition on the correlation
function (1.1). The conformal partial waves $G_{\Delta,\ell}$ are completely
fixed by symmetry in terms of $\Delta$ and $\ell$, of the scaling dimensions
$\Delta_{i}$ of the external operators (which may be distinct or not) and of
the space-time dimension $d$ (possibly including non-integer dimensions, as
our results are analytic in $d\geq 3$). We obtain
$\boxed{G_{\Delta,\ell}(p_{1},p_{2},p_{3})=\sum_{m=0}^{\ell}C_{\Delta,\ell,m}\mathcal{C}_{m}^{(d-3)/2}(\cos\theta)V_{\Delta,\ell,m}^{[12]}\left(\frac{p_{1}^{2}}{s},\frac{p_{2}^{2}}{s}\right)V_{\Delta,\ell,m}^{[34]}\left(\frac{p_{3}^{2}}{s},\frac{p_{4}^{2}}{s}\right),}$
(1.4)
where $\theta$ is the scattering angle, defined in terms of the momenta
$p_{i}$ in eq. (2.10), and $\mathcal{C}^{(d-3)/2}_{m}(\cos\theta)$ a
Gegenbauer polynomial of degree $m$ that appears in the ordinary spin partial
wave expansion in $d$ dimensions. The numerical factor $C_{\Delta,\ell,m}\geq
0$ is given in eq. (2.35), and the “vertex functions”
$V_{\Delta,\ell,m}^{[ab]}$ are defined in section 3, depending on which
ordering is understood in the definition (1.1). This result admits a simple
diagrammatic representation given in figure 1. One observes a factorization in
the sense that the dependence on the external CFT data (the scaling dimensions
$\Delta_{i}$) and on the “invariant masses” $p_{i}^{2}$ is entirely contained
in the vertex functions $V_{\Delta,\ell,m}^{[ab]}$. Note that we have used the
evident notation $p_{4}^{2}=(p_{1}+p_{2}+p_{3})^{2}$.
Figure 1: Diagrammatic representation of the conformal partial wave (1.4), in
which each line corresponds to a local primary operator of the CFT carrying a
certain momentum. Note that the vertex functions $V_{\Delta,\ell,m}^{[ab]}$
only depend on the scaling dimensions, on the spin (the total spin $\ell$ and
its projection $m$ onto a reference direction), and on the momenta of the
lines attached to it.
Sections 2 and 3 of this work are dedicated to the derivation of the momentum-
space conformal partial waves (1.4). This derivation does not rely on solving
a Casimir equation as in ordinary position space: in fact, since the conformal
Ward identities are second-order differential equations with respect to the
momenta $p_{i}$, one cannot make use of invariant cross-ratios in momentum
space. Instead, the 4-point function depends on six Lorentz-invariant
quantities (taken here to be all four $p_{i}^{2}$, $s$ and $\cos\theta$), and
solving a differential equation in as many variables appears like a formidable
task. This disadvantage is however counterbalanced by two elements that are
unique to the Minkowski momentum-space representation of correlation
functions:
* •
The orthogonality of momentum eigenstates, that allows to factorize the
conformal partial wave into a product of 3-point functions, up to the
contraction of Lorentz indices.
* •
The ability to use a reference frame, the center-of-mass frame, in which the
2-point function of the intermediate operator can be decomposed into
polarizations that transform under irreducible representations of the rotation
group $\text{SO}(d-1)$.
These points have been put forward in a recent derivation of position-space
conformal blocks [24], as well as in Mack’s classification of all the
representations of the Lorentzian conformal group in $d=4$ [30]. The operator
product expansion in general and the conformal partial wave expansion (1.2) in
particular are nothing but the use of a Hilbert space completeness relation,
which in a CFT can be made explicit thanks to the two properties listed above.
In fact, the correlation function (1.1) computes the overlap between two
states of the theory: one state obtained by acting on the vacuum with the
product of operators $[\phi_{3}\phi_{4}]$, and the other state obtained acting
with $[\phi_{1}\phi_{2}]^{\dagger}$. The partial wave (1.4) corresponds to the
projection of this overlap of states onto a conformal family consisting of a
primary operator and of all of its descendants.
A corollary of this observation is that each conformal partial wave is
positive in the forward scattering limit $p_{1}\to-p_{4}$, $p_{2}\to-p_{3}$
provided that $\phi_{1}=\phi_{4}$ and $\phi_{2}=\phi_{3}$. In this case the
vertex functions are complex conjugate of each other,
$V_{\Delta,\ell,m}^{[12]}=\big{(}V_{\Delta,\ell,m}^{[34]}\big{)}^{*}$, and the
Gegenbauer polynomial evaluated at zero scattering angle ($\cos\theta=1$) is
positive, hence $G_{\Delta,\ell}\geq 0$. The positivity of this configuration
and its decomposition into conformal partial waves have actually been used in
the derivation of sum rules for anomaly coefficients [31, 32].
It should be emphasized that the conformal partial wave expansion described in
this work relies heavily on the Hamiltonian evolution being unitary, i.e.
given by $\exp(iHt)$ where $t$ is the Lorentzian time and $H$ a Hamiltonian
operator bounded below. In a Euclidean theory, where this evolution is
replaced by $\exp(-H\tau)$ with Euclidean “time” $\tau$, the OPE only makes
sense for correlation functions that are time-ordered,111If one uses radial
quantization instead, then this applies to any configuration which is
equivalent to a radially-ordered one under a global conformal symmetry
transformation, which in practice extends to most of the space of
configurations. The conformal bootstrap works precisely because it uses
configurations that can be expanded into two or more distinct OPEs by
different choices of quantization surfaces. a property that is lost upon
Fourier transform. For this reason, the conformal partial wave expansion (1.2)
differs at a fundamental level from other decompositions of the momentum-space
4-point function, such as the decomposition into Witten exchange diagrams in
the case of a holographic theory (usually formulated in Mellin space [33, 34,
35]), or the decomposition into symmetric Polyakov blocks expressed directly
in Euclidean momentum space [36, 37].222Instead, there is an interesting
connection between the conformal partial waves expansion described in this
work and the Polyakov-Regge blocks recently introduced [38], through the
generalization in momentum space of the concept of CFT dispersion relations
[39]. We leave the exploration of this venue for future work. For the same
reason, it is difficult to compare our conformal partial wave expansion with
recent results on 4-point correlators in Euclidean momentum space [40, 41, 42,
43, 44].
On the other hand, eqs. (1.2) and (1.4) are immediately suitable for
comparison with results obtained using ordinary Feynman diagram computations
in theories that are both perturbative and conformal. While the goal of this
work is not to study such situations systematically, we present in section 4 a
simple example that involves the real part of the scalar box integral in four
dimensions. This box integral appears in the 4-point function of the composite
operator $\Phi^{2}$ in the theory of a free boson $\Phi$, or in that of the
gauge-invariant operator $\operatorname{tr}(\Phi^{i}\Phi^{i})$ in
$\mathcal{N}=4$ supersymmetric Yang-Mills. The general form of this loop
integral with off-shell momenta is known thanks to its dual conformal [45] or
Yangian invariance [46].
Besides perturbative CFT, the conformal partial waves (1.4) will certainly
find applications in other contexts. One of their salient features is the
presence of double zeroes at scaling dimension of double-trace operators, a
property reminiscent of the “double-twist” functional methods for the
conformal bootstrap [47, 48, 49, 50, 51, 52]. However, it should be emphasized
that the correlator (1.1) is ill-suited for writing a bootstrap equation, as
it does not have crossing symmetry built in. Microcausality in Minkowski
momentum space implies certain analyticity properties of the correlation
functions that might eventually be used to place bounds on the CFT data, but
the relevant methods remain to be developed. Similarly, our conformal partial
wave expansion could also find use in the cosmological bootstrap program [53,
54, 55, 56, 57, 58, 59, 60], even though the connection between correlators in
Minkowski momentum space and observables in de Sitter remains to be
established (see ref. [61] for hints in this direction). For now, we provide
the decomposition (1.2) as a tool and encourage the community to find its own
applications.
## 2 Spin eigenstates and completeness relation
The core of the computation of the conformal partial waves is the formulation
of a Hilbert space completeness relation. In this section, we assemble the
different elements, construct a complete basis of states, and detail its
dependence on the kinematic invariants of the 4-point function. The
computation of the overlap of these states with the pairs of operators
$[\phi_{1}\phi_{2}]$ and $[\phi_{3}\phi_{4}]$ is discussed instead in section
3.
### 2.1 Kinematics of the 4-point function
We begin with a detailed description of the kinematics of the 4-point
function, as it will play a central role in what follows. As seen in eq.
(1.1), momentum-space correlation functions involving $n$ operators are always
proportional to a delta function imposing momentum conservation, and hence
depend non-trivially on $n-1$ momenta only. It is convenient to introduce the
double-bracket notation
$\left\langle
0\right|\phi_{1}(p_{1})\cdots\phi_{n}(p_{n})\left|0\right\rangle\equiv(2\pi)^{d}\delta^{d}(p_{1}+\ldots+p_{n})\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\phi_{1}(p_{1})\cdots\phi_{n}(p_{n})\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}.$
(2.1)
Whenever this notation is being used, it is understood that one can trade one
of the momenta for the sum of the others, say $p_{n}=-(p_{1}+\ldots+p_{n-1})$.
For the 4-point function, this mean that there are 3 independent momenta, and
hence 6 Lorentz invariants.333We are always working in $d\geq 3$ dimensions.
In $d=2$, three vectors cannot be linearly independent and there is one less
invariant In analogy with scattering amplitudes, one can use the “invariant
masses”
$p_{1}^{2},\quad p_{2}^{2},\quad p_{3}^{2},\quad
p_{4}^{2}=(p_{1}+p_{2}+p_{3})^{2},$ (2.2)
as well as the Mandelstam variables
$s=(p_{1}+p_{2})^{2},\qquad t=(p_{1}+p_{3})^{2},\qquad u=(p_{2}+p_{3})^{2},$
(2.3)
subject to the constraint $s+t+u=\sum p_{i}^{2}$. Unlike scattering
amplitudes, however, we do not require the $p_{i}^{2}$ to take any special
value. In fact, we even want to be agnostic about their sign: the momenta
$p_{i}$ might be space-like, time-like, or even light-like. However, the
specific ordering of the operators in eq. (1.1) does imply a constraint on the
momentum flowing between the pairs $[\phi_{1}\phi_{2}]$ and
$[\phi_{3}\phi_{4}]$: defining
$p\equiv-p_{1}-p_{2}=p_{3}+p_{4},$ (2.4)
the correlation function (1.1) vanishes by the spectral condition whenever $p$
lies outside the forward light cone, i.e. unless
$p^{0}>0,\qquad\text{and}\qquad s=p^{2}>0.$ (2.5)
We will assume henceforth that these two conditions are fulfilled.
With $s$ strictly positive, one can rescale all dimensionful quantities by
$s$, including the 4-point function (1.1) whose overall scaling dimension is
fixed by its operator content and by the space-time dimension. Since we chose
in the definition (1.2) to factor out precisely the appropriate power of $s$,
$G_{\Delta,\ell}$ is dimensionless. It is a function of the rescaled invariant
masses
$w_{i}\equiv\frac{p_{i}^{2}}{s},$ (2.6)
and of a fifth dimensionless variable that we define as follows: we introduce
two new linear combinations of the momenta,
$q_{12}^{\mu}\equiv\frac{(p\cdot p_{2})p_{1}^{\mu}-(p\cdot
p_{1})p_{2}^{\mu}}{\sqrt{(p_{1}\cdot
p_{2})^{2}-p_{1}^{2}p_{2}^{2}}},\qquad\qquad q_{34}^{\mu}\equiv\frac{(p\cdot
p_{3})p_{4}^{\mu}-(p\cdot p_{4})p_{3}^{\mu}}{\sqrt{(p_{3}\cdot
p_{4})^{2}-p_{3}^{2}p_{4}^{2}}},$ (2.7)
which have the property of being both orthogonal to $p$, and normalized in
units of $s$,
$q_{12}\cdot p=q_{34}\cdot p=0,\qquad\qquad q_{12}^{2}=q_{34}^{2}=-s.$ (2.8)
The definitions (2.7) are non-singular as long as neither $p_{1}$ and $p_{2}$,
nor $p_{3}$ and $p_{4}$ are colinear; the special configuration in which some
of these momenta are colinear can be eventually reached as a limit of the
general case. We now define $\theta$ to be the angle between these two
vectors,
$\cos\theta\equiv-\frac{q_{12}\cdot q_{13}}{s},$ (2.9)
or in terms of the Mandelstam invariants
$\boxed{\cos\theta=\frac{s(u-t)-(p_{1}^{2}-p_{2}^{2})(p_{3}^{2}-p_{4}^{2})}{\sqrt{(s-p_{1}^{2}-p_{2}^{2})^{2}-4p_{1}^{2}p_{2}^{2}}\sqrt{(s-p_{3}^{2}-p_{4}^{2})^{2}-4p_{3}^{2}p_{4}^{2}}}.}$
(2.10)
$\theta$ corresponds in fact to the scattering angle in a $2\to 2$ inelastic
process in which the four “particles” have distinct masses (possibly including
negative masses squared). In the center-of-mass frame, working for
definiteness in $d=3$, one can always choose
$\displaystyle p$ $\displaystyle=\sqrt{s}\left(1,0,0\right),$ $\displaystyle
q_{12}$ $\displaystyle=\sqrt{s}\left(0,1,0\right),$ (2.11) $\displaystyle
q_{34}$ $\displaystyle=\sqrt{s}\left(0,\cos\theta,\sin\theta\right),$
which in terms of momenta $p_{i}$ corresponds to
$\displaystyle p_{1}$
$\displaystyle=-\frac{\sqrt{s}}{2}\left(1+w_{1}-w_{2},\Omega_{12},0\right),$
$\displaystyle p_{2}$
$\displaystyle=-\frac{\sqrt{s}}{2}\left(1-w_{1}+w_{2},-\Omega_{12},0\right),$
(2.12) $\displaystyle p_{3}$
$\displaystyle=\frac{\sqrt{s}}{2}\left(1+w_{3}-w_{4},-\Omega_{34}\cos\theta,-\Omega_{34}\sin\theta\right),$
where we have defined
$\Omega_{12}=\sqrt{(1-w_{1}-w_{2})^{2}-4w_{1}w_{2}},\qquad\Omega_{34}=\sqrt{(1-w_{3}-w_{4})^{2}-4w_{3}w_{4}}.$
(2.13)
The quantities under the square roots are non-negative for any configuration
of momenta. Note that $\Omega_{12}$ and $\Omega_{34}$ will play an important
role later. The center-of-mass configuration is illustrated in figure 2.
Figure 2: A possible configuration of momenta in the center-of-mass frame
(energy along the vertical axis). The scattering angle $\theta$ corresponds to
the angle between the planes spanned by $(p_{1},p_{2})$ and $(p_{3},p_{4})$,
or equivalently between the vectors $q_{12}$ and $q_{34}$. In this particular
example $p_{1}$ is time-like (backward directed), while $p_{2}$, $p_{3}$ and
$p_{4}$ are all space-like, as seen by their position relative to the light
cone (dotted lines); the covariant definition (2.10) is valid independently of
the sign of the $p_{i}^{2}$.
### 2.2 Polarization tensors
The Hilbert space completeness relation relies on a basis of momentum
eigenstates that can be constructed out of a single primary operator insertion
acting on the vacuum, for all primaries of the theory. In the case of scalar
external operators, only scalar and traceless symmetric tensor operators enter
the OPE. In order to avoid dealing with the contraction of tensor indices, and
more importantly to obtain the factorized result (1.4), it is necessary to
introduce a complete orthogonal basis of polarization tensors
$\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}$, after which one could write
$\mathcal{O}^{\mu_{1}\ldots\mu_{\ell}}(p)\left|0\right\rangle\equiv\sum_{m}\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}\mathcal{O}^{(\ell,m)}(p)\left|0\right\rangle.$
(2.14)
The goal of this section is to provide the explicit construction of such a
basis, using only the momenta at hand, so that its transformation under
special conformal transformations can later be determined explicitly. Such a
construction is most easily done using spinor variables [62], but then it
depends explicitly on the space-time dimension $d$. For scalar external
operators, we will show that a $d$-independent construction can be performed.
The starting point is to realize that, thanks to the ability of going to the
center-of-mass frame (2.11), the polarization tensors
$\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}$ can be decomposed into products of
the vector $p$ and of invariant tensors of the group of spatial rotations
$\text{SO}(d-1)$. An orthogonal basis of such polarization tensors is given by
$\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}\equiv
s^{-(\ell-m)/2}\left[\varepsilon_{\perp}^{(\mu_{1}\ldots\mu_{m}}p^{\mu_{m+1}}\cdots
p^{\mu_{\ell})}-\text{traces}\right],$ (2.15)
where the indices on the right-hand side are symmetrized, and the tensors
$\varepsilon_{\perp}^{\mu_{1}\ldots\mu_{n}}$ form themselves a basis of
traceless, symmetric tensors orthogonal to $p$,
$p_{\mu_{1}}\varepsilon_{\perp}^{\mu_{1}\ldots\mu_{n}}=0.$ (2.16)
This means that the tensors $\varepsilon_{\perp}^{\mu_{1}\ldots\mu_{n}}$ only
have non-zero entries when all indices are spatial.
In our case, one can even go further and construct explicitly the tensors
$\varepsilon_{\perp}^{\mu_{1}\ldots\mu_{m}}$ out of the momenta at hand. The
reason is that we are only interested in polarization tensors that live in a
plane spanned by a pair of momenta, say $p_{1}$ and $p_{2}$, or equivalently
$p$ and $q_{12}$. Any polarization overlapping with a state created by two
scalar operators acting on the vacuum is of this form. Using the projector
onto the directions transverse to $p$,
$\eta_{\perp}^{\mu\nu}\equiv\eta^{\mu\nu}-\frac{p^{\mu}p^{\nu}}{p^{2}},$
(2.17)
one can take
$\varepsilon_{\perp}^{\mu_{1}\ldots\mu_{m}}(p,q)\equiv\sum_{n=0}^{\lfloor
m/2\rfloor}\frac{m!}{n!(m-2n)!}\frac{1}{2^{2n}\left(\frac{d-3}{2}+m-n\right)_{n}}\frac{\eta_{\perp}^{(\mu_{1}\mu_{2}}\cdots\eta_{\perp}^{\mu_{2n-1}\mu_{2n}}q^{\mu_{2n+1}}\cdots
q^{\mu_{m})}}{s^{m/2-n}}.$ (2.18)
This is a covariant definition in terms of any pair of momenta $p$ and $q$
that satisfy $p\cdot q=0$, $p^{2}=-q^{2}=s$. Similarly, a covariant definition
of the more general tensors $\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}$ is
directly obtained plugging eq. (2.18) into (2.15). The final result is most
easily expressed with the help of an auxiliary polarization vector $z^{\mu}$
satisfying $z^{2}=0$, in terms of which
$z_{\mu_{1}}\cdots
z_{\mu_{\ell}}\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(p,q)=\frac{(z\cdot
p)^{\ell-m}(z\cdot
q)^{m}}{s^{\ell/2}}{}_{2}F_{1}\left(-\frac{m}{2},-\frac{m-1}{2};\frac{5-d}{2}-m;\frac{(z\cdot
p)^{2}}{(z\cdot q)^{2}}\right).$ (2.19)
Note that the hypergeometric series terminates both for even and odd $m$
(either one of the first two parameters is a negative integer), so that the
right-hand side is in fact a homogeneous polynomial of degree $\ell$ in
$z\cdot p$ and $z\cdot q$. The hypergeometric form is nevertheless convenient
for its compactness, and it shows explicitly that the definition (2.19) is
analytic in the space-time dimension $d$. It also gives a simple proof of the
identities
$\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(p,q)=(-1)^{m}\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(p,-q)=(-1)^{\ell-m}\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(-p,q)=(-1)^{\ell}\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(-p,-q).$
(2.20)
The orthogonality of the basis follows by construction. Its normalization is
given by
$\varepsilon_{m^{\prime}\mu_{1}\ldots\mu_{\ell}}(p,q)\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(p,q)=\delta_{m^{\prime}m}(-1)^{m}\mathcal{N}_{\ell,m}.$
(2.21)
where
$\mathcal{N}_{\ell,m}=\frac{m!(\ell-m)!}{\ell!}\frac{(d-2+2m)_{\ell-m}}{2^{\ell-m}\left(\frac{d-2}{2}+m\right)_{\ell-m}}\frac{(d-3)_{m}}{2^{m}\left(\frac{d-3}{2}\right)_{m}}\geq
0.$ (2.22)
Moreover, if one considers two distinct polarization tensors defined with
respect to two reference vectors $q$ and $q^{\prime}$, both orthogonal to the
same vector $p$, then one one obtains the identity
$\varepsilon_{m^{\prime}\mu_{1}\ldots\mu_{\ell}}(p,q)\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(p,q^{\prime})=\delta_{m^{\prime}m}(-1)^{m}\mathcal{N}_{\ell,m}\frac{m!}{(d-3)_{m}}\,\mathcal{C}_{m}^{(d-3)/2}(\cos\theta),$
(2.23)
where $\theta$ is the angle between $q$ and $q^{\prime}$, and
$\mathcal{C}_{m}^{(d-3)/2}$ is a Gegenbauer polynomial. This identity will
play a central role in the computation of the conformal partial waves.
### 2.3 Spin eigenstates and normalization
We show next how these polarization tensors naturally appear in the 2-point
function of primary operators, and how this leads to the construction of an
orthogonal basis of spin eigenstates. The 2-point function of traceless,
symmetric tensor operators in momentum space satisfies [63]
$\displaystyle\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}^{\mu_{1}\ldots\mu_{\ell}}(-p)\mathcal{O}^{\nu_{1}\ldots\nu_{\ell}}(p)\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}$
$\displaystyle=s^{\Delta-d/2}\Theta(p^{0})\Theta(p^{2})B_{\Delta,\ell}\sum_{n=0}^{\ell}\frac{(-1)^{n}2^{n}\ell!}{n!(\ell-n)!}\frac{\left(\frac{d}{2}-\Delta\right)_{n}}{\left(2-\Delta-\ell\right)_{n}}$
$\displaystyle\quad\qquad\times\left[\frac{p^{(\mu_{1}}p^{\nu_{1}}\cdots
p^{\mu_{n}}p^{\nu_{n}}\eta^{\mu_{n+1}\nu_{n+1}}\cdots\eta^{\mu_{\ell}\nu_{\ell})}}{s^{n}}-\text{traces}\right],$
(2.24)
where
$B_{\Delta,\ell}=\frac{(4\pi)^{(d+2)/2}\left(\Delta-1\right)_{\ell}}{2^{2\Delta+1}\Gamma\left(\Delta-\frac{d-2}{2}\right)\Gamma(\Delta+\ell)}.$
(2.25)
This result is the unique solution (up to a normalization constant) to the
conformal Ward identities for the 2-point function. Alternatively, it can be
obtained from the direct Fourier transform of the position-space 2-point
function in Lorentzian signature. The Heaviside $\Theta$-functions appearing
in eq. (2.24) indicate that the 2-point function vanishes whenever $p$ lies
outside the forward light cone.
To relate eq. (2.24) with the polarization tensors described above, we
consider again its projection onto the plane spanned by $p$ and a vector $q$
orthogonal to it. This amounts to replacing
$\eta^{\mu\nu}\rightarrow\frac{p^{\mu}p^{\nu}-q^{\mu}q^{\nu}}{s}$ (2.26)
in eq. (2.24). A bit of combinatorics allows then to rewrite the 2-point
function (2.24) as
$\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}^{\mu_{1}\ldots\mu_{\ell}}(-p)\mathcal{O}^{\nu_{1}\ldots\nu_{\ell}}(p)\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}=s^{\Delta-d/2}\sum_{m=0}^{\ell}\frac{B_{\Delta,\ell,m}}{\mathcal{N}_{\ell,m}}\,\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(-p,-q)\varepsilon_{m}^{\nu_{1}\ldots\nu_{\ell}}(p,q),$
(2.27)
where
$B_{\Delta,\ell,m}\equiv\frac{(\Delta-\ell-d+2)_{\ell-m}}{(\Delta+m-1)_{\ell-m}}B_{\Delta,\ell}\geq
0.$ (2.28)
Requiring that the 2-point function defines a positive inner product, one
actually recovers the well-known unitarity bounds $\Delta\geq(d-2)/2$ for
scalars, and $\Delta-\ell\geq d-2$ for traceless, symmetric tensors of spin
$\ell$. Note that in the generic case the 2-point function is a sum of
$\ell+1$ polarizations, the exception being spinning operators that saturate
the unitarity bound: when $\Delta-\ell=d-2$, all $B_{\Delta,\ell,m}$ with
$m<\ell$ vanish: conserved currents only have transverse polarizations
$m=\ell$.
This construction suggests to define the spin eigenstates
$\left|\mathcal{O}^{(\ell,m)}(p{\,|\,}q)\right\rangle\equiv\mathcal{N}_{\ell,m}^{-1}\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(p,q)\mathcal{O}_{\mu_{1}\ldots\mu_{\ell}}(p)\left|0\right\rangle.$
(2.29)
The notation indicates that this is a state carrying momentum $p$, with total
spin $\ell$, and component $m$ along a direction that is defined by a
reference vector $q$. These states form an orthogonal basis,
$\big{\langle}\mathcal{O}^{(\ell^{\prime},m^{\prime})}(-p^{\prime}{\,|\,}-q)\big{|}\mathcal{O}^{(\ell,m)}(p{\,|\,}q)\big{\rangle}=(2\pi)^{d}\delta^{d}(p^{\prime}-p)\delta_{\ell^{\prime}\ell}\delta_{m^{\prime}m}\mathcal{N}_{\ell,m}^{-1}B_{\Delta,\ell,m}s^{\Delta-d/2}.$
(2.30)
They realize the decomposition (2.14) envisioned at the beginning of this
section. Note that an equivalent definition of the spin eigenstates can be
obtained more generically by group-theoretical considerations, according to
which $m$ labels the representation of the state under the little group that
preserves the momentum $p$ [30, 24]. The advantage of our explicit
construction in terms of the momentum $p$ and of the reference vector $q$ is
that it is immediately suited to examine the transformation properties of the
polarization tensors under special conformal transformations, as developed
later in section 3.1.
### 2.4 Completeness relation
We are now equipped with a complete basis of states that can appear in the OPE
of two scalar operators acting on the vacuum. Writing down a completeness
relation amounts to inverting the 2-point function (2.30), and this is now a
trivial matter since the states are readily orthogonal. The null states
corresponding to the longitudinal polarizations ($m<\ell$) of conserved
currents can be safely excluded, as we shall see in section 3.3 that their
overlap with two operators acting on the vacuum is zero. Similarly, it is
sufficient to consider states that carry a momentum $p$ inside the forward
light cone $\mathcal{V}_{+}$, as the states are otherwise null. We write
therefore
$\mathds{1}=\left|0\right\rangle\left\langle
0\right|+\sum_{\mathcal{O}}\sum_{m=0}^{\ell}B_{\Delta,\ell,m}^{-1}\mathcal{N}_{\ell,m}\int\limits_{\mathcal{V}_{+}}\frac{d^{d}k}{(2\pi)^{d}}(k^{2})^{d/2-\Delta}\left|\mathcal{O}^{(\ell,m)}(k{\,|\,}q)\right\rangle\left\langle\mathcal{O}^{(\ell,m)}(-k{\,|\,}-q)\right|,$
(2.31)
where the sum is over all primary operators of the theory. Spin
representations other than traceless symmetric tensors have been omitted, as
they do not appear in the scalar 4-point function. One can also choose
different reference vectors $q$ and $q^{\prime}$ for the two states, in which
case eq. (2.23) leads to the more general completeness relation
$\displaystyle\mathds{1}=\left|0\right\rangle\left\langle
0\right|+\sum_{\mathcal{O}}\sum_{m=0}^{\ell}$ $\displaystyle
B_{\Delta,\ell,m}^{-1}\mathcal{N}_{\ell,m}\frac{m!}{(d-3)_{m}}\mathcal{C}_{m}^{(d-3)/2}(\cos\theta)$
$\displaystyle\times\int\limits_{\mathcal{V}_{+}}\frac{d^{d}k}{(2\pi)^{d}}(k^{2})^{d/2-\Delta}\left|\mathcal{O}^{(\ell,m)}(k{\,|\,}q)\right\rangle\left\langle\mathcal{O}^{(\ell,m)}(-k{\,|\,}-q^{\prime})\right|,$
(2.32)
where $\theta$ is the angle between the vectors $q$ and $q^{\prime}$.
It is important to remember that, in the derivation of these completeness
relations, we made the constraining assumption that the Lorentz indices of the
traceless, symmetric operators only live in the subspace of the
$d$-dimensional Minkowski space spanned by the momenta $p$ and $q$
(respectively $q^{\prime}$). This is sufficient when working with the
kinematics of the scalar 4-point functions, provided that we use $q_{12}$ and
$q_{34}$ defined in eq. (2.7) in place of $q$ and $q^{\prime}$. Plugging the
completeness relation (2.32) in the correlator (1.1), and resolving the
trivial delta functions, one arrives therefore at
$\displaystyle G(p_{1},p_{2},p_{3})$
$\displaystyle=\sum_{\mathcal{O}}s^{d/2-\Delta}\sum_{m=0}^{\ell}C_{\Delta,\ell,m}\mathcal{C}_{m}^{(d-3)/2}(\cos\theta)$
$\displaystyle\qquad\times\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}{}[\phi_{1}(p_{1})\phi_{2}(p_{2})]\mathcal{O}^{(\ell,m)}(p{\,|\,}q_{12})\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\mathcal{O}^{(\ell,m)}(-p{\,|\,}-q_{34})[\phi_{3}(p_{3})\phi_{4}(p_{4})]\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}.$
(2.33)
Note that the vacuum state does not contribute for $s>0$. This is now
precisely in the form of eqs. (1.2) and (1.4), where the vertex functions are
given by the 3-point correlators, and we have defined
$C_{\Delta,\ell,m}=B_{\Delta,\ell,m}^{-1}\mathcal{N}_{\ell,m}\frac{m!}{(d-3)_{m}},$
(2.34)
or explicitly
$\boxed{C_{\Delta,\ell,m}=\frac{2^{2\Delta-\ell+1}(m!)^{2}(\ell-m)!(d-2+2m)_{\ell-m}\Gamma\left(\Delta-\frac{d-2}{2}\right)\Gamma(\Delta+\ell)}{(4\pi)^{(d+2)/2}\ell!\left(\frac{d-2}{2}+m\right)_{\ell-m}\left(\frac{d-3}{2}\right)_{m}\left(\Delta-1\right)_{m}(\Delta-\ell-d+2)_{\ell-m}}.}$
(2.35)
This coefficient diverges in $d=3$ for $m>0$, but the Gegenbauer polynomial
$\mathcal{C}_{m}^{(d-3)/2}$ vanishes in the same case: the product of the two
is in fact well-defined in the limit $d\to 3$, and it is proportional to the
Chebyshev polynomial $T_{m}(\cos\theta)$.
In summary, using polarization tensors defined in terms of the kinematic
variables, we have been able to write down a decomposition of the 4-point
function into conformal partial waves, and to reduce the computation of these
partial wave to a pair of 3-point functions, which we are going to discuss
next.
## 3 Vertex functions
In this section we describe the computation of the 3-point functions appearing
in eq. (2.33). This computations relies essentially on the results of ref.
[64], but the projection onto spin eigenstates is new. At first this looks
like a major complication, as the standard approach relying on solving
conformal Ward identities requires to determine the transformation of the
polarization tensors under special conformal symmetry. We shall see
nonetheless that the results take a particularly simple form for the highest-
spin function in each conformal family ($m=\ell$), and that the lower spin
functions can then be determined recursively.
### 3.1 Conformal Ward identities
The starting point is to strip the 3-point correlation functions from its
dynamical information, the OPE coefficient, and from an overall power of $s$
fixed by scaling symmetry. We write
$\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}{}[\phi_{1}(p_{1})\phi_{2}(p_{2})]\mathcal{O}^{(\ell,m)}(p{\,|\,}q_{12})\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}\equiv\lambda_{12\mathcal{O}}\,s^{(\Delta_{1}+\Delta_{2}+\Delta-2d)/2}V_{\Delta,\ell,m}^{[12]}\left(\frac{p_{1}^{2}}{s},\frac{p_{2}^{2}}{s}\right).$
(3.1)
$V_{\Delta,\ell,m}^{[12]}$ on the right-hand side is a function of two
dimensionless variables only. It is precisely the vertex function that enters
in the conformal partial wave (1.4). All information about scale and Poincaré
symmetry is already included in this equation and in the definition (2.19) of
the polarization tensors. What remains to be determined is actually contained
in the action of special conformal transformations, together with a boundary
condition that can be traced back to Euclidean position space. Note that the
vertex function $V_{\Delta,\ell,m}^{[34]}$ in eq. (1.4) is simply related to
$V_{\Delta,\ell,m}^{[12]}$ by hermitian conjugation and a change in the labels
$1\to 4$ and $2\to 3$. We shall therefore focus exclusively on
$V_{\Delta,\ell,m}^{[12]}$ in this whole section.
Working in Minkowski space, there is no choice but to use the infinitesimal
form of special conformal transformation, as their exponentiated form does not
preserve the causal structure of space-time. At the level of correlation
functions, this means solving a partial differential equation: the Ward
identity for special conformal transformations in momentum space is a second-
order differential equation that can be written as [65, 66]
$\widehat{K}_{12}^{\alpha}\,\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}{}[\phi_{1}(p_{1})\phi_{2}(p_{2})]\mathcal{O}^{\mu_{1}\ldots\mu_{\ell}}(-p_{1}-p_{2})\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}=0$
(3.2)
where $\widehat{K}_{12}^{\alpha}$ is the differential operator
$\widehat{K}_{12}^{\alpha}=\sum_{i=1}^{2}\left[p_{i}^{\beta}\frac{\partial^{2}}{\partial
p_{i\alpha}\partial
p_{i}^{\beta}}-\frac{1}{2}p_{i}^{\alpha}\frac{\partial^{2}}{\partial
p_{i\beta}\partial p_{i}^{\beta}}-(\Delta_{i}-d)\frac{\partial}{\partial
p_{i\alpha}}\right].$ (3.3)
Note that this differential equation is valid no matter what the ordering
between the operators $\phi_{1}$ and $\phi_{2}$ is: the difference in
orderings only means different boundary conditions, which will be discussed
later in section 3.2.
The Ward identity (3.2) does not apply directly to the vertex function
$V_{\Delta,\ell,m}^{[12]}$, but rather to the correlator with the primary
operator $\mathcal{O}^{\mu_{1}\ldots\mu_{\ell}}$ acting on the vacuum. In
order to include the polarization tensors and the power of $s$ in this
equation, we write, in accordance with eq. (2.14),
$\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}{}[\phi_{1}(p_{1})\phi_{2}(p_{2})]\mathcal{O}^{\mu_{1}\ldots\mu_{\ell}}(-p_{1}-p_{2})\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}=\lambda_{12\mathcal{O}}\,s^{(\Delta_{1}+\Delta_{2}+\Delta-2d)/2}\sum_{m=0}^{\ell}\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(p,q_{12})V_{\Delta,\ell,m}^{[12]}(w_{1},w_{2}).$
(3.4)
We also introduce an external polarization vector $z^{\mu}$ as in eq. (2.19),
and use the shorthand notation
$\varepsilon_{m}\equiv z_{\mu_{1}}\cdots
z_{\mu_{\ell}}\varepsilon_{m}^{\mu_{1}\ldots\mu_{\ell}}(p,q_{12}).$ (3.5)
It is straightforward (if tedious) to compute the action of
$\widehat{K}_{12}^{\alpha}$ on the polarization tensors, as they are
explicitly defined in terms of $p_{1}$ and $p_{2}$. In doing so, we encounter
derivatives of the hypergeometric function (2.19), and we find it convenient
to introduce
$\varepsilon^{\prime}_{m}\equiv\frac{\partial\varepsilon_{m}}{\partial(z\cdot
q_{12})},$ (3.6)
which describes a traceless symmetric tensor with $\ell-1$ indices. The Ward
identity (3.2) is now a simple vector equation, whose components can be
resolved taking its scalar product with various reference vectors. For
instance, the Ward identity given by $p\cdot\widehat{K}_{12}$ becomes
$\displaystyle\sum_{m=0}^{\ell}\varepsilon_{m}\bigg{\\{}\Omega_{12}^{m}\left[(1+w_{1}-w_{2})\widehat{D}_{1}+(1-w_{1}+w_{2})\widehat{D}_{2}\right]\Omega_{12}^{-m}$
$\displaystyle+\frac{(\ell-m)(d-2+\ell+m)}{2}$
$\displaystyle\bigg{\\}}V_{\Delta,\ell,m}^{[12]}(w_{1},w_{2})=0.$ (3.7)
If one introduce an auxiliary vector $r$ orthogonal to both $p$ and $q_{12}$,
the Ward identity $r\cdot\widehat{K}_{12}$ gives
$\displaystyle\sum_{m=0}^{\ell}\bigg{\\{}\varepsilon^{\prime}_{m}\Omega_{12}^{-1}\widehat{D}_{0}+\varepsilon^{\prime}_{m-1}\frac{m(\Delta-
m-d+2)(d-3+\ell+m)}{(d-3+2m)(d-5+2m)}$
$\displaystyle-\varepsilon^{\prime}_{m+1}\frac{(\ell-m)(\Delta+m-1)}{m+1}$
$\displaystyle\bigg{\\}}V_{\Delta,\ell,m}^{[12]}(w_{1},w_{2})=0.$ (3.8)
Finally, the remaining content of the Ward identity can be obtained from the
scalar product $q_{12}\cdot\widehat{K}_{12}$, which after subtraction of eq.
(3.8) leads to
$\displaystyle\sum_{m=0}^{\ell}\bigg{\\{}$
$\displaystyle\varepsilon_{m}\Omega_{12}^{m}\left(\widehat{D}_{1}-\widehat{D}_{2}\right)\Omega_{12}^{-m}$
$\displaystyle-\varepsilon^{\prime}_{m}\frac{z\cdot
p}{\Omega_{12}}(\Delta-\ell-d+2)-\varepsilon^{\prime}_{m-1}\frac{z\cdot
p}{\Omega_{12}^{2}}\frac{m}{d-5+2m}\widehat{D}_{0}\bigg{\\}}V_{\Delta,\ell,m}^{[12]}(w_{1},w_{2})=0.$
(3.9)
Several elements of eqs. (3.7), (3.8) and (3.9) must still be explained: the
quantity $\Omega_{12}=\sqrt{(1-w_{1}-w_{2})^{2}-4w_{1}w_{2}}$ was originally
defined in eq. (2.13); $\widehat{D}_{1}$ and $\widehat{D}_{2}$ are elliptic
differential operators of the Appell $F_{4}$ type,
$\displaystyle\widehat{D}_{1}$
$\displaystyle=w_{1}(1-w_{1})\frac{\partial^{2}}{\partial
w_{1}^{2}}-2w_{1}w_{2}\frac{\partial^{2}}{\partial w_{1}\partial
w_{2}}-w_{2}^{2}\frac{\partial^{2}}{\partial w_{2}^{2}}$
$\displaystyle\quad+\left[c_{1}-(a+b+1)w_{1}\right]\frac{\partial}{\partial
w_{1}}-(a+b+1)w_{2}\frac{\partial}{\partial w_{2}}-ab,$ (3.10)
$\displaystyle\widehat{D}_{2}$
$\displaystyle=w_{2}(1-w_{2})\frac{\partial^{2}}{\partial
w_{2}^{2}}-2w_{1}w_{2}\frac{\partial^{2}}{\partial w_{1}\partial
w_{2}}-w_{1}^{2}\frac{\partial^{2}}{\partial w_{1}^{2}}$
$\displaystyle\quad+\left[c_{2}-(a+b+1)w_{2}\right]\frac{\partial}{\partial
w_{2}}-(a+b+1)w_{1}\frac{\partial}{\partial w_{1}}-ab,$ (3.11)
with
$\displaystyle a$ $\displaystyle=\frac{d+\Delta-\Delta_{1}-\Delta_{2}+m}{2},$
$\displaystyle c_{1}$ $\displaystyle=1+\frac{d}{2}-\Delta_{1},$ $\displaystyle
b$ $\displaystyle=\frac{2d-\Delta-\Delta_{1}-\Delta_{2}+m}{2},$ $\displaystyle
c_{2}$ $\displaystyle=1+\frac{d}{2}-\Delta_{2};$ (3.12)
the remaining differential operator $\widehat{D}_{0}$ is given by
$\displaystyle\widehat{D}_{0}$
$\displaystyle=(1-w_{1}+w_{2})\left[-2w_{1}\frac{\partial}{\partial
w_{1}}+(\Delta_{1}-d+1)\right]$
$\displaystyle\quad-(1+w_{1}-w_{2})\left[-2w_{2}\frac{\partial}{\partial
w_{2}}+(\Delta_{2}-d+1)\right].$ (3.13)
The interpretation of this system of partial differential equations for
$\ell+1$ functions of 2 variables $w_{1}$ and $w_{2}$ appears difficult at
first. This is worsened by the fact that we have not expressed the conformal
Ward identities in the complete basis $\\{\varepsilon_{m}\\}$ of traceless,
symmetric tensors, but have instead appealed to the additional tensors
$\varepsilon^{\prime}_{m}$. Nevertheless, the system contains a lot of
redundancies, due in particular to the closure of the conformal algebra in
momentum space, and we will see in the next sections how a simple solution
emerges.
### 3.2 Highest-spin functions in various orderings
The first key observation to be made is that the Ward identities (3.7) and
(3.9) contain a closed system of equations for the function
$V_{\Delta,\ell,\ell}^{[12]}(w_{1},w_{2})$, i.e. the highest-spin component
$m=\ell$. The tensors $(z\cdot p)\varepsilon^{\prime}_{i}$ appearing in eq.
(3.9) are orthogonal to $\varepsilon_{\ell}$, the latter being totally
transverse while the former not. When projecting against $\varepsilon_{\ell}$,
these two equations give therefore the system
$\displaystyle\left[(1+w_{1}-w_{2})\widehat{D}_{1}+(1-w_{1}+w_{2})\widehat{D}_{2}\right]\Omega_{12}^{-\ell}V_{\Delta,\ell,\ell}^{[12]}(w_{1},w_{2})$
$\displaystyle=0,$ (3.14)
$\displaystyle\left(\widehat{D}_{1}-\widehat{D}_{2}\right)\Omega_{12}^{-\ell}V_{\Delta,\ell,\ell}^{[12]}(w_{1},w_{2})$
$\displaystyle=0,$ (3.15)
which is equivalent to
$\widehat{D}_{1}\left[\Omega_{12}^{-\ell}V_{\Delta,\ell,\ell}^{[12]}(w_{1},w_{2})\right]=\widehat{D}_{2}\left[\Omega_{12}^{-\ell}V_{\Delta,\ell,\ell}^{[12]}(w_{1},w_{2})\right]=0.$
(3.16)
As already stated, $\widehat{D}_{1}$ and $\widehat{D}_{2}$ form a system of
generalized hypergeometric differential equations of the Appell $F_{4}$ type,
with a particular solution being $F_{4}(a,b;c_{1},c_{2};w_{1},w_{2})$. General
solutions to such systems can be constructed from linear combinations of the
four functions [67]
$\begin{array}[]{r@{\hspace{18mm}}r}w_{1}^{\Delta_{1}-d/2}w_{2}^{\Delta_{2}-d/2}F_{\Delta_{1}\Delta_{2};\Delta,\ell}(w_{1},w_{2}),\hskip
51.21495pt&w_{1}^{\Delta_{1}-d/2}F_{\widetilde{\Delta}_{1}\Delta_{2};\Delta,\ell}(w_{1},w_{2}),\\\
w_{2}^{\Delta_{2}-d/2}F_{\Delta_{1}\widetilde{\Delta}_{2};\Delta,\ell}(w_{1},w_{2}),\hskip
51.21495pt&F_{\widetilde{\Delta}_{1}\widetilde{\Delta}_{2};\Delta,\ell}(w_{1},w_{2}),\end{array}$
(3.17)
where we have used $\widetilde{\Delta}_{i}=d-\Delta_{i}$ and introduced the
following notation for the Appell $F_{4}$ function:
$F_{\Delta_{a}\Delta_{b};\Delta,\ell}(w_{1},w_{2})\equiv
F_{4}\left(\begin{array}[]{c}\frac{\Delta_{a}+\Delta_{b}-\Delta+\ell}{2},\leavevmode\nobreak\
\frac{\Delta_{a}+\Delta_{b}+\Delta+\ell-d}{2}\\\
\Delta_{a}-\frac{d}{2}+1,\leavevmode\nobreak\
\Delta_{b}-\frac{d}{2}+1\end{array};w_{1},w_{2}\right).$ (3.18)
Note that the function $F$ satisfies the identities
$F_{\Delta_{1}\Delta_{2};\Delta,\ell}(w_{1},w_{2})=F_{\Delta_{1}\Delta_{2};\widetilde{\Delta},\ell}(w_{1},w_{2})=F_{\Delta_{2}\Delta_{1};\Delta,\ell}(w_{2},w_{1}).$
(3.19)
An equivalent basis of solutions is obtained after the change of variables
$(w_{1},w_{2})\to(w_{1}/w_{2},1/w_{2})$, in terms of the four functions
$\begin{array}[]{r@{\hspace{8mm}}r}w_{1}^{\Delta_{1}-d/2}w_{2}^{(\Delta_{2}-\Delta_{1}-\Delta-\ell)/2}F_{\Delta_{1}\Delta;\Delta_{2},\ell}\left(\frac{w_{1}}{w_{2}},\frac{1}{w_{2}}\right),\hskip
22.76219pt&w_{2}^{(\Delta_{1}+\Delta_{2}-\Delta-\ell-d)/2}F_{\widetilde{\Delta}_{1}\Delta;\Delta_{2},\ell}\left(\frac{w_{1}}{w_{2}},\frac{1}{w_{2}}\right),\\\
w_{1}^{\Delta_{1}-d/2}w_{2}^{(\Delta_{2}-\Delta_{1}+\Delta-\ell-d)/2}F_{\Delta_{1}\widetilde{\Delta};\Delta_{2},\ell}\left(\frac{w_{1}}{w_{2}},\frac{1}{w_{2}}\right),\hskip
22.76219pt&w_{2}^{(\Delta_{1}+\Delta_{2}+\Delta-\ell-2d)/2}F_{\widetilde{\Delta}_{1}\widetilde{\Delta};\Delta_{2},\ell}\left(\frac{w_{1}}{w_{2}},\frac{1}{w_{2}}\right).\end{array}$
(3.20)
The highest-spin vertex function $V_{\Delta,\ell,\ell}^{[12]}(w_{1},w_{2})$ is
therefore a linear combination of the functions (3.17), or of the functions
(3.20), multiplied by $\Omega_{12}^{\ell}$. What linear combination depends on
the ordering of the operators in the 3-point function (3.1). We examine below
two distinct cases.
#### Wightman function
When $\phi_{1}$ and $\phi_{2}$ are out-of-time-order, the Wightman function
(3.1) has been computed in ref. [64] (see also ref. [68]). The key observation
is that the Euclidean OPE fixes the behavior of the Wightman 3-point function
in a neighborhood of $p_{1}^{2}=p^{2}=0$, i.e. around $w_{1}=0$ and
$w_{2}\to-\infty$. Among the solutions to the conformal Ward identities only
the first function of the list (3.20) satisfies the constraint.
The normalization of the 3-point function can also be determined from the
results of ref. [64], which are given in terms of traceless, symmetric tensors
built out of $p$ and $p_{1}$. The comparison can be made between the term of
order $(z\cdot p_{1})^{\ell}$ in that tensor, and the same power coming from
the term $(z\cdot q_{12})^{\ell}$ in
$\varepsilon_{\ell}^{\mu_{1}\ldots\mu_{\ell}}(p,q_{12})$, using
$q_{12}^{\mu}=-2\Omega_{12}^{-1}p_{1}^{\mu}+\ldots$ We obtain finally444Note
that we replaced the generic label $[12]$ with $12$ for the Wightman function.
Similarly, we use $\operatorname{T}[12]$ for time-ordered product given later
in eq. (3.27).
$\boxed{\begin{aligned}
V_{\Delta,\ell,\ell}^{12}(w_{1},w_{2})&=\frac{i^{\ell}(4\pi)^{d+2}2^{-(\Delta_{1}+\Delta_{2}+\Delta+\ell+2)}(\Delta-1)_{\ell}\left[(1-w_{1}-w_{2})^{2}-4w_{1}w_{2}\right]^{\ell/2}}{\Gamma\left(\Delta_{1}-\frac{d}{2}+1\right)\Gamma\left(\Delta-\frac{d}{2}+1\right)\Gamma\left(\frac{\Delta_{2}-\Delta_{1}+\Delta+\ell}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}-\Delta+\ell}{2}\right)}\\\
&\quad\times
w_{1}^{\Delta_{1}-d/2}(-w_{2})^{(\Delta_{2}-\Delta_{1}-\Delta-\ell)/2}F_{\Delta_{1}\Delta;\Delta_{2},\ell}\left(\frac{w_{1}}{w_{2}},\frac{1}{w_{2}}\right).\end{aligned}}$
(3.21)
This expression is valid when $p_{1}$ is time-like and backward-directed (the
Wightman function vanishes otherwise by the spectral condition), and when
$p_{2}$ is space-like, implying $w_{1}>0$ and $w_{2}<0$. The case of time-like
$p_{2}$ can be obtained in terms of the functions in the list (3.17) following
the analysis of ref. [64]. Note that the normalization of OPE coefficients
that we use here is such that the analytic continuation to Euclidean position
space corresponds to the 3-point function
$\displaystyle z_{\mu_{1}}\cdots z_{\mu_{\ell}}$ $\displaystyle\left\langle
0\right|\phi_{1}(x_{1})\phi_{2}(x_{2})\mathcal{O}^{\mu_{1}\ldots\mu_{\ell}}(0)\left|0\right\rangle$
$\displaystyle=\frac{\lambda_{12\mathcal{O}}\left[x_{2}^{2}(z\cdot
x_{1})-x_{1}^{2}(z\cdot
x_{2})\right]^{\ell}}{|x_{1}|^{\Delta_{1}-\Delta_{2}+\Delta+\ell}|x_{2}|^{\Delta_{2}-\Delta_{1}+\Delta+\ell}|x_{1}-x_{2}|^{\Delta_{1}+\Delta_{2}-\Delta+\ell}}.$
(3.22)
#### Time-ordered product
In the case where the 3-point function (3.1) contains a time-ordered product
of the operators $\phi_{1}$ and $\phi_{2}$, the general structure is more
complicated, as both $p_{1}$ and $p_{2}$ can be either space-like or time-
like. We focus here on the case where they are both space-like, i.e.
$w_{1},w_{2}<0$. In this case, the Euclidean OPE condition only gives
information about the behavior at $p^{2}=0$ ($w_{1},w_{2}\to-\infty$). It
implies that the highest-spin vertex function is a linear combination of the
first two functions in the list (3.20),
$\displaystyle V_{\Delta,\ell,\ell}^{\operatorname{T}[12]}(w_{1},w_{2})$
$\displaystyle=(-w_{2})^{(\Delta_{1}+\Delta_{2}-\Delta-\ell-d)/2}\left[(1-w_{1}-w_{2})^{2}-4w_{1}w_{2}\right]^{\ell/2}$
$\displaystyle\quad\times\left[A\,\left(\frac{w_{1}}{w_{2}}\right)^{\Delta_{1}-d/2}F_{\Delta_{1}\Delta;\Delta_{2},\ell}\left(\frac{w_{1}}{w_{2}},\frac{1}{w_{2}}\right)+B\,F_{\widetilde{\Delta}_{1}\Delta;\Delta_{2},\ell}\left(\frac{w_{1}}{w_{2}},\frac{1}{w_{2}}\right)\right].$
(3.23)
The coefficients $A$ and $B$ can be fixed by symmetry arguments and by the
relation to the Wightman function. The singularities of the time-ordered
product have been analyzed in ref. [69], where it was found that there is a
branch-point singularity at $w_{1}=0$ when $\Delta_{1}<\frac{d}{2}$, and that
the coefficient of this singularity is related to the Wightman function by a
multiplicative factor, giving
$\lim_{w_{1}\to
0}(-w_{1}-i\epsilon)^{d/2-\Delta_{1}}V_{\Delta,\ell,\ell}^{\operatorname{T}[12]}(w_{1},w_{2})=\frac{1}{2i\sin\left[\pi\left(\frac{d}{2}-\Delta_{1}\right)\right]}\lim_{w_{1}\to
0_{+}}w_{1}^{d/2-\Delta_{1}}V_{\Delta,\ell,\ell}^{12}(w_{1},w_{2}).$ (3.24)
This implies
$A=\frac{i^{\ell-1}(4\pi)^{d+1}\Gamma\left(\frac{d}{2}-\Delta_{1}\right)(\Delta-1)_{\ell}}{2^{\Delta_{1}+\Delta_{2}+\Delta+\ell+1}\Gamma\left(\Delta-\frac{d}{2}+1\right)\Gamma\left(\frac{\Delta_{2}-\Delta_{1}+\Delta+\ell}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}-\Delta+\ell}{2}\right)}.$
(3.25)
The coefficient $B$ can be fixed in turn by imposing the exchange symmetry
between the operators $\phi_{1}$ and $\phi_{2}$, which is built into the time-
ordered product but does not appear obvious in eq. (3.23). This constraint can
be resolved by going from the basis of Appell $F_{4}$ functions (3.20) to the
symmetric basis (3.17). We see that the result is symmetric under the
simultaneous exchange $w_{1}\leftrightarrow w_{2}$ and
$\Delta_{1}\leftrightarrow\Delta_{2}$ if and only if
$B=\frac{\Gamma\left(\Delta_{1}-\frac{d}{2}\right)\Gamma\left(\frac{\Delta_{2}-\Delta_{1}+\Delta+\ell}{2}\right)\Gamma\left(\frac{\Delta-\Delta_{1}-\Delta_{2}+\ell+d}{2}\right)}{\Gamma\left(\frac{d}{2}-\Delta_{1}\right)\Gamma\left(\frac{\Delta_{1}-\Delta_{2}+\Delta+\ell}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}+\Delta+\ell-d}{2}\right)}\,A,$
(3.26)
after which we arrive at
$\boxed{\begin{aligned}
V_{\Delta,\ell,\ell}^{\operatorname{T}[12]}(w_{1},w_{2})&=\frac{i^{\ell-1}(4\pi)^{d+1}2^{-(\Delta_{1}+\Delta_{2}+\Delta+\ell+1)}(\Delta-1)_{\ell}\left[(1-w_{1}-w_{2})^{2}-4w_{1}w_{2}\right]^{\ell/2}}{\Gamma\left(\frac{\Delta_{1}-\Delta_{2}+\Delta+\ell}{2}\right)\Gamma\left(\frac{\Delta_{2}-\Delta_{1}+\Delta+\ell}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}-\Delta+\ell}{2}\right)\Gamma\left(\frac{\Delta_{1}+\Delta_{2}+\Delta+\ell-d}{2}\right)}\\\
&\quad\times\bigg{[}f_{\Delta_{1}\Delta_{2};\Delta,\ell}(-w_{1})^{\Delta_{1}-d/2}(-w_{2})^{\Delta_{2}-d/2}F_{\Delta_{1}\Delta_{2};\Delta,\ell}(w_{1},w_{2})\\\
&\quad\qquad+f_{\Delta_{1}\widetilde{\Delta}_{2};\Delta,\ell}(-w_{1})^{\Delta_{1}-d/2}F_{\Delta_{1}\widetilde{\Delta}_{2};\Delta,\ell}(w_{1},w_{2})\\\
&\quad\qquad+f_{\widetilde{\Delta}_{1}\Delta_{2};\Delta,\ell}(-w_{2})^{\Delta_{2}-d/2}F_{\widetilde{\Delta}_{1}\Delta_{2};\Delta,\ell}(w_{1},w_{2})\\\
&\quad\qquad+f_{\widetilde{\Delta}_{1}\widetilde{\Delta}_{2};\Delta,\ell}F_{\widetilde{\Delta}_{1}\widetilde{\Delta}_{2};\Delta,\ell}(w_{1},w_{2})\bigg{]},\end{aligned}}$
(3.27)
where we have defined
$f_{\Delta_{a}\Delta_{b};\Delta,\ell}=\frac{\Gamma\left(\frac{d}{2}-\Delta_{a}\right)\Gamma\left(\frac{d}{2}-\Delta_{b}\right)\Gamma\left(\frac{\Delta_{a}+\Delta_{b}+\Delta+\ell-d}{2}\right)}{\Gamma\left(1-\frac{\Delta_{a}+\Delta_{b}-\Delta+\ell}{2}\right)}.$
(3.28)
This result is analytic in the scaling dimensions $\Delta_{1},\Delta_{2}$,
with the exception of poles at $\Delta_{1},\Delta_{2}=\frac{d}{2}+n$ with
integer $n$. This means that the result (3.27) is valid in all generality, and
not only in the regime $\Delta_{1}<\frac{d}{2}$ used to fix the coefficient
$A$. The apparent divergences when $\Delta_{1},\Delta_{2}=\frac{d}{2}+n$ are
reminiscent of the anomalies discussed in refs. [70, 71]: they appear
precisely when the powers of $w_{1}$ and $w_{2}$ are integer, in which case
the terms of the different hypergeometric series can be mixed. Resolving these
special cases by analytic continuation in $\Delta_{i}$ (or in $d$), one finds
cancellations between the coefficients $f_{\Delta_{a}\Delta_{b};\Delta,\ell}$,
so that the vertex functions remain in fact finite, however with the
appearance of logarithms of $w_{1}$ and $w_{2}$. An example realizing the case
$\Delta_{1}=\Delta_{2}=\frac{d}{2}$ is presented below in section 4.3.
#### Other orderings
Besides the Wightman function and the time-ordered product, other orderings of
the operators can be envisioned, but they can all be related to the two
results (3.21) and (3.27): ordinary commutators are nothing but the difference
between two Wightman functions; causal commutators (i.e. advanced or retarded)
are similarly related to the sum or difference between the time-ordered
product and the Wightman function, and as such they coincide with the former
when both momenta $p_{1}$ and $p_{2}$ are space-like; the anti-time-ordered
product is complex conjugated to the time-ordered one. In order to obtain a
complete picture of all the possible vertex functions, one needs of course to
go beyond the kinematics assumptions made in deriving eqs. (3.21) and (3.27).
Causal commutators would be a useful tool to do so, as they have well-
understood domains of analyticity in terms of complex momenta. We leave this
task for future work.
One important property that is shared by all orderings is the vanishing of the
highest-spin vertex function when the intermediate operator takes a double-
trace dimension, i.e. when $\Delta=\Delta_{1}+\Delta_{2}+\ell+2n$ with integer
$n$. This ensures the triviality of Gaussian theories in momentum space, and
places on equal footing the contributions of single- and double-trace
operators in large-$N$ theories. It also implies that the conformal partial
wave expansion (1.2) shares the “dispersive” property of the double
discontinuity in position space [38].
### 3.3 Lower spin functions
With the knowledge of the highest-spin vertex function, the computation of all
remaining vertex functions $V_{\Delta,\ell,m}^{[12]}$ with $m<\ell$ does not
require solving more partial differential equations, nor making assumptions
about boundary conditions, as eq. (3.8) gives a simple two-step algebraic
recursion relation: since the tensors $\varepsilon^{\prime}_{m}$ are
orthogonal to each other, we get
$\boxed{\begin{aligned}
V_{\Delta,\ell,m}^{[12]}(w_{1},w_{2})&=\frac{m+1}{(\ell-m)(\Delta+m-1)}\\\
&\quad\times\bigg{[}\frac{1}{\sqrt{(1-w_{1}-w_{2})^{2}-4w_{1}w_{2}}}\widehat{D}_{0}V_{\Delta,\ell,m+1}^{[12]}(w_{1},w_{2})\\\
&\quad\qquad+\frac{(m+2)(\Delta-
m-d)(d-1+\ell+m)}{(d-1+2m)(d+1+2m)}\,V_{\Delta,\ell,m+2}^{[12]}(w_{1},w_{2})\bigg{]},\end{aligned}}$
(3.29)
using the first-order differential operator $\widehat{D}_{0}$ defined in eq.
(3.13). The last line must be dropped in the first step of the recursion,
namely when $m=\ell-1$; equivalently, the recursion relation can be defined in
terms of the functions $V_{\Delta,\ell,\ell}^{[12]}(w_{1},w_{2})$ computed
above and of $V_{\Delta,\ell,\ell+1}^{[12]}(w_{1},w_{2})=0$. Note that the
operator $\widehat{D}_{0}$ is odd under the permutation $1\leftrightarrow 2$,
which is consistent with the fact that the Gegenbauer polynomials
$\mathcal{C}_{m}^{(d-3)/2}(\cos\theta)$ is similarly odd under
$\cos\theta\to-\cos\theta$ for odd $m$: in this way the exchange symmetry
$1\leftrightarrow 2$ of the time-ordered product is preserved for all $m$.
Derivatives of Appell $F_{4}$ functions can be again expressed in terms of
linear combinations of $F_{4}$ functions, so one could in principle write down
closed-form formulae for the vertex functions
$V_{\Delta,\ell,m}^{[12]}(w_{1},w_{2})$ with the different operator orderings.
This exercise is not particularly enlightening, though, and the recursion
relation (3.29) seems simple enough to provide the most efficient
implementation of the vertex functions. In fact, since $\widehat{D}_{0}$ is a
first-order differential operator that only raises powers of $w_{1}$ and
$w_{2}$, the vertex functions with $m<\ell$ inherit the analytic structure of
the highest-spin function.
The case in which the operator $\mathcal{O}^{\mu_{1}\ldots\mu_{\ell}}$ is a
conserved current deserves special attention. It turns out that the
differential operator $\widehat{D}_{0}$ annihilates
$V_{\Delta,\ell,\ell}^{[12]}(w_{1},w_{2})$ whenever the unitarity bound is
saturated ($\Delta=d-2+\ell$) and the two operators $\phi_{1}$ and $\phi_{2}$
have identical scaling dimensions ($\Delta_{1}=\Delta_{2}$), two conditions
that must be met for $\mathcal{O}$ to be conserved. This implies that
$V_{\Delta,\ell,\ell-1}^{[12]}(w_{1},w_{2})=0$. Moreover, the factor $(\Delta-
m-d)$ in eq. (3.29) ensures that the second step of the recursion is trivial
as well, and so $V_{\Delta,\ell,\ell-2}^{[12]}(w_{1},w_{2})=0$. It follows
immediately that all vertex functions with $m<\ell$ vanish in this case. This
is not a surprise: conserved currents only admit transverse polarizations,
while any longitudinal polarization defines a null state. This property is
essential in the conformal partial wave expansion, as it implies that one does
not need to sum over $m<\ell$ for conserved currents, and hence that the
singularities in the definition (2.35) of the coefficient $C_{\Delta,\ell,m}$
never occurs.
### 3.4 Special kinematic limits
We will finally examine several special kinematic limits in which the vertex
functions admits a simpler form, and compare the conformal partial waves in
these limits with results already existing in the literature.
#### Light-like limit
The first interesting case is the limit $w_{1},w_{2}\to 0$ of the vertex
function with a time-ordered product. As the Appell $F_{4}$ function is
analytic around the point $(w_{1},w_{2})=(0,0)$, one can see from eq. (3.27)
that the limit exists if $\Delta_{1},\Delta_{2}>\frac{d}{2}$ (the opposite
situation is discussed below). In this case, the recursion relation (3.29)
becomes an algebraic equation,
$\displaystyle V_{\Delta,\ell,m}^{\operatorname{T}[12]}(0,0)$
$\displaystyle=\frac{m+1}{(\ell-m)(\Delta+m-1)}\bigg{[}(\Delta_{1}-\Delta_{2})V_{\Delta,\ell,m+1}^{[12]}(0,0)$
$\displaystyle\hskip 71.13188pt+\frac{(m+2)(\Delta-
m-d)(d-1+\ell+m)}{(d-1+2m)(d+1+2m)}\,V_{\Delta,\ell,m+2}^{\operatorname{T}[12]}(0,0)\bigg{]},$
(3.30)
which is solved by
$\displaystyle V_{\Delta,\ell,m}^{\operatorname{T}[12]}(0,0)$
$\displaystyle=V_{\Delta,\ell,\ell}^{\operatorname{T}[12]}(0,0)\frac{(-2)^{\ell-m}\ell!}{m!(\ell-m)!}\frac{\left(\frac{\Delta_{2}-\Delta_{1}+\Delta-\ell-d+2}{2}\right)_{\ell-m}}{\left(\Delta-1+m\right)_{\ell-m}}$
$\displaystyle\quad\times{}_{3}F_{2}\left(\begin{array}[]{c}-(\ell-m),\leavevmode\nobreak\
d-\Delta-1+m,\leavevmode\nobreak\ \frac{d-2+2m}{2}\\\
\frac{\Delta_{1}-\Delta_{2}+d-\Delta-\ell+2m}{2},\leavevmode\nobreak\
d-2+2m\end{array};1\right).$ (3.33)
Remarkably, the recursion relation only depends on the difference between the
external scaling dimensions, $\Delta_{1}-\Delta_{2}$, and not on their sum. In
fact, when $\Delta_{1}=\Delta_{2}(\equiv\Delta_{\phi})$, all vertex functions
with odd $\ell-m$ vanish identically, and one obtains
$V_{\Delta,\ell,\ell-2n}^{\operatorname{T}[\phi\phi]}(0,0)=\frac{(-1)^{n}\ell!}{2^{2n}n!(\ell-2n)!}\frac{\left(\frac{\Delta-\ell-d+2}{2}\right)_{n}}{\left(\frac{d-1}{2}+\ell-2n\right)_{n}\left(\frac{3-\Delta-\ell}{2}\right)_{n}}\,V_{\Delta,\ell,\ell}^{\operatorname{T}[\phi\phi]}(0,0).$
(3.34)
The conformal partial waves (1.4) can then be written as
$G_{\Delta,\ell}(w_{i}=0,\cos\theta)=\frac{2^{2\Delta+1}\Gamma\left(\Delta-\frac{d}{2}+1\right)\Gamma(\Delta+\ell)}{(4\pi)^{(d+2)/2}(\Delta-1)_{\ell}}\left[V_{\Delta,\ell,\ell}^{[\phi\phi]}(0,0)\right]^{2}g_{\Delta,\ell}(\cos\theta),$
(3.35)
where $g_{\Delta,\ell}$ is a polynomial of degree $\ell$ in $\cos\theta$,
defined by
$g_{\Delta,\ell}(\cos\theta)=\sum_{n=0}^{\ell/2}\frac{(-1)^{n}\ell!(2n)!}{2^{\ell+2n+1}(n!)^{2}}\frac{d-3+2\ell-4n}{\left(2-\frac{d}{2}-\ell\right)_{n}\left(\frac{d-3}{2}\right)_{\ell-n+1}}\frac{\big{(}\frac{2-\Delta-\ell}{2}\big{)}_{n}\big{(}\frac{2-\widetilde{\Delta}-\ell}{2}\big{)}_{n}}{\big{(}\frac{3-\Delta-\ell}{2}\big{)}_{n}\big{(}\frac{3-\widetilde{\Delta}-\ell}{2}\big{)}_{n}}\mathcal{C}_{\ell-2n}^{(d-3)/2}(\cos\theta).$
(3.36)
These are precisely the polynomials defined in ref. [63], where the light-like
limit of the conformal partial wave expansion was studied. The
$g_{\Delta,\ell}$ have the remarkable property of interpolating continuously
between the Gegenbauer polynomials $\mathcal{C}_{\ell}^{(d-3)/2}(\cos\theta)$
when $\Delta-\ell$ saturates the unitarity bound, and
$\mathcal{C}_{\ell}^{(d-2)/2}(\cos\theta)$ when $\Delta-\ell\to\infty$. Using
$V_{\Delta,\ell,\ell}^{\operatorname{T}[\phi\phi]}(0,0)=\frac{i^{\ell-1}(4\pi)^{d+1}2^{-(2\Delta_{\phi}+\Delta+\ell+1)}\Gamma\left(\Delta_{\phi}-\frac{d}{2}\right)^{2}\Gamma\left(\frac{\Delta+\ell+d}{2}-\Delta_{\phi}\right)(\Delta-1)_{\ell}}{\Gamma\left(\frac{\Delta+\ell}{2}\right)^{2}\Gamma\left(\Delta_{\phi}-\frac{\Delta-\ell}{2}\right)\Gamma\left(\Delta_{\phi}+\frac{\Delta+\ell-d}{2}\right)\Gamma\left(\Delta_{\phi}+\frac{\Delta-\ell}{2}-d+1\right)},$
(3.37)
one can also verify that the multiplicative coefficient in eq. (3.35) also
matches the definition of $G_{\Delta,\ell}(\cos\theta)$ in ref. [63].555Eq.
(1.14) in ref. [63] contains a typo: the numerator is missing a factor of
$\Gamma(\Delta+\ell)$ (corrected in the arXiv version).
#### Scattering amplitude and form factor
An alternative, maybe more interesting situation is the asymptotic limit
$w_{i}\to 0$ in the case $\Delta_{i}<\frac{d}{2}$. The time-ordered vertex
function (3.27) diverges in this case, but one obtains a well-defined limit by
multiplying it with the appropriate powers of $w_{i}$. The peculiarity of this
limit is that it allows to define objects that have an interesting field-
theoretical interpretation: consider
$F(s,t,u)\equiv\left[\prod_{i=1}^{3}\lim_{p_{i}^{2}\to
0_{-}}(-p_{i}^{2})^{d/2-\Delta_{i}}\right]\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\operatorname{T}[\phi_{1}(p_{1})\phi_{2}(p_{2})\phi_{3}(p_{3})\phi_{4}(-p_{1}-p_{2}-p_{3})]\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}},$
(3.38)
as well as the subsequent limit $p_{4}^{2}=s+t+u\to 0$,
$A(s,t)\equiv\lim_{p_{4}^{2}\to 0_{-}}(-p_{4}^{2})^{d/2-\Delta_{4}}F(s,t,u).$
(3.39)
$F$ and $A$ have been interpreted respectively as a “form factor” and as an
“amplitude” [69]: on the one hand, they are related to the time-ordered
correlation function in eq. (3.38) by a procedure similar to the LSZ reduction
in massive quantum field theory; on the other hand, from a careful analysis of
the singularities in the Fourier transform from position space, one can show
that they admit a conformal partial wave expansion that is reminiscent of the
“unitarity cuts” method relating higher-point amplitudes to lower-point ones.
This latter property follows from the fact that the form factor is related to
the same limit of the partially-time-ordered function,
$F(s,t,u)\sim\left[\prod_{i=1}^{3}\lim_{p_{i}^{2}\to
0_{-}}(-p_{i}^{2})^{d/2-\Delta_{\phi}}\right]\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\operatorname{T}[\phi_{1}(p_{1})\phi_{2}(p_{2})]\operatorname{T}[\phi_{3}(p_{3})\phi_{4}(-p_{1}-p_{2}-p_{3})]\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}},$
(3.40)
provided that the limit is taken with $p_{1}$, $p_{2}$ backward-directed, and
$p_{3}$ forward-directed. The equivalence is up to a phase involving the
dilatation operator (i.e. it depends on the scaling dimension $\Delta$ of the
intermediate operator in the conformal partial wave). We do not want to re-
derive here the results of ref. [69], but we can directly make use of the
right-hand side of the equivalence (3.40) to provide a simple computation of
the form factor $F$ and of the amplitude $A$ using our conformal partial
waves, up to the aforementioned phase.
The computation involves two distinct vertex functions,
$V_{\Delta,\ell,m}^{\operatorname{T}[12]}$ and the complex conjugate of
$V_{\Delta,\ell,m}^{\bar{\operatorname{T}}[34]}$.
$V_{\Delta,\ell,m}^{\operatorname{T}[12]}$ is evaluated with both $w_{1}$ and
$w_{2}\to 0$. In this case the computation proceeds as in the light-like limit
discussed above: the recursion relation (3.29) for the vertex function is
again algebraic, though it differs from eq. (3.30) by the replacement
$\Delta_{1,2}\to d-\Delta_{1,2}$. One arrives at
$\displaystyle\lim_{w_{1},w_{2}\to
0_{-}}(-w_{1})^{d/2-\Delta_{1}}(-w_{2})^{d/2-\Delta_{2}}V_{\Delta,\ell,m}^{\operatorname{T}[12]}(w_{1},w_{2})$
$\displaystyle\quad=\frac{i^{\ell-1}(4\pi)^{d}\Gamma\left(\frac{d}{2}-\Delta_{1}\right)\Gamma\left(\frac{d}{2}-\Delta_{2}\right)(\Delta-1)_{\ell}\left(\frac{\Delta_{1}-\Delta_{2}+\Delta-\ell-d+2}{2}\right)_{\ell-m}\sin\left[\pi\frac{\Delta_{1}+\Delta_{2}-\Delta+\ell}{2}\right]}{2^{\Delta_{1}+\Delta_{2}+\Delta+\ell-1}\Gamma\left(\frac{\Delta_{1}-\Delta_{2}+\Delta+\ell}{2}\right)\Gamma\left(\frac{\Delta_{2}-\Delta_{1}+\Delta+\ell}{2}\right)\left(\Delta-1+m\right)_{\ell-m}}$
$\displaystyle\quad\quad\times\frac{(-2)^{\ell-m}\ell!}{m!(\ell-m)!}{}_{3}F_{2}\left(\begin{array}[]{c}-(\ell-m),\leavevmode\nobreak\
d-\Delta-1+m,\leavevmode\nobreak\ \frac{d-2+2m}{2}\\\
\frac{\Delta_{2}-\Delta_{1}+d-\Delta-\ell+2m}{2},\leavevmode\nobreak\
d-2+2m\end{array};1\right).$ (3.43)
For $V_{\Delta,\ell,m}^{\bar{\operatorname{T}}[34]}$, consider first the
highest-spin vertex function as defined in eq. (3.23), which in the limit
$w_{3}\to 0$ becomes
$\displaystyle\lim_{w_{3}\to 0_{-}}$
$\displaystyle(-w_{3})^{d/2-\Delta_{3}}V_{\Delta,\ell,\ell}^{\bar{\operatorname{T}}[34]}(w_{3},w_{4})$
$\displaystyle=A^{*}(-w_{4})^{(\Delta_{3}+\Delta_{4}-\Delta+\ell-2)/2}(1-w_{4})^{1-\Delta_{3}}$
$\displaystyle\quad\times{}_{2}F_{1}\left(\tfrac{\Delta-\ell-\Delta_{3}-\Delta_{4}+2}{2},\tfrac{\Delta_{4}-\Delta_{3}+\Delta-\ell-d+2}{2};\Delta-\tfrac{d}{2}+1;\frac{1}{w_{4}}\right).$
(3.44)
where $A^{*}$ is the complex conjugate of the coefficient $A$ given in eq.
(3.25), upon replacement $1\to 3$ and $2\to 4$. Applying the recursion
relation (3.29) leads to
$\lim_{w_{3}\to
0_{-}}(-w_{3})^{d/2-\Delta_{3}}V_{\Delta,\ell,m}^{\operatorname{T}[12]}(w_{1},w_{2})=A^{*}(-w_{4})^{(\Delta_{3}+\Delta_{4}-\Delta+\ell-2)/2}(1-w_{4})^{1-\Delta_{3}}v_{\Delta,\ell,m}\left(\frac{1}{w_{4}}\right),$
(3.45)
where we have defined
$\displaystyle v_{\Delta,\ell,m}(z)=\sum_{j=0}^{\ell-m}$
$\displaystyle\frac{2^{\ell-m}\ell!}{m!j!(\ell-
m-j)!}\frac{\left(\frac{d}{2}-1+m\right)_{\ell-
m-j}\left(\Delta-\ell-d+2+j\right)_{\ell-m-j}}{\left(d-2+2m\right)_{\ell-
m-j}}$
$\displaystyle\times\frac{\left(\frac{\Delta-\ell-\Delta_{3}-\Delta_{4}+2}{2}\right)_{j}\left(\frac{\Delta_{4}-\Delta_{3}+\Delta-\ell-d+2}{2}\right)_{j}}{\left(2-\Delta-\ell\right)_{\ell-m}\left(\Delta-\frac{d}{2}+1\right)_{j}}$
$\displaystyle\times
z^{j}\,{}_{2}F_{1}\left(\tfrac{\Delta-\ell-\Delta_{3}-\Delta_{4}+2}{2}+j,\tfrac{\Delta_{4}-\Delta_{3}+\Delta-\ell-d+2}{2}+j;\Delta-\frac{d}{2}+1+j;z\right).$
(3.46)
The conformal partial waves corresponding to the form factor follow simply
from plugging the vertex functions (3.43) and (3.45) in eq. (1.4). This gives
a closed-form expression for distinct $\Delta_{i}$ that was lacking in ref.
[69], as well as a new representation of the result in the case of identical
operators ($\Delta_{i}=\Delta_{\phi}$): the function
$f_{\Delta,\ell}(z,\cos\theta)$ computed in appendix B of ref. [69] using a
method based on the conformal Casimir equations can be written
$\displaystyle f_{\Delta,\ell}(z,\cos\theta)$
$\displaystyle=z^{(\Delta-\ell-2\Delta_{\phi})/2}(1-z)^{1-\Delta_{\phi}}$
$\displaystyle\quad\times\sum_{n=0}^{\ell/2}\frac{(-1)^{n}\ell!}{2^{\ell}n!}\frac{\left(\frac{d-3}{2}+\ell-2n\right)\left(\frac{\Delta-\ell-d+2}{2}\right)_{n}}{\left(\frac{d-3}{2}\right)_{\ell-n+1}\left(\frac{3-\Delta-\ell}{2}\right)_{n}}\mathcal{C}_{\ell-2n}^{(d-3)/2}(\cos\theta)v_{\Delta,\ell,\ell-2n}(z)$
(3.47)
in terms of $z=1/w_{4}=p_{4}^{2}/s$. Similarly, for the scattering amplitude
(3.25) one recovers the polynomials $g_{\Delta,\ell}$ of eq. (3.36) in the
case of identical external operators, as well as a new set of polynomials
obtained by plugging the vertex function (3.43) twice in eq. (1.4) when the
external operators are distinct.
These examples of special kinematics are far from being exhaustive, but they
illustrate how our results can be efficiently used in various specific
configurations. This concludes the derivation of the vertex functions, and we
move next to a complete physical example where the conformal partial wave
expansion are put to good use.
## 4 Example: the scalar box integral in 4 dimensions
We present in this section an example in which the conformal partial wave
expansion can be explicitly compared with a known 4-point function. We take
the theory of a free, complex scalar field $\Phi$, and consider the time-
ordered 4-point function involving the composite operator $\Phi^{2}$ and its
complex conjugate $\overline{\Phi}^{2}$,
$\left\langle
0\right|\operatorname{T}[\Phi^{2}(p_{1})\overline{\Phi}^{2}(p_{2})\Phi^{2}(p_{3})\overline{\Phi}^{2}(p_{4})]\left|0\right\rangle.$
(4.1)
This correlation function is given in terms of a single Feynman diagram shown
in figure 3. In $d=4$ dimensions, the integral over the loop momentum can be
performed in arbitrary kinematics, and the result is a Bloch-Wigner
dilogarithm [72, 73, 74]. The simplicity of this result can be traced back to
presence of a dual conformal symmetry at the level of the integrand [45, 46].
Figure 3: The scalar box diagram corresponding to the 4-point function (4.1)
in the theory of a free complex scalar field. The solid lines are propagators
of the field $\Phi$, with arrows indicating the charge flow. The external,
dashed lines correspond to the composite operators $\Phi^{2}$ and
$\overline{\Phi}^{2}$. The dual graph in terms of the variables
$x_{i}-x_{i+1}=p_{i}$ is shown in red, with dotted lines.
### 4.1 CFT optical theorem
The 4-point function (4.1) in itself does not admit a decomposition into
conformal partial waves: the time-ordered product runs over all 4 operators,
and as such it does not respect the structure of the correlator (1.1).
However, the real part of that function does, by an identity that is sometimes
referred to as the “CFT optical theorem” [31, 32] (see also ref. [75]): by a
combinatoric identity involving the time-ordered product and its hermitian
conjugate $\operatorname{\overline{T}}$, it can be shown that
$\displaystyle\left\langle
0\right|\operatorname{T}[\Phi^{2}(p_{1})\overline{\Phi}^{2}(p_{2})\Phi^{2}(p_{3})\overline{\Phi}^{2}(p_{4})]\left|0\right\rangle$
$\displaystyle+\left\langle
0\right|\operatorname{\overline{T}}[\Phi^{2}(p_{1})\overline{\Phi}^{2}(p_{2})\Phi^{2}(p_{3})\overline{\Phi}^{2}(p_{4})]\left|0\right\rangle$
$\displaystyle=\left\langle
0\right|\operatorname{T}[\Phi^{2}(p_{1})\overline{\Phi}^{2}(p_{2})]\operatorname{\overline{T}}[\Phi^{2}(p_{3})\overline{\Phi}^{2}(p_{4})]\left|0\right\rangle,$
(4.2)
provided that the following conditions on the momenta and Mandelstam
invariants are fulfilled:
$p_{i}^{2}<0,\qquad\qquad s>0,\qquad\qquad t,u<0.$ (4.3)
The right-hand side of this optical theorem is precisely in the form of eq.
(1.1), and we can therefore define
$G(s,w_{i},\cos\theta)=\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\operatorname{T}[\Phi^{2}(p_{1})\overline{\Phi}^{2}(p_{2})]\operatorname{\overline{T}}[\Phi^{2}(p_{3})\overline{\Phi}^{2}(p_{4})]\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}.$
(4.4)
At the same time, the left-hand side of eq. (4.2) corresponds to (twice) the
real part of the scalar box integral. This can be compared with the known
results that give [74, 45]666Our real part of the box integral differs from
the usual conventions by a multiplicative factor of $2^{10}\pi^{8}$ that has
to do with the use of the standard CFT normalization of operators in position
space: on the one hand, the free field propagator in 4 dimensions is
$\mathopen{\hbox{${\langle}$}\kern-1.94444pt\leavevmode\hbox{${\langle}$}}\operatorname{T}[\Phi(p)\overline{\Phi}(-p)]\mathclose{\hbox{${\rangle}$}\kern-1.94444pt\leavevmode\hbox{${\rangle}$}}=4\pi^{2}i/p^{2}$,
accounting for an overall factor of $(4\pi^{2})^{4}$; on the other hand, each
vertex involving the properly-normalized CFT operator
$\frac{1}{\sqrt{2}}\Phi^{2}$ gives rise to a factor of $\sqrt{2}$,
contributing in total another factor of 4 to our box integral.
$G(s,w_{i},\cos\theta)=\frac{(2\pi)^{7}}{su}\frac{\log\left(\frac{1-1/z}{1-1/\bar{z}}\right)}{z-\bar{z}}.$
(4.5)
$z$ and $\bar{z}$ are related to the dual conformal cross ratios, given in
terms of $p_{i}=x_{i}-x_{i+1}$ by the two equations
$z\bar{z}=\frac{x_{12}^{2}x_{34}^{2}}{x_{13}^{2}x_{24}^{2}}=\frac{p_{1}^{2}p_{3}^{2}}{su},\qquad\qquad(1-z)(1-\bar{z})=\frac{x_{14}^{2}x_{23}^{2}}{x_{13}^{2}x_{24}^{2}}=\frac{p_{2}^{2}p_{4}^{2}}{su}.$
(4.6)
### 4.2 IR divergences
In spite of the relative simplicity of (4.5) in terms of $z$ and $\bar{z}$,
the dependence on $w_{i}$ and $\cos\theta$ is quite intricate: determining $z$
and $\bar{z}$ requires solving a quadratic equation involving the Mandelstam
invariant $u$, which is itself given by
$\displaystyle u=-\frac{s}{2}\Big{[}\,$ $\displaystyle
1-w_{1}-w_{2}-w_{3}-w_{4}-(w_{1}-w_{2})(w_{3}-w_{4})$
$\displaystyle-\cos\theta\sqrt{(1-w_{1}-w_{2})^{2}-4w_{1}w_{2}}\sqrt{(1-w_{3}-w_{4})^{2}-4w_{3}w_{4}}\Big{]}.$
(4.7)
A comparison with the conformal partial wave expansion at generic kinematics
is quite difficult to achieve. Instead, we will focus on the IR divergences of
this box integral when some or all of the $p_{i}^{2}$ approach zero. In the
simplest case where all “masses” are identical ($w_{i}=w$), we have
$z,\bar{z}=\frac{1}{2}\left(1\pm\sqrt{1+\frac{8w^{2}}{(1-4w)(1-\cos\theta)}}\right),$
(4.8)
and, as $w\to 0$,
$G(s,w_{i}=w,\cos\theta)=-\frac{512\pi^{7}}{s^{2}(1-\cos\theta)}\log\left(\frac{2w^{2}}{1-\cos\theta}\right)+\ldots$
(4.9)
The real part of the box integral has a logarithmic divergence in $w$, and
moreover a pole in the forward limit $\cos\theta\to 1$.
To perform the comparison with the conformal partial waves, one would like to
expand $G$ in spin partial waves first, and write
$G(s,w_{i},\cos\theta)=s^{-2}\sum_{\ell=0}^{\infty}\left(\ell+\tfrac{1}{2}\right)a_{\ell}(w_{i})P_{\ell}(\cos\theta),$
(4.10)
or equivalently, using the orthogonality of Legendre polynomials,
$a_{\ell}(w_{i})=\int_{-1}^{1}d\cos\theta\,s^{2}G(s,w_{i},\cos\theta).$ (4.11)
Note that the asymptotic limit in eq. (4.9) does not commute with the partial
wave expansion, since the pole in $\cos\theta$ is not integrable. One must
work instead with generic $w$, and take the limit $w\to 0$ at the end. To do
so, note that $G$ can be rewritten as
$G(s,w_{i},\cos\theta)=-\frac{64\pi^{7}}{s^{2}\Omega_{12}\Omega_{34}}\frac{\partial}{\partial\cos\theta}\left[\log\left(\frac{1-1/z}{1-1/\bar{z}}\right)\right]^{2}.$
(4.12)
The angular integral can now be performed using integration by parts after
taking the IR limit in the logarithm. For identical $w_{i}=w$, the limit $w\to
0$ gives
$G(s,w_{i}=w,\cos\theta)=-\frac{256\pi^{7}}{s^{2}}\frac{\partial}{\partial\cos\theta}\left[\log\left(\frac{2w^{2}}{1-\cos\theta}\right)+\ldots\right]^{2},$
(4.13)
and thus
$a_{\ell}(w_{i}=w)=1024\pi^{7}[\log(-w)+H_{\ell}]^{2}+\ldots$ (4.14)
where $H_{\ell}$ is the $\ell$-th harmonic number, and the ellipsis indicates
that we have neglected terms of order $w\log(-w)$ and higher. For each
individual partial wave the leading IR divergence is now a double logarithm in
$w$.
To study a slightly more general form, one can also consider the situation in
which the masses are equal two-by-two, say $w_{4}=w_{1}$ and $w_{3}=w_{2}$,
and take the limit $w_{1}\to 0$ while keeping $w_{2}$ generic. Using eq.
(4.12), one arrives at
$a_{\ell}(w_{i})=\frac{256\pi^{7}}{(1-w_{2})^{2}}\left[\log(-w_{1})+\log(-w_{2})-2\log(1-w_{2})+2H_{\ell}\right]^{2}+\ldots,$
(4.15)
neglecting terms that vanish as $w_{1}\to 0$. We shall now see in the next
section how the partial waves (4.14) and (4.15) are matched one-to-one by the
conformal partial waves.
### 4.3 Conformal partial wave expansion
To compute the conformal partial waves, one simply needs to apply the recipe
given in sections 2 and 3 to the 4-point function (4.4). We work with $d=4$,
and with scaling dimensions of the external operator that are all equal,
$\Delta_{i}=2$. Since the theory is non-interacting, it is straightforward to
determine the spectrum of primary operators that enter the OPE of $\Phi^{2}$
and $\overline{\Phi}^{2}$, as these must be composites of the free field.
There are two families of such operators:
* •
Operators of the schematic form
$[\overline{\Phi}^{2}\square^{n}\partial^{\ell}\Phi^{2}]$, that have spin
$\ell$ and scaling dimensions $\Delta=4+\ell+2n$. As these are precisely
double-trace dimensions, the vertex functions involving them vanish. These
operators do not contribute to the conformal partial wave expansion in
momentum space.
* •
Operators of the form $[\overline{\Phi}\partial^{\ell}\Phi]$, with spin $\ell$
and scaling dimensions $\Delta=2+\ell$. There is one such operator for every
spin $\ell\geq 0$, and each of them is a conserved current saturating the
unitarity bound, with the exception of the scalar operator
$[\overline{\Phi}\Phi]$.
Therefore, the conformal partial wave decomposition (1.2) is a sum over
integers labeled by the spin $\ell$, with each term given by single
polarization $m=\ell$. Plugging the operator data in the coefficient
$C_{\Delta,\ell,\ell}$ of eq. (2.35), one gets
$G(s,w_{i},\cos\theta)=s^{-2}\sum_{\ell=0}^{\infty}\left(\ell+\tfrac{1}{2}\right)P_{\ell}(\cos\theta)a_{\ell}(w_{i}),$
(4.16)
where now
$a_{\ell}(w_{i})=\lambda_{\Phi^{2}\overline{\Phi}^{2}[\overline{\Phi}\partial^{\ell}\Phi]}^{2}\frac{2^{3\ell}(\ell!)^{4}}{\pi^{3}(2\ell)!}\,V_{2+\ell,\ell,\ell}^{\operatorname{T}[\phi\phi]}(w_{1},w_{2})V_{2+\ell,\ell,\ell}^{\operatorname{T}[\phi\phi]}(w_{3},w_{4})^{*}.$
(4.17)
The computation of the vertex functions is slightly more complicated, as one
cannot simply plug the relevant scaling dimensions in eq. (3.27): the
coefficients $f_{\Delta_{1}\Delta_{2};\Delta,\ell}$ diverge precisely when
$\Delta_{i}=\frac{d}{2}$. Fortunately, it is straightforward to regularize the
computation of these vertex functions by analytic continuation in $d$: the
free scalar theory can be defined in any $d\neq 4$, and we shall see that the
limit $d\to 4$ ends up to be finite, due to cancellations among the different
terms in eq. (3.27).
In generic $d$, the scaling dimension of $\Phi^{2}$ and $\overline{\Phi}^{2}$
is $\Delta_{i}=d-2$, and the conserved currents entering the OPE have
$\Delta=d-2+\ell$. The coefficients $f_{\Delta_{1}\Delta_{2};\Delta,\ell}$
appearing in eq. (3.27) take the values
$f_{\Delta_{1}\Delta_{2};\Delta,\ell}=\Gamma(d-3+\ell)\Gamma\left(\tfrac{4-d}{2}\right),\quad
f_{\widetilde{\Delta}_{1}\widetilde{\Delta}_{2};\Delta,\ell}=\ell!\Gamma\left(\tfrac{d-4}{2}\right),\quad
f_{\Delta_{1}\widetilde{\Delta}_{2};\Delta,\ell}=f_{\widetilde{\Delta}_{1}\Delta_{2};\Delta,\ell}=0.$
(4.18)
Since the combination
$f_{\Delta_{1}\Delta_{2};\Delta,\ell}+f_{\widetilde{\Delta}_{1}\widetilde{\Delta}_{2};\Delta,\ell}$
vanishes in the limit $d\to 4$, the individual divergences cancel out and the
powers $(-w_{i})^{\Delta_{i}-d/2}$ turn into logarithms. The general structure
of the vertex function is still quite complicated, as it involves Appell
$F_{4}$ functions and derivatives of them with respect to their parameters.
But it is simple to obtain the limit in which one of their argument $w_{i}$ is
small. For instance, as $w_{1}\to 0$, we get
$V_{2+\ell,\ell,\ell}^{\operatorname{T}[\phi\phi]}(w_{1},w_{2})=8\pi^{5}i^{\ell+1}\frac{(2\ell)!}{2^{2\ell}(\ell!)^{3}}\frac{1}{1-w_{2}}\left[\log(-w_{1})+\log\left(\frac{-w_{2}}{(1-w_{2})^{2}}\right)+2H_{\ell}+\ldots\right].$
(4.19)
This function already bears resemblance to the partial wave decomposition
obtained above. Assuming all identical $w_{i}=w$, one gets
$a_{\ell}(w_{i}=w)=256\pi^{7}\frac{(2\ell)!}{2^{\ell}(\ell!)^{2}}\lambda_{\Phi^{2}\overline{\Phi}^{2}[\overline{\Phi}\partial^{\ell}\Phi]}^{2}\left[\log(-w)+H_{\ell}+\ldots\right]^{2},$
(4.20)
while with $w_{4}=w_{1}$ and $w_{3}=w_{2}$ we have
$a_{\ell}(w_{i})=64\pi^{7}\frac{(2\ell)!}{2^{\ell}(\ell!)^{2}}\lambda_{\Phi^{2}\overline{\Phi}^{2}[\overline{\Phi}\partial^{\ell}\Phi]}^{2}\left[\log(-w_{1})+\log(-w_{2})-2\log(1-w_{2})+2H_{\ell}+\ldots\right]^{2}.$
(4.21)
These coefficients match precisely eqs. (4.14) and (4.15) provided that
$\lambda_{\Phi^{2}\overline{\Phi}^{2}[\overline{\Phi}\partial^{\ell}\Phi]}^{2}=\frac{2^{\ell+2}(\ell!)^{2}}{(2\ell)!}.$
(4.22)
This is indeed the known expression for the OPE coefficients in the complex
scalar field theory in $d=4$ dimensions. This example gives therefore a non-
trivial verification of our general results in a specific conformal field
theory.
## 5 Conclusions
In this work, we have presented a computation of the conformal partial waves
in momentum space, with an emphasis on their factorization properties: they
are given as a finite sum over vertex functions and polynomials in the
scattering angle. The terms of this sum are in one-to-one correspondence with
the ordinary spin partial waves. The vertex functions are simple to obtain
using a recursion relation descending from the highest-spin function, and the
latter are expressed in terms of Appell $F_{4}$ double hypergeometric
functions, which are straightforward to evaluate numerically. In some
particular cases where the scaling dimensions of the external operators take
(half-)integer values, the vertex function must be obtained by analytic
continuation: we gave an example where this situation happens in section 4.
The outcome of this work is a new mathematical tool for the study of conformal
field theory, which we hope will be useful in the future. Some directions that
are certainly worth investigating are the possibility to use partial wave
unitarity to put constraints on the CFT data, in the spirit of the S-matrix
bootstrap [76], as well as the prospect of writing CFT dispersion relations
directly in momentum space [39, 38].
We also believe that our computation of the conformal partial waves
illustrates the power of Minkowski momentum-space techniques in CFT. It is
well known that the orthogonality of momentum eigenstates allows to factorize
the computation of correlation functions: after all, this is the reason why
the majority of quantum field theory results are obtained in momentum space.
What is certainly less appreciated is the structural simplicity of the
operator product expansion in momentum space. While 3-point function are
admittedly complicated, the fact that they depend by momentum conservation on
only two momenta is very powerful: they can always be thought to live on a
2-dimensional slice of Minkowski space, and their spin can be projected onto
this subspace.777As already mentioned, this also implies that the ideal
framework to work with momentum-space 3-point functions is in terms of spinor
variables [62]. For this reason, the dependence on the scattering angle can
always be factorized in the operator product expansion in momentum space.
In fact, it is a simple exercise to generalize the computation of conformal
partial waves to correlation functions involving more than four points, at
least as long as one is interested in the comb channel [77, 78, 79, 80, 81].
Similarly, we believe that the inclusion of spinning external operators in the
computation could be potentially simpler in momentum space than it is for
ordinary conformal blocks in position space (see ref. [82], as well as the
connection with the Mellin-space representation in refs. [83, 84]).
Finally, several properties of our conformal partial waves could (and should)
be studied in the future. First and foremost, we have not touched the question
of the convergence of the momentum space OPE. This convergence is actually
guaranteed in a distributional sense, since CFT correlation functions in
Minkowski space are tempered distributions and so are their Fourier
transforms, and the position-space OPE can be shown to converge
distributionally [85]. Nevertheless, the situation can be quite subtle, as
illustrated by the solution to this problem in $d=2$ dimensions [86]:
depending on the kinematics, the OPE might or might not converge in a point-
like manner.
Other interesting properties of the momentum-space conformal partial wave
expansion have to do with the remarkable fact that our results are valid in
any space-time dimension $d\geq 3$. This would allow to study them in the
limit $d\to\infty$ [87, 88], or to examine whether recursion relations between
various space-time dimensions can be found [89, 90, 91]. The analyticity in
$d$ also implies the ability to study momentum-space correlation functions in
non-integer dimensions, where unitarity is known to be broken [92, 93].
Similarly, if one projects the OPE onto ordinary spin partial waves by working
at fixed $m$, then the result is not only analytic in $d$ but also in the spin
$\ell$ of the exchanged operator: in this case one can imagine formulating an
OPE inversion formula in momentum space [94, 95]. It would be interesting in
any case to understand the concept of light-ray operators in the same context
[96, 97, 98]. We leave the study of these interesting properties for future
work.
## Acknowledgments
The author would like to thank the organizers of the workshop “Cosmology Meets
CFT Correlators 2020” at the National Taiwan University for their hospitality,
as well as the workshop’s participants for numerous stimulating discussions.
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|
an algorithm that constructs $\mathsf{qGR}_{k}(X,Y)$ with a delay of at most
$k^{2}$ characters in $\tilde{\mathcal{O}}(k^{3})$ amortised time per
character and $\tilde{\mathcal{O}}(k^{2})$ space. The delay means that at the
moment when the $y$-th character of $Y$ arrives, the algorithm knows
$\mathsf{qGR}_{k}(X[\mathinner{.\,.\allowbreak}y^{\prime}],Y[\mathinner{.\,.\allowbreak}y^{\prime}])$,
where $|y-y^{\prime}|\leq k^{2}$.
###### Proof.
If $\ell\leq k^{2}$, construct $\mathsf{qGR}_{k}(X,Y)$ via 5.18 in
$\tilde{\mathcal{O}}(k^{3})$ (total) time and $\tilde{\mathcal{O}}(k^{2})$
space. Otherwise, partition the strings into non-overlapping blocks
$X=X_{1}\cdots X_{p}$ and $Y=Y_{1}\cdots Y_{p}$ so that
$|X_{i}|=|Y_{i}|=k^{2}$ for all $i\in[1\mathinner{.\,.\allowbreak}p)$ and
$|X_{p}|=|Y_{z}|=\ell\bmod k^{2}$. Suppose that we have computed
$\mathsf{qGR}_{k}(X_{1}\cdots X_{i-1},Y_{1}\cdots Y_{i-1})$ for
$i\in[1\mathinner{.\,.\allowbreak}p)$. Compute $\mathsf{qGR}_{k}(X_{1}\cdots
X_{i},Y_{1}\cdots Y_{i})$ in the following manner: first, compute
$\mathsf{qGR}_{k}(X_{i},Y_{i})$ in $\mathcal{O}(k^{5})$ time and
$\mathcal{O}(k^{2})$ space via 5.18, and then compute
$\mathsf{qGR}_{k}(X_{1}\cdots X_{i},Y_{1}\cdots Y_{i})$ in
$\tilde{\mathcal{O}}(k^{5})$ time and $\tilde{\mathcal{O}}(k^{2})$ space via
Lemma 5.21. ∎
### 5.4 Products of Greedy Alignments
See 3.12
See 3.13
###### Proof.
We proceed by induction on $|X|+|Y|+|Z|$. In the base case, when at least one
of the strings $X,Y,Z$ is empty, we set $\mathcal{A}^{X,Y}$ and
$\mathcal{A}^{Y,Z}$ to be any greedy optimal alignments of $X,Y$ and $Y,Z$,
respectively, so that $\mathsf{cost}(\mathcal{A}^{X,Y})=\mathsf{ed}(X,Y)$ and
$\mathsf{cost}(\mathcal{A}^{Y,Z})=\mathsf{ed}(Y,Z)\leq\mathsf{ed}(X,Y)+\mathsf{ed}(X,Z)\leq\mathsf{ed}(X,Y)+\mathsf{cost}(\mathcal{A}^{X,Z})$.
Moreover, it easy to check that $\mathcal{A}^{X,Z}$ is a product of
$\mathcal{A}^{X,Y}$ and $\mathcal{A}^{Y,Z}$ because two out of these three
alignments simply delete all characters of the non-empty string.
In the inductive step, we assume that all strings $X,Y,Z$ are non-empty, and
we consider several cases.
1. 1.
$\mathbf{X[1]=Z[1]=Y[1]}$.
We recurse on $X^{\prime}=X[2\mathinner{.\,.\allowbreak}]$,
$Y^{\prime}=Y[2\mathinner{.\,.\allowbreak}]$,
$Z^{\prime}=Z[2\mathinner{.\,.\allowbreak}]$, and
$\mathcal{A}^{X^{\prime},Z^{\prime}}=\mathcal{A}^{X,Z}_{[2\mathinner{.\,.\allowbreak}],[2\mathinner{.\,.\allowbreak}]}$,
which is greedy due to $X[1]\simeq_{\mathcal{A}^{X,Z}}Z[1]$. This yields
greedy alignments
$\mathcal{A}^{X^{\prime},Y^{\prime}}\\!\\!,A^{Y^{\prime},Z^{\prime}}$ of cost
at most $d^{\prime}=2\mathsf{cost}(\mathcal{A}^{X,Z})+\mathsf{ed}(X,Y)=d$. We
extend them so that $X[1]\simeq_{\mathcal{A}^{X,Y}}Y[1]$ and
$Y[1]\simeq_{\mathcal{A}^{Y,Z}}Z[1]$, obtaining alignments of cost up to $d$.
2. 2.
$\mathbf{X[1]=Z[1]\neq Y[1]}$.
In this case, we have
$\mathsf{ed}(X,Y)>\min(\mathsf{ed}(X[2\mathinner{.\,.\allowbreak}],Y[2\mathinner{.\,.\allowbreak}]),\mathsf{ed}(X[2\mathinner{.\,.\allowbreak}],Y),\mathsf{ed}(X,Y[2\mathinner{.\,.\allowbreak}]))$.
1. (a)
If
$\mathsf{ed}(X,Y)>\mathsf{ed}(X[2\mathinner{.\,.\allowbreak}],Y[2\mathinner{.\,.\allowbreak}])$,
we recurse on $X^{\prime}=X[2\mathinner{.\,.\allowbreak}]$,
$Y^{\prime}=Y[2\mathinner{.\,.\allowbreak}]$,
$Z^{\prime}=Z[2\mathinner{.\,.\allowbreak}]$, and
$\mathcal{A}^{X^{\prime},Z^{\prime}}=\mathcal{A}^{X,Z}_{[2\mathinner{.\,.\allowbreak}],[2\mathinner{.\,.\allowbreak}]}$,
which is greedy due to $X[1]\simeq_{\mathcal{A}^{X,Z}}Z[1]$. This yields
greedy alignments
$\mathcal{A}^{X^{\prime},Y^{\prime}}\\!\\!,A^{Y^{\prime},Z^{\prime}}$ of cost
at most $d^{\prime}=2\mathsf{cost}(\mathcal{A}^{X,Z})+\mathsf{ed}(X,Y)-1=d-1$.
We extend them so that $X[1]\sim_{\mathcal{A}^{X,Y}}Y[1]$ and
$Y[1]\sim_{\mathcal{A}^{Y,Z}}Z[1]$, obtaining alignments of cost up to $d$.
2. (b)
If $\mathsf{ed}(X,Y)>\mathsf{ed}(X[2\mathinner{.\,.\allowbreak}],Y)$, we
recurse on $X^{\prime}=X[2\mathinner{.\,.\allowbreak}]$, $Y^{\prime}=Y$,
$Z^{\prime}=Z[2\mathinner{.\,.\allowbreak}]$, and
$\mathcal{A}^{X^{\prime},Z^{\prime}}=\mathcal{A}^{X,Z}_{[2\mathinner{.\,.\allowbreak}],[2\mathinner{.\,.\allowbreak}]}$,
which is greedy due to $X[1]\simeq_{\mathcal{A}^{X,Z}}Z[1]$. This yields
greedy alignments
$\mathcal{A}^{X^{\prime},Y^{\prime}}\\!\\!,A^{Y^{\prime},Z^{\prime}}$ of cost
at most $d^{\prime}=2\mathsf{cost}(\mathcal{A}^{X,Z})+\mathsf{ed}(X,Y)-1=d-1$.
We extend them so that $\mathcal{A}^{X,Y}$ deletes $X[1]$ and
$\mathcal{A}^{Y,Z}$ deletes $Z[1]$, obtaining alignments of cost up to $d$.
3. (c)
If $\mathsf{ed}(X,Y)>\mathsf{ed}(X,Y[2\mathinner{.\,.\allowbreak}])$, we
recurse on $X^{\prime}=X$, $Y^{\prime}=Y[2\mathinner{.\,.\allowbreak}]$,
$Z^{\prime}=Z$, and $\mathcal{A}^{X^{\prime},Z^{\prime}}=\mathcal{A}^{X,Z}$.
This yields greedy alignments
$\mathcal{A}^{X^{\prime},Y^{\prime}}\\!\\!,A^{Y^{\prime},Z^{\prime}}$ of cost
at most $d^{\prime}=2\mathsf{cost}(\mathcal{A}^{X,Z})+\mathsf{ed}(X,Y)-1=d-1$.
We extend them so that $\mathcal{A}^{X,Y}$ and $\mathcal{A}^{Y,Z}$ both delete
$Y[1]$, obtaining alignments of cost up to $d$.
3. 3.
$\mathbf{X[1]\neq Z[1]=Y[1]}$.
1. (a)
If $\mathcal{A}^{X,Z}$ deletes $X[1]$, we recurse on
$X^{\prime}=X[2\mathinner{.\,.\allowbreak}]$, $Y^{\prime}=Y$, $Z^{\prime}=Z$,
and
$\mathcal{A}^{X^{\prime},Z^{\prime}}=\mathcal{A}^{X,Z}_{[2\mathinner{.\,.\allowbreak}],[1\mathinner{.\,.\allowbreak}]}$.
This yields greedy alignments
$\mathcal{A}^{X^{\prime},Y^{\prime}}\\!\\!,A^{Y^{\prime},Z^{\prime}}$ of cost
$\leq
d^{\prime}=2\mathsf{cost}(\mathcal{A}^{X,Z})-2+\mathsf{ed}(X^{\prime},Y^{\prime})\leq
d-1$. We derive $\mathcal{A}^{Y,Z}=\mathcal{A}^{Y^{\prime},Z^{\prime}}$ and
extend $\mathcal{A}^{X^{\prime},Y^{\prime}}$ so that $\mathcal{A}^{X,Y}$
deletes $X[1]$, obtaining an alignment of cost up to $d$.
2. (b)
If $\mathcal{A}^{X,Z}$ deletes $Z[1]$, we recurse on $X^{\prime}=X$,
$Y^{\prime}=Y[2\mathinner{.\,.\allowbreak}]$,
$Z^{\prime}=Z[2\mathinner{.\,.\allowbreak}]$, and
$\mathcal{A}^{X^{\prime},Z^{\prime}}=\mathcal{A}^{X,Z}_{[1\mathinner{.\,.\allowbreak}],[2\mathinner{.\,.\allowbreak}]}$.
This yields greedy alignments
$\mathcal{A}^{X^{\prime},Y^{\prime}}\\!\\!,A^{Y^{\prime},Z^{\prime}}$ of cost
$\leq
d^{\prime}=2\mathsf{cost}(\mathcal{A}^{X,Z})-2+\mathsf{ed}(X^{\prime},Y^{\prime})\leq
d-1$. We extend them so that $\mathcal{A}^{X,Y}$ deletes $Y[1]$ and
$Y[1]\simeq_{\mathcal{A}^{Y,Z}}Z[1]$, obtaining alignments of cost up to $d$.
3. (c)
If $X[1]\sim_{\mathcal{A}^{X,Z}}Z[1]$, we recurse on
$X^{\prime}=X[2\mathinner{.\,.\allowbreak}]$,
$Y^{\prime}=Y[2\mathinner{.\,.\allowbreak}]$,
$Z^{\prime}=Z[2\mathinner{.\,.\allowbreak}]$, and
$\mathcal{A}^{X^{\prime},Z^{\prime}}=\mathcal{A}^{X,Z}_{[2\mathinner{.\,.\allowbreak}],[2\mathinner{.\,.\allowbreak}]}$.
This yields greedy alignments
$\mathcal{A}^{X^{\prime},Y^{\prime}}\\!\\!,A^{Y^{\prime},Z^{\prime}}$ of cost
$\leq
d^{\prime}=2\mathsf{cost}(\mathcal{A}^{X,Z})-2+\mathsf{ed}(X^{\prime},Y^{\prime})\leq
d-1$. We extend them so that $X[1]\sim_{\mathcal{A}^{X,Y}}Y[1]$ and
$Y[1]\simeq_{\mathcal{A}^{Y,Z}}Z[1]$, obtaining alignments of cost up to $d$.
4. 4.
$\mathbf{X[1]\neq Z[1]\neq Y[1]}$. (Note that this case allows both
$X[1]=Y[1]$ and $X[1]\neq Y[1]$.)
1. (a)
If $\mathcal{A}^{X,Z}$ deletes $X[1]$, we recurse on
$X^{\prime}=X[2\mathinner{.\,.\allowbreak}]$,
$Y^{\prime}=Y[2\mathinner{.\,.\allowbreak}]$, $Z^{\prime}=Z$, and
$\mathcal{A}^{X^{\prime},Z^{\prime}}=\mathcal{A}^{X,Z}_{[2\mathinner{.\,.\allowbreak}],[1\mathinner{.\,.\allowbreak}]}$.
This yields greedy alignments
$\mathcal{A}^{X^{\prime},Y^{\prime}}\\!\\!,A^{Y^{\prime},Z^{\prime}}$ of cost
at most
$d^{\prime}=2\mathsf{cost}(\mathcal{A}^{X,Z})-2+\mathsf{ed}(X^{\prime},Y^{\prime})\leq
d-2$. We extend them so that $X[1]\sim_{\mathcal{A}^{X,Y}}Y[1]$ and
$\mathcal{A}^{Y,Z}$ deletes $Y[1]$, obtaining alignments of cost up to $d-1$.
2. (b)
If $\mathcal{A}^{X,Z}$ deletes $Z[1]$, we recurse on $X^{\prime}=X$,
$Y^{\prime}=Y$, $Z^{\prime}=Z[2\mathinner{.\,.\allowbreak}]$, and
$\mathcal{A}^{X^{\prime},Z^{\prime}}=\mathcal{A}^{X,Z}_{[1\mathinner{.\,.\allowbreak}],[2\mathinner{.\,.\allowbreak}]}$.
This yields greedy alignments
$\mathcal{A}^{X^{\prime},Y^{\prime}}\\!\\!,A^{Y^{\prime},Z^{\prime}}$ of cost
$\leq d^{\prime}=2\mathsf{cost}(\mathcal{A}^{X,Z})-2+\mathsf{ed}(X,Y)=d-2$. We
derive $\mathcal{A}^{X,Y}=\mathcal{A}^{X^{\prime},Y^{\prime}}$ and extend
$\mathcal{A}^{Y^{\prime},Z^{\prime}}$ so that $\mathcal{A}^{Y,Z}$ deletes
$Z[1]$, obtaining an alignment of cost up to $d-1$.
3. (c)
If $X[1]\sim_{\mathcal{A}^{X,Z}}Z[1]$, we recurse on
$X^{\prime}=X[2\mathinner{.\,.\allowbreak}]$,
$Y^{\prime}=Y[2\mathinner{.\,.\allowbreak}]$,
$Z^{\prime}=Z[2\mathinner{.\,.\allowbreak}]$, and
$\mathcal{A}^{X^{\prime},Z^{\prime}}=\mathcal{A}^{X,Z}_{[2\mathinner{.\,.\allowbreak}],[2\mathinner{.\,.\allowbreak}]}$.
This yields greedy alignments
$\mathcal{A}^{X^{\prime},Y^{\prime}}\\!\\!,A^{Y^{\prime},Z^{\prime}}$ of cost
$\leq
d^{\prime}=2\mathsf{cost}(\mathcal{A}^{X,Z})-2+\mathsf{ed}(X^{\prime},Y^{\prime})\leq
d-2$. We extend them so that $X[1]\sim_{\mathcal{A}^{X,Y}}Y[1]$ and
$Y[1]\sim_{\mathcal{A}^{Y,Z}}Z[1]$, obtaining alignments of cost up to $d-1$.
In all the cases above, $\mathcal{A}^{X,Y}$ is greedy because
$\mathcal{A}^{X^{\prime},Y^{\prime}}$ is greedy and $\mathcal{A}^{X,Y}$
matches $X[1]$ with $Y[1]$ whenever $X[1]=Y[1]$. Similarly,
$\mathcal{A}^{Y,Z}$ is greedy because $\mathcal{A}^{Y^{\prime},Z^{\prime}}$ is
greedy and $\mathcal{A}^{Y,Z}$ matches $Y[1]$ with $Z[1]$ whenever
$Y[1]=Z[1]$. Moreover, $\mathcal{A}^{X,Z}$ is a product of $\mathcal{A}^{X,Y}$
and $\mathcal{A}^{Y,Z}$ because each alignment starts with $(1,1)$ and since
$\mathcal{A}^{X^{\prime},Z^{\prime}}$ is a product of
$\mathcal{A}^{X^{\prime},Y^{\prime}}$ and
$\mathcal{A}^{Y^{\prime},Z^{\prime}}$. ∎
Assume that we are given three strings $X,Y,Z$. Let $d=\mathsf{ed}(X,Y)+2k$,
and define $\mathcal{G}_{X}=\mathsf{GR}_{d}(X,Y)$ and
$\mathcal{G}_{Z}=\mathsf{GR}_{d}(Y,Z)$. We show that given $\mathcal{G}_{X}$
and $\mathcal{G}_{Z}$, we can compute an optimal alignment between $X$ and $Z$
efficiently if its cost is at most $k$. Let $M=\\{(x,z):\exists y\text{ such
that }(x,y)\in\mathcal{M}_{d}(X,Y)\text{ and
}(y,z)\in\mathcal{M}_{d}(Y,Z)\\}$.
###### Lemma 5.23.
If $\mathsf{ed}(X,Z)\leq k$, then $M\subseteq\mathcal{M}_{k}(X,Z)$.
###### Proof.
Suppose that $(x,y)\in\mathcal{M}_{d}(X,Y)$ and
$(y,z)\in\mathcal{M}_{d}(Y,Z)$. For a proof by contradiction, suppose that
$X[x]\not\sim_{\mathcal{A}}Z[z]$ for some
$\mathcal{A}\in\mathsf{GA}_{k}(X,Z)$. Then, there exists
$(x^{\prime},z^{\prime})\in\mathcal{A}$ such that either $x^{\prime}\leq x$
and $z^{\prime}>z$, or $x^{\prime}>x$ and $z^{\prime}\leq z$. By symmetry,
without loss of generality, we consider the first alternative. By Lemma 3.13,
$\mathcal{A}$ is a product $\mathcal{A}^{X,Y}\in\mathsf{GA}_{d}(X,Y)$ and
$\mathcal{A}^{Y,Z}\in\mathsf{GA}_{d}(Y,Z)$. According to Definition 3.12, this
means that there exists $y^{\prime}$ such that
$(x^{\prime},y^{\prime})\in\mathcal{A}^{X,Y}$ and
$(y^{\prime},z^{\prime})\in\mathcal{A}^{Y,Z}$. If $y^{\prime}\leq y$, then
$(y,z),(y^{\prime},z^{\prime})\in\mathcal{A}^{Y,Z}$ implies that
$\mathcal{A}^{Y,Z}$ deletes $Z[z]$, contradicting
$(y,z)\in\mathcal{M}(\mathcal{A}^{Y,Z})$. Similarly, if $y^{\prime}>y$, then
$(x,y),(x^{\prime},y^{\prime})\in\mathcal{A}^{X,Y}$ implies that
$\mathcal{A}^{X,Y}$ deletes $Y[y]$, contradicting
$(x,y)\in\mathcal{M}(\mathcal{A}^{X,Y})$. This completes the proof that
$X[x]\sim_{\mathcal{A}}Z[z]$ for every $\mathcal{A}\in\mathsf{GA}_{k}(X,Z)$.
Due to $X[x]=Y[y]=Z[z]$, we also have $X[x]\simeq_{\mathcal{A}}Z[z]$ for every
$\mathcal{A}\in\mathsf{GA}_{k}(X,Z)$, i.e., $(x,y)\in\mathcal{M}_{k}(X,Z)$
holds as claimed. ∎
###### Lemma 5.24.
If $\mathsf{ed}(X,Y)=\mathcal{O}(k)$ and
$\mathcal{G}_{X},\mathcal{G}_{Z}\neq\bot$, $\mathsf{E}^{M}(X,Z)$ can be
computed in $\tilde{\mathcal{O}}(k^{3})$ time and $\tilde{\mathcal{O}}(k^{2})$
space.
###### Proof.
Let us first explain how $X^{M}$ can be constructed. Consider a position
$x\in[1\mathinner{.\,.\allowbreak}|X|]$ such that $(x,z)\notin M$ for every
$z\in[1\mathinner{.\,.\allowbreak}|Z|]$. If
$X^{\mathcal{M}_{d}(X,Y)}[x]\neq\\#$, then
$X^{M}[x]=X[x]=X^{\mathcal{M}_{d}(X,Y)}[x]$. Otherwise,
$(x,y)\in\mathcal{M}_{d}(X,Y)$ for some
$y\in[1\mathinner{.\,.\allowbreak}|Y|]$. By Lemma 5.23, we have
$(y,z)\notin\mathcal{M}_{d}(Y,Z)$ for every
$z\in[1\mathinner{.\,.\allowbreak}|Z|]$. Hence,
$Y^{\mathcal{M}_{d}(Y,Z)}[y]\neq\\#$, and we have
$X[x]=Y[y]=Y^{\mathcal{M}_{d}(Y,Z)}[y]$. In words, non-dummy characters of
$X^{M}$ can be retrieved from non-dummy characters of
$X^{\mathcal{M}_{d}(X,Y)}$ and non-dummy characters of
$Y^{\mathcal{M}_{d}(Y,Z)}$.
Thus, for each dummy segment $[\ell\mathinner{.\,.\allowbreak}r)$ in
$X^{\mathcal{M}_{d}(X,Y)}$ and its counterpart
$[\ell^{\prime}\mathinner{.\,.\allowbreak}r^{\prime})$ in
$Y^{\mathcal{M}_{d}(X,Y)}$, we need to set
$X^{M}[\ell\mathinner{.\,.\allowbreak}r):=Y^{\mathcal{M}_{d}(Y,Z)}[\ell^{\prime}\mathinner{.\,.\allowbreak}r^{\prime})$.
Such a copy-paste operation can be implemented using Proposition 4.2(g)(i), in
$\mathcal{O}(k^{2}\log^{4}n)$ time per dummy segment. We can also keep track
of the dummy segments in $X^{M}$ within the same procedure. The algorithm for
$Z^{M}$ is symmetric, and thus we can construct $\mathsf{D}(X^{M}Z^{M})$ along
with the dummy segments in $\tilde{\mathcal{O}}(k^{3})$ time and
$\tilde{\mathcal{O}}(k^{2})$ space. Finally, we build
$\mathsf{RS}_{\\#}(X^{M})$ and $\mathsf{RS}_{\\#}(Z^{M})$ in $\mathcal{O}(k)$
time. ∎
###### Corollary 5.25.
Given $\mathcal{G}_{X}$ and $\mathcal{G}_{Z}$. If
$\mathsf{ed}(X,Y)=\mathcal{O}(k)$, then we can compute
$\min(k+1,\mathsf{ed}(X,Z))$ in $\tilde{\mathcal{O}}(k^{3})$ time and
$\tilde{\mathcal{O}}(k^{2})$ space.
###### Proof.
If $\mathcal{G}_{X}$ or $\mathcal{G}_{Z}$ equals to $\bot$, then
$\mathsf{ed}(X,Z)>k$ by Lemma 3.13. Otherwise, construct $\mathsf{E}^{M}(X,Z)$
using Lemma 5.24. Finally, we pass $k$ and $\mathsf{E}^{M}(X,Z)$ to the
algorithm of Corollary 5.5, and we return the resulting value $d$.
Note that $M$ is a non-crossing matching of $X,Z$, so $d=k+1$ is correctly
returned if $\mathsf{ed}(X,Z)>k$. If $\mathsf{ed}(X,Z)\leq k$, then Lemma 5.23
implies $M\subseteq\mathcal{M}_{k}(X,Z)$, and thus $d=\mathsf{ed}(X,Z)$ holds
as claimed.
The overall runtime and space complexity are dominated by the procedure of
Lemma 5.24. ∎
###### Remark 5.26.
Using Proposition 5.12 instead of Corollary 5.5, we could construct
$\mathsf{GR}_{k}(X,Z)$.
###### Corollary 5.27.
Consider three strings $X,Y,Z$. Let $d=\mathsf{ed}(X,Y)+2k$, and assume that
we are given $\mathsf{qGR}_{d}(X,Y)$ and $\mathsf{qGR}_{d}(Y,Z)$. If
$\mathsf{ed}(X,Y)=\mathcal{O}(k)$, then we can compute
$\min(k+1,\mathsf{ed}(X,Z))$ in $\tilde{\mathcal{O}}(k^{3})$ time and
$\tilde{\mathcal{O}}(k^{2})$ space.
###### Proof.
We compute $\mathcal{G}_{X}=\mathsf{GR}_{d+1}(\$_{1}X,\$_{2}Y)$ and
$\mathsf{qGR}_{d+1}(\$_{2}Y,\$_{3}Z)$ in $\tilde{\mathcal{O}}(k^{2})$ time and
space via Corollary 5.17, and apply Corollary 5.25 to compute
$\min(k+2,\mathsf{ed}(\$_{1}X,\$_{2}Z))-1=\min(k+1,\mathsf{ed}(X,Z))$ in
$\tilde{\mathcal{O}}(k^{3})$ time and $\tilde{\mathcal{O}}(k^{2})$ space. ∎
## 6 Edit Distance Sketches
### 6.1 CGK embedding
In this section, we prove Proposition 3.16 based on the CGK embedding
introduced in [10]. Recall that the Hamming distance between the embeddings of
two strings $X,Y\in\Sigma^{\leq n}$ is bounded in terms of the edit distance
$\mathsf{ed}(X,Y)$, which allows using Hamming distance sketches to
approximate edit distance.
###### Definition 6.1 (CGK embedding [10]).
Consider an alphabet $\Sigma$, a sentinel character $\bot\notin\Sigma$, and a
2-independent family $\mathcal{H}$ of hash functions $h:\Sigma\to\\{0,1\\}$.
For an integer $n\in\mathbb{Z}_{+}$, a (uniformly random) sequence
$R\in\mathcal{H}^{3n}$, and a string $S\in\Sigma^{\leq n}$, the _CGK walk_
$W_{\mathsf{CGK}}(S)=(s_{t})_{t=1}^{3n+1}$ and the _CGK embedding_
$\mathsf{CGK}(S)\in(\Sigma\cup\\{\bot\\})^{3n}$ are defined by the following
algorithm:
Input: An integer $n\in\mathbb{Z}_{+}$, a string $S\in\Sigma^{\leq n}$.
Randomness: A sequence $R\in\mathcal{H}^{3n}$ of 2-independent hash functions
$\Sigma\to\\{0,1\\}$.
Output: The CGK walk $W_{\mathsf{CGK}}(S)=(s_{t})_{t=1}^{3n+1}$ and the CGK
embedding $\mathsf{CGK}(S)$.
1 $s_{1}:=1$;
2 for _$t:=1$ to $3n$_ do
3 if _$s_{t}\leq|S|$_ then
4 $\mathsf{CGK}(S)[t]:=S[s_{t}]$;
5 $s_{t+1}:=s_{t}+R_{t}(S[s_{t}])$;
6
7 else
8 $\mathsf{CGK}(S)[t]:=\bot$;
9 $s_{t+1}:=s_{t}$;
10
11
Algorithm 2 The CGK algorithm
Recall from Definition 3.14 that an $m$-step walk over $S$ is complete if
$s_{m+1}=|S|+1$.
###### Fact 6.2 ([10, Theorem 4.1]).
Each $S\in\Sigma^{\leq n}$ satisfies $\Pr_{R}[W_{\mathsf{CGK}}(S)\text{ is
complete}]\geq 1-e^{-\Omega(n)}$.
The following result summarizes the central property of the CGK embedding.
###### Fact 6.3 ([10, Theorem 4.3]).
For every $X,Y\in\Sigma^{\leq n}$ and every constant $c>0$, the embeddings
$\mathsf{CGK}(X)$, $\mathsf{CGK}(Y)$ satisfy
$\Pr_{R\in\mathcal{H}^{3n}}[\mathsf{hd}(\mathsf{CGK}(X),\mathsf{CGK}(Y))>c\cdot\mathsf{ed}(X,Y)^{2}]<\frac{12}{\sqrt{c}}$.
If $W_{\mathsf{CGK}}(X)$ and $W_{\mathsf{CGK}}(Y)$ are complete, Definition
3.15 yields an edit-distance alignment of $X,Y$, which we call the _CGK
alignment_ of $X,Y$. that they induce an edit-distance alignment
###### Fact 6.4 (see also [10, Theorem 4.2]).
If $W_{\mathsf{CGK}}(X)$ and $W_{\mathsf{CGK}}(Y)$ are complete for some
$X,Y\in\Sigma^{\leq n}$ and $R\in\mathcal{H}^{3n}$, then the CGK alignment of
$X,Y$ belongs to
$\mathsf{GA}_{\mathsf{hd}(\mathsf{CGK}(X),\mathsf{CGK}(Y))}(X,Y)$.
###### Proof.
Let $\mathcal{A}$ be the CGK alignment of $X,Y$, i.e., the zip alignment of
$W_{\mathsf{CGK}}(X)=(x_{t})_{t=1}^{3n+1}$ and
$W_{\mathsf{CGK}}(Y)=(y_{t})_{t=1}^{3n+1}$. Consider
$(x_{t},y_{t})\in\mathcal{B}(\mathcal{A})$, with
$t\in[1\mathinner{.\,.\allowbreak}3n+1]$ chosen so that $t=3n+1$ or
$(x_{t+1},y_{t+1})\neq(x_{t},y_{t})$. If $t=3n+1$, then
$(x_{t},y_{t})=(|X|+1,|Y|+1)$ by the assumption that $W_{\mathsf{CGK}}(X)$ and
$W_{\mathsf{CGK}}(Y)$ are complete. If $(x_{t+1},y_{t+1})=(x_{t}+1,y_{t}+1)$,
then we have $\mathsf{CGK}(X)[t]=X[x_{t}]\neq Y[y_{t}]=\mathsf{CGK}(Y)[t]$.
The remaining possibility $x_{t+1}-x_{t}\neq y_{t+1}-y_{t}$ holds only in the
following three cases: if $x_{t}=|X|+1$ and $y_{t}\leq|Y|$ (when
$\mathsf{CGK}(X)[t]=\bot\neq\mathsf{CGK}(Y)[t]=Y[y_{t}]$), if $x_{t}\leq|X|$
and $y_{t}=|Y|+1$ (when
$X[x_{t}]=\mathsf{CGK}(X)[t]\neq\bot=\mathsf{CGK}(Y)[t]$), or if
$x_{t}\leq|X|$, $y_{t}\leq|Y|$, and $R_{t}(X[x_{t}])\neq R_{t}(Y[y_{t}])$
(when $\mathsf{CGK}(X)[t]=X[x_{t}]\neq Y[y_{t}]=\mathsf{CGK}(Y)[t]$). Overall,
we have $X[x_{t}]\neq Y[y_{t}]$ whenever
$(x_{t},y_{t})\in[1\mathinner{.\,.\allowbreak}|X|]\times[1\mathinner{.\,.\allowbreak}|Y|]$,
and $\mathsf{CGK}(X)[t]\neq\mathsf{CGK}(Y)[t]$ whenever
$t\in[1\mathinner{.\,.\allowbreak}3n]$. Consequently,
$\mathcal{A}\in\mathsf{GA}_{\mathsf{hd}(\mathsf{CGK}(X),\mathsf{CGK}(Y))}(X,Y)$.
∎
The final property of the CGK alignment required in this work is that its
width is within $\mathcal{O}(\mathsf{ed}(X,Y))$ with good probability. For
this, we first prove a fact about random walks:
###### Fact 6.5.
Let $(w_{i})_{i\geq 0}$ be an unbiased lazy random walk (that is, $w_{0}=0$,
and, for every $i\geq 1$, we have $\Pr[w_{i+1}=w_{i}\mid
w_{0},\ldots,w_{i}]=\frac{1}{2}$, $\Pr[w_{i+1}=w_{i}+1\mid
w_{0},\ldots,w_{i}]=\Pr[w_{i+1}=w_{i}-1\mid w_{0},\ldots,w_{i}]=\frac{1}{4}$).
Then, for every $m,\ell\in\mathbb{Z}_{+}$, we have
$\Pr[\max_{i=0}^{m}|w_{i}|\geq\ell]\leq\frac{m}{\ell^{2}}$.
###### Proof.
By the reflection principle [44, Excercise 2.10], we have
$\Pr[\max_{i=0}^{m}|w_{i}|\geq\ell]\leq
2\Pr[|w_{m}|\geq\ell]=2\Pr[w_{m}^{2}\geq\ell^{2}]$. By Markov’s inequality,
$2\Pr[w_{m}^{2}\geq\ell^{2}]\leq\frac{2\mathbb{E}[w_{m}^{2}]}{\ell^{2}}=\frac{2\mathbb{E}[(w_{1}-w_{0})^{2}+\cdots+(w_{m}-w_{m-1})^{2}]}{\ell^{2}}=\frac{m}{\ell^{2}}.\qed$
###### Corollary 6.6.
For every constant $\delta\in(0,1)$, there exists a constant $c$ such that for
sufficiently large $n$, all strings $X,Y\in\Sigma^{\leq n}$, and their CGK
alignment $\mathcal{A}$, the probability over $R\in\mathcal{H}^{3n}$ that
$W_{\mathsf{CGK}}(X),W_{\mathsf{CGK}}(Y)\text{ are complete,
}\mathcal{A}\in\mathsf{GA}_{c\cdot\mathsf{ed}(X,Y)^{2}}(X,Y)\text{, and
}0pt(\mathcal{A})\leq c\cdot\mathsf{ed}(X,Y)$ is at least $1-\delta$.
###### Proof.
By Fact 6.2, $\Pr[W_{\mathsf{CGK}}(S)\text{ is
incomplete}]\leq\frac{\delta}{4}$ holds for every $S\in\Sigma^{\leq n}$ and
sufficiently large $n=\Omega(\log\frac{1}{\delta})$. By Fact 6.3, there is a
constant $c$ such that
$\Pr[\mathsf{hd}(\mathsf{CGK}(X),\mathsf{CGK}(Y))>c\cdot\mathsf{ed}(X,Y)^{2}]\leq\frac{\delta}{4}$.
We may take arbitrarily large $c$, so we shall also assume that
$c\geq\frac{4}{\delta}$. Moreover, by Fact 6.4, if $W_{\mathsf{CGK}}(X)$ and
$W_{\mathsf{CGK}}(Y)$ are complete and
$\mathsf{hd}(\mathsf{CGK}(X),\mathsf{CGK}(Y))\leq c\cdot\mathsf{ed}(X,Y)^{2}$,
then $\mathcal{A}\in\mathsf{GA}_{c\cdot\mathsf{ed}(X,Y)^{2}}(X,Y)$.
To bound $0pt(\mathcal{A})$, let us analyze the CGK walks
$W_{\mathsf{CGK}}(X)=(x_{t})_{t=1}^{3n+1}$ and
$W_{\mathsf{CGK}}(Y)=(y_{t})_{t=1}^{3n+1}$. Let $T$ be the first step such
that $x_{T}=|X|+1$, $y_{T}=|Y|+1$, or $T=3n+1$. Note that it suffices to bound
$\Pr[\max_{t=1}^{T}|x_{t}-y_{t}|\geq ck]$, where $k=\mathsf{ed}(X,Y)$.
Let $A=\\{1\\}\cup\\{t+1:t\in[1\mathinner{.\,.\allowbreak}T)\text{ and
}X[x_{t}]\neq Y[y_{t}]\\}$. Observe that $x_{t}-y_{t}=x_{t-1}-y_{t-1}$ holds
for $t\in[1\mathinner{.\,.\allowbreak}T]\setminus A$, so we may focus on
bounding $\Pr[\max_{t\in A}|x_{t}-y_{t}|\geq ck]$. Let
$A=\\{t_{0},\ldots,t_{a}\\}$ with $t_{0}<\cdots<t_{a}$, and let
$d_{i}=x_{t_{i}}-y_{t_{i}}$ for $i\in[0\mathinner{.\,.\allowbreak}a]$. Observe
that $(d_{i})_{i=0}^{a}$ is an $a$-step unbiased lazy random walk and that
$a\leq\mathsf{hd}(\mathsf{CGK}(X),\mathsf{CGK}(Y))$. Moreover, if we extend
$(d_{i})_{i=0}^{a}$ to an infinite unbiased lazy random walk
$(d_{i})_{i=0}^{\infty}$, then Fact 6.5 yields $\Pr[\max_{i=0}^{\lfloor
ck^{2}\rfloor}|d_{i}|\geq
ck]\leq\frac{ck^{2}}{c^{2}k^{2}}=\frac{1}{c}\leq\frac{\delta}{4}$. Therefore,
$\Pr\left[\max_{i=0}^{a}|d_{i}|\geq
ck\right]\leq\Pr\left[\mathsf{hd}(\mathsf{CGK}(X),\mathsf{CGK}(Y))>ck^{2}\right]+\Pr\left[{\max_{i=0}^{\lfloor
ck^{2}\rfloor}|d_{i}|\geq ck}\right].$
Consequently, if $\mathsf{CGK}(X)$ and $\mathsf{CGK}(Y)$ are complete,
$\mathsf{hd}(\mathsf{CGK}(X),\mathsf{CGK}(Y))\leq ck^{2}$, and the random walk
satisfies $\Pr\big{[}{\max_{i=0}^{\lfloor ck^{2}\rfloor}|d_{i}|<ck}\big{]}$,
then the alignment $\mathcal{A}$ satisfies the lemma. The total probability of
the complementary events is at most $\delta$, so this completes the proof. ∎
To complete the proof of Proposition 3.16 (repeated below), it remains to
reduce the number of random bits using Nisan’s pseudorandom generator [50].
See 3.16
###### Proof.
Let $c_{\ref{cor:CGK}}$ and $n_{\ref{cor:CGK}}$ be the constant and threshold
(respectively) of Corollary 6.6 for $\delta_{\ref{cor:CGK}}=\frac{\delta}{2}$.
If $n<\max(\frac{10}{\delta},n_{\ref{cor:CGK}})$, we set
$\mathsf{W}(n,r,S)=(\min(t,|S|+1))_{t=1}^{3n+1}$ for all $S\in\Sigma^{\leq n}$
so that $\mathsf{W}(n,r,S)$ is trivially a $3n$-step complete walk over $S$.
Now, consider strings $X,Y\in\Sigma^{\leq n}$ and the zip alignment
$\mathcal{A}_{\mathsf{W}}$ of $\mathsf{W}(n,r,X)$ and $\mathsf{W}(n,r,Y)$.
Observe that $\mathcal{A}_{\mathsf{W}}\in\mathsf{GA}_{0}(X,Y)$ if $X=Y$ and
$\mathcal{A}_{\mathsf{W}}\in\mathsf{GA}_{n}(X,Y)$ otherwise. Moreover,
$0pt(\mathcal{A}_{\mathsf{W}})=\big{|}|X|-|Y|\big{|}\leq\mathsf{ed}(X,Y)$.
Consequently, the claimed conditions are (deterministically) satisfied for
$c\geq n_{\ref{cor:CGK}}$. The construction algorithm uses $\mathcal{O}(\log
n)$ bits and $\mathcal{O}(n)$ time.
If $n\geq\max(\frac{10}{\delta},n_{\ref{cor:CGK}})$, we first develop an
algorithm $\mathsf{W}^{\prime}$ that uses $\Theta(n\log\sigma)$ random bits,
interpreted as a sequence $R\in\mathcal{H}^{3n}$. Specifically, each hash
function $R_{t}$ is specified by an $\lceil\log\sigma\rceil$-bit integer
$h_{t}$ so that $R_{t}(a)$ counts (modulo 2) the set bits in $a\textsf{ xor
}h_{t}$.
We set $\mathsf{W}^{\prime}(n,R,S)=(\max(t-3n+|S|,s_{t}))_{t=1}^{3n+1}$ based
on the CGK walk $W_{\mathsf{CGK}}(S)=(s_{t})_{t=1}^{3n+1}$ for all
$S\in\Sigma^{\leq n}$. Observe that this modification guarantees that
$\mathsf{W}^{\prime}(n,R,S)$ is a complete walk over $S$ and that
$\mathsf{W}^{\prime}(n,R,S)=W_{\mathsf{CGK}}(S)$ if $W_{\mathsf{CGK}}(S)$ is
already complete. Consequently, by Corollary 6.6, the claimed conditions are
satisfied with probability at least
$1-\delta_{\ref{cor:CGK}}=1-\frac{\delta}{2}$ for $c\geq c_{\ref{cor:CGK}}$.
The construction algorithm uses $\mathcal{O}(\log n)$ bits, costs
$\mathcal{O}(n)$ time, reads the random bits $(h_{t})_{t=1}^{3n+1}$ from left
to right, and outputs the elements of $\mathsf{W}^{\prime}(n,R,S)$ when
required.
In order to use Nisan’s pseudorandom generator [50], we need to argue that we
only care about properties testable using $\mathcal{O}(\log n)$-bit algorithms
with one-way access to the randomness $R$. As in [10], our testers are given
two (read-only) strings $X,Y\in\Sigma^{\leq n}$ and a stream of random bits
representing $R$ (the sequence $(h_{t})_{t=1}^{3n}$). Observe that the
following properties of the zip alignment $\mathcal{A}^{\prime}$ of
$\mathsf{W}^{\prime}(n,R,X)=(x_{t})_{t=1}^{3n+1}$ and
$\mathsf{W}^{\prime}(n,R,Y)=(y_{t})_{t=1}^{3n+1}$ are testable in
$\mathcal{O}(\log n)$ bits and $\mathcal{O}(n)$ time:
* •
whether $\mathcal{A}^{\prime}$ is greedy;
* •
whether $\mathsf{cost}(\mathcal{A}^{\prime})\leq k$ for a given integer
$k\in[0\mathinner{.\,.\allowbreak}n]$;
* •
whether $0pt(\mathcal{A}^{\prime})\leq w$ for a given integer
$w\in[0\mathinner{.\,.\allowbreak}n]$.
All these testers simply construct triples $(t,x_{t},y_{t})$ for subsequent
$t\in[1\mathinner{.\,.\allowbreak}3n+1]$.
In this setting, Nisan’s pseudorandom generator [50], given a sequence $r$ of
$\mathcal{O}(\log^{2}n)$ random bits, constructs a sequence $\mathsf{PRG}(r)$
of pseudorandom bits in $\mathcal{O}(1)$ amortized time per bit, using
$\mathcal{O}(\log^{2}n)$ bits of working space. Moreover, for every tester,
the probabilities of accepting a given input $\mathcal{I}$ with randomness $R$
and $\mathsf{PRG}(r)$ differ by at most $\frac{1}{n^{2}}$ [10, Theorem 5].
Consequently, setting
$\mathsf{W}(n,r,S)=\mathsf{W}^{\prime}(n,\mathsf{PRG}(r),S)$, we can guarantee
that the claimed condition is satisfied with probability at least
$1-\frac{1}{2}\delta-\frac{(2n+3)}{n^{2}}\geq 1-\delta$. The streaming
construction algorithm takes $\mathcal{O}(\log^{2}n)$ bits and
$\mathcal{O}(n\log n)$ time, dominated by the generation of $\mathsf{PRG}(r)$.
∎
### 6.2 Context encoding
For a string $S\in\Sigma^{*}$, define
$\overline{\mathsf{maxLZ}}(S)=\max_{[\ell\mathinner{.\,.\allowbreak}r)\subseteq[1\mathinner{.\,.\allowbreak}|S|]}|\mathsf{LZ}(\overline{S[\ell\mathinner{.\,.\allowbreak}r)})|$.
###### Observation 6.7.
Consider a string $S\in\Sigma^{*}$ and an integer $k\in\mathbb{Z}_{+}$. If
$\overline{\mathsf{maxLZ}}(S[\ell\mathinner{.\,.\allowbreak}r))\leq k$ holds
for a non-empty fragment $S[\ell\mathinner{.\,.\allowbreak}r)$, then:
1. (a)
$\overline{\mathsf{maxLZ}}(S[\ell\mathinner{.\,.\allowbreak}r])\leq k$ if and
only if $\mathsf{LZ}(\overline{S[\ell\mathinner{.\,.\allowbreak}r]})\leq k$;
2. (b)
$\overline{\mathsf{maxLZ}}(S[\ell+1\mathinner{.\,.\allowbreak}r))\leq k$;
3. (c)
$\overline{\mathsf{maxLZ}}(S[\ell\mathinner{.\,.\allowbreak}r])\leq k+1$.
###### Proof.
The first two parts follow from the fact that the size of the
$\mathsf{LZ}$-factorisation of a prefix of a string is bounded by the size of
the $\mathsf{LZ}$-factorisation of the string itself. The third claim follows
from the optimality of the LZ77 parsing among all LZ-like parsings (4.1). ∎
###### Definition 6.8 ((Double) Context).
Consider a string $S\in\Sigma^{*}$ and an integer $k\in\mathbb{Z}_{+}$. For a
position $p\in[1\mathinner{.\,.\allowbreak}|S|]$, we define the _context_
$C_{k}(S,p)$ as the longest prefix $S[p\mathinner{.\,.\allowbreak}q)$ of
$S[p\mathinner{.\,.\allowbreak}|S|]$ such that
$\overline{\mathsf{maxLZ}}(S[p\mathinner{.\,.\allowbreak}q))\leq k$ and the
_double context_ $C^{2}_{k}(S,p)$ as the longest prefix
$S[p\mathinner{.\,.\allowbreak}q)$ of $S[p\mathinner{.\,.\allowbreak}|S|]$
such that $\overline{\mathsf{maxLZ}}(S[p\mathinner{.\,.\allowbreak}r))\leq k$
and $\overline{\mathsf{maxLZ}}(S[r\mathinner{.\,.\allowbreak}q))\leq k$ for
some $r\in[p\mathinner{.\,.\allowbreak}q]$.
For integers $t<v$, let $\mu(t,v)$ denote the integer
$w\in[t\mathinner{.\,.\allowbreak}v)$ divisible by the largest power of 2.
Input: An integer $k\in\mathbb{Z}_{\geq 0}$, a string $S\in\Sigma^{*}$, and a
complete walk $(s_{t})_{t=1}^{m+1}$ over $S$.
Output: The string $\mathsf{CE}_{k}(W)[1\mathinner{.\,.\allowbreak}m]$.
1 $\mathsf{CE}_{k}(W):=\bot^{m}$, where $\bot=(\mathsf{LZ}(\varepsilon),0)$;
2 $s_{0}:=0;\;\mu_{0}:=0;\;u:=0$;
3 for _$t:=1$ to $m$_ do
4 if _$s_{t}\leq|S|$_ then
5
$\mu_{t}:=\mu(t,\min\\{v\in[1\mathinner{.\,.\allowbreak}m+1]:s_{v}=s_{t}+|C_{k}(S,s_{t})|\\})$;
6 if _$\mu_{t} >\mu_{t-1}$_ then
7
$\mathsf{CE}_{k}(W)[t]:=\left(\mathsf{LZ}\left(\overline{C^{2}_{k}(S,s_{t})}\right),s_{t}-s_{u}\right)$;
8 $u:=t$;
9
10
11
12 return _$\mathsf{CE}_{k}(W)$_ ;
Algorithm 3 Function $\mathsf{CE}$
We say that a position $S[s]$ is covered by a fragment
$S[\ell\mathinner{.\,.\allowbreak}r)$ if $\ell\leq s<r$.
###### Lemma 6.9.
Consider $\mathsf{CE}_{k}(W)$ constructed for an integer $k\in\mathbb{Z}_{\geq
0}$ and an $m$-complete walk $W=(s_{t})_{t=1}^{m+1}$ over $S\in\Sigma^{*}$.
Each position $s\in[1\mathinner{.\,.\allowbreak}|S|]$ satisfies
$1\leq|\\{t\in[1\mathinner{.\,.\allowbreak}m]:\mathsf{CE}_{k}(W)[t]\neq\bot\text{
and }C_{k}(S,s_{t})\text{ covers }S[s]\\}|\leq\\\
\leq|\\{t\in[1\mathinner{.\,.\allowbreak}m]:\mathsf{CE}_{k}(W)[t]\neq\bot\text{
and }C^{2}_{k}(S,s_{t})\text{ covers }S[s]\\}|\leq\mathcal{O}(\log m)$
###### Proof.
Let us first bound the covering number from below. Consider an index
$t\in[1\mathinner{.\,.\allowbreak}m]$ such that $s=s_{t}$, and let
$t^{\prime}\in[1\mathinner{.\,.\allowbreak}m]$ be the smallest index such that
$\mu_{t^{\prime}}=\mu_{t}$. Note that
$s_{\mu_{t^{\prime}}}<s_{t^{\prime}}+|C_{k}(S,s_{t^{\prime}})|$ and
$s_{\mu_{t}}\geq s_{t}$ by definition of $\mu$ and monotonicity of the walk
$W$. Due to $\mu_{t^{\prime}}=\mu_{t}$, this implies $s_{t^{\prime}}\leq
s_{t}<s_{t^{\prime}}+|C_{k}(S,s_{t^{\prime}})|$, i.e., that $S[s_{t}]$ is
covered by $C_{k}(S,s_{t^{\prime}})$. Due to $\mu_{t^{\prime}-1}\neq\mu_{t}$,
this guarantees $\mathsf{CE}_{k}(S)[t^{\prime}]\neq\bot$.
As for the upper bound, note that the indexes
$t\in[1\mathinner{.\,.\allowbreak}m]$ with $\mathsf{CE}_{k}(S)[t]\neq\bot$
have distinct values $\mu_{t}$. Let us further classify them into
$\mathcal{O}(\log n)$ groups depending on the largest power of two dividing
$\mu_{t}$. Consider two indexes $t<t^{\prime}$ in the same group. Since
$\mu_{t}<\mu_{t^{\prime}}$ are divisible by the same largest power of two,
there is a number $\nu\in(\mu_{t}\mathinner{.\,.\allowbreak}\mu_{t^{\prime}})$
divisible by a strictly larger power of two. By definition of $\mu_{t}$, we
have $s_{\nu}\geq s_{t}+|C_{k}(S,s_{t})|$ and, by definition of
$\mu_{t^{\prime}}$, we have $s_{\nu}<s_{t^{\prime}}$. Consequently,
$s_{t^{\prime}}>s_{t}+|C_{k}(S,s_{t})|$, i.e., the contexts $C_{k}(S,s_{t})$
and $C_{k}(S,s_{t^{\prime}})$ are disjoint. By monotonicity of
$\overline{\mathsf{maxLZ}}$, this also means that
$s_{t}+|C^{2}_{k}(S,s_{t})|\leq s_{t^{\prime}}+|C_{k}(S,s_{t^{\prime}})|$. In
particular, each position $s\in[1\mathinner{.\,.\allowbreak}|S|]$ is covered
by at most one context $C_{k}(S,s_{t})$ and at most two double contexts
$C^{2}_{k}(S,s_{t})$ for $t\in[1\mathinner{.\,.\allowbreak}m]$ belonging to a
single group. ∎
Let $\mathcal{A}$ be an alignment between $X,Y\in\Sigma^{*}$. We define
$\mathcal{B}_{X}(\mathcal{A})=\\{x:(x,y)\in\mathcal{B}(\mathcal{A})\\}\cap[1\mathinner{.\,.\allowbreak}|X|]$.
Moreover, for two alignments $\mathcal{A},\mathcal{A}^{\prime}$ between
$X,Y\in\Sigma^{*}$, we define $\Delta_{X}(\mathcal{A},\mathcal{A}^{\prime})$
to contain $x\in[1\mathinner{.\,.\allowbreak}|X|]$ unless
$(x,y)\in\mathcal{M}(\mathcal{A})\cap\mathcal{M}(\mathcal{A}^{\prime})$ for
some $y\in[1\mathinner{.\,.\allowbreak}|Y|]$.
###### Lemma 6.10.
Let $b\in\mathbb{Z}_{+}$ and $\mathcal{A},\mathcal{A}^{\prime}$ be greedy
alignments of strings $X$ and $Y$. The positions in
$\Delta_{X}(\mathcal{A},\mathcal{A}^{\prime})$ can be covered by at most
$\mathsf{cost}(\mathcal{A})+\frac{1}{b}\mathsf{cost}(\mathcal{A}^{\prime})$
contexts $C_{0pt(\mathcal{A})+0pt(\mathcal{A}^{\prime})+2b}(X,\cdot)$.
Moreover, if $b>\mathsf{cost}(\mathcal{A}^{\prime})$, then the positions in
$\Delta_{X}(\mathcal{A},\mathcal{A}^{\prime})$ can be covered by contexts
$C_{0pt(\mathcal{A})+0pt(\mathcal{A}^{\prime})+2b}(X,x)$ with
$x\in\mathcal{B}_{X}(\mathcal{A})$.
###### Proof.
Let $w=0pt(\mathcal{A})$ and $w^{\prime}=0pt(\mathcal{A}^{\prime})$. Without
loss of generality, we may trim the longest common prefix of $X$ and $Y$; this
is feasible because both $\mathcal{A}$ and $\mathcal{A}^{\prime}$ match the
common prefix, so $\Delta_{X}(\mathcal{A},\mathcal{A}^{\prime})$ only contains
positions following the prefix. Moreover, we assume that $X\neq\varepsilon$;
otherwise $\Delta_{X}(\mathcal{A},\mathcal{A}^{\prime})=\emptyset$ and the
lemma is trivial.
In the remaining case, we are guaranteed that
$1\in\mathcal{B}_{X}(\mathcal{A})$. Let
$\mathcal{B}_{X}(\mathcal{A})=\\{x_{1},\ldots,x_{m}\\}$, where
$1=x_{1}<\cdots<x_{m}$, and let $x_{m+1}=|X|+1$. This yields a decomposition
$X=X_{1}\cdots X_{m}$ into $m$ non-empty substrings
$X_{i}:=X[x_{i}\mathinner{.\,.\allowbreak}x_{i+1})$. The alignment
$\mathcal{A}^{\prime}$ further yields a decomposition $Y=Y_{1}\cdots Y_{m}$
into $m$ (possibly empty) substrings
$Y_{i}:=Y[y_{i}\mathinner{.\,.\allowbreak}y_{i+1})$ so that
$X_{i}\sim_{\mathcal{A}^{\prime}}Y_{i}$ and $|x_{i}-y_{i}|\leq w^{\prime}$ for
$i\in[1\mathinner{.\,.\allowbreak}m]$. Moreover, the cost of
$\mathcal{A}^{\prime}$ can be expressed as
$\mathsf{cost}(\mathcal{A}^{\prime})=\sum_{i=1}^{m}c^{\prime}_{i}$, where
$c^{\prime}_{i}$ denotes the cost of $\mathcal{A}^{\prime}$ restricted to
$X_{i}$ and $Y_{i}$.
###### Claim 6.11.
For every $i\in[1\mathinner{.\,.\allowbreak}m]$, the set
$\Delta_{X}(\mathcal{A},\mathcal{A}^{\prime})\cap[x_{i}\mathinner{.\,.\allowbreak}x_{i+1})$
can be covered by at most $\lceil\frac{1}{b}(c^{\prime}_{i}+1)\rceil$ contexts
$C_{w+w^{\prime}+2b}(X,\cdot)$ including $C_{w+w^{\prime}+2b}(X,x_{i})$.
###### Proof.
Let
$q_{i}=\max(\Delta_{X}(\mathcal{A},\mathcal{A}^{\prime})\cap[x_{i}\mathinner{.\,.\allowbreak}x_{i+1}))$.
We may assume that $q_{i}\geq x_{i}+w+w^{\prime}$; otherwise, the claim holds
trivially. We shall prove that
$X[x_{i}+1\mathinner{.\,.\allowbreak}q_{i}-(w+w^{\prime})]$ can be decomposed
into at most $2c^{\prime}_{i}+1$ phrases, each of which is a single character
or has another occurrence at most $w+w^{\prime}$ positions to the right.
Accounting for $X[x_{i}]$, this yields a decomposition of
$X[x_{i}\mathinner{.\,.\allowbreak}q_{i}-(w+w^{\prime})]$ into at most
$2c^{\prime}_{i}+2$ such phrases. The $\overline{\mathsf{maxLZ}}$ measure of
the concatenation of every $2b$ subsequent phrases and $w+w^{\prime}$
following single characters does not exceed $w+w^{\prime}+2b$. Hence,
$X[x_{i}\mathinner{.\,.\allowbreak}q_{i}]$ can be covered by
$\lceil\frac{1}{b}(c^{\prime}_{i}+1)\rceil$ contexts
$C_{w+w^{\prime}+2b}(X,\cdot)$ including $C_{w+w^{\prime}+2b}(X,x_{i})$.
Thus, it remains to construct the decomposition of
$X[x_{i}+1\mathinner{.\,.\allowbreak}q_{i}-(w+w^{\prime})]$ into phrases. By
definition of $\mathcal{B}_{X}(\mathcal{A})$, we have
${X[x_{i}+1\mathinner{.\,.\allowbreak}x_{i+1})\simeq_{\mathcal{A}}Y[x_{i}+1+d\mathinner{.\,.\allowbreak}x_{i+1}+d)}$
for some shift $d\in[-w\mathinner{.\,.\allowbreak}w]$. We consider two cases.
In the first case, we assume that $(x_{i}+1,\bar{y})\in\mathcal{A}^{\prime}$
holds for some $\bar{y}\geq x_{i}+1+d$. The greedy nature of
$\mathcal{A}^{\prime}$ then guarantees that if $(x,y)\in\mathcal{A}^{\prime}$
with $x\in[x_{i}+1\mathinner{.\,.\allowbreak}q_{i})$, then $y>x+d$. We
decompose $X[x_{i}+1\mathinner{.\,.\allowbreak}q_{i}-(w+w^{\prime})]$ (which
is contained in $X_{i}$) into maximal phrases
$X[x\mathinner{.\,.\allowbreak}x^{\prime}]$ satisfying
$X[x\mathinner{.\,.\allowbreak}x^{\prime}]\simeq_{\mathcal{A}^{\prime}}Y[x+d^{\prime}\mathinner{.\,.\allowbreak}x^{\prime}+d^{\prime}]$
for some $d^{\prime}\in(d\mathinner{.\,.\allowbreak}w^{\prime}]$ and remaining
single characters (deleted or substituted by $\mathcal{A}^{\prime}$). Observe
that this yields at most $1+c^{\prime}_{i}$ phrases and at most
$c^{\prime}_{i}$ single characters. Moreover, we have
$X[x\mathinner{.\,.\allowbreak}x^{\prime}]\simeq_{\mathcal{A}^{\prime}}Y[x+d^{\prime}\mathinner{.\,.\allowbreak}x^{\prime}+d^{\prime}]\simeq_{\mathcal{A}}X[x+(d^{\prime}-d)\mathinner{.\,.\allowbreak}x^{\prime}+(d^{\prime}-d)]$
due to $x_{i}+1\leq x\leq x+(d^{\prime}-d)$ and $x^{\prime}+(d^{\prime}-d)\leq
x^{\prime}+(w^{\prime}+w)\leq q_{i}$. Hence, each phrase
$X[x\mathinner{.\,.\allowbreak}x^{\prime}]$ has another occurrence located
$d^{\prime}-d\in[1\mathinner{.\,.\allowbreak}w+w^{\prime}]$ positions to the
right.
In the second case, we assume that $(x_{i}+1,\bar{y})\in\mathcal{A}^{\prime}$
holds for some $\bar{y}\leq x_{i}+1+d$. The greedy nature of
$\mathcal{A}^{\prime}$ then guarantees that if $(x,y)\in\mathcal{A}^{\prime}$
with $x\in[x_{i}+1\mathinner{.\,.\allowbreak}q_{i})$, then $y<x+d$. We
decompose $Y[x_{i}+1+d\mathinner{.\,.\allowbreak}q_{i}-(w+w^{\prime})+d]$
(which is contained in $Y_{i}$) into maximal phrases
$Y[y\mathinner{.\,.\allowbreak}y^{\prime}]$ satisfying
$Y[y\mathinner{.\,.\allowbreak}y^{\prime}]\simeq_{\mathcal{A}^{\prime}}X[y-d^{\prime}\mathinner{.\,.\allowbreak}y^{\prime}-d^{\prime}]$
for some $d^{\prime}\in[-w\mathinner{.\,.\allowbreak}d)$ and remaining single
characters (deleted or substituted by $\mathcal{A}^{\prime}$). Observe that
this yields at most $1+c^{\prime}_{i}$ phrases and at most $c^{\prime}_{i}$
single characters. Moreover, we have
$Y[y\mathinner{.\,.\allowbreak}y^{\prime}]\simeq_{\mathcal{A}^{\prime}}X[y-d^{\prime}\mathinner{.\,.\allowbreak}y^{\prime}-d^{\prime}]\simeq_{\mathcal{A}}Y[y+(d-d^{\prime})\mathinner{.\,.\allowbreak}y+(d-d^{\prime})]$
due to $x_{i}+1\leq x_{i}+1+d-d^{\prime}\leq y-d^{\prime}$ and
$y^{\prime}-d^{\prime}\leq q_{i}-(w+w^{\prime})+d-d^{\prime}\leq q_{i}$.
Hence, each phrase $Y[y\mathinner{.\,.\allowbreak}y^{\prime}]$ has another
occurrence located $d-d^{\prime}\in[1\mathinner{.\,.\allowbreak}w+w^{\prime}]$
characters to the right. Since
$Y[x_{i}+1+d\mathinner{.\,.\allowbreak}q_{i}+d]=X[x_{i}+1\mathinner{.\,.\allowbreak}q_{i}]$,
this decomposition of
$Y[x_{i}+1+d\mathinner{.\,.\allowbreak}q_{i}-(w+w^{\prime})+d]$ gives an
analogous decomposition of
$X[x_{i}+1\mathinner{.\,.\allowbreak}q_{i}-(w+w^{\prime})]$. ∎
The set $\Delta_{X}(\mathcal{A},\mathcal{A}^{\prime})$ can be covered by
$\sum_{i=1}^{m}\lceil\frac{1}{b}(c^{\prime}_{i}+1)\rceil\leq
m+\frac{\mathsf{cost}(\mathcal{A}^{\prime})}{b}\leq\mathsf{cost}(\mathcal{A})+\frac{\mathsf{cost}(\mathcal{A}^{\prime})}{b}$
contexts $C_{w+w^{\prime}+2b}(X,\cdot)$. If
$b\geq\mathsf{cost}(\mathcal{A}^{\prime})+1$, then the contexts starting at
positions in $\mathcal{B}_{X}(\mathcal{A})$ are sufficient. ∎
###### Lemma 6.12.
Let $\mathcal{A}_{W}$ be the zip alignment of $m$-complete walks
$W_{X}=(x_{t})_{t=1}^{m+1}$, $W_{Y}=(y_{t})_{t=1}^{m+1}$ over strings
$X,Y\in\Sigma^{*}$, and let $k^{\prime}\in\mathbb{Z}_{+}$. If
$\mathcal{A}_{W}$ is greedy, then $\mathcal{B}_{X}(\mathcal{A}_{W})\subseteq
P_{X}$, where $P_{X}$ is the set of positions
$x\in[1\mathinner{.\,.\allowbreak}|X|]$ such that $X[x]$ is covered by
$C^{2}_{k^{\prime}}(X,x_{t})$ for some $t\in[1\mathinner{.\,.\allowbreak}m]$
with
$\bot\neq\mathsf{CE}_{k^{\prime}}(W_{X})[t]\neq\mathsf{CE}_{k^{\prime}}(W_{Y})[t]$.
Moreover, for every positive integer $k\geq\mathsf{ed}(X,Y)$ such that
$k^{\prime}\geq 0pt(\mathcal{A}_{W})+5k$:
* •
every $\mathcal{A}\in\mathsf{GA}_{k}(X,Y)$ satisfies
$\Delta_{X}(\mathcal{A},\mathcal{A}_{W})\subseteq P_{X}$,
* •
$\mathsf{hd}(\mathsf{CE}_{k^{\prime}}(X),\mathsf{CE}_{k^{\prime}}(Y))\leq
c(k+\frac{1}{k}\mathsf{cost}(\mathcal{A}_{W}))\log m$ holds for a sufficiently
large constant $c$.
###### Proof.
Consider a position $x\in\mathcal{B}_{X}(\mathcal{A}_{W})$. By Lemma 6.9,
there is an index $t\in[1\mathinner{.\,.\allowbreak}m]$ such that
$\mathsf{CE}_{k^{\prime}}(W_{X})[t]\neq\bot$ and $C^{2}_{k^{\prime}}(X,x_{t})$
covers $X[x]$. To conclude that $x\in P_{X}$, it remains to prove
$\mathsf{CE}_{k^{\prime}}(W_{X})[t]\neq\mathsf{CE}_{k^{\prime}}(W_{Y})[t]$.
For a proof by contradiction, suppose that
$\mathsf{CE}_{k^{\prime}}(W_{X})[t]=\mathsf{CE}_{k^{\prime}}(W_{Y})[t]$. This
implies $C^{2}_{k^{\prime}}(X,x_{t})=C^{2}_{k^{\prime}}(Y,y_{t})$. Since
$\mathcal{A}_{W}$ is greedy, we derive
$X[x_{t}\mathinner{.\,.\allowbreak}x_{t}+|C^{2}_{k^{\prime}}(X,x_{t})|)\simeq_{\mathcal{A}_{W}}Y[y_{t}\mathinner{.\,.\allowbreak}y_{t}+|C^{2}_{k^{\prime}}(Y,y_{t})|)$,
which contradicts $x\in\mathcal{B}_{X}(\mathcal{A}_{W})$.
In the remainder of the proof, we assume that $k\geq\mathsf{ed}(X,Y)$ and
$k^{\prime}\geq 0pt(\mathcal{A}_{W})+5k$. Next, consider a greedy alignment
$\mathcal{A}$ with $\mathsf{cost}(\mathcal{A})\leq k$ and a position
$x\in\Delta_{X}(\mathcal{A},\mathcal{A}_{W})$. Due to $k^{\prime}\geq
0pt(\mathcal{A}_{W})+0pt(\mathcal{A})+4k$, Lemma 6.10 shows that there exists
a position $r\in\mathcal{B}_{X}(\mathcal{A}_{W})$ such that
$C_{k^{\prime}}(X,r)$ covers $X[x]$ and, by Lemma 6.9, there exists a position
$t\in[1\mathinner{.\,.\allowbreak}m]$ such that
$\mathsf{CE}_{k^{\prime}}(X)[t]\neq\bot$ and $C_{k^{\prime}}(X,x_{t})$ covers
$X[r-1]$. Since
$\overline{\mathsf{maxLZ}}(X[x_{t}\mathinner{.\,.\allowbreak}r)),\overline{\mathsf{maxLZ}}(X[r\mathinner{.\,.\allowbreak}x])\leq
k^{\prime}$, we conclude that both $r$ and $x$ are covered by
$C^{2}_{k^{\prime}}(X,x_{t})$. We shall prove that
$\mathsf{CE}_{k^{\prime}}(X)[t]\neq\mathsf{CE}_{k^{\prime}}(Y)[t]$. For a
proof by contradiction, suppose that
$\mathsf{CE}_{k^{\prime}}(W_{X})[t]=\mathsf{CE}_{k^{\prime}}(W_{Y})[t]$, which
implies $C^{2}_{k^{\prime}}(X,x_{t})=C^{2}_{k^{\prime}}(Y,y_{t})$. Let us fix
the smallest $t^{\prime}\in[t\mathinner{.\,.\allowbreak}m+1]$ such that
$(x_{t^{\prime}},y_{t^{\prime}})\in\mathcal{B}(\mathcal{A}_{W})$. Observe that
$X[x_{t}\mathinner{.\,.\allowbreak}x_{t^{\prime}})\simeq_{\mathcal{A}_{W}}Y[y_{t}\mathinner{.\,.\allowbreak}y_{t^{\prime}})$
and, by the greedy nature of $\mathcal{A}_{W}$,
$X[x_{t}\mathinner{.\,.\allowbreak}x_{t^{\prime}})=Y[y_{t}\mathinner{.\,.\allowbreak}y_{t^{\prime}})$
is the longest common prefix of $X[x_{t}\mathinner{.\,.\allowbreak}]$ and
$Y[y_{t}\mathinner{.\,.\allowbreak}]$. At the same time, due to
$x_{t}<r\in\mathcal{B}_{X}(\mathcal{A}_{W})$, we have $x_{t^{\prime}}\leq r$,
so $X[x_{t}\mathinner{.\,.\allowbreak}x_{t^{\prime}}]$ is a prefix of
$C^{2}_{k^{\prime}}(X,x_{t})$. However,
$X[x_{t}\mathinner{.\,.\allowbreak}x_{t^{\prime}}]$ is not a prefix
$Y[y_{t}\mathinner{.\,.\allowbreak}]$, so $C^{2}_{k^{\prime}}(X,x_{t})$ is not
a prefix of $Y[y_{t}\mathinner{.\,.\allowbreak}]$ and
$C^{2}_{k^{\prime}}(X,x_{t})\neq C^{2}_{k^{\prime}}(Y,y_{t})$. The
contradiction completes the proof.
Finally, we bound
$\mathsf{hd}(\mathsf{CE}_{k^{\prime}}(X),\mathsf{CE}_{k^{\prime}}(Y))$ using
several claims. Consider a set $M$ of indices
$t\in[1\mathinner{.\,.\allowbreak}m]$ such that
$\mathsf{CE}_{k^{\prime}}(X)[t]$ and $\mathsf{CE}_{k^{\prime}}(Y)[t]$ differ
on the first coordinate.
###### Claim 6.13.
If $t\in M$ satisfies $\mathsf{CE}_{k^{\prime}}(X)[t]\neq\bot$, then
$[x_{t-1}\mathinner{.\,.\allowbreak}x_{t}+|C^{2}_{k^{\prime}}(X,x_{t})|]\cap(\mathcal{B}_{X}(\mathcal{A}_{W})\cup\\{|X|+1\\})\neq\emptyset.$
###### Proof.
For a proof by contradiction, suppose that
$[x_{t-1}\mathinner{.\,.\allowbreak}x_{t}+|C^{2}_{k^{\prime}}(X,x_{t})|]\cap(\mathcal{B}_{X}(\mathcal{A}_{W})\cup\\{|X|+1\\})=\emptyset$.
Let $L$ be the length of the longest common prefix of
$X[x_{t}\mathinner{.\,.\allowbreak}]$ and
$Y[y_{t}\mathinner{.\,.\allowbreak}]$. Note that
$(x_{t}+L,y_{t}+L)\in\mathcal{B}(\mathcal{A}_{W})$, so
$x_{t}+L\in\mathcal{B}_{X}(\mathcal{A}_{W})\cup\\{|X|+1\\}$. Consequently,
$L>|C^{2}_{k^{\prime}}(X,x_{t})|\geq|C_{k^{\prime}}(X,x_{t})|$, and therefore
$X[x_{t}\mathinner{.\,.\allowbreak}x_{t}+|C^{2}_{k^{\prime}}(X,x_{t})|]=Y[y_{t}\mathinner{.\,.\allowbreak}y_{t}+|C^{2}_{k^{\prime}}(Y,y_{t})|]$
and
$X[x_{t}\mathinner{.\,.\allowbreak}x_{t}+|C_{k^{\prime}}(X,x_{t})|]=Y[y_{t}\mathinner{.\,.\allowbreak}y_{t}+|C_{k^{\prime}}(Y,y_{t})|]$.
In particular, $C^{2}_{k^{\prime}}(X,x_{t})=C^{2}_{k^{\prime}}(Y,y_{t})$.
If $t=1$, then we note that
$\mathsf{CE}_{k^{\prime}}(X)[t]\neq\bot\neq\mathsf{CE}_{k^{\prime}}(Y)[t]$, so
$C^{2}_{k^{\prime}}(X,x_{t})=C^{2}_{k^{\prime}}(Y,y_{t})$ contradicts $t\in
M$.
In the following, we assume that $t\geq 2$. Due to
$(x_{t-1},y_{t-1})\notin\mathcal{B}(\mathcal{A}_{W})$, either
$(x_{t-1},y_{t-1})=(x_{t},y_{t})$, or $(x_{t-1},y_{t-1})=(x_{t}-1,y_{t}-1)$
and $X[x_{t}-1]=Y[y_{t}-1]$. In either case, we have
$X[x_{t-1}\mathinner{.\,.\allowbreak}x_{t-1}+|C_{k^{\prime}}(X,x_{t-1})|]=Y[y_{t-1}\mathinner{.\,.\allowbreak}y_{t-1}+|C_{k^{\prime}}(Y,y_{t-1})|]$
due to
$X[x_{t-1}\mathinner{.\,.\allowbreak}x_{t}+|C_{k^{\prime}}(X,x_{t})|]=Y[y_{t-1}\mathinner{.\,.\allowbreak}y_{t}+|C_{k^{\prime}}(Y,y_{t})|]$
and by 6.7. Consequently,
$\min\\{u\in[1\mathinner{.\,.\allowbreak}m]:x_{u}=x_{t-1}+|C_{k^{\prime}}(X,x_{t-1})|\\}=\min\\{u\in[1\mathinner{.\,.\allowbreak}m]:y_{u}=y_{t-1}+|C_{k^{\prime}}(Y,y_{t-1})|\\}$
and, by a similar reasoning,
$\min\\{u\in[1\mathinner{.\,.\allowbreak}m]:x_{u}=x_{t}+|C_{k^{\prime}}(X,x_{t})|\\}=\min\\{u\in[1\mathinner{.\,.\allowbreak}m]:y_{u}=y_{t}+|C_{k^{\prime}}(Y,y_{t})|\\}$.
Hence, the assumption $\mathsf{CE}_{k^{\prime}}(X)[t]\neq\bot$ implies
$\mathsf{CE}_{k^{\prime}}(Y)[t]\neq\bot$. Therefore,
$C^{2}_{k^{\prime}}(X,x_{t})=C^{2}_{k^{\prime}}(Y,y_{t})$ contradicts $t\in
M$. ∎
###### Claim 6.14.
There are $\mathcal{O}((k+\frac{1}{k}\mathsf{cost}(\mathcal{A}_{W}))\log m)$
positions $t\in M$ such that $\mathsf{CE}_{k^{\prime}}(X)[t]\neq\bot$.
###### Proof.
Let $\mathcal{O}$ be an optimum greedy alignment between $X$ and $Y$. Due to
$k^{\prime}\geq 0pt(\mathcal{O})+0pt(\mathcal{A}_{W})+4k$, from Lemma 6.10 it
follows that $\Delta_{X}(\mathcal{O},\mathcal{A}_{W})$ can be covered by
$\mathcal{O}(\mathsf{cost}(\mathcal{O})+\frac{1}{k}\mathsf{cost}(\mathcal{A}_{W}))=\mathcal{O}(k+\frac{1}{k}\mathsf{cost}(\mathcal{A}_{W}))$
contexts $C_{k^{\prime}}(X,\cdot)$. Let us fix such the smallest family
$\mathcal{C}$ of contexts covering
$\mathcal{B}_{X}(\mathcal{A}_{W})\subseteq\Delta_{X}(\mathcal{O},\mathcal{A}_{W})$.
Define a set
$R_{X}=\\{|X|\\}\cup\bigcup_{X[q\mathinner{.\,.\allowbreak}r]\in\mathcal{C}}\\{q-1,r+1\\}$
and note that $|R_{X}|=\mathcal{O}(|\mathcal{C}|)$. We shall prove that, if
$t\in M$ and $\mathsf{CE}_{k^{\prime}}(X)[t]\neq\bot$, then
$C^{2}_{k^{\prime}}(X,x_{t})$ covers at least one position in $R_{X}$. This is
sufficient to derive the claim because, by Lemma 6.9, every position in $X$ is
covered by at most $\mathcal{O}(\log m)$ double contexts
$C^{2}_{k^{\prime}}(X,x_{t})$ with $\mathsf{CE}_{k^{\prime}}(X)[t]\neq\bot$.
By 6.13, we have
$[x_{t-1}\mathinner{.\,.\allowbreak}x_{t}+|C^{2}_{k^{\prime}}(X,x_{t})|]\cap(\mathcal{B}_{X}(\mathcal{A}_{W})\cup\\{|X|+1\\})\neq\emptyset$.
If
$\\{|X|+1\\}\in[x_{t-1}\mathinner{.\,.\allowbreak}x_{t}+|C^{2}_{k^{\prime}}(X,x_{t})|]$,
then $|X|+1=x_{t}+|C^{2}_{k^{\prime}}(X,x_{t})|$. Hence,
$C^{2}_{k^{\prime}}(X,x_{t})$ covers the position $|X|\in R_{X}$. Thus, we may
assume that
$[x_{t-1}\mathinner{.\,.\allowbreak}x_{t}+|C^{2}_{k^{\prime}}(X,x_{t})|]$
contains a position in $\mathcal{B}_{X}(\mathcal{A}_{W})$. Note that the
fragment $X[q\mathinner{.\,.\allowbreak}r]\in\mathcal{C}$ covering that
position in $\mathcal{B}_{X}(\mathcal{A}_{W})$ satisfies $r\geq x_{t-1}$ and
$q\leq x_{t}+|C^{2}_{k^{\prime}}(X,x_{t})|$. In particular, $r+1\geq x_{t}$
and $q-1<x_{t}+|C^{2}_{k^{\prime}}(X,x_{t})|$. Now, if
$C^{2}_{k^{\prime}}(X,x_{t})$ covers $q-1$ or $r+1$, then we are done.
Otherwise, $q-1<x_{t}$ and $r+1\geq x_{t}+|C^{2}_{k^{\prime}}(X,x_{t})|$, so
$q\leq x_{t}$ and $r\geq x_{t}+|C^{2}_{k^{\prime}}(X,x_{t})|$, i.e.,
$C^{2}_{k^{\prime}}(X,x_{t})$ is contained in
$X[q\mathinner{.\,.\allowbreak}r]$ and, since $\overline{\mathsf{maxLZ}}$ is
monotone, $\overline{\mathsf{maxLZ}}(C^{2}_{k^{\prime}}(X,x_{t}))\leq
k^{\prime}$. Consequently,
$C^{2}_{k^{\prime}}(X,x_{t})=C_{k^{\prime}}(X,x_{t})$ is a suffix of $X$, so
$C^{2}_{k^{\prime}}(X,x_{t})$ covers the position $|X|\in R_{X}$. ∎
A symmetric argument shows that there are
$\mathcal{O}((k+\frac{1}{k}\mathsf{cost}(\mathcal{A}_{W}))\log m)$ positions
$t\in M$ such that $\mathsf{CE}_{k^{\prime}}(Y)[t]\neq\bot$. Consequently,
$|M|=\mathcal{O}((k+\frac{1}{k}\mathsf{cost}(\mathcal{A}_{W}))\log m)$. We
bound $\mathsf{hd}(\mathsf{CE}_{k^{\prime}}(X),\mathsf{CE}_{k^{\prime}}(Y))$
using the following claim.
###### Claim 6.15.
$\mathsf{hd}(\mathsf{CE}_{k^{\prime}}(X),\mathsf{CE}_{k^{\prime}}(Y))\leq
2|M|$.
###### Proof.
Let
$M^{\prime}=\\{t\in[1\mathinner{.\,.\allowbreak}m]:\mathsf{CE}_{k^{\prime}}(X)[t]\neq\mathsf{CE}_{k^{\prime}}(Y)[t]\\}$.
Note that $M\subseteq M^{\prime}$, so the claim is equivalent to
$|M^{\prime}\setminus M|\leq|M|$. For every $t\in M^{\prime}\setminus M$, we
shall prove that $t$ is not the leftmost position in $M^{\prime}$ and that the
preceding position in $M^{\prime}$ belongs to $M$.
The assumption $t\in M^{\prime}\setminus M$ implies
$\mathsf{CE}_{k^{\prime}}(X)[t]\neq\bot\neq\mathsf{CE}_{k^{\prime}}(Y)[t]$. We
define $t^{\prime}$ as the largest position in
$[1\mathinner{.\,.\allowbreak}t)$ such that
$\mathsf{CE}_{k^{\prime}}(X)[t^{\prime}]\neq\bot$ or
$\mathsf{CE}_{k^{\prime}}(Y)[t^{\prime}]\neq\bot$. To see that $t^{\prime}$ is
well defined, note that $t>1$ and $\mathsf{CE}_{k^{\prime}}(X)[1]\neq\bot$.
Now, suppose that $t^{\prime}\notin M$. Consequently, we have
$\mathsf{CE}_{k^{\prime}}(X)[t^{\prime}]\neq\bot\neq\mathsf{CE}_{k^{\prime}}(Y)[t^{\prime}]$.
Due to
$\mathsf{CE}_{k^{\prime}}(X)[t^{\prime}+1\mathinner{.\,.\allowbreak}t)=\bot^{t-t^{\prime}-1}$,
Lemma 6.9 implies that $C_{k^{\prime}}(X,x_{t^{\prime}})$ covers $X[x_{t}-1]$,
that is,
$\overline{\mathsf{maxLZ}}(X[x_{t^{\prime}}\mathinner{.\,.\allowbreak}x_{t}))\leq
k^{\prime}$, and hence $X[x_{t^{\prime}}\mathinner{.\,.\allowbreak}x_{t}]$ is
a prefix of $C^{2}_{k^{\prime}}(X,x_{t^{\prime}})$. Moreover,
$C^{2}_{k^{\prime}}(X,x_{t^{\prime}})=C^{2}_{k^{\prime}}(Y,y_{t^{\prime}})$
implies
$X[x_{t^{\prime}}\mathinner{.\,.\allowbreak}x_{t}]\simeq_{\mathcal{A}_{W}}Y[y_{t^{\prime}}\mathinner{.\,.\allowbreak}y_{t^{\prime}}+x_{t}-x_{t^{\prime}}]$
by the greedy nature of $\mathcal{A}_{W}$. As
$(x_{t},y_{t})\in\mathcal{A}_{W}$, we conclude that
$y_{t}=y_{t^{\prime}}+x_{t}-x_{t^{\prime}}$. At the same time, since
$\mathsf{CE}_{k^{\prime}}(X)[t^{\prime}+1\mathinner{.\,.\allowbreak}t)=\bot^{t-t^{\prime}-1}=\mathsf{CE}_{k^{\prime}}(Y)[t^{\prime}+1\mathinner{.\,.\allowbreak}t)$,
we have
$\mathsf{CE}_{k^{\prime}}(X)[t]=(\mathsf{LZ}(\overline{C^{2}_{k^{\prime}}(X,x_{t})}),x_{t}-x_{t^{\prime}})$
and
$\mathsf{CE}_{k^{\prime}}(Y)[t]=(\mathsf{LZ}(\overline{C^{2}_{k^{\prime}}(Y,y_{t})}),y_{t}-y_{t^{\prime}})$.
Thus, $\mathsf{CE}_{k^{\prime}}(X)[t]=\mathsf{CE}_{k^{\prime}}(Y)[t]$, which
contradicts $t\in M^{\prime}$.
Consequently, $t^{\prime}\in M$. In this case,
$M^{\prime}\cap[t^{\prime}\mathinner{.\,.\allowbreak}t]=\\{t^{\prime},t\\}$,
so $t^{\prime}$ is the position of $M^{\prime}$ preceding $t$. Our goal was to
show that such a position exists and belongs to $M$, so this completes the
proof of the claim. ∎
Overall, we conclude that
$\mathsf{hd}(\mathsf{CE}_{k^{\prime}}(X),\mathsf{CE}_{k^{\prime}}(Y))\leq
2|M|=\mathcal{O}((k+\frac{1}{k}\mathsf{cost}(\mathcal{A}_{W}))\log m)$. ∎
###### Lemma 6.16.
Given integers $n\geq k\geq 1$, a seed $r$ of $\mathcal{O}(\log^{2}n)$ random
bits, and streaming access to a string $S\in\Sigma^{\leq n}$, the string
$\mathsf{CE}_{k}(\mathsf{W}(S,n,r))$ can be computed in
$\tilde{\mathcal{O}}(k)$ space and $\tilde{\mathcal{O}}(nk)$ time.
###### Proof.
Let us start with an auxiliary subroutine:
###### Claim 6.17.
There is a streaming algorithm that computes $\mathsf{D}(C_{k}(S,p))$ for
subsequent positions $p\in[1\mathinner{.\,.\allowbreak}|S|]$. The algorithm
uses $\tilde{\mathcal{O}}(k)$ space and $\tilde{\mathcal{O}}(nk)$ time.
Moreover, if $C_{k}(S,p)=S[p\mathinner{.\,.\allowbreak}q)$, then
$\mathsf{D}(C_{k}(S,p))$ is reported while the algorithm processes $S[q]$ (or
the end-of-string token if $q=|S|+1$).
###### Proof.
The algorithm maintains a fragment $S[p\mathinner{.\,.\allowbreak}q)$
satisfying $\overline{\mathsf{maxLZ}}(S[p\mathinner{.\,.\allowbreak}q))\leq k$
and the encoding $\mathsf{D}(S[p\mathinner{.\,.\allowbreak}q))$. Initially,
$S[p\mathinner{.\,.\allowbreak}q)=S[1\mathinner{.\,.\allowbreak}1)=\varepsilon$.
In each iteration, we read $S[q]$, compute
$\mathsf{D}(S[p\mathinner{.\,.\allowbreak}q])$ (using Proposition 4.2(i)), and
check whether $|\mathsf{LZ}(\overline{S[p\mathinner{.\,.\allowbreak}q]})|\leq
k$ (using Proposition 4.2(f)). By 6.7(a), this condition is equivalent to
$\overline{\mathsf{maxLZ}}(S[p\mathinner{.\,.\allowbreak}q])\leq k$. If
$\overline{\mathsf{maxLZ}}(S[p\mathinner{.\,.\allowbreak}q])\leq k$, we
discard $\mathsf{D}(S[p\mathinner{.\,.\allowbreak}q))$ and increment $q$.
Otherwise, we are guaranteed that
$S[p\mathinner{.\,.\allowbreak}q)=C_{k}(S,p)$, so we output
$\mathsf{D}(S[p\mathinner{.\,.\allowbreak}q))$, compute
$\mathsf{D}(S[p+1\mathinner{.\,.\allowbreak}q))$ (using Proposition 4.2(g)),
discard $\mathsf{D}(S[p\mathinner{.\,.\allowbreak}q))$ and
$\mathsf{D}(S[p\mathinner{.\,.\allowbreak}q])$, and increment $p$. By 6.7(b),
we are guaranteed that the invariant
$\overline{\mathsf{maxLZ}}(S[p\mathinner{.\,.\allowbreak}q))\leq k$ remains
satisfied. In the special case of $q=|S|+1$, we proceed as if
$\overline{\mathsf{maxLZ}}(S[p\mathinner{.\,.\allowbreak}q])>k$.
By 6.7(c), we store $\mathsf{D}(X)$ only for strings $X$ satisfying
$|\overline{\mathsf{maxLZ}}(X)|\leq k+1$, so the space usage and the per-
iteration running time is $\tilde{\mathcal{O}}(k)$. Each iteration increments
either $p$ or $q$, so the algorithm reads $S$ in a streaming fashion and the
amortized running time is $\tilde{\mathcal{O}}(k)$ per character. ∎
We maintain two instances of the algorithm of 6.17 and two instances of the
algorithm of Proposition 3.16. We feed the first instance of 6.17 with the
input stream $S$, obtaining $\mathsf{D}(C_{k}(S,q))$ for subsequent positions
$q\in[1\mathinner{.\,.\allowbreak}|S|]$. Upon retrieving
$\mathsf{D}(C_{k}(S,q))$, we extract $S[q]$ using Proposition 4.2(b) and
forward $S[q]$ to the first instance of Proposition 3.16, which lists indices
$v\in[1\mathinner{.\,.\allowbreak}3n]$ such that $s_{v}=q$. We also forward
$S[q]$ to the second instance of 6.17, obtaining $\mathsf{D}(C_{k}(S,p))$ for
all subsequent positions $p$ such that
$\mathsf{D}(C_{k}(S,p))=S[p\mathinner{.\,.\allowbreak}q)$. Upon retrieving
$\mathsf{D}(C_{k}(S,p))$, we extract $S[p]$ using Proposition 4.2(b) and
forward $S[p]$ to the second instance of Proposition 3.16, which lists indices
$t\in[1\mathinner{.\,.\allowbreak}3n]$ such that $s_{t}=q$. For each such
position $t$, we have $C^{2}_{k}(S,s_{t})=C_{k}(S,p)C_{k}(S,q)$. We compute
$\mu_{t}=\mu(t,\min\\{v\in[1\mathinner{.\,.\allowbreak}3n]:s_{v}=q\\})$ based
on the output of the first instance of Proposition 3.16. If
$\mu_{t}>\mu_{t-1}$, we construct
$\mathsf{LZ}\left(\overline{C^{2}_{k}(S,s_{t})}\right)$ using Proposition
4.2(i) and Proposition 4.2(f).
By Propositions 4.2, 3.16 and 6.17, the algorithm uses
$\tilde{\mathcal{O}}(k)$ space and $\tilde{\mathcal{O}}(nk)$ time. ∎
### 6.3 Applications of Hamming distance sketches
Let us start by reminding the fingerprints (sketches) for testing string
equality.
###### Fact 6.18 (see e.g. [38]).
There exists a fingerprint $\psi$ (parameterized by an integer
$n\in\mathbb{Z}_{+}$, a threshold $\delta$ with $1\geq\delta\geq
n^{-\mathcal{O}(1)}$, an alphabet
$\Sigma=[0\mathinner{.\,.\allowbreak}n^{\mathcal{O}(1)})$, and a seed of
$\mathcal{O}(\log n)$ random bits) such that:
1. 1.
The fingerprint $\psi(S)$ of a string $S\in\Sigma^{\leq n}$ takes
$\mathcal{O}(\log\delta^{-1})$ bits. Given streaming access to $S$, it can be
constructed in $\mathcal{O}(|S|)$ time using $\mathcal{O}(\log n)$ bits of
space.
2. 2.
For all strings $X,Y\in\Sigma^{\leq n}$, we have
$\Pr[\psi(X)=\psi(Y)]\leq\delta$ if $X\neq Y$ (and $\psi(X)=\psi(Y)$
otherwise).
For two equal-length strings $X,Y$, the set of _mismatch positions_ is defined
as $\mathsf{MP}(X,Y)=\\{i\in[1\mathinner{.\,.\allowbreak}|X|]:X[i]\neq
Y[i]\\}$ and the _mismatch information_
$\mathsf{MI}(X,Y)=\\{(i,X[i],Y[i]):i\in\mathsf{MP}(X,Y)\\}$. Below, we adapt
the Hamming sketches of [16] to large alphabets.
###### Theorem 6.19.
For every constant $\delta\in(0,1)$, there exists a sketch
$\mathsf{sk}^{H}_{k}$ (parameterized by integers $n\geq k\geq 1$, an alphabet
$\Sigma=[0\mathinner{.\,.\allowbreak}\sigma)$, and a seed of
$\mathcal{O}(\log(n\log\sigma))$ random bits) such that:
1. 1.
The sketch $\mathsf{sk}^{H}_{k}(S)$ of a string $S\in\Sigma^{n}$ takes
$\mathcal{O}(k\log(n\sigma))$ bits. Given streaming access to $S$, it can be
constructed in $\mathcal{O}(n\log(n\sigma)\log(n\log\sigma))$ time using
$\mathcal{O}(k\log(n\sigma))$ bits of space.
2. 2.
There exists a decoding algorithm that, given $\mathsf{sk}^{H}_{k}(X)$ and
$\mathsf{sk}^{H}_{k}(Y)$ for strings $X,Y\in\Sigma^{n}$, with probability at
least $1-\delta$ either returns $\mathsf{MI}(X,Y)$ or certifies that
$\mathsf{hd}(X,Y)>k$. The algorithm uses
$\mathcal{O}(k\log(n\sigma)\log^{2}(n\log\sigma))$ time and
$\mathcal{O}(k\log(n\sigma))$ bits of space.
###### Proof.
The construction of [16] satisfies the required conditions provided that
$\sigma=n^{\mathcal{O}(1)}$. Henceforth, we assume without loss of generality
that $\sigma$ is a power of two satisfying $\sigma\geq n\log\sigma$.
We interpret each character in $\Sigma$ as a block of
$b=\lceil\log_{n\log\sigma}\sigma\rceil$ characters in
$[0\mathinner{.\,.\allowbreak}n\log\sigma)$. Moreover, we consider the
fingerprints $\psi$ of Fact 6.18 with
$\delta_{\ref{fact:kr}}=\frac{\delta}{2n}$ and
$n_{\ref{fact:kr}}=\sigma_{\ref{fact:kr}}=n\log\sigma$. This construction uses
$\mathcal{O}(\log n_{\ref{fact:kr}})=\mathcal{O}(\log(n\log\sigma))$ random
bits and produces fingerprints of
$\mathcal{O}(\log\delta^{-1}_{\ref{fact:kr}})=\mathcal{O}(\log n)$ bits.
Given a string $S\in\Sigma^{n}$, we define a string
$\bar{S}[1\mathinner{.\,.\allowbreak}nb]$ so that
$\bar{S}[ib-j+1]=(S[i][j],\psi(S[i]))$ for
$i\in[1\mathinner{.\,.\allowbreak}|S|]$ and
$j\in[1\mathinner{.\,.\allowbreak}b]$. Consequently, we set
$\mathsf{sk}^{H}_{k}(S):=\mathsf{sk}^{H}_{\bar{k}}(\bar{S})$ using the the
sketch of [16] with $\bar{\delta}=\frac{\delta}{2}$, $\bar{n}=nb$,
$\bar{k}=kb$, and $\bar{\sigma}\leq n^{\mathcal{O}(1)}\log\sigma$. Note that
this is feasible since $\bar{\sigma}\leq
n^{\mathcal{O}(1)}\leq\bar{n}^{\mathcal{O}(1)}$ if $n\leq\log\sigma$ and
$\bar{\sigma}\leq\log^{\mathcal{O}(1)}\sigma\leq(\log_{\log^{2}\sigma}\sigma)^{\mathcal{O}(1)}\leq
b^{\mathcal{O}(1)}\leq\bar{n}^{\mathcal{O}(1)}$ otherwise. This construction
uses $\mathcal{O}(\log\bar{n})=\mathcal{O}(\log(n\log\sigma))$ further random
bits and produces a sketch of
$\mathcal{O}(\bar{k}\log\bar{n})=\mathcal{O}(kb\log(nb))=\mathcal{O}(k\log_{n\log\sigma}\sigma\log(n\log\sigma)=\mathcal{O}(k\log\sigma)$
bits.
The auxiliary string $\bar{S}$ is constructed in
$\mathcal{O}(\bar{n})=\mathcal{O}(n\log\sigma)$ time using
$\mathcal{O}(\log(n\sigma))$ bits of space. The stream representing $\bar{S}$
is passed to the encoding algorithm of [16], which takes
$\mathcal{O}(\bar{n}\log^{2}\bar{n})=\mathcal{O}(n\log_{n\log\sigma}\sigma\log^{2}(n\log\sigma))=\mathcal{O}(n\log(n\sigma)\log(n\log\sigma))$
time and $\mathcal{O}(\bar{k}\log\bar{n})=\mathcal{O}(k\log\sigma)$ bits of
space.
The decoding algorithm, given $\mathsf{sk}^{H}_{\bar{k}}(\bar{X})$ and
$\mathsf{sk}^{H}_{\bar{k}}(\bar{Y})$, runs the decoding algorithm of [16]. If
the latter certifies $\mathsf{hd}(\bar{X},\bar{Y})>\bar{Y}$, we certify that
$\mathsf{hd}(X,Y)>k$. Otherwise, we interpret the output as
$\mathsf{MI}(\bar{X},\bar{Y})$. For each position
$i\in[1\mathinner{.\,.\allowbreak}|X|]$ such that
$(ib-b\mathinner{.\,.\allowbreak}ib]\subseteq\mathsf{MP}(\bar{X},\bar{Y})$ we
retrieve $X[i][j]$ and $Y[i][j]$ for each
$j\in[1\mathinner{.\,.\allowbreak}b]$ from
$(ib-j+1,\bar{X}[ib-j+1],\bar{Y}[ib-j+1])\in\mathsf{MI}(\bar{X},\bar{Y})$, and
then we output $(i,X[i],Y[i])\in\mathsf{MI}(X,Y)$.
As for correctness, with at most $\delta_{\ref{fact:kr}}+n\bar{\delta}=\delta$
probability loss, we may assume that the decoder of [16] is successful and
that, for all $i\in[1\mathinner{.\,.\allowbreak}|X|]$, we have
$\psi(X[i])=\psi(Y[i])$ if and only if $X[i]=Y[i]$. The latter assumption
yields
$\mathsf{MP}(\bar{X},\bar{Y})=\bigcup_{i\in\mathsf{MP}(X,Y)}(ib-b\mathinner{.\,.\allowbreak}ib]$
and thus $\mathsf{hd}(\bar{X},\bar{Y})=b\cdot\mathsf{hd}(X,Y)$. Hence, we
correctly certify $\mathsf{hd}(X,Y)>k$ if
$\mathsf{hd}(\bar{X},\bar{Y})>\bar{k}$, and we correctly reconstruct
$\mathsf{MI}(X,Y)$ otherwise.
The decoder uses
$\mathcal{O}(\bar{k}\log^{3}\bar{n})=\mathcal{O}(k\log_{n\log\sigma}\sigma\log^{3}(n\log\sigma))=\mathcal{O}(k\log\sigma\log^{2}(n\log\sigma))$
time and $\mathcal{O}(\bar{k}\log\bar{n})=\mathcal{O}(k\log\sigma)$ bits of
space. ∎
Next, consider an alphabet $\hat{\Sigma}:=\Sigma\times\mathbb{Z}_{\geq 0}$ and
a function $\pi:\hat{\Sigma}\to\mathbb{Z}_{\geq 0}$ defined so that
$\pi((a,v))=v$ for $(a,v)\in\hat{\Sigma}$ and
$\pi(S)=\sum_{i=1}^{|S|}\pi(S[i])$ for $S\in\hat{\Sigma}^{*}$. For two strings
$X,Y\in\hat{\Sigma}^{*}$ of the same length, we define the _prefix mismatch
information_
$\mathsf{PMI}(X,Y)=\\{(i,\pi(X[1\mathinner{.\,.\allowbreak}i)),\pi(Y[1\mathinner{.\,.\allowbreak}i))):i\in\mathsf{MP}(X,Y)\\}.$
###### Proposition 6.20.
For every constant $\delta\in(0,1)$, there exists a sketch
$\mathsf{sk}^{P}_{k}$ (parameterized by integers $n\geq k\geq 1$, an alphabet
$\hat{\Sigma}=[1\mathinner{.\,.\allowbreak}\sigma]\times[0\mathinner{.\,.\allowbreak}n^{\mathcal{O}(1)}]$,
and a seed of $\mathcal{O}(\log(n\log\sigma))$ random bits) such that:
1. 1.
The sketch $\mathsf{sk}^{P}_{k}(S)$ of a string $S\in\hat{\Sigma}^{n}$ takes
$\mathcal{O}(k\log^{2}n)$ bits. Given streaming access to $S$, it can be
constructed in $\mathcal{O}(n\log n\log(n\log\sigma)+n\log^{2}n)$ time using
$\mathcal{O}(k\log^{2}n+\log n\log(n\log\sigma))$ bits of space.
2. 2.
There exists a decoding algorithm that, given the sketches
$\mathsf{sk}^{P}_{k}(X)$ and $\mathsf{sk}^{P}_{k}(Y)$ of strings
$X,Y\in\hat{\Sigma}^{n}$ satisfying $\pi(X),\pi(Y)<n$, with probability at
least $1-\delta$ either returns $\mathsf{PMI}(X,Y)$ or certifies that
$\mathsf{hd}(X,Y)>k$. The algorithm uses $\mathcal{O}(k\log^{4}n)$ time and
$\mathcal{O}(k\log^{2}n)$ bits of space.
###### Proof.
Let us fix a complete binary tree $\mathcal{T}$ with $n$ leaves, numbered with
$[1\mathinner{.\,.\allowbreak}n]$ in the left-to-right order, and let
$v_{1},\ldots,v_{2n-1}$ denote the nodes of $\mathcal{T}$ in the pre-order.
For each node $v_{i}\in\mathcal{T}$, let
$[p_{i}\mathinner{.\,.\allowbreak}q_{i})\subseteq[1\mathinner{.\,.\allowbreak}n]$
be the indices of the leaves in the subtree of $v_{i}$. Consider the
fingerprints $\psi$ of Fact 6.18 parameterized by
$\delta_{\ref{fact:kr}}=\frac{\delta}{4n-2}$,
$n_{\ref{fact:kr}}=\mathcal{O}(n\log(n\sigma))$, and
$\sigma_{\ref{fact:kr}}=2$: Given a string $S\in\hat{\Sigma}^{n}$, we define a
string $\mathcal{T}(S)[1\mathinner{.\,.\allowbreak}2n)$ so that
$\mathcal{T}(S)[i]=(\pi(S[p_{i}\mathinner{.\,.\allowbreak}q_{i})),\psi(S[p_{i}\mathinner{.\,.\allowbreak}q_{i})))$
for every $i\in[1\mathinner{.\,.\allowbreak}2n)$, where $\psi$ expands each
character of $\hat{\Sigma}$ into a sequence of $\mathcal{O}(\log(n\sigma))$
bits. We set
$\mathsf{sk}^{P}_{k}(S):=\mathsf{sk}^{H}_{k_{\ref{thm:skH}}}(\mathcal{T}(S))$,
where $\mathsf{sk}^{H}_{k_{\ref{thm:skH}}}$ is the sketch of Theorem 6.19 with
$\delta_{\ref{thm:skH}}=\frac{\delta}{2}$, $n_{\ref{thm:skH}}=2n-1$,
$k_{\ref{thm:skH}}=\min(k\lceil{\log(2n)}\rceil,n_{\ref{thm:skH}})$, and
$\sigma_{\ref{thm:skH}}\leq
2^{\mathcal{O}(\log(n/\delta_{\ref{fact:kr}}))}\leq n^{\mathcal{O}(1)}$.
This construction uses $\mathcal{O}(\log
n_{\ref{fact:kr}}+\log(n_{\ref{thm:skH}}\log\sigma_{\ref{thm:skH}}))=\mathcal{O}(\log(n\log\sigma))$
random bits and produces sketches of
$\mathcal{O}(k_{\ref{thm:skH}}\log(n_{\ref{thm:skH}}\sigma_{\ref{thm:skH}}))=\mathcal{O}(k\log^{2}n)$
bits.
The encoding algorithm transforms $S$ into $\mathcal{T}(S)$ and feeds
$\mathcal{T}(S)$ into the encoding procedure of Theorem 6.19. Construction of
$\mathcal{T}(S)$ is organized in $\mathcal{O}(\log n)$ layers, each
responsible for nodes $v_{i}\in\mathcal{T}$ at a fixed level. The intervals
$[p_{i}\mathinner{.\,.\allowbreak}q_{i})$ corresponding to these nodes are
disjoint so, at any time, a layer produces a single character
$\mathcal{T}(S)[i]$ and, by Fact 6.18, spends $\mathcal{O}(n\log\sigma)$
amortized time and uses $\mathcal{O}(\log n)$ bits of space to process a
single character $S[j]$. Since the tree $\mathcal{T}$ is linearized in the
post-order fashion, all levels can read the string $S$ with a common left-to-
right pass and outputting $\mathcal{T}(S)$ does not require any buffers.
Overall, construction of $\mathcal{T}(S)$ takes $\mathcal{O}(n\log
n\log(n\log\sigma))$ time and uses $\mathcal{O}(\log n\log(n\log\sigma))$ bits
of space. The encoder of Theorem 6.19 takes
$\mathcal{O}(n_{\ref{thm:skH}}\log(n_{\ref{thm:skH}}\sigma_{\ref{thm:skH}})\log(n_{\ref{thm:skH}}\log\sigma_{\ref{thm:skH}}))=\mathcal{O}(n\log^{2}n)$
time and uses
$\mathcal{O}(k_{\ref{thm:skH}}\log(n_{\ref{thm:skH}}\sigma_{\ref{thm:skH}}))=\mathcal{O}(k\log^{2}n)$
bits of space.
The decoding algorithm first retrieves
$\mathsf{MI}(\mathcal{T}(X),\mathcal{T}(Y))$ from
$\mathsf{sk}^{H}_{k_{\ref{thm:skH}}}(\mathcal{T}(X))$ and
$\mathsf{sk}^{H}_{\ref{thm:skH}}(\mathcal{T}(Y))$ using the decoder of Theorem
6.19. If the output of that subroutine can be interpreted as
$\mathsf{MI}(\mathcal{T}(X),\mathcal{T}(Y))$, then we use 6.21 below to
retrieve $\mathsf{MI}(X,Y)$. If the size of the obtained set does not exceed
$k$, we return the set. In the remaining cases, we certify that
$\mathsf{hd}(X,Y)>k$.
###### Claim 6.21.
For every $X,Y\in\hat{\Sigma}^{n}$, the set $\mathsf{PMI}(X,Y)$ can be
extracted from $\mathsf{MI}(\mathcal{T}(X),\mathcal{T}(Y))$ in time and space
$\mathcal{O}(\mathsf{hd}(X,Y)\log n)$ with success probability at least
$1-\frac{\delta}{2}$.
###### Proof.
We assume that, for every $i\in[1\mathinner{.\,.\allowbreak}2n)$, the equality
$\psi(X[p_{i}\mathinner{.\,.\allowbreak}q_{i}))=\psi(Y[p_{i}\mathinner{.\,.\allowbreak}q_{i}))$
implies
$X[p_{i}\mathinner{.\,.\allowbreak}q_{i})=Y[p_{i}\mathinner{.\,.\allowbreak}q_{i})$.
By Fact 6.18, each of the implications fails with probability at most
$\delta_{\ref{fact:kr}}$, so the overall failure probability can be bounded by
$\delta_{\ref{fact:kr}}\cdot(2n-1)=\frac{\delta}{2}$.
Our assumption yields
$\mathsf{MP}(\mathcal{T}(X),\mathcal{T}(Y))=\\{i\in[1\mathinner{.\,.\allowbreak}2n):[p_{i}\mathinner{.\,.\allowbreak}q_{i})\cap\mathsf{MP}(X,Y)\neq\emptyset\\}.$
In particular, $\mathsf{MP}(X,Y)=\\{p_{i}:v_{i}\text{ is a leaf and
}i\in\mathsf{MP}(\mathcal{T}(X),\mathcal{T}(Y))\\}$. Thus, it remains to
describe how to extract $\pi(X[1\mathinner{.\,.\allowbreak}p))$ and
$\pi(Y[1\mathinner{.\,.\allowbreak}p))$ for each $p\in\mathsf{MP}(X,Y)$; by
symmetry, we focus on $\pi(X[1\mathinner{.\,.\allowbreak}p))$. For this, we
process subsequent nodes $v_{i}$ on the path from the root of $\mathcal{T}$ to
the leaf $v_{j}$ such that $\\{p\\}=[p_{j}\mathinner{.\,.\allowbreak}q_{j})$
maintaining $\pi(X[1\mathinner{.\,.\allowbreak}p_{i}))$. Note that the values
$\pi(X[p_{i}\mathinner{.\,.\allowbreak}q_{i}))$ can be extracted from
$\mathsf{MI}(\mathcal{T}(X),\mathcal{T}(Y))$ due to
$p\in[p_{i}\mathinner{.\,.\allowbreak}q_{i})$.
If $v_{i}$ is the root, then
$\pi(X[1\mathinner{.\,.\allowbreak}p_{i}))=\pi(\varepsilon)=0$. If $v_{i}$ is
the left child of $v_{i^{\prime}}$, then $p_{i}=p_{i^{\prime}}$, so
$\pi(X[1\mathinner{.\,.\allowbreak}p_{i}))=\pi(X[1\mathinner{.\,.\allowbreak}p_{i^{\prime}}))$
has already been computed. If $v_{i}$ is the right child of $v_{i^{\prime}}$,
then $q_{i}=q_{i^{\prime}}$, so
$\pi(X[1\mathinner{.\,.\allowbreak}p_{i}))+\pi(X[p_{i}\mathinner{.\,.\allowbreak}q_{i}))=\pi(X[1\mathinner{.\,.\allowbreak}q_{i}))=\pi(X[1\mathinner{.\,.\allowbreak}q_{i^{\prime}}))=\pi(X[1\mathinner{.\,.\allowbreak}p_{i^{\prime}}))+\pi(X[p_{i^{\prime}}\mathinner{.\,.\allowbreak}q_{i^{\prime}}))$.
Consequently, $\pi(X[1\mathinner{.\,.\allowbreak}p_{i}))$ can be retrieved
from $\pi(X[1\mathinner{.\,.\allowbreak}p_{i^{\prime}}))$, which has already
been computed, as well as $\pi(X[p_{i}\mathinner{.\,.\allowbreak}q_{i}))$ and
$\pi(X[p_{i^{\prime}}\mathinner{.\,.\allowbreak}q_{i^{\prime}}))$, which are
available in $\mathsf{MI}(\mathcal{T}(X),\mathcal{T}(Y))$. When this process
reaches $v_{j}$, it results in the sought value
$\pi(X[1\mathinner{.\,.\allowbreak}p))=\pi(X[1\mathinner{.\,.\allowbreak}p_{j}))$.
∎
It remains to analyze correctness of the decoding algorithm. The decoder of
Theorem 6.19 fails with probability at most
$\delta_{\ref{thm:skH}}=\frac{\delta}{2}$ and the procedure of 6.21 fails with
probability at most $\frac{\delta}{2}$. Thus, with at most $\delta$
probability loss, we may assume that both calls are successful.
If $\mathsf{hd}(\mathcal{T}(X),\mathcal{T}(Y))\leq k_{\ref{thm:skH}}$, then
the decoder of Theorem 6.19 retrieves
$\mathsf{MI}(\mathcal{T}(X),\mathcal{T}(Y))$ and the procedure of 6.21 results
in $\mathsf{MI}(X,Y)$. Depending on whether
$\mathsf{hd}(X,Y)=|\mathsf{MI}(X,Y)|\leq k$ or not, our decoding algorithm
thus correctly returns $\mathsf{MI}(X,Y)$ or certifies that
$\mathsf{hd}(X,Y)>k$.
Otherwise, the algorithm of Theorem 6.19 certifies
$\mathsf{hd}(\mathcal{T}(X),\mathcal{T}(Y))>k_{\ref{thm:skH}}$, and the whole
decoding procedure certifies $\mathsf{hd}(X,Y)>k$. This is correct because of
$\mathsf{hd}(\mathcal{T}(X),\mathcal{T}(Y))\leq\mathsf{hd}(X,Y)\lceil{\log(2n)}\rceil$.
To see this, observe that
$[p_{i}\mathinner{.\,.\allowbreak}q_{i})\cap\mathsf{MP}(X,Y)=\emptyset$
implies
$X[p_{i}\mathinner{.\,.\allowbreak}q_{i})=Y[p_{i}\mathinner{.\,.\allowbreak}q_{i})$
and $\mathcal{T}(X)[i]=\mathcal{T}(Y)[i]$. However, every leaf of
$\mathcal{T}$ has at most $\lceil{\log(2n)}\rceil$ ancestors (including
itself). Consequently, for every $j\in\mathsf{MP}(X,Y)$, there are at most
$\lceil{\log(2n)}\rceil$ nodes $v_{i}$ such that
$j\in[p_{i}\mathinner{.\,.\allowbreak}q_{i})$. ∎
### 6.4 Edit Distance Sketches
We are now ready to show the main result of this section.
See 3.17
###### Proof.
Let $c_{\ref{prp:alg}}$ be the constant of Proposition 3.16 for
$\delta_{\ref{prp:alg}}=\frac{\delta}{3}$, and let $c_{\ref{lem:bcgk}}$ be the
constant of Lemma 6.12. We shall use $\mathsf{CE}_{k^{\prime}}(W)$ with
$k^{\prime}=(c_{\ref{prp:alg}}+5)k$ and $W=\mathsf{W}(n,r,S)$, where $r$ is a
random seed of $\mathcal{O}(\log^{2}n)$ bits. Observe that the alphabet of
$\mathsf{CE}_{k^{\prime}}(W)$ can be interpreted as
$[0\mathinner{.\,.\allowbreak}n^{\mathcal{O}(k)})\times[0\mathinner{.\,.\allowbreak}n]$
because the $|\mathsf{LZ}(\overline{C^{2}_{k^{\prime}}(S,s)})|=\mathcal{O}(k)$
for each $s\in[1\mathinner{.\,.\allowbreak}|S|]$. We shall use Theorems 6.19
and 6.20 with $n_{\ref{thm:skH}}=n_{\ref{prp:mpi}}=3n$,
$\delta_{\ref{thm:skH}}=\delta_{\ref{prp:mpi}}=\tfrac{\delta}{3}$,
$k_{\ref{thm:skH}}=k_{\ref{prp:mpi}}=\min(n_{\ref{thm:skH}},\lfloor
c_{\ref{lem:bcgk}}(1+c_{\ref{prp:alg}})k\log(3n)\rfloor)$, and
$\sigma_{\ref{thm:skH}}=\sigma_{\ref{prp:mpi}}=n^{\mathcal{O}(k)}.$
The sketch $\mathsf{sk}^{E}_{k}(S)$ of a string $S\in\Sigma^{\leq n}$ consists
of $|S|$ as well as the sketches
$\mathsf{sk}^{H}_{k_{\ref{thm:skH}}}(\mathsf{CE}_{k^{\prime}}(W))$ and
$\mathsf{sk}^{P}_{k_{\ref{prp:mpi}}}(\mathsf{CE}_{k^{\prime}}(W))$. This
construction uses
$\mathcal{O}(\log^{2}n+\log(n_{\ref{thm:skH}}\log\sigma_{\ref{thm:skH}})+\log(n_{\ref{prp:mpi}}\log\sigma_{\ref{prp:mpi}}))=\mathcal{O}(\log^{2}n+\log(nk\log
n))=\mathcal{O}(\log^{2}n)$
random bits and produces sketches of bit-size
$\mathcal{O}(k_{\ref{thm:skH}}\log(n_{\ref{thm:skH}}\sigma_{\ref{thm:skH}})+k_{\ref{prp:mpi}}\log^{2}n_{\ref{prp:mpi}})=\mathcal{O}(k\log
n\log(n^{\mathcal{O}(k)})+k\log^{3}n)=\mathcal{O}(k^{2}\log^{3}n).$
The encoding algorithm uses Lemma 6.16 to transform the input stream
representing $S$ to an auxiliary stream representing
$\mathsf{CE}_{k^{\prime}}(W)$, which we forward to the encoders constructing
$\mathsf{sk}^{H}_{k_{\ref{thm:skH}}}(\mathsf{CE}_{k^{\prime}}(S))$ and
$\mathsf{sk}^{P}_{k_{\ref{prp:mpi}}}(\mathsf{CE}_{k^{\prime}}(W))$. Thus, it
takes
$\tilde{\mathcal{O}}(nk^{\prime}+n_{\ref{thm:skH}}\log\sigma_{\ref{thm:skH}}+n_{\ref{prp:mpi}})=\tilde{\mathcal{O}}(nk)$
time and uses
$\tilde{\mathcal{O}}(k^{\prime}+k_{\ref{thm:skH}}\log\sigma_{\ref{thm:skH}}+k_{\ref{prp:mpi}})=\tilde{\mathcal{O}}(k^{2})$
space.
##### Decoding Algorithm
Let $\mathcal{A}_{\mathsf{W}}$ be the zip alignment of walks
$W_{X}=(x_{t})_{t=1}^{3n+1}=\mathsf{W}(n,r,X)$ and
$W_{Y}=(y_{t})_{t=1}^{3n+1}=\mathsf{W}(n,r,Y)$ over strings
$X,Y\in\Sigma^{\leq n}$.
Given sketches $\mathsf{sk}^{E}_{k}(X),\mathsf{sk}^{E}_{k}(Y)$, we run the
decoders for
$\mathsf{sk}^{H}_{k_{\ref{thm:skH}}}(\mathsf{CE}_{k^{\prime}}(W_{X})),\mathsf{sk}^{H}_{k_{\ref{thm:skH}}}(\mathsf{CE}_{k^{\prime}}(W_{Y}))$
and
$\mathsf{sk}^{P}_{k_{\ref{prp:mpi}}}(\mathsf{CE}_{k^{\prime}}(W_{X})),\mathsf{sk}^{P}_{k_{\ref{prp:mpi}}}(\mathsf{CE}_{k^{\prime}}(W_{Y}))$.
We certify $\mathsf{ed}(X,Y)>k$ if either procedure certifies
$\mathsf{hd}(\mathsf{CE}_{k^{\prime}}(W_{X}),\mathsf{CE}_{k^{\prime}}(W_{Y}))>k_{\ref{thm:skH}}=k_{\ref{prp:mpi}}$.
Otherwise, the decoder interprets the outputs as
$\mathsf{MI}(\mathsf{CE}_{k^{\prime}}(W_{X}),\mathsf{CE}_{k^{\prime}}(W_{Y}))$
and
$\mathsf{PMI}(\mathsf{CE}_{k^{\prime}}(W_{X}),\mathsf{CE}_{k^{\prime}}(W_{Y}))$,
respectively. In this case, we construct $\mathsf{E}^{M}(X,Y)$ for
$M=\\{(x,y)\in\mathcal{M}_{X,Y}(\mathcal{A}_{\mathsf{W}}):x\in P_{X}\text{ or
}y\in P_{Y}\\}$, where $P_{X}$ and $P_{Y}$ are defined in the statement of
Lemma 6.12. Finally, we compute $\mathsf{GR}_{k}(X,Y)$ using Proposition 5.12
or $\min(\mathsf{ed}(X,Y),k+1)$ using Corollary 5.5.
###### Claim 6.22.
Given
$\mathsf{MI}(\mathsf{CE}_{k^{\prime}}(W_{X}),\mathsf{CE}_{k^{\prime}}(W_{Y}))$
and
$\mathsf{PMI}(\mathsf{CE}_{k^{\prime}}(W_{X}),\mathsf{CE}_{k^{\prime}}(W_{Y}))$,
the encoding $\mathsf{E}^{M}(X,Y)$ can be constructed in
$\tilde{\mathcal{O}}(k^{3})$ time using $\tilde{\mathcal{O}}(k^{2})$ space.
Moreover, $|\mathsf{LZ}(X^{M}Y^{M})|=\tilde{\mathcal{O}}(k^{2})$.
###### Proof.
We proceed in two phases. In the first phase, we construct the compressed
representation $\mathsf{D}(X^{\prime})$ of a string
$X^{\prime}\in(\Sigma\cup\\{\\#\\})^{|X|}$ such that $X^{\prime}[x]=X[x]$ if
$x\in P_{X}$ and $X^{\prime}[x]=\\#$ otherwise. We initialize
$X^{\prime}:=\\#^{|X|}$ (via $\mathsf{LZ}(\\#^{|X|})$ using Proposition
4.2(h)). Next, we iterate over positions
$t\in[1\mathinner{.\,.\allowbreak}3n]$ such that
$\bot\neq\mathsf{CE}_{k^{\prime}}(W_{X})[t]\neq\mathsf{CE}_{k^{\prime}}(W_{Y})[t]$.
We retrieve $x_{t}$ and
$\mathsf{LZ}\big{(}\overline{C^{2}_{k^{\prime}}(X,x_{t})}\big{)}$ from the
mismatch information, convert
$\mathsf{LZ}\big{(}\overline{C^{2}_{k^{\prime}}(X,x_{t})}\big{)}$ to
$\mathsf{D}(C^{2}_{k^{\prime}}(X,x_{t}))$ (Proposition 4.2(h)), and update
$\mathsf{D}(X^{\prime})$, setting
$X^{\prime}[x_{t}\mathinner{.\,.\allowbreak}x_{t}+|C^{2}_{k^{\prime}}(X,x_{t})|):=C^{2}_{k^{\prime}}(X,x_{t})$
(Proposition 4.2(g)(i)). We also symmetrically construct the compressed
representation $\mathsf{D}(Y^{\prime})$ of a string
$Y^{\prime}\in(\Sigma\cup\\{\\#\\})^{|Y|}$ such that $Y^{\prime}[y]=Y[y]$ if
$y\in P_{Y}$ and $Y^{\prime}[y]=\\#$ otherwise.
In the second phase, we convert $\mathsf{D}(X^{\prime})$ to
$\mathsf{D}(X^{M})$. Here, the goal is make sure that $X^{M}[x]=\\#$ not only
for $x\in P_{X}$, but also when $X[x]\simeq_{\mathcal{A}_{\mathsf{W}}}Y[y]$
for some $y\in P_{Y}$. For this, we iterate over dummy segments
$Y^{\prime}[y\mathinner{.\,.\allowbreak}y^{\prime})$. By Lemma 6.12,
$\mathcal{A}_{\mathsf{W}}$ matches $Y[y\mathinner{.\,.\allowbreak}y^{\prime})$
to a fragment of $X[x\mathinner{.\,.\allowbreak}x^{\prime})$. Hence, we shall
identify the shift $x-y$ and set
$X^{M}[x\mathinner{.\,.\allowbreak}x^{\prime}):=\\#^{x^{\prime}-x}$
(Proposition 4.2(h)(g)(i)). Let
$\mathsf{MP}_{\mathsf{CE}}:=\mathsf{MP}(\mathsf{CE}_{k^{\prime}}(W_{X}),\mathsf{CE}_{k^{\prime}}(W_{Y}))$.
Define $u,v\in[1\mathinner{.\,.\allowbreak}3n+1]$ so that $u=v=3n+1$ if
$\pi(\mathsf{CE}_{k^{\prime}}(W_{Y})[1\mathinner{.\,.\allowbreak}t])\neq
y^{\prime}$ for all $t\in\mathsf{MP}_{\mathsf{CE}}$; otherwise, $u$ is the
smallest index in $\mathsf{MP}_{\mathsf{CE}}$ with
$\pi(\mathsf{CE}_{k^{\prime}}(W_{Y})[1\mathinner{.\,.\allowbreak}u])=y^{\prime}$,
whereas $v$ is the smallest index $\mathsf{MP}_{\mathsf{CE}}$ with
$\pi(\mathsf{CE}_{k^{\prime}}(W_{Y})[1\mathinner{.\,.\allowbreak}v))=\pi(\mathsf{CE}_{k^{\prime}}(W_{Y})[1\mathinner{.\,.\allowbreak}u))$.
We claim that
$x-y=\pi(\mathsf{CE}_{k^{\prime}}(W_{X})[1\mathinner{.\,.\allowbreak}v))-\pi(\mathsf{CE}_{k^{\prime}}(W_{Y})[1\mathinner{.\,.\allowbreak}v))$.
If $y^{\prime}\leq|Y|$, then $y^{\prime}-1\notin P_{Y}$ and $y^{\prime}\in
P_{Y}$. Hence, $Y[y^{\prime}]$ is covered by $C^{2}_{k^{\prime}}(Y,y_{t})$ for
some $t\in\mathsf{MP}_{\mathsf{CE}}$ with
$\mathsf{CE}_{k^{\prime}}(W_{X})[t]\neq\bot$, whereas $Y[y^{\prime}-1]$ is not
covered by $C^{2}_{k^{\prime}}(Y,y_{t})$. Hence, there is
$t\in\mathsf{MP}_{\mathsf{CE}}$ with $y_{t}=y^{\prime}$, and therefore
$y_{u}=y^{\prime}$. Similarly, $y_{u}=y^{\prime}$ holds if $y^{\prime}=|Y|+1$.
Let $w\in[1\mathinner{.\,.\allowbreak}3n]$ be the smallest index such that
$\mathsf{CE}_{k^{\prime}}(W_{Y})[w]\neq\bot$ and $C^{2}_{k^{\prime}}(Y,y_{t})$
covers $Y[y^{\prime}-1]$ (such an index exists by Lemma 6.9). Due to
$y^{\prime}-1\in P_{Y}$, we have
$\mathsf{CE}_{k^{\prime}}(W_{X})[w]=\mathsf{CE}_{k^{\prime}}(W_{Y})[w]$, and
hence $x-y=x_{w}-y_{w}$. Definition of $w$ further yields
$\mathsf{CE}_{k^{\prime}}(W_{Y})[w+1\mathinner{.\,.\allowbreak}v)=\bot^{u-w-1}$,
and thus
$\pi(\mathsf{CE}_{k^{\prime}}(W_{Y})[1\mathinner{.\,.\allowbreak}v))=y_{w}$.
Moreover, since
$\pi(\mathsf{CE}_{k^{\prime}}(W_{Y})[1\mathinner{.\,.\allowbreak}t))=y_{w}$
for $t\in[w+1\mathinner{.\,.\allowbreak}v]$, we have
$\pi(\mathsf{CE}_{k^{\prime}}(W_{X})[1\mathinner{.\,.\allowbreak}v))=x_{w}$ by
definition of $v$. This completes the proof that
$x-y=\pi(\mathsf{CE}_{k^{\prime}}(W_{X})[1\mathinner{.\,.\allowbreak}v))-\pi(\mathsf{CE}_{k^{\prime}}(W_{Y})[1\mathinner{.\,.\allowbreak}v))$.
To derive $\mathsf{E}^{M}(X,Y)$, it suffices to convert
$\mathsf{D}(Y^{\prime})$ to $\mathsf{D}(Y^{M})$ (symmetrically), and to
construct $\mathsf{D}(X^{M}Y^{M})$ using Proposition 4.2(i). Since
$\overline{\mathsf{maxLZ}}(C^{2}_{k^{\prime}}(X,x_{t})),\overline{\mathsf{maxLZ}}(C^{2}_{k^{\prime}}(Y,y_{t}))=\mathcal{O}(k)$
holds for all $t\in[1\mathinner{.\,.\allowbreak}3n+1]$ and since
$|\mathsf{MP}_{\mathsf{CE}}|=\mathcal{O}(k\log n)$, the
$\overline{\mathsf{maxLZ}}(\cdot)$ measure of all intermediate strings is
$\tilde{\mathcal{O}}(k^{2})$. Consequently, the $\tilde{\mathcal{O}}(k)$
applications of Proposition 4.2 cost $\tilde{\mathcal{O}}(k^{3})$ time and use
$\tilde{\mathcal{O}}(k^{2})$ space. ∎
To complete the complexity analysis, observe that the decoding procedure of
Theorem 6.19 uses
$\mathcal{O}(k_{\ref{thm:skH}}\log(n_{\ref{thm:skH}}\sigma_{\ref{thm:skH}})\log^{2}(n_{\ref{thm:skH}}\log\sigma_{\ref{thm:skH}}))=\tilde{\mathcal{O}}(k^{2})$
time and
$\mathcal{O}(k_{\ref{thm:skH}}\log(n_{\ref{thm:skH}}\sigma_{\ref{thm:skH}}))=\tilde{\mathcal{O}}(k^{2})$
bits of space. The procedure of Proposition 6.20 uses
$\mathcal{O}(k_{\ref{prp:mpi}}\log^{2}n_{\ref{prp:mpi}})=\tilde{\mathcal{O}}(k)$
bits of space and costs
$\mathcal{O}(k_{\ref{prp:mpi}}\log^{4}n_{\ref{prp:mpi}})=\tilde{\mathcal{O}}(k)$
time. Finally, due to $|\mathsf{LZ}(X^{M}Y^{M})|=\tilde{\mathcal{O}}(k^{2})$
and $|\mathcal{B}_{k}(X,Y)|=\mathcal{O}(k^{5})$ (Lemma 3.11), the algorithm of
Proposition 5.12 uses $\tilde{\mathcal{O}}(k^{2})$ space and costs
$\tilde{\mathcal{O}}(k^{5})$ time (dominating the overall decoding
complexity). If we only aim to retrieve $\min(\mathsf{ed}(X,Y),k+1)$, the
algorithm of Corollary 5.5 takes $\tilde{\mathcal{O}}(k^{2})$ time and space
(in which case the overall decoding uses $\tilde{\mathcal{O}}(k^{2})$ space
and costs $\tilde{\mathcal{O}}(k^{3})$ time.
It remains to argue that the decoding algorithm is correct. With
$\delta_{\ref{prp:alg}}=\frac{\delta}{3}$ probability loss, we may assume that
$\mathcal{A}_{\mathsf{W}}\in\mathsf{GA}_{c_{\ref{prp:alg}}(\mathsf{ed}(X,Y))^{2}}(X,Y)$
and $0pt(\mathcal{A}_{\mathsf{W}})\leq c_{\ref{prp:alg}}\mathsf{ed}(X,Y)$.
With $\delta_{\ref{thm:skH}}+\delta_{\ref{prp:mpi}}=\frac{2\delta}{3}$ further
probability loss, we may assume that the decoders of Theorems 6.19 and 6.20
are successful. If $\mathsf{ed}(X,Y)>k$, then the correctness follows from
Propositions 5.12 and 5.5 because
$M\subseteq\mathcal{M}_{X,Y}(\mathcal{A}_{\mathsf{W}})$ is a non-crossing
matching. Otherwise, Proposition 3.16 guarantees
$\mathsf{cost}(\mathcal{A}_{\mathsf{W}})\leq c_{\ref{prp:alg}}k^{2}$ and
$0pt(\mathcal{A}_{\mathsf{W}})\leq c_{\ref{prp:alg}}k$ so, in particular,
$k^{\prime}\geq 0pt(\mathcal{A}_{\mathsf{W}})+5k$. By Lemma 6.12, we thus have
$\mathsf{hd}(\mathsf{CE}_{k^{\prime}}(W_{X}),\mathsf{CE}_{k^{\prime}}(W_{X}))\leq
c_{\ref{lem:bcgk}}(1+c_{\ref{prp:alg}})k\log(3n)$. Hence, the decoders of
Theorems 6.19 and 6.20 report the mismatch information. Lemma 6.10 further
implies that $\Delta_{X}(\mathcal{A},\mathcal{A}_{\mathsf{W}})\subseteq P_{X}$
and $\Delta_{Y}(\mathcal{A},\mathcal{A}_{\mathsf{W}})\subseteq P_{Y}$ for all
$\mathcal{A}\in\mathsf{GA}_{k}(X,Y)$. In particular,
$\mathcal{M}_{X,Y}(\mathcal{A})\subseteq M$, and therefore
$\mathcal{M}_{k}(X,Y)\subseteq M$. Consequently, the algorithms of
Propositions 5.12 and 5.5 correctly compute $\mathsf{GR}_{k}(X,Y)$ and
$\min(\mathsf{ed}(X,Y),k+1)$, respectively. ∎
Next, we boost the success probability and strengthen the sketches so that we
can retrieve $\mathsf{qGR}_{k}(X,Y)$ instead of $\mathsf{GR}_{k}(X,Y)$.
###### Corollary 6.23.
There is a sketch $\mathsf{sk}^{\mathsf{q}}_{k}$ (parametrized by
$\delta\in(0,\frac{1}{2})$, integers $n\geq k\geq 1$, an alphabet
$\Sigma=[0\mathinner{.\,.\allowbreak}n^{\mathcal{O}(1)})$, and a seed of
$\mathcal{O}(\log^{2}n\log(1/\delta))$ random bits) such that:
1. (a)
The sketch $\mathsf{sk}^{\mathsf{q}}_{k}(S)$ of a string $\Sigma^{\leq n}$
takes $\mathcal{O}(k^{2}\log^{3}n\log(1/\delta))$ bits. Given streaming access
to $S$, it can be constructed in $\tilde{\mathcal{O}}(nk\log(1/\delta))$ time
using $\tilde{\mathcal{O}}(k^{2}\log(1/\delta))$ space.
2. (b)
There exists a decoding algorithm that, given
$\mathsf{sk}^{\mathsf{q}}_{k}(X),\mathsf{sk}^{\mathsf{q}}_{k}(Y)$ for
$X,Y\in\Sigma^{\leq n}$, with probability at least $1-\delta$ computes
$\mathsf{qGR}_{k}(X,Y)$. The algorithm takes
$\tilde{\mathcal{O}}(k^{5}\log(1/\delta))$ time and uses
$\tilde{\mathcal{O}}(k^{2}\log(1/\delta))$ space.
###### Proof.
We shall use $\mu=\mathcal{O}(\log(1/\delta))$ sketches
$\mathsf{sk}^{E}_{k_{\ref{thm:ske}}}$ of 3.17 with
$\delta_{\ref{thm:ske}}=\frac{1}{3}$, $n_{\ref{thm:ske}}=n+1$,
$k_{\ref{thm:ske}}=k+1$, and independent seeds. For each of the $\mu$ sketches
$\mathsf{sk}^{E}_{k_{\ref{thm:ske}}}$, the sketch
$\mathsf{sk}^{\mathsf{q}}_{k}(S)$ contains
$\mathsf{sk}^{E}_{k_{\ref{thm:ske}}}(\$_{1}S)$,
$\mathsf{sk}^{E}_{k_{\ref{thm:ske}}}(\$_{2}S)$, where
$\$_{1},\$_{2}\notin\Sigma$ are two distinct symbols. This construction uses
$\mathcal{O}(\mu\log^{2}n_{\ref{thm:ske}})=\mathcal{O}(\log^{2}n\log(1/\delta))$
random bits and produces sketches of $\mathcal{O}(\mu
k^{2}_{\ref{thm:ske}}\log^{3}n_{\ref{thm:ske}})=\mathcal{O}(k^{2}\log^{3}n\log(1/\delta))$
bits.
The encoding algorithm calls $2\mu$ instances of the encoding algorithm of
3.17. Hence, it uses $\tilde{\mathcal{O}}(\mu
k^{2}_{\ref{thm:ske}})=\mathcal{O}(k^{2}\log(1/\delta))$ space and costs
$\tilde{\mathcal{O}}(\mu
n_{\ref{thm:ske}}k_{\ref{thm:ske}})=\tilde{\mathcal{O}}(nk\log(1/\delta))$
time.
The decoding algorithm, given
$\mathsf{sk}^{\mathsf{q}}_{k}(X),\mathsf{sk}^{\mathsf{q}}_{k}(Y)$ for
$X,Y\in\Sigma^{\leq n}$, runs the decoder of 3.17 for
$\mathsf{sk}^{E}_{k_{\ref{thm:ske}}}(\$_{1}X),\mathsf{sk}^{E}_{k_{\ref{thm:ske}}}(\$_{1}Y)$
for each of the $\mu$ sketches $\mathsf{sk}^{E}_{k_{\ref{thm:ske}}}$. This
yields $\mu$ candidates for
$\mathsf{GR}_{k_{\ref{thm:ske}}}(\$_{1}X,\$_{2}Y)$, which we convert to
candidates for $\mathsf{qGR}_{k}(X,Y)$ using Corollary 5.17. Finally, we
determine the majority answer among the $\mu$ candidates. The equality test
uses Proposition 4.2(i)(c) to compare two candidates for
$\mathsf{D}(X^{M}Y^{M})$.
Recall that the decoding procedures of 3.17 use $\tilde{\mathcal{O}}(\mu
k^{5}_{\ref{thm:ske}})=\tilde{\mathcal{O}}(k^{5}\log(1/\delta))$ time and
$\tilde{\mathcal{O}}(\mu
k^{2}_{\ref{thm:ske}})=\tilde{\mathcal{O}}(k^{2}\log(1/\delta))$ space. The
applications of Corollary 5.17 and the equality tests take
$\tilde{\mathcal{O}}(\mu
k^{2}_{\ref{thm:ske}})=\tilde{\mathcal{O}}(k^{2}\log(1/\delta))$ time and
space. The entire decoding algorithm uses
$\tilde{\mathcal{O}}(k^{2}\log(1/\delta))$ space and $\tilde{\mathcal{O}}(\mu
k^{5}_{\ref{thm:ske}})=\tilde{\mathcal{O}}(k^{5}\log(1/\delta))$ time.
As for correctness, since the $\mu$ sketches
$\mathsf{sk}^{E}_{k_{\ref{thm:ske}}}$ are independent, by the Chernoff bound,
the majority answer is wrong with probability at most
$\exp(-\mathcal{O}(\mu))$. Setting $\mu=\mathcal{O}(\log(1/\delta))$ (with a
sufficiently large constant factor) guarantees a success probability of
$1-\delta$. ∎
## 7 Pattern Matching with $k$ Edits
In this section, we present solutions for pattern matching with $k$ edits in
the semi-streaming and streaming settings.
### 7.1 Periodicity under Edit Distance
We start by recalling combinatorial properties of strings periodic under the
edit distance.
See 3.2
###### Claim 7.1.
Suppose that a string $X$ is a prefix of a string $Y$, where $|X|<|Y|\leq
2|X|$. If $X$ is $k$-periodic with $k$-period $Q$, $|Q|\leq|X|/128k$, then
either $Y$ is not $k$-periodic, or $Y$ is $k$-periodic with $k$-period $Q$.
###### Proof.
Suppose by contradiction that $Y$ is $k$-periodic with $k$-period
$Q^{\prime}\neq Q$. Let $q=|Q|$ and $q^{\prime}=|Q^{\prime}|$. Assume first
$q\leq q^{\prime}$. Fix an alignment of the smallest cost between $Y$ and a
prefix of $(Q^{\prime})^{\infty}$. It induces an alignment
$\mathcal{A}^{\prime}$ of cost at most $2k$ between $X$ and a prefix of
$(Q^{\prime})^{\infty}$ and hence generates a partition $X=X_{1}X_{2}\ldots
X_{z}$, where each $X_{i}$, $1\leq i\leq z-1$, is aligned with $Q^{\prime}$
and $X_{z}$ is aligned with a prefix of $Q^{\prime}$. From $|Q|\leq|X|/128k$
we obtain $|X|\geq 128k$ and therefore
$z\geq(|X|-2k)/q^{\prime}\geq\frac{128-2}{128}|X|/(|Y|/128k)\gg 20k$
Consider fragments $X_{1}X_{2}X_{3}X_{4}$, $X_{5}X_{6}X_{7}X_{8}$, and so on.
The total number of such fragments is at least $(20k-3)/4>4k$, and at least
$2k+1$ of them are not edited under $\mathcal{A}^{\prime}$. Fix an optimal
alignment $\mathcal{A}$ between $X$ and a prefix $Q^{\infty}$. The cost of
$\mathcal{A}$ is bounded from above by $2k$, and therefore there is at least
one fragment $X_{4i+1}X_{4i+2}X_{4i+3}X_{4i+4}$ that is not edited. Consider
one such fragment $F$. On the one hand,
$F=Q^{\prime}Q^{\prime}Q^{\prime}Q^{\prime}$. On the other hand,
$F=\mathsf{suff}(Q)Q^{j}\mathsf{pref}(Q)$, where $\mathsf{suff}(Q)$ and
$\mathsf{pref}(Q)$ are some suffix and some prefix of $Q$, respectively.
Suppose first that $q^{\prime}$ is a multiple of $q$. In this case,
$Q^{\prime}=\left(Q[\ell+1\mathinner{.\,.\allowbreak}q]Q[1\mathinner{.\,.\allowbreak}\ell]\right)^{r}$,
where $\ell=q-|\mathsf{suff}(Q)|$ and an integer $r$, which contradicts the
fact that $Q^{\prime}$ is primitive. Otherwise, consider the copy of $Q$ that
contains $F[q^{\prime}]$. Consider also a substring $QQQ$ of $F$ formed by the
copy of $Q$ that contains $F[2q^{\prime}]$, and the preceding and succeeding
copies of $Q$. We then obtain that there is an occurrence of $Q$ in $QQQ$ that
is not aligned with any copy of $Q$ (otherwise, $q^{\prime}$ is a multiple of
$q$), and therefore $Q$ is not primitive, a contradiction.
The case $q>q^{\prime}$ can be treated analogously. ∎
Note that 7.1 implies in particular that a string can have at most one
$k$-period.
### 7.2 Semi-streaming Algorithm
We first present a deterministic algorithm for pattern matching with $k$ edits
in the semi-streaming setting.
#### 7.2.1 Preprocessing Stage
Consider a set $\Pi$ of $\mathcal{O}(\log m)$ prefixes $P_{i}$ of $P$
initialised to contain $P$ itself as well as the prefixes of length $2^{\ell}$
for all $0\leq\ell\leq\lfloor{\log|P|}\rfloor$. Order the prefixes by lengths,
and consider two consecutive prefixes $P^{\prime},P^{\prime\prime}$. If
$P^{\prime}$ is $k$-periodic with $k$-period $Q^{\prime}$ while
$P^{\prime\prime}$ is not $k$-periodic, we add two more prefixes to $\Pi$.
Namely, if $\ell$ be the maximum integer such that
$P[1\mathinner{.\,.\allowbreak}\ell]$ is $k$-periodic with $k$-period
$Q^{\prime}$, add to $\Pi$ the prefixes $P[1\mathinner{.\,.\allowbreak}\ell]$
and $P[1\mathinner{.\,.\allowbreak}\ell+1]$. Note that
$P[1\mathinner{.\,.\allowbreak}\ell+1]$ is not $k$-periodic by 7.1.
Let $\Pi=\\{P_{1},P_{2},\ldots,P_{z}\\}$ be the resulting set of prefixes. We
assume that the patterns are ordered in the ascending order of their lengths.
During the preprocessing step, for each $i$ such that $P_{i}$ is $k$-periodic,
we compute its $k$-period $Q_{i}$. We use notation $\ell_{i}=P_{i}$ and
$q_{i}=|Q_{i}|$ (if defined). Importantly, we do not have to store $Q_{i}$
explicitly, we can simply memorize its endpoints in $P_{i}$ which takes
$\mathcal{O}(\log m)$ extra space in total. We also store, for each of the
$\mathcal{O}(k)$ rotations $D$ of $Q_{i}$ that can be a difference of a chain
of $k$-edit occurrences of $P_{i}$ (Corollary 3.5), the encodings
$\mathsf{qGR}_{32k}(D,D)$ and
$\mathsf{qGR}_{30k}(P[\ell_{i-1}+1\mathinner{.\,.\allowbreak}\ell_{i}],D^{\infty}[1\mathinner{.\,.\allowbreak}\Delta_{i}])$,
where $\Delta_{i}=\ell_{i}-\ell_{i-1}+k$.
#### 7.2.2 Main Stage
The main stage of the algorithm starts after we have preprocessed the pattern.
During the main stage, we receive the text as a stream, one character of a
time. We exploit the following result:
###### Fact 7.2 (cf. [59]).
Given a read-only string $X$ of length $m$ and a streaming string $Y$. There
is a dynamic programming algorithm that correctly identifies all prefixes
$Y^{\prime}$ of $Y$ within edit distance at most $k\leq m$ from $X$, as well
as $\mathsf{ed}(Y^{\prime},X)$ itself. The algorithm takes $\mathcal{O}(km)$
time and $\mathcal{O}(k)$ space besides the space required to store $X$.
##### Chains of $k$-edit occurrences.
During the main stage of the algorithm, we store the following information.
Let $r$ be the newly arrived position of the text $T$. For each $2\leq i\leq
z$, consider all $k$-edit occurrences of $P_{i-1}$ in
$T[r-\ell_{i}-k+1\mathinner{.\,.\allowbreak}r]$. We call such occurrences
_active_. We denote the set of right endpoints of the active occurrences by
$\mathsf{aOCC}_{k}^{E}(P_{i-1},T)$. By Corollaries 3.3 and 3.5,
$\mathsf{aOCC}_{k}^{E}(P_{i-1},T)$ forms $\mathcal{O}(k^{3})$ chains. For each
chain $\mathcal{C}$, we store the following information:
1. 1.
The leftmost position $lp$ and the size $|\mathcal{C}|$ of $\mathcal{C}$;
2. 2.
An integer $\mathsf{ed}(\mathcal{C})$ equal to the smallest edit distance from
$P_{i-1}$ to a suffix of $T[1\mathinner{.\,.\allowbreak}r]$ for every endpoint
$r\in\mathcal{C}$;
3. 3.
If $|\mathcal{C}|\geq 2$, we store the shift $\Delta$ of the difference
$D=Q_{i-1}[1+\Delta\mathinner{.\,.\allowbreak}q_{i-1}]Q_{i-1}[1\mathinner{.\,.\allowbreak}q_{i-1}]$
of $\mathcal{C}$.
If $p^{\ast}$ is the first position added to $\mathcal{C}$ (it can be
different from $lp$ as we will update chains to contain only active
occurrences), then at the position $(p^{\ast}+1)$ we start running the dynamic
programming algorithm for $T[p^{\ast}+1\mathinner{.\,.\allowbreak}]$ and
$P[\ell_{i-1}+1\mathinner{.\,.\allowbreak}\ell_{i}]$ (Fact 7.2).
Furthermore, consider the moment when we detect the second position in
$\mathcal{C}$ (if it exists) and hence the difference $D$ of the chain.
Starting from this moment, for every newly added position $p\in\mathcal{C}$,
at the position $(p+1)$ we start computing the greedy encoding
$\mathsf{qGR}_{32k}(T[p+1\mathinner{.\,.\allowbreak}p+\Delta_{i}],D^{\infty}[1\mathinner{.\,.\allowbreak}\Delta_{i}])$.
We continue running the algorithm until either the computation is over or a
new position in the chain is detected. In the end, we compute the encoding for
the rightmost position in the chain.
##### Detecting new $k$-edit occurrences of $P_{i}$.
We now explain how to detect new $k$-edit occurrences of the prefixes $P_{i}$.
Let $r$ be the latest arrived position of $T$. If $i=1$, then since $k\geq 1$,
$r\in\mathsf{aOCC}_{k}^{E}(P_{1},T)$. Below we consider three possible cases
for $i\geq 2$: $P_{i-1}$ is $k$-periodic, $P_{i}$ is not $k$-periodic;
$P_{i-1}$ is not $k$-periodic; $P_{i-1}$ and $P_{i}$ are $k$-periodic.
###### Case 1: $P_{i-1}$ is $k$-periodic, $P_{i}$ is not $k$-periodic.
By construction, in this case $\ell_{i-1}+1=\ell_{i}$. The position
$r\in\mathsf{aOCC}_{k}^{E}(P_{i},T)$ iff one of the following conditions is
satisfied:
1. 1.
The smallest edit distance from $P_{i-1}$ to a suffix of
$T[1\mathinner{.\,.\allowbreak}r]$ is at most $k-1$ (this corresponds to the
case when the last character of $P_{i}$ is deleted in an optimal alignment of
a suffix of $T[1\mathinner{.\,.\allowbreak}r]$ and $P_{i}$);
2. 2.
The smallest edit distance from $P_{i-1}$ to a suffix of
$T[1\mathinner{.\,.\allowbreak}r-1]$ is at most $k-1$ and $P_{i}[\ell_{i}]\neq
T[r]$ (this corresponds to the case when the last character of $P_{i}$ is
substituted for $T[r]$ in an optimal alignment of a suffix of
$T[1\mathinner{.\,.\allowbreak}r]$ and $P_{i}$);
3. 3.
The smallest edit distance from $P_{i-1}$ to a suffix of
$T[1\mathinner{.\,.\allowbreak}r-1]$ is at most $k$ and $P_{i}[\ell_{i}]=T[r]$
(this corresponds to the case when the last character of $P_{i}$ is matched
with $T[r]$ in an optimal alignment of a suffix of
$T[1\mathinner{.\,.\allowbreak}r]$ and $P_{i}$);
4. 4.
The smallest edit distance from $P_{i}$ to a suffix of
$T[1\mathinner{.\,.\allowbreak}r-1]$ is at most $k-1$ (this corresponds to the
case when $T[r]$ is deleted in an optimal alignment of a suffix of
$T[1\mathinner{.\,.\allowbreak}r]$ and $P_{i}$).
We can decide which of the conditions is satisfied, and therefore whether
$r\in\mathsf{aOCC}_{k}^{E}(P_{i},T)$ in $\mathcal{O}(k^{3})$ time using
$\mathsf{aOCC}_{k}^{E}(P_{i-1},T)$ and $\mathsf{aOCC}_{k}^{E}(P_{i},T)$.
Moreover, we can compute the smallest edit distance from $P_{i}$ to a suffix
of $T[1\mathinner{.\,.\allowbreak}r]$ if it is bounded by $k$.
For the next two cases, we will use the following simple observation that
follows from Fact 2.3:
###### Observation 7.3.
Let $\mathsf{ed}_{i-1}(r^{\prime})$ be the smallest edit distance from
$P_{i-1}$ to a suffix of $T[1\mathinner{.\,.\allowbreak}r^{\prime}]$, and
define
$d=\min_{\begin{subarray}{c}r^{\prime}\in\mathsf{aOCC}_{k}^{E}(P_{i-1},T)\\\
r^{\prime}\in[r+1-\Delta_{i},r+1-\Delta_{i}+2k]\end{subarray}}\\{\mathsf{ed}_{i-1}(r^{\prime})+\mathsf{ed}(P[\ell_{i-1}+1\mathinner{.\,.\allowbreak}\ell_{i}],T[r^{\prime}+1\mathinner{.\,.\allowbreak}r])\\}$
(2)
The smallest edit distance from $P_{i}$ to $T[1\mathinner{.\,.\allowbreak}r]$
is equal to $d$ if $d\leq k$ and is larger than $k$ otherwise.
It follows that to decide whether $r\in\mathsf{aOCC}_{k}^{E}(P_{i},T)$, it
suffices to compute the value $\min\\{d,k+1\\}$, where $d$ is as defined
above.
###### Case 2: $P_{i-1}$ is not $k$-periodic.
In this case, $\mathsf{aOCC}_{k}^{E}(P_{i-1},T)$ is stored as
$\mathcal{O}(k^{3})$ chains of size one. Therefore, we can find the positions
$r^{\prime}$ from Eq. 2 in $\mathcal{O}(k^{3})$ time. Moreover, for each
position $r^{\prime}$, we run the dynamic programming algorithm for
$T[r^{\prime}+1\mathinner{.\,.\allowbreak}]$ and
$P[\ell_{i-1}+1\mathinner{.\,.\allowbreak}\ell_{i}]$, which outputs the edit
distance between $T[r^{\prime}+1\mathinner{.\,.\allowbreak}r]$ and
$P[\ell_{i-1}+1\mathinner{.\,.\allowbreak}\ell_{i}]$ if it is at most $k$. As
we also know the smallest edit distance between $P_{i-1}$ and a suffix of
$T[1\mathinner{.\,.\allowbreak}r^{\prime}]$, we can compute $d$ in
$\mathcal{O}(k^{3})$ time.
###### Case 3: $P_{i-1}$ and $P_{i}$ are $k$-periodic.
We can identify all positions $r^{\prime}$ from Eq. 2 in $\mathcal{O}(k^{3})$
time. (It suffices to check each of the $\mathcal{O}(k^{3})$ chains that we
store for $P_{i-1}$). We must now test each of these positions. Consider a
position $r^{\prime}$ and let $\mathcal{C}$ be the chain containing it. It
suffices to compute the edit distance between
$P[\ell_{i-1}+1\mathinner{.\,.\allowbreak}\ell_{i}]$ and
$T[r^{\prime}+1\mathinner{.\,.\allowbreak}r]$ as we already know the smallest
edit distance from $P_{i-1}$ to a suffix of
$T[1\mathinner{.\,.\allowbreak}r^{\prime}]$. If $|\mathcal{C}|=1$, the
distance has been computed by the dynamic programming algorithm. Otherwise, we
use quasi-greedy encodings. On a high level, our goal is to compute the edit
distance between $\pi=P[\ell_{i-1}\mathinner{.\,.\allowbreak}\ell_{i}]$ and
$\tau=T[r^{\prime}+1\mathinner{.\,.\allowbreak}r]$ via a string
$\mu=D^{\infty}[1\mathinner{.\,.\allowbreak}\Delta_{i}]$, where $D$ is the
difference of $\mathcal{C}$ and $\Delta_{i}=\ell_{i}-\ell_{i-1}+k$.
###### Lemma 7.4.
There is $\mathsf{ed}(\pi,\mu)\leq 26k$.
###### Proof.
As $P_{i-1}$ and $P_{i}$ are $k$-periodic, by 7.1 we obtain that
$P_{i}=P[1\mathinner{.\,.\allowbreak}\ell_{i}]$ is $k$-periodic with
$k$-period $Q_{i}=Q_{i-1}$, that is, there is a prefix of $Q_{i}^{\infty}$
such that the edit distance between it and $P_{i}$ is at most $2k$. By Fact
2.3, there is a substring $Q_{i}^{\infty}[r\mathinner{.\,.\allowbreak}t]$ such
that $|r-\ell_{i-1}|\leq 2k$ and $|t-\ell_{i}|\leq 2k$ and
$\mathsf{ed}(Q_{i}^{\infty}[r\mathinner{.\,.\allowbreak}t],\pi)\leq 2k$. By
the triangle inequality, we obtain that
$\mathsf{ed}(Q_{i}^{\infty}[\ell_{i-1}+1\mathinner{.\,.\allowbreak}\ell_{i}],\pi)\leq
6k$. Let $a=\ell_{i-1}-7k\pmod{q_{i}}$ and $b=\ell_{i-1}+7k\pmod{q_{i}}$. By
Corollary 3.5, $D$ is a rotation of $Q_{i-1}=Q_{i}$ with shift $\Delta$, where
$\Delta\in[a-3k,b+3k]$ if $a\leq b$ and $\Delta\in[0,b+3k]\cup[a-3k,q_{i})$ if
$a>b$. It follows that
$D^{\infty}=Q_{i}^{\infty}[s\mathinner{.\,.\allowbreak}]$, where
$|s-\ell_{i-1}|\leq 10k$. As
$\mu=D^{\infty}[1\mathinner{.\,.\allowbreak}\Delta_{i}]=Q_{i}^{\infty}[s\mathinner{.\,.\allowbreak}s+\Delta_{i}-1]$,
by the triangle inequality we obtain
$\mathsf{ed}(\mu,Q_{i}^{\infty}[\ell_{i-1}+1\mathinner{.\,.\allowbreak}\ell_{i}])\leq
20k$. Applying the triangle inequality one more time, we obtain the claim. ∎
Let $\mathcal{G}_{P}=\mathsf{qGR}_{30k}(\pi,\mu)$ and
$\mathcal{G}_{T}=\mathsf{qGR}_{30k}(\mu,\tau)$. By Corollary 5.27, knowing
$\mathcal{G}_{P}$ and $\mathcal{G}_{T}$ is sufficient to compute the edit
distance between $\pi$ and $\tau$. Note that we do not know $\mathcal{G}_{T}$
yet, we must compute it using the available information. Let $p$ be the
rightmost position in $\mathcal{C}$.
1. 1.
Recall that at the position $(p+1)$ we launched an algorithm that is computing
$\mathsf{qGR}_{32k}(T[p+1\mathinner{.\,.\allowbreak}p+\Delta_{i}],D^{\infty}[1\mathinner{.\,.\allowbreak}\Delta_{i}])$
with a delay of $k$ characters (Corollary 5.22). We have $p+\Delta_{i}\geq
r^{\prime}+\Delta_{i}\geq r$. Therefore, upon reaching $r$, we can use the
memory of the algorithm to compute
$\mathcal{G}_{T,1}=\mathsf{qGR}_{32k}(T[p+1\mathinner{.\,.\allowbreak}r],D^{\infty}[1\mathinner{.\,.\allowbreak}r-p])$
in $\tilde{\mathcal{O}}(k^{5})$ time and $\tilde{\mathcal{O}}(k^{2})$ space
(Lemma 5.21).
2. 2.
By the definition of chains,
$T[r^{\prime}+1\mathinner{.\,.\allowbreak}p]=D^{j}$ for some integer $j$. By
Lemma 5.21, we can use $\mathsf{qGR}_{32k}(D,D)$ computed during the
preprocessing step to compute
$\mathcal{G}_{T,2}=\mathsf{qGR}_{32k}(T[r^{\prime}+1\mathinner{.\,.\allowbreak}p],D^{j})$
in $\tilde{\mathcal{O}}(k^{5})$ time and $\tilde{\mathcal{O}}(k^{2})$ space.
By applying Lemma 5.21 again, we can compute
$\mathsf{qGR}_{32k}(T[r^{\prime}+1\mathinner{.\,.\allowbreak}p],D^{j}D^{\infty}[1\mathinner{.\,.\allowbreak}r-p])$
in $\tilde{\mathcal{O}}(k^{5})$ time and $\tilde{\mathcal{O}}(k^{2})$ space.
3. 3.
Finally, we compute via Corollary 5.22 the encoding
$\mathsf{qGR}_{30k}(\varepsilon,D^{\infty}[r-p+1\mathinner{.\,.\allowbreak}\Delta_{i}])$,
where $\varepsilon$ is the empty string and $\Delta_{i}-(r-p)\leq 2k$. We then
apply 5.20 to compute
$\mathsf{qGR}_{30k+(\Delta_{i}-(r-p))}(T[r^{\prime}+1\mathinner{.\,.\allowbreak}p],D^{j}D^{\infty}[1\mathinner{.\,.\allowbreak}r-p])$
and further Lemma 5.21 to compute
$\mathsf{qGR}_{30k}(T[r^{\prime}+1\mathinner{.\,.\allowbreak}p],D^{\infty}[1\mathinner{.\,.\allowbreak}\Delta_{i}])=\mathcal{G}_{T}$.
##### Updating the chains.
When we detect a new $k$-edit occurrence of $P_{i}$, we must decide if it
should be added to some existing chain or if we must create a new chain for
this occurrence.
To this end, for each $1\leq i\leq z$, we consider $\mathcal{O}(k)$ of its
rotations of $Q_{i}$ that can be the difference of a chain of $k$-edit
occurrences of $P_{i}$ in $T$ (Corollary 3.5). For each rotation $R$, we run a
constant-space and linear-time deterministic pattern matching algorithm [54].
The algorithm processes the text $T$ as a stream and if there is an occurrence
$T[\ell\mathinner{.\,.\allowbreak}r]$ of the rotation reports it while reading
$T[r]$. The algorithm uses $\mathcal{O}(1)$ space and $\mathcal{O}(1)$
amortised time per character of $T$.
Suppose that we detect a new right endpoint $r$ of a $k$-edit occurrence
$T[\ell\mathinner{.\,.\allowbreak}r]$ of $P_{i}$. We must decide whether $r$
belongs to an existing chain of $k$-edit occurrences of $P_{i}$ or starts a
new one. In order to do this, we first find the chain $\mathcal{C}$ that
contains $r-q_{i}+1$ if it exists by checking each chain in turn. We then
check that the smallest edit distance from a suffix of
$T[1\mathinner{.\,.\allowbreak}r]$ to $P_{i}$ equals to
$\mathsf{ed}(\mathcal{C})$ and that $T[r-q_{i}+1\mathinner{.\,.\allowbreak}r]$
is equal to the difference of the chain. (Recall that we run the exact pattern
matching algorithm for each rotation of $Q_{i}$ that can be the difference of
a chain so that both checks can be performed in $\mathcal{O}(1)$ time).
If these conditions are not satisfied, we create a new chain that contains $r$
only. Otherwise, we add $r$ to $\mathcal{C}$ (i.e., increment the size of
$\mathcal{C}$). To finalize the update of the chains, we must delete all
$k$-edit occurrences that become inactive: for each $i$ and for each chain of
$k$-edit occurrences of $P_{i}$, we “delete” the first $k$-edit occurrence if
it starts before $r-\ell_{i}-k+1$. To “delete” a $k$-edit occurrence, we
simply update the endpoints of the first occurrence in the chain and the total
number of occurrences in the chain.
#### 7.2.3 Analysis
We summarize the results of this section:
###### Theorem 7.5.
Assume a read-only pattern $P$ of length $m$ and a streaming text $T$ of
length $n$. There is a deterministic algorithm that finds the set
$\mathsf{OCC}_{k}(P,T)$ using $\tilde{\mathcal{O}}(k^{5})$ space and
$\tilde{\mathcal{O}}(k^{6})$ amortised time per character of the text $T$.
###### Proof.
Let us first upper bound the space complexity of the algorithm. For each
$i=1,\ldots,z=\mathcal{O}(\log m)$, we store the set
$\mathsf{aOCC}_{k}(P_{i},T)$ as $\mathcal{O}(k^{3})$ chains. For each chain,
we launch the dynamic programming algorithm (Fact 7.2), which takes
$\tilde{\mathcal{O}}(k^{2})$ space and Corollary 5.22 that takes
$\tilde{\mathcal{O}}(k^{2})$ space. The pattern matching algorithms for the
rotations of $Q_{i}$ take $\tilde{\mathcal{O}}(k)$ space in total. Finally,
testing if a position of the text is the rightmost position of a $k$-edit
occurrence of $P_{i}$ requires $\tilde{\mathcal{O}}(k^{2})$ space.
We now show the time bound. Updating the chains takes $\tilde{\mathcal{O}}(1)$
time. At any time, we run $\tilde{\mathcal{O}}(k^{3})$ instances of Corollary
5.22 that takes $\mathcal{O}(k^{3})$ amortised time per character. To test
each position $r$, we spend $\tilde{\mathcal{O}}(k\cdot k^{3})$ time. ∎
### 7.3 Streaming Algorithm
We now modify the algorithm for the semi-streaming model to develop a fully
streaming algorithm. W.l.o.g., assume $k\leq m$ and take $\delta=1/n^{c}$ for
$c$ large enough.
#### 7.3.1 Preprocessing
We define the prefixes $P_{i}$ and their periods $Q_{i}$ exactly in the same
way as in Section 7.2. Recall that $\ell_{i}=|P_{i}|$ and $q_{i}=|Q_{i}|$. For
every $i>1$, we store the following information, where all sketches
$\mathsf{sk}^{\mathsf{q}}$ (6.23) are parametrized by probability $\delta$,
maximal length $\Delta_{i}$, the alphabet of $P$ and $T$, and a seed of
$\mathcal{O}(\log^{2}n\log(1/\delta))$ random bits:
1. 1.
$P[\ell_{i}]$ and the sketch
$\mathsf{sk}^{\mathsf{q}}_{k}(P[\ell_{i-1}\mathinner{.\,.\allowbreak}\ell_{i}])$;
2. 2.
For each of rotation $D$ of $Q_{i}$ that can be a difference of a chain of
$k$-edit occurrences of $P_{i}$, the encoding
$\mathsf{qGR}_{30k}(P[\ell_{i-1}+1\mathinner{.\,.\allowbreak}\ell_{i}],D^{\infty}[1\mathinner{.\,.\allowbreak}\Delta_{i}])$,
where $\Delta_{i}=\ell_{i}-\ell_{i-1}+k$;
3. 3.
For each of rotation $D$ of $Q_{i}$ that can be a difference of a chain of
$k$-edit occurrences of $P_{i}$, the sketches
$\mathsf{sk}^{\mathsf{q}}_{32k}(D)$ and
$\mathsf{sk}^{\mathsf{q}}_{32k}(D[1\mathinner{.\,.\allowbreak}\Delta_{i}\pmod{q_{i}}])$.
#### 7.3.2 Main Stage
As in Section 7.2, for each $i$, we store $\mathsf{aOCC}_{k}^{E}(P_{i-1},T)$
in $\mathcal{O}(k^{3})$ chains. For each chain $\mathcal{C}$, we store the
leftmost position $lp$ in it, its size $|\mathcal{C}|$, and the smallest edit
distance, $\mathsf{ed}(\mathcal{C})$, from a suffix of
$T[1\mathinner{.\,.\allowbreak}lp]$ to $P_{i-1}$. If $|\mathcal{C}|\geq 2$, we
also store its difference (defined by the shift of the rotation of $Q_{i-1}$).
If $p^{\ast}$ is the first position added to $\mathcal{C}$, then at the
position $(p^{\ast}+1)$, we start running the streaming algorithm of 6.23(a)
for computing the sketch
$\mathsf{sk}^{\mathsf{q}}_{k}(T[p^{a}st+1\mathinner{.\,.\allowbreak}p^{a}st+\Delta_{i}])$.
Furthermore, consider the moment when we detect the second position in
$\mathcal{C}$ (if it exists) and hence the difference $D$ of the chain.
Starting from this moment, for every newly added position $p\in\mathcal{C}$,
at the position $(p+1)$ we start computing the quasi-greedy encoding
$\mathsf{qGR}_{32k}(T[p+1\mathinner{.\,.\allowbreak}p+\Delta_{i}],D^{\infty}[1\mathinner{.\,.\allowbreak}\Delta_{i}])$
as follows: Assume that we have computed
$\mathsf{qGR}_{32k}(T[p+1\mathinner{.\,.\allowbreak}p+\ell\cdot
q_{i}],D^{\infty}[1\mathinner{.\,.\allowbreak}\ell\cdot q_{i}])$. Suppose
first that $(\ell+1)\cdot q_{i}\leq\Delta_{i}$. While reading $T[p+\ell\cdot
q_{i}+1\mathinner{.\,.\allowbreak}p+(\ell+1)\cdot q_{i}]$, we compute
$\mathsf{sk}^{\mathsf{q}}_{32k}(T[p+\ell\cdot
q_{i}+1\mathinner{.\,.\allowbreak}p+(\ell+1)\cdot q_{i}])$ again via the
algorithm of 6.23(a). We then use this sketch and
$\mathsf{sk}^{\mathsf{q}}_{32k+1}(D)$ to compute
$\mathsf{qGR}_{32k}(T[p+\ell\cdot
q_{i}+1\mathinner{.\,.\allowbreak}p+(\ell+1)\cdot q_{i}],D)$ (6.23(b)) and
then $\mathsf{qGR}_{32k}(T[p+1\mathinner{.\,.\allowbreak}p+(\ell+1)\cdot
q_{i}],D^{\infty}[1\mathinner{.\,.\allowbreak}(\ell+1)\cdot q_{i}])$ (Lemma
5.21). If $(\ell+1)\cdot q_{i}>\Delta_{i}$, we use the sketches
$\mathsf{sk}^{\mathsf{q}}_{32k}(T[p+\ell\cdot
q_{i}\mathinner{.\,.\allowbreak}p+\Delta_{i}])$ and
$\mathsf{sk}^{\mathsf{q}}_{32k}(D[1\mathinner{.\,.\allowbreak}\Delta_{i}\pmod{q_{i}}])$,
the rest is analogous. We continue running the algorithm until either the
computation is completed or a new $k$-edit occurrence in the chain has been
detected. In other words, in the end we compute the encoding for the rightmost
position in the chain.
##### Detecting new $k$-edit occurrences of $P_{i}$.
We now explain how we modify the algorithm for detecting new $k$-edit
occurrences of the prefixes $P_{i}$. The algorithm for Case 1 does not change.
Instead of the dynamic programming algorithm in Case 2, we use
$\mathsf{sk}^{\mathsf{q}}_{k}$ and use 6.23(b) and then Corollary 5.27 to
compute the edit distance. It remains to explain how we modify the algorithm
for Case 3.
We exploit Eq. 2 again. We can find all positions $r^{\prime}$ in
$\mathcal{O}(k^{3})$ time. To test a position $r^{\prime}$, it suffices to
compute the edit distance between
$P[\ell_{i-1}+1\mathinner{.\,.\allowbreak}\ell_{i}]$ and
$T[r^{\prime}+1\mathinner{.\,.\allowbreak}r]$. Let $\mathcal{C}$ be the chain
with difference $D$ that contains $r^{\prime}$. If $|\mathcal{C}|=1$, we use
the edit distance sketch, and otherwise the quasi-greedy encodings. By Lemma
7.4,
$\mathsf{ed}(P[\ell_{i-1}\mathinner{.\,.\allowbreak}\ell_{i}],D^{\infty}[1\mathinner{.\,.\allowbreak}\Delta_{i}])\leq
26k$ and hence by Corollary 5.27, the edit distance between
$P[\ell_{i-1}+1\mathinner{.\,.\allowbreak}\ell_{i}]$ and
$T[r^{\prime}+1\mathinner{.\,.\allowbreak}r]$ can be computed from the
encodings $\mathcal{G}_{P}=\mathsf{qGR}_{30k}(\pi,\mu)$ and
$\mathcal{G}_{T}=\mathsf{qGR}_{30k}(\mu,\tau)$ for
$\pi=P[\ell_{i-1}\mathinner{.\,.\allowbreak}\ell_{i}]$,
$\mu=D^{\infty}[1\mathinner{.\,.\allowbreak}\Delta_{i}]$, and
$\tau=T[r^{\prime}+1\mathinner{.\,.\allowbreak}r]$. $\mathcal{G}_{P}$ was
computed during the preprocessing step and we store it explicitly. Hence, we
only need to explain how to compute $\mathcal{G}_{T}$. Let $p$ be the
rightmost position in the chain $\mathcal{C}$.
1. 1.
Recall that $T[r^{\prime}+1\mathinner{.\,.\allowbreak}p]=D^{j}$ for
$j=(p-r^{\prime})/q_{i}$. We first compute $\mathsf{qGR}_{32k}(D,D)$ from
$\mathsf{sk}^{\mathsf{q}}_{32k}(D,D)$ via 6.23(b), and then
$\mathcal{G}_{T,1}=\mathsf{qGR}_{32k}(T[r^{\prime}+1\mathinner{.\,.\allowbreak}p],D^{j})=\mathsf{qGR}_{32k}(D^{j},D^{j})$
in $\tilde{\mathcal{O}}(k^{5})$ time and $\tilde{\mathcal{O}}(k^{2})$ space as
in Section 7.2.
2. 2.
At the position $(p+1)$ we launched the streaming algorithm computing
$\mathsf{qGR}_{32k}(T[p+1\mathinner{.\,.\allowbreak}p+\Delta_{i}],D^{\infty}[1\mathinner{.\,.\allowbreak}\Delta_{i}])$
with a delay of $q_{i}$ characters. We have $p+\Delta_{i}\geq
r^{\prime}+\Delta_{i}\geq r$. Therefore, at a position $p+\ell\cdot q_{i}$,
where $\ell=\lfloor(r-p+1)/q_{i}\rfloor$, the algorithm computes
$\mathcal{G}_{T,2}=\mathsf{qGR}_{32k}(T[p+1\mathinner{.\,.\allowbreak}p+\ell\cdot
q_{i}],D^{\infty}[1\mathinner{.\,.\allowbreak}\ell\cdot q_{i}])$. The
algorithm then continues to compute the sketch
$\mathsf{sk}^{\mathsf{q}}_{32k}(T[p+\ell\cdot
q_{i}+1\mathinner{.\,.\allowbreak}r])$. We use this sketch and the sketch
$\mathsf{sk}^{\mathsf{q}}_{32k}(D[1\mathinner{.\,.\allowbreak}\Delta_{i}\pmod{q_{i}}])$
to compute $\mathcal{G}_{T,3}=\mathsf{qGR}_{30k}(T[p+\ell\cdot
q_{i}+1\mathinner{.\,.\allowbreak}r],D[1\mathinner{.\,.\allowbreak}\Delta_{i}\pmod{q_{i}}])$
via 6.23(b) and 5.20.
3. 3.
We finally concatenate $\mathcal{G}_{T,1}$ and $\mathcal{G}_{T,2}$ to obtain
$\mathsf{qGR}_{32k}(T[r^{\prime}+1\mathinner{.\,.\allowbreak}r^{\prime}+(\ell+j)\cdot
q_{i}],D^{\infty}[1\mathinner{.\,.\allowbreak}(\ell+j)\cdot q_{i}])$, and then
$\mathsf{qGR}_{32k}(T[r^{\prime}+1\mathinner{.\,.\allowbreak}r^{\prime}+(\ell+j)\cdot
q_{i}],D^{\infty}[1\mathinner{.\,.\allowbreak}(\ell+j)\cdot q_{i}])$ and
$\mathcal{G}_{T,3}$ to obtain $\mathcal{G}_{T}$ via Lemma 5.21 (note that the
difference of lengths of strings in $\mathcal{G}_{T,3}$ is bounded by $2k$).
##### Updating the chains.
When we detect a new $k$-edit occurrence of $P_{i}$, we must decide if it
should be added to some existing chain or if we must create a new chain for
this occurrence. We use the algorithm of Section 7.2, but replace the
constant-space pattern matching algorithm with the streaming pattern matching
algorithm [52] that for a rotation of $Q_{i}$ takes $\tilde{\mathcal{O}}(1)$
space and $\tilde{\mathcal{O}}(1)$ time per character and retrieves all its
occurrences correctly with probability at least $1-\delta$.
#### 7.3.3 Analysis
We summarize the results of this section:
###### Theorem 7.6.
Given a pattern $P$ of length $m$ and a text $T$ of length $n$. There is a
streaming algorithm that finds the set $\mathsf{OCC}_{k}^{E}(P,T)$ using
$\tilde{\mathcal{O}}(k^{5})$ space and $\tilde{\mathcal{O}}(k^{8})$ amortised
time per character of the text $T$. The algorithm computes
$\mathsf{OCC}_{k}^{E}(P,T)$ correctly with high probability.
###### Proof.
By Corollary 3.5, for a fixed $i$ only $\mathcal{O}(k)$ rotations of $Q_{i}$
can be a difference of a chain of occurrences of $P_{i}$. The sketch
$\mathsf{sk}^{\mathsf{q}}_{30k+1}(\cdot)$ takes $\tilde{\mathcal{O}}(k^{2})$
space (6.23(a)) and $\mathsf{qGR}_{30k}(\cdot,\cdot)$
$\tilde{\mathcal{O}}(k^{2})$ space as well (Corollary 5.17). Therefore, the
information computed during the preprocessing stage occupies
$\tilde{\mathcal{O}}(k^{3})$ space. During the main stage, we store
$\tilde{\mathcal{O}}(k^{3})$ chains. For each chain, we run the algorithm of
6.23(a) that takes $\tilde{\mathcal{O}}(k^{2})$ space. The algorithm than
computes the quasi-greedy encoding (6.23(b)) takes
$\tilde{\mathcal{O}}(k^{2})$ space as well. In total, the information we store
for the chains occupies $\tilde{\mathcal{O}}(k^{5})$ space. When checking for
new occurrences, we apply Lemmas 5.21 and 5.27, which require an overhead of
$\tilde{\mathcal{O}}(k^{2})$ space. Finally, the streaming pattern matching
algorithms for the rotations of $Q_{i}$ that can be differences of occurrences
of $P_{i}$ take $\tilde{\mathcal{O}}(k)$ space in total. The space bound
follows.
At any time, we run $\tilde{\mathcal{O}}(k^{3})$ instances of the algorithm of
6.23(a) that takes $\tilde{\mathcal{O}}(k)$ amortised time per character. In
addition, for every character we run $\tilde{\mathcal{O}}(k^{3})$ instances of
the algorithms of Lemmas 5.21 and 5.27 taking $\tilde{\mathcal{O}}(k^{8})$
time in total. The pattern matching algorithms for the rotations of $Q_{i}$
take $\tilde{\mathcal{O}}(k)$ time per character.
Note that the only probabilistic procedures in the algorithm are streaming
pattern matching [52] and that of 6.23(b) that computes quasi-greedy
encodings. These procedures are called $\mathrm{poly}(n,k)=\mathrm{poly}(n)$
times. By choosing the constant $c$ in $\delta=1/n^{c}$ large enough, we can
guarantee that the algorithm is correct with high probability by the union
bound. ∎
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# Communication games, sequential equilibrium, and mediators
Ivan Geffner
Cornell University Supported in part by NSF grants IIS-1703846 and
IIS-0911036, ARO grant W911NF-17-1-0592, MURI grant W911NF-19-1-0217 from the
ARO, and a grant from Open Philanthropy.Declarations of interest: none. Joseph
Y. Halpern
Cornell University11footnotemark: 1 22footnotemark: 2
We consider _$k$ -resilient sequential equilibria_, strategy profiles where no
player in a coalition of at most $k$ players believes that it can increase its
utility by deviating, regardless of its local state. We prove that all
$k$-resilient sequential equilibria that can be implemented with a trusted
mediator can also be implemented without the mediator in a synchronous system
of $n$ players if $n>3k$. In asynchronous systems, where there is no global
notion of time and messages may take arbitrarily long to get to their
recipient, we prove that a $k$-resilient sequential equilibrium with a
mediator can be implemented without the mediator if $n>4k$. These results
match the lower bounds given by Abraham, Dolev, and Halpern [2008b] and
Geffner and Halpern [2023] for implementing a Nash equilibrium without a
mediator (which are easily seen to apply to implementing a sequential
equilibrium) and improve the results of Gerardi [2004], who showed that, in
the case that $k=1$, a sequential equilibrium can be implemented in
synchronous systems if $n\geq 5$.
## 1 Introduction
A standard technique for dealing with complicated problems in distributed
computing, cryptography, and game theory is to show that the problem can be
solved under the assumption that there is a _trusted mediator_ , a third party
to whom all agents in the system can communicate, and then showing that we can
_implement_ the mediator; that is, the agents just communicating among
themselves (using what economists call _cheap talk_) can simulate what the
mediator does. In the distributed computing and cryptography literature, the
focus has been on _secure multiparty computation_ Goldreich et al. [1987]; Yao
[1982]. Given $n$ agents in which each agent $i$ has input $x_{i}$, the goal
is to compute some function $f(x_{1},\ldots,x_{n})$ without revealing anything
about the agents’ inputs besides what can be deduced from the output of the
function itself. With a mediator, this is trivial: each agent $i$ just tells
the mediator $x_{i}$, and the mediator tells the agents
$f(x_{1},\ldots,x_{n})$. It is known that, as long as no more than $k$ agents
are malicious, this mediator can be implemented if $n>3k$.
Game theory adds incentives to the mix. While in the work done in
cryptography, it is simply assumed that malicious agents will do what they can
to bring down the system, in the game-theoretic setting, it is assumed that
agents will deviate from a strategy only if it is in their best interests to
do so. To make this precise, in the game theory literature, three games were
considered: an _underlying_ normal-form game $\Gamma$, a game $\Gamma_{d}$
with a mediator, where, after communicating with the mediator, players make a
move in $\Gamma$ and get the same payoffs that they do in $\Gamma$, and a
_cheap-talk game_ $\Gamma_{\mathit{CT}}$, where players just talk to each
other and then make a move in the underlying game. Forges [1990] and Barany
[1992] showed that a Nash equilibrium (NE) in the mediator game $\Gamma_{d}$
could be _implemented_ by a NE in the cheap-talk game if $n\geq 4$ in the
sense that the distribution over action profiles in the NE in $\Gamma_{d}$ is
the same as that in the NE in $\Gamma_{\mathit{CT}}$.
As is well known, Nash equilibrium considers only deviations on the
_equilibrium path_ (situations that arise with positive probability if the NE
is played). In extensive-form games such as $\Gamma_{\mathit{CT}}$, Nash
equilibrium does not always describe what intuitively would be reasonable
play. For example if $A$ tells $B$ to give her $1 or she will destroy the
world, $B$ giving $1 to $A$ and $A$ not destroying the world is a NE. However,
this equilibrium is based on a non-credible threat. Would $A$ really destroy
the world if she doesn’t get $1? For this reason, game theorists have
considered solution concepts such as _sequential equilibrium_ Kreps and Wilson
[1982], where players cannot increase their utility by deviating even off the
equilibrium path (see Section 13 for a formal definition).
Not surprisingly, the question of whether a sequential equilibrium with a
mediator can be implemented has been considered before in the game theory
literature. Ben-Porath [2003] claimed that a sequential equilibrium could be
implemented in $\Gamma_{\mathit{CT}}$ if $n>3$ provided that there is a
_punishment strategy_ (a way for players to punish players who are caught
cheating—see Section 3 for a formal definition);111Unfortunately, there is a
serious error in Ben-Porath’s proof; see Abraham et al. [2008b]; Ben-Porath
[2021]. Gerardi [2004] showed that, in the case that $k=1$, a sequential
equilibrium can be implemented if $n\geq 5$.
Nash and sequential equilibria incentivize individual participants to follow
the given strategy since otherwise they would get a lower expected payoff.
However, if players can form coalitions and deviate in a coordinated way, then
a coalition of players may have incentive to deviate in a Nash or sequential
equilibrium. For instance, consider a normal-form game for $n$ players in
which players can play either $0$ or $1$. If at least $n-1$ players play $0$,
they all get $0$ utility, but if two or more players play $1$, these players
get a utility of $1$ and everyone else gets a utility of $-1$. Clearly,
playing $0$ is a Nash equilibrium of this game. However, if two players can
deviate together, they can increase their payoff by playing $1$ instead of
$0$. Designing mechanisms that incentivize not only individuals, but also
coalitions to act as intended has become increasingly important, especially in
Internet applications (e.g., blockchain), where the same person can maintain
several identities. Not surprisingly, coalitions have received significant
attention recently in the literature (see, e.g., Eyal and Sirer [2014]; Heller
[2010]). Abraham, Dolev, Gonen and Halpern [2006] (ADGH from now on)
introduced the notion of _$k$ -resilient Nash equilibrium_, which is a
strategy profile in which no coalition of up to $k$ players can increase their
payoff by deviating. They also generalized Forges and Barany’s results by
showing that any $k$-resilient Nash equilibrium in $\Gamma_{d}$ can be
implemented in $\Gamma_{\mathit{CT}}$ if the number of players $n$ satisfies
$n>3k$, and proved that the $n>3k$ bound is tight Abraham et al. [2008b]. More
precisely, if $n\leq 3k$, then there exists a game $\Gamma_{d}$ for $n$
players and a mediator and a $k$-resilient Nash equilibrium $\vec{\sigma}$ in
$\Gamma_{d}$ that can’t be implemented in $\Gamma_{\mathit{CT}}$.
Two natural questions follow from these results. First, whether we can
generalize Gerardi’s result and give necessary and sufficient conditions on
$n$ and $k$ to implement a $k$-resilient sequential equilibrium (which is
defined analogously to $k$-resilient Nash equilibrium), and second, whether
the $n\geq 5$ upper bound given by Gerardi is actually tight. It is easy to
check that the $n>3k$ lower bound given in Abraham et al. [2008b] for Nash
equilibrium applies without change to $k$-resilient sequential equilibria. In
particular, if $k=1$, it shows that at least $4$ players are required to
implement a sequential equilibrium, but it was not known whether $n\geq 5$ is
necessary. Here, we show that we can always implement a $k$-resilient
sequential equilibrium in a game with a mediator and $n$ players if $n>3k$,
matching the lower bound given in Abraham et al. [2008b] and improving
Gerardi’s result for $k=1$.
Following Gerardi [2004] and Gerardi and Myerson [2007], we also relate our
results to two other solution concepts in normal-form games: _correlated
equilibrium_ Aumann [1987] and _communication equilibrium_ Forges [1986];
Myerson [1986]. Both of these concepts can be understood in terms of games
with mediators. A correlated equilibrium is an equilibrium in a game with a
mediator where the players do not talk to the mediator; the mediator simply
tells the agent which strategy to play. A _communication equilibrium_ in a
Bayesian game is an equilibrium in a game with a mediator where the players
can tell the mediator their types, and the mediator then tells the players
what strategy to play (see Appendix A for formal definitions). Our results
give a characterization of the set $SE_{k}(\Gamma_{\mathit{CT}})$ of outcomes
that can be implemented with $k$-resilient sequential equilibria in the cheap-
talk game $\Gamma_{\mathit{CT}}$; if $\Gamma$ is a normal-form game, then
$SE_{k}(\Gamma_{\mathit{CT}})$ is the set of $k$-resilient correlated
equilibria of $\Gamma$, while if $\Gamma$ is a Bayesian game,
$SE_{k}(\Gamma_{\mathit{CT}})$ is the set of $k$-resilient communication
equilibria of $\Gamma$.
Sequential equilibrium involves describing not only what the players do at
each point in an extensive-form game, but describing what their beliefs are,
even off the equilibrium path. It must be shown that players are always best
responding to their beliefs. The key difficulty in our proof involves finding
appropriate beliefs that are consistent with the equilibrium strategy. The
proof given by Gerardi involves an extensive case analysis for both
consistency and sequential rationality, which makes it hard to generalize to
other settings. Our proof introduces an interesting new technique. We show
that all strategies in $\Gamma_{\mathit{CT}}$ (even those that are not an
equilibrium) admit a consistent _$k$ -paranoid belief system_, where all
coalitions $K$ of at most $k$ players always believe that, if other players
deviated, they did so by sending inappropriate messages only to players in
$K$. That is, all coalitions of size at most $k$ believe that the remaining
players are being truthful among themselves. We show that, given a
$k$-resilient Nash equilibrium $\vec{\sigma}$, we can extend it to a
$k$-resilient sequential equilibrium by constructing a $k$-paranoid belief
system. The idea is that, given these $k$-paranoid beliefs, players in a
coalition $K$ will not believe that there is anything that they can do to
prevent the remaining players from playing their part of the equilibrium. We
then apply these results to the $k$-resilient Nash equilibrium of Abraham et
al. [2006] to get our result.
All of the results presented so far assume a _synchronous_ setting:
communication proceeds in atomic rounds, and all messages sent during round
$r$ are received by round $r+1$. In an _asynchronous_ setting, there are no
rounds and messages sent by the players may take arbitrarily long to get to
their recipients. Asynchrony is a standard assumption in the distributed
computing and cryptography literature, precisely because many systems that
practitioners care about, such as markets or the internet, are asynchronous in
practice. Considering asynchronous systems can have significant implications
for how players will play a game. For instance, in an online second-price
auction, the seller can benefit from inserting fake transactions whose value
is between that of the highest and second-highest bid immediately after a new
highest bid is received, thus increasing the second-highest price at no cost
Roughgarden [2020]. This type of attack can be carried out only in
asynchronous or _partially synchronous_ systems (where there is an upper bound
on how long messages take to arrive), since in a synchronous system, all bids
are received at the same time. (In a synchronous system, the seller would have
to guess what the players will bid in order to benefit from a fake
transaction.)
This example shows that asynchrony and partial synchrony allow players to
influence the communication pattern in ways that they can’t in a synchronous
setting. Other examples include the fact that, in a fully asynchronous system,
a player that didn’t send a message is indistinguishable from a player whose
message is delayed (which means that players can safely deviate by not sending
messages for a while, which they cannot do in a synchronous system), and that,
in a partially synchronous system, players may read other players’ messages
before sending their own and adapt their strategy accordingly. As a result,
the number of honest players required to implement a given functionality is
greater in an asynchronous setting. For instance, the upper bound for
asynchronous secure computation is $n>4k$ Ben-Or et al. [1993], as opposed to
$n>3k$ bound in synchronous systems proved by Ben-Or, Goldwasser, and
Wignerson [1988].
The simplicity of our approach allows us to generalize our main result to the
asynchronous setting with very little work. Given a $k$-resilient sequential
equilibria $\vec{\sigma}$ in an asynchronous game with $n$ players and a
mediator, we extend implementation given by Abraham et al. [2019b] of
asynchronous $k$-resilient Nash equilibria in $\Gamma_{\mathit{CT}}$ for
$n>4k$ by constructing a consistent $k$-paranoid belief system. Reasoning
analogous to that used in the synchronous case shows that the resulting
strategy and belief system form a $k$-resilient sequential equilibria. In the
asynchronous setting, this result is also optimal, since it matches the $n>4k$
lower bound given by Geffner and Halpern [2023]. We believe that the
techniques presented in this paper can be applied to design $k$-resilient
sequential equilibria in many other scenarios as well.
The rest of the paper is organized as follows. In Section 2, we provide the
concepts and definitions required to understand the main results and its
proofs (some of the most basic definitions can be found in Appendix A). In
Section 3, we state the main result and give a short outline of its proof. The
details are given in Section 4. In Section 5, we provide our characterization
of $SE_{k}(\Gamma_{CT})$. In Section 6, we carefully define asynchronous
systems and extend the results of Section 6 to the asynchronous case; these
results are further extended in Section 7 assuming that players can send
arbitrarily precise real numbers in their messages (rather than just rational
numbers, which is what we assume up to this point).
## 2 $k$-resilient equilibrium
In this section, we extend the different types of equilibria that appear in
the literature (which are discussed in Appendix A) to account for deviations
of coalitions of at most $k$ players. For future reference, in normal-form
games, we use $n$ to denote the number of players, $P$ to denote the set of
players, $A$ to denote the set of possible action profiles, and $U$ to denote
the tuple of utility functions. In Bayesian games, we use $T$ to denote the
possible type profiles and $q$ to denote the prior distribution over $T$.
Moreover, in extensive-form games, we use $G$ to denote the game tree, $M$ to
denote the function that outputs which player moves at each node, and $R$ to
denote the tuple of equivalence relations between the nodes in which two nodes
$v$ and $w$ are equivalent according to the $i$th relation if $i$ cannot
distinguish these between $v$ and $w$ (i.e., if $v$ and $w$ are in the same
partition). The full definitions can be found in Appendix A.
Intuitively, traditional notions of equilibria guarantee that no individual
player can increase her own payoff by deviating from the proposed strategy.
For each of these equilibria we consider also its $k$-resilient variant, which
states that no coalition of at most $k$ players can jointly increase their
payoff by deviating, even if they do so in a coordinated way. We also consider
the notion of _strong $k$-resilience_, which guarantees that no individual
player inside the coalition can increase its own payoff, even at the expense
of other coalition members (as opposed to the original notion, in which no
member of the coalition can be worse than by following the proposed strategy).
### 2.1 $k$-resilient Nash, sequential, Bayesian Nash, and communication
equilibrium
We begin by extending the definition of Nash equilibrium (Definition 11) to
$k$-resilient Nash equilibrium:
###### Definition 1.
In a normal-form game $\Gamma$, a (mixed) strategy profile
$\vec{\sigma}=(\sigma_{1},\ldots,\sigma_{n})$ is a _$k$ -resilient Nash
equilibrium_ (resp., _strongly $k$-resilient Nash equilibrium_) if, for all
coalitions $K$ such that $|K|\leq k$, and all strategies
$\vec{\sigma}^{\prime}_{K}\in\Delta(A_{K})$
$u_{i}(\vec{\sigma}_{K},\vec{\sigma}_{-K})\geq
u_{i}(\vec{\sigma}^{\prime}_{K},\vec{\sigma}_{-K})$
for some (resp., for all) $i\in K$.
We can similarly extend the notion of correlated equilibrium (see Definition
12 of Appendix A) to $k$-resilient correlated equilibrium:
###### Definition 2.
Given a normal-form game $\Gamma=(n,A,U)$, a distribution $p\in\Delta(A)$ is a
_$k$ -resilient correlated equilibrium_ (resp., _strongly $k$-resilient
correlated equilibrium_) if, for all subsets $K\subseteq P$ such that $|K|\leq
k$, and all action profiles $\vec{a}^{\prime}_{K},\vec{a}^{\prime\prime}_{K}$
for players in $K$ such that $p(\vec{a}^{\prime}_{K})>0$,
$\sum_{\vec{a}\in\Delta(A):\vec{a}_{K}=\vec{a}^{\prime}_{K}}u_{i}(\vec{a}^{\prime}_{K},\vec{a}_{-K})p(\vec{a}\mid\vec{a}_{K}=\vec{a}^{\prime}_{K})\geq\sum_{\vec{a}\in\Delta(A):\vec{a}_{K}=\vec{a}^{\prime}_{K}}u_{i}(a^{\prime\prime}_{K},a_{-K})p(\vec{a}\mid\vec{a}_{K}=\vec{a}^{\prime}_{K})$
for some (resp., for all) $i\in K$.
Simply put, a distribution $p\in\Delta(A)$ is a $k$-resilient correlated
equilibrium if no subset of at most $k$ players would be better off by
deviating if they knew that the action profile used was sampled according to
$p$ (and they knew their components of the profile), even if they could
coordinate their deviations.
The definition of $k$-resilient Nash equilibrium in extended-form games is
equivalent. However, the extension of sequential equilibrium (Definition 13)
to $k$-resilient sequential equilibrium requires additional definitions
Abraham et al. [2019a].
###### Definition 3.
Let $\Gamma=(P,G,M,U,R)$ be an extensive-form game and $K\subseteq P$ be a
subset of players. We define the equivalence relation $\sim_{K}$ by
$v\sim_{K}v^{\prime}$ iff $v\sim_{i}v^{\prime}$ for all $i\in K$.222The
$\sim_{i}$ relation defines player $i$’as information sets in an extensive-
form game. Se Appendix A for the formal definition. The equivalence classes of
$\sim_{K}$ are called _$K$ -information sets_.
Intuitively, two nodes are related according to $\sim_{K}$ if they are
indistinguishable for all players $i\in K$. With this, we can extend the
notion of belief systems to account for coalitions of players that share
information between themselves.
###### Definition 4.
A _$k$ -belief system_ $b$ in a game $\Gamma$ is a function that maps each
$K$-information set $I$ such that $K\subseteq P$ and $|K|\leq k$ to a
distribution over the nodes in $I$.
We can define what it means for a $k$-belief system to be consistent with
$\vec{\sigma}$ just as in the case of standard belief systems. We say that a
$k$-belief system $b$ is consistent with strategy $\vec{\sigma}$ if there
exists a sequence $\vec{\sigma}^{1},\vec{\sigma}^{2},\ldots$ of completely-
mixed strategies that converges to $\vec{\sigma}$ such that the beliefs
induced by Bayes’ rule converge to $b$. With these definitions in hand, we can
find the desired generalization of sequential equilibrium.
###### Definition 5.
A pair $(\vec{\sigma},b)$ consisting of a strategy profile $\vec{\sigma}$ and
a $k$-belief system $b$ consistent with $\vec{\sigma}$ is a _$k$ -resilient
sequential equilibrium_ (resp., _strongly $k$-resilient sequential
equilibrium_) if, for all $K\subseteq P$ with $|K|\leq k$, all $K$-information
sets $I$, and all strategies $\tau_{K}$ for players in $K$ (w.r.t.
$\sim_{K}$),
$u_{i}(\vec{\sigma},I,b)\geq u_{i}((\tau_{K},\vec{\sigma}_{-K}),I,b)$
for some (resp., for all) $i\in K$.
Note that the notions of $1$-resilient correlated equilibrium and
$1$-resilient sequential equilibrium are equivalent to the standard notions of
correlated equilibrium and sequential equilibrium, respectively. Note that, in
games with a mediator $d$, since the mediator never deviates from its
strategy, a belief system $b$ is consistent with strategy
$\vec{\sigma}+\sigma_{d}$ (where $\vec{\sigma}+\sigma_{d}$ means that players
play strategy profile $\vec{\sigma}$ and the mediator plays strategy
$\sigma_{d}$) if there exists a sequence
$\vec{\sigma}^{1}+\sigma_{d},\vec{\sigma}^{2}+\sigma_{d},\ldots$ of
completely-mixed strategies that converges to $\vec{\sigma}+\sigma_{d}$ such
that the beliefs induced by Bayes’ rule converge to $b$. The definition of
$k$-resilient sequential equilibrium in mediator games is identical to the one
given in Definition 5, using this definition of consistent beliefs.
Intuitively, this means that the players believe that the mediator never
deviates from its protocol.
In a Bayesian game with type space $T$ (see Appendix A.2 for the formal
definitions), a _correlated strategy profile_ is a map
$\mu:T\rightarrow\Delta(A)$. Note that all distributions over strategy
profiles can be viewed as correlated strategy profiles, but the converse is
not true. For instance, in a game with two players in which
$A_{i}=T_{i}=\\{0,1\\}$ for all $i$, a correlated strategy profile may consist
of both players playing action $t_{1}+t_{2}\bmod 2$; this cannot be
represented by a strategy profile, since players must independently choose
their action given their type.
The expected utility of player $i$ in a coalition $K$ when playing (a possibly
correlated) strategy profile $\vec{\mu}$, where $T_{K}$ is the set of possible
type profiles of the players in $K$, is
$u_{i}^{K}(\vec{\mu})=\sum_{\vec{t}_{K}\in
T_{K}}q(\vec{t}_{K})\sum_{\vec{t}\in
T}q(\vec{t}\mid\vec{t}_{K})u_{i}(\vec{\mu}(\vec{t})),$
where $u_{i}(\vec{\mu}(\vec{t}))$ denotes the expected utility of player $i$
when an action profile is chosen according to $\mu(\vec{t})$. Intuitively, we
are assuming that players in $K$ can share their types, which is why we
condition on $\vec{t}_{K}$. With this, we can extend Bayesian Nash equilibrium
(see Definition 14 in Appendix A) to $k$-resilient Bayesian Nash equilibrium:
###### Definition 6.
In a Bayesian game $\Gamma=(P,T,q,A,U)$, a strategy profile
$\vec{\mu}:=(\mu_{1},\ldots,\mu_{n})$ is a _$k$ -resilient Bayesian Nash
equilibrium_ (resp., _strongly $k$-resilient Bayeisan Nash equilibrium_) if,
for all coalitions $K$ with $|K|\leq k$ and all maps
$\mu^{\prime}_{K}:T_{K}\rightarrow\Delta(A_{K})$,
$u_{i}^{K}(\vec{\mu})\geq u_{i}^{K}(\vec{\mu}_{-K},\mu^{\prime}_{K})$
for some (resp., for all) $i\in K$.
We can similarly generalize communication equilibrium (see Definition 15 of
Appendix A)) to $k$-resilient communication equilibrium:
###### Definition 7.
A correlated strategy profile $\mu:T\rightarrow\Delta(A)$ is a _$k$ -resilient
communication equilibrium_ (resp., _strongly $k$-resilient communication
equilibrium_ of Bayesian Game $\Gamma=(P,T,q,A,U)$ if, for all $K\subseteq P$
such that $|K|\leq k$, all $\vec{t}_{K}\in T_{K}$, all $\psi:T_{K}\rightarrow
T_{K}$, and all $\varphi:A_{K}\rightarrow A_{K}$, we have that
$\sum_{\vec{t}_{-K}\in T_{-K}}\sum_{\vec{a}\in
A}q(t_{-K},\vec{t}_{K})\mu(\vec{a}\mid\vec{t}_{-K},\vec{t}_{K})u_{i}(\vec{t}_{-K},\vec{t}_{K},\vec{a})\geq$
$\sum_{\vec{t}_{-K}\in T_{-K}}\sum_{\vec{a}\in
A}q(t_{-K},\vec{t}_{K})\mu(\vec{a}\mid\vec{t}_{-K},\psi(\vec{t}_{K}))u_{i}(\vec{t}_{-K},\vec{t}_{K},\vec{a}_{-K},\varphi(\vec{a}_{K}))$
for some (resp., for all) $i\in K$.
As in the case of the standard communication equilibrium, we can identify a
correlated strategy profile $\mu$ with a canonical game with a mediator in
which each player $i$ must send its type $t_{i}$ to the mediator, then the
mediator samples an action profile $\vec{a}$ from $\mu(\vec{t})$ and sends
$a_{i}$ to each player $i$, and then each player $i$ plays whatever it
received from the mediator. In Definition 7, the functions $\psi$ and
$\varphi$ represent possible deviations for players in $K$ in this canonical
game; $\psi$ describes how they might lie to the mediator about their type (by
sending type profile $\psi(\vec{t}_{K})$ instead of their actual type profile
$\vec{t}_{K}$), while $\varphi$ represents how they might deviate in what they
do (playing strategy profile $\varphi(\vec{a}_{K})$ instead of the strategy
profile $\vec{a}_{K}$ suggested by the mediator). Definition 7 says that $\mu$
is a $k$-resilient (resp., strongly $k$-resilient) communication equilibrium
if none of these deviations is profitable for all players (resp., for some
player) in $K$.
### 2.2 Cheap-talk games
Given a normal-form game $\Gamma=(n,A,U)$ or a Bayesian game
$\Gamma=(P,T,q,A,U)$, consider an extension $\Gamma_{\mathit{CT}}$ in which
players can communicate among themselves before playing an action in $\Gamma$.
We call this extension a _cheap-talk game_. More precisely, in a cheap-talk
game $\Gamma_{\mathit{CT}}$, players are able to exchange messages and play an
action in $\Gamma$ (although they may play an action in $\Gamma$ at most
once). The payoff of each player is defined by the utilities $U$ of $\Gamma$,
given the action profile played in $\Gamma$. In this paper we always assume
that messages are _authenticated_ , so that each recipient knows who sent the
message, and messages are never corrupted or modified.
We can view cheap-talk games as possibly infinite extensive-form games in
which the nodes are the _global histories_ $\vec{h}=(h_{1},\ldots,h_{n})$ of
the players, and the actions associated with an edge involve either performing
an internal computation (e.g., tossing a random coin or updating the local
state with the messages received), sending messages, or playing an action in
the underlying game. A global history $\vec{h}$ is just a tuple of local
histories, where the local history $h_{i}$ of player $i$ contains all internal
computations, all messages sent by $i$ and received by $i$, along with their
time stamps (in the synchronous case) or the number of times that $i$ was
scheduled since the beginning of the game (in the asynchronous case). Two
global histories $\vec{h}=(h_{1},\ldots,h_{n})$ and
$\vec{h}^{\prime}=(h_{1}^{\prime},\ldots,h_{n}^{\prime})$ are
indistinguishable by player $i$ if $h_{i}=h^{\prime}_{i}$. Analogously,
$\vec{h}$ and $\vec{h}^{\prime}$ are indistinguishable by a coalition $K$ of
players if $\vec{h}_{K}=\vec{h}^{\prime}_{K}$.
A strategy profile $\vec{\sigma}$ in a cheap-talk game $\Gamma_{\mathit{CT}}$
_induces_ a strategy profile $\vec{\tau}$ in $\Gamma$ if the outcome that
results from playing $\vec{\sigma}$ in $\Gamma_{\mathit{CT}}$ is identical to
that that results from playing $\vec{\tau}$ in $\Gamma$. If $\Gamma$ is a
normal-form game, the outcome that results from playing $\vec{\sigma}$ is the
distribution over action profiles in $\Gamma$ that results when $\vec{\sigma}$
is played in $\Gamma_{\mathit{CT}}$. If $\Gamma$ is a Bayesian game, the
outcome is a map from type profiles to distributions over action profiles
(i.e., a correlated strategy profile). A strategy profile $\vec{\sigma}$
_implements_ a strategy profile $\vec{\sigma}^{\prime}$ if $\vec{\sigma}$ and
$\vec{\sigma}^{\prime}$ induce the same strategy profiles in $\Gamma$.
Unless specified otherwise, we assume that all communication in cheap-talk
games is _synchronous_ , which means that it proceeds in rounds and that all
messages sent in a given round by each player are guaranteed to be received by
their recipient at the beginning of the next round. The analysis of
_asynchronous_ communication is done in section 6.
These definitions are illustrated by the following example.
###### Example 1.
Consider a Bayesian game $\Gamma$ with two players. Each player
$i\in\\{1,2\\}$ has a type $t_{i}\in\\{0,1\\}$ and can play actions in
$\\{0,1\\}$. If the action $a_{i}$ played by player $i$ satisfies
$a_{i}=t_{1}t_{2}$, then that player gets a utility of 1; otherwise, she gets
0\. Consider a strategy profile $\vec{\sigma}$ for the cheap-talk extension
$\Gamma_{\mathit{CT}}$ of $\Gamma$ in which each player sends its type to the
other player in the first round, then the players each locally compute
$t_{1}t_{2}$ and play it in the underlying game. It is easy to check that
$\vec{\sigma}$ is a Bayesian Nash equilibrium (both players get the maximum
possible payoff), and that it implements the correlated strategy profile that
maps a given type profile $(t_{1},t_{2})$ to the distribution that assigns
probability $1$ to $(t_{1}t_{2},t_{1}t_{2})$ and $0$ to everything else.
## 3 The main result
Given a normal-form or Bayesian game $\Gamma$, let $\Gamma_{\mathit{CT}}$ be
its cheap-talk extension and $\Gamma_{d}$ be its cheap-talk extension, where
players can also communicate with a trusted mediator $d$. Let
$SE_{k}(\Gamma_{\mathit{CT}})$ (resp., $SE_{k}(\Gamma_{d})$) denote the set of
possible strategies in $\Gamma$ induced by $k$-resilient sequential equilibria
in $\Gamma_{\mathit{CT}}$ (resp., $\Gamma_{d})$. Note that if $\Gamma$ is a
normal-form game, $SE_{k}(\Gamma_{\mathit{CT}})$ is a set of distributions
over action profiles, while if $\Gamma$ is a Bayesian game,
$SE_{k}(\Gamma_{\mathit{CT}})$ is a set of correlated strategy profiles. We
define $SE_{k}^{S}$ equivalently but for strong $k$-resilient sequential
equilibria instead.
In the rest of this paper, we consider only equilibria in which each action
profile is sampled with rational probability. The reason for this restriction
is that it is a standard assumption in distributed systems that messages have
finite length and that players cannot perform arbitrarily large operations.
This means that agents cannot encode real numbers into messages and they
cannot perform operations on real numbers, in general. If we relax these
constraints and allow players to send messages and operate with arbitrary real
numbers, our results can be extended to all equilibria inside the convex hull
of rational equilibria, using techniques due to Gerardi [2004]. These
techniques are described in Section 7.
Our main result shows that we can implement a (strongly) $k$-resilient
equilibrium with a mediator and $n$ players using cheap talk if $n>3k$.
###### Theorem 1.
If $\Gamma=(P,T,q,A,U)$ is a Bayesian game for $n$ players and $n>3k$, then
$SE_{k}(\Gamma_{\mathit{CT}})=SE_{k}(\Gamma_{d})$ and
$SE_{k}^{S}(\Gamma_{\mathit{CT}})=SE_{k}^{S}(\Gamma_{d})$.
As shown in Section 5, the sets $SE_{k}(\Gamma_{d})$ and
$SE_{k}^{S}(\Gamma_{d})$ have a relatively simple characterization. For
instance, it can be shown that if $\Gamma$ is a normal-form game, then
$SE_{k}(\Gamma_{d})$ is the set of $k$-resilient correlated equilibria of
$\Gamma$; and if $\Gamma$ is a Bayesian game, then $SE_{k}(\Gamma_{d})$ is the
set of $k$-resilient communication equilibria of $\Gamma$. (Gerardi [2004]
proved these results for $k=1$.)
### 3.1 Outline of the proof
The proof of Theorem 1 is divided into two parts. We first describe a
relatively simple strategy profile $\vec{\sigma}$ in $\Gamma_{\mathit{CT}}$
that implements the desired strategy profile in $\Gamma_{d}$ and tolerates
deviations of up to $k$ players. Moreover, with $\vec{\sigma}$, no subset of
at most $k$ players can sabotage the joint computation or learn anything other
than what they were originally supposed to learn. These security properties of
$\vec{\sigma}$ guarantee that it is a $k$-resilient Nash equilibrium.
The next step in the proof is extending $\vec{\sigma}$ to a $k$-resilient
sequential equilibrium $\vec{\sigma}^{seq}$. The key idea for doing this is
showing that there exists a belief system $b$ such that all coalitions $K$ of
at most $k$ players believe that all players that deviated from the protocol
did so by sending inappropriate messages only to players in $K$; that is, each
coalition $K$ of size at most $k$ believes that if $i,j\not\in K$, then $i$
and $j$ always send each other the messages that they are supposed to send
according to $\vec{\sigma}$. If players have belief system $b$, then all
coalitions $K$ with $|K|\leq k$ believe, regardless of their local history,
that the remaining players will always compute their part of the sequential
equilibrium correctly, since $\vec{\sigma}$ tolerates deviations by up to $k$
players and all the players not in $K$ get the messages that they are supposed
to get according to $\vec{\sigma}$ from all players except possibly those in
$K$ (and thus do not observe deviations by more than $k$ players). Moreover,
in the construction of $\vec{\sigma}$, if fewer than $k$ players deviate, all
players eventually receive enough information to compute their own action,
even if they don’t take part in the computation. These two properties are
enough to guarantee that players in $K$ cannot gain by deviating during the
communication phase, since (they believe that) nothing they can do will affect
what the remaining players play in the underlying game, and they can’t learn
any additional information.
Constructing a strategy profile that implements a given sequential equilibrium
with a mediator is in general nontrivial, since players must not only jointly
compute the action that the mediator sends to each player, but this must be
done in such a way that $i$ does not learn anything from the communication
phase other than its own action $a_{i}$. For if $i$ was able to get such
information, $i$ might be tempted to deviate (since it might have more
information about what the utility of deviating would be).
There are well-known techniques in distributed computing for constructing
protocols with the appropriate security properties. The standard solution
(which we also use) involves distributing the information about the necessary
computations between the players in such a way that each player knows only a
part or _share_ of that information. Players can thus jointly compute the
mediator’s local history and simulate everything that the mediator would do,
without leaking any information. More precisely, if a player $i$ would send
its type $t_{i}$ to the mediator in the mediator game, in the cheap-talk game
it instead shares $t_{i}$ among all players in such a way that no player (or
coalition of players) can learn anything about it. When all the types have
been distributed, players can then compute the actions that the mediator would
have sent to each player in the game with the mediator. However, each player
$i$ does not directly compute its own action $a_{i}$, but rather its share of
every action $a_{j}$. Then each player $i$ sends the share of each action
$a_{j}$ to player $j$. This guarantees that only $j$ learns $a_{j}$.
There are two standard primitives that we use in this construction,
_verifiable secret sharing_ (VSS) and _circuit computation_ (CC). Roughly
speaking, VSS takes as input a value $v$, and distributes to each player $i$ a
share $s^{i}_{v}$ of $v$. If we are interested in $k$-resilience, we can
arrange for these shares to have the property that knowing a subset of at most
$k$ shares gives no information about $v$, while knowing strictly more than
$k$ suffices to reconstruct $v$. CC is used to distribute the shares of the
output of a function for which each of the players has shares of all of its
inputs. More precisely, given a function $f:D^{t}\rightarrow D$ and $t$ values
$v_{1},\ldots,v_{t}$ such that each player $i$ knows only its own share
$s_{v_{j}}^{i}$ of $v_{j}$, for $j=1,\ldots,t$, the CC procedure takes as
inputs the shares $s_{v_{1}}^{i},\ldots,s_{v_{t}}^{i}$ of each player $i$, and
allows player $i$ to compute its share $s^{i}$ of $f(v_{1},\ldots,v_{t})$ in
such a way that no player learns any information about $v_{1},\ldots,v_{t}$.
Therefore, VSS allows players to share private values in a secret way, and CC
allows players to generate (shares of) new values without leaking any
information about all values previously shared. We define VSS and CC formally
in Appendix B, and show that, if $n>3k$, then it is possible to implement VSS
and CC in a $k$-resilient way, so that they both tolerate deviations by up to
$k$ players. In the next section, we prove Theorem 1 assuming that we can
implement VSS and CC.
## 4 Proof of Theorem 1
In this section, we prove Theorem 1 for the case of $k$-resilient sequential
equilibrium. The case of strongly $k$-resilient sequential equilibrium is
similar.
### 4.1 Constructing a $k$-resilient Nash equilibrium
By definition, $SE_{k}(\Gamma_{\mathit{CT}})\subseteq SE_{k}(\Gamma_{d})$. It
thus remains to show that a $k$-resilient sequential equilibrium with a
mediator can be implemented by a $k$-resilient sequential equilibrium in the
cheap-talk game. Let $Com_{k}(\Gamma)$ be the set of $k$-resilient
communication equilibria of $\Gamma$. As we noted in Section 3,
$SE_{k}(\Gamma_{d})=Com_{k}(\Gamma)$, and thus
$SE_{k}(\Gamma_{\mathit{CT}})\supseteq SE_{k}(\Gamma_{d})$ reduces to showing
that all $k$-resilient communication equilibria $\mu$ of $\Gamma$ can be
implemented with cheap talk without a mediator. We begin by constructing a
strategy profile $\vec{\sigma}$ in $\Gamma_{\mathit{CT}}$ that is a
$k$-resilient Nash equilibrium and induces the same correlated strategy
profile as $\mu$.
Given $\mu$, consider the canonical protocol $\vec{\sigma}_{can}$ for $n$
players and a mediator in which each player $i$ sends its type $t_{i}$ to the
mediator, and the mediator then computes the type profile $\vec{t}$ of the
players using the messages received, replacing the types of players that
didn’t send theirs by a default type $\bot$. (Since we are working in
synchronous systems for this part of the proof, the mediator can tell if a
player did not send his type.) The mediator then samples $\vec{a}\in\Delta(A)$
according to the distribution $\mu(\vec{t})$ and sends $a_{i}$ to each player
$i$. If $i$ sent its true type at the beginning of the game and receives an
action $a_{i}$ from the mediator, it plays $a_{i}$. Otherwise, it plays the
best response assuming that the remaining players are playing their part of
$\mu(\vec{t})$ (note that the best response is well-defined since both $\mu$
and the distribution $q$ of type profiles are common knowledge). It is
straightforward to check that $\vec{\sigma}_{can}$ induces the same correlated
strategy profile as $\mu$, and that $\vec{\sigma}_{can}$ is a $k$-resilient
sequential equilibrium. The details are left to the reader.
The idea for constructing $\vec{\sigma}$ is that instead of having a mediator
that receives the types of the players and then computes $\mu$, the players
perform these tasks by themselves using VSS and CC to help them simulate the
mediator: each player $i$, instead of sending its type $t_{i}$ to the
mediator, shares it among all the players using a $k$-resilient VSS
implementation. After the number of rounds needed to perform VSS, players
compute $(a_{1},\ldots,a_{n}):=\mu(t_{1},\ldots,t_{n})$ with CC, using as
input the shares of each of the types $t_{i}$. Each player $i$ then sends to
each player $j$ $i$’s share of $a_{j}$. Thus, each player $i$ can compute
$a_{i}$ without learning anything about the other players’ actions.
Note that, since $\vec{a}$ is sampled from $\mu(\vec{t})$, players are
required to randomize in a coordinated way. The problem of jointly sampling an
element from a distribution with rational probabilities over a finite set can
be reduced to the problem of sampling an integer from $[N]:=\\{1,\ldots,N\\}$
for some $N$, as the following proposition shows.
###### Proposition 1.
Given a correlated strategy profile $\mu$ with rational parameters, there
exists an integer $N$ and a function $\mu^{*}:T\times[N]\rightarrow A$ such
that, if $X_{\vec{t}}$ is the distribution over action profiles obtained from
playing $\mu^{*}(\vec{t},r)$ when $r$ is uniformly sampled from $[N]$, then
$X_{\vec{t}}=\mu(\vec{t})$ for all $\vec{t}\in T$.
###### Proof.
Fix an integer $N$ such that, for all $\vec{t}$ and all action profiles
$\vec{a}$, the probability $\mu(\vec{t})(\vec{a})$ is an integer multiple of
$1/N$. For each $\vec{t}$, partition $N$ into subsets
$C_{1}^{\vec{t}},C_{2}^{\vec{t}},\ldots,C_{m}^{\vec{t}}$ such that
$|C_{i}^{\vec{t}}|$ is proportional to the probability of
$\mu(\vec{t})(\vec{a}^{i})$ for some indexing $\vec{a}^{1},\ldots,\vec{a}^{m}$
of the action profiles. We take $\mu^{*}(\vec{t},r)$ to be $\vec{a}^{i}$ if
$r\in C_{i}^{\vec{t}}$ for some $i$. It is straightforward to check that
$\mu^{*}$ satisfies the desired properties. ∎
Players can thus compute the desired action profile using a deterministic
function $\mu^{*}$, given that they are able to jointly sample an integer
$r\in N$ uniformly at random.
We next present a formal construction of the protocol $\vec{\sigma}$ described
above. This protocol is an adaptation of the multiparty computation protocol
of Ben-Or, Goldwasser, and Wigderson [1988] to a setting in which the output
of the function must be sampled from a common distribution.
1. Phase 1:
Each player $i$ generates a random number $r_{i}$ between $1$ and $N$
uniformly at random. Then $i$ shares $r_{i}$ and its type $t_{i}$ using a
$k$-resilient VSS procedure. If a player $i$ does not share $t_{i}$ or
$r_{i}$, then $t_{i}$ and $r_{i}$ are taken to be a default type $\bot\in
T_{i}$ and $0$, respectively.
2. Phase 2:
For each player $j$, let $r_{j}^{i}$ be $i$’s share of $r_{j}$ and let
$t_{j}^{i}$ be $i$’s share of $t_{j}$. Each player $i$ initiates a
$k$-resilient CC of the functions $\mu^{*}_{1},\ldots,\mu^{*}_{n}$, where
$\mu^{*}_{j}(\vec{t},\vec{r})$ is the $j$th component of $\mu^{*}(\vec{t},r)$
and $r:=r_{1}+\ldots+r_{n}\bmod N$. Player $i$’s inputs for each of these CC
instances are the shares $r_{1}^{i},\ldots,r_{n}^{i}$ and
$t_{1}^{i},\ldots,t_{n}^{i}$ computed in Phase 1.
3. Phase 3:
For each player $i$, let $o_{j}^{i}$ be $i$’s output of the CC instance of
$\mu^{*}_{j}$ (this output is $i$’s share of $\mu^{*}_{j}(\vec{t},\vec{r})$).
For each player $j$, player $i$ sends $o_{j}^{i}$ to $j$.
4. Phase 4:
If $i$ shared its true type in Phase 1 and $i$ receives at least $2k+1$ shares
$o_{i}^{j}$ from different players such that every subset of those shares of
size $k+1$ defines the same secret $a_{i}$, $i$ plays $a_{i}$ and terminates.
Otherwise, $i$ plays a best response given its local history, assuming that
all the remaining players shared their true type and play their part of the
communication equilibrium. The optimal response depends on which was the value
$t_{i}$ that $i$ shared in Phase 1 (note that $t_{i}$ is 0 if $i$ didn’t
successfully share a type), and which shares of $o_{i}$ $i$ received from
other players. Note that these values give a posterior distribution over the
possible types of the other players and the actions that they played since $i$
is assuming that they share their true type $\vec{t}_{-i}$ and play their part
of the communication equilibrium with input $\vec{t}$.
Since the VSS and CC primitives are $k$ resilient, the protocol above is
explicitly defined in all scenarios in which at most $k$ players deviate.
However, there might be histories in which more than $k$ players deviate and
other players are not able to continue because either they didn’t receive
enough messages or the messages they received were inconsistent. (For example,
players will not able to jointly compute the functions $u_{i}^{*}$ in Phase 2
if insufficiently many players send messages. This means they won’t be able to
start Phase 3.) We extend the protocol to cover all these cases using the
following convention: if a player is not able to continue because the protocol
does not explicitly state what to do in a certain situation, the player does
not take part in subsequent phases. For instance, if a player $i$ is not able
to compute the functions $u_{i}^{*}$ in Phase 2, then $i$ does not take part
in Phase 3. Note, however, that the action played at the end of Phase 4 is
always specified even if a player doesn’t complete some of the previous
phases.
###### Proposition 2.
If $n>3k$, $\vec{\sigma}$ is a $k$-resilient Nash equilibrium and induces the
same correlated strategy profile as $\mu$.
###### Proof.
Clearly $\vec{\sigma}$ induces the same correlated strategy profile as $\mu$
by construction. The fact that $\vec{\sigma}$ is a $k$-resilient NE follows
directly from the properties of $k$-resilient VSS and $k$-resilient CC: no
matter what a coalition of at most $k$ players does, the remaining $n-k$
players are guaranteed to terminate all their VSS and CC instances correctly.
Since honest players choose $r_{i}$ uniformly at random, the number
$r:=r_{1}+\ldots+r_{n}\bmod N$ is also randomly distributed in $[N]$
regardless of what a coalition of at most $k$ players decides to share. This
guarantees that the actions $a_{1},\ldots,a_{n}$ encoded by the shares
$(o_{1}^{1},\ldots,o_{1}^{n}),\ldots,(o_{n}^{1},\ldots,o_{n}^{n})$ of honest
players are distributed according to $\mu(\vec{t})$. The correctness of the
output of each player follows from the fact that if a player $i$ waits until
receiving at least $2k+1$ shares that define the same secret $a_{i}$, at least
$k+1$ of those shares were sent by honest players and thus define the correct
value $a_{i}$ (note that if $n>3k$, all players are guaranteed to be able to
reconstruct their action $a_{i}$ in Phase 4 even if a coalition of $k$ players
deviates, since at least $2k+1$ players are honest). To complete the proof,
note that the secrecy properties of the $k$-resilient VSS and CC guarantee
that no coalition of at most $k$ players can learn anything besides their own
actions, and thus that playing the action $a_{i}$ computed in Phase 4 is
optimal. ∎
### 4.2 Constructing a $k$-resilient sequential equilibrium
Proposition 2 shows that, for all $\mu\in Com_{k}(\Gamma)$, there exists a
strategy $\vec{\sigma}$ in $\Gamma_{\mathit{CT}}$ such that $\vec{\sigma}$ is
a $k$-resilient Nash equilibrium that implements $\mu$. Our aim is to extend
$\vec{\sigma}$ to a $k$-resilient sequential equilibrium. The difficulty in
constructing such an extension is due to the fact that the $k$-resilience of
VSS and CC guarantees that no coalition of at most $k$ players can prevent the
remaining players from carrying out the VSS and CC computations if no other
player deviates (which is all that is needed to show that the construction
gives a $k$-resilient Nash equilibrium). However, it is not clear what the
coalition can accomplish starting at a point off the equilibrium path where
they detect that other players are deviating as well.
For instance, consider a normal-form game $\Gamma$ for $n$ players such that
the set of actions for each player is $[n]$. The payoffs are as follows:
$\vec{a}$, if at least $n-k$ coordinates of an action profile $\vec{a}$ are
different, then all players get a payoff of $0$; otherwise, all players get a
payoff of $1$. Consider the $k$-resilient correlated equilibrium $p$ in which
all possible permutations of $[n]$ are played with equal probability, giving
all players a payoff of 0. Suppose that players play the protocol
$\vec{\sigma}$ described in Section 4.1. If at least $n-k$ of the players are
honest, it is guaranteed by construction that all of them will play different
actions by the end of the protocol, and thus that all players will get a
payoff of 0 regardless of what the remaining $k$ players do. However, consider
the case that a coalition $K$ of at most $k$ players follows the protocol but
at some point detects an inconsistency among their local histories. Since
players in $K$ were following the protocol faithfully, it must be the case
that one of the remaining $n-k$ players deviated. If this leads the players in
$K$ to believe that there is a chance that at least two of the remaining
players played the same action $a\in[n]$, it is worthwhile for players in $K$
to deviate and play $a$ at the end of the protocol. If this is the case, then
we do not have a sequential equilibrium.
Our approach to showing that $\mu$ can be implemented with a $k$-resilient
sequential equilibrium is to show that there exists a belief system such that
all coalitions $K$ of size at most $k$ believe, regardless of their local
history, that
* (a)
the remaining players will successfully complete their part of $\vec{\sigma}$,
no matter what the players in $K$ do;
* (b)
players in $K$ cannot infer anything about the action profile $\vec{a}$ being
played other than their own part $\vec{a}_{K}$ (and $\mu$).
It is important to stress that property (b) does in fact depend on the beliefs
of players in $K$: If some player $i$ in $K$ receives a message $\bot$ from a
player $j$ that was not supposed to be sent on the equilibrium path, if $i$
believes that $j$ sends that message only when $j$ has type $t_{j}$ (whether
or not this is actually the case), then $i$ will believe that it has acquired
important information when it receives $\bot$, and adjust its play
accordingly. However, if $i$ believes that $j$ chose that message uniformly at
random, then $i$ will not adjust its play. The following proposition shows
that if properties (a) and (b) are satisfied, then $\vec{\sigma}$ is
$k$-sequentially rational (i.e., it is never in the interest of coalitions of
size at most $k$ to deviate from $\vec{\sigma}$).
###### Proposition 3.
If $b$ is a belief system compatible with $\vec{\sigma}$ that satisfies
properties (a) and (b), then $(\vec{\sigma},b)$ is a $k$-resilient sequential
equilibrium.
###### Proof.
If $b$ satisfies property (a), then at all nodes in the game tree (including
nodes off the equilibrium path), no matter what the players in $K$ do, the
remaining players will eventually terminate their part of $\vec{\sigma}$ and
thus sample an action profile $\vec{a}\in\mu(\vec{t})$ and play their part
$\vec{a}_{-K}$. Property (a) also guarantees that by the end of Phase 4 (and
the end of the communication phase), players not in $K$ will send the shares
of $\vec{a}_{K}$ to players in $K$, which means that players in $K$ believe
that they will always learn $\vec{a}_{K}$. Property (b) guarantees that
players in $K$ won’t learn anything besides their own actions. Properties (a)
and (b) together imply that players in $K$ believe that they cannot prevent
the remaining players from playing their part of an action profile $\vec{a}$
sampled according to $\mu$, and that players in $K$ will eventually learn
$\vec{a}_{K}$ and nothing else. Since $\mu$ is a $k$-resilient communication
equilibrium, it is always a best response for the players in $K$ to share
their own type in Phase 1, as long as they can continue the basic protocol.
Similar arguments show that following the basic protocol if they can (i.e.,
they receive messages enough consistent with the protocol) is a best response
in phases 2 and 3. Moreover, it also follows from properties (a) and (b) that,
since the players in $K$ cannot influence the outcome, doing nothing is a best
response if they cannot continue the basic protocol (in all phases). It
remains to show that the action played in Phase 4 is optimal. By construction,
this reduces to showing that if a coalition $K$ of players with $|K|\leq k$
shared their true type in Phase 1 and were able to compute their actions
$a_{i}$ in Phase 4, it is optimal for them to play $a_{i}$ (note that
otherwise $i$ is already best responding given that the belief system $b$
satisfies properties (a) and (b)). This follows from the fact that, by
properties (a) and (b), players in $K$ believe that all remaining players
computed $\vec{a}$ and thus played $\vec{a}_{-K}$. Since $\mu$ is a
$k$-resilient communication equilibrium, it is then always in the interest of
players in $K$ to play $\vec{a}_{K}$. ∎
Proposition 3 shows that Theorem 1 reduces to finding a belief system $b$
compatible with $\vec{\sigma}$ that satisfies properties (a) and (b). We start
by showing how such a belief system can be constructed for a fixed subset $K$
with $|K|\leq k$. Consider the sequence
$\vec{\sigma}^{1},\vec{\sigma}^{2},\ldots$ of strategy profiles in which
player $i$ proceeds as follows with $\vec{\sigma}^{m}_{i}$: Given local
history $h_{i}$, $i$ does exactly what it would have done with $\vec{\sigma}$,
except that if it is supposed to send a message $msg$ to another player $j$,
if $j\in K$, with probability $1/m$ it replaces $msg$ by a random message
$msg^{\prime}$ sampled according to some fixed distribution that gives all
messages positive probability, and sends $msg^{\prime}$ instead. If $j\not\in
K$, $i$ replaces $msg$ with $msg^{\prime}$ with probability $m^{-m}$. That is,
players play as in $\vec{\sigma}$, except that they may lie to players in $K$
with a low probability, and to players outside of $K$ with a much lower
probability. By construction, each $\vec{\sigma}^{m}$ is completely mixed and
the belief $b$ induced by the limits of the beliefs in
$\vec{\sigma}^{1},\vec{\sigma}^{2},\ldots$ satisfies the properties (a) and
(b) described above for the set $K$. To see this, note that property (a) is
satisfied because players in $K$ believe that the remaining $n-k$ players are
being truthful between themselves (note that the probability of anyone lying
to a player not in $K$ is negligible) and thus, since the VSS and CC
implementations are $k$-resilient, that they can carry out all the necessary
computations without the help of players in $K$. Moreover, since players in
$K$ believe that if they get a message incompatible with $\vec{\sigma}$ (i.e.,
a message that could not have been according to $\vec{\sigma}$, given their
joint histories), then they are getting a message drawn from some fixed
distribution, they learn no useful information about the action profile being
played by getting this message. Property (b) follows.
This example shows how to construct a $k$-belief system compatible with
$\vec{\sigma}$ that satisfies properties (a) and (b) for a particular subset
$K$ with $|K|\leq k$, but it unfortunately does not satisfy these properties
for all such sets simultaneously. Note that it is critical that players in $K$
believe that they are the only ones being lied to, and also that the lies sent
to players in $K$ convey no information since they do not depend on the
senders’ local histories (in the example, they are sampled from a fixed
distribution). The next definitions generalizes these ideas:
###### Definition 8.
Given history $h_{i}$ of player $i$, a message $msg$ in $h_{i}$ from $i$ to
$j$ is a _lie_ if it is incompatible with $\vec{\sigma}$. A subset $K$ of
players is _truthful relative to $\vec{\sigma}$ in global history $\vec{h}$_
if, in $\vec{h}$, no player in $K$ ever _lies_ (i.e., sends a message that is
a lie) to another player in $K$.
Note that since the local history $h_{i}$ of player $i$ includes all of $i$’s
randomization up to the point where it is supposed to send a message, given
local history $h_{i}$, there is a unique message sent by $i$ to $j$ that is
compatible with $\vec{\sigma}$, even if $\vec{\sigma}$ is a mixed strategy
(intuitively, when $i$ sends a message to $j$, all of $i$’s coins have already
been tossed and their outcomes appear in $i$’s history). A message from player
$i$ to player $j$ in local history $h_{i}$ is _correct_ if it is not a lie
(relative to the protocol $\vec{\sigma}$).
###### Definition 9 ($k$-paranoid belief system).
Given a $k$-belief system $b$ and the local histories $\vec{h}_{K}$ of a
subset $K$ of players, a global history $\vec{h}$ is _$b$ -consistent_ with
$\vec{h}_{K}$ if the probability according to $b$ that players have global
history $\vec{h}$ conditional on players in $K$ having history $\vec{h}_{K}$
is non-zero. Given a protocol $\vec{\sigma}$, a $k$-belief system $b$ is _$k$
-paranoid_ if
* (1)
for all subsets $K\subseteq[n]$ with $|K|\leq k$ and all histories
$\vec{h}_{K}$ of players in $K$, in all histories $\vec{h}$ that are
$b$-consistent with $\vec{h}_{K}$, the set $\overline{K}$ of players is
truthful relative to $\vec{\sigma}$ (where $\overline{K}$ is the complement of
$K$);
* (2)
there exists a sequence $\vec{\sigma}^{1},\vec{\sigma}^{2},\ldots$ of
protocols that induces $b$ such that the lies relative to $\vec{\sigma}$ sent
in $\vec{\sigma}^{1},\vec{\sigma}^{2},\ldots$ do not depend on the players’
local histories.
Intuitively, a $k$-belief system is $k$-paranoid if all subsets of at most $K$
players believe, regardless of their local history, that they are the only
ones being lied to, and also that the lies received convey no information. Our
earlier discussion provides a proof of the following proposition.
###### Proposition 4.
Let $\vec{\sigma}$ be the protocol presented in Section 4.1. If $b$ is a
$k$-paranoid belief system consistent with $\vec{\sigma}$, then $b$ satisfies
properties (a) and (b).
Thus, by Proposition 3, Theorem 1 reduces to find a $k$-paranoid belief system
$b$ that is consistent with $\vec{\sigma}$. The first candidate for generating
such a belief system is the sequence
$\vec{\sigma}^{1},\vec{\sigma}^{2},\ldots$ of strategy profiles mentioned
earlier, except that with strategy $\sigma_{i}^{m}$, rather than player $i$
lying to player $j$ with probability $1/m$ if $j$ is in some fixed set $K$ and
with probability $m^{-m}$ if $j$ is not in $K$, now $i$ lies to all players
with probability $1/m$. We might hope that if players detect that they
received inappropriate messages, they would assume that they are the only ones
being lied to, since lies are highly unlikely. For instance, if a player $i$
receives four inconsistent messages, we would hope that $i$ would believe that
these are the only lies. Unfortunately, this is not necessarily the case. If
an earlier lie by some other player could result in players sending these
messages to $i$ according to $\vec{\sigma}$, then $i$ would instead believe
that there was only one lie. To prevent this, we want players to believe that
earlier lies are much less likely than later lies. It turns out that if we do
this, then we can get the result that we want.
###### Proposition 5.
For all strategies $\vec{\sigma}$ in $\Gamma_{\mathit{CT},syn}$, there exists
a $k$-paranoid belief system $b$ that is consistent with $\vec{\sigma}$.
###### Proof.
Consider the sequence
$\vec{\sigma}^{1},\vec{\sigma}^{2},\vec{\sigma}^{3},\ldots$ of strategy
profiles where in $\vec{\sigma}^{n}$ players act exactly as they do in
$\vec{\sigma}$, except that whenever $i$ would send a message $msg$ to another
player $j$, with probability $m^{-\left((2n)^{-r}\right)}$ (where $r$ is the
current round of communication), $i$ replaces this message by a message
$msg^{\prime}$ chosen according to some fixed distribution and sends
$msg^{\prime}$ instead. Intuitively, these are protocols where each player
lies to other players with a probability that (greatly) increases with the
round number and that decreases with $m$. It is straightforward to check that
$\vec{\sigma}^{1},\vec{\sigma}^{2},\vec{\sigma}^{3},\ldots$ converges to
$\vec{\sigma}$. Let $b$ be the belief system induced by the sequence. We want
to show that $b$ is $k$-paranoid. By construction, $b$ satisfies property (2)
of Definition 9 since lies in
$\vec{\sigma}^{1},\vec{\sigma}^{2},\vec{\sigma}^{3},\ldots$ are sampled from a
fixed distribution. To show that it also satisfies property (1), we need some
preliminary lemmas.
Given a global history $\vec{h}$ of $\vec{\sigma}$, let $L(\vec{h})$ be the
sequence $(\ell_{1},\ell_{2},\ldots,\ell_{R})$, where $R$ is the total number
of communication rounds in $\vec{h}$ and $\ell_{r}$ is the number of lies sent
at round $r$ in $\vec{h}$.
###### Lemma 1.
If $P^{m}$ is the probability on global histories induced by
$\vec{\sigma}^{m}$, and $\vec{h}$ and $\vec{h}^{\prime}$ are global history
profiles such that $L(\vec{h})<L(\vec{h}^{\prime})$ in lexicographical order
(the smaller sequence is the one that has the smaller value in the first
position in which they differ), then
$\lim_{m\rightarrow\infty}\frac{P^{m}(\vec{h}^{\prime})}{P^{m}(\vec{h})}=0$.
###### Proof.
If $L(\vec{h})=(\ell_{1},\ell_{2},\ldots,\ell_{R})$, then
$\log(P^{m}(\vec{h}))=-\log(m)\left(\sum_{r=1}^{n}\ell_{r}(2n)^{-r}\right)+\Theta(1),$
where $\Theta(1)$ denotes an expression that is bounded by some constant.
Thus, if $L(\vec{h})=(\ell_{1},\ell_{2},\ldots,\ell_{R})$ and
$L(\vec{h}^{\prime})=(\ell^{\prime}_{1},\ldots,\ell^{\prime}_{R^{\prime}})$,
we must show that if $L(\vec{h})<L(\vec{h}^{\prime})$, then
$\sum_{r=1}^{R^{\prime}}\ell_{r}(2n)^{-r}<\sum_{r=1}^{R^{\prime}}\ell^{\prime}_{r}(2n)^{-r}$,
which is equivalent to showing that
$\sum_{r=1}^{\max(R,R^{\prime})}(\ell^{\prime}_{r}-\ell_{r})(2n)^{-r}>0$
(taking the elements beyond the end of a sequence to be 0).
Since each player sends exactly one message to each other player, we have that
$0\leq\ell_{i}\leq n$. Therefore, if $r^{*}$ is the first round such that
$\ell^{\prime}_{r}\not=\ell_{r}$, we have that
$\sum_{r=1}^{\max(R,R^{\prime})}(\ell^{\prime}_{r}-\ell_{r})(2n)^{-r}\geq(2n)^{-r^{*}}-n\sum_{r=r^{*}+1}^{\max(R,R^{\prime})}(2n)^{-r}=(2n)^{-r^{*}}\left(1-n\sum_{r=1}^{\max(R,R^{\prime})-r^{*}}(2n)^{-r}\right),$
and
$1-n\sum_{r=1}^{\max(R,R^{\prime})-r^{*}}(2n)^{-r}>1-n\sum_{r>0}(2n)^{-r}>1-n\sum_{r>0}2^{-r}n^{-1}=0$
∎
###### Lemma 2.
If $|K|\leq k$ and $\vec{h}_{K}^{*}$ is a local history of the players in $K$,
then for every global history profile $\vec{h}$ such that $\overline{K}$ is
not truthful in $\vec{h}$,
$\lim_{m\rightarrow\infty}P^{m}(\vec{h}\mid\vec{h}_{K}^{*})=0.$
###### Proof.
Suppose that for some global history $\vec{h}$,
$P^{m}(\vec{h}\mid\vec{h}_{K}^{*})>0$ and $\overline{K}$ is not truthful in
$\vec{h}$. Since
$P^{m}(\vec{h}\mid\vec{h}_{K}^{*})=\frac{P^{m}(\vec{h})}{\sum_{\vec{h}^{\prime}:\vec{h}^{\prime}_{K}=\vec{h}_{K}^{*}}P^{m}(\vec{h}^{\prime})},$
by Lemma 1, this result reduces to finding a global history $\vec{h}^{\prime}$
such that $\vec{h}^{\prime}_{K}=\vec{h}_{K}$ and
$L(\vec{h}^{\prime})<L(\vec{h})$. This history can be constructed as follows:
suppose that $r$ is the first round in $\vec{h}$ where a player
$i\in\overline{K}$ lies to another player $j\in\overline{K}$. Consider a
history $\vec{h}^{\prime}$ in which players in $K$ perform the same internal
computations and send exactly the same messages as in $\vec{h}$, but players
in $\overline{K}$ only do so until round $r-1$. In round $r$, players in
$\overline{K}$ perform exactly the same internal computations as in $\vec{h}$
and send the same messages to players in $K$, but they send to other players
in $\overline{K}$ the _correct_ messages (i.e., the messages players in
$\overline{K}$ were supposed to send according to the protocol). Notice that
since a player $i$’s local history includes the random bits that the players
needs, there is a unique correct message, even if player $i$ is uses a
randomized protocol. From round $r+1$ on, players in $\overline{K}$ send
players in $K$ exactly the same messages as in $\vec{h}$. Clearly, by
construction, $\vec{h}^{\prime}_{K}=\vec{h}_{K}$. Let
$L(\vec{h})=(\ell_{1},\ell_{2},\ldots,\ell_{R})$ and
$L(\vec{h}^{\prime})=(\ell^{\prime}_{1},\ell^{\prime}_{2},\ldots,\ell^{\prime}_{R})$.
Then $\ell_{i}=\ell^{\prime}_{i}$ for all $i<r$, since players have the same
local histories in $\vec{h}$ and $\vec{h}^{\prime}$ for the first $r-1$ rounds
of communication, and they also send exactly the same messages. A similar
argument shows that, for all $r^{\prime}\geq r$, the players in $K$ tell
identical lies in round $r^{\prime}$ in both $h$ and $\vec{h}^{\prime}$.
Finally, note that strictly fewer lies are told by players in $\overline{K}$
in round $r$ of $\vec{h}^{\prime}$ than in round $r$ of $\vec{h}$, since the
players in $\overline{K}$ have exactly the same local history in both
$\vec{h}$ and $\vec{h}^{\prime}$, send exactly the same messages to players in
$K$ in both, but in $\vec{h}^{\prime}$ they tell no lies to each other
whereas, by assumption, there is at least one such lie told in $\vec{h}$.
Thus, $\ell^{\prime}_{r}<\ell_{r}$, and $L(\vec{h}^{\prime})<L(\vec{h})$ as
desired. ∎
This shows that $b$ satisfies property (1) of Definition 9, and thus that $b$
is a $k$-paranoid belief system consistent with $\vec{\sigma}$, completing the
proof of Proposition 5. ∎
## 5 Characterization of $SE_{k}(\Gamma_{d})$ and $SE_{k}^{S}(\Gamma_{d})$
In this section we characterize the sets $SE_{k}(\Gamma_{d})$, and
$SE_{k}^{S}(\Gamma_{d})$ (note that, by Theorem 1, the characterization of
$SE_{k}(\Gamma_{\mathit{CT}})$ and $SE_{k}^{S}(\Gamma_{\mathit{CT}})$ is
identical if $n>3$.) If $\Gamma$ is a normal-form game, let $CE_{k}(\Gamma)$
denote the set of possible distributions over action profiles induced by a
$k$-resilient correlated equilibrium in $\Gamma$; if $\Gamma$ is a Bayesian
game, let $Com_{k}(\Gamma)$ denote the set of possible maps from type profiles
to distributions over action profiles induced by $k$-resilient communication
equilibria in $\Gamma$. We also denote by $CE_{k}^{S}(\Gamma)$ the set of
possible distributions over action profiles induced by a strongly
$k$-resilient correlated equilibrium in $\Gamma$. $Com_{k}^{S}(\Gamma)$ and
$SE_{k}^{S}(\Gamma)$ are defined analogously.
The first characterization shows that, if $\Gamma$ is a normal-form game, then
$SE_{k}(\Gamma_{d})$ is the set of $k$-resilient correlated equilibria of
$\Gamma$ and $SE_{k}^{S}(\Gamma_{d})$ is the set of strongly $k$-resilient
correlated equilibria of $\Gamma$.
###### Proposition 6.
If $\Gamma=(P,A,U)$ is a normal-form game for $n$ players, then
$SE_{k}(\Gamma_{d})=CE_{k}(\Gamma)$ and
$SE_{k}^{S}(\Gamma_{d})=CE_{k}^{S}(\Gamma)$ for all $k\leq n$.
###### Proof.
Clearly, the distribution over actions induced by a $k$-resilient sequential
equilibrium (resp., strongly $k$-resilient sequential equilibrium)
$\vec{\sigma}$ in $\Gamma_{d}$ is also a $k$-resilient correlated equilibrium
(resp., strongly $k$-resilient correlated equilibrium) of $\Gamma$, for
otherwise there exists a coalition $K$ of at most $k$ players such that all
(resp., some) members of the coalition can increase their utility by deviating
from $\vec{\sigma}$ by playing a different action in the underlying game. For
the opposite inclusion, observe that all $k$-resilient (resp., strongly
$k$-resilient) correlated equilibria $p$ of $\Gamma$ can be easily implemented
with a mediator the mediator samples an action profile $\vec{a}$ following
distribution $p$ and gives $a_{i}$ to each player $i$. Then each player $i$
plays whatever is sent by the mediator. ∎
Theorem 1 and Proposition 6 together imply the following corollary:
###### Corollary 1.
If $\Gamma=(P,A,U)$ is a normal-form game for $n$ players and $n>3k$, then
$SE_{k}(\Gamma_{\mathit{CT}})=CE_{k}(\Gamma)$ and
$SE_{k}^{S}(\Gamma_{\mathit{CT}})=CE_{k}^{S}(\Gamma)$.
For Bayesian games, we can show that the set of sequential equilibria in
$\Gamma_{d}$ is equal to the set of communication equilibria of $\Gamma$.
###### Proposition 7.
If $\Gamma=(P,T,q,A,U)$ is a Bayesian game for $n$ players and $n\geq k$, then
$SE_{k}(\Gamma_{d})=Com_{k}(\Gamma)$ and
$SE_{k}^{S}(\Gamma_{d})=Com_{k}^{S}(\Gamma)$ .
###### Proof.
Suppose that $\vec{\sigma}$ is a $k$-resilient (resp., strongly $k$-resilient)
sequential equilibrium of $\Gamma_{d}$. Then the correlated strategy profile
$\mu$ induced by $\vec{\sigma}$ in $\Gamma$ must be a $k$-resilient (resp.,
strongly $k$-resilient) communication equilibrium. If $\mu$ is not a
$k$-resilient (resp., strongly $k$-resilient) communication equilibrium, then
there exists a coalition $K$ of players with $|K|\leq k$ and two functions
$\psi:T_{K}\rightarrow T_{K}$ and $\varphi:A_{K}\rightarrow A_{K}$ such that
the inequality of Definition 7 does not hold for some (resp., for all) $i\in
K$. It follows that if agents in $\Gamma_{d}$ $K$ play $\vec{\sigma}$ as if
they had type profile $\psi(\vec{t}_{K})$ instead of their true types, and
then play action $\varphi(\vec{a}_{K})$ instead of the action profile
$\vec{a}_{K}$, the utility of all (resp., some) agents would strictly
increase, which contradicts the assumption that $\vec{\sigma}$ is a
$k$-resilient (resp., strongly $k$-resilient) sequential equilibrium of
$\Gamma_{d}$. For the opposite inclusion, recall from the discussion after
Definition 7 that we can identify a correlated strategy profile $\mu$ in
$\Gamma$ with a canonical strategy $\vec{\sigma}$ in $\Gamma_{d}$ in which
players tell the mediator their type and the mediator computes which action
profile they should play according to $\mu$. If $\mu$ is a $k$-resilient
(resp., strongly $k$-resilient) communication equilibrium, then the canonical
strategy $\vec{\sigma}$ is a $k$-resilient (resp., strongly $k$-resilient)
sequential equilibrium of $\Gamma_{d}$ that induces $\mu$ (see the discussion
after Definition 7 for details). ∎
Theorem 1 and Proposition 7 together imply the following:
###### Corollary 2.
If $\Gamma=(P,T,q,A,U)$ be a Bayesian game for $n$ players and $n>3k$, then
$SE_{k}(\Gamma_{\mathit{CT}})=Com_{k}(\Gamma)$ and
$SE_{k}^{S}(\Gamma_{\mathit{CT}})=Com_{k}^{S}(\Gamma)$ .
## 6 The asynchronous setting
Up to this point, as in the game-theory literature, we assumed that the
communication between players proceeded in synchronous rounds. That is, at
each round $t$, all the players send messages to each other player (we
identify not sending a message with sending a special message $\bot$) and
these messages are received by their intended recipients before the beginning
of round $t+1$. The assumption that we identify not sending a message with
sending $\bot$ is made without loss of generality—player $j$ can tell if
player $i$ has not sent her a message. Moreover, it is typically assumed when
analyzing games with mediators and communication games that it is common
knowledge when the communication phase ends and that, after it ends, all the
players simultaneously move in the underlying game. We call this the
_synchronous_ setting.
In this section, we consider an _asynchronous_ setting that is quite standard
in the distributed computing literature. In the asynchronous setting, there is
no global notion of time, and messages may take arbitrary amounts of time to
be delivered (although we do assume that all messages are eventually
delivered). Thus, we can no longer identify not sending a message with sending
$\bot$; if $j$ has not received a message from player $i$, $j$ is not sure if
this is because $i$ did not send $j$ a message or if $i$ sent a message that
has not yet been delivered. For ease of exposition, we assume that message
delivery is under the control of a _scheduler_ , who also decides when each
player gets to move, with the guarantee that all players eventually get to
move if they want to move. More precisely, in an asynchronous setting, the
players and the scheduler alternate turns. During a player’s turn, it receives
all the messages delivered to it by the scheduler since its last turn; it may
also perform additional computation and send messages to other players (note
that these messages are not received until they are delivered by the scheduler
to their recipients). When the player finishes all necessary operations and
sends all the necessary messages, it ends its turn, at which point it become’s
the scheduler’s turn. During its turn, the scheduler can deliver messages that
have been sent but not yet delivered, or it may schedule a player. If the
scheduler schedules player $i$, the scheduler’s turn ends and $i$’s turn
begins.
What internal computations a player $i$ performs, what messages $i$ sends, and
to which players $i$ sends messages during its turn is dictated by $i$’s
strategy, which is a function from $i$’s local history to operations and
messages (i.e., a strategy tells a player what to do as a function of its
local history). Note that $i$’s local history is simply an ordered list
containing, for each time that $i$ was scheduled, the messages that $i$ sent
and received, and the internal computations that $i$ performed. (We can
represent this as a game, in which case we would identify each information set
of $i$ with a local history.) Note that $i$ does not in general know how many
times another player $j$ has been scheduled since $i$’s last turn.
The scheduler’s behavior is also encoded in its strategy, which describes
which messages the scheduler delivers and which players are scheduled, as a
function of the scheduler’s local history. The scheduler’s local history is
again an ordered list, which includes which players have been scheduled up to
that point and which messages have been delivered. Note that the scheduler
does not know the contents of the messages sent by the players, but it can
still identify the messages by, for instance, labelling them by the order they
were sent. For example, an entry in the scheduler’s local history could be
“delivered player 3’s fourth message”. The only constraint on the scheduler’s
strategy is that it must eventually deliver all messages that have been sent
and, starting from any local history, it must eventually schedule all players
that have not terminated. (We assume that “done” is a signal that a player can
send the scheduler that indicates that it has terminated.)
If $\Gamma_{\mathit{CT}}$ is asynchronous, the payoff of the players may
depend on the strategy $\sigma_{e}$ of the scheduler. This means that the
definition of implementation and of the solution concepts must take the
scheduler into account. We extend the definitions of Nash equilibrium,
correlated equilibrium, sequential equilibrium, and communication equilibrium
by requiring that the relevant inequality holds for all choices of
$\sigma_{e}$. For example a strategy profile $\vec{\sigma}$ is a Nash
equilibrium in an asynchronous setting if, for all $i\in P$, all strategies
$\tau_{i}$ for player $i$, and all schedulers $\sigma_{e}$,
$u_{i}(\vec{\sigma},\sigma_{e})\geq
u_{i}(\tau_{i},\vec{\sigma}_{-i},\sigma_{e})$. Since the action profile played
might depend on the scheduler, a strategy $\vec{\sigma}$ in
$\Gamma_{\mathit{CT}}$ might induce more than one strategy in $\Gamma$ (see
Example 2 below). A strategy profile $\vec{\sigma}$ for $\Gamma_{\mathit{CT}}$
_implements a set $S$ of strategy profiles in $\Gamma$_ in an asynchronous
setting if (a) for every scheduler $\sigma_{e}$ in $\Gamma_{\mathit{CT}}$, the
outcome obtained when playing $\vec{\sigma}$ with scheduler $\sigma_{e}$ is
the same as that obtained when playing some strategy profile in $S$, and (b)
for every strategy profile $\vec{\tau}$ in $S$, there exists a scheduler
$\sigma_{e}$ such that the outcome obtained when playing $\vec{\sigma}$ with
$\sigma_{e}$ is the same as that obtained when playing $\vec{\tau}$.
Intuitively, $\vec{\sigma}$ implements $S$ if the set of outcomes that result
from playing $\vec{\sigma}$ with different choices of scheduler are precisely
those that result from playing strategy profiles in $S$. As in the synchronous
setting, if $\Gamma$ is a normal-form game, we take the strategy profiles in
$S$ to consist of distributions over action profiles in $\Gamma$, while if
$\Gamma$ is a Bayesian game, the strategy profiles in $S$ are correlated
strategy profiles.
In cheap-talk games in synchronous systems, we assumed (as is standard in the
literature) that the game tree is finite and at the last move on a path, the
players play an action in the underlying game; we take the utility of a leaf
to be the utility of the corresponding action profile in the normal-form game.
We cannot assume this in asynchronous systems. Indeed, in asynchronous
systems, to allow for players who deviate from an equilibrium, we must
consider game trees with infinite paths. For example, a player $i$ may wait
forever for a message that never arrives, because the sender deviated and
never sent it, so $i$ may never play an action in the underlying game. We must
thus define what the payoffs are on infinite paths and at the leaf of a finite
path where some players do not play an action in the underlying game. This
amounts to treating players who have not played an action as, in fact, having
played some action. The two main approaches for doing this are the _default-
move approach_ Abraham et al. [2019b], where if player $i$ never plays an
action in the underlying game, its action is sampled from a default
distribution, and the _Aumann and Hart_ (AH) approach Aumann and Hart [2003],
where the action played by $i$ is some function of $i$’s local history. The AH
approach essentially assumes that players can leave a “will” defining the
action in the underlying game that they should play if they never actually
play an action in the underlying game while playing the cheap-talk game. In
the asynchronous setting (with both the AH and the default move approach), we
say that a strategy profile $\vec{\sigma}$ in $\Gamma_{\mathit{CT}}$
_implements_ a strategy profile $\vec{\tau}$ in $\Gamma$ if the set of
strategy profiles implemented by $\vec{\sigma}$ (for different choices of
scheduler) is $\\{\vec{\tau}\\}$ (thus, playing $\vec{\sigma}$ results in the
same outcome no matter what the scheduler does).
The following example illustrates some of the new subtleties that asynchrony
introduces.
###### Example 2.
Let $\Gamma$ be a normal-form game for $n$ players where the set of actions is
$\\{1,2,\ldots,n\\}$, and let $\Gamma_{d,asyn}$ be a cheap-talk extension of
$\Gamma$ in the asynchronous setting, where there is a trusted mediator $d$
and the AH approach is used. Suppose that $\vec{\sigma}$ is the strategy
profile that proceeds as follows: each player sends an empty message to the
mediator when they are first scheduled. The mediator waits until it receives a
message, and then sends a message to all players with the index of the player
whose message was received first. When a player $j$ receives a message from
the mediator with a number $i\in\\{1,2,\ldots,n\\}$, player $j$ plays action
$i$. If player $j$ never receives a message from the mediator, then according
to its will, $j$ plays action $j$. The player whose index appears the most
often in the resulting action profile receives a payoff of 1 (if there are
ties, they all get 1), while the remaining players get 0.
This game can be viewed as a race. The player whose message gets to the
mediator first receives a payoff of 1 while the rest receive 0. Note that,
regardless of the scheduler, all players play the same action in $\Gamma$.
However, the scheduler decides which action is played. Thus, the strategy
$\vec{\sigma}$ in the example implements the set of strategies
$\\{(1,1\ldots,1),(2,2,\ldots,2),\ldots,(n,n,\ldots,n)\\}$. This strategy is
not a Nash equilibrium. To see this, consider the scheduler $\sigma_{e}$ that
schedules the players sequentially (first player 1, then player 2, and so on).
If player $i$ sends two messages to the mediator and is the only player to do
so, then the scheduler delivers $i$’s first message before it delivers any
other player’s message. If more than one player sends two messages to the
mediator, the schedulers chooses one of these players at random and delivers
her message first. The remaining messages are delivered in some random order.
With this scheduler, the players benefit by deviating and sending two messages
to the mediator instead of just one. Thus, $\vec{\sigma}$ is not a Nash
equilibrium. A similar argument can be used to show that, for all games
$\Gamma$, if $\vec{\sigma}$ is a $k$-resilient Nash equilibrium in
$\Gamma_{\mathit{CT}}$, then the payoffs of the players when playing
$\vec{\sigma}$ cannot depend on the scheduler (see Abraham et al. [2019b]).
This example shows that in asynchronous systems, the set of possible
deviations is much larger than in synchronous systems. However, there are
cases where controlling the scheduling of the messages and players does not
give that much power to an adversary. For instance, if we consider an
asynchronous cheap-talk extension of Example 1, the strategy profile in which
both agents send their type to each other the first time they are scheduled
and play action $t_{1}t_{2}$ is a Nash equilibrium (note that this is almost
equivalent to the strategy proposed in Example 1, however there is no notion
of “rounds” in an asynchronous setting). Many of the results that hold in the
synchronous case also hold in the asynchronous case, but may require a larger
proportion of the players to be honest. For instance, asynchronous multiparty
secure computation tolerates deviations by at most a quarter of the agents,
while synchronous multiparty secure computation tolerates deviations by at
most a third Ben-Or et al. [1988, 1993]. Our results hold for similar bounds.
(An intuitive explanation of why there is a difference between the thresholds
for the synchronous and asynchronous settings can be found in Appendix B.)
### 6.1 Main results
Given a normal-form or Bayesian game $\Gamma$, let $\Gamma_{\mathit{CT},asyn}$
and $\Gamma_{d,asyn}$ be the asynchronous equivalents of
$\Gamma_{\mathit{CT}}$ and $\Gamma_{d}$ respectively (i.e., the asynchronous
cheap-talk and mediator extensions of $\Gamma$). As in Section 3, let
$SE_{k}(\Gamma_{\mathit{CT},asyn})$ (resp.,
$SE_{k}^{S}(\Gamma_{\mathit{CT},asyn})$) denote the set of possible strategy
profiles in $\Gamma$ induced by $k$-resilient sequential equilibria in
$\Gamma_{\mathit{CT},asyn}$. However, since in the asynchronous setting the
outcome may depend on the scheduler (see the discussion in Section 6), if
$\Gamma$ is a normal-form game, then $SE_{k}(\Gamma_{\mathit{CT},asyn})$ is a
set of sets of strategy profiles, and if $\Gamma$ is a Bayesian game, then
$SE_{k}(\Gamma_{\mathit{CT},asyn})$ is a set of sets of correlated strategy
profiles.
In the asynchronous setting we have equivalent results to those in the
synchronous setting.
###### Theorem 2.
If $\Gamma=(P,T,q,A,U)$ is a Bayesian game for $n$ players and $n>4k$, then
$SE_{k}(\Gamma_{\mathit{CT},asyn})=SE_{k}(\Gamma_{d,asyn})$ and
$SE_{k}^{S}(\Gamma_{\mathit{CT},asyn})=SE_{k}^{S}(\Gamma_{d,asyn})$ with both
the default move and AH approaches.
Note that the only difference with respect to Theorem 1, is that in the
asynchronous setting we require that $n>4k$ instead of $n>3k$. This difference
comes down to the fact that, in the asynchronous setting, players cannot tell
if a message has not been received because the player didn’t send it or
because the message is being delayed by the scheduler. In fact, this is the
reason why most of the distributed computing primitives in asynchronous
systems are resilient to deviations of up to a fourth of the players, as
opposed to a third in the synchronous case (see Appendix B for more details).
As in the synchronous case, $SE_{k}(\Gamma_{\mathit{CT},asyn})$ and
$SE_{k}^{S}(\Gamma_{\mathit{CT},asyn})$ have a relatively simple
characterization. This is described in Section 6.3.
### 6.2 Proof of Theorem 2 (outline)
The proof of Theorem 2 is quite similar to the proof of Theorem 1. Given a
strategy profile $\vec{\sigma}^{\prime}+\sigma^{\prime}_{d}\in
SE_{k}(\Gamma_{d,asyn})$, we first construct a $k$-resilient Nash equilibrium
$\vec{\sigma}$ that implements $\vec{\sigma}^{\prime}+\sigma^{\prime}_{d}$ and
guarantees that, if less than $k$ players deviate, all players will eventually
be able to compute the action that they should play, and then we construct a
$k$-paranoid belief system $b$ that is consistent with $\vec{\sigma}$ just as
in the proof of Theorem 1. An argument similar to the one used in the
synchronous case shows that $(\vec{\sigma},b)$ is indeed a $k$-resilient
sequential equilibrium. The only major difference between the synchronous and
the asynchronous case is that, in an asynchronous system, constructing a
$k$-resilient Nash equilibrium $\vec{\sigma}$ that implements
$\vec{\sigma}^{\prime}+\sigma^{\prime}_{d}$ is far more complicated than in a
synchronous system (the reason for this can be found below), and is beyond the
scope of this work. In this paper, we only provide a very high-level overview
of the construction used in Geffner and Halpern [2021] and Abraham et al.
[2019b] that satisfies all the properties we need. As in the case of Theorem
1, we prove Theorem 2 for the case of $k$-resilient sequential equilibrium.
The case of strongly $k$-resilient equilibrium is analogous.
As discussed in Section 3, in general, there is no simple description of
$SE_{k}(\Gamma_{d,asyn})$; thus, the proof of Theorem 2 consists of showing
that arbitrary interactions $\vec{\sigma}^{\prime}+\sigma^{\prime}_{d}$ with a
trusted mediator in the asynchronous setting can be emulated by a strategy
$\vec{\sigma}$ without the mediator that tolerates deviations of coalitions of
size at most $k$. We proceed as in the proof of Theorem 1: we first construct
a $k$-resilient Nash equilibrium $\vec{\sigma}$ that implements
$\vec{\sigma}^{\prime}+\sigma^{\prime}_{d}$, and then we construct a
$k$-paranoid belief system to extend the Nash equilibrium to a $k$-resilient
sequential equilibrium $\vec{\sigma}_{seq}$. The main problem is that the
construction of a $k$-resilient Nash equilibrium is not as simple as in the
synchronous setting. The communication between the players and the mediator
can be arbitrarily convoluted (as opposed to the players just sending their
types and receiving their suggested action), and it may depend on the
scheduler. This means that, to be sure that all possible cases are considered,
players in $\vec{\sigma}$ must somehow be able to simulate all possible
scheduling orders for the players and the messages in
$\vec{\sigma}^{\prime}+\sigma_{d}$. (In a given run, the particular order that
they choose must depend on the actual scheduler in $\vec{\sigma}$.) The
simulation is quite nontrivial, but the high-level idea is that each player
$i$ in $\vec{\sigma}$ simulates its counterpart in
$\vec{\sigma}^{\prime}+\sigma^{\prime}_{d}$, except that whenever it would
send a message $msg$ in $\vec{\sigma}^{\prime}+\sigma^{\prime}_{d}$, $i$
shares $msg$ using VSS. Moreover, each player simultaneously plays its part
simulating the mediator, which means using a _consensus protocol_ (see
Appendix C) to guarantee that the nonfaulty players agree on when the mediator
is scheduled (there exist $t$-resilient consensus protocols in asynchronous
systems if $n>3t$ Abraham et al. [2008a]) and using CC to compute which
messages it receives, which messages it sends, and what the contents of those
messages are. With this procedure, each player is able to compute its share of
every message sent by the mediator. If that message was supposed to be sent to
player $j$, then each player $i$ sends its share of the message to $j$ in
order for $j$ to be able to reconstruct it. The full construction can be found
in Geffner and Halpern [2021]. It satisfies a property called _$k$
-bisimulation_, which states the following:
* (a)
For all coalitions $K$ of size at most $k$, all strategies $\vec{\tau}_{K}$
for players in $K$, and all schedulers $\sigma_{e}$, there exists a strategy
$\vec{\tau}^{\prime}_{K}$ and a scheduler $\sigma_{e}^{\prime}$ such that
$(\vec{\sigma}_{-K},\tau_{K},\sigma_{e})$ and
$((\vec{\sigma}^{\prime}_{-K},\tau^{\prime}_{K})+\sigma^{\prime}_{d},\sigma^{\prime}_{e})$
are identically distributed.
* (b)
For all coalitions $K$ of size at most $k$, all strategies
$\vec{\tau}^{\prime}_{K}$ for players in $K$, and all schedulers
$\sigma^{\prime}_{e}$, there exists a strategy $\vec{\tau}_{K}$ and a
scheduler $\sigma_{e}$ such that $(\vec{\sigma}_{-K},\tau_{K},\sigma_{e})$ and
$((\vec{\sigma}^{\prime}_{-K},\tau^{\prime}_{K})+\sigma^{\prime}_{d},\sigma^{\prime}_{e})$
are identically distributed.
Intuitively, if $\vec{\sigma}$ $k$-bisimulates
$\vec{\sigma}^{\prime}+\sigma^{\prime}_{d}$, then all possible distributions
over action profiles in $\vec{\sigma}$ when a coalition of at most $k$ players
deviates also occur in $\vec{\sigma}^{\prime}+\sigma^{\prime}_{d}$ when the
same coalition deviates (possibly in a different way). In particular, if a
coalition $K$ of players with $|K|\leq k$ can increase their payoff in
$\vec{\sigma}$ by deviating, they could already do so in
$\vec{\sigma}^{\prime}+\sigma^{\prime}_{d}$, which contradicts the assumption
that $\vec{\sigma}^{\prime}+\sigma^{\prime}_{d}$ is a $k$-resilient sequential
equilibrium. This implies that $\vec{\sigma}$ is a $k$-resilient Nash
equilibrium that implements $\vec{\sigma}^{\prime}+\sigma^{\prime}_{d}$ (a
result shown by Abraham, Dolev, Geffner, and Halpern [2019b]).
In the proof of Theorem 2 we use a slightly modified variant of Geffner and
Halpern’s construction. In this variant, each player $i$ terminates only if
all messages sent in the past are correct according to $\vec{\sigma}$ and
$i$’s local history. If $i$ ever lied to some player $j$, $i$ sends the
correct message before terminating. The purpose of this is that even if a
player $j$ is not able to proceed with $\vec{\sigma}$ (for instance, because
it received incorrect messages), $j$ would still believe that in the future it
will still be able to reconstruct the (simulated) mediator’s messages as long
as the remaining players are able to continue. Additionally, as in the
synchonous case, if players cannot continue with the protocol (for instance,
because they received incorrect messages), they do nothing. It is easy to
check that this variant still $k$-bisimulates
$\vec{\sigma}^{\prime}+\sigma^{\prime}_{d}$ and, thus, $\vec{\sigma}$ is also
a $k$-resilient Nash equilibrium that implements
$\vec{\sigma}^{\prime}+\sigma^{\prime}_{d}$.
The rest of the proof of Theorem 2 consists of finding an appropriate
$k$-paranoid belief system consistent with $\vec{\sigma}$. If these beliefs
exist then, as in the synchronous case, no coalition $K$ of up to $k$ players
would have an incentive to deviate from the proposed protocol since, according
to their beliefs, the remaining players would still be able to compute their
part of the action profile $\vec{a}$ (which is sampled from a communication
equilibrium), and thus it is optimal for players in $K$ to play their part of
$\vec{a}$ as well. Note that, since the protocol states that players should
send the correct messages at the very end of the protocol, with any belief
system consistent with $\vec{\sigma}$, players in $K$ always believe that they
will be able to compute their part of $\vec{a}$ if they wait long enough,
regardless of their local history. This means that proving Theorem 2 reduces
to proving an analogue of Proposition 5 for asynchronous systems:
###### Proposition 8.
For all strategies $\vec{\sigma}$ in $\Gamma_{\mathit{CT},asyn}$, there exists
a $k$-paranoid belief system $b$ that is consistent with $\vec{\sigma}$.
###### Proof.
Consider a sequence $\vec{\sigma}^{1},\vec{\sigma}^{2},\ldots$ of strategy
profiles defined just as in the proof of Proposition 5, and consider a
randomized scheduler $\sigma_{e}$ such that all valid _schedule prefixes_
(i.e., ordered list of players scheduled and messages delivered so far) occur
with positive probability. For instance, $\sigma_{e}$ could be a scheduler
that at each point in time either delivers a message or chooses a player to be
scheduled uniformly at random. Consider the belief system $b$ induced by the
previous sequence of strategies with scheduler $\sigma_{e}$. An argument
similar to that given in Lemmas 1 and 2 shows that $b$ is a $k$-paranoid
belief system, as desired. ∎
### 6.3 Characterization of $SE_{k}(\Gamma_{d,asyn})$ and
$SE_{k}^{S}(\Gamma_{d,asyn})$
In the asynchronous setting, the description of $SE_{k}(\Gamma_{d,asyn})$ is a
bit more convoluted than those of Section 5. Reasoning similar to that used in
the proof of Proposition 6 shows that for a fixed scheduler, a $k$-resilient
(resp., strongly $k$-resilient) sequential equilibrium in
$SE_{k}(\Gamma_{d,asyn})$ induces a $k$-resilient (resp., strongly
$k$-resilient) correlated equilibrium in $\Gamma$. Thus,
$SE_{k}(\Gamma_{d,asyn})\subseteq\mathcal{P}(CE_{k}(\Gamma))$ and
$SE_{k}^{S}(\Gamma_{d,asyn})\subseteq\mathcal{P}(CE_{k}^{S}(\Gamma))$, where
$\mathcal{P(S)}$ denotes the power set of $S$ (i.e., the set of all subsets of
$S$). The next proposition gives a precise description of
$SE_{k}(\Gamma_{d,asyn})$ and $SE_{k}^{S}(\Gamma_{d,asyn})$.
###### Proposition 9.
Given a set $S$ of strategies, let $\mathcal{P}_{=}(S)$ the set of nonempty
subsets $S^{\prime}$ of $S$ such that every element of $S^{\prime}$ gives the
same expected utility to all players. If $\Gamma=(P,A,U)$ is a normal-form
game for $n$ players and $n\geq k$, then
$SE_{k}(\Gamma_{d,asyn})=\mathcal{P}_{=}(CE_{k}(\Gamma))$ and
$SE_{k}^{S}(\Gamma_{d,asyn})=\mathcal{P}_{=}(CE_{k}^{S}(\Gamma))$.
###### Proof.
In games with a mediator, we write $\sigma_{d}$ to denote a generic strategy
for the mediator, and $\vec{\sigma}+\sigma_{d}$ to denote a strategy profile
for the players and the mediator. To show that
$SE_{k}(\Gamma_{d,asyn})\subseteq\mathcal{P}_{=}(CE_{k}(\Gamma))$ and
$SE_{k}^{S}(\Gamma_{d,asyn})\subseteq\mathcal{P}_{=}(CE_{k}^{S}(\Gamma))$,
suppose by way of contradiction that some strategy $\vec{\sigma}+\sigma_{d}$
in $SE_{k}(\Gamma_{d,asyn})$ (resp., $SE_{k}^{S}(\Gamma_{d,asyn})$) induces a
set $S$ of strategies such that, for some player $i$, there exist two
strategies $\vec{\tau},\vec{\tau}^{\prime}\in S$ induced by schedulers
$\sigma_{e}$ and $\sigma_{e}^{\prime}$, respectively, such that
$u_{i}(\vec{\tau})<u_{i}(\vec{\tau}^{\prime})$. Consider a scheduler
$\sigma_{e}^{\prime\prime}$ that does the following: it first schedules player
$i$. If $i$ sends a message to itself when it is first scheduled, then
$\sigma_{e}$ acts like $\sigma^{\prime}_{e}$, and otherwise it acts like
$\sigma_{e}$. By construction, when the scheduler is
$\sigma^{\prime\prime}_{e}$, $i$ gains by deviating from $\sigma_{i}$ and
sending a message to itself when it is first scheduled. (This construction
will not work if $\sigma_{i}$ already requires $i$ to send a message to
itself; in this case, we construct $\sigma^{\prime\prime}_{e}$ so that the
signal from $i$ to the scheduler is encoded differently, for example, by $i$
sending two messages to itself.) This shows that
$SE_{k}(\Gamma_{d,asyn})\subseteq\mathcal{P}_{=}(CE_{k}(\Gamma))$ and
$SE_{k}^{S}(\Gamma_{d,asyn})\subseteq\mathcal{P}_{=}(CE_{k}^{S}(\Gamma))$.
To prove the opposite inclusions, since we consider only equilibria with
rational probabilities, the set $CE_{k}(\Gamma)$ (resp., $CE_{k}^{S}(\Gamma)$)
is countable, as is any subset $S\subseteq CE_{k}(\Gamma)$ (resp., any subset
$S\subseteq CE_{k}^{S}(\Gamma)$). Given
$S=\\{\vec{\tau}^{1},\vec{\tau}^{2},\ldots\\}$ such that for all $k$ and
players $i$ and $j$, $u_{i}(\vec{\tau}^{k})=u_{j}(\vec{\tau}^{k})$, consider
the following strategy $\vec{\sigma}+\sigma_{d}$. Each player sends an empty
message to the mediator when it is first scheduled. Let $N$ be the number of
times the mediator that has been scheduled before receiving a message from
some player. The mediator samples an action profile $\vec{a}$ from
$\vec{\tau}^{N}$ (or from $\vec{\tau}^{1}$ if $S$ is finite and has fewer than
$N$ elements), and sends $a_{i}$ to each player $i$. Players play whatever
they receive from the mediator. It is straightforward to check that for each
strategy $\vec{\tau}$ in $S$, there is a scheduler $\sigma_{e}$ that induces
$\vec{\tau}$ when the players and the mediator play $\vec{\sigma}+\sigma_{d}$.
To check that $\vec{\sigma}+\sigma_{d}$ is indeed a $k$-resilient (resp.,
strongly $k$-resilient) correlated equilibrium, note that all strategies in
$S$ are $k$-resilient (resp., strongly $k$-resilient) correlated equilibria of
$\Gamma$ that give the same utility to each player, and thus (a) after
determining the number of times $t$ that the mediator is scheduled before
receiving the first message, no coalition of $k$ or less players can benefit
from playing an action profile different from the one suggested by the
mediator, and (b) the expected utility of each player is independent of $t$.
This shows that $\vec{\sigma}+\sigma_{d}$ is indeed a $k$-resilient (resp.,
strongly $k$-resilient) correlated equilibrium. ∎
Theorem 2 and Proposition 9 together imply:
###### Corollary 3.
If $\Gamma=(P,A,U)$ is a normal-form game for $n$ players and $n>4k$, then
$SE_{k}(\Gamma_{\mathit{CT},asyn})=\mathcal{P}_{=}(CE_{k}(\Gamma))$ and
$SE_{k}^{S}(\Gamma_{\mathit{CT},asyn})=\mathcal{P}_{=}(CE_{k}^{S}(\Gamma))$
with both the DM and AH approaches.
For Bayesian games, we do not believe that there is a simple description of
$SE_{k}(\Gamma_{d,asyn})$. To understand why, note that a key step in the
proofs of Theorems 1 and 2 is for players to send their types to the mediator.
In asynchronous systems, the mediator cannot distinguish between players that
don’t send their type to the mediator and players that send it but the receipt
of the message is delayed by the scheduler. Nevertheless, if players can
somehow punish those that never send their type to the mediator, we can
guarantee that it is optimal for all players to send a type to the mediator,
and thus we can extend the constructions used in the proof of Propositions 9
and 7 to describe $SE_{k}(\Gamma_{d,asyn})$ and $SE_{k}^{S}(\Gamma_{d,asyn})$.
###### Definition 10 (Ben-Porath [2003]; Abraham et al. [2006]).
If $\Gamma=(P,T,q,S,U)$ is a Bayesian game for $n$ players and $\mu\in
Com_{k}(\Gamma)$ or $\mu\in Com_{k}^{S}(\Gamma)$, then a _$k$ -punishment
equilibrium_ with respect to $\mu$ is a $k$-resilient Bayesian Nash
equilibrium $\vec{\tau}$ such that $u_{i}(\mu)>u_{i}(\vec{\tau})$ for all
players $i$.
Intuitively, a $k$-punishment equilibrium with respect to $\mu$ is a strategy
profile that is a $k$-resilient Bayesian Nash equilibrium which, if played by
at least $n-k$ players, results in all players being worse off than they would
be with $\mu$. If a $k$-punishment equilibrium exists, then we have the
following.
###### Proposition 10.
If $\Gamma=(P,T,q,S,U)$ is a Bayesian game, $S$ is a subset of $k$-resilient
(resp., strongly $k$-resilient) communication equilibria of $\Gamma$ such
that, for all players $i$ and all types $t_{i}$ of $i$, the expected utility
of $i$ given $t_{i}$ is the same for all $\mu$ in $S$ (i.e., $u_{i}(\mu\mid
t_{i})=u_{i}(\mu^{\prime}\mid t_{i})$ for all $\mu,\mu^{\prime}\in S$), and
there exists a $k$-punishment equilibrium $\vec{\tau}$ with respect to some
$\mu$ in $S$, then $S\in SE_{k}(\Gamma_{d,asyn})$ (resp., $S\in
SE_{k}^{S}(\Gamma_{d,asyn})$) with the AH approach or with the default move
approach if, for each player $i$, $i$’s default move is $\tau_{i}$.
Note that if $\vec{\tau}$ is a $k$-punishment equilibrium with respect to some
$\mu$ in $S$, then it is a $k$-punishment equilibrium with respect to all
$\mu\in S$, since all $\mu$ give the same utility to the players.
###### Proof.
This proof follows the same lines as the proof of Proposition 9, with the only
difference that the players must tell the mediator their type at the beginning
of the game. The punishment strategy is required to force players to tell
their true type; if players do not tell report a type, the remaining players
will play their part of the punishment strategy (note that in Proposition 9, a
punishment strategy is not needed since players have no types and thus the
mediator requires no inputs). Moreover, this type must be their true type
since all strategies in $S$ are $k$-resilient communication equilibria of
$\Gamma$. Since we are considering only distributions with rational
probabilities, the set $S$ is countable. Let $S=\\{\mu^{1},\mu^{2},\ldots\\}$.
Consider a strategy $\vec{\sigma}+\sigma_{d}$ in which each player sends its
type to the mediator when it is first scheduled and the mediator waits until
it receives the type of each player. Let $N$ be the number of times that the
mediator is scheduled before receiving all the types. The mediator samples an
action profile $\vec{a}$ from $\mu^{N}$ (or from $\mu^{1}$ if $S$ is finite
and has fewer than $N$ elements), and sends $a_{i}$ to each player $i$. If a
player sent its true type to the mediator, it plays whatever it receives from
the mediator. Otherwise, it plays the optimal action conditioned on what it
sent and what it receives from the mediator. If player $i$ never receives an
action, it plays $\tau_{i}$ (either because it is the default action or
because $i$ leaves it in its will). As in the proof of Proposition 9, it is
easy to check that $\vec{\sigma}$ induces $S$.
We next show that $\vec{\sigma}$ is a $k$-resilient (resp., strongly
$k$-resilient) sequential equilibrium for some belief system $b$. Consider a
sequence of strategy profiles $\vec{\sigma}^{1},\vec{\sigma}^{2},\ldots$ such
that, in $\vec{\sigma}^{N}$, each player $i$ acts as follows. The first time
$i$ is scheduled, it sends its type $t_{i}$ to the mediator with probability
$1-1/N$ (instead of sending its type to the mediator with probability 1, as in
$\vec{\sigma}$). Moreover, each time it is scheduled, $i$ sends a random
message to each other player $j$ with probability $1/N$. Whenever a player $i$
receives an action $a_{i}$ from the mediator, it plays $a_{i}$. Otherwise, it
plays $\vec{\tau}(t_{i})$. By construction, the strategy profiles
$\vec{\sigma}^{1}+\sigma_{d},\vec{\sigma}^{2}+\sigma_{d},\ldots$ are
completely mixed and converge to $\vec{\sigma}$. Let $b$ be the belief system
induced by $\vec{\sigma}^{1}+\sigma_{d},\vec{\sigma}^{2}+\sigma_{d},\ldots$.
According to $b$, since all messages received from other players are randomly
generated, they can be ignored. Thus, $i$’s utility depends only on the
message received by the mediator. If player $i$ receives a message from the
mediator, then it must be that every player sent its type and therefore, if
$i$ sent its true type, it is optimal for $i$ to play the action received. If
$i$ did not report its true type, by construction $i$ also plays its optimal
action. On the other hand, if $i$ never receives a message from the mediator,
then according to its beliefs (as characterized by $b$), it must be that no
one else got a message and thus everyone else is going to play their part of
$\vec{\tau}$. Since $\vec{\tau}$ is a $k$-resilient Nash equilibrium, playing
$\tau_{i}$ is optimal. This shows that $(\vec{\sigma},b)$ is a $k$-resilient
(resp., strongly $k$-resilient) sequential equilibrium. ∎
Proposition 10 implies the following:
###### Corollary 4.
If $\Gamma=(P,T,q,S,U)$ is a Bayesian game and $\mu$ is a $k$-resilient
(resp., strongly $k$-resilient) correlated equilibrium of $\Gamma$ such that
there exists a $k$-punishment equilibrium $\vec{\tau}$ w.r.t. $\mu$, then
$\\{\mu\\}\in SE_{k}(\Gamma_{d,asyn})$ (resp., $\\{\mu\\}\in
SE_{k}^{S}(\Gamma_{d,asyn})$) with the AH approach or with the default move
approach if the default move is $\vec{\tau}$.
## 7 Extending the set of equilibria
Up to now we have considered only distributions over action profiles that
involve rational probabilities. We can extend our result to some extent to
distributions described by real numbers. Specifically, suppose that players
can send messages and operate with real numbers, and an equilibrium
$e=\sum_{i}\lambda_{i}e_{i}$ is a convex combination of countable rational
equilibria $e_{i}$. If players could jointly sample a real number $r\in[0,1)$
uniformly at random, they could easily implement $e$: after sampling
$r\in[0,1]$ uniformly at random, they play their part of the rational
equilibrium $e_{m}$ such that $\sum_{i=1}^{m-1}\lambda_{i}\leq
r<\sum_{i=1}^{m}\lambda_{i}$. (Our earlier results show that $e_{m}$ can be
implemented.) Note that this procedure guarantees that players implement
$e_{m}$ with probability $\lambda_{m}$, as desired.
The following is a $k$-resilient strategy $\vec{\tau}$ that allows players to
jointly sample $r$ in the synchronous setting if $n>3k$:
1. 1.
Each player privately generates a random number $r_{i}\in[0,1)$ and broadcasts
it using Bracha’s $k$-resilient broadcast protocol Bracha [1987]. If $i$
doesn’t broadcast a number, $r_{i}$ is assumed to be $0$.
2. 2.
Players take $r:=frac(r_{1}+\ldots+r_{n})$, where $frac(x)$ is the fractional
part of $x$.
In the asynchronous setting, this process is harder because players cannot
distinguish players that didn’t broadcast a value from players that are being
delayed by the scheduler. Moreover, players cannot agree on using only a
subset $C$ of the values shared since players might want to influence the
computation of $C$ in such a way that the number $r$ chosen is beneficial for
them. However, if $n>4k$, each player can generate $r_{i}$ as in the
synchronous setting, and then compute $r$ using a $k$-resilient secure
computation Ben-Or et al. [1993].
## 8 Conclusion
We have shown that, for all Bayesian games $\Gamma$ for $n$ players, all
$k$-resilient sequential equilibria in $\Gamma_{d}$ can also be implemented in
$\Gamma_{\mathit{CT}}$ if $n>3k$ in the synchronous setting or $n>4k$ in the
asynchronous setting. These results are optimal since it follows from Abraham
et al. [2008b] and Geffner and Halpern [2023] that if $n\leq 3k$ in the
synchronous case or $n\leq 4k$ in the asynchronous case, there are
$k$-resilient sequential equilibria in $\Gamma_{d}$ that cannot be implemented
even with a $k$-resilient Nash equilibrium.
Our results also allow us to characterize the sets
$SE_{k}(\Gamma_{\mathit{CT}})$ of all possible strategies in $\Gamma$ induced
by $k$-resilient sequential equilibria of $\Gamma_{ACT}$, generalizing results
of Gerardi [2004]. If players have perfect information (i.e. $\Gamma$ is a
normal-form game), then $SE_{k}(\Gamma_{\mathit{CT}})$ is the set of
$k$-resilient correlated equilibria of $\Gamma$; if $\Gamma$ is a Bayesian
game, then $SE_{k}(\Gamma_{\mathit{CT}})$ is the set of $k$-resilient
communication equilibria. In the asynchronous setting, we show that
$SE_{k}(\Gamma_{\mathit{CT}})=SE_{k}(\Gamma_{d})$, but characterizing the set
$SE_{k}(\Gamma_{d})$ of $k$-resilient sequential equilibria with a trusted
mediator is still an open problem. However, if $\Gamma$ has a $k$-punishment
equilibrium, then $SE_{k}(\Gamma_{d})$ is also the set of $k$-resilient
communication equilibria of $\Gamma$.
If $\Gamma$ has a $k$-punishment equilibrium, Abraham, Dolev, Gonen, and
Halpern [2006] and Abraham, Dolev, Geffner, and Halpern [2019b] showed that
all $k$-resilient Nash equilibria in $\Gamma_{d}$ could be implemented in
$\Gamma_{\mathit{CT}}$ if $n>2k$ in the synchronous setting or $n>3k$ in the
asynchronous setting respectively. The question of whether these results can
be extended to $k$-resilient sequential equilibria is still open.
## Appendix A Game theory: basic definitions
In this section, we review the basic concepts and definitions that are needed
for this paper. We begin by presenting several types of games that are common
in the literature, each of them with its own settings. For each of these
games, we also introduce some of the _solution concepts_ that we are
interested in. A solution concept describes what rational agents should do in
a game, according to some theory of rationality. Typically, a solution concept
determines for each game a set of _strategy profiles_ , where a strategy
profile $\vec{s}=(s_{1},\ldots,s_{n})$ is a tuple consisting of one strategy
for each agent, and a strategy is a description of how each player should
move/act in all “situations” in the given game. (As we shall see, what counts
as a situation depends on the type of game we consider.)
### A.1 Nash, Correlated, and Sequential Equilibrium
A normal-form game $\Gamma$ is a tuple $(P,A,U)$, where $P=\\{1,\ldots,n\\}$
is the set of _players_ , $A=A_{1}\times\cdots A_{n}$, where $A_{i}$ is the
set of possible actions for player $i\in P$, and $U=(u,\ldots,u_{n})$ is an
$n$-tuple of _utility_ functions $u_{i}:A\rightarrow\mathbb{R}$, again, one
for each player $i\in P$. A _pure strategy profile_ $\vec{a}$ is an $n$-tuple
of actions $(a_{1},\ldots,a_{n})$, with $a_{i}\in A_{i}$. A _mixed strategy_
for player $i$ is an element of $\Delta(A_{i})$, the set of probability
distributions on $A_{i}$. We extend $u_{i}$ to mixed strategy profiles
$\sigma=(\sigma_{1},\ldots,\sigma_{n})$ by defining
$u_{i}(\sigma)=\sum_{(a_{1},\ldots,a_{n})\in
A}\sigma_{1}(a_{1})\ldots\sigma_{n}(a_{n})u_{i}(a_{1},\ldots,a_{n})$: that is,
the sum over all pure strategy profiles $\vec{a}$ of the probability of
playing $\vec{a}$ according to $\sigma$ times the utility of $\vec{a}$ to $i$.
###### Definition 11.
In a normal-form game $\Gamma$, a (mixed) strategy profile
$\vec{\sigma}=(\sigma_{1},\ldots,\sigma_{n})$ is a _Nash equilibrium_ if, for
all $i\in P$ and strategies $\sigma_{i}^{\prime}\in\Delta(A_{i})$,
$u_{i}(\sigma_{i},\vec{\sigma}_{-i})\geq
u_{i}(\sigma^{\prime}_{i},\vec{\sigma}_{-i}).$
###### Definition 12.
In a normal-form game $\Gamma=(P,A,U)$, a distribution $p\in\Delta(A)$ is a
_correlated equilibrium_ (CE) if, for all $i\in P$ and all
$a^{\prime}_{i},a^{\prime\prime}_{i}\in A_{i}$ such that
$p(a^{\prime}_{i})>0$,
$\sum_{\vec{a}\in\Delta(A):a_{i}=a^{\prime}_{i}}u_{i}(a^{\prime}_{i},\vec{a}_{-i})p(\vec{a}\mid
a_{i}=a^{\prime}_{i})\geq\sum_{\vec{a}\in\Delta(A):a_{i}=a^{\prime}_{i}}u_{i}(a_{i}^{\prime\prime},\vec{a}_{-i})p(\vec{a}\mid
a_{i}=a^{\prime}_{i}).$
Intuitively, for a distribution $p$ over action profiles to be a correlated
equilibrium, it cannot be worthwhile for player $i$ to deviate from $a_{i}$ to
$a_{i}^{\prime}$ if $i$ knows only that $\vec{a}$ is sampled from distribution
$p$ (and $a_{i}$). Note that all Nash equilibria are correlated equilibria as
well.
An _extensive-form game_ $\Gamma$ is a tuple $(P,G,M,U,R)$, where
* •
$P=\\{1,\ldots,n\\}$ is the set of players.
* •
$G$ is a game tree in which each node represents a possible state of the game
and each of the edges going out of a node an action that a player can perform
in the state corresponding to that node. (In the sequel, we refer to the edges
as actions.)
* •
$M$ is a function that associates with each non-leaf node of $G$ (we let
$G^{\circ}$ denote the non-leaf nodes) a player in $P$ that describes which
player moves at each of these nodes; if $M(v)=i$, then all the edges going out
of node $v$ must correspond to actions that player $i$ can play.
* •
$U=(u_{1},\ldots,u_{n})$ is an $n$-tuple of utility functions from the leaves
of $G$ to $\mathbb{R}$.
* •
$R=(\sim_{1},\ldots,\sim_{n})$ is an $n$-tuple of equivalence relations on the
nodes of $G^{\circ}$, one for each player. Intuitively, if
$v\sim_{i}v^{\prime}$, then nodes $v$ and $v^{\prime}$ are indistinguishable
to player $i$. Clearly, if $v\sim_{i}v^{\prime}$ for some $i\in[n]$, then the
set of possible actions that can be performed at $v$ and $v^{\prime}$ must be
identical and $M(v)=M(v^{\prime})$ (for otherwise, $i$ would have a basis for
distinguishing $v$ and $v^{\prime}$).
The equivalence relation $\sim_{i}$ induces a partition of $G^{\circ}$ into
equivalence classes called _information sets_ (of player $i$). It is more
standard in game theory to define $\sim_{i}$ as an equivalence relation only
on the nodes where $i$ moves. We define it on all non-leaf nodes because if
$K$ is a subset of players, we are then able to define $\sim_{K}$ as the the
intersection of $\sim_{i}$ for $i\in K$. Intuitively, $v\sim_{K}v^{\prime}$ if
the agents in $K$ cannot distinguish $v$ from $v^{\prime}$, even if they pool
all their information together. We use the equivalence relation $\sim_{K}$
when defining $k$-resilient sequential equilibrium.
A _(pure) strategy_ $s_{i}$ for player $i$ in an extensive-form game $\Gamma$
is a function that maps each node $v$ where $i$ moves to an action that can be
performed at $v$, with the constraint that if two nodes are indistinguishable
to $i$, then $s_{i}$ must choose the same action on both. More precisely, if
$v\sim_{i}v^{\prime}$ then $s_{i}(v)=s_{i}(v^{\prime})$. Each pure strategy
profile $\vec{s}$ induces a unique path from the root of the tree to some leaf
$\ell\in G\setminus G^{\circ}$ where, given node $v$ on the path, the next
node in the path is the node reached from $v$ by the edge (action)
$s_{M(v)}(v)$. The payoff of player $i$ given strategy $\vec{\sigma}$ is given
by $u_{i}(\ell)$, although we also write $u_{i}(\vec{\sigma})$ for simplicity.
A _behavioral strategy_ for player $i$ allows randomization. More precisely, a
behavioral strategy is a function $s_{i}$ that maps each node $v$ such that
$M(v)=i$ to a distribution over $i$’s possible actions at $v$ (again, with the
requirement that $s_{i}(v)=s_{i}(v^{\prime})$ if $v\sim_{i}v^{\prime}$). We
denote by $u_{i}(\vec{\sigma})$ the expected payoff of player $i$ if players
play (behavioral) strategy profile $\vec{\sigma}$.
Nash equilibrium is defined for extensive-form games just as it is in
strategic-form games. Given a NE $\vec{\sigma}$, the _equilibrium path_
consists of the nodes of $G$ that can be reached with positive probability
using $\vec{\sigma}$. It is well known that in extensive-form games, Nash
equilibrium does not always describe what intuitively would be a reasonable
play. For instance, consider the following game for two players in which
$(p_{1},p_{2})$ describes the payoff of players 1 and 2 respectively:
In this case, the strategy profile in which player 1 plays $B$ at node $a$ and
player 2 plays $L$ at node $b$ and $R$ at $c$ is a Nash equilibrium. However,
the reason that it is better for player 1 to play $B$ is that if she plays
$A$, then player 2 would play $L$ and they would both get a utility of $-5$.
This means that player 1 is being influenced by an irrational threat, since if
1 plays $A$, it is in player 2’s best interest to play $R$ instead of $L$. In
order to avoid these situations, we can extend the notion of Nash equilibrium
to _subgame-perfect Nash equilibrium_ , in which, roughly speaking, the
strategy must be a Nash equilibrium, not only in $\Gamma$, but in all the of
subgames of $\Gamma$ as well. In the example above, the only subgame-perfect
equilibrium is the one given by $\sigma_{1}(a)=A$, $\sigma_{2}(b)=R$ and
$\sigma_{2}(c)=R$.
Unfortunately, subgame-perfect equilibrium may not be well-defined if players
have nontrivial information sets. Consider the following game, where $b$ and
$b^{\prime}$ are in the same information set for player 2, as are $c$ and
$c^{\prime}$.
At node $b^{\prime}$, player 2 would play $L$; at $b$, he would play $R$. For
player 2 to decide its move, it must have a belief about the probability of
being at each node of the information set, and this belief must be consistent
with the strategy $\vec{\sigma}$ being played. For example, if
$\sigma_{1}(a)=B$, then if player 2 is in information set
$\\{b,b^{\prime}\\}$, then 2 would be sure that the true node is $b^{\prime}$.
We capture the players’ beliefs by using a _belief system_ $b$, which is a
function from information sets $I$ to a probability distribution over nodes in
$I$. Intuitively, if $I$ is an information set for player $i$, then $b$
represents how likely it is for $i$ to be at each of the nodes of $I$. Note
that if $\vec{\sigma}$ is _completely mixed_ , which means that all actions
are taken with positive probability, then $b$ is uniquely determined by Bayes’
rule. More generally, we say that a belief system $b$ is _consistent_ with a
strategy $\vec{\sigma}$ if there exists a sequence
$\vec{\sigma}^{1},\vec{\sigma}^{2},\ldots$ of completely-mixed strategies that
converges to $\vec{\sigma}$ such that the beliefs induced by Bayes’ rule
converge to $b$. Given an extensive-form game $\Gamma=(P,G,M,U,R)$, a strategy
profile $\vec{\sigma}$, a belief system $b$, and an information set $I$ for
player $i$, let $u_{i}(\vec{\sigma},I,b)$ denote $i$’s expected utility
conditional on being in information set $I$ and having belief system $b$, if
$\vec{\sigma}$ is played.
###### Definition 13 (Kreps and Wilson [1982]).
A pair $(\vec{\sigma},b)$ consisting of a strategy profile $\vec{\sigma}$ and
a belief system $b$ consistent with $\vec{\sigma}$ is a _sequential
equilibrium_ if, for all players $i$, all information sets $I$ of player $i$,
and all strategies $\tau_{i}$ for player $i$,
$u_{i}(\vec{\sigma},I,b)\geq u_{i}((\tau_{i},\vec{\sigma}_{-i}),I,b).$
Even though it is standard to define a sequential equilibrium as a pair
$(\vec{\sigma},b)$ consisting of strategy profile and a belief system, for
convenience we also say that a strategy profile $\vec{\sigma}$ is a sequential
equilibrium if there exists a belief system $b$ such that $(\vec{\sigma},b)$
is a sequential equilibrium.
### A.2 Bayesian games
In all of the previous definitions, the utility of each player is assumed to
be common knowledge (_perfect information_). However, this is not always the
case. Bayesian games capture this idea by assuming that each player $i$ has a
type $t_{i}\in T_{i}$ sampled from a distribution $q\in\Delta(T)$, where
$T=(T_{1},\ldots,T_{n})$, and that the utility $u_{i}$ of $i$ is not only a
function of the action profile being played, but also of its type $t_{i}$.
Formally, a _Bayesian games_ is a tuple $(P,T,q,A,U)$, where, as in normal-
form games, $P$, $A$, and $U$ are the set of players, their actions, and their
utility functions, respectively; $T$ is the set of possible type profiles, and
$q$ is a distribution in $\Delta(T)$.
A _strategy_ in a Bayesian game for player $i$ is a map
$\mu_{i}:T_{i}\rightarrow\Delta(A_{i})$. Intuitively, a strategy in a Bayesian
game tells player $i$ how to choose its action given its type. Since the
distribution $q$ is common knowledge, given a strategy profile $\vec{\mu}$ in
$\Gamma$, the expected utility of a player $i$ is
$u_{i}(\vec{\mu})=\sum_{t_{i}\in T_{i}}q(t_{i})\sum_{\vec{t}}q(\vec{t}\mid
t_{i})u_{i}(\vec{\mu}(\vec{t})),$ (1)
where $u_{i}(\vec{\mu}(\vec{t}))$ denotes the expected utility of player $i$
when an action profile is chosen according to $(\mu(\vec{t}))$. This allows us
to define Bayesian Nash equilibrium as follows:
###### Definition 14.
In a Bayesian Game $\Gamma=(P,T,q,A,U)$, a strategy profile
$\vec{\mu}:=(\mu_{1},\ldots,\mu_{n})$ is a _(Bayesian) Nash equilibrium_ if,
for all players $i$ and all strategies $u^{\prime}_{i}$ for $i$,
$u_{i}(\vec{\mu})\geq u_{i}(\vec{\mu}_{-i},\mu^{\prime}_{i}).$
We can generalize correlated equilibrium to Bayesian games as follows. We can
view a correlated equilibrium as a distribution $p\in\Delta(A)$ such that if a
trusted mediator samples an action profile $\vec{a}\in p$ and sends action
$a_{i}$ to each player $i$, it is always better for $i$ to play $a_{i}$ rather
than something else. In a Bayesian game, the mediator instead samples the
action profile from a distribution that depends on the type profile. More
precisely, suppose that players send their types to a trusted mediator, the
mediator samples an action profile $\vec{a}$ from a distribution
$\mu(\vec{t})$ that depends on the type profile $\vec{t}$ received, and then
sends action $a_{i}$ to each player $i$. We say that $\mu$ is a _communication
equilibrium_ if it is optimal for the players to (a) tell their true type to
the mediator, and (b) play the action sent by the mediator. The following
definitions make this precise.
###### Definition 15.
Forges [1986]; Myerson [1986] A correlated strategy profile
$\mu:T\rightarrow\Delta(A)$ is a _communication equilibrium_ of Bayesian Game
$\Gamma=(P,T,q,A,U)$ if, for all $i\in P$, all $t_{i}\in T_{i}$, all
$\psi:T_{i}\rightarrow T_{i}$ and all $\varphi:A_{i}\rightarrow A_{i}$, we
have that
$\sum_{\vec{t}_{-i}\in T_{-i}}\sum_{\vec{a}\in
A}q(t_{-i},t_{i})\mu(\vec{a}\mid\vec{t}_{-i},t_{i})u_{i}(\vec{t}_{-i},t_{i},\vec{a})\geq$
$\sum_{\vec{t}_{-i}\in T_{-i}}\sum_{\vec{a}\in
A}q(t_{-i},t_{i})\mu(\vec{a}\mid\vec{t}_{-i},\psi(t_{i}))u_{i}(\vec{t}_{-i},t_{i},\vec{a}_{-i},\varphi(a_{i})).$
We can combine the notions of extensive-form game and Bayesian game in the
obvious way to get _extensive-form Bayesian games_ : we start with an
extensive form game, add a type space $T$ and a commonly known distribution
$q$ on $T$, and then have the utility function depend on the type profile as
well the leaf reached. We leave formal details to the reader.
## Appendix B VSS and CC: a review
In this section, we present the two main primitives used to construct the
strategies required for the proof of Theorem 1. The purpose of these
primitives is that agents are able to distribute information between them
(verifiable secret sharing), and compute any relevant data from the
information shared without learning anything besides the data itself (circuit
computation).
Intuitively, verifiable secret sharing is designed to distribute between the
players a given piece of information $y$ in some finite field $\mathbb{F}_{q}$
in such a way that each agent $i$ knows only a part $y_{i}$ of the secret that
does not reveal any information about the secret by itself. In fact, with
$k$-resilient verifiable secret sharing, even if $k$ agents collude and put
their information together, they still cannot deduce anything about the shared
secret, but any subset of $k+1$ agents can compute the secret with no error
given their pieces of information. To achieve these properties, the idea is
that the agent $i$ that knows the secret $y$ shares it using Shamir’s secret-
sharing scheme Shamir [1979]: $i$ gives each agent $j$ a value $y_{j}$ such
that there exists a polynomial $p$ of degree $k$ such that $p(j)=y_{j}$ for
each $j$ and $p(0)=y$. It is easy to check that, if $p$ is chosen uniformly at
random, then the values $y_{i}$ received by any subset of $k$ agents is
uniformly distributed in $\mathbb{F}_{q}^{k}$, and thus convey no information
about $y$. However, any subset of $k+1$ agents can reconstruct the secret $y$
by interpolating their $k+1$ points. Note that, since up to $k$ agents may
misreport their values, players cannot just simply reconstruct the secret
using the first $k+1$ points they receive. In fact, they must wait until
receiving at least $2k+1$ points that lie on the same polynomial $p$ of degree
$k$. In that case, even if $k$ players misreport their values, at least $k+1$
of the points in $p$ are guaranteed to be reported by honest players, which
ensures that $p$ defines the correct secret. At a high level, this is why this
primitive requires that the number of players $n$ satisfies $n>3k$: even if
$k$ players misreport their values, at least $2k+1$ of those values are shared
by honest players, and thus all players are guaranteed to eventually find
$2k+1$ points that lie on the same polynomial of degree $k$, which allows them
to reconstruct the secret correctly.
In order to share a secret using Shamir’s secret-sharing scheme, it is
unfortunately not enough to have the sender choose a polynomial $p$ uniformly
at random such that $p(0)=y$ and send each agent $i$ the value of $p(i)$. This
protocol is vulnerable when the sender is malicious: it could generate $n$
points that do not lie on the same polynomial, and agents could never
reconstruct the secret. In order to guarantee that the shared points define
some secret, players have to follow a non-trivial protocol that checks the
validity of the points without leaking information about the underlying
secret. In the synchronous setting, Ben-Or, Goldwasser, and Wigderson [1988]
provide a $k$-resilient VSS protocol if $n>3k$; in the asynchronous setting,
Ben-Or, Canetti, and Goldreich [1993] provide a $k$-resilient VSS protocol if
$n>4k$. Intuitively, in asynchronous systems, $n>4k$ is necessary since $k$ of
the $4k+1$ players might be delayed arbitrarily (and are indistinguishable
from deviating players that didn’t send a message), and $k$ might send the
wrong values, which leaves $2k+1$ points to interpolate without error. A more
rigorous description of the properties of VSS can be found in Appendix B.1.
In circuit computation, players generate the shares of either the sum or
product of two shared secrets without leaking information about the secrets
themselves. More precisely, suppose that secrets $y,z\in\mathbb{F}_{q}$ are
shared among the players. Then $k$-resilient circuit computation that allows
each player to compute its share of $y+z$ or $yz$ without leaking any
information about $y$ or $z$ to any coalition of at most $k$ players. By
successively using this primitive, players can compute the shares of any
secret that is the output of a circuit with addition and multiplication gates
given the inputs. This, in turn, allows agents to compute the shares of the
output of any function whose domain and range are finite, since, without loss
of generality, we can then take the domain and range to be $\mathbb{F}_{q}$
for some $q$ sufficiently large, and any function
$f:\mathbb{F}_{q}\rightarrow\mathbb{F}_{q}$ can be viewed as a polynomial on
$\mathbb{F}_{q}$, and so involves only addition and multiplication. An
implementation of a $k$-resilient circuit computation protocol is given by
Ben-Or, Goldwasser, and Wigderson [1988] in the case of synchronous systems,
and by Ben-Or, Canetti, and Goldreich [1993] in the case of asynchronous
systems. A more detailed description of the properties of CC can be found in
Appendix B.2.
### B.1 Verifiable secret sharing
Verifiable secret sharing allows a player (the sender) to securely distribute
shares of a private value $v$ among all players. A protocol $\vec{\sigma}$ is
a _$k$ -resilient implementation of VSS in synchronous systems_ if, for all
senders $s\in[n]$, all coalitions $K\subseteq[n]$ of at most $k$ players, and
all strategies $\vec{\tau}_{K}$ for players in $K$, the following holds of
$(\vec{\sigma}_{-K},\vec{\tau}_{K})$:
* (a)
All players not in $K$ terminate.
* (b)
If $s^{i}$ is $i$’s output, then there exists a polynomial $r$ of degree $k$
such that $r(i)=s^{i}$ for all $i\not\in K$. Moreover, if the sender $s$ is
not in $K$, $r(0)$ is its input $v$.
* (c)
If the sender $s$ is not in $K$, the distribution over histories $h_{K}^{v}$
of players in $K$ when $s$ shares $v$ is the same for all
$v\in\mathbb{F}_{q}$.
Properties (a) and (b) guarantee that, regardless of what a coalition of at
most $k$ players does, all honest players terminate and their outputs are
consistent even when the sender is deviating from the protocol, which means
that even if the sender decides not to share an input or not send messages at
all, all honest players eventually agree on some shares. Property (c)
guarantees that no coalition of $k$ players can learn anything about the
secret being shared if the sender is not in the coalition.
In asynchronous systems, properties (a) and (b) cannot be guaranteed
simultaneously, since a player that sends no messages is indistinguishable
from a player that is being delayed by the scheduler. In this case we have
that:
* (a1)
If the sender is not in $K$, all players not in $K$ terminate.
* (a2)
If a player $i\not\in K$ terminates, all players not in $K$ terminate.
* (b)
If $S$ is the subset of players not in $K$ that terminate and $s^{i}$ is the
output of player $i\in S$, then there exists a polynomial $r$ of degree $k$
such that $r(i)=s^{i}$ for all $i\in S$. Moreover, if the sender $s$ is not in
$K$, $r(0)$ is its input $v$.
* (c)
If the sender $s$ is not in $K$, then for all schedulers, the distribution
over histories $h_{K}^{v}$ of players in $K$ when $s$ shares $v$ is the same
for all $v\in\mathbb{F}_{q}$.
Note that, in the asynchronous setting, a player might not terminate if the
|
# Language Guided Adversarial Purification
###### Abstract
Adversarial purification using generative models demonstrates strong
adversarial defense performance. These methods are classifier and attack-
agnostic, making them versatile but often computationally intensive. Recent
strides in diffusion and score networks have improved image generation and, by
extension, adversarial purification. Another highly efficient class of
adversarial defense methods known as adversarial training requires specific
knowledge of attack vectors, forcing them to be trained extensively on
adversarial examples. To overcome these limitations, we introduce a new
framework, namely Language Guided Adversarial Purification (LGAP), utilizing
pre-trained diffusion models and caption generators to defend against
adversarial attacks. Given an input image, our method first generates a
caption, which is then used to guide the adversarial purification process
through a diffusion network. Our approach has been evaluated against strong
adversarial attacks, proving its effectiveness in enhancing adversarial
robustness. Our results indicate that LGAP outperforms most existing
adversarial defense techniques without requiring specialized network training.
This underscores the generalizability of models trained on large datasets,
highlighting a promising direction for further research.
Index Terms— Adversarial purification, Language guidance, Diffusion
## 1 Introduction
The use of deep neural networks, especially within the realm of computer
vision, has ushered in transformative advancements in various applications.
Despite these strides, a consistent vulnerability is the susceptibility of
such models to adversarial perturbations [1]. These perturbations, often
imperceptible, can fool even the most sophisticated neural networks, causing
them to misclassify inputs. Addressing this alarming vulnerability has become
a research imperative, leading to a rapidly growing body of literature
dedicated to understanding and defending against these adversarial threats [2,
3].
Fig. 1: Illustration of LGAP. A pre-trained image-captioning model (BLIP)
generates captions for input images, providing a textual representation of the
visual content. Leveraging the generated captions, purified images are created
via the diffusion model. The red dashed lines represent the adversarial image
input, while the green dotted lines indicate the resulting purified image.
Historically, adversarial training, introduced by Goodfellow et al. [1], has
been posited as an effective defense strategy. This approach, which integrates
adversarial examples into the training phase, aims to strengthen models
against specific adversarial attacks. However, its efficacy is often limited
to the spectrum of attacks encountered during training, thereby leaving models
vulnerable to novel adversarial strategies. This constraint underscores the
necessity for alternative defensive paradigms.
Given their inherent capability to generate or transform data, generative
models have recently been explored as potential tools for adversarial
purification [4, 5, 6]. Within this domain, diffusion models have emerged as
particularly promising candidates. Recent studies, as exemplified by Nie et
al. [7] and Carlini et al. [8], have harnessed the potential of score-based
and diffusion models towards purification of adversarial samples.
Primarily, the adversarial purification techniques have focussed only on the
image modality, despite promising performance of diffusion models in multi-
modal tasks such as text-to-image generation [9]. Thus, in our work, we
investigate the impact of language towards the robustness of vision models.
Our research focuses on defensive strategy based on vision and language models
trained on large datasets. By leveraging the capabilities of such models
trained jointly on language and vision tasks, we propose a novel framework of
Language Guided Adversarial Purification (LGAP), as illustrated in Figure 1.
This novel framework, which seamlessly integrates a caption generator and a
pre-trained diffusion model with a classifier, leverages the inherent
generalisability of these models to purify an adversarial input. To the best
of our knowledge, language based adversarial purification has not been
addressed in the literature.
We conduct elaborate empirical evaluations across benchmark datasets,
including ImageNet [10], CIFAR-10 [11] and CIFAR-100 [11]. The results of
evaluation against $L_{\infty}$ norm attacks corroborate the robustness of our
framework. Notably, for the ImageNet, our method reveals better performance
compared to previous techniques.
## 2 Related Works
Diffusion models in image generation: The landscape of image generation has
been revolutionized by diffusion models. Rooted in the foundational works of
Sohl-Dickstein et al. [12] and later extended by Song et al. [13] and Ho et
al. [14], these models have exhibited unparalleled prowess in generating high-
quality image samples. Song et al. [15] further advanced this domain by
combining generative learning mechanisms with stochastic differential
equations, thereby broadening the horizon of diffusion models.
Language-image pretraining: A significant milestone in deep learning,
language-image pretraining bridges the gap between textual and visual data.
Pioneering models such as CLIP [16] and BLIP [17] have leveraged vast amounts
of text and image data to jointly train vision and language models,
demonstrating tremendous progress in multi-modal tasks.
Adversarial training: The foundational work of Madry et al. [2] established
adversarial training as a robust method for safeguarding neural networks from
known adversarial attacks. While the effectiveness of the method is well-
recognized, its scalability and adaptability have been enhanced through
inspirations from metric learning [18] and self-supervised paradigms [19].
However, the computational demands of adversarial training has spurred
research into more efficient training methods [20, 21].
Adversarial purification: Generative models have emerged as a pioneer in the
adversarial purification realm. Initial endeavors, such as those by Samangouei
et al. [4], harnessed GANs for purification. Subsequent innovations leaned on
energy-based models (EBMs) to refine the purification process using Langevin
dynamics [22]. Notably, the intersection of score networks and diffusion
models with adversarial purification has been explored recently, with
promising results against benchmark adversarial attacks [6, 7].
## 3 Proposed Method
We propose a novel defense strategy against adversarial attacks on
classification models by leveraging language guidance in diffusion models for
adversarial purification. For a clean sample $\mathbf{x}$ with label $y$, and
a target neural network $f_{\boldsymbol{\theta}}$, the adversary aims to
produce $\mathbf{x}_{\text{adv}}$ by introducing adversarial perturbations.
This results in a prediction $f_{\boldsymbol{\theta}}(\mathbf{x}_{adv})$ that
differs from the original prediction $f_{\boldsymbol{\theta}}(\mathbf{x})=y$.
The underlying premise of the proposed method is to preprocess the input
$\mathbf{x}$ through a diffusion model conditioned on a caption to remove any
adversarial perturbations before feeding it to $f_{\boldsymbol{\theta}}$. We
first discuss the caption generation followed by purification using diffusion
model.
### 3.1 Image captioning
For image captioning, we use a caption generator from BLIP [17]. BLIP has a
multi-modal encoder-decoder architecture which consists of three major
components a unimodal encoder for generating image and text embeddings, an
image-grounded text encoder that computes cross attention and self-attention
between the two encodings to give a multimodal representation of image text
pair, and an image-grounded text decoder that uses casual self-attention to
give the text caption. We use the unimodal encoder and image-grounded text
decoder to generate the captions. Given an input $\mathbf{x}$, the captions
are generated as,
$\text{{Caption}}_{\text{BLIP}}=\text{{BLIP}}(\mathbf{x}).$
We show some sample captions in Figure 2. We can see that the captions for the
clean samples (top row) contains the true label. In the second row,
adversarial samples are given and the classifier’ prediction is incorrect.
Here, truck is classified as ship. However, the BLIP caption still contains
the true label truck, though the caption is not the same as that of clean
sample. Thus, using these captions can condition the diffusion models with
true semantics which can enhance purification of the adversarial images. Next,
we discuss the diffusion based purification.
### 3.2 Diffusion purification process
Latent diffusion process In a standard diffusion model [14], the diffusion
process can be defined as:
${\bf{x}_{t}}=\sqrt{1-\beta_{t}}\cdot{\bf{x}_{t-1}}+\sqrt{\beta_{t}}\cdot\boldsymbol{\epsilon}_{t}$
where $\beta_{t}\in(0,1)$ is the variance schedule, ${\bf{x}_{t}}$ is the
noisy sample, and $\boldsymbol{\epsilon}_{t}$ is the noise at time step $t$.
In Latent Diffusion Models [9], this process is applied in latent space:
$\displaystyle\mathbf{z}_{0}=\mathcal{E}(\mathbf{x})$
$\displaystyle{\bf{z}_{t}}=\sqrt{1-\beta_{t}}\cdot{\bf{z}_{t-1}}+\sqrt{\beta_{t}}\cdot{\boldsymbol{\epsilon}_{t}}$
where $\mathbf{z}_{0}$ is the latent vector obtained from the encoder
$\mathcal{E}$ and $\mathbf{z}_{t}$ is the noisy latent vector at time step
$t$.
Reverse process in latent space In the reverse process, the aim is to recover
$\mathbf{z}_{0}$ from $\mathbf{z}_{T}$ given a sequence of noise terms
$\boldsymbol{\epsilon}_{t}$. Mathematically, this is defined as:
${\mathbf{z}_{t}}=g_{\theta}({\mathbf{z}_{t+1}},t,{\boldsymbol{\epsilon}_{t}})$
where ${g_{\theta}}$ is a parameterized model. Additionally, ${g_{\theta}}$ is
conditioned on text by augmenting ${g_{\theta}}$ architecture with cross
attention layers. Since our goal is to leverage the BLIP generated captions,
we condition the diffusion model as:
${\mathbf{z}_{t}}=g_{\theta}({\mathbf{z}_{t+1}},t,{\boldsymbol{\epsilon}_{t}},\mathbf{C})$
where $\mathbf{C}={\tau_{\theta}}(\text{{Caption}}_{\text{BLIP}})$, and
$\tau_{\theta}$ is text encoder. Since BLIP is a powerful model, the
likelihood that it correctly identifies the image is high. This gives a better
guidance to diffusion model compared to image-only case.
Final image reconstruction and training Finally, the reconstructed image
$\hat{\mathbf{x}}$ can be obtained from the reconstructed latent
representation ${\mathbf{z}}_{0}$ as,
$\hat{\mathbf{x}}=\mathcal{D}({\mathbf{z}}_{0})$, where $\mathcal{D}$ is the
decoder. Given model $f_{\boldsymbol{\theta}}$, clean image $\mathbf{x}$, its
corresponding pre-processed sample $\hat{\mathbf{x}}$ and labels $y$, we
optimize for,
$\arg\min_{\boldsymbol{\theta}}\frac{1}{n}\sum_{i=1}^{n}\mathcal{L}_{CE}(f_{\boldsymbol{\theta}}(\hat{\mathbf{x}}_{i}),{y}_{i})$,
where $\mathcal{L}_{CE}$ is the cross-entropy loss and $n$ is the number of
samples.
In contrast to adversarial training of several epochs with adversarial
samples, we only need a few epochs of finetuning with pre-processed clean
samples. Further, compared to score or diffusion-based purification, which
extensively trains these models, we only need minimal training of the
classifier.
## 4 Experiments and Results
### 4.1 Experimental settings
Datasets and network architectures: Our experimental evaluation involves three
datasets, namely CIFAR-10 [11], CIFA-100 [11] and ImageNet [10]. We utilize
the base models from RobustBench [23] model zoo for CIFAR-10 and ImageNet. For
CIFAR-100 we train the model following Yoon et al. [6]. We compare our
approach against other adversarial purification strategies on CIFAR-10,
adhering to their experimental configurations. We also evaluate our method
against preprocessor blind attacks on ImageNet. Regarding classifier
architectures, we opt for two prevalent models: ResNet-50 [24] for ImageNet
and WideResNet-28-10 [25] for CIFAR-10 and CIFAR-100. We fine-tune the
WideResNet on images generated from the diffusion network for 15 epochs. We
utilize Adam optimizer with a $10^{-3}$ learning rate. For generating
captions, we use pre-trained BLIP [17] with default hyperparameters, and for
the diffusion process, we use a pre-trained latent diffusion model from [9]
with default parameters except for the noise parameter t. We set t to 0.5 for
CIFAR-10, and CIFAR-100 and 0.1 for ImageNet. We will be releasing the code
soon.
Fig. 2: Purified samples given by LGAP. The first, second, and third rows contain clean, adversarial, and purified samples. The BLIP generated captions are given on the right, and the predicted label is on top of the image. Methods | Accuracy(%) | Architecture
---|---|---
| Natural | Robust |
Raw WideResNet | 95.80 | 0.00 | WRN-28-10
Adv. purification methods | | |
LGAP | 90.03 | 71.68 | WRN-28-10
Yoon et al. [6]* | | |
($\sigma$ = 0.1)* | 93.09 | 85.45 | WRN-28-10
($\sigma$ = 0.25)* | 86.14 | 80.24 | WRN-28-10
Hill et al. [26]* | 84.12 | 78.91 | WRN-28-10
Shi et al. [5]* | 96.93 | 63.10 | WRN-28-10
Du et al. [27]* | 48.7 | 37.5 | WRN-28-10
Grathwohl et al. [22]* | 75.5 | 23.8 | WRN-28-10
Song et al. [3]* | | |
Natural + PixelCNN | 82 | 61 | ResNet-62
AT + PixelCNN | 90 | 70 | ResNet-62
Adv training methods | | |
Madry et al. [2]* | 87.3 | 70.2 | ResNet-56
Dong et al. [28]* | 84.98 | 51.29 | ResNet-18
Table 1: Results for preprocessor blind PGD attack for CIFAR-10, within an
$L_{\infty}$ $\epsilon$-ball, where $\epsilon$ = 8/255. Data sourced from
existing literature is indicated by an asterisk *.
Adversarial attacks: We test our algorithm against preprocessor blind PGD
attacks, in which the adversary has complete visibility into the classifier
but is uninformed about the purification model. We also evaluate our algorithm
against strong adaptive attack, which involves more complex scenarios due to
our purification algorithm’s iterative nature through neural networks,
potentially leading to obfuscated gradients. To rigorously test our defense
mechanism, we use potent adaptive attacks, such as Backward Pass
Differentiable Approximation (BPDA) [29] and its variations. We experiment
with the basic form of BPDA, where the purification function is approximated
as the identity function. We further validate its robustness using Expectation
Over Time (EOT) attacks [29].
Method | Accuracy
---|---
| Natural | Robust
LGAP | 58.71 | 39.82
Yoon et al.[6]* | 77.83 | 43.21
Adversarial training methods | |
Madry et al.[2]* | 59.58 | 25.47
Li et al.[30]* | 61.01 | 28.88
Table 2: Preprocessor blind PGD attack for CIFAR-100, $\epsilon$ = $8$/$255$. Data sourced from existing literature is indicated by an asterisk *. Methods | Accuracy(%) | Architecture
---|---|---
Attacks | Natural | Robust |
Adv. Purification Methods | | |
LGAP | 90.30 | |
BPDA 40+EOT | | 44.96 | WRN-28-10
BPDA | | 62.63 | WRN-28-10
Yoon et al. [6] ($\sigma$ = 0.25)* | | |
BPDA 40+EOT | 86.14 | 70.01 | WRN-28-10
Yoon et al. [6] ($\sigma$ = 0.0)* | | |
BPDA | 90.60 | 76.87 | WRN-28-10
Hill et al. [26]* | | |
BPDA 50+EOT | 84.12 | 54.9 | WRN-28-10
Song et al. [3]* | | |
BPDA | 95.00 | 9 | ResNet-62
Yang et al. [31]* | | |
BPDA 1000 | 94.8 | 40.8 | ResNet-18
(+AT. p = 0.4 ->0.6) | 88.7 | 55.1 | WRN-28-10
(+AT. p = 0.6 ->0.8) | 91 | 52.9 | WRN-28-10
Approx. Input | 89.4 | 41.5 | ResNet-18
Shi et al. [5]* | | |
Classifier PGD 20 | 91.89 | 53.58 | WRN-28-10
Adv. training methods | | |
Madry et al. [2]* | 87.3 | 45.8 | ResNet-18
Carmon et al. [32]* | 89.67 | 63.1 | WRN-28-10
Table 3: Adaptive attacks for CIFAR-10, $\epsilon$ = 8/255.
### 4.2 Comparison with state of the art
The results for preprocessor blind setup shown in Table 1 on CIFAR10 show that
our method gives better robust performance than most previous methods,
specifically adversarial training methods, while maintaining comparable
performance on natural images. Our method achieves a robust accuracy of
71.68%, which clearly outperforms seven out of ten methods including two
adversarial defense methods and five adversarial purification methods. A
snapshot of adversarial samples and their corresponding purified images is
given in Figure 2.
We further extend our evaluation to the CIFAR-100 dataset, with the robust
performance comparisons listed in Table 2. Unlike other methods, such as the
one by Yoon et al., which demands training a score network and noise parameter
tuning, our method, LGAP delivers competitive results with substantially lower
computational overhead.
Table 3 shows robust accuracy of our method against BPDA attack for CIFAR-10.
Our method outperforms most previous techniques of adversarial purification
and adversarial training. The gap in accuracy between our method and some
recent techniques remains owing to other methods training the purification
model on CIFAR10. Yoon et al. and Hill et al. which show better robust
performance, train diffusion and EBM networks on CIFAR-10 for 200,000
iterations [6, 26]. Whereas our method requires no such training.
Method | Accuracy | Architecture
---|---|---
Attacks | Natural | Robust |
Undefended | 76.76 | 0 | ResNet-50
LGAP | 69.09 | | ResNet-50
AA | | 57.12 |
BPDA-40 | | 45.31 |
PGD-10 | | 52.73 |
Adv training methods | | |
Salman et al. [33]* | 64.02 | | ResNet-50
AA | | 34.96 |
Wong et al. [21]* | 55.62 | | ResNet-50
AA | | 26.24 |
Table 4: Preprocessor blind attacks for ImageNet, $\epsilon$ = $4$/$255$.
Table 4 shows the robust performance of our method for ImageNet. Due to the
high computational cost of some attacks, we evaluate on a fixed set of 2048 as
robust accuracy does not change much on the sampled subset compared to the
whole subset [7]. We can see that even against strong adaptive attack such as
BPDA-40, LGAP attains an accuracy of 45.31% demonstrating the efficacy of the
proposed method. The enhanced performance of the method can be attributed to
the diffusion model trained on ImageNet. Similarly, a diffusion model trained
on CIFAR-10 is expected to yield improved results when applied to CIFAR-10
classification.
## 5 Conclusion
Our method addressed key limitations in adversarial defense by introducing a
language-guided purification approach. Unlike traditional methods, which
require extensive computational resources and specific attack knowledge, our
method leverages pre-trained diffusion models and caption generators. This
reduces computational overhead and enhances scalability. Empirical tests show
our approach is robust, outperforming conventional methods in several metrics,
despite trailing some diffusion-based methods. Notably, this performance is
achieved with minimal training and do not require adversarial samples or
training the score or diffusion networks, thus broadening the method’s
applicability and setting a new efficiency standard. Our method underscores
the generalizability of deep learning models trained on large datasets,
pointing to avenues for future research, especially in model generalizability.
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|
11institutetext: P. Sopasakis 22institutetext: KU Leuven, Department of
Electrical Engineering (ESAT), STADIUS Center for Dynamical Systems, Signal
Processing and Data Analytics, Kasteelpark Arenberg 10, 3001 Leuven, Belgium.
Tel.: +32-486-928034
22email<EMAIL_ADDRESS>33institutetext: H. Sarimveis
44institutetext: School of Chemical Engineering, National Technical University
of Athens, 9 Heroon Polytechneiou Street, 15780 Zografou Campus, Athens,
Greece.
Tel.: +30-210-7723237
44email<EMAIL_ADDRESS>55institutetext: P. Macheras 66institutetext:
Department of Pharmacy, University of Athens, Panepistimiopolis Zografou,
15784 Athens, Greece.
Tel: +30-210-7274026
66email<EMAIL_ADDRESS>77institutetext: A.
Dokoumetzidis88institutetext: Department of Pharmacy, University of Athens,
Panepistimiopolis Zografou, 15784 Athens, Greece.
Tel: +30-210-7274122
88email<EMAIL_ADDRESS>
# Fractional Calculus in Pharmacokinetics
Pantelis Sopasakis Haralambos Sarimveis Panos Macheras Aristides
Dokoumetzidis
###### Abstract
We are witnessing the birth of a new variety of pharmacokinetics where non-
integer-order differential equations are employed to study the time course of
drugs in the body: this is dubbed “fractional pharmacokinetics”. The presence
of fractional kinetics has important clinical implications such as the lack of
a half-life, observed, for example with the drug amiodarone and the associated
irregular accumulation patterns following constant and multiple-dose
administration. Building models that accurately reflect this behaviour is
essential for the design of less toxic and more effective drug administration
protocols and devices. This article introduces the readers to the theory of
fractional pharmacokinetics and the research challenges that arise. After a
short introduction to the concepts of fractional calculus, and the main
applications that have appeared in literature up to date, we address two
important aspects. First, numerical methods that allow us to simulate
fractional order systems accurately and second, optimal control methodologies
that can be used to design dosing regimens to individuals and populations.
###### Keywords:
Fractional pharmacokinetics Numerical methods Drug Administration Control Drug
Dosing
††journal: J. Pharmacokinet. Pharmacodyn.
## Introduction
### Background
Diffusion is one of the main mechanisms of various transport processes in
living species and plays an important role in the distribution of drugs in the
body. Processes such as membrane permeation, dissolution of solids and
dispersion in cellular matrices are considered to take place by diffusion.
Diffusion is typically described by Fick’s law, which, in terms of the
pharmacokinetics of drugs, gives rise to exponential washout curves that have
a characteristic time scale, usually expressed as a half-life. However, in the
last few decades, strong experimental evidence has suggested that this is not
always true and diffusion processes that deviate from this law have been
observed. These are either faster (super-diffusion) or slower (sub-diffusion)
modes of diffusion compared to the regular case West and Deering (1994);
Ionescu et al (2017).
Such types of diffusion give rise to kinetics that are referred to as
anomalous to indicate the fact that they stray from the standard diffusion
dynamics West and Deering (1994). Moreover, anomalous kinetics can also result
from reaction-limited processes and long-time trapping. Anomalous kinetics
introduces memory effects into the distribution process that need to be
accounted for to correctly describe it. An distinctive feature of anomalous
power-law kinetics is that it lacks a characteristic time scale contrary to
exponential kinetics. A mathematical formulation that describes such anomalous
kinetics is known as fractal kinetics Kopelman (1988); Macheras (1996);
Pereira (2010) where explicit power functions of time in the form of time-
dependent coefficients are used to account for the memory effects replacing
the original rate constants. In pharmacokinetics several datasets have been
characterised by power laws, epmpirically Wise (1985); Tucker et al (1984),
while the first article that utilised fractal kinetics in pharmacokinetics was
Macheras (1996). Molecules that their kinetics presents power law behaviour
incude those distributed in deeper tissues, such as amiodarone Weiss (1999)
and bone-seeking elements, such as calcium, lead, strontium and plutonium
Macheras (1996); Phan et al (2006).
An alternative theory to describe anomalous kinetics employs fractional
calculus Sokolov et al (2002); Podlubny (1999), which introduces derivatives
and integrals of fractional order, such as half or three quarters. Although
fractional calculus was introduced by Leibniz more than 300 years ago, it is
only within the last couple of decades that real-life applications have been
explored Magin (2004a, b, c). It has been shown that differential equations
with fractional derivatives (FDEs) describe experimental data of anomalous
diffusion more accurately. Although fractional-order derivatives were first
introduced as a novel mathematical concept with unclear physical meaning,
nowadays, a clear connection between diffusion over fractal spaces (such as
networks of capillaries) and fractional-order dynamics has been established
Butera and Paola (2014); Chen et al (2010); Metzler et al (1994); Copot et al
(2014). In particular, in anomalous diffusion the standard assumption that the
mean square displacement $\langle x^{2}\rangle$ is proportional to time does
not hold. Instead, it is $\langle x^{2}\rangle\sim t^{\nicefrac{{2}}{{d}}}$,
where $d\neq 2$ is the associated fractal dimension Gmachowski (2015); Klafter
and Sokolov (2011); Eirich (1990).
Fractional kinetics Dokoumetzidis and Macheras (2011) was introduced in the
pharmaceutical literature in Dokoumetzidis and Macheras (2009) and the first
example of fractional pharmacokinetics which appeared in that article was
amiodarone, a drug known for its power-law kinetics Tucker et al (1984). Since
then, other applications of fractional pharmacokinetics have appeared in the
literature. Kytariolos et al. Kytariolos et al (2010) presented an application
of fractional kinetics for the development of nonlinear in vitro-in vivo
correlations. Popović et al. have presented several applications of fractional
pharmacokinetics to model drugs, namely for diclofenac Popović et al (2010),
valproic acid Popović et al (2011), bumetanide Popović et al (2013) and
methotrexate Popović et al (2015). Copot et al. have further used a fractional
pharmacokinetics model for propofol Copot et al (2013). In most of these cases
the fractional model has been compared with an equivalent ordinary PK model
and was found superior. FDEs have been proposed to describe drug response too,
apart from their kinetics. Verotta has proposed several alternative fractional
PKPD models that are capable of describing pharmacodynamic times series with
favourable properties Verotta (2010). Although these models are empirical,
i.e., they have no mechanistic basis, they are attractive since the memory
effects of FDEs can link smoothly the concentration to the response with a
variable degree of influence, while the shape of the responses generated by
fractional PKPD models can be very flexible and parsimonious (modelled using
few parameters).
Applications of FDEs in pharmacokinetics fall in the scope of the newly-coined
discipline of mathematical pharmacology Van der Graaf et al (2015) to which
this issue is dedicated. Mathematical pharmacology utilises applied
mathematics, beyond the standard tools used commonly in pharmacometrics, to
describe drug processes in the body and assist in controlling them.
This paper introduces the basic principles of fractional calculus and FDEs,
reviews recent applications of fractional calculus in pharmacokinetics and
discusses their clinical implications. Moreover, some aspects of drug dosage
regimen optimisation based on control theory are presented for the case of
pharmacokinetic models that follow fractional kinetics. Indeed, in clinical
pharmacokinetics and therapeutic drug monitoring, dose optimisation is carried
out usually by utilising Bayesian methodology. Optimal control theory is a
powerful approach that can be used to optimise drug administration, which can
handle complicated constraints and has not been used extensively for this
task. The case of fractional pharmacokinetic models is of particular interest
for control theory and poses new challenges.
A problem that has hindered the applications of fractional calculus is the
lack of efficient general-purpose numerical solvers, as opposed to ordinary
differential equations. However, in the past few years significant progress
has been achieved in this area. This important issue is discussed in the paper
and some techniques for numerical solution of FDEs are presented.
### Fractional derivatives
Derivatives of integer order $n\in\mathbb{N}$, of functions
$f:\mathbb{R}\to\mathbb{R}$ are well defined and their properties have been
extensively studied in real analysis. The basis for the extension of such
derivatives to real orders $\alpha\in\mathbb{R}$ commences with the definition
of an integral of order $\alpha$ which hinges on Cauchy’s iterated ($n$-th
order) integral formula and gives rise to the celebrated Riemann-Liouville
integral ${{}_{0}^{\mathrm{rl}}\mathrm{I}}^{\alpha}_{t}$ Hennion and Hanert
(2013):
$\displaystyle{{}_{0}^{\mathrm{rl}}\mathrm{I}}^{\alpha}_{t}f(t)={}_{0}\mathrm{D}_{t}^{-\alpha}f(t)=\tfrac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-\tau)^{\alpha-1}f(\tau)\mathrm{d}\tau,$
(1)
where $\Gamma$ is the Euler gamma function. In Eq. (1) we assume that $f$ is
such that the involved integral is well-defined and finite. This is the case
if $f$ is continuous everywhere except at finitely many points and left-
bounded, i.e., $\\{f(x);0\leq x\leq t\\}$ is bounded for every $t\geq 0$. Note
that for $\alpha\in\mathbb{N}$,
${{}_{0}^{\mathrm{rl}}\mathrm{I}}^{\alpha}_{t}f$ is equivalent to the ordinary
$\alpha$-th order integral. Hereafter, we focus on derivatives of order
$\alpha\in[0,1]$ as only these are of interest to date in the study of
fractional pharmacokinetics.
The left-side subscript $0$ of the ${}_{0}\mathrm{D}_{t}^{\alpha}$ and
${{}_{0}^{\mathrm{rl}}\mathrm{I}}_{t}^{\alpha}$ operators, denotes the lower
end of the integration limits, which in this case has been assumed to be zero.
However, alternative lower bounds can be considered leading to different
definitions of the fractional derivative with slightly different properties.
When the lower bound is $-\infty$, the entire history of the studied function
is accounted for, which is considered preferable in some applications and is
referred to as the Weyl definition Magin (2004a).
It is intuitive to define a fractional-order derivative of order $\alpha$ via
${}_{0}\mathrm{D}_{t}^{\alpha}=\mathrm{D}^{1}{\,}_{0}\mathrm{D}_{t}^{\alpha-1}=\mathrm{D}^{1}\,{{}_{0}^{\mathrm{rl}}\mathrm{I}}^{\alpha-1}$,
(where $\mathrm{D}^{1}$ is the ordinary derivative of order $1$), or
equivalently
$\displaystyle{}_{0}\mathrm{D}_{t}^{\alpha}f(t)=\tfrac{\mathrm{d}}{\mathrm{d}t}\left[\tfrac{1}{\Gamma(1-\alpha)}\int_{0}^{t}\frac{f(\tau)}{(t-\tau)^{\alpha}}\mathrm{d}\tau\right]$
(2)
This is the Riemann-Liouville definition — one of the most popular constructs
in fractional calculus. One may observe that the fractional integration is
basically a convolution between function $f$ and a power law of time, i.e.,
${}_{0}\mathrm{D}_{t}^{-\alpha}f(t)=t^{\alpha-1}\ast f(t)$, where $\ast$
denotes the convolution of two functions. This explicitly demonstrates the
memory effects of the studied process. Fractional derivatives possess
properties that are not straightforward or intuitive; for example, the half
derivative of a constant $f(t)=\lambda$ with respect to $t$ does not vanish
and instead is
${}_{0}\mathrm{D}_{t}^{\nicefrac{{1}}{{2}}}\lambda=\nicefrac{{\lambda}}{{\sqrt{\pi
t}}}$.
Perhaps the most notable shortcoming of the Riemann-Liouville definition with
the $0$ lower bound is that when used in differential equations it gives rise
to initial conditions that involve the fractional integral of the function and
are difficult to interpret physically. This is one of the reasons the Weyl
definition was introduced, but this definition may not be very practical for
most applications either, as it involves an initial condition at time
$-\infty$ Magin (2004a); Samko et al (1993).
A third definition of the fractional derivative, which is referred to as the
Caputo definition is preferable for most physical processes as it involves
explicitly the initial condition at time zero. The definition is:
$\displaystyle{}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{\alpha}f(t)=\tfrac{1}{\Gamma(1-\alpha)}\int_{0}^{t}\frac{\tfrac{\mathrm{d}f(\tau)}{\mathrm{d}\tau}}{(t-\tau)^{\alpha}}\mathrm{d}\tau$
(3)
where the upper-left superscript $\mathrm{c}$ stands for Caputo. The Caputo
derivative is interpreted as
${}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{\alpha}={{}_{0}^{\mathrm{rl}}\mathrm{I}}^{\alpha-1}\mathrm{D}^{1}$,
that is $\mathrm{D}^{1}$ and ${{}_{0}^{\mathrm{rl}}\mathrm{I}}^{\alpha-1}$ are
composed in the opposite way compared to the Riemann-Liouville definition. The
Caputo derivative gives rise to initial value problems of the form
$\displaystyle{}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{\alpha}x(t)$
$\displaystyle=F(x(t)),$ (4a) $\displaystyle x(0)$ $\displaystyle=x_{0}.$ (4b)
This definition for the fractional derivative, apart from the more familiar
initial conditions, comes with some properties similar to those of integer-
order derivatives, one of them being that the Caputo derivative of a constant
is in fact zero. Well-posedeness conditions, for the existence and uniqueness
of solutions, for such fractional-order initial value problems are akin to
those of integer-order problems Deng and Deng (2014),(Podlubny, 1999, Chap.
3). The various definitions of the fractional derivative give different
results, but these are not contradictory since they apply for different
conditions and it is a matter of choosing the appropriate one for each
specific application. All definitions collapse to the usual derivative for
integer values of the order of differentiation.
A fourth fractional derivative definition, which is of particular interest is
the Grünwald-Letnikov, which allows the approximate discretisation of
fractional differential equations. The Grünwald-Letnikov derivative, similar
to integer order derivatives, is defined via a limit of a fractional-order
difference. Let us first define the Grünwald-Letnikov difference of order
$\alpha$ and step-size $h$ of a left-bounded function $f$ as
$\displaystyle{{}^{\mathrm{gl}}\Delta}^{\alpha}_{h}f(t)=\tfrac{1}{h^{\alpha}}\sum_{j=0}^{\left\lfloor\nicefrac{{t}}{{h}}\right\rfloor}c_{j}^{\alpha}f(t-jh),$
(5)
with $c^{\alpha}0=1$ and for $j\in\mathbb{N}_{>0}$
$\displaystyle c_{j}^{\alpha}=(-1)^{j}\prod_{i=0}^{j-1}\frac{\alpha-i}{i+1}.$
(6)
The Grünwald-Letnikov difference operator leads to the definition of the
Grünwald-Letnikov derivative of order $\alpha$ which is defined as (Samko et
al, 1993, Section 20)
$\displaystyle{{}^{\mathrm{gl}}\mathrm{D}}_{t}^{\alpha}f(t)=\lim_{h\to
0^{+}}{{}^{\mathrm{gl}}\Delta}^{\alpha}_{h}f(t),$ (7)
insofar as the limit exists. This definition is of great importance in
practice, as it allows us to replace ${{}^{\mathrm{gl}}\mathrm{D}}_{t}$ with
${{}^{\mathrm{gl}}\Delta}_{h}^{\alpha}$ provided that $h$ is adequately small,
therefore it serves as an Euler-type discretisation of fractional-order
continuous time dynamics. By doing so, we produce a discrete-time yet
infinite-dimensional approximation of the system since
${{}^{\mathrm{gl}}\Delta}_{h}^{\alpha}f(t)$ depends on $f(t-jh)$ for all
$j\in\mathbb{N}_{>0}$. However, we may truncate this series up to a finite
history $\nu$, defining
$\displaystyle{{}^{\mathrm{gl}}\Delta}_{h,\nu}^{\alpha}f(t)=\tfrac{1}{h^{\alpha}}\sum_{j=0}^{\min\\{\nu,\left\lfloor\nicefrac{{t}}{{h}}\right\rfloor\\}}c_{j}^{\alpha}f(t-jh).$
(8)
This leads to discrete-time finite-memory approximations which are
particularly useful as we shall discuss in what follows.
## Fractional Kinetics
The most common type of kinetics encountered in pharmaceutical literature are
the so called “zero order” and “first order”. Here the “order” refers to the
order of linearity and is not to be confused with the order of
differentiation, i.e., a zero order process refers to a constant rate and a
first order to a proportional rate. The fractional versions of these types of
kinetics are presented below and take the form of fractional-order ordinary
differential equations. Throughout this presentation the Caputo version of the
fractional derivative is considered for reasons already explained.
### Zero-order kinetics
The simplest kinetic model is the zero-order model where it is assumed that
the rate of change of the quantity $q$, expressed in [mass] units, is constant
and equal to $k_{0}$, expressed in [mass]/[time] units. Zero-order systems are
governed by differential equations of the form
$\displaystyle\frac{\mathrm{d}q}{\mathrm{d}t}=k_{0}.$ (9)
The solution of this equation with initial condition $q(0)=0$ is
$\displaystyle q(t)=k_{0}t.$ (10)
The fractional-order counterpart of such zero-order kinetics can be obtained
by replacing the derivative of order $1$ by a derivative of fractional order
$\alpha$, that is
$\displaystyle{}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{\alpha}q(t)=k_{0,f},$ (11)
where $k_{0,f}$ is a constant with units [mass]/[time]α. The solution of this
equation for initial condition $q(0)=0$ is a power law for $t>0$ Podlubny
(1999)
$\displaystyle q(t)=\frac{k_{0,f}}{\Gamma(\alpha+1)}t^{\alpha}.$ (12)
### First-order kinetics
The first-order differential equation, where the rate of change of the
quantity $q$ is proportional to its current value, is described by the
ordinary differential equation
$\displaystyle\frac{\mathrm{d}q}{\mathrm{d}t}=-k_{1}q(t).$ (13)
Its solution with initial condition of $q(0)=q_{0}$ is given by the classical
equation of exponential decay
$\displaystyle q(t)=q_{0}\exp(-k_{1}t).$ (14)
Likewise, the fractional-order analogue of such first-order kinetics is
derived by replacing $\nicefrac{{\mathrm{d}}}{{\mathrm{d}t}}$ by the
fractional-order derivative ${}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{\alpha}$
yielding the following fractional differential equation
$\displaystyle{}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{\alpha}q(t)=-k_{1,f}q(t),$
(15)
where $k_{1,f}$ is a constant with [time]-α units. The solution of this
equation can be found in most books or papers of the fast growing literature
on fractional calculus Podlubny (1999) and for initial condition $q(0)=q_{0}$
it is
$\displaystyle q(t)=q_{0}\mathcal{E}_{\alpha}(-k_{1,f}t^{\alpha}),$ (16)
where $\mathcal{E}_{\alpha}$ is the Mittag-Leffler function Podlubny (1999)
which is defined as
$\displaystyle\mathcal{E}_{\alpha}(t)=\sum_{k=0}^{\infty}\frac{t^{k}}{\Gamma(\alpha
k+1)}.$ (17)
The function $\mathcal{E}_{\alpha}(t)$ is a generalisation of the exponential
function and it collapses to the exponential when $\alpha=1$, i.e.,
$\mathcal{E}_{1}(t)=\exp(t)$. Alternatively, Eq. (16) can be reparametrised by
introducing a time scale parameter with regular time units,
$\tau_{f}=k_{1,f}^{-1/\alpha}$ and then becomes
$q(t)=q_{0}\mathcal{E}_{\alpha}(-(t/\tau_{f})^{\alpha})$. The solution of Eq.
(15) basically means that the fractional derivative of order $\alpha$ of the
function $\mathcal{E}_{\alpha}(t^{\alpha})$ is itself a function of the same
form, exactly like the classic derivative of an exponential is also an
exponential.
It also makes sense to restrict $\alpha$ to values $0\leq\alpha\leq 1$, since
for values of a larger than $1$ the solution of Eq. (15) is not monotonic and
negative values for $q$ may appear, therefore, it is not applicable in
pharmacokinetics and pharmacodynamics. Based on these elementary equations the
basic relations for the time evolution in drug disposition can be formulated,
with the assumption of diffusion of drug molecules taking place in
heterogeneous space. The simplest fractional pharmacokinetic model is the one-
compartment model with i.v. bolus administration and the concentration, c, can
be expressed by Eq. (16) divided by a volume of distribution, as
$\displaystyle c(t)=c_{0}\mathcal{E}_{\alpha}(-k_{1,f}t^{\alpha}),$ (18)
with $\alpha\leq 1$ and $c_{0}$ is the ratio [dose]/[apparent volume of
distribution]. This equation for times $t\ll 1$ behaves as a stretched
exponential, i.e., as ${\sim}\exp(-k_{1,f}t^{\alpha}/\Gamma(1+\alpha))$, while
for large values of time it behaves as a power-law,
${\sim}t^{-\alpha}/\Gamma(1-\alpha)$ (see Figure 1) Mainardi (2014). This
kinetics is, therefore, a good candidate to describe various datasets
exhibiting power-law-like kinetics due to the slow diffusion of the drug in
deeper tissues. Moreover, the relevance of the stretched exponential (Weibull)
function with the Mittag-Leffler function probably explains the successful
application of the former function in describing drug release in heterogeneous
media Papadopoulou et al (2006). Eq. (18) is a relationship for the simplest
case of fractional pharmacokinetics. It accounts for the anomalous diffusion
process, which may be considered to be the limiting step of the entire
kinetics. Classic clearance may be considered not to be the limiting process
here and is absent from the equation.
Figure 1: Amount-time profile according to Eq. (16) for $\alpha=0.5$ (solid
curve) along with its approximation at values of $t$ close to $0$ by a
stretched exponential function $\exp(-t^{\nicefrac{{1}}{{2}}}/\Gamma(1.5))$
(bottom dashed curve) and an approximation at high values of $t$ by a power
function $t^{-\nicefrac{{1}}{{2}}}/\Gamma(0.5)$ (top dashed-dotted curve).
Note that the time axis is logarithmic. The evolution of the amount of drug
starts as a stretched exponential and eventually shapes as a power function.
### The Laplace transform for FDEs
Fractional differential equations (FDEs) can be easily written in the Laplace
domain since each of the fractional derivatives can be transformed similarly
to the ordinary derivatives, as follows, for order $\alpha\leq 1$:
$\displaystyle\mathcal{L}\\{{}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{\alpha}f(t)\\}=s^{\alpha}F(s)-s^{\alpha-1}f(0^{+}),$
(19)
where $F(s)$ is the Laplace transform of $f(t)$ Podlubny (1999). For
$\alpha=1$, Eq. (19) reduces to the classic well-known expression for ordinary
derivatives, that is, $\mathcal{L}\\{\dot{f}(t)\\}=sF(s)-f(0^{+})$.
Let us take as an example the following simple FDE
$\displaystyle{}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{1/2}q(t)=-q(t),$ (20)
with initial value $q(0)=1$ which can be written in the Laplace domain, by
virtue of Eq. (19), as follows
$\displaystyle
s^{\nicefrac{{1}}{{2}}}Q(s)-s^{-\nicefrac{{1}}{{2}}}q(0)=-Q(s),$ (21)
where $Q(s)=\mathcal{L}\\{q(t)\\}(s)$ is the Laplace transform of $q$. After
simple algebraic manipulations, we obtain
$\displaystyle Q(s)=\frac{1}{s+\sqrt{s}}.$ (22)
By applying the inverse Laplace transform to Eq. (22) the closed-form
analytical solution of this FDE can be obtained; this involves a Mittag-
Leffler function of order one half. In particular,
$\displaystyle
q(t)=\mathcal{E}_{\nicefrac{{1}}{{2}}}(-t^{\nicefrac{{1}}{{2}}})=e^{t}(1+\operatorname{erf}(-\sqrt{t})).$
(23)
Although more often than not it is easy to transform an FDE in the Laplace
domain, it is more difficult to apply the inverse Laplace transform so as to
solve it explicitly for the system variables so that an analytical solution in
the time domain is obtained. However, it is possible to perform that step
numerically using a numerical inverse Laplace transform (NILT) algorithm De
Hoog et al (1982) as described above.
## Fractional Pharmacokinetics
### Multi-compartmental models
A one-compartment pharmacokinetic model with i.v. bolus administration can be
easily fractionalised as in Eq. (15) by changing the derivative on the left-
hand-side of the single ODE to a fractional order. However, in
pharmacokinetics and other fields where compartmental models are used, two or
more ODEs are often necessary and it is not as straightforward to
fractionalise systems of differential equations, especially when certain
properties such as mass balance need to be preserved.
When a compartmental model with two or more compartments is built, typically
an outgoing mass flux becomes an incoming flux to the next compartment. Thus,
an outgoing mass flux that is defined as a rate of fractional order, cannot
appear as an incoming flux into another compartment, with a different
fractional order, without violating mass balance Dokoumetzidis et al (2010a).
It is therefore, in general, impossible to fractionalise multi-compartmental
systems simply by changing the order of the derivatives on the left hand side
of the ODEs.
One approach to the fractionalisation of multi-compartment models is to
consider a common fractional order that defined the mass transfer from a
compartment $i$ to a compartment $j$: the outflow of one compartment becomes
an inflow to the other. This implies a common fractional order known as a
commensurate order. In the general case, of non-commensurate-order systems, a
different approach for fractionalising systems of ODEs needs to be applied.
In what follows, a general form of a fractional two-compartment system is
considered and then generalised to a system of an arbitrary number of
compartments, which first appeared in Dokoumetzidis et al (2010b). A general
ordinary linear two-compartment model, is defined by the following system of
linear ODEs,
$\displaystyle\frac{\mathrm{d}q_{1}(t)}{\mathrm{d}t}$
$\displaystyle=-k_{12}q_{1}(t)+k_{21}q_{2}(t)-k_{10}q_{1}(t)+u_{1}(t),$ (24a)
$\displaystyle\frac{\mathrm{d}q_{2}(t)}{\mathrm{d}t}$
$\displaystyle=k_{12}q_{1}(t)-k_{21}q_{2}(t)-k_{20}q_{2}(t)+u_{2}(t),$ (24b)
where $q_{1}(t)$ and $q_{2}(t)$ are the mass or molar amounts of material in
the respective compartments and the $k_{ij}$ constants control the mass
transfer between the two compartments and elimination from each of them. The
notation convention used for the indices of the rate constants is that the
first corresponds to the source compartment and the second to the target one,
e.g. $k_{12}$ corresponds to the transfer from compartment 1 to 2, $k_{10}$
corresponds to the elimination from compartment 1, etc. The units of all the
$k_{ij}$ rate constants are (1/[time]). $u_{i}(t)$ are input rates in each
compartment which may be zero, constant or time dependent. Initial values for
$q_{1}$ and $q_{2}$ have to be considered also, $q_{1}(0)$ and $q_{2}(0)$,
respectively.
In order to fractionalise this system, first the ordinary system is
integrated, obtaining a system of integral equations and then the integrals
are fractionalised as shown in Dokoumetzidis et al (2010b). Finally, the
fractional integral equations, are differentiated in an ordinary way. The
resulting fractional system contains ordinary derivatives on the left hand
side and fractional Riemann-Liouville derivatives on the right hand side:
$\displaystyle\frac{\mathrm{d}q_{1}(t)}{\mathrm{d}t}$
$\displaystyle=-k_{12,f}{\,}_{0}\mathrm{D}_{t}^{1-\alpha_{12}}q_{1}(t)+k_{21,f}{\,}_{0}\mathrm{D}_{t}^{1-\alpha_{21}}q_{2}(t)-k_{10,f}{\,}_{0}\mathrm{D}_{t}^{1-\alpha_{10}}q_{1}(t)+u_{1}(t),$
(25a) $\displaystyle\frac{\mathrm{d}q_{2}(t)}{\mathrm{d}t}$
$\displaystyle=k_{12,f}{\,}_{0}\mathrm{D}_{t}^{1-\alpha_{12}}q_{1}(t)-k_{21,f}{\,}_{0}\mathrm{D}_{t}^{1-\alpha_{21}}q_{2}(t)-k_{20,f}{\,}_{0}\mathrm{D}_{t}^{1-\alpha_{20}}q_{2}(t)+u_{2}(t),$
(25b)
where $0<\alpha_{ij}\leq 1$ is a constant representing the order of the
specific process. Different values for the orders of different processes may
be considered, but the order of the corresponding terms of a specific process
are kept the same when these appear in different equations, e.g., there can be
an order $\alpha_{12}$ for the transfer from compartment 1 to 2 and a
different order $\alpha_{21}$ for the transfer from compartment 2 to 1, but
the order for the corresponding terms of the transfer, from compartment 1 to
2, $\alpha_{12}$, is the same in both equations. Also the index $f$ in the
rate constant was added to emphasise the fact that these are different to the
ones of Eq. (24) and carry units [time]-α.
It is convenient to rewrite the above system Eq. (25) with Caputo derivatives.
An FDE with Caputo derivatives accepts the usual type of initial conditions
involving the variable itself, as opposed to RL derivatives which involve an
initial condition on the derivative of the variable, which is not practical.
When the initial value of $q_{1}$ or $q_{2}$ is zero then the respective RL
and Caputo derivatives are the same. This is convenient since a zero initial
value is very common in compartmental analysis. When the initial value is not
zero, converting to a Caputo derivative is possible, for the particular term
with a non-zero initial value. The conversion from a RL to a Caputo derivative
of the form that appears in Eq. (25) is done with the following expression:
$\displaystyle{}_{0}\mathrm{D}_{t}^{1-\alpha_{ij}}q_{i}(t)={}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{1-\alpha_{ij}}q_{i}(t)+\frac{q_{i}(0)t^{\alpha_{ij}-1}}{\Gamma(\alpha_{ij})}.$
(26)
Summarising the above remarks about initial conditions, we may identify three
cases: (i) The initial condition is zero and then the derivative becomes a
Caputo by definition. (ii) The initial condition is non-zero, but it is
involved in a term with an ordinary derivative so it is treated as usual.
(iii) The initial condition is non-zero and is involved in a fractional
derivative which means that in order to present a Caputo derivative, an
additional term, involving the initial value appears, by substituting Eq.
(26). Alternatively, a zero initial value for that variable can be assumed,
with a Dirac delta input to account for the initial quantity for that
variable. So, a general fractional model with two compartments, Eq. (25), was
formulated, where the fractional derivatives can always be written as Caputo
derivatives. It is easy to generalise the above approach to a system with an
arbitrary number of $n$ compartments as follows
$\displaystyle\frac{\mathrm{d}q_{i}(t)}{\mathrm{d}t}=-k_{i0}{\,}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{1-\alpha_{i0}}q_{i}(t)-\sum_{j\neq
i}k_{ij}{\,}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{1-\alpha_{ij}}q_{i}(t)+\sum_{j\neq
i}k_{ji}{\,}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{1-\alpha_{ji}}q_{j}(t)+u_{i}(t),$
(27)
for $i=1,\ldots,n$, where Caputo derivatives have been considered throughout
since, as explained above, this is feasible. This system of Eqs. (27) is too
general for most purposes as it allows every compartment to be connected with
every other. Typically the connection matrix would be much sparser than that,
with most compartments being connected to just one neighbouring compartment
while only a few “hub” compartments would have more than one connections. The
advantage of the described approach of fractionalisation is that each
transport process is fractionalised separately, rather than fractionalising
each compartment or each equation. Thus, processes of different fractional
orders can co-exist since they have consistent orders when the corresponding
terms appear in different equations. Also, it is important to note that
dynamical systems of the type (27) do not suffer from pathologies such as
violation of mass balance or inconsistencies with the units of the rate
constants.
As mentioned, FDEs can be easily written in the Laplace domain. In the case of
FDEs of the form of Eq. (27), where the fractional orders are $1-\alpha_{ij}$,
the Laplace transform of the Caputo derivative becomes
$\displaystyle\mathcal{L}\\{{}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{1-\alpha_{ij}}q_{i}(t)\\}=s^{1-\alpha_{ij}}Q_{i}(s)-s^{-\alpha_{ij}}q_{i}(0).$
(28)
An alternative approach for fractionalisation of non-commensurate fractional
pharmacokinetic systems has been proposed in Popović et al (2013), where the
conditions that the pharmacokinetic parameters need to satisfy for the mass
balance to be preserved have been defined.
### A one-compartment model with constant rate input
After the simplest fractional pharmacokinetic model with one-compartment and
i.v. bolus of Eq. (18), the same model with fractional elimination, but with a
constant rate input is considered Dokoumetzidis et al (2010b). Even in this
simple one-compartment model, it is necessary to employ the approach of
fractionalising each process separately, described above, since the constant
rate of infusion is not of fractional order. That would have been difficult if
one followed the approach of changing the order of the derivative of the left
hand side of the ODE, however here it is straightforward. The system can be
described by the following equation
$\displaystyle\frac{\mathrm{d}q(t)}{\mathrm{d}t}=k_{01}-k_{10,f}{\,}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{1-\alpha}q(t),$
(29)
with $q(0)=0$ and where $k_{01}$ is a zero-order input rate constant, with
units [mass]/[time], $k_{10,f}$ is a rate constant with units [time]-α and
$\alpha$ is a fractional order less than $1$. Eq. (29) can be written in the
Laplace domain as
$\displaystyle
sQ(s)-q(0)=\frac{k_{01}}{s}-k_{10,f}(s^{1-\alpha}Q(s)-s^{-\alpha}q(0)).$ (30)
Since $q(0)=0$, Eq. (30) can be solved to obtain
$\displaystyle Q(s)=\frac{k_{01}s^{\alpha-2}}{s^{\alpha}+k_{10,f}}.$ (31)
By applying the following inverse Laplace transform formula (Equation (1.80)
in Podlubny (1999), page 21):
$\displaystyle\mathcal{L}^{-1}\left\\{\frac{s^{\mu-\nu}}{s^{\mu}+k}\right\\}=t^{\nu-1}\mathcal{E}_{\mu,\nu}(-kt^{\mu}),$
(32)
where $\mathcal{E}_{\mu,\nu}$ is the Mittag-Leffler function with two
parameters. For $\mu=\alpha$ and $\nu=2$, we obtain
$\displaystyle q(t)=k_{01}t\mathcal{E}_{\alpha,2}(-k_{10,f}t^{\alpha}).$ (33)
In Theorem 1.4 of Podlubny (1999), the following expansion for the Mittag-
Leffler function is proven to hold asymptotically as $|z|\to\infty$:
$\displaystyle\mathcal{E}_{\mu,\nu}(z)=-\sum_{k=1}^{p}\frac{z^{-k}}{\Gamma(\nu-\mu)}+\mathcal{O}(|z|^{1-p}).$
(34)
Applying this formula for Eq. (33) and keeping only the first term of the sum
since the rest are of higher order, the limit of Eq. (33) for $t$ going to
infinity can be shown that it is $\lim_{t\to\infty}q(t)=\infty$ for all
$0<\alpha<1$ Dokoumetzidis et al (2010b).
The fact that the limit of $q(t)$ in Eq. (33) diverges as $t$ goes to
infinity, for $\alpha<1$, means that unlike the classic case, for $\alpha=1$ —
where (33) approaches exponentially the steady state
$\nicefrac{{k_{01}}}{{k_{10,f}}}$, for $\alpha<1$ — there is infinite
accumulation. In Figure 2 a plot of (33) is shown for $\alpha=0.5$
demonstrating that in the fractional case the amount grows unbounded. In the
inset of Figure 2 the same profiles are shown a 100-fold larger time span,
demonstrating the effect of continuous accumulation.
The lack of a steady state under constant rate administration which results to
infinite accumulation is one of the most important clinical implications of
fractional pharmacokinetics. It is clear that this implication extents to
repeated doses as well as constant infusion, which is the most common dosing
regimen, and can be important in chronic treatments. In order to avoid
accumulation, the constant rate administration must be adjusted to a rate
which decreases with time. Indeed, in Eq. (29), if the constant rate of
infusion $k_{01}$ is replaced by the term $f(t)=k_{01}t^{1-\alpha}$, Hennion
and Hanert (2013) the solution of the resulting FDE is, instead of Eq. (33),
the following
$\displaystyle
q(t)=k_{01}\Gamma(\alpha)t^{\alpha}\mathcal{E}_{\alpha,\alpha+1}(-k_{10,f}t^{\alpha}).$
(35)
The drug quantity $q(t)$ in Eq. (35) converges to the steady state
$\Gamma(\alpha)\nicefrac{{k_{01}}}{{k_{10,f}}}$ as time goes to infinity,
while for the special case of $\alpha=1$, the steady state is the usual
$\nicefrac{{k_{01}}}{{k_{10,f}}}$. Similarly, for the case of repeated doses,
if a steady state is intended to be achieved, in the presence of fractional
elimination of order $\alpha$, then the usual constant rate of administration,
e.g., a constant daily dose, needs to be replaced by an appropriately
decreasing rate of administration. As shown in Hennion and Hanert (2013), this
can be either the same dose, but given at increasing inter-dose intervals,
i.e.,
$T_{i}=(T_{i-1}^{\alpha}+\alpha\Delta\tau^{\alpha})^{\nicefrac{{1}}{{\alpha}}}$,
where $T_{i}$ is the time of the $i$-th dose and $\Delta\tau$ is the inter-
dose interval of the corresponding kinetics of order $\alpha=1$; or a
decreasing dose given at constant intervals, i.e.,
$q_{0,i}=\nicefrac{{q_{0}}}{{\alpha}}((i+1)^{\alpha}-i^{\alpha})$. In this
way, an ever decreasing administration rate is implemented which compensates
the decreasing elimination rate due to the fractional kinetics.
Figure 2: Amount-time profiles for $\alpha=0.5$: (solid line) amount of drug
versus time according to Eq. (33) with constant infusion where there is
unbounded accumulation of drug, (dashed line) time-profile according to Eq.
(35), with power-law infusion where the amount of drug approaches a steady
state. (Inset) The same profiles for $100$ times longer time span.
### A two-compartment i.v. model
Based on the generalised approach for the fractionalisation of compartmental
models, which allows mixing different fractional orders, developed above, a
two-compartment fractional pharmacokinetic model is considered, shown
schematically in Figure 3. Compartment 1 (central) represents general
circulation and well perfused tissues while compartment 2 (peripheral)
represents deeper tissues. Three transfer processes (fluxes) are considered:
elimination from the central compartment and a mass flux from the central to
the peripheral compartment, which are both assumed to follow classic kinetics
(order 1), while a flux from the peripheral to the central compartment is
assumed to follow slower fractional kinetics accounting for tissue trapping
Petráš and Magin (2011); Dokoumetzidis and Macheras (2009).
The system is formulated mathematically as follows:
$\displaystyle\frac{\mathrm{d}q_{1}(t)}{\mathrm{d}t}$
$\displaystyle=-(k_{12}+k_{10})q_{1}(t)+k_{21,f}{\,}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{1-\alpha}q_{2}(t),$
(36a) $\displaystyle\frac{\mathrm{d}q_{2}(t)}{\mathrm{d}t}$
$\displaystyle=k_{12}q_{1}(t)-k_{21,f}{\,}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{1-\alpha}q_{2}(t),$
(36b)
where $\alpha<1$ and initial conditions are $q_{1}(0)=q_{1,0}$, $q_{2}(0)=0$
which account for a bolus dose injection and no initial amount in the
peripheral compartment. Note, that it is allowed to use Caputo derivatives
here since the fractional derivatives involve only terms with $q_{2}$ for
which there is no initial amount, which means that Caputo and RL derivatives
are identical.
Figure 3: A fractional two-compartment PK model with an i.v. bolus.
Elimination from the central compartment and a mass flux from the central to
the peripheral compartment, which are both assumed to follow classic kinetics
(order 1), while a flux from the peripheral to the central compartment is
assumed to follow slower fractional kinetics, accounting for tissue trapping
(dashed arrow).
Applying the Laplace transform, to the above system the following algebraic
equations are obtained:
$\displaystyle sQ_{1}(s)-q_{1}(0)$
$\displaystyle=-(k_{12}+k_{10})Q_{1}(s)+k_{21}(s^{1-\alpha}Q_{2}(s)-s^{-\alpha}q_{2}(0)),$
(37a) $\displaystyle sQ_{2}(s)-q_{2}(0)$
$\displaystyle=k_{12}Q_{1}(s)-k_{21,f}(s^{1-\alpha}Q_{2}(s)-s^{-\alpha}q_{2}(0)).$
(37b)
Solving for $Q_{1}(s)$ and $Q_{2}(s)$ and substituting the initial conditions,
$\displaystyle Q_{1}(s)$
$\displaystyle=\frac{q_{1,0}(s^{\alpha}+k_{21,f})}{(s+k_{12}+k_{10})(s^{\alpha}+k_{21,f})-k_{12}k_{21,f}}$
(38a) $\displaystyle Q_{2}(s)$
$\displaystyle=\frac{q_{1,0}s^{\alpha-1}k_{12}}{(s+k_{12}+k_{10})(s^{\alpha}+k_{21,f})-k_{12}k_{21,f}}$
(38b)
Using the above expression for $Q_{1}$ and $Q_{2}$, Eq. (38a) and Eq. (38b),
respectively, can be used to simulate values of $q_{1}(t)$ and $q_{2}(t)$ in
the time domain by an NILT method. Note, that primarily, $q_{1}(t)$ is of
interest, since in practice we only have data from this compartment. The
output for $q_{1}(t)$ from this numerical solution may be combined with the
following equation
$\displaystyle c(t)=\frac{q_{1}(t)}{V},$ (39)
where $c$ is the drug concentration in the blood and $V$ is the apparent
volume of distribution. Eq. (39) can be fitted to pharmacokinetic data in
order to estimate parameters $V$, $k_{10}$, $k_{12}$, $k_{21,f}$ and $\alpha$.
The closed-form analytical solution of Eq. (36), can be expressed in terms of
an infinite series of generalised Wright functions as demonstrated in the book
by Kilbas et al. Kilbas et al (2006), but these solutions are hard to
implement and apply in practice. Moloni in Moloni (2015) derived analytically
the inverse Laplace transform of $Q_{1}(s)$ as:
$\displaystyle q_{1}(t)$
$\displaystyle=q_{1,0}\sum_{n=0}^{\infty}(-1)^{n}k_{21,f}^{n}\sum_{l=0}^{n}\frac{k_{10}^{l}n!}{(n-l)!l!}\Big{(}t^{l+\alpha
n}\mathcal{E}_{1,l+\alpha n+1}^{n+1}(-(k_{10}+k_{12})t)$
$\displaystyle+t^{l+\alpha(n+1)}\mathcal{E}_{1,l+\alpha(n+1)+1}^{n+1}(-(k_{10}+k_{12})t)\Big{)},$
(40a) while for $Q_{2}$ the inverse Laplace transform works out to be Moloni
(2015): $\displaystyle
q_{2}(t)=q_{1,0}k_{12}\sum_{n=0}^{\infty}(-1)^{n}k_{21,f}^{n}\sum_{l=0}^{n}\frac{k_{10}^{l}n!}{(n-l)!l!}t^{l+\alpha
n}\mathcal{E}_{1,l+\alpha n+2}^{n+1}(-(k_{10}+k_{12})t).$ (40b)
Figure 4: Concentration-time profile of amiodarone in the central compartment.
The solid line corresponds to the fractional-order pharmacokinetic model of
Eq. (36) with the parameter estimates: $\alpha=0.587$,
$k_{10}=1.49\,\mathrm{days}^{-1}$, $k_{12}=2.95\,\mathrm{days}^{-1}$,
$k_{21,f}=0.48\,\mathrm{days}^{-\alpha}$ and
$\nicefrac{{q_{0}}}{{V}}=4.72\,\nicefrac{\mathrm{ng}}{\mathrm{ml}}$. The
circles correspond to experimental measurements.
An application of the fractional two compartment model, the system of Eq. (36)
to amiodarone PK data was presented in Dokoumetzidis et al (2010b). Amidarone
is an antiarrhythmic drug known for its anomalous, non-exponential
pharmacokinetics, which has important clinical implications due to the
accumulation pattern of the drug in long-term administration. The fractional
two-compartment model of the previous section was used to analyse an
amiodarone i.v. data-set which first appeared in Holt et al (1983) and
estimates of the model parameters were obtained. Analysis was carried out in
MATLAB while the values for $q_{1}(t)$ of Eq. (40a) were simulated using a
NILT algorithm De Hoog et al (1982) from the expression of $Q_{1}(s)$ in the
Laplace domain. In Figure 4 the model-based predictions are plotted together
the data demonstrating good agreement for the 60 day period of this study. The
estimated fractional order was $\alpha=0.587$ and the non-exponential
character of the curve is evident, while the model follows well the data both
for long and for short times, unlike empirical power-laws which explode at
$t=0$.
## Numerical methods for fractional-order systems
For simple fractional-order models, as we discussed previously, there may
exist closed-form analytical solutions which involve the one-parameter or two-
parameters Mittage-Leffler function, or are given by more intricate analytical
expressions such as Eq. (40). Interestingly, even for simple analytical
solutions such as Eq. (16), the Mittag-Leffler function itself is evaluated by
solving an FDE numerically Kaczorek (2011). This fact, without discrediting
the value of analytical solutions, necessitates the availability of reliable
numerical methods that allow us to simulate and study fractional-order
systems. The availability of accurate discrete-time approximations of the
trajectories of such systems is important not only for simulating, but also
for the design of open-loop or closed-loop administration strategies, based on
control theory Sopasakis et al (2015). Time-domain approximations are less
parsimonious than $s$-domain ones, but are more suitable for control
applications as we discuss in the next section.
There can be identified four types of solutions for fractional-order
differential equations: (i) closed-form analytical solutions, (ii)
approximations in the Laplace $s$-domain using integer-order rational transfer
functions, (iii) numerical approximation schemes in the discrete time domain
and (iv) numerical inverse Laplace transforms. Each of these comes with
certain advantages as well as limitations; for example, closed-form analytical
solutions are often not available, while the inverse Laplace function requires
an explicit closed form for $u_{i}(t)$ so it cannot be used for administration
rates that are defined implicitly or are arbitrary signals. Approximations in
the $s$-domain are powerful modelling tools, but they fail to provide error
bounds on the concentration of the administered drug in the time domain which
are necessary in clinical practice.
In regard to closed-form analytical solutions, when available, they involve
special functions such as the Mittag-Leffler function
$\mathcal{E}_{\alpha,\beta}(t)=\sum_{k=0}^{\infty}{t^{k}}/{\Gamma(\alpha
k+\beta)}$ whose evaluation calls, in turn, for some numerical approximation
scheme. Analytical closed-form solutions of fractional differential equations
are available for pharmacokinetic systems Verotta (2010). Typically for the
evaluation of this function we resort to solving an FDE numerically Garrappa
(2015); Seybold and Hilfer (2009); Gorenflo et al (2002) because the series in
the definition of $\mathcal{E}_{\alpha,\beta}(t)$ converges rather slowly and
no exact error bounds as available so as to establish meaningful termination
criteria.
### Transfer functions and integer-order approximations
Fractional-order systems, like their integer-order counterparts, can be
modelled in the Laplace $s$-domain via transfer functions, that is,
$\displaystyle F_{ij}(s)=\frac{Q_{j}(s)}{U_{i}(s)},$ (41)
where $Q_{j}(s)$ and $U_{i}(s)$ are the Laplace transforms of the drug
quantity $q_{j}(t)$ and the administration rate $u_{i}(t)$. If the
pharmacokinetics is described by linear fractional-order models such as the
ones discussed above, $F_{ij}$ will be a fractional-order rational function (a
quotient of polynomials with real exponents).
Rational approximations aim at approximating such transfer functions — which
involve terms of the form $s^{\alpha}$, $\alpha\in\mathbb{R}$ — by ordinary
transfer functions of the form
$\displaystyle\tilde{F}_{ij}(s)=\frac{P_{ij}(s)}{S_{ij}(s)},$ (42)
where $P_{ij}$ and $S_{ij}$ are polynomials and the degree of $P_{ij}$ is no
larger than the degree of $S_{ij}$. For convenience with notation, we
henceforth drop the subscripts $ij$.
Padé Approximation: The Padé approximation of order $[m/n]$,
$m,n\in\mathbb{N}$, at a point $s_{0}$ is rather popular and leads to rational
functions with $\deg P=m$ and $\deg S=n$ Silva et al (2006).
Matsuda-Fujii continuous fractions approximation: This method consists in
interpolating a function $F(s)$, which is treated as a black box, across a set
of logarithmically spaced points Matsuda and Fujii (1993). By letting the
selected points be $s_{k}$, $k=0,1,2,\dots$, the approximation is written as
the continued fractions expansion
$F(s)=\alpha_{0}+\frac{s-s_{0}}{\alpha_{1}+\frac{s-s_{1}}{\alpha_{2}+\frac{s-s_{2}}{\alpha_{3}+\dots}}}$
(43)
where, $\alpha_{i}=\upsilon_{i}(s_{i})$, $\upsilon_{0}(s)=H(s)$ and
$\upsilon_{i+1}(s)=\frac{s-s_{i}}{\upsilon_{i}(s)-\alpha_{i}}$.
Oustaloup’s method: Oustaloup’s method is based on the approximation of a
function of the form:
$H(s)=s^{\alpha},$ (44)
with $\alpha>0$ by a rational function
$\widehat{H}(s)=c_{0}\prod_{k=-N}^{N}\frac{s+\omega_{k}}{s+\omega^{\prime}_{k}},$
(45)
within a range of frequencies from $\omega_{b}$ to $\omega_{h}$ Oustaloup et
al (2000). The Oustaloup method offers an approximation at frequencies which
are geometrically distributed about the characteristic frequency
$\omega_{u}=\sqrt{\omega_{b}\omega_{h}}$ — the geometric mean of $\omega_{b}$
and $\omega_{h}$. The parameters $\omega_{k}$ and $\omega_{k}^{\prime}$ are
determined via the design formulas Petráš (2011)
$\displaystyle\omega_{k}^{\prime}=\omega_{b}\left(\frac{\omega_{h}}{\omega_{b}}\right)^{\frac{k+N+0.5(1+\alpha)}{2N+1}},$
(46a)
$\displaystyle\omega_{k}=\omega_{b}\left(\frac{\omega_{h}}{\omega_{b}}\right)^{\frac{k+N+0.5(1-\alpha)}{2N+1}},$
(46b) $\displaystyle
c_{0}=\left(\frac{\omega_{h}}{\omega_{b}}\right)^{-\frac{r}{2}}\prod_{k=-N}^{N}\frac{\omega_{k}}{\omega_{k}^{\prime}}.$
(46c)
Parameters $\omega_{b}$, $\omega_{h}$ and $N$ are design parameters of the
Oustaloup method.
There are a few more methods which have been proposed in the literature to
approximate fractional-order systems by rational transfer functions such as
Charef et al (1992); Carlson and Halijak (1964), as well as data-driven system
identification techniques Gao and Liao (2012).
### Time domain approximations
Several methods have been proposed which attempt to approximate the solution
to a fractional-order initial value problem in the time domain.
Grünwald-Letnikov: This is the method of choice in the discrete time domain
where ${{}^{\mathrm{gl}}\mathrm{D}}_{t}^{\alpha}x(t)$ is approximated by its
discrete-time finite-history variant
$({{{}^{\mathrm{gl}}\Delta}}^{\alpha}_{h,\nu}x)_{k}$ which is proven to have
bounded error with respect to $({{{}^{\mathrm{gl}}\Delta}}_{h}^{\alpha}x)_{k}$
Sopasakis and Sarimveis (2017). The boundedness of this approximation error is
a singular characteristic of this approximation method and is suitable for
applications in drug administration where it is necessary to guarantee that
the drug concentration in the body will remain within certain limits Sopasakis
and Sarimveis (2017). As an example, system (15) is approximated (with
sampling time $h$) by the discrete-time linear system
$\displaystyle\tfrac{1}{h^{\alpha}}\sum_{j=0}^{\min\\{\nu,\left\lfloor\nicefrac{{t}}{{h}}\right\rfloor\\}}c_{j}^{\alpha}q_{k+1-j}=-k_{1,f}q_{k},$
(47)
where $q_{k}=q(kh)$.
The discretisation of the Grünwald-Letnikov derivative suffers from the fact
that the required memory grows unbounded with the simulation time. The
truncation of the Grünwald-Letnikov series up to some finite memory gives rise
to viable solution alorithms which can be elegantly described using triangular
strip matrices as described in Podlubny (2000) and are available as a MATLAB
toolbox.
Approximation by parametrisation: Approximate time-domain solutions can be
obtained by assuming a particular parametric form for the solution. Such a
method was proposed by Hennion and Hanert Hennion and Hanert (2013) where
$q(t)$ is approximated by finite-length expansions of the form
$\sum_{j=0}^{N}A_{j}\phi_{j}(t)$, where $\phi_{j}$ are Chebyshev polynomials
and $A_{j}$ are constant coefficients. By virtue of the computability of
fractional derivatives of $\phi_{j}$, the parametric approximation is plugged
into the fractional differential equation which, along with the initial
conditions, yields a linear system. What is notable in this method is that
expansions as short as $N=10$ lead to very low approximation errors. Likewise,
other parametric forms can be used. For example, Zainal and Kılıçman (2014)
used a Fourier-type expansion and Kumar and Agrawal (2006) used piecewise
quadratic functions.
Numerical integration methods: Fractional-order initial value problems can be
solved with various numerical methods such as the Adams-Bashforth-Moulton
predictor-corrector (ABMPC) method Zayernouri and Matzavinos (2016) and
fractional linear multi-step methods (FLMMs) Lubich (1986); Garrappa (2015).
These methods are only suitable for systems of FDEs in the form
$\displaystyle{}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{\gamma}x(t)$
$\displaystyle=f(t,x(t)),$ (48a) $\displaystyle x^{(k)}(0)$
$\displaystyle=x_{0,k},\,k=0,\dots,m-1$ (48b)
where $\gamma$ is a rational, and $m=\lceil\gamma\rceil$. Typically in
pharmacokinetics we encounter cases with $0<\gamma\leq 1$, therefore, we will
have $m=1$.
Let us give an example on how this applies to the two-compartment model we
presented above. In order to bring Eq. (36) in the above form, we need to find
a rational approximation of two derivatives, $1-\alpha$ and $1$. If we can
find a satisfying rational approximation of $1-a\approxeq p/q$, then the first
order derivative follows trivially. Now, Eq. (36) can be written as
$\displaystyle{}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{\gamma}x_{0}$
$\displaystyle=x_{1}$ (49a)
$\displaystyle{}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{\gamma}x_{1}$
$\displaystyle=x_{2}$ (49b) $\displaystyle\ \,\vdots$
$\displaystyle{}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{\gamma}x_{q-1}$
$\displaystyle=-(k_{12}+k_{10})x_{0}{+}k_{21,f}x_{q+p}{+}u$ (49c)
$\displaystyle{}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{\gamma}x_{q}$
$\displaystyle=x_{q+1}$ (49d) $\displaystyle\ \,\vdots$
$\displaystyle{}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{\gamma}x_{2q-1}$
$\displaystyle=k_{12}x_{0}-k_{21,f}x_{q+p}$ (49e)
with initial conditions $x_{0}(0)=q_{1}(0)$, $x_{q}(0)=q_{2}(0)$ and
$x_{i}(0)=0$ for $i\notin\\{1,q\\}$, and $\gamma=1/q$. This system is in fact
a linear fractional-order system for which closed-form analytical solutions
are available (Kaczorek, 2011, Thm. 2.5). In particular, Eq. (49) can be
written concisely in the form
$\displaystyle{}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{\gamma}\mathbf{x}(t)$
$\displaystyle=A\mathbf{x}(t)+Bu(t)$ (50a) $\displaystyle\mathbf{x}(0)$
$\displaystyle=\mathbf{x}_{0},$ (50b)
where $\mathbf{x}=(x_{0},x_{1},\ldots,x_{2q-1})$ and
$\mathbf{x}_{0}=(x_{0}(0),\ldots,x_{2q-1}(0))$ and matrices $A$ and $B$ can be
readily constructed from the above dynamical equations. This fractional-order
initial value problem has the closed-form analytical solution (Kaczorek, 2011,
Thm. 2.5)
$\displaystyle\mathbf{x}(t)=\mathcal{E}_{\gamma}(At^{\gamma})\mathbf{x}_{0}+\int_{0}^{t}\sum_{k=0}^{\infty}\frac{A^{k}(t-\tau)^{k\gamma}}{\Gamma((k+1)\gamma)}Bu(\tau)\mathrm{d}\tau.$
(51)
Evidently, the inherent complexity of this closed-form analytical solution —
which would require the evaluation of slowly-converging series — motivates and
necessitates the use of numerical methods.
The number of states of system (49) is $2q$, therefore, the rational
approximation should aim at a small $q$. Yet another reason to choose small
$q$ is that small values of $\gamma$ render the system hard to simulate
numerically because they increase the effect of and dependence on the memory.
Adams-Bashforth-Moulton predictor-corrector (ABMPC): Methods of the ABMPC type
have been generalised to solve fractional-order systems. The key idea is to
evaluate $({{{}_{0}^{\mathrm{rl}}\mathrm{I}}}^{\gamma}f)(t,x(t))$ by
approximating $f$ with appropriately selected polynomials. Solutions of Eq.
(48) satisfy the following integral representation
$\displaystyle
x(t)=\sum_{k=0}^{m-1}x_{0,k}\frac{t^{k}}{k!}+({{{}_{0}^{\mathrm{rl}}\mathrm{I}}}^{\gamma}f)(t,x(t)),$
(52)
where the first term on right hand side will be denoted by $T_{m-1}(t).$ This
is precisely an integral representation of Eq. (48). The integral on the right
hand side of the previous equation can be approximated, using an uniformly
spaced grid $t_{n}=nh$, by
$\frac{h^{\gamma}}{\gamma(\gamma+1)}\sum_{j=0}^{n+1}a_{j,n+1}f(t_{j})$ for
suitable coefficients $a_{j,n+1}$ Diethelm et al (2002). The numerical
approximation of the solution of Eq. (48) is
$\displaystyle x(t_{n+1})$
$\displaystyle=T_{m-1}(t_{n+1})+\tfrac{h^{\gamma}}{\Gamma(\gamma+2)}f(t_{n+1},x_{p}(t_{n+1}))+\sum_{j=1}^{n}a_{j,n+1}f(t_{j},x(t_{j})).$
(53a) The equation above is usually referred to as the corrector formula and
$x_{p}(t_{n+1})$ is given by the predictor formula $\displaystyle
x_{P}(t_{n+1})=T_{m-1}(t_{n})$
$\displaystyle+\tfrac{1}{\Gamma(\gamma)}\sum_{j=0}^{n}b_{j,n+1}f(t_{j},x(t_{j})).$
(53b)
Unfortunately, the convergence error of ABMPC when $0<\gamma<1$ is
$\mathcal{O}(h^{1+\gamma})$, therefore, a rather small step size $h$ is
required to attain a reasonable approximation error. A modification of the
basic predictor-corrector method with more favourable computational cost is
provided in Garrappa (2010) for which the MATLAB implementation fde12 is
available.
Lubich’s method: Fractional linear multi-step methods (FLMM) Lubich (1986) are
a generalisation of linear multi-step methods (LMM) for ordinary differential
equations. The idea is to approximate the Riemann-Liouville fractional-order
integral operator (1) with a discrete convolution, known as a convolution
quadrature, as
$\displaystyle({{}_{0}^{\mathrm{rl}}\mathrm{I}}_{h}^{\gamma}f)(t)\approxeq
h^{\gamma}\sum_{j=0}^{n}\omega_{n-j}f(t_{j})+h^{\gamma}\sum_{j=0}^{s}w_{n,j}f(t_{j}),$
(54)
for $t_{j}=jh$, where ($w_{n,j}$) and ($\omega_{n}$) are independent of $h$.
Surprisingly, the latter can be constructed from any linear multistep method
for arbitrary fractional order $\gamma$ Lubich (1986). FLMM constructed this
way will inherit the same convergence rate and at least the same stability
properties as the original LMM method Lubich (1985).
A MATLAB implementation of Lubich’s method, namely flmm2 Garrappa (2015) which
is based on Garrappa (2015) is available. However, these methods do not
perform well for small $\gamma$. According to Herceg et al (2017), for the
case of amiodarone, it is reported that values of $\gamma$ smaller than $0.1$
give poor results and often do not converge, while using a crude approximation
given by $\gamma=1/5$, flmm2 was shown to outperform fde12 in terms of
accuracy and stability at bigger step sizes $h$.
### Numerical inverse Laplace
The inverse Laplace transform of a transfer function $F(s)$ — on the Laplace
$s$-domain — is given by the complex integral
$\displaystyle
f(t)=\mathcal{L}^{-1}\\{F\\}(t)=\lim_{T\rightarrow\infty}\frac{1}{2\pi
i}\int_{\sigma-iT}^{\sigma+iT}e^{st}F(s)\mathrm{d}s,$ (55)
where $\sigma$ is any real number larger than the real parts of the poles of
$F$. The numerical inverse Laplace (NILT) approach aims at approximating the
above integral numerically. Such numerical methods apply also to cases where
$F$ is not rational as it is the case with fractional-order systems (55).
In the approach of De Hoog et al., the integral which appears in Eq. (55) is
cast as a Fourier transform which can then be approximated by a Fourier series
followed by standard numerical integration (e.g., using the trapezoid rule) De
Hoog et al (1982). Though quite accurate for a broad class of functions, these
methods are typically very demanding from the computational point of view. An
implementation of the above method is available online Hollenbeck (1998).
In special cases analytical inversion can be done by means of Mittag-Leffler
function Lin and Lu (2013); Kexue and Jigen (2011). A somewhat different
approach is taken by Valsa and Brančik (1998), where authors approximate
$e^{st}$, the kernel of the inverse Laplace transformation, by a function of
the form $\tfrac{e^{st}}{1+e^{-2a}e^{2st}}$ and choose $a$ appropriately so as
to achieve an accurate inversion.
In general, numerical inversion methods can achieve high precision, but they
are not suitable for control design purposes, especially for optimal control
problems. Moreover, there is not one single method that gives the most
accurate inversion for all types of functions. An overview of the most popular
inversion methods used in engineering practice is given in Hassanzadeh and
Pooladi-Darvish (2007).
## Drug administration for fractional pharmacokinetics
Approaches for drug administration scheduling can be classified according to
the desired objective into methods where (i) we aim at stabilising the
concentration in certain organs or tissues towards certain desired values (set
points) Sopasakis et al (2014) and (ii) the drug concentration needs to remain
within certain bounds which define a therapeutic window Sopasakis et al
(2015).
Another level of classification alludes to the mode of administration where we
identify the (i) continuous administration by, for instance, intravenous
continuous infusion, (ii) bolus administration, where the medicine is
administered at discrete time instants Sopasakis et al (2015); Rivadeneira et
al (2015) and (iii) oral administration where the drug is administered both at
discrete times and from a discrete set of dosages (e.g., tablets) Sopasakis
and Sarimveis (2012).
Drug administration is classified according to the way in which decisions are
made in regard to how often, at what rate and/or what amount of drug needs to
be administered to the patient. We can identify (i) open-loop administration
policies where the patient follows a prescribed dosing scheme without
adjusting the dosage and (ii) closed-loop administration where the dosage is
adjusted according to the progress of the therapy Sopasakis et al (2014). The
latter is suitable mainly for hospitalised patients who are under monitoring
and where drug concentration measurements are available or the effect of the
drug can be quantified. Such is the case of controlled anaesthesia Krieger and
Pistikopoulos (2014). However, applications of closed-loop policies extend
beyond hospitals, such as in the case of glucose control in diabetes Favero et
al (2014).
Despite the fact that optimisation-based methods are well-established in
numerous scientific disciplines along with their demonstrated advantages over
other approaches, to date, empirical approaches remain popular Kannikeswaran
et al (2016); Fukudo et al (2017); De Ocenda et al (2016); Savic et al (2017).
In the following sections we propose decision making approaches for optimal
drug administration using as a benchmark a fractional two-compartment model
using, arbitrarily and as a benchmark, the parameters values of amiodarone. We
focus on the methodological framework rather than devising an administration
scheme for a particular compound. We discuss three important topics in optimal
administration of compounds which are governed by fractional pharmacokinetics.
Next, we formulate the drug administration problem as an optimal control
problem we prescribe optimal therapeutic courses to individuals with known
pharmacokinetic parameters. Lastly, we design optimal administration
strategies for populations of patients whose pharmacokinetic parameters are
unknown or inexactly known. We present an advanced closed-loop controlled
administration methodology based on model predictive control.
### Individualised administration scheduling
In this section we present an optimal drug administration scheme based on the
two-compartment fractional model of Eq. (36) and assuming that the drug is
administered into the central compartment continuously. We will show that the
same model and the same approach can be modified to form the basis for bolus
administration.
In order to state the optimal control problem for optimal drug administration
we first need to discretise the two-compartment model (36) with a (small)
sampling time $t_{c}$
$\displaystyle
t_{c}^{-1}(x_{k+1}-x_{k})=Ax_{k}+F\,{{}^{\mathrm{gl}}\Delta}^{1-a}_{t_{c},\nu}x_{k}+Bu_{k}$
(56)
where $x_{k}=\left[q_{1}(kt_{c})\,\,q_{2}(kt_{c})\right]^{\prime}$ and $u_{k}$
is the administration rate at the discrete time $k$. The left hand side of
(56) corresponds to the forward Euler approximation of the first-order
derivative, and we shall refer to $t_{c}=10^{-2}\,\mathrm{days}$ as the
control sampling time. On the right hand side of (56),
${{}^{\mathrm{gl}}\Delta}^{1-a}_{t_{c},\nu}$ has replaced the fractional-order
operator ${}_{0}^{\mathrm{c}}\mathrm{D}_{t}^{1-\alpha}$. Matrices $A,F$ and
$B$ are
$\displaystyle A=\begin{bmatrix}-(k_{12}+k_{10})&0\\\ k_{21,f}&0\\\
\end{bmatrix},F=\begin{bmatrix}0&k_{21,f}\\\ 0&-k_{21,f}\\\
\end{bmatrix},B=\begin{bmatrix}1\\\ 0\\\ \end{bmatrix}.$ (57)
The discrete-time dynamic equations of the system can now be stated as
$\displaystyle
x_{k+1}=x_{k}+t_{c}\bigg{(}Ax_{k}+\frac{F}{t_{c}^{1-a}}\sum_{j=0}^{\nu}c_{j}^{1-a}x_{k-j}+Bu_{k}\bigg{)},$
(58)
where $c_{j}^{\alpha}=(-1)^{j}\binom{\alpha}{j}$. By augmenting the system
with past values as $\mathbf{x}_{k}=(x_{k},x_{k-1},\dots,x_{k-\nu+1})$ we can
rewrite Eq. (58) as a finite-dimension linear system
$\displaystyle\mathbf{x}_{k+1}=\hat{A}\mathbf{x}_{k}+\hat{B}u_{k}.$ (59)
Matrices $\hat{A}$ and $\hat{B}$ are straightforward to derive and are given
in Sopasakis and Sarimveis (2017). The therapeutic session will last for
$N_{d}=Nt_{c}=7\,\mathrm{days}$ in total, where $N$ is called the prediction
horizon. Since it is not realistic to administer the drug to the patient too
frequently, we assume that the patient is to receive their treatment every
$t_{d}=0.5\,\mathrm{days}$. The administration schedule must ensure that the
concentration of drug in all compartments never exceeds the minimum toxic
concentration limits while tracking the prescribed reference value as close as
possible. To this aim we formulate the constrained optimal control problem
Bertsekas (2017)
$\displaystyle\min_{u_{0},\ldots,u_{N_{d}-1}}J$
$\displaystyle=\sum_{k=0}^{\nicefrac{{N_{d}}}{{t_{c}}}+1}(x_{\mathrm{ref},k}-x_{k})^{\prime}Q(x_{\mathrm{ref},k}-x_{k})$
(60a) subject to the constraints $\displaystyle\mathbf{x}_{k+1}$
$\displaystyle=\hat{A}\mathbf{x}_{k}+\hat{B}u_{j},\text{ for }\ kt_{c}=jt_{d}$
(60b) $\displaystyle\mathbf{x}_{k+1}$
$\displaystyle=\hat{A}\mathbf{x}_{k},\text{ otherwise}$ (60c) $\displaystyle
0$ $\displaystyle\leq x_{k}\leq 0.5$ (60d) $\displaystyle 0$
$\displaystyle\leq u_{j}\leq 0.5$ (60e) for $k=0,\ldots,N$;
$j=0,\ldots,N_{d}-1$.
In the above formulation $x_{\mathrm{ref},k}$ is the desired drug
concentration at time $k$ and operator ′ denotes vector transposition. Any
deviation from set point is penalised by a weight matrix $Q$, where here we
chose $Q=\mathrm{diag}([0\;1])$. Note that we are tracking only the second
state. Our underlying GL model has a relative history of
$t_{c}\nu=5\,\mathrm{days}$. Optimal drug concentrations are denoted by
$u^{\star}_{k}$, for $k=0,\dots,N_{d}-1$ and they correspond to dosages
administered intravenously at times $kt_{d}$. In the optimal control
formulation we have implicitly assumed that $t_{d}$ is an integer multiple of
$t_{c}$, which is not restrictive since $t_{c}$ can be chosen arbitrarily. In
Figure 5 we present the pharmacokinetic profile of a patient following the
prescribed optimal administration course.
Finally, the problem in Eq. (60) is a standard quadratic problem with simple
equality and inequality constraints that can be solved at low computational
complexity. Such problems can be easily formulated in YALMIP Löfberg (2004) or
ForBES Stella et al (2017) for MATLAB or CVXPY Diamond and Boyd (2016) for
Python.
Figure 5: Drug administration scheduling via optimal control. The behaviour of
the controlled system was simulated using the ABMPC method (dashed red line).
The predicted pharmacokinetic profile using the Grünwald-Letnikov
approximation (solid blue line) and fde12 are indistinguishable. The set-point
on $q_{2}$ (dashed orange line) and the maximum allowed concentration (black
solid lines) are shown in the figure.
The optimal control framework offers great flexibility in using arbitrary
system dynamics, constraints on the administration rate and the drug
concentration in the various compartments, and cost functions which encode the
administration objectives.
The quadratic function which we proposed in Eq. (60) is certainly not the only
admissible choice. For instance, other possible choices are
1. 1.
$J=\sum_{k=0}^{\nicefrac{{N_{d}}}{{t_{c}}}+1}(x_{\mathrm{ref},k}-x_{k})^{\prime}Q(x_{\mathrm{ref},k}-x_{k})+\beta
u_{k}^{2}$, where we also penalise the total amount of drug that is
administered throughout the prediction horizon,
2. 2.
$J=\sum_{k=0}^{\nicefrac{{N_{d}}}{{t_{c}}}+1}\mathrm{dist}^{2}(x_{k},[\underline{$x$},\bar{x}])$,
where $[\underline{$x$},\bar{x}]$ is the therapeutic window and
$\mathrm{dist}^{2}(\cdot,[\underline{$x$},\bar{x}])$ is the squared distance
function defined as
$\mathrm{dist}^{2}(x_{k},[\underline{$x$},\bar{x}])=\min_{z_{k}}\\{\|x_{k}-z_{k}\|^{2};\
\underline{$x$}\leq z_{k}\leq\bar{x}\\}$.
### Administration scheduling for populations
In the previous section, we assumed that the pharmacokinetic parameters are
known aiming at an individualised dose regimen. When designing an
administration schedule for a population of patients — without the ability to
monitor the distribution of the drug or the progress of the therapy — then $J$
becomes a function of the pharmacokinetic parameters (in our case $k_{10}$,
$k_{12}$, $k_{21,f}$ and $\alpha$). That said, $J$ becomes a random quantity
which follows an — often unknown or inexactly known — probability
distribution. In order to formulate an optimal control problem we now need to
extract a characteristic value out of the random quantity $J$. There are two
popular ways to do so leading to different problem statements.
First, we may consider the maximum value of $J$, $\max J$, over all possible
values of $k_{10}$, $k_{12}$, $k_{21,f}$ and $\alpha$. This leads to robust or
minimax problem formulations which consist in solving Bertsekas (2017)
$\displaystyle\min_{u_{0},\ldots,u_{N_{d}-1}}\max_{k_{10},k_{12},k_{21,f},\alpha}J(u_{0},\ldots,u_{N_{d}-1},k_{10},k_{12},k_{21,f},\alpha),$
(61)
subject to the system dynamics and constraints. In Eq. (61), the maximum is
taken with respect to the worst-case values of $k_{10}$, $k_{12}$, $k_{21,f}$
and $\alpha$, that is, their minimum and maximum values. Apparently, the
minimax approach does not make use of any probabilistic or statistical
information which is typically available for the model parameters. As a
result, it is likely to be overly conservative and lead to poor performance.
Figure 6: Drug administration to a population of $100$ (randomly generated)
individuals by stochastic optimal control. The yellow thin lines correspond to
the predicted individual responses to the administration and the black lines
represent the average response.
On the other hand, in the stochastic approach we minimise the Bertsekas and
Shreve (1996) expected cost $\mathbb{E}J$
$\displaystyle\min_{u_{0},\ldots,u_{N_{d}-1}}\mathbb{E}_{k_{10},k_{12},k_{21,f},\alpha}\
J(u_{0},\ldots,u_{N_{d}-1},k_{10},k_{12},k_{21,f},\alpha),$ (62)
subject to the system dynamics and constraints. Open-loop stochastic optimal
control methodologies have been proposed for the optimal design of regimens
under uncertainty for classical integer-order pharmacokinetics Schumitzky et
al (1983); Lago (1992); Bayard et al (1994), yet, to the best of our
knowledge, no studies have been conducted for the effectiveness of stochastic
optimal control for fractional pharmacodynamics.
The expectation in Eq. (62) can be evaluated empirically given a data-set of
estimated pharmacokinetic parameters
$(k_{10}^{(i)},k_{12}^{(i)},k_{21,f}^{(i)},\alpha^{(i)})_{i=1}^{L}$ by
minimising the sample average of $J$, that is
$\displaystyle\min_{u_{0},\ldots,u_{N_{d}-1}}\tfrac{1}{L}\sum_{i=1}^{L}J(u_{0},\ldots,u_{N_{d}-1},k_{10}^{(i)},k_{12}^{(i)},k_{21,f}^{(i)},\alpha^{(i)}).$
(63)
The effect of $L$ on the accuracy of the derived optimal decisions is
addressed in Campi et al (2009) where probabilistic bounds are provided.
Simulation results with the stochastic optimal control methodology on a
population of $100$ patients using $L=20$ are shown in Figure 6. Note that in
both the minimax and the stochastic approach, a common therapeutic course
$u_{0},\ldots,u_{N_{d}-1}$ is sought for the whole population of patients.
In stochastic optimal control it is customary to adapt the state constraints
in a probabilistic context requiring that they be satisfied with certain
probability, that is
$\displaystyle\mathrm{P}[x_{k}\leq x_{\max}]\geq 1-\beta,$ (64)
where $\beta\in[0,1]$ is a desired level of confidence. Such constraints are
known as chance constraints or probabilistic constraints. Such problems,
unless restrictive assumptions are imposed on the distribution of $x_{k}$,
lead to nonconvex optimisation problems and solutions can be approximated by
Monte Carlo sampling methods (Shapiro et al, 2009, Sec. 5.7.1) or by means of
convex relaxations known as ambiguous chance constraints where (64) are
replaced by $\mathrm{AV@R}_{\beta}[x_{k}]\leq x_{\max}$, where
$\mathrm{AV@R}_{\beta}[x_{k}]$ is the average value-at-risk of $x_{k}$ at
level $\beta$ (Shapiro et al, 2009, Sec. 6.3.4).
The minimax and the expectation operator are the two “extreme” choices with
the former relating to complete lack of probabilistic information and the
latter coming with the assumption of exact knowledge of the underlying
distribution of pharmacokinetic parameters. Other operators can be chosen to
account for imperfect knowledge of that distribution, therefore, bridging the
gap between minimax and stochastic optimal control. Suitable operators for
optimal control purposes are the coherent risk measures which give rise to
risk-averse optimal control which is the state of the art in optimsation under
uncertainty Herceg et al (2017).
### Model Predictive Control
Model predictive control fuses an optimal control open-loop decision making
process with closed-loop control when feedback is available. Following its
notable success and wide adoption in the industry, MPC has been proposed for
several cases of drug administration for drugs that follow integer-order
kinetics such as erythropoietin Gaweda et al (2008), lithium ions Sopasakis et
al (2015), propofol for anaesthesia Ionescu et al (2008) and, most
predominantly, insulin administration to diabetic patients Schaller et al
(2016); Hovorka et al (2004); Toffanin et al (2013); Parker et al (1999).
In MPC, at every time instant, we solve an optimal control problem which
produces a plan of future drug dosages over a finite prediction horizon by
minimising a performance index $J$. The future distribution of the drug in the
organism is predicted with a dynamical model such as the ones based on the
Grünwald-Letnikov discretisation presented in the previous sections. Out of
the planned sequence of dosages, the first one is actually administered to the
patient. At the next time instant, the drug concentration is measured and the
same procedure is repeated in a fashion known as receding horizon control
Rawlings and Mayne (2009).
A fractional variant of the classical PID controller, namely a PIλDμ
fractional-order controller, has been proposed in Sopasakis and Sarimveis
(2014) for the controlled administration. However, the comparative advantage
of MPC is that it can inherently account for administration rate, drug amount
and drug concentration constraints which are of capital importance. In
Sopasakis et al (2015); Sopasakis and Sarimveis (2017), the truncated
Grünwald-Letnikov approximation serves as the basis to formulate MPC
formulations with guaranteed asymptotic stability and satisfaction of the
constraints at all time instants. We also single out impulsive MPC, which was
proposed in Sopasakis et al (2015), which is particularly well-suited for
applications of bolus administration where the patient is injected the
medication (e.g., intravenously), thus abruptly increasing the drug
concentration. In such cases, it is not possible to achieve equilibrium, but
instead the objective becomes to keep the concentration within certain
therapeutic limits. Model predictive control can further be combined with
stochastic Patrinos et al (2014); Sopasakis et al (2017) and risk-averse
Herceg et al (2017) optimal control aiming at a highly robust administration
that is resilient to the inexact knowledge of the pharmacokinetic parameters
of the patient as well as potential time varying phenomena (e.g., change in
the PK/PD parameter values due to illness, drug-drug interactions and
alternations due to other external influences).
Furthermore, in feedback control scenarios, state information is often
partially available. For example, when a multi-compartment model is used,
concentration measurement from only a single compartment is available. As an
example, in the case of multi-compartment physiologically based models, we
would normally only have information from the blood compartment. In such
cases, a state observer can be used to produce estimates of concentrations or
amounts of drug in compartments where we do not have access. A state observer
is a dynamical system which, using observations of (i) those concentrations
that can be measures and (ii) amount or rate of administered drug at every
time instant, produces estimates $\hat{q}_{i}$ of those amounts/concentrations
of drug to which we do not have access. State observers are designed so that
$\hat{q}_{i}(t)-q_{i}(t)\to 0$ as $t\to\infty$, that is, although at the
beginning (at $t=0$) we only have an estimate $\hat{q}_{i}(0)$ of $q_{i}(0)$,
we shall eventually obtain better results which converge to the actual
concentrations. State observers, such as the Kalman filter, have been
successfully employed in the administration of compounds following integer-
order dynamics Sopasakis et al (2014) including applications of artificial
pancreas Patek et al (2007); Wang et al (2014).
Observers are, furthermore, employed to filter out measurement errors and
modelling errors. When the system model is inexact, the system dynamics in Eq.
(59) becomes
$\displaystyle\mathbf{x}_{k+1}=\hat{A}\mathbf{x}_{k}+\hat{B}u_{k}+d_{k},$ (65)
where $d_{k}$ serves as a model mismatch term. We may then assume that $d_{k}$
follows itself some dynamics often as trivial as $d_{k+1}=d_{k}$. We may then
formulate the following linear dynamical system with state
$(\mathbf{x}_{k},d_{k})$
$\displaystyle\begin{bmatrix}\mathbf{x}_{k+1}\\\
d_{k+1}\end{bmatrix}=\begin{bmatrix}\hat{A}&I\\\
0&I\end{bmatrix}\begin{bmatrix}\mathbf{x}_{k}\\\
d_{k}\end{bmatrix}+\begin{bmatrix}\hat{B}\\\ 0\end{bmatrix}u_{k},$ (66)
and build a state observer for $(\mathbf{x}_{k},d_{k})$ jointly. The estimates
$(\hat{\mathbf{x}}_{k},\hat{d}_{k})$ are then fed to the MPC. This leads to an
MPC formulation known as offset-free MPC Sopasakis et al (2014) that can
control systems with biased estimates of the pharmacokinetic parameters.
Model predictive control is a highly appealing approach because it can account
for imperfect knowledge of the pharmacokinetic parameters of the patient,
measurement noise, partial availability of measurements and constraints while
it decides the amount of administered drug by optimising a cost function thus
leading to unparalleled performance. Moreover, the tuning of MPC is more
straightforward compared to other control approaches such as PID or fractional
PID. In MPC the main tuning knob is the cost function $J$ of the involved
optimal control problem which reflects and encodes exactly the control
objectives.
## Conclusions
Fractional kinetics offers an elegant description of anomalous kinetics, i.e.,
non-exponential terminal phases, the presence of which has been acknowledged
in pharmaceutical literature extensively. The approach offers simplicity and a
valid scientific basis, since it has been applied in problems of diffusion in
physics and biology. It introduces the Mittag-Leffler function which describes
power law behaved data well, in all time scales, unlike the empirical power-
laws which describe the data only for large times. Despite the mathematical
difficulties, fractional pharmacokinetics offer undoubtedly a powerful and
indispensable approach for the toolbox of the pharmaceutical scientist.
Solutions of fractional-order systems involve Mittag-Leffler-type functions,
or other special functions whose numerical evaluation is nontrivial. Several
approximation techniques have been proposed as we outlined above for
fractional systems with different levels of accuracy, parsimony and
suitability for optimal administration scheduling or control. In particular,
those based on discrete time-domain approximations of the system dynamics with
bounded approximation error, such as the truncated Grünwald-Letnikov
approximation, are most suitable for optimal control applications. There
nowadays exist software that implement most of the algorithms that are
available in the literature and facilitate their practical use. Further
research effort needs to be dedicated in deriving error bounds in the time
domain for available approximation methods that will allow their adoption in
optimal control formulations.
The optimal control framework is suitable for the design of administration
courses both to individuals as well as to populations where the intra-patient
variability of pharmacokinetic parameters needs to be taken into account. In
fact, stochastic optimal control offers a data-driven decision making solution
that enables us to go from sample data to administration schemes for
populations. Optimal control offers great flexibility in formulating optimal
drug dosing problems and different problem structures arise naturally for
different modes of administration (continuous, bolus intravenous, oral and
more). At the same time, further research is necessary to make realistic
assumptions and translate them into meaningful optimisation problems.
MPC methodologies are becoming popular for controlled drug administration.
Yet, their potential for fractional-order pharmacokinetics and their related
properties, especially in regard to stochastic systems and the
characterisation of invariant sets, needs to be investigated.
Overall, at the intersection of fractional systems theory, pharmacokinetics,
numerical analysis, optimal control and model predictive control, spring
numerous research questions which are addressed in the context of mathematical
pharmacology.
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|
d\eta+2\beta^{\prime}\int_{0}^{B}e^{2\beta^{\prime}\eta}4h^{3}G_{B}^{\varepsilon}\partial_{\eta}G_{B}^{\varepsilon}d\eta$
$\displaystyle\quad=-\beta^{\prime}\int_{0}^{B}\int_{-\pi}^{\pi}\left(e^{2\beta^{\prime}\eta}\sin\phi(\varphi_{B}^{\varepsilon})^{2}-2e^{2\beta^{\prime}\eta}4h^{3}G_{B}^{\varepsilon}\sin\phi\varphi_{B}^{\varepsilon}\right)d\phi
d\eta$
$\displaystyle\quad=-\beta^{\prime}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}\sin\phi(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi
d\eta.$
The right-hand side of equation (4.3) can be estimated by
$\displaystyle\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}S_{2}(\varphi^{\varepsilon}_{B}-4h^{3}G^{\varepsilon}_{B})d\phi
d\eta\leq\frac{1}{4}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi
d\eta+\|e^{\beta^{\prime}\eta}S_{2}\|_{L^{2}([0,B]\times[-\pi,\pi))}^{2}.$
Taking the above inequalities into (4.3), we obtain
$\displaystyle\frac{3}{4}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi
d\eta+\frac{1}{2}\int_{0}^{B}\int_{-\pi}^{\pi}(F(\varepsilon;\eta)-2\beta^{\prime})\sin\phi
e^{2\beta^{\prime}\eta}(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi
d\eta$
$\displaystyle\quad+\frac{1}{2}\int_{\sin\phi<0}|\sin\phi|(\varphi_{B}^{\varepsilon})^{2}(0,\cdot)d\phi\leq\frac{1}{2}\int_{\sin\phi>0}\sin\phi\varphi_{b}^{2}d\phi+\|e^{\beta^{\prime}\eta}S_{2}\|_{L^{2}([0,B]\times[-\pi,\pi))}^{2}.$
For $\varepsilon$ sufficiently small, for example
$\varepsilon\leq(4-3\delta)/16$, inequality (3.11) implies for any
$\eta\in\mathbb{R}_{+}$,
$\displaystyle|F(\varepsilon;\eta)-2\beta^{\prime}|\leq
2\beta^{\prime}+\frac{4}{4-3\delta}\varepsilon\leq 1+\frac{1}{4}=\frac{5}{4}.$
(4.32)
With this, the previous inequality implies (16).
Moreover, we can get from equation (4.8) that
$\displaystyle\int_{0}^{B}e^{2\beta^{\prime}\eta}|\partial_{\eta}^{2}G_{B}^{\varepsilon}|^{2}d\eta$
$\displaystyle\quad\leq\int_{0}^{B}e^{2\beta^{\prime}\eta}|F(\varepsilon;\eta)\partial_{\eta}G_{B}^{\varepsilon}|^{2}d\eta+\int_{0}^{B}e^{2\beta^{\prime}\eta}|\langle\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon}\rangle|^{2}d\eta+\int_{0}^{B}e^{2\beta^{\prime}\eta}|S_{1}|^{2}d\eta$
$\displaystyle\quad\leq\int_{0}^{B}|F(\varepsilon;\eta)|^{2}e^{2\beta^{\prime}\eta}\langle\sin\phi(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})\rangle
d\eta+\int_{0}^{B}\left(\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi\cdot\int_{-\pi}^{\pi}d\phi\right)d\eta$
$\displaystyle\qquad+\int_{0}^{B}e^{2\beta^{\prime}\eta}|S_{1}|^{2}d\eta$
$\displaystyle\quad\leq\|F\|_{L^{\infty}(\mathbb{R}_{+})}^{2}2\pi\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi
d\eta+2\pi\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi
d\eta$
$\displaystyle\qquad+\int_{0}^{B}e^{2\beta^{\prime}\eta}|S_{1}|^{2}d\eta$
$\displaystyle\quad\leq
2\pi\left(\frac{4}{4-3\delta}\varepsilon+1\right)\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi
d\eta+\int_{0}^{B}e^{2\beta^{\prime}\eta}|S_{1}|^{2}d\eta.$
For $\varepsilon$ sufficiently small, for example
$\varepsilon\leq(4-3\delta)/16$, the above inequality becomes
$\displaystyle\int_{0}^{B}e^{2\beta^{\prime}\eta}|\partial_{\eta}^{2}G_{B}^{\varepsilon}|^{2}d\eta$
$\displaystyle\quad\leq\frac{5}{2}\pi\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi
d\eta+\int_{0}^{B}e^{2\beta^{\prime}\eta}|S_{1}|^{2}d\eta$
$\displaystyle\quad\leq
10\pi\int_{\sin\phi>0}\sin\phi\varphi_{b}^{2}d\phi+20\pi\|e^{\beta^{\prime}\eta}S_{2}\|_{L^{2}([0,B]\times[-\pi,\pi))}^{2}+\|e^{\beta^{\prime}\eta}S_{1}\|_{L^{2}([0,B])}^{2},$
and hence is uniformly bounded in $B$. Let
$C_{1}:=10\pi\int_{\sin\phi>0}\sin\phi\varphi_{b}^{2}d\phi+20\pi\|e^{\beta^{\prime}\eta}S_{2}\|_{L^{2}([0,B]\times[-\pi,\pi))}^{2}+\|e^{\beta^{\prime}\eta}S_{1}\|_{L^{2}([0,B])}^{2}$.
Using the bounded condition $\partial_{\eta}G_{B}^{\varepsilon}(B)=0$, we have
for any $\eta\in[0,B]$,
$\displaystyle|\partial_{\eta}G_{B}^{\varepsilon}(\eta)|$
$\displaystyle=-\left|\int_{\eta}^{B}\partial_{s}^{2}G_{B}^{\varepsilon}(s)ds\right|=\left|\int_{\eta}^{B}e^{-\beta^{\prime}s}e^{\beta^{\prime}s}\partial_{s}^{2}G_{B}^{\varepsilon}(s)ds\right|$
$\displaystyle\leq\left(\int_{\eta}^{B}e^{-2\beta^{\prime}s}ds\right)^{\frac{1}{2}}\int_{0}^{B}e^{2\beta^{\prime}s}|\partial_{s}^{2}G_{B}^{\varepsilon}(s)|^{2}ds$
$\displaystyle\leq
C_{1}\frac{\sqrt{e^{-2\beta^{\prime}\eta}-e^{-2\eta^{\prime}B}}}{\sqrt{2\beta^{\prime}}}\leq
C_{1}(2\beta^{\prime})^{-\frac{1}{2}}e^{-\beta^{\prime}\eta}.$
Using the boundary condition $G_{B}^{\varepsilon}(0)=0$, we have
$\displaystyle|G_{B}^{\varepsilon}(\eta)|$
$\displaystyle=\left|\int_{0}^{\eta}\partial_{s}G_{B}^{\varepsilon}(s)ds\right|\leq\int_{0}^{\eta}|\partial_{s}G_{B}^{\varepsilon}(s)|ds\leq
C_{1}(2\beta^{\prime})^{-\frac{1}{2}}\int_{0}^{\eta}e^{-\beta^{\prime}s}ds$
$\displaystyle\leq\sqrt{2}C_{1}(\beta^{\prime})^{-\frac{3}{2}}(1-e^{-\beta^{\prime}\eta})\leq\sqrt{2}C_{1}(\beta^{\prime})^{-\frac{3}{2}}.$
### 4.4. Existence on the half line
Next we pass to the limit $B\to\infty$ in (4.8)-(4.9) to show the existence
for system (4.1)-(4.2) on the half line.
###### Proof of Theorem 13.
By the uniform estimate (4.29), $G_{B}^{\varepsilon}\in L^{\infty}_{\rm
loc}([0,\infty))$. Hence there exists a subsequence such that
$\displaystyle
G_{B}^{\varepsilon}\rightharpoonup^{*}G^{\varepsilon},\quad\text{weakly star
in }L_{\rm loc}^{\infty}([0,\infty)),$ (4.33) $\displaystyle
G_{B}^{\varepsilon}\rightharpoonup G^{\varepsilon},\quad\text{weakly in
}L_{\rm loc}^{2}([0,\infty)).$
Moreover, by the estimate (16),
$\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon}$ is uniformly bounded in
$L^{2}_{\rm loc}([0,\infty);L^{2}([-\pi,\pi)))$, hence
$\displaystyle\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon}\rightharpoonup\varphi^{\varepsilon}-4h^{3}G^{\varepsilon},\quad\text{weakly
in }L^{2}_{\rm loc}([0,\infty);L^{2}([-\pi,\pi))),$
and so
$\displaystyle\varphi_{B}^{\varepsilon}\rightharpoonup\varphi^{\varepsilon},\quad\text{weakly
in }L^{2}_{\rm loc}([0,\infty);L^{2}([-\pi,\pi))).$ (4.34)
With (4.33) and (4.34), we can pass to the limit in (4.8)-(4.9) with boundary
conditions (4.10)-(4.11) and obtain that
$(G^{\varepsilon},\varphi^{\varepsilon})$ solve system (4.1)-(4.2) in the
sense of distributions and satisfy the boundary conditions (4.3)-(4.4).
Moreover, we can pass to the limit $B\to\infty$ in (4.30) and obtain
$\displaystyle\partial_{\eta}G^{\varepsilon}=\langle\sin\phi\varphi^{\varepsilon}(\eta,\cdot)\rangle$
(4.35)
for $\eta\in\mathbb{R}_{+}$ almost everywhere. By the semi-lower continuity of
the $L^{2}$ norm, we can take $\liminf$ in (16) and use
$\displaystyle\liminf_{B\to\infty}\int_{0}^{B}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi_{B}^{\varepsilon}-4h^{3}G_{B}^{\varepsilon})^{2}d\phi
d\eta\geq\int_{0}^{\infty}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi^{\varepsilon}-4h^{3}G^{\varepsilon})^{2}d\phi
d\eta$
and obtain (13). ∎
### 4.5. Decay properties of the solutions
We can use the above derived uniform estimates to show that solutions to
system (4.1)-(4.2) decay to constants as $\eta\to\infty$. We set
$\displaystyle
C_{2}:=4\int_{\sin\phi>0}\sin\phi\varphi_{b}^{2}d\phi+8\|e^{\beta^{\prime}\eta}S_{2}\|_{L^{2}([0,\infty)\times[-\pi,\pi))}^{2}.$
Then
$\displaystyle\int_{0}^{\infty}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi^{\varepsilon}-4h^{3}G^{\varepsilon})^{2}d\phi
d\eta\leq C_{2}.$ (4.36)
By the relation (4.35), we get
$\displaystyle\int_{0}^{\infty}e^{2\beta^{\prime}\eta}|\partial_{\eta}G^{\varepsilon}|^{2}d\eta$
$\displaystyle=\int_{0}^{\infty}e^{2\beta^{\prime}\eta}\left|\int_{-\pi}^{\pi}\sin\phi(\varphi^{\varepsilon})d\phi\right|^{2}d\eta=\int_{0}^{\infty}e^{2\beta^{\prime}\eta}\left|\int_{-\pi}^{\pi}\sin\phi(\varphi^{\varepsilon}-4h^{3}G^{\varepsilon})d\phi\right|^{2}d\eta$
$\displaystyle\leq\int_{0}^{\infty}e^{2\beta^{\prime}\eta}\left(\int_{-\pi}^{\pi}\sin^{2}\phi
d\phi\right)\left(\int_{-\pi}^{\pi}(\varphi^{\varepsilon}-4h^{3}G^{\varepsilon})^{2}d\phi\right)d\eta$
$\displaystyle=\pi\int_{0}^{\infty}\int_{-\pi}^{\pi}e^{2\beta^{\prime}\eta}(\varphi^{\varepsilon}-4h^{3}G^{\varepsilon})^{2}d\phi\leq\pi
C_{2}.$
Therefore, $\lim_{\eta\to\infty}G^{\varepsilon}(\eta)$ exists and is finite.
To show this, we can take $M_{2}>M_{1}>0$ to be some large numbers and
$\displaystyle|G^{\varepsilon}(M_{1})-G^{\varepsilon}(M_{2})|$
$\displaystyle=\left|\int_{M_{1}}^{M_{2}}\partial_{\eta}G^{\varepsilon}d\eta\right|\leq\int_{M_{1}}^{M_{2}}e^{-\beta^{\prime}\eta}e^{\beta^{\prime}\eta}|\partial_{\eta}G^{\varepsilon}|d\eta$
$\displaystyle\leq\left(\int_{M_{1}}^{M_{2}}e^{-2\beta^{\prime}\eta}d\eta\right)^{\frac{1}{2}}\left(\int_{M_{1}}^{M_{2}}e^{2\beta^{\prime}\eta}|\partial_{\eta}G^{\varepsilon}|^{2}d\eta\right)^{\frac{1}{2}},$
$\displaystyle=\sqrt{(e^{-2\beta^{\prime}M_{1}}-e^{-2\beta^{\prime}M_{2}})}\pi\sqrt{C_{2}/(2\beta^{\prime})}.$
Since for $M_{1},M_{2}$ sufficiently large,
$e^{-2\beta^{\prime}M_{1}}-e^{-2\beta^{\prime}M_{2}}$ is sufficiently small.
$\lim_{\eta\to\infty}G^{\varepsilon}(\eta)$ exists. Let
$G^{\varepsilon}_{\infty}=\lim_{\eta\to\infty}G^{\varepsilon}(\eta)$. Then we
take $M_{1}=\eta$, $M_{2}=\infty$ in the above inequality and obtain
$\displaystyle|G^{\varepsilon}(\eta)-G^{\varepsilon}_{\infty}|\leq\pi\sqrt{C_{2}/(2\beta^{\prime})}e^{-\beta^{\prime}\eta}.$
Moreover, taking $M_{1}=0$, $M_{2}=\infty$ gives
$\displaystyle|G^{\varepsilon}_{\infty}|\leq\pi\sqrt{C_{2}/(2\beta^{\prime})}.$
Due to the exponentially convergence of $h$, we have
$\displaystyle|4h^{3}(\eta)G^{\varepsilon}(\eta)-4h_{\infty}^{3}G^{\varepsilon}_{\infty}|$
$\displaystyle\leq
4h^{3}(\eta)|G^{\varepsilon}(\eta)-G^{\varepsilon}_{\infty}|+4|h^{3}(\eta)-h_{\infty}^{3}||G^{\varepsilon}_{\infty}|\leq
Ce^{-\alpha\eta}+Ce^{-\beta^{\prime}\eta}$ $\displaystyle\leq
2Ce^{-\beta^{\prime}\eta},$ (4.37)
taking $1>\alpha>\beta^{\prime}$.
To show the exponential convergence of $\varphi^{\varepsilon}$, we use the
formulas (A.4), (A.5) and (20). In the case $\sin\phi>0$, we have
$\displaystyle|\varphi^{\varepsilon}(\eta,\phi)-4h_{\infty}^{3}G_{\infty}^{\varepsilon}|$
$\displaystyle=\left|\int_{0}^{\eta}\frac{4h^{3}(\xi)G^{\varepsilon}(\xi)+S_{2}(\xi,\phi^{\prime}(\phi,\eta,\xi))}{\sin\phi^{\prime}(\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}d\xi-4h^{3}_{\infty}G_{\infty}^{\varepsilon}\right|$
$\displaystyle\leq\int_{0}^{\eta}\frac{|4h^{3}(\xi)G^{\varepsilon}(\xi)-4h_{\infty}^{3}G^{\varepsilon}_{\infty}|}{\sin\phi^{\prime}(\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}d\xi$
$\displaystyle\quad+\int_{0}^{\eta}\frac{|S_{2}(\xi,\phi^{\prime}(\phi,\eta,\xi))|}{\sin\phi^{\prime}(\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}d\xi$
$\displaystyle\quad+\left|1-\int_{0}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}d\xi\right||4h_{\infty}^{3}G_{\infty}^{\varepsilon}|$
$\displaystyle\quad\leq
2C\int_{0}^{\eta}\frac{e^{-\beta^{\prime}\xi}}{\sin\phi^{\prime}(\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}d\xi+Ce^{-\int_{0}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}$
$\displaystyle\quad\leq
2C\int_{0}^{\eta}\frac{1}{1-\beta^{\prime}\sin\phi^{\prime}(\phi,\eta,\xi)}de^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}-\frac{1}{2}\beta^{\prime}\xi}+Ce^{-\int_{0}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}$
$\displaystyle\quad\leq\frac{2C}{1-\beta^{\prime}}(e^{-\beta^{\prime}\eta}-e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}})+Ce^{-\int_{0}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}$
$\displaystyle\quad\leq\frac{3C}{1-\beta^{\prime}}e^{-\beta^{\prime}\eta},$
where we use $\beta^{\prime}<1/2$ and $\sin\phi^{\prime}\leq 1$ in the above
inequality. One can also check for $\sin\phi<0$, there exists positive
constants $C>0$,
$|\varphi^{\varepsilon}(\eta,\phi)-4h_{\infty}^{3}G_{\infty}^{\varepsilon}|\leq
Ce^{-\beta^{\prime}\eta},$
which follows a similar estimate as the case for $\sin\phi>0$ above and the
details are similar to the proof of (3.47).
### 4.6. Uniqueness
In order to show the uniqueness, let
$(G_{1}^{\varepsilon},\varphi_{1}^{\varepsilon})$ and
$(G_{2}^{\varepsilon},\varphi_{2}^{\varepsilon})$ be two solutions of the
problem (4.1)-(4.4). Let $g:=G_{1}^{\varepsilon}-G_{2}^{\varepsilon}$ and
$\psi:=\varphi_{1}^{\varepsilon}-\varphi_{2}^{\varepsilon}$, then they satisfy
$\displaystyle\partial_{\eta}^{2}g+F(\varepsilon;\eta)\partial_{\eta}g+\langle\psi-4h^{3}g\rangle=0,$
(4.38)
$\displaystyle\sin\phi\partial_{\eta}\psi+F(\varepsilon;\eta)\cos\phi\partial_{\phi}\psi+\psi-4h^{3}g=0,$
(4.39)
with boundary conditions
$\displaystyle g(0)=0,\quad\psi(0,\phi)=0,\quad\text{for any }\sin\phi>0.$
Moreover, as $\eta\to\infty$, $G_{1}^{\varepsilon}\to G_{1,\infty}$,
$G_{2}^{\varepsilon}\to G_{2,\infty}$ and $\psi\to
4h_{\infty}^{3}(G_{1,\infty}-G_{2,\infty})$, where $G_{1,\infty}$,
$G_{2,\infty}$ are constants.
We multiply the first equation by $4h^{3}g$ and the second by $\psi$, and
integrate over $[0,\infty)\times[-\pi,\pi)$ to get
$\displaystyle\int_{0}^{\infty}4h^{3}|\partial_{\eta}g|^{2}d\eta+\int_{0}^{\infty}\partial_{\eta}(4h^{3})g\partial_{\eta}gd\eta-\int_{0}^{\infty}F(\varepsilon;\eta)4h^{3}g\partial_{\eta}gd\eta+\frac{1}{2}\int_{0}^{\infty}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\cos\phi\partial_{\phi}\psi^{2}d\phi
d\eta$
$\displaystyle\quad+\int_{-\pi}^{0}|\sin\phi|\psi^{2}(0,\cdot)d\phi+\int_{0}^{\infty}\int_{-\pi}^{\pi}(\psi-4h^{3}g)^{2}d\phi
d\eta=0.$ (4.40)
From the relation (4.35), we have
$\partial_{\eta}g=\langle\sin\phi\varphi\rangle$. Thus
$\displaystyle-\int_{0}^{\infty}F(\varepsilon;\eta)4h^{3}g\partial_{\eta}gd\eta+\frac{1}{2}\int_{0}^{\infty}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\cos\phi\partial_{\phi}\psi^{2}d\phi
d\eta$
$\displaystyle\quad=-\int_{0}^{\infty}\int_{-\pi}^{\pi}F(\varepsilon;\eta)4h^{3}g\sin\phi\varphi
d\phi
d\eta+\frac{1}{2}\int_{0}^{\infty}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\sin\phi\psi^{2}d\phi
d\eta$
$\displaystyle\quad=-\frac{1}{2}\int_{0}^{\infty}\int_{-\pi}^{\pi}F(\varepsilon;\eta)\sin\phi(\psi-4h^{3}g)^{2}d\phi
d\eta.$
The spectral assumption implies
$\displaystyle\int_{0}^{\infty}4h^{3}|\partial_{\eta}g|^{2}d\eta+\int_{0}^{\infty}\partial_{\eta}(4h^{3})g\partial_{\eta}gd\eta\geq
0.$
Therefore, we get from (4.6) that
$\displaystyle\int_{-\pi}^{0}|\sin\phi|\psi^{2}(0,\cdot)d\phi+\int_{0}^{\infty}\int_{-\pi}^{\pi}\left(1+\frac{1}{2}F(\varepsilon;\eta)\right)(\psi-4h^{3}g)^{2}d\phi
d\eta=0.$
For $\eta$ sufficiently small, $1+\tfrac{1}{2}F(\varepsilon;\eta)>0$ for any
$\eta\in\mathbb{R}_{+}$, hence the above equation implies $\psi=4h^{3}g$
almost everywhere on $\mathbb{R}_{+}\times[-\pi,\pi)$ and $\psi(0,\phi)=0$ for
almost every $\phi\in[-\pi,\pi)$. Hence
$\partial_{\eta}g=\langle\sin\phi\psi\rangle=\langle\sin\phi
4h^{3}g\rangle=0$, which with $g(0)=0$ implies $g=0$ on $\mathbb{R}_{+}$. We
also have $\psi=4h^{3}g=0$. Hence system (4.38)-(4.39) only has trival
solution and the solution to the problem (4.1)-(4.4).
∎
###### Remark 17.
Compared to the flat boundary case without geometric corrections [6], the
effect of the geometric corrections could be seen from (4.32). for
$\varepsilon$ not sufficiently small, one cannot get a $L^{2}$ estimate on
$\|e^{\beta^{\prime}\eta}(\varphi^{\varepsilon}-4h^{3}G^{\varepsilon})\|_{L^{2}(\mathbb{R}_{+}\times[-\pi,\pi])}$.
However, for $\varepsilon$ sufficiently small, the geometric correction
contributes to a small error of this $L^{2}$ norm so does not affect the
boundness of this norm, which is crucial for proving Theorem 13.
## 5\. Convergence of the diffusive limit
### 5.1. Errors of the approximation
Next we calculate the approximation errors. Using systems (2.71)-(2.79) and
(2.56)-(2.67), we can calculate
$\displaystyle\mathcal{R}_{1}\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}^{\varepsilon}+\bar{T}_{k}^{\varepsilon}),\sum_{k=0}^{N}\varepsilon^{k}(\psi_{k}^{\varepsilon}+\bar{\psi}_{k}^{\varepsilon})\right)$
$\displaystyle\quad=\varepsilon^{2}\Delta\sum_{k=0}^{N}\varepsilon^{k}(T_{k}^{\varepsilon}+\bar{T}_{k}^{\varepsilon})+\left\langle\sum_{k=0}^{N}\varepsilon^{k}(\psi_{k}^{\varepsilon}+\bar{\psi}_{k}^{\varepsilon})+\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}^{\varepsilon}+\bar{T}_{k}^{\varepsilon})\right)^{4}\right\rangle$
$\displaystyle\quad=\varepsilon^{2}\Delta\sum_{k=0}^{N}\varepsilon^{k}T_{k}^{\varepsilon}+\sum_{k=0}^{N}\varepsilon^{k}\langle\psi_{k}^{\varepsilon}\rangle+\partial_{\eta}^{2}\sum_{k=0}^{N}\varepsilon^{k}\bar{T}_{k}^{\varepsilon}+F(\varepsilon;\eta)\partial_{\eta}\sum_{k=0}^{N}\varepsilon^{k}\bar{T}_{k}^{\varepsilon}-\frac{\chi_{0}\varepsilon^{2}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\sum_{k=0}^{N}\varepsilon^{k}\bar{T}_{k}^{\varepsilon}$
$\displaystyle\qquad+\left\langle\sum_{k=0}^{N}\varepsilon^{k}\bar{\psi}_{k}^{\varepsilon}-\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}^{\varepsilon}+\bar{T}_{k}^{\varepsilon})\right)^{4}\right\rangle$
$\displaystyle\quad=\mathcal{R}_{1}\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k}^{\varepsilon},\sum_{k=0}^{N}\varepsilon^{k}\psi_{k}^{\varepsilon}\right)+\left\langle\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k}^{\varepsilon}\right)^{4}\right\rangle$
$\displaystyle\qquad+\sum_{k=0}^{N}\varepsilon^{k}(\partial_{\eta}^{2}\bar{T}_{k}^{\varepsilon}-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\bar{T}_{k}^{\varepsilon}+\langle\bar{\psi}_{k}^{\varepsilon}-\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k)\rangle+\frac{1}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{k-2}^{\varepsilon})$
$\displaystyle\qquad+\frac{\varepsilon^{N+1}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{N-1}^{\varepsilon}+\frac{\varepsilon^{N+2}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{N}^{\varepsilon}-\sum_{k=N+1}^{4N}\varepsilon^{k}\langle\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k)\rangle$
$\displaystyle\quad=\mathcal{R}_{1}\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k},\sum_{k=0}^{N}\varepsilon^{k}\psi_{k}\right)-\sum_{k=0}^{N}\varepsilon^{k}\langle\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k)-\mathcal{C}(T^{\varepsilon},k)\rangle$
$\displaystyle\qquad-\sum_{k=N+1}^{4N}\varepsilon^{k}\langle\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k)-\mathcal{C}(T^{\varepsilon},k)\rangle$
$\displaystyle\qquad+\frac{\varepsilon^{N+1}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{N-1}^{\varepsilon}+\frac{\varepsilon^{N+2}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{N}^{\varepsilon}+\sum_{k=0}^{N}\varepsilon^{k}\langle\mathcal{C}(\bar{T}^{\varepsilon}+P^{\varepsilon},k)-\mathcal{C}(P^{\varepsilon},k)\rangle$
$\displaystyle\qquad+\sum_{k=0}^{N}\varepsilon^{k}(\partial_{\eta}^{2}\bar{T}_{k}^{\varepsilon}-\frac{\varepsilon}{(1-\varepsilon\eta)}\partial_{\eta}\bar{T}_{k}^{\varepsilon}+\frac{1}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{k-2}^{\varepsilon}+\langle\bar{\psi}_{k}^{\varepsilon}-\mathcal{C}(\bar{T}^{\varepsilon}+P^{\varepsilon},k)+\mathcal{C}(P^{\varepsilon},k)\rangle),$
(5.1)
and
$\displaystyle\mathcal{R}_{2}\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}^{\varepsilon}+\bar{T}_{k}^{\varepsilon}),\sum_{k=0}^{N}\varepsilon^{k}(\psi_{k}^{\varepsilon}+\bar{\psi}_{k}^{\varepsilon})\right)$
$\displaystyle\quad=\varepsilon\beta\cdot\nabla\sum_{k=0}^{N}\varepsilon^{k}(T_{k}^{\varepsilon}+\bar{T}_{k}^{\varepsilon})+\sum_{k=0}^{N}\varepsilon^{k}(\psi_{k}^{\varepsilon}+\bar{\psi}_{k}^{\varepsilon})-\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}^{\varepsilon}+\bar{T}_{k}^{\varepsilon})\right)^{4}$
$\displaystyle\quad=\varepsilon\beta\cdot\nabla\sum_{k=0}^{N}\varepsilon^{k}\psi_{k}^{\varepsilon}+\sum_{k=0}^{N}\varepsilon^{k}\psi_{k}^{\varepsilon}+\sin\phi\partial_{\eta}\sum_{k=0}^{N}\varepsilon^{k}\bar{\psi}_{k}^{\varepsilon}-\frac{\varepsilon}{1-\varepsilon\eta}\cos\phi\partial_{\phi}\sum_{k=0}^{N}\varepsilon^{k}\bar{\psi}_{k}^{\varepsilon}$
$\displaystyle\qquad-\frac{\varepsilon}{1-\varepsilon\eta}\cos\phi\partial_{\theta}\sum_{k=0}^{N}\varepsilon^{k}\bar{\psi}_{k}^{\varepsilon}+\sum_{k=0}^{N}\varepsilon^{k}\bar{\psi}_{k}^{\varepsilon}-\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}^{\varepsilon}+\bar{T}_{k}^{\varepsilon})\right)^{4}$
$\displaystyle\quad=\mathcal{R}_{2}\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k}^{\varepsilon},\sum_{k=0}^{N}\varepsilon^{k}\psi_{k}^{\varepsilon}\right)+\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k}^{\varepsilon}\right)^{4}$
$\displaystyle\qquad+\sum_{k=0}^{N}\varepsilon^{k}(\sin\phi\partial_{\eta}\bar{\psi}_{k}^{\varepsilon}-\frac{\varepsilon}{1-\varepsilon\eta}\cos\phi\partial_{\phi}\bar{\psi}_{k}^{\varepsilon}+\bar{\psi}_{k}^{\varepsilon}-\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k)-\frac{1}{(1-\varepsilon\eta)}\cos\phi\partial_{\theta}\bar{\psi}_{k-1})$
$\displaystyle\qquad-\frac{\varepsilon^{N+1}}{1-\varepsilon\eta}\cos\phi\partial_{\theta}\bar{\psi}_{N}^{\varepsilon}-\sum_{k=N+1}^{4N}\varepsilon^{k}\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k)$
$\displaystyle\quad=\mathcal{R}_{2}\left(\sum_{k=0}^{N}\varepsilon^{k}T_{k}^{\varepsilon},\sum_{k=0}^{N}\varepsilon^{k}\psi_{k}^{\varepsilon}\right)-\sum_{k=0}^{N}\varepsilon^{k}(\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k)-\mathcal{C}(T^{\varepsilon},k))-\sum_{k=N+1}^{4N}\varepsilon^{k}(\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k)-\mathcal{C}(T^{\varepsilon},k))$
$\displaystyle\qquad-\frac{\varepsilon^{N+1}}{1-\varepsilon\eta}\cos\phi\partial_{\theta}\bar{\psi}_{N}^{\varepsilon}+\sum_{k=0}^{N}\varepsilon^{k}(\mathcal{C}(\bar{T}^{\varepsilon}+P^{\varepsilon},k)-\mathcal{C}(P^{\varepsilon},k))$
$\displaystyle\qquad+\sum_{k=0}^{N}\varepsilon^{k}(\sin\phi\partial_{\eta}\bar{\psi}_{k}^{\varepsilon}-\frac{\varepsilon}{1-\varepsilon\eta}\cos\phi\partial_{\phi}\bar{\psi}_{k}^{\varepsilon}-\frac{1}{1-\varepsilon\eta}\partial_{\theta}\bar{\psi}^{\varepsilon}_{k-1}+\bar{\psi}_{k}^{\varepsilon}-\mathcal{C}(\bar{T}^{\varepsilon}+P^{\varepsilon},k)+\mathcal{C}(P^{\varepsilon},k)).$
(5.2)
By the definition (2.56) of
$(\bar{T}_{0}^{\varepsilon},\bar{\psi}_{0}^{\varepsilon})$, we get (we drop
the superscripts $\varepsilon$ in the calculations below)
$\displaystyle E_{0}^{0}:=$
$\displaystyle\partial_{\eta}^{2}\bar{T}_{0}-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\bar{T}_{0}+\langle\bar{\psi}_{0}-(\bar{T}_{0}+T_{0}(0))^{4}+T_{0}^{4}(0)\rangle$
$\displaystyle=$
$\displaystyle\partial_{\eta}^{2}(\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty}))-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}(\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty}))+\langle\chi(\tilde{\psi}_{0}-\tilde{\psi}_{0,\infty})-(\bar{T}_{0}+T_{0}(0))^{4}+T_{0}^{4}(0)\rangle$
$\displaystyle=$
$\displaystyle(\tilde{T}_{0}-\tilde{T}_{0,\infty})\partial_{\eta}^{2}\chi+2\partial_{\eta}\chi\partial_{\eta}\tilde{T}_{0}+\chi\partial_{\eta}^{2}\tilde{T}_{0}-\frac{\varepsilon}{(1-\varepsilon\eta)}\partial_{\eta}\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty})$
$\displaystyle-\frac{\varepsilon}{(1-\varepsilon\eta)}\chi\chi_{0}\partial_{\eta}\tilde{T}_{0}+\langle\chi(\tilde{\psi}_{0}-\tilde{\psi}_{0,\infty})-(\bar{T}_{0}+T_{0}(0))^{4}+T_{0}^{4}(0)\rangle$
$\displaystyle=$
$\displaystyle(\tilde{T}_{0}-\tilde{T}_{0,\infty})\partial_{\eta}^{2}\chi+2\partial_{\eta}\chi\partial_{\eta}\tilde{T}_{0}-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty})+\chi(\partial_{\eta}^{2}\tilde{T}_{0}+F(\varepsilon;\eta)\partial_{\eta}\tilde{T}_{0}+\langle\tilde{\psi}_{0}-\tilde{\psi}_{0,\infty}\rangle)$
$\displaystyle-\langle(\bar{T}_{0}+T_{0}(0))^{4}-T_{0}^{4}(0)\rangle$
$\displaystyle=$
$\displaystyle(\tilde{T}_{0}-\tilde{T}_{0,\infty})\partial_{\eta}^{2}\chi+2\partial_{\eta}\chi\partial_{\eta}\tilde{T}_{0}-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty})$
$\displaystyle+\chi\langle\tilde{T}_{0}^{4}-\tilde{T}_{0,\infty}^{4}\rangle-\langle(\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty})+\tilde{T}_{0,\infty})^{4}-\tilde{T}_{0,\infty}^{4}\rangle,$
(5.3)
where we use the fact that $\partial_{\eta}^{2}\chi=0$ and
$\chi\chi_{0}=\chi$. We can also calculate
$\displaystyle E_{0}^{1}:=$
$\displaystyle\sin\phi\partial_{\eta}\bar{\psi}_{0}-\frac{\varepsilon}{1-\varepsilon\eta}\cos\phi\partial_{\phi}\bar{\psi}_{0}+\bar{\psi}_{0}-(\bar{T}_{0}+T_{0}(0))^{4}+T_{0}^{4}(0)$
$\displaystyle=$
$\displaystyle\sin\phi(\tilde{\psi}_{0}-\tilde{\psi}_{0,\infty})\partial_{\eta}\chi+\chi(\sin\phi\partial_{\eta}\tilde{\psi}_{0}+F(\varepsilon;\eta)\cos\phi\partial_{\phi}\tilde{\psi}_{0}+\tilde{\psi}_{0}-\tilde{T}_{0}^{4})$
$\displaystyle+\chi(\tilde{T}_{0}^{4}-\tilde{T}_{0,\infty}^{4})-((\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty})+\tilde{T}_{0,\infty})^{4}-\tilde{T}_{0,\infty}^{4})$
$\displaystyle=$
$\displaystyle\sin\phi(\tilde{\psi}_{0}-\tilde{\psi}_{0,\infty})\partial_{\eta}\chi+\chi(\tilde{T}_{0}^{4}-\tilde{T}_{0,\infty}^{4})-((\chi(\tilde{T}_{0}-\tilde{T}_{0,\infty})+\tilde{T}_{0,\infty})^{4}-\tilde{T}_{0,\infty}^{4}).$
(5.4)
Similarly, by (2.67), we get
$\displaystyle E_{k}^{0}:=$
$\displaystyle\partial_{\eta}^{2}\bar{T}_{k}-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\bar{T}_{k}+\frac{1}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{k-2}+\langle\bar{\psi}_{k}-\mathcal{C}(\bar{T}+P,k)+\mathcal{C}(P,k)\rangle$
(5.5) $\displaystyle=$
$\displaystyle\partial_{\eta}^{2}(\chi(\tilde{T}_{k}-\tilde{T}_{k,\infty}))-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}(\chi(\tilde{T}_{k}-\tilde{T}_{k,\infty}))+\frac{1}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{k-2}$
$\displaystyle+\langle\chi(\tilde{\psi}_{k}-\tilde{\psi}_{k,\infty})-4(P_{0}+\bar{T}_{0})^{3}(P_{k}+\bar{T}_{k})+4P_{0}^{3}P_{k}\rangle-\langle\mathcal{E}(P+\bar{T},k-1)-\mathcal{E}(P,k-1)\rangle$
$\displaystyle=$
$\displaystyle(\tilde{T}_{k}-\tilde{T}_{k,\infty})\partial_{\eta}^{2}\chi+2\partial_{\eta}\chi\partial_{\eta}\tilde{T}_{k}-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\chi(\tilde{T}_{k}-\tilde{T}_{k,\infty})$
$\displaystyle+\chi\bigg{\langle}\partial_{\eta}^{2}\tilde{T}_{k}+F(\varepsilon;\eta)\partial_{\eta}\tilde{T}_{k}+\tilde{\psi}_{k}-4\tilde{T}_{0}^{3}\tilde{T}_{k}-4(\tilde{T}_{0}^{3}-P_{0}^{3})(P_{k}-P_{k}(0))-\mathcal{E}(\bar{T}+P,k-1)$
$\displaystyle+\mathcal{E}(P,k-1)+\frac{\chi_{0}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{k-2}\bigg{\rangle}+\chi\langle
4\tilde{T}_{0}^{3}\tilde{T}_{k}-4\tilde{T}_{0,\infty}^{3}\tilde{T}_{k,\infty}\rangle$
$\displaystyle-\langle
4(P_{0}+\bar{T}_{0})^{3}(P_{k}+\bar{T}_{k})-4P_{0}^{3}P_{k}-\chi(4(\tilde{T}_{0}^{3}-P_{0}^{3})(P_{k}-P_{k}(0)))\rangle$
$\displaystyle-(1-\chi)\langle\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1)\rangle+\frac{1-\chi}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{k-2}$
(5.6) $\displaystyle=$
$\displaystyle(\tilde{T}_{k}-\tilde{T}_{k,\infty})\partial_{\eta}^{2}\chi+2\partial_{\eta}\chi\partial_{\eta}\tilde{T}_{k}-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\chi(\tilde{T}_{k}-\tilde{T}_{k,\infty})+\chi\langle
4\tilde{T}_{0}^{3}\tilde{T}_{k}-4\tilde{T}_{0,\infty}^{3}\tilde{T}_{k,\infty}\rangle$
$\displaystyle-\langle
4(P_{0}+\bar{T}_{0})^{3}(P_{k}+\bar{T}_{k})-4P_{0}^{3}P_{k}-\chi(4(\tilde{T}_{0}^{3}-P_{0}^{3})(P_{k}-P_{k}(0)))\rangle$
$\displaystyle-(1-\chi)\langle\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1)\rangle+\frac{1-\chi}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{k-2}$
(5.7)
and
$\displaystyle E_{k}^{1}:=$
$\displaystyle\sin\phi\partial_{\eta}\bar{\psi}_{k}-\frac{\varepsilon}{1-\varepsilon\eta}\cos\phi\partial_{\phi}\bar{\psi}_{k}-\frac{1}{1-\varepsilon\eta}\cos\phi\partial_{\theta}\bar{\psi}_{k-1}+\bar{\psi}_{k}-\mathcal{C}(\bar{T}+P,k)+\mathcal{C}(P,k)$
$\displaystyle=$
$\displaystyle\sin\phi\partial_{\eta}\chi(\tilde{\psi}_{k}-\tilde{\psi}_{k,\infty})+\chi\bigg{(}\sin\phi\partial_{\eta}\tilde{\psi}_{k}+F(\varepsilon;\eta)\cos\phi\partial_{\eta}\tilde{\psi}_{k}-\frac{\chi_{0}}{(1-\varepsilon\eta)}\partial_{\theta}\bar{\psi}_{k-1}$
$\displaystyle+\tilde{\psi}_{k}-4\tilde{T}_{0}^{3}\tilde{T}_{k}-4(\tilde{T}_{0}^{3}-P_{0}^{3})(P_{k}-P_{k}(0))-\mathcal{E}(\bar{T}+P,k-1)+\mathcal{E}(P,k-1)\bigg{)}$
$\displaystyle+\chi(4\tilde{T}_{0}^{3}\tilde{T}_{k}-4\tilde{T}_{0,\infty}^{3}\tilde{T}_{k,\infty})-(4(P_{0}+\bar{T}_{0})^{3}(P_{k}+\bar{T}_{k})-4P_{0}^{3}P_{k}-\chi(4(\tilde{T}_{0}^{3}-P_{0}^{3})(P_{k}-P_{k}(0))))$
$\displaystyle-(1-\chi)(\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1))-\frac{1-\chi}{1-\varepsilon\eta}\cos\phi\partial_{\theta}\bar{\psi}_{k-1},$
$\displaystyle=$
$\displaystyle\sin\phi\partial_{\eta}\chi(\tilde{\psi}_{k}-\tilde{\psi}_{k,\infty})+\chi(4\tilde{T}_{0}^{3}\tilde{T}_{k}-4\tilde{T}_{0,\infty}^{3}\tilde{T}_{k,\infty})$
$\displaystyle-(4(P_{0}+\bar{T}_{0})^{3}(P_{k}+\bar{T}_{k})-4P_{0}^{3}P_{k}-\chi(4(\tilde{T}_{0}^{3}-P_{0}^{3})(P_{k}-P_{k}(0))))$
$\displaystyle-(1-\chi)(\mathcal{E}(\bar{T}+P,k-1)-\mathcal{E}(P,k-1))-\frac{1-\chi}{1-\varepsilon\eta}\cos\phi\partial_{\theta}\bar{\psi}_{k-1}.$
(5.8)
Using the formulas (5.1) and (5.1), we get from the above equations
$\displaystyle\mathcal{R}_{1}(T^{a,\varepsilon},\psi^{a,\varepsilon})$
$\displaystyle\quad=\varepsilon^{N+1}\Delta T_{N-1}+\varepsilon^{N+2}\Delta
T_{N}^{\varepsilon}-\sum_{k=0}^{N}\varepsilon^{k}\langle\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k)-\mathcal{C}(T^{\varepsilon},k)-\mathcal{C}(\bar{T}^{\varepsilon}+P^{\varepsilon},k)+\mathcal{C}(P^{\varepsilon},k)\rangle$
$\displaystyle\qquad-\sum_{k=N+1}^{4N}\varepsilon^{k}\langle\mathcal{C}(T+\bar{T},k)\rangle+\frac{\varepsilon^{N+1}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{N-1}^{\varepsilon}+\frac{\varepsilon^{N+2}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{N}^{\varepsilon}+\sum_{k=0}^{N}\varepsilon^{k}E_{k}^{0},$
(5.9)
and
$\displaystyle\mathcal{R}_{2}(T^{a,\varepsilon},\psi^{a,\varepsilon})$
$\displaystyle=\varepsilon^{N+1}\beta\cdot\nabla\psi_{N}^{\varepsilon}-\sum_{k=0}^{N}\varepsilon^{k}(\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k)-\mathcal{C}(T^{\varepsilon},k)-\mathcal{C}(\bar{T}^{\varepsilon}+P^{\varepsilon},k)+\mathcal{C}(P^{\varepsilon},k))$
$\displaystyle\quad-\sum_{k=N+1}^{4N}\varepsilon^{k}(\mathcal{C}(T^{\varepsilon}+\bar{T}^{\varepsilon},k))-\frac{\varepsilon^{N+1}}{1-\varepsilon\eta}\cos\phi\partial_{\theta}\bar{\psi}_{N}^{\varepsilon}+\sum_{k=0}^{N}\varepsilon^{k}E_{k}^{1}.$
(5.10)
### 5.2. Estimation on the approximation errors
The approximation errors calculated are expected to be small for large $N$.
Here we estimate on the terms in (5.1)-(5.1) and prove the following lemma.
###### Lemma 18.
Assume the boundary data in (1.3)-(1.4) satisfy $T_{b}\in C^{2}(\Omega)$,
$\psi_{b}\in C^{1}(\Gamma_{-})$. Let
$(T^{a,\varepsilon},\psi^{a,\varepsilon})$ be the approximate solution
constructed in section 2. Assume the solution $\tilde{T}_{0}^{\varepsilon}$ to
(2.56) satisfy the spectral assumption (A) and
$\tilde{T}_{0}^{\varepsilon}\geq a$ for some constant $a>0$. Then for
sufficient large $N$,
$\displaystyle\|\mathcal{R}_{1}(T^{a},\psi^{a})\|_{L^{2}(\Omega)},~{}\|\mathcal{R}_{1}(T^{a},\psi^{a})\|_{L^{\infty}(\Omega)}\leq
C\varepsilon^{N+1}+(N+2)Ce^{-\frac{\lambda\delta}{4\varepsilon}},$ (5.11)
$\displaystyle\|\mathcal{R}_{2}(T^{a},\psi^{a})\|_{L^{2}(\Omega\times\mathbb{S}^{2})},~{}\|\mathcal{R}_{2}(T^{a},\psi^{a})\|_{L^{\infty}(\Omega\times\mathbb{S}^{2})}\leq
C\varepsilon^{N+1}+(N+2)Ce^{-\frac{\lambda\delta}{4\varepsilon}}.$ (5.12)
for some constant $0\leq\lambda\leq 1/2$, and $C>0$ a constant independent of
$\varepsilon$.
###### Proof.
The proof of the lemma follows from the decay property of solutions for the
nonlinear Milne and linear Milne problems (Theorem 4 and Theorem 13). Below we
give the estimates on the terms related to geometric corrections. Other terms
also appear in the flat case and the estimates can be found in [6]. Since the
solutions to Milne problems with geometric corrections decay to constant
solutions exponentially, same as the Milne problem without geometric
corrections in [6], these terms can be estimated in the same way.
First by Theorem 4 and Theorem 13, there exists a constant $0\leq\lambda<1/2$
such that
$\displaystyle|\tilde{T}_{k}^{\varepsilon}(\eta)-\tilde{T}_{k,\infty}^{\varepsilon}|\leq
Ce^{-\lambda\eta},\quad|\tilde{\psi}_{k}^{\varepsilon}(\eta)-\tilde{\psi}_{k,\infty}^{\varepsilon}|\leq
Ce^{-\lambda\eta}.$
By the definition of $\bar{T}_{k}^{\varepsilon},\bar{\psi}_{k}^{\varepsilon}$
The terms involving $\partial_{\theta}^{2}$ in (5.1)-(5.1) can be estimated by
$\displaystyle\left|\frac{\varepsilon^{N+1}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{N-1}^{\varepsilon}\right|+\left|\frac{\varepsilon^{N+2}}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{N}^{\varepsilon}\right|+\left|\frac{\varepsilon^{N+1}}{1-\varepsilon\eta}\cos\phi\partial_{\theta}\bar{\psi}_{N}^{\varepsilon}\right|\leq
C\varepsilon^{N+1},$
since $\bar{T}_{k}^{\varepsilon},\bar{\psi}_{k}^{\varepsilon}$ are bounded
functions. For the term $E_{0}^{0}$ given in (5.1), the term related to
geometric correction can be estimated by
$\displaystyle\left|\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\chi(\tilde{T}_{0}^{\varepsilon}-\tilde{T}_{0,\infty}^{\varepsilon})\right|\leq\frac{C\varepsilon}{1-\varepsilon\eta}1_{\tfrac{1}{4}\delta\leq\varepsilon\eta\leq\tfrac{3}{8}\delta}e^{-\lambda\eta}\leq\frac{C}{1-\tfrac{3}{8}\delta}e^{-\frac{\lambda\delta}{4\varepsilon}}.$
The term $E_{0}^{1}$ is the same as the case without geometric correction [6].
For $E_{k}^{0}$, the related terms are
$\displaystyle\left|-\frac{\varepsilon}{1-\varepsilon\eta}\partial_{\eta}\chi(\tilde{T}_{k}^{\varepsilon}-\tilde{T}_{k,\infty}^{\varepsilon})\right|+\left|\frac{1-\chi}{(1-\varepsilon\eta)^{2}}\partial_{\theta}^{2}\bar{T}_{k-2}^{\varepsilon}\right|$
$\displaystyle\quad\leq\frac{C\varepsilon}{1-\varepsilon\eta}1_{\tfrac{1}{4}\delta\leq\varepsilon\eta\leq\tfrac{3}{8}\delta}e^{-\lambda\eta}+\frac{1}{(1-\varepsilon\eta)^{2}}1_{\tfrac{1}{4}\delta\leq\varepsilon\eta\leq\tfrac{3}{8}\delta}e^{-\lambda\eta}$
$\displaystyle\quad\leq\frac{C}{1-\tfrac{3}{8}\delta}e^{-\frac{\lambda\delta}{4\varepsilon}}+\frac{C}{(1-\tfrac{3}{8}\delta)^{2}}e^{-\frac{\lambda\delta}{4\varepsilon}}.$
The geometric correction related term in $E_{k}^{1}$ can be estimated
similarly by
$\displaystyle\left|-\frac{1-\chi}{1-\varepsilon\eta}\cos\phi\partial_{\theta}\bar{\psi}_{k-1}\right|\leq\frac{C}{1-\tfrac{3}{8}\delta}e^{-\frac{\lambda\delta}{4\varepsilon}}.$
Combing the above estimates, we can obtain
∎
### 5.3. Proof of Theorem 1
The proof of Theorem 1 is based on Banach fixed point theorem by showing that
system (1.1)-(1.2) has a unique solution around
$(T^{a,\varepsilon},\psi^{a,\varepsilon})$ and is given in [6, section 4]. A
crucial step in the proof is based on the following lemma.
###### Lemma 19.
Let $(T^{a,\varepsilon},\psi^{a,\varepsilon})$ be the approximate solution
constructed in the section 2. Assuming the spectral assumption (A) holds for
the solution $\tilde{T}_{0}^{\varepsilon}$ of the nonlinear Milne problem
(2.56) and $\tilde{T}_{0}^{\varepsilon}\geq a$ for some constant $a>0$, and
$T^{a,\varepsilon}$ and $\tilde{T}^{\varepsilon}_{0}$ are positive. The the
following inequality holds
$\displaystyle-\int_{\Omega}4(T^{a,\varepsilon})^{3}g\Delta
gdx=\int_{\Omega}4(T^{a,\varepsilon})^{3}|\nabla
g|^{2}dx-\int_{\Omega}\nabla(4(T^{a,\varepsilon})^{3})\cdot g\nabla
gdx\geq\kappa\int_{\Omega}|\nabla g|^{2}dx-C\|g\|_{L^{2}(\Omega)}^{2},$ (5.13)
for any function $g$ satisfying $g(0)=0$ and for some constants $\kappa>0$,
$C>0$ depending on $M$, where $M<1$ is the constant of the spectral
assumption.
###### Proof.
Note that
$T^{a,\varepsilon}=\sum_{k=0}^{N}\varepsilon^{k}(T_{k}^{\varepsilon}+\bar{T}_{k}^{\varepsilon})$
where
$\bar{T}_{k}^{\varepsilon}=\chi(1-r)(\tilde{T}_{k}^{\varepsilon}-\tilde{T}_{k,\infty}^{\varepsilon})$.
In the domain $1-r>\tfrac{3}{8}\delta$, $\chi(1-r)=0$ and
$T^{a,\varepsilon}=\sum_{k=0}^{N}\varepsilon^{k}T_{k}^{\varepsilon}$, which
only contain the interior approximations. Since
$\|T_{k}^{\varepsilon}\|_{C^{s}(\Omega)}$ is bounded for any $s>0$ and
$k=1,\ldots,N$,
$\displaystyle\int_{\Omega\cap\\{1-r>\tfrac{3}{8}\delta\\}}4(T^{a,\varepsilon})^{3}|\nabla
g|^{2}dx-\int_{\Omega\cap\\{1-r>\tfrac{3}{8}\delta\\}}\nabla(4(T^{a,\varepsilon})^{3})\cdot
g\nabla gdx$
$\displaystyle\quad=\int_{\Omega\cap\\{1-r>\tfrac{3}{8}\delta\\}}4(T^{a,\varepsilon})^{3}|\nabla
g|^{2}dx-\int_{\Omega\cap\\{1-r>\tfrac{3}{8}\delta\\}}2(T^{a,\varepsilon})^{3/2}\nabla
g\cdot 6(T^{a,\varepsilon})^{1/2}\nabla T^{a,\varepsilon}gdx$
$\displaystyle\quad\geq\int_{\Omega\cap\\{1-r>\tfrac{3}{8}\delta\\}}4(T^{a,\varepsilon})^{3}|\nabla
g|^{2}dx-\frac{1}{2}\int_{\Omega\cap\\{1-r>\tfrac{3}{8}\delta\\}}4(T^{a,\varepsilon})^{3}|\nabla
g|^{2}dx-\frac{1}{2}\int_{\Omega\cap\\{1-r>\tfrac{3}{8}\delta\\}}36T^{a,\varepsilon}|\nabla
T^{a,\varepsilon}|^{2}g^{2}dx$
$\displaystyle\quad\geq\int_{\Omega\cap\\{1-r>\tfrac{3}{8}\delta\\}}2(T^{a,\varepsilon})^{3}|\nabla
g|^{2}dx-C\|g\|_{L^{2}(\Omega)}^{2}.$ (5.14)
In the domain $1-r\leq\frac{3}{8}\delta$, boundary layer effects play a role.
First we split the integral as
$\displaystyle\int_{\Omega\cap\\{1-r\leq\tfrac{3}{8}\delta\\}}4(T^{a,\varepsilon})^{3}|\nabla
g|^{2}dx-\int_{\Omega\cap\\{1-r\leq\tfrac{3}{8}\\}}\nabla(4(T^{a,\varepsilon})^{3})\cdot
g\nabla gdx$
$\displaystyle\quad=\int_{\Omega\cap\\{1-r\leq\tfrac{3}{8}\delta\\}}4(T^{a,\varepsilon})^{3}|\partial_{\eta}T^{a,\varepsilon}|^{2}dx+\int_{\Omega\cap}=:I_{1}+I_{2}.$
Since $\|\nabla_{x^{\prime}}(T^{a})\|_{L^{2}(\Omega)}$ is bounded, we can
estimate $I_{1}$ the same as (5.14):
$\displaystyle
I_{1}\geq\int_{\Omega\cap\\{x_{1}\leq\tfrac{3}{8}\\}}2(T^{a})^{3}|\nabla_{x^{\prime}}g|^{2}dx-C\|g\|_{L^{2}(\Omega)}^{2}.$
(5.15)
To estimate $I_{2}$, we use the spectral assumption (A). The composite
approximate solution $(T^{a},\psi^{a})$ is close to the solution
$(\tilde{T}_{0},\tilde{\psi}_{0})$ of the nonlinear Milne problem (2.56).
Using the equation
$\displaystyle(T^{a})^{3}=\left(\sum_{k=0}^{N}\varepsilon^{k}(T_{k}+\bar{T}_{k})\right)^{3}=(T_{0}+\bar{T}_{0})^{3}+\varepsilon
G,$
where
$G=3(T_{0}+\bar{T}_{0})^{2}\sum_{k=1}^{N}\varepsilon^{k-1}(T_{k}+\bar{T}_{k})+6(T_{0}+\bar{T}_{0})^{2}(\sum_{k=1}^{N}\varepsilon^{k-1}(T_{k}+\bar{T}_{k}))^{2}+3(T_{0}+\bar{T}_{0})(\sum_{k=1}^{N}\varepsilon^{k-1}(T_{k}+\bar{T}_{k}))^{2}$,
we can rewrite $I_{2}$ as
$\displaystyle I_{2}$
$\displaystyle=\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}\int_{0}^{\frac{3}{8\varepsilon}}4(T_{0}+\bar{T}_{0})^{3}+4\varepsilon
G)|\partial_{\eta}g|^{2}d\eta
dx^{\prime}-\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}\int_{0}^{\frac{3}{8\varepsilon}}\partial_{\eta}(4(T_{0}+\bar{T}_{0})^{3}+4\varepsilon
G)g\partial_{\eta}gd\eta dx^{\prime}$
$\displaystyle=\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}(4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}-\partial_{\eta}(4\tilde{T}_{0}^{3})g\partial_{\eta}g)d\eta+\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}(4(T_{0}+\bar{T}_{0})^{3}-4\tilde{T}_{0}^{3})|\partial_{\eta}g|^{2}d\eta$
$\displaystyle\quad-\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}\partial_{\eta}(4(T_{0}+\bar{T}_{0})^{3}-4\tilde{T}_{0}^{3})g\partial_{\eta}gd\eta
dx^{\prime}+\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}\varepsilon
4(G|\partial_{\eta}g|^{2}-\partial_{\eta}Gg\partial_{\eta}g)d\eta$
$\displaystyle=:I_{21}+I_{22}+I_{23}+I_{24}.$
The spectral assumption (A) implies
$\displaystyle I_{21}$
$\displaystyle=\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}(4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}-\partial_{\eta}(4\tilde{T}_{0}^{3})g\partial_{\eta}g)d\eta$
$\displaystyle\geq\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}(4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}-\frac{1}{2}(4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}+36\tilde{T}_{0}|\partial_{\eta}\tilde{T}_{0}|^{2}g^{2}))d\eta$
$\displaystyle\geq\frac{1}{2\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}(4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}-36\tilde{T}_{0}|\partial_{\eta}\tilde{T}_{0}|^{2}g^{2})d\eta$
$\displaystyle\geq\frac{1-M}{2\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}d\eta.$
For $I_{22}$, since
$\bar{T}_{0}=T_{0}+\chi(\varepsilon\eta)(\tilde{T}_{0}-T_{0}(0))$, it holds
that
$\displaystyle(T_{0}+\bar{T}_{0})^{3}-\tilde{T}_{0}^{3}$
$\displaystyle=(T_{0}+\chi(\varepsilon\eta)(\tilde{T}_{0}-T_{0}(0))-\tilde{T}_{0})(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2})$
$\displaystyle=((T_{0}-T_{0}(0))-(1-\chi(\varepsilon\eta))(\tilde{T}_{0}-\bar{T}_{0}))(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2})$
$\displaystyle=(\partial_{x_{1}}T_{0}(\xi)\varepsilon\eta-(1-\chi(\varepsilon\eta))(\tilde{T}_{0}-\bar{T}_{0}))(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2}).$
Since we are considering the integration over
$x_{1}=\varepsilon\eta\in[0,\tfrac{3}{8}\delta]$ and
$(1-\chi(\varepsilon\eta))$ is supported on
$[\tfrac{1}{4}\delta,\tfrac{3}{8}\delta]$,
$\displaystyle
I_{22}=\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3}{8\varepsilon}}(4(T_{0}+\bar{T}_{0})^{3}-4\tilde{T}_{0}^{3})|\partial_{\eta}g|^{2}d\eta\leq\frac{3\delta
C}{8\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{\delta}{2\varepsilon}}|\partial_{\eta}g|^{2}d\eta.$
For $I_{23}$, due to
$\displaystyle\partial_{\eta}((T_{0}+\bar{T}_{0})^{3}-\tilde{T}_{0}^{3})$
$\displaystyle=\partial_{\eta}(T_{0}+\chi(\varepsilon\eta)(\tilde{T}_{0}-T_{0}(0))-\tilde{T}_{0})(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2})$
$\displaystyle\quad+(T_{0}+\chi(\varepsilon\eta)(\tilde{T}_{0}-T_{0}(0))-\tilde{T}_{0})\partial_{\eta}(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2})$
$\displaystyle=\varepsilon\chi^{\prime}(\varepsilon\eta)(\tilde{T}_{0}-{T}_{0}(0))(\tilde{T}_{0}-T_{0}(0))-\tilde{T}_{0})(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2})$
$\displaystyle\quad+(\chi(\varepsilon\eta)-1)\partial_{\eta}\tilde{T}_{0}(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2})$
$\displaystyle\quad+(\partial_{x_{1}}T_{0}(\xi)\varepsilon\eta-(1-\chi(\varepsilon\eta))(\tilde{T}_{0}-\bar{T}_{0}))\partial_{\eta}(\tilde{T}_{0}^{2}+\tilde{T}_{0}(T_{0}+\bar{T}_{0})+(T_{0}+\bar{T}_{0})^{2}),$
with consideration of $\varepsilon\eta\in[0,\tfrac{3}{8}\delta]$ and and
$(1-\chi(\varepsilon\eta))$ being supported on
$[\tfrac{1}{4}\delta,\tfrac{3}{8}\delta]$, it holds that
$\displaystyle I_{23}$
$\displaystyle=-\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}\partial_{\eta}(4(T_{0}+\bar{T}_{0})^{3}-4\tilde{T}_{0}^{3})g\partial_{\eta}gd\eta\leq\frac{C}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}(\varepsilon|g||\partial_{\eta}g|+\frac{3\delta}{8}|g||\partial_{\eta}g|)d\eta$
$\displaystyle\leq\frac{3\delta
C}{8\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}(g^{2}+|\partial_{\eta}g|^{2})d\eta.$
For $I_{24}$, we have
$\displaystyle
I_{24}=\frac{1}{\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}\varepsilon
4(G|\partial_{\eta}g|^{2}-\partial_{\eta}Gg\partial_{\eta}g)d\eta\leq\frac{C}{\varepsilon}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}(|g|^{2}+|\partial_{\eta}g|^{2})d\eta.$
Combing the above estimates gives
$\displaystyle I_{2}$
$\displaystyle\geq\frac{1-M}{2\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}d\eta-\frac{3\delta
C}{8\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}(|g|^{2}+|\partial_{\eta}g|^{2})d\eta$
$\displaystyle\quad-\frac{C}{\varepsilon}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}(|g|^{2}+|\partial_{\eta}g|^{2})d\eta.$
By the assumption of the lemma, $\tilde{T}_{0}\geq a$, hence
$4\tilde{T}_{0}^{3}\geq 4a^{3}$ for some constant $a>0$. We can take
sufficiently small $\varepsilon$ and $\delta$ such that
$\varepsilon<(1-M)a^{3}/C$ and $3\delta C/8\leq(1-M)/8$, and we get from the
above inequality
$\displaystyle
I_{2}\geq\frac{1-M}{4\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}d\eta.$
Combing this with (5.14) and (5.15) implies
$\displaystyle-\int_{\Omega}4(T^{a})^{3}g\Delta gdx$
$\displaystyle=\int_{\Omega}4(T^{a})^{3}|\nabla
g|^{2}dx-\int_{\Omega}\nabla(4(T^{a})^{3})\cdot g\nabla gdx$
$\displaystyle\geq\int_{\Omega\cap\\{x_{1}>\tfrac{3}{8}\delta\\}}2(T^{a})^{3}|\nabla
g|^{2}dx+\int_{\Omega\cap\\{x_{1}\leq\tfrac{3}{8}\delta\\}}2(T^{a})^{3}|\nabla_{x^{\prime}}g|^{2}dx$
$\displaystyle\quad+\frac{1-M}{4\varepsilon^{2}}\int_{\mathbb{T}^{2}}dx^{\prime}\int_{0}^{\frac{3\delta}{8\varepsilon}}4\tilde{T}_{0}^{3}|\partial_{\eta}g|^{2}d\eta-C\|g\|_{L^{2}(\Omega)}^{2}$
$\displaystyle\geq\kappa\|\nabla
g\|_{L^{2}(\Omega)}^{2}-C\|g\|_{L^{2}(\Omega)}^{2},$
where $\kappa=\min\\{2a^{3},(1-M)a^{3}\\}$, which finishes the proof. ∎
With the above lemma, Theorem 1 can be proved in the same way as [6, Theorem
1].
## Appendix A Transport equation with geometric correction in half-space
Consider the equation in $(\eta,\phi)\in\mathbb{R}_{+}\times[-\pi,\pi)$:
$\displaystyle\sin\phi\partial_{\eta}f+F(\eta)\cos\phi\partial_{\phi}f=H(\eta,\phi),$
(A.1) $\displaystyle f(0,\phi)=h(\phi),\quad\text{for }\sin\phi>0.$ (A.2)
The following lemma was proved in [10, Pages 1512-1514].
###### Lemma 20.
There exists a solution to the above problem given by
$\displaystyle f(\eta,\phi)=\mathcal{A}h(\phi)+\mathcal{T}H(\eta,\phi),$ (A.3)
with $\mathcal{A}$, $\mathcal{H}$ are defined in the following: for
$\sin\phi>0$,
$\displaystyle\mathcal{A}h(\phi)=h(\phi^{\prime}(\phi,\eta,0))e^{-\int_{0}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\xi)}d\xi},\quad\mathcal{T}H(\eta,\phi)=\int_{0}^{\eta}\frac{h(\xi,\phi^{\prime}(\phi,\eta,\xi))}{\sin\phi^{\prime}(\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(\phi,\eta,\rho)}d\rho}d\xi,$
(A.4)
for $\sin\phi<0$ and $|E(\phi,\eta)|\leq e^{-V(\varepsilon;\infty)}$,
$\displaystyle\mathcal{A}h(\phi)$
$\displaystyle=0,\quad\mathcal{T}H(\eta,\phi)=\int_{\eta}^{\infty}\frac{h(\xi,-\phi^{\prime}(-\phi,\eta,\xi))}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi,$
(A.5)
and for $\sin\phi<0$ and $|E(\phi,\eta)|\geq e^{-V(\varepsilon;\infty)}$,
$\displaystyle\mathcal{A}h(\phi)=h(\phi^{\prime}(-\phi,\eta,0))e^{-\int_{0}^{\eta_{+}}+\int_{\eta}^{\eta_{+}}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\xi)}d\xi},$
$\displaystyle\mathcal{T}H(\eta,\phi)=\int_{0}^{\eta_{+}}\frac{h(\xi,\phi^{\prime}(-\phi,\eta,\xi))}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{-\int_{\xi}^{\eta_{+}}+\int_{\eta}^{\eta_{+}}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi$
$\displaystyle\qquad\qquad\quad+\int_{\eta}^{\eta_{+}}\frac{h(\xi,-\phi^{\prime}(-\phi,\eta,\xi))}{\sin\phi^{\prime}(-\phi,\eta,\xi)}e^{\int_{\xi}^{\eta}\frac{1}{\sin\phi^{\prime}(-\phi,\eta,\rho)}d\rho}d\xi.$
(A.6)
Moreover, $\mathcal{A}$ and $\mathcal{H}$ satisfies
$\displaystyle\|e^{\beta\eta}\mathcal{A}h\|_{L^{\infty}(\mathbb{R}_{+}\times[-\pi,\pi))}\leq\|h\|_{L^{\infty}((0,\pi))},\quad\text{for
any }0\leq\beta\leq 1,$ (A.7)
$\displaystyle\|e^{\beta\eta}\mathcal{T}H\|_{L^{\infty}(\mathbb{R}_{+}\times[-\pi,\pi))}\leq
C\|e^{\beta\eta}H\|_{L^{\infty}(\mathbb{R}_{+}\times[-\pi,\pi))},\quad\text{for
any }0\leq\beta\leq\tfrac{1}{2}.$ (A.8)
## Appendix B Existence for the linearized Milne problem without geometric
correction
Consider the following linear Milne problem on the half-line:
$\displaystyle\partial_{\eta}G+\langle\varphi-4h^{3}G\rangle=S_{1},$ (B.1)
$\displaystyle\sin\phi\partial_{\eta}\varphi+\varphi-4h^{3}G=S_{2},$ (B.2)
with boundary conditions
$\displaystyle
G(0)=G_{b},\quad\varphi(0,\phi)=\varphi_{b}(\phi),\quad\text{for }\sin\phi>0.$
(B.4)
The following theorem was proved in [6, Theorem 2].
###### Theorem 21.
Assume $h$ satisfies the spectral assumption (A). Assume $S_{1}=S_{1}(\eta)$,
$S_{2}S_{2}(\eta,\phi)$ satisfy $|S_{1}(\eta)|,|S_{2}(\eta,\phi)|\leq
C_{S}e^{-\beta\eta}$ for any $\eta\in\mathbb{R}_{+}$, $\phi\in[-\pi,\pi)$.
Then there exists a unique bounded solution $(G,\varphi)\in L^{2}_{\rm
loc}(\mathbb{R}_{+})\times L^{2}(\mathbb{R}_{+}\times[-\pi,\pi))$ to system
(B.1)-(B.2) with boundary conditions (B.4). Moreover there exists a constant
$G_{\infty}\in\mathbb{R}$ such that
$\displaystyle|G(\eta)-G_{\infty}|\leq
Ce^{-\beta\eta},\quad|\varphi(\eta,\phi)-4h^{3}_{\infty}G_{\infty}|\leq
Ce^{-\beta\eta},\quad\text{for any
}\eta\in\mathbb{R}_{+},~{}\phi\in[-\pi,\pi),$ (B.5)
where $C>0$ is a positive constant depending on $\beta$ and
$\int_{\sin\phi>0}\sin\phi\varphi_{b}^{2}d\phi$.
The theorem was proved in [6] by first showing the existence on bounded
interval and then using uniform estimate to extend the solution to the half-
line. Consider system (B.1)-(B.2) on the interval $\eta\in[0,B]$ with the
following boundary conditions at $\eta=B$,
$\displaystyle\partial_{\eta}G(B)=0,\quad\varphi(B,\phi)=\varphi(B,-\phi),\quad\text{for
}\sin\phi>0.$ (B.6)
The following theorem holds.
###### Theorem 22.
Let the assumptions of Theorem 22 hold. Then there exists a unique solution
$(G,\varphi)\in C^{2}([0,B])\times C^{1}([0,B]\times[-\pi,\pi))$ to system
(B.1)-(B.2) with boundary condtions (B.4)-(B.6).
## References
* [1] A. Bensoussan, J. L. Lions, and G. C. Papanicolaou, Boundary layers and homogenization of transport processes, Publications of the Research Institute for Mathematical Sciences, 15 (1979), pp. 53–157.
* [2] J. Clouët and R. Sentis, Milne problem for non-grey radiative transfer, Kinetic & Related Models, 2 (2009), p. 345.
* [3] L. C. Evans, Partial differential equations and monge-kantorovich mass transfer, Current developments in mathematics, 1997 (1997), pp. 65–126.
* [4] M. Ghattassi, X. Huo, and N. Masmoudi, On the diffusive limits of radiative heat transfer system i: Well-prepared initial and boundary conditions, SIAM Journal on Mathematical Analysis, 54 (2022), pp. 5335–5387.
* [5] , Diffusive limits of the steady state radiative heat transfer system: Boundary layers, accepted for publication in Journal de Mathématiques Pures et Appliquées, (2023).
* [6] , Stability of the nonlinear milne problem for radiative heat transfer system, Archive for Rational Mechanics and Analysis. arXiv preprint arXiv:2207.10769, (2023).
* [7] G.-M. Gie, M. Hamouda, and R. Temam, Boundary layers in smooth curvilinear domains: parabolic problems, Discrete & Continuous Dynamical Systems, 26 (2010), p. 1213.
* [8] G.-M. Gie, C.-Y. Jung, and R. Temam, Recent progresses in boundary layer theory, Discrete & Continuous Dynamical Systems, 36 (2016), p. 2521.
* [9] S. Ukai, Solutions of the boltzmann equation, in Studies in Mathematics and its Applications, vol. 18, Elsevier, 1986, pp. 37–96.
* [10] L. Wu and Y. Guo, Geometric correction for diffusive expansion of steady neutron transport equation, Communications in Mathematical Physics, 336 (2015), pp. 1473–1553.
|
# Non-Standard Interactions of Supernova Neutrinos
and Mass Ordering Ambiguity at DUNE
Sudip Jana<EMAIL_ADDRESS>Max-Planck-Institut für Kernphysik,
Saupfercheckweg 1, 69117 Heidelberg, Germany Yago Porto
<EMAIL_ADDRESS>Centro de Ciências Naturais e Humanas, Universidade
Federal do ABC, 09210-170, Santo André, SP, Brazil
###### Abstract
We show that non-standard neutrino interactions (NSI) can notably modify the
pattern of resonant flavor conversion of neutrinos within supernovae and
significantly impact the neutronization burst signal in forthcoming
experiments such as the Deep Underground Neutrino Experiment (DUNE). The
presence of NSI can invert the energy levels of neutrino matter eigenstates
and even induce a new resonance in the inner parts close to the proto-neutron
star. We demonstrate how DUNE can use these new configurations of energy
levels to have sensitivity to NSIs down to $\mathcal{O}(0.1)$. We also
elucidate how the effect may result in a puzzling confusion of normal and
inverted mass orderings by highlighting the emergence or vanishing of the
neutronization peak, which distinguishes between the two mass orderings.
Potential implications are analyzed thoroughly.
_Introduction_.– In recent decades, extensive efforts and data from solar,
atmospheric, reactor, and accelerator neutrino experiments have provided
robust evidence for neutrino oscillations, indicating the presence of neutrino
masses and mixing. However, the origin of these phenomena remains
unestablished, leaving room for potential new physics, particularly in the
form of non-standard neutrino interactions (NSIs). First introduced by
Wolfenstein in 1978 Wolfenstein:1977ue , NSIs have been the subject of intense
scrutiny since then, providing an avenue for exploring new aspects of neutrino
physics. These interactions involve higher-dimensional operators with
neutrinos and matter, as represented by the equations:
$\displaystyle\mathcal{L}_{\mathrm{NC}}=-2\sqrt{2}G_{F}\sum_{f,P,\alpha,\beta}\varepsilon_{\alpha\beta}^{f,P}\left(\bar{\nu}_{\alpha}\gamma^{\mu}P_{L}\nu_{\beta}\right)\left(\bar{f}\gamma_{\mu}Pf\right)$
(1)
$\displaystyle\mathcal{L}_{\mathrm{CC}}=-2\sqrt{2}G_{F}\sum_{f,P,\alpha,\beta}\varepsilon_{\alpha\beta}^{f,P}\left(\bar{\nu}_{\alpha}\gamma^{\mu}P_{L}\ell_{\beta}\right)\left(\bar{f}\gamma_{\mu}Pf^{\prime}\right)$
where $\varepsilon$ represents the strength of NSI relative to the weak scale,
$P\in{P_{L},P_{R}}$ indicates chirality projection operators and the sum is
over matter fermions $f,f^{\prime}\in{e,u,d}$. Such NSIs modify the matter
potential experienced by neutrinos, adding substantial intricacy to the
determination of neutrino oscillation parameters. For instance, the existence
of NSI introduces ambiguity in the determination of $\theta_{12}$ from the
solar neutrino data Miranda:2004nb . Furthermore, NSI effects have been shown
to alleviate the tension between solar and KamLAND data, as it flattens the
solar neutrino spectrum at high energies ($>3$ MeV) and generates larger day-
night asymmetry Maltoni:2015kca . In this study, we advocate for the
utilization of supernova neutrinos to investigate the influence of these NSI
on the precise determination of neutrino oscillation parameters.
Figure 1: A simplified picture of flavor conversions of supernova neutrinos in
presence of NSI.
During a supernova (SN) explosion, as the progenitor’s core collapses to form
a neutron star, about $10^{53}$ erg of gravitational binding energy is
released in the form of neutrinos Colgate:1966ax ; Arnett:1966 ; Bethe:1985sox
; Wilson:1985 ; Janka:2017vlw . Originating in the SN core with energies in
the tens of MeV, these neutrinos travel through the stellar mantle and
envelope, undergoing modifications in mixing and oscillations through
neutrino-matter interactions111Matter effects can also influence the evolution
of high-energy astrophysical neutrinos with energies greater than 100 TeV
Dev:2023znd . Wolfenstein:1977ue ; Mikheyev:1985zog ; Mikheev:1986wj ;
Mikheev:1986if . The consequences of neutrino propagation in the SN-dense
environment offer significant potential for probing new physics, and these
effects are observable by flavor-sensitive Earth-based detectors Valle:1987gv
; Nunokawa:1997ct ; Nunokawa:1998vh ; Esteban-Pretel:2007zkv ;
deGouvea:2019goq ; Tang:2020pkp ; Jana:2022tsa ; Jana:2023ufy ;
dosSantos:2023skk ; Bendahman:2023hjj . A galactic SN, a rare event in the
Milky Way occurring once or twice per century, was last observed in 1987 from
the Large Magellanic Cloud, 51 Kpc away Tammann:1994ev ; Kamiokande-II:1987idp
; Bionta:1987qt ; Alekseev:1988gp . Neutrino detectors recorded about 25
$\bar{\nu}_{e}$ events during that event, significantly advancing our
understanding of core-collapse processes and neutrino emission Olsen:2021uvt ;
Li:2023ulf ; DedinNeto:2023hhp ; Fiorillo:2023frv . Future detectors,
especially the Deep Underground Neutrino Experiment (DUNE), are anticipated to
capture hundreds of thousands of events in various channels during a galactic
supernova IceCube:2011cwc ; DarkSide20k:2020ymr ; Hyper-Kamiokande:2021frf ;
KM3NeT:2021moe ; JUNO:2023dnp ; DUNE:2020zfm , with DUNE’s unique capability
to detect a clean $\nu_{e}$ signal in the critical first 40 ms after core
bounce, known as the neutronization burst phase DUNE:2020zfm ; Cuesta:2023nnt
; Zhu:2018rwc . In this article, we investigate the features of neutrino
emission in the neutronization burst phase, elucidate the anticipated resonant
flavor conversion within the stellar envelope triggered by NSI [cf.Fig 1] and
discuss the crucial role of DUNE in exploring new physics impacting supernova
neutrinos.
_Dynamics of Resonant Flavor Conversions_.–
Figure 2: Configuration of energy levels for neutrinos and antineutrinos for
NO (left) and IO (right). For each panel, neutrino lines are shown for
positive $V_{e}$, while antineutrinos are plotted with negative $V_{e}$. The
location of the MSW resonances is indicated by the letters $L$ and $H$. Solid
lines represent the effective matter eigenstates, while the dashed lines
follow the track of flavor states. Matter eigenstates mix with different
flavors at different locations; this is shown by the flavor tags in red, blue,
and green colors.
Understanding the flavor composition of SN neutrinos reaching the Earth during
the neutronization burst phase requires monitoring how the initial fluxes
generated in the core evolve while traveling outward within the star (see
Appendix for details). In the dense environment where neutrinos propagate
inside the star, vacuum oscillations are suppressed. However, the density
variation in the way out of the SN instigates a flavor evolution dictated by a
time (or space) dependent Hamiltonian that can trigger Mikheyev-Smirnov-
Wolfenstein (MSW) resonant flavor conversion in specific layers of the matter
profile Wolfenstein:1977ue ; Mikheyev:1985zog ; Mikheev:1986wj ;
Mikheev:1986if ; Mikheev:1987jp .
Assuming that neutrinos propagate radially outwards, the evolution equation
for the flavor state $\nu=(\nu_{e},\nu_{\mu},\nu_{\tau})^{T}$ is
$i\frac{d}{dr}\nu=\mathcal{H}\nu.$ (2)
$\mathcal{H}$ is the Hamiltonian in the flavor basis given by
$\mathcal{H}=\frac{1}{2E}U\left(\begin{array}[]{ccc}0&0&0\\\ 0&\Delta
m_{21}^{2}&0\\\ 0&0&\Delta
m_{31}^{2}\end{array}\right)U^{\dagger}+V_{e}\left(\begin{array}[]{ccc}1&0&0\\\
0&0&0\\\ 0&0&0\end{array}\right),$ (3)
where $E$ and $U$ denote the neutrino energy and the Pontecorvo-Maki-Nakagawa-
Sakata (PMNS) matrix. The matter potential due to charged current interactions
can be expressed as
$V_{e}=\sqrt{2}G_{F}\frac{\rho}{m_{N}}Y_{e},$ (4)
where $Y_{e}$ is the electron number fraction and $\rho$ is the matter density
(see Fig. 7 in the Appendix for the profile of $\rho$ and $Y_{e}$). The
antineutrino Hamiltonian is similar to $\mathcal{H}$, with the only difference
being that the matter potential inverts sign $\bar{V}_{e}=-V_{e}$.
In dense matter, $\rho>10^{5}$ $\text{g/cm}^{3}$, $V_{e}$ is much bigger than
all other matrix elements in Eq. 3, suppressing mixing and making $\nu_{e}$
the heaviest effective eigenstate in matter, $\nu_{e}\approx\nu_{3}^{m}$ for
normal ordering (NO) and $\nu_{e}\approx\nu_{2}^{m}$ for inverted ordering
(IO), the superscript $m$ denotes an effective eigenstate in matter. In a
vacuum, however, $\nu_{e}$ is mostly associated with $\nu_{1}$. Therefore, as
matter density decreases, so does $V_{e}$, and $\nu_{e}$ will transit from
$\nu_{3}^{m}$($\nu_{2}^{m})$ to $\nu_{1}^{m}$. In doing so, it has to cross
energy levels twice for NO
($\nu_{3}^{m}\rightarrow\nu_{2}^{m}\rightarrow\nu_{1}^{m}$) and once for IO
($\nu_{2}^{m}\rightarrow\nu_{1}^{m}$). The level crossing diagram is shown in
Fig. 2. The red dashed line follows the evolution of the electron flavor,
$\nu_{e}$ on the right of $V_{e}=0$ and $\bar{\nu}_{e}$ on the left. Note that
$\bar{\nu}_{e}\approx\bar{\nu}_{1}^{m}$ in the inner regions, and there is no
level crossing for antineutrinos in NO, while there is one level crossing in
IO: $\bar{\nu}_{3}^{m}\rightarrow\bar{\nu}_{1}^{m}$. While $\nu_{e}$ and
$\bar{\nu}_{e}$ are effective eigenstates in the SN environment, $\nu_{\mu}$,
$\nu_{\tau}$, and their antiparticles are not effective eigenstates due to
near maximal vacuum mixing. In this situation, it is convenient to diagonalize
the $\nu_{\mu}-\nu_{\tau}$ subspace and work with the effective eigenstates
$\nu^{\prime}_{\mu}$, $\nu^{\prime}_{\tau}$, and their antiparticles. The
primed states have constant energy levels as a function of $V_{e}$, as shown
in Fig. 2.
The crossing $\nu_{e}\leftrightarrow\nu^{\prime}_{\tau}$ is called
$H-$resonance and occurs when
$V_{e}(\rho_{H})\approx\frac{\Delta m_{31}^{2}}{2E}\cos\theta_{13},$ (5)
which corresponds to higher densities compared to the $L-$resonance,
$\nu_{e}\leftrightarrow\nu^{\prime}_{\mu}$, that happens when
$V_{e}(\rho_{L})\approx\frac{\Delta m_{21}^{2}}{2E}\cos\theta_{12}.$ (6)
Note that Eq. 5 is satisfied for neutrinos in NO and for antineutrinos in IO
($\Delta m^{2}_{31}=-|\Delta m^{2}_{31}|$ and $V_{e}$ is negative) while Eq. 6
is satisfied only for neutrinos in both orderings. Using the profile in Fig.
7, and oscillation parameters from Esteban:2018ppq , the $H-$resonance happens
at $r\sim 10^{4}$ Km ($\rho_{H}$ in the range $10^{3}-10^{4}$
$\text{g/cm}^{3}$) and the $L-$resonance at $r\sim 10^{5}$ Km ($\rho_{L}\sim
1-10$ $\text{g/cm}^{3}$). Efficient resonant conversion only happens if
neutrinos cross resonance layers adiabatically Dighe:1999bi . We will assume
perfect adiabaticity of the $H-$ and $L-$resonances so that transitions
between distinct energy eigenstates are negligible. Moreover, the produced
flux during the neutronization burst phase is mostly
$\nu_{e}\approx\nu_{3}^{m}(\nu_{2}^{m})$ for NO (IO). Due to adiabaticity, the
initial eigenstates are preserved and reach vacuum as $\nu_{3}(\nu_{2})$.
Consequently, for NO only $|U_{e3}|^{2}\approx 0.02$ of the initial $\nu_{e}$
flux, $F_{e}^{i}$, survives and reaches Earth, and some amount of $\nu_{e}$,
($1-|U_{e3}|^{2})F_{x}^{i}\approx 0.98F_{x}^{i}$, comes from the conversion of
non-electron flavors, $F_{\mu}^{i}=F_{\tau}^{i}=F_{x}^{i}$:
$F_{e}^{NO}=|U_{e3}|^{2}F_{e}^{i}+(1-|U_{e3}|^{2})F_{x}^{i}\approx
0.02F_{e}^{i}+0.98F_{x}^{i}.$ (7)
Similarly for IO,
$F_{e}^{IO}=|U_{e2}|^{2}F_{e}^{i}+(1-|U_{e2}|^{2})F_{x}^{i}\approx
0.3F_{e}^{i}+0.7F_{x}^{i}.$ (8)
Due to $F_{e}^{i}$ being approximately ten times $F_{x}^{i}$ at the
neutronization peak, the $\nu_{e}$ flux is greater in the Inverted Ordering
(IO) than in the Normal Ordering (NO) during the initial 20 milliseconds of
neutrino emission. This disparity enhances the visibility of the peak in
neutrino detectors for the IO scenario. In the following sections, we will
analyze signals derived from flux equations (7) and (8) in DUNE and discuss
the presence or absence of the peak as a discriminative criterion between the
two mass orderings.
Figure 3: Configuration of energy levels for neutrinos and antineutrinos for
NO (left) and IO (right) with the inclusion of
$\varepsilon_{\tau\tau}^{d}>0.33$. See text for details.
_Effect of NSI on the Energy Levels_.– The introduction of NSIs has a
pronounced impact on the configuration of energy levels and the flavor
composition of SN neutrinos detected on Earth. To demonstrate this effect, an
additional energy term involving NSI parameters must be incorporated into the
Hamiltonian mentioned in Eq. 3. Our focus here is on NSI specifically
influencing $\nu_{\tau}$, potentially originating from interactions with
electrons, $u$-quarks, and $d$-quarks. Recent systematic investigations
indicate the feasibility of obtaining substantial values for
$\varepsilon_{\tau\tau}$ in various radiative neutrino mass models
Babu:2019mfe . It is important to note that the approach to $\nu_{\mu}$ NSI is
analogous and yields numerically equivalent results.
The effective strength of the NSI parameter
$\varepsilon^{\text{eff}}_{\tau\tau}$ can be expressed in relation to
interactions involving electrons, $u$-quarks, and $d$-quarks as follows:
$\varepsilon^{\text{eff}}_{\tau\tau}=\varepsilon_{\tau\tau}^{e}+(2\varepsilon_{\tau\tau}^{u}+\varepsilon_{\tau\tau}^{d})\frac{n_{p}}{n_{e}}+(\varepsilon_{\tau\tau}^{u}+2\varepsilon_{\tau\tau}^{d})\frac{n_{n}}{n_{e}}.$
(9)
In various models of neutrino mass, new interactions emerge, giving rise to
diverse NSI scenarios where $\varepsilon_{\tau\tau}^{u}$,
$\varepsilon_{\tau\tau}^{d}$, or $\varepsilon_{\tau\tau}^{e}$ may be
exclusively generated, or in certain instances, combinations of two NSIs or
all three NSIs. For instance, a scalar leptoquark with Standard Model gauge
quantum numbers ($3,2,1/6$), denoted as $\tilde{R_{2}}$, can induce only
$\varepsilon_{\tau\tau}^{d}$, while the leptoquark ${R_{2}}$ with charges
($3,2,7/6$) exclusively leads to $\varepsilon_{\tau\tau}^{u}$. On the other
hand, the leptoquark ${S_{3}}$ with charges ($\bar{3},3,1/3$) can give rise to
both $\varepsilon_{\tau\tau}^{d}$ and $\varepsilon_{\tau\tau}^{u}$. Further
elaboration on these scenarios can be found in the detailed discussion
provided in reference Babu:2019mfe . To optimize the signal, we exclusively
consider quark NSIs here. Although leptonic NSIs may have some impact, they
are expected to be less significant compared to quark NSIs, and therefore,
they do not affect the phenomenology we are concentrating on. If only
$\varepsilon_{\tau\tau}^{d}$ exhibits a non-zero value, and under the
condition of charge neutrality ($n_{p}=n_{e}$), we obtain:
$\varepsilon^{\text{eff}}_{\tau\tau}=\varepsilon_{\tau\tau}^{d}\left(\frac{2-Y_{e}}{Y_{e}}\right).$
(10)
Considering non-zero values of $\varepsilon_{\tau\tau}^{u}$, we obtain:
$\varepsilon^{\text{eff}}_{\tau\tau}=\varepsilon_{\tau\tau}^{u}\left(\frac{1+Y_{e}}{Y_{e}}\right).$
(11)
Now, we focus on the region of the SN matter profile where $H-$ and $L-$
resonance occur, specifically for $r>10^{4}$ Km. In this particular zone,
$Y_{e}=0.5$ (see Fig. 7 in the Appendix). The existence of either
$\varepsilon_{\tau\tau}^{d}$ or $\varepsilon_{\tau\tau}^{u}>0.33$ indicates
that the effective matter potential for $\nu_{\tau}$ surpasses that of
$\nu_{e}$, thereby modifying the dynamics of flavor conversion. The
Hamiltonian in the presence of NSI is expressed as follows:
$\mathcal{H}^{NSI}=\frac{1}{2E}U\left(\begin{array}[]{ccc}0&0&0\\\ 0&\Delta
m_{21}^{2}&0\\\ 0&0&\Delta
m_{31}^{2}\end{array}\right)U^{\dagger}+V_{e}\left(\begin{array}[]{ccc}1&0&0\\\
0&0&0\\\ 0&0&3\varepsilon^{u,d}_{\tau\tau}\end{array}\right).$ (12)
For $\varepsilon^{u,d}_{\tau\tau}>0.33$, $\nu_{\tau}$ is the heaviest
eigenstate in matter, $\nu_{\tau}\approx\nu_{3}^{m}(\nu_{2}^{m})$ for NO (IO).
Hence, the main difference from the standard case is that $\nu_{e}$ starts as
the second heaviest eigenstate, $\nu_{e}\approx\nu_{2}^{m}(\nu_{1}^{m})$ for
NO (IO), see Fig. 3. For NO, $\nu_{e}$ crosses level once through a
$L-$resonance, $\nu_{2}^{m}\rightarrow\nu_{1}^{m}$, at densities given by Eq.
6. While for IO, $\nu_{e}$ is already produced as $\nu_{1}^{m}$ and reaches
vacuum mostly as $\nu_{1}$ without crossing levels. For antineutrinos,
$\bar{\nu}_{e}\approx\bar{\nu}_{2}^{m}(\bar{\nu}_{1}^{m})$, and crosses levels
only for NO, via $H-$resonance. In this case, however, the potential of
$\nu_{\tau}$, $V_{\tau}=\varepsilon_{\tau\tau}V_{e}$, is higher than $V_{e}$,
so the difference $V_{e}-V_{\tau}$ is negative. The resonance condition is
found by modifying Eq. 5 with
$V_{e}\rightarrow(1-\varepsilon_{\tau\tau})V_{e}$. Neutrinos fail to satisfy
this new condition in NO. However, antineutrinos in NO can fulfill this
condition since their potentials exhibit an inverted sign, denoted as
$\bar{V_{e}}=-V_{e}$. Furthermore, the eigenstates are not represented in the
primed basis, as the potential for $\nu_{\tau}$ significantly surpasses the
mixing terms, distinctly separating $\nu_{\mu}$ and $\nu_{\tau}$.
Assuming perfect adiabaticity of $H-$ and $L-$ resonances, we find the final
flux for both orderings:
$F_{e}^{NO}(\varepsilon_{\tau\tau}^{u,d}>0.33)=|U_{e2}|^{2}F_{e}^{i}+(1-|U_{e2}|^{2})F_{x}^{i}\approx
0.3F_{e}^{i}+0.7F_{x}^{i},$ (13)
and,
$F_{e}^{IO}(\varepsilon_{\tau\tau}^{u,d}>0.33)=|U_{e1}|^{2}F_{e}^{i}+(1-|U_{e1}|^{2})F_{x}^{i}\approx
0.7F_{e}^{i}+0.3F_{x}^{i}.$ (14)
It is noteworthy that Eq.13 is equivalent to Eq.8 since, in both instances,
$\nu_{e}$ is generated as $\nu_{2}^{m}$. Consequently, the neutronization
burst phase in the NO scenario with NSI and the standard IO scenario without
NSI are indistinguishable. In the next sections, we analyze the signals from
Eq. 13 and Eq. 14 at DUNE, emphasizing potential confusion in discerning
between the two mass orderings in the presence of NSI.
_New Resonances triggered by NSI_.–
Figure 4: Configuration of energy levels for neutrinos and antineutrinos for
NO (left) and IO (right) with the inclusion of
$0.1<\varepsilon^{d}_{\tau\tau}<0.33$ and
$\varepsilon^{d}_{e\tau}>10^{-5}$.The MSW resonance locations are marked by
the letters $L$ and $H$, while the NSI-triggered resonance location is marked
by the letter $C$.
We will now explore scenarios in which NSI can induce an additional resonance,
designated as the $C$-resonance, alongside the existing $L-$ and $H-$
resonances. While examining NSI within the range
$0.1<\varepsilon_{\tau\tau}^{u,d}<0.33$, we find that it influences the flavor
evolution of neutrinos during the neutronization burst phase. Despite such
values of $\varepsilon_{\tau\tau}^{u,d}$ not modifying the energy level
pattern in the region $r>10^{3}$ Km where $L-$ and $H-$resonances occur
($Y_{e}\approx 0.5$), it does alter the energy levels inside the iron core
($r<10^{3}$ Km) where neutrinos are produced ($0.2<Y_{e}<0.5$). In this
configuration, the $C-$ resonance emerges inside the iron core, as illustrated
in Fig. 4.
Figure 5: Expected number of $\nu_{e}$ events per bin of $5$ ms at DUNE in the
time interval from $-5$ to $40$ ms corresponding to the neutronization burst
phase for NO (left) and IO (right). $0$ ms represents the time of core bounce.
Blue lines are computed from fluxes in Eqs. 7 and 8, assuming the distance
between the SN and Earth to be $10$ Kpc. Green lines are computed from fluxes
modified by the presence of $\varepsilon_{\tau\tau}^{d}>0.33$ in Eqs. 13 and
14. The similarity between the green line on the left panel and the blue line
on the right panel displays the degeneracy between the NO$+$NSI and standard
IO cases. Red lines are computed from fluxes modified by the presence of
$0.1<\varepsilon_{\tau\tau}<0.33$ and $10^{-6}<\varepsilon_{e\tau}<10^{-5}$.
For $\varepsilon_{e\tau}$ in such interval, counting rates are intermediate
between the NSI case with $\varepsilon_{\tau\tau}>0.33$ and the standard
scenario. For the totally adiabatic $C-$resonance
($\varepsilon_{e\tau}>10^{-5}$), results are degenerate with the ones for
$\varepsilon_{\tau\tau}>0.33$.
In order to examine the emergence of the $C-$resonance, we rewrite Eq. 12 by
incorporating the expression for $\varepsilon_{\tau\tau}$ from Eq. 10 for
$d$-quark NSI. Here, we concentrate exclusively on the matter potential term,
which dominates over the vacuum term in regions of high core densities:
$\mathcal{H}^{NSI}\approx V_{e}\left(\begin{array}[]{ccc}1&0&0\\\ 0&0&0\\\
0&0&\varepsilon_{\tau\tau}^{d}\left(\frac{2-Y_{e}}{Y_{e}}\right)\end{array}\right).$
(15)
Note that our focus here is on $d$-quark NSI. Nevertheless, the qualitative
outcomes remain similar for $u$-quark NSI due to the resemblance between Eq.
10 and Eq. 11. In the subsequent section, we analyze the consequential effects
by concurrently considering NSIs involving both $d$-quark and $u$-quark. Using
the $Y_{e}$ values from Fig. 7 (See Appendix for details), a level crossing
occurs within the range of $50$ to $1000$ Km for
$0.1<\varepsilon^{d}_{\tau\tau}<0.33$ as $Y_{e}$ transitions from $0.2$ to
$0.5$. Prior to the crossing point,
$\varepsilon_{\tau\tau}^{d}(2-Y_{e})/Y_{e}>1$, rendering $\nu_{\tau}$ as the
heaviest state, and $\nu_{e}$ as the second heaviest:
$\nu_{e}\approx\nu_{2}^{m}(\nu_{1}^{m})$ for NO (IO). Subsequent to the
crossing point, the scenario inverts with
$\varepsilon_{\tau\tau}^{d}(2-Y_{e})/Y_{e}<1$, causing the energy levels of
$\nu_{e}$ and $\nu_{\tau}$ to intersect, as depicted in Fig. 4. This crossing
can result in resonant conversion if we introduce off-diagonal
NSI222Significant flavor off-diagonal NSI can resolve discrepancies in the
determination of the standard CP-phase $\delta_{CP}$ between the NO$\nu$A and
T2K long-baseline accelerator experiments Chatterjee:2020kkm ; Denton:2020uda
. $\varepsilon_{e\tau}$ and couple $\nu_{e}$ and $\nu_{\tau}$ states:
$\mathcal{H}^{NSI}\approx
V_{e}\left(\begin{array}[]{ccc}1&0&\varepsilon_{e\tau}^{d}\left(\frac{2-Y_{e}}{Y_{e}}\right)\\\
0&0&0\\\
\varepsilon_{e\tau}^{d}\left(\frac{2-Y_{e}}{Y_{e}}\right)&0&\varepsilon_{\tau\tau}^{d}\left(\frac{2-Y_{e}}{Y_{e}}\right)\end{array}\right).$
(16)
In such a scenario, the resonance condition can be expressed as
$\varepsilon_{\tau\tau}^{d}\left(\frac{2-Y_{e}}{Y_{e}}\right)=1,$ (17)
and can be satisfied by both neutrinos and antineutrinos simultaneously.
Furthermore, Eq. 17 is independent of matter density, energy, and mass
orderings. The adiabaticity condition is now formulated and can be expressed
as
$\gamma_{C}=\left|\frac{4\left(\mathcal{H}_{e\tau}^{NSI}\right)^{2}}{\dot{\mathcal{H}}_{\tau\tau}^{NSI}-\dot{\mathcal{H}}_{ee}^{NSI}}\right|\approx\left|\frac{16V_{e}(\varepsilon_{e\tau}^{d})^{2}}{Y_{e}\dot{Y_{e}}(1+\varepsilon_{\tau\tau}^{d})^{3}}\right|>1.$
(18)
Eq. 18 is satisfied for very small values of off-diagonal NSI:
$\varepsilon_{e\tau}>10^{-5}$. The resonance is partially adiabatic for
$\varepsilon_{e\tau}$ values within the range of $10^{-6}$ to $10^{-5}$. In
the limit of total adiabaticity, the produced $\nu_{e}$ reaches vacuum as
$\nu_{2}(\nu_{1})$ for NO (IO). The scenario is analogous to the one stated in
the last section by Eq. 13 and Eq. 14. When adiabaticity is totally violated,
the results are identical to the standard case in Eq. 7 and Eq. 8. Partial
adiabaticity provides fluxes, which are intermediary between the standard
scenario and the $\varepsilon^{d}_{\tau\tau}>0.33$ scenario. In the next
section, we analyze these signals in DUNE.
Figure 6: Projected sensitivities for NSI $\varepsilon_{\tau\tau}$ at DUNE.
For the totally adiabatic $C-$resonance, we consider
$\varepsilon_{e\tau}>10^{-5}$. Global-fit bound on NSI, as shown by the
hatched region, is adopted from Ref. Coloma:2023ixt .
We want to stress here that the search for $\varepsilon_{\tau\tau}^{u,d}$
during the neutronization burst phase is limited to values of
$\mathcal{O}(0.1)$. Conversely, in later phases of SN neutrino emission, when
$Y_{e}$ can attain values below $\mathcal{O}(0.1)$, smaller values of
$\varepsilon_{\tau\tau}^{u,d}$ could potentially alter observables. Note that
for $Y_{e}\ll 1$, Eq. 10 and Eq. 11 transforms as:
$\varepsilon_{\tau\tau}^{eff}\approx
2\frac{\varepsilon_{\tau\tau}^{d}}{Y_{e}}\hskip 14.22636pt\text{and}\hskip
14.22636pt\varepsilon_{\tau\tau}^{eff}\approx\frac{\varepsilon_{\tau\tau}^{u}}{Y_{e}}.$
(19)
Consequently, $\varepsilon_{\tau\tau}^{u,d}$ can be as small as $Y_{e}\sim
O(0.01)$, while maintaining the $\varepsilon^{eff}_{\tau\tau}$ value of order
one necessary to satisfy the resonance criterion in Eq. 17. Therefore,
sensitivity to $\varepsilon^{u,d}_{\tau\tau}$ can be improved to
$\mathcal{O}(0.01)$ by further investigating flavor conversion that occurs
during other phases of neutrino emission.
Signal analysis at DUNE.– Here, we investigate the expected SN neutrino signal
spectrum at DUNE under both the standard scenario and the consideration of NSI
effects. The graphical representation of the analysis, depicted in Fig. 5,
illustrates the projected number of $\nu_{e}$ events per 5 ms interval within
the time range of -5 to 40 ms (with 0 ms denoting the moment of core bounce).
The scenarios for Normal Ordering (NO) and Inverted Ordering (IO) are
presented on the left and right sides, respectively, assuming a distance of 10
Kpc between the supernova (SN) and Earth. For detailed information regarding
the computation of DUNE event spectra, see the Appendix. The initial neutrino
fluxes ($F_{e}^{i}$ and $F_{x}^{i}$) are taken from a simulation of a 15 solar
mass progenitor garching , as outlined in deGouvea:2019goq ; Jana:2022tsa .
The standard case, represented by the blue lines in Fig. 5 [cf. Eq. 7 and Eq.
8], highlights a notable feature—the visibility of the neutronization peak,
which distinguishes between the two mass orderings. The green lines are
computed from Eq. 13 and Eq. 14 which includes NSI
($\varepsilon_{\tau\tau}^{d}>0.33$). Crucially, as evident from Eq. 8 and Eq.
13, along with Fig. 5, the introduction of NSI leads to a scenario where
perfect confusion arises between the two mass orderings (green lines on the
left panel and blue lines on the right panel are identical).
DUNE can distinguish between the standard case and signals characterized by
$\varepsilon^{d}_{\tau\tau}>0.33$, as depicted in each panel of Fig. 5. We
analyze it using the $\chi^{2}$ estimator,
$\chi^{2}=\min_{\xi}\left(\sum_{i=1}^{n}2\left[(1+\xi)F_{i}-D_{i}+D_{i}\ln\left(\frac{D_{i}}{(1+\xi)F_{i}}\right)\right]\right).$
(20)
$F_{i}$ and $D_{i}$ are the number of $\nu_{e}$ events in the $i$-th time bin
for $\varepsilon_{\tau\tau}^{d}>0.33$ and $\varepsilon_{\tau\tau}^{d}=0$,
respectively. The parameter $\xi$ is allowed to vary in the range $-1<\xi\leq
1000$ with no penalty and we select the minimum to make our derived
sensitivity as much as possible independent of the overall normalization. We
find for NO,
$\chi^{2}(NO)\approx 27,$ (21)
and, for IO,
$\chi^{2}(IO)\approx 11.$ (22)
Notably, $\chi^{2}(NO)$ surpasses $\chi^{2}(IO)$ by a considerable margin.
This discrepancy is primarily attributed to the more pronounced
differentiation between the green and blue lines in Fig. 5 for NO, arising
from the presence (green) or absence (blue) of the peak around $\sim 5$ ms.
Conversely, for IO, both lines incorporate the peak, and the blue line could
mimic the green one even without NSI, contingent on different spectral
parameters of the supernova. We want to stress that the green lines in Fig. 5
also represent the case of NSI in the range
$0.1<\varepsilon_{\tau\tau}^{d}<0.33$ with adiabatic $C$-resonance
($\varepsilon_{e\tau}^{d}>10^{-5}$). Partial adiabaticity of the $C$-resonance
($10^{-6}<\varepsilon_{e\tau}^{d}<10^{-5}$) provides counting rates between
the standard scenario and the $\varepsilon^{d}_{\tau\tau}>0.33$ scenario; see
red lines in Fig. 5. We exclusively analyze scenarios with $d$-quark NSI.
However, for completeness, Fig. 6 displays projected sensitivities considering
the presence of both $d$-quark and $u$-quark NSIs.
NSI affecting the flavor evolution of SN neutrinos can also be studied in the
electron antineutrino ($\bar{\nu}_{e}$) channel using other future experiments
such as Hyper-Kanmiokande Hyper-Kamiokande:2021frf and JUNO JUNO:2023dnp .
Although these experiments will certainly provide more statistics than DUNE,
their observations will lack some specific features of the time signal that
are present in the $\nu_{e}$ channel (such as the peak at $5$ ms) that can be
crucial in diagnosing the presence of NSI. The quark NSI we discussed, which
is capable of inducing resonant flavor conversion of supernova neutrinos, may
also be subject to complementary tests. For instance, it could produce a
Glashow-like resonance feature detectable by neutrino telescopes Babu:2022fje
; Babu:2019vff ; Huang:2021mki or be examined in future collider experiments,
potentially resolving existing degeneracies Babu:2020nna .
Conclusions.– We have shown that the NSIs have a profound impact on supernova
neutrino flavor conversion, significantly impacting the neutronization burst
signals. The presence of NSI can invert the energy levels of neutrino matter
eigenstates and even induce a new resonance in the inner parts close to the
proto-neutron star. We showcase how the forthcoming experiments, such as DUNE,
can exploit these altered energy level configurations to achieve sensitivity
to NSIs at the order of $\mathcal{O}(0.1)$. Furthermore, we elaborate on how
this phenomenon may lead to a puzzling ambiguity between normal and inverted
mass orderings, and we thoroughly analyze its potential implications.
Acknowledgments.– We are grateful to André de Gouvêa for an illuminating
discussion and email correspondence. We also thank Pedro A. N. Machado and
Manibrata Sen for useful discussions. We wish to acknowledge the Center for
Theoretical Underground Physics and Related Areas (CETUP*) and the Institute
for Underground Science at SURF for hospitality and for providing a
stimulating environment. The work of YP was supported by the São Paulo
Research Foundation (FAPESP) Grant No. 2023/10734-3 and 2023/01467-1 and by
the National Council for Scientific and Technological Development (CNPq) Grant
No. 151168/2023-7. S.J. thanks Centro de Ciências Naturais e Humanas at
Universidade Federal do ABC and ICTP South American Institute for Fundamental
Research for their warm hospitality during the completion of this work.
## Appendices
### .1 Neutronization burst phase
Figure 7: Status of the matter density $\rho$ and and electron number fraction
$Y_{e}$ at $4.37$ ms after core bounce. Serpico:2011ir ; Tang:2020pkp .
The SN neutronization burst phase encompasses the period of $\sim 40$ ms after
core bounce when the $\nu_{e}$ luminosity dominates over all other flavors,
$\bar{\nu}_{e}$, $\nu_{x}$ and $\bar{\nu}_{x}$, with $x=\mu,\tau$ ($\nu_{e}$
luminosity can be $10-100$ times higher than luminosity in other flavors, see
deGouvea:2019goq ; Jana:2022tsa ). During the core collapse, the inner core is
compressed and reaches nuclear densities, while the matter falling above it
bounces back and launches a shock wave, which dissociates nuclei into its
component nucleons as it travels outwards. The capture of electrons in the
environment by the dissociated protons, $e^{-}+p\rightarrow n+\nu_{e}$, is
responsible for generating the large $\nu_{e}$ burst during the neutronization
burst phase. Theoretical uncertainties in the calculation of the
neutronization fluxes are believed to be as small and $10\%$ Serpico:2011ir ;
Wallace:2015xma ; OConnor:2018sti ; Kachelriess:2004ds and uncertainties
related to neutrino-neutrino refraction Mirizzi:2015eza ; Capanema:2024hdm
can also be avoided. Therefore, the burst phase presents a great opportunity
to search for new physics with SN neutrinos.
Neutrinos are produced in the region $r<100$ Km, which is opaque due to very
high densities ($\rho>10^{11}$ $\text{g/cm}^{3}$), and, at $r\sim 100$ Km,
they start free streaming. The efficient electron capture at the production
region reduces its electron number fraction, $Y_{e}=n_{e}/(n_{p}+n_{n})$,
where $n_{e}$, $n_{p}$ and $n_{n}$ are the electron, proton, and neutron
number densities, respectively, to levels below the one found in the envelope
($Y_{e}=0.5$). Precisely, this difference between $Y_{e}$ in inner and outer
layers might produce observable consequences for non-zero NSI. Further details
can be seen in Fig. 7, where we plot $\rho$ and $Y_{e}$ at $\sim 5$ ms
Fischer:2009af ; Tang:2020pkp . This instant is representative of the matter
profile during the neutronization peak, the most distinct feature of the burst
phase.
### .2 DUNE: technical details
The Deep Underground Neutrino Experiment (DUNE) will be composed of four-time
projection chambers, each with $10$ Kton of liquid argon, placed underground
in the Long-Baseline Neutrino Facility (LNBF) in South Dakota, United States
DUNE:2020zfm ; Cuesta:2023nnt . DUNE will primarily detect neutrinos with
energies of GeV and higher coming from a beam produced at Fermilab.
Nevertheless, DUNE is also sensitive to SN neutrinos in the range from about
$5$ MeV to tens of MeV via the charged-current interaction
$\nu_{e}+{}^{40}\mathrm{Ar}\rightarrow{}^{40}\mathrm{~{}K}^{*}+e^{-},$ (23)
that separates the electron flavor and has a much higher cross-section than
the elastic scattering,
$\nu_{e,\mu,\tau}+e^{-}\rightarrow\nu_{e,\mu,\tau}+e^{-}$, which is the
current main detection channel for $\nu_{e}$ Scholberg:2012id ; Zhu:2018rwc .
We compute the event spectrum for $\nu_{e}$ at DUNE as Capozzi:2018rzl
$\frac{dN_{\nu_{e}}}{dE_{r}}=\frac{N_{\mathrm{Ar}}}{4\pi R^{2}}\int
dE_{\nu_{e}}F_{\nu_{e}}\left(E_{\nu_{e}}\right)\sigma_{\nu_{e}+\mathrm{Ar}}\left(E_{\nu_{e}}\right)W\left(E_{r},E_{\nu_{e}}\right)$
(24)
where $N_{\mathrm{Ar}}$ is the number of target ${}^{40}\mathrm{Ar}$ nuclei in
the chambers, $R$ is the distance between the SN and the Earth, $F_{\nu_{e}}$
is the $\nu_{e}$ flux leaving the SN with energy $E_{\nu_{e}}$,
$\sigma_{\nu_{e}+\mathrm{Ar}}$ is the cross section for the interaction in Eq.
23 generated by MARLEY Gardiner:2018zfg , $W$ is the gaussian energy
resolution with
$\sigma_{E}/\text{MeV}=0.11\sqrt{E_{r}/\text{MeV}}+0.02E_{r}/\text{MeV}$ and
$E_{r}$ is the reconstructed electron energy.
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|
# Error terms for the motives of discriminant complements and a Cayley-
Bacharach theorem
Ishan Banerjee
###### Abstract
In this paper we prove under some simplifying hypotheses questions of Picoco
and Levinson-Ullery on Cayley-Bacharach sets. Our results imply that, under
suitable hypotheses Cayley-Bacharach sets lie on curves of low degree. We then
use these results to estimate error terms to the normalized motive of the
space of smooth degree $d$ hypersurfaces in $\mathbb{P}^{n}$ as $d$ grows to
infinity. The error term can be expressed in terms of a certain ‘sum over
points’ on plane cubic curves and the associated Hodge structure can be
expressed in terms of the cohomology of the moduli space of elliptic curves.
We also prove convergence of the motive of degree $d$ hypersurfaces in
$\mathbb{P}^{n}$ as $n$ grows to infinity as well as other results on
discriminant complements of high dimensional varieties.
## 1 Introduction
Let $X$ be a smooth projective variety over a field $\mathbb{F}$. Let
$\mathcal{L}$ be an ample line bundle on $X$. Let
$\Sigma(X,\mathcal{L}):=\\{f\in H^{0}(X,\mathcal{L})|\exists p\in
X,f(p)=0,df(p)=0\\}.$
We call $\Sigma(X,\mathcal{L})$ the discriminant variety associated to $X$ and
$\mathcal{L}$, it consists of those sections of $\mathcal{L}$ which define a
singular hypersurface in $X$. Let
$U(X,\mathcal{L})=H^{0}(X,\mathcal{L})\setminus\Sigma(X,\mathcal{L}).$
Theorems of Vakil-Wood and Poonen mply that after some suitable normalization,
the motive of $U(X,\mathcal{L}^{j})$ converges (in an appropriate sense) as
$j\to\infty$ (see [16] and [14]).
In this paper we study the _rate of convergence_ of the motive of
$U(X,\mathcal{L}^{j})$. Our main contributions in this regard are:
1. 1.
An improvement of known rates of convergence for all $X$ and $\mathcal{L}$
(Theorem 1.6). Interestingly enough, our results suggest that the rate of
convergence grows faster with the increase in the dimension of $X$, as in the
upper bounds we establish on the ‘error term’ decrease as $\dim X\to\infty$.
2. 2.
In the special case of $\mathbb{P}^{n}$, we obtain a leading error term (see
Theorem 1.9 ) for the convergence of $U(\mathbb{P}^{n},\mathcal{O}(d))$ as
$d\to\infty$.
3. 3.
We establish upper bounds on the difference between the motive of
$U(X,\mathcal{O}(d))$ and a special value of the motivic zeta function of $X$
for all smooth projective varieties $X$ (see Theorem 1.7). Our bound depends
only on the dimension of $X$. Our bounds improve as the dimension of $X$
increases. As a consequence we establish (for $d\geq 3$ ) that after a
suitable normalisation, the motive of $U(\mathbb{P}^{n},\mathcal{O}(d))$
converges to a certain limit of motivic zeta functions as $n\to\infty$ (see
Theorem 1.7). It is important to note here that we do not assume that
$d\to\infty$ or take some kind of limit in which the line bundle $\mathcal{L}$
becomes more and more ample.
In order to establish the above results, we prove several theorems about
Cayley Bacharach sets in $\mathbb{P}^{n}$ which are of independent interest
(the definition of Cayley-Bacharach sets will be recalled in Section 2). Our
results regarding Cayley-Bacharach sets are as follows:
1. 1.
We show that under certain hypotheses, Cayley-Bacharach subsets of
$\mathbb{P}^{n}$ are forced to lie on curves of low degree (see Theorem 1.13
). This is partial progress on a question of Picoco(see Question 1.1 of [13] )
2. 2.
We establish Conjecture 1.2 of [11] under certain additional hypotheses
(Theorem 1.15 ). The conjecture states that a subset $Z$ of $\mathbb{P}^{n}$
that is Cayley-Bacharach for $\mathcal{O}(r)$, that satisfies
$|Z|\leq(d+1)r+1$ lies on a plane configuration of dimension $d$.
### 1.1 Grothendieck ring of varieties
In this section, will briefly discuss the Grothendieck ring of varieties. We
largely follow section 1 of [16] where the Grothendieck ring of varieties is
discussed in more detail.
The Grothendieck ring of varieties $\mathcal{M}$ over the field $\mathbb{F}$
(also denoted $K_{0}(Var_{\mathbb{F}})$) is defined as follows, in terms of
generators and relations.
1. 1.
Given a finite type $\mathbb{F}$ scheme $X$, we have an element
$[X]\in\mathcal{M}$. Isomorphic schemes define the same element of
$\mathcal{M}$.
2. 2.
The collection of all $[X]$ generate $\mathcal{M}$ as an abelian group.
3. 3.
Given any finite type scheme $X$ and a closed subscheme $Y$ with complement
$U=X\setminus Y$, we have the relation
$[X]=[U]+[Y].$
4. 4.
Given $[X],[Y]\in\mathcal{M}$ we define $[X][Y]=[X\times Y]$. This gives
$\mathcal{M}$ the structure of a ring.
Let $\mathbb{L}=[\mathbb{A}^{1}].$ Let
$\mathcal{M}_{\mathbb{L}}=\mathcal{M}[\mathbb{L}^{-1}]$. There is an
increasing filtration on $\mathcal{M}$, with varieties of dimension
$\leq\ell$, generating the $\ell$th piece of the filtration. We will call this
filtration the dimensional filtration on $\mathcal{M}$. The filtration extends
to a filtration on $\mathcal{M}_{\mathbb{L}}$. We denote the completion of
$\mathcal{M}_{\mathbb{L}}$ with respect to this filtration by
$\hat{\mathcal{M}_{\mathbb{L}}}$.
###### Remark 1.
In this paper we will often have formulae where an element $x\in\mathcal{M}$
is multiplied by a vector space over the base field $\mathbb{F}$. These
formulae are to be understood in the following sense- we identify a vector
space $V$ with $\mathbb{L}^{\dim V}$. Any expression of the form $xV$ where
$x\in\mathcal{M}$ is simply shorthand for $x\mathbb{L}^{\dim V}$. Similarly an
expression of the form $xV^{-1}$ is understood to mean $x\mathbb{L}^{-\dim
V}$.
Let us now assume $\mathbb{F}=\mathbb{C}$. Let $\mathcal{M}^{Hdg}$ denote the
Grothendieck group of mixed Hodge structures . Let
$\mathbb{L}^{Hdg}=[H^{2}_{c}(\mathbb{A}^{1},\mathbb{Q})]$. We define
$\mathcal{M}_{\mathbb{L}}^{Hdg}=\mathcal{M}^{Hdg}[(\mathbb{L}^{Hdg})^{-1}].$
There is an increasing filtration on $\mathcal{M}^{Hdg}$ where the $\ell$th
piece of the filtration is generated by Hodge structures of weight $\leq\ell$.
The filtration extends to a filtration on $\mathcal{M}_{\mathbb{L}}^{Hdg}$. We
denote the completion of $\mathcal{M}_{\mathbb{L}}^{Hdg}$ with respect to this
filtration $\hat{\mathcal{M}_{\mathbb{L}}}^{Hdg}$
There is a specialisation map $\phi:\mathcal{M}\to\mathcal{M}^{Hdg}$ defined
on smooth proper generators by
$[X]\mapsto\sum_{k}(-1)^{k}H^{k}_{c}(X,\mathbb{Q}).$ The map extends to a map
$\mathcal{M}_{\mathbb{L}}\to\mathcal{M}_{\mathbb{L}}^{Hdg}$ and a map
$\hat{\mathcal{M}_{\mathbb{L}}}\to\hat{\mathcal{M}_{\mathbb{L}}}^{Hdg}.$
There is a further specialisation of the map
$\hat{\mathcal{M}_{\mathbb{L}}}\to\hat{\mathcal{M}_{\mathbb{L}}}^{Hdg}$ to a
map
$e:\hat{\mathcal{M}_{\mathbb{L}}}\to\mathbb{Z}[x,y][[x^{-1},y^{-1}]]$
defined on generators by
$e(X)=\sum_{(p,q)\in\mathbb{N}^{2}}\sum_{k\in\mathbb{N}}(-1)^{k}h^{p,q}(H^{k}_{c}(X,\mathbb{Q}))x^{p}y^{q}.$
Here $h^{p,q}$ denotes the $(p,q)^{th}$ Hodge number of the mixed Hodge
structure $H^{k}_{c}(X,\mathbb{Q})$. We will call the polynomial $e(X)$ the
Serre polynomial of $X$.
#### Motivic zeta function
Let $X$ be a quasi-projective variety over $\mathbb{F}$. For $n>0$ we define
$\textrm{Sym}^{n}X$ to be the $n^{th}$ symmetric power of $X.$ For $n=0$, we
set $\textrm{Sym}^{0}X$ to be $\textrm{Spec}\mathbb{F}$.
We define the Kapranov motivic zeta function of $X$,
$Z_{X}(t)\in\mathcal{M}_{\mathbb{L}}[[t]]$ by
$Z_{X}(t)=\sum_{k=0}^{\infty}[\textrm{Sym}^{k}X]t^{k}.$
We define
$\zeta_{X}(s)=Z_{X}(\mathbb{L}^{-s}).$
For $s>\dim X$, $\zeta_{x}(s)\in\hat{\mathcal{M}}_{\mathcal{L}}.$
#### 1.1.1 A note on brackets
In this paper we will deal with many large formulae in the Grothendieck ring.
As a result we have decided to omit the square brackets in our formulae when
referring to the class of a variety, i.e. we will refer to $[X]$ as $X$. This
is because keeping the brackets in the notation would lead to a huge number of
brackets and be quite difficult to parse.
### 1.2 Meta problems for convergence
In this subsection we describe several meta problems about convergence of a
sequence in the completion of a filtered ring. Theorems 1.6, 1.9, and 1.7 will
be instances of these metaproblems. This subsection is in a strict sense not
essential to the rest of the paper and the main results of the paper will
still be completely understandable without this subsection. However we include
this section for motivation and to place some of our results in a wider
context.
Let $R$ be a ring with an increasing bi-infinite filtration $F^{l}$. Assume
that
$\cap_{\ell\in\mathbb{Z}}F^{\ell}R=0,\cup_{\ell\in\mathbb{Z}}F^{\ell}R=R$. Let
$\hat{R}$ be the completion of $R$ with respect to the filtration.
Now we define a list of meta problems regarding convergence in $\hat{R}$ in
increasing order of difficulty.
###### Problem 1.1.
Let $((x_{n}))$ be a sequence in $\hat{R}$. Show that $\lim_{n\to\infty}x_{n}$
exists.
###### Problem 1.2.
Let $((x_{n}))$ be a sequence in $\hat{R}$. Show that $\lim_{n\to\infty}x_{n}$
exists. Give an explicit description of the limit.
###### Problem 1.3.
Let $((x_{n}))$ be a sequence in $\hat{R}$. Show that
$\lim_{n\to\infty}x_{n}=L$. Find $\ell_{n}\in\mathbb{Z},\ell_{n}\to\infty$
such that $x_{n}-L\in F^{-\ell_{n}}\hat{R}$.
###### Problem 1.4.
Let $((x_{n}))$ be a sequence in $\hat{R}$. Show that
$\lim_{n\to\infty}x_{n}=L$. Find $\ell_{n}\in\mathbb{Z},\ell_{n}\to\infty$
such that $x_{n}-L\in F^{-\ell_{n}}\hat{R}$ and for $n\gg 0$ $x_{n}-L\not\in
F^{-\ell_{n}+1}\hat{R}$.
One could just as well propose a variant of Problem 4 where we require
$x_{n}-L\not\in F^{-\ell_{n}+C}\hat{R}$ where $C\in\mathbb{R}$ is some
constant not depending on $n$.
###### Problem 1.5.
Let $((x_{n}))$ be a sequence in $\hat{R}$ such that
$\lim_{n\to\infty}x_{n}=L$ Construct a sequence $y_{n}$ such that
$x_{n}-L=y_{n}+z_{n}$, where $z_{n}$ is much smaller than $y_{n}$.
Of course Problem 5 as stated is quite imprecise and one would need to decide
what it means for $z_{n}$ to be much smaller than $y_{n}$. For example, if the
$y_{n}$ we construct are all units one could ask that,
$\lim_{n\to\infty}\frac{z_{n}}{y_{n}}=0$.
### 1.3 Statement of results on convergence
We will adopt the following notation:
1. 1.
Let $X$ be a scheme over a field $\mathbb{F}$. For a closed subscheme
$Y\subseteq X$ we denote the ideal sheaf of $Y$ by $I_{Y}$.
2. 2.
Let $\mathcal{F}$ be a coherent sheaf on $X$. We define
$h^{i}(X,\mathcal{F}):=\dim_{\mathbb{F}}H^{i}(X,\mathcal{F}).$
3. 3.
We define $N_{X}^{*}$ to be the conormal sheaf of $Y$ in $X$ it is the
pullback of the sheaf $I_{Y}$ to $Y$.
4. 4.
Given $X$ and $Y$ as above we define $N_{X}(Y)$ to be the subscheme of $X$
defined by the sheaf $I^{2}_{Y}$, i.e. the first infinitesmal neighbourhood of
$Y$.
Given $x,y\in\hat{\mathcal{M}}_{\mathbb{L}}$, we will say $x=y$ up to
dimension $r$ if $x-y$ lies in the $F^{r}\hat{\mathcal{M}}_{\mathbb{L}}$,
where $F^{r}$ refers to the dimension filtration on
$\hat{\mathcal{M}}_{\mathbb{L}}$.
Let $X$ be a smooth variety over a field $\mathbb{F}$, $\mathcal{L}$ an ample
line bundle bundle on $X$. Then Proposition 3.7 of [16] states that
$\lim_{d\to\infty}\frac{U(X,\mathcal{L}^{d})}{H^{0}(X,\mathcal{L}^{d})}=\zeta_{X}^{-1}(\dim
X+1).$
This can be seen as as an instance of Problem 1. However the proof of
Proposition 3.7 in [16] actually gives more. Let $m(d)$ be the largest integer
such that for all zero dimensional reduced subschemes $Z\subseteq X$ of length
$\leq m(d)$, the map $H^{0}(X,\mathcal{L}^{d})\to
H^{0}(N_{X}(Z),\mathcal{L}^{d})$ is surjective. Then the proof of Proposition
3.7 in [16] actually implies that
$\frac{U(X,\mathcal{L}^{d})}{H^{0}(X,\mathcal{L}^{d})}-\zeta_{X}^{-1}(n+1)$
lies in $F^{-m(d)}\hat{\mathcal{M}_{\mathcal{L}}}$. We improve this result to
the following:
###### Theorem 1.6.
Let $X,\mathcal{L},m(d)$ be as above. Then there exists $C>0$ such that for
all $d\gg 0$,
$\frac{U(X,\mathcal{L}^{d})}{H^{0}(X,\mathcal{L}^{d})}=\zeta_{X}^{-1}(n+1)$
up to dimension $-m(d)(\dim X)+C$.
###### Remark 2.
The constant $C$ in the above theorem depends on $X$ and $\mathcal{L}$, it is
related to the dimensions of the spaces of curves in $X$ such that the line
bundle $\mathcal{L}$ restricts to a low degree. However, it is fairly
straightforward to adapt the proof to obtain a constant $C^{\prime}$,
satisfying $C<C^{\prime}$ where $C^{\prime}$ depends only on the dimension of
$X$.
We note that the improvement is greater for varieties of high dimension. This
motivates us to study whether there are bounds on the difference
$\frac{U(X,\mathcal{L}^{d})}{H^{0}(X,\mathcal{L}^{d})}-\zeta_{X}^{-1}(\dim
X+1)$
that improve with the dimension of $X$ and are valid for small values of $d$.
###### Theorem 1.7.
Let $X$ be a smooth projective variety, let $\mathcal{L}$ be a _very_ ample
line bundle. Then for any $d\geq 3$,
$\frac{U(X,\mathcal{L}^{d})}{H^{0}(X,\mathcal{L}^{d})}=\zeta_{X}^{-1}(\dim
X+1),$
up to dimension $\sqrt{\dim X}/3$.
As a corollary we have:
###### Corollary 1.8.
Let $X_{n}$ be a sequence of smooth projective varieties such that $\dim
X_{n}\to\infty$ and $\lim_{n\to\infty}\zeta_{X}^{-1}(\dim X_{n}+1)$ exists.
Let $\mathcal{L}_{n}$ be a very ample line bundle on $X_{n}$. Let $d_{n}\geq
3$ be a sequence of positive integers. Then
$\lim_{n\to\infty}\frac{U(X_{n},\mathcal{L}_{n}^{d_{n}})}{H^{0}(X_{n},\mathcal{L}_{n}^{d_{n}})}=\lim_{n\to\infty}\zeta_{X_{n}}^{-1}(\dim
X_{n}+1).$
In particular
$\lim_{n\to\infty}\frac{U(\mathbb{P}^{n},\mathcal{O}(d_{n}))}{H^{0}(\mathbb{P}^{n},\mathcal{O}(d_{n}))}=\prod_{k=1}^{\infty}(1-\mathbb{L}^{-k}).$
Finally we identify the leading error term for the sequence
$\frac{U(\mathbb{P}^{n},\mathcal{O}(d))}{H^{0}(\mathbb{P}^{n},\mathcal{O}(d))}$,
where $n$ is fixed and $d$ goes to infinity, providing a complete answer to
Problem 5 in this instance.
###### Theorem 1.9.
Let $n\geq 2$. Then for $d\gg n$, there exists $\epsilon>0$.
$\frac{U(\mathbb{P}^{n},\mathcal{O}(d))}{H^{0}(\mathbb{P}^{n},\mathcal{O}(d))}-Z_{\mathbb{P}^{n}}^{-1}(n+1)=Y(d)+Z(d)$
where:
1. 1.
$\dim Z(d)\leq-(\frac{3n}{2}+\epsilon)d$.
2. 2.
For $d\equiv 1\mod{4}$ , the virtual Hodge structure of $Y(d)$ has highest
weight term $E_{k}(\mathbb{L}^{Hdg})^{n^{2}-k(n+1)+1}$ in $K_{0}(MHS),$ where
$E_{d}$ is the highest weight term of the virtual Hodge structure
corresponding to $H^{1}(\mathcal{M}_{1,1},H_{k})$. Here $\mathcal{M}_{1,1}$ is
the moduli space of elliptic curves, $H_{k}$ is the Hodge bundle corresponding
to $k^{th}$ tensor power of the relative canonical bundle and
$k=\frac{3d+1}{2}$.
$Y(d)$ is explicitly defined in Section 6.
###### Remark 3.
$Y(d)$ has an explicit description in terms of configurations of points on
smooth plane cubic curves a precise description is given in Section 6 (defined
just before Proposition 6.14) and its virtual Hodge structure may be readily
computed from that description using the results of [7].
### 1.4 Motivation for our results on convergence
Our motivation for the results proven above comes from two different sources,
homological stability and number theory. We will first discuss some motivation
related to homological stability.
#### 1.4.1 Homological stability
It was established by [15] that the spaces $U(\mathbb{P}^{n},\mathcal{O}(d))$
satisfy homological stability. Later on Aumonier in [1] and Das-Howe in [3]
established homological stability for the sequence of spaces
$U(X,\mathcal{L}^{d}$. However, it was known (and this is even mentioned in
[15]) that the range of homological stability given in [15] is not optimal. It
was of interest therefore to see where and how homological stability fails and
explain the first class in
$H^{*}(U(\mathbb{P}^{n},\mathcal{O}(d)),\mathbb{Q})$ that is not stable.
Related to the above is the desire to establish ‘secondary homological
stability’-like results for the family of spaces
$U(\mathbb{P}^{n},\mathcal{O}(d)).$ For instance the paper [5] establishes
that.
It would be interesting to establish some analogue of these results in the
case of $U(\mathbb{P}^{n},\mathcal{O}(d)),$ however there are several
difficulties in trying to do so, first of all there are no maps between
different $U(\mathbb{P}^{n},\mathcal{O}(d))$ for different values of $d$ and
secondly the nature of the spaces $U(\mathbb{P}^{n},\mathcal{O}(d))$ is very
different from the kind of spaces people have established ‘secondary
stability’ for; secondary stability results are known in the setting of
configuration spaces on manifolds and other more topological and less
algebraic geometric spaces as compared to discriminant complements. The
techniques used to establish such results are much more topological than the
ones in this paper.
In the end we were unable to establish such a ‘secondary stability’ theorem
for the homology of the spaces $U(\mathbb{P}^{n},\mathcal{O}(d))$ and this
paper does not really deal with results at the level of homology.
One may in fact combine the techniques of this paper with those of [15] to
conclude that $H^{k}(U(\mathbb{P}^{n},\mathcal{O}(d),\mathcal{O}(d))\cong
H^{k}(GL_{n+1}(\mathbb{C}),\mathbb{Q}),$ for $k\leq\frac{nd}{2}$. However we
do not include this result as we do not believe this is optimal either and we
do not how to obtain the optimal answer. In some sense our failure to get such
a result is a consequence of some of the differences between cohomology and
the Grothendieck ring, in the Grothendieck ring setting, if a variety $X$
admits a stratification with understandable pieces, we understand the class of
$[X]$ in the Grothendieck ring. However if a variety $X$ has a stratification
with understandable pieces, it is still not clear what the cohomology of $X$
is, the best we can do is obtain a spectral sequence whose $E_{1}$ page is
related to the cohomology of the strata. However it is often not clear in
these situations what the differentials in this spectral sequence are.
As a result we were unable to adapt some of the arguments involved in proving
Theorem 1.9 to a cohomological setting. It still may very well be possible to
establish a cohomological analogue of Theorem 1.9 but it would require
somewhat different techniques than we have used/ are aware of.
However it is possible to establish that for fixed $d\geq 3$,
$H^{k}(U(\mathbb{P}^{n},\mathcal{O}(d)))\cong H^{k}(GL_{n+1}(\mathbb{C}))$ for
$k\leq f(n)$ where $f(n)$ is some function of $n$ that grows to $\infty$. This
is a consequence of combining the techniques involved in proving Theorem 1.7
with those of [15]. //
We believe it should be possible to get a cohomological analogue of Theorem
1.7 as well.
While we have not established any secondary stability results for the sequence
of spaces $U(\mathbb{P}^{n},\mathcal{O}(d))$, we have established an analogue
of secondary stability in the setting of $K_{0}(Var_{\mathbb{F}});$ by the
results of [16], we have
$\lim_{d\to\infty}\frac{U(\mathbb{P}^{n},\mathcal{O}(d))}{H^{0}(\mathbb{P}^{n},\mathcal{O}(d))}\to\frac{1}{Z_{\mathbb{P}^{n}}(n+1)}$
Theorem 1.9 refines this result to give us an estimate of the difference
between the limit and the elements of the sequence. Furthermore the proof of
Theorem 1.9 identifies this leading error term as arising from the failure of
the proof of [16] to give an equality between
$\frac{1}{Z_{\mathbb{P}^{n}}(n+1)}$ and
$\frac{U(\mathbb{P}^{n},\mathcal{O}(d))}{H^{0}(\mathbb{P}^{n},\mathcal{O}(d))}$;
the proof of Theorem 1.13 in [16] crucially involves the fact that if $Z$ is a
collection of points in $X$ whose length is small compared to $d$, then the
vanishing of $1$-jets at the points of $Z$ impose linearly independent
conditions. This fails when the length of $Z$ is comparable to $d$ and our
error term $Y(d)$ is related to (and in fact defined in terms) of the variety
parametrizing collections of points where this linear independence just begins
to fail.
#### 1.4.2 Arithmetic statistics
A second source of motivation comes from number theory, in particular from
arithmetic statistics. In many situations in arithmetic statistics, one is
interested in establishing asymptotic expressions for the number of some
quantity of arithmetic interest. For instance as a prototype, one has Roberts
conjecture (established in [2]) which says that the number of degree $3$
extensions of $\mathbb{Q}$ of discriminant $\leq X$ is
$aX+bX^{\frac{5}{6}}+o(X^{\frac{5}{6}})$.
We view Theorem 1.9 as a Theorem of this kind, we have essentially established
that the class of the discriminant complement $U(X,\mathcal{L})$ is
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))Z_{\mathbb{P}^{n}}^{-1}(n+1)+H^{0}(\mathbb{P}^{n},\mathcal{O}(d))Y(d)+\textrm{
lower order terms},$
where lower order is to be understood in terms of the dimension filtration.
A natural question to ask here is if Theorem 1.9 and 1.7 have consequences
with regard to point counting over finite fields. More precisely, one may ask
if one has a result of the following form: Let $\mathbb{F}=\mathbb{F}_{q}$.
Let $X,\mathcal{L},d$ be as in Theorem 1.7. Is there a bound on
$\frac{\\#U(X,\mathcal{L}^{d})}{\\#3H^{0}(X,\mathcal{L}^{d})}-\\#Z_{X}^{-1}(-n-1)$
analogous to Theorem 1.9? Does
$\lim_{n\\\
to\infty}\\#\frac{U(\mathbb{P}^{n},\mathcal{O}(d))}{H^{0}(\mathbb{P}^{n},\mathcal{O}(d))}$
exist and equal
$\prod_{i=1}^{\infty}(1-\frac{1}{q^{i}})?$
Does one have
$\\#\frac{(U(\mathbb{P}^{n},\mathcal{O}(d)))}{H^{0}(\mathbb{P}^{n},\mathcal{O}(d))}=\\#Z_{\mathbb{P}^{n}}(-n-1)+\\#Y(d)+\textrm{
lower order terms}?$
We believe that the answer to all the above questions is yes and we are
currently working on solving the above questions. The reason why point
counting formulae like the above do not immediately follow from Theorems 1.7
and 1.9 is that the functor $\\#$ is not continuous with respect to the
dimension filtration, varieties of small dimension can have an arbitrary
number of points. In addition to this some of the methods we use are not well
adapted to the setting of point counts, e.g. the sum over ordered partitions
constructions that we use involve a very large number of terms that may cause
problems when added up. However we believe that we may combine our methods
with that of [14] to obtain the above results.
### 1.5 Cayley-Bacharach sets
In this section, we will define what Cayley-Bacharach sets are and prove a few
basic propositions about them. This may appear to be quite unrelated to the
material of the previous section, however it is not as the results of the
previous section rely on our results on Cayley-Bacharach sets.
In this section, we assume $\mathbb{F}$ to be an algebraically closed field.
Let $X$ be a projective variety over $\mathbb{F}$. Let $\mathcal{L}$ be a line
bundle on $X$.
###### Definition 1.10.
A zero dimensional reduced subscheme $Z\subseteq X$ is said to be Cayley-
Bacharach for $\mathcal{L}$ if given $f\in H^{0}(X,\mathcal{L})$, if $f$
vanishes on all but one point of $Z$ then $f$ vanishes on all of $Z$.
We will now reformulate the above definition. Given any $Z^{\prime}\subseteq
Z\subseteq X$ we have restriction maps $H^{0}(X,\mathcal{L})\to
H^{0}(Z,\mathcal{L})\to H^{0}(Z^{\prime},\mathcal{L})$. We hence have a map
$\textrm{Im}(H^{0}(X,\mathcal{L})\to
H^{0}(Z,\mathcal{L}))\to\textrm{Im}(H^{0}(X,\mathcal{L})\to
H^{0}(Z^{\prime},\mathcal{L})).$
Given a zero dimensional scheme $Z$, we will use $|Z|$ to denote the length of
$Z$.
###### Proposition 1.11.
A zero dimensional reduced subscheme $Z\subseteq X$ is Cayley-Bacharach for
$\mathcal{L}$ iff for all $Z^{\prime}\subseteq Z$, $|Z\setminus
Z^{\prime}|=1$, the restriction map
$\textrm{Im}(H^{0}(X,\mathcal{L})\to
H^{0}(Z,\mathcal{L}))\to\textrm{Im}(H^{0}(X,\mathcal{L})\to
H^{0}(Z^{\prime},\mathcal{L}))$ (1)
is an isomorphism.
###### Proof.
If $Z^{\prime}\subseteq Z$, then the homomorphism in (1) is always surjective.
For it to be to an isomorphism is equivalent therefore to it having zero
kernel. But if $f$ is in the kernel of (1), it is an element of
$H^{0}(X,\mathcal{L})$ that restricts to $0$ on $Z^{\prime}$. If $Z$ is
Cayley-Bacharach this implies that $f$ rstricts to $0$ on $Z$ and hence the
kernel of the (1) is trivial. Conversely if the kernel of (1) is trivial, then
any $f$ restricting to zero on $Z^{\prime}$ restricts to $0$ on $Z$, and hence
if the kernel of (1) is trivial for all choices of $Z^{\prime}$, then $Z$ is
Cayley-Bacharach for $\mathcal{L}$. ∎
### 1.6 Results on Cayley-Bacharach sets
In[13], Picoco asks the following question.
###### Question 1.12.
Let $e,d>0$. Suppose $Z\subseteq\mathbb{P}^{n}$ is a reduced set of points and
that $|Z|<e(d-e+3)-1$. Then is it true that if $Z$ is Cayley-Bacharach for
$\mathcal{O}(d)$, it lies on a curve of degree $<e$?
While we do not know if Picoco’s question has an affirmative answer we have
made the following partial result.
###### Theorem 1.13.
Let $d\gg e\geq 1$. Let $Z$ be a Cayley Bacharach subset of $\mathbb{P}^{n}$
for $\mathcal{O}(d)$. Then there exists a function $f$, depending on $n$ but
not on $d$ such that if
$|Z|<ed-f(e),$
$Z$ lies on a curve of degree $\leq e$.
In [11](Conjecture 1.2 in the paper) Levinson and Ullery made the following
conjecture
###### Conjecture 1.14.
Let $d\geq 1$. Suppose $Z\subseteq\mathbb{P}^{n}$ is a Cayley-Bacharach set
for $\mathcal{O}(d)$. Suppose $|Z|\leq(e+1)d+1$ Then $Z$ is contained in a
union of positive dimensional linear subspaces $L_{1},\dots L_{k}$ of
$\mathbb{P}^{n}$ such that $\sum_{i=1}^{k}\dim L_{i}\leq e$.
###### Theorem 1.15.
Let $d\gg e$. Then Conjecture 1.14 is true.
One can replace $d\gg e$ in 1.15 with $d$ greater than a certain quartic
polynomial in $e$, by going through the proof. The proof of Theorem 1.15
(assuming Theorem 1.13 ) is fairly straightforward and is essentially a
consequence of the fact that an integral curve of degree $e$ lies in an $e$
dimensional linear subspace of projective space.
### 1.7 Acknowledgements
I would like to thank Eduard Looijenga and Madhav Nori for helpful and
clarifying discussions. I’d like to thank Jake Levinson, Brooke Ullery and
David Stapleton for discussing some of their ideas with me. I’d like to thank
Ronno Das, and Aaron Landesmann for suggestion on how to improve the paper.
I’d like to thank Jordan Ellenberg for a question that led to Theorem 1.7.
Finally, I would like to thank Joshua Mundinger for extensive comments on
several drafts of this paper.
## 2 Cayley-Bacharach subsets lie on curves
In this section we will prove Theorems 1.13 and 1.15 . Our strategy to prove
Theorem 1.13 is to induct on the dimension of the ambient space. In the case
of $\mathbb{P}^{2}$ (the base case of our induction), we have the following
theorem of Lopez and Pirola (this is a subcase of Lemma 2.5 of [12]).
###### Proposition 2.1.
Let $Z\subseteq\mathbb{P}^{2}$ be a Cayley Bacharach set for $\mathcal{O}(d)$.
Let $e\geq 1$. If $|Z|\leq e(d-e+3)-1$, then $Z$ lies on a curve of degree
$\leq e-1$.
Our induction strategy will involve linearly projecting onto smaller
dimensional projective spaces. We will need the following propositions in our
proof, which deal with projections and Cayley Bacharach sets.
###### Proposition 2.2.
Let $n\geq 2$. Let $Z\subseteq\mathbb{P}^{n}$ be a Cayley-Bacharach set for
$\mathcal{O}(d)$. Let $P\in\mathbb{P}^{n}\setminus Z$. Let
$H\cong\mathbb{P}^{n-1}$ be a hyperplane not containing $P$. Let
$\pi:\mathbb{P}^{n}\setminus\\{P\\}\to H$ be the projection map. Suppose that
the scheme theoretic image of $Z$, $\pi(Z)$ is reduced. Then $\pi(Z)$ is a
Cayley-Bacharach set for $\mathcal{O}(d)$.
###### Proof.
We note that given a linear projection $\pi$, defined by projecting from a
point $P$ and a zero dimensional reduced subscheme $Z$, $\pi(Z)$; the
conditions that $P$ does not lie on a line joining any two of the points in
$Z$ and that $\pi(Z)$ is reduced are equivalent.
Without loss of generality, we may assume $P=[0:0:\dots:0:1]$ and $H$ is the
hyperplane defined by $X_{n+1}=0.$ Then the projection map $\pi$ is defined by
$[x_{1}:x_{2}:\dots:x_{n}:x_{n+1}]\mapsto[x_{1}:x_{2}:\dots:x_{n}].$ Let
$Z=\\{P_{1},P_{2},\dots,P_{m}\\}.$ Let
$\pi(Z)=\\{\pi(P_{1}),\pi(P_{2}),\dots,\pi(P_{m})\\}$. Let $f\in
H^{0}(\mathbb{P}^{n-1},\mathcal{O}(d))$ be a section vanishing at
$\\{\pi(P_{1}),\dots,\pi(P_{m-1})\\}.$ Such an $f$ is given by a degree $d$
homogenous polynomial $p$ in the variables $X_{1}\dots,X_{n}$. However the $p$
can be thought of as a polynomial in the variables $X_{1}\dots,X_{n},X_{n+1}$
as well, with no dependence on $X_{n+1}$. We observe that for any
$Q\in\mathbb{P}^{n}-\\{[0:\dots:0:1]\\}$, $p(Q)=0$ iff $p(\pi(Q))=0$. Hence
$p(P_{i})=0$ for $i\in\\{1,\dots m-1\\}.$ But since $Z$ is Cayley-Bacharach
for $\mathcal{O}(d)$ this implies $p$ vanishes on $P_{m}$ as well. But this
implies $p$ vanishes on $\pi(P_{m})$. Hence $\pi(Z)$ is Cayley-Bacharach for
$\mathcal{O}(d)$. ∎
###### Proposition 2.3.
Let $n\geq 3$. Let $P_{1},P_{2}$ be two distinct points in $\mathbb{P}^{n}$.
Let $H$ be a hyperplane in $\mathbb{P}^{n}$ such that $P_{1},P_{2}\not\in H$.
Let $C_{1},C_{2}$ be two curves in $H$ such that $\textrm{deg}(C_{i})\leq
e_{i}$. Let $S_{i}$ be the cone over $C_{i}$ with apex $P_{i}$. Then
$S_{1}\cap S_{2}=C\cup\Gamma$, where $C$ is a curve and $\Gamma$ is zero
dimensional. Furthermore $\deg(C)+|\Gamma|\leq e_{1}e_{2}$.
###### Proof.
We begin by noting that a cone over an irreducible curve is irreducible and
that cones with different apexes are distinct subvarieties. This immediately
implies that $S_{1}$ and $S_{2}$ have no irreducible components in common and
$S_{1}\cap S_{2}$ is a lower dimensional subvariety of $\mathbb{P}^{n}$ and
hence is of the form $C\cup\Gamma$, where $C$ is a curve and $\Gamma$ is zero
dimensional.
We note that $\deg(S_{i})=\deg(C_{i})$. We now apply the generalised Bezout’s
theorem of [4] (bottom of page 223), which states that the total degree of
$S_{1}\cap S_{2}$ (i.e. $\deg C+|\Gamma|$) is less than or equal to
$\deg(S_{1})\deg(S_{2})\leq e_{1}e_{2}$. ∎
###### Proposition 2.4.
Let $d\gg e>0$. Let $Z\subseteq\mathbb{P}^{n}$ be a Cayley-Bacharach set for
$\mathcal{O}(d)$. Let $C\subseteq\mathbb{P}^{n}$ be a reduced curve. Let
$\Gamma\subseteq\mathbb{P}^{n}$ be a finite set of points disjoint from $C$.
Suppose $\deg(C)+|\Gamma|\leq e^{2}$. If $Z\subseteq C\cup\Gamma$, then
$Z\subseteq C$.
###### Proof.
For a fixed value of $\deg(C)+|\Gamma|$, there exists $d_{0}$ such that for
$d>d_{0}$, the map $H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to
H^{0}(C\cup\Gamma,\mathcal{O}(d))$ is surjective. Suppose $P\in Z\cap\Gamma$,
and that $P\not\in C$. Then there exists a section $f\in
H^{0}(C\cup\Gamma,\mathcal{O}(d))$ that is nonzero at $P$ and vanishes on
$C\cup\Gamma\setminus\\{P\\}$. Since the map
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to H^{0}(C\cup\Gamma,\mathcal{O}(d))$ is
surjective, we can find a polynomial $p\in
H^{0}(\mathbb{P}^{n},\mathcal{O}(d))$ that restricts to $f$. Such a $p$
vanishes on $Z\setminus\\{P\\}$ and doesn’t vanish on $P$. This contradicts
the Cayley-Bacharach property of $Z$. Hence no such $p$ exists, i.e. $Z$ is
contained in $C$. ∎
###### Proposition 2.5.
Let $d\gg e>0$. Let $C\subseteq\mathbb{P}^{n}$ be a reduced curve of degree
$e$. Let $Z\subseteq C$ be a Cayley-Bacharach set for $\mathcal{O}(d)$. We
assume that $Z$ is not contained in any curve of degree $<e$. Then there is a
positive real valued increasing function $f$ not depending on $Z$ or $d$ such
that $|Z|\geq ed-f(e)$.
###### Proof.
The proof of this proposition is essentially Riemann-Roch for curves. However,
we first need to do some work to convert the given curve $C$ into a smooth
curve to which we can apply the Riemann Roch theorem.
We first note that a Cayley-Bacharach set must contain at least $d$ points
(see for instance [11]). Furthermore there is a function $\phi_{1}$, such that
any curve $C_{0}$ of degree $e$ satisfies
$|\textrm{Sing}(C_{0})|\leq\phi_{1}(e)$ . Hence for $d\gg e$, $Z$ is not
contained in $\textrm{Sing}(C)$. Let $C_{1},\dots,C_{k}$ be the irreducible
components of $C$. We note that $k\leq e$.
We can similarly conclude that for any irreducible component $C_{i}$ of $C$,
the intersection $Z\cap(C_{i}\setminus\textrm{Sing}(C))$ is nonempty. If this
were not the case, $Z$ would be a Cayley-Bacharach set contained in the union
of a curve $C^{\prime}=\cup_{j\neq i}C_{j}$ and a finite set,
$\textrm{Sing}C_{i}$. But for $d\gg e$,we may apply the argument of
Proposition 2.4 to conclude $Z\subseteq C^{\prime}$ which contradicts the fact
that $Z$ is not contained in a curve of degree $<e$.
Let $\pi:\tilde{C}\to C$ be a resolution of singularities. Let
$\tilde{Z}=\pi^{-1}(Z)\cup\pi^{-1}(\textrm{Sing}C)$. Note that there is some
function $\phi_{2}$ such that $|\tilde{Z}|-|Z|<\phi_{2}(e)$. Let $P\in
Z\setminus\textrm{Sing}(C)$. Let $\tilde{P}=\pi^{-1}(P)$. Suppose there exists
$\tilde{f}\in H^{0}(\tilde{C},\mathcal{O}(d))$ vanishing at
$\tilde{Z}-\tilde{P}$, such that $\tilde{f}(\tilde{P})\neq 0$. Then
$\tilde{f}$ is the pullback of a section $f\in H^{0}(C,\mathcal{O}(d))$
vanishing at $Z\setminus\\{P\\}$ and not vanishing at $P$. For $d\gg e$,
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))$ surjects onto $H^{0}(C,\mathcal{O}(d))$
and hence $f$ is the restriction of some element of
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))$. This contradicts the Cayley-Bacharach
property of $Z$. Hence no $f$ and consequently no such $\tilde{f}$ can exist.
We may now apply the Riemann Roch theorem to the curve $\tilde{C}$.
Let $\tilde{C}_{1},\dots,\tilde{C}_{k}$ be the corresponding components of
$\tilde{C}$.
Let $\tilde{P}\in\pi^{-1}((Z\cap C_{i})\setminus\textrm{Sing}(C))$. Let
$\pi(\tilde{P})=P$. We note that any $\tilde{f}\in
H^{0}(\tilde{C}_{i},\mathcal{O}(d))$ that vanishes on
$\pi^{-1}(\textrm{Sing}(C))\cup\pi^{-1}(Z)\setminus\\{P\\}$ is automatically
the pullback of a section $f\in H^{0}(C,\mathcal{O}(d))$ vanishing at
$Z\setminus\\{P\\}$ and hence $\tilde{f}$ is forced to vanish at $P$. Let
$Y_{i}=\pi^{-1}(\textrm{Sing}(C))\cup\pi^{-1}(Z)$. We have established that
the restriction map $H^{0}(\tilde{C}_{i},\mathcal{O}(d))\to
H^{0}(Y_{i},\mathcal{O}(d))$ is not surjective. Let $|Y_{i}|$ denote the line
bundle on $\tilde{C}_{i}$ associated with the divisor $Y_{i}$. Thus,
$H^{1}(\tilde{C}_{i},\mathcal{O}(d)\otimes|Y_{i}|^{\vee})\neq 0.$
By Riemann-Roch this implies that
$d\deg C_{i}-|\pi^{-1}(Z\cap C_{i})|-|\pi^{-1}(\textrm{Sing}(C)\cap
C_{i})|\leq\textrm{genus(C)}.$ (2)
If we sum (2) over all $i$ we obtain $d\deg(C)-\sum_{i}(|\pi^{-1}(Z\cap
C_{i})|-|\pi^{-1}(\textrm{Sing}(C)\cap
C_{i})|)\leq\sum_{i}\textrm{genus}(C_{i}).$ Hence
$\sum_{i}(|\pi^{-1}(Z\cap C_{i})|\geq
d\deg(C)-\sum_{i}\textrm{genus}(C_{i})-\sum|\pi^{-1}(\textrm{Sing}(C)\cap
C_{i})|).$ (3)
We note that $|\pi^{-1}(Z\cap C_{i})|-|Z\cap C_{i}|\leq\phi_{2}(e)$, with
$\phi_{2}$ as defined above. Hence, $|Z\cap C_{i}|\geq d(\deg
C_{i})-\phi_{2}(e)-\phi_{1}(e)$. Furthermore $\sum_{i}|Z\cap C_{i}|-|Z|\leq
k\textrm{Sing}(C)\leq e\phi_{1}(e).$
Combining this with 3 we have
$|Z|\geq
d\deg(C)-\sum_{i}(\textrm{genus}(C_{i}))-\sum_{i}|\pi^{-1}(\textrm{Sing}(C)\cap
C_{i})|)-e\phi_{2}(e)$ $\geq
d\deg(C)-\sum_{i}(\textrm{genus}(C_{i}))-2e\phi_{2}(e).$
Now we note that the genus of each individual $C_{i}$ is bounded in terms of
$e$ and $n$ by the Castelnuovo bound. Let us call this bound $\phi_{3}(e)$.
This gives us
$|Z|\geq d\deg(C)-e(\phi_{3}(e)+2\phi_{2}(e)).$
We let $f(e)=\max_{j\leq e}(1,j(\phi_{3}(j)+2\phi_{2}(j)))$. This establishes
the result. ∎
We now prove Theorem 1.13.
###### Proof of Theorem 1.13.
We will prove the theorem by induction on $n$. For $n=2$, the Theorem follows
from Proposition 2.1. Let us assume the Theorem for Cayley-Bacharach sets in
$\mathbb{P}^{n}$ for some $n\geq 2$. Let $Z\subseteq{\mathbb{P}^{n+1}}$ be a
Cayley Bacharach set for $\mathcal{O}(d)$, with $|Z|<ed-f(e)$. Let
$P_{1},P_{2}$ be two distinct points disjoint from $Z$ such that no line
passing through two of the points of $Z$ contains $P_{1}$ or $P_{2}$. Let $H$
be a hyperplane in $\mathbb{P}^{n+1}$ not containing $P_{1}$ or $P_{2}$. We
project $Z$ linearly from $P_{i}$ onto $H$ to obtain a set $Z_{i}$. The set
$Z_{i}$ is Cayley-Bacharach by Proposition 2.2. By the induction hypothesis
$Z_{i}$ lies on a curve $C_{i}$ of degree $e_{i}\leq e$. Therefore $Z$ lies on
the intersection of $S_{1}$ and $S_{2}$, where $S_{i}$ is the cone over
$C_{i}$ with apex $P_{i}$. By Proposition 2.3 $S_{1}\cap S_{2}=C\cup\Gamma$
where $\deg(C)+|\Gamma|\leq e_{1}e_{2}$. By Proposition 2.4, $Z\subseteq C$.
Let us replace $C$ by the curve of minimal degree containing $Z$. By
Proposition 2.5, $|Z|\geq\deg(C)d-f(\deg(C))$. But our assumption implies that
$|Z|\leq ed-f(e)$. Since $d\gg e$, this implies $\deg(C)\leq e$.
∎
We now move on to the proof of Theorem 1.15.
###### Proof of Theorem 1.15.
Let $Z$ be the given Cayley-Bacharach set. By Theorem 1.13, for $d\gg e$ and
$(e+1)d+1\leq ed-f(e)$ $Z$ is contained in a curve $C$ of degree at most
$e+1$. We remind the reader that an integral curve of degree $e$ is contained
in an $e$ dimensional linear subspace and an integral curve of degree $e$ that
is not contained in an $e-1$ dimensional subspace is a rational normal curve.
As a result the only curves of degree $e+1$ that are not contained in a union
of linear subspaces whose dimensions sum to $\leq e$ are unions of rational
normal curves. Hence for our purposes we may assume $Z$ is a Cayley- Bacharach
set for $\mathcal{O}(d)$ that is contained in a union of $k$ rational normal
curves $C_{i}$ with $\deg(C_{i})=r_{i}$ and $\sum r_{i}=e+1$, as the result
follows in all other cases.
. Suppose two rational normal curves $C_{i}$ and $C_{j}$ intersect at $k$
points. Then by the properties of rational normal curves one may conclude that
their union $C_{i}\cup C_{j}$ is contained in a subspace of dimension $\deg
C_{i}+\deg C_{j}+1-k$. So if $k>1$ for any pair $(i,j)$ we may replace
$P_{i},P_{j}$ by the span of $P_{i}$ and $P_{j}$ to obtain a new sequence of
linear subspaces whose dimensions sum to $\leq e$.
Hence we may assume $|C_{i}\cap C_{j}|\leq 1$ as the result follows in all
other cases.
Furthermore we claim that the intersection graph of the $C_{i}$ is a forest.
This is established as follows:
If
$|C_{i}\cap C_{j}|=|C_{j}\cap C_{k}|=|C_{i}\cap C_{k}|=1$
, then $\dim\textrm{span}P_{i}\cup P_{j}\cup P_{k}<\dim P_{i}+\dim P_{j}+\dim
P_{k}$ unless $|C_{i}\cap C_{j}\cap C_{k}|\neq 0$. A similar argument implies
that there are no cycles in the intersection graph, and hence it is a forest.
Let $d(C_{i})$ denote the degree of the vertex $C_{i}$ in the intersection
graph. We claim that $|Z\cap C_{i}|+d(C_{i})\geq r_{i}d+2$. For suppose that
$|Z\cap C_{i}|+d(C_{i})<r_{i}d+2$. Let $S$ be the set of points in $C_{i}$
that are also in some $C_{j}$ for $j\neq i$. Let $C_{i}^{\circ}=C_{i}\setminus
S$, i.e. the interior of $C_{i}$ in $C$. Let $p\in Z\cap C_{i}^{\circ}$. We
note that $|Z\cap C_{i}^{\circ}\setminus{p}|+|S|\leq|Z\cap
C_{i}|-1+d(C_{i})\leq r_{i}d$, by assumption. However, given any set of $\leq
r_{i}d$ points not containing $p$ in $C_{i}$, there is a section of $f\in
H^{0}(C_{i},\mathcal{O}(d))$ vanishing at those points and not at $p$ (Here we
use the fact that $C_{i}$ is isomorphic to $\mathbb{P}^{1}$). We may extend
$f$ from $C_{i}$ to $C$, by defining it to be zero outside $C_{i}$. Then $f$
vanishes on $Z\setminus\\{p\\}$ but not at $p$. This contradicts the Cayley-
Bacharach property of $Z$. Hence $|Z\cap C_{i}^{\circ}|+d(C_{i})\geq
r_{i}d+2$.
By summing this inequality over $i$, we obtain
$|Z|+\sum_{i}(d(C_{i}))\geq\sum_{i}|Z\cap
C_{i}^{\circ}|+\sum_{i}d(C_{i})\geq\sum_{i}r_{i}d+2k=(e+1)d+2k.$
By the above inequality, to establish that $|Z|\geq d(e+1)$ suffices to
establish that $2k-\sum_{i}d(C_{i})\geq 1$. We note that $\sum_{i}d(C_{i})$ is
twice the number of edges in the intersection graph. As a result,
$2k-\sum_{i}d(C_{i})$ is twice the Euler characteristic of the intersection
graph. Since the intersection graph is a forest, the Euler characteristic is
always $\geq 1$.
∎
## 3 Preparatory lemmas
The purpose of this section is to state and prove Lemma 3.1 which will be used
later on in the paper to prove our theorems on convergence of the motives of
discriminant complements. It is essentially a corollary of Theorem 1.13 .
###### Lemma 3.1.
Let $d\gg e>0$. Let $f$ be as in Theorem 1.13 Let $Z\subseteq\mathbb{P}^{n}$
be a reduced zero dimensional subscheme. Suppose $2|Z|\leq ed-f(e)$. Suppose
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))\neq 0$. Then there is a function $f$ such
that if $2|Z|\leq ed-f(e)$, then there is a subset $Z^{\prime}\subseteq Z$ and
a curve $C$ of degree $\leq e$ such that
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z^{\prime}}(d))=h^{1}(N_{\mathbb{P}^{n}}(C),I^{2}_{Z^{\prime}}(d))$.
Furthermore the curve of minimal degree satisfying the above conditions is
unique.
Before embarking on the proof of lemma, we will need a few propositions.
###### Proposition 3.2.
Assume that $\mathbb{F}$ is algebraically closed. Let
$Z\subseteq\mathbb{P}^{n}$ be a reduced zero dimensional subscheme. Suppose
that the restriction map $H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to
H^{0}(Z,\mathcal{O}(d))$ is not surjective. Then $Z$ has a subset $Z^{\prime}$
that is Cayley-Bacharach for $\mathcal{O}(d)$.
###### Proof.
Let $Z^{\prime}\subseteq Z$ be such that
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to H^{0}(Z^{\prime},\mathcal{O}(d))$
is not surjective and for any proper subset $Z^{\prime\prime}\subseteq
Z^{\prime}$, the restriction map
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to
H^{0}(Z^{\prime\prime},\mathcal{O}(d))$
is surjective. Clearly, such a $Z^{\prime}$ exists as the restriction map is
surjective for singleton sets. Let us now take any subset
$Z^{\prime\prime}\subseteq Z^{\prime}$ such that $|Z^{\prime}\setminus
Z^{\prime\prime}|=1$.
Since $H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to
H^{0}(Z^{\prime},\mathcal{O}(d))$ is not surjective,
$\dim Im(H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to
H^{0}(Z^{\prime},\mathcal{O}(d)))<\dim
H^{0}(Z^{\prime},\mathcal{O}(d)))=|Z^{\prime}|.$
However $Im(H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to
H^{0}(Z^{\prime},\mathcal{O}(d)))$ surjects onto
$Im(H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to
H^{0}(Z^{\prime\prime},\mathcal{O}(d)))=H^{0}(Z^{\prime\prime},\mathcal{O}(d))$
which is of dimension $|Z^{\prime\prime}|=|Z^{\prime}|-1$. However, since the
dimension of the source is at most $|Z^{\prime}|-1$, the restrcition map is
forced to be an isomorphism, which establishes that $Z^{\prime}$ is Cayley
Bacharach.
∎
###### Proposition 3.3.
Assume that $\mathbb{F}$ is algebraically closed. Let
$Z\subseteq\mathbb{P}^{n}$ be a reduced zero dimensional subscheme. Suppose
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))\neq 0$. Then $Z$ has a subset $Z^{\prime}$
that is Cayley-Bacharach for $\mathcal{O}(\lceil(d/2)\rceil)$.
###### Proof.
We observe that Theorem 1.1 of [6] of implies that if
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))\neq 0$, then
$h^{1}(\mathbb{P}^{n},I_{Z}(\lceil(d/2)\rceil)\neq 0$. Associate to the short
exact sequence
$0\to
I_{Z}(\lceil(d/2)\rceil)\to\mathcal{O}(\lceil(d/2)\rceil)\to\mathcal{O}_{Z}(\lceil(d/2)\rceil)\to
0$
, i we have a long exact sequence in cohomology, part of which is as follows:
$H^{0}(\mathbb{P}^{n},\mathcal{O}(\lceil(d/2)\rceil))\to
H^{0}(Z,\mathcal{O}(\lceil(d/2)\rceil))\to
H^{1}(\mathbb{P}^{n},I_{Z}(\lceil(d/2)\rceil))\to
H^{1}(\mathbb{P}^{n},\mathcal{O}(\lceil(d/2)\rceil)))$
But $H^{1}(\mathbb{P}^{n},\mathcal{O}(\lceil(d/2)\rceil)))\neq 0,$. As a
result the fact that $h^{1}(\mathbb{P}^{n},I_{Z}(\lceil(d/2)\rceil)\neq
0$implies that the map $H^{0}(\mathbb{P}^{n}\mathcal{O}(\lceil(d/2)\rceil))\to
H^{0}(Z\mathcal{O}(\lceil(d/2)\rceil))$ can not be surjective.
We then apply Proposition 3.2 to conclude that $Z$ has a subset $Z^{\prime}$
that is Cayley-Bacharach for $\mathcal{O}(\lceil(d/2)\rceil)$. ∎
###### Proposition 3.4.
Let $e,n>0$. Let $Z\subseteq\mathbb{P}^{n}$ be a reduced set of points. Let
$C$ be a curve of degree $e$. Let $d\gg e,n$. Suppose
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))\neq 0$. Let $Z^{\prime}=Z\cap C$. Suppose
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))>h^{1}(\mathbb{P}^{n},I^{2}_{Z^{\prime}}(d)).$
Let $Z^{\prime\prime}=Z\setminus Z^{\prime}$. Then
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d-2e))\to
H^{0}(N(Z^{\prime\prime}),\mathcal{O}(d-2e))$ is not surjective.
###### Proof.
Let $D\subseteq\mathbb{P}^{n}$ be a divisor containing $C$ not containing any
point of $Z^{\prime\prime}$. Let $|D|$ denote the degree of $D$. Suppose for
the sake of contradiction that $H^{0}(\mathbb{P}^{n},\mathcal{O}(d-2|D|)$
surjects onto $H^{0}(N(Z^{\prime\prime}),\mathcal{O}(d-2|D|)$. Let $f\in
H^{0}(\mathbb{P}^{n},|D|)$ be a section whose zero locus is $D$. We have a
subspace
$f^{2}\otimes H^{0}(\mathbb{P}^{n},\mathcal{O}(d-2|D|)\subseteq
H^{0}(\mathbb{P}^{n},\mathcal{O}(d))$
that surjects onto $H^{0}(Z^{\prime\prime},\mathcal{O}(d))$ and maps to $0$ in
$H^{0}(N(Z^{\prime}),\mathcal{O}(d))$. We note that
$H^{0}(N(Z),\mathcal{O}(d))\cong H^{0}(N(Z^{\prime}),\mathcal{O}(d))\oplus
H^{0}(N(Z^{\prime\prime}),\mathcal{O}(d)).$ Some elementary linear algebra
then implies that the codimension of the image of
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))$ in $H^{0}(N(Z),\mathcal{O}(d))$ is
equal to the codimension of the image of
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))$ in
$H^{0}(N(Z^{\prime}),\mathcal{O}(d))$, since the map surjects onto the other
factor. This contradicts the assumption that
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))>h^{1}(\mathbb{P}^{n},I^{2}_{Z^{\prime}}(d)).$
∎
We now proceed to prove Lemma 3.1
###### Proof of Lemma 3.1.
Let us first prove the lemma under the assumption that our base field
$\mathbb{F}$ is algebraically closed. We begin by applying Proposition 3.2 to
conclude that $Z$ contains a subset $Z_{0}$ that is Cayley Bacharach for
$\mathcal{O}(\lceil d/2\rceil)$. We then apply Theorem 1.13 to conclude that
there is a curve $C$ that contains $Z_{0}$. Let $Z^{\prime}=Z\cap C$. If
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z^{\prime}}(d))$
we are done and $C$ is our required curve. If not, we proceed as follows:
1. 1.
By Proposition 3.4 $Z^{\prime\prime}=Z\setminus Z^{\prime}$ must satisfy
$h^{1}(\mathbb{P}^{n},I^{2}_{Z^{\prime\prime}}(d-2e)>0$. Hence it must contain
a Cayley Bacharach subset $Z^{\prime\prime\prime}$ for $\mathcal{O}(\lceil
d/2\rceil-e)$, which is contained on a curve $C^{\prime}$.
2. 2.
We replace our original curve $C$ by $C\cup C^{\prime}$ and replace
$Z^{\prime}$ by $Z\cap(C\cup C^{\prime})$.
3. 3.
We repeat steps 1 and 2 until
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z^{\prime}}(d))$.
Since $|Z^{\prime}|$ increases every time we perform step $2$ and
$|Z^{\prime}|<|Z|$ this process terminates.
We then have our required curve $C$ and subset $Z^{\prime}$.
If our base field $\mathbb{F}$ is not algebraically closed, we may pass to the
algebraic closure $\bar{\mathbb{F}}$ and then find a curve $C$ defined over
$\bar{\mathbb{F}}$ containing $Z$ and of degree $\leq e$. We will prove that
$C$ is defined over $\mathbb{F}$ which will complete the proof of the lemma.
Let $g\in Gal(\bar{\mathbb{F}},\mathbb{F})$. We claim that $g(C)=C$. If
$g(C)\neq C$, there is some irreducible component $C_{i}\subseteq C$ such that
$g(C_{i})\neq C_{i}$. But $Z\cap C_{i}\subseteq|g(C_{i})\cap
C_{i}|\leq\deg(C_{i})^{2}\leq e^{2}.$ However, $Z\cap C_{i}\geq e^{2}$ (see
proof of Proposition 2.4). Hence for $d\gg e$, $g(C_{i})=C_{i}$ and hence $C$
is defined over $\mathbb{F}$. This completes the proof of existence.
Let us now turn to proving uniqueness. It suffices to prove uniqueness over an
algebraically closed field. Suppose that $C_{1},C_{2}$ are curves of the same
degree, with $\deg(C_{i})\leq e\ll d$, such that
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z_{i}}(d))=l$
where $Z_{i}=Z\cap C_{i}$ and no curve of strictly smaller degree satisfies
the conditions of the lemma. Let $W=Z_{1}\cap Z_{2}$. Suppose
$h^{1}(\mathbb{P}^{n},I^{2}_{W}(d))<h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=l$. We
will show that this implies that
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))>h^{1}(\mathbb{P}^{n},I^{2}_{Z_{i}}(d))$
which contradicts our assumption. Let $Z_{i}^{\prime}=Z_{i}\setminus W$. Let
$U\subseteq H^{0}(Z_{1}^{\prime}\cup Z_{2}^{\prime}\cup W,\mathcal{O}(d))$ be
the image of $H^{0}(\mathbb{P}^{n},\mathcal{O}(d))$ which is of codimension
$l$. Let $\pi_{i}:H^{0}(Z,\mathcal{O}(d))\to H^{0}(Z_{i},\mathcal{O}(d))$ be
the restriction map. Let $\pi:H^{0}(Z,\mathcal{O}(d))\to
H^{0}(W,\mathcal{O}(d))$ be the restriction map. By our assumption,
$\pi_{i}(U)$ is of codimension $l$ in $H^{0}(Z_{i},\mathcal{O}(d))$. Therefore
$U\subseteq\pi_{i}^{-1}(\pi_{i})(U)$ and both are of codimension $l$ in
$H^{0}(Z,\mathcal{O}(d))$. Hence we have
$U=\pi_{1}^{-1}(\pi_{1}(U))=\pi_{2}^{-1}(\pi_{2}(U)).$
The only way the above equation can hold is if $U=\pi^{-1}(\pi(U))$ and as a
result
$\pi(U)$ is also of codimension $l$, which contradicts
$h^{1}(\mathbb{P}^{n},I^{2}_{Z_{0}}(d))<h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=l$.
Hence
$h^{1}(\mathbb{P}^{n},I^{2}_{Z_{0}}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=l>0$.
However in this case we note that $Z_{0}\subseteq C_{1}\cap C_{2}$. Let
$C_{1}\cap C_{2}=C\cup\Gamma$ where $C$ is a reduced curve and $\Gamma$ is a
finite set of points. Either $h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap
C}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$, in which case $C$ is a curve of
smaller degree than $C_{1}$ or $C_{2}$ satisfying the conditions of the lemma.
Otherwise $h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap
C}(d))<h^{1}(\mathbb{P}^{n},I^{2}_{Z_{0}}(d))$ However, this implies that
$Z_{0}\setminus(Z\cap C)\subseteq\Gamma$ contains a Cayley Bacharach set for
$\mathcal{O}(d-2\deg C)$ by the combination of Proposition 3.4 and Proposition
3.3. But $|\Gamma|\leq e^{2}$ and a Cayley Bacharach set for
$\mathcal{O}(d-2\deg C)$ must contain at least $d-2e$ points. Since $d\gg e$,
we are done.
∎
## 4 Proof of Theorem 1.6
In this section we prove Theorem 1.6.
We will begin by reviewing some material in section 2 of [16].
###### Definition 4.1.
Let $S$ be a nonempty finite set. An ordered partition $\lambda$ of the set
$S$ is a tuple $(A_{1},\dots,A_{m})$, where the $A_{i}$ are nonempty disjoint
subsets of $S$ such that $A_{1}\cup\dots\cup A_{m}=S$. The empty set
$\varnothing$ has a single ordered partition called the empty partition and
also denoted $\varnothing$. We will denote the ordered partition of
$\\{1,\dots,n\\}$ with only one set by $[n]$.
We can associate two numbers to an ordered partition $\lambda$. If
$\lambda=(A_{1},A_{2},\dots.A_{m})$ is an ordered partition of a nonempty
finite set $S$, we define $|\lambda|=|S|$ and $||\lambda||=m$. For the empty
partition $\varnothing$, we define $||\varnothing||=|\varnothing|=0$.
In what follows we will have to consider sums over ordered partitions of a
certain length. By $\sum_{|\lambda|=k}f(\lambda)$ we mean that we sum the
values of $f(\lambda)$ as $\lambda$ ranges over all ordered partitions of
$\\{1,\dots,k\\}$.
###### Definition 4.2.
. Let $X$ be a quasiprojective variety over $\mathbb{F}$. Let $S$ be a
nonempty finite set. Let $\lambda=(A_{1},A_{2},\dots,A_{m})$ be an ordered
partition of $S$. We define
$\textrm{Sym}^{\lambda}(X):=\prod_{i=1}^{m}\textrm{Sym}^{|A_{i}|}(X)$.
There is a quotient map $\pi:X^{|S|}\to\textrm{Sym}^{\lambda}(X)$ We define
the $\lambda$ configuration space of $X$,
$w_{\lambda}(X)\subseteq\textrm{Sym}^{\lambda}(X)$ to be
$\pi(X^{|S|}\setminus\Delta)$, where $\Delta$ is the big diagonal.
We define
$\textrm{Sym}^{\varnothing}(X)=w_{\varnothing}(X)=\textrm{Spec}\textrm{
}\mathbb{F}$.
If $\lambda$ is an ordered partition and $X$ is a quasiprojective variety, any
$Z\in w_{\lambda}(X)$ defines a reduced zero dimensional subscheme of $X$
which we will denote $\textrm{supp}(Z)$. For any subvariety $Y\subseteq X$, we
say $Z\subseteq Y$ if $\textrm{supp}(Z)\subseteq Y$.
There is a forgetful covering map $w_{\lambda}(X)\to w_{|\lambda|}(X)$ defined
by $Z\mapsto\textrm{supp}(Z).$
###### Definition 4.3.
Let $X$ be a quasi projective variety over $\mathbb{F}$. Let $\mathcal{L}$ be
a line bundle on $X$. We define
$W_{\geq\lambda}(X)=\\{(f,Z)\in H^{0}(X,\mathcal{L})\times
w_{\lambda}(X)|Z\subseteq\textrm{Sing}(f)\\}.$
For $N\geq 0$, we define
$W_{\lambda,\geq N}(X):=\\{(f,Z)\in W_{\geq\lambda}(X)||\textrm{Sing}(f)|\geq
N\\}.$
We define
$W_{\lambda}(X):=\\{(f,Z)\in
W_{\geq\lambda}(X)||\lambda|=|\textrm{Sing}(f)|\\}.$
Our notation does not involve the line bundle $\mathcal{L}$. We do not believe
this will cause confusion as the line bundle $\mathcal{L}$ will be fixed
throughout. We note that $W_{\varnothing}(X)=U(X,\mathcal{L})$ the
discriminant complement.
###### Definition 4.4.
Let $S_{1}\subseteq S_{2}$ be two nonempty finite sets. Let
$\lambda_{i}=(A_{1}^{i},A_{2}^{i},\dots,A^{i}_{m_{i}})$ be ordered partitions
of $S_{i}$. We say $\lambda_{1}$ is a subpartition of $\lambda_{2}$, denoted
$\lambda_{1}\subseteq\lambda_{2}$, if there is an increasing function
$f:\\{1,\dots,m_{1}\\}\to\\{1,\dots,m_{2}\\}$ such that $A_{j}^{1}\subseteq
A_{f(j)}^{2}$ for all $j\in\\{1,\dots,m_{1}\\}$. If this is the case we may
form an ordered partition of $S_{3}=S_{1}\setminus S_{2}$ as follows. We
define $B_{i}=A_{i}^{2}-A_{j}^{1}$ if $i=f(j)$ for $j\in\\{1,2\dots,m_{1}\\}$,
otherwise we define $B_{i}=A_{i}$. We define $\lambda_{3}$ to be the partition
corresponding to the tuple $(B_{1},\dots B_{m_{2}})$ after deleting empty
sets.
In what follows we will need to sum over all subpartitions of a given
partition $\lambda$. By $\sum_{\alpha\subseteq\lambda}f(\alpha)$ we mean the
sum of $f(\alpha)$ where $\alpha$ ranges over all subpartitions of $\lambda$.
###### Definition 4.5.
We assume $X$ is a quasi projective variety and $\mathcal{L}$ is a very ample
line bundle on it. We define the _forced positive dimensional singularities_
subset of $w_{\lambda}(X)$ denoted $w_{\lambda}^{p}(X)$ by
$w_{\lambda}^{p}(X)=\\{Z\in w_{\lambda}(X)|\textrm{if }f\in
H^{0}(X,\mathcal{L}),Z\subseteq\textrm{Sing}(f),\textrm{ then
}\dim\textrm{Sing}(f)\geq 1\\}.$
Let $w_{\lambda}^{n}(X)=w_{\lambda}(X)\setminus w_{\lambda}^{p}(X).$
Let $W_{\geq\lambda}^{p}(X)=\\{(f,Z)\in W_{\geq\lambda}(X)|Z\in
w_{\lambda}^{p}(X)\\}.$ Let
$W_{\geq\lambda}^{n}(X)=W_{\geq\lambda}(X)\setminus W_{\geq\lambda}^{p}(X).$
Let $W_{\lambda,\geq N}^{p}(X)=\\{(f,Z)\in W_{\lambda,\geq N}(X)|Z\in
w_{\lambda}^{p}(X)\\}.$ Let $W_{\lambda,\geq N}^{n}(X)=W_{\lambda,\geq
N}(X)\setminus W_{\lambda,\geq N}^{p}(X).$
###### Proposition 4.6.
Let $X,\lambda$ be as above. The set $w_{\lambda}^{p}(X)$ is a constructible
subset of $w_{\lambda}(X).$
###### Proof.
Let $j$ be a positive integer. Let
$w_{\lambda,j}(X)=\\{(Z,f_{1},\dots,f_{j})|Z\in w_{\lambda}(X),f_{i}\in
H^{0}(X,\mathcal{L}),Z\subseteq\textrm{Sing}(f_{i})\\}.$
There is a natural forgetful map $w_{\lambda,j}(X)\to w_{\lambda}(X)$. Let
$w_{\lambda,j}^{p}(X)=\\{(Z,f_{1},\dots f_{j})|f_{i}\textrm{form a basis for
}H^{0}(X,I^{2}_{Z}\otimes\mathcal{L})\\}.$
Clearly $w_{\lambda}^{p}(X)$ is the union of the images of
$w_{\lambda,j}^{p}(X)$. Hence it suffices to establish that the
$w_{\lambda,j}^{p}(X)$ are constructible.
Let $E_{\lambda,j}(X)=\\{(P,Z,f_{1},\dots,f_{j})\in X\times
w_{\lambda^{j}}(X)|P\in\cap\textrm{Sing}(f_{i})\\}.$ There is an obvious
forgetful map $\pi:E_{\lambda_{j}}(X)\to w_{\lambda,j}(X)$. Consider a
flattening stratification of $S_{1},\dots S_{k}$ $w_{\lambda,j}$ with respect
to $\pi.$ We note that $(Z,f_{1},\dots,f_{j})\in w_{\lambda,j}^{p}(X)$ iff the
$f_{i}$ form a basis of $H^{0}(X,I^{2}_{Z}\otimes\mathcal{L})$ and if
$\pi^{-1}((Z,f_{1},\dots,f_{j}))$ is positive dimensional. However the
dimension of $\pi^{-1}((Z,f_{1},\dots,f_{j}))$ is constant on each stratum.
Hence $w_{\lambda,j}^{p}(X)$ is just the union of a collection of strata,
intersected with the set of $(Z,f_{1},\dots,f_{j})$ where the $f_{i}$ form a
basis of $H^{0}(X,I^{2}_{Z}\otimes\mathcal{L})$. However the latter is clearly
a constructible set as is each stratum. Hence $w_{\lambda,j}^{p}(X)$ is the
union of an intersection of constructible sets and is hence constructible. ∎
###### Remark 4.
The p and n in the notation $w_{\lambda}^{p},w_{\lambda^{n}}$ is intended to
mean positive singularities and not positive dimensional singularities. These
variants of the $w_{\lambda}$ are largely used in the final section of this
paper and are necessary because we will often want to disregard elements of
$w_{\lambda}^{p}$ for various technical reasons. The necessity of using the
$w_{\lambda}^{n}$ is almost exclusive to Proposition 6.16 and the reader is
advised to not pay too much attention to it at a first reading.
###### Proposition 4.7.
Let $N$ be a positive integer. The following equations are true, where both
sides are treated as part of $\mathcal{M}_{\mathbb{L}}$
$W_{\varnothing}=\sum_{|\lambda|\leq
N}(-1)^{||\lambda||}W_{\geq\lambda}-\sum_{|\lambda|\leq
N}(-1)^{||\lambda||}W_{\lambda,\geq N+1}.$
$W_{\varnothing}=\sum_{|\lambda|\leq
N}(-1)^{||\lambda||}W^{n}_{\geq\lambda}-\sum_{|\lambda|\leq
N}(-1)^{||\lambda||}W^{n}_{\lambda,\geq N+1}.$
###### Proof.
The first equation follows from the proof of Proposition 3.7 in [16]. The
second follows immediately from the first and the fact that
$W_{\geq\lambda}-W_{\lambda,\geq
N-|\lambda|+1}=W^{n}_{\geq\lambda}-W^{n}_{\lambda,\geq N+1}.$
∎
###### Proposition 4.8.
Let $Z$ be a constructible subset of $w_{[n]}(X)$. Let
$\pi_{\lambda}:w_{\lambda}(X)\to w_{[|\lambda|]}(X)$ be the covering map. Let
$\phi:\mathcal{M}\to\mathcal{M}_{Hdg}$ denote the specialisation homomorphism.
Then
$\phi(\sum_{|\lambda|=n}(-1)^{||\lambda||}(\pi_{\lambda}^{-1}(Z)))=H^{*}_{c}(Z,\pm\mathbb{Q})$
, where $\pm\mathbb{Q}$ is the restriction of the alternating sheaf on
$w_{[n]}(X)$.
Also, for $k>3$,
$H^{*}_{c}(w_{k}(\mathbb{P}^{1}),\pm\mathbb{Q})=H^{*}_{c}(w_{k}(\mathbb{A}^{1}),\pm\mathbb{Q})=0.$
###### Proof.
Let $S_{\lambda}$ be the local system $(\pi_{\lambda})_{*}\mathbb{Q}$. Then by
[3] $\sum_{|\lambda|=n}(-1)^{||\lambda||}S_{\lambda}=\pm\mathbb{Q}$ as a
virtual representation. Thus,
$\sum_{|\lambda|=n}(-1)^{||\lambda||}H^{*}_{c}(Z,S_{\lambda})=H^{*}_{c}(Z,\pm\mathbb{Q})$.
However
$H^{*}_{c}(Z,S_{\lambda})=H^{*}_{c}(\pi_{\lambda}^{-1}(Z),\mathbb{Q})$. This
completes the proof. The equality
$H^{*}_{c}(w_{k}(\mathbb{P}^{1}),\pm\mathbb{Q})=H^{*}(w_{k}(\mathbb{A}^{1}),\pm\mathbb{Q})=0,$
is established in Lemma 2 of [17]. ∎
We define
$w_{\lambda}^{l}(X)=\\{Z\in
w_{\lambda}(X)|h^{0}(X,I^{2}_{Z}(d))+h^{0}(N(Z),\mathcal{O}(d))=l+h^{0}(X,\mathcal{O}(d))\\}.$
We note that the $w_{\lambda}^{l}(X)$ are constructible, this follows
immediately from the fact that $\\{Z\in
w_{\lambda}(X)|h^{0}(X,I^{2}_{Z}(d))+h^{0}(N(Z),\mathcal{O}(d))\geq
l+h^{0}(X,\mathcal{O}(d))\\}$ is closed, which is a consequence of
semicontinuity of cohomology.
###### Proposition 4.9.
Let $X$ be a smooth projective variety of dimension $n$. Let $\lambda$ be an
ordered partition. Then,
$[W_{\geq\lambda}(X)]=H^{0}(X,\mathcal{L})\sum_{l=0}^{\infty}[w_{\lambda}^{l}(X)]\mathbb{L}^{l-|\lambda|(n+1)}.$
###### Proof.
Let $\pi:W_{\lambda}(X)\to w_{\lambda}(X)$ be the projection map. We note that
$w_{\lambda}(X)$ is the disjoint union of $w_{\lambda}^{l}(X)$. Therefore
$W_{\lambda}(X)$ is the disjoint union of $\pi^{-1}(w_{\lambda}^{l}(X)),$ and
hence $[W_{\lambda}(X)]=\sum_{l=0}^{\infty}[\pi^{-1}(w_{\lambda}^{l}(X))].$
The map $\pi:\pi^{-1}(w_{\lambda}^{l}(X))\to w_{\lambda}^{l}(X)$ is a vector
bundle of dimension $h^{0}(X,\mathcal{L})-|\lambda|(n+1)+l$ by Grauert’s
theorem (see for instance p. 288 Cor. 12.9 [9] ). Hence
$[\pi^{-1}(w_{\lambda}^{l}(X))]=H^{0}(X,\mathcal{L})[w_{\lambda}^{l}(X)]\mathbb{L}^{l-|\lambda|(n+1)}.$
∎
###### Proposition 4.10.
Let $X$ be a smooth projective variety of dimension $n$. Let $C\subseteq X$ be
a reduced curve of degree $e$. Let $Z\subseteq C$ be a reduced set of points.
Let $d\gg\deg C$. Then
$h^{1}(X,I^{2}_{Z}(d))\leq\max(2|Z|-ed,0)+(n-1)\max(|Z|-ed,0)-k.$
###### Proof.
This follows immediately from Corollary 1.1.25 in [10]. ∎
Let $\psi_{e,d}(l)$ denote the minimum $m$ such that
$l\geq\max(2m-ed,0)+(n-1)\max(m-ed,0)-k.$
###### Proposition 4.11.
Let $X$ be a smooth projective variety of dimension $n\geq 2$. Let $l\geq 1$.
Let $d\gg e>0$. Let $k$ be as in Proposition 4.10.
Let $\lambda$ be an ordered partition such that $e\geq|\lambda|/d$. Then
$\dim w_{\lambda}^{l}(X)\leq\max_{e>0}(|\lambda|(n)-(n-1)\psi_{e,d}(l))+K$
where $K=\max_{r\leq e}\dim Chow_{r}(X)$ where $Chow_{r}(X)$ is the Chow
variety of reduced curves of degree $r$ in $X$. Let $r_{0}$ denote the minimal
degree of any curve in $X$.
Let $(nr_{0}+1)d>N>r_{0}nd$ Then, there exists $C$ such that
$\dim W_{\lambda,\geq N}^{l}(X)\leq h^{0}(X,\mathcal{O}(d))-r_{0}nd+K.$
###### Proof.
By Lemma 3.1 any $Z\in w_{\lambda}^{l}(X)$ must be of the form $Z^{\prime}\cup
Z^{\prime\prime}$ where $Z^{\prime}=Z\cap C$ for some curve $C$ of degree
$e^{\prime}\leq e$. Furthermore
$l=h^{1}(X,I^{2}_{Z}(d))=h^{1}(X,I^{2}_{Z^{\prime}}(d))$. By Proposition 4.10,
this implies that $|Z^{\prime}|\geq\psi_{e,d}(l)$.
As a result,
$\dim
w_{\lambda}^{l}(X)\leq\max_{|\lambda|/d>e>0}(|\lambda|(n)-(n-1)\psi_{e,d}(l))+\max_{|\lambda|/d>e>0}\dim
Chow(e).$
∎
We now prove Theorem 1.6, The essential idea is as follows- we use the
techniques of [16] to argue that the difference between
$\frac{U(X,\mathcal{L})}{H^{0}(X,\mathcal{L})}-\zeta^{-1}(\dim X+1)$ is a
certain sum of contributions corresponding to configurations of points
contained in $w_{\lambda}^{l}(X)$ where $l\geq 1$. We then use Proposition
4.11 to bound this contribution.
###### Proof of Theorem 1.6.
By Proposition 4.7
$U(X,\mathcal{L})=W_{\varnothing}(X)=\sum(-1)^{||\lambda||}(W_{\geq\lambda}(X)-W_{\lambda,\geq
N+1}).$
By Proposition 4.11 the terms $W_{\lambda,\geq N+1})=0$ up to dimension
$-m(d)(\dim X)+C$. Also for $||\lambda||\leq N,l\geq 1$, $w_{\lambda}^{l}$
lies in $F$. Hence we may apply Proposition and conclude that
$W_{\geq\lambda}=w_{\lambda}\mathbb{L}^{-|\lambda|(n+1)}$ upto dimension
$-m(d)(\dim X)+C$.
This implies that
$W_{\varnothing}(X)=\sum_{\lambda}(-1)^{||\lambda||}w_{\lambda}(X)\mathbb{L}$,
up to dimension $-m(d)(\dim X)+C$.. But by Proposition 3.7 of [16] the right
hand side of this last equation equal $Z_{X}^{-1}(\dim X+1)$.
∎
## 5 Proof of Theorem 1.7
In this section we will prove Theorem 1.7. The proof of Theorem 1.7 is similar
to that of Theorem 1.6 in that we use Proposition 4.7 and provide bounds on
the dimensions of $w^{l}_{\lambda}(X)$. However our assumptions are different.
This section is strictly speaking independent of Sections 2 and 3 , and we do
not use any results on Cayley-Bacharach sets.
We define the projective hull of a subscheme $Y\subseteq\mathbb{P}^{n}$,
denoted $phull(Y)$ to be the intersection of all projective subspaces of
$\mathbb{P}^{n}$ containing $Y$.
###### Proposition 5.1.
Let $n,m\geq 0$. Let $Z\subseteq\mathbb{P}^{n}$ be a reduced zero dimensional
subscheme. SUppose $|Z|\geq m+2$. Suppose that $phull(Z)$ in $\mathbb{P}^{n}$
is $m$ dimensional. Let $d\geq 3$. Then the kernel of the restriction map
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to H^{0}(N(Z),\mathcal{O}(d))$ is of
codimension $\geq(m+1)(n+1)+n-1$.
###### Proof.
We claim that it suffices to prove the Proposition in the special case when
$Z=\\{P_{0},\dots,P_{m},P\\}$ where the projective hull of the $P_{i}$ is $m$
dimensional, and $P$ is in $phull(\\{P_{0},\dots,P_{m}\\}).$ To see this,
first note that if $Z^{\prime}\subseteq Z$ the kernel of
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to H^{0}(N(Z),\mathcal{O}(d))$ is
contained in the kernel of $H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to
H^{0}(N(Z^{\prime}),\mathcal{O}(d)).$ As a result, to establish the
Proposition for $Z$, it suffices to establish it for any subset
$Z^{\prime}\subseteq Z$. We then note that given any $Z$ whose projective hull
is $m$ dimensional, we can find a $Z_{ind}\subseteq Z$ such that
$|Z_{ind}|=m+1$ and $phull(Z_{ind})=phull(Z)$. We now consider any point $P\in
Z\setminus Z_{ind}$ and note that $Z^{\prime}=Z_{ind}\cup\\{P\\}$ is a subset
of $Z$ satisfying the conditions of the claim and that to prove the
Proposition for $Z$ it suffices to prove it for $Z^{\prime}$.
In light of the above paragraph, we now assume $Z=\\{P_{0},\dots,P_{m},P\\},$
where $phull(P_{0},\dots,P_{m})$ is $m$ dimensional and contains $P$. We may
further assume that $P_{i}=[0:0\dots:0:1:0\dots 0],$ here the $1$ is in the
$i+1^{st}$ position and $P=[x_{0}:\dots x_{m}:0\dots:0].$ This is because
there is an element $g\in PGL_{n+1}(\mathbb{F})$ satisfying
$g(P_{i})=[0:0\dots:0:1:0\dots 0].$
We now claim that it suffices to prove the proposition in the case where
$P=[1:x:0\dots,:0]$. to see this, note that we may assume without loss of
generality that if $P=[x_{0}:x_{1}:\dots,x_{m}:0\dots:0],$ $x_{0}\neq 0$ and
$x_{1}\neq 0$. For $t\in\mathbb{A}^{1}$, let $Z_{t}$ be the reduced zero
dimensional subscheme $\\{P_{1},\dots P_{m}.P_{t}\\}$, where
$P_{t}=[x_{0}:x_{1}:tx_{2},\dots tx_{m}:0\dots 0].$ The $Z_{t}$ form a family
of subschemes over $\mathbb{A}^{1},$ with the original subscheme $Z$ being
equal to $Z_{1}$. Furthermore, given $t\neq 0$, there is a $g\in
PGL_{n+1}(\mathbb{F})$ such that $g(Z_{t})=Z_{1}$. This has the consequence
that the dimension of the kernel of the restriction of
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to H^{0}(N(Z_{t}),\mathcal{O}(d))$ is
independent of $t$ as long as $t\neq 0$. The kernel of
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to H^{0}(N(Z_{t}),\mathcal{O}(d))$ is
isomorphic to $H^{0}(\mathbb{P}^{n},I^{2}_{Z_{t}}(d)).$ Semicontinuity now
implies that $h^{0}(\mathbb{P}^{n},I^{2}_{Z_{0}}(d))\geq
h^{0}(\mathbb{P}^{n},I^{2}_{Z_{1}}(d)).$ As a result it suffices to prove the
proposition for $Z_{0}$. However $Z_{0}$ is of the required form.
To complete the proof, we directly compute the bound on the codimension of the
kernel when $Z$ is of the form $\\{P_{0},\dots,P_{m},P\\}$ where
$P_{i}=[0:0\dots:0:1:0\dots 0],$ and $P=[1:x:0\dots:0]$. Let $f$ be a section
of $H^{0}(\mathbb{P}^{n},\mathcal{O}(d))$, we may think of $f$ as a homogenous
polynomial. Then requiring $f$ to vanish to second order at $P_{i}$ implies
that the coefficients of $X_{i}^{d}$ and $X_{j}X_{i}^{d-1}$ are zero. Let
$a_{k,r}$ be the coefficient to $X_{k}X_{1}^{r}X_{0}{d-1-r}$. Requiring that
the partial derivatives of $f$ vanish at $P$ is equivalent to requiring that
for $k\in\\{2,\dots n\\}$,
$\sum_{r=0}^{d}x^{r}a_{k,r}=0.$
We note that the above linear relations are linearly independent and hence the
kernel of $H^{0}(\mathbb{P}^{,}\mathcal{O}(d))\to H^{0}(N(Z),\mathcal{O}(d))$
is of codimension at least $(m+1)(n+1)+n-1$.
∎
The above proof gives us a little more information than the statement of
Proposition 5.1. We record this information in the following Proposition.
###### Proposition 5.2.
Let $m,n\geq 1$. Let $d\geq 3$ Let $P_{i}=[0:\dots
1::0\dots:0]\in\mathbb{P}^{n}$, where the $1$ is in the $i+1^{th}$ position.
Let $P\in phull\\{P_{0},\dots,P_{m}\\}$ Let $V_{i}^{j}$ be any basis of
$T_{P_{i}}(\mathbb{P}^{n})$. Let $V^{j}$ be a basis of $T_{P}\mathbb{P}^{n}$.
Then requiring a polynomial $f\in H^{0}(\mathbb{P}^{n},\mathcal{O}(d))$ to
satisfy
$f(P_{i})=0,\frac{\partial}{\partial
V_{i}^{j}}(f)(P_{i})=0,\frac{\partial}{\partial v^{j}}f(p)=0,$
imposes $(m+1)(n+1)+n-1$ linearly independent conditions on $f$.
###### Proof.
We omit the proof as this is already present in the proof of Proposition 5.1.
∎
###### Proposition 5.3.
Let $n,m\geq 0$. Let $Z\subseteq X\subseteq\mathbb{P}^{n}$ be a reduced set of
points, $m+1<|Z|<\dim X$. Let $d\geq 3$. Let $\dim phull(Z)=m$. Then the
kernel of the restriction map $H^{0}(X,\mathcal{O}(d))\to
H^{0}(N(Z),\mathcal{O}(d))$ is of codimension $\geq(m+1)(\dim X+1)+\dim X-1$.
###### Proof.
This proof is similar to that of Proposition 5.1 . It suffices to prove the
proposition in the case where $Z=\\{P_{0},\dots,P_{m},P\\}$ where
$P_{i}=[0:\dots:0:1:\dots:0]$ where there is a $1$ in the $i+1$th position.
This is by an argument identical to that in Proposition 5.1.
Let $V_{i}^{j}$ be a basis of $T_{P_{i}}X$. Let $v^{j}$ be a basis of
$T_{P}X$.It suffices to prove that requiring a polynomial $f$ to satisfy at
$f(P_{i})=0,$ for all $i$, $f(P_{i})=0$ and that
$\partial_{v_{i}^{j}}(f)(P_{i})$, $\partial_{v_{i}}(f)(P)$ impose linearly
independent conditions on $f$. However, this follows from Proposition 5.2,
since in that proof we show that a larger set of linear equations impose
linearly independent conditions on $f$. ∎
###### Proposition 5.4.
Let $n\geq 0$. Let $X$ be a smooth projective variety of dimension $n$. Let
$\lambda$ be an ordered partition, $|\lambda|\leq\sqrt{n}/3$. Let $l\geq 1$.
Then
$\dim w_{\lambda}^{l}(X)+l-(n+1)(|\lambda|)\leq-\sqrt{n}/{3}$
and
$\dim W_{\lambda,\geq N}^{l}(X)\leq h^{0}(X,\mathcal{O}(d))-\sqrt{n}/{3}.$
In addition we have $\dim W_{\lambda,\geq N}^{0}(X)\leq
h^{0}(X,\mathcal{O}(d))-\sqrt{n}/{3}.$
###### Proof.
Let $l\geq 1$. Let $Z\in w_{\lambda}^{l}(X)$. Then by definition,
$h^{0}(X,I^{2}_{Z}(d))=h^{0}(X,\mathcal{O}(d))-|Z|(\dim X+1)+l$. Let $\dim
phull(Z)=m$. Then by Proposition 5.3, $|Z|(\dim X+1)-l\geq(m+1)(\dim X+1)-l$.
But a simple dimension count shows that the space of $|\lambda|$ points in $X$
contained in an $m$ dimensional subspace is of dimension $\leq(m+1)\dim
X+(|\lambda|-m-1)m$.
This implies that $\dim w_{\lambda}^{l}(X)\leq\max_{m}(m+1)\dim
X+(|\lambda|-m-1)m.$
We will now bound the quantity $(m+1)\dim X+(|\lambda|-m-1)m+l-(\dim
X+1)|\lambda|$. By the above we have
$(m+1)\dim X+(|\lambda|-m-1)m+l-(\dim X+1)|\lambda|$
$\leq(\dim X+1)|\lambda|-l-\dim X+1+(|\lambda|-m-1)m+l-(\dim X+1)|\lambda|$
$=-\dim X+1+(|\lambda|-m-1)m\leq-\dim X+1+(|\lambda|)^{2}\leq-\dim
X+1+\frac{\dim X}{9}$
$\leq-\frac{8}{9}\dim X+1\leq-\frac{\sqrt{\dim X}}{3}.$
As a result
$\dim w_{\lambda}^{l}(X)+l-(n+1)(|\lambda|)\leq-\sqrt{n}/{3}$
. The inequality
$\dim W_{\lambda,\geq N+1}^{l}(X)\leq h^{0}(X,\mathcal{O}(d))-\sqrt{n}/{3}$
follows similarly to the above for $l\geq 1$ and is immediate if $l=0$.
This implies the result. ∎
We note here that the proof of the above Proposition gives us a better bound
on the dimension of $w_{\lambda}^{l}(X)$ than we claimed in the result.
However we are unable to utilise this improved bound. The sticking factor is
that we only have the bound $\dim W_{\lambda,\geq N}^{0}(X)\leq
h^{0}(X,\mathcal{O}(d))-N$, which we believe to be tight.
###### Proof of Theorem 1.7.
Let $N=\sqrt{n}/3$. By Proposition 4.7
$\frac{W_{\varnothing}(X)}{H^{0}(X,\mathcal{O}(d))}=\sum_{|\lambda|\leq
N}(-1)^{||\lambda||}\frac{W_{\geq\lambda}(X)}{H^{0}(X,\mathcal{O}(d))}-\frac{W_{\lambda,\geq
N+1}(X)}{H^{0}(X,\mathcal{O}(d)}.$
By Proposition 4.7 the right hand side equals
$\sum_{|\lambda|\leq N}w_{\lambda}^{0}(X)\mathbb{L}^{-|\lambda|(\dim
X+1)}+\sum_{|\lambda|\leq N}w_{\lambda}^{l}(X)(\mathbb{L}^{-|\lambda|)(\dim
X+1)+l}-\sum_{|\lambda|\leq N}\frac{W_{\lambda,\geq
N+1}(X)}{H^{0}(X,\mathcal{O}(d))}.$
By Proposition 4.7 the first term equals $Z_{X}^{-1}(\dim
X+1)-\sum_{|\lambda|\leq N}w_{\lambda}^{l}(X)\mathbb{L}^{-|\lambda|(\dim
X+1)}-\sum_{|\lambda|>N}(-1)^{||\lambda||}w_{|\lambda}(X)\mathbb{L}^{-|\lambda|(\dim
X+1)}=Z_{X}^{-1}(\dim X+1)$ up to dimension $-N$. By Proposition 5.4 the
latter two terms are of dimension $\leq-N$ which implies the result. ∎
## 6 Proof of Theorem 1.9
In this section we prove Theorem 1.9 . The proof of Theorem 1.9 is similar to
that of Theorems 1.6 and 1.7 , albeit a bit more involved.
In this section we will exclusively work with $X=\mathbb{P}^{n}$. As a result
we will drop the $\mathbb{P}^{n}$ from our notation, i.e. we will refer to
$w_{\lambda}(\mathbb{P}^{n})$ as $w_{\lambda}$,
$W_{\geq\lambda}(\mathbb{P}^{n})=W_{\geq\lambda}$ etc. However if the argument
of $w_{\lambda},$ is not $\mathbb{P}^{n}$ we will explicitly write down what
the space is.
The following basic proposition will be used repeatedly in this section.
###### Proposition 6.1.
Let $C$ be a smooth curve in $\mathbb{P}^{n}$. Then for $d\gg\deg C$,
$h^{0}(N(C),\mathcal{O}(d))=h^{0}(C,\mathcal{O}(d))+h^{0}(C,N_{C}^{*}(d)).$
###### Proof.
Let $I$ denote the ideal sheaf of $C$. Consider the following chain of
inclusions of sheaves on $\mathbb{P}^{n}$-
$0\subseteq I^{2}(d)\subseteq I(d)\subseteq\mathcal{O}(d).$
We will break the above exact sequence into short exact sequences. We have a
short exact sequence
$0\to I^{2}(d)\to I(d)\to I/I^{2}(d)\to 0.$
Under the assumption that $d\gg\deg(C)$, Serre vanishing implies that
$H^{1}(\mathbb{P}^{n},I^{2}(d))=0$. As a result,
$h^{0}(\mathbb{P}^{n},I(d))=h^{0}(\mathbb{P}^{n},I^{2}(d))+h^{0}(\mathbb{P}^{n},I/I^{2}(d)).$
We also have the short exact sequence $0\to
I(d)\to\mathcal{O}(d)\to\mathcal{O}/I(d).$ Since $d\gg\deg(C)$,
$h^{0}(\mathbb{P}^{n},\mathcal{O}(d))=h^{0}(\mathbb{P}^{n},I(d))+h^{0}(\mathbb{P}^{n},\mathcal{O}/I(d)).$
Finally we have the short exact sequence $0\to
I^{2}(d)\to\mathcal{O}(d)\to\mathcal{O}/I^{2}(d).$ Since $d\gg\deg(C)$,
$h^{0}(\mathbb{P}^{n},\mathcal{O}(d))=h^{0}(\mathbb{P}^{n},I^{2}(d))+h^{0}(\mathbb{P}^{n},\mathcal{O}/I^{2}(d)).$
As a result, we have
$h^{0}(\mathbb{P}^{n},\mathcal{O}(d)/I^{2}(d))=h^{0}(\mathbb{P}^{n},\mathcal{O}/I(d))+h^{0}(\mathbb{P}^{n},I/I^{2}(d)).$
But we now note that $\mathcal{O}/I^{2}$ is the push forward of the structure
sheaf of $N(C)$ and $\mathcal{O}/I$ is the pushforward of the structure sheaf
of $C$. As a result
$h^{0}(\mathbb{P}^{n},\mathcal{O}(d)/I^{2}(d))=h^{0}(N(C),\mathcal{O}(d))$
and
$h^{0}(\mathbb{P}^{n},\mathcal{O}(d)/I(d))=h^{0}(C,\mathcal{O}(d)).$
Finally, we note that $I/I^{2}$ is the pushforward of the conormal sheaf of
$C$ and as a result $h^{0}(\mathbb{P}^{n},I/I^{2}(d))=h^{0}(C,N^{*}(d))$. This
gives us the required result.
∎
###### Definition 6.2.
Let $\textrm{Chow}(r)$ be the Chow variety of degree $d$ curves in
$\mathbb{P}^{n}$. Let $M\subseteq\mathbb{P}^{n}$ be a constructible subset of
$Chow(r)$. Let $\lambda$ be an ordered partition. We define
$w_{\lambda}^{M,l}\subseteq w_{\lambda}^{l}$ as consisting of all $Z\in
w_{\lambda}$ such that there exists $C\in M$ satisfying
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap C}(d))$
and furthermore there is no curve $C^{\prime}$ of degree less than $r$ such
that
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap
C^{\prime}}(d)).$
###### Proposition 6.3.
Let $d,r,l\geq 0$. Let $\lambda$ be an ordered partition. Let $M$ be a
constructible subset of $Chow(r)$. Assume $d\gg r.$ The subset
$w_{\lambda}^{M,l}$ is constructible and hence defines an element in
$\mathcal{M}$.
###### Proof.
Let
$\tilde{w}_{\lambda}^{M,l}:=\\{(Z,C)\in w_{\lambda}^{M,l}\times
M|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap
C}(d))\\}.$
We note that since $d\gg r$, by Proposition 3.1 for any $Z\in
w_{\lambda}^{M,l}$, there exists a unique $(Z,C)\in\tilde{w}_{\lambda}^{M,l}$
mapping onto it under the projection, thus it suffices to establish that
$\tilde{w}_{\lambda}^{M,l}$ is constructible. Consider $X=w_{\lambda}\times
M\times\mathbb{P}^{n}$. The variety $X$ has two ideal sheaves on it
$\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ corresponding to the subvarieties
$E_{1}=\\{(Z,C,P)\in\tilde{w}_{\lambda}^{M,l}\times\mathbb{P}^{n}|P\in Z\\}$
and $E_{2}=\tilde{w}_{\lambda}^{M,l}\times\mathbb{P}^{n}|P\in Z\cap C\\}$. Let
$\pi:\tilde{w}_{\lambda}^{M,l}\times\mathbb{P}^{n}\to\tilde{w}_{\lambda}^{M,l}$
denote the projection. Now we may consider a stratification $S_{i}$ of
$\tilde{w}_{\lambda}^{M,l}$ such that the sheaves $\pi_{*}\mathcal{F}_{i}(d)$
are flat on each stratum. The semicontinuity theorem then implies that one may
further refine the stratification $S_{i}$ to a stratification $S_{i}^{\prime}$
on which $h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap C}(d))$ and
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ are constant. We then note that
$\tilde{w}_{\lambda}^{M,l}$ is precisely the subset of $X$ where
$h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap C}(d))$ and
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ are equal to $l$. Thus
$\tilde{w}_{\lambda}^{M,l}$ is a union of strata and is thus constructible. ∎
Let $\pi:W_{\geq\lambda}\to w_{\lambda}$ denote the natural projection. Let
$W_{\geq\lambda}^{M,l}(\mathbb{P}^{n})=\pi^{-1}(w_{\lambda}^{M,l})$. Let
$W_{\lambda}^{M,l}=W_{\geq\lambda}^{M,l}\cap W_{\lambda}$. We define
$\bar{w}_{\lambda}^{M,l}=\\{Z\in w_{\lambda}|\exists C\in
M,h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap C}(d))=l\\}.$
We note that
$w_{\lambda}^{M,l}\subseteq\bar{w}_{\lambda}^{M,l}.$
We start by decomposing the varieties $w_{\lambda}^{l}$ into pieces.
###### Proposition 6.4.
Let $l\geq 1$.
$w_{\lambda}^{l}=\sum_{r=1}^{\infty}w_{\lambda}^{\textrm{Chow}(r),l}.$
###### Proof.
We note that the sum is finite, for $r$ sufficiently large
$w_{\lambda}^{\textrm{Chow}(r),l}$ is empty.
It then suffices to prove that the collection
$w_{\lambda}^{\textrm{Chow}(r),l}$ partitions $w_{\lambda}^{l}$. This follows
from Proposition 3.1. ∎
###### Proposition 6.5.
Let $e\geq 1$. Let $\mathcal{A}$ be a finite collection of constructible
subsets defining a partition of $\textrm{Chow}(e).$ Then
$w_{\lambda}^{\textrm{Chow}(e),l}=\sum_{M\in\mathcal{A}}w_{\lambda}^{M,l}.$
###### Proof.
This follows from the fact that a partition of $\textrm{Chow}(e)$ induces a
partition of $w_{\lambda}^{\textrm{Chow}(e),l}$. ∎
We now prove a dimension bound on several of the $w_{\lambda}^{M,l}$.
###### Proposition 6.6.
Let $e\geq 4$. Let $M\subseteq\textrm{Chow}(e)$ be a constructible set. Let
$n\geq 2,l\geq 1$. Then there exists $\epsilon>0$ such that,
$\dim(W_{\geq\lambda}^{M,l}-W^{M,l}_{\geq\lambda,\geq N+1})\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d.$
Furthermore $\dim
w_{\lambda}^{M,l}\leq|\lambda|(n)-l-(\frac{3n}{2}+\epsilon)d$.
###### Proof.
Let $M\subseteq\textrm{Chow}(e)$. We will first prove that $\dim
w_{\lambda}^{M,l}\leq|\lambda|(n)-l-(\frac{3n}{2}+\epsilon)d$. We note that
$\dim w_{\lambda}^{M,l}\leq\dim\tilde{w}_{\lambda}^{M,l}$. By Proposition 4.10
any $(Z,C)\in w_{\lambda}^{M,l}$ must have $|Z\cap C|\geq\psi_{e,d}(l)$.
Thus
$\dim w_{\lambda}^{M,l}\leq\dim\tilde{w}_{\lambda}^{M,l}\leq\dim
M+(|\lambda|-\psi_{e,d}(l))(n)+\psi_{e,d}(l).$
An inspection of the function $\psi_{e,d}$ then implies that for $e\geq 4$ and
$d\gg 0$, $\dim w_{\lambda}^{M,l}\leq|\lambda|(n)-l-(\frac{3n}{2}+\epsilon)d.$
Since
$W_{\geq\lambda,l}^{M}=w_{\lambda}^{M,l}H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\mathbb{L}^{-|\lambda|(n+1)+l}$,
the result follows.
∎
###### Proposition 6.7.
Let $d_{0}\geq 1$. Let $M\subseteq Chow(d_{0})$ be a constructible set. Then
$w_{\lambda}^{M,l}\subseteq\bar{w}_{\lambda}^{M,l}$. Furthermore
$\bar{w}_{\lambda}^{M,l}\setminus
w_{\lambda}^{M,l}=\cup_{d>d_{0},l^{\prime}>l}w_{\lambda}^{M^{\prime}(d),l^{\prime}},$
where $M^{\prime}(d)$ is the subset of $Chow(d)$ consisting of reducible
curves that contain a curve in $M$.
###### Proof.
This follows from Lemma 3.1. ∎
###### Proposition 6.8.
Let $X$ and $Y$ be algebraic varieties over $\mathbb{F}$. Then
$w_{\lambda}(X\amalg
Y)=\sum_{\alpha\subseteq\lambda}w_{\alpha}(X)w_{\lambda\setminus\alpha}(Y).$
###### Proof.
This follows immediately from the fact that $w_{\lambda}(X\amalg Y)$ is
partitioned into subsets isomorphic to $w_{\alpha}(X)\times
w_{\lambda\setminus\alpha}(Y)$ as $\alpha$ ranges over all subpartitions of
$\lambda$. ∎
###### Proposition 6.9.
Let $X$ be an algebraic variety. Let $Y$ be a subvariety of $X$. Suppose
$[X],[Y]$ are polynomials in $\mathbb{L}$. Then for any $\lambda$,
$w_{\lambda}(X),w_{\lambda}(Y)$ and $w_{\lambda}(X\setminus Y)$ are
polynomials in $\mathbb{L}$.
###### Proof.
For $X=\mathbb{A}^{n}$,
$[\textrm{Sym}^{\lambda}(\mathbb{A}^{n})]=\mathbb{L}^{|\lambda|n}$. Let $X$
and $Y$ be such that $Y\subseteq X$ and $w_{\lambda}(X)$, $w_{\lambda}(Y)$ can
always be expressed as a polynomial in $\mathbb{L}$.
We claim that if $U=X\setminus Y$ then $w_{\lambda}(U)$ can also be expressed
as a polynomial in $\mathbb{L}$ for all $\lambda$. Suppose for the sake of
contradiction that $\lambda_{0}$ is an ordered partition such that
$w_{\lambda_{0}}(U)$ is not a polynomial in $\mathbb{L}$ and furthermore for
all $\lambda^{\prime}\subseteq\lambda_{0}$, $w_{\lambda^{\prime}}(U)$ is a
polynomial in $\mathbb{L}$. Then by Proposition 6.8,
$w_{\lambda_{0}}(U)=w_{\lambda_{0}}(X)-\sum_{\lambda^{\prime}\subsetneq\lambda_{0}}w_{\lambda^{\prime}}(U)w_{\lambda_{0}\setminus\lambda^{\prime}}(Y)$
which is a polynomial in $\mathbb{L}$.
Hence the class of varieties such that $w_{\lambda}(X)$ is always a polynomial
contains $\mathbb{A}^{n}$ and is closed under differences and unions. This
completes the proof of the proposition.
∎
In the next proposition we describe the possible reduced $0$\- dimensional
subschemes $Z$ contained in a curve $C$ of low degree ($\leq 3$) such that
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=l>0$.
###### Proposition 6.10.
1. 1.
Let $T\subseteq\mathbb{P}^{n}$ be a smooth rational normal curve of degree
$r\leq 3$. Let $Z\subseteq T$ be a reduced zero dimensional subscheme. Let
$d\gg r$. Then, $h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ depends only on $|Z|$.
Furthermore,
$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-\max(2|Z|-rd,0)|+(n-1)\max(|Z|-d,0)\leq
K$ for some constant $K$.
2. 2.
Let $d\gg e$. Let $C_{1},C_{2}$ be disjoint reduced curves of degree $\leq e$.
Let $Z=Z_{1}\amalg Z_{2}$, with $Z_{i}\subseteq C_{i}$. Then
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z_{1}}(d))+h^{1}(\mathbb{P}^{n},I^{2}_{Z_{2}}(d))$.
3. 3.
Let $L_{1},L_{2}$ be two coplanar lines. Let $Z=Z_{1}\amalg Z_{2}$ such that
$Z_{i}\subseteq L_{i}$ and $Z\cap L_{1}\cap L_{2}=\varnothing$. Then
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ only depends on $|Z_{1}|$ and $|Z_{2}|$
and furthermore,
$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-h^{1}(N(L_{1}),I^{2}_{Z_{1}}(d))-h^{1}(N(L_{2}),I^{2}_{Z_{2}}(d))|\leq
2(n+1)$.
4. 4.
Let $L_{1},L_{2}$ be two coplanar lines. Let $\\{p\\}=L_{1}\cap L_{2}$ Let
$Z=Z_{1}\amalg Z_{2}\amalg\\{p\\}$ such that $Z_{i}\subseteq L_{i}$ and
$Z_{i}\cap L_{1}\cap L_{2}=\varnothing$. Then Then
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ only depends on $|Z_{1}|$ and $|Z_{2}|$
and furthermore,
$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-h^{1}(N(L_{1}),I^{2}_{Z_{1}}(d))-h^{1}(N(L_{2}),I^{2}_{Z_{2}}(d))|\leq
4(n+1).$
5. 5.
Let $L,C$ be a line and a conic intersecting at one point, transversally.Let
$Z=Z_{1}\amalg Z_{2}$ such that $Z_{1}\subseteq L,Z_{2}\subseteq C$ and $Z\cap
L\cap C=\varnothing$. Then $h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ only depends
on $|Z_{1}|$ and $|Z_{2}|$ and furthermore,
$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-h^{1}(N(L),I^{2}_{Z_{1}}(d))-h^{1}(N(C),I^{2}_{Z_{2}}(d))|\leq
2(n+1).$
6. 6.
Let $L,C$ be a line and a conic intersecting at one point, transversally.Let
$Z=Z_{1}\amalg Z_{2}\amalg\\{p\\}$ such that $Z_{1}\subseteq L,Z_{2}\subseteq
C$ and $Z_{i}\cap L\cap C=\varnothing$. Then
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ only depends on $|Z_{1}|$ and $|Z_{2}|$
and furthermore,
$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-h^{1}(N(L),I^{2}_{Z_{1}}(d))-h^{1}(N(C),I^{2}_{Z_{2}}(d))|\leq
4(n+1)$.
7. 7.
Let $L,C$ be a line and a conic intersecting at one point,
nontransversally.Let $Z=Z_{1}\amalg Z_{2}$ such that $Z_{1}\subseteq
L,Z_{2}\subseteq C$ and $Z\cap L\cap C=\varnothing$.
Then $h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ only depends on $|Z_{1}|$ and
$|Z_{2}|$ and furthermore,
$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-h^{1}(N(L),I^{2}_{Z_{1}}(d))-h^{1}(N(C),I^{2}_{Z_{2}}(d))|\leq
4(n+1).$
8. 8.
Let $L,C$ be a line and a conic intersecting at one point,
nontransversally.Let $Z=Z_{1}\amalg Z_{2}\amalg\\{p\\}$ such that
$Z_{1}\subseteq L,Z_{2}\subseteq C$ and $Z_{i}\cap L\cap C=\varnothing$. Then
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ only depends on $|Z_{1}|$ and $|Z_{2}|$
and furthermore,
$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-h^{1}(N(L),I^{2}_{Z_{1}}(d))-h^{1}(N(C),I^{2}_{Z_{2}}(d))|\leq
8(n+1)$.
9. 9.
Let $L,C$ be a line and a conic intersecting at two points. Let $Z_{1}=Z\cap
L,Z_{2}=Z\cap C$. Then $h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ only depends on
$|Z_{1}|,|Z_{2}|,|Z_{1}\cap Z_{2}|$ and furthermore,
$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-h^{1}(N(L),I^{2}_{Z_{1}}(d))-h^{1}(N(C),I^{2}_{Z_{2}}(d))|\leq
8(n+1)$.
10. 10.
Let $L_{1},L_{2},L_{3}$ be three lines meeting in the following configuration.
Let $Z\subseteq L_{1}\cup L_{2}\cup L_{3}$. Let $Z_{i}=Z\cap L_{i}$. Then
$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-\sum
h^{1}(N(L_{i}),I^{2}_{Z_{i}}(d))|\leq 12(n+1).$
Furthermore, $h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ depends only on the
configuration of the lines as well as $|Z_{i}|,|Z_{i},\cap Z_{j}|,|Z_{i}\cap
Z_{j}\cap Z_{k}|$.
11. 11.
Let $E$ be a plane cubic in $\mathbb{P}^{n}$. Let $Z\subseteq\mathbb{P}^{n}$.
Then
$h^{1}(\mathbb{P}^{n},I^{2}_{E}(d))=h^{1}(E,\mathcal{O}(d)-2Z)+h^{1}(E,N^{*}\otimes(\mathcal{O}(d)-Z))$.
Since $E$ is a complete intersection, $N^{*}$ splits and the term
$h^{1}(E,N^{*}\otimes(\mathcal{O}(d)-Z))=0$, if $|Z|<3d-5$. The first term
$h^{1}(\mathcal{O}(d)-2Z)=\begin{cases}0&\text{if}\
2|Z|<3d\text{or}2|Z|=3d\text{and}2Z\neq\mathcal{O}(d)\\\ 1,&\text{if}\
2Z=\mathcal{O}(d)\\\ 2|Z|-3d,&\text{otherwise}\end{cases}$ (4)
###### Proof.
We begin with the proof of (1). By Proposition 6.1,
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(N(T),I^{2}_{Z}(d))=h^{1}(T,\mathcal{O}(d)-2|Z|)+h^{1}(T,N^{*}\otimes(\mathcal{O}(d)-|Z|))$
Here $N^{*}$ is the conormal bundle of $T$ in $\mathbb{P}^{n}$, and $|Z|$ is
the line bundle on $T$ corresponding to $Z$. Since $T$ is isomorphic to
$\mathbb{P}^{1}$, the above two $h^{1}$ terms only depend on $|Z|$ and it is
also easy to see that
$|h^{1}(T,\mathcal{O}(d)-2|Z|)+h^{1}(T,N^{*}\otimes(\mathcal{O}(d)-|Z|))$
$-\max(2|Z|-rd,0)|+(n-1)\max(|Z|-d,0)\leq K$
for some constant $K$.
We move on to the proof of (2). We note that since $d\gg e$, the restriction
map
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))\to H^{0}(N(C_{1}\cup
C_{2}),\mathcal{O}(d))$
is surjective (by Serre vanishing). Since $C_{1}$ and $C_{2}$ are disjoint,
$N(C_{1}\cup C_{2})=N(C_{1})\cup N(C_{2})$ and also
$H^{0}(N(C_{1}\cup C_{2}),\mathcal{O}(d))\cong
H^{0}(N(C_{1}),\mathcal{O}(d))\oplus H^{0}(N(C_{2}),\mathcal{O}(d)).$
The map
$H^{0}(N(C_{1}\cup C_{2}),\mathcal{O}(d))\to H^{0}(N(Z),\mathcal{O}(d))$
is then the direct sum of the maps
$H^{0}(N(C_{1}),\mathcal{O}(d))\to H^{0}(N(Z_{1}),\mathcal{O}(d))$
and
$H^{0}(N(C_{2}),\mathcal{O}(d))\to H^{0}(N(Z_{2}),\mathcal{O}(d)).$
We note that $h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))$ is the codimension of the
image of first map and $h^{1}(\mathbb{P}^{n},I^{2}_{Z_{1}}(d))$ (resp.
$h^{1}(\mathbb{P}^{n},I^{2}_{Z_{2}}(d))$) is the codimension of the second
(resp. third) map. Since codimension is additive over direct sums, we have the
required equality.
We now move on to the proofs of (3) and (4). Since $d\gg 0$,
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(N(L_{1}\cup L_{2}),I^{2}_{Z}(d))$.
We note that we have a map $\pi:N(L_{1})\amalg N(L_{2})\to N(L_{1}\cup
L_{2})$. There is a short exact sequence of sheaves on $N(L_{1}\cup L_{2})$,
$0\to I^{2}_{Z}(d)\to\pi_{*}\pi^{*}I^{2}_{Z}(d)\to\mathcal{F}\to 0,$
where the quotient sheaf $\mathcal{F}$ is supported on $N(p)$, where $p$ is
the intersection point of the two lines. Associated to the above short exact
sequence of sheaves we have a long exact sequence in cohomology, a part of
which is as follows:
$H^{0}(N(L_{1}\cup L_{2}),\mathcal{F})\to H^{1}(N(L_{1}\cup
L_{2}),I^{2}(d))\to H^{1}(N(L_{1}\cup L_{2}),\pi_{*}\pi^{*}I^{2}_{Z}(d))\to
0.$
The fact that the above long exact sequence is exact at the last term is due
to the fact that $H^{1}(N(L_{1}\cup L_{2}),\mathcal{F})=0$ as its support is
$0$ dimensional. Furthermore, $H^{0}(N(L_{1}\cup L_{2}),\mathcal{F})\cong
H^{0}(N(p),I^{2}_{Z\cap p}(d))$. Hence $h^{0}(N(p),I^{2}_{Z\cap p}(d))\leq
n+1$ (it is either $0$ or $n+1$ depending on whether $p\in Z$).
We note that
$H^{1}(N(L_{1}\cup L_{2}),\pi_{*}\pi^{*}I^{2}_{Z}(d))\cong
H^{1}(N(L_{1}),I^{2}_{Z\cap L_{1}}(d))\oplus H^{1}(N(L_{2}),I^{2}_{Z\cap
L_{2}}(d)).$
Let
$M=Im(H^{0}(N(L_{1}\cup L_{2}),\mathcal{F})\to H^{1}(N(L_{1}\cup
L_{2}),I^{2}(d))).$
Then by the long exact sequence we have that
$h^{1}(N(L_{1}\cup L_{2}),I^{2}_{Z}(d))=h^{1}(N(L_{1}),I^{2}_{Z\cap
L_{1}}(d))+h^{1}(N(L_{2}),I^{2}_{Z\cap L_{2}}(d))+\dim_{\mathbb{F}}M.$
We note that $\dim_{\mathbb{F}}M\leq h^{0}(\mathcal{F})\leq n+1$
We further note that
$0\leq h^{1}(N(L_{i}),I^{2}_{Z\cap
L_{i}}(d))-h^{1}(N(L_{i}),I^{2}_{Z_{i}}(d))\leq n+1.$
Hence
$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-h^{1}(N(L_{1}),I^{2}_{Z_{1}}(d))-h^{1}(N(L_{2}),I^{2}_{Z_{2}}(d))|\leq(n+1)$
in the case when $p\not\in Z$ (the situation of case (3)) and
$|h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))-h^{1}(N(L_{1}),I^{2}_{Z_{1}}(d))-h^{1}(N(L_{2}),I^{2}_{Z_{2}}(d))|\leq
3(n+1)$
in the case when $p\in Z$ (the situation of case (4)).
To finish the proof of (3) and (4) we must show that $\dim M$ only depends on
$|Z_{1}|$ and $|Z_{2}|$. Since $\dim M$ is also equal to
$h^{0}(N(L_{1}\cup L_{2}),\mathcal{F})-\dim\textrm{ker}H^{0}(N(L_{1}\cup
L_{2}),\mathcal{F})\to H^{1}(N(L_{1}\cup L_{2}),I^{2}(d))$ $=h^{0}(N(L_{1}\cup
L_{2}),\mathcal{F})-\dim Im(H^{0}(N(L_{1}\cup
L_{2}),\pi_{*}\pi^{*}(I^{2}(Z(d))))\to H^{0}(N(L_{1}\cup
L_{2}),\mathcal{F})),$
it suffices to show that
$\dim(Im(H^{0}(N(L_{1}\cup L_{2}),\pi_{*}\pi^{*}(I^{2}(Z(d))))\to
H^{0}(N(L_{1}\cup L_{2}),\mathcal{F}))$
depends only on $|Z_{1}|$ and $|Z_{2}|$. In case $p\in Z$, $H^{0}(N(L_{1}\cup
L_{2}),\mathcal{F})=0$, so this is immediate. This establishes (4). We will
now complete the proof of (3).
The map
$H^{0}(N(L_{1}\cup L_{2}),\pi_{*}\pi^{*}(I^{2}(Z(d))))\to H^{0}(N(L_{1}\cup
L_{2}),\mathcal{F})$
is isomorphic to the map
$H^{0}(N(L_{1}),I^{2}_{Z_{1}}(d))\oplus H^{0}(N(L_{2}),I^{2}_{Z_{2}}(d))\to
H^{0}(N(p),\mathcal{O}(d))$
given by restriction. It suffices to show that the image of
$H^{0}(N(L_{1}),I^{2}_{Z_{1}}(d))\to H^{0}(N(p),\mathcal{O}(d))$ is dependent
on $|Z_{1}|$ only. This happens in all but one case. We record what happens
below without proof since it is elementary and just a computation.
If $2|Z_{1}|+1\leq d$, the map $H^{0}(N(L_{1}),I^{2}_{Z_{1}}(d))\to
H^{0}(N(p),\mathcal{O}(d))$ is surjective. We note that if $2|Z_{1}|\geq d+1$
and $|Z_{1}|<d$ the image consists of all vectors vanishing at $p$ and whose
first derivative along the tangent direction to $L_{1}$ also vanishes at $p$.
If $|Z_{1}|\geq d$, $H^{0}(N(L),I^{2}_{Z_{1}}(d))=0$, so the image is also
zero. The remaining case is when $2|Z_{1}|=d,$ In this situation the image is
constrained by the fact that any $f\in H^{0}(N(L_{1}),I^{2}_{Z_{1}}(d))$is
such that $(f(p),\partial_{v}f(p))$ is forced to lie in some fixed one
dimensional linear subspace.
Now the only way that the image of
$H^{0}(N(L_{1}),I^{2}_{Z_{1}}(d))\oplus H^{0}(N(L_{2}),I^{2}_{Z_{2}}(d))\to
H^{0}(N(p),\mathcal{O}(d))$
can have different dimensions for the same value of $|Z_{1}|,|Z_{2}|$ is if
either $2|Z_{1}|=d$ or $2|Z_{2}|=d$.
Assume that $2|Z_{1}|=d$ and $2|Z_{2}|+1\leq d$. Then in this case the map is
surjective, so the dimension of the image is fixed. Similarly, if $2|Z_{1}|=d$
and $|Z_{2}|\geq d$, the image is of dimension $n$ and dependent only on the
sizes of $Z_{1},Z_{2}$. If $2|Z_{1}|=d$ and $2|Z_{2}|\geq d+1$ one gets that
the image is still surjective. In the case when $2|Z_{1}|=2|Z_{2}|=d$ the map
is still surjective and in particular only depends on the sizes of $|Z_{1}|$
and $|Z_{2}|$.
The proof of (5)- (10) are almost identical to that of (3) and (4), so we omit
them. Case (11) is a standard application of Riemann-Roch for an elliptic
curve.
∎
###### Proposition 6.11.
Let $M$ be one of the following:
1. 1.
The space of twisted cubic curves in $\textrm{Chow}(3)$.
2. 2.
The space of skew triples of lines in $\textrm{Chow}(3)$.
3. 3.
The space of pairs of a line and a disjoint conic in $\textrm{Chow}(3)$.
4. 4.
The space of pairs of a line intersecting a conic at a point in
$\textrm{Chow}(3)$.
5. 5.
The space of pairs of a line intersecting a conic at two points in
$\textrm{Chow}(3)$.
6. 6.
The space of smooth planar conics in $\textrm{Chow}(2)$.
7. 7.
The space of skew lines in $\textrm{Chow}(2)$.
8. 8.
The space of coplanar lines in $\textrm{Chow}(2)$.
9. 9.
$\textrm{Chow}(1)$.
Then there exists $\epsilon>0$
$\sum_{|\lambda|\leq N}(-1)^{||\lambda||}W^{M}_{\geq\lambda}=0,$
up to dimension
$-(\frac{3n}{2}+\epsilon)d+h^{0}(\mathbb{P}^{n},\mathcal{O}(d))$. Furthermore
for $k\leq N$,
$\sum_{|\lambda|=k}(-1)^{||\lambda||}w^{M,l}_{\lambda}=0$
up to dimension $-(\frac{3n}{2}+\epsilon)d-l+|\lambda|n$. .
Furthermore for $k\leq N$,
$\sum_{|\lambda|=k}(-1)^{||\lambda||}w^{M,l,n}_{\lambda}=0,$
up to dimension $-(\frac{3n}{2}+\epsilon)d-l+|\lambda|n$.
###### Proof.
The proof of the Proposition 6.11 in these 9 separate cases are almost
identical. However there are some minor differences. We will prove the
Proposition 6.11 in detail in the cases numbered 1,2 and 4. We will indicate
how the proof is to be adjusted in all remaining cases.
Let us begin with the proof in the case when $M$ is the moduli space of
twisted cubic curves, i.e. case (1).
Let $T$ be a fixed twisted cubic curve. Then,
$\bar{w}_{\lambda}^{M,l}=Mw_{\lambda}^{T,l}$
since the family of twisted cubics over $M$ is Zariski locally-trivial. Note
that
$\bar{w}_{\lambda}^{T,l}=\pi_{\lambda}^{-1}(\bar{w}_{|\lambda|}^{M,l}).$
By Proposition 6.10 $\bar{w}_{|\lambda|}^{\\{T\\},l}$ consists of pairs
$(Z,T)$ and $|Z\cap T|=k_{1}$ (where $k_{1}$ is some number depending on $l$
necessarily bigger than $3$ for $d$ big enough). Hence
$\bar{w}_{\lambda}^{T,l}=\sum_{\alpha\subseteq\lambda,|\alpha|=k_{1}}w_{\alpha}(T)w_{\lambda-\alpha}(\mathbb{P}^{n}\setminus
T)$
By Proposition 6.9 $w_{\alpha}(T)w_{\lambda-\alpha}(\mathbb{P}^{n}\setminus
T)$ is a polynomial in $\mathbb{L}$. Thus it suffices to establish that the
image of the above class in $K_{0}(MHS)$ is $0$. By Proposition 4.8 the image
of
$\sum_{|\lambda|=k_{1}+k_{2}}(-1)^{||\lambda||}\sum_{\alpha\subseteq\lambda,|\alpha|=k_{1},|\lambda|=k_{1}+k_{2}}w_{\alpha}(T)w_{\lambda-\alpha}(\mathbb{P}^{n}\setminus
T)$
in $K_{0}(MHS)$ is the class of $H_{*}^{c}(w_{k_{1}}(T)\times
w_{k_{2}}(\mathbb{P}^{n}\setminus T),\pm\mathbb{Q})$. by Lemma 2 in [17] is
$0$, which establishes that
$\sum_{|\lambda|=k_{1}+k_{2}}\bar{w}_{\lambda}^{T,l}(\mathbb{P}^{n})=0.$
By Proposition 6.7
$\bar{w}_{\lambda}^{M,l}(\mathbb{P}^{n})-w_{\lambda}^{M,l}(\mathbb{P}^{n})\subseteq\cup_{d=1}^{\infty}w_{\lambda}^{M^{\prime}(d),l}(\mathbb{P}^{n}),$
where $M^{\prime}(d)$ is the moduli space of degree $d$ curves strictly
containing a twisted cubic. By Proposition 6.6 this is of dimension
$\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d-(n+1)|\lambda|-l.$
Hence
$\sum_{|\lambda|=k_{1}+k_{2}}w_{\lambda}^{T,l}(\mathbb{P}^{n})=0,$
up to dimension
$\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(-\frac{3n}{2}-\epsilon)d-(n+1)||\lambda||-l.$
Note that in case (1), for a fixed value of $l$, $w_{\lambda}^{M,l,n}$ is
either equal to $w_{\lambda}^{M,l}$ or is $0$. Hence
$\sum_{|\lambda|=k_{1}+k_{2}}w_{\lambda}^{T,l,n}(\mathbb{P}^{n})=0,$
up to dimension
$h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}-\epsilon)d-(n+1)|\lambda|-l.$
Since $W_{\geq\lambda}^{M}=\sum_{l=0}^{\infty}W_{\geq\lambda}^{M,l}$ and
$W_{\geq\lambda}^{M,l}=w_{\geq\lambda}^{M,l}\mathbb{L}^{h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-|\lambda|(n+1)+l}$
case (1) follows.
We will now deal with case when $M$ is the space of skew triples of lines.
Claim: Let $L_{1},L_{2},L_{3}$ be a fixed skew triple of lines. Let
$Z\subseteq L_{1}$ be a subset of size $n$. Let
$f(n)=h^{1}(N(L),I^{2}_{Z}(d))$ (One can check that this quantity only depends
on $|Z|$ ). Let $S_{l}=\\{(a,b,c)|f(a)+f(b)+f(c)=l\\}$. One can check that
$S_{l}$ is a finite set.
Firstly, since any family of skew triples of lines is Zariski locally trivial,
$\sum_{|\lambda|=k}(-1)^{||\lambda||}\bar{w}_{\lambda}^{M,l}(\mathbb{P}^{n})=M\sum_{|\lambda|=k}(-1)^{||\lambda||}\bar{w}_{\lambda}^{L_{1}\cup
L_{2}\cup L_{3},l}(\mathbb{P}^{n}).$
By Proposition 6.10, this equals
$\sum_{|\lambda|=k}(-1)^{||\lambda||}\sum_{(k_{1},k_{2},k_{3})\in
S_{l}}\sum_{\begin{subarray}{c}\alpha_{3}\subseteq\alpha_{2}\subseteq\alpha_{1}\subseteq\lambda\\\
(|\alpha_{1}|,|\alpha_{2}|,|\alpha_{3}|)=(k_{1},k_{2},k_{3})\end{subarray}}\bar{w}_{\alpha_{3}}(L_{3})\bar{w}_{\alpha_{2}-\alpha_{3}}(L_{2})\bar{w}_{\alpha_{1}-\alpha_{2}}(L_{1})\bar{w}_{\lambda-\alpha_{1}}(\mathbb{P}^{n}-L_{1}\cup
L_{2}\cup L_{3}).$
By Proposition 6.9 the right hand side of the above equation is a polynomial
in $\mathbb{L}$. By Proposition 4.8, the image of the right hand side in
$K_{0}(MHS)$ is a sum of terms of the form $H_{*}^{c}(w_{k_{1}}(L_{1})\times
w_{k_{2}}(L_{2})\times w_{k_{3}}(L_{3}))$which by Proposition in [17] is zero.
This implies
$\sum_{|\lambda|=k}(-1)^{||\lambda||}\bar{w}_{\lambda}^{M,l}(\mathbb{P}^{n})=0$
. As in case(1)
$\sum_{|\lambda|=k}(-1)^{||\lambda||}\bar{w}_{\lambda}^{M,l}(\mathbb{P}^{n})-w_{\lambda}^{M,l}(\mathbb{P}^{n})$
is of dimension less than equal to that of
$\cup_{d=1}^{\infty}w_{\lambda}^{M^{\prime}(d),l}(\mathbb{P}^{n}),$ where
$M^{\prime}(d)$ is the moduli space of degree $d$ curves strictly containing a
skew triple of lines. By Proposition 6.6 this is $\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}-\epsilon)d-(n+1)||\lambda||-l.$
Case (4) follows.
The proofs of the other cases are nearly identical. In each case we prove that
$\bar{w}_{\lambda}^{M,l}=M\bar{w}_{\lambda}^{\\{C\\},l}$ where $C\in M$, since
$M$ is Zariski locally trivial in all of the cases. We then use Proposition
6.10 to explicitly describe $\bar{w}_{|\lambda|}^{\\{C\\},l}$. In each of our
cases $H^{*}(\bar{w}_{|\lambda|}^{\\{C\\},l},\pm\mathbb{Q})=0$. As a result by
Proposition 4.8
$\sum_{|\lambda|=n}(-1)^{||\lambda||}\bar{w}_{\lambda}^{\\{C\\},l}$ is mapped
to $0$ in $K_{0}(MHS).$
We then note that
$\bar{w}_{\lambda}^{\\{C\\},l}=\pi_{\lambda}^{-1}\bar{w}_{|\lambda|}^{\\{C\\},l}$
is seen to be a polynomial in $\mathbb{L}$. Hence
$\sum_{|\lambda|=n}(-1)^{||\lambda||}\bar{w}_{\lambda}^{\\{C\\},l}=0\in\mathcal{M}$.
We then note that
$\sum_{|\lambda|=n}(-1)^{||\lambda||}(\bar{w}_{\lambda}^{M,l}-w_{\lambda}^{M,l})=\sum_{l}\sum_{M^{\prime},|\lambda|=n}(-1)^{||\lambda||}w_{\lambda}^{M^{\prime},l}.$
where $M^{\prime}$ ranges over the moduli spaces of curves strictly containing
a curve in $M$. However either $M^{\prime}$ consists of curves of degree $\geq
4$ and hence $w_{\lambda}^{M^{\prime},l}$ is of dimension $\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}-\epsilon)d-(n+1)|\lambda|-l$
by Proposition 6.6, or $M^{\prime}$ is a case that has already been considered
on this list and hence has dimension $\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}-\epsilon)d-(n+1)|\lambda|-l$.
∎
###### Proposition 6.12.
Let $M\subseteq Chow(3)$ be contained in the subset parametrizing three lines.
Then there exists $\epsilon>0$, such that for
$N=\lceil(\frac{3n}{2}+\epsilon)d\rceil$, and $l>0$,
$\sum_{|\lambda|<N}(-1)^{||\lambda||}w_{\lambda}^{M,l}$ is of dimension
$\leq-l+|\lambda|n+-(\frac{3n}{2}+\epsilon)d$
###### Proof.
We note that we may express $M$ as a disjoint union of constructible sets
$M=\amalg_{i=1}^{k}M_{i}$ such that for each $i$, and for each triple of lines
$\\{L_{1},L_{2},L_{3}\\}$ in $M_{i}$ the triple has the same configuration,
i.e. the number of points of intersection of the lines remain constant in
$L_{i}$. It suffices to prove the Proposition for each $M_{i}$ separately.
Thus in what follows we assume that each triple of lines in $M$ has the same
configuration and $M$ is irreducible. We note that under this assumption the
family of triples of lines over $M$ is Zariski locally trivial.
By Proposition 6.7
$\bar{w}_{\lambda}^{M,l}-w_{\lambda}^{M,l}\subseteq\cup_{d^{\prime}\geq
4,l^{\prime}>l}w_{\lambda}^{Chow(d^{\prime}),l^{\prime}}$
and by Proposition 6.6 the right hand side of the above expression is of
dimension $\leq-l+|\lambda|n+-(\frac{3n}{2}+\epsilon)d$. Therefore it suffices
to prove that
$\sum_{|\lambda|<N}(-1)^{||\lambda||}\bar{w}_{\lambda}^{M,l}$
is of dimension $\leq-l+|\lambda|n+-(\frac{3n}{2}+\epsilon)d$.
Let $A=\mathbb{Z}^{7}/S_{3}$ where $S_{3}$ denotes the symmetric group of
three letters and the action of $S_{3}$ on $\mathbb{Z}^{7}$ is induced from
the natural action of $S_{3}$ on the power set of
$\\{1,2,3\\}\setminus\varnothing$. A configuration of points
$Z\subseteq\mathbb{P}^{n}$ and a set of 3 lines $\\{L_{1},L_{2},L_{3}\\}$
naturally gives us $|Z\cap\\{L_{1},L_{2},L_{3}\\}|:=(|Z\cap L_{1}|,|Z\cap
L_{2}|,|Z\cap L_{3}|,|Z\cap L_{1}\cap L_{2}|,\dots|Z\cap L_{1}\cap L_{2}\cap
L_{3}|)$ which is a well defined element of $A$.
Given $a\in A$, let
$\bar{w}_{\lambda}^{M}(a)=\\{(Z,\\{L_{1},L_{2},L_{3}\\})\in
w_{\lambda}^{M}||Z\cap\\{L_{1},L_{2},L_{3}\\}|=a\\}.$
We may now further stratify each $\bar{w}_{\lambda}^{M,l}$ as the disjoint
union of subspaces $\bar{w}_{\lambda}^{M,l}\cap\bar{w}_{\lambda}^{M}(a)$ where
$a\in A$.
However by Proposition 6.10, $\bar{w}_{\lambda}^{M,l}(a)$ is either all of
$\bar{w}_{\lambda}^{M}(a)$ or it is empty. We note that since the family of
triples of lines over $M$ is Zariski locally trivial,
$\bar{w}_{\lambda}^{M}(a)$ is also a locally trivial family over $M$.
Thus since Zariski locally trivial families split into prosucts in
$\mathcal{M}$, we have the following equality:
$[\bar{w}_{\lambda}^{M}(a)]=[M][w_{\lambda}^{c}(a)],$
where $c$ is a point in $M$. Thus it suffices to establish that for a fixed
$c\in M$ and $a\in A$
$\sum_{|\lambda|<N}(-1)^{||\lambda||}w_{\lambda}^{c}(a)=0$.
Since the above expression is polynomial in $\mathbb{L}$, it suffices to prove
that its image in $K_{0}(MHS)$ is $0$. By Proposition 4.8
$\sum_{|\lambda|=r}(-1)^{||\lambda||}w_{\lambda}^{c}(a)$ is mapped to
$H_{*}^{c}(w_{[r]}^{c}(a))$. The topological space $w_{[r]}^{c}(a)$ has a
factor of the form $w_{[r]}^{c}(\mathbb{A}^{1})$ . However by Proposition 4.8
$H_{*}^{c}(w_{[r]}^{c}(\mathbb{A}^{1}),\pm\mathbb{Q})=0$ and thus
$H_{*}^{c}(w_{[r]}^{c}(a))=0$. This concludes the proof. ∎
###### Definition 6.13.
Let $\epsilon>0$. Let $M$ denote the moduli space of smooth plane cubics in
$\mathbb{P}^{n}$. Let $N=(\frac{3n}{2}+\epsilon)d$. Let
$Y(d):=\sum_{l>0}\sum_{|\lambda|<N}((-1)^{||\lambda||})w_{\lambda}^{M,l}\mathbb{L}^{n|\lambda|}(\mathbb{L}^{l}-1).$
We note that a priori this definition of $Y(d)$ depends on our choice of
$\epsilon>0$. However if $\epsilon$ is sufficiently small, different values of
$\epsilon$ will only change the value of $Y(d)$ in very high codimension, and
this is an ambiguity that is acceptable for our purposes.
###### Proposition 6.14.
There exists $\epsilon>0$ such that: Up to dimension
$h^{0}(\mathbb{P}^{n}\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d$,
$Y(d)=\sum_{l>0}\sum_{\lambda}((-1)^{||\lambda||})\bar{w}_{\lambda}^{M,l}\mathbb{L}^{n|\lambda|}(\mathbb{L}^{l}-1).$
Let us now assume $d$ is even. Let $k_{1}(l)=\frac{3d+l}{2}$.
Then up to dimension
$h^{0}(\mathbb{P}^{n}\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d$,
$\sum_{l>0}\sum_{\lambda}((-1)^{||\lambda||})\bar{w}_{\lambda}^{M,l}\mathbb{L}^{n|\lambda|}(\mathbb{L}^{l}-1)$
$=\sum_{l>0,l\textrm{
odd}}\sum_{|\lambda|<N}\sum_{\alpha\subseteq\lambda,|\alpha|=k_{1}(l)}((-1)^{||\lambda||})\\{(Z_{1},Z_{2},C)|C\in
M,Z_{1}\in w_{\alpha}(C),Z_{2}\in
w_{\lambda\setminus\alpha}(\mathbb{P}^{n}\setminus
C)\\}\mathbb{L}^{-(n+1)|\lambda|}(\mathbb{L}^{l}-1).$
###### Proof.
Let $M$ denote the space of plane cubics in $\mathbb{P}^{n}$. We note that
$Y(d)-\sum_{l>0}\sum_{\lambda}((-1)^{||\lambda||})\bar{w}_{\lambda}^{M,l}\mathbb{L}^{n|\lambda|}(\mathbb{L}^{l}-1),$
equals
$\sum_{l>0}\sum_{\lambda}((-1)^{||\lambda||})w_{\lambda}^{M,l}-\bar{w}_{\lambda}^{M,l}\mathbb{L}^{n|\lambda|}(\mathbb{L}^{l}-1).$
By Proposition 6.6 and Proposition 6.7 each difference
$w_{\lambda}^{M,l}-\bar{w}_{\lambda}^{M,l}\mathbb{L}^{n|\lambda|}(\mathbb{L}^{l}-1)$
is of dimension $\leq
h^{0}(\mathbb{P}^{n}\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d$. Thus the entire
sum is of dimension $\leq
h^{0}(\mathbb{P}^{n}\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d.$
If $d$ is even and $l<\frac{3d}{2}$ we have
$barw_{\lambda}^{M,l}=\sum_{\alpha\subseteq\lambda,|\alpha|=k_{1}(l)}((-1)^{||\lambda||})\\{(Z_{1},Z_{2},C)|C\in
M,Z_{1}\in w_{\alpha}(C),Z_{2}\in
w_{\lambda\setminus\alpha}(\mathbb{P}^{n}\setminus C)\\},$
by Proposition $\ref{h1list}(11).$ To see this note that under the assumption
$l<\frac{3d}{2}$, it must be the case that $h^{1}(N(C),I^{2}_{Z\cap
C}(d))=h^{1}(C,\mathcal{O}(d)-2|Z\cap C|)$. Since $C$ is of genus $1$,
$h^{1}(C,\mathcal{O}(d)-2|Z\cap C|)=h^{0}(C,2|Z\cap C|-\mathcal{O}(d))=2|Z\cap
C|-3d$ (by our parity assumption $3d\neq 2|Z\cap C|$ and the case when
$2|Z\cap C|-\mathcal{O}(d)$ is the trivial line bundle does not arise).
We note that we may disregard the terms when $l\geq\frac{3d}{2}$ because they
contribute in too high a codimension.
Thus we have our desired equality. ∎
###### Proposition 6.15.
Let $M$ be the space of plane cubics in $\mathbb{P}^{n}$. Let $d\equiv
1\mod{4}$. Let $k_{0}=\frac{3d+1}{2}$. Then for $\epsilon$ sufficiently small
(the definition of $Y(d)$ implicitly depends on an $\epsilon$), $Y(d)$ has the
same highest weight term as
$H^{*}(\mathcal{M}_{1,1},H_{k_{0}})H^{*}(PGL_{n+1}(\mathbb{C}))\mathbb{L}^{-k_{0}(n+1)+1}$
in $K_{0}(MHS)$. The weight of this highest weight term is
$2(-k_{0}(n+1)+1)+k_{0}+1+(n^{2}-1)$.
###### Proof.
By Proposition 6.14, up to dimension $d(\frac{3n}{2}+\epsilon)$ $Y(d)$
$=\sum_{l>0,l\textrm{
odd}}\sum_{|\lambda|<N}\sum_{\alpha\subseteq\lambda,|\alpha|=k_{1}(l)}((-1)^{||\lambda||})\\{(Z_{1},Z_{2},C)|C\in
M,Z_{1}\in w_{\alpha}(C),Z_{2}\in
w_{\lambda\setminus\alpha}(\mathbb{P}^{n}\setminus
C)\\}\mathbb{L}^{-(n+1)|\lambda|}(\mathbb{L}^{l}-1).$
To finish the proof of the Proposition it suffices to compute the image of the
above expression in $K_{0}(MHS)$ . We note that the image equals
$\sum_{l,k}H^{*}(w_{[k]}^{M,l},\pm\mathbb{Q})\mathbb{L}^{-n|\lambda|+l}$.
For $l=1$, $\sum_{k}H^{*}(w_{[k]}^{M,1},\pm\mathbb{Q})\mathbb{L}^{-nk+l}$ has
highest weight term equal to that of $H^{*}(w_{k_{0}}^{M,1},\pm\mathbb{Q})$,
where $k_{0}=\frac{3d+1}{2}$, there are other terms corresponding to higher
values of $k$, but we will establish (after first dealing with the leading
term)that these have lower weight. This highest weight term corresponds to the
situation where we have $k_{0}$ points on a plane cubic $C$, in this situation
being singular on such a collection of points imposes $(k_{0})(n+1)-1$ linear
conditions.
Now $w_{k_{0}}^{M,1}=\\{(Z,C)\in w_{k_{0}}\times M|Z\subseteq C\\}$(by
Proposition 6.10) and hence
$w_{k_{0}}^{M,1}/PGL_{n+1}(\mathbb{C})=\mathcal{M}_{1,k}$. So we have,
$H^{*}(w_{k_{0}}^{M,1},\pm\mathbb{Q})=H^{*}(PGL_{n+1},\mathbb{Q})\sum_{l,n}H^{*}(\mathcal{M}_{1,k_{0}},\pm\mathbb{Q})$.
By the results of Section 5 of [7] (also see Corollary 2.8 in [8] ) the
highest weight term of this equals that of
$H^{*}(PGL_{n+1}(\mathbb{C}),\mathbb{Q})H^{*}(\mathcal{M}_{1,1},H_{k_{0}}),$
we note that this then implies that
$H^{*}(w_{[k_{0}]}^{M,1},\pm\mathbb{Q})\mathbb{L}^{-nk_{0}+1}$ has the
required highest weight, see [7] for instance.
We will now establish that for $k>k_{0}$, the term
$H^{*}(w_{[k]}^{M,1},\pm\mathbb{Q})\mathbb{L}^{-(n+1)k+l}$ has lower weight
than for $k=k_{0}$. To see this let $\bar{w}_{[k]}^{M,1}=\\{(Z,C)\in
w_{[k]}|h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap C}(d))=1\\}$. We note that
$\bar{w}_{[k]}^{M,1}\setminus w_{[k]}^{M,1}$ is of small dimension by
Proposition 6.6. Thus it suffices to establish that
$H^{*}(\bar{w}_{[k]}^{M,1},\pm\mathbb{Q})\mathbb{L}^{-nk+l}$ has lower weight.
But we note that by Proposition 6.10 $\bar{w}_{[k]}^{M,1}=\\{(Z,C)\in
w_{[k]}\times C||Z\cap C|=k_{0}\\}.$ Now we have a fibration $w_{[k]}^{M,1}\to
M$ with fiber isomorphic to $w_{[k_{0}]}(C)\times
w_{k-k_{0}}(\mathbb{P}^{n}\setminus C)$. We have a further fibration (in the
orbifold sense) $M\to M/PGL_{n+1}(\mathbb{C}).$ The fiber of the composite map
$w_{[k]}^{M,1}\to M/PGL_{n+1}(\mathbb{C})$ is seen to be a copy of
$PGL_{n+1}(\mathbb{C})\times w_{[k_{0}]}(C)\times
w_{k-k_{0}}(\mathbb{P}^{n}\setminus C).$
We may thus use the Leray spectral sequence to bound the weights of
$H_{*}^{c}(w_{[k]}^{M,1}),\pm\mathbb{Q})$ in terms of the dimension of
$M/PGL_{n+1}(\mathbb{C})$ (which is complex 1 dimensional and non compact) and
the weights of $H_{*}^{c}PGL_{n+1}(\mathbb{C})\times w_{[k_{0}]}(C)\times
w_{k-k_{0}}(\mathbb{P}^{n}\setminus C))$. The highest weight term of the
latter is $((n+1)^{2}-1)+k_{0}+1+(2n-1)(k-k_{0})+1$ (this uses a computation
of the Borel-Moore homology of configuration spaces- see Corollary 7.2.6 of
[3] for instance).
Thus the highest weight term of $H_{*}^{c}(w_{[k]}^{M,1}),\pm\mathbb{Q})$ is
at most $1+((n+1)^{2}-1)+k_{0}+1+(2n-1)(k-k_{0})$. Thus the highest weight of
$H_{*}^{c}(w_{[k]}^{M,1}),\pm\mathbb{Q})\mathbb{L}^{-k(n+1)+1}$
is at most $1+((n+1)^{2}-1)+k_{0}+1+(2n-1)(k-k_{0})+2-2k(n+1)$. If we subtract
this quantity from $2(-k_{0}(n+1)+1)+k_{0}+1+((n+1)^{2}-1)$ we obtain:
$3(k-k_{0})-1$
which is always positive for $k>k_{0}$. Thus the possible weights are lower
than the highest term.
We note that for $l>1$ a similar computation shows that all the corresponding
terms $H^{*}(w_{[k]}^{M,l})\mathbb{L}^{-nk+l}$ are of smaller weight and we
may ignore them for the purpose of finding the highest weight term.
This establishes the result.
∎
###### Proposition 6.16.
Let $d\gg n$. Let $L$ be a fixed line in $\mathbb{P}^{n}$. Then there exists,
$\epsilon>0$ such that for $N=(\frac{3n}{2}+\epsilon)d$, we have the following
inequalities/equalities.
1. 1.
$\\{(f,Z)\in W_{\lambda,\geq N}^{l,n}|\dim\textrm{Sing}(f)=0\\}$
is of dimension $\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(-\frac{3n}{2}-\epsilon)d$
2. 2.
$W_{\lambda,\geq N}^{0,n}$is of dimension $\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(-\frac{3n}{2}-\epsilon)d$
3. 3.
$\\{(f,Z)\in W_{\lambda,\geq N}^{l,n}|\textrm{Sing}(f)\textrm{contains a curve
of degree}\geq 3\\}$
is of dimension $\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(-\frac{3n}{2}-\epsilon)d$.
4. 4.
$W_{\lambda,\geq N}^{l,n}=W_{\lambda,\geq
N}^{l,n}\cap\\{(f,Z)|C\subseteq\textrm{Sing}(f),C\textrm{ is a curve of degree
}\leq 2\\}$ $=\\{(f,L,Z)|(f,Z)\in W_{\lambda,\geq
N}^{l,n},L\subseteq\textrm{Sing}(f),L\textrm{ is a line}\\}$
$+\\{(f,C,Z)|(f,Z)\in W_{\lambda,\geq
N}^{l,n},C\subseteq\textrm{Sing}(f),C\textrm{is an irreducible conic}\\}$
$-\\{(f,L_{1},L_{2},Z)|(f,Z)\in W_{\lambda,\geq
N}^{l,n},L_{i}\subseteq\textrm{Sing}(f)L_{i}\textrm{are distinct lines}\\},$
up to dimension $\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(-\frac{3n}{2}-\epsilon)d.$
5. 5.
$\sum_{|\lambda|<N}(-1)^{||\lambda||}\\{(f,L_{1},L_{2},Z)|(f,Z)\in
W_{\lambda,\geq N}^{l,n},L_{i}\subseteq\textrm{Sing}(f)\\}=0$ up to dimension
$\leq h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(-\frac{3n}{2}-\epsilon)d$.
6. 6.
$\sum_{|\lambda|<N}(-1)^{||\lambda||}\\{(f,C,Z)|(f,Z)\in W_{\lambda,\geq
N}^{l,n},C\subseteq\textrm{Sing}(f)\\}=0$ up to dimension $\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(-\frac{3n}{2}-\epsilon)d$.
7. 7.
$\sum_{|\lambda|<N}(-1)^{||\lambda||}\\{(f,L,Z)|(f,Z)\in W_{\lambda,\geq
N}^{l,n},L\subseteq\textrm{Sing}(f)L\textrm{ is a line}\\}=0$ up to dimension
$\leq h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(-\frac{3n}{2}-\epsilon)d$.
###### Proof.
We begin by proving (1). It suffices to prove (1) in the case when
$|\lambda|=N$. By Proposition 3.1 if $l>0$, then associated to any $Z\in
W_{\geq\lambda}^{n,l}$ is a unique reduced curve $C$ such that
$l=h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap
C}(d))$. Furthermore, if the components of $C$ are $C_{1},\dots C_{k}$ then
$\frac{\deg(C_{k})d-g(C_{k})}{2}\leq|Z\cap C_{k}|$ as otherwise $2|Z|\cap
C_{k}$ imposes linearly independent conditions which is not possible (see
Proposition 3.1). Also, there is some constant $K$ depending on $\deg(C_{k})$
such that if $|Z\cap C_{k}|>\deg(C_{k})d+K,$ a section singular at $Z$ is
singular on all of $C_{k}$. if that happens $Z\not\in W_{\geq\lambda}^{n,l}$.
Thus we must have $|Z\cap C_{k}|<\deg(C_{k})d+K$.
Furthermore there is some number $K^{\prime}$ such that ,
$|l-(n-1)\sum_{k}(|Z\cap C_{k}|)|\leq K^{\prime}$(Refer to Proposition 4.10).
We can therefore bound both the dimension of the space of such $Z$ and $l$,
which gives us (after a computation):
,
$\dim\\{(f,Z)|\dim\textrm{Sing}(f)=0\\}\cap W_{\lambda,\geq N+1}^{l,n}\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(-\frac{3n}{2}-\epsilon)d.$
Part (2) is immediate, since
$\dim W_{\lambda,\geq N+1}^{0,n}\leq\dim W_{\geq[N]}^{0,n}\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-N.$
Let us now proceed to prove part (3). We first begin with a claim:
Claim: For any fixed curve $C$ of degree $\geq 3$ and a fixed curve
$C^{\prime}$ of degree $r^{\prime}$ possibly empty, there exists
$\epsilon^{\prime}>0$, such that
$\dim\\{(f,Z)\in
W_{\geq\lambda}^{l,n}|C\subseteq\textrm{Sing}(f),h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap(C\cup
C^{\prime})}(d))\\}\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}+\epsilon^{\prime})d$
, here we require $\epsilon^{\prime}$ to be independent of
$C^{\prime},r^{\prime}$.
To prove the claim, we let
$A=\\{(f,Z)|\textrm{Sing}(f)\textrm{contains a curve of degree}\geq 3\\}\cap
W_{\lambda,\geq N}^{l,n}$
and $\tilde{A}=\\{(f,Z,C)|(f,Z)\in A,C\textrm{ is a curve of degree}\geq
3,C\subseteq\textrm{Sing}(f)\\}.$ We note that $\dim\tilde{A}\geq\dim A$. Then
$\dim\tilde{A}\leq\sup_{r\geq 3,C\textrm{is a curve of degree }r}(\dim
Chow(r)+\dim\\{(f,Z)|C\subseteq\textrm{Sing}(f)\\}\cap W_{\lambda,\geq
N}^{l,n}).$
Let us first argue that the claim implies (4). We will establish the claim
afterwards. Let $B_{r^{\prime}}(C)$ be the subset of
$W_{\geq\lambda}^{l,n}\times Chow(r^{\prime})$ consisting of all
$(f,Z,C^{\prime})$ such that $C\subseteq\textrm{Sing}(f)$ and $C^{\prime}$ is
the curve of minimal degree such that
$h^{1}(\mathbb{P}^{n},I^{2}_{Z}(d))=h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap(C\cup
C^{\prime})}(d))\\}.$
Now we use Proposition 3.1 to argue that
$\dim\\{(f,Z)|C\subseteq\textrm{Sing}(f)\\}\cap W_{\lambda,\geq
N+1}^{l,n})\leq\sup_{r^{\prime}}(\dim Chow(r^{\prime})+\dim
B_{r^{\prime}}(C).$
Note that in the above two expressions, the supremums are actually maximums
for a given $|\lambda|$, there are only finitely many $r,r^{\prime}$ for which
the sets $B_{r^{\prime}}(C)$ can be nonempty. Further, we note that for $d\gg
0$, $\epsilon/10d\gg\dim Chow(r)$ for $r$ in the relevant range. Thus if we
establish that $\dim B_{r^{\prime}}(C)\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d$ we would be
done.
This concludes the proof of (4) assuming the claim, so we will now proceed to
establish the claim.
For this we note the following- for $d\gg\deg(C)$ the space $\\{f\in
H^{0}(\mathbb{P}^{n},\mathcal{O}(d))|C\subseteq\textrm{Sing}(f)\\}$ is of
codimension $h^{0}(N(C),\mathcal{O}(d))$ in
$H^{0}(\mathbb{P}^{n},\mathcal{O}(d))$. We know that
$h^{0}(N(C),\mathcal{O}(d))\approx n\deg Cd$ by Proposition 4.10. For
$(f,Z)\in W_{\geq\lambda}^{C\cup C^{\prime},l,n}$, let $Z_{1}=Z\cap
C$,$Z_{2}=Z\cap C^{\prime}\setminus Z_{1}$ $Z_{3}=Z\setminus(Z_{1}\cup
Z_{2})$. We now have the following inequalities:
1. 1.
$\deg(C)d\geq|Z_{1}|$.
2. 2.
We note that
$|2|Z_{2}|-\deg(C^{\prime})d-h^{0}(\mathbb{P}^{n},I^{2}_{Z_{2}}(d))|$ is
bounded by a constant independent of $d$.
3. 3.
$\\{f|Z\cup C\subseteq\textrm{Sing}(f)\\}$ is of dimension
$h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-h^{0}(N(C),\mathcal{O}(d))-h^{1}(\mathbb{P}^{n},I^{2}_{Z^{\prime}}(d))-(n+1)|Z_{3}|-(n+1)|Z_{2}|$
in .
4. 4.
But the dimension of the space of $Z$ such that $(f,Z)\in
W_{\geq\lambda}^{C\cup C^{\prime},l,n}$ is less than
$|Z_{1}|+|Z_{2}|+n|Z_{3}|$.
5. 5.
Hence the dimension of
$\\{(f,Z)\in W_{\geq\lambda}^{C\cup
C^{\prime},l,n}|C\subseteq\textrm{Sing}(f)\\}$
$\leq|Z_{1}|+|Z_{2}|+n|Z_{3}|+h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-h^{0}(N(C),\mathcal{O}(d))+h^{1}(\mathbb{P}^{n},I^{2}_{Z_{2}}(d))-(n+1)|Z_{3}|-(n+1)|Z_{2}|$
$\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+\deg(C)d-n|Z_{2}|-h^{0}(N(C),\mathcal{O}(d))+h^{1}(\mathbb{P}^{n},I^{2}_{Z_{2}}(d))$
$\approx
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(n-1)\deg(C)d-n|Z_{2}|+h^{1}(\mathbb{P}^{n},I^{2}_{Z_{2}}(d))$
$\leq h^{0}(\mathbb{P}^{n},\mathcal{O}(d))+(n-1)\deg(C)d$
$\leq h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d.$
For part (4), we note that by (1),
$W_{\lambda,\geq N+1}^{l,n}=X$
up to dimension
$h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d,$ where
$X:=\\{(f,Z)\in W_{\lambda,\geq N+1}^{l,n}|\textrm{Sing}(f)\textrm{contains a
curve of degree}\leq 2\textrm{and does not contain a curve of degree}\geq
3\\}.$
Standard inclusion-exclusion techniques then imply that the motive of
$X=\\{(f,L,Z)|(f,Z)\in W_{\lambda,\geq
N+1}^{l,n},L\subseteq\textrm{Sing}(f)\\}+\\{(f,C,Z)|(f,Z)\in W_{\lambda,\geq
N+1}^{l,n},C\subseteq\textrm{Sing}(f)\\}$ $-\\{(f,L_{1},L_{2},Z)|(f,Z)\in
W_{\lambda,\geq N+1}^{l,n},L_{i}\subseteq\textrm{Sing}(f)\\}.$
up to
dimension$h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d$.
We will omit the proof of parts (5) and (6) because their proofs are similar
to that of part (7) (which we will not omit) and in fact a bit simpler. For
part (7), we note that the family of lines over $Chow(1)$ is Zariski locally
trivial. Hence $\\{(f,L,Z)|(f,Z)\in W_{\lambda,\geq
N+1}^{l,n},L\subseteq\textrm{Sing}(f)\\}$ defines a locally trivial family
over $Chow(1)$ and thus,
$\\{(f,L,Z)|(f,Z)\in W_{\lambda,\geq
N+1}^{l,n},L\subseteq\textrm{Sing}(f)\\}=Chow(1)\\{(f,Z)|(f,Z)\in
W_{\lambda,\geq N+1}^{l,n},L\subseteq\textrm{Sing}(f)\\}$
in $\mathcal{M}$ (Zariski locally trivial families split into products in
$\mathcal{M}$).
We now make the following claim.
Claim:
$\sum_{|\lambda|=k}(-1)^{||\lambda||}\\{(f,Z)|(f,Z)\in W_{\lambda,\geq
N+1}^{l,n},L\subseteq\textrm{Sing}(f)\\}$
$=\sum_{|\lambda|=k}(-1)^{||\lambda||}\sum_{\alpha\subseteq\lambda}w_{\alpha}(L)w_{\lambda\setminus\alpha}(\mathbb{P}^{n}\setminus
L)\mathbb{L}^{-(n+1)(|\lambda|-|\alpha|-h^{0}(N(L),\mathcal{O}(d))}$
up to dimension
$h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d$, i.e. the sum
is well approximated by the corresponding sum when we ignore the fact that
points in $\mathbb{P}^{n}\setminus L$ need not impose linearly independent
conditions on $H^{0}(\mathbb{P}^{n},I^{2}_{L}(d))$.
The proof of the claim is essentially a dimension computation similar to those
we’ve already seen, if a collection of points $Z$ fails to impose linearly
independent conditions on $H^{0}(\mathbb{P}^{n},I^{2}_{L}(d))$, then a large
number of points of $Z$ lie on a curve, either the curve is of degree $\geq 2$
and the dimension of the resulting error is small (see Proposition 6.6) or the
curve is a line and the alternating sum vanishes for the same reason as in
Proposition 6.11. We will first establish (7) assuming the claim.
To do so we must simply establish that
$\sum_{|\lambda|<N}(-1)^{||\lambda||}\sum_{\alpha\subseteq\lambda}w_{\alpha}(L)w_{\lambda\setminus\alpha}(\mathbb{P}^{n}\setminus
L)\mathbb{L}^{-(n+1)(|\lambda|-|\alpha|-h^{0}(N(L),\mathcal{O}(d))}=0$
up to dimension
$h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-(\frac{3n}{2}+\epsilon)d.$ It is clear
that the sum is a polynomial in $\mathbb{L}$ and thus it suffices to establish
that its image in $K_{0}(MHS)$ is of sufficiently low weight. Furthermore by
Proposition 4.8 the image of the above expression in $K_{0}(MHS)$ is
$\mathbb{L}^{-h^{0}(N(L),\mathcal{O}(d))}\sum_{k<N}\sum_{k^{\prime}\leq
k_{1}}H_{*}^{c}(w_{k_{1}}(\mathbb{P}^{1})\times
w_{k-k_{1}}(\mathbb{P}^{n}\setminus\mathbb{P}^{1}),\pm\mathbb{Q})\mathbb{L}^{-(k-k_{1})(n+1)}.$
To proceed with the proof of (7) we will need the following identity:
$\sum_{k=0}^{\infty}H_{*}^{c}(w_{k}(\mathbb{P}^{1}),\pm\mathbb{Q})=0.$
To establish this we note that
$\sum_{k=0}^{\infty}H_{*}^{c}(w_{k}(\mathbb{P}^{1}),\pm\mathbb{Q})$ is the
image of
$\sum_{\lambda}(-1)^{||\lambda||}w_{\lambda}(\mathbb{P}^{1})=Z_{\mathbb{P}^{1}}^{-1}(0)=0$.
Thus we have established the previous identity.
We also note that in the sum
$\sum_{k=0}^{\infty}H_{*}^{c}(w_{k}(\mathbb{P}^{1}),\pm\mathbb{Q})$ the only
relevant terms are for $k=0,1,2$ as all other terms vanish by Lemma 1.2 of
[17].
We now note that
$\mathbb{L}^{-h^{0}(N(L),\mathcal{O}(d))}\sum_{k<N}\sum_{k^{\prime}\leq
k}H_{*}^{c}(w_{k^{\prime}}(\mathbb{P}^{1})\times
w_{k-k^{\prime}}(\mathbb{P}^{n}\setminus\mathbb{P}^{1}),\pm\mathbb{Q})\mathbb{L}^{-(k-k^{\prime})(n+1)}$
$=\mathbb{L}^{-h^{0}(N(L),\mathcal{O}(d))}\sum_{k^{\prime}=0}^{2}\sum_{k=k^{\prime}}^{N}H_{*}^{c}(w_{k^{\prime}}(\mathbb{P}^{1})\times
w_{k-k^{\prime}}(\mathbb{P}^{n}\setminus\mathbb{P}^{1}),\pm\mathbb{Q})\mathbb{L}^{-(k-k^{\prime})(n+1)}$
$=\mathbb{L}^{-h^{0}(N(L),\mathcal{O}(d))}\sum_{k^{\prime}=0}^{2}H_{*}^{c}(w_{k^{\prime}}(\mathbb{P}^{1}),\pm\mathbb{Q})\sum_{k^{\prime\prime}=0}^{N-k^{\prime}-1}H_{*}^{c}(w_{k^{\prime\prime}}(\mathbb{P}^{n}\setminus\mathbb{P}^{1}),\pm\mathbb{Q})\mathbb{L}^{-(k^{\prime\prime})(n+1)}$
$=\mathbb{L}^{-h^{0}(N(L),\mathcal{O}(d))}\sum_{k^{\prime}=0}^{2}H_{*}^{c}(w_{k^{\prime}}(\mathbb{P}^{1}),\pm\mathbb{Q})\sum_{k^{\prime\prime}=0}^{N-1}H_{*}^{c}(w_{k^{\prime\prime}}(\mathbb{P}^{n}\setminus\mathbb{P}^{1}),\pm\mathbb{Q})\mathbb{L}^{-(k^{\prime\prime})(n+1)}$
$-\mathbb{L}^{-h^{0}(N(L),\mathcal{O}(d))}\sum_{k^{\prime}=0}^{2}H_{*}^{c}(w_{k^{\prime}}(\mathbb{P}^{1}),\pm\mathbb{Q})\sum_{k^{\prime\prime}=N-k^{\prime}-1}^{N-1}H_{*}^{c}(w_{k^{\prime\prime}}(\mathbb{P}^{n}\setminus\mathbb{P}^{1}),\pm\mathbb{Q})\mathbb{L}^{-(k^{\prime\prime})(n+1)}.$
By our previous computation the first term is 0. The second term is easily
seen to be of dimension $\leq-(\frac{3n}{2}+\epsilon)d$. This establishes the
proof of (7) assuming the claim. We will now prove the claim.
Let
$X_{\lambda,j,l}=\\{(f,Z)|(f,Z)\in W_{\lambda,\geq
N}^{l,n},L\subseteq\textrm{Sing}(f),|Z\cap L|=j\\}$
Let
$Y_{\lambda,j,l}=\sum_{|\alpha|=j,\alpha\subseteq\lambda}w_{\alpha}(L)w_{\lambda\setminus\alpha}(\mathbb{P}^{n}\setminus
L)\mathbb{L}^{-(n+1)(|\lambda|-|\alpha|)-h^{0}(N(L),\mathcal{O}(d))}$
We will establish that
$\sum_{|\lambda|<N}(-1)^{||\lambda||}X_{\lambda,j,l}-Y_{\lambda,j,l}$ is of
dimension $\leq h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-N$ which will immediately
give us the claim.
We may further stratify $X_{\lambda,j,l}$ into the pieces
$X_{\lambda,j,l,k}=\\{(f,Z)\in
X_{\lambda,j,l}|h^{1}(\mathbb{P}^{n},I^{2}_{Z\cup
L}(d))=h^{0}(N(L),\mathcal{O}(d))+(n+1)(|\lambda|-j)+k\\},$
i.e. $X_{\lambda,j,l,k}$ consists of those $Z$ such that vanishing to order 2
on $Z\cup L$ imposes $k$ fewer linear conditions than expected. Let
$Y_{\lambda,j,l,k}=X_{\lambda,j,l,k}\mathcal{L}^{-}k$. We note that
$\sum_{k}Y_{\lambda,j,l,k}=Y_{\lambda,j,l}$.
We now note that it suffices to establish that
$\sum_{|\lambda|<N}(-1)^{||\lambda||}X_{\lambda,j,l,k}-Y_{\lambda,j,l,k}$ is
of dimension $\leq h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-N$. For $k=0$, the
above is exactly $0$.
Let us assume $k>0$. We then note that by an argument similar to that of
Proposition 3.1 there must exist a curve $C^{\prime}$ such that
$h^{1}(\mathbb{P}^{n},I^{2}_{Z\cup L}(d))=h^{1}(N(C^{\prime}\cup
L),I^{2}_{Z\cap(C)\cup L}(d)).$ Further more the above curve $C^{\prime}$ is
uniquely determined by $Z$ assuming it is of minimal degree.
Let
$X_{\lambda,j,l,k}^{r}\subseteq X_{\lambda,j,l,k}\times Chow(r)$
consist of all $(f,Z,C)$ such that
$h^{1}(\mathbb{P}^{n},I^{2}_{Z\cup L}(d))=h^{1}(N(C\cup L),I^{2}_{Z\cap(C)\cup
L}(d)),$
and $C$ is of minimal degree. Thus we have that
$X_{\lambda,j,l,k}=\sum_{r}X_{\lambda,j,l,k}^{r}$
Let $Y_{\lambda,j,l,k}^{r}=X_{\lambda,j,l,k}^{r}\mathbb{L}^{-k}$. Note that
$\sum_{k,r}Y_{\lambda,j,l,k}^{r}=Y_{\lambda,j,l}.$
We now consider the difference $X_{\lambda,j,l,k}^{r}-Y_{\lambda,j,l,k}^{r}$.
It suffices to establish for all fixed values of $j,l,k,r$ that
$\sum_{|\lambda|<N}(-1)^{||\lambda||}X_{\lambda,j,l,k}^{r}-Y_{\lambda,j,l,k}^{r}$
is of dimension $\leq h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-N$.
It is an easy computation to note that for $r\geq 2$ the above difference is
of dimension $\leq h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-N$. Also for $r=0$,
the above term always vanishes. Thus the only case that we must consider is
when $r=1$. Let $M_{1}$ denote the space of all lines in $\mathbb{P}^{n}$
intersecting $L$ at a single point. Let $M_{2}$ denote the space of all lines
in $\mathbb{P}^{n}$ not intersecting $L$. Let
$X_{\lambda,j,l,k}^{M_{i}}=\\{(f,Z,L^{\prime})\in
X_{\lambda,j,l,k}^{1},L^{\prime}\in M_{i}\\}.$
Let
$Y_{\lambda,j,l,k}^{M_{i}}=X_{\lambda,j,l,k}^{M_{i}}\mathbb{L}^{-k}.$
Let
$\bar{X}_{\lambda,j,l,k}^{M_{i}}=\\{(f,Z,L^{\prime})\in
X_{\lambda,j}|h^{1}(L^{\prime}\cup L,I^{2}_{Z\cap(L^{\prime})\cup
L}(d))=k\\}.$
Let
$\bar{Y}_{\lambda,j,k}^{M_{i}}=\bar{X}_{\lambda,j,k}^{M_{i}}\mathbb{L}^{-k}.$
We claim that:
$\bar{X}_{\lambda,j,l,k}^{M_{i}}-X_{\lambda,j,l,k}^{M_{i}}$ and
$\bar{Y}_{\lambda,j,l,k}^{M_{i}}-Y_{\lambda,j,l,k}^{M_{i}}$ is of dimension
$\leq h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-N.$ This follows from the fact that
these differences are contained in the union of $X_{\lambda,j,l,k}^{r}$ for
$r\geq 2$ and we have already remarked that these have dimension $\leq
h^{0}(\mathbb{P}^{n},\mathcal{O}(d))-N.$.
Thus it suffices to establish that
$\sum(-1)^{||\lambda||}\bar{X}_{\lambda,j,l,k}^{M_{i}}=\sum(-1)^{||\lambda||}barY_{\lambda,j,l,k}^{M_{i}}=0.$
To do so we will explicitly describe the sets
$\bar{X}_{\lambda,j,l,k}^{M_{1}}$ and $\bar{X}_{\lambda,j,l,k}^{M_{2}}$.
We note that we have a Zariski locally trivial bundle
$\bar{X}_{\lambda,j,l,k}^{M_{2}}\to Chow(1)$ defined by
$(f,Z,L^{\prime})\mapsto L^{\prime}$. Let us denote the fibre of this map by
$F_{\lambda,j,l,k}^{M_{2}}$, it is clear that
$X_{\lambda,j,l,k}^{M_{2}}=Chow(1)F_{\lambda,j,l,k}^{M_{2}}$ The fibre over
$L^{\prime}$ consist of those $Z$ such that $|Z\cap L|=j$ and
$h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap L^{\prime}\cup
L}(d))=k+h^{1}(\mathbb{P}^{n},I^{2}_{L}(d))$ or equivalently
$h^{1}(\mathbb{P}^{n},I^{2}_{Z\cap}{d})=k$. But as is established in
Proposition 6.10, this just implies that $|Z\cap L|=f(k)$ for some number
$f(k)$ depending on $k$. We may then apply Proposition 6.9 to conclude that
$F_{\lambda,j,l,k}^{M_{2}}$ is a polynomial in $\mathbb{L}$ and it suffices to
establish that the image of
$\sum_{|\lambda|<N}(-1)^{||\lambda||}\bar{F}_{\lambda,j,l,k}^{M_{2}}$ is zero
in $K_{0}(MHS)$.
But by Proposition 4.8 the image of
$\sum(-1)^{||\lambda||}\bar{F}_{\lambda,j,l,k}^{M_{i}}$ in $K_{0}(MHS)$ is
$\sum_{i=0}^{N}H_{*}^{c}(w_{j}(L)\times w_{f(k)}(L^{\prime})\times
w_{i-j-f(k)}(\mathbb{P}^{n}\setminus L\setminus L^{\prime}),\pm,\mathbb{Q}),$
where in all relevant cases $f(k)\geq\frac{d}{2}$ which since $d$ is assumed
to be large is greater than $2$. Thus by Lemma 1.2 in [17] the above
cohomology group vanishes and we have the desired equality.
Similarly
$\sum_{|\lambda|<N}(-1)^{||\lambda||}\bar{X}_{\lambda,j,l,k}^{M_{1}}=0.$
and
|
Beam Instabilities
G. Rumolo
CERN, Geneva, Switzerland
When a beam propagates in an accelerator, it interacts with both the external
fields and the self-generated electromagnetic fields. If the latter are
strong enough, the interplay between them and a perturbation in the beam
distribution function can lead to an enhancement of the initial perturbation,
resulting in what we call a beam instability. This unstable motion can
be controlled with a feedback system, if available,
or it grows, causing beam degradation and loss.
Beam instabilities in particle accelerators have been
studied and analysed in detail since the late 1950s. The subject owes its relevance to the fact
that the onset of instabilities usually determines the performance of an
accelerator. Understanding and suppressing the underlying sources and
mechanisms is therefore the key to overcoming intensity
limitations, thereby pushing forward the performance reach of a machine.
§ INTRODUCTION
The motion of charged particles forming a beam in an accelerator can be studied either individually or taking into account the electromagnetic interaction
between them.
In the former case, the beam is regarded as a collection of non-interacting
particles and the forces acting on them, i.e. the driving terms in each
particle's equations of motion, are fully prescribed by the
accelerator design. The study of the single-particle dynamics is then
complicated by all non-linear components of the applied electromagnetic fields.
In practice, this description is sufficient as long as additional electromagnetic
fields caused by the presence of the whole beam of particles are not strong enough
to perturb significantly the motion imparted by the external fields.
In many applications, however, beams carrying a high charge
(high intensity) and densely packed in a tiny phase space (high brightness)
are required, for which the electromagnetic fields created
by the interaction of the beam with the external environment need to be included when solving the particles' motion.
Under unfavourable conditions, these electromagnetic fields act back on the beam distribution
itself in a closed loop, such as to enhance a however small initial perturbation.
This situation eventually leads to an instability. The most general example of an instability loop is schematically
illustrated in Fig. <ref>.
Schematic of the closed loop through which a beam can become unstable under
the effect of self-generated electromagnetic fields.
The block labelled `Interaction between beam and external environment' has been
willingly left vague, as any further specification depends on the type of problem
being modelled. In the most frequent case, which will also be the subject of this article, the interaction between beam and external
environment will be purely electromagnetic, so that it can be expressed in terms
of Maxwell's equations with the beam as source term and boundary conditions given
by the accelerator devices through which the beam propagates. Another case that is frequently the object
of study is when the beam generates an
electron or ion cloud that acts back on the beam itself and potentially destabilizes it. In this case, the interaction of
the beam with the environment needs to be described with all of the physical processes
leading to the cloud formation. The additional electric field from the cloud can then
be evaluated through Poisson's equation and used as a driving term in the equations
of motion of the beam particles.
In practice, a beam becomes unstable when, as a result of the loop described above, at least a moment of its six-dimensional (6D) phase space
distribution, $\psi(x,y,z,x',y',\delta)$, exhibits an exponential growth (e.g. typically the
mean positions $\langle x\rangle$, $\langle y\rangle$,
$\langle z\rangle$ or the standard deviations $\sigma_x$, $\sigma_y$, $\sigma_z$),
resulting in beam loss or emittance growth. Assuming an arbitrary observation point $s_0$ along the trajectory
of a beam inside an accelerator, described through the coordinate $s$, the full 6D phase space can usually be decomposed into transverse and longitudinal phase spaces. The 4D transverse space is described by the two pairs of conjugate variables $(x, x',y, y')$, i.e. the offsets from the nominal orbit in the horizontal and vertical directions (horizontal is the direction in which the beam is bent), and the relative divergences from the nominal orbit, $x'={\rm d}x/{\rm d}s$ and $y'={\rm d}y/{\rm d}s$. The longitudinal plane is described by the conjugate pair $(z,\delta)$, i.e. a space coordinate proportional to the delay in the arrival time at the selected location with respect to the synchronous particle, $z=-c\tau$ (the minus sign is chosen such that particles arriving before the synchronous particle have a positive $z$), and the relative longitudinal momentum deviation from the nominal momentum, $\delta=\delta p/p_0$. As an example of instability detection, the onset of a transverse instability can be easily revealed by the signal captured from a beam position monitor (BPM). A phase of exponential growth can be observed, usually followed by saturation and decay either due to non-linearities or because of beam loss. Figure <ref> shows an example of horizontal BPM signals from two different bunches during the store of a train of 72 bunches with 25 ns spacing in the CERN-Proton Synchrotron (PS). The signal in Fig. <ref>(a) is basically BPM noise and represents a stable bunch, while that in Fig. <ref> is from an unstable one. This also highlights how the unstable beam oscillation is eventually associated with a certain amount of beam loss and is damped after the loss occurs.
(a) (b)
Examples of stable (a) and unstable (b) signals from a BPM. The
signal from a beam current transformer (BCT) is also sketched, showing how the stable
beam does not suffer from any intensity loss, while a sharp intensity decrease is associated
with the rise of the instability.
The interest in studying coherent beam instabilities arises from the fact that the onset of a beam instability usually determines the maximum beam intensity that a machine can store/accelerate (i.e. its performance limitation). Understanding the type of instability limiting the performance, and its underlying mechanism, is essential because it allows the source and possible measures to mitigate/suppress the effect to be identified, or providing the specifications of an active feedback system to prevent the instability. Beam instabilities occur in both linear and circular machines and can equally affect the longitudinal plane or the transverse plane. Coherent instabilities can affect the beam on different scales. For example, a typical multibunch instability exhibits an excitation pattern extending over different bunches in a train and depends on a long-range coupling agent. Nevertheless, in some cases the unstable motion of subsequent bunches does not appear as coupled, because the instability can be just the consequence of a certain mechanism that builds up along the bunch train, but visibly affects only the last bunches of a train (e.g. an electron cloud). In a pure single-bunch instability, usually the coupling happens between head and tail of the same bunch. In this case, the mechanism that drives the instability only needs to act on the short range.
In the following sections, we will first set the mathematical framework to address the problem of beam instabilities driven by self-generated electromagnetic fields (wake functions and impedances) and we will then apply these concepts to reduced models (one- or two-particle) to explain the physics of some of the most frequent instability mechanisms in particle accelerators. The reference that will be followed throughout this article is [1].
§ THE LONGITUDINAL PLANE
Let us consider two ultra-relativistic charged particles ($q_1$ and $q_2$, travelling at $v\approx c$, or equivalently $\gamma\gg 1$) going through an accelerator structure, separated by a distance $|z|$ ($z=-c\tau$, with $\tau$ expressing the delay between the arrival times of the two particles at an arbitrary location). The leading particle will be our source and the trailing particle will be the witness. Since both particles are travelling basically at the speed of light, causality imposes that the leading particle cannot be affected by the trailing particle. As long as source and witness move in a perfectly conducting chamber, the witness does not feel any force from the source. However, when the source encounters a discontinuity, the electromagnetic field produced to satisfy the boundary conditions (wake field) can effectively reach the witness particle and affect its motion. In this process, the source loses energy, while the witness feels a net force all along an effective length, $L$, of the discontinuity/structure/device that caused the wake. Figure <ref> shows a simple sketch of how the situation could look like after a source has gone through a cavity-like object and modes are trapped after its passage.
Wake field from a source particle potentially affecting a witness travelling at distance $z$ behind the source
In fact, geometric discontinuities are not the only possible origin of wake fields. For example, in a chamber with finite conductivity the induced current from a source particle is delayed and can significantly act back on witness particles within a certain distance range. Generally, electromagnetic boundary conditions other than a perfect electrical conductor (PEC) can generate wake fields.
The longitudinal wake function associated with a certain accelerator object (able to create a wake field) is defined as the integrated longitudinal force ($q_2E_s(s,z)$) acting on the witness particle along the effective length $L$ of the object (i.e. its energy change, $\Delta E_2$), normalized by the source and witness charges:
\begin{equation}
W_{||}(z)=-\frac{\int_0^L E_s(s,z) \, {\rm d}s}{q_1} = -\frac{\Delta E_2}{q_1q_2}.
\label{longwake}
\end{equation}
The minus sign is also introduced in the definition, so that $W(0)=-\Delta E_1/q_1^2$ is defined positive (the source particle can only lose energy, $\Delta E_1<0$). The beam loading theorem also proves that the wake function is discontinuous in $z=0$, with $W_{||}(0^-)=2\cdot W_{||}(0)$. Intuitively, this theorem states that a particle travelling at the speed of light can only see half of its own wake. Besides, causality imposes that $W_{||}(0^+)=0$, and actually $W_{||}(z)=0$ for $z>0$. In a global energy balance, the energy lost by the source, $\Delta E_1$, splits into
* Electromagnetic energy of the modes that may remain trapped in the object. This is then partly dissipated on the lossy walls or into purposely designed inserts or higher order mode (HOM) absorbers. Partly, it can be potentially transferred to the trailing particles (or the same particle over successive turns), possibly feeding into an instability.
* Electromagnetic energy of modes that propagate down the beam chamber (above cut-off), which will be eventually lost on surrounding lossy materials.
The energy loss of a beam is very important, because the fraction lost on the beam environment causes equipment heating (with consequent outgassing and possible damage), while the part associated with long-lived wake fields can feed into both longitudinal and transverse instabilities. The calculation of the energy loss will be the subject of the next subsection.
The wake function of an accelerator object is basically its Green function in the time domain (i.e. the electromagnetic response of the object to a pulse excitation). Therefore, it is very useful for macroparticle models and simulations, because it can be used to describe the driving terms in the single-particle equations of motion, as we will see in one of the next subsections. However, we can also describe this response as a transfer function in the frequency domain.
This gives the definition of longitudinal beam coupling impedance of the object under study:
\begin{equation}
Z_{||}(\omega)=\int_{-\infty}^{\infty} W_{||}(z) \exp\left(-\frac{{\rm i}\omega z}{c}\right)\frac{{\rm d}z}{c}.
\label{longimped}
\end{equation}
Typical longitudinal wake/impedance pairs are described as resonators and are displayed in Fig. <ref>. The wake function is a damped oscillation with a discontinuity in $z=0$, while the beam coupling impedance spectrum exhibits a peak at the specific oscillation frequency. The width of the peak relates to the lifetime of the oscillation in the time domain before becoming fully damped, distinguishing between a narrowband and a broadband resonator, as shown in top and bottom of Fig. <ref>, respectively. In more complex cases, several modes can be excited in the object and the beam coupling impedance is a combination of several peaks like those shown in the single-resonance examples depicted in Fig. <ref>. For example, a pill-box cavity with walls having finite conductivity and attached to a vacuum chamber left and right (Fig. <ref>(a)) can resonate on all its characteristic modes determined by its geometry. The width of the excited peaks will be narrower for the modes below the cut-off frequency of the chamber (as the decay is purely determined by the resistive losses), while they will be broader for the peaks above cut-off, as additional losses come from the propagation of these modes into the chamber. This is visible in Fig. <ref> (simulations done with CST® Particle Studio Suite).
Wake functions (left) and beam coupling impedances (right) for narrowband (top) and broadband (bottom) resonator objects.
(a) (b)
Pill-box cavity: a 3D longitudinal cut of the simulated cavity (a) and the obtained
longitudinal beam coupling impedance (b). The cut-off frequency of the beam chamber is shown with a vertical dashed line. Courtesy of C. Zannini.
In beam physics, broadband impedances, such that the associated wake functions decay over the length of one particle bunch, are only responsible for intrabunch (head–tail) coupling, potentially leading to single-bunch instabilities. Conversely, narrowband impedances, associated with long-lived wake functions decaying over the length of a bunch train or several turns, cause bunch-to-bunch or multiturn coupling, leading to multibunch or multiturn instabilities.
§.§ Energy loss
By using the concepts so far introduced, we can easily derive an analytical expression for the energy lost by a bunch with line density $\lambda(z)$ (see Fig. <ref>) when it goes through a structure characterized by a wake function $W_{||}(z)$ or beam coupling impedance $Z_{||}(\omega)$. The energy change $\Delta E(z)$ of the witness slice $e\lambda(z) \, {\rm d}z$ can be expressed as the integral of the contributions from the wake functions generated by all the preceding source slices, $e\lambda(z') \, {\rm d}z'$. Integrating $\Delta E(z)$ over the whole bunch provides the total energy loss of the bunch:
\begin{equation}
\Delta E = \int_{-\hat{z}}^{\hat{z}}\Delta E(z) \, {\rm d}z = - \int_{-\hat{z}}^{\hat{z}} e\lambda(z) \int_z^{\hat{z}}e\lambda(z')W_{||}(z-z') \, {\rm d}z' \, {\rm d}z.
\label{bunchenloss1}
\end{equation}
By using the Parseval identity and the convolution theorem, the energy loss can be easily written in terms of bunch spectrum $\Lambda(\omega)$ and beam coupling impedance:
\begin{equation}
\Delta E = -\frac{e^2}{2\pi}\int_{-\infty}^{\infty}\Lambda^*(\omega)\left[\Lambda(\omega)Z_{||}(\omega)\right] \, {\rm d}\omega=-\frac{e^2}{2\pi}\int_{-\infty}^{\infty}|\Lambda(\omega)|^2\mathrm{Re}\left[Z_{||}(\omega)\right] \, {\rm d}\omega.
\label{bunchenloss2}
\end{equation}
Sketch of the bunch and line density. Source and witness slices are also highlighted
In the last expression, we also took into account that, since $W_{||}(z)$ is a real function, $\mathrm{Re}[Z_{||}(\omega)]$ and $\mathrm{Im}[Z_{||}(\omega)]$ are even and odd functions of $\omega$, respectively. Since Eq. (<ref>) represents the total energy lost by the bunch over a single pass through the object with beam coupling impedance $Z_{||}(\omega)$, it can also be interpreted as the bunch energy loss per turn in a circular accelerator (again due to a single object with beam coupling impedance $Z_{||}(\omega)$, or the total energy loss per turn if $Z_{||}(\omega)$ represents instead the total longitudinal beam coupling impedance modelling the whole ring). However, this is rigorously true only as long as the wake function is short enough lived to be fully damped after one turn, so that subsequent passages of the bunch are not coupled through the wake.
In fact, defining $C$ as the circumference of the ring, Eqs. (<ref>) and (<ref>) can be generalized to the case of a bunch going through a structure that keeps memory of previous passages, assuming that its longitudinal distribution does not change in time:
\begin{equation}
\Delta E = \int_{-\hat{z}}^{\hat{z}}\Delta E(z) \, {\rm d}z = - \int_{-\hat{z}}^{\hat{z}} e\lambda(z) \int_z^{\hat{z}}e\lambda(z')\sum_{k=-\infty}^{\infty}W_{||}(kC+z-z') \, {\rm d}z' \, {\rm d}z.
\label{bunchenloss1multi}
\end{equation}
Applying the identity
\begin{equation}
\sum_{k=-\infty}^{\infty}W_{||}(kC+z-z')=\frac{\omega_0}{2\pi}\sum_{p=-\infty}^{\infty}Z_{||}(p\omega_0)\exp\left[-\frac{{\rm i}p\omega_0(z-z')}{c}\right],
\end{equation}
in which $\omega_0=2\pi c/C$ is the revolution frequency, we can easily recast Eq. (<ref>) in the following form:
\begin{equation}
\Delta E = -\frac{e^2\omega_0}{2\pi}\sum_{p=-\infty}^{\infty}|\Lambda(p\omega_0)|^2\mathrm{Re}\left[Z_{||}(p\omega_0)\right].
\label{bunchenloss2multi}
\end{equation}
Equation (<ref>) is very powerful, because it can be applied to the full beam circulating in an accelerator ring and can be used for calculating the total beam energy loss per turn. In this case, we would simply need to replace $\Lambda(\omega)$, the Fourier transform of the single-bunch distribution, with the Fourier transform of the full beam signal, $\Lambda_B(\omega)$. For example, we could assume the beam to be a train of $M$ bunches covering only a fraction of the full circumference ($M<h$, $h$ being the harmonic number of the accelerator) with spacing between bunches $\tau_b=2\pi/(h\omega_0)$:
\begin{equation}
\lambda_B(z)=\sum_{n=0}^{M-1}\lambda(z-nc\tau_b) \;\; \stackrel{\displaystyle\mathcal{F}}{\iff} \;\; \Lambda_B(\omega)=\Lambda(\omega)\sum_{n=0}^{M-1}\exp\left(-{\rm i}\omega\tau_b\right).
\end{equation}
Summing up the terms in the expression of the beam spectrum, we obtain
\begin{equation}
\Lambda_B(\omega)=\Lambda(\omega)\exp\left[\-\frac{{\rm i}\omega\tau_b(M-1)}{2}\right]\cdot\frac{\displaystyle\sin\left(\frac{M\omega\tau_b}{2}\right)}{\displaystyle\sin\left(\frac{\omega\tau_b}{2}\right)},
\end{equation}
which can be finally inserted into Eq. (<ref>), yielding
\begin{equation}
\Delta E = \frac{e^2\omega_0}{2\pi}\sum_{p=-\infty}^{\infty}|\Lambda(p\omega_0)|^2\mathrm{Re}\left[Z_{||}(p\omega_0)\right]
\cdot \left[\frac{1-\displaystyle\cos\left(\frac{2\pi Mp}{h}\right)}{1-\displaystyle\cos\left(\frac{2\pi p}{h}\right)}\right].
\end{equation}
The terms in the summation above are maximum for $p=k\cdot h$, as the ratio in brackets becomes equal to $M^2$. This means that narrowband impedances peaked around multiples of the harmonic number of the accelerator are the most efficient to drain energy from the beam, and consequently the associated objects suffer from beam-induced heating. However, this type of impedances, usually associated with the RF systems and their HOMs, need to be avoided in accelerator design by either detuning them or including HOM absorbers. In fact, they not only cause equipment heating, but potentially lead to important instabilities (e.g. the Robinson instability, see the next subsection, or transverse coupled bunch instabilities).
The total energy loss per turn associated with the global accelerator impedance needs to be compensated for by the RF system, so that the average stable phase shifts by an amount $\langle\Delta \Phi_s\rangle$ given by
\begin{equation}
\sin\langle\Delta\Phi_s\rangle = \frac{\Delta E}{MN_beV_m},
\label{phaseshift}
\end{equation}
where $N_b$ is the number of particles per bunch and $V_m$ is the applied RF voltage.
§.§ The Robinson instability
To study instabilities, the effect of wake fields (or impedances) must be formally introduced in the equation of motion of the beam particles. Resorting to the concepts introduced at the beginning of this section, we can write the equation of motion of any single particle in the witness slice $\lambda(z) \, {\rm d}z$ under the effect of the force from the RF system and that associated with the wake, which can extend to several turns:
\begin{equation}
\frac{{\rm d}^2z}{{\rm d}t^2} + \frac{\eta e V_{\mathrm{RF}}(z)}{m_0\gamma C} = \frac{\eta e^2}{m_0\gamma C}\int_{z}^{\infty}\sum_{k=0}^{\infty}\lambda(z'+kC, t)W_{||}(z-z'-kC) \, {\rm d}z'.
\label{motion}
\end{equation}
Equation (<ref>) is very general and can be used in macroparticle tracking programs, which solve it for each macroparticle, determining self consistently the full beam evolution $\lambda(z,t)$. It is to be noted that both the integral and the summation in the above equation can be formally extended from $-\infty$, as the wake function vanishes for positive values of $z$ due to causality.
In the following, to illustrate the most basic mechanism of longitudinal instability, i.e. the Robinson instability, we will make use of these simplifications:
* The bunch is assumed to be point-like (carrying the full bunch charge $N_be$) and feels an external linear focusing force (i.e. in absence of the wake forces, it would execute linear synchrotron oscillations with synchrotron frequency $\omega_s$).
* The bunch additionally feels the effect of the multiturn wake from an impedance source distributed over the ring circumference $C$ (the analysis would not change if the impedance source had been lumped at one ring location, and in reality both the external voltage and the impedance source should be localized, making Eq. (<ref>) de facto time discrete).
In this case, the equation of motion (<ref>) reduces to
\begin{equation}
\frac{{\rm d}^2z}{{\rm d}t^2} +\omega_s^2z= \frac{N_b\eta e^2}{m_0\gamma C}\sum_{k=0}^{\infty}W_{||}\left[z(t)-z(t-kT_0)-kC\right].
\label{Rob-motion}
\end{equation}
First of all, we assume that the wake function can be linearized on the scale of the synchrotron oscillation (i.e. the wake function does not exhibit abrupt changes over a half-bucket length):
\begin{equation}
W_{||}\left[z(t)-z(t-kT_0)-kC\right]\approx W_{||}(kC)+W'_{||}(kC)\cdot\left[z(t)-z(t-kT_0)-kC\right].
\end{equation}
We can use the above expansion in Eq. (<ref>). The term $\sum_k W_{||}(kC)$ only contributes to a constant term in the solution of the equation of motion, shifting the centre of the synchrotron oscillation from the bucket centre to a certain $z_0$. This term represents the stable phase shift that compensates for the energy loss introduced by the wake and will be neglected in the following. The dynamic term proportional to $z(t)-z(t-kT_0) \approx kT_0 \, {\rm d}z/{\rm d}t$, instead, is a friction-like term in the equation of the harmonic oscillator and, under certain conditions, can lead to an instability. Going to the frequency domain then yields
\begin{equation}
\omega^2 - \omega_s^2 = -\frac{{\rm i}N_b\eta e^2}{m_0\gamma C^2}\sum_{p=-\infty}^{\infty}\left[p\omega_0Z_{||}(p\omega_0)-(p\omega_0+\omega)Z_{||}(p\omega_0+\omega)\right].
\end{equation}
At this point, assuming that the wake only introduces a small deviation from the nominal synchrotron frequency, we can easily write the complex frequency shift, which results in a real part (synchrotron frequency shift) and an imaginary part (growth/damping rate):
\begin{equation}
\left\{\begin{array}{l}
\Delta\omega_s=\mathrm{Re}(\omega-\omega_s)=\displaystyle\left(\frac{e^2}{m_0c^2}\right)\frac{N_b\eta}{2\gamma T_0^2\omega_s}\sum_{p=-\infty}^{\infty}\left[p\omega_0\mathrm{Im}Z_{||}(p\omega_0)-(p\omega_0+\omega_s)\mathrm{Im}Z_{||}(p\omega_0+\omega_s)\right],\\[0.7cm]
\tau^{-1}=\mathrm{Im}(\omega-\omega_s)=\displaystyle\left(\frac{e^2}{m_0c^2}\right)\frac{N_b\eta}{2\gamma T_0^2\omega_s}\sum_{p=-\infty}^{\infty}(p\omega_0+\omega_s)\mathrm{Re}Z_{||}(p\omega_0+\omega_s).
\end{array}
\right.
\label{shifts}
\end{equation}
The possibility of having an instability is related to a positive value of $\tau$ in the second of the equations in (<ref>). This is determined by the sign of $\eta$ and that of the weighted summation on $\mathrm{Re}Z_{||}$, which are the only two terms that can admit both signs.
A relevant situation that can be studied in further detail is when the impedance has a spectrum peaked at a frequency $\omega_r$ close to the RF frequency $h\omega_0$, or to a multiple of it (i.e. associated with the cavity fundamental mode or with a HOM). In this case, out of the infinite summation only two terms will dominate the right-hand side of the equation for the growth/damping rate:
\begin{equation}
\tau^{-1}=\mathrm{Im}(\omega-\omega_s)\approx\displaystyle\left(\frac{e^2}{m_0c^2}\right)\frac{N_b\eta h\omega_0}{2\gamma T_0^2\omega_s}\left[\mathrm{Re}Z_{||}(h\omega_0+\omega_s)-\mathrm{Re}Z_{||}(h\omega_0-\omega_s)\right].
\end{equation}
Stability requires that $\eta$ and the variation of $\mathrm{Re}Z_{||}(\omega)$ around $n\omega_0 $ have different signs. Figure <ref> shows that, assuming $\omega_s$ to be small with respect to the width of the resonance peak, this can be achieved differently according to whether $\omega_r$ is below or above $n\omega_0$. In particular, when $h\omega_0<\omega_r$ (Fig. <ref>(a)), the term $\left[\mathrm{Re}Z_{||}(h\omega_0+\omega_s)-\mathrm{Re}Z_{||}(h\omega_0-\omega_s)\right]$ is positive and therefore $\eta$ needs to be negative for stability (i.e. the machine should be operating below transition). Otherwise, for $h\omega_0>\omega_r$ (Fig. <ref>(b)), stability is guaranteed only above transition.
(a) (b)
Sketch of the two possible situations for the Robinson instability
Other types of impedances can also cause instabilities through the Robinson mechanism, following the general equations (<ref>).
However, a smooth broadband impedance with no narrow structures on the $\omega_0$ scale cannot give rise to an instability, because
\begin{equation}
\sum_{p=-\infty}^{\infty}(p\omega_0+\omega_s)\mathrm{Re}Z_{||}(p\omega_0+\omega_s) \rightarrow \frac{1}{\omega_0}\int_{-\infty}^{\infty}\omega\mathrm{Re}Z_{||}(\omega) \, {\rm d}\omega \rightarrow 0.
\end{equation}
Physically, this could be expected, because the absence of structure on $\omega_0$ scale in the spectrum implies that the wake has fully decayed over one turn and, therefore, the driving term in the equation of motion (<ref>) also vanishes.
To summarize, the Robinson instability affects a single bunch under the action of a multiturn wake field. It is characterized by a term of coherent synchrotron tune shift (the first of the equations (<ref>)) and an unstable rigid bunch dipole oscillation (growth rate given by the second of the equations (<ref>) under the conditions explained above). It does not involve higher order moments of the bunch longitudinal phase space distribution. Other important collective effects can affect a bunch in a beam, for instance:
* Potential well distortion, resulting in synchronous phase shift, bunch lengthening or shortening, synchrotron tune shift/spread.
* Coupled bunch instabilities.
* Higher order mode and mode-coupling single-bunch instabilities (e.g. microwave instability).
* Coasting beam instabilities (e.g. negative-mass instability).
To be able to study these effects, more refined modes of the beam are needed (e.g. the kinetic model described by the Vlasov equation or macroparticle simulations), but this is beyond the scope of this introductory article.
§ THE TRANSVERSE PLANE
We can start from the same system we have used in the previous section to introduce the concept of longitudinal wake function. We consider two ultra-relativistic charged particles, $q_1$ and $q_2$, going through an accelerator structure, with the trailing (witness) particle at a distance $z$ from the leading (source) one. In an axisymmetric structure (or simply with a top–bottom and left–right symmetry) a source particle travelling on axis cannot induce net transverse forces on a witness particle also following on axis. A symmetry breaking has to be introduced to drive transverse effects, and at the first order there are two options, i.e. offset the source or the witness (see Fig. <ref>). The transverse (horizontal or vertical) dipolar wake function associated with a certain accelerator object (able to create a wake field) is defined as the integrated transverse force from an offset source ($q_2\cdot [\vec{E}(s,z) + \vec{v}\times \vec{B}(s,z)]_{x,y}$) acting on the witness particle along the effective length of the object, normalized by the source and witness charges and by the offset of the source charge, $\Delta x_1$ or $\Delta y_1$ (see Fig. <ref>, top):
\begin{equation}
\begin{array}{l}
\displaystyle W_{Dx}(z)=-\frac{\int_0^L \left[\vec{E}(s,z) + \vec{v}\times\vec{B}(s,z)\right]_x \, {\rm d}s}{q_1\Delta x_1} = -\left(\frac{E_0}{q_1q_2}\right)\frac{\Delta x'_2}{\Delta x_1},\\[5mm]
\displaystyle W_{Dy}(z)=-\frac{\int_0^L \left[\vec{E}(s,z) + \vec{v}\times\vec{B}(s,z)\right]_y \, {\rm d}s}{q_1\Delta y_1} = -\left(\frac{E_0}{q_1q_2}\right)\frac{\Delta y'_2}{\Delta y_1}.
\end{array}
\label{longwake2}
\end{equation}
Wake field from a source particle potentially affecting a witness travelling at distance $z$ behind the source: dipolar (top) and quadrupolar (bottom).
The transverse (horizontal or vertical) quadrupolar wake function associated with a certain accelerator object (able to create a wake field) is defined as the integrated transverse force from an on-axis source ($q_2\cdot [\vec{E}(s,z) + \vec{v}\times \vec{B}(s,z)]_{x,y}$) acting on an offset witness particle along the effective length of the object, normalized by the source and witness charges and by the offset of the witness charge, $\Delta x_2$ or $\Delta y_2$ (see Fig. <ref>, bottom):
\begin{equation}
\begin{array}{l}
\displaystyle W_{Qx}(z)=-\frac{\int_0^L \left[\vec{E}(s,z) + \vec{v}\times\vec{B}(s,z)\right]_x \, {\rm d}s}{q_1\Delta x_2} = -\left(\frac{E_0}{q_1q_2}\right)\frac{\Delta x'_2}{\Delta x_2},\\[5mm]
\displaystyle W_{Qy}(z)=-\frac{\int_0^L \left[\vec{E}(s,z) + \vec{v}\times\vec{B}(s,z)\right]_y \, {\rm d}s}{q_1\Delta y_2} = -\left(\frac{E_0}{q_1q_2}\right)\frac{\Delta y'_2}{\Delta y_2}.
\end{array}
\label{longwake3}
\end{equation}
For most objects of interest, it can be seen that the wake functions so defined do not depend on the source or witness offsets, provided the offsets are much smaller than the transverse size of the object. For larger offsets, coupling and/or higher order non-linear terms can become important and may need to be taken into account to describe correctly the particle dynamics. Both the dipolar and quadrupolar wake functions in $z=0$, $W_{Dx,Dy}(0)$ and $W_{Qx,Qy}(0)$, must vanish because for $z=0$ source and witness particles are travelling together and they can only mutually interact through space charge, which is not included in this framework. Besides, $W_{Dx,Dy}(0^-)$ is generally negative, because trailing particles tend to be deflected toward the source particle (offset and kick have the same sign). The sign of the quadrupolar wake functions in $0^-$ depends on the geometry and properties of the surrounding environment. As we also discussed for the longitudinal wake function, the condition that transverse wake functions must vanish for $z>0$ due to causality also holds.
The transverse wake function of an accelerator object is very useful for macroparticle models and simulations, because it relates source or witness perturbations to the associated kicks on trailing particles:
\begin{equation}
\left\{\begin{array}{l}
\displaystyle \Delta x'_2(z)=-\left(\frac{q_1q_2}{E_0}\right)\left[ W_{Dx}(z)\Delta x_1 + W_{Qx}(z)\Delta x_2\right], \\[5mm]
\displaystyle \Delta y'_2(z)=-\left(\frac{q_1q_2}{E_0}\right)\left[ W_{Dy}(z)\Delta y_1 + W_{Qy}(z)\Delta y_2\right].
\end{array}
\right.
\end{equation}
We can also describe the interaction as a transfer function in the frequency domain, which defines the transverse beam coupling impedance (dipolar and quadrupolar) of the object under study:
\begin{equation}
\left\{\begin{array}{l}
\displaystyle Z_{Dx,Dy}(\omega)={\rm i}\int_{-\infty}^{\infty}W_{Dx,Dy}(z)\exp\left(\frac{{\rm i}\omega z}{c}\right)\frac{{\rm d}z}{c},\\[5mm]
\displaystyle Z_{Qx,Qy}(\omega)={\rm i}\int_{-\infty}^{\infty}W_{Qx,Qy}(z)\exp\left(\frac{{\rm i}\omega z}{c}\right)\frac{{\rm d}z}{c}.
\end{array}
\right.
\label{transv-imp}
\end{equation}
Similarly to the longitudinal plane, in the transverse plane typical wake/impedance pairs are also represented by resonators, which have a peaked structure in the frequency domain and are damped oscillations in the time domain. Another important example of transverse impedance is the wall impedance. Figure <ref> depicts the dipolar impedance spectrum for a simple cylindrical chamber with a wall of finite thickness $t$, finite conductivity $\sigma$ and radius $b$. This impedance extends over a very wide range of frequencies and exhibits different behaviours that can be intuitively understood as described in the following. At low frequencies, such that the penetration depth of the electromagnetic fields into the chamber is much larger than the wall thickness, i.e. $\delta(\omega)=\sqrt{2/(\mu_0\sigma\omega)}\gg t$, the beam can only see the induced charges on the inner surface of the chamber, associated with a constant imaginary part of the impedance (betatron tune shift by extra defocusing) and basically zero real part (no losses). At intermediate frequencies, the electromagnetic interaction between the beam and the conducting pipe happens through a decreasing $\delta(\omega)$ and the impedance becomes about proportional to the penetration depth and decays like $\sqrt{\omega}$. At high frequency, a point is reached at which the electromagnetic fields can become trapped within the penetration depth, generating a resonant peak.
Wall dipolar impedance for a cylindrical pipe of radius $b$ and thickness $t$, as illustrated. Courtesy of N. Mounet.
Corresponding to the different frequency ranges, the wake function also exhibits different behaviours on different distance ranges from the source charge. In the medium–long range (usually coupled bunch and/or multiturn), the wake function is characterized by a monotonic decay slowly converging to zero, while in the short range (typically responsible for single-bunch effects) the high-frequency resonance peak dominates and a damped oscillatory behaviour is found. The wall wake function is displayed in Fig. <ref>, in which the short-range part has also been zoomed in order to highlight the different behaviour. The switch between the two regimes obviously depends on the frequency at which the peak is actually located.
It can be demonstrated that the transverse impedance of a resistive wall is inversely proportional to $b^3$, while its longitudinal counterpart is inversely proportional to $b$. That is why the transverse effects due to a resistive wall are in general more severe than the corresponding longitudinal ones. In particular, the transverse resistive wall impedance is responsible for coupled bunch instabilities and determines which damping time a feedback system needs in order to be able to efficiently counteract this effect. Furthermore, resistive wall effects become especially important in machines with low-emittance beams, which have chambers with very small radii and might require coatings with low-conductivity materials to avoid other effects.
§.§ The rigid bunch instability
Repeating the same procedure used for the longitudinal plane, to study instabilities in the transverse plane, the effect of wake fields must be formally introduced in the equations of transverse motion of the beam particles. Using the concepts introduced at the beginning of this section, we can write the equations of the transverse motion of any single particle in the witness slice $\lambda(z) \, {\rm d}z$ under the effect of the focusing force from the external magnets and that associated with a distributed wake, which can extend to several turns:
\begin{equation}
\left\{\begin{array}{l}
\displaystyle \frac{{\rm d}^2x}{{\rm d}s^2} + K_x(s)x = -\frac{e^2}{m_0c^2\gamma C}\cdot\\[3mm]
\displaystyle\;\;\;\;\;\;\cdot\sum_{k=0}^{\infty}\int_{z}^{\infty}\lambda(z'+kC,s)\left[\langle x \rangle(z'+kC,s)W_{Dx}(z-z'-kC) + x W_{Qx}(z-z'-kC) \right] \, {\rm d}z',\\[8mm]
\displaystyle\frac{{\rm d}^2y}{{\rm d}s^2} + K_y(s)y = -\frac{e^2}{m_0c^2\gamma C}\cdot\\[3mm]
\displaystyle\;\;\;\;\;\;\cdot\sum_{k=0}^{\infty}\int_{z}^{\infty}\lambda(z'+kC,s)\left[\langle y \rangle(z'+kC,s)W_{Dy}(z-z'-kC) + y W_{Qy}(z-z'-kC) \right] \, {\rm d}z'.
\end{array}
\right.
\label{transv-motion}
\end{equation}
Wall dipolar wake function for a cylindrical pipe of radius $b$ and thickness $t$. The short range has been zoomed in to highlight the oscillatory behaviour of the function at short distances from the source. Courtesy of N. Mounet.
Here $K_{x,y}(s)$ are the external focusing coefficients as from Hill's equation and $\langle x, y\rangle(z,s)$ represent the average $x$ and $y$ across the ${\rm d}N_b(z)=\lambda(z,s) \, {\rm d}z$ particles contained in the thin slice ${\rm d}z$. Equation (<ref>) is very general and can be used in macroparticle tracking programs, which solve it for each macroparticle, determining self consistently the evolution of $\lambda(z,s)$ as well as of $\langle x, y\rangle(z,s)$. As in the longitudinal case, all integrals and summations in the above equations can be formally extended from $-\infty$, as the wake functions vanish for positive values of $z$ due to causality.
In the following, to illustrate the most basic mechanism of transverse instability, i.e. the rigid bunch instability, we will make use of a set of simplifying assumptions:
* The bunch is point-like, carries a charge $N_be$ and feels an external linear force (i.e. it would execute linear betatron oscillations in absence of the wake forces).
* Longitudinal motion is neglected.
* Smooth approximation (constant focusing).
* Distributed wake, only dipolar and only in the $y$ direction.
The equation of motion of this one-particle beam can then be written
\begin{equation}
\displaystyle\frac{{\rm d}^2y}{{\rm d}s^2} + \left(\frac{\omega_{\beta}}{c}\right)^2y=-\left(\frac{N_be^2}{m_0c^2\gamma C}\right)\sum_{k=0}^{\infty}y(s+kC)W_{Dy}(-kC).
\end{equation}
Transforming into the frequency domain and applying the Poisson sum formula, we obtain
\[
\displaystyle\omega^2 - \omega_{\beta}^2 = \frac{N_be^2}{m_0\gamma C}\sum_{k=-\infty}^{\infty}\exp(ik\omega T_0)W_{Dy}(kC)=
\]
\begin{equation}
\displaystyle\;\;\;=-\frac{iN_be^2}{m_0\gamma CT_0}\sum_{p=-\infty}^{\infty}Z_y(p\omega_0+\omega) \;\; .
\end{equation}
Assuming that the effect of the wake results in a small deviation from the betatron tune, we can derive from the above equation a simple estimate of the real frequency shift (tune shift) and the imaginary frequency shift (growth/damping rate):
\begin{equation}
\begin{array}{l}
\displaystyle \frac{\mathrm{Re}(\omega-\omega_{\beta})}{\omega_0}=\Delta Q_y \approx \frac{N_be^2\langle\beta_y\rangle}{4\pi m_0\gamma c C}\sum_{p=-\infty}^{\infty}\mathrm{Im}\left[Z_{Dy}(p\omega_0+\omega_{\beta})\right], \\[6mm]
\displaystyle \mathrm{Im}(\omega-\omega_{\beta})=\tau^{-1}\approx -\frac{N_be^2\langle\beta_y\rangle}{2m_0\gamma C^2}\sum_{p=-\infty}^{\infty}\mathrm{Re}\left[Z_{Dy}(p\omega_0+\omega_{\beta})\right].
\end{array}
\label{rigid-eqs}
\end{equation}
With the given definition of the transverse impedance (i.e. including the imaginary unit in Eqs. (<ref>)), the tune shift is found to depend only on the imaginary part of the impedance, while the growth/damping rate depends only on its real part. It is interesting to note that the tune shift can also be expressed in the following well-known compact form:
\begin{equation}
\Delta Q_y = \frac{1}{4\pi}\left[\langle\beta_y\rangle \frac{eI_b\mathrm{Im}(Z_{Dy}^{\mathrm{eff}})}{E} \right]\rightarrow\frac{1}{4\pi}\oint\beta_y(s)\Delta k_y(s) \, {\rm d}s,
\end{equation}
where $\Delta k_y(s)$ is the distributed quadrupolar error due to the wake, which can be written as the relative energy kick from the wake $\Delta E_y/E$ divided by the circumference $C$. Furthermore, besides the tune shift, the presence of the wake introduces an imaginary part of the betatron frequency shift, which, if positive, can result in a beam instability. In particular, the summation in the second of the equations (<ref>) can be positive or negative, because $\mathrm{Re}[Z_{Dy}(\omega)]$ is an odd function. Unlike the case of the Robinson instability in the longitudinal plane, here the sign of the imaginary frequency shift is solely determined by the sign of this summation. In a first noteworthy case, similarly to what we discussed in the longitudinal case, we can assume the transverse impedance to be peaked at a frequency $\omega_r$ close to $h\omega_0$ (e.g. RF cavity fundamental mode or a HOM). If we define the tune $Q_y=n_y + \Delta_{\beta y}$ with $-0.5<\Delta_{\beta y}<0.5$ and we make use of the property of the real part of the impedance to be an odd function of $\omega$, we can easily reduce the summation at the right-hand side of the second of the equations (<ref>) to the sum of its two leading terms alone ($p$ such that $p+n_y=h$):
\begin{equation}
\tau^{-1}\approx -\frac{N_be^2\langle\beta_y\rangle}{2m_0\gamma C^2}\left(\mathrm{Re}[Z_{Dy}(h\omega_0+\Delta_{\beta y}\omega_0)] - \mathrm{Re}[Z_{Dy}(h\omega_0-\Delta_{\beta y}\omega_0)] \right).
\label{stab-res}
\end{equation}
Figure <ref> illustrates the two possible situations that can occur, corresponding to the case of positive $\Delta_{\beta y}$, i.e. tune below the half integer. If $\omega_r>h\omega_0$ (Fig. <ref>(a)), the right-hand side of Eq. (<ref>) is negative and the bunch will be stable. Conversely, if $\omega_r<h\omega_0$
(Fig. <ref>(b)), the right-hand side of Eq. (<ref>) is positive, entailing an instability.
Obviously, the situation is reversed when the tune is above the half integer ($\Delta_{\beta y}<0$).
(a) (b)
Sketch of the two possible situations for the rigid bunch instability with resonator impedance
Another interesting case is when the impedance is of resistive wall type, i.e. strongly peaked in the very low frequency range (diverging for $\omega\rightarrow 0$ in the approximation of thick wall). Then we can distinguish the two situations depicted in Fig. <ref>, i.e. fractional tune below or above the half integer. In the former case, the two leading terms of the summation can be expressed as
\begin{equation}
\tau^{-1}\approx -\frac{N_be^2\langle\beta_y\rangle}{2m_0\gamma C^2}\left(\mathrm{Re}[Z_{Dy}(\Delta_{\beta y}\omega_0)] - \mathrm{Re}[Z_{Dy}((1-\Delta_{\beta y})\omega_0)] \right),
\label{stab-resb}
\end{equation}
which is negative (see Fig. <ref>, top plot), ensuring beam stability. In the latter case, i.e. for fractional tune above the half integer, we obtain
\begin{equation}
\tau^{-1}\approx -\frac{N_be^2\langle\beta_y\rangle}{2m_0\gamma C^2}\left(\mathrm{Re}[Z_{Dy}((1+\Delta_{\beta y})\omega_0)] -\mathrm{Re}[Z_{Dy}(-\Delta_{\beta y}\omega_0)] \right).
\label{stab-res2}
\end{equation}
As is visible from Fig. <ref>, bottom plot, this is positive, leading in any case to an instability. This is the reason why most of the running machines are usually operated with a fractional part of the tunes below 0.5, although, in practice, tunes above the half integer can be used, if the resistive wall instability is Landau damped or efficiently suppressed with a feedback system.
Sketch of the two possible situations for the rigid bunch instability with resistive wall impedance
§.§ Strong head–tail instability and transverse-mode coupling
Making now one step further in the description of the mechanisms of transverse instability, the case of the strong head–tail instability, also called transverse mode coupling instability (TMCI), can be illustrated by means of a simple two-particle model. It is assumed that the beam consists of two macroparticles, each having a charge of $\Np_be/2$. They perform synchrotron oscillations of the same frequency and amplitude, but with opposite phase. During half a synchrotron period $T_s=2\pi/\omega_s$, particle with index 1 is leading and thus performing free betatron oscillations, while the trailing particle indexed 2 feels the wake field generated by particle 1. Thus, for $0 < s < \pi c/\omega_s$, assuming zero chromaticity, and therefore no dependence of the frequency of the transverse oscillation on the longitudinal parameters, the equations of motion for the two macroparticles are simply written as
\begin{align}
y_1''+\left(\frac{\omega_\beta}{c}\right)^2 y_1 &= 0 \label{EQ:2ParticleEquationOfMotionLeading}, \\
y_2''+\left(\frac{\omega_\beta}{c}\right)^2 y_2 &= \left(\frac{e^2}{m_0c^2}\right)\frac{\Np_b \wake_0}{2 \gammarel C}\,y_1(s),
\label{EQ:2ParticleEquationOfMotionTrailing}
\end{align}
where $y_1$ denotes the vertical position of particle 1 and $y_2$ the position of particle 2, and the focusing term in Hill's equation has been written as $K_y=\left({\omega_\beta}/{c}\right)^2$ with $\omega_\beta$ denoting the (vertical) betatron frequency. Note that it is assumed here that the wake field $\wake_0$ (integrated over the machine circumference $C$) is constant but vanishes before the beam passage in the consecutive turn. This corresponds practically to the case of a broadband impedance.
The stability of the two-particle system is analysed in the following. The solution for the free betatron oscillation of Eq. (<ref>) can be written as
\begin{equation}
\tilde y_1(s)=\tilde y_1 (0) \exp{\left(\frac{-{\rm i} \omega_\beta s}{c}\right)},
\end{equation}
where the complex phasor $\tilde y_{1}(s)$
\begin{equation}
\tilde y_{1}(s)=y_{1}(s)+{\rm i}\frac{c}{\omega_\beta}y'_{1}(s)
\end{equation}
has been introduced. Inserting the solution for $\tilde y_1(s)$ into Eq. (<ref>) leads to the solution for $\tilde y_2(s)$
\begin{equation}
\tilde y_2(s)={\tilde y_2 (0) \exp\!{\left(\!-\frac{{\rm i} \omega_\beta s}{c}\right)}} +{{\rm i} \frac{\Np_b e^2 \wake_0}{4 m_0 \gammarel c C \omega_\beta} \left[\frac{c}{\omega_\beta}\tilde y_1^*(0) \sin{\left( \frac{\omega_\beta s}{c}\right)\!+ \tilde y_1 (0)\,s\exp\!{\left(\!-\frac{{\rm i} \omega_\beta s}{c}\right)}} \right]},
\label{EQ:PositionTrailingParticle}
\end{equation}
which consists of the free betatron oscillation term and a driven oscillation term. For the further analysis, the position of the two particles is evaluated at $s=\pi c/\omega_s$, i.e. after half the synchrotron period. Since the betatron frequency is typically much larger than the synchrotron frequency, i.e. $\omega_\beta\gg\omega_s$, the second term on the right-hand side of Eq. (<ref>) is small compared to the last term. Thus, the solutions of the equations of motion can be written in matrix form:
\begin{equation}
\left(\begin{array}{c}\tilde y_1 \\\tilde y_2 \end{array}\right)_{s=\pi c/\omega_s}=\exp{\left(\!-\frac{{\rm i} \pi \omega_\beta}{\omega_s}\right)}\cdot\left(\begin{array}{cc}1 & 0 \\ {\rm i} \Upsilon & 1 \end{array}\right)\cdot\left(\begin{array}{c}\tilde y_1 \\\tilde y_2 \end{array}\right)_{s=0},
\end{equation}
where the positive dimensionless parameter $\Upsilon$ has been defined as
\begin{equation}
\Upsilon=\frac{\pi \Np_b e^2 \wake_0}{4 m_0 \gammarel C \omega_\beta \omega_s}.
\end{equation}
During the second half of the synchrotron period, i.e. $\pi c/\omega_s < s < 2\pi c/\omega_s$, the two particles exchange their roles and now particle 2 is leading, while particle 1 is feeling the wake. Thus, the equations of motion have to be exchanged and, by combining the transformations over the two half synchrotron periods, the transformation matrix for the full synchrotron period can be finally obtained as
\begin{align}
\left(\begin{array}{c}\tilde y_1 \\\tilde y_2 \end{array}\right)_{s=2\pi c/\omega_s}&=\exp{\left(\!-\frac{{\rm i} 2\pi \omega_\beta}{\omega_s}\right)}\cdot\left(\begin{array}{cc}1 & {\rm i} \Upsilon \\ 0 & 1 \end{array}\right)\cdot\left(\begin{array}{cc}1 & 0 \\ {\rm i} \Upsilon & 1 \end{array}\right)\cdot\left(\begin{array}{c}\tilde y_1 \\\tilde y_2 \end{array}\right)_{s=0}\\
&=\exp{\left(\!-\frac{{\rm i} 2\pi \omega_\beta}{\omega_s}\right)}\cdot\left(\begin{array}{cc}1 - \Upsilon^2 & {\rm i}\Upsilon \\ {\rm i}\Upsilon & 1 \end{array}\right)\cdot\left(\begin{array}{c}\tilde y_1 \\\tilde y_2 \end{array}\right)_{s=0}.
\end{align}
The stability of the system is determined by the eigenvalues of the transformation matrix. The characteristic equation for the two eigenvalues $\lambda_{\pm}$ yields
\begin{equation}
\lambda_{\pm}=\left(1-\Upsilon^2/2\right)\pm\sqrt{\left(1-\Upsilon^2/2\right)^2-1}.
\label{EQ:2ParticleModelEigenvalueEquation}
\end{equation}
Since the product of the two eigenvalues is equal to 1, the condition for stability requires that they should be purely imaginary exponentials, i.e.
\begin{equation}
\lambda_{+}\cdot\lambda_{-}=1 ~ \Rightarrow ~ \lambda_{1,2}=\exp{(\pm {\rm i}\upsilon)}.
\end{equation}
Inserting this back into Eq. (<ref>) yields finally
\begin{equation}
\lambda_{+}+\lambda_{-}=2-\Upsilon^2 ~ \Rightarrow ~ \sin{\left(\frac{\upsilon}{2} \right)}=\frac{\Upsilon}{2}.
\end{equation}
Stability requires that $\upsilon$ should be real, which in turn is satisfied only if $\Upsilon\leq 2$. Therefore, the condition for stability written in terms of wake, machine and beam parameters reads
\begin{equation}
\Upsilon = \frac{\pi \Np_b e^2 \wake_0}{4m_0\gammarel C \omega_\beta\omega_s}\leq2.
\end{equation}
The threshold intensity for the onset of the strong head–tail instability in the two-particle model is thus obtained as
\begin{equation}
\Np_{b,\text{thr}}=\frac{8}{\pi e^2}\frac{p_0 \omega_s}{\beta_y}\left(\frac{C}{\wake_0}\right).
\label{EQ:TMCIthreshold2ParticleModel}
\end{equation}
From Eq. (<ref>), we can deduce the following main features of this instability. The intensity threshold:
* is proportional to the momentum $p_o$, i.e. bunches with higher energy are more stable;
* scales proportionally with the synchrotron frequency $\omega_s$, i.e. faster synchrotron motion helps increasing the stability range;
* is inversely proportional to the beta function at the location of the impedance source, which is expected because the strength of a kick is always proportional to the beta function at the kick location;
* is inversely proportional to the integrated wake field around the ring per unit length $\wake_0/C$, which means that a larger wake will decrease the intensity threshold.
The evolution of the centre of charge of the beam in the two-particle model is obtained by the sum of $\tilde y_1 + \tilde y_2$, which is found as
\begin{equation}
\begin{split}
\left(\tilde y_1 + \tilde y_2\right)(s) &= \exp\left[-{\rm i}\left(\omega_\beta \mp \frac{\upsilon \omega_s}{2 \pi} \right)\frac{s}{c}\right] \sum_{m=-\infty}^{\infty}C_m \exp{\left(-\frac{{\rm i} m \omega_s s}{c} \right)},\\
C_m &= 2 {\rm i} \Upsilon \frac{1\pm (-1)^m}{(2\pi m \mp \upsilon)^2}\left( 1\mp e^{\pm {\rm i} \upsilon /2}\right)
\end{split}
\end{equation}
with the amplitude coefficients $C_m$ for the oscillation modes with the mode number $m$. The oscillation frequencies of these
modes are given by
\begin{equation}
\begin{cases}
\Omega_+ = \omega_\beta+m\omega_s+\upsilon\omega_s/2\pi,\quad m~\text{even},\\
\Omega_- = \omega_\beta+m\omega_s-\upsilon\omega_s/2\pi,\quad m~\text{odd}.
\end{cases}
\end{equation}
Thus, as a function of the beam intensity the modes are shifting in frequency through the dependence on $\upsilon$. Figure <ref> shows the frequencies of these modes for $m=0$ and $m=-1$ as a function of $\Upsilon$. The two modes merge at $\Upsilon=2$ and the oscillation frequency becomes imaginary, i.e. the beam becomes unstable and exhibits exponential growth. This is illustrated by plotting also the imaginary part of the oscillation frequencies. The strong head–tail instability is therefore also called TMCI.
Frequency spectrum of the centre of charge motion as a function of the parameter $\Upsilon$ as predicted by the two-particle model.
Beyond the two-particle model, several analytical formalisms have been developed for describing the TMCI. Good agreement between the different approaches is obtained when assuming a broadband resonator $\BBimpedanceVertical$ as driving impedance,
\begin{equation}
\BBimpedanceVertical (\omega)=\frac{\ResonatorFrequency}{\omega}\frac{\shuntImpedance}{\displaystyle 1+{\rm i}\QualityFactor \left(\frac{\omega}{\ResonatorFrequency} - \frac{\ResonatorFrequency}{\omega} \right)},
\end{equation}
where $\ResonatorFrequency$ is the resonance angular frequency, $\QualityFactor$ is the resonator quality factor and $\shuntImpedance$ is the resonator shunt impedance (in $\Omega$/m). In the long bunch regime, i.e. $\blength>\pi/\ResonatorFrequency$, the TMCI threshold can be obtained for example from the quasi-coasting beam approach using the peak values of bunch current and momentum spread, which yields
\begin{equation}
N_{b,\mathrm{thr}}^{\scriptscriptstyle \text{TMC}}=\frac{16\sqrt{2}}{3\pi} \frac{C |\eta| \emitlong}{ \langle\beta_y\rangle e c}\frac{\ResonatorFrequency}{|\BBimpedanceVertical|}\left(1+\frac{ Q'_y\,\omegarev }{\eta\,\ResonatorFrequency} \right), \label{EQ:TMCIthreshold}
\end{equation}
where $C$ is the machine circumference, $|\BBimpedanceVertical|$ is the peak value of the broadband resonator impedance and $\omegarev$ is the angular revolution frequency. Note that in comparison to the instability threshold obtained with the two-particle model in Eq. (<ref>), the TMCI intensity threshold depends here in addition to the synchrotron tune (through the slip factor $\eta$) also on the longitudinal emittance $\emitlong$. Furthermore, the threshold can be raised by operating the machine with positive (negative) chromaticity above (below) transition. For a real bunch under the effect of a generic impedance, modes usually exhibit a more complicated shift pattern, which can be calculated via the Vlasov equation or can be found through macroparticle simulations. Examples of more complicated mode shift pictures for a short bunch (a) and a long Super Proton Synchrotron (SPS) bunch (b) under the effect of a broadband impedance are shown in Fig. <ref>.
Examples of mode shifts for a short bunch ((a), courtesy of A. Chao) and for a long bunch ((b), courtesy of B. Salvant). The left plot contains both the results of an analytical calculation (solid red lines) and those from macroparticle simulations (white lines).
Apart from the TMCI, it can be demonstrated that, including a non-zero value of chromaticity in the previous analysis, the individual coherent modes of a single bunch are intrinsically unstable (head–tail instabilities), even below the threshold for which they couple and give rise to TMCI. In particular, the main mode (mode $m=0$, corresponding to the betatron frequency) is naturally unstable below transition with positive chromaticity or above transition with negative chromaticity. Correspondingly, all higher order modes, $m\geq 1$, are naturally unstable below transition with negative chromaticity or above transition with positive chromaticity. Since, in practice, the most dangerous mode for a bunch is mode $m=0$, which is associated with the fastest rise time, accelerators are always operated with settings such as to keep this mode stable, while the other modes, slowly unstable, are damped through other mechanisms. That is the reason why low-energy accelerators operating below transition energy do not need chromaticity correction and can operate with their natural chromaticity (usually negative) without having problems of beam stability. On the other hand, high-energy accelerators operating above transition energy need sextupoles to correct chromaticity and stabilize the otherwise unstable mode $m=0$. Accelerators crossing transition need to make a chromaticity jump upon transition crossing, such as to ensure that the conditions for stabilizing mode $m=0$ are fulfilled at all times during the cycle.
§ FINAL REMARKS
Although all the mechanisms for beam instability reviewed in this article might seem to define sharp instability boundaries and thin parameter ranges for the operation of accelerators, in real life beam stability is eased by some other mechanisms not included in the simple models analysed in this article:
* Spreads of the beam characteristic frequencies and the possible associated non-linearities have a natural stabilizing action through Landau damping. Examples are momentum spread and synchrotron frequency spread in the longitudinal plane, or chromaticity and amplitude detuning in the transverse plane.
* Active feedback systems are routinely employed to control/suppress instabilities. The principle is that the onset of a beam coherent motion is detected through a pick up, which sends a signal to a kicker that acts back on the beam to damp the motion before it can cause any degradation. Most of the running accelerators rely on this type of device, which is especially efficient against coupled-bunch instabilities. For single-bunch effects, especially in machines operating with short bunches, bandwidth and power requirements can be very stringent, potentially putting a technological limit to the feasibility of the system.
Furthermore, nowadays there is also a constant effort to identify, monitor and control impedance sources in present or future machines. In particular, impedance localization and reduction techniques are applied to running accelerators as well as for the design of new accelerators to extend their performance reach or ensure a smooth operation with the desired (target) parameter sets.
§ ACKNOWLEDGEMENTS
The author would like to thank H. Bartosik, G. Iadarola, K. Li, E. Métral, N. Mounet, B. Salvant, R. Tomás and C. Zannini for their invaluable help and the input/material kindly provided for the preparation of this article.
[1] A. W. Chao, Physics of collective beam instabilities in high energy accelerators (Wiley Series in Beam Physics and Accelerator Technology) (Wiley & Sons Inc., New York, 1993).
|
11institutetext: Tallinn University of Technology
# The Structure of Concurrent Process Histories
Chad Nester This research was supported by the ESF funded Estonian IT Academy
research measure (project 2014-2020.4.05.19-0001).
###### Abstract
We identify the algebraic structure of the material histories generated by
concurrent processes. Specifically, we extend existing categorical theories of
resource convertibility to capture concurrent interaction. Our formalism
admits an intuitive graphical presentation via string diagrams for proarrow
equipments.
## 1 Introduction
Concurrent systems are abundant in computing, and indeed in the world at
large. Despite the large amount of attention paid to the modelling of
concurrency in recent decades (e.g., [12, 19, 22, 20, 1]), a canonical
mathematical account has yet to emerge, and the basic structure of concurrent
systems remains elusive.
In this paper we present a basic structure that captures what we will call the
_material_ aspect of concurrent systems: As a process unfolds in time it
leaves behind a material history of effects on the world, like the way a slug
moving through space leaves a trail of slime. This slime is captured in a
natural way by _resource theories_ in the sense of [5], in which morphisms of
symmetric monoidal categories – conveniently expressed as string diagrams –
are understood as transformations of resources.
$\includegraphics[height=85.35826pt,align=c]{figs/process-slime.png}\hskip
28.45274pt\leftrightsquigarrow\hskip
28.45274pt\includegraphics[height=85.35826pt,align=c]{figs/slug.png}$
From the resource theoretic perspective, objects of a symmetric monoidal
category are understood as collections of resources, with the unit object
denoting the empty collection and the tensor product of two collections
consisting of their combined contents. Morphisms are understood as ways to
transform one collection of resources into another, which may be combined
sequentially via composition, and in parallel via the tensor product. For
example, the process of baking bread might generate the following material
history:
meaning that the baking process involved kneading dough and baking it in an
oven to obtain bread (and also the oven).
This approach to expressing the material history of a process has many
advantages: It is general, in that is assumes minimal structure; canonical, in
that monoidal categories are well-studied as mathematical objects; and
relatively friendly, as it admits an intuitive graphical calculus (string
diagrams). However, it is unable to capture the interaction between components
of a concurrent process. For example, consider our hypothetical baking process
and suppose that the kneading and baking of the dough are handled by separate
subsystems, with control of the dough being handed to the baking subsystem
once the kneading is complete. Such interaction of parts is a fundamental
aspect of concurrency, but is not expressible in this framework – we can only
describe the effects of the system as a whole.
We remedy this by extending a given resource theory to allow the decomposition
of material histories into concurrent components. Specifically, we augment the
string diagrams for symmetric monoidal categories with _corners_ , through
which resouces may flow between different components of a transformation.
$\includegraphics[height=85.35826pt,align=c]{figs/component-clouds.png}\hskip
28.45274pt\leftrightsquigarrow\hskip
28.45274pt\includegraphics[height=85.35826pt,align=c]{figs/composite-
clouds.png}$
Returning to our baking example, we might express the material history of the
kneading and baking subsystems _separately_ with the following diagrams, which
may be composed horizontally to obtain the material history of the baking
process as a whole.
These augmented diagrams denote cells of a single object double category
constructed from the original resource theory. The corners make this double
category into a proarrow equipment, which turns out to be all the additional
structure we need in order to express concurrent interaction.
From only this structure, we obtain a theory of exchanges – a sort of minimal
system of behavioural types – that conforms to our intuition about how such
things ought to work remarkably well. For example, if we begin with a compact
closed resouce theory, then duals may be understood as _debits_ , and the
corresponding theory of exchanges allows us to work with debits as accountants
do, treating our material histories as a sort of ledger [21].
Our approach to these concurrent material histories retains the aforementioned
advantages of the resource-theoretic perspective: We lose no generality, since
our construction applies to any resource theory; It is canonical, with
proarrow equipments being a fundamental structure in formal category theory –
although not usually seen in such concrete circumstances; Finally, it remains
relatively friendly, since the string diagrams for monoidal categories extend
in a natural way to string diagrams for proarrow equipments [13].
### 1.1 Contributions and Related Work
_Related Work_. Monoidal categories are ubiquitous – if often implicit – in
theoretical computer science. An example from the theory of concurrency is
[18], in which monoidal categories serve a purpose similar to their purpose
here. String diagrams for monoidal categories seem to have been invented
independently a number of times, but until recently were uncommon in printed
material due to technical limitations. The usual reference is [15]. We credit
the resource-theoretic interpretation of monoidal categories and their string
diagrams to [5]. Double categories first appear in [7]. Free double categories
are considered in [6] and again in [9]. The idea of a proarrow equipment first
appears in [26], albeit in a rather different form. Proarrow equipments have
subsequently appeared under many names in formal category theory (see e.g.,
[24, 11]). String diagrams for double categories and proarrow equipments are
treated precisely in [13]. We have been inspired by work on message passing
and behavioural types, in particular [3], from which we have adopted our
notation for exchanges. Finally, our observations concerning accounting build
on the work of [21].
_Contributions_. Our main contribution is the resource-theoretic
interpretation certain proarrow equipments, which we call _cornerings_ , and
the observation that they capture exactly the structure of concurrent process
histories. Our mathematical contributions are minor, most significantly the
identification of crossing cells in the free cornering of a resource theory
and the corresponding Lemma 2, which we believe to be novel. We do not claim
the other lemmas of the paper as significant mathematical contribuitions.
Instead, they serve to flesh out the structure of the free cornering, serving
as evidence that our interpretation is a good one.
### 1.2 Organization and Prerequisites
_Prerequisites_. This paper is largely self-containted, but we assume some
familiarity with category theory, in particular with monoidal categories and
their string diagrams. Some good references are [17, 23, 10].
_Organization_. In Section 2 we review the resource-theoretic interpretation
of symmetric monoidal categories. We continue by reviewing the theory of
double categories in Section 3, specialized to the single object case. In
Section 4 we introduce cornerings of a resource theory, in particular the free
such cornering, and exhibit the existence of crossing cells in the free
cornering. In Section 5 we show how the free cornering of a resource theory
inherits its resource-theoretic interpretation while enabling the concurrent
decomposition of resource transformations. In Section 6 we consider the case
in which the original resource theory is given by a compact closed category,
as in the theory of double-entry bookkeeping. In Section 7 we conclude and
consider directions for future work.
## 2 Monoidal Categories as Resource Theories
Symmetric strict monoidal categories can be understood as theories of resource
transformation. Objects are interpreted as collections of resources, with
$A\otimes B$ the collection consisting of both $A$ and $B$, and $I$ the empty
collection. Arrows $f:A\to B$ are understood as ways to transform the
resources of $A$ into those of $B$. We call symmetric strict monoidal
categories _resource theories_ when we have this sort of interpretation in
mind.
For example, let $\mathfrak{B}$ be the free symmetric strict monoidal category
with generating objects
$\\{\texttt{bread},\texttt{dough},\texttt{water},\texttt{flour},\texttt{oven}\\}$
and with generating arrows
mix : water ⊗flour →dough
knead : dough →dough
bake : dough ⊗oven →dough ⊗oven
subject to no equations. $\mathfrak{B}$ can be understood as a resource theory
of baking bread. The arrow mix represents the process of combining water and
flour to form a bread dough, knead represents kneading dough, and bake
represents baking dough in an oven to obtain bread (and an oven).
The structure of symmetric strict monoidal categories provides natural
algebraic scaffolding for composite transformations. For example, consider the
following arrow of $\mathfrak{B}$:
$(\texttt{bake}\otimes
1_{\texttt{dough}});(1_{\texttt{bread}}\otimes\sigma_{\texttt{oven},\texttt{dough}};\texttt{bake})$
of type
$\texttt{dough}\otimes\texttt{oven}\otimes\texttt{oven}\to\texttt{bread}\otimes\texttt{bread}\otimes\texttt{oven}$
where $\sigma_{A,B}:A\otimes B\stackrel{{\scriptstyle\sim}}{{\to}}B\otimes A$
is the braiding. This arrow describes the transformation of two units of dough
into loaves of bread by baking them one after the other in an oven.
It is often more intuitive to write composite arrows like this as string
diagrams: Objects are depicted as wires, and arrows as boxes with inputs and
outputs. Composition is represented by connecting output wires to input wires,
and we represent the tensor product of two morphisms by placing them beside
one another. Finally, the braiding is represented by crossing the wires
involved. For the morphism discussed above, the corresponding string diagram
is:
Notice how the topology of the diagram captures the logical flow of resources.
Given a pair of parallel arrows $f,g:A\to B$ in some resource theory, both $f$
and $g$ are ways to obtain $B$ from $A$, but they may not have the same effect
on the resources involved. We explain by example: Consider the parallel arrows
$1_{\texttt{dough}},\texttt{knead}:\texttt{dough}\to\texttt{dough}$ of
$\mathfrak{B}$. Clearly these should not be understood to have the same effect
on the dough in question, and this is reflected in $\mathfrak{B}$ by the fact
that they are not made equal by its axioms. Similarly, knead and
$\texttt{knead}\circ\texttt{knead}$ are not equal in $\mathfrak{B}$, which we
understand to mean that kneading dough twice does not have the same effect as
kneading it once, and that in turn any bread produced from twice-kneaded dough
will be different from once-kneaded bread in our model.
Consider a hypothetical resource theory constructed from $\mathfrak{B}$ by
imposing the equation $\texttt{knead}\circ\texttt{knead}=\texttt{knead}$. In
this new setting we understand kneading dough once to have the same effect as
kneading it twice, three times, and so on, because the corresponding arrows
are all equal. Of course, the sequence of events described by knead is not the
one described by $\texttt{knead}\circ\texttt{knead}$: In the former the dough
has been kneaded only once, while in the latter it has been kneaded twice. The
equality of the two arrows indicates that these two different processes would
have the same effect on the dough involved. We adopt as a general principle in
our design and understanding of resource theories that transformations should
be equal if and only if they have the same effect on the resources involved.
For the sake of further illustration, observe that by naturality of the
braiding maps the following two resource transformations are equal in
$\mathfrak{B}$:
Each transformation gives a method of baking two loaves of bread. On the left,
two batches of dough are mixed and kneaded before being baked one after the
other. On the right, first one batch of dough is mixed, kneaded and baked and
only then is the second batch mixed, kneaded, and baked. Their equality tells
us that, according to $\mathfrak{B}$, the two procedures will have the same
effect, resulting in the same bread when applied to the same ingredients with
the same oven.
## 3 Single Object Double Categories
In this section we set up the rest of our development by presenting the theory
of _single object double categories_ , being those double categories
$\mathbb{D}$ with exactly one object. In this case $\mathbb{D}$ consists of a
_horizontal edge monoid_ $\mathbb{D}_{H}=(\mathbb{D}_{H},\otimes,I)$, a
_vertical edge monoid_ $\mathbb{D}_{V}=(\mathbb{D}_{V},\otimes,I)$, and a
collection of _cells_
where $A,B\in\mathbb{D}_{H}$ and $X,Y\in\mathbb{D}_{V}$. Given cells
$\alpha,\beta$ where the right boundary of $\alpha$ matches the left boundary
of $\beta$ we may form a cell $\alpha|\beta$ – their _horizontal composite_ –
and similarly if the bottom boundary of $\alpha$ matches the top boundary of
$\beta$ we may form $\frac{\alpha}{\beta}$ – their _vertical composite_ – with
the boundaries of the composite cell formed from those of the component cells
using $\otimes$. We depict horizontal and vertical composition, respectively,
as in:
and
Horizontal and vertical composition of cells are required to be associative
and unital. We omit wires of sort $I$ in our depictions of cells, allowing us
to draw horizontal and vertical identity cells, respectively, as in:
and
Finallly, the horizontal and vertical identity cells of type $I$ must coincide
– we write this cell as $\square_{I}$ and depict it as empty space, see below
on the left – and vertical and horizontal composition must satisfy the
interchange law. That is,
$\frac{\alpha}{\beta}|\frac{\gamma}{\delta}=\frac{\alpha|\gamma}{\beta|\delta}$,
allowing us to unambiguously interpret the diagram below on the right:
Every single object double category $\mathbb{D}$ defines strict monoidal
categories $\mathbf{V}\mathbb{D}$ and $\mathbf{H}\mathbb{D}$, consisting of
the cells for which the $\mathbb{D}_{H}$ and $\mathbb{D}_{V}$ valued
boundaries respectively are all $I$, as in:
and
That is, the collection of objects of $\mathbf{V}\mathbb{D}$ is
$\mathbb{D}_{H}$, composition in $\mathbf{V}\mathbb{D}$ is vertical
composition of cells, and the tensor product in $\mathbf{V}\mathbb{D}$ is
given by horizontal composition:
In this way, $\mathbf{V}\mathbb{D}$ forms a strict monoidal category, which we
call the category of vertical cells of $\mathbb{D}$. Similarly,
$\mathbf{H}\mathbb{D}$ is also a strict monoidal category (with collection of
objects $\mathbb{D}_{V}$) which we call the _horizontal cells_ of
$\mathbb{D}$.
## 4 Cornerings and Crossings
Next, we define cornerings, our primary technical device. In particular we
discuss the free cornering of a resource theory, which we show contains
special crossing cells with nice formal properties. Tersely, a cornering of a
resource theory $\mathbb{A}$ is a single object proarrow equipment with
$\mathbb{A}$ as its vertical cells. Explicitly:
###### Definition 1
Let $\mathbb{A}$ be a symmetric strict monoidal category. Then a _cornering of
$\mathbb{A}$_ is a single object double category $\mathbb{D}$ such that:
1. (i)
The vertical cells of $\mathbb{D}$ are $\mathbb{A}$. That is, there is an
isomorphism of categories $\mathbf{V}\mathbb{D}\cong\mathbb{A}$.
2. (ii)
For each $A$ in $\mathbb{A}_{0}\cong\mathbb{D}_{H}$, there are distinguished
elements $A^{\circ}$ and $A^{\bullet}$ of $\mathbb{D}_{V}$ along with
distinguished cells of $\mathbb{D}$
called _$\circ$ -corners_ and _$\bullet$ -corners_ respectively, which must
satisfy the _yanking equations_ :
Intuitively, $A^{\circ}$ denotes an instance of $A$ moving from left to right,
and $A^{\bullet}$ denotes an instance of $A$ moving from right to left (see
Section 5).
Of particular interest is the free cornering of a resource theory:
###### Definition 2
Let $\mathbb{A}$ be a resource theory. Then the _free cornering of
$\mathbb{A}$_, written
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$,
is the free single object double category defined as follows:
* •
The horizontal edge monoid
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}_{H}=(\mathbb{A}_{0},\otimes,I)$
is given by the objects of $\mathbb{A}$.
* •
The vertical edge monoid
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}_{V}=(\mathbb{A}_{0}\times\\{\circ,\bullet\\})^{*}$
is the free monoid on the set $\mathbb{A}_{0}\times\\{\circ,\bullet\\}$ of
polarized objects of $\mathbb{A}$ – whose elements we write $A^{\circ}$ and
$A^{\bullet}$.
* •
The generating cells consist of corners for each object $A$ of $\mathbb{A}$ as
above, subject to the yanking equations, along with a vertical cell
${\text{}^{\ulcorner}_{\llcorner}\\!{f}\\!_{\lrcorner}^{\urcorner}}$ for each
morphism $f:A\to B$ of $\mathbb{A}$ subject to equations as in:
For a precise development of free double categories see [9]. In brief: cells
are formed from the generating cells by horizontal and vertical composition,
subject to the axioms of a double category in addition to any generating
equations. The free cornering is free both in the sense that it is freely
generated, and in the sense that for any cornering $\mathbb{D}$ of
$\mathbb{A}$ there is exactly one double functor
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}\to\mathbb{D}$
that sends corner cells to corner cells and restricts to the identity on
$\mathbb{A}\cong\mathbf{V}\mathbb{D}$. That is, diagrams in
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
have a canonical interpretation in any cornering of $\mathbb{A}$.
###### Proposition 1
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
is a cornering of $\mathbb{A}$.
###### Proof
Intuitively
$\mathbf{V}\,{\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}\cong\mathbb{A}$
because in a composite vertical cell every wire bent by a corner must
eventually be un-bent by the matching corner, which by yanking is the
identity. The only other generators are the cells
${\text{}^{\ulcorner}_{\llcorner}\\!{f}\\!_{\lrcorner}^{\urcorner}}$, and so
any vertical cell in
${\text{}^{\ulcorner}_{\llcorner}\\!{A}\\!_{\lrcorner}^{\urcorner}}$ can be
written as
${\text{}^{\ulcorner}_{\llcorner}\\!{g}\\!_{\lrcorner}^{\urcorner}}$ for some
morphism $g$ of $\mathbb{A}$. A more rigorous treatment of corner cells can be
found in [13], to the same effect.
∎
Before we properly explain our interest in
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
we develop a convenient bit of structure: _crossing cells_. For each $B$ of
${\text{}^{\ulcorner}_{\llcorner}\\!{A}\\!_{\lrcorner}^{\urcorner}}_{H}$ and
each $X$ of
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}_{V}$
we define a cell
of
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
inductively as follows: In the case where $X$ is $A^{\circ}$ or $A^{\bullet}$,
respectively, define the crossing cell as in the diagrams below on the left
and right, repectively:
in the case where $X$ is $I$, define the crossing cell as in the diagram below
on the left, and in the composite case define the crossing cell as in the
diagram below on the right:
We prove a technical lemma:
###### Lemma 1
For any cell $\alpha$ of
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
we have
###### Proof
By structural induction on cells of
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$.
For the $\circ$-corners we have:
and for the $\bullet$-corners, similarly:
the final base cases are the
${\text{}^{\ulcorner}_{\llcorner}\\!{f}\\!_{\lrcorner}^{\urcorner}}$ maps:
There are two inductive cases. For vertical compositon, we have:
Horizontal composition is similarly straightforward, and the claim follows by
induction.
∎From this we obtain a “non-interaction” property of our crossing cells,
similar to the naturality of braiding in symmetric monoidal categories:
###### Corollary 1
For cells $\alpha$ of
$\mathbf{V}\,{\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
and $\beta$ of
$\mathbf{H}\,{\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$,
the following equation holds in
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$:
These crossing cells greatly aid in the legibility of diagrams corresponding
to cells in
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$,
but also tell us something about the categorical structure of
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$,
namely that it is a monoidal double category in the sense of [25]:
###### Lemma 2
If $\mathbb{A}$ is a symmetric strict monoidal category then
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
is a monoidal double category. That is,
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
is a pseudo-monoid object in the strict 2-category $\mathbf{V}\mathsf{DblCat}$
of double categories, lax double functors, and vertical transformations.
###### Proof
We give the action of the tensor functor on cells:
This defines a pseudofunctor, with the component of the required vertical
transformation given by exchanging the two middle wires as in:
Notice that $\otimes$ is strictly associative and unital, in spite of being
only pseudo-functorial.
∎
## 5 Concurrency through Cornering
We next proceed to extend the resource-theoretic interpretation of some
symmetric strict monoidal category $\mathbb{A}$ to its free cornering
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$.
Interpret elements of
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}_{V}$
as _$\mathbb{A}$ -valued exchanges_. Each exchange $X_{1}\otimes\cdots\otimes
X_{n}$ involves a left participant and a right participant giving each other
resources in sequence, with $A^{\circ}$ indicating that the left participant
should give the right participant an instance of $A$, and $A^{\bullet}$
indicating the opposite. For example say the left participant is Alice and the
right participant is Bob. Then we can picture the exchange $A^{\circ}\otimes
B^{\bullet}\otimes C^{\bullet}$ as:
$\texttt{Alice}\rightsquigarrow\includegraphics[height=28.45274pt,align=c]{figs/alice-
bob-swap-stack-
example.png}\mathrel{\mathchoice{\reflectbox{$\displaystyle\rightsquigarrow$}}{\reflectbox{$\textstyle\rightsquigarrow$}}{\reflectbox{$\scriptstyle\rightsquigarrow$}}{\reflectbox{$\scriptscriptstyle\rightsquigarrow$}}}\texttt{Bob}$
Think of these exchanges as happening _in order_. For example the exchange
pictured above demands that first Alice gives Bob an instance of $A$, then Bob
gives Alice an instance of $B$, and then finally Bob gives Alice an instance
of $C$.
We interpret cells of
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
as _concurrent transformations_. Each cell describes a way to transform the
collection of resources given by the top boundary into that given by the
bottom boundary, via participating in $\mathbb{A}$-valued exchanges along the
left and right boundaries. For example, consider the following cells of
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathfrak{B}}\\!_{\lrcorner}^{\urcorner}}$:
From left to right, these describe: A procedure for transforming water into
nothing by mixing it with flour obtained by exchange along the right boundary,
then sending the resulting dough away along the right boundary; A procedure
for transforming an oven into an oven, receiving flour along the right
boundary and sending it out the left boundary, then receiving dough along the
left boundary, which is baked in the oven, with the resulting bread sent out
along the right boundary; Finally, a procedure for turning flour into bread by
giving it away and then receiving bread along the left boundary. When we
compose these concurent transformations horizontally in the evident way, they
give a transformation of resources in the usual sense, i.e., a morphism of
$\mathbb{A}\cong\mathbf{V}\,{\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$:
We understand equality of cells in
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
much as we understand equality of morphisms in a resource theory: two cells
should be equal in case the transformations they describe would have the same
effect on the resources involved. In this way, cells of
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
allow us to break a transformation into many concurrent parts. Note that with
the crossing cells, it is possible to exchange resources “across” cells.
Consider the category
$\mathbf{H}\,{\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
of horizontal cells. If the vertical cells
$\mathbf{V}\,{\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
are concerned entirely with the transformation of resources, then our
interpretation tells us that the horizontal cells are concerned entirely with
exchange. Just as isomorphic objects in
$\mathbf{V}\,{\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}\cong\mathbb{A}$
can be thought of as equivalent collections of resources – being freely
transformable into each other – we understand isomorphic objects in
$\mathbf{H}\,{\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
as _equivalent exchanges_. For example, There are many ways for Alice to give
Bob an $A$ and a $B$: Simultaneously, as $A\otimes B$; one after the other, as
$A$ and then $B$; or in the other order, as $B$ and then $A$. While these are
different sequences of events, they achieve the same thing, and are thus
equivalent. Similarly, for Alice to give Bob an instance of $I$ is equivalent
to nobody doing anything. Formally, we have:
###### Lemma 3
In
$\mathbf{H}\,{\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
we have for any $A,B$ of $\mathbb{A}$:
1. (i)
$I^{\circ}\cong I\cong I^{\bullet}$.
2. (ii)
$A^{\circ}\otimes B^{\circ}\cong B^{\circ}\otimes A^{\circ}$ and
$A^{\bullet}\otimes B^{\bullet}\cong B^{\bullet}\otimes A^{\bullet}$.
3. (iii)
$(A\otimes B)^{\circ}\cong A^{\circ}\otimes B^{\circ}$ and $(A\otimes
B)^{\bullet}\cong A^{\bullet}\otimes B^{\bullet}$
###### Proof
1. (i)
For $I\cong I^{\circ}$, consider the $\circ$-corners corresponding to $I$:
we know that these satisfy the yanking equations:
which exhibits an isomorphism $I\cong I^{\circ}$. Similarly, $I\cong
I^{\bullet}$. Thus, we see formally that exchanging nothing is the same as
doing nothing.
2. (ii)
The $\circ$-corner case is the interesting one: Define the components of our
isomorphism to be:
and
then for both of the required composites we have:
and so $A^{\circ}\otimes B^{\circ}\cong B^{\circ}\otimes A^{\circ}$. Similarly
$A^{\bullet}\otimes B^{\bullet}\cong B^{\bullet}\otimes A^{\bullet}$. This
captures formally the fact that if Alice is going to give Bob an $A$ and a
$B$, it doesn’t really matter which order she does it in.
3. (iii)
Here it is convenient to switch between depicting a single wire of sort
$A\otimes B$ and two wires of sort $A$ and $B$ respectively in our string
diagrams. To this end, we allow ourselves to depict the identity on $A\otimes
B$ in multiple ways, using the notation of [4]:
Then the components of our isomorphism $(A\otimes B)^{\circ}\cong
A^{\circ}\otimes B^{\circ}$ are:
and
and, much as in (ii), it is easy to see that the two possible composites are
both identity maps. Similarly, $(A\otimes B)^{\bullet}\cong(A^{\bullet}\otimes
B^{\bullet})$. This captures formally the fact that giving away a collection
is the same thing as giving away its components.
∎
For example, we should be able to compose the cells on the left and right
below horizontally, since their right and left boundaries, respectively,
indicate equivalent exchanges:
Our lemma tells us that there will always be a canonical isomorphism, as above
in the middle, making composition possible.
It is worth noting that we _do not_ have $A^{\circ}\otimes B^{\bullet}\cong
B^{\bullet}\otimes A^{\circ}$:
###### Observation 1
There is a morphism $d^{\circ}_{\bullet}:A^{\circ}\otimes B^{\bullet}\to
B^{\bullet}\otimes A^{\circ}$ in one direction, defined by
but there is need not be a morphism in the other direction, and this is not in
general invertible. In particular,
$\mathbf{H}\,{\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
is monoidal, but need not be symmetric.
This observation reflects formally the intuition that if I receive some
resources before I am required to send any, then I can send some of the
resources that I receive. However, if I must send the resources first, this is
not the case. In this way,
$\mathbf{H}\,{\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
contains a sort of causal structure.
## 6 Compact Closure and Accounting
In this section we consider the case where our resource theory $\mathbb{A}$ is
compact closed. From the standpoint of accounting, string diagrams in resource
theories are like _ledgers_ , recording the material history of the resources
they concern [21]. The _double-entry_ method of accounting insists that every
change to a ledger consists of a matching credit (positive change) and debit
(negative change) so that the system of accounts represented by the ledger
remains _balanced_. This serves as a kind of integrity check: given a ledger
we may attempt to _balance_ it by matching credits with debits and cancelling
them out, with our ledger being balanced if and only if all entries may be
cancelled in this way.
While the credits and debits of double-entry accounting are usually positive
and negative integers, the technique makes sense in the context of any compact
closed resource theory. The unit arrows $\eta_{A}:I\to A\otimes A^{*}$ create
matching credits and debits, and the cancellative process of balancing is
performed via the counit arrows $\varepsilon_{A}:A^{*}\otimes A\to I$.
Following the usual convention, we depict the units and counits, respectively,
as:
and
and then their defining equations are:
and
For example, there is a well-known compact closed category $\mathbb{Z}$ whose
set of objects is the group of differences construction of the integers[14] in
which there is a morphism between two such objects if and only if the
corresponding integers are equal. This category captures integer valued
double-entry accounting in the sense discussed above – extending its treatment
in [8].
$\mathbb{A}$ being compact closed has effects on the structure of
${\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$.
To begin, we extend Lemma 3 with a formal version of the idea that Alice
giving Bob $A^{*}$ ought to be equivalent to Bob giving Alice $A$.
###### Lemma 4
If $\mathbb{A}$ is compact closed, then in
$\mathbf{H}\,{\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
we have $A^{\circ}\cong(A^{*})^{\bullet}$ and
$A^{\bullet}\cong(A^{*})^{\circ}$.
###### Proof
For $A^{\circ}\cong(A^{*})^{\bullet}$, the two halves of the isomorphism are
given by:
and
then we have:
and
as required. $A^{\bullet}\cong(A^{*})^{\circ}$ similarly.
∎
Next, consider the way that debits allow us to violate causality in everyday
life: By incurring a debit I can trade something away _before I have it_. This
is also reflected in our development. For example, let us augment our resource
theory $\mathfrak{B}$ of baking with an object $1 which we will use to
represent currency. We also require a dual object $\texttt{\$1}^{*}$ along
with a cap $\eta_{\texttt{\$1}}$ and cap $\varepsilon_{\texttt{\$1}}$
satisfying the equations above. Consider the following cell of the free
cornering of this augmented resource theory:
Here our process must bake bread from flour and water, but has no flour, and
must purchase it for $1 along the left boundary. However it also has no money,
so it incurs a debit (via the cap) in order to purchase the flour. This done,
the bread is baked, and then sold along the right boundary in exchange for two
instances of $1, one of which is used to balance the debit (via the cup), with
the other kept as profit. If we compose our cell with a suitable flour seller
and bread buyer on the left and right respectively, for example
and
then we obtain the following material history:
While in Observation 1 we saw that
$\mathbf{H}\,{\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
is not symmetric in general, the presence of dual objects $A^{*}$ in
$\mathbb{A}$ makes it so. This is enough to make
$\mathbf{H}\,{\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
itself compact closed.
###### Lemma 5
If $\mathbb{A}$ is compact closed, so is
$\mathbf{H}\,{\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$.
###### Proof
We show that
$\mathbf{H}\,{\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
is both rigid and symmetric, beginning with the latter. The inverse to
$d^{\circ}_{\bullet}$ of Observation 1 is:
making
$\mathbf{H}\,{\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
symmetric. Now, define the dual $X^{*}$ of an object $X$ inductively:
$I^{*}=I$, $(A^{\circ})^{*}=A^{\bullet}$, $(A^{\bullet})^{*}=A^{\circ}$, and
$(X\otimes Y)^{*}=X^{*}\otimes Y^{*}$. It suffices to give the unit and counit
for the $A^{\circ}$ and $A^{\bullet}$ cases. In one of them we have:
and
which are easily seen to satisfy the required equations:
and
For the other, the unit and counit are given by
and
which again satisfy the required equations as in:
and
For $I$ both unit and counit are given by $1_{I}$, and the inductive case is
as in:
and
∎
In fact, if $\mathbb{A}$ is compact closed then
$\mathbf{H}\,{\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
and $\mathbb{A}$ are equivalent as categories. We omit the proof of this,
which is both rather technical and, we feel, not particularily enlightenting.
The main idea is that using Lemma 4 we can obtain from any arrow of
$\mathbf{H}\,{\text{}^{\ulcorner}_{\llcorner}\\!{\mathbb{A}}\\!_{\lrcorner}^{\urcorner}}$
an equivalent one in which all exchanges flow from left to right ($\circ$
polarity), which is just a morphism of $\mathbb{A}$.
## 7 Conclusions and Future Work
We have shown how to decompose the material history of a process into
concurrent components by working in the free cornering of an appropriate
resource theory. We have explored the structure of the free cornering in light
of this interpretation and found that it captures formally many of our
intuitions – in particular those concerning accounting and debt. We do not
claim to have solved all problems in the modelling of concurrency, but we feel
that our formalism captures the material aspect of concurrent systems very
well. Thus, we have identified the algebraic structure of a fundamental aspect
of concurrent systems.
There is much future work, and we list a few directions that we feel are
particularly promising. Our work is inspired by the message passing logic of
[3], which has its categorical semantics in _linear actegories_. Any cornering
defines an actegory – although not quite a _linear_ actegory – and we
speculate that cornerings are equivalent to some class of actegories, which
would connect our work to the literature on behavioural types. Another
direction for furture work is to connect our material histories to a theory of
concurrent processes – the slugs to our slime – with the goal of a formalism
accounting for both. The category of spans of reflexive graphs, interpreted as
open transition systems, seems especially promising here[16]. Finally, our
observations about accounting are in part motivated by an interest in smart
contracts, which would benefit from a rigorous formal treatment[2].
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* [2] N. Atzei, M. Bartolietti, and T. Cimoli. A survey of attacks on ethereum smart contracts. In POST 2017, volume 10204 of LNCS, pages 164–186, 2017.
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* [11] Marco Grandis and Robert Pare. Adjoint for double categories. Cahiers de Topologie et Géométrie Différentielle Catégoriques, 45(3):193–240, 2004.
* [12] C. A. R. Hoare. Communicating sequential processes. Commun. ACM, 21(8):666–677, 1978.
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* [15] André Joyal and Ross Street. The geometry of tensor calculus, i. Advances in Mathematics, 88(1):55 – 112, 1991.
* [16] P. Katis, N. Sabadini, and R.F.C Walters. Span(graph): A categorical algebra of transition systems. In Intnational Conference on Algebraic Methodology and Software Technology, pages 307–321. Springer, Berlin, Heidelberg, 1997.
* [17] S. Mac Lane. Categories for the Working Mathematician. Springer, 1971.
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|
# Spin-Phonon interaction in quasi 2D- $Cr_{2}Te_{3}$
Gurupada Ghorai School of Physical Sciences, National Institute of Science
Education and Research (NISER) Bhubaneswar, An OCC of Homi Bhabha National
Institute, Jatni-752050, Odisha, India. Kalyan Ghosh School of Physical
Sciences, National Institute of Science Education and Research (NISER)
Bhubaneswar, An OCC of Homi Bhabha National Institute, Jatni-752050, Odisha,
India. Abhilash Patra School of Physical Sciences, National Institute of
Science Education and Research (NISER) Bhubaneswar, An OCC of Homi Bhabha
National Institute, Jatni-752050, Odisha, India. Prasanjit Samal School of
Physical Sciences, National Institute of Science Education and Research
(NISER) Bhubaneswar, An OCC of Homi Bhabha National Institute, Jatni-752050,
Odisha, India. Kartik Senapati School of Physical Sciences, National
Institute of Science Education and Research, An OCC of Homi Bhabha National
Institute, Jatni, Odisha - 752050, India. Center for Interdisciplinary
Sciences (CIS), NISER Bhubaneswar, HBNI, Jatni-752050, Odisha, India. Pratap
K. Sahoo<EMAIL_ADDRESS>School of Physical Sciences, National
Institute of Science Education and Research, An OCC of Homi Bhabha National
Institute, Jatni, Odisha - 752050, India. Center for Interdisciplinary
Sciences (CIS), NISER Bhubaneswar, HBNI, Jatni-752050, Odisha, India.
(August 28, 2024)
###### Abstract
Spin-phonon interaction plays an important role in 2D magnetic materials and
motivates the development of next-generation spin- and charge-dependent
microelectronic devices. Understanding the spin-phonon interaction by tuning
the growth parameter of single crystal Cr${}_{2}Te_{3}$, a robust quasi-2D
room temperature magnetic material, is crucial for spintronic devices. The
synthesis of single crystal 2D Cr${}_{2}Te_{3}$ flakes on a Si substrate from
co-deposited thin film by plasma annealing techniques is a significant
achievement. The temperature dependence and polarization-resolved Raman
spectroscopy with support of density functional theory classified lattice
symmetry operations were used to identify the phonon modes to investigate the
spin/electron-phonon interactions in Cr${}_{2}Te_{3}$. The mean-field theory
model in single crystal Cr${}_{2}Te_{3}$ is employed to quantify the spin-
phonon interaction and correlate with in-plane and out-of-plane magnetic
behavior. The observation of a positive correlation between phonon mode
frequency and spin-phonon interaction strength in single crystal
Cr${}_{2}Te_{3}$ can be a potential candidate for spintronic applications.
Keyword
2D-Cr22Te3, TEM, SQUID, Raman spectroscopy, DFT.
## I INTRODUCTION
The thermal transport and spin properties of the 2D magnetic materials are
crucial for spintronic devices[1]. Although spin-phonon, electron-phonon, and
phonon-phonon interactions are offered for study for high-performance spin-
based thermal management devices[2, 3]. This recommends establishing the
influence of spin-phonon interaction on the physical phenomena of the 2D
magnet. The spin-phonon solid interaction in the 2D $CrSiTe_{3}$ was proposed
from the temperature-dependent I-R spectroscopy and super-exchange mechanism
influenced in the Si-Te stretching and Te displacement in temperature-
dependent frequency study[4, 5]. The unconventional spin-phonon interaction
vis Dzyaloshinskii–Moriya interaction(DMI) in $Y_{2}Ir_{2}O_{7}$ crystal by
temperature-dependent infrared spectroscopy was also reported[6]. The spin-
phonon interaction in the bulk crystal of yttrium iron garnet (YIG) with
enhanced interaction strength at higher phonon frequency was investigated [7].
However, spin-phonon coupling is a primary need to tackle how spin affects
phonon thermal transport. To address these coupling issues, we are interested
in exploring the spin-phonon interaction of $Cr_{2}Te_{3}$. The 2D magnetic
order in chromium trihalide materials was first discovered[8]. The 2D magnetic
materials could have been lead to one of the spintronics materials by
utilizing charge and spin for applications ranging from molecular quantum
devices and sensing to ultrathin high-density data storage devices[9, 10]. In
a similar line, the intrinsic 2D layer-dependent ferromagnetism properties in
CrGe2 were also investigated [11]. The structural properties using scanning
tunneling microscopy, bandgap, and electronic properties were calculated using
DFT calculation of $CrGeTe_{3}$ materials in ref[12]. Although three.
Theoretically, it has been suggested that layered materials like
$CrXTe_{3}(X=Si,Ge,Sn)$ show 2D ferromagnetic or antiferromagnetic behavior
dependent on their exchange or superexchange interaction strength[13, 14]. It
has been proposed that $Cr_{2}Te_{3}$ is a tunable curie temperature up to
room temperature and suitable for magnetic memory device applications[15]. So,
for magnetic-based applications of $Cr_{2}Te_{3}$, there should be clear
evidence of the magnetic properties of such material, which are influenced by
defect spins in the lattice. It has been demonstrated that the 2D
$Cr_{2}Te_{3}$ materials show ferrimagnet or ferromagnet properties, which are
dependent on the strength of the nearest neighbor exchange interaction[16].
The (001) oriented crystal grown at a temperature of 400∘C at a rate of 1.7
nm/min on Al${}_{2}O_{3}$(0001) using molecular beam epitaxial(MBE) reveals
the local electronic and magnetic structure of Cr ions, which was demonstrated
the edge of energy of Cr L2,3 using X-ray magnetic circular dichroism (XMCD)
and soft x-ray absorption spectroscopy[17]. However, the magnetic structure of
$Cr_{2}Te_{3}$ is not yet fully understood, because of the complicated
coupling of the spins in three distinct Cr sites. Curie temperature and
magnetic properties vary with various defect sites in the 2D magnetic
materials.
To understand the spin-phonon interaction, we have synthesized a single
crystal $Cr_{2}Te_{3}$, a quasi 2D magnetic material by co-deposited Cr and Te
thin film followed by thermal annealing. The $A_{g}$ and three $E_{g}$ phonon
modes were analyzed from the temperature-dependent Raman study. Different
vibrational modes and frequencies were obtained using DFT calculations and
compared with the experimental results. The anharmonicity of Raman modes can
be well-fitted with Balkanski’s model above the critical temperature. The
ferromagnetic transition with temperature matched well with the Raman
anharmonicity, which can be explained in terms of spin-phonon interaction. The
results can be understood in terms of magnon-phonon coupling.
## II Experimental method
The commercially available high-purity (99.999$\%$, Sigma Aldrich) metal
pieces of Chromium(Cr) and Tellurium(Te) are co-evaporated using the e-beam
evaporation techniques and thermal evaporation technique in a base vacuum of
2$\times 10^{-7}$ mbar to produce thin films of CrxTey on the clean Si
substrate. The deposition rate of Cr and Te were 0.1 Å/s and 0.4Å/s,
respectively, and the thickness of 40 nm was measured by a quartz crystal
thickness monitor attached to the deposition chamber. The Co-deposited samples
were annealed at 500∘C inside of quartz tube of the plasma-enhanced chemical
vapor deposition (PECVD) system at 120 W plasma power with the ambient of
forming gas for two hours. The surface micrographs were collected using a
Field emission of scanning electron microscopy (FESEM) system (SIGMA ZEISS),
and the elemental mapping and percentage of composition were estimated using
Apex energy dispersive spectroscopy (EDS) attached to the FESEM. The XRD
pattern of the $Cr_{2}Te_{3}$ crystalline phases was obtained using a Rigaku
diffractometer with $CuK_{\alpha}$ radiation ($\lambda=1.54056$ Å) at room
temperature with a grazing incidence X-ray diffraction (GIXRD) approach at a
grazing angle of 1.0 degree with a step size of 0.02deg. The local crystalline
structural properties of the 2D-$Cr_{2}Te_{3}$ were verified using a high-
resolution transmission electron microscope (HRTEM) of a 200 KV energetic
electron source. The high-resolution confocal Raman scattering with a spatial
resolution of 2-4 microns was used to understand the vibrational modes of
$Cr_{2}Te_{3}$. The temperature-dependent Raman spectrum of $Cr_{2}Te_{3}$
flake was measured using a Linkam cooling stage. The specimens were mounted on
a sample holder fixed to a cold head in a vacuum with a constant liquid
nitrogen flow to cool the chamber down to 80K. A half-wave plate in an
incident light path was fixed to study the polarization-dependent Raman
spectra. We also calculated the Raman peaks position, and modes of vibration
were obtained theoretically using density functional theory (DFT). The
temperature dependence of the Raman spectrum of $Cr_{2}Te_{3}$ flakes was
measured using a Horiba Raman microscope equipped with a Linkam cooling stage.
The specimens were mounted on a non-background sample holder fixed to a cold
head in a vacuum, and the liquid nitrogen was used to cool the chamber up to
80K. The superconducting quantum interference device (SQUID) system of Quantum
Design was used to investigate the magnetic properties of $Cr_{2}Te_{3}$ in
the temperature range of 5-250K.
## III Results and discussions
The GIXRD measurement at a grazing angle of 1∘ was used to obtain the
crystalline properties and the orientation of the $Cr_{2}Te_{3}$ growth. The
GIXRD spectra of the pristine and 120W plasma, 500∘C annealed samples are
shown in figure 1(a). The pristine sample shows a weak XRD peak at 27.56∘,
corresponding to the lattice plane of Te (101) (JCPDS # 36-1452)[18, 19]. The
120W plasma annealed sample shows three XRD peak positions at 2$\theta$ values
of 14.54∘, 29.68∘, and 44.52∘ corresponding to the lattice plane of (002),
(004), and (006), respectively for the crystalline plane of $Cr_{2}Te_{3}$.
The above lattice plans describe the Trigonal family of $Cr_{2}Te_{3}$, with
the space group of 163 $(P\bar{3}1c)$, with lattice parameters of a=6.78 Å,
b=6.78 Å, c=12.06 Å, $\alpha$=90, $\beta$=90 and $\gamma$=120 (PDF#71-2245),
which matches well with an earlier report[20, 17]. Figure 1(b) and (c) shows
the FESEM surface morphology of the pristine and 120 W annealed sample. The
pristine sample shows a very smooth surface. It has been observed that after
annealing the pristine at 500∘C in 120W plasma for 2 hours, the CrTe thin
films grew as individual flakes with clear grain boundaries, as shown in
figure 1(c), which is a 2D $Cr_{2}Te_{3}$. To further clarify the crystalline
growth of quasi 2D $Cr_{2}Te_{3}$, we have prepared a lamella for TEM study
from a single flake. The lamella was prepared using a focused ion beam (FIB)
system using 30 keV Ga beam followed by in situ polishing by low energy Ar
ion. The lamella provides the cross-sectional view of 40 nm thin flake as
shown in figure 1(d), which is basically the c-axis of the 2D-$Cr_{2}Te_{3}$
crystalline plane. The high-resolution TEM with lattice arrangements and
corresponding selected area electron diffraction (SAED) patterns of
2D-Cr${}_{2}Te_{3}$ is shown in figure 1(e). The yellow parallel line shows
the (001) plane of the Cr${}_{2}Te_{3}$ with interplanar separation d001 of
3.03 Å. The schematic image along the (010) zone axis and the lattice spacing
of d001 is shown in figure 1(f). The XRD and HRTEM images strongly suggest the
crystalline growth of quasi-2D $Cr_{2}Te_{3}$ along the (001) plane.
Figure 1: (a) GIXRD data of the pristine, applied 120 W plasma power samples.
(b) Schematic illustration of crystal structure of $Cr_{2}Te_{3}$. (c) FESEM
image of co-deposited films, (d) Surface morphology image of $Cr_{2}Te_{3}$
sample after annealed $500^{0}C$ with the presence of 120 W plasma power.
### III.1 Raman spectra of Cr2Te3 from density functional theory
Raman spectroscopy is a versatile tool to understand the electron-phonon or
spin-phonon interaction in 2D materials. Depending on the crystal orientation
the vibrational modes can be suppressed or enhanced which may affect the spin-
phonon interaction. The observation of Raman modes in the case of
Cr${}_{2}Te_{3}$ strongly depends on the crystalline quality and synthesis
process [xx-xx]. In order to identify different Raman modes of
Cr${}_{2}Te_{3}$ , we have used the density functional theory[21, 22] using
the Vienna Ab-initio Simulation Package (VASP).[23, 24, 25, 26]. We have used
the recommended Perdew-Burke-Ernzerhof[27] (PBE) PAW pseudo potentials in all
the calculations along with the plane wave basis cutoff of 500 eV and set the
value of EDIFF = $1.0\times 10^{-8}$ eV to represent an energy convergence
criterion. For the Brillouin zone integration, a $\Gamma$-centered ($12\times
12\times 6$) k-point sampling is used. The Hubbard correction method within
semilocal DFT[28], i.e., GGA(PBE)+U is used for both structure optimization
and studying the Raman spectra. The value of U=3.0 eV[29] for Cr atoms is set
to take into account the orbital localization. The unit cell of Cr2Te3 with
P-31c space group is optimized with the above given set up and obtained the
lattice constants as a=b=6.99 Å, and c=12.65 Å, respectively. To get the Raman
active modes, the dielectric matrix is calculated for each mode using the
density functional perturbation theory(DFPT) as implemented in the VASP
package. The intensity of each mode is calculated following the Ref. [30] and
using the raman-sc[31] python package. The Raman spectra of Cr2Te3 is shown in
figure 2(a). The Lorentzian broadening width of 5 cm-1 is used to match with
the experimental spectra. Six dominating peaks corresponding to more intense
peaks are found at 40, 73, 102, 123, 144, and 190 cm-1. For comparison, the
experimental Raman peaks are shown in figure 2(b). The peaks at 102, 123, and
144 cm-1 are closer to experimental peaks at 106, 128, and 146 cm-1. One can
observe that two modes at 40 and 190 cm-1 are not observed in the experimental
Raman spectra. The Raman modes observed from the experiment and DFT
calculation are tabulated in Table 1.
Figure 2: (a) Polarisation dependent Raman at the temperature 150K where blue
color code for unpolarised(XX) condition and red color curve for polarised(XY)
Raman spectrum. (b) The theoretical DFT calculation of the Raman spectrum of
$Cr_{2}Te_{3}$ single crystal. The atomic displacements of the Raman active
modes (the color code of atoms is the same as Fig 2(a)). The out-of-plane
$A_{g}$ vibrational modes are depicted on the left panel (top view and side
view), while the in-plane$E_{g}$ vibrational modes are on the right (top view
only).
It should be noted that Cr2Te3 belongs to the point group of D3d(-3m). Based
on the point groups of the character table, the deduced Raman active modes for
Cr2Te3 are 4A1g+9Eg. The Raman-active mode intensity is related to the Raman
tensor and the direction of incident and scattered light as:
$I_{m}^{R}=\frac{d\sigma}{d\Omega}\propto\sum_{m}|\vec{e}_{s}\cdot
Rt_{m}\cdot\vec{e}_{i}|^{2}$. Where Rtm is the Raman tensor of mth active mode
and $\vec{e}_{s}$ ( $\vec{e}_{i}$) is polarization vectors of scattered
(incoming) light[32, 33]. The Raman tensors correspond to active Raman modes
A1g, and Eg are as follows,
A${}_{1g}(D{{}_{3d}}(-3m)):$ $\begin{pmatrix}a&0&0\\\ 0&a&0\\\
0&0&b\end{pmatrix}$ E${}_{g}(D{{}_{3d}}(-3m)):\begin{pmatrix}c&0&0\\\
0&-c&d\\\ 0&d&0\end{pmatrix},\begin{pmatrix}0&-c&-d\\\ -c&0&0\\\
-d&0&0\end{pmatrix}.$
We use the notation XX and YX to represent unpolarized and polarized
scattering configurations, respectively. In the case of unpolarized
configuration XX, both the incoming and scattering polarization vectors are
$\vec{e_{i}}$=(1, 0, 0)=$\vec{e_{s}}$ and for YX, $\vec{e_{i}}$=(1, 0, 0),
$\vec{e_{s}}=(0,1,0)$. So, under unpolarized configuration XX, I(A1g) = a2,
and I(Eg) = c2. Under polarized condition YX, I(A1g) = 0, and I(Eg) = c2. So,
in cross polarization condition, the intensity of A1g mode vanishes, and Eg
peaks remain unchanged.
Table 1: The experimental and theoretical frequencies, and vibrational modes of strain-free R3 Cr${}_{2}Te_{3}$. The ‘$||$’ and ‘$\perp$’ represent the parallel and cross polarization conditions, respectively. The ‘$-$’ represents peaks that were not clearly observed. mode | $A_{g}$ | $E_{g}$ | $A_{g}$ | $E_{g}$ | $E_{g}$ | $A_{g}$
---|---|---|---|---|---|---
EXPT‘$||$’ | - | 96 | 106 | 128 | 146 | -
EXPT‘$\perp$’ | - | 96 | - | 128 | 146 | -
Theory | 40 | 73 | 102 | 123 | 144 | 190
The Raman active modes for $Cr_{2}Te_{3}$ belong to the trigonal lattice
structure with 163 space group[34]. For this space group, two types of active
Raman vibration modes are observed, i.e non-degenerate $E_{g}$ and degenerate
$A_{g}$[34]. So the $A_{g}$ mode is the out-plane lattice vibration and
$E_{g}$ is in-plane atomic vibration of the lattice.
We have studied the polarization Raman spectra at low temperatures (80K) to
best match the theoretical Raman modes. Compared to theoretical parameters
where unpolarized configuration XX, I(A1g) = a2, and I(Eg) = c2, the
experimental spectra shows four Raman peaks at $96cm^{-1}$, $106cm^{-1}$,
$128cm^{-1}$ and $146cm^{-1}$. For the theoretical polarized condition YX,
I(A1g) = 0, and I(Eg) = c2, the experimental spectra shows three Raman peaks
at $96cm^{-1}$, $128cm^{-1}$ and $146cm^{-1}$. The above observation clearly
illustrates the condition and provides the information that the $A_{g}$ is
suppressed and other peaks correspond to $E_{g}$ Raman modes. This procedure
quantifies the modes of vibration in $Cr_{2}Te_{3}$ quasi 2D materials. The
schematic configuration of the atomic vibration in the lattice atoms of
$Cr_{2}Te_{3}$ is shown in figure 2(c, d, e, f, g, h). The direction of
vibration is shown in red arrows and the Cr and Te atoms are shown in magenta
and yellow, respectively. Figure 2(c, d, e) shows the three configurations as
A1g, A2g, A3g for out-of-plane vibration in the c-axis, in which the light
interact ‘$||$’ to the lattice plane. Figure 2(f, g, h) represents three
configurations as E1g, E2g, E3g for in-plane vibration in the XY plane, in
which polarized light interacts $\perp$ to the lattice plane.
### III.2 Temperature dependent Raman study:
The interaction of phonon with spin and magnon can be understood from the
temperature dependant Raman spectroscopy. The interpretation of Raman spectra
of soft modes in a ferromagnetic material can be understood from its over-
damped line shape and the temperature dependence near the transition
temperature ($T_{N}$).It has been observed that Raman spectroscopy a unique
tool to probe spin-phonon coupling(s) which is strongly influenced by
elementary excitations in multiferroic materials [7]. For a better
understanding of the phonon modes in Cr2Te3, systematic Raman spectra were
collected in the temperature range of 80-300 K and analyzed their evolution
using Lorentzian multifunction. The temperature dependant Raman spectra of
Cr2Te3, sample synthesized at 120 W is shown in Fig. 3(a). Two distinct peaks
at 125 cm-1 and 144 cm-1 corresponding to $E_{2g}$ and $E_{3g}$, respectively,
are observed with peak intensities varying with temperature. A 2D contour plot
shown in Fig 3(b) indicates the distribution of Raman peak intensities within
the frequency range of 80-165 cm-1. One can clearly perceive that both $E_{g}$
mode intensities are maximum within the temperature range of 100-165 K and
then reduced at the higher temperature. The evolution of parameters like peak
positions, full width at half maximum (FWHM), and integrated intensity as a
function of temperature are evaluated to understand the strength of the spin-
phonon interaction and the phonon dynamics of the system.
Figure 3: (a) Temperature dependent Raman spectrum of 120 W plasma power
samples in the temperature range of 82K to 273K. (b) 2D contour, in the range
of 80 to 270 cm-1. Figure 4: Magnetisation measurement (a) The MH loop
measurement parallel the ab plane configuration of 120 W plasma sample. (b)
The MH loop parallel to c axis of the 120 W plasma sample. (c) The MT
measurement with applied 300 Oe field parallel and perpendicular to the ab
plane of 120 W plasma used sample.(d) are the Exchange bias field of the above
four configurations.
### III.3 Magnetization study:
In the magnetically ordered system, both Raman line shift and magnetic moments
deviate at similar temperatures due to the strong spin-phonon interaction. In
order to correlate that, the magnetic properties of the material, hysteresis
loop, zero field cooled (ZFC), and field cooled (FC), measurements were
performed with both perpendicular and parallel applied external magnetic
fields with respect to the sample plane. In Figure 4(a), the magnetic moments
(ZFC and FC) are plotted as a function of temperature for the 120 W plasma
power sample. The brown curve represents the case of a perpendicular applied
field of 300 Oe to the ab plane, while the pink curve represents the case of
an applied field parallel to the ab plane. The inset of Figure 4 shows the
first order derivative of M-T, and the TC is determined to be $\approx 162K$.
However, the ZFC curve of the parallel external magnetic field for the 120 W
plasma power samples shows an increase in the magnetic moment at the
transition temperature. This suggests the presence of a well-defined
paramagnetic to AFM phase transition. In the case of $Cr_{2}Te_{3}$, the
magnetic behavior is complex due to the coexistence of strong FM and weak AFM
interactions within the system. The system exhibits spin frustration at lower
temperatures, leading to the emergence of a re-entrant spin glass phase.
Previous studies by Roy et al. and Hashimoto et al. have reported the presence
of both FM and AFM interactions in $Cr_{2}Te_{3}$, with the stronger FM
interactions dominating above the TC (165K)[35]. The existence of a more
complex magnetic order below the TC has also been predicted. The magnetic
field-dependant magnetization M-H measurement of the 120W plasma power sample
was performed to confirm the FM and perpendicular anisotropy. The M-H loops
are shown in Figure 4(b) and (c). Figure 4(d) shows the exchange bias field
from the M-H loop as a function of temperature.
It has been observed that the exchange field decreases with increasing
temperature, indicating the presence of FM and AFM coupling in the sample. The
observation of exchange bias in $Cr_{2}Te_{3}$ coupled with the ferromagnet
CdTe suggests the presence of an AFM component in $Cr_{2}Te_{3}$, likely
arising from c-axis Cr atom defects in the lattice[36]. The existence of both
ferromagnet or ferrimagnet properties in $Cr_{2}Te_{3}$ dependent on the
nearest neighbor spin coupling energy of the Cr sites [16]. It has been
reported that the ferromagnetic order enhanced perpendicular magnetic
anisotropy. Depending on the spin orientations by the applied magnetic field,
the net magnetization in both directions is different. So, the magnetization
property will depend on the direction of the applied magnetic field to the
c-axis of the $Cr_{2}Te_{3}$. The unit cell of $Cr_{2}Te_{3}$ contains
alternating layers of Cr and Te atoms and has CrI vacancies in every second
metal layer. The CrI atoms in the partially occupied layers are sandwiched
between the CrIII layers and have no close-by neighbors in the a-b plane,
whereas the CrII and CrIII atoms make up the fully occupied metal layers. The
crystal growth resembles a hexagonal lattice in the (001) plane, indicating
high crystalline quality as observed from XRD measurements.
### III.4 Spin-phonon interaction in $Cr_{2}Te_{3}$
The Raman shift of the most prominent peaks, $E_{2g}$(125 cm-1) and
$E_{3g}$(144 cm-1), as a function of temperature are shown in Figure 5(a) and
(b). The Balkanski model fits the data well, indicating the presence of strong
electron-phonon interactions. The FWHM and integrated intensity of the
$E_{2g}$(125 cm-1) and $E_{3g}$(144 cm-1) peaks were studied as a function of
temperature, as depicted in Figure 5(c) and (d). Anomalous behavior near the
transition temperature of 166 K indicates that all Raman peaks correspond to
the quasi-2D $Cr_{2}Te_{3}$ film. Magnetization measurement of $Cr_{2}Te_{3}$
confirms the presence of a strong ferromagnetic component with a weak
antiferromagnetic component. The temperature-dependent behavior of the
magnetic moment suggests the induction of a magnetic moment below the
transition temperature, owing to the influence of nearest neighboring spin-
spin interactions. The excellent agreement between the temperature-dependent
Raman spectroscopy and magnetization measurements, and theoretical
calculations provides a comprehensive understanding of the spin-phonon
interaction in $Cr_{2}Te_{3}$, which is rucial for potential applications in
spintronics and contributes to the understanding of the material’s behavior
and properties.
Figure 5: Temperature dependent Raman spectrum, (a) The Raman peak position of
E2g (125 cm-1) and (b) E3g (145 cm-1) mode of vibration with temperature and
fitted equation(1) for showing the anharmonicity. (c) FWHM with the function
of temperature of two most prominent peaks of E2g and E3g modes, respectively.
(d) Integrated intensity of Raman peaks as a function of temperature of two
most prominent peaks of E2g and E3g modes. The extra Raman peak shift below
the transition temperature of both peaks (e) E2g and (f) E3g and fitting
function equation 4.
In the absence of spin order above the transition point, the temperature-
induced phonon position shift follows a three-phonon scattering model that
takes into account the anharmonicity contributions leading to softening or
hardening of some Raman modes. The occupation probabilities of the phonons are
responsible for the temperature dependence Raman shift and linewidth
broadening. In the case of the phonon decay process, the optical phonon at
$\gamma$ point with energy $\hbar\omega_{c2}(0)$ decays into two acoustic
phonons from the same branch, keeping both energy and momentum conserved. So
the decay phonon has $\hbar\omega_{c2}(0)$/2 energy with equal and opposite
momentum. This process can be expressed as the temperature dependence Raman
shift for the optical phonon frequency $\omega_{c2}(T)$ as follows (Klemens’s
model):
$\omega_{c2}(T)=\omega_{0}-C(1+\frac{2}{\exp\frac{\hbar\omega}{k_{B}T}-1})$
(1)
Here, $\omega_{0}$ represents the phonon frequency at 0 K, $\hbar$ is the
reduced Planck constant, $k_{B}$ is the Boltzmann constant, T is the
temperature, and C is the three phonon interaction constant. The first term
represents the initial phonon frequency, while the second term accounts for
anharmonic contributions due to phonon decay. This can be understood by
fitting parameters such as C and $\omega_{0}$. It has been observed that the
temperature dependence Raman shift fitted well with Klemens’s model up to
162K. The anomaly below 162K does not follow the three-phonon decay process.
According to Klemens’s model, the decayed acoustic phonon should have a
constant velocity below the critical temperature depending on the thermal
conductivity of the materials. However, the upturn behavior indicates the
increase of the phonon velocity. By further decreasing the temperature below
162 K the Raman shift deviats from the expected anharmonic tendency indicating
an additional scattering mechanism involved. Such a mechanism can be
attributed to the coupling of the lattice to spin fluctuations. e.g., spin-
phonon interaction. Revisiting the M-T data of $Cr_{2}Te_{3}$, it has been
observed that the paramagnetic to FM transition occurs at 162K, indicating the
role of spin below TC of 162K. Fennie et al. suggested that for an FM material
spin coupled with optical phonon below transition temperature [37]. The
optical phonon frequency $\omega{{}^{\prime}}$ can be considered as the
contribution of the anharmonic behavior and an extra term due to the deviation
due to spin-phonon interaction ($\lambda_{sp}$), which can be written as :
$\omega{{}^{\prime}}=\omega_{c2}(T)+\Delta{\omega_{sp}}$ (2)
Where $\omega_{c2}(T)$ is the Raman frequency shift as mentioned in the
Klemens model and $\Delta\omega_{sp}$ is the deviation due to spin-phonon
interaction from the Klemens model fitting. Finally, the spin-phonon coupling
parameter $\lambda_{sp}$ for $Cr_{2}Te_{3}$ can be calculated using this
method. This effect is similar to that observed in other magnetic materials,
and it is associated with phonon renormalization induced by magnetic ordering
due to the coupling between magnetic ordering and crystal lattice. Considering
the N-N Heisenberg spin system, the magnetic energy of the FM system can be
correlated with the frequency shift of lattice vibrations by introducing the
proportional constant as spin-phonon coupling parameter $\lambda_{sp}$, which
depends on the derivatives of the exchange constants with respect to the
coordinates of the magnetic ions. In the absence of magnetostriction effects
and electronic states renormalization, the contribution of spin-phonon
coupling $(\lambda_{sp}$ of the $k^{th}$ position is approximately given by[7,
38]:
$(\Delta{\omega_{k}})_{sp}=-\sum_{i,j}\lambda_{sp}<S_{i}.S_{j}>$ (3)
where $<S_{i}.S_{j}>$ is the spin correlation function. This contribution,
considering only first-neighbor interactions and a molecular field
approximation, can be considered in the case of the lattice vibration, as
$(\Delta{\omega_{p}})=-\lambda_{sp}S^{2}\phi(T)$ (4)
where $\Delta\omega_{p}$ indicates the deviation of Raman frequency,
$\lambda_{sp}$ is the coefficients of the spin-phonon coupling,
$<S_{i}.S_{i+1}>$ denotes the average for adjacent spins (where S can be taken
as 3/2 for Cr spins); and $\phi$(T) is the order parameter given by
$\phi(T)=1-(T/T_{\theta})^{\gamma}$ where $T_{\theta}$ is the transition
temperature. The temperature dependence function of $\phi(T)$ is the
normalized order parameter, which is proportional to the square of the average
magnetization M(T).
Figure 6: (a) Above the TC the lattice vibration are observed due to Laser
shining. (b) Below the transition temperature lattice vibration perturbed the
exchange interaction.
The quasi-2D ferromagnetism in $Cr_{2}Te_{3}$ is intriguing due to its high
spin-phonon interaction, especially above its paramagnetic region. Klemens’s
model well fits the temperature dependence Raman frequency till TC,
$\approx$162K and anomalies below that, as shown in Figure 5(a) and (b),
indicating anharmonicity and spin-phonon interactions. Figure 5(e) and (f)
fitted with the equations 3 and 4 in the temperature range of 80 – 160 K. The
spin-phonon interaction parameter ($\lambda_{sp}$) is calculated from the
fitting and found to be 0.47$\pm$0.03 cm-1. This value confirms the presence
of a strong spin-phonon interaction, which is responsible for the large phonon
dispersion. Below the transition temperature, the order parameter is well-
fitted, and the $\gamma$ value is determined to be 2.61$\pm$0.04. The value of
$\lambda_{sp}$ is comparable with the literature and indicates the colinear
spin arrangement describes the magnetostructural transition below TC. The
temperature-dependent Raman concluded that the anomalies below TC is the
signature of spin-phonon interaction arising from the paramagnetic to FM
transition. The larger spin-phonon interaction implies that atoms with
stronger bonds have a greater influence on the magnetic ordering with colinear
spin states. We attribute that below TC the structural fluctuation rearranges
the spins and tends to frustrate spin states with larger magnetization. A
schematic model, as shown in Figure 6(a) shows the normal lattice vibration
and Figure 6(b) illustrates the lattice vibration with spin-spin interaction
and distortion of the exchange coupling due to spin-phonon interaction due to
laser shining. Both the Raman modes and the magnetic moment as a function of
temperature strongly suggest the interaction of spin momentum with lattice
phonons below TC due to nearest neighbor spin-spin interactions.
## IV Conclusions
The quasi 2D FM $Cr_{2}Te_{3}$ flakes were synthesized and optimized from co-
evaporated thin films and subsequent plasma treatment. The FESEM and EDX
analysis confirms the surface flake-like structures. The XRD and HRTEM confirm
the (001) oriented single crystal $Cr_{2}Te_{3}$ flakes with c-axis van-der-
wall gap of 2.95 Å. The magnetic behavior of the 2D $Cr_{2}Te_{3}$ has been
investigated using SQUID measurements with both perpendicular and parallel
applied magnetic fields to the c-axis of the flakes. A mixed magnetic phase
with strong FM and weak AFM components has been confirmed from M-H and M-T
measurements. The polarization-dependent Raman analysis of the thin flakes
provides insights into the modes of lattice vibration. Additionally, the DFT
calculations support the experimental frequency of the Raman modes. This
strengthens the identification of the Raman modes and supports the reliability
of the experimental observations. The observed Raman frequency, FWHM, and
integrated intensity from the temperature-dependent Raman measurements
indicate the transition temperature. The anomalies below the TC suggest the
strong spin-phonon interaction parameter ($\lambda_{sp}$) of 0.47$\pm$0.03
cm-1. The combination of experimental techniques, along with theoretical
calculations, has provided a comprehensive understanding of the structural,
magnetic, and vibrational properties of the 2D $Cr_{2}Te_{3}$ flakes and
strengthened the understanding of strong spin-phonon interaction mechanisms.
The spin-phonon phenomena of 2D ferromagnetic $Cr_{2}Te_{3}$ thin flakes can
be suitable for 2D magnetic spintronic device applications.
###### Acknowledgements.
The authors acknowledge the National Institute of Science Education and
Research Bhubaneswar, DAE, India for supporting this work through the project
RIN-4001.
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|
# Jovian sodium nebula and Io plasma torus S+ and brightnesses 2017 – 2023:
insights into volcanic vs. sublimation supply
###### Abstract
We present first results derived from the largest collection of
contemporaneously recorded Jovian sodium nebula and Io plasma torus (IPT) in
[S II] 6731 Å images assembled to date. The data were recorded by the
Planetary Science Institute’s Io Input/Output observatory (IoIO) and provide
important context to Io geologic and atmospheric studies as well as the Juno
mission and supporting observations. Enhancements in the observed emission are
common, typically lasting 1 – 3 months, such that the average flux of material
from Io is determined by the enhancements, not any quiescent state. The
enhancements are not seen at periodicities associated with modulation in solar
insolation of Io’s surface, thus physical process(es) other than insolation-
driven sublimation must ultimately drive the bulk of Io’s atmospheric escape.
We suggest that geologic activity, likely involving volcanic plumes, drives
escape.
JGR: Space Physics
Planetary Science Institute 1700 East Fort Lowell, Suite 106 Tucson, AZ
85719-2395, USA Center for Space Physics Boston University Boston, MA 02155,
USA University Of Colorado, Boulder Boulder, CO 80309, USA
Jeffrey P<EMAIL_ADDRESS>
A large set of Jovian sodium nebula and Io plasma torus S+ images provides
context for Io and Jovian magnetospheric studies
Enhancements in Na and S+ emission last 1 – 3 months, ruling out insolation-
driven sublimation as their driver
Volcanic plumes likely play a key role in atmospheric escape
## Plain Language Summary
The Planetary Science Institute’s Io Input/Output observatory (IoIO) is
composed almost entirely of off-the-shelf parts popular with amateur
astronomers. IoIO uses special filters to isolate emission from two gasses
found around Jupiter: neutral sodium and ionized sulfur. The sodium is thrown
out from Io in a vast cloud called the Jovian sodium nebula. The ionized
sulfur collects into the Io plasma torus (IPT), a ring-shaped structure
centered around Jupiter that wobbles around Io’s orbital path. These gasses
ultimately come from Jupiter’s highly volcanic moon, Io. We see the Na nebula
and IPT brighten frequently. This demonstrates that the majority of the
material leaving Io comes from whatever drives the frequent brightening
events, with volcanic plumes likely playing a key role. Our results challenge
a widely held belief in the scientific community, that the majority of the
material in the Na nebula and IPT comes from Io’s tenuous global atmosphere,
which is fed by the sublimation of surface frosts and is relatively stable in
time. Our dataset also provides important context for NASA’s Juno mission and
supporting observations that focus on Io volcanism, the material’s likely
source, and Io’s magnetosphere, the material’s ultimate destination.
## 1 Introduction
One of the first hints that Io was somehow releasing material into the Jovian
magnetospheric environment in large amounts compared to the other Galilean
satellites came from spectroscopic observations of sodium D1 (5896 Å) and D2
(5890 Å) emissions that were time variable, broad, and in a ratio suggesting
optically thick gas [RA. Brown Chaffee (1974)]. kupo76 conducted spectroscopic
studies of S+ in the [S II] 6717 Å and 6731 Å doublet in the orbital plane of
Io and detected extended emission corotational with Jupiter. As Voyager I
approached Jupiter, Extreme ultraviolet (EUV) emission from S III, S IV, and O
III resolved into a torus-like structure encircling Jupiter, dubbed the Io
plasma torus [<]IPT;¿broadfoot79. The potential source of the material
escaping Io was first hinted at when Io itself was also seen to be
intermittently bright at a particular orbital phase angle in the 3 – 5 $\mu$m
region of the infrared spectrum, with volcanic activity being one of several
possible explanations offered [Witteborn . (1979)]. Volcanic activity on Io
was subsequently unambiguously confirmed by Voyager 1 images of plumes and
volcanic surface features [Morabito . (1979), Smith . (1979)]. This volcanism
gives rise, either directly or indirectly, to an SO2 dominated atmosphere
[Pearl . (1979), Kumar (1979), de Pater . (2021)] with minor constituents,
including NaCl [Lellouch . (2003), McGrath . (2004), Moullet . (2010), Redwing
. (2022)].
Io’s atmosphere undergoes several reactions with material in the IPT,
resulting in detectable effects. For majority species S and O, change
exchange, sputtering, and electron impact ionization are important processes
for removing atmospheric material [<]e.g.,¿mcgrath87, smyth88a, smyth88,
thomas04, schneider07, dols08, dols12, smith22, resulting in a roughly torus-
shaped neutral cloud confined to Io’s orbital plane and mapped in the EUV at O
I 1304 Å [Koga . (20181)]. IPT electron impact ionization of this neutral
cloud is the primary process by which the IPT receives new material, with
direct ionization in Io’s atmosphere providing only a minor component [Dols .
(2008), Dols . (2012)]. The canonical value of $\sim$1 ton s-1 of material
flowing into the IPT from Io has been estimated using the IPT’s total EUV
power output and a simple geometric model of the EUV emission region
[<]e.g.,¿broadfoot79, schneider07.
The path that sodium-bearing material takes as it escapes Io’s atmosphere is
different, thanks to the low ionization potential of any sodium-containing
molecule, NaX. For these molecules, impact ionization and charge exchange
processes are very efficient [<]e.g.,¿schneider07. Pickup NaX ions generated
in Io’s exosphere that promptly neutralize and dissociate create a directional
feature, the “jet,” that points radially outward from Io and flaps up and down
in synchrony with the IPT [Pilcher . (1984), Wilson Schneider (1999), Burger .
(1999), see also animations accompanying Figure 1]. A more extended structure,
dubbed the “stream,” has a similar radial morphology and behavior relative to
IPT modulation, but extends for several hours in Jovian local time downstream
of Io’s position. This comes from NaX+ ionized in Io’s exosphere by IPT plasma
and swept downstream in the plasma flow before they neutralize [Schneider .
(1991), Wilson Schneider (1994)]. IPT NaX+ ions that dissociate produce
neutral fragments that are ejected from the IPT at an average of $\sim$70 km
s-1, which is the Jovian corotational velocity at the IPT. This velocity is
above Jupiter’s escape velocity. Thus, neutral Na is detected at distances
$>$500 Jovian radii (Rj) from Jupiter [Mendillo . (1990)]. All these neutral
sodium emission features are known collectively as the Jovian sodium nebula
and are well-described by Monte Carlo modeling techniques [Wilson . (2002)].
The Jovian sodium nebula has been the subject of several long-term studies
using ground-based coronagraphic techniques [Mendillo . (2004), Yoneda .
(2009), Yoneda . (2015), Roth . (2020), Morgenthaler . (2019), and this work].
Except for the study presented here, long-term observations of the IPT from
ground-based observatories have been limited. The longest continuous study to
date covered a full Jovian opposition using a spectroscopic technique [ME.
Brown Bouchez (1997)]. Ground-based IPT imaging campaigns using coronagraphic
techniques have typically lasted a few weeks per opposition, though some have
extended over several oppositions [Schneider Trauger (1995), Woodward .
(2000), Nozawa . (2004), Kagitani . (2020)]. These ground-based observations
concentrate on the bright [S II] emissions of the 6717 Å, 6731 Å doublet,
which are excited by IPT thermal electrons. A significant amount of structure
is seen in high-resolution IPT images: a dense “ribbon” near Io’s orbit is
separated by a gap from the more disk-like “cold torus,” closer to Jupiter
[<]e.g.¿schneider95. There is evidence that diffusion proceeds inward from the
ribbon to the cold torus [Herbert . (2008)]. Long-term spectro-imaging
observations of EUV emission of the IPT have been conducted from Voyager,
Cassini, and Hisaki [Broadfoot . (1979), Steffl . (2004), Yoshioka . (2014)].
This emission, known as the “warm torus,” is more extended radially and
vertically than the ribbon. The EUV emissions are excited by suprathermal
electrons and have been used to study radial transport in the IPT, providing
evidence that the total residence time of material in the IPT is 20 – 80 days
[Bagenal Delamere (2011), Hess . (2011), Copper . (2016), Tsuchiya . (2018)].
Once material has left the IPT, it rapidly spirals outward in a few days
[Bagenal Delamere (2011)]. Thus, long-term monitoring of the IPT, such as that
presented here, provides critical context to any study of Jupiter’s broader
magnetosphere, such as that conducted by NASA’s Juno mission [Bolton . (2017)]
and supporting observations [Orton . (2020), Orton . (2022)].
In this work, we present the first results of a combined Jovian sodium nebula
and IPT monitoring campaign, conducted since March 2017 by the Planetary
Science Institute’s Io Input/Output observatory (IoIO). The coronagraphic
observations are described in §2 and §3 provides the methodology used to
reduce the data. Section 4 presents the primary results of our study, which is
a time history of the surface brightnesses of the Na nebula and IPT (Figure
3.3). This time history shows 1 – 2 brightness enhancements per 7-month
observing season, each lasting 1 – 3 months, such that emissions are seldom
found in a quiescent state. In §5 we compare our results to previous studies,
noting that although none of the previous workers reported such frequent
activity in the Na nebula and IPT, all are consistent with it. In §6, we use
the IoIO data to rule out solar insolation-driven sublimation of Io’s surface
frosts as the primary driver of material from Io’s atmosphere, showing instead
that geologic processes must be involved. We then review the existing evidence
that connects enhancements in material escape from Io’s atmosphere with
volcanic plume activity and discuss implications for the transport of
material. A summary and concluding remarks are provided in §7. In §8, we
suggest additional uses for the IoIO dataset, including providing support for
current and planned missions to Jupiter.
## 2 Observations
All observations presented here were conducted with the Planetary Science
Institute’s Io Input/Output observatory (IoIO). IoIO consists of a 35 cm
Celestron telescope feeding a custom-built coronagraph, described by
morgenthaler19. Since the publication of that work, both the observatory
hardware and control software have been upgraded, enabling fully robotic
acquisition of Jovian sodium nebula and IPT [S II] on- and off-band images,
regular photometric observations of burnashev85 spectrophotometric standard
stars in all filters, and observations of telluric sodium foreground emission.
Bias, dark and sky flat images are also periodically recorded. Since
2017-03-09, IoIO has contemporaneous recorded Na 5890 Å nebula and IPT [S II]
6731 Å observations on over 550 nights, with over 2300 Na images and over 8300
[S II] images collected. The observatory has been operated on another
$\sim$500 nights in support of other Planetary Science Institute projects
[<]e.g.,¿adams23DPS and pilot studies, increasing the number of
spectrophotometric calibrations and time coverage of telluric Na emission,
which provides a time-variable foreground emission [<]e.g.,¿plane18.
## 3 Data Reduction
### 3.1 General Considerations
All IoIO data are reduced pipeline-style using the software enumerated in the
Open Research section. The burnashev85 spectrophotometric observations show
each filter in our filter library provides stable zero points and extinction
coefficients over the length of our study, modulo random nightly variations
due to variation in atmospheric transparency, which we ignore in our current
analyses, using instead the biweight location [Tukey (1977)], of all the
measurements of each filter. The biweight location, $\zeta_{biloc}$, is
defined as:
$\zeta_{biloc}=M+\frac{\sum_{|u_{i}|<1}\
(x_{i}-M)(1-u_{i}^{2})^{2}}{\sum_{|u_{i}|<1}\ (1-u_{i}^{2})^{2}}$ (1)
where $x$ is the data, $M$ is the median of the data. The quantity $u_{i}$ is:
$u_{i}=\frac{(x_{i}-M)}{c*MAD}$ (2)
where $c$ is a tuning constant, set to 9 in our case, and $MAD$ is the median
absolute deviation:
$MAD=\mathrm{median}(|(x_{i}-\overline{x}|)$ (3)
The biweight location is a more robust statistical measure of the central
location of a distribution than the median, particularly for data not
distributed as a Gaussian [Beers . (1990)]. Surface brightnesses are expressed
in rayleighs (R), where:
$1~{}R={10^{6}\over 4\pi}~{}\mathrm{photons~{}cm^{-2}~{}sec^{-1}~{}sr^{-1}}$
(4)
Astrometric solutions of our images, together with high-quality JPL HORIZONS
ephemerides [Giorgini . (1996)] enable high-precision alignment of on- and
off-band images before subtraction of the off-band images. Subtraction of the
off-band images effectively removes Jupiter’s scattered continuum light from
the on-band images. When astrometric solutions using field stars fail, the
position of Jupiter on the coronagraph neutral density (ND) filter is used to
establish the astrometric center of the image, with the clock angle determined
by the previous successfully solved image. As expected from our stellar
calibrations, we found the ratio between our on- and off-band sky flats gave
stable results over the lifetime of the project. Thus, we used the biweight
location of all ratios to scale the off-band images before subtraction from
the on-band. Sample reduced images are shown in Figure 1.
Figure 1: Sample IoIO Na nebula (left column) and IPT [S II] 6731 Å (right
column) images. The top row shows images recorded just before the large 2018
enhancements in Na nebula and IPT emission (see Figure 3.3), the middle row
images recorded near the 2018 Na nebula and IPT peaks in emission and the
bottom row images recorded near the 2022 peaks. Animations are provided in the
on-line journal.
We note that our calibration procedure is a significant improvement over the
technique used by morgenthaler19, which relied on the image of Jupiter through
the ND filter for surface brightness calibration. As discovered after the
installation of a larger and filter wheel in 2019, Jupiter’s detected
brightness is subject to an unexpected Fabry Pérot effect between the narrow-
band and ND filters, with each narrow-band filter providing a different
magnitude of effect. Our current procedure avoids this issue by using the
stellar and flat-field calibrations described above.
In order to establish a time-sequence of the Na nebula and IPT brightnesses,
we first rotate the images reduced by the procedure above into the plane of
the IPT centrifugal equator using the relation:
$\alpha=-A\times\ cos(\lambda_{\mathrm{III}}-P)$ (5)
where $\alpha$ is the angle between the Jovian rotational axis and the
perpendicular to the IPT centrifugal equator, $\lambda_{\mathrm{III}}$ is the
sub-observer System III longitude, $A$ is the amplitude of the oscillation of
the centrifugal equator and $P$ is the $\lambda_{\mathrm{III}}$ longitude of
the intersection of the magnetic and equatorial planes. For this work, we used
$A$ = 6.8∘ [Moirano . (2021)] and $P$ = 290.8∘ [Connerney . (1998)]. Values of
$A$ = 6.3∘ [Phipps Bagenal (2021)] and $P$ = 286.61∘ [Connerney . (2018)]
could also be used, and would result in trivial differences in our extracted
surface brightnesses.
### 3.2 Na Nebula
As shown by previous work (§1) and the Na nebula animations accompanying
Figure 1 in the online journal, the bulk of the bright jet and stream emission
follow Io in its 42 hour orbit and flap up and down with each 9.925 hr Jovian
rotation. To minimize the effects of this high variability when extracting
surface brightnesses from individual Na nebula images, we rotate each image by
$\alpha$, as described above, and divide the resulting image into horizontal
apertures distributed vertically from the IPT centrifugal plane, as shown in
Figure 2. The ND filter and area beyond the edge of the narrow-band filters
are masked, as are pixels with values above the non-linear point of the CCD,
with a larger mask area applied around Galilean satellites. The average
surface brightness in each aperture is calculated by totaling the individual
surface brightnesses of the unmasked pixels and dividing by the total number
of unmasked pixels. The final surface brightness for a given distance from the
IPT centrifugal plane is the average of the surface brightnesses of the pair
of apertures located at that distance above and below the plane.
Figure 2: Sample Na nebula image illustrating reduction steps described in
§3.2. The blue lines indicate boundaries between apertures used to extract
surface brightness values as a function of vertical distance from the IPT
centrifugal plane. The boundaries between apertures are defined by the
following vertical distances from the IPT centrifugal plane: 4 Rj – 8 Rj, 8 Rj
– 16 Rj, and 16 Rj – 32 Rj, with the average distances from the plane of each
pair of apertures used for subsequent identification (e.g., see legend of
Figure 3.3, top). Masked areas are shown in white.
#### 3.2.1 Removal of Telluric Sodium Contamination
Telluric sodium emission provides a time-varying and, at times, substantial
field-filling component to our Na nebula images. We attempted to remove this
emission using an empirical model constructed from our multi-year dataset of
telluric sodium emission observations. The model accounted for airmass
effects, solar scattering angle, and seasonal effects. However, after
subtraction of the model, the time sequence of Na nebula surface brightnesses
was still quite noisy. Thus, we instead subtract the average surface
brightness of emission $>$32 Rj above and below the centrifugal plane from the
extracted surface brightnesses of each image. As a result, the variation
induced by telluric emission was greatly diminished. A final step in the Na
nebula reduction is to compute the biweight location of all of the
measurements at each distance on each night. The results are plotted together
with 21-point moving median filter in Figure 3.3 (top).
A byproduct of our telluric sodium removal technique is to induce an
intensity-dependent error of the order $\sim$ 5 R – $\sim$ 25 R in our quoted
Na nebula surface brightnesses. This is because our telluric removal procedure
effectively assumes that the brightness of the Na nebula is zero at the edge
of the IoIO FOV. However, as shown by larger-field images [Mendillo . (2004),
Yoneda . (2009), Yoneda . (2015)], the emission $>$30 Rj above and below
Jupiter’s equatorial plane varies from $<$ 5 R to $\sim$ 25 R, depending on
whether or not the nebula is enhanced. This effect could be corrected using a
model of the Na nebula emission, however, doing so would not affect the
results of our current study.
### 3.3 IPT
As shown by comparing Figure 1 (lower right) to the IPT image in Figure 3,
rotating by Equation 5 provides a natural coordinate system for extracting
brightness values of the ansas (edges, Latin: “handles”) of the IPT. As
described in §1, the [S II] 6731 Å ansas primarily capture the IPT ribbon
emission. We extract the average surface brightness from each ribbon feature
using the two-step process shown graphically in Figure 3. Specifically,
starting from the rotated image, we define ansa extraction regions that extend
radially 4.75 Rj to 6.75 Rj from Jupiter and $\pm$1.25 Rj above and below the
IPT centrifugal equator (white boxes). Radial profiles of the emission in the
white boxes are shown in the bottom row. These profiles are generally well-fit
by a Gaussian plus sloping continuum of the form:
$f(x)=Ae^{-{(x-x_{0})^{2}}\over{2\sigma^{2}}}+P_{0}+P_{1}x$ (6)
where $A$ is the peak surface brightness of the Gaussian component of the
radial profile, $x_{0}$ is the ribbon radial distance from Jupiter, $\sigma$
is the width of the ribbon, and $P_{0}$ and $P_{1}$ are the coefficients of
the linear background. The $\pm$1-$\sigma$ limits of these Gaussians are used
to define the radial limits of the region used to extract vertical profiles,
shown in the top row, outside plots. Equation 6, with $P_{1}$ = 0, is used to
fit the vertical profiles. The Gaussian component of this function is
integrated to arrive at an average ribbon intensity of $A\sigma\sqrt{2\pi}$.
This is converted to an average surface brightness by dividing by $\sigma$.
Occasionally, the data are of high enough quality and the torus configured
such that cold torus is resolved. This is the case for the dawn (left) ansa
and results in a small peak inside of the ribbon. We ignore the effect of this
feature on our fits, since the simple sloping continuum plus Gaussian provides
an adequate a foundation for determining the region over which to extract
vertical profiles. As shown in the Figure, the vertical profiles are well-
described by Equation 6, with $P_{1}$ = 0. The total area of the Gaussian
components of each fit is then used to establish the average surface
brightness of each ribbon. If an extraction area contains saturated pixels
from any nearby Galilean satellite, it is excluded from the analysis. Fits
that result in ribbon peak positions outside of the range 5.0 Rj – 6.5 Rj or
peak widths outside the range 0.1 Rj – 0.5 Rj are discarded. In this way, our
extractions are able to adjust for varying observing conditions and the
intrinsic variability in the IPT ansa morphology [<]e.g.,¿schneider95 and
reliably discard pathological cases. The time history of the average ribbon
surface brightnesses, together with a 21-point running median filter of the
dusk ribbon points are plotted in Figure 3.3 (bottom). On timescales of weeks
to months, all other parameters of the fits roughly scale with ribbon surface
brightness, except for the radial peak positions of the ribbon. This behavior
is expected because all of the parameters of the fits, except the radial peak
positions, are sensitive to the total amount of material in the IPT, whereas
the radial positions of the ribbon are determined by physical effects outside
of the IPT [<]e.g.,¿[see also §8]barbosa83, ip83.
Figure 3: Graphical depiction of [S II] ribbon surface brightness extraction
process described in §3.3. An IPT image rotated into the reference frame of
the IPT centrifugal equator is shown in the top, middle panel. Radial profiles
of the ansas are shown in the bottom row, fit by Equation 6. The 1-$\sigma$
limits indicated on these plots define the edges of regions between which the
vertical profiles are computed (top row, outer plots). The average surface
brightness of each ribbon is the integral of the Gaussian component of the fit
of the corresponding vertical profile.
We note that, as derived, the absolute values of the dawn ribbon surface
brightness values shown in Figure 3.3 (bottom) are artificially low because,
at the blueshift of the dawn ribbon, IPT emission falls outside of the central
bandpass of the [S II] 6731 Å filter as measured in a collimated beam. This
effect can be corrected using a velocity-dependent IPT map, the [S II] filter
transmission curve, and consideration of the effects of the telescope’s F/11
light paths on the filter’s total transmission. However, making these
corrections would not affect the results of our current study.
Top: Time sequence of surface brightnesses in Jovian sodium nebula at three
distances from the plasma torus equatorial plane. Bottom: Time sequence of IPT
ribbon average surface brightnesses. A running median filter, 21 points wide,
is applied to each Na aperture and the dusk ribbon brightnesses.
## 4 Results
We anticipate the IoIO data will be very useful for correlative studies for
observations focusing on Io and the effect that material escaping Io has on
Jupiter’s magnetosphere, such as afforded by NASA’s Juno mission [Bolton .
(2017)] and supporting observations [Orton . (2020), Orton . (2022)]. To that
effect, the surface brightness points shown in Figure 3.3 have been archived
at Zenodo [Morgenthaler . (2023)]. In this paper, we focus on what the data
themselves can say about the physical processes that drive material escape
from Io’s atmosphere.
Our 6-year time sequence of Jovian Na nebula and IPT [S II] 6731 Å ribbon
brightnesses (Figure 3.3) shows considerable modulation in each emission line
as a function of time. During each $\sim$7-month observing window, at least 1
– 2 enhancements, each lasting 1 – 3 months are seen. Very little time is
spent in a quiescent state. Visual inspection of Figure 3.3 reveals that the
average values of the Na nebula and IPT surface brightnesses are determined by
the enhancements, rather than any quiescent value. To quantify this finding we
compute the tukey77 biweight distribution (Equation 1) of the measurements
presented in Figure 3.3. Recall from §3.1 that the biweight distribution is a
more robust statistical measure of the central location of a distribution than
the median or average. The biweight distribution values of the Na nebula
points are 80 R, 50 R and 15 R in the 6 Rj, 12 Rj, and 24 Rj apertures,
respectively. Compare this to low values of approximately 30 R, 20 R, and 5 R.
Similarly, the biweight distributions of the ribbon brightnesses are 50 R and
90 R for dawn and dusk, respectively, with minima of 15 R and 30 R.
Visual inspection of Figure 3.3 also shows a quasi-contemporaneous
relationship between the Na nebula and IPT enhancements. For instance, the
relative timing between the peaks of the 2018 Na nebula and IPT enhancements
is significantly different than that seen in 2020. And the fall 2022 Na nebula
enhancement is particularly bright compared to other Na nebula enhancements,
yet the IPT enhancement during that time period is particularly weak compared
to other years. This type of behavior has not been reported before.
We discuss the implications of our results in §6. But first, we compare our
results to those of previous studies, which provide valuable context to our
discussions.
## 5 Comparison to previous studies
IoIO occupies a unique niche in sensitivity, which ideally suits it to study
of the modulation of material flow from Io into the broader Jovian
magnetosphere. The 35 cm telescope aperture of IoIO was chosen to be
comparable to the smallest apertures that have successfully imaged the IPT
[Nozawa . (2004)]. This has allowed us to reliably capture, at modest cost, a
6-year history of the modulation in the IPT [S II] 6731 Å ribbon brightnesses
(presented here) and positions (to be presented in a subsequent work). Our
equipment choice limited the FOV of the instrument to 0.4∘, which is much
smaller than the 2.5∘ – 7∘ FOV of long-term previous coronagraphic Na nebula
studies [Mendillo . (2004), Yoneda . (2009), Yoneda . (2015), Roth . (2020)].
However, the narrower FOV of IoIO affords it much greater sensitivity to
emission close to Io, as evident by comparing the left columns of our Figure 1
to Figure 1 of mendillo04 and Figures 2 of yoneda09, yoneda15. This feature of
the IoIO Na nebula observations will allow us to conduct detailed
morphological studies of the jet and stream in future work.
### 5.1 Sodium-related studies
The outer portions of the IoIO FOV overlap with the inner portions of the
images recorded by other wide-field Na nebula studies [Mendillo . (2004),
Yoneda . (2009), Yoneda . (2015), Roth . (2020)], allowing direct comparison.
For instance, the peak intensity of the fall 2022 Na nebula enhancement
detected by IoIO roughly compares to the peak intensities in the 2007 and 2015
enhancements captured by yoneda09, yoneda15. The roth20 study is useful, since
it provides a time-history of Na nebula brightnesses measured with the same
coronagraph used in the yoneda15 work for a 4-month interval in 2017 during
which no enhancement was reported. Nevertheless, modulation at the $\sim$10 R
level in daily values ($\sim$5 R in the half-month averages) is seen. This is
comparable to the variation seen in the IoIO dataset during the 2018
enhancement. The greater sensitivity of the IoIO coronagraph to emission
closer to Jupiter makes variation of this magnitude much easier to detect.
This implies that periods formerly identified as quiescent in the yoneda15
dataset may, in fact, contain enhancements. By that interpretation, the period
highlighted in the roth20 appears to be capturing the low point between two
enhancements.
Spectroscopic observations conducted at the Lick observatory over the entire
1995 Jovian opposition [ME. Brown (1994), ME. Brown Bouchez (1997)] are also
useful for comparison. This work captured an enhancement in both the Na 5890 Å
doublet and [S II] 6731 Å (see also Sections 5.2 – 5.3). The 10′′ spectrograph
slit was aligned along the centrifugal equator, with peak emission averaged
along the slit reaching levels of 400 R – 800 R. In order to compare to our
data, we extend the aperture extraction procedure outlined in §3.2 using
apertures progressively closer to the centrifugal plane following the same
geometric sequence. We stopped decreasing the aperture size when the emission
brightness increased by $<$10%. The resulting aperture extended 0.5 Rj above
and below the centrifugal plane and resulted in peak brightnesses of 200 R –
300 R during the years 2017 – 2021 and 630 R in 2022. This suggests that the
1995 Na enhancement captured by brown97 was comparable in size to the 2022
enhancement shown in Figure 3.3 (upper panel).
Also important to mention are the Galileo dust detector measurements acquired
1996 – 2003 [Krüger, Geissler . (2003)]. There is evidence that the dust comes
from Io [Graps . (2000), Krüger, Horányi Grün (2003)], is composed almost
entirely of NaCl [Postberg . (2006)] and has its origin from Io volcanic
plumes [Krüger, Geissler . (2003)]. Further evidence shows that NaCl+ is an
important pathway for Na escape from Io’s atmosphere [Grava . (2014), Schmidt
. (2023)]. This suggests that variation seen in our Na nebula dataset and
others should be echoed in the Galileo dust detector data. krueger03 used a
simple geometric model of dust emission from Io to translate dust detector
count rates into the flux of dust from Io (their Figure 2). As discussed by
krueger03, the Galileo orbit precluded continuous measurements of the dust
streams before mid 2000. However, beginning after this time, there was a
large, well-covered enhancement that lasted $\sim$6 months. Subsequent
enhancements in the calculated Io dust flux last $\sim$1 – $\sim$3 months and
have smaller amplitudes than the 2000 enhancement. The magnitudes of the
enhancements are 1 – 4 orders of magnitude, which is much larger than those
seen in sodium nebula data. Full treatment of the reasons for the difference
in magnitude seen between the different measurement methods is beyond the
scope of this work. Rather, we point out that the durations of the
enhancements in the derived dust flux from Io is comparable to those observed
in the Jovian sodium nebula [ME. Brown Bouchez (1997), Yoneda . (2009), Yoneda
. (2015), and this work].
### 5.2 IPT studies
Previous studies of IPT [S II] 6731 Å emission show peak ribbon brightness
values in the $\sim$100 R – $\sim$1000 R range in individual measurements
[<]e.g.,¿morgan85a, oliversen91, jockers92, woodward94, schneider95, thomas01,
nozawa04_no_SII, yoneda09, schmidt18. As described in §3.3, the values shown
in Figure 3.3 are the average surface brightness of each ribbon derived using
a two-step Gaussian fitting procedure, with one Gaussian used to isolate
emission in the radial direction and one to compute the average surface
brightness in the vertical. To convert from averages over the Gaussian
functions to peak values, we multiply by $2\pi$, one factor of $\sqrt{2\pi}$
for the integral over the vertical Gaussian and another factor of
$\sim\sqrt{2\pi}$ to account for the summation between the $\pm 1\sigma$
limits of the radial Gaussian. Following our 21-point moving averages, this
yields peak values of $\sim$200 R – $\sim$900 R, with individual points
ranging from $\sim$50 R to $\sim$1200 R. We take this to be good agreement
with previous studies and therefore independent validation of our stellar
calibration procedure.
The only other published study of IPT [S II] 6731 Å emission lasting more than
a few weeks during a single Jovian opposition is the companion of the
spectroscopic Na nebula observations collected in 1995 at the Lick
observatory, discussed in §5.1 [ME. Brown (1994), ME. Brown Bouchez (1997)].
That study captured an IPT enhancement that lasted $\sim$2.5 months. The
emission pre- and post enhancement was $\sim$200 R and the emission was
$\sim$400 R at its peak. The factor of $\sim$2 difference between the pre/post
and peak values is comparable to the broad enhancement partially captured at
the beginning of the 2019 Jovian opposition. In other words, within the range
observed over our 6-year study.
Also useful for comparison are the two long-term studies of the warm torus
that have been conducted in the EUV, one by Cassini and one by Hisaki
[<]e.g.,¿steffl04a, yoshikawa17. Comparison of the surface brightnesses seen
in the EUV warm torus observations to the surface brightness of the [S II]
6731 Å observations of the ribbon region would require detailed IPT modeling
that is beyond the scope of this work. Thus, we limit our discussion to the
duration of the enhancements. During the duration of its observations, Cassini
captured two enhancements in emissions from ionization states of S and O
lasting of order 1 – 3 months, one in late 2000, the other in early 2001.
During its multi-year observing campaign Hisaki saw one large enhancement that
lasted $\sim$3 months in 2015. Smaller amplitude modulations in the Hisaki
data have also been noted [Roth . (2020)]. These enhancement durations are
comparable to those seen in the IoIO data.
### 5.3 Contemporaneous Na nebula and IPT studies
Two previous studies reported contemporaneous Na nebula and IPT enhancements
[ME. Brown Bouchez (1997), Tsuchiya . (2018)]. These studies, and related
work, concentrated on the detailed behavior of the observed emission during
the enhancements and the implications for physical processes occurring within
the IPT and broader Jovian magnetosphere [<]e.g.,¿yoshikawa17, kimura18,
hikida18, yoshioka18, tsuchiya18, hikida20, roth20, tao16a, tao16b, tao18,
tao21. Such in-depth study of individual enhancements is beyond the scope of
our current work. Rather, we note for comparison to our data, that for both
the 1995 and 2015 enhancements, there was delay of $\sim$4 weeks between the
peak in the Na nebula and IPT S+ emissions, even though in the 2015 case, the
S+ emissions were from the ribbon region and detected via the [S II] 6731 Å
line and in the 2015 case, the emissions were from the warm torus and detected
via the S II 765 Å line. Because different regions of the torus were studied
in the two cases, comparison of the relative strengths of the IPT enhancements
requires modeling that is beyond the scope of this effort. Thus, we are not
currently able to use these studies to corroborate our observation that the
relative strengths of the Na nebula and S+ enhancements can vary significantly
with time.
## 6 Discussion
Our study has a unique combination of sensitivity, cadence and duration that
has enabled it to determine that the Na nebula and IPT are frequently in
states of enhancement and that the enhancements in the two species have a
quasi-contemporaneous relationship. When interpreted within the context of
previous studies, the former result allows us to rule out solar insolation-
driven sublimation as the primary mechanism driving Io atmospheric escape; the
later provides insights into the most likely mechanism – volcanism – and the
subsequent path sodium- and sulfur-containing materials take through and out
of Io’s atmosphere. Our discussion begins with atmospheric escape.
### 6.1 Response of sodium nebula and IPT to Io atmospheric escape
As reviewed in §1, Io’s atmosphere is removed by interaction with the IPT via
charge exchange, sputtering, and electron impact ionization to fill the
neutral clouds on timescales of hours [<]e.g.,¿smyth88, dols08, dols12,
smith22. The apertures used to extract Na nebula surface brightnesses (Figure
2) are primarily filled by sodium traveling near IPT’s $\sim$70 km s-1
corotation velocity. The residence time of this material in the IoIO FOV is
$\sim$11 hours. Furthermore, we have chosen the apertures that integrate over
the effect of Jupiter’s $\sim$10 hr rotation period and Io’s $\sim$40 hr
orbit. Thus, to the accuracy of $\sim$1 day, the Na nebula surface
brightnesses shown in Figure 3.3 (top panel) provide a good indicator of the
modulations in the escape rate of sodium-bearing material from Io’s
atmosphere.
The response of the IPT to Io atmospheric escape is somewhat more complicated
than that of the Na nebula. The neutral clouds described in the previous
paragraph are shaped by interaction with the IPT through the processes of
impact ionization and charge-exchange [<]e.g.,¿smyth03. Impact ionization
results in the addition of new material and proceeds on timescales of $\sim$ 1
day [Smyth Marconi (2003)]. The residence time of plasma in the IPT, is 20 –
80 days, with the shorter residence times corresponding to times of higher
total plasma density [Bagenal Delamere (2011), Hess . (2011), Copper . (2016),
Tsuchiya . (2018)]. Thus, when there is an enhancement in the escape of
sulfur-bearing material from Io’s atmosphere, the peak in the IPT S+ 6731 Å
ribbon will lag by an amount dependent on the IPT plasma density. A model is
being developed that could, in principle, calculate the precise IPT ribbon
response [D. Coffin . (2020), DA. Coffin . (2022), DA. Coffin Withers (2023),
Nerney Bagenal (2020), Nerney . (2022), Nerney . (2023)], but its completion
and application to the IoIO dataset is beyond the scope of our current
project. Thus, we take the $\sim$4 week delay between the peaks in Na nebula
and IPT S+ emission in the previous two studies that captured contemporaneous
enhancements [<]§5.3;¿brown97, tsuchiya18 as indicative of the plasma
transport time during a typical IPT enhancement. Four weeks is also similar to
the transport time deduced from the larger IPT enhancement captured by Cassini
[Steffl . (2004), Copper . (2016)].
### 6.2 Interpretation of quasi-contemporaneity of Na nebula and IPT
enhancements
Within the context of the discussion in Sections 5.3 and 6.1, we can now offer
an interpretation the quasi-contemporaneous nature of the Na nebula and IPT
enhancements seen in Figure 3.3. In all cases except 2020, each major Na
nebula enhancement has a companion enhancement seen in the IPT nebula that is
delayed by $\sim$4 weeks. This $\sim$4 week delay is consistent with that seen
in previous studies and is indicative of simultaneous release of sodium- and
sulfur-bearing material from Io’s atmosphere. In 2020, the delay between the
Na nebula and IPT enhancements peaks is almost twice as long, however, the
profiles of both enhancements are more complicated than the enhancements in
other years: the Na nebula enhancement has a shoulder on its trailing edge and
the IPT enhancement appears to consist of a broad, main peak, followed by a
small, sharp peak. Thus, we offer the suggestion that in mid 2020, there are
two overlapping sets of Na nebula and IPT enhancements, with the earlier set
being larger than the later. A similar relationship may exist between other,
smaller enhancements, such as the shoulder on the early 2020 Na nebula peak
and the small IPT peak in mid 2019. We discuss the implications of the
variation in the relative Na nebula and IPT peak _sizes_ seen in each
contemporaneous pair (e.g., fall 2022 being the most extreme) in §6.6.
### 6.3 Ruling out solar insolation-driven sublimation
The current paradigm holds that the bulk of the escaping material from Io’s
atmosphere is supplied by Io’s global sublimation atmosphere
[<]e.g.,¿schneider07, dols08, dols12. This paradigm suggests that, in the
absence of some other perturbing effect on Io’s atmosphere, the variations
seen in the Na nebula and IPT should be dominated by variations in solar
insolation, that is, Io’s 42 hour orbit or Jupiter’s 12 year orbit
[<]e.g.,¿depater20_post_eclipse, tsang12. Enhancement in the escape of
material from Io’s atmosphere may also be modulated by Jupiter’s magnetic
rotational period (9.925 hr) due to Io’s apparent motion within the IPT
[<]e.g.,¿smyth11. Even when considering the timescales of the responses of the
Na nebula and IPT to material escape from Io’s atmosphere, discussed in
Sections 6.1 – 6.2, solar insolation and magnetic periodicities are not
compatible with the behavior of the enhancements seen in Figure 3.3. Thus, we
argue that one or more other physical mechanisms are driving atmospheric
escape during enhancements. Since enhancements dominate the average supply of
material from Io’s atmosphere (Sections 4, 6.1), the process(es) driving
enhancements provide the bulk of the material in the Jovian sodium nebula and
IPT.
### 6.4 The case for volcanism
The initial claim of a link between Io volcanism and material release from
Io’s atmosphere was made using Jovian sodium nebula images recorded with a
cadence of weeks to months and disk-integrated infrared observations [Mendillo
. (2004)]. A subsequent study using Jovian sodium nebula observations recorded
with a near-nightly cadence and a much more extensive set of disk-resolved Io
infrared observations, has failed to validate that initial claim [Roth .
(2020)]. In this work, we take a different approach and use the time behavior
of material release from Io’s atmosphere to suggest the most likely driver for
atmospheric escape is volcanic plumes.
Recall from §1, that based on EUV brightness of the IPT, $\sim$1 ton s-1 of
material must be flowing into it from Io’s atmosphere. We can now attribute
this amount of flux to the mechanism(s) that cause enhancements. Io’s
atmosphere is itself tenuous and, without resupply from the surface or
subsurface, cannot act as a reservoir for supplying enhancements in escape
that last weeks to months. We have ruled out variation due to solar
insolation-induced sublimation and magnetospherically enhanced escape of Io’s
global atmosphere, in §6.3. Thus, geologic processes of some sort must
ultimately be involved in the dominant process(es) of atmospheric escape from
Io’s atmosphere.
In §6.2, we showed that enhancements in the escape of sodium and sulfur-
bearing material from Io’s atmosphere occur simultaneously. This simultaneity,
together with the geologic nature of the processes driving escape imply that
there is a single geographical location responsible for driving atmospheric
escape in each pair of enhancements, with that location not necessarily being
the same for each pair. Finally, we note that behavior of the Na nebula and
IPT surface brightnesses appears to be stochastic, with large and small
enhancements interleaved. The picture that emerges is that geologic activity
at discrete sites on Io results in enhancements in the escape of sodium- and
sulfur-bearing materials that last 1 – 3 months, with 1 – 2 enhancements seen
during each 7 month Jovian opposition observing window and with the relative
amount of material escape varying between individual events.
Of the known geologic processes on Io that match the above criteria, volcanism
is the most likely to result in the stochastic perturbation of Io’s
atmosphere, via processes involving plumes. Observational support for Io
volcanic plume-driven atmospheric escape comes from the correlation between
plumes observed by Galileo [P. Geissler . (2004), PE. Geissler McMillan
(2008)] and enhancements in the Jovian dust streams [Krüger, Geissler .
(2003)]. As reviewed in §5.1, the dust streams are composed almost entirely of
NaCl and come from Io [Graps . (2000), Krüger, Horányi Grün (2003), Postberg .
(2006)]. Furthermore, NaCl+ has been shown to be an important pathway for Na
escape from Io [Grava . (2014), Schmidt . (2023)]. Based on this evidence, it
is plausible to suggest that volcanic plumes play a key role in the supply of
the Jovian sodium nebula. We use the very large Jovian dust stream enhancement
that peaked in early September 2000 and the IPT enhancement observed by
Cassini that peaked about a month later as observational evidence of the
connection between IPT enhancements and volcanic plumes [Krüger, Geissler .
(2003), Steffl . (2004), see also §5.2].
The difficulty with suggesting that volcanic plumes themselves are responsible
for launching material out of Io’s atmosphere is that plume vent velocities
are far below Io’s gravitational escape velocity, even when considering local
atmospheric heating by plume dynamics [<]e.g.,¿schneider07, mcdoniel17,
depater20_post_eclipse, redwing22. Thus, if plumes are implicated in material
escape from Io, the mechanism must be indirect. Plume models show that shocks
in the plume canopy impede upward flow of material, redirecting it outward and
toward the surface [Zhang . (2003), Zhang . (2004)]. zhang03 suggest that the
material that is redirected forcibly toward the surface enhances sublimation
of SO2 frost over a large area. Sublimation of SO2 frosts by hot
surface/subsurface lavas would provide a similar localized enhancement in Io’s
SO2 atmosphere. This SO2 would then be available to interact with the IPT via
the pathways of sputtering, electron impact ionization and change exchange.
The difficulty with this scenario is that it does not provide a comparable
mechanism for enhancing the escape of NaCl, since NaCl has a much higher
sublimation temperature than SO2. However, the amount of NaCl provided by the
plume itself and/or NaCl lofted from Io’s surface by sublimation may be
sufficient to drive escape, when considering interaction with the IPT over a
large area (see also §6.5).
Perhaps more a more plausible path of escape for both SO2 and NaX is to
consider the ability of plumes to loft material to sufficient altitude to
enable enhanced interaction between IPT plasma and the tops of plumes. An
extension of the zhang03, zhang04 model of Pele’s plume shows that by
including interaction between the top of the plume and the IPT, better
agreement is found with the distribution of material seen on the surface
[McDoniel . (2015), McDoniel . (2017), McDoniel . (2019)]. The mcdoniel19 work
provides theoretical validation of the ability of plume tops and IPT plasma to
interact, but stops short of a full quantitative calculation of the amount of
material that could be removed by this interaction. Thus, although there is
observational evidence that connects volcanic plumes to enhancements in the
escape of sodium- and sulfur-bearing material from Io’s atmosphere, current
state-of-the-art theoretical calculations have not been able to determine the
exact pathway taken by the material to Io’s exosphere.
### 6.5 Implications for the transport of sodium- and sulfur-bearing material
through Io’s atmosphere
Atacama Large Millimeter/submillimeter Array (ALMA) observations of Io’s
atmosphere reveal collections of hot NaCl, KCl and SO2 gases, that are
interpreted as plumes [Redwing . (2022)]. Redwing . note that the highest
column density collections of alkali and SO2 gasses are consistently _not_
found to be coincident with each other (Figure 9 of that work). As discussed
by Redwing ., these results are difficult to explain given that SO2, the
primary volatile on Io, is expected to be associated with all plume activity.
These results are also in apparent conflict with our result that sodium- and
sulfur-bearing material are consistently seen to escape at the same time,
implying a common geographic source (Sections 6.2 – 6.4). In this Section, we
discuss some mechanisms that might contribute to this effect, including those
not considered by Redwing ..
redwing22 provide two potential reasons for the lack of spatial coincidence
between alkali and SO2 plumes in Io’s atmosphere which rely on the difference
in vaporization pressures of these materials. (1) SO2 gas is produced
primarily by hot lava vaporizing frost deposits, with these deposits found
primarily at low- to mid-latitudes. Alkalis sublime at a much higher
temperature and therefore will not be released into Io’s atmosphere by this
effect. (2) The alkalis observed by ALMA are released in the plumes of high-
temperature volcanoes. Redwing . note that these high-temperature plumes
should also produce SO2 but that these alkali-producing volcanoes are
consistently located at high latitudes where atmospheric temperatures may be
low enough to freeze SO2 within the plumes. In this way, (1) explains why SO2
plumes appear in the absence of alkali plumes (low latitudes) and (2) suggests
that SO2 is always collocated with the high-latitude alkali plumes, however,
this high-latitude SO2 is largely invisible to ALMA because it is in solid
form. Evidence of solid-phase transport of SO2 through Io’s atmosphere comes
from Galileo detection of very high mass-to-charge ratio ions, interpreted as
clusters of SO2 molecules, or “snowflakes,” when it flew over Io’s north pole
Frank Paterson (2002). This explanation requires that the SO2 gas in the
plumes at low latitudes not contribute significantly to SO2 escape, and
requires NaCl to be primarily sourced from the polar plumes, possibly by the
mechanism discussed in the next paragraph. Plume models have yet to be
constructed that would test this “snowflakes” hypothesis.
Another way to “hide” SO2 and/or NaCl from ALMA is to ionize them. The
behavior of Io’s auroral Na, O, SO2, and SO emission while Io transitions to
and from eclipse behind Jupiter provides evidence that (a) photoionization is
the primary mechanism for producing SO${}_{2}^{+}$ and (b) SO${}_{2}^{+}$
plays an important role on the pathway of NaCl escape from Io’s atmosphere via
charge-exchange Schmidt . (2023). Io’s polar atmosphere is exposed to the sun
for longer periods of time than that above the rest of Io, thus increasing the
average rate of photoionization at the poles. Furthermore, Io’s collisional
atmosphere is thinner in these regions <e.g.,¿walker10, providing more access
of plume material to the exosphere, where IPT-driven escape processes are the
most efficient. This suggests that plumes at the poles will have enhanced
escape (see also §6.6). Like the SO2 “snowflake” atmospheric transport
mechanism suggested in the previous paragraphs, this mechanism favors the
NaCl-rich high-latitude plumes detected by ALMA as the source of the sodium-
and sulfur-bearing material contributing to the Na nebula and IPT
enhancements.
For logical completeness, we also consider the suggestion that NaCl-containing
volcanic dust and ash (“dust bunnies”) may transport NaCl through Io’s
atmosphere in a form not visible to ALMA. To simultaneously explain the ALMA
and IoIO data, this would imply that these large particles are driven through
the atmosphere by the SO2 plumes at low latitudes and that those particles are
quickly charged by interaction with the IPT and removed from ALMA’s FOV by
Jupiter’s magnetic field. Importantly, under this interpretation, the high-
latitude NaCl plumes seen by ALMA would not contribute to escape and the SO2
from these plumes must still be invisible to ALMA, necessitating one or more
of the mechanisms in the previous paragraphs.
In support of the “dust bunny” hypothesis, we recall the observational
evidence we used to connect plumes to the Jovian sodium nebula
<§6.4;¿krueger03, grava14, schmidt23. Krüger, Geissler . found that dust
stream enhancements were likely associated with the plumes of volcanoes such
as Pele, Tvashtar, a region near the north pole, and a region south of Karei
<now known as Grian;¿geissler08. The plume deposits of these eruptions are
primarily SO2 rich, with minor contributions from silicate ash PE. Geissler .
(1999); P. Geissler . (2004). The association between atmospheric escape of
sodium- and sulfur-bearing material and these large, SO2-rich plumes is
suggestive that the large atmospheric disturbances caused by these plumes may
be the key to driving escape. The lack of association between SO2-dominated
plumes and NaCl emission in the ALMA data then becomes support for transport
of NaCl through the atmosphere in these plumes in a form such as dust or ash.
Finally, we suggest that there may be no need to “hide” SO2 from ALMA in the
regions where bright NaCl emission is seen. In the ALMA observation of NaCl
and SO2 that has the highest resolution, relatively faint concentrations of
SO2 gas are seen in the vicinity of the brightest NaCl emission <see
2016-07-26 in Figure 9 of¿redwing22. More detailed analysis of the ALMA data
would be needed to determine if the NaCl and SO2 seen in this observation is
consistent with a single geographic source. Plume models could be used to
determine if the amount of SO2 is, in fact, less than what is expected for the
range volcanic activity that has been observed on Io <see, e.g. review
by¿depater21. Additional high-resolution ALMA images, ideally recorded
contemporaneously with IoIO data, would also be useful. If the NaCl to SO2
ratio were to be found to be reasonable in NaCl plumes, this could imply that
atmospheric escape is primarily tied to NaCl-rich and/or high-latitude plumes
and that the collections of high column density SO2 gas seen at lower
latitudes in the ALMA data contribute at most a minor, steady amount to the
IPT (e.g., baseline in Figure 3.3, bottom panel.)
### 6.6 Processes that modulate Na nebula and IPT enhancement sizes
Having established that volcanic plumes are the likely precipitating agent for
material escape from Io’s atmosphere and discussed possible pathways of
material transport through Io’s atmosphere, we return to our discussion of the
quasi-contemporaneous nature of the Na nebula and IPT enhancements, begun in
§6.2, to offer possible causes for the modulation seen in the _sizes_ of Na
nebula and IPT enhancements. These suggested causes divide into four general
categories: (1) variation in the content of sodium- and sulfur-bearing
material in volcanic plumes, such as produced by different magmas <e.g.,
see¿redwing22; (2) processes involving the interaction of the plumes with the
atmosphere, such as shocks <§6.4;¿zhang03, zhang04 (3) modulation in material
transport through the atmosphere <§6.5;¿schmidt23; and (4) variation in the
efficiency of escape. In this Section, we concentrate on category (4), because
understanding effects in this category greatly enhances the ability to use
Jovian sodium nebula and IPT data to make progress understanding physical
effects in categories (1) – (3).
One of the scenarios discussed in §6.5 suggested that enhanced SO2 ionization
in Io’s polar regions may play a role in enhancing NaCl escape Schmidt .
(2023). This would favor the Tvashtar plume (63∘N 124∘W) over that of Pele
(19∘S 255∘W) or Surt (45∘N, 336∘W) for the source of the very large Jovian
dust stream enhancement observed by Galileo in late 2000, even though Surt,
active at the time, had a much larger infrared output and Pele was the most
active large plume during the Galileo era Marchis . (2002); Porco . (2003);
Krüger, Geissler . (2003); P. Geissler . (2004). Because SO2 ionization and
subsequent rapid dissociation Huddleston . (1998) may be enhanced in the polar
regions Schmidt . (2023), the efficiency of S and O production, and thus
escape, may be enhanced at the poles as well.
Because NaCl+ has an important role in the pathway of Na escape from Io’s
atmosphere <Sections 5.1, 6.4, 6.5;¿schmidt23, sodium-bearing material in Io’s
anti-Jovian equatorial exosphere may have an exaggerated escape efficiency
over that of sulfur-bearing material. The sodium “jet” was seen to be rooted
in the anti-Jovian equatorial region of Io’s exosphere the one time it has
been imaged in sufficient spatial detail for detection near Io Burger .
(1999). Burger . offered two hypotheses to explain this behavior: (i) Material
was being injected into the exosphere from below at this location, e.g., by
volcanism. (ii) Ionization is enhanced in this region due to Io equatorial
auroral activity <e.g.,¿roesler99, geissler99, roth14. Case (ii) would provide
a mechanism for enhancing material flow into the jet, over material flow to
the IPT, since the jet is formed by prompt neutralization of pickup ions
within Io’s exosphere Schneider . (1991); Wilson Schneider (1994), whereas
direct ionization of material in Io’s exosphere has been shown to be a minor
contributor to the influx of plasma to the IPT <§1;¿dols08, dols12. The prompt
neutralization of the sodium forming the jet also explains why it is not
expected to be seen rooted at the sub-Jovian auroral spot: the initial
gyration of the ions directs them into Io’s surface. We note that hypotheses
(i) and (ii) are not necessarily mutually exclusive: the “jet” may always be
rooted in the region of the anti-Jovian equatorial auroral spot, but its
response to atmospheric escape may be exaggerated if that atmospheric escape
is located in that region.
Finally, dust, which acquires a negative charge Zook . (1996), enhances the
initial escape of Na-bearing material on Io’s sub-Jovian hemisphere Grava .
(2021). As the dust particles are destroyed and the constituent molecules and
atoms are released, they acquire a positive charge and join the “jet” feature.
A positive identification between a Na nebula enhancement and a plume on the
sub-Jovian hemisphere, could thus potentially be used in support of the “dust
bunnies” hypothesis discussed in §6.5.
### 6.7 Toward a connection with Io infrared observations
Since the early 1980s, synoptic Io infrared observations have been used to
help understand Io volcanic processes and their geologic implications <e.g.,
see review by¿depater21. As noted in §6.4, these infrared observations have
been compared to sodium nebula observations in an attempt to establish a
correlation Mendillo . (2004) which has not stood the test of time Roth .
(2020). The initial attempt to connect infrared indicators of Io volcanic
activity to enhancements in the sodium nebula made the implicit assumption
that the brightest infrared events should be correlated to the brightest
nebula images. Here we suggest that instead, the dimmer infrared events may be
more likely to be correlated to the brighter sodium nebula enhancements.
One of the fundamental results of our study is that the Jovian sodium nebula
and IPT show contemporaneous enhancements of varying relative amplitudes that
last 1 – 3 months (Figure 3.3; Sections 4, 6.1 – 6.2). A long-term study of
the time variability of Io’s hotspots found that they divided into two groups:
those with persistent activity and those that exhibited sudden brightening,
followed by a steady decay de Kleer de Pater (2016). For those hotspots that
exhibited sudden brightening events, the brighter the event, the shorter the
decay <see, e.g., Figure 11 of¿dekleer16. The hot spots with decay times of
order 1 month or longer were dimmer, by a factor of 5 or more, than the
brightest outbursts. If we make the very simplistic assumption that in a
volcano where plume activity is found, plume activity will persist over
roughly the same time period as infrared activity, we can support the argument
that infrared hotspots exhibiting eruption phases lasting 1 – 3 months are the
more likely to be correlated to atmospheric release events. Thus, dim infrared
outbursts may be the most likely to be correlated with enhancements in
material release from Io’s atmosphere.
## 7 Summary and Conclusions
We have used IoIO, an observatory composed almost entirely off-the-shelf
equipment <§2 and¿morgenthaler19 to collect the largest set of
contemporaneously recorded Jovian sodium nebula and Io plasma torus (IPT) in
[S II] 6731 Å images assembled to date (see examples in Figure 1 and
accompanying animations). Using simple image analysis techniques (§3), we
construct a time history of the brightnesses of the Na nebula and IPT [S II]
emission (Figure 3.3). Qualitative inspection of this Figure shows 1 – 2
enhancements in the Na nebula and IPT [S II] emission per $\sim$7-month
observing window, such that a quiescent state of emission is rare (§4). The
minimum and maximum surface brightness values seen in the IoIO Na nebula and
IPT images compare favorably with previous studies (§5.1 – 5.2). Most large
IPT enhancements peak $\sim$4-weeks after the corresponding enhancement in the
Na nebula, as seen in previous studies (§5.3) and as expected from plasma
transport within the IPT (§6.1). The exception to this, seen in mid 2020, is
likely caused by the overlap of multiple enhancements (§6.2). We rule out
sublimation as the primary driver of material escape from Io’s atmosphere in
§6.3. This is our most definitive result.
Having ruled out sublimation as the primary driver of atmospheric escape from
Io, we show that geologic activity in some form, likely volcanic plumes,
drives escape, either directly or indirectly (§6.4). In light of other
published results, this has implications on the transport of material through
Io’s atmosphere (§6.5). In §6.6, we review the processes that can modulate the
relative sizes of contemporaneous Na nebula and IPT enhancements, focusing on
processes that might modulate the efficiency of Io atmospheric escape as a
function of geographic location. Finally, in §6.7, we note that Io’s dimmer
infrared outbursts have durations and time profiles similar to Na nebula and
IPT enhancements, suggesting that it these 1 – 3 month-long infrared outbursts
may be the more likely to show correlation with the release of material from
Io’s atmosphere.
In conclusion, our work shows that off-the-shelf equipment with minimal
customization, together with simple analysis techniques can be used to collect
data that provides valuable insights into the processes which produce material
on Io’s surface, transport it through its atmosphere, and release it into
Jupiter’s broader magnetospheric environment.
## 8 Future Plans
We have pointed out the existing observational evidence that links plume
activity on Io to atmospheric escape in §6.4. Further confirmation of this
link may be accomplished by accumulating additional contemporaneous IoIO
observations of the Na nebula and IPT together with ALMA observations of Io’s
atmosphere – currently there is only overlap during March 2018 Redwing .
(2022). Disk-integrated observations conducted by the NOrthern Extended
Millimetre Array (NOEMA) interferometer of the Institut de Radioastronomie
Millimetrique (IRAM), such as those conducted by roth20, may also be useful.
Continued theoretical work on the effect that plumes have on Io’s atmosphere
and exosphere, as well as the interaction between the exosphere and the IPT is
also needed <e.g.,¿bloecker18, mcdoniel19, dols08, dols12, dols23,
adeloye23DPS. These observational and theoretical studies can also be useful
to help differentiate between the hypotheses offered in 6.5 concerning
material transport through Io’s atmosphere.
Continued disk-resolved observations of Io IR activity, such as those carried
out at the Keck and Gemini telescopes de Kleer . (2019) will also be
interesting as they might lead to validation of the correlation that was
initially claimed between Na nebula, IPT and IR brightnesses Mendillo .
(2004); Yoshikawa . (2017); Yoshioka . (2018); Tao . (2018); Koga . (20182),
but has subsequently proven elusive de Kleer . (2016); Roth . (2020).
We are also planning to conduct more detailed analysis of the IoIO images. For
instance, the IoIO images of the Na nebula contain three distinct features –
the “banana,” “jet,” and “stream” – that can be used to estimate the neutral
sodium source rate from Io Wilson . (2002). The IPT ribbon positions, which
are detectable with IoIO (Figure 3, lower panels), are related to the dawn-
dusk electric field, which is modulated by a combination of material flow
toward the magnetotail and solar wind pressure Barbosa Kivelson (1983); Ip
Goertz (1983). When combined with the analysis presented here, the IPT ribbon
positions retrieved from the IoIO data will provide a significant amount of
information regarding the production of material on Io and its subsequent flow
through and out of Jupiter’s magnetosphere.
Finally, the unique sensitivity of IoIO to Na nebula and IPT [S II] 6731 Å
enhancements, together with reliable robotic operation and $<$24 hour
turnaround for pipeline reduction ideally suits it to provide real-time alerts
of enhancements in the departure of material from Io’s atmosphere. These can
inform planned observations of Io from both ground- and space-based platforms.
In particular, nearly all of the plasma found in Jupiter’s magnetosphere comes
from Io and makes its way through the IPT in 20 – 80 days before rapidly
spiraling out through the rest of the magnetosphere Bagenal Delamere (2011);
Hess . (2011); Copper . (2016); Tsuchiya . (2018). The modulations seen in the
lower panel of Figure 3.3 therefore precede modulation in plasma density
throughout the Jovian magnetosphere, a feature that can be used to enhance the
science operations of the Juno mission Bolton . (2017) and supporting
observations Orton . (2020, 2022). NASA’s Europa Clipper Howell Pappalardo
(2020) and ESA’s JUICE Grasset . (2013) missions will benefit from planned
IoIO observations, because of the record of exogenic material impinging on
Europa, Ganymede and Callisto during those missions. Also, because
enhancements in the Jovian dust streams can induce detector fatigue in Europa
Clipper’s SUrface Dust Analyzer <SUDA;¿goode23, IoIO observations can be used
to inform SUDA operations while Europa Clipper is sampling the broader Jovian
magnetospheric environment and thus optimize detector performance for that
mission’s primary target.
## Open Research Section
The following software was used in this project: Astrometry.net Lang . (2010),
AstroPy Astropy Collaboration . (2022), Astroquery Ginsburg . (2019),
BigMultiPipe Morgenthaler (2022), Burnashev Morgenthaler (20232), CCDMultiPipe
Morgenthaler (20233), ccdproc Craig . (2017), IoIO control software
Morgenthaler (20231), matplotlib Hunter (2007), moviepy Zulko . (2021), NumPy
Oliphant (2006); Harris . (2020), photutils Bradley . (2022), precisionguide
Morgenthaler (20234), Python 3 Van Rossum Drake (2009), reproject Robitaille .
(2020), SciPy Virtanen . (2020), specutils Earl . (2022)
The reduction products used to create Figure 3.3 are archived with Zenodo
Morgenthaler . (2023).
###### Acknowledgements.
IoIO is hosted at the San Pedro Valley Observatory, near Benson, Arizona, and
has benefited greatly from the services provided by observatory manager, Dean
Salman. Gavin Nelson, Andy Sorenson, Elisabeth Adams, and the staff of
Starizona have also provided invaluable assistance in observatory maintenance.
The authors are grateful for the insightful comments received during the
manuscript review process. This work has been supported by NSF grants 1616928
and 2109219 and NASA grant 80NSSC22K0317 to the Planetary Science Institute
and NASA grant 80NSSC20K0559 to Boston University.
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# On polynomial solutions of PDE
Anna R. Gharibyan, Hakop A. Hakopian
###### Abstract
In this paper we prove that the PDE $p(D)f=q,$ where $p$ and $q$ are
multivariate polynomials, has a solution in the space of polynomials of total
degree not exceeding ${n+s},$ where $n$ is the degree of $q$ and $s$ is the
zero order of $O=(0,\ldots,0)$ for $p.$
Key words: PDE with constant coefficients, bivariate polynomial, $s$-fold zero
of a polynomial.
Mathematics Subject Classification (2010): 35E20.
## 1 Introduction
Let us start with the bivariate case. Denote by $\Pi_{n}$ the space of
biariate polynomials of total degree at most $n:$
$\Pi_{n}=\\{\sum_{i+j\leq
n}a_{ij}x^{i}y^{j}\\},\quad\dim\Pi_{n}=1/2(n+1)(n+2).$
Thus $\Pi_{0}=\\{c:c=const.\\},\dim\Pi_{0}=1$ and
$\Pi_{-1}=\\{0\\},\dim\Pi_{-1}=0.$
We set
$\Pi=\cup_{n\geq 0}\Pi_{n}.$
Denote also by $\overline{\Pi}_{n}$ the space homogeneous polynomials of total
degree $n:$
$\overline{\Pi}_{n}=\\{\sum_{i+j=n}a_{ij}x^{i}y^{j}\\},\quad\dim\overline{\Pi}_{n}=n+1.$
For a polynomial $p$ denote by $p(D)$ the respective differentiation operator:
$p(D)=p(\frac{\partial}{\partial x},\frac{\partial}{\partial y}).$
###### Definition 1.1.
Suppose that $p\in\Pi_{n},\ p(x,y)=\sum_{i+j\leq n}a_{ij}x^{i}y^{j}$. Then the
$k$th homogeneous layer of $p$ we denote by
$p_{(k)}=\sum_{i+j=k}a_{ij}x^{i}y^{j}.$
Denote also by $p^{\uparrow}$ the upper nonzero homogeneous layer of $p$ and
by $p_{\downarrow}$ the lower nonzero homogeneous layer of $p.$
###### Definition 1.2.
The point $O=(0,0)$ is called an $m$-fold zero for $p$ if the lower
homogeneous layer of $p(x)$ is the $m$th one. A point $a\in\mathbb{C}^{2}$ is
called an $m$-fold zero for $p$ if the lower homogeneous layer of $p(x+a)$ is
the $m$th one.
Let $V$ and $W$ be finite dimensional linear spaces and $\mathcal{L}$ be a
linear operator $\mathcal{L}:V\rightarrow W.$ In the sequel we will use the
following formula
$\dim(Im\mathcal{L})=\dim V-\dim(ker\mathcal{L}).$ (1.1)
Let us bring the following result:
###### Theorem 1.3 ( Th. 5, [1]; Th. 6, [2]).
Assume that a polynomial $f_{0}$ is a solution of the PDE $p(D)f=0.$ Then the
partial derivatives of $f_{0}$ are solutions too.
###### Corollary 1.4.
Suppose that $p(x,y)=\sum_{i+j\leq n}a_{ij}x^{i}y^{j}$: The homogeneous PDE
$p(D)f=0$ has nonzero polynomial solution if and only if $f=1$ is its
solution, i.e., $p(D)1=a_{00}=0$:
In the sequel we will use the following
###### Theorem 1.5 (Th. 2.1, [3]).
Suppose that the origin $O=(0,0)$ is an $s$-fold zero of $p\in\Pi_{n}.$ Then
the homogeneous PDE $p(D)f=0$ has exactly $D_{k}$ linearly independent
solutions in $\Pi_{k},$ where $D_{k}$ is the $k$th partial sum of the
following number series
$\sum_{i=0}^{\infty}d_{i}=1+2+\cdots+s+\cdots+s+\cdots.$
Note that in the case $s=0,$ i.e., $p(0,0)\neq 0,$ this result states that the
PDE $p(D)f=0$ has no solution except $f=0.$ Of course this statement coincides
with Corollary 1.4.
## 2 Main result
We are going to prove the following
###### Theorem 2.1.
Assume that $q\in\Pi_{m}$ and the origin $O=(0,0)$ is an $s$-fold zero of
$p\in\Pi.$ Then we have that the PDE
$p(D)f=q$
has a polynomial solution from $\Pi_{m+s}.$
In view of Theorem 1.5 we get
###### Corollary 2.2.
Assume that the origin $O=(0,0)$ is an $s$-fold zero of $p\in\Pi$ and
$q\in\Pi_{m}.$ Then we have that the solutions of PDE
$p(D)f=q,$ (2.1)
belonging to $\Pi_{k},\ k\geq m+s,$ form an affine space of dimension
$\sigma=\frac{s(2k-s+1)}{2}.$
Thus the solutions of PDE (2.1) can be represented as
$f=f_{0}+\sum_{i=1}^{\sigma}\lambda_{i}f_{i},$
where $f_{0}\in\Pi_{m+s}$ is a solution of PDE (2.1) and $f_{i},\
i=1,...,\sigma,$ are the linearly independent solutions of the homogeneous PDE
$p(D)f=0$ in $\Pi_{k}.$
Let us start the consideration with the case when both $p$ and $q$ are
homogeneous. Note that the following is a particular case $s=n$ of Theorem
3.1.
###### Proposition 2.3.
Assume that $p\in\overline{\Pi}_{n}$ and $q\in\overline{\Pi}_{m}.$ Then the
PDE
$p(D)f=q$ has a solution from $\overline{\Pi}_{n+m}.$
###### Proof.
Let us verify that in this case the number of linearly independent solutions
of PDE $p(D)f=0$ in $\overline{\Pi}_{n+m}$ is $n.$ Indeed, according to
Theorem 1.5, the number of linearly independent solutions of PDE $p(D)f=0$ in
$\Pi_{n+m}$ is:
$1+2+...+n+mn,$
while in $\Pi_{n+m-1}$ is:
$1+2+...+n+(m-1)n:$
Then consider the linear operator
$\mathcal{L}:\overline{\Pi}_{n+m}\rightarrow\overline{\Pi}_{m}$, given by
$\mathcal{L}f=p(D)f.$ What we verified above means that
$dim(ker\mathcal{L})=n.$ Now, by using the formula (1.1), we get
$dim(Im\mathcal{L})=dim\overline{\Pi}_{n+m}-dim(ker\mathcal{L})=m+n+1-n=m+1=dim\overline{\Pi}_{m}.$
Therefore the operator $\mathcal{L}$ is on $\overline{\Pi}_{n+m}$ and the
equation (2.1) has a solution. ∎
###### Second proof of Proposition 2.3.
By using Theorem 1.5, we proved Proposition 2.3, which states the polynomial
solability of the PDE $p(D)f=q$ in the case when $p$ and $q$ are homogeneous
polynomials. Now, let us establish the same result in other way, which will be
important in the proof of the result in the case of polynomials of more
variables.
Assume that $p\in\overline{\Pi}_{n}$ and $q\in\overline{\Pi}_{m}:$
$p(x,y)=a_{0}x^{n}+a_{1}x^{n-1}y+...+a_{n}y^{n},$
$q(x,y)=b_{0}x^{m}+b_{1}x^{m-1}y+...+b_{m}y^{m}.$
Let us find a solution $f\in\overline{\Pi}_{n+m}$ of the PDE
$p(D)f=q.$ (2.2)
Suppose that
$f(x,y)=\gamma_{0}x^{n+m}+\gamma_{1}x^{n+m-1}y+\cdots+\gamma_{n+m}y^{n+m}.$
Now, the PDE (3.2) looks like
$(a_{0}x^{n}+a_{1}x^{n-1}y+...+a_{n}y^{n})(D)(\gamma_{0}x^{n+m}+\gamma_{1}x^{n+m-1}y+...+\gamma_{n+m}y^{n+m})$
$=b_{0}x^{m}+b_{1}x^{m-1}y+...+b_{m}y^{m}$
By equating the coefficients of $x^{m},x^{m-1}y,\ldots,y^{m}$ in the left and
right hand sides of the equation we get
$\frac{(n+m)!0!}{m!0!}a_{0}\gamma_{0}+\frac{(n+m-1)!1!}{m!0!}a_{1}\gamma_{1}+...+\frac{m!n!}{m!0!}a_{n}\gamma_{n}=b_{0},$
$\frac{(n+m-1)!1!}{(m-1)!1!}a_{0}\gamma_{1}+\frac{(n+m-2)!2!}{(m-1)!1!}a_{1}\gamma_{2}+...+\frac{(m-1)!(n+1)!}{(m-1)!1!}a_{n}\gamma_{n+1}=b_{1},$
$\vdots$
$\frac{n!m!}{0!m!}a_{0}\gamma_{m}+\frac{(n-1)!(m+1)!}{0!m!}a_{1}\gamma_{m+1}+...+\frac{0!(n+m)!}{0!m!}a_{n}\gamma_{n+m}=b_{m},$
respectively.
If $a_{0}\neq 0$, then $\gamma_{m+1},\gamma_{m+2},...,\gamma_{m+n},$ are $n$
free variables in above linear system. Thus the solutions of system form an
affine space of dimension $n.$
If $a_{0}=0$ and $a_{1}\neq 0$, then it is easily seen that $\gamma_{0}$
becomes a free variable instead of $\gamma_{m+1}$ and again we have $n$ free
variables.
In the general case, when $a_{0}=a_{1}=\cdots a_{k-1}=0$ and $a_{k}\neq 0,$
the $n$ free variables are $\gamma_{0},\ldots,\gamma_{k-1}$ and
$\gamma_{m+k+1},\ldots,\gamma_{m+n}.$ ∎
Thus in the above considered case, when $p\in\overline{\Pi}_{n}$ and
$q\in\overline{\Pi}_{m},$ we get another proof of the fact that the solutions
of the PDE (3.2) form an affine space of dimension $n.$
Now consider a particular case $s=0$ of Theorem 3.1 and Corollary 2.2:
###### Proposition 2.4.
Suppose that $p\in\Pi_{n},\ q\in\Pi_{m}$ and the free term of $p$ is not zero:
$a_{00}\neq 0.$ Then we have that the PDE
$p(D)f=q$ (2.3)
has a unique solution $f_{0}\in\Pi_{m}.$ The polynomial $f_{0}$ is the only
solution of the PDE (2.3) also in each $\Pi_{k},\ k\geq m.$
###### Proof.
Since $p(D)1=a_{00}\neq 0$, we get from Corollary 1.4, that the PDE $p(D)f=0$
has a unique polynomial solution $f=0$:
Then consider the linear operator $\mathcal{L}:\Pi_{m}\rightarrow\Pi_{m}$
given by $\mathcal{L}f=p(D)f.$ Above we verified that
$ker\mathcal{L}=\\{0\\}$: Now, by using the formula (1.1), we get that
$dim(Im\mathcal{L})=dim\Pi_{m}-dim(ker\mathcal{L})=dim\Pi_{m}.$
Thus the equation (2.3) has a unique solution in $\Pi_{m}$ denoted by $f_{1}.$
Now, concerning the other polynomial spaces, assume by way of contradiction,
that the equation (2.3) has another solution, denoted by $f_{2},\ f_{2}\neq
f_{1},$ in $\Pi_{k},$ where $k>m.$ Then we get that $f_{1}-f_{2}$ is a nonzero
polynomial solution of the PDE $p(D)f=0,$ which contradicts Corollary 1.4. ∎
Now we are in a position to present
###### Proof of Theorem 3.1.
Let us use induction on $m.$ As the first step of induction consider the case
$m=-1,\ q\in\Pi_{-1},$ i.e., $q=0$ (the next step is $m=0,\ q\in\Pi_{0},$
i.e., $q=const.\neq 0).$ In this case the PDE $p(D)f=0$ has a polynomial
solution $0\in\Pi_{m+s},\ \forall s\geq 0.$
Now assume that the PDE
$p(D)f=q$ (2.4)
has a solution if $q\in\Pi_{m+s-1}.$ Let us prove that it has a solution
assuming that $q\in\Pi_{m+s}.$ Now assume that the lower homogeneous layer of
$p$ is the $s$th layer: $p_{\downarrow}=p_{(s)}.$
Next assume that $f\in\Pi_{m+s}$ and $f^{\uparrow}=f_{(m+s)}$ is the upper
homogeneous layer of $f.$ Note that the case $f_{(m+s)}=0$ follows from the
induction hypothesis.
Consider the following PDE with homogeneous polynomials $p_{(s)}$ and
$q_{(m)}:$
$p_{(s)}(D)f=q_{(m)}.$
According to Theorem 2.3 this equation has a solution
$\hat{f}\in\overline{\Pi}_{m+s}:$
$p_{(s)}(D)\hat{f}=q_{(m)}.$ (2.5)
Next let us seek for a solution of the PDE (2.4) in the form $f=g+\hat{f}:$
$p(D)(g+\hat{f})=q.$ (2.6)
It is easily seen that, in view of (2.5), this PDE is equivalent to
$p(D)(g)=r,$ (2.7)
where $r=q-p(D)(\hat{f}).$
Note that $p_{\downarrow}=p_{(s)}$ readily implies that
$p(D)(\hat{f})\in\Pi_{m}.$ Since also $r\in\Pi_{m}$ we obtain that
$r\in\Pi_{m}$ too.
Next let us verify that $r\in\Pi_{m-1}.$ It is enough to show that
$r_{(m)}=0.$ To this purpose, by using (2.5), we obtain that
$r_{(m)}=q_{(m)}-[p(D)(\hat{f})]_{(m)}=q_{(m)}-p_{(s)}(D)(\hat{f})=q_{(m)}-q_{(m)}=0.$
Now, in view of $r\in\Pi_{m-1},$ let us use the induction hypothesis. Hence,
we get that the PDE (2.7) has a solution denoted by $g_{0},$ where
$g_{0}\in\Pi_{m+s-1}.$ Therefore $f=g_{0}+\hat{f}$ is a solution of the PDE
(2.4). Note also that this implies that $\hat{f}=f_{m+s}.$ ∎
## 3 The case of more than two variables
Let us use stamdard multivariate notation.
For $x=(x_{1},\ldots,x_{k})\in\mathbb{C}^{k}$ and
$\alpha=(\alpha_{1},\ldots,\alpha_{k}\in mathbbZ_{+}^{k}$ denote
$x^{\alpha}=x_{1}^{\alpha_{1}}\cdots
x_{k}^{\alpha_{k}},\quad|\alpha|=\alpha_{1}+\cdots+\alpha_{k},\quad\alpha!=\alpha_{1}!\cdots\alpha_{k}!.$
Denote by $\Pi_{n}^{(k)}$ the space polynomials of $k$ variables of total
degree at most $n,$
$\Pi_{n}=\\{\sum_{|\alpha|\leq
n}a_{\alpha}x^{\alpha}\\},\quad\dim\Pi_{n}^{(k)}=\binom{n+k}{k}.$
We set
$\Pi^{(k)}=\cup_{n\geq 0}\Pi_{n}^{(k)}.$
Denote also by $\overline{\Pi}_{n}^{(k)}$ the space homogeneous polynomials of
total degree $n,$ for which we have that
$\overline{\Pi}_{n}=\\{\sum_{|\alpha|=n}a_{\alpha}x^{\alpha}\\},\quad\dim\overline{\Pi}_{n}^{(k)}=\binom{n+k-1}{k-1}.$
The following result holds for polynomials of $k$ variables:
###### Theorem 3.1.
Assume that $q\in\Pi_{m}^{(k)}$ and the origin $O=(0,\ldots,0)$ is an $s$-fold
zero of $p\in\Pi^{(k)}.$ Then we have that the PDE
$p(D)f=q$
has a polynomial solution from $\Pi_{m+s}^{(k)}.$
It is clear from the previous section that all we need is to generalize
Proposition 2.3 to the case of $k$ variables:
###### Proposition 3.2.
Assume that $p\in\overline{\Pi}_{n}^{(k)}$ and $q\in\overline{\Pi}_{m}^{(k)}.$
Then the PDE
$p(D)f=q$ (3.1)
has a solution from $\overline{\Pi}_{n+m}^{(k)}.$
###### Proof.
Assume that $p\in\overline{\Pi}_{n}$ and $q\in\overline{\Pi}_{m}:$
$p(x)=\sum_{|\alpha|=n}a_{\alpha}x^{\alpha},$
$q(x)=\sum_{|\alpha|=m}b_{\alpha}x^{\alpha}.$
Let us find a solution $f\in\overline{\Pi}_{n+m}$ of the PDE
$p(D)f=q.$ (3.2)
Suppose that
$f(x)=\sum_{|\alpha|=m+n}\gamma_{\alpha}x^{\alpha}.$
Now, the PDE (3.2) looks like
$\sum_{|\alpha|=n}a_{\alpha}x^{\alpha}(D)\sum_{|\alpha|=m+n}\gamma_{\alpha}x^{\alpha}=\sum_{|\alpha|=m}b_{\alpha}x^{\alpha}.$
By equating the terms with $x^{\alpha_{0}},\ |\alpha_{0}|\leq m,$ in the left
and right hand sides of the equation, we get that
$\sum_{|\alpha|=n}a_{\alpha}x^{\alpha}(D)\gamma_{\alpha+\alpha_{0}}x^{\alpha+\alpha_{0}}=b_{\alpha_{0}}x^{\alpha_{0}}.$
For the respective coefficients we have that
$\sum_{|\alpha|=n}(\alpha+\alpha_{0})!a_{\alpha}\gamma_{\alpha+\alpha_{0}}=\alpha_{0}!b_{\alpha_{0}}.$
(3.3)
By the change of unknowns $\gamma^{\prime}_{\alpha}=\alpha!\gamma_{\alpha}$ we
get the linear system
$\sum_{|\alpha|=n}a_{\alpha}\gamma_{\alpha+\alpha_{0}}^{\prime}=\alpha_{0}!b_{\alpha_{0}}.$
Let us verify that the main matrix of this system has full rank.
Note that in the row of the matrix with $\alpha_{0}=(0,\ldots,0)$ we have the
coefficients of the polynomial $p(x),$ while in the row with any $\alpha_{0},$
where $|\alpha_{0}|\leq m,$ we have the coefficients of the polynomial
$x^{\alpha}p(x).$
Thus if the rows of the main matrix are linearly dependent then so are the
polynomials $x^{\alpha}p(x),\ |\alpha_{0}|\leq m,$ which evidently takes place
only if $p(x)\equiv 0.$
Therefore we have that the linear system (3.3) also is a full rank linear
system, where we have $\binom{m+k-1}{k-1}$ equations and
$\binom{n+m+k-1}{k-1}$ unknowns.
Thus we have that the system is consistent. Moreover, in the solution of the
linear system we have exactly
$\sigma(n,m,k):=\binom{n+m+k-1}{k-1}-\binom{m+k-1}{k-1}$
free variables. ∎
## REFERENCES
* [1] H. Hakopian, A multivariate analog of fundamental theorem of algebra and Hermite interpolation, Constructive Theory of Functions, (B. Bojanov, ed.), Proceedings of the international conference, Varna, 2002, Darba, Sofia, 2003, 1-18.
* [2] H. Hakopian and M. Tonoyan, On a multivariate theory, Approximation Theory, A Volume Dedicated to Blagovest Sendov, (B. Bojanov, ed.), Darba, Sofia, 2002, pp.212–230.
* [3] Navasard, Vardanyan, On constant coefficient PDE systems and intersection multiplicities, Proceedings of YSU. Physical and Mathematical Sciences, 54 no. 2 (2020), 108–114.
|
# Notes on conformal anomaly, nonlocal effective action and the metamorphosis
of the running scale
A. O. Barvinsky<EMAIL_ADDRESS>Theory Department, Lebedev Physics
Institute, Leninsky Prospect 53, Moscow 119991, Russia Institute for
Theoretical and Mathematical Physics, Moscow State University, Leninskie Gory,
GSP-1, Moscow, 119991, Russia W. Wachowski<EMAIL_ADDRESS>Theory
Department, Lebedev Physics Institute, Leninsky Prospect 53, Moscow 119991,
Russia
###### Abstract
We discuss the structure of nonlocal effective action generating the conformal
anomaly in classically Weyl invariant theories in curved spacetime. By the
procedure of conformal gauge fixing, selecting the metric representative on a
conformal group orbit, we split the renormalized effective action into
anomalous and Weyl invariant parts. A wide family of thus obtained anomalous
actions is shown to include two special cases of Riegert–Fradkin–Tseytlin and
Fradkin–Vilkovisky actions. Both actions are shown to be contained in the
first three orders of the curvature expansion for a generic one-loop effective
action obtained by covariant perturbation theory. The complementary Weyl
invariant part of the action is given by the “conformization” of the full
effective action—restricting its argument to the conformally invariant
representative of the orbit of the conformal group. This is likely to resolve
a long-standing debate between the proponents of the Riegert action and
adherents of the perturbation expansion for the effective action with typical
nonlocal logarithmic form factors. We derive the relation between quantum
stress tensors on conformally related metric backgrounds, which generalizes
the known Brown-Cassidy equation to the case of nonzero Weyl tensor, and
discuss applications of this relation in the cosmological model driven by
conformal field theory. We also discuss the issue of renormalization group
running for the cosmological and gravitational coupling constants and show
that it exhibits a kind of a metamorphosis to the nonlocal form factors of the
so-called partners of the cosmological and Einstein terms—nonlocal curvature
squared terms of the effective action.
###### Contents
1. 1 Introduction
2. 2 Conformal gauge fixing
1. 2.1 Riegert–Fradkin–Tseytlin gauge
2. 2.2 Fradkin-Vilkovisky gauge
3. 3 Conformal anomaly and covariant curvature expansion
1. 3.1 Cubic order
2. 3.2 Conformal resummation: Fradkin–Vilkovisky anomaly action
3. 3.3 The problem of double poles and global conformal transformations
4. 4 Stress tensor in conformally related spacetimes
1. 4.1 Conformal anomaly from the divergent part of the effective action
2. 4.2 Minimal form of Wess-Zumino action and a-theorem
3. 4.3 Renormalized stress tensors
5. 5 Conformally flat spacetime
1. 5.1 Anomaly driven cosmology
6. 6 Renormalization group and the metamorphosis of the running scale
7. 7 Conclusions
To the memory of Stanley Deser
## 1 Introduction
The status of local Weyl anomalies is widely considered to be fully settled in
current literature. However, the issue of their relevance to concrete physical
effects, as opposed to a mere criterion of consistency at the quantum level of
the classically Weyl invariant theories, often remains a subject of the
debate. The manifestation of the conformal anomaly in physical applications
usually occurs within the effective action formalism, and there is extending
over years debate on the structure of this action, taking place between the
pioneers of the conformal anomaly and adherents of perturbation theory. The
nature of this debate consists in a seemingly contradictory difference between
the known expression for the anomaly action and the form of the nonlocal
effective action obtained by Feynman diagrammatic technique.
As is well known, the one-loop conformal anomaly for classically Weyl
invariant 4-dimensional theory having in Euclidean curved spacetime the
covariantly renormalized effective action $\varGamma[\,g_{\mu\nu}]$ reads as
[1, 2, 3, 4, 5, 6, 7]
$\displaystyle\langle\,T^{\mu}_{\mu}\,\rangle\equiv\frac{2\,g_{\mu\nu}}{\sqrt{g}}\frac{\delta\varGamma}{\delta
g_{\mu\nu}}=\frac{1}{16\pi^{2}}\big{(}\alpha C^{2}+\beta E+\gamma\Box
R\big{)},$ (1.1)
$\displaystyle\;E=R_{\mu\nu\alpha\gamma}R^{\mu\nu\alpha\gamma}-4R_{\mu\nu}R^{\mu\nu}+R^{2},$
(1.2)
where $\sqrt{g}E$ denotes the Gauss–Bonnet density, $C_{\mu\nu\alpha\beta}$ is
the Weyl tensor, $C^{2}=C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}$, and
$\alpha$, $\beta$ and $\gamma$ are the numerical coefficients depending on the
spin of the quantum field.111We work in Euclidean signature spacetime, and our
notations are
$R^{\alpha}{}_{\beta\mu\nu}=\partial_{\mu}\varGamma^{\alpha}_{\nu\beta}-\cdots$,
$R_{\mu\nu}=R^{\alpha}{}_{\mu\alpha\nu}$,
$\Box=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$. For simplicity we do not include in
the anomaly the contribution $F_{\mu\nu}^{2}$ of the vector gauge field and
$\varphi^{4}$-contribution of the self-interacting conformal scalar field. The
anomalous action $\varGamma_{A}[\,g_{\mu\nu}]$ generating this anomaly was
first derived in the nonlocal form by Riegert [8] and by Fradkin and Tseytlin
[9] in the local form of the conformal Wess-Zumino action involving an
auxiliary scalar field—the dilaton responsible for intetwining two conformally
related metrics. The nonlocal form of the Riegert–Fradkin–Tseytlin (RFT)
action reads as
$\displaystyle\varGamma_{A}[\,g\,]$ $\displaystyle=\frac{1}{64\pi^{2}}\int
d^{4}x\,\sqrt{g}\,\Big{(}\alpha\,C^{2}+\frac{\beta}{2}\mathcal{E}_{4}\Big{)}\frac{1}{\Delta}_{4}\mathcal{E}_{4}$
$\displaystyle-\frac{1}{32\pi^{2}}\Big{(}\frac{\gamma}{6}+\frac{\beta}{9}\Big{)}\,\int
d^{4}x\,\sqrt{g}R^{2},$ (1.3)
where
${\cal E}_{4}\equiv E-\tfrac{2}{3}\Box R,$ (1.4)
$\Delta_{4}$ denotes the so-called Paneitz operator [10]
$\Delta_{4}=\Box^{2}+2R^{\mu\nu}\nabla_{\mu}\nabla_{\nu}-\frac{2}{3}R\,\Box+\frac{1}{3}(\nabla^{\mu}R)\,\nabla_{\mu}$
(1.5)
and $1/\Delta_{4}$ implies its inverse—the notation for the operation of
acting by its Green’s function ${\cal G}(x,y)$ on a generic test function
$\psi(y)$, $\Delta_{4}{\cal G}(x,y)=\delta(x,y)$,
$\tfrac{1}{\Delta}_{4}\psi(x)=\int d^{4}y\,{\cal G}(x,y)\,\psi(y)$.
Some time after the invention of the RFT action the attention to it was drawn
by Antoniadis, Mazur and Mottola due to several applications in gravity theory
[11, 12], but this caused a serious criticism [13] of the expression (1.3) in
view of its drastic structural difference from the renormalized effective
action built within perturbation theory in powers of spacetime curvature. This
expansion begins with [3]
$\displaystyle\varGamma_{\rm ren}$ $\displaystyle=\frac{1}{32\pi^{2}}\int
dx\,\sqrt{g}\Big{[}-\alpha\,C_{\mu\nu\alpha\beta}\ln\Big{(}\\!-\frac{\Box}{\mu^{2}}\Big{)}C^{\mu\nu\alpha\beta}$
$\displaystyle-\frac{\gamma}{6}R^{2}\,\Big{]}+O(\Re^{3}),$ (1.6)
$\Re$ collectively denoting here the Riemann, Ricci and scalar curvature, and
does not at all resemble the form of (1.3). This criticism was maintained by
objections against short distance behavior of stress tensor correlation
functions generated by the RFT action, which were shown to contradict the
conformal Ward identities for these correlator [14]. Another criticism was
associated with the objections against the double pole structure of the
Green’s function of the operator (1.5), $\sim 1/\Box^{2}$ [15]. Although these
objections were disclaimed in [16] by explicit calculations of $\langle
TTT\rangle$-correlators, the question might still be hovering unsettled in the
literature [17].
The goal of this paper will be to discuss the status of the effective action
responsible for the generation of the Weyl anomaly. To begin with we will
focus on a wide variety of nonlocal anomalous actions by including the RFT
action in their functional family. The idea of this construction is similar to
gauge fixing applied to the ambiguity of the conformal split of the metric
argument of the action functional, which was suggested rather long ago in
[18]. The resulting class of anomaly actions will be parameterized by the
conformal gauge selecting the representative on the orbit of the local
conformal group. We will explicitly demonstrate that the difference between
the members of this class is a Weyl invariant functional—a point of departure
between various suggestions for the anomalous action. Two particular gauges
will be considered, one of them exactly corresponding to the RFT action (1.3)
and another associated with the Weyl invariant nonlocal rescaling of the
metric field suggested by Fradkin and Vilkovisky. This rescaling, which is
directly applicable in asymptotically flat spacetimes, was designed as a
remedy against the trace anomaly [19]—the analogue of the Yamabe problem of a
local Weyl transformation to the metric with a vanishing scalar curvature.
Then we show how the Fradkin–Vilkovisky version of the anomaly action arises
in the first three orders of the covariant curvature expansion for a generic
one-loop effective action. We discuss the associated mechanism of partial
summation of scalar curvature terms of this expansion [20] along with the
double pole problem for the Green function of the Paneitz operator (1.5).
Lack of uniqueness of the anomaly action defined only up to a Weyl invariant
functional raises, of course, the question of its incompleteness in concrete
applications. This also poses the question of whether the RFT action or its
modifications within the above class provides an optimal description of the
physical problem in question. For example, it is well known that in two
dimensions the stress tensor trace anomaly and the associated nonlocal
Polyakov action are fully responsible for the Hawking radiation of the two
dimensional black holes [21]. On the contrary, in higher dimensions the
anomaly action is insufficient to describe this phenomenon. Still there is a
strong belief [11, 12, 16] that at distances of the horizon scale gravity
theory is essentially modified due to large infrared effects of the conformal
mode described by the action (1.3). These effects might dominate macroscopic
physics at such scales, like for instance the near black hole horizon behavior
of quantum stress tensor [22], the contribution to the scalar sector of
gravitational waves [23] or dynamical vacuum energy in effective theory of
gravity [24]. Though it is not entirely clear how complete is the setup in
these problems, there are physical situations when the conformal mode really
runs the whole show, and we consider as a direct application of (1.3) two
examples of such a situation. These are the calculation of the metric stress
tensor in a generic conformally flat spacetime [25] and the Friedmann metric
cosmology driven by the trace anomaly of conformal invariant fields [26], the
latter playing important role in the model of initial conditions for
inflationary cosmology [27, 28].
A related issue in the problem of nonlocal effective action is the question of
renormalization group (RG) running of the cosmological and gravitational
constants. Though the issue of running scale and its relation to the
cosmological constant problem has already become a byword in current
literature, it becomes increasingly clearer that this running should not be
interpreted in the usual sense of RG theory [29, 30]. The notion of “scale” is
so ambiguous in physics that its running nature actually looses universality
when addressing various physical setups, like for example associating
cosmological inflation with RG running [31]. Serious arguments against running
nature of the cosmological and gravitational couplings in [29, 30] have led to
the notion of cosmological constant partners [32] interpreted in [33] in terms
of separation of scales or decoupling of heavy modes [34, 35]. Still, it is
customary to have nontrivial solutions of RG equations in renormalizable
gravity models [36, 37] with running scale dependent $\varLambda$ and $G$.
Therefore a natural question arises how these solutions have to be interpreted
when the tadpole structure of the covariant cosmological and Einstein terms
preclude them from their actual dependence on the momentum [30].
So one of the goals of this paper is an attempt to clarify this issue within a
special version of the notion of the “scale”. Looking forward to the final
conclusion, we might formulate the suggestion for the notions of running
$\varLambda$ and $G$ couplings as their conversion or metamorphosis into their
nonlocal partners similar to those introduced by J. Donoghue in [32]. Within
perturbation scheme the cosmological and Einstein terms start manifesting
themselves as nonlocal curvature squared terms very different from their
original form.
The paper is organized as follows. In Sect. 2 we decompose the quantum
effective action into anomalous and Weyl invariant parts by imposing the
conformal gauge for the choice of the representative on the orbit of the
conformal group. This allows one to build the whole class of nonlocal
anomalous actions, functionally parameterized by the choice of this gauge and
including the RFT action (1.3) and the Fradkin–Vilkovisky action suggested in
[20]. Sect. 3 contains the discussion of the covariant curvature expansion of
[38, 39, 40] and the way how it contains the anomalous action in the lowest
orders of this expansion. In particular, it is shown that the
Fradkin–Vilkovisky version of this action performs a resummation of the
covariant curvature series in powers of the Ricci scalar [20]. In Sect. 4 we
give a direct and, apparently, not very well known derivation from the RFT
action of the vacuum stress-tensor behavior at the orbit of the conformal
group—a good example of direct applicability of (1.3). Here we also comment on
the application of the anomalous conformal Wess-Zumino action to the a-theorem
[41, 42] and present the generalization of the Brown–Cassidy formula [25] for
the stress tensor to the case of a nonzero Weyl tensor, see Eq.(4.34).
Applications of the anomaly action in conformally flat spacetime are presented
in Sect. 5. It is shown how this action underlies the construction of the
inflation scenario starting from the cosmological initial state in the form of
the mircocanonical density matrix [27, 28, 43, 44, 45], recently reviewed in
[46]. Important feature of this application is the value of the Casimir vacuum
energy which is also determined by the coefficients of the anomalous trace
(1.1) [47, 48, 49, 50].
In Sect.6 we discuss the problem of scale dependence of the gravitational and
cosmological constants related to the ideas of [29, 30, 32] and [51, 52]. Here
we show that in the UV regime the RG analysis of the cosmological and Einstein
terms strongly points out to the conversion of their scale dependence into the
nonlocal form factors of their UV partners represented by curvature squared
terms with dimensionless nonlocal coefficients. We call this phenomenon a
metamorphosis of the running scale, which we derive by using a special scaling
operator. In IR domain the same analysis leads to the low energy partners
depending on mass scale of the theory. These nonlocal partners were suggested
in [32] by J. Donoghue for the cosmological constant term and blueprinted for
the Einstein term in [51, 52] in the form of the long distance modification of
Einstein gravity.
In the concluding section we briefly recapitulate the above observations and
dwell on related potential problems and applications. We start by discussing
the role of Weyl anomaly in the problem of cosmological initial conditions for
the inflation scenario driven by a conformal field theory [46]. This scenario
motivates introduction of numerous conformal higher spin (CHS) fields whose
Weyl anomaly is generated only in the one-loop approximation and, thus,
acquires a kind of nonperturbative status. Then we discuss the uniqueness for
the nonlocal scaling operator used for the derivation of the above
metamorphosis phenomenon. In particular, we show that in the curvature squared
terms of the action it is nearly uniquely determined due to general covariance
of the theory, though in Lorentz symmetry violating models like Hořava gravity
[53] it may be rather ambiguous.
## 2 Conformal gauge fixing
The splitting of the renormalized effective action of a classically
conformally invariant theory into the anomaly part $\varGamma_{A}$ generating
the trace anomaly (1.1) and the Weyl invariant part $\varGamma^{\rm conf}$,
$g_{\mu\nu}\delta\varGamma^{\rm conf}/\delta g_{\mu\nu}=0$,
$\varGamma_{\rm ren}=\varGamma_{A}+\varGamma^{\rm conf},$ (2.1)
is obviously not unique and admits the freedom
$\varGamma_{A}\to\varGamma_{A}+W^{\rm conf},\qquad\varGamma^{\rm
conf}\to\varGamma^{\rm conf}-W^{\rm conf},$ (2.2)
with an arbitrary conformally invariant functional $W^{\rm conf}$,
$g_{\mu\nu}\frac{\delta W^{\rm conf}}{\delta g_{\mu\nu}}=0.$ (2.3)
The freedom in the choice of $W^{\rm conf}[\,g_{\mu\nu}]$ arises as a
functional integration constant for the first order variational equation that
can be written down for $\varGamma_{A}[\,g_{\mu\nu}]$ or for the renormalized
effective action $\varGamma[\,g_{\mu\nu}]\equiv\varGamma_{\rm
ren}[\,g_{\mu\nu}]$. At the orbit of the conformal group passing through the
metric $g_{\mu\nu}$—the argument of the effective action—and parameterized by
the local conformal parameter $\sigma=\sigma(x)$,
$g_{\mu\nu}=e^{2\sigma}\bar{g}_{\mu\nu},$ (2.4)
the renormalized action $\varGamma_{\rm ren}[\,e^{\sigma}\bar{g}\,]$ satisfies
the equation
$\frac{\delta\varGamma_{\rm
ren}[e^{2\sigma}\bar{g}]}{\delta\sigma}=\frac{\sqrt{g}}{16\pi^{2}}\big{(}\alpha
C^{2}+\beta E+\gamma\Box
R\big{)}\Big{|}_{g_{\mu\nu}=e^{\sigma}\bar{g}_{\mu\nu}},$ (2.5)
which can be integrated to give conformal Wess-Zumino action [9]
$\displaystyle\Delta\varGamma[\,\bar{g},\sigma\,]\equiv\varGamma_{\rm
ren}[\,g\,]-\varGamma_{\rm ren}[\,\bar{g}\,]$
$\displaystyle\qquad=\frac{1}{16\pi^{2}}\int
d^{4}x\,\sqrt{\bar{g}}\left\\{\Big{[}\alpha\bar{C}^{2}+\beta\bar{\mathcal{E}}_{4}\Big{]}\sigma+2\beta\sigma\bar{\Delta}_{4}\sigma\right\\}$
$\displaystyle\qquad-\frac{1}{32\pi^{2}}\Big{(}\frac{\gamma}{6}+\frac{\beta}{9}\Big{)}\,\int
d^{4}x\,\Big{(}\sqrt{g}R^{2}-\sqrt{\bar{g}}\bar{R}^{2}\Big{)},$ (2.6)
where the two metrics $g_{\mu\nu}$ and $\bar{g}_{\mu\nu}$ are related by the
equation (2.4), all barred quantities are built in terms of $\bar{g}_{\mu\nu}$
and $\bar{\Delta}_{4}$ is the barred version of the fourth-order Paneitz
operator (1.5). This expression $\varGamma_{\rm ren}-\bar{\varGamma}_{\rm
ren}=\varGamma_{A}-\bar{\varGamma}_{A}$ can also be rewritten in the other
form
$\displaystyle\varGamma_{A}[\,g\,]-\varGamma_{A}[\,\bar{g}\,]$
$\displaystyle\qquad=\frac{1}{16\pi^{2}}\int
d^{4}x\sqrt{g}\,\left\\{\,\Big{[}\,\alpha\,C^{2}+\beta\mathcal{E}_{4}\Big{]}\,\sigma-2\beta\,\sigma\Delta_{4}\sigma\,\right\\}$
$\displaystyle\qquad-\frac{1}{32\pi^{2}}\Big{(}\frac{\gamma}{6}+\frac{\beta}{9}\Big{)}\,\int
d^{4}x\,\Big{(}\sqrt{g}R^{2}-\sqrt{\bar{g}}\bar{R}^{2}\Big{)},$ (2.7)
if one takes into account two important properties of the Paneitz
operator—Weyl invariance of its densitized form,
$\sqrt{\bar{g}}\,\bar{\Delta}_{4}=\sqrt{g}\,\Delta_{4},$ (2.8)
and the finite conformal transformation of ${\cal E}_{4}$—the Gauss-Bonnet
density modified by $\sqrt{g}\,\Box R$ term (1.4),
$\sqrt{g}\,\mathcal{E}_{4}=\sqrt{\bar{g}}\,\bar{\mathcal{E}}_{4}+4\sqrt{\bar{g}}\,\bar{\Delta}_{4}\sigma.$
(2.9)
These two properties are consistent with each other because the last equation
should obviously remain valid under the interchange of $g_{\mu\nu}$ and
$\bar{g}_{\mu\nu}$ accompanied by flipping the sign of $\sigma$.
There is also the third form of the Wess-Zumino action, which will be given
below in Eq.(4.24). It exists for a special renormalization converting to zero
the coefficient $\gamma$ of the $\Box R$ term in (1.1), and underlies the
proof of the so-called $a$-theorem for the monotonic RG flow of the
coefficient $a=\beta/16\pi^{2}$ of the topological term in the trace anomaly
[41, 42].
Modulo a nonvanishing conformal anomaly all points on the orbit of the
conformal group (2.4) are physically equivalent, and this typical situation of
a broken local gauge invariance can be managed by introducing the gauge
condition which uniquely selects $\bar{g}_{\mu\nu}$ as the representative of
the equivalence class of metrics (2.4). If we denote this gauge condition as
$\chi[\bar{g}]=0$ then this representative should be uniquely selected by the
solution of the equation for the conformal parameter $\sigma$,
$\chi[\,\bar{g}\,]=\chi[\,ge^{-2\sigma}]=0,$ (2.10)
this solution being a functional of the metric
$\varSigma_{\chi}[\,\bar{g}\,]$, labelled by the gauge symbol $\chi$,
$\sigma=\varSigma_{\chi}[\,\bar{g}\,].$ (2.11)
The representative of the conformal orbit $\bar{g}_{\mu\nu}[g]$ as a
functional of a given metric $g_{\mu\nu}$ (through which the orbit is passing)
becomes Weyl invariant,
$\bar{g}_{\mu\nu}[\,g\,]\equiv
g_{\mu\nu}\,e^{-2\varSigma_{\chi}[\,\bar{g}\,]},\quad
g_{\alpha\beta}\frac{\delta\bar{g}_{\mu\nu}[\,\bar{g}\,]}{\delta
g_{\alpha\beta}}=0,$ (2.12)
because under any local Weyl rescaling $g_{\mu\nu}\to e^{2\sigma}g_{\mu\nu}$
the conformal parameter transforms as
$\varSigma_{\chi}[g]\to\varSigma_{\chi}[g]+\sigma$ in view of the identity
$\chi\big{[}ge^{-\varSigma_{\chi}[g]}\big{]}\equiv 0$, so that
$\delta_{\sigma}\varSigma_{\chi}[\,\bar{g}\,]=\sigma,$ (2.13)
where $\delta_{\sigma}$ is the operator of the conformal variation
$\delta_{\sigma}\equiv 2\int
d^{4}x\;\sigma(x)\,g_{\mu\nu}(x)\frac{\delta}{\delta g_{\mu\nu}(x)}.$ (2.14)
For the uniqueness of such conformal gauge fixing procedure (in spacetime and
at least in some finite domain of the space of metrics) the Faddeev–Popov
operator $Q_{\chi}=Q_{\chi}(x,y)$, corresponding to the gauge $\chi[g]$,
$\delta_{\omega}\chi(x)=\int d^{4}y\,Q_{\chi}(x,y)\,\omega(y)$, should be
nondegenerate.
Thus, the terms of (2.7)
$\displaystyle W^{\rm
conf}[\,g\,]=\varGamma_{A}[\,\bar{g}\,]+\frac{1}{32\pi^{2}}\Big{(}\frac{\gamma}{6}+\frac{\beta}{9}\Big{)}\\!\int
d^{4}x\sqrt{\bar{g}}\,\bar{R}^{2}$ (2.15)
taken at $\bar{g}_{\mu\nu}[\,g\,]$ can be considered as an irrelevant Weyl
invariant integration “constant”, while the rest of the terms can be
identified with the anomaly action after the substitution of
$\sigma=\varSigma_{\chi}[\,g\,]$. This set of anomaly actions
$\varGamma_{A}[\,g\,]\equiv\varGamma_{\chi}[\,g\,]$ parameterized and labelled
by conformal gauge conditions $\chi$ reads as
$\displaystyle\varGamma_{\chi}[\,g\,]=\frac{1}{16\pi^{2}}\int
d^{4}x\,\sqrt{g}\,\Big{\\{}\big{(}\alpha
C^{2}+\beta\mathcal{E}_{4}\big{)}\varSigma_{\chi}$
$\displaystyle\;\;\;\;-2\beta\varSigma_{\chi}\Delta_{4}\varSigma_{\chi}\Big{\\}}-\frac{1}{32\pi^{2}}\Big{(}\frac{\gamma}{6}+\frac{\beta}{9}\Big{)}\\!\int
d^{4}x\,\sqrt{g}R^{2}.$ (2.16)
The difference between various members of this set is, of course, a Weyl
invariant functional. For two arbitrary conformal gauges one has
$\displaystyle\varGamma_{\chi_{1}}-\varGamma_{\chi_{2}}$
$\displaystyle=\frac{1}{16\pi^{2}}\int
d^{4}x\,\sqrt{g}\,\big{(}\varSigma_{\chi_{1}}-\varSigma_{\chi_{2}}\big{)}$
$\displaystyle\times\Big{[}\alpha\,C^{2}+\beta\mathcal{E}_{4}-2\beta\Delta_{4}\big{(}\varSigma_{\chi_{1}}+\varSigma_{\chi_{2}}\big{)}\Big{]}.$
(2.17)
Conformal variation of this expression is vanishing, because of the
transformation law (2.13) for $\varSigma_{1,2}$, Weyl invariance of the
density $\sqrt{g}\,C^{2}$ and the relation (2.9) which in the infinitesimal
form reads as
$\delta_{\sigma}\Big{[}\sqrt{g}\,\mathcal{E}_{4}\Big{]}=4\sqrt{g}\,\Delta_{4}\sigma,$
(2.18)
so that using all the above properties
$\delta_{\sigma}\big{(}\varGamma_{\chi_{1}}-\varGamma_{\chi_{2}}\big{)}=0$.
Note that with our definition of the anomaly action (2.16) the way it enters
the full quantum action can be represented as
$\varGamma[\,g\,]=\varGamma_{\chi}[\,g\,]+\varGamma[\,\bar{g}\,]+\frac{1}{32\pi^{2}}\Big{(}\frac{\gamma}{6}+\frac{\beta}{9}\Big{)}\int
d^{4}x\sqrt{\bar{g}}\,\bar{R}^{2},$ (2.19)
where $\bar{g}_{\mu\nu}[\,g\,]=e^{-2\varSigma_{\chi}[\,g\,]}g_{\mu\nu}$
### 2.1 Riegert–Fradkin–Tseytlin gauge
An obvious choice of the conformal gauge associated with the Gauss–Bonnet
density and the Branson curvature is the Riegert–Fradkin–Tseytlin gauge
$\chi_{{\vphantom{L}}_{\rm RFT}}[\,\bar{g}\,]\equiv\bar{\mathcal{E}}_{4}=0.$
(2.20)
It can be imposed for topologically simple spacetime manifolds with a
vanishing bulk part of the Euler characteristics (see Eq.(2.38) and footnote 3
below), to which in particular belongs asymptotically flat spacetime to be
mainly considered throughout the paper. The advantage of this gauge is that it
is exactly solvable due to the transformation law for the Branson curvature
(2.9). Applying this gauge and using Eq. (2.8) we obtain a linear equation on
$\varSigma_{{\vphantom{L}}_{\rm RFT}}$ which has a solution in terms of the
inverse Paneitz operator
$\varSigma_{{\vphantom{L}}_{\rm
RFT}}=\frac{1}{4}\,\frac{1}{\Delta}_{4}\mathcal{E}_{4}.$ (2.21)
Formally substituting this expression to (2.16) we obtain exactly the RFT
action (1.3).
This RFT action and the inverse Paneitz operator are well defined and exist in
asymptotically flat spacetime under Dirichlet boundary conditions at infinity
when treated within perturbation theory in powers of the curvatures whose
collection is denoted below as $\Re$. Indeed, in this case
$\frac{1}{\Delta}_{4}=\frac{1}{\Box^{2}}+O(\Re),$ (2.22)
and this operator works well when it is applied to the functions of the
Branson curvature type $\sim\mathcal{E}_{4}$. Because of the double-pole
nature of the operator $1/\Box^{2}$ its action on generic functions may be
badly defined due to infrared divergences, but when the function is
represented by the total derivative structure it generates, when acted upon by
$1/\Box^{2}$, well defined multipole expansion valid in four dimensions at
spacetime infinity [39].222 As discussed in [39], the operator $1/\Box^{n}$ in
$D$-dimensional space with $D<2n$ is ill defined unless the functions it acts
upon are of the form $\partial_{\alpha_{1}}...\partial_{\alpha_{m}}j(x)$,
$m=2n-D+1$ with the function $j(x)$ having an asymptotic behavior
$j(x)=O(1/|x|^{D})$, $|x|\to\infty$. This property can be explained by the
fact that in the multipole expansion of
$\tfrac{1}{\Box}\partial_{\alpha_{1}}...\partial_{\alpha_{m}}j(x)$ the first
few multipoles vanish, which improves the fall-off properties of the result at
infinity and makes possible a repeated action by $1/\Box$. But the
Gauss–Bonnet density and $\sqrt{g}\,\Box R$ are both locally a total
derivative which makes $1/\Delta_{4}$ well defined in the expression (2.21)
for $\varSigma_{{\vphantom{L}}_{\rm RFT}}$. This in fact implies the
invertibility of the Faddeev–Popov operator in this gauge, which up to
coefficient coincides with the Paneitz operator, $Q_{{\vphantom{L}}_{\rm
RFT}}=4\Delta_{4}$, and thus guarantees local uniqueness of conformal gauge
fixing procedure.
Moreover, the above observation serves as a repudiation of the harmful role of
double poles in the RFT action that was claimed in [15]. Absence of infrared
dangerous double poles is explicit in the lowest order of the curvature
expansion for $\varSigma_{{\vphantom{L}}_{\rm RFT}}$ which reads
$\varSigma_{{\vphantom{L}}_{\rm RFT}}=-\frac{1}{6\,\Box}R+O(\Re^{2}),$ (2.23)
in view of the fact that the Gauss–Bonnet density is quadratic in the
curvature $\sqrt{g}E=O(\Re^{2})$. Higher orders of this expansion are also
safe because of the total derivative nature of $\sqrt{g}\,E$. Regarding the
lowest order quadratic in curvature part, with the above approximation for
$\varSigma_{{\vphantom{L}}_{\rm RFT}}$ it equals
$\varGamma_{{\vphantom{L}}_{\rm RFT}}[\,g\,]=-\frac{\gamma}{192\pi^{2}}\int
d^{4}x\sqrt{g}\,R^{2}+O(\Re^{3}),$ (2.24)
because all the terms depending on the parameter $\beta$ completely cancel
out, and what remains coincides with the last quadratic term of (1.6). This
coincidence fully matches with the linear in curvature part of the trace
anomaly (1.1) (its $\gamma$-term) generated by the quadratic action (1.6).
Indeed, the conformal transformation of its nonlocal Weyl term contributes
only to $O(\Re^{2})$-part of the anomaly due to the fact that only its form
factor $\ln(-\Box/\mu^{2})$ is not Weyl invariant, and the whole $\gamma$-term
of the anomaly entirely comes from the $R^{2}$-part of (1.6).
### 2.2 Fradkin-Vilkovisky gauge
Another conformal gauge arises in context of conformal off-shell extension of
Einstein gravity suggested in [19] and corresponds to the 4-dimensional
version of the Yamabe problem. The representative of the conformal group orbit
is chosen to be the metric with a vanishing scalar curvature
$\chi_{{\vphantom{L}}_{\rm FV}}[\,\bar{g}\,]=\bar{R},$ (2.25)
which implies a nonlinear but still explicitly solvable equation for
$\varSigma_{{\vphantom{L}}_{\rm FV}}$ ,
$R[e^{-2\varSigma_{{\vphantom{L}}_{\rm
FV}}}g_{\mu\nu}]=e^{3\varSigma_{{\vphantom{L}}_{\rm
FV}}}\big{(}R-6\,\Box\big{)}\,e^{-\varSigma_{{\vphantom{L}}_{\rm FV}}}=0.$
(2.26)
This solution reads
$\displaystyle\varSigma_{{\vphantom{L}}_{\rm
FV}}=-\ln\Big{(}1+\frac{1}{6}\frac{1}{\Box-R/6}R\Big{)},$ (2.27)
$\displaystyle\lim\limits_{|x|\to\infty}e^{-\varSigma_{{\vphantom{L}}_{\rm
FV}}}=1$ (2.28)
in terms of the inverse of the conformal second order operator
$\Box-\tfrac{1}{6}R$ subject to zero boundary conditions at infinity. This
inverse operator also admits covariant curvature expansion and in the lowest
order yields the function $\varSigma_{{\vphantom{L}}_{\rm FV}}$ coinciding
with that of the RFT gauge (2.23),
$\varSigma_{{\vphantom{L}}_{\rm FV}}=\varSigma_{{\vphantom{L}}_{\rm
RFT}}+O(\Re^{2}),$ (2.29)
and, therefore, generates in the quadratic order the same expression for the
anomaly action
$\varGamma_{{\vphantom{L}}_{\rm FV}}=\varGamma_{{\vphantom{L}}_{\rm
RFT}}+O(\Re^{3}).$ (2.30)
Using Eqs. (2.17) and (2.21) it is easy to see that the difference between RFT
and FV actions is given by the exact expression
$\displaystyle\varGamma_{{\vphantom{L}}_{\rm
RFT}}-\varGamma_{{\vphantom{L}}_{\rm FV}}$
$\displaystyle=\frac{1}{16\pi^{2}}\int
d^{4}x\,\sqrt{g}\big{(}\varSigma_{{\vphantom{L}}_{\rm
RFT}}-\varSigma_{{\vphantom{L}}_{\rm FV}}\big{)}$
$\displaystyle\times\Big{[}\,\alpha\,C^{2}+2\beta\,\Delta_{4}\big{(}\varSigma_{{\vphantom{L}}_{\rm
RFT}}-\varSigma_{{\vphantom{L}}_{\rm FV}}\big{)}\Big{]},$ (2.31)
bilinear in the local Weyl squared term and conformally invariant nonlocal
functional
$\displaystyle\varSigma_{{\vphantom{L}}_{\rm
RFT}}-\varSigma_{{\vphantom{L}}_{\rm FV}}$
$\displaystyle=\frac{1}{4}\,\frac{1}{\Delta}_{4}\mathcal{E}_{4}$
$\displaystyle+\ln\Big{(}1+\frac{1}{6}\frac{1}{\Box-R/6}R\Big{)}=O(\Re^{2}).$
(2.32)
Therefore within perturbation theory these two actions remain coinciding even
in the cubic order and become different only starting from the fourth order in
the curvature.
Perturbatively both terms of (2.32) produce similar nonlocal structures of
tree-like nature, that is the terms characteristic of the tree-level
approximation in field theory. Such terms are composed of the powers of
inverse d’Alembertians acting on the curvature tensor structures or on the
products of similar nonlocal tensor structures built according to the same
pattern. However, taken separately as exact entities they have essentially
different types of nonlocality. RFT action formalism involves the Green’s
function of the fourth order Paneitz operator, whereas the FV version of the
action is based on the Green’s function of the second order operator
$\Box-\tfrac{1}{6}R$. Both operators are conformally covariant, but the Weyl
transformation of $\Box-\tfrac{1}{6}R$ is different from (2.8)
$\Box-\tfrac{1}{6}R=e^{-3\sigma}\,\big{(}\bar{\Box}-\tfrac{1}{6}\bar{R}\big{)}\,e^{\sigma},\qquad
g_{\mu\nu}=e^{2\sigma}\bar{g}_{\mu\nu}.$ (2.33)
Moreover, FV action formalism involves a special logarithmic nonlinearity
absent in RFT gauge fixing. The action of the Paneitz operator derivatives in
(2.16) can destroy this logarithmic structure, but the
$\varSigma_{{\vphantom{L}}_{\rm FV}}C^{2}$-term in
$\varGamma_{{\vphantom{L}}_{\rm FV}}$ still contains it intact.
A further comparison of the RFT and FV actions can be done along the lines of
their “naturalness”. RFT gauge (2.20) is based on structures organically
belonging to the conformal anomaly formalism in the sense that it involves the
same fundamental objects—the Branson curvature $\mathcal{E}_{4}$ and the
relevant Paneitz operator $\Delta_{4}$ which are immanently present in the
flow of the anomalous action along the conformal group orbit (2.6). One could
even interpret this gauge as the one providing the extremum of $\beta$-terms
in this expression with respect to the variation of the orbit parameter
$\sigma$. This interpretation is, however, erroneous because $g_{\mu\nu}$,
$\bar{g}_{\mu\nu}$ and $\sigma$ cannot be treated as independent variables in
Eq. (2.6).
On the contrary, FV gauge (2.25) uses a somewhat extraneous entity—the scalar
curvature—which is singled out only by the fact that it turns out to be the
bearer of the metric conformal mode. As the result the advantage of FV gauge
is that it does not involve higher than second order derivatives and does not
produce double pole nonlocalities. Another advantage is that the equation
(2.19) disentangling the FV anomaly action from the full effective action
becomes in view of $\bar{R}=0$ much simpler
$\varGamma[\,g\,]=\varGamma_{{\vphantom{L}}_{\rm
FV}}[\,g\,]+\varGamma[\,\bar{g}\,]\,\big{|}_{\bar{g}_{\mu\nu}[\,g\,]},$ (2.34)
where $\bar{g}_{\mu\nu}[\,g\,]=e^{-2\varSigma_{{\vphantom{L}}_{\rm
FV}}[\,g\,]}g_{\mu\nu}$, which is obviously consistent with the fact that
$\varGamma_{{\vphantom{L}}_{\rm FV}}[\,\bar{g}\,]=0$ because
$\varSigma_{{\vphantom{L}}_{\rm FV}}[\,\bar{g}\,]\equiv 0$.
As compared to the FV version, among technical disadvantages of the RFT gauge
and the action is the presence of fourth order derivatives of the Paneitz
operator. Due to this the RFT version turns out to be vulnerable from the
viewpoint of possible generalizations. For example, a modification of the
gauge (2.20) by the additional Weyl squared term, $\chi_{{\vphantom{L}}_{\rm
RFT}}\to\chi_{{\vphantom{L}}_{\rm RFT}}+aC^{2}$ would not work, because the
relevant modification $\varSigma_{{\vphantom{L}}_{\rm
RFT}}\to\varSigma_{{\vphantom{L}}_{\rm RFT}}+a\,(2\Delta_{4})^{-1}C^{2}$ is
badly defined for the reasons described above in the footnote 2—the additional
term should have a total derivative structure.
The generalization to spacetimes with nontrivial topology is also not
straightforward, because the condition (2.20) should not contradict
nonvanishing Euler number of the manifold, which for compact manifolds without
a boundary reads $e_{E}=\tfrac{1}{32\pi^{2}}\int d^{4}x\sqrt{g}\,E(x)$. Say,
for a compact manifold of a finite volume $V=\int d^{4}x\,\sqrt{g}$ the gauge
(2.20) can be chosen to be
$\chi(\bar{g})=\sqrt{\bar{g}}\,\Big{(}\bar{E}-\frac{2}{3}\bar{\Box}\bar{R}-32\pi^{2}\frac{e_{E}}{\bar{V}}\Big{)},$
(2.35)
but this leads to a nonlinear integro-differential equation for the relevant
$\varSigma$
$\displaystyle 4\sqrt{g}\Delta_{4}\varSigma$
$\displaystyle=\sqrt{g}\bigg{(}E-\frac{2}{3}\Box
R-32\pi^{2}\frac{e^{-4\varSigma}}{\langle\,e^{-4\varSigma}\,\rangle}\frac{e_{E}}{V}\bigg{)},$
(2.36) $\displaystyle\langle\,e^{-4\varSigma}\,\rangle$
$\displaystyle\equiv\frac{1}{V}\int d^{4}x\,\sqrt{g}e^{-4\varSigma},$ (2.37)
which apparently can be solved analytically only by perturbations in
$e_{E}/V$.
Unless stated otherwise, below we consider asymptotically flat spacetime with
a trivial topology, whose Euler characteristics should be modified by the
boundary term. For generic 4-dimensional manifolds with a smooth boundary it
reads
$e_{E}=\tfrac{1}{32\pi^{2}}\Big{(}\int_{\cal
M}d^{4}x\sqrt{g}\,E(x)+\int_{\cal\partial
M}d^{3}x\sqrt{\gamma}\varOmega(x)\Big{)},$ (2.38)
where $\gamma=\det\gamma_{ab}$ and $\gamma_{ab}$ is the induced metric on
$\partial{\cal M}$. For asymptotically flat case due to the contribution of
$\partial{\cal M}$ at infinity $|\,x\,|\to\infty$ it equals $1$, so that
everywhere in what follows the bulk part of the Euler characteristics is
$\tfrac{1}{32\pi^{2}}\int d^{4}x\sqrt{g}\,E(x)\equiv
e^{\prime}_{E}=e_{E}-1=0$.333I am grateful for this observation to M.Duff.
Explicit and simple expression for the boundary term of the Euler
characteristics in the 4-dimensional case can be found in [36],
$\varOmega=\tfrac{1}{4}R_{a\perp b\perp}K^{ab}+16\det K^{a}_{b}$, where
$K_{ab}=\nabla_{a}n_{b}$ is the extrinsic curvature of the boundary, and
$\perp$ denotes the projection on the outward pointing normal vector
$n^{\mu}$. The last term in $\varOmega$ exactly reproduces the value of the
Euler number $e_{E}=1$ for flat and asymptotically flat spaces [54].
## 3 Conformal anomaly and covariant curvature expansion
Despite the diversity of nonlocal structures of RFT and FV versions of anomaly
action, neither of them seem to appear in conventional perturbation theory for
quantum effective action. The covariant form of this perturbation theory in
curved spacetime (1.6) was pioneered in [3], but its logarithmic nonlocal
formfactor did not resemble the nonlocal operators of the RFT action (1.3).
Here we show how in spite of these discrepancies the anomaly action originates
from covariant perturbation theory of [38, 39, 40].
This perturbation theory arose as a concrete implementation of the ideas of
[3] as an expansion in powers of covariant tensors of spacetime and fibre
bundle curvatures and other covariant background field objects. This expansion
is completely equivalent to standard Feynman diagrammatic technique and
represents its resummation converting the original perturbation series in
noncovariant odjects, like matter and metric field perturbations on top of
flat and empty spacetime background, into the series in powers of covariant
fields strengths denoted collectively below by $\Re$ and including spacetime
and fibre bundle curvature.
To be more specific, consider the theory with the inverse propagator on top of
the nontrivial field background $\hat{F}(\nabla)=F^{A}_{B}(\nabla)$, hat
denoting the matrix structure of the operator acting in the space of fields
$\varphi=\varphi^{A}(x)$ with a generic spin-tensor index $A$ and
$\nabla=\nabla_{\mu}$ denoting the covariant derivative with respect to the
corresponding fibre bundle connection,
$\hat{F}(\nabla)=\Box+\hat{P}-\frac{\hat{1}}{6}R,\qquad\Box=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}.$
(3.1)
This operator is characterized by the “curvatures”—metric Riemann tensor with
its Ricci contractions, fibre bundle curvature $\hat{\cal R}_{\mu\nu}$
determining the commutator of covariant derivatives,
$[\nabla_{\mu},\nabla_{\nu}]\,\varphi=\hat{\cal R}_{\mu\nu}\,\varphi$, and the
potential term $\hat{P}$ (the term $-\tfrac{\hat{1}}{6}R$ is disentangled from
the operator potential for reasons of convenience),
$\Re=(R^{\mu}{}_{\nu\alpha\beta},R_{\mu\nu},R,\hat{\cal R}_{\mu\nu},\hat{P}).$
(3.2)
In covariant perturbation theory the one-loop effective action gets expanded
in powers of these curvatures
$\varGamma=\frac{1}{2}{\rm Tr}\ln
F(\nabla)=\\!\\!\\!\\!\stackrel{{\scriptstyle\text{local power
div}}}{{\overbrace{\varGamma_{0}+\varGamma_{1}}}}\\!\\!\\!\\!\\!\\!+\,\varGamma_{2}+\varGamma_{3}+O(\Re^{4}),$
(3.3)
where $\varGamma_{n}\sim\Re^{n}$. Within dimensional regularization of
$2\omega$-dimensional spacetime, $\omega\to 2$, the zeroth and first order
terms of the expansion represent pure power divergences (note that we consider
the case of a massless theory, or the theory where the mass matrix is included
in the potential term $\hat{P}$ and treated by perturbations), so that these
two terms are annihilated by the regularization, while the second order term
is given by the expression [38]
$\displaystyle\varGamma^{(2)}_{\rm dim\;reg}$
$\displaystyle=-\frac{\Gamma(2-\omega)\Gamma(\omega+1)\Gamma(\omega-1)}{2(4\pi)^{\omega}\Gamma(2\omega+2)}\,\mu^{4-2\omega}$
$\displaystyle\times\int dx\,\sqrt{g}\,{\rm
tr}\,\Big{\\{}R_{\mu\nu}(-\Box)^{\omega-2}R^{\mu\nu}\hat{1}$
$\displaystyle-\frac{1}{18}(4-\omega)(\omega+1)R(-\Box)^{\omega-2}R\hat{1}$
$\displaystyle-\frac{2}{3}(2-\omega)(2\omega+1)\,\hat{P}(-\Box)^{\omega-2}R$
$\displaystyle+2(4\omega^{2}-1)\,\hat{P}(-\Box)^{\omega-2}\hat{P}$
$\displaystyle+(2\omega+1)\hat{\cal R}_{\mu\nu}(-\Box)^{\omega-2}\hat{\cal
R}^{\mu\nu}\Big{\\}},$ (3.4)
where $\omega=\tfrac{d}{2}\to 2$. Here ${\rm tr}$ denotes the matrix trace and
the concrete coefficients implement the originally conjectured structure of
dimensionally regularized effective action Lagrangian,
$\Re(-\Box)^{\omega-2}\Re$, that was blueprinted in [3]. What is important and
should be especially emphasized is that
$\Box=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$ means here the full covariant
d’Alembertian acting on a respective scalar $R$, tensor $R_{\mu\nu}$ or
spintensor $\hat{\cal R}_{\mu\nu}$ and $\hat{P}$ objects.
For brevity we will consider the case of a single conformal scalar field with
$\hat{1}=1$, $\hat{P}=0$, $\hat{\cal R}_{\mu\nu}=0$ and the following values
of the trace anomaly coefficients444The coefficients have the opposite sign to
those of $b=-\alpha/16\pi^{2}$ and $b^{\prime}=-\beta/16\pi^{2}$ in [4],
because in our case the stress tensor is defined with respect to the Euclidean
effective action $\varGamma=-i\varGamma_{L}$ in contrast to the definition of
$T^{\mu\nu}=2g^{-1/2}\delta\varGamma_{L}/\delta g_{\mu\nu}$ in the Lorentzian
signature spacetime of [4]. Comparison with [5] should also take into account
another sign of the stress tensor defined by the variation with respect to the
contravariant metric.
$\alpha=-\frac{1}{120},\quad\beta=\frac{1}{360},\quad\gamma=-\frac{1}{180},$
(3.5)
for which the action (3.4) takes the form—a particular case of (1.6),
$\displaystyle\varGamma^{(2)}_{\rm ren}$
$\displaystyle=\frac{1}{32\pi^{2}}\int
dx\,\sqrt{g}\,\Big{\\{}\frac{1}{60}\Big{[}R_{\mu\nu}\gamma(-\Box)R^{\mu\nu}$
$\displaystyle-\frac{1}{3}R\gamma(-\Box)R\Big{]}+\frac{R^{2}}{1080}\Big{\\}}$
$\displaystyle=\frac{1}{32\pi^{2}}\int
dx\,\sqrt{g}\,\Big{\\{}\frac{1}{120}C_{\mu\nu\alpha\beta}\gamma(-\Box)C^{\mu\nu\alpha\beta}$
$\displaystyle+\frac{R^{2}}{1080}\Big{\\}}+O(\Re^{3}).$ (3.6)
Here $\gamma(-\Box)$ is the nonlocal formfactor (in minimal subtraction scheme
with $\ln(4\pi)$ and Euler constants absorbed in $\mu$)
$\gamma(-\Box)=\ln\Big{(}\\!-\frac{\Box}{\mu^{2}}\Big{)}-\frac{16}{15},$ (3.7)
and the transition to the last line is valid up to the higher order terms in
curvature and based on the nonlocal generalization of the identity
$\int d^{4}x\,\sqrt{g}\,C^{2}=2\int
d^{4}x\,\sqrt{g}\,(R_{\mu\nu}R^{\mu\nu}-\tfrac{1}{3}R^{2})$ (3.8)
derived in [39, 40] by integration by parts and use of the nonlocal
representation of the Riemann tensor in terms of the Ricci one (see footnote 5
below).
The first term of this action is obviously conformal invariant in quadratic
order, so that the linear in curvature part of the anomaly originates from the
last term which is the RFT (or FV) action (2.24) in the quadratic
approximation with $\gamma=-1/180$. Thus, the RFT or FV action is fully
recovered in this approximation from perturbation theory and, as expected,
turns out to be local.
### 3.1 Cubic order
Quadratic order of the covariant curvature expansion is, in fact, a trivial
generalization of the flat space expressions for self-energy operators of
Feynman diagrammatic technique, because $\ln(-\Box/\mu^{2})$ is just a
straightforward replacement of the typical momentum space formfactor
$\ln(p^{2}/\mu^{2})$ by its position space version. At higher orders the
situation becomes much more complicated and usually represented in terms of
correlators of stress-tensor and other observables, written down in momentum
space representation, see [55, 56, 57] for the treatment of generic conformal
field theories. These correlators are, of course, contained in the effective
action expanded in curvatures which, for reasons of general covariance, we
prefer to consider in coordinate representation.
In this representation the effective action becomes for each order $N$ in the
curvature a sum of nonlocal monomials
$\int d^{4}x_{1}\cdots
d^{4}x_{N}\,F(x_{1},\ldots,x_{N})\nabla...\nabla\Re(x_{1})...\Re(x_{N})$ (3.9)
with nonlocal multiple-point coefficients and covariant derivatives somehow
acting on the product of curvatures at their various points. The absence of
convenient and generally covariant momentum space representation makes us to
work in coordinate representation and invent a special language which would
simplify the formalism and make it manageable [38, 39, 40]. This language is
based on the operator representation of nonlocal formfactors,
$\displaystyle F(x_{1},\ldots,x_{N})$
$\displaystyle=\varGamma(\nabla_{1},\ldots,\nabla_{N})$
$\displaystyle\times\delta(x_{1},x_{2})\,\delta(x_{1},x_{2})\cdots\delta(x_{1},x_{N}),$
(3.10)
where $\varGamma(\nabla_{1},\ldots,\nabla_{N})$ is the operator valued
function of $N$ independent covariant derivatives such that each $\nabla_{i}$
is acting on its own $x_{i}$. This allows one to write the orders of
perturbation theory as
$\displaystyle\varGamma^{(N)}$
$\displaystyle=\frac{1}{2(4\pi)^{2}}\int\\!d^{4}x\,\sqrt{g}\sum\limits_{M}\varGamma_{M}(\nabla_{1},\ldots,\nabla_{N})$
$\displaystyle\times I_{M}(x_{1},\ldots,x_{N})\,\Big{|}_{\\{x\\}=x},$ (3.11)
where summation runs over all invariant monomials in curvatures of a given
$n$-th order
$I_{M}(x_{1},\ldots,x_{N})\sim\nabla\cdots\nabla\Re(x_{1})\cdots\Re(x_{N})$
(3.12)
and after the action of all independent derivatives on their arguments all
these arguments $\\{x\\}=(x_{1},\ldots x_{N})$ have to be identified.
In the cubic order for the full set of curvatures (3.2) there are 29 such
invariant structures built of these curvatures and their covariant derivatives
with all indices fully contracted with each other. Moreover, in view of the
scalar (no free indices) nature of the formfactors and the formal identity
$\nabla_{1}+\nabla_{2}+\nabla_{3}=0$ (reflecting the possibility of
integration by parts without surface terms, which is a counterpart to the
momentum conservation in Feynman diagrams) the formfactors of
$\varGamma^{(3)}$ can be written down as functions of three d’Alembertians
$\Box_{1}$, $\Box_{2}$ and $\Box_{3}$ independently acting on three arguments
of $I_{M}(x_{1},x_{2},x_{3})$. Thus, cubic order reads as
$\displaystyle\varGamma^{(3)}$
$\displaystyle=\frac{1}{2(4\pi)^{2}}\int\\!dx\,\sqrt{g}\sum\limits^{29}_{M=1}\varGamma_{M}(\Box_{1},\Box_{2},\Box_{3})$
$\displaystyle\times I_{M}(x_{1},x_{2},x_{3})\,\Big{|}_{\\{x\\}=x}.$ (3.13)
The list of cubic invariants and their formfactors is presented in [40, 18,
20]. It is very long and, as its details are not necessary for our purposes,
we will not fully present it here. We only give the general structure of the
nonlocal formfactors of these invariants. It reads as a sum of three different
groups of terms
$\displaystyle\varGamma_{M}(\Box_{1},\Box_{2},\Box_{3})$
$\displaystyle=A_{M}\,\varGamma(\Box_{1},\Box_{2},\Box_{3})$
$\displaystyle+\sum_{1\leq
i<k}^{3}\frac{D^{ik}_{M}}{(\Box_{i}-\Box_{k})}\ln\frac{\Box_{i}}{\Box_{k}}+B_{M}.$
(3.14)
Here $\varGamma(\Box_{1},\Box_{2},\Box_{3})$ is the fundamental cubic
formfactor corresponding to the triangular Feynman graph of massless theory
with unit vertices [58],
$\displaystyle\varGamma(\Box_{1},\Box_{2},\Box_{3})$
$\displaystyle=\int\limits_{\alpha\geq
0}\frac{d^{3}\alpha\,\delta(1-\alpha_{1}-\alpha_{2}-\alpha_{3})}{\alpha_{1}\alpha_{2}(-\Box_{3})+\alpha_{1}\alpha_{3}(-\Box_{2})+\alpha_{2}\alpha_{3}(-\Box_{1})},$
(3.15)
which cannot be reduced to an elementary function. The operator-valued
coefficients $A_{M}$, $B_{M}$ and $D_{M}^{ik}$ are rational functions of three
$\Box$-arguments with a polynomial numerator $P(\Box_{1},\Box_{2},\Box_{3})$
and the denominator containing together with the product
$\Box_{1}\Box_{2}\Box_{3}$ also the powers of a special quadratic form of
these arguments $D$,
$\displaystyle
A_{M},\;D^{ik}_{M},\;B_{M}\sim\frac{P(\Box_{1},\Box_{2},\Box_{3})}{\Box_{1}\Box_{2}\Box_{3}\,D^{L}},\;L\leq
6,$ (3.16) $\displaystyle
D={\Box_{1}}^{2}\\!+{\Box_{2}}^{2}\\!+{\Box_{3}}^{2}\\!-\\!2\Box_{1}\Box_{2}\\!-\\!2\Box_{1}\Box_{3}\\!-\\!2\Box_{2}\Box_{3}.$
(3.17)
In this cubic order of the curvature expansion the conformal anomaly (1.1),
which is quadratic in curvatures, was explicitly derived by the direct
variation of the metric in [59]. Though this derivation has demonstrated
nontrivial localization of the nonlocal terms under straightforward tracing
the metric variational derivative, it still remained rather technical and not
very illuminating because it has not revealed the anomalous part of the
action. It turns out, however, that the transition to another basis of
curvature invariants, suggested in [20, 18], explicitly disentangles this
part.
### 3.2 Conformal resummation: Fradkin–Vilkovisky anomaly action
The recovery of the anomaly part of the action and its conformal invariant
part is based on a simple idea that the latter should consist of the series of
Weyl invariant structures. The construction of Weyl invariants can be done by
the gauge fixing procedure of the above type—choosing the representative
metric on the group orbit by imposing the conformal gauge. Obviously the set
of invariants surviving after imposing this gauge will be minimal if the gauge
would explicitly annihilate the maximum number of invariants in their original
full set. For this reason the FV gauge (2.25) is much easier to use for the
separation of the total set of invariants into the Weyl type ones and those
which vanish when the gauge is enforced. As $R$ is one of the curvatures in
the set of $\Re$ the FV gauge is more useful for the purpose of such a
separation than the RFT gauge (2.20) which nonlinearly intertwines all the
curvatures. Intuitively it is also clear because $R$, in contrast to
$C^{\alpha}{}_{\beta\mu\nu}$, is a bearer of the conformal mode.
In the purely metric sector such a separation is attained by the transition to
the new curvature basis [20],
$\Re=\big{(}\,R^{\mu}_{\;\nu\alpha\beta},R_{\mu\nu},R\,\big{)}\to\tilde{\Re}=\big{(}\,C^{\alpha}_{\;\;\beta\mu\nu},R\,\big{)},$
(3.18)
via expressing Ricci tensor in terms of the Weyl tensor and the Ricci
scalar555In fact, the original basis and the curvature expansion of [38, 39,
40] consisted of $R_{\mu\nu}$ and $R$ because in asymptotically flat Euclidean
spacetime Riemann tensor can be expressed as nonlocal power series in the
Ricci tensor,
$R_{\alpha\beta\mu\nu}=\frac{1}{\Box}\big{(}\nabla_{\mu}\nabla_{\alpha}R_{\nu\beta}-\nabla_{\nu}\nabla_{\alpha}R_{\mu\beta}\big{)}-(\alpha\leftrightarrow\beta)+O(\Re^{2})$,—the
corollary of contracted Bianchi identity.. This expression follows from the
contracted Bianchi identity which for the Weyl tensor reads as
$\nabla^{\beta}\nabla^{\alpha}C_{\alpha\mu\beta\nu}=\frac{1}{2}\Box
R_{\mu\nu}-\frac{1}{6}\nabla_{\mu}\nabla_{\nu}R-\frac{g_{\mu\nu}}{12}\Box
R+O(\Re^{2}).$ (3.19)
This equation can be solved by iterations for Ricci tensor in terms of
nonlocal series in powers of two objects—Ricci scalar $R$ and the new
traceless (and up to quadratic order transverse) tensor $C_{\mu\nu}$ which is
itself a nonlocal derivative of Weyl,
$C_{\mu\nu}=\frac{2}{\Box}\nabla^{\beta}\nabla_{\alpha}C^{\alpha}_{\;\;\mu\beta\nu}.$
(3.20)
The resulting series begins with
$R_{\mu\nu}=C_{\mu\nu}+\frac{1}{3}\nabla_{\mu}\nabla_{\nu}\frac{1}{\Box}R+\frac{1}{6}g_{\mu\nu}R+O(\tilde{\Re}^{2}).$
(3.21)
Effective action reexpansion imples the transition from
$I_{M}(x_{1},\ldots,x_{n})$ to a new basis of invariants
$\tilde{I}_{M}(x_{1},...x_{n})\sim\nabla...\nabla\tilde{\Re}(x_{1})...\tilde{\Re}(x_{n}),$
(3.22)
which can be separated in the set of monomials $I_{C}(x_{1},\ldots,x_{n})$
involving only $C_{\mu\nu}$ and the set of monomials
$I_{R}(x_{1},\ldots,x_{n})$ containing at least one scalar curvature factor,
$\displaystyle I_{C}(x_{1},...x_{n})\sim$ $\displaystyle\nabla...\nabla
C(x_{1})...C(x_{n}),$ (3.23) $\displaystyle I_{R}(x_{1},...x_{n})\sim$
$\displaystyle\nabla...\nabla R(x_{1})C(x_{2})...C(x_{n}),$
$\displaystyle\nabla...\nabla R(x_{1})R(x_{2})C(x_{3})...C(x_{n}),\;...$
(3.24)
Expansion in the new basis of invariants implies, of course, the transition to
a new set of their relevant formfactors
$\varGamma_{M}(\nabla_{1},...\nabla_{n})\to\varGamma_{C}(\nabla_{1},...\nabla_{n}),\varGamma_{R}(\nabla_{1},...\nabla_{n}),$
(3.25)
and the new expansion takes the form
$\varGamma=W+\varGamma_{R},$ (3.26)
where $W$ is the Weyl and $\varGamma_{R}$ is the mixed Weyl–Ricci scalar parts
of the whole expansion, which we write in abbreviated form (omitting multiple
spacetime arguments and the operation of equating them)
$\displaystyle W$
$\displaystyle=\frac{1}{32\pi^{2}}\int\\!d^{4}x\,\sqrt{g}\sum\limits_{n,C}\varGamma_{C}^{(n)}I_{C}^{(n)},$
(3.27) $\displaystyle\varGamma_{R}$
$\displaystyle=\frac{1}{32\pi^{2}}\int\\!d^{4}x\,\sqrt{g}\sum\limits_{n,R}\varGamma_{R}^{(n)}I_{R}^{(n)}.$
(3.28)
Note that $W$ and its Weyl basis invariants are not Weyl invariant, because
apart from Weyl tensors they contain covariant derivatives and nontrivial
formfactors which do not possess conformal invariance properties.
The main statement on the conformal decomposition of the effective action of
[20] is that
$\varGamma[\,g\,]=\varGamma_{{\vphantom{L}}_{\rm
FV}}[\,g\,]+W[\,\bar{g}\,]\,\Big{|}_{\,\bar{g}_{\mu\nu}=e^{-2\varSigma_{{\vphantom{L}}_{\rm
FV}}[\,g\,]}g_{\mu\nu}},$ (3.29)
where $\varGamma_{{\vphantom{L}}_{\rm FV}}[\,g\,]$ is exactly the FV anomaly
action introduced above666One can check that the last four lines of Eq. (24)
in [20] form exact expression for $\varGamma_{{\vphantom{L}}_{\rm FV}}[\,g\,]$
by taking into account that the function $Z$ in this equation coincides with
$-\varSigma_{{\vphantom{L}}_{\rm FV}}$ and satisfies the equation $\Box
Z+\tfrac{1}{2}(\nabla Z)^{2}=\tfrac{1}{3}R$.. Conformally invariant part is
obtained by the “conformization” of $W$, while the rest of the effective
action is exhausted by the Fradkin-Vilkovisky anomaly action. Invariant
meaning of this representation is that the Ricci part of the full action is
not independent, but fully determined by the anomaly and Weyl parts of the
action. This representation looks as the realization of Eq. (2.34) within
perturbation theory in curvatures. This result is likely to resolve a long-
standing debate between the proponents of the Riegert action and adherents of
the flat space perturbation expansion for the effective action with typical
nonlocal logarithmic form factors of the form (3.7). Note that these form
factors do not contribute to the anomaly even though their coefficients are
directly related to its expression (1.1). Rather they become Weyl invariant
under the substitution of $\bar{g}_{\mu\nu}$ as their functional argument.
Validity of the representation (3.29) was checked in the cubic order
approximation for the effective action in [20]. The transition to the new
basis of invariants in the second order leads to (see the second line of Eq.
(3.6)),
$\displaystyle W^{(2)}[\,g\,]$ $\displaystyle=\frac{1}{32\pi^{2}}\int
dx\,\sqrt{g}\frac{1}{120}C_{\mu\nu\alpha\beta}\,\gamma(-\Box)C^{\mu\nu\alpha\beta},$
(3.30) $\displaystyle\varGamma^{(2)}_{R}[\,g\,]$
$\displaystyle=\frac{1}{32\pi^{2}}\int dx\,\sqrt{g}\frac{1}{1080}R^{2},$
(3.31)
whereas in the third order it results in a great simplification of the “Ricci
scalar” formfactors $\varGamma_{R}^{(3)}$ as compared to the original
ones—they become much simpler and, moreover, in their expressions of the form
(3.14) the coefficients $A,D^{ik}_{M},B_{M}$ of (3.16) completely loose powers
of the function $D$ in the denominator. Thus, modulo the contributions of
$\ln(\Box_{i}/\Box_{k})/(\Box_{i}-\Box_{k})$ the formfactors
$\varGamma_{R}^{(3)}$ acquire the tree-level structure. The terms with these
factors get, however, completely absorbed with accuracy $O(\Re^{4})$ by the
replacement $W^{(2)}[\,g_{\mu\nu}\,]\to W^{(2)}[\,\bar{g}_{\mu\nu}\,]$ in view
of the following relation [40]
$\displaystyle W^{(2)}[\,g\,]-W^{(2)}[\,\bar{g}\,]$ $\displaystyle\sim\int
dx\,\sqrt{g}\,C_{\mu\nu\alpha\beta}\Big{[}\ln(-\Box)-\ln(-\bar{\Box})\Big{]}C^{\mu\nu\alpha\beta}$
$\displaystyle=\int
dx\,\sqrt{g}\frac{\ln(\Box_{1}/\Box_{2})}{\Box_{1}-\Box_{2}}[\Box_{2}-\bar{\Box}_{2}]C_{1\,\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}_{2}+O(\Re^{4}),$
$\displaystyle\Box_{2}-\bar{\Box}_{2}\sim\Re_{3}+O(\Re^{2}),$ (3.32)
where the right hand side is the set of relevant cubic order terms with the
above factor acting on two Weyl tensors out of three curvatures in $RCC$-type
invariants. What remains in the sector of cubic $I^{(3)}_{R}$-invariants is
the set of tree-like nonlocal form factors which comprise the curvature
expansion of FV action up to $\tilde{\Re}^{3}$ order inclusive. This
observation done in [20] can be formalized as the following sequence of
identical transformations
$\displaystyle\varGamma[\,g\,]=W^{(2+3)}[\,g\,]+\varGamma^{(2+3)}_{R}[\,g\,]+O(\Re^{4})=W^{(2+3)}[\,\bar{g}\,]$
$\displaystyle\;\;\;\;+\mathop{\underbrace{\varGamma^{(2+3)}_{R}[\,g\,]+\\!\big{(}W^{(2)}[\,g\,]-W^{(2)}[\,\bar{g}\,]\big{)}}}_{{\varGamma^{(2+3)}_{{\vphantom{L}}_{\rm
FV}}+O(\Re^{4})}}+O(\Re^{4}),$ (3.33)
where the group of the last three terms forms Fradkin–Vilkovisky anomaly
action expanded with $\tilde{\Re}^{3}$-accuracy. Explicitly the cubic part of
$\varGamma_{{\vphantom{L}}_{\rm FV}}$ for the model of a single conformal
scalar field with (3.5) reads [20]
$\displaystyle\varGamma^{(3)}_{{\vphantom{L}}_{\rm FV}}$
$\displaystyle=-\frac{1}{32\pi^{2}}\int
dx\,\sqrt{g}\bigg{\\{}\frac{1}{19440}\Big{(}\frac{2}{\Box}_{3}-\frac{\Box_{1}}{\Box_{2}\Box_{3}}\Big{)}R_{1}R_{2}R_{3}$
$\displaystyle+\frac{1}{1620\,\Box_{2}\Box_{3}}C_{1}^{\alpha\beta}\nabla_{\alpha}R_{2}\nabla_{\beta}R_{3}$
$\displaystyle+\frac{1}{540}\Big{(}\frac{4}{\Box}_{2}-\frac{1}{\Box}_{3}-\frac{2\,\Box_{1}}{\Box_{2}\Box_{3}}-\frac{\Box_{3}}{\Box_{1}\Box_{2}}\Big{)}C_{1}^{\mu\nu}C_{2\,\mu\nu}R_{3}$
$\displaystyle+\frac{1}{135}\Big{(}\frac{1}{\Box_{1}\Box_{2}}-\frac{2}{\Box_{2}\Box_{3}}\Big{)}\nabla^{\mu}C_{1}^{\nu\alpha}\nabla_{\nu}C_{2\,\mu\alpha}R_{3}$
$\displaystyle-\frac{1}{135\,\Box_{1}\,\Box_{2}\Box_{3}}\nabla_{\alpha}\nabla_{\beta}C_{1}^{\mu\nu}\nabla_{\mu}\nabla_{\nu}C_{2}^{\alpha\beta}R_{3}\bigg{\\}}\bigg{|}_{\,\\{x\\}=x},$
(3.34)
where $C_{\mu\nu}$ is the “Weyl” part (3.20) of Ricci tensor (3.21)
### 3.3 The problem of double poles and global conformal transformations
The expression (3.34) shows that in the cubic order the anomalous effective
action is free from double pole nonlocal terms. For the FV action this is
obviously true to all orders of the curvature expansion, since all its tree
type nonlocalities originate from the Green’s function of the conformal scalar
operator $\Box-\tfrac{1}{6}R$. However, for the RFT action double poles
formally appear starting from the fourth order in the curvature because the
metric variation of $\varSigma_{\chi}=\varSigma_{\rm RTF}$ in (2.16) leads to
the action of the inverse Paneitz operator upon the square of the Weyl tensor
$C^{2}=C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}$ due to a formal variational
rule
$\int d^{4}x\,\sqrt{g}\,C^{2}\delta\varSigma_{\rm RFT}=\int
d^{4}x\,\sqrt{g}\,(\Delta_{4}^{-1}C^{2})\delta(\ldots).$ (3.35)
This operation is not well defined, because $C^{2}$ is not a total derivative
and the repeated action of $1/\Box$ upon generic test functions in four
dimensions leads to IR divergent integrals—see footnote 2. In the cubic order
of $\varGamma_{\rm RFT}$ this problem does not arise because of the extra
$\Box$ factor in $\Box R$, as it was checked in [16] by explicit calculations
of $\langle\,TTT\,\rangle$ correlators, but one is not granted to be free from
this difficulty for higher order correlators.
In fact this is a typical situation of IR divergences in two dimensions, where
the kernel of $1/\Box$ has a logarithmic dependence at infinity, and the
correlators of undifferentiated conformal fields $\phi$ are UV divergent,
while the correlators $\langle\partial\phi(x)\partial\phi(y)\cdots\rangle$
stay well defined. Apparently, the same property in four dimensions also
underlies absence of unitarity in dipole theories with $1/\Box^{2}$-type
propagators recently discussed in [60]. The mechanism of transition from
operators to their derivatives in shift symmetric theories actually helps to
justify the RFT action as a source of well defined stress tensor correlators
and extend the validity of results in [16] to all higher orders.
This follows from the observation that the Paneitz operator reads
$\sqrt{g}\Delta_{4}=\partial_{\mu}\Big{[}\sqrt{g}\big{(}\nabla^{\mu}\nabla_{\nu}+2R^{\mu\nu}-\frac{2}{3}Rg^{\mu\nu}\big{)}\Big{]}\partial_{\nu}$
(3.36)
and, therefore, perturbatively on the flat space background can be represented
as
$\sqrt{g}\Delta_{4}=\tilde{\Box}^{2}+V,\;\tilde{\Box}=\delta^{\mu\nu}\partial_{\mu}\partial_{\nu},\;V=\overrightarrow{\partial_{\mu}V^{\mu\nu}\partial_{\nu}},$
(3.37)
where the perturbation $V=O(\Re)$ has a special form—another differential
operator $V^{\mu\nu}$ sandwiched between two derivatives with all derivatives
acting to the right (which is indicated by the arrow). Within perturbation
theory in powers of $V$ the action of the inverse operator on a generic test
function $\psi$—scalar density—could have been understood as the expansion
$\displaystyle\phi=\frac{1}{\sqrt{g}\Delta_{4}}\psi$
$\displaystyle=\sum\limits_{n=0}^{\infty}\frac{(-1)^{n}}{\tilde{\Box}^{2}}\left(V\frac{1}{\tilde{\Box}^{2}}\right)^{n}\psi$
$\displaystyle=\sum\limits_{n=0}^{\infty}\frac{(-1)^{n}}{\tilde{\Box}^{2}}\left(\overrightarrow{\partial_{\mu}V^{\mu\nu}}\frac{1}{\tilde{\Box}^{2}}\partial_{\nu}\right)^{n}\psi,$
(3.38)
where we deliberately permuted the factors of $\partial_{\nu}$ and
$1/\tilde{\Box}^{2}$ using their formal commutativity in order to provide the
action of $1/\tilde{\Box}^{2}$ on the total derivative function. Thus all
terms of this expansion except the first one become infrared finite. The first
term $(1/\tilde{\Box}^{2})\psi$, however, makes this function $\phi$ ill
defined. On the contrary, its derivative $\partial_{\alpha}\phi$ becomes
consistent if one understands the first term of the expansion as
$(1/\tilde{\Box}^{2})\partial_{\alpha}\psi$, so that the prescription for the
operation of $\partial_{\alpha}(1/\sqrt{g}\Delta_{4})$ on a generic non-
derivative type test function reads as
$\partial_{\alpha}\frac{1}{\sqrt{g}\Delta_{4}}\psi=\sum\limits_{n=0}^{\infty}\frac{(-1)^{n}}{\tilde{\Box}^{2}}\partial_{\alpha}\left(\overrightarrow{\partial_{\mu}V^{\mu\nu}}\frac{1}{\tilde{\Box}^{2}}\partial_{\nu}\right)^{n}\psi.$
(3.39)
With this prescription the term $C^{2}\varSigma_{\rm RFT}$ in the RFT action
becomes perturbatively well defined to all orders of expansion. Indeed, this
term with $\varSigma_{\rm RFT}$ given by (2.21) and on account of total
derivative structure $\sqrt{g}(E-\tfrac{2}{3}\,\Box
R\big{)}=\partial_{\alpha}E^{\alpha}$ can be rewritten by integration by parts
as
$4\\!\int d^{4}x\sqrt{g}\,C^{2}\varSigma_{\rm RFT}=-\int
d^{4}x\,\sqrt{g}E^{\alpha}\,\partial_{\alpha}\frac{1}{\sqrt{g}\Delta_{4}}\big{(}\sqrt{g}\,C^{2}\big{)}$
(3.40)
with the above prescription (3.39). This confirms a well defined nature of all
multiple point correlators of stress tensor generated by RFT action.
Finally, it is worth discussing the effective action behavior under global
conformal transformations with $\sigma_{0}={\rm const}$. Higher order
curvature terms of the effective action scale as negative powers of
$e^{\sigma_{0}}$ and therefore are irrelevant in the IR limit. In [61] this
was a main argument in favor of a dominant role of the Wess–Zumino action
(2.6) in this limit because $\Delta\varGamma[g,\sigma]$ behaves linearly in
$\sigma_{0}$ (or logarithmically in the distance). Indeed,
$\displaystyle\Delta\varGamma[g,\sigma+\sigma_{0}]$
$\displaystyle=\Delta\varGamma[g,\sigma]$
$\displaystyle+\sigma_{0}\Big{(}\frac{\gamma}{32\pi^{2}}\int
d^{4}x\,\sqrt{g}\,C^{2}+\beta\,e^{\prime}_{E}\Big{)},$ (3.41)
where $e^{\prime}_{E}$ is the Euler characteristics of the manifold modulo its
boundary contribution (see footnote 3). Note, however, that this behavior
cannot be captured within the nonlocal RTF form of the anomaly action (1.3)
because it is valid only under Dirichlet boundary conditions for the Green’s
function of $\Delta_{4}$ (which would be violated by the $\sigma_{0}$-shift).
In other words, the expression (1.3) lacks the contribution of the zero mode
of the Paneitz operator, which on the contrary is explicitly featuring in
(3.41). For compact manifolds with possibly nontrivial topology global Weyl
transformations would not contradict boundary conditions, and these
transformations will obviously show up in the generalized RFT gauge (2.35) as
an ambiguity of the solution for Eq. (2.37),
$\varSigma\to\varSigma+\sigma_{0}$.
## 4 Stress tensor in conformally related spacetimes
Equations (2.19) and (2.34) show that the anomalous action makes sense as an
object specifying the difference of effective actions on conformally related
metrics and other fields. Outside of this context this action, being a subject
of shifting by an arbitrary conformal invariant functional $W^{\rm
conf}[\,g\,]$, as in Eq. (2.2), is not very instructive because such a shift
can include essential physical information on conformally invariant degrees of
freedom. Anomaly action $\varGamma_{\chi}$, or it would be better to say, the
Wess–Zumino type action (2.6)—the generating functional of
$\varGamma_{\chi}$—is really useful in situations when the physics of a
conformally related spacetime with the metric $\bar{g}_{\mu\nu}$ is fully
known. Then the effective action at $g_{\mu\nu}$ can be completely recovered
from the knowledge of the Weyl anomaly.
The simplest situation belongs to the class of conformally flat spacetimes
when $\bar{g}_{\mu\nu}$ can be associated with flat metric for which all the
metric field invariants are vanishing and $\varGamma[\,\bar{g}\,]$ is either
exactly zero or calculable for quantum matter fields in flat spacetime. In
particular, the fundamental observable which can then be obtained is the UV
renormalized expectation value of the stress tensor of classically conformally
invariant fields,
$\sqrt{g}\,\big{\langle}\,T^{\alpha\beta}\big{\rangle}=2\frac{\delta\varGamma_{\rm
ren}}{\delta g_{\alpha\beta}}$ (4.1)
provided $\langle\,\bar{T}^{\alpha\beta}\rangle=0$ or known from flat space
physics. Here we derive from (2.6) the expression for the difference of
(densitized) stress tensors
$\sqrt{g}\,\langle\,T^{\alpha}_{\beta}\rangle-\sqrt{\bar{g}}\,\langle\,\bar{T}^{\alpha}_{\beta}\rangle$,
which for a conformally flat spacetime coincides with a well-known
Brown–Cassidy expression [25] and generalizes it to the case of a nonvanishing
Weyl tensor.
### 4.1 Conformal anomaly from the divergent part of the effective action
To derive the behavior of the renormalized stress tensor on the conformal
group orbit we, first, have to trace the origin of conformal anomaly as the
result of subtracting UV divergences from covariantly regularized effective
action, $\varGamma_{\rm ren}=\varGamma_{\rm reg}-\varGamma_{\infty}$. In
dimensional regularization, $\varGamma_{\rm reg}={}^{(d)}\\!\varGamma$, these
divergences are given by
$\displaystyle\varGamma_{\infty}$
$\displaystyle=-\frac{1}{16\pi^{2}\epsilon}\int d^{d}x\,\sqrt{g}\,a_{2}$
$\displaystyle=\frac{1}{16\pi^{2}\epsilon}\int
d^{d}x\,\sqrt{g}\,\big{(}\alpha\,{}^{(4)}\\!C^{2}+\beta\,{}^{(4)}\\!E\,\big{)},$
(4.2)
where $\epsilon=4-d$, ${}^{(4)}\\!C^{2}$ and ${}^{(4)}\\!E$ are the four-
dimensional invariants formally continued to $d$-dimensions and $a_{2}$ is the
relevant second Schwinger–DeWitt coefficient of the corresponding heat kernel
expansion for the inverse propagator of the theory [62, 63, 64],
$\displaystyle\;a_{2}=-\big{(}\alpha\,{}^{(4)}\\!C^{2}+\beta\,{}^{(4)}E+\gamma\,\Box
R\big{)},$ (4.3)
$\displaystyle{}^{(4)}C^{2}=R_{\mu\nu\alpha\beta}^{2}-2R_{\mu\nu}^{2}+\frac{1}{3}R^{2},$
(4.4)
$\displaystyle{}^{(4)}\\!E=R_{\mu\nu\alpha\beta}^{2}-4R_{\mu\nu}^{2}+R^{2}.$
(4.5)
This structure of $a_{2}$ follows from the local conformal invariance of the
pole residue of $\varGamma_{\infty}$ at $d=4$ and associated with the
integrability (or conformal Wess–Zumino) condition for a conformal anomaly. It
includes the topological Gauss–Bonnet density $\sqrt{g}E$, Weyl tensor squared
and the total derivative $\Box R$ terms.
Conformal anomaly arises as a contribution of the conformal transformation of
the one-loop counterterm (4.2) subtracted from the regularized effective
action
$\sqrt{g}\,\big{\langle}\,T^{\alpha}_{\alpha}\big{\rangle}=-2g_{\alpha\beta}\frac{\delta\varGamma_{\infty}}{\delta
g_{\alpha\beta}},$ (4.6)
because the regularized (but not yet renormalized by counterterm subtracting)
action $\varGamma_{\rm reg}$ is assumed to be conformally invariant777Or the
Weyl invariance violation of dimensionally regularized $\varGamma_{\rm reg}$
is proportional to $(d-4)^{2}$ as it happens for spin one case [4], so that it
does not contribute to the residue of the simple pole in dimensionality.. The
$\Box R$ term does not contribute to the divergences but it appears in the
conformal anomaly in view of the conformal transformation of the Weyl squared
term continued to $d$ dimensions. Moreover, within the above subtraction
scheme its coefficient $\gamma$ in the anomaly turns out to be determined by
the coefficient $\alpha$ of the Weyl term [4].
Indeed, introduce conformally covariant Weyl tensor in $d$ dimensions
$\displaystyle{}^{(d)}\\!C_{\mu\nu\alpha\beta}$
$\displaystyle=R_{\mu\nu\alpha\beta}+2P_{\beta[\mu}g_{\nu]\alpha}-2P_{\alpha[\mu}g_{\nu]\beta},$
(4.7) $\displaystyle{}^{(d)}C^{\mu}{}_{\nu\alpha\beta}$
$\displaystyle={}^{(d)}\\!\bar{C}^{\mu}{}_{\nu\alpha\beta},$ (4.8)
which is written down in terms of the Schouten tensor
$P_{\mu\nu}\equiv\frac{1}{d-2}\Big{(}R_{\mu\nu}-\frac{Rg_{\mu\nu}}{2(d-1)}\Big{)}.$
(4.9)
In view of the relation between the square of Weyl tensors
${}^{(d)}C^{2}\equiv{}^{(d)}C_{\mu\nu\alpha\beta}^{2}$ and
$C^{2}\equiv{}^{(4)}\\!C^{2}_{\mu\nu\alpha\beta}$ (both formally continued to
$d$ dimensions) [25]
${}^{(4)}\\!C^{2}={}^{(d)}\\!C^{2}-\frac{\epsilon}{2}\big{(}E-C^{2}-\tfrac{1}{9}R^{2}\big{)}+O(\epsilon^{2})$
(4.10)
one has
$\displaystyle\frac{\delta}{\delta g_{\mu\nu}}\int
d^{d}x\,\sqrt{g}\,C^{2}=\frac{\delta}{\delta g_{\mu\nu}}\int
d^{d}x\,\sqrt{g}\,{}^{(d)}\\!C^{2}$
$\displaystyle\;\;\;\;+\frac{\epsilon}{2}\frac{\delta}{\delta g_{\mu\nu}}\int
d^{4}x\,\sqrt{g}\,\big{(}C^{2}+\tfrac{1}{9}R^{2}\big{)}+O(\epsilon^{2}).$
(4.11)
Then, since the tensor ${}^{(d)}C_{\mu\nu\alpha\beta}$ is conformally
covariant in any dimension, $g_{\mu\nu}(\delta/\delta g_{\mu\nu})\int
d^{d}x\,\sqrt{g}\,^{(d)}C^{2}=-\tfrac{\epsilon}{2}\sqrt{g}\,{}^{(d)}C^{2}$, we
have
$\frac{1}{\epsilon}g_{\mu\nu}\frac{\delta}{\delta g_{\mu\nu}}\int
d^{d}x\,\sqrt{g}C^{2}=-\frac{1}{2}\sqrt{g}\Big{(}C^{2}+\frac{2}{3}\Box
R\Big{)}+O(\epsilon).$ (4.12)
Using this in (4.6) one recovers the $C^{2}$ and the $\Box R$ terms in the
expression for the anomaly
$\sqrt{g}\,\big{\langle}T^{\alpha}_{\alpha}\big{\rangle}=-\frac{1}{16\pi^{2}}\sqrt{g}\,a_{2},$
(4.13)
with the parameter $\gamma$ related to the coefficient $\alpha$ of the Weyl
squared term [4]
$\gamma=\frac{2}{3}\alpha.$ (4.14)
This simple expression for the trace anomaly in terms of the second
Schwinger–DeWitt coefficient also follows from the zeta-function
regularization [65].
The Gauss–Bonnet part of the anomaly follows from the conformal variation of
the ${}^{(4)}\\!E$-term in the divergent part of the action. Just like $\Box
R$, as the residue of the pole in $\varGamma_{\infty}$ the integral of
$\sqrt{g}{}^{(4)}\\!E$ at least naively does not contribute to the stress
tensor, because in four dimensions this integral is a constant Euler
characteristics of the manifold. But in a covariant renormalization procedure
the coefficient of $1/\epsilon$ in $\varGamma_{\infty}$ cannot be treated
other than as a $d$-dimensional object, so that $\int
d^{d}x\sqrt{g}{}^{(4)}\\!E$ is no longer a topological invariant, and its
metric variation is nontrivial. Therefore, rewriting, similarly to (4.10), the
dimensionally continued Gauss–Bonnet density in terms of ${}^{(d)}\\!C^{2}$,
$\displaystyle{}^{(4)}\\!E$
$\displaystyle=R_{\mu\nu\alpha\beta}^{2}-4R_{\mu\nu}^{2}+R^{2}$
$\displaystyle={}^{(d)}\\!C^{2}-(2-3\epsilon)\big{(}R_{\mu\nu}^{2}-\tfrac{1}{3}R^{2}\big{)}+O(\epsilon^{2}),$
(4.15)
one has
$\displaystyle\frac{1}{\epsilon}\frac{\delta}{\delta g_{\alpha\beta}}\int
d^{d}x\,\sqrt{g}\,{}^{(4)}\\!E=$
$\displaystyle-\sqrt{g}\big{(}\,\tfrac{1}{2}W^{\alpha\beta}+^{(3)}\\!\\!H^{\alpha\beta}$
$\displaystyle+2R_{\mu\nu}C^{\mu\alpha\nu\beta}\big{)}+O(\epsilon),$ (4.16)
where the two new tensors arise
$\displaystyle\\!\\!\\!{}^{(3)}\\!H^{\alpha\beta}=R^{\alpha\mu}R^{\beta}_{\mu}-\frac{2}{3}RR^{\alpha\beta}-\frac{1}{2}g^{\alpha\beta}R_{\mu\nu}^{2}+\frac{1}{4}g^{\alpha\beta}R^{2},$
(4.17) $\displaystyle W^{\alpha\beta}=\lim\limits_{\epsilon\to
0}\frac{1}{\epsilon}\left(4\,{}^{(d)}\\!C^{\alpha}{}_{\mu\nu\lambda}{}^{(d)}\\!C^{\beta\mu\nu\lambda}-g^{\alpha\beta}\,{}^{(d)}\\!C^{2}\right).$
(4.18)
The limit to $d=4$ for the tensor $W^{\alpha\beta}$ is regular here because at
$d=4$ there is the important identity
$4\,{}^{(4)}\\!C^{\alpha}{}_{\mu\nu\lambda}{}^{(4)}\\!C^{\beta\mu\nu\lambda}=g^{\alpha\beta}{}^{(4)}\\!C^{2}$
(4.19)
—it can be proven by antisymmetrization over five indices in the four-
dimensional spacetime [66]. Tensors ${}^{(3)}H^{\alpha\beta}$ and
$W^{\alpha\beta}$ have the following traces
${}^{(3)}\\!H^{\alpha}_{\alpha}=\tfrac{1}{3}R^{2}-R_{\mu\nu}^{2}=\tfrac{1}{2}(E-C^{2}),\quad
W_{\alpha}^{\alpha}=C^{2}.$ (4.20)
Thus from (4.1) and (4.20) we have the relation
$\frac{2}{\epsilon}g_{\alpha\beta}\frac{\delta}{\delta g_{\alpha\beta}}\int
d^{d}x\,\sqrt{g}\,{}^{(4)}\\!E=-\sqrt{g}{}^{(4)}\\!E+O(\epsilon),$ (4.21)
which recovers the contribution of $E$-term in the conformal anomaly (4.13)
with the expression (4.3) for $a_{2}$.
### 4.2 Minimal form of Wess-Zumino action and a-theorem
Of course there is a big ambiguity in the above analytic continuation of the
coefficients relating 4-dimensional objects to their $d$-dimensional
counterparts. This ambiguity reduces to the renormalization by finite
4-dimensional counterterms $\int d^{4}x\sqrt{g}\,R_{\mu\nu\alpha\beta}^{2}$,
$\int d^{4}x\sqrt{g}\,R_{\mu\nu}^{2}$ and $\int d^{4}x\sqrt{g}\,R^{2}$ among
which in view of the total-derivative nature of the Gauss-Bonnet density only
one counterterm can additionally break Weyl invariance and change the
coefficient $\gamma$ of the $\Box R$ term in the conformal anomaly. This is
because the combination $\int d^{4}x\,\sqrt{g}(C^{2}-E)=2\int
d^{4}x\,\sqrt{g}(R_{\mu\nu}^{2}-\tfrac{1}{3}R^{2})$ is Weyl invariant, and
such a counterterm can be chosen as the square of the curvature scalar,
satisfying
$g_{\mu\nu}\frac{\delta}{\delta g_{\mu\nu}}\int
d^{4}x\sqrt{g}\,R^{2}=-6\sqrt{g}\,\Box R.$ (4.22)
Therefore this finite local counterterm can be used to alter the coefficient
$\gamma$ and, in particular, put it to zero by a special finite
renormalization which we will denote by a subscript $\rm Ren$,
$\varGamma_{\rm ren}[\,g\,]\to\varGamma_{\rm Ren}[\,g\,]\equiv\varGamma_{\rm
ren}[\,g\,]+\frac{\gamma}{192\pi^{2}}\int d^{4}x\sqrt{g}\,R^{2}.$ (4.23)
Regularization and subtraction scheme dependence of $\gamma$-coefficient
manifests itself in the violation of the relation (4.14) for the dimensionally
regularized electromagnetic vector field [5], but ultimately does not change
the physics of the theory because of the locality of the covariant counterterm
$\int d^{4}x\sqrt{g}\,R^{2}$, whose subtraction point should be determined
from the comparison with the observable value of its coupling constant. In the
cosmological example considered below the above renormalization (4.23)
corresponds to fixing the coupling constant in the Starobinsky $R^{2}$-model
[43].
The renormalization (4.23) has an important consequence – with $\gamma=0$ the
terms with quartic derivatives of $\sigma$, contained in the combination
$\tfrac{\beta}{16\pi^{2}}\int
d^{4}x\,\big{(}4\sqrt{\bar{g}}\,\sigma\bar{\Delta}_{4}\sigma-\frac{1}{9}\sqrt{g}\,R^{2}\big{)}$
of (2.6), completely cancel out, and the resulting minimal Wess-Zumino action
does not acquire extra hihger-derivative degrees of freedom,
$\displaystyle\varGamma_{\rm Ren}[\,g\,]-\varGamma_{\rm
Ren}[\,\bar{g}\,]=\frac{\alpha}{16\pi^{2}}\int
d^{4}x\,\sqrt{\bar{g}}\,\bar{C}_{\mu\nu\alpha\beta}^{2}\sigma$
$\displaystyle\quad+\frac{\beta}{16\pi^{2}}\int
d^{4}x\,\sqrt{\bar{g}}\,\Big{\\{}\bar{E}\,\sigma-4\,\big{(}\bar{R}^{\mu\nu}-\tfrac{1}{2}\bar{g}^{\mu\nu}\bar{R}\,\big{)}\,\partial_{\mu}\sigma\,\partial_{\nu}\sigma$
$\displaystyle\quad-4\,\bar{\Box}\sigma\,(\bar{\nabla}^{\mu}\sigma\,\bar{\nabla}_{\mu}\sigma)-2\,(\bar{\nabla}^{\mu}\sigma\,\bar{\nabla}_{\mu}\sigma)^{2}\Big{\\}}.$
(4.24)
This minimal version of the action for the dilaton field $\sigma$ was
discussed in [41] and used in the derivation of the $a$-theorem in [42, 67] –
monotonically decreasing coefficient $a=\beta/16\pi^{2}$ in the RG flow of the
theory from UV to IR domains. This theorem is based on the sign of the last
quartic interaction term for this field, related to the cross section of the
forward $2\to 2$ dilaton scattering which should be positive in unitary
theory, its unitarity being related to the absence of higher-derivative ghosts
in (4.24).
### 4.3 Renormalized stress tensors
The behavior of the stress tensor on the orbit of the conformal group can be
obtained by using the commutativity of the following functional variations
$\left[\,g_{\mu\nu}(y)\frac{\delta}{\delta
g_{\mu\nu}(y)},g_{\beta\gamma}(x)\frac{\delta}{\delta
g_{\alpha\gamma}(x)}\right]=0,$ (4.25)
which allows one to write
$\displaystyle\frac{\delta}{\delta\sigma(y)}\sqrt{g}\left\langle
T^{\alpha}_{\beta}(x)\right\rangle=2g_{\beta\gamma}(x)\frac{\delta}{\delta
g_{\alpha\gamma}(x)}\frac{\delta\varGamma_{\rm
ren}}{\delta\sigma(y)}\bigg{|}_{g_{\mu\nu}=e^{2\sigma}\bar{g}_{\mu\nu}}$
$\displaystyle\qquad=g_{\beta\gamma}(x)\frac{\delta}{\delta
g_{\alpha\gamma}(x)}\sqrt{g}(y)\left\langle
T^{\mu}_{\mu}(y)\right\rangle\Big{|}_{g_{\mu\nu}=e^{2\sigma}\bar{g}_{\mu\nu}}.$
(4.26)
Bearing in mind that $g_{\beta\gamma}\delta/\delta
g_{\alpha\gamma}=\bar{g}_{\beta\gamma}\delta/\delta\bar{g}_{\alpha\gamma}$ at
fixed $\sigma$ and functionally integrating this relation over $\sigma$ one
has
$\sqrt{g}\,\big{\langle}T^{\alpha}_{\beta}\big{\rangle}-\sqrt{\bar{g}}\,\big{\langle}\bar{T}^{\alpha}_{\beta}\big{\rangle}=2\bar{g}_{\beta\gamma}\frac{\delta}{\delta\bar{g}_{\alpha\gamma}}\Delta\varGamma[\,\bar{g},\sigma\,],$
(4.27)
where $\Delta\varGamma[\,\bar{g},\sigma\,]=\varGamma_{\rm
ren}-\bar{\varGamma}_{\rm ren}$ is given by (2.6).
Before calculating this difference by the metric variation of
$\Delta\varGamma[\bar{g},\sigma]$ it is instructive to obtain it directly from
the divergent part of the action as it was done in [25]. Note that
$\varGamma_{\rm ren}-\bar{\varGamma}_{\rm
ren}=-(\varGamma_{\infty}-\bar{\varGamma}_{\infty})$ because $\varGamma_{\rm
reg}$ does not contribute to the anomaly (see footnote 7). Therefore,
$\sqrt{g}\,\big{\langle}\,T^{\alpha}_{\beta}\big{\rangle}\,\Big{|}_{\,\bar{g}}^{\,g}=-2\,g_{\beta\gamma}\frac{\delta\varGamma_{\infty}}{\delta
g_{\alpha\gamma}}\,\Big{|}_{\,\bar{g}}^{\,g}$ (4.28)
To calculate the contribution of the ${}^{(4)}C^{2}$-term in
$\varGamma_{\infty}$ we rewrite it in terms of ${}^{(d)}C^{2}$ and use Eq.
(4.11). This leads to the contribution of the first term of this equation
$\displaystyle\frac{\delta}{\delta g_{\mu\nu}}\int
d^{d}x\,\sqrt{g}\,{}^{(d)}C^{2}=-\frac{\epsilon}{2}\sqrt{g}\,W^{\mu\nu}-4\sqrt{g}\,{}^{(d)}B^{\mu\nu},$
(4.29)
$\displaystyle{}^{(d)}B^{\mu\nu}=\Big{(}\frac{1}{d-2}R_{\alpha\beta}+\nabla_{(\alpha}\nabla_{\beta)}\Big{)}C^{\mu\alpha\nu\beta},$
(4.30)
where the tensor $W^{\mu\nu}$ is defined by Eq. (4.18) and
${}^{(d)}\\!B^{\mu\nu}$ is the $d$-dimensional Bach tensor. Assembling this
with the second term of Eq. (4.11) we get on the orbit of the conformal group
$\frac{1}{\epsilon}g_{\beta\gamma}\frac{\delta}{\delta g_{\alpha\gamma}}\int
d^{d}x\,\sqrt{g}\,^{(4)}\\!C^{2}\,\Big{|}_{\,\bar{g}}^{\,g}\\\
=-\sqrt{g}\Big{[}\,\frac{4}{\epsilon}{}^{(d)}\\!B^{\alpha}_{\beta}+\frac{1}{18}{}^{(1)}\\!H^{\alpha}_{\beta}\,\Big{]}_{\,\bar{g}}^{\,g}+O(\epsilon),$
(4.31)
where the tensor ${}^{(1)}\\!H^{\alpha}_{\beta}$ is given by the equation
$\displaystyle{}^{(1)}\\!H^{\alpha}_{\beta}=\frac{1}{\sqrt{g}}g^{\alpha\gamma}\frac{\delta}{\delta
g^{\beta\gamma}}\int d^{4}x\,\sqrt{g}R^{2}$
$\displaystyle=-\frac{1}{2}\delta^{\alpha}_{\beta}R^{2}+2RR^{\alpha}_{\beta}+2\delta^{\alpha}_{\beta}\Box
R-2\nabla^{\alpha}\nabla_{\beta}R,$ (4.32)
and we took into account that the both tensor densities
$\sqrt{g}\,W^{\alpha}_{\beta}$ and $\sqrt{g}\,B^{\alpha}_{\beta}$ in four
dimensions are invariant on the conformal orbit. Outside of four dimensions
the Bach tensor density transforms on this orbit as (here as above
$g_{\mu\nu}=e^{2\sigma}\bar{g}_{\mu\nu}$)
$\sqrt{g}\,{}^{(d)}\\!B^{\alpha}_{\beta}\,\Big{|}_{\bar{g}}^{g}=-\frac{\epsilon}{2}\sqrt{\bar{g}}\big{(}\bar{R}^{\mu\nu}+2\bar{\nabla}^{(\mu}\bar{\nabla}^{\nu)}\big{)}\big{(}\sigma\bar{C}^{\alpha}{}_{\mu\beta\nu}\big{)}+O(\epsilon^{2}),$
(4.33)
which makes the first term on the right hand side of (4.31) well defined at
$d\to 4$. Note that the expression
$\sqrt{\bar{g}}(\bar{R}^{\mu\nu}+2\bar{\nabla}^{(\mu}\bar{\nabla}^{\nu)}\big{)}\big{(}\sigma\bar{C}^{\alpha}{}_{\mu\beta\nu})$
treated as a functional of independent $\bar{g}_{\mu\nu}$ and $\sigma$ is Weyl
invariant under local conformal transformations of the barred metric. This can
be easily inferred from the invariance of Eq.(4.33) under the interchange
$g_{\mu\nu}\leftrightarrow\bar{g}_{\mu\nu}$ and $\sigma\to-\sigma$ or directly
checking the conformal transformation of $\bar{g}_{\mu\nu}$ (with a fixed
scalar $\sigma$).
The contribution of Gauss–Bonnet term to the stress tensor behavior on the
conformal orbit is obtained from using (4.1)–(4.17). Collecting this
contribution with the contribution (4.31) of the Weyl tensor squared part we
finally have
$\displaystyle\sqrt{g}\,\big{\langle}T^{\alpha}_{\beta}\big{\rangle}\Big{|}_{\,\bar{g}}^{\,g}=-\frac{\alpha}{4\pi^{2}}\sqrt{\bar{g}}\,\big{(}\bar{R}^{\mu\nu}+2\bar{\nabla}^{(\mu}\bar{\nabla}^{\nu)}\big{)}\big{(}\sigma\bar{C}^{\alpha}{}_{\mu\beta\nu}\big{)}+\frac{1}{8\pi^{2}}\sqrt{g}\,\Big{[}\,\beta\,^{(3)}\\!H^{\alpha}_{\beta}+\frac{\alpha}{18}\,^{(1)}\\!H^{\alpha}_{\beta}+2\beta
R^{\mu\nu}C^{\alpha}{}_{\mu\beta\nu}\,\Big{]}_{\,\bar{g}}^{\,g}.$ (4.34)
This is a generalization of the Brown–Cassidy formula to the case of a nonzero
Weyl tensor. The first term of this expression is Weyl invariant in view of
the above remark and can be represented by its unbarred version.
The check of consistency of this formula with the original expression for the
conformal anomaly is trivial in view of
${}^{(3)}\\!H^{\alpha}_{\alpha}=(E-C^{2})/2$,
${}^{(1)}\\!H^{\alpha}_{\alpha}=6\Box R$ and tracelessness of the Weyl tensor,
$\sqrt{g}\,\big{\langle}T^{\alpha}_{\alpha}\big{\rangle}\Big{|}_{\,\bar{g}}^{\,g}=\frac{\sqrt{g}}{16\pi^{2}}\\!\Big{[}\beta
E-\beta C^{2}+\tfrac{2\alpha}{3}\Box
R\Big{]}_{\bar{g}}^{g}=-\frac{\sqrt{g}\,a_{2}}{16\pi^{2}}\Big{|}_{\,\bar{g}}^{\,g},$
(4.35)
where the last equality follows from the conformal invariance of the density
$\sqrt{g}\,C^{2}$ and from the relation (4.14) between the coefficients
$\gamma$ and $\alpha$, $\alpha=\tfrac{3}{2}\gamma$.
The recovery of (4.34) from the direct variation of the Wess–Zumino action
(4.27) goes as follows. We use metric variational formulae
$\displaystyle\frac{\delta}{\delta g_{\alpha\beta}}\\!\int
d^{4}x\,\sqrt{g}\,C^{2}\sigma=-2\sqrt{g}\big{(}R_{\mu\nu}\\!+2\nabla_{(\mu}\\!\nabla_{\nu)}\big{)}\big{(}\sigma
C^{\alpha\mu\beta\nu}\big{)},$ (4.36) $\displaystyle\frac{\delta}{\delta
g_{\alpha\beta}}\int
d^{4}x\,\sqrt{g}\,\mathcal{E}_{4}\sigma=\sqrt{g}\,\Delta^{\alpha\beta}\sigma,$
(4.37) $\displaystyle\frac{\delta}{\delta g_{\alpha\beta}}\int
d^{4}x\,\sqrt{g}\,\varphi\Delta_{4}\sigma=-\frac{\sqrt{g}}{2}D^{\alpha\beta}[\varphi,\sigma],$
(4.38)
which hold for generic scalar test functions $\sigma$ and $\varphi$ with the
differential operator $\Delta^{\alpha\beta}$ acting on $\sigma$,
$\displaystyle\Delta_{\alpha\beta}$
$\displaystyle=\frac{1}{3}(g_{\alpha\beta}\Box-\nabla_{\alpha}\nabla_{\beta})\,\Box$
$\displaystyle+\left[\,2(g_{\alpha\beta}P_{\mu\nu}-g_{\alpha\mu}P_{\beta\nu}-g_{\alpha\nu}P_{\beta\mu})+\frac{8}{3}g_{\mu\nu}P_{\alpha\beta}\right.$
$\displaystyle\left.+2Pg_{\alpha\mu}g_{\beta\nu}-\frac{5}{3}Pg_{\alpha\beta}g_{\mu\nu}-2W_{\alpha\mu\beta\nu}\right]\nabla^{\mu}\nabla^{\nu}$
$\displaystyle+\big{(}\,g_{\alpha\beta}g_{\mu\nu}-g_{\alpha\mu}g_{\beta\nu}-g_{\alpha\nu}g_{\beta\mu}\big{)}(\nabla^{\mu}P)\nabla^{\nu},$
(4.39)
and the bilinear form $D^{\alpha\beta}(\varphi,\sigma)$,
$\displaystyle D_{\alpha\beta}[\varphi,\sigma]$
$\displaystyle=-\frac{1}{2}g_{\alpha\beta}\Box\varphi\,\Box\sigma-2\sigma_{\alpha\beta}\Box\varphi$
$\displaystyle+2\sigma_{\alpha}\Box\varphi_{\beta}-\frac{1}{3}g_{\alpha\beta}\sigma_{\mu}\Box\varphi^{\mu}-\frac{2}{3}\varphi_{\mu(\alpha\beta)}\sigma^{\mu}$
$\displaystyle+\left[\,2W_{\alpha\mu\beta\nu}+\frac{1}{3}(g_{\mu\nu}R_{\alpha\beta}-g_{\alpha\mu}g_{\beta\nu}R)\right]\varphi^{(\mu}\sigma^{\nu)}$
$\displaystyle+\frac{1}{3}\big{(}\,4\varphi_{\alpha\mu}\sigma^{\mu}_{\beta}-g_{\alpha\beta}\varphi_{\mu\nu}\sigma^{\mu\nu}\big{)}+\big{(}\,\varphi\Leftrightarrow\sigma\,\big{)},$
(4.40)
where $\varphi_{\alpha}\equiv\nabla_{\alpha}\varphi$,
$\sigma_{\alpha\beta}\equiv\nabla_{\beta}\nabla_{\alpha}\sigma$,
$\varphi_{\alpha\beta\gamma}\equiv\nabla_{\gamma}\nabla_{\beta}\nabla_{\alpha}\varphi$,
etc. Note that the trace of $\Delta^{\alpha\beta}$ coincides with the Paneitz
operator, $g_{\alpha\beta}\Delta^{\alpha\beta}=\Delta_{4}$, which matches with
the conformal variation (2.9), and the bilinear form
$D^{\alpha\beta}(\varphi,\sigma)$ is traceless in view of conformal invariance
of $\sqrt{g}\Delta_{4}$.
Using these relations we get from (4.27) and (2.6)
$\displaystyle\sqrt{g}\,\big{\langle}T^{\alpha}_{\beta}\big{\rangle}\,\Big{|}_{\,\bar{g}}^{\,g}=-\frac{\alpha}{4\pi^{2}}\sqrt{g}\big{(}R^{\mu\nu}+2\nabla^{(\mu}\nabla^{\nu)}\big{)}\big{(}\sigma
C^{\alpha}{}_{\mu\beta\nu}\big{)}$
$\displaystyle+\frac{\sqrt{g}}{8\pi^{2}}\\!\Big{(}2\beta\Delta^{\alpha}_{\beta}\sigma+\beta
D^{\alpha}_{\beta}[\sigma,\sigma]\Big{)}+\sqrt{g}\left(\tfrac{\gamma}{12}+\tfrac{\beta}{18}\right){}^{(1)}\\!H^{\alpha}_{\beta}\,\Big{|}_{\,\bar{g}}^{\,g}.$
(4.41)
The term in the first line here coincides with its barred version in
(4.34)—this easily follows from the relation (4.36) where the integrand can be
identically replaced by the barred one. The
$\frac{\gamma}{12}{}^{(1)}\\!H^{\alpha}_{\beta}$ term here matches with the
$\frac{\alpha}{18}{}^{(1)}\\!H^{\alpha}_{\beta}$ term of (4.34) in view of the
relation $\alpha=\tfrac{3}{2}\gamma$. And finally, the identity holds
$\displaystyle\sqrt{g}\,\Big{[}{}^{(3)}\\!H^{\alpha}_{\beta}+\tfrac{1}{18}{}^{(1)}\\!H^{\alpha}_{\beta}$
$\displaystyle+2R^{\mu\nu}C^{\alpha}{}_{\mu\beta\nu}\Big{]}_{\,\bar{g}}^{\,g}$
$\displaystyle=\sqrt{g}\Big{(}2\Delta^{\alpha}_{\beta}\sigma+D^{\alpha}_{\beta}[\sigma,\sigma]\Big{)},$
(4.42)
which completely reconciles the two expressions (4.34) and (4.41) for the
stress tensor behavior on the orbit of the conformal group.
## 5 Conformally flat spacetime
The generalization (4.34) of Brown-Cassidy formula to the case of a
nonvanishing Weyl tensor might be not very useful, because in the general case
not much can be said about $\langle\,T^{\alpha}_{\beta}\rangle\,|_{\bar{g}}$.
Therefore we will restrict ourselves with the case of the conformally flat
spacetime for which the conformal transformation of the metric can lead to the
metric $\bar{g}_{\mu\nu}$ of flat spacetime, where
$\langle\,\bar{T}^{\alpha}_{\beta}\rangle$ is either zero or can be obtained
from flat space physics. Interestingly, in this case the parameter of the
conformal transformation $\sigma$ making this transition satisfies the
equation
$\Delta_{4}\,\sigma=\frac{1}{4}\mathcal{E}_{4}$ (5.1)
and in asymptotically flat case with Dirichle boundary conditions has a unique
solution (2.21), $\sigma=\varSigma_{\rm RFT}$. This, apparently not very well
known fact, can be proven by using the equation for the conformal
transformation of the four-dimensional Schouten tensor (4.8)
($g_{\mu\nu}=e^{2\sigma}\bar{g}_{\mu\nu}$)
$P_{\mu\nu}-\bar{P}_{\mu\nu}=-\sigma_{\mu\nu}-\sigma_{\mu}\sigma_{\nu}+\frac{1}{2}\sigma_{\alpha}\sigma^{\alpha}g_{\mu\nu},$
(5.2)
where $\sigma_{\mu}\equiv\nabla_{\mu}\sigma$ and
$\sigma_{\mu\nu}\equiv\nabla_{\nu}\nabla_{\mu}\sigma$. Assuming that
$\bar{g}_{\mu\nu}$ is flat space metric with $\bar{P}_{\mu\nu}=0$,
differentiating twice and again using this relation to express $P_{\mu\nu}$ in
terms of the derivatives of $\sigma$ one has
$\nabla^{\mu}\nabla^{\nu}\left(P_{\mu\nu}+\sigma_{\mu\nu}+\sigma_{\mu}\sigma_{\nu}-\frac{1}{2}\sigma_{\alpha}\sigma^{\alpha}g_{\mu\nu}\right)\\\
=\Delta_{4}\sigma-\frac{1}{4}\mathcal{E}_{4}=0,$ (5.3)
whence it follows that the conformal invariant metric (2.12) in the RFT gauge
(2.20) is actually the flat space one when the Weyl tensor is zero
$\bar{R}^{\alpha}_{\;\;\beta\mu\nu}=0,\quad\bar{g}_{\mu\nu}=e^{-2\varSigma_{\rm
RFT}[\,g\,]}g_{\mu\nu}\big{|}_{\,C_{\alpha\beta\mu\nu}=0}.$ (5.4)
Note that $\bar{g}_{\mu\nu}$ here is not automatically diagonal unit matrix
$\delta_{\mu\nu}$, because this is the invariant statement which is valid in
any coordinate system.
### 5.1 Anomaly driven cosmology
Applications of the conformal anomaly in the cosmological context have a long
history, see for example [26, 68, 9, 69, 70, 71, 72]. In particular, cosmology
with the Friedman–Robertson–Walker (FRW) metric represents the situation when
the anomalous action $\Delta\varGamma[\bar{g},\sigma]$ entirely determines the
physics of the field model and via effective equations of motion produces a
nontrivial back reaction of quantum matter on the dynamical metric background.
The most interesting example is, perhaps, the case when
$\varGamma[\,\bar{g}\,]$ in (2.6) nontrivially contributes to this back
reaction effect rather than just serves as an inert flat space background.
This is the spatially closed cosmology driven by a conformal field theory
(CFT) from the initial state in the form of a special microcanonical density
matrix, which was orginally suggested in [27] and recently reviewed in [46].
With the density matrix defined as the projector on the space of solutions of
the Wheeler–DeWitt equations [28, 73] the statistical sum in this model has a
representation of the Euclidean quantum gravity (EQG) path integral
$Z=\int D[\,g_{\mu\nu},\phi\,]\;e^{-S[\,g_{\mu\nu},\phi\,]},$ (5.5)
where integration runs over the metric $g_{\mu\nu}$ and matter fields $\phi$
which are periodic on the Euclidean spacetime of topology $S^{1}\times S^{3}$
with the time $\tau$ compactified to a circle $S^{1}$.
When the classical action $S[\,g_{\mu\nu},\phi\,]$ is dominated by numerous
CFT fields $\varPhi$ with their action
$S_{C\\!F\\!T}[\,g_{\mu\nu},\varPhi\,]$, the statistical sum can be
approximated by the contribution of the saddle point of this integral. This is
the extremum of the total action including the tree-level gravitational
Einstein–Hilbert action $S_{EH}[\,g_{\mu\nu}]$ and the effective action
$\varGamma[\,g_{\mu\nu}]$ of these CFT fields888Disregarding the graviton
loops can be justified by the domination of conformal fields outnumbering the
metric, and retaining the Einstein–Hilbert term obviously follows from the
fact that this term with renormalized gravitational and cosmological constants
is anyway induced from the quantum conformal sector.,
$\displaystyle\varGamma_{\rm tot}[\,g_{\mu\nu}]$
$\displaystyle=S_{EH}[\,g_{\mu\nu}]+\varGamma[\,g_{\mu\nu}],$ (5.6)
$\displaystyle e^{-\varGamma[\,g_{\mu\nu}]}$ $\displaystyle=\int
D\varPhi\,e^{-S_{C\\!F\\!T}[\,g_{\mu\nu},\varPhi\,]}.$ (5.7)
Choosing as $g_{\mu\nu}$ the FRW metric with the scale factor $a(\tau)$ and
the lapse function $N$ ($\varOmega^{2}_{(3)}$ is the metric of the
3-dimensional sphere of a unit radius),
$ds^{2}=N^{2}d\tau^{2}+a^{2}d\varOmega^{2}_{(3)}=a^{2}(\tau)\big{(}d\eta^{2}+d\varOmega^{2}_{(3)}\big{)},$
(5.8)
one immediately finds that in terms of the conformal time variable $\eta$,
related to the Euclidean time $\tau$ by the relation $d\eta=d\tau/a(\tau)$,
this metric is conformally equivalent to the metric $\bar{g}_{\mu\nu}\equiv
g_{\mu\nu}^{EU}$ of the Einstein static universe with spatial sections—the
3-dimensional spheres of some constant radius $a_{0}$,
$\displaystyle
d\bar{s}^{2}=a_{0}^{2}\big{(}d\eta^{2}+d\varOmega^{2}_{(3)}\big{)}\equiv
g_{\mu\nu}^{EU}dx^{\mu}dx^{\nu},$ (5.9) $\displaystyle
ds^{2}=e^{2\sigma}d\bar{s}^{2},\quad
g_{\mu\nu}=e^{2\sigma}g_{\mu\nu}^{EU},\quad\sigma=\ln\frac{a}{a_{0}}.$ (5.10)
Therefore the CFT effective action expresses in terms of the same action on a
static Einstein universe $\varGamma[\,g_{\mu\nu}^{EU}]\equiv\varGamma_{EU}$
and Wess–Zumino action (2.6) with the above conformal parameter $\sigma$
$\varGamma[\,g_{\mu\nu}]=\Delta\varGamma[\,g_{\mu\nu}^{EU},\sigma\,]+\varGamma_{EU}.$
(5.11)
The calculation of $\varGamma_{EU}$ is strongly facilitated by the static
nature of the background, but it still yields a nontrivial result in view of
compactification of time on $S^{1}$. To begin with, note that although
$g_{\mu\nu}^{EU}$ explicitly depends on the size $a_{0}$ of $S^{3}$, the value
of $\varGamma_{EU}$ is $a_{0}$-independent for a fixed period of the conformal
time $\eta=\oint d\eta$. This follows from the invariance of the effective
action under global conformal transformations (3.41) for conformally flat
spacetimes with zero bulk part of the Euler characteristics (which is the case
of $S^{1}\times S^{3}$). This also can be confirmed by using scaling
properties of the conformal fields. Indeed, the energies of conformal quanta
on a static spacetime scale as $1/a_{0}$ and their Hamiltonian reads,
$\hat{H}=\sum_{\omega}\frac{\omega}{a_{0}}\Big{(}\hat{a}^{\dagger}_{\omega}\hat{a}_{\omega}\pm\frac{1}{2}\Big{)},$
(5.12)
where summation runs over all quantum numbers (and spins) of the energies
$\omega/a_{0}$ of all field oscillator modes on a static 3-dimensional sphere
of the radius $a_{0}$ and $\hat{a}^{\dagger}_{\omega}$ and $\hat{a}_{\omega}$
are the relevant creation-annihilation operators ($\pm$ signs correspond to
bosons or fermions). The path integral over (anti)periodic conformal (fermion)
boson fields with a period ${\cal T}=\oint d\tau N$ on a static metric
background is exactly calculable and equals the equilibrium statistical sum at
the temperature $1/{\cal T}$ which expresses as a function of the conformal
time period $\eta={\cal T}/a_{0}$
$e^{-\varGamma_{EU}}=\int
D\varPhi\,e^{-S_{C\\!F\\!T}[\,g_{\mu\nu}^{EU},\varPhi\,]}\\\ ={\rm
Tr}\,e^{-{\cal T}\hat{H}}=\exp\big{(}-\eta E_{\rm vac}-F(\eta)\big{)}.$ (5.13)
Here $F(\eta)$ is the free energy of the gas of conformal particles and
$E_{\rm vac}$ is a UV divergent Casimir energy which should be covariantly
renormalized
$\displaystyle F(\eta)$ $\displaystyle=\sum_{\omega}\Big{[}\pm\,\ln\big{(}1\mp
e^{-\omega\eta}\big{)}\,\Big{]},$ (5.14) $\displaystyle E_{\rm vac}$
$\displaystyle=\Big{(}\sum_{\omega}\frac{\pm\,\omega}{2}\Big{)}_{\rm ren}.$
(5.15)
Thus, the dependence on $a_{0}$ is absorbed into the dependence on $\eta$
which should be fixed under the rescaling of $a_{0}$. Note that it is $\eta$
that should be kept fixed under the global conformal transformation which
simultaneously rescales the lapse function $N$ and $a_{0}$ in the definition
of the conformally invariant $\eta=\oint d\tau\,N/a_{0}$.
Remarkably, the covariant renormalization of the vacuum Casimir energy $E_{\rm
vac}$ also follows from the behavior of the effective action on the orbit of
the conformal group. The Einstein universe extending from $-\infty$ to
$+\infty$ in $\eta$ is mapped to flat space by the transition to the radial
coordinate $\rho$
$\eta\mapsto\rho=a_{0}e^{\eta},\quad-\infty<\eta<+\infty,\quad
0\leq\rho<\infty,$ (5.16)
with the conformal relation between the two metrics
$\displaystyle ds^{2}_{EU}=e^{2\sigma}ds_{\rm
flat}^{2},\quad\sigma=-\eta=\ln\frac{a_{0}}{\rho},$ (5.17) $\displaystyle
ds_{\rm flat}^{2}=d\rho^{2}+\rho^{2}d\varOmega^{2}_{(3)}.$ (5.18)
For the vacuum state (the limit $\eta\to\infty$ and $F(\eta)\to 0$ in Eq.
(5.13)) $\varGamma_{EU}\to E_{\rm vac}\eta$. On the other hand, from Eq. (2.6)
with the above expression for $\sigma$
$\displaystyle\Delta\varGamma[\,g_{\rm
flat},\sigma\,]=\frac{\beta}{8\pi^{2}}\int d^{4}x\sqrt{g_{\rm
flat}}\big{(}\Box_{\rm flat}\sigma\big{)}^{2}$
$\displaystyle\qquad\qquad\quad-\frac{1}{32\pi^{2}}\Big{(}\frac{\gamma}{6}+\frac{\beta}{9}\Big{)}\,\int
d^{4}x\sqrt{g_{EU}}R^{2}_{EU}.$ (5.19)
Bearing in mind that $\Box_{\rm flat}\sigma=-2/\rho^{2}$, $\int
d^{4}x\sqrt{g_{\rm flat}}\\!\mapsto 2\pi^{2}\int d\rho\,\rho^{3}$,
$R_{EU}=6/a_{0}^{2}$ and $\int d^{4}x\sqrt{g_{EU}}\mapsto
2\pi^{2}a_{0}^{4}\int d\eta$, one has
$\displaystyle\varGamma_{EU}-\varGamma_{\rm flat}=\Delta\varGamma[\,g_{\rm
flat},\sigma\,]$
$\displaystyle=\beta\\!\int\frac{d\rho}{\rho}-\Big{(}\frac{3}{8}\gamma+\frac{\beta}{4}\Big{)}\\!\int
d\eta$ $\displaystyle=\frac{3}{4}\,\Big{(}\beta-\frac{\gamma}{2}\Big{)}\int
d\eta.$ (5.20)
Therefore, under an obvious assumption that $\varGamma_{\rm flat}=0$ one has
$E_{\rm vac}=\frac{3}{4}\,\Big{(}\beta-\frac{\gamma}{2}\Big{)}.$ (5.21)
In other words, after covariant renormalization by covariant counterterms the
Casimir energy gets the value compatible with the behavior of the renormalized
effective action on the conformal group orbit (or with the Brown–Cassidy
formula for the vacuum stress tensor). This compatibility was indeed checked
by direct renormalization of the UV divergent sum over field modes in (5.15)
[47, 48, 49, 50].
Let us now turn to the contribution of the conformal transformation from the
generic FRW metric to that of the static Einstein universe in (5.11). To begin
with we use the freedom of finite renormalization (4.23) which reduces the
theory to the case of anomaly (1.1) with $\gamma=0$ and, in particular,
renders $E_{\rm vac}=\tfrac{3}{4}\beta$. In the cosmological context this
freedom corresponds to the adjustment of the coupling constant of the
Starobinsky $R^{2}$-action [68] which plays an important role in inflation
theory and the dark energy model. Then, with $\gamma=0$ and $\sigma$ given by
(5.17) the Wess–Zumino term in (5.11) takes the form [27]
$\varGamma_{\rm Ren}[\,g\,]-\varGamma_{\rm
Ren}[\,g_{EU}]=\frac{3\beta}{2}\oint d\tau\,N\left(\frac{a^{\prime
2}}{a}-\frac{a^{\prime 4}}{6\,a}\right),$ (5.22)
when written down in terms of the original FRW coordinates with the notation
for the invariant time derivative $a^{\prime}=da/Nd\tau$. Note that the result
is again independent of the constant $a_{0}$ because it contains only
differentiated $\sigma$ and, moreover, it does not involve higher order
derivatives of $a(\tau)$. The last property is entirely due to the fact of
$\gamma$ being renormalized to zero and due to the cancellation of higher
derivative terms in the minimal form of Wess-Zumino action (4.24).
Now we assemble together the Einstein-Hilbert action (with the reduced Planck
mass $M_{\rm P}=1/\sqrt{8\pi G}$ and the cosmological constant $\varLambda$),
the action on the Einstein universe space (5.13) and (5.22). This leads to the
total effective action on the generic Euclidean FRW background periodic in
Euclidean time with the period $\eta$ measured in units of the conformal time
$\displaystyle\varGamma_{\rm tot}[\,a,N\,]=6\pi^{2}M_{P}^{2}\oint
d\tau\,N\bigg{\\{}-aa^{\prime 2}-a+\frac{\varLambda}{3}a^{3}$
$\displaystyle+\frac{\beta}{4\pi^{2}M_{P}^{2}}\left(\frac{a^{\prime
2}}{a}-\frac{a^{\prime 4}}{6a}+\frac{1}{2a}\right)\bigg{\\}}+F(\eta),$ (5.23)
$\displaystyle\eta=\oint\frac{d\tau N}{a}.$ (5.24)
Here the contribution of the conformal anomaly and Casimir energy (5.21) (with
$\gamma=0$) are both weighted by the parameter $\beta$ of the topological term
in the conformal anomaly. The free energy of the gas of conformal particles
$F(\eta)$ is a function of the effective (“comoving”) temperature of this gas
– the inverse of the circumference $\eta$ of the cosmological instanton
(5.24). Despite essentially non-stationary metric background this gas stays in
equilibrium state because of scaling properties of its particles and produces
back reaction on the Friedmann metric background.
Applications of the action (5.23) have been considered in the number of papers
[27, 43, 44] and recently reviewed in [46]. Physics of the CFT driven
cosmology is entirely determined by this effective action and the effective
(Euclidean) Friedmann equation. The latter follows from the action by varying
the lapse $N(\tau)$ and expressing the Hubble factor in terms of the energy
density. In cosmic type gauge $N=1$, $\dot{a}=da/d\tau$, it reads
$\displaystyle\frac{1}{a^{2}}-\frac{\dot{a}^{2}}{a^{2}}=\frac{\varepsilon}{3M_{\pm}^{2}\big{(}\varepsilon\big{)}},$
(5.25)
$\displaystyle\varepsilon=M_{P}^{2}\varLambda+\frac{1}{2\pi^{2}a^{4}}\sum_{\omega}\frac{\omega}{e^{\eta\omega}-1},$
(5.26) $\displaystyle
M_{\pm}^{2}(\varepsilon)=\frac{M_{P}^{2}}{2}\Big{(}1\pm\sqrt{1-\beta\varepsilon/6\pi^{2}M_{P}^{4}}\,\Big{)},$
(5.27)
where the total energy density $\varepsilon$ includes the cosmological
constant contribution and the radiation density of conformal field modes
distributed over Planckian spectrum with the comoving temperature $1/\eta$.
The nonlinear effect of the Weyl anomaly manifests itself in the effective
Planck mass squared explicitly depending on $\varepsilon$ which takes two
possible values $M_{\pm}^{2}(\varepsilon)$.999To avoid mixup of the signs in
$M_{\pm}^{2}$ and sign factors associated with the statistics of conformal
$\omega$-modes we present here the radiation spectrum only for bosonic case.
These equations should be amended by the expression for the conformal time
period that interpolates between the turning points of the solution with
$\dot{a}(\tau)=0$. Note that the right hand side of the Friedmann equation
does not contain Casimir energy density – it turns out to be fully screened
due to the dynamical effect of the Weyl anomaly. This is the result of the
finite renormalization (4.23) leading to a particular value of the anomaly
coefficient of $\Box R$, $\gamma=0$.
For the choice of $+$ sign in $M_{\pm}^{2}$ the solutions of this quantum
Friedmann equation turn out to be the so-called garlands – the cosmological
instantons of $S^{1}\times S^{3}$ topology, which have the periodic scale
factor $a(\tau)$ oscillating on $S^{1}$ between maximal and minimal values
$a_{\pm}$ [27]. These instantons serve as initial conditions for the
cosmological evolution in the physical Lorentzian spacetime. This evolution
follows from $a(\tau)$ by the analytic continuation $a_{L}(t)=a(\tau_{+}+it)$,
$(da_{L}/dt)^{2}=-\dot{a}^{2}$, to the complex plane of the Euclidean time at
the turning point with the maximal scale factor $a_{+}=a(\tau_{+})$. It can
incorporate a finite inflationary stage if the model is generalized to the
case when a primordial cosmological constant is replaced by the potential of
the inflaton field $\phi$, $\varLambda\to V(\phi)/M_{P}^{2}$, staying in the
slow-roll regime during the inflationary stage101010Alternatively, the role of
inflaton can be played by Ricci curvature in the Starobinsky $R^{2}$-model,
the coupling of the $R^{2}$ term being subject to the renormalization
respecting the zero value of $\alpha$ in the total Weyl anomaly [43]. and
decaying in the end of inflation by a usual exit scenario [43, 44]. The energy
scale of inflation – its Hubble parameter $H\sim\sqrt{\varLambda/3}$ turns out
to be bounded from above by $\sqrt{2}\pi M_{P}/\sqrt{\beta}$, so that to solve
the problem of hierarchy between the Planck and inflation scales one needs
$\beta\gg 1$ which matches with the previously adopted assumption that
numerous conformal fields drastically outnumber all other fields and dominate
over their loop corrections.
For the negative sign in $M_{\pm}^{2}$ the solutions represent vacuum
$S^{4}$-instantons of the no-boundary type with the vanishing minimal value of
the scale factor $a_{-}=0$. They correspond to the diverging
$\eta\sim\int_{0}^{a_{+}}da/a\dot{a}\to\infty$ or zero temperature. These
solutions, however, do not contribute to the statistical sum because of their
infinitely positive action $\varGamma_{\rm tot}\to+\infty$ — the quantum
effect of the trace anomaly which flips the sign of the negative tree-level
action of the Hartle-Hawking instantons [74] and sends it to $+\infty$ [27].
Thus the CFT cosmology scenario is free from the infrared catastrophe of the
no-boundary quantum state which would imply that the origin of an infinitely
big Universe is infinitely more probable than that of a finite one.
## 6 Renormalization group and the metamorphosis of the running scale
This section has essentially discussion nature and is associated with the
covariant perturbation theory of the above type. One of the motivations for
this discussion is that, in spite of a widespread concept of running
cosmological and gravitational constants, which is especially popular within
the asymptotic safety approach, there is a very profound and persuading
criticism of this concept [30]. It is based on numerous arguments of the
tadpole structure of the cosmological and Einstein terms, on concrete results
for graviton scattering amplitudes [29] which cannot be interpreted in terms
of a universal scaling of $\varLambda$ and $G$, etc.
At the same time in renormalizable gravity models with multiple couplings the
solution of the full set of RG equations includes running cosmological and
gravitational constants [36]. So the question arises how to interpret their
running scale. Here is the attempt to do this in terms of the covariant
curvature expansion developed in [38, 39, 40].
We start with the classical action which is the sum of local curvature
invariants of growing dimensionality $(4+m)$ in units of the mass
$S[\,g_{\mu\nu}]=\sum\limits_{m,N}\varLambda^{(m)}_{N}\int
d^{4}x\,\sqrt{g}\,\Re^{(4+m)}_{N}(x).$ (6.1)
They are monomials of $N$-th order in curvature tensors which are acted upon
by covariant derivatives
$\displaystyle\Re^{(m)}_{N}(x)=\mathop{\underbrace{\nabla...\nabla}_{m-2N}}\mathop{\overbrace{\Re(x)...\Re(x)}^{N}},$
(6.2) $\displaystyle{\rm
dim}\;\Re^{(m)}_{N}(x)\equiv\big{[}\,\Re^{(m)}_{N}(x)\,\big{]}=m.$ (6.3)
The curvature monomials enter the action with coupling constants
$\varLambda^{(m)}_{N}$ of the decreasing (with growing $m$) dimensionality
$[\,\varLambda^{(m)}_{N}\,]=d-m,\quad m=0,1,\ldots.$ (6.4)
Summation in (6.1) can run over finite set of terms providing the
renormalizability of the theory, or formally extended to the infinite set in
the framework of generalized RG theory with infinite set of couplings
$\\{\varLambda\\}=\varLambda^{(m)}_{N}$.
Within covariant perturbation theory the full metric is decomposed as a sum of
the flat spacetime metric $\tilde{g}_{\mu\nu}$ and the perturbation
$h_{\mu\nu}$
$g_{\mu\nu}=\tilde{g}_{\mu\nu}+h_{\mu\nu},$ (6.5)
so that each curvature invariant becomes expanded as an infinite series in
powers of $h_{\mu\nu}$ forming a new set of $h$-monomials on the flat space
background
$\displaystyle\int
d^{4}x\,\sqrt{g}\,\Re^{(m)}_{N}=\sum\limits_{M=N}^{\infty}\int
d^{4}x\,\sqrt{\tilde{g}}\,I_{M}^{(m)}(h),$ $\displaystyle
I_{M}^{(m)}(h)\propto\mathop{\underbrace{\tilde{\nabla}...\tilde{\nabla}}_{m}}\mathop{\overbrace{h(x)...h(x)}^{M}}.$
(6.6)
Then in the notations of the covariant perturbation theory the calculation of
the renormalized effective action leads to the same sequence of monomials
acted upon by the operator form factors
$\varGamma_{n}^{(i)}\big{(}\\{\varLambda\\},\tilde{\nabla}_{1},...\tilde{\nabla}_{1}\big{)}$
which make them nonlocal, $\\{\varLambda\\}$ denoting the full set of
couplings (6.4). Within dimensional regularization these renormalized coupling
constants get rescaled by the normalization parameter $\mu$ and expressed in
terms of their dimensionless analogues $\lambda^{(m)}_{N}(\mu)$
$\varLambda^{(m)}_{N}=\mu^{d-m}\lambda^{(m)}_{N}(\mu),$ (6.7)
and the perturbation theory form factors also express as the functions of
dimensionless arguments
$\varGamma_{M}^{(m)}\big{(}\\{\varLambda\\},\tilde{\nabla}_{1},...\tilde{\nabla}_{M}\big{)}=\mu^{d-m}\gamma_{M}^{(m)}\Big{(}\\{\lambda(\mu)\\},\tfrac{\tilde{\nabla}_{1}}{\mu},...\tfrac{\tilde{\nabla}_{M}}{\mu}\Big{)}$
(6.8)
Correspondingly the effective action becomes
$\displaystyle\varGamma[\,g_{\mu\nu}]=\sum\limits_{(m)}\mu^{d-m}\sum\limits_{M=0}^{\infty}\int
d^{d}x\,\sqrt{\tilde{g}}$
$\displaystyle\times\gamma_{M}^{(m)}\big{(}\\{\lambda(\mu)\\},\tfrac{\tilde{\nabla}_{1}}{\mu},...\tfrac{\tilde{\nabla}_{M}}{\mu}\big{)}I_{M}^{(m)}(h_{1},h_{2},...h_{M})\,\big{|}_{\,\\{x\\}=x},$
(6.9)
where $I_{M}^{(m)}(h_{1},h_{2},...h_{M})$ is the analogue of the invariant
(6.6) with split spacetime arguments. A typical assumption of the RG theory
that the renormalized action is independent of the running scale then leads to
the set of equations for $\lambda^{(m)}_{N}(\mu)$ with the beta functions
following from the residues of spacetime dimension poles in the formfactors
$\varGamma_{M}^{(m)}\big{(}\\{\lambda(\mu)\\},\\{\tilde{\nabla}/\mu\\}\big{)}$,
$\mu\frac{d}{d\mu}\varGamma[\,g_{\mu\nu}]=0\to\mu\frac{d}{d\mu}\lambda^{(m)}_{N}(\mu)=\beta^{(m)}_{N}(\mu)\big{(}\\{\lambda(\mu)\\}\big{)}.$
(6.10)
A critical step now consists in the choice of the running scale which could
probe the high energy limit of the theory and embrace a simultaneous scaling
of all formfactors and invariant monomials of (6.9). Then the replacement of
the parameter $\mu$ by this scale will identically bring the effective action
to the form explicitly revealing its UV limit. The choice of this scaling
object can be very different depending on the concrete physical setup. If the
theory has a dimensional scalar field $\phi$ with a nonvanishing and slowly
varying mean value it would be natural to identify RG normalization $\mu$ with
$\phi$. This would lead to the nontrivially “running” in $\phi$ of the
cosmological and Einstein terms, $\varLambda\to\varLambda(\phi)$ and $G\to
G(\phi)$, (amended of course by a gradient expansion series in derivatives of
$\phi$), but of course these terms acquire the interpretation of the Coleman-
Weinberg type potential and nonminimal coupling of $\phi$ to the scalar
curvature.
We, however, are interested in the UV scaling of all derivatives
$\tilde{\nabla}\to\infty$, which in momentum space representation of
scattering amplitudes is conventionally represented by the high energy
Mandelstam invariants or some other combinations of external momenta. In the
coordinate representation of the covariant perturbation theory of [38, 39, 40]
the role of this scale should be played by some operator. So we suggest as a
candidate for this object the following nonlocal operator $\tilde{D}$ which
also formally tends to infinity in the limit of $\tilde{\nabla}\to\infty$ and
in fact embraces a simultaneous scaling of all invariant monomials in (6.9),
$\tilde{D}\equiv\Big{(}-\sum_{N=1}^{\infty}\tilde{\Box}_{N}\Big{)}^{1/2},\quad\tilde{\Box}_{N}\equiv\tilde{g}^{\mu\nu}\tilde{\nabla}_{\mu}\tilde{\nabla}_{\nu}.$
(6.11)
Though being very formal, this operator is well defined in each $N$-th
monomial order because it becomes truncated to the finite sum when acting on
the monomial of $N$ perturbations $h_{1},...h_{N}$, and for $N=0$ it is just
zero because of its action on an independent of $x$ constant,
$\tilde{D}_{N}\equiv\bigg{(}-\sum_{M=1}^{N}\tilde{\Box}_{M}\bigg{)}^{1/2},\quad\tilde{D}_{0}=0.$
(6.12)
In the UV domain $\tilde{\nabla}_{n}\to\infty$, when
$\tilde{\nabla}_{n}/\tilde{D}_{N}=O(1)$, $n\leq N$, the formfactors in each
$N$-th order become after the replacement $\mu\to\tilde{D}$ the functions of a
single operator variable $\tilde{D}_{N}$,
$\displaystyle\mu^{4-m}\gamma_{N}^{(m)}\big{(}\lambda(\mu)\,\big{|}\,\tfrac{\tilde{\nabla}_{1}}{\mu},...\tfrac{\tilde{\nabla}_{N}}{\mu}\big{)}\,\Big{|}_{\,\mu\to\tilde{D}_{N}}\;\to$
$\displaystyle\;(\tilde{D}_{N})^{4-m}\gamma_{N}^{(m)}\big{(}\lambda(\tilde{D}_{N})\,\big{|}\,O(1)\big{)}\equiv(\tilde{D}_{N})^{4-m}\lambda_{N}^{(m)}\big{(}\tilde{D}_{N}\big{)},$
(6.13)
and the expansion of the formally independent of $\mu$ action takes the form
$\displaystyle\varGamma[\,g_{\mu\nu}]\,\Big{|}_{\,\mu\to\tilde{D}}\to\,\sum\limits_{m}\sum\limits_{N=0}^{\infty}\int
d^{4}x\,\sqrt{\tilde{g}}$
$\displaystyle\times\,(\tilde{D}_{N})^{4-m}\lambda_{N}^{(m)}(\tilde{D}_{N})\,I_{N}^{(m)}(h_{1},h_{2},...h_{N})\,\Big{|}_{\,\\{x\\}=x}.$
(6.14)
The next step consists in the recovery of the covariant form of the expansion
in terms of the original spacetime curvature. Curiously, despite the fact that
the covariant perturbation theory of [38, 39, 40] is rather often being
referred to in literature, subtle details of this step are usually disregarded
which leads to confusing statements on the ambiguity of this procedure,
dependence on the gauge by which the metric perturbation $h_{\mu\nu}$ is
related to the curvature [33], etc. At the same time, this procedure is
unique, provided that one does not treat $\tilde{g}_{\mu\nu}$ and
$\tilde{\nabla}_{\mu}$ as Cartesian $\delta_{\mu\nu}$ and $\partial_{\mu}$,
but rather proceeds in generic coordinate system and uses the only invariant
statements that the curvature of the tilded metric is vanishing
$\tilde{R}^{\alpha}_{\;\;\beta\mu\nu}=0$. This is the covariant equation for
$\tilde{g}_{\mu\nu}$ in terms of the curved metric $g_{\mu\nu}$ and its
curvature $R^{\alpha}_{\;\;\beta\mu\nu}$, whose solution exists as
perturbation expansion in $R^{\alpha}_{\;\;\beta\mu\nu}$ and also requires
imposing the gauge [38, 39]. But the result of substituting this solution back
into manifestly noncovariant (double field) series (6.6) is gauge independent
because of the implicit invariance of the left hand side of (6.6).
In the convenient DeWitt type gauge
$\tilde{\nabla}^{\nu}h_{\mu\nu}-\tfrac{1}{2}\nabla_{\mu}h=O[\,h^{2}]$,
$h\equiv\tilde{g}^{\alpha\beta}h_{\alpha\beta}$, the solution for $h_{\mu\nu}$
and $\tilde{\nabla}_{\mu}$ in terms of $g_{\mu\nu}$ and $\nabla_{\mu}$ reads
in the lowest order as [38, 39]
$h_{\mu\nu}=-\frac{2}{\Box}R_{\mu\nu}+O[\,\Re^{2}],\quad\tilde{\nabla}_{\mu}=\nabla_{\mu}+O[\,\Re\,].$
(6.15)
Using this in (6.14) we get the replacement of $h$-monomials by the covariant
curvature monomials along with the replacement of $\tilde{D}_{N}$ by $D_{N}$,
$\displaystyle I_{N}^{(m)}(h_{1},h_{2},...h_{N})\to$
$\displaystyle\qquad\quad\frac{1}{\Box_{1}...\Box_{N}}\Re^{(m+2N)}_{N}(x_{1},...x_{N})+O[\,\Re^{N+1}],$
(6.16) $\displaystyle\tilde{D}_{N}\to D_{N}+O[\,\Re\,],$ (6.17)
where $D_{N}$ is obviously defined by (6.12) in terms of full-fledged
covariant d’Alembertians $\Box=g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}$, and we
reabsorb the coefficient $(-2)^{n}$ into the symbolic definition of the $N$-th
order covariant monomial – the analogue of the local $\Re^{(m)}_{N}(x)$, see
Eq. (6.2), with split $N$ spacetime arguments
$\Re^{(m)}_{N}(x_{1},...x_{N})=\mathop{\underbrace{\nabla...\nabla}_{m-2N}}\Re(x_{1})...\Re(x_{N}),\,N\geq
1.$ (6.18)
For $N=0$ this monomial can be defined as an irrelevant constant bringing no
contribution in the UV limit.
Thus the UV limit of the effective action takes the form
$\displaystyle\varGamma[\,g_{\mu\nu}]\to\int
d^{4}x\,\sqrt{g}\sum\limits_{m,N\geq
0}^{\infty}\\!\frac{\lambda_{N}^{(m)}(D_{N})(D_{N})^{4-m}}{\Box_{1}...\Box_{N}}$
$\displaystyle\qquad\qquad\qquad\times\,\Re^{(m+2N)}_{N}(x_{1},\cdots
x_{N})\,\Big{|}_{\,\\{x\\}=x},$ (6.19)
where we remind that the dimensionless formfactors $\lambda_{N}^{(m)}(D_{N})$
follow from the running RG couplings of the theory $\lambda_{N}^{(m)}(\mu)$ by
the replacement of $\mu$ with the operator $D_{N}$.
Let us consider application of this result to the cosmological constant sector
involving the metric invariants of dimensionality $m=0$ and
$\varLambda^{(4)}_{0}=\varLambda/16\pi G$. This classical cosmological term
gives rise to the infinite set of zero dimension invariants
$\displaystyle\int d^{4}x\,\sqrt{g}=\sum\limits_{n=0}^{\infty}\int
d^{4}x\,\sqrt{\tilde{g}}\,I_{n}^{(0)}(\tilde{g},h),$ (6.20)
$\displaystyle\;\;I_{0}^{(0)}(\tilde{g},h)=1,\;\;I_{1}^{(0)}(\tilde{g},h)=-\tfrac{1}{2}h,$
$\displaystyle\;\;I_{2}^{(0)}(\tilde{g},h)=\tfrac{1}{4}h^{2}-\tfrac{1}{2}h_{\mu\nu}^{2},\ldots$
(6.21)
(indices are contracted by the flat metric and
$h=\tilde{g}^{\mu\nu}h_{\mu\nu}$), whereas at the quantum level they generate
the sequence of high energy $m=0$ structures of (6.19)
$\int\\!d^{4}x\sqrt{g}\sum\limits_{N=2}^{\infty}\lambda_{N}^{(0)}(D_{N})\frac{(D_{N})^{4}}{\Box_{1}...\Box_{N}}\,\Re^{(2N)}_{N}(x_{1},...\,x_{N})\Big{|}_{\\{x\\}=x},$
(6.22)
where the zeroth order term is zero in view of $D_{0}=0$ (see Eq.(6.11) and
the first order term is also absent due to its tadpole (total derivative)
nature – remember that $D_{1}=(-\Box_{1})^{1/2}$ and
$D_{1}^{4}/\Box_{1}=\Box_{1}$ is acting on
$\Re^{(2)}_{1}(x_{1})$.111111Important caveat is necessary here concerning the
annihilation of the total derivative terms. The surface terms at infinity
should be vanishing, which is equivalent to a good IR behavior of the nonlocal
form factor $\lambda_{1}^{(0)}(D_{1})$ at $\Box\to 0$. We will assume this
property basing on the maximum logarithmic singularity of
$\lambda_{1}^{(0)}(D_{1})$ which is a function of $\log(-\Box)$ solving the RG
equation. The same also applies to integrations by parts considered in what
follows. Otherwise, the procedure of subtracting the boundary terms, like the
Gibbons-Hawking surface action at asymptotically flat infinity, will be
needed, which we briefly discuss below.
The expansion starts at $N=2$ with the term which has the following structure
$\displaystyle 4\sum\int
d^{4}x\,\sqrt{g}\,\Re^{(2)}(x)\,\lambda_{2}^{(0)}\big{(}\sqrt{-2\Box}\big{)}\,\Re^{(2)}(x)$
$\displaystyle\;\;=\int
d^{4}x\sqrt{g}\Big{(}R_{\mu\nu}F_{1}(\Box)R^{\mu\nu}+RF_{2}(\Box)R\Big{)}\\!+\\!O[\,\Re^{3}].$
(6.23)
Here we took into account that the set of invariants
$\Re^{(4)}_{2}(x_{1},x_{2})$ can be represented as a sum of terms factored out
into the products of Ricci tensors and Ricci scalars with some
coefficients121212Bilinear in Riemann curvature terms under the integration
sign also reduce to bilinear combinations of $R_{\mu\nu}$ and $R$ by using the
expression for Riemann tensor in terms of the Ricci one [38, 39], see footnote
5. $a$ and $b$,
$\Re^{(4)}_{2}(x_{1},x_{2})=aR_{\mu\nu}(x_{1})\,R^{\mu\nu}(x_{2})+bR(x_{1})R(x_{2}),$
(6.24)
and also used an obvious corollary of integration by parts
$\displaystyle\int
d^{4}x\sqrt{g}\,F(\Box_{1},\Box_{2})\Re(x_{1})\Re(x_{2})\,\Big{|}_{\,\\{x\\}=x}$
$\displaystyle\qquad\qquad=\int d^{4}x\sqrt{g}\,\Re(x)F(\Box,\Box)\Re(x).$
(6.25)
Remarkable feature of the expression (6.23) is that the power-law operator
factors in $(D_{N})^{4}/\Box_{1}...\Box_{N}$ at $N=2$ completely cancelled out
to give the dimensionless formfactors $F_{1}(\Box)$ and $F_{2}(\Box)$ which
originate as linear combinations of relevant running
$\lambda_{2}^{(0)}\big{(}\sqrt{-2\Box}\big{)}$ obtained by solving the RG
equation. Even more remarkable is the fact that this is a nonlocal term which
is quadratic in the curvature even though it has originated from the sector of
cosmological term expanded in the series of zero dimension invariants. This is
what can be called as metamorphosis to high-energy partners of the
cosmological constant suggested by J.Donoghue in [32]. Their structure is a
direct corollary of the dimensionality arguments within RG approach. The
arising form factors of the curvature squared terms are the descendants of RG
running couplings of the zero dimension invariants which participate in the
decomposition of the cosmological constant term.
In fact, the same structure (6.23) gets reproduced for the contribution of any
dimension $m$ in the expansion (6.19). For even dimensionality131313For the
set of 2-dimensional curvatures $\Re$ only even dimensions $m$ enter the
expansion (6.19), but this can always be generalized to the case of odd-
dimensional “curvatures”, like for example the extrinsic curvature in Hořava
gravity models., $m\to 2m$, this can be easily demonstrated by decomposing any
$(2m+4)$-dimensional quadratic invariant as this was done above
$\Re^{(2m+4)}_{2}(x_{1},x_{2})=\\!\\!\\!\\!\sum_{m_{1}+m_{2}=2m}\\!\\!\\!\\!\Re^{(m_{1}+2)}_{1}(x_{1})\Re^{(m_{2}+2)}_{1}(x_{2}).$
(6.26)
Using this in (6.19) one has complete cancellation of the dimensional factor
$(D_{2})^{4-2m}/\Box^{2}\sim\Box^{-m}$ in the expression
$\displaystyle\int
d^{4}x\,\sqrt{g}\\!\sum\limits_{m_{1}+m_{2}=2m}\Re^{(m_{1}+2)}_{1}(x)$
$\displaystyle\qquad\times\frac{\lambda_{2}^{(2m)}(D_{2})(D_{2})^{4-m_{1}-m_{2}}}{\Box^{2}}\,\Re^{(m_{2}+2)}_{1}(x)$
$\displaystyle=\\!\int
d^{4}x\sqrt{g}\Big{(}R_{\mu\nu}F_{1}(\Box)R^{\mu\nu}+RF_{2}(\Box)R\Big{)}\\!+O[\,\Re^{3}].$
(6.27)
Noting that with
$\Re^{(m+2)}_{1}=\mathop{\overbrace{\nabla\cdots\nabla}^{m}}\Re^{(2)}_{1}$
this follows from integration by parts and the use of various corollaries of
contracted Bianchi identity
($\nabla^{\nu}R_{\mu\nu}=\tfrac{1}{2}\nabla_{\mu}R$, etc.),
$\displaystyle\int
d^{4}x\,\sqrt{g}\\!\sum\limits_{m_{1}+m_{2}=2m}\mathop{\underbrace{\nabla\cdots\nabla}_{m_{1}}}\Re(x)F(\Box)\mathop{\underbrace{\nabla\cdots\nabla}_{m_{2}}}\Re(x)$
$\displaystyle=\int
d^{4}x\,\sqrt{g}\Big{(}R_{\mu\nu}\,\Box^{m}F_{1}(\Box)R^{\mu\nu}+R\,\Box^{m}F_{2}(\Box)R\Big{)}$
$\displaystyle\qquad+O[\,\Re^{3}].$ (6.28)
Here the operators $F_{1}(\Box)$ and $F_{2}(\Box)$ have the same dimension as
$F(\Box)$ and originate from $F(\Box)$ by the algebra of contracting the
indices of covariant derivatives. Using this relation in the left hand side of
(6.27) one gets the right hand side with completely cancelled powers of
$\Box$.
Thus, Eq.(6.27) with $m=2$ implies the conversion of the gravitational
coupling constant into the dimensionless formfactors of the Einstein term
partners. These partners have the same structure as the cosmological term
partners quadratic in curvatures. This is again the metamorphosis of RG
running of the form $1/16\pi G(\mu)=\mu^{2}\lambda^{(2)}_{2}(\mu)\to
F_{1,2}(\Box)$.
Note that all this takes place in the UV limit where all curvatures in their
monomials are rapidly varying in spacetime with their derivatives
$\nabla\to\infty$. At intermediate energies, when the mass scale $M$ surfaces
up, the scaling (6.13) ceases to make sense and roughly should be replaced
with $D\sim M$, and instead of (6.23) one gets exactly the cosmological
constant partners of Donoghue [32] which have the structure of
$M^{4}\\!\int d^{4}x\,\sqrt{g}\Big{(}R_{\mu\nu}\frac{F_{1}^{\rm
part}(\Box)}{\Box^{2}}R^{\mu\nu}+R\,\frac{F_{2}^{\rm
part}(\Box)}{\Box^{2}}R\Big{)}.$ (6.29)
The dimensionless form factors $F^{\rm part}_{1,2}(\Box)$ here are
accumulating loop corrections with nonlocal logarithmic structures of the form
$F^{\rm part}(\Box)\sim\ln\frac{M^{2}-\Box}{M^{2}}.$ (6.30)
Note that these partners are still in high-energy domain $-\Box\geq M^{2}$,
but they are subdominant as compared to the leading contribution (6.23) with
dimensionless form factors which incorporate the logarithmically running
solutions of RG equations. This is because the partners (6.29) are suppressed
by power law factors $M^{4}/\Box^{2}$. Exact form of these formfactors at
intermediate scales was derived at one-loop order in [33] for rather generic
theory of massive fields by using the heat kernel technique of [38, 39]. In IR
domain $-\Box\ll M^{2}$ they are of course expandable in local gradient series
reflecting the decoupling phenomenon [34, 35, 33].
Similarly, the gravitational constant partner in IR reads as
$M^{2}\int
d^{4}x\,\sqrt{g}\Big{(}R_{\mu\nu}\frac{F_{1}(\Box)}{\Box}R^{\mu\nu}+R\,\frac{F_{2}(\Box)}{\Box}R\Big{)},$
(6.31)
which reminds the construction of the nonlocal action for long-distance
modifications of gravity theory in [51, 52]. This differs from the
cosmological constant partner by another powers of $M$ and the power of $\Box$
in the denominator.
One should be more careful at this point – while the case of (6.31) is well
defined in asymptotically flat spacetime, the cosmological constant partner
(6.29) is IR divergent for the reasons discussed above. The action of
$\tfrac{1}{\Box^{2}}$ is not well defined in four dimensions (or,
equivalently, $\int d^{4}x\sqrt{g}(\tfrac{1}{\Box}\Re)^{2}$ is IR divergent),
so that the perturbation expansion in the dimension zero sector should be
critically reconsidered. To trace the origin of this difficulty note that the
first three terms of the cosmological term expansion (6.20) are divergent,
whereas a similar expansion for the Einstein term becomes well defined only
after the subtraction of the Gibbons-Hawking surface term
$\int_{\infty}d^{3}\sigma^{\mu}\big{(}\partial_{\mu}h-\partial^{\nu}h_{\mu\nu}\big{)}$
at the infinity of asymptotically flat spacetime. Due to this subtraction we
can write for the integral of the invariant $\Re^{(2)}_{1}(x)=-R(x)$, weighted
in the Einstein action by $\varLambda^{(2)}_{1}=1/16\pi G$, a legitimate
expansion (6.6) starting with the quadratic order in $h_{\mu\nu}$,
$\displaystyle\int
d^{4}x\sqrt{g}(-R)-\int_{\infty}d^{3}\sigma^{\mu}\big{(}\partial_{\mu}h-\partial^{\nu}h_{\mu\nu}\big{)}$
$\displaystyle\qquad\qquad\quad\quad\;=\sum\limits_{M=2}^{\infty}\int
d^{4}x\sqrt{\tilde{g}}\,I_{M}^{(2)}(\tilde{g},h),$ (6.32) $\displaystyle
I_{2}^{(2)}(\tilde{g},h)=\\!-\tfrac{1}{4}h_{\mu\nu}\tilde{\Box}h^{\mu\nu}\\!+\tfrac{1}{8}h\tilde{\Box}h-\tfrac{1}{2}\big{(}\tilde{\nabla}^{\nu}h_{\mu\nu}\\!-\tfrac{1}{2}\tilde{\nabla}_{\mu}\tilde{h}\big{)}^{2}.$
(6.33)
Then the above calculational strategy leads to the effective action (6.31)
whose tree level IR limit should match low energy physics with the Planck mass
cutoff $M^{2}$ and the form factors $F_{1}(0)=1$ and $F_{2}(0)=1/2$. This tree
level answer up to $\Re^{3}$-corrections directly corresponds to the above
expression for $I_{2}^{(2)}(\tilde{g},h)$ with $h_{\mu\nu}$ given by Eq.
(6.15) in terms of the curved space metric $g_{\mu\nu}$ [51, 52].
To the best of our knowledge, no such subtraction is known for cosmological
term expansion (6.20), so that its rigorous treatment is still to be done. It
is interesting if new structures can be generated by the regularization of
this IR behavior. Apparently, this should be based on the analogue of the
Graham-Fefferman construction for asymptotically AdS spaces [75, 76] and
deserves further studies.
In any case, the UV behavior of both cosmological and gravitational constant
partners, which should be not sensitive to IR problems, is determined by
curvature squared terms (6.23) with running dimensionless “couplings”. Their
|
# Vortex Fermi Liquid and Strongly Correlated Quantum Bad Metal
Nayan Myerson-Jain Department of Physics, University of California, Santa
Barbara, CA 93106 Chao-Ming Jian Department of Physics, Cornell University,
Ithaca, New York 14853, USA Cenke Xu Department of Physics, University of
California, Santa Barbara, CA 93106
###### Abstract
The semiclassical description of two-dimensional ($2d$) metals based on the
quasiparticle picture suggests that there is a universal threshold of the
resistivity: the resistivity of a $2d$ metal is bounded by the so called Mott-
Ioffe-Regal (MIR) limit, which is at the order of $h/e^{2}$. If a system
remains metallic while its resistivity is beyond the MIR limit, it is referred
to as a “bad metal”, which challenges our theoretical understanding as the
very notion of quasiparticles is invalidated. The description of the system
becomes even more challenging when there is also strong correlation between
the electrons. Partly motivated by the recent experiment on transition metal
dichalcogenides moiré heterostructure, we seek for understanding of strongly
correlated bad metals whose resistivity far exceeds the MIR limit. For some
strongly correlated bad metals, though a microscopic description based on
electron quasiparticles fails, a tractable dual description based on the
“vortex of charge” is still possible. We construct a concrete example of such
strongly correlated bad metals where vortices are fermions with a Fermi
surface, and we demonstrate that its resistivity can be exceptionally large at
zero temperature. And when extra charge $\delta n_{e}$ is doped into the
system away from half-filling, a small Drude weight proportional to $(\delta
n_{e})^{2}$ will emerge in the optical conductivity .
## I Introduction
The most rudimentary description of a metal relies on the notion of
quasiparticles, $i.e.$ an electron near the Fermi surface can be well
approximated as a wave packet between two consecutive elastic scatterings with
impurities. This picture requires that $l_{\mathrm{m}}k_{F}>1$, where
$l_{\mathrm{m}}$ is the mean free path of the electrons from scattering with
the impurities Emery and Kivelson (1995). When $l_{\mathrm{m}}k_{F}\sim 1$,
the resistivity of a two dimensional system is of the order of $h/e^{2}$,
which is also known as the Mott-Ioffe-Regal (MIR) limit. The common wisdom is
that, for noninteracting electrons, when the resistivity of a $2d$ metal
exceeds the MIR limit, not only would the rudimentary description of the
system fail, the system would actually become an insulator due to the Anderson
localization. The potential metal-insulator transition (MIT) of a
noninteracting $2d$ electron system Evers and Mirlin (2008) (within certain
symmetry class such as the symplectic) should happen when the resistivity is
of order $h/e^{2}$.
In strongly interacting electron systems, the universal threshold of
resistivity $h/e^{2}$ still appears to hold. For electrons at half-filling (on
average one electron per site) on a lattice, the competition between the
interaction and kinetic energy can lead to an interaction-driven MIT between a
metal and a Mott insulator phase. When the insulator is a particular type of
spin liquid phase, this MIT can be understood through a parton construction
Lee and Lee (2005); Senthil (2008), and the total resistivity follows the
Ioffe-Larkin rule Ioffe and Larkin (1989) $\rho=\rho_{b}+\rho_{f}$, where
$\rho_{b}$ and $\rho_{f}$ are resistivity from the bosonic and fermionic
partons respectively. $\rho_{f}$ is a smooth function across the MIT, and at
low temperature $\rho_{f}$ mostly arises from disorder, which is expected to
be small if we assume weak disorder. Hence at the MIT, the critical
resistivity is mostly dominated by the bosonic parton $\rho_{b}$. The critical
resistivity $\rho_{b}$ is expected to be of order $h/e^{2}$ (though in the DC
limit $\rho_{b}$ may acquire an extra factor of $7\sim 8$, based on analytical
evaluation in certain theoretical limit Witczak-Krempa et al. (2012)) 111Here
we also note that a weak disorder, or Umklapp process is needed to remove the
logarithmic divergence of conductivity due to thermal fluctuation, which is
predicted by hydrodynamics Delacrétaz (2020); Kovtun (2012)..
Hence in most noninteracting as well as strongly interacting systems that we
have understood, the resistivity of a $2d$ metallic system should be roughly
bounded by the MIR limit. Hence if a $2d$ system remains metallic while its
resistivity far exceeds the MIR limit, it challenges our theoretical
understanding. These exotic metals are referred to as “bad metals” Emery and
Kivelson (1995). The recent experiment on transition metal dichalcogenides
(TMD) revealed the existence of a novel interaction-driven MIT Li et al.
(2021), where the universal MIR limit is violated: the DC critical resistivity
at the MIT exceeds the MIR limit by nearly two orders of magnitude. The system
is supposedly modelled by an extended Hubbard model of spin-1/2 electrons on a
triangular moiré lattice Wu et al. (2018); Tang et al. (2019), but the
experimental finding is qualitatively beyond the previous theory of MIT. A few
recent theoretical proposals Xu et al. (2022); Kim et al. (2022) were made in
order to understand this exotic MIT. The experiment mentioned above only
revealed a critical point whose resistivity is clearly beyond the MIR limit.
Given the current experimental finding and the strongly interacting nature of
the system, it is natural to ask, can there also be a stable bad metal phase
of strongly correlated electrons, whose properties can be evaluated in a
tractable way?
In this work we discuss the construction of a quantum bad metal state with
longitudinal transport only; the electrical resistivity $\rho_{e}$ of the
state can far exceed the MIR limit even with weak disorder, at zero and low
temperature. It is worth noting that the phenomenology of the state we
construct is different from the original example of “bad metal” (hole doped
cuprates) discussed in Ref. Emery and Kivelson, 1995, where the resistivity
increases with temperature monotonically and exceeds the MIR limit at high
temperature; while the resistivity of our “quantum bad metal” remains finite
and large at zero temperature, and clearly violates the MIR bound. Our
construction is formulated through the dual degrees of freedom of “charge
vortex”. The particle-vortex duality has a long history Peskin (1978);
Dasgupta and Halperin (1981); Fisher and Lee (1989). This duality was
originally discussed for bosons, but recent developments have generalized the
duality to fermion-vortex duality Son (2015); Wang and Senthil (2015a);
Metlitski and Vishwanath (2016); Mross et al. (2016); Kachru et al. (2017), as
well as Chern-Simons matter theory to free Dirac or Majorana fermion duality
Aharony (2016); Seiberg et al. (2016); Hsin and Seiberg (2016); Aharony et al.
(2017); Metlitski et al. (2017); Chen et al. (2018); Son et al. (2019); Jian
et al. (2019). And since the particle-vortex duality is still a “strong-weak”
duality, when the charges are strongly correlated which invalidates a
perturbative description based on quasiparticles, the vortices are weakly
interacting through the dual gauge field, which facilitates a rudimentary
description.
Hence one way to construct a quantum bad metal for charges is to drive the
vortices into a good metal. The vortices can naturally form a good metal as
long as (1) the vortex is a fermion, and (2) the fermionic vortices form a
Fermi surface with a finite density of states. In the next section we will
discuss how exactly a vortex becomes a fermion in our construction, and how to
derive the charge responses of the system from the physics of vortices. We
would like to clarify that we are not the first to investigate correlated
electrons as vortex liquid. Besides the more well-known interpretation of
composite fermions as “vortex liquid” in the context of half-filled Landau
level (and similarly for charged bosons at filling 1) Halperin et al. (1993);
Read (1998); Son (2015); Wang and Senthil (2016a, b); a metallic phase with
anomalously large conductivity that emerges in amorphous thin film also
motivated discussions of exotic physics of superconductor vortices Galitski et
al. (2005); Wu and Phillips (2006); Kapitulnik et al. (2019). We will compare
our construction with the previous works.
## II construction of the quantum bad metal
### II.1 General considerations
Before we detail our construction, some general considerations can already be
made.
(1) As was pointed out in previous literatures, at least for charged bosons,
the product of the conductivity of the charges and the conductivity of the
vortices is a constant Fisher et al. (1990); Gazit (2015), $i.e.$
$\sigma_{e}\sim 1/\sigma_{v}$. If this relation still (at least approximately)
holds in our construction, it implies that if the vortex conductivity
$\sigma_{v}$ follows the standard behavior of a good metal at finite
temperature, then the resistivity $\rho_{e}(T)$ of charge should decrease with
$T$, at least below certain characteristic energy scale.
(2) A charge vortex can generally be viewed as a point defect with circulating
vorticity of charge current. A charge vortex must become an anti-vortex under
spatial reflection $\mathcal{P}$. This is because the electric current
circulation will reverse its orientation under reflection. If the vortices
form a Fermi surface, in general it would break $\mathcal{P}$, as a Fermi
surface usually is not invariant under the particle-hole transformation. The
same observation can be made for time-reversal $\mathcal{T}$: since charge
density is invariant under $\mathcal{T}$, time-reversal would reverse the
direction of electric current circulation. We will discuss later how to
preserve $\mathcal{P}$ and $\mathcal{T}$ in our construction, by enforcing
certain particle-hole symmetry of the fermionic vortices.
(3) As was pointed out in Ref. Wang and Senthil, 2016a, b, the Wiedemann-Franz
law $\kappa\sim T\sigma$ should generally be strongly violated in a vortex
liquid, as the vortices carry entropy, but no charge. In the state we
construct this is still true, the modified Wiedmann-Franz law should be
$\kappa\sim LT\sigma_{v}\sim LT\rho_{e}$. The Lorenz number $L$ is about
$\kappa/(T\sigma_{e})\sim\rho_{e}^{2}$, which can be exceedingly larger than
an ordinary metal. Here we remind the readers that our state has longitudinal
transport only, and it can have both large resistivity and thermal
conductivity.
(4) For a strongly interacting electron system, the relaxation of the electric
current is pretty much independent from the relaxation of a single particle.
Hence the physics of a strongly interacting electron liquid may be only
captured by some hydrodynamical description without microscopic particles
Delacrétaz (2020); Kovtun (2012); Hartnoll (2014); Hartnoll and Mackenzie
(2021); Lucas and Fong (2018); Hartnoll et al. (2016), as hydrodynamics is
defined at a much larger length scale. But since the particle-vortex duality
is a strong-weak duality, the interaction between electron density becomes
$\displaystyle\sum_{i,j}V_{i,j}n_{i}n_{j}\sim\int d^{2}x\
\frac{1}{g}(\vec{\nabla}\times\vec{a})^{2}$ (1)
in the dual picture, where $g$ can be viewed as the gauge coupling of the
gauge charges (vortices), and also the charge compressibility $\kappa_{e}$.
The stronger the charge interaction is, the weaker is the bare gauge coupling
of the vortices. The common “patch theory” for analyzing the RG flow of a
Fermi surface coupled with a U(1) gauge field predicts that the gauge coupling
would flow to a strongly coupled fixed point eventually Nayak and Wilczek
(1994a, b); Lee (2009); Metlitski and Sachdev (2010); Mross et al. (2010). But
this patch theory breaks down when there is disorder, as disorder would mix
different patches in the momentum space. But at least when the bare gauge
coupling $g$ is weak enough (which corresponds to a strong charge density-
density interaction), there should be a sufficient window for the gauge
coupling to be viewed as a perturbation, and the momentum of the vortices can
be transferred to the photons, and then relax through disorder before “feeding
back” to the vortices. Hence in this sense we can view the dual vortex system
as an approximate vortex Fermi liquid.
Figure 1: The resistivity $\rho(T)$ as a function of temperature at low
temperature, computed using the composition formula Eq. 17. We have chosen
$\sigma_{b}=\exp(-\Delta_{b}/T)$ where $\Delta_{b}=1$, and $\sigma_{f}\sim
200$. $\rho$ is measured in unit of $h/e^{2}$ in this plot. We can also give
$\sigma_{f}$ a Fermi-liquid like temperature dependence, the plot remains
qualitatively unchanged.
### II.2 Quantum Bad metal at half-filling on a lattice
The system we begin with is a strongly interacting electron system with half-
filling (one electron per site on average) on a lattice, later we will discuss
what happens when the system is doped away from half-filling. We start with
the standard “$\mathrm{SU}(2)$ slave rotor” theory for the electron operator
Affleck et al. (1988); Dagotto et al. (1988); Wen and Lee (1996); Lee et al.
(1998):
$\displaystyle c_{\uparrow}$ $\displaystyle=$ $\displaystyle
f_{\uparrow}z_{1}-f^{\dagger}_{\downarrow}z^{\dagger}_{2},$ (2) $\displaystyle
c_{\downarrow}$ $\displaystyle=$ $\displaystyle
f_{\downarrow}z_{1}+f^{\dagger}_{\uparrow}z^{\dagger}_{2}.$ (4)
Here $(f_{\uparrow},f_{\downarrow})$ is a fermionic spinon doublet (fermionic
partons) that carries spin-1/2, and $(z_{1},z_{2})$ are slave rotors (bosonic
partons) carrying the electric charge. This formalism can maximally host a
SU(2) charge transformation and a SU(2) gauge transformation, and both
transformations can be made explicit by rewriting Eq. 4 in a matrix form (see
$e.g.$ Ref. Hermele, 2007; Ran et al., 2008 and references therein) 222An even
more complete parton construction can accommodate an SO(4) gauge symmetry Xu
and Sachdev (2010). But for our purpose it suffices to assume all SU(2)
transformations, including the spin symmetry are broken down to U(1). In fact,
on a lattice with frustration, both the charge SU(2) and the gauge SU(2)
transformations are broken down to U(1) by the most natural mean field states
of $z_{\alpha}$ and $f_{\alpha}$. The assignment of the electric charge
symmetry $\mathrm{U}(1)_{e}$, gauge symmetry $\mathrm{U}(1)_{g}$ and spin
symmetry $\mathrm{U}(1)_{s}$ on the partons is
$\displaystyle\mathrm{U}(1)_{e}$ $\displaystyle:$ $\displaystyle
z_{1}\rightarrow e^{\mathrm{i}\theta_{e}}z_{1},\ \ \ z_{2}\rightarrow
e^{-\mathrm{i}\theta_{e}}z_{2},\ \ \ f_{\alpha}\rightarrow f_{\alpha};$ (5)
$\displaystyle\mathrm{U}(1)_{g}$ $\displaystyle:$ $\displaystyle
z_{1}\rightarrow e^{\mathrm{i}\theta_{g}}z_{1},\ \ \ z_{2}\rightarrow
e^{\mathrm{i}\theta_{g}}z_{2},\ \ \ f_{\alpha}\rightarrow
e^{-\mathrm{i}\theta_{g}}f_{\alpha},$ (7) $\displaystyle\mathrm{U}(1)_{s}$
$\displaystyle:$ $\displaystyle z_{\alpha}\rightarrow z_{\alpha},\ \ \
f_{\uparrow}\rightarrow e^{\mathrm{i}\theta_{s}}f_{\uparrow},\ \ \
f_{\downarrow}\rightarrow e^{-\mathrm{i}\theta_{s}}f_{\downarrow}.$ (9)
The system being at half-filling implies that the total rotor number of
$z_{1}$ and $z_{2}$ are equal: $\sum_{i}n_{1,i}=\sum_{i}n_{2,i}$. There is a
dynamical $\mathrm{U}(1)_{g}$ gauge field $a_{\mu}$ that couples to both
$z_{\alpha}$ and $f_{\alpha}$. The U(1) gauge constraint demands that on every
site $i$,
$\sum_{\alpha}n_{\alpha,i}+1=\sum_{\alpha}f^{\dagger}_{\alpha,i}f_{\alpha,i}$.
A detailed discussion of the physical meaning of the partons introduced can be
found in Ref. Ran et al., 2008.
This slave rotor parton construction allows us to construct many states of the
strongly interacting electron system which are difficult to visualize using
free or weakly interacting electrons. For example, if the bosonic partons
$z_{\alpha}$ are in a trivial bosonic Mott insulator state with
$n_{1,i}=n_{2,i}=0$ (meaning
$\sum_{\alpha}f^{\dagger}_{\alpha,i}f_{\alpha,i}=1$ on each site), the system
becomes a Mott insulator of electrons with a charge gap, and the spin physics
of the Mott insulator depends on the state of $f_{\alpha}$. Various spin
liquid states can be designed and classified depending on the mean field band
structure of $f_{\alpha}$ Wen (2002).
The many-body state of electrons is determined by the states of the partons.
In the last decade the study of symmetry protected topological states (SPT)
significantly broadened our understanding of the states of matter Chen et al.
(2013, 2012), which also allows us to construct even more novel states of
electrons using the partons. We first use $z_{\alpha}$ to define two other
composite bosonic fields: $\phi_{e}=z^{\dagger}_{1}z_{2}$,
$\phi_{g}=z_{1}z_{2}$. $\phi_{e}$ and $\phi_{g}$ carry charge $(2,0)$ and
$(0,2)$ under $\left(\mathrm{U}(1)_{e},\mathrm{U}(1)_{g}\right)$. Then we
drive the composite fields $\phi_{e}$ and $\phi_{g}$ into a bosonic SPT state
with $\mathrm{U}(1)_{e}$ and $\mathrm{U}(1)_{g}$ symmetries, which is the bSPT
state for two flavors of bosons constructed in Ref. Levin and Senthil, 2004.
The physics of this bSPT state is analogous to the quantum spin Hall
insulator: the vortex of $\phi_{e}$ carries charge of $\phi_{g}$, and vice
versa. If we follow the Chern-Simons description of the bSPT Lu and Vishwanath
(2012), this state is
$\displaystyle\mathcal{L}_{\mathrm{bSPT}}=\frac{\mathrm{i}K^{IJ}}{2\pi}\tilde{a}_{I}\wedge
d\tilde{a}_{J}+\frac{\mathrm{i}2}{2\pi}\tilde{a}_{1}\wedge
da+\frac{\mathrm{i}2}{2\pi}\tilde{a}_{2}\wedge dA^{e},$ (10)
where $K^{IJ}$ takes the same form as the Pauli matrix $\sigma^{x}$. Here
$\ast d\tilde{a}_{1}$ and $\ast d\tilde{a}_{2}$ are the dual of the currents
of $\phi_{g}$ and $\phi_{e}$ respectively. This bSPT state of the rotors also
has a particle-hole symmetry of the bosonic rotors, and in this bSPT state the
expectation value of the total rotor number of both $z_{1}$ and $z_{2}$ is
zero. Please note that the total rotor number being zero does not imply a
trivial vacuum state, as the rotor number (just like a spin $S^{z}$ operator)
can take both positive and negative values.
A bSPT state is gapped, and also nondegenerate, hence it is safe to integrate
out the bosonic degree of freedom, and obtain the response to the gauge
fields. After integrating out $\tilde{a}_{1,2}$ from Eq. 10, we obtain:
$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{F}(f_{\alpha},a_{\mu})+\frac{4\mathrm{i}}{2\pi}a\wedge
dA^{e}+\frac{1}{g}(\vec{\nabla}\times\vec{a})^{2}+\cdots.$ (11)
One can also introduce the external gauge field for the spin symmetry
$\mathrm{U}(1)_{s}$, but it won’t have a nontrivial response from the bSPT
state.
The mutual Chern-Simons term in the last term of Eq. 11 fundamentally changes
the physics of the system in the following way:
The electric charge current, which is defined as
$J^{e}=\delta\mathcal{L}/\delta A_{e}$, is identified as
$\displaystyle J^{e}=e\frac{4}{2\pi}\ast da,$ (12)
meaning the flux of $a_{\mu}$ now carries electric charge $4e$. Hence the bSPT
state so constructed turns the gauge field $a_{\mu}$ into the dual of the
charge current in the sense of the particle-vortex duality Peskin (1978);
Dasgupta and Halperin (1981); Fisher and Lee (1989), and turns the gauge
charge of $a_{\mu}$ into the charge vortex. The fermionic parton $f_{\alpha}$,
which carries gauge charge $1$ under $a_{\mu}$, now automatically becomes the
vortex of the electric charge, as when a charge (now the flux of $a_{\mu}$)
circulates the gauge charged $f_{\alpha}$, it would accumulate a Berry’s
phase. It is worth noting that, for more general bSPT states of $z_{\alpha}$
with only mutual $\mathrm{U}(1)_{e}\times\mathrm{U}(1)_{g}$ Chern-Simons
response, the electric charge carried by the flux of $a_{\mu}$ has to be an
integer multiple of $4e$. Hence, the bSPT described above is the minimal non-
trivial choice.
Now we take the long wavelength limit, and integrate out both $f_{\alpha}$ and
$a_{\mu}$; we also choose the temporal gauge with $a_{\tau}=0$. The response
Lagrangian in terms of $A^{e}$ is
$\displaystyle\mathcal{L}_{\mathrm{res}}=\sum_{\omega}\frac{1}{2}\frac{4}{\pi^{2}}\frac{\omega^{2}}{\Pi_{f}(\omega)}|\vec{A}^{e,t}(\omega)|^{2}.$
(13)
$\vec{A}^{e,t}$ is the transverse component of $\vec{A}^{e}$.
$\Pi_{f}(\omega)$ is the polarization of $f_{\alpha}$, and it should be
proportional to $\mathrm{i}\omega\sigma_{f}(\omega)$ after analytic
continuation Lee and Nagaosa (1992), where $\sigma_{f}(\omega)$ is the
conductivity of the fermionic parton ($i.e.$ also the vortex) $f_{\alpha}$.
This implies that the electrical resistivity of the system should be
$\displaystyle\rho_{e}(\omega)=\frac{\pi^{2}}{4}\sigma_{f}(\omega)=\frac{\pi^{2}}{4}\left(\frac{\sigma_{0}}{1-\mathrm{i}\omega\tau_{v}}\right).$
(14)
We note that here $\rho_{e}$ is measured with unit $\hbar/e^{2}$; $\sigma_{f}$
is computed in the convention that $f_{\alpha}$ carries charge 1. Here we can
evaluate the conductivity of the good metal of $f_{\alpha}$ using the
rudimentary Drude formula. $\sigma_{0}$ is the conductivity of the
$f_{\alpha}$ in the DC limit. It was shown recently that even when a Fermi
surface is coupled to a dynamical gauge field, the response to the gauge field
is still exactly the same as what is computed by the Drude theory (at least
when there is no disorder) Shi et al. (2022). We also exploit the fact that,
when the electron density has a strong interaction, the bare gauge coupling
becomes weak, and the photon-vortex interaction remains perturbative at least
within a large window of scale. An analysis of the fermions interacting with
gauge field in a disordered environment can be found in Ref. Kumar et al.,
2022.
In Eq.14, $\sigma_{0}$ can be rather large, namely the vortices form a good
metal, when there is a finite Fermi surface of $f_{\alpha}$ and the disorder
is weak. In this case the electrical resistivity of the system can be far
beyond the MIR limit, $i.e.$ the system is a very bad metal.
Now we investigate the spatial reflection symmetry $\mathcal{P}$ of our
system. And let us use $\mathcal{P}_{y}:y\rightarrow-y$ as an example. We
assume that $c_{\alpha}$ changes up to a sign under $\mathcal{P}_{y}$, then
this leads to the transformation of $f_{\alpha}$, $z_{\alpha}$:
$\displaystyle\mathcal{P}_{y}:~{}~{}f_{\alpha}\rightarrow(\sigma^{x})_{\alpha\beta}f_{\alpha}^{\dagger},\
\ \ z_{\alpha}\rightarrow(\sigma^{x})_{\alpha\beta}z^{\dagger}_{\beta}.$ (15)
The $\mathrm{U}(1)_{e}$ and $\mathrm{U}(1)_{g}$ charges are even and odd under
$\mathcal{P}_{y}$ respectively. This means that $\tilde{a}_{1,2}$ transform
oppositely under $\mathcal{P}$, and the bSPT state preserves $\mathcal{P}$
based on Eq. 10. In order to ensure the reflection symmetry, we also need the
band structure of $f_{\alpha}$ to satisfy
$\varepsilon_{\uparrow}(k_{x},k_{y})=-\varepsilon_{\downarrow}(k_{x},-k_{y})$.
Our construction also preserves a (special) time-reversal symmetry
$\mathcal{T}$ defined as following: the electron operator is still invariant
under $\mathcal{T}$ (up to an extra sign). We can choose the following
transformations of $z_{\alpha}$ and $f_{\alpha}$ to ensure the desired
transformation of the electrons:
$\displaystyle\mathcal{T}:~{}~{}z_{\alpha}\rightarrow(\sigma^{x})_{\alpha\beta}z^{\dagger}_{\beta},\
\ \ f_{\alpha}\rightarrow(\sigma^{x})_{\alpha\beta}f^{\dagger}_{\beta}.$ (16)
As we can see, the $\mathrm{U}(1)_{e}$ and $\mathrm{U}(1)_{g}$ charges are
again even and odd under $\mathcal{T}$ respectively. This means that
$\tilde{a}_{1,2}$ transform oppositely under $\mathcal{T}$, and the bSPT state
preserves $\mathcal{T}$ based on Eq. 10. In order to ensure the time-reversal
symmetry, we also need the band structure of $f_{\alpha}$ to satisfy
$\varepsilon_{\uparrow}(\vec{k})=-\varepsilon_{\downarrow}(-\vec{k})$. More
precisely, here the time-reversal is a product between the particle-hole
transformation and a spatial-inversion 333Without the spatial-inversion
transformation, the time-reversal transformation alone would demand
$\varepsilon_{\uparrow}(\vec{k})=-\varepsilon_{\downarrow}(\vec{k})$, which
would make the Fermi pockets of the two spinon bands overlap with each other,
and lead to a potential exciton instablity due to nesting. . A more realistic
time-reversal symmetry for electrons with $\mathcal{T}^{2}=-1$ can be defined
and preserved if we introduce another orbital flavor to the electrons.
So far we have ignored the conductivity of the bosonic partons, which is valid
when the energy scale is much smaller than the gap of the bosons $\Delta_{b}$.
With finite frequency $\omega$, the bosons will also make two new nonzero
contributions to the longitudinal response of $A^{e}$. The first of which is a
simple addition to the response Lagrangian of the boson polarization $i.e.$
$\mathcal{L}^{b}_{\mathrm{res}}\sim\frac{1}{2}\Pi_{b}(\omega)|\vec{A}^{e,t}(\omega)|^{2}$,
for which $\Pi_{b}(\omega)\rightarrow\mathrm{i}\omega\sigma_{b}(\omega)$ after
analytic continuation. This only modifies the charge conductivity by shifting
it a value of $\sigma_{b}(\omega)$. The second, more interesting contribution,
is that a longitudinal term will be generated for the internal gauge field
$a_{\mu}$ as well. In principle $\Pi_{b}(\omega)$ can be different for
$A^{e}_{\mu}$ and $a_{\mu}$, but without loss of generality we assume that
they are the same. Since the internal gauge field and electromagnetic gauge
field transform differently under $\mathcal{P}$ and $\mathcal{T}$ and so do
the $\phi_{g}$ and $\phi_{e}$ current, there cannot be mixed terms between $a$
and $A^{e}$ that would lead to mutual longitudinal response in the effective
Lagrangian. Then eventually the electrical conductivity of the system follows
the following composition rule
$\displaystyle\sigma_{e}=\sigma_{b}+\frac{4}{\pi^{2}}\left(\frac{1}{\sigma_{f}+\sigma_{b}}\right).$
(17)
In this equation, $\sigma_{e}$ is measured in the unit of $e^{2}/\hbar$;
$\sigma_{b}$ and $\sigma_{f}$ are computed in the convention of charge $1$ and
$\hbar=1$. This composition is very different from the Ioffe-Larkin rule Ioffe
and Larkin (1989), and we expect this composition rule to be valid for
temperature far smaller than the gap of the bosonic parton, $i.e.$ when the
thermally activated bosonic partons are very dilute. In Fig. 1 we plot
$\rho_{e}(T)=1/\sigma_{e}(T)$ measured in unit of $h/e^{2}$ using the
composition rule Eq. 17.
Another quantity of interest is the charge compressibility. The
compressibility can be computed from the charge density-density correlation
function which can be attained by reading off the $A_{\tau}(q)A_{\tau}(-q)$
term in $\mathcal{L}_{\text{res}}$ at vanishing frequency after integrating
out all the matter and the internal gauge fields. As we discussed before, one
contribution to the compressibility is proportional to the gauge coupling $g$,
which is evident after integrating out $\vec{a}$ from Eq. 11. Eventually the
compressibility should involve the bare charge density-density interaction,
after being renormalized by integrating out the fermions $f_{\alpha}$. In the
limit $\omega\rightarrow 0$ and large gap of bosonic rotors, the total
compressibility is given by
$\displaystyle\kappa_{e}(\mathbf{q})=\frac{1}{(\kappa_{0})^{-1}+\frac{\pi^{2}}{4}\frac{\Pi_{f}(q)}{|\mathbf{q}|^{2}}},$
(18)
where $\kappa_{0}$ is the “bare” compressibility of the system when the gauge
field $a_{\mu}$ is not coupled to any matter fields, hence $\kappa_{0}$ is
proportional to the gauge coupling $g$: $\kappa_{0}=g(4/\pi^{2})$. If we
choose a simple quadratic dispersion or a circular Fermi surface for the
vortices, then the result for the fermionic vortex polarization at zero
frequency is well-known to be Nave et al. (2007); Lee and Nagaosa (1992)
$\Pi_{f}(\mathbf{q})\sim|\mathbf{q}|^{2}/(12\pi m_{v})$ where $m_{v}$ is the
effective mass of the fermionic vortices. Like the charge conductivity, this
gives us a composition rule for the charge compressibility in terms of linear
response functions for the different species of partons. For the more general
case of a finite boson gap, this composition rule involves both the boson
compressibility and boson polarization $\Pi_{b}\sim\chi_{b}|\mathbf{q}|^{2}$:
$\displaystyle\kappa_{e}(\mathbf{q})=\kappa_{b}(\mathbf{q})+\frac{1}{(\kappa_{0})^{-1}+(\chi_{b}+\chi_{f})\pi^{2}/4}.$
(19)
Here, $\chi_{b}$ is the magnetic susceptibility of the bosons. We expect this
composition rule to hold at small finite temperature much below the boson gap.
### II.3 Physics at weak doping
Figure 2: The optical conductivity $\mathrm{Re}[\sigma(\omega)]$ near
$\omega=0$ at half-filling, and with small doping $\delta n_{e}$.
We would also like to consider the effects of weak charge doping $\delta
n_{e}$ away from one electron per site. As the charge density is now bound
with the internal gauge flux through the mutual CS term in Eq. 11, weakly
doping the system corresponds to adding a background $\mathrm{U}(1)_{g}$ gauge
field with small average magnetic flux. Since the vortices (spinon
$f_{\alpha}$) form a good metal, their conductivity may still be evaluated
through the semiclassical Drude theory. Using the semiclassical equation of
motion with magnetic field, it is straight forward to compute the Drude
conductivity of $f_{\alpha}$:
$\displaystyle\sigma_{f}(\omega)=\bigg{(}\frac{\frac{1}{\tau_{v}}-i\omega}{\omega_{c}^{2}+(\frac{1}{\tau_{v}}-i\omega)^{2}}\bigg{)}\frac{1}{\tau_{v}}\sigma_{0}.$
(20)
$\omega_{c}$ is the cyclotron frequency of the vortices
$\omega_{c}=\bar{b}/m_{v}$, and $\bar{b}\sim\delta n_{e}$ is the average flux
seen by the fermionic vortices. Note that the conductivity
$\sigma_{f}(\omega)$ of $f_{\alpha}$ only contains a longitudinal component
given by the expression above. The Hall conductivities from $f_{\uparrow}$ and
$f_{\downarrow}$ cancel each other due to the time-reversal symmetry
$\mathcal{T}$ which involves a particle-hole transformation of the spinons
$f_{\alpha}$. From the relation between the charge conductivity and vortex
conductivity in Eq. 17, if the boson gap is taken to infinity, we can extract
the main interesting piece of the charge conductivity, which is given by
$\displaystyle\sigma_{e}(\omega)\sim\frac{4}{\pi^{2}\sigma_{f}(\omega)}=\frac{4\tau_{v}}{\pi^{2}\sigma_{0}}\bigg{[}\frac{\omega_{c}^{2}}{\frac{1}{\tau_{v}}-i\omega}+\bigg{(}\frac{1}{\tau_{v}}-i\omega\bigg{)}\bigg{]}.$
(21)
At zero doping, the optical conductivity does not have a Drude weight; but the
presence of added charge has introduced a new Drude peak to the optical
conductivity with Drude weight:
$\displaystyle
D\sim\frac{2\omega_{c}^{2}\tau_{v}}{\pi^{2}\sigma_{0}}\sim(\delta n_{e})^{2},$
(22)
and the Drude weight is proportional the square of the doped charge density,
contrary to the ordinary Drude theory where the Drude weight is linear with
the charge density.
The DC resistivity now takes the form of
$\displaystyle\rho_{e}=1/\sigma_{e}(0)=\frac{\pi^{2}\sigma_{0}}{4}\frac{1}{1+\tau_{v}^{2}\omega_{c}^{2}}.$
(23)
The Lorenz number, defined as $L=\kappa/(T\sigma_{e})$ becomes
$\displaystyle
L=\frac{\kappa}{T\sigma_{e}}\sim\frac{L_{0}\sigma_{f}}{\sigma_{e}}\sim\frac{\pi^{2}}{4}\frac{L_{0}\sigma_{0}^{2}}{(1+\tau_{v}^{2}\omega_{c}^{2})^{2}}.$
(24)
Here $\kappa$ represents the thermal conductivity. Both the resistivity, and
Lorenz number decrease with the doped charge density. Combing with the
emergence of the Drude weight under doping, it suggests that doping would
eventually drive the the system more like a normal metal. Here we have ignored
the thermal conductivity arising from the gauge bosons, which also transports
heat without charge, hence it also contributes to the violation of the
Wiedemann-Franz law.
### II.4 Nearby Phases
(1) Spin liquid Mott insulator, and metal
The quantum bad metal state constructed above can be driven to a Mott
insulator which is also a $\mathrm{U}(1)$ spin liquid with a Fermi surface of
spinon $f_{\alpha}$, by driven $(z_{1},z_{2})$ into a trivial bosonic Mott
insulator, with zero rotor number of $z_{1}$ and $z_{2}$. In the Mott
insulator, $f_{\alpha}$ still has Fermi pockets, but the gauge flux of
$a_{\mu}$ no longer carries any nontrivial quantum number. This is one of the
most studied spin liquid states in the literature Lee and Lee (2005);
Motrunich (2005); Ran et al. (2007), with potential applications to a variety
of materials.
The quantum phase transition between a bSPT state and a trivial Mott insulator
of the boson is described by the $N=2$ QED Grover and Vishwanath (2013); Lu
and Vishwanath (2012), and this theory is part of a web of duality involving
also the easy-plane deconfined quantum critical point Xu and You (2015); Mross
et al. (2016); Hsin and Seiberg (2016); Potter et al. (2017); Wang et al.
(2017); Senthil et al. (2019). The original theory of the bSPT-MI transition
is now coupled to an extra dynamical gauge field $a_{\mu}$. When there is no
disorder, the dynamics of $a_{\mu}$ is overdamped by the Fermi surface of
$f_{\alpha}$, then we do not expect the gauge field $a_{\mu}$ to change the
infrared fate of the transition. The presence of disorder may complicate the
nature of the bSPT-MI transition.
The quantum bad metal phase can also be driven into an ordinary metal phase by
condensing either $z_{1}$ or $z_{2}$. The $\mathrm{U}(1)_{g}$ gauge field will
be gapped out by the Higgs mechanism, and the spinon operator $f_{\alpha}$
becomes the electron operator due to the condensate of, $e.g.$ $z_{1}$,
similar to the previous theory of interaction-driven MIT Lee and Lee (2005);
Senthil (2008).
(2) Charge-$4e$ superconductor
Starting with the quantum bad metal, we can also drive the spinon $f_{\alpha}$
into a trivial insulator without special topological response, likely through
a Lifshitz transition where the Fermi pockets shrink to zero. Then the action
of $a_{\mu}$ is just the ordinary Maxwell term, which describes a photon
phase. The monopole which creates and annihilates the gauge flux is prohibited
here as the gauge flux carries charge-$4e$ as we discussed before, and the
electric charge is a conserved quantity. The photon phase of the gauge field
$a_{\mu}$ is also dual to the condensate of its flux, $i.e.$ a condensate of
charge$-4e$, or in other words a charge$-4e$ superconductor.
(3) $Z_{2}$ spin-charge topological order
We can also consider a situation where the fermionic vortices $f_{\alpha}$
form a “superconductor”, $i.e.$ the Cooper pair of $f_{\alpha}$ condenses.
This condensation will gap out $a_{\mu}$ through a Higgs mechanism, and break
$a_{\mu}$ down to a $Z_{2}$ gauge field, which supposedly forms a $Z_{2}$
topological order. Like all the $Z_{2}$ topological orders, here there are
three types of anyons with mutual semionic statistics. One type of anyon is
$f_{\alpha}$, another is the half-flux of $a_{\mu}$. Since the flux of
$a_{\mu}$ carries charge-$4e$, the half-flux of $a_{\mu}$ carries charge-$2e$.
### II.5 Other Constructions
One can also construct a similar quantum bad metal phase starting with a
charge-$2e$ spin-singlet superconductor. Let us assume there are two flavors
of bosons, $b_{1}$ and $b_{2}$, which carry charge $\pm 2e$ respectively. We
take the following parton treatment for $b_{\alpha}$:
$\displaystyle b_{1}=\psi_{1}f,\ \ \ b_{2}=\psi_{2}f,$ (25)
where all the partons are complex fermions. Apparently there is also a gauge
$U(1)_{g}$ shared by the partons. $\psi_{1,2}$ carries electric charge $\pm
2$, and gauge charge $+1$ of a dynamical internal gauge field; $f$ carries
gauge charge $-1$ of the internal gauge field. We now consider the following
state: $f$ again forms a band structure which is a good metal; $\psi_{\alpha}$
forms a quantum “psudospin” Hall insulator, in the sense that the flavor index
of $\psi_{\alpha}$ is viewed as a pseudospin index. After integrating out
$\psi_{\alpha}$, a mutual Chern-Simons term between the external EM field
$A^{e}$ and the internal gauge field $a$ is generated, with the same form as
Eq. 11. We assume that there is no other conserved charges other than the
charge $\mathrm{U}(1)$. The charge response of this construction can be
evaluated following the steps of the previous section.
The bSPT state is one of the states that the bosonic partons $(z_{1},z_{2})$
can form that make the charge vortex a fermion. There are other options which
achieve the similar effect, if we allow topological degeneracy. For example,
$z_{1}$ and $z_{2}$ can each form a bosonic fractional quantum Hall state with
Hall conductivity $\pm 1/(2k)$ where $k$ is an integer. Although there is a
bit subtlety of integrating out a topological order, suppose we can do this,
the response mutual CS theory in Eq. 11 would have level $2/k$. The rest of
the discussion follows directly.
We would like to compare our state with other “vortex liquids” discussed in
previous literature, for example the well-known Dirac vortex liquid in the
context of half-filled Landau level Son (2015); Wang and Senthil (2016b). The
electrical conductivity tensor of the system reads
$\sigma_{ij}=\delta_{ij}\sigma^{\ast}+\frac{e^{2}}{2h}\epsilon_{ij}$, where
$\sigma^{\ast}\sim 1/\sigma_{v}$, and $\sigma_{v}$ is the conductivity of the
Dirac composite fermions (the vortices). Although the longitudinal
conductivity of the system can be small when $\sigma_{v}$ is large, the
longitudinal resistivity would still be small due to the nonzero Hall
conductivity. Hence in the simplest experimental set-up where the transport is
measured along the $\hat{x}$ direction while the $\hat{y}$ direction of the
sample has an open boundary, the measured longitudinal resistivity along the
$x$ direction would be small.
We can also put $c_{\uparrow}$ in the Dirac vortex liquid of the half-filled
Landau level; and $c_{\downarrow}$ in the time-reversal conjugate state of
$c_{\uparrow}$. In this case in order to correctly extract the longitudinal
electrical resistivity, we need to introduce both the external electromagnetic
field, and a “spin gauge field” $A^{s}$:
$\displaystyle\mathcal{L}$ $\displaystyle=$
$\displaystyle\mathcal{L}_{D}(\psi_{1},a_{1})+\frac{\mathrm{i}}{4\pi}a_{1}\wedge
A_{1}+\frac{\mathrm{i}}{8\pi}A_{1}\wedge dA_{1},$ (26) $\displaystyle+$
$\displaystyle\mathcal{L}_{D}(\psi_{2},a_{2})-\frac{\mathrm{i}}{4\pi}a_{2}\wedge
A_{2}-\frac{\mathrm{i}}{8\pi}A_{2}\wedge dA_{2}.$ (28)
$\psi_{1,2}$ are the composite Dirac fermions, and their Lagrangian
$\mathcal{L}_{D}$ should in general have a nonzero chemical potential. We have
introduced two external gauge fields $A_{1}=A^{e}+A^{s}$, and
$A_{2}=A^{e}-A^{s}$. The system does not have a net charge Hall response, but
there is a spin-Hall effect, $i.e.$ there is a mutual CS term between the
electromagnetic field and the spin gauge field. The existence of the spin Hall
response will again lead to a small longitudinal electrical resistivity when
$\psi_{1,2}$ are good metals. By contrast, in our construction presented in
the previous section, there is only longitudinal transport, hence a large
vortex conductivity would ensure a large longitudinal electrical resistivity.
Another vortex liquid with fermionic vortex was discussed in Ref. Galitski et
al., 2005, aiming to understand the observed metallic state with an
anomalously large conductivity between the superconductor and insulator in
amorphous thin films. There the vortex is turned into a fermion through manual
flux attachment.
Vortices can also play an important role in quantum magnets. Exotic quantum
spin liquid states were also constructed through fermionic spin vortices in
previous literature Alicea et al. (2005a, b, 2006); Hermele (2009); Wang and
Senthil (2015b). These works generally use two approaches to generate
fermionic vortices: one can either introduce fermionic partons for the vortex
operator, or turn a vortex into a fermion through flux attachment when the
vortex sees a background magnetic field (dual of fractional spin density). In
our work, instead of granting existing vortices fermionic statistics, an
interpretation of the fermionic partons as charge vortices naturally emerges
due to the topological physics of the bosonic sector.
## III Summary
We present a construction of a strongly interacting quantum bad metal phase,
$i.e.$ at zero and low temperature the resistivity is finite but exceedingly
larger than the MIT limit $h/e^{2}$, by making the charge vortices a good
metallic phase with a vortex Fermi surface. In this construction the charge
vortex is naturally a fermion by driving the charge degree of freedom into a
bosonic symmetry protected topological state. The quantum bad metal so
constructed has the following features: (1) its resistivity can be exceedingly
larger than the MIR limit; (2) a small Drude weight proportional to $(\delta
n_{e})^{2}$ emerges under weak charge doping away from half-filling (one
electron per unit cell); (3) like previously discussed vortex liquids, our
construction should also have strong violation of the Wiedemann-Franz law. We
also demonstrated that this quantum bad metal phase is next to a charge-$4e$
superconductor, a $Z_{2}$ spin-charge topological order, the Mott insulator
phase which is also a well-studied spin liquid, and a normal metal phase.
The exoticness of the state we constructed is of “quantum nature”, as strictly
speaking a bSPT state that our construction strongly relies on is only sharply
defined at zero temperature. At high temperature all the partons will be
confined, and our state no longer enjoys a tractable description in terms of
the dual weakly interacting vortices. Hence our state is different from the
original example of bad metal discussed in Ref. Emery and Kivelson, 1995,
$i.e.$ the cuprates materials with hole doping, where the resistivity of the
system in each $2d$ layer reaches the threshold of bad metal at finite
temperature.
The authors thank Matthew Fisher and Steve Kivelson for very helpful
discussions. We also thank Umang Mehta and Xiao-Chuan Wu for participating in
the early stage of the work. This work is supported by the NSF Grant No.
DMR-1920434, and the Simons Investigator program. C.-M. J. is supported by a
faculty startup grant at Cornell University
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|
1]Physics Department, Bar-Ilan University, Ramat-Gan, 52900, Israel
2]Institute of Nanotechnology and Advanced Materials, Bar-Ilan University,
Ramat-Gan, 52900, Israel
# Laplace’s first law of errors applied to diffusive motion
Omer Hamdi Stanislav Burov Eli Barkai [ [
###### Abstract
In biological, glassy, and active systems, various tracers exhibit Laplace-
like, i.e., exponential, spreading of the diffusing packet of particles. The
limitations of the central limit theorem in fully capturing the behaviors of
such diffusive processes, especially in the tails, have been studied using the
continuous time random walk model. For cases when the jump length distribution
is super-exponential, e.g., a Gaussian, we use large deviations theory and
relate it to the appearance of exponential tails. When the jump length
distribution is sub-exponential the packet of spreading particles is described
by the big jump principle. We demonstrate the applicability of our approach
for finite time, indicating that rare events and the asymptotics of the large
deviations rate function can be sampled for large length scales within a
reasonably short measurement time.
## 1 Introduction
††footnotetext: Invited paper for the topical issue in the European Physical
Journal B: New Trends in Statistical Physics of Complex Systems: Theoretical
and Experimental Approaches.
Laplace’s two laws of error are milestones in statistics. The first was
published in $1774$ [1] and states that the frequency of an error could be
expressed as an exponential of the magnitude of the error, in absolute value.
The second law of errors, from $1778$ [2], states that the frequency of the
error is an exponential of a quadratic function of the error. In the context
of diffusing particles in one dimension, and for a packet starting at the
origin, the probability density of the particles is $P(x,t)$. The first law
states $P(x,t)\propto\exp(-\mbox{const}\ |x|)$ and the second is the more
familiar law, $P(x,t)\propto\exp(-\mbox{const}\ x^{2})$. In $1923$ Wilson [3]
discussed some of the history of the problem. He noted that the second law is
typically called the normal or Gauss law; however, despite Gauss’s well-known
precocity, he probably did not make this discovery before he was two years
old† †††Gauss utilized the least squares method of error estimation for the
discovery of the lost dwarf planet, Ceres. [4, 5].. Indeed, it was Laplace who
promoted the central limit theorem, putting a firm mathematical basis for the
second law. The normal distribution has since been used in all fields of
science, while the first law, as far as we know, was not in the spotlight for
an extended period.
However, using single-molecule tracking data and computer-generated
trajectories of a variety of tracers diffusing in disordered media, in more
recent years, the first law experienced a revival, as it was used to fit a
large body of experimental data [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 22, 23, 24]. These observations are related to the diffusing
diffusivity models [25, 26, 21, 27] and a phenomena known as Brownian yet non-
Gaussian diffusion [14, 28, 15, 29, 30, 31], and Fickian yet non-Gaussian
diffusion [16, 17, 18, 19, 20]. By now, the exponential decay of
$P(x,t)\propto\exp(-\mbox{const}\ |x|)$ is well documented. In some cases,
only the large $x$ limit is described by this law, while in others, the
Laplace law, as a fitting procedure, holds for all $x$ [14]. A large body of
phenomenological models was used to describe these behaviors, for example, by
assuming that the diffusion constant $D(t)$ is a stochastic process.
Chaudhuri, Berthier, and Kob [22] analyzed four systems, focusing on particle
displacements near glass and jamming transitions, highlighting behaviors like
sticking (caging) and rapid jumps between basins. This type of dynamics is
described by the continuous time random walk (CTRW) [32, 33, 34, 35, 36, 37,
38, 39, 40, 41, 42, 43, 44, 45], which will also be analyzed in this paper.
Using specific waiting time distributions, they showed how the basic CTRW
model can be used to predict exponential tails, in accordance with Laplace’s
first law. This theory was later advanced by Wang, Burov, and Barkai [46, 47],
showing that it holds for very general settings. They have shown that, for any
distribution of waiting times, which is analytical for short waiting times,
and for any jump length distribution, which decays faster than exponential,
Laplace’s first law holds at the tails of $P(x,t)$. The two above-mentioned
conditions are expected to hold in many systems. The analysis was based on
large deviations [48, 49, 50, 51, 52, 53, 54, 55] arguments, and a saddle
point approximation [56], namely to put things in a historical context, the
technique used is an extension of the Laplace method for solving integrals
[1].
Schramma et al. [6] discovered that chloroplasts, which are components of
plant cells, adapt to dim light by moving in a manner closely resembling the
caged dynamics observed in supercooled liquids or colloidal suspensions near
the glass transition. This movement demonstrates exponential tails, fitting
both the CTRW and the diffusing diffusivity models.
The connection between Laplace diffusion and CTRW in polymer nanocomposites
was further explored by Hu et al [57]. They demonstrated the ability to
manipulate the emergence of exponential tails in contrast to Gaussian
diffusion by modifying the strength of the disorder.
In this manuscript, we first provide a more detailed analysis of CTRW dynamics
using three tools. We will then conclude the paper with a broader perspective
on the Laplace-like behavior, i.e., exponential decay of spreading particles.
We start with a study of the CTRW model for the case when the jump length
distribution is either sub or super-exponential. In the former case large
deviations theory holds, while for the latter the big jump principle is valid
(see below). This transition is related to the fact that for sub-exponential
jump length distribution, the cumulant generating function diverges, and thus
the standard Laplace-Cramers-Daniles tool of saddle point approximation of
large deviations theory does not hold. We show how the big-jump principle is
related to an extension of the Laplace method of solving integrals. In
Laplace’s method, close to the saddle point, an analytic i.e., quadratic
function is used, and hence, eventually, a Gaussian integral is computed,
while the original integral is non-Gaussian. For the sub-exponential
distribution of jump lengths, we find a similar extremum, but with non-
analytical features. Namely, the quadratic expansion close to the extremum is
invalid. Finally, we present the Edgeworth expansion to approximate the CTRW
probability density [58]. This approximation provides corrections for the
central limit theorem (CLT). It deals with a long time limit, and not very
large $x$, while the large deviations theory and the big-jump principle cover
the behavior of the tails of the density. To study these effects, we
constructed a numerical tool, to sample finite time and finite $x$ propagators
$P(x,t)$ using CTRW.
## 2 Model
In the CTRW model, waiting times between jump events are independent
identically distributed (IID) random variables with a probability density
function (PDF) $\psi(\tau)$. The process starts at time $t=0$, and then the
particle waits at its initial position $x=0$, for a random time $\tau$ drawn
from $\psi(\tau)$. At time $t=\tau$, the random walker jumps to a new
position, the duration of the jump being negligible. The jump lengths are also
IID random variables, with $f(\chi)$ as the PDF. The process is then renewed,
namely, a second waiting time is drawn, followed by a spatial jump, etc. The
goal is to find the PDF $P(x,t)$ of finding the particle at $x$ at time $t$.
The CTRW process is called a semi-Markovian process and the statistics for the
number of jumps can be analyzed utilizing the renewal theory [59, 60].
We focus on exponentially distributed waiting times, $\psi(\tau)=\exp(-\tau)$,
with the mean time between jumps set to unity. It follows that the number of
jumps in the time interval $(0,t)$ is described by Poisson statistics. The
jump length PDF $f\left(\chi\right)$ is:
$f\left(\chi\right)=\widetilde{N}\exp\left(-\alpha^{\beta}|\chi|^{\beta}\right).$
(1)
The mean jump length is zero, and the symmetry of $f\left(\chi\right)$ sets
the symmetry of $P(x,t)$, i.e., $P(x,t)=P(-x,t)$. The variance of the jump
length is set to be $1$. Here we have
$\alpha=\sqrt{\Gamma(3/\beta)/\Gamma(1/\beta)}$, and the normalization
$\widetilde{N}=\beta\sqrt{\Gamma(3/\beta)}/2\Gamma^{3/2}(1/\beta)$, with
$\Gamma(\dots)$ being the Gamma function. The exponent $\beta>0$ is an
important parameter in this study. For $\beta<1$ $(\beta>1)$ we have sub (or
super) exponential decay of the jump length distribution. The case $\beta=2$
corresponds to Gaussian statistics for the jump length PDF.
The probability of jumping $n$ times during time $t$ is $t^{n}\exp(-t)/n!$
hence
$P(x,t)=\sum_{n=0}^{\infty}\frac{{\rm e}^{-t}t^{n}}{n!}\phi(x|n),$ (2)
$\phi(x|n)$ is the PDF of finding the particle at $x$, conditioned on
performing exactly $n$ jumps. In the previous equation, the summation is
performed over all the possible number of jumps that can occur during the
process. Using the fact that jump lengths are IID random variables, the
characteristic function of $\phi(x|n)$
$\int_{-\infty}^{\infty}{\rm e}^{ikx}\phi(x|n){\rm d}x=\langle{\rm
e}^{ik(\chi_{1}+...\chi_{n})}\rangle=\widetilde{f}^{n}(k),$ (3)
is expressed in terms of the Fourier transform [61] of $f(\chi)$, namely
$\widetilde{f}(k)=\int_{-\infty}^{\infty}\exp(ik\chi)f(\chi){\rm d}\chi$. The
Fourier transform of $P(x,t)$, i.e., $\widetilde{P}(k,t)$, is obtained from
Eq.(2) and Eq.(3)
$\widetilde{P}(k,t)=\exp\left[-t\left(1-\widetilde{f}(k)\right)\right].$ (4)
The goal is to find the inverse Fourier transform of the expression above. The
CLT is valid in the limit $t\to\infty$ and $x\to\infty$ while the ratio
$x/\sqrt{t}$ is finite. We utilize the expansion of $\widetilde{f}(k)$ around
$k\to 0$. Since the jump length distribution is an even function,
$\widetilde{f}(k)\sim 1-\sigma^{2}k^{2}/2$, and according to our notation
$\sigma^{2}=1$. Therefore Eq. (4) yields
$P(x,t)\sim\frac{\exp\left(-\frac{x^{2}}{2t}\right)}{\sqrt{2\pi t}}.$ (5)
Namely, the central limit theorem holds.
## 3 Large Deviations $\beta\ >\ 1$
Capturing the essence of the limiting scaling law in Eq. (5) is but one aspect
of the large deviations Theory. This theory also delves into analyzing a
different yet significant limit. The inverse Fourier transform of
$\widetilde{P}(k,t)$ in Eq. (4), while changing variable $ik=u$, reads
$\displaystyle P(x,t)$ $\displaystyle=$ $\displaystyle\frac{1}{2\pi
i}\int_{-i\infty}^{i\infty}\exp{\left[-xK(u)\right]}{\rm d}u,$ (6)
where
$\displaystyle K(u)$ $\displaystyle=$ $\displaystyle
u+\frac{1}{q}[1-\hat{f}(u)],$ (7)
$q\equiv x/t$, and $\hat{f}(u)=\int_{-\infty}^{\infty}\exp(u\chi)f(\chi){\rm
d}\chi$ is the moment generating function. Note that $\hat{f}(u)$ diverges for
$\beta<1$, leading to the divergence of the integral in Eq. (6). Therefore, in
this section, we focus on super-exponential PDFs, i.e., $\beta>1$.
Large deviations theory is based on the saddle point approximation and
considers the case of large $x$ limit of $P(x,t)$ in Eq. (6). Strictly
speaking, here the scaling is $x\to\infty$ and $t\to\infty$ while the ratio
$q=x/t$ is kept finite. Nevertheless, we show that the approximate solution
for $P(x,t)$ works reasonably well for large $x$ while $t$ is kept finite.
The problem of finding $P(x,t)$ is solved by using the following steps: First,
find the $u$ that satisfies $K^{\prime}(u)=0$ and term this $u$ as $u_{0}$.
Then, expand $K(u)$ in Eq. (6), in the vicinity of $u_{0}$, up to a quadratic
term. The obtained Gaussian integral yields
$P(x,t)\sim\frac{\exp\left[-xK(u_{0})\right]}{\sqrt{2\pi|xK^{\prime\prime}(u_{0})|}}.$
(8)
The approximation for large x can be obtained through numerical methods.
Namely, we numerically find $u$ that satisfy $K^{\prime}(u)=0$, and insert it
into Eq. (8). We call this solution method the numerical large deviations
method (Numerical-LDT). Below, we discuss how to obtain the exact form of
$P(x,t)$ and compare it to the solution obtained via Numerical-LDT.
Using $K^{\prime}(u)=0$, and the definition of $\hat{f}(u)$, the equation for
$u_{0}$ reads
$1-\frac{1}{q}\int_{-\infty}^{\infty}\chi f(\chi)\exp(u_{0}\chi){\rm
d}\chi=0.$ (9)
We treat this equation in two limits, small and large $q=x/t$. For small $q$
the value of integral in Eq. (9), must be small. Since the integral in Eq. (9)
is a growing function of $u$, the limit of small $q$ corresponds to small
values of $u$. Expanding $\exp(u\chi)$ and using the fact that the variance of
jump lengths equals $1$ we find $u_{0}\sim q\ll 1$, and $K(u_{0})\sim
u_{0}+\frac{1}{q}[1-\hat{f}(u_{0})]\sim q/2$. It then follows that for $q\ll
1$, $P(x,t)\propto\exp[-x^{2}/(2t)]$. This is the standard prediction of the
CLT, as delineated in Eq. (5), hence the more interesting case is $q\gg 1$.
We use Eq. (9) in the large $q$ limit, considering large $u$, to find $u_{0}$.
For that aim, we need to find $\hat{f}(u)$ when $u$ is large. With a change of
variable $\chi=u^{\frac{1}{\beta-1}}\xi$, we obtain
$\hat{f}(u)=\widetilde{N}u^{\frac{1}{\beta-1}}\int_{-\infty}^{\infty}\exp\left[u^{\frac{\beta}{\beta-1}}\left(\xi-\alpha^{\beta}|\xi|^{\beta}\right)\right]{\rm
d}\xi.$ (10)
Since $u$ is large this integral is solved using Laplace’s method
$\hat{f}(u)\sim\widetilde{N}C_{2}u^{\frac{1}{2}\frac{2-\beta}{\beta-1}}\exp\left(C_{1}u^{\frac{\beta}{\beta-1}}\right),$
(11)
where $C_{1}=(\beta-1)(\beta\alpha)^{\frac{\beta}{1-\beta}}$, and
$C_{2}=\sqrt{2\pi/(\beta-1)(\beta\alpha^{\beta})^{\frac{1}{\beta-1}}}$. This
expression is exact for the Gaussian case $\beta=2$. Eq. (9) for $u_{0}$ is
written as $0=1-\frac{1}{q}\hat{f^{\prime}}(u)$, and using Eq. (11) we find
that $u_{0}$ satisfies
$\displaystyle
0=1-\frac{1}{q}\frac{\beta\widetilde{N}C_{1}C_{2}}{\beta-1}u_{0}^{\frac{4-\beta}{2\beta-2}}\exp\left(C_{1}u_{0}^{\frac{\beta}{\beta-1}}\right).$
(12)
Note that in the case of $\beta=4$, this approximation breaks down. For that
reason, in this work, we focused on the range $1<\beta<4$ and the transition
to $\beta<1$, where the big jump principle holds. To solve Eq. (12) we utilize
the Lambert function $W(x)$ [62] which satisfies
$W(x)\exp\left[W(x)\right]=x$. The branch of the Lambert function that is
relevant to our study is the well-documented principal branch $W_{0}(x)$.
Solving Eq. (12) we obtain
$u_{0}\sim
C_{3}W_{0}\left[C_{4}q^{\frac{2\beta}{4-\beta}}\right]^{\frac{\beta-1}{\beta}}.$
(13)
For large values of the argument, the Lambert function is expressed in terms
of logarithmic functions, i.e.,
$W_{0}(z)\underset{z\to\infty}{\sim}\ln(z)-\ln(\ln(z))$, hence $u_{0}$ is
growing with $x$, however very slowly. This holds true only when $\beta<4$,
while the opposite case demands further study. The constants are:
$\displaystyle
C_{3}=\beta\alpha\left(\frac{2\beta(\beta-1)}{4-\beta}\right)^{\frac{1-\beta}{\beta}}\quad,\quad$
$\displaystyle
C_{4}=\frac{2(\beta(\beta-1))^{\frac{4}{4-\beta}}}{4-\beta}\left(\frac{\alpha^{2}}{\sqrt{2\pi}\widetilde{N}(\beta)}\right)^{\frac{2\beta}{4-\beta}}.$
(14)
Using $\exp[aW(x)]=(x)^{a}/[W(x)]^{a}$, we obtain
$K(u_{0})=u_{0}+\frac{1}{q}[1-\hat{f}(u_{0})]$. Therefore, according to Eq.
(8), we have that
$P(x,t)\sim\frac{\exp\left\\{-t\left(\frac{|x|}{t}Z\left(\frac{|x|}{t}\right)+1\right)\right\\}}{\sqrt{2\pi
K^{{}^{\prime\prime}}(u_{0})}},$ (15)
where
$Z(q)=\frac{c_{3}W_{0}\left[c_{4}q^{\frac{2\beta}{4-\beta}}\right]-\alpha\left(\frac{2\beta(\beta-1)}{4-\beta}\right)^{\frac{1}{\beta}}}{W_{0}\left[c_{4}q^{\frac{2\beta}{4-\beta}}\right]^{\frac{1}{\beta}}}.$
(16)
Using $W_{0}(x)\sim\ln(x)$ we find
$P(x,t)\sim\frac{\exp\left\\{-t\left[\kappa\frac{|x|}{t}\log\left(\frac{|x|}{t}\right)^{1-1/\beta}+1\right]\right\\}}{\sqrt{2\pi
K^{{}^{\prime\prime}}(u_{0})}},$ (17)
where $\kappa=\beta\alpha/(\beta-1)^{1-1/\beta}$. Thus, for large $x$, the
packet is decaying according to Laplace’s first law, with logarithmic
corrections which are typically hard to detect in experiments. The result in
Eq. (17) was found in a more general setting in [46]. We also note a mistake
in Eq. (9) in [46] the later equation corresponds to Eq. (16). Finally, we do
not advocate the practical use of Eq. (17) beyond an asymptotic result. The
Lambert functions in Eq. (16) are needed unless, $x/t$ is astronomically
large. To see this consider the asymptotic expansion
$W_{0}(x)\sim\ln(x)-\ln(\ln(x))$ the second term is one percent of the first
when $x\sim 10^{280}$. In other words, the leading term, in applications not
sampling extremely rare events is insufficient. We now show how the theory is
applicable for finite times, using numerically exact $P(x,t)$.
Figure 1: Comparative analysis of the CTRW exact solution (circles), our
approximation derived from Eq. (15) (blue line), numerical LDT derived using
Eq. (8) (dashed black), and CLT predictions (orange line), log-plotted for two
time scales (0.3,3), and various $\beta$ values. Notably, the nearly
exponential tails are a dominant feature in both time scales, with an
observable convergence to the CLT regime over time. The numerical LDT shows
remarkable fitting in the tails and also exhibits a strong performance in the
central part as time progresses. The approximation we propose in Eq.(15)
demonstrates a close match with the numerical data, however, deviations become
more pronounced for large time values as $\beta\rightarrow 1$. This trend,
indicative of slower convergence at smaller $\beta$ values, is further
analyzed and discussed in the following section.
### 3.1 CTRW Propagator
Sampling the “rare events” of the CTRW, particularly the tails of the PDF,
poses significant challenges in trajectory simulations. To address this, we
utilize the convolution theorem of the Fourier transform to compute
$\phi(x|n)$ in Eq (2). More specifically, in Fourier space, the equation
$\hat{\phi}(k|n)=\widetilde{f}^{n}(k)$ holds true. This leads to the recursive
relation: $\phi(x|n)=f\ast f\dots\ast f=\phi(x|n-1)\ast f$, where $*$ is the
convolution operator. For the case of $N=2$, this translates to
$\phi(x|2)=f\ast
f=\int_{-\infty}^{\infty}f(\chi^{\prime})f(x-\chi^{\prime})d\chi^{\prime}$.
This equation for $\phi(x|2)$ represents the integration over the probability
of making a first step to a certain point $\chi^{\prime}$ and then a
subsequent step to $x$.
In Fig. 1, we conduct a comparison between this numerically exact solution
(CTRW), our derived approximation in Eq. (15), and the numerical LDT
calculated using Eq. (8), juxtaposed with the predictions of the CLT. For
short time scales, our derived approximation and the numerical LDT demonstrate
excellent agreement in the tails, where the CLT fails. As the time scale
increases, the numerical LDT consistently aligns for all values of $x$ and
$\beta$, whereas our derived approximation, though precise for $\beta=2$
(Gaussian), and closely fitting for $\beta=3/2$, converges more gradually when
$\beta=1.1$, nearing the critical transition for $\beta=1$ that was studied in
[63]. For this case, when $\beta=1$, exponential-like tails are still
exhibited but are not described by Eq. (17).
Upon further examination of Fig. 1, it becomes apparent that for $t\gg 1$, the
application of the CLT or the Edgeworth expansion is also viable, a topic we
will discuss shortly.
Figure 2: Convergence of the CTRW results (dashed-dotted) for increasing time
scales, $t=1,5,15$, towards the numerical LDT, as derived in Eq. (8) (blue
stars). The theory demonstrates exceptional accuracy, with the CTRW aligning
with our numerical LDT across all values of $\beta$ for long times.
Additionally, the linear-like nature of the tails is observed, aligning with
the expected Laplace’s exponential tails. We also plot the asymptotical rate
function (ARF) from Eq. (19) (shown as a black line). Notably, for $\beta=2$,
the ARF aligns perfectly with the predictions, and for $\beta=1.5$, we achieve
a close fit. However, as we approach the regime of the big jump principle, for
$\beta=1.1$, convergence is slow, as we later describe.
### 3.2 Rate Function
The rate function is an important concept in the large deviations literature
as it holds the main characteristics of the propagator, describing both the
typical and rare events. It is obtained in the limit of $t\rightarrow\infty$,
but $x/t$ finite:
$\displaystyle\mathcal{I}(x/t)=\lim_{t\to\infty}-\frac{1}{t}\ln{\left[P(x,t)\right]}.$
(18)
In our model, we can extract the rate function from Eq. (8). Taking into
consideration the fact that $u_{0}$ is a function of $q=x/t$, see Eq. (9),
i.e. $q$ is the scaling parameter of the rate function:
$\displaystyle\mathcal{I}(q)=qK(u_{0})\approx\begin{dcases*}q^{2}/2&, $q\ll
1$\\\ |q|Z(|q|)+1&, $q\gg 1$ .\end{dcases*}$ (19)
The rate function in the small $q$ limit exhibits the quadratic CLT behavior,
whereas the large $q$ behavior is what we call the asymptotic rate function
(ARF). In Fig. 2, we demonstrate how our model effectively captures both the
central part, corresponding to the CLT, and the Laplace(exponential)-like
tails, corresponding to the large $q$ limit in Eq. (19), as $Z(q)$ is a slowly
varying function. It becomes evident that, as time progresses, the numerical
solution (CTRW) converges to our numerical LDT rate function. Moreover,
although the ARF theory is valid, as expected, we observe an issue of slow
convergence for values of $\beta$ near 1.
Figure 3: Slow convergence of the asymptotical rate function (ARF) as
formulated in Eq. (19), to the numerical LDT obtained using Eq. (8), for the
case $\beta=1.1$. As we approach $\beta=1$ from above, the convergence of our
approximation (ARF) becomes increasingly gradual. This slowing of convergence
is attributed to the proximity to the transition point. At this critical
juncture, the moment-generating function becomes non-existent, and the system
transitions into the big jump principle (BJP) regime.
### 3.3 Slow Convergence of the ARF
The issue of slow convergence of the ARF, as observed in Fig. 2, becomes
increasingly pronounced as we mentioned before. More specifically, we focus on
the scenario where our approximation, formulated in Eq. (19), under the
umbrella of LDT $(\beta>1)$, is nearing the critical transition point that
occurs for $\beta=1$. For this value of $\beta$, the moment-generating
function ceases to exist and our approximation becomes invalid.
In Fig. 3, we highlight how, for $\beta=1.1$, the ARF converges slowly to the
numerical LDT, as $q=x/t\rightarrow\infty$. This gradual convergence is a
direct consequence of our approximation for $u_{0}$, and the numerical
solution for $u_{0}$ eliminates this problem. In Fig. 4 we demonstrate how our
approximation of $u_{0}$ worsens as $\beta$ approaches 1, leading to the slow
convergence of the rate function and the propagator in such cases.
Figure 4: Approximation of $u_{0}$ (colored lines), and numerically computed
$u_{0}$ (corresponding colored symbols). While for $\beta=2$ our approximation
is exact, as $\beta\rightarrow 1$ the approximation works well only when $x\gg
t$.
## 4 Edgeworth Expansion
In the context of the summation of n IID random variables, the Edgeworth
expansion [64, 65, 66] provides corrections to the CLT. As a reminder, we
denote the PDF of finding the particle at $x$, conditioned on performing $n$
jumps, as $\phi(x|n)$. The leading order of the Edgeworth expansion when $n$
is large, is:
$\phi(x|n)\sim\frac{\exp(-x^{2}/2n)}{\sqrt{2\pi
n}}\left[1+\frac{\kappa_{4}\mathrm{He}_{4}(x/\sqrt{n})}{4!\ n}+\cdots\right]$
(20)
where $\kappa_{4}$ is the fourth cumulant of $f(\chi)$, namely
$\kappa_{4}=m_{4}-3\sigma^{2}$, and $m_{4}$ is the forth moment of $f(\chi)$.
In our example $\sigma=1$ while
$m_{4}=\Gamma(1/\beta)\Gamma(5/\beta)/\Gamma(3/\beta)^{2}$,
$\mathrm{He}_{4}(\xi)=\xi^{4}-6\xi^{2}+3$. In Eq. (20) the correction term is
proportional to the Hermite polynomial.
It is tempting to use the Edgeworth expansion in the summation presented in
Eq. (2). However, if $\kappa_{4}=0$, namely when $f(\chi)$ is Gaussian
corresponding to $\beta=2$, the correction term vanishes. We introduce a
modified Edgeworth expansion, where $t$ is the large parameter. We use the
small $k$ expansion $\widetilde{f}(k)=1-k^{2}/2+m_{4}k^{4}/4!+\cdots$ in Eq.
(4)
$\widetilde{P}(k,t)=\exp\left[-\frac{tk^{2}}{2}+\frac{tm_{4}k^{4}}{4!}+\cdots\right].$
(21)
Further expanding
$\widetilde{P}(k,t)=\exp\left(-\frac{tk^{2}}{2}\right)\left[1+k^{4}\frac{m_{4}t}{4!}+\cdots\right].$
(22)
Performing the inverse Fourier transform we obtain
$P(x,t)=\frac{\exp\left(-x^{2}/2t\right)}{\sqrt{2\pi
t}}\left[1+\frac{m_{4}}{4!\
t}\mathrm{He}_{4}\left(\frac{x}{\sqrt{t}}\right)+\cdots\right],$ (23)
Eq. (23) has the same structure as the original expansion in Eq. (20), where
we replaced large $n$ with large $t$. The difference is that now the
correction is proportional to the fourth moment of the distribution $f(\chi)$
and not to its cumulant. The former is always non-negative while the latter
can be negative zero or positive. Therefore, we see that the fluctuations in
the number of jumps have a significant effect.
The contribution of the Hermite polynomials within our model can be
demonstrated through a straightforward rearrangement of $P(x,t)$ from Eq.
(23), denoted as $\mathcal{R}(x,t)$. By defining the central limit theorem
component as $\mathrm{CLT}=\exp\left(-x^{2}/2t\right)/\sqrt{2\pi t}$, we
obtain the following form:
$\displaystyle\mathcal{R}(x,t)=\frac{4!t}{m_{4}}\frac{P(x,t)-CLT}{CLT},$ (24)
where, at sufficiently long times, this expression approaches the fourth
Hermite polynomial. This is shown in Fig. 5 for different values of $\beta$,
where $\mathcal{R}(x,t)$ is plotted versus the scaled variable $x/\sqrt{t}$.
The Edgeworth expansion uses the diffusive scaling $x/\sqrt{t}$ while the
large deviations theory relies on the scaled variable $x/t$. Both contribute
to the leading order of the central limit theorem when $x/t\ll 1$. Note that
when using the Edgeworth approach in the limit $x/\sqrt{t}\gg 1$, we have from
the Hermite function a correction term to the Gaussian $P(x,t)$ that can be
written as $tm_{4}(x^{4}/t^{4})$. This results in the same scaling $q=x/t$ as
found in the rate function $P(x,t)\propto\exp[-t{\cal{I}}(x/t)]$. Thus, as
expected, the Edgeworth expansion and large deviations theory are not
conflicting. They consider different limits of the problem. The large
deviations theory is valid for $\beta>1$ as it demands finite cumulants. The
convergence properties of the Edgeworth expansion and their $\beta$ dependence
are left for future work.
Figure 5: We showcase the leading order of the Edgeworth expansion by
plotting, for $t=10$ the rearranged result from the numerical CTRW as outlined
in Eq.(24) (colored lines). This is compared against the fourth Hermite
polynomial $\mathrm{He}_{4}(x/\sqrt{t})=(x/\sqrt{t})^{4}-6(x/\sqrt{t})^{2}+3$.
The excellent agreement is observed regardless of the value of $\beta$,
thereby confirmming the validity of this expansion for large time scales.
## 5 Big Jump Principle
### 5.1 Sum of IID Random Variables
The big jump principle (BJP) [67, 68, 69, 70, 71, 72, 73] describes how, in a
heavy-tailed process with a sub-exponential distribution of jump lengths, a
single “big” jump among a series of $n$ IID random variables can dominate the
asymptotic characteristics of the process. This principle connects the sum
$x=\sum_{i=1}^{n}\chi_{i}$ to the largest displacement
$\chi_{\text{max}}=\max{\chi_{1},\dots,\chi_{n}}$ [69]. Specifically, this
principle provides the asymptotic equality:
$\displaystyle\int_{\chi_{\text{large}}}^{\infty}\phi(x|n)dx$ $\displaystyle=$
$\displaystyle\text{Prob}(x>\chi_{\text{large}})=$ (25)
$\displaystyle\text{Prob}(\chi_{\text{max}}>\chi_{\text{large}}),$
in the limit of $\chi_{\text{large}}\rightarrow\infty$. To further elucidate,
we calculate:
Prob
$\displaystyle\left(\chi_{\text{max}}>\chi_{\text{large}}\right)=1-\text{Prob}\left(\chi_{i}<\chi_{\text{large}}\right)^{n}$
(26)
$\displaystyle\quad=1-\left[1-\int_{\chi_{\text{large}}}^{\infty}f(\chi)d\chi\right]^{n}.$
Taking the derivative with respect to $\chi_{\text{large}}$, we obtain the PDF
of $x$. Then, by approximating
$\int_{\chi_{\text{large}}}^{\infty}f(\chi)d\chi$ as approximately zero, which
is a leading-order approximation, we get the asymptotic result [74]:
$\displaystyle\phi(x|n)_{\beta<1}\approx nf(x).$ (27)
It is important to note that, in contrast with the CLT, which is applicable
primarily when $n$ is significantly large, Eq. (27) is valid for all values of
$n$, including instances when $n=2$. We will discuss the BJP for CTRW later,
but for now, our focus is on $\phi(x|n)$, specifically in the case where the
number of jumps, $n$, equals 2.
Figure 6: The graph of $g(y)=-|1-y|^{\beta}-|y|^{\beta}$, for different values
of $\beta$. It is evident that for $\beta>1$, the function is smooth with a
single global maximum at $y_{0}=1/2$. For $\beta=1$, the function is piecewise
linear, and for $\beta<1$, it exhibits two non-analytical cusps at $y=0$, and
$y=1$. This example serves as a foundational demonstration of the transition
from BJP $(\beta<1)$ to LDT $(\beta>1)$, while $\beta=1$ is a special
transition point [63].
### 5.2 Big Jump Principle for $n=2$
An understanding of the big jump principle is best demonstrated through the
simple yet comprehensive scenario of two jumps. Employing the convolution
theorem as seen earlier; $\phi(x|2)=f\ast
f=\widetilde{N}^{2}\int_{-\infty}^{\infty}\exp\left(-\alpha^{\beta}|\chi^{\prime}|^{\beta}-\alpha^{\beta}|x-\chi^{\prime}|^{\beta}\right)d\chi^{\prime}$,
we introduce the scaling variable $y=\frac{\chi^{\prime}}{x}$, and obtain the
following equation:
$\displaystyle\phi(x|2)=\widetilde{N}^{2}|x|\int_{-\infty}^{\infty}{\rm
e}^{\alpha^{\beta}|x|^{\beta}g(y)}dy,$ (28)
where we define $g(y)\equiv-|1-y|^{\beta}-|y|^{\beta}$. In Fig. 6 we
demonstrate the transition in the analytical properties of $g(y)$. For
$\beta>1$ where $f(\chi)$ is super-exponential, $g(y)$ is a smooth function
with a single global maximum at $y_{0}=1/2$, suitable for the Laplace method
of solving integrals. Conversely, as shown in Fig. 6, for $\beta<1$, $g(y)$
displays a pair of cusps, and two distinct maxima are observed. We proceed to
examine these cases. Beginning with the case of $\beta>1$, one can approximate
$g(y)\approx g(y_{0})+\frac{1}{2}g^{{}^{\prime\prime}}(y_{0})(y-y_{0})^{2}$,
facilitating a direct Gaussian integration:
$\displaystyle\phi(x|2)_{\beta>1}\approx\widetilde{N}^{2}|x|\sqrt{\frac{\pi
2^{\beta-2}}{\beta(\beta-1)\alpha^{\beta}|x|^{\beta}}}{\rm
e}^{-\frac{1}{2}^{\beta-1}\alpha^{\beta}|x|^{\beta}}.$
This result diverges as $\beta$ approaches 1. The critical case of $\beta=1$
marks a transition, where the cumulant generating function of $\phi(x|n)$,
exists only for $\beta>1$, and a piecewise linear function is displayed. At
$\beta=1$, we can exactly solve Eq. (28), yielding:
$\phi(x|2)=\frac{\sqrt{2}+2|x|}{4}{\rm
e}^{-\sqrt{2}|x|}=\left(\frac{1+\sqrt{2}|x|}{2}\right)f(x)$, which is not
described by either the BJP or the LDT.
However, for $\beta<1$, where $f(\chi)$ is sub-exponential, $g(y)$ exhibits
two non-analytical cusps, and a slightly modified version of the Laplace
method of solving integrals is needed. Due to the symmetry of $g(y)$, it is
easy to show that the main contributions from the two maxima to the
integration are identical. Opting for the left cusp, around $y=0$, we
approximate $g(y)\approx-1-|y|^{\beta}$, and insert it into Eq. (28), to
derive: $\phi(x|2)\approx\widetilde{N}^{2}|x|\int_{-\infty}^{\infty}{\rm
e}^{-\alpha^{\beta}|x|^{\beta}\left(1+|y|^{\beta}\right)}dy=2f(x)$, aligning
with the BJP.
Now, we calculate corrections to the BJP which are particularly important when
$\beta\to 1$. The improved approximation is given by
$g(y)\approx-1-|y|^{\beta}+\beta y+\frac{1}{2}\beta(1-\beta)y^{2}$, which we
can plug into Eq. (28) to obtain
$\displaystyle\phi(x|2)\approx 2\widetilde{N}^{2}|x|$
$\displaystyle\times\int_{-\infty}^{\infty}{\rm
e}^{\alpha^{\beta}|x|^{\beta}(-1-|y|^{\beta})}\big{(}1+{\rm d}_{1}(x)y+{\rm
d}_{2}(x)y^{2}\big{)}dy,$
where ${\rm d}_{1}(x)=\alpha^{\beta}|x|^{\beta}\beta$ vanishes due to
symmetry. The solution of Eq. (5.2) yields
$\phi(x|2)\approx 2\left(1+\frac{{\rm d}_{2}(x)}{x^{2}}\right)f(x),$ (31)
where
${\rm
d}_{2}(x)=\frac{1}{2}\alpha^{\beta}|x|^{\beta}\beta\left(1-\beta+\alpha^{\beta}|x|^{\beta}\beta\right).$
(32)
The first term of Eq. (31) is the BJP, as detailed in Eq. (27), while ${\rm
d}_{2}(x)/x^{2}$, serves as the correction term. The correction becomes
significant as $\beta$ approaches $1$ from below, since the magnitude of ${\rm
d}_{2}(x)/x^{2}$ is of the same order as the leading term of the BJP. In Fig.
7, we compare the numerically obtained $\phi(x|2)$, the leading term from Eq.
(27), and the correction term we have formulated in Eq. (31). This comparison
emphasizes the importance of additional correction terms as $\beta\to 1$ and
demonstrates their redundancy for $\beta\ll 1$.
Figure 7: The big jump principle for the pedagogical case of two jumps. The
figure shows the numerically calculated $\phi(x|2)$ (colored symbols), the
leading approximation $(2f(x))$ (colored lines), and the corrected form,
derived in Eq. (31) (colored dashed lines) for $\beta=0.2,0.7,0.9$, are
displayed on a semi-logarithmic scale. While the BJP is effective at the far
tails, it becomes apparent that convergence slows as $\beta$ nears the
critical value of 1.
### 5.3 BJP for CTRW
As we progress with the big jump principle, integrating Eq. (27) into Eq. (2)
leads us to a straightforward relation [63]:
$\displaystyle P(x,t)\approx\langle n_{t}\rangle f(x),$ (33)
where $\langle n_{t}\rangle$ represents the average number of jumps at time
$t$. This relationship holds true for every distribution of waiting times. In
this manuscript, our focus is specifically on exponential waiting times,
corresponding to Poisson statistics. Namely, we have $\langle n\rangle=t$,
hence $P(x,t)\approx tf(x)$. This signifies a pivotal behavior change: instead
of exhibiting exponential tails, the propagator will now display sub-
exponential characteristics. Thus, the universality observed in the super-
exponential case (for $\beta>1$) disappears for $\beta<1$. Similarly, if
$f(x)$ has a power law tail, Eq. (33) is still valid, and $P(x,t)$ will decay
as a power law for large x. In Fig. 8, we showcase this transition, observing
how the propagator, $P(x,t)$, shifts from sub-exponential features to
exponential, Laplace-like characteristics.
Figure 8: The propagator $P(x,t)$ is shown for different values of $\beta$
(shown in legend). The transition of $P(x,t)$ from exponential to Laplace-like
characteristics is associated with the large deviations theory. When
$\beta>1$, we observe a transition to sub-exponential behavior as per the big
jump principle when $\beta<1$. Circles denote the numerically obtained CTRW
data, while solid lines represent our approximations. For $\beta<1$, Eq. (33)
is employed, and for $\beta>1$, Eq. (15) is utilized.
## 6 Discussion
This research has centered on applying large deviations theory (LDT) to the
study of diffusive motion in the context of continuous time random walks with
exponential waiting times. A notable aspect of LDT is its traditional use in
probing rare events, specifically within the large $x$ limit of the
probability density function $P(x,t)$. While rate functions, a key component
in LDT, are often perceived as challenging to sample in empirical settings
[75, 76], our study suggests otherwise. By focusing on short time scales and
large $x$ values, we demonstrate that the nearly exponential decay of $P(x,t)$
\- albeit with logarithmic corrections - is more accessible for sampling than
previously assumed.
This accessibility is particularly pertinent in single tracer experiments,
where the microscopic time scale, characterized by the mean time between
jumps, is not significantly smaller than the observation period $t$. Under
these conditions, a universal exponential decay in the particle density is
observed. Additionally, our investigation delved into the realm of the big
jump principle, particularly its interplay and transition to LDT. The BJP is
critical for understanding heavy-tailed processes in which significant
fluctuations, or “big jumps”, dominate the characteristics of the random walk.
This principle becomes particularly relevant when the jump length distribution
is sub-exponential, leading to sub-exponential behavior in the propagator [74,
63, 77, 78, 79, 80]. The transition from the BJP regime $(\beta<1)$ to the LDT
regime $(\beta>1)$ highlights how different statistical frameworks govern the
behavior of diffusive processes under varying conditions. These findings are
crucial as they extend the utility of LDT and BJP beyond theoretical confines,
making them practical tools for capturing statistical behaviors of particle
dispersion that deviate from the normal distribution, inherent in various
physical systems.
While the CLT remains a cornerstone in statistical physics, our study
illustrates that deviations from CLT predictions, particularly in the tails,
are theoretically significant and experimentally observable. Furthermore, the
employment of the Edgeworth expansion has provided valuable corrections to the
CLT, leading to a more nuanced understanding of the propagator’s behavior in
different regimes. This approach has bridged the gap between the Gaussian
central, and the non-Gaussian tails of the distribution, capturing the essence
of diffusive dynamics more comprehensively.
The understanding of Laplace’s first law is not limited to the CTRW framework.
In Refs. [81, 29], a many-body scenario was studied, employing mainly
simulations. One example is the Hitchhiker model [81], a framework based on
experimental findings that the size of molecules may fluctuate due to
diffusion, aggregation, and fragmentation processes, resulting in exponential-
like tails for $P(x,t)$. Similarly, Ref. [29] argues that polymerization
processes lead to the demonstrations of Brownian yet non-Gaussian diffusion.
### 6.1 Open Questions
In the CTRW framework, a significant challenge is presented by the
intermediate regime, where neither the central limit theorem nor the large
deviations theory/big jump principle effectively approximates the results.
Rusciano et al.[16] introduced a scaling parameter, assuming exponential
statistics; $\exp{(-|x|/\lambda(t))}$, where $\lambda(t)\sim t^{1/3}$. An
opposing comment was quickly raised by Berthier et al.[82], yet the
phenomenon, particularly the non-trivial exponent $1/3$, was observed in
several other experiments [57, 23, 16, 24]. More specifically, in Ref. [57],
both the scaling parameter and the Lambert functions, as derived in this
manuscript, are demonstrated. Furthermore, in Ref. [14], a similar scaling
parameter, $\lambda(t)\sim t^{1/2}$, is proposed, adding further intrigue to
the subject. It remains unclear whether the experimental and numerical
evidence for $\lambda(t)\sim t^{b}$ represents merely a fitting issue due to
insufficient data, or if it describes some deep physics not fully understood
by the authors of this manuscript.
Chaudhuri et al. [22] proposed that the waiting time PDF is a combination of
two exponentials, with the first jump in the random walk being drawn from a
less frequent PDF compared to the subsequent jumps. Under certain conditions,
this results in more pronounced exponential tails. As mentioned in our model,
for the sake of mathematical simplicity and broad applicability, we employed a
Poisson process, while other works investigated a wider variety of processes
[47]. However, a practical question that remains largely unanswered is the
impact of non-exponential waiting times on the observations made in this
study. Similarly, the effect of changing the dimensionality of the problem on
our results is yet to be explored.
Another interesting approach to the tails of CTRW was recently pioneered by
Sokolov and Pacheco [55]. They use the rate function representation of the
temporal process (waiting times) and the rate function representation of the
spatial process (jumps) to study the properties of the rate function of
$P(x,t)$.
Finally, delving beyond sub- and super-exponential jump lengths, we still
expect Laplace-like tails, for example, the case of CTRW on a lattice [47].
One may find similar results to those presented in this manuscript, though the
Lambert function used extensively here, is not found to be generic. This
implies that the characterization of the rate function for more general jump
length distributions might yield further insights.
Acknowledgements: We thank Lucianno Defaveri for insightful discussions. We
also acknowledge the support of Israel Science Foundation’s grants 1614/21,
and 2796/20. †† *<EMAIL_ADDRESS>*<EMAIL_ADDRESS>**
<EMAIL_ADDRESS>
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|
# Context Quality Matters in Training Fusion-in-Decoder
for Extractive Open-Domain Question Answering
Kosuke Akimoto Kunihiro Takeoka Masafumi Oyamada
NEC Corporation
{kosuke_a, k_takeoka<EMAIL_ADDRESS>
###### Abstract
Retrieval-augmented generation models augment knowledge encoded in a language
model by providing additional relevant external knowledge (context) during
generation. Although it has been shown that the quantity and quality of
context impact the performance of retrieval-augmented generation models during
inference, limited research explores how these characteristics affect model
training. This paper explores how context quantity and quality during model
training affect the performance of Fusion-in-Decoder (FiD), the state-of-the-
art retrieval-augmented generation model, in extractive open-domain question
answering tasks. Experimental results suggest that FiD models overfit to
context quality during training and show suboptimal performance when evaluated
on different context quality. Through the experimental results, we also reveal
FiD models trained with different context quality have different cross-
attention distribution patterns. Specifically, as context quality during
training increases, FiD models tend to attend more uniformly to each passage
in context. Finally, based on these observations, we propose a method to
mitigate overfitting to specific context quality by introducing bias to the
cross-attention distribution, which we demonstrate to be effective in
improving the performance of FiD models on different context quality.
## 1 Introduction
Recently, large-scale pre-trained language models have achieved impressive
performance in the field of Natural Language Generation, which includes tasks
that require real-world knowledge, e.g., closed-book question answering and
common sense reasoning Brown et al. (2020). However, these models are still
prone to generate factually incorrect outputs known as hallucinations Ji et
al. (2023), particularly when dealing with rare entities Mallen et al. (2023).
Also, they cannot handle new information that arises after their training
phase Kasai et al. (2022).
In order to address these challenges, retrieval-augmented generation models
have been recently proposed Izacard and Grave (2021b); Lewis et al. (2020).
These models draw inspiration from retrieval-based extractive open-domain
question answering methods Chen et al. (2017) and utilize additional relevant
external knowledge (e.g., a Wikipedia article about an entity in a given
question) during generation to augment knowledge encoded in a language model.
Retrieval-augmented generation models have demonstrated effectiveness in
knowledge-intensive tasks Petroni et al. (2021) such as question answering and
fact checking Hofstätter et al. (2022), and have been reported to reduce
hallucinations in dialogue tasks Shuster et al. (2021).
The external knowledge given to the models is called a context, and it is
usually obtained through information retrieval systems Lin et al. (2022).
Multiple passages, typically up to 100, are often used collectively as a
single context to ensure the high recall of relevant information. This
strategy addresses the limitations of retrieval systems, which may return
irrelevant passages and fail to capture relevant information in the top
results. When dealing with contexts composed of multiple passages, we can
define their quantity (the number of passages in the context) and quality (the
proportion of relevant passages in the context). Since the context quantity
and quality vary depending on model configuration or application, e.g., the
performance of the retrieval system and the computational resources available,
understanding how these characteristics impact the model performance becomes
an important research question.
Indeed, during the inference phase, it has been shown that the quantity and
quality of contexts impact the performance of retrieval-augmented generation
models. For example, Izacard and Grave (2021b) showed that increasing the
number of top-ranked retrieved passages used as a context during inference
improves the performance of their model in the question answering task, and
Weller et al. (2022) found that the model prediction is distracted more
strongly as the proportion of conflicting misinformation in the context
increases.
However, regarding the training phase, it is not yet fully understood how
these context characteristics impact the performance of the trained models.
Limited research suggests that increasing the number of retrieved passages
used as a context during training improves question answering performance
Izacard and Grave (2021b) and reduces memorization Chen et al. (2022). Still,
the impact of quantity and quality of contexts is mixed in these studies, as
relevant passages are typically biased towards higher rank in the retrieval
result, and simply increasing the number of top-ranked passages changes both
the quantity and quality of the context.
In this paper, we focused on extractive open-domain question answering tasks
and investigated the impact of context quantity and quality on the training of
Fusion-in-Decoder (FiD) Izacard and Grave (2021b), a state-of-the-art
retrieval-augmented generation model. We demonstrate that context quality
during training affects the performance of the trained model. As far as our
knowledge, this work is the first attempt to explicitly control context
quality and investigate its effect on training of retrieval-augmented
generation models.
Key insights obtained through our experiments are as follows:
* •
FiD models overfit to context quality during training, resulting in
deteriorated performance when evaluated on a different quality of context.
* •
FiD models overfit less to context quantity compared to context quality.
* •
FiD models trained with different context qualities show different patterns of
cross-attention probability. As context quality during training increases, the
trained models tend to attend more uniformly to each passage in context and
vice versa.
Based on these observations, we propose a method to mitigate the overfitting
of a trained FiD model to specific context quality without additional training
by controlling the selectivity of its cross-attention distribution. We present
an empirical analysis demonstrating the proposed method’s effectiveness in
improving the performance of a trained FiD model when deployed in environments
with different context quality than those used during training.
## 2 Experimental Setup
In this section, we describe the task (§2.1) and model architecture (§2.2)
used in our experiments, and we define quality and quantity that we used in
this paper (§2.3, 2.4).
### 2.1 Task and Dataset
This study focuses on the extractive open-domain question answering task,
where models have to extract answers from retrieved documents. We conducted
experiments on two standard benchmark datasets of the task:
* •
Natural Questions Kwiatkowski et al. (2019) contains questions submitted to
Google Search engine. We use the open-domain version of this dataset presented
by Lee et al. (2019).
* •
TriviaQA Joshi et al. (2017) contains questions authored by trivia
enthusiasts. Following Lee et al. (2019), we use the unfiltered set of the
dataset.
For each dataset, following Izacard and Grave (2021b), we used top-100
passages retrieved by DPR Karpukhin et al. (2020)111The retrieved passages
were obtained from the published dataset at
https://github.com/facebookresearch/FiD.. As an evaluation metric, we computed
the exact match (EM) between a ground-truth answer and predicted answer
generated by greedy decoding222If a question has multiple ground-truth
answers, EM is computed as one if the predicted answer matches any one of the
ground-truth answers.. We evaluated the performance on the development set of
each dataset.
### 2.2 Model
In our experiments, we focused on Fusion-in-Decoder (FiD) Izacard and Grave
(2021b), a state-of-the-art architecture for retrieval-augmented generation
model. FiD is extended from sequence-to-sequence models, such as T5 Raffel et
al. (2020), and consists of a Transformer encoder $E$ and decoder $D$ Vaswani
et al. (2017).
Given a question $q$ and its context $c=\\{p_{i}\\}_{i=1}^{N}$, where $p_{i}$
is the $i$-th passage of the context, a FiD model converts each passage
$p_{i}$ to $\tilde{p}_{i}$ by a template "question: {q} title: {t} context:
{c}". Here, {q}, {t}, and {c} are respectively replaced by $q$, the title of
$p_{i}$, and the main text of $p_{i}$. Then, a FiD model independently encodes
each converted passage $\tilde{p}_{i}$ by the encoder $E$ and feeds their
concatenation to the decoder $D$ to get predicted answer $a$ as follows:
$a=D([E(\tilde{p}_{1});...;E(\tilde{p}_{n})]).$ (1)
We followed standard practice and trained FiD models by minimizing a cross-
entropy loss of a ground-truth answer. As the position of a passage is not
considered while encoding, the prediction of a FiD model is insensitive to the
order of the passages. Thus we did not perform any special shuffling or
ordering of the passages during training and evaluation. We used
t5-base333https://huggingface.co/t5-base Raffel et al. (2020) to initialize
the model. See Appendix A for other implementation details of training and
inference of FiD models.
### 2.3 Relevant and Irrelevant Passage
In this paper, we adopt the same definition of relevant and irrelevant passage
as in Li et al. (2022). More specifically, a passage is relevant to a question
if it logically entails an answer to the question, and it is irrelevant if it
does not.
However, in our open-domain setting, no ground-truth annotation of passage
relevance exists for Natural Questions and TriviaQA. As discussed by Li et al.
(2022), a simple rule that determines a passage that contains a ground-truth
answer as a relevant passage is insufficient to filter out irrelevant passages
that contain the answer but do not entail the answer. Since accurately
determining whether a passage is relevant or not is crucial for estimating
context quality, we applied an additional rule to extract relevant passages.
We fed a pair of a question and a passage to a pre-trained question answering
model444To annotate a passage of a question $q$ in dataset $\mathcal{D}$, we
used FiD models trained on a subset of $\mathcal{D}$ that did not contain $q$.
See Appendix C for more details., and deemed the passage relevant if the
predicted answer matched a ground-truth answer to the question555We discarded
those passages that contained a ground-truth answer but did not pass the
additional filter, and we did not use those passage in our experiments..
Following Li et al. (2022), we considered a passage that did not contain any
ground-truth answers to the question as irrelevant.
We respectively denote the set of relevant and irrelevant passages of question
$q$ by $R(q)$ and $\bar{R}(q)$, and we omit $(q)$ when it is not necessary.
### 2.4 Context Quality and Quantity
For a question $q$ and a context $c=\\{p_{i}\\}_{i=1}^{N}$ of $N$ passages, we
define context quality and quantity as follows:
* •
Context quality is defined as the proportion of passages in $c$ that is
relevant to $q$, i.e. $\frac{|R(q)|}{N}$.
* •
Context quantity is the number of passages in $c$, i.e. $N$.
## 3 Case Studies
In this section, we describe our experiments to investigate how context
quantity and quality during training affect the performance of FiD models.
Throughout the experiments in this section, we created various training and
evaluation environments with controlled context quantity or quality by
sampling $n^{+}$ relevant passages from $R(q)$ and $n^{-}$ irrelevant passages
from $\bar{R}(q)$ for each question $q$, without replacement666See Appendix B
for more details of the experimental design.. In the rest of this paper, we
define $n=n^{+}+n^{-}$ as the number of total passages and
$k=\frac{n^{-}}{n^{+}}$ as the ratio of irrelevant passages to relevant ones.
We will use subscripts $\cdot_{\text{train}}$ and $\cdot_{\text{eval}}$
respectively to denote the values of training and evaluation environments if
required.
### 3.1 Effect of Context Quality during Training
Setting: To investigate the effect of context quality during model training,
we created training and evaluation environments with the same context quantity
but different context qualities. More specifically, for each total number of
passages (i.e., context quantity) $n$ in $\\{10,25,60\\}$, we varied the value
of $n^{+}$ among $\\{1,2,3\\}$ (Natural Question) or $\\{1,2,3,5,10\\}$
(TriviaQA) to obtain environments with different context qualities.
Figure 1: Performance of FiD models on Natural Questions with varying training
context quality. Panels represent different evaluation environments with
different $(n_{\text{eval}}^{+},n_{\text{eval}})$ pairs, and a red dashed line
shows the context quality of the corresponding evaluation environment. Red
stars represent the best-performed models in the corresponding evaluation
environments. Dotted lines show models trained on the same context quantity
$n_{\text{train}}$.
Result: Figure 1 shows the performance of models with different training
context qualities777See Figure 6,7 and Table 7,8 in Appendix D for full
results.. We can obtain the following observations from the figure: (i) For a
given evaluation context quality, models trained with similar context quality
showed the highest performance. (ii) There was a trend of monotonically
decreasing performance as training context quality deviated further from
evaluation context quality. (iii) Difference of context quantity had a
negligible impact in the above trends.
Insight: FiD models overfit to context quality during training, and suboptimal
performance can be obtained when evaluated on different context qualities.
### 3.2 Effect of Context Quantity during Training
Setting: To investigate the effect of context quantity during model training,
we created training and evaluation environments with the same context quality
but different context quantities. More specifically, for each ratio $k$ among
$\\{1,5,20\\}$, we varied the value of $n^{+}$ among $\\{1,2,3\\}$ (Natural
Questions) or $\\{1,2,3,5,8,10\\}$ (TriviaQA) to change context
quantity888Since target questions were only guaranteed to have at least 64
irrelevant passages, we did not conduct experiments in environments with
$kn^{+}>64$..
Result: Figure 2 shows the performance of models with different training
context quantities999See Figure 8, 9 and Table 9, 10 in Appendix D for full
results.. As can be seen in the figure, the influence of context quantity on
model training was generally less significant than that of context quality
(§3.1). However, we observed a more significant influence for smaller
$k_{\text{train}}$ (higher context quality) in comparison to larger
$k_{\text{train}}$ (lower context quality), especially in the cases where the
training context quantity was small. One possible explanation for this
behavior is that the impact of noise in the annotation of relevant (or
irrelevant) passages disturbs the actual context quality, and this effect is
magnified in such cases due to limited context quantities. Nevertheless, our
experiments did not reveal a consistent trend in performance changes due to
varying context quantity. Hence, the results indicate that context quantity’s
influence is relatively insignificant compared to context quality’s.
Insight: Training context quantity has less influence on model performance
compared to context quality.
### 3.3 Effect of Mixed Context Quality during Training
Generally, in practical uncontrolled settings, a training dataset for FiD
models may consist of questions with different context qualities. Thus, we
conducted experiments with mixed context qualities to investigate whether a
similar overfitting phenomenon occurs for a training dataset with multiple
context qualities.
Setting: We created three environments for each dataset. Specifically, context
quantity was set to $n=10$, and $n^{+}$ was varied among $\\{1,2,3\\}$ for
Natural Questions, and context quantity was set to $n=25$, and $n^{+}$ was
varied among $\\{2,5,10\\}$ for TriviaQA. Then, we uniformly mixed each subset
of the three environments and trained FiD models in each of them101010We mixed
environments by randomly selectig the value of $n^{+}$ and sampling passages
accordingly for each question and training step.. Performance in a mixed
environment was computed by averaging the performance in each constituting
environment.
Figure 2: Performance of FiD models on TriviaQA with varying training context
quantity. Panels represent different evaluation environments with different
$(n^{+}_{\text{eval}},k_{\text{eval}})$ pairs, and a red dashed line shows the
context quantity of the corresponding evaluation environment. Dotted lines
show models trained on the same context quality
$\frac{1}{1+k_{\text{train}}}$.
Result: Table 1 shows model performance for each pair of training and
evaluation environment in Natural Questions111111See Table 5 for the result in
TriviaQA.. High scores at diagonal elements of the table show that the models
performed best or as well as the best model when they were evaluated in the
same mixture of environments as one in training. For example, the models
trained in the uniform mixture of all environments performed best only when
they were evaluated on the same mixture. It suggests that covering all context
qualities during evaluation is insufficient for optimal performance, and the
distribution of the context qualities also matters to the performance.
Table 1: Performance of FiD models trained in each mixture of environments on
Natural Questions. The color intensity indicates the relative performance
within the same evaluation mixture of environments (column). Checkmarks
indicate which environment included in each mixture.
${n^{+}}_{\text{eval}}(\rightarrow)$ ${n^{+}}_{\text{train}}(\downarrow)$ | 1 | ✓ | | | | ✓ | ✓ | ✓
---|---|---|---|---|---|---|---|---
2 | | ✓ | | ✓ | | ✓ | ✓
3 | | | ✓ | ✓ | ✓ | | ✓
1 | 2 | 3 | | | | | | | |
✓ | | | | 69.8 | 82.9 | 88.6 | 85.7 | 79.2 | 76.4 | 80.4
| ✓ | | | 65.0 | 85.5 | 92.2 | 88.9 | 78.6 | 75.3 | 80.9
| | ✓ | | 53.9 | 83.4 | 93.0 | 88.2 | 73.5 | 68.7 | 76.8
| ✓ | ✓ | | 62.3 | 85.2 | 92.5 | 88.8 | 77.4 | 73.7 | 80.0
✓ | | ✓ | | 69.1 | 84.0 | 90.0 | 87.0 | 79.5 | 76.5 | 81.0
✓ | ✓ | | | 69.7 | 83.9 | 89.6 | 86.8 | 79.6 | 76.8 | 81.1
✓ | ✓ | ✓ | | 68.3 | 84.6 | 90.7 | 87.6 | 79.5 | 76.4 | 81.2
Insight: FiD models overfit to the distribution of context qualities during
training.
### 3.4 Effect of Context Quality during Training on Model’s Cross-attention
As we discussed in §3.1, FiDs trained on different context qualities may
overfit to each quality, and they perform differently in the same evaluation
environment. We hypothesize that overfitting to different context quality
occurs due to changes in how a model selects relevant passages since lower
context quality may force the model to concentrate more on selecting passages
and vice versa. Thus, as a first step to investigate which aspect of model
inference is affected by different context qualities during training, we
analyzed how patterns of cross-attention varied among FiD models trained on
different context qualities (§3.4.1). Then, we conducted intervention
experiments to validate that different patterns of cross-attention explain
part of the overfitting to context quality (§3.4.2).
#### 3.4.1 Investigation on Patterns of Cross-attention Probability
Setting: We denote cross-attention probability from the $l$-th decoder token
to the $j$-th token of the $i$-th input passage $\tilde{p}_{i}$ at the $k$-th
decoder layer by $c^{(k)}_{ijl}$. Following Izacard and Grave (2021a), we
computed cross-attention probability from the first decoder token,
$c^{(k)}_{ij1}$, and we computed aggregated cross-attention probability
$\tilde{c}_{i}^{(k)}=\sum_{j}c_{ij1}^{(k)}$ for each passage $\tilde{p}_{i}$.
We conducted the following two analyses:
* (i)
We analyzed how much cross-attention probability was allocated to relevant
passages at each layer, i.e., $\sum_{i\in\\{i|p_{i}\in
R\\}}\tilde{c}_{i}^{(k)}$.
* (ii)
We analyzed the difference between the distribution of cross-attention
probability to relevant passages, i.e., $\\{\tilde{c}_{i}^{(k)}|i\in R\\}$,
and that to irrelevant passages, i.e.,
$\\{\tilde{c}_{i}^{(k)}|i\in\bar{R}\\}$.
We focused our analyses on FiD models trained for Natural Questions in §3.1
with the following settings: $(n,n^{+})\in\\{(10,1),(10,2),(10,3)\\}$. Note
that these models were trained with the same context quantity but different
qualities. We analyzed these models in two evaluation environments of Natural
Questions with the following settings: $(n^{+},n^{-})\in\\{(3,7),(3,57)\\}$.
Figure 3: Distribution of cross-attention probability to each relevant or
irrelevant passage at Layer 9. A similar trend can be seen in other higher
layers. Red vertical dashed lines represent uniform cross-attention
probability, i.e., $\frac{1}{N}$ if context quantity is $N$. Table 2: Cross-
attention probability allocated to relevant passages at each layer. High and
Low respectively represent high and low context quality.
| Evaluation environment $(n^{+},n^{-})$
---|---
| | $(3,7)$
---
High
| $(3,57)$
---
Low
| Training environment $(n,n^{+})=(10,n^{+})$
$n^{+}$ | | 3
---
High
| 2
---
| 1
---
Low
| 3
---
High
| 2
---
| 1
---
Low
Layer 1 | 31.5 | 31.9 | 31.0 | 5.4 | 5.5 | 5.3
Layer 2 | 31.8 | 32.3 | 32.1 | 5.4 | 5.6 | 5.6
Layer 3 | 33.7 | 34.5 | 33.6 | 6.1 | 6.3 | 6.0
Layer 4 | 32.5 | 33.4 | 32.3 | 5.7 | 6.0 | 5.7
Layer 5 | 32.4 | 33.7 | 33.3 | 5.6 | 6.0 | 6.0
Layer 6 | 33.9 | 35.4 | 36.2 | 6.1 | 6.6 | 7.2
Layer 7 | 41.1 | 45.6 | 47.4 | 8.5 | 10.7 | 13.7
Layer 8 | 40.8 | 44.1 | 45.8 | 8.6 | 10.3 | 13.2
Layer 9 | 47.4 | 51.9 | 56.3 | 11.3 | 15.1 | 21.8
Layer 10 | 48.2 | 52.9 | 55.5 | 10.8 | 13.9 | 20.4
Layer 11 | 45.6 | 49.8 | 53.4 | 10.1 | 12.6 | 19.0
Layer 12 | 39.1 | 41.2 | 42.9 | 9.5 | 11.4 | 15.7
Result: Table 2 shows the cross-attention probability that was allocated to
relevant passages at each layer (Analysis (i)). As shown in the table, in both
evaluation environments, FiD models trained with lower context quality
attended more strongly to relevant passages, especially at higher layers that
are closer to the output layer121212The result may also suggest that a
function to select relevant passages is learned at higher layers, which are
consistent with the claims made by Tenney et al. (2019) that higher layers
capture task-specific information..
A similar trend is also observed in Figure 3 that shows the distribution of
cross-attention probability to a relevant or irrelevant passage for each model
(Analysis (ii)). Models trained with lower context quality showed more long-
tailed distribution for relevant passages, and there was a more significant
difference between the distribution for relevant and irrelevant passages,
which suggests they are trained to attend more selectively to relevant
passages. On the contrary, the distribution is relatively closer to uniform
distribution for models trained with higher context quality.
We conjecture that this excessive selectivity of models trained in a low-
quality environment may explain their relatively lower performance in a high-
quality environment (§3.2), because such excessive selectivity makes the model
overlook necessary information in ignored relevant passages and, as a result,
fail to correctly answer the questions. It may be the case that, when
evaluated in a high-quality environment (i.e., where the majority of passages
are relevant to the question), it is more optimal for the model to examine all
passages more uniformly without being overly selective. This claim is
empirically supported by the result of our experiments in §4.2.
Insight: FiD models trained with different context quality show different
levels of selectivity w.r.t. allocation of cross-attention probability. Models
trained with lower context quality attend more selectively to relevant
passages.
#### 3.4.2 Intervention Experiment
Results in §3.4.1 suggests that overfitting of FiD models to different context
quality is due to different level of selectivity of their cross-attention. We
validate this claim by intervening on the cross-attention probability of these
models during inference.
Setting: We intervened on the cross-attention probability of FiD models so
that the ratio of cross-attention probability to a relevant passage $p_{i}\in
R$ and an irrelevant passage $p_{j}\in\bar{R}$ to be $r$ for all layers.
Intuitively, the model completely ignores irrelevant passages when $r=0$,
whereas the model attends uniformly to all passages when $r=1$. More
specifically, for each decoder layer $k$, we converted original cross-
attention probability $c_{ijl}^{(k)}$ into intervened version
${c^{\prime}}_{ijl}^{(k)}$ as follows:
${c^{\prime}}_{ijl}^{(k)}=\frac{w_{i}^{(k)}}{\sum_{j^{\prime}}c_{ij^{\prime}l}^{(k)}}c_{ijl}^{(k)},$
(2)
where $w_{i}^{(k)}=\frac{1}{n^{+}+rn^{-}}$ if $p_{i}\in R$ and
$w_{i}^{(k)}=\frac{r}{n^{+}+rn^{-}}$ if $p_{i}\in\bar{R}$. We selected $r$
from $\\{1,0.1,0\\}$ and conducted experiments in the same models and
evaluation environments as in §3.4.1.
Result: Figure 4 shows model performance with and without intervention on
cross-attention probability. In both evaluation environments with lower and
higher context quality, the difference in the performance of the models
decreased, and the intervention mitigated the effect of overfitting to context
quality.
Insight: The result suggests that the difference of cross-attention
probability as described in §3.4.1 is one element that explains the
overfitting of FiD models to context quality.
## 4 Adapting Models to Different Context Quality
Figure 4: Model performance under intervention on cross-attention probability.
"No" represents a setting without intervention.
While FiD models overfit to context quality during training as shown in §3, it
is not desirable to train a dedicated model for each target environment that
has potentially different context qualities from each other. Thus, in this
section, we propose a method to mitigate the effect of overfitting and adapt
an already trained FiD model to an environment with different context quality.
### 4.1 Proposed Method
Based on the insights in §3.4 that shows the overfitting to context quality
occurs due to the different levels of selectivity to relevant passages, we
propose to change sharpness of distribution of cross-attention probability
during inference. More specifically, we introduce temperature parameter $T$
$(T>0)$ and compute total cross-attention probability from the $l$-th decoder
token to the $i$-th passage at the $k$-th layer as follows:
$w_{il}^{(k)}=\left(\text{softmax}\left[\frac{\log(\sum_{j}c_{1jl})}{T},...,\frac{\log(\sum_{j}c_{Njl})}{T}\right]\right)_{i}.$
(3)
Then, we use Equation (2) to convert cross-attention probability as in
§3.4.2131313We use $w_{il}^{(k)}$ instead of $w_{i}^{(k)}$ in Equation (2)
when decoding the $l$-th token. Note that the cross-attention probability is
sequentially computed for each decoder token during inference.. Intuitively,
the model attends more uniformly as $T$ becomes larger, which simulates the
overfitting effect of a FiD model trained with higher context quality and vice
versa.
Note that our proposed temperature parameter does not change the set of input
passages and can be tuned complementary with other existing hyperparameter
that changes the set of input passages, e.g., the number of input passages.
### 4.2 Experiment
Figure 5: Top panels: Performance of FiD models on Natural Questions with
adaptation by the proposed method (solid lines) and without adaptation (dotted
lines). Bottom panels: Optimal temperature parameter $T^{*}$ selected for each
model. Multiple $T^{*}$ were selected for some context qualities, i.e.,
training environments, because we selected $T^{*}$ for each of the three
models trained with different random seeds for each training environment.
Panels represent different evaluation environments with different
$(n_{\text{eval}}^{+},n_{\text{eval}})$ pairs, and a red dashed line shows the
context quality of the corresponding evaluation environment.
To validate the effectiveness of our proposed method, we adapted models
trained in §3.1 by the proposed method and evaluated their performance on
evaluation environments with different context qualities where
$n_{\text{eval}}^{+}=3$ and $n_{\text{eval}}\in\\{10,25,60\\}$. Since the
temperature parameter $T$ has to be tuned, we conducted 2-fold cross-
validation. Specifically, we split the evaluation data into two folds and
searched optimal temperature parameter $T^{*}\in\\{0.125,0.25,0.5,1,2,4,8\\}$
based on the EM score on one fold, and then, we used a model adapted with
$T^{*}$ to evaluate performance on the other fold141414We used the single
temperature parameter $T^{*}$ for all questions in a fold instead of using a
different temperature parameter for each question. Predicting the optimal
temperature parameter for each input question is an interesting direction of
future works..
Figure 5 shows the performance of FiD models with and without adaptation by
the proposed method151515See Figure 10 in Appendix D for the result in
TriviaQA.. As shown in the figure, the proposed method improved the
performance of the models in environments with different context qualities
compared to those during training, and it reduced the effect of overfitting to
context quality. Also, $T^{*}$ increased in the case of lower context
qualities during training, and vice versa, which corroborates our finding that
more uniform cross-attention, corresponding to higher $T^{*}$ , is effective
when context quality in evaluation is higher than one in training.
## 5 Related Works
Retrieval-augmented Generation Models: Lewis et al. (2020) introduced
retrieval augmentation approach, firstly originating in the field of
extractive open-domain question answering Chen et al. (2017), to sequence-to-
sequence models and validated its effectiveness in knowledge-intensive tasks.
Contemporary work by Min et al. (2020) applied the retrieval augmentation
approach to the task of ambiguous question answering. Izacard and Grave
(2021b) proposed Fusion-in-Decoder (FiD), in which each retrieved passage is
independently encoded and then jointly input to the decoder. FiD achieves high
scalability for the number of passages and effectively aggregates information
by jointly using all passages in the decoder.
Recently, the retrieval-augmented language generation approach has received
attention, including its incorporation into the language model pretraining
Borgeaud et al. (2022); Izacard et al. (2022); Zhang et al. (2022b), dialogue
generation Komeili et al. (2022), and code generation Parvez et al. (2021).
The approach has also been shown effective in improving inference of pre-
trained language models (e.g., GPT-3 Brown et al. (2020)) without additional
training Lazaridou et al. (2022); Mallen et al. (2023). For a comprehensive
survey on this evolving field, refer to Yu et al. (2022).
Effect of Context Characteristics on Retrieval-augmented Models: Several
studies have investigated how context characteristics affect inference of
retrieval-augmented generation models. For example, increasing the number of
top-ranking passages used as the context has been found to improve performance
in question answering Izacard and Grave (2021b) and response/prose generation
Zhang et al. (2022b), while a higher proportion of false information in the
context degrades performance in question answering Weller et al.
(2022)161616Du et al. (2022) reported similar results in fact verification,
while their focus is not on language generation models.. Liu et al. (2023)
found that the performance of language models on multi-document question
answering is influenced by the position of a relevant document.
However, limited knowledge is available regarding the impact of context
characteristics on the training of retrieval-augmented geneartion models.
Notably, a few existing research suggest that the model performance in
question answering improves by providing more top-ranking passages during
training Izacard and Grave (2021b) or by randomly masking top-ranking passages
during training Zhang et al. (2022a), and that the model’s memorization
behavior is reduced by increasing recall of relevant information in the
context during training Longpre et al. (2021); Chen et al. (2022).
## 6 Conclusion
In this paper, we investigate how context quality and quantity affect the
training of FiD models in extractive open-domain question answering tasks. We
show that FiD models tend to overfit to the context quality during training,
resulting in degraded performance when evaluated in environments with
different context qualities. Additionally, our research reveals that the
overfitting to context quality is partially explained by different patterns in
the model’s cross-attention probability. Based on these observations, we
propose changing the selectivity of the cross-attention probability to
mitigate the effect of overfitting to context quality. The results of this
paper suggest a broad spectrum of future work, including more sophisticated
adaptation methods and investigations of the effect of other context
characteristics on the training of retrieval-augmented generation models.
## 7 Limitations
In this study, we investigated how the quality and quantity of context affect
the training of FiD models in extractive open-domain question answering tasks.
Our experiments revealed for the first time that context quality significantly
impacts FiD models’ training and that FiD models tend to overfit to context
quality of the training data. The implications of our findings suggest that
various context characteristics similarly affect the training of retrieval-
augmented generation models, potentially leading to issues such as
overfitting.
However, our experiments have several limitations that reduce the
generalizability of our findings:
Task:
Firstly, in this paper, we only focused on the extractive open-domain question
answering task, and it is unclear whether similar results can be obtained in
other tasks such as dialogue generation, fact verification, code generation,
and summarization.
Model Architecture:
Secondly, our analysis only targeted FiD models, and it is unclear whether
different architectures such as RAG Lewis et al. (2020) and Internet-augmented
language models Lazaridou et al. (2022) produce similar results. Also, it is
an interesting direction of future work to conduct similar investigations on
non-generative retrieval-augmented models such as FiE Kedia et al. (2022).
Model Size:
Thirdly, our experiments are focused on only t5-base, and it is unclear how
scaling model size changes the behavior of overfitting to context quality.
Characteristic of Context:
Lastly, coverage of our analysis is limited to quality and quantity, and
further research is required to investigate the effect of other context
characteristics. For example, in the field of extractive question answering,
it has been shown that models may overfit to answer positions in the contexts
Ko et al. (2020), be misled by adversarially inserted sentence Jia and Liang
(2017); Jiang and Bansal (2019), and be susceptible to whether an answer is in
the most similar sentence in the context Sugawara et al. (2018). These
findings suggest that those context characteristics may also affect retrieval-
augmented generation models.
Other than that, our experiments involved the automatic annotation of relevant
and irrelevant passages, which may limit the accuracy of our analysis. Future
studies should incorporate human annotation to ensure the high quality of the
annotation. Also, passages with similar relevant information can impact models
differently due to qualitative factors such as readability and writing style.
Nevertheless, as quantitatively evaluating these factors poses challenges, our
study did not conduct a fine-grained analysis regarding these aspects.
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## Appendix A Implementation Details
### A.1 Training and Evaluation Data
We trained FiD models with the original train set of each dataset, and we
further split the original train set into $\mathcal{D}_{\text{train}}$ for
training and $\mathcal{D}_{\text{dev}}$ for evaluating performance during
training. To train models in a strictly extractive task environment, we
excluded questions for which no retrieved passage contained any of their
ground-truth answers. For evaluation, we used the original development set of
each dataset as evaluation data $\mathcal{D}_{\text{eval}}$. For a fair
comparison, we used the same set of questions with at least 3 (Natural
Questions) or 10 (TriviaQA) relevant passages and at least 64 irrelevant
passages during training or evaluation. Statistics of the datasets are shown
in Table 3.
### A.2 Details of FiD Training and Inference
Our model implementation of FiD, including the loss function for training, is
based on the official implementation by Izacard and Grave
(2021b)171717https://github.com/facebookresearch/FiD. For both training and
inference, we used transformers (Ver. 4.23.1) Wolf et al. (2020). We trained
the models with Seq2SeqTrainer provided in transformers. Hyperparameters for
Seq2SeqTrainer used in our experiments are listed in Table 4 and other
hyperparameters were set to default values. Since we trained models with 32
A100 GPUs, the effective batch size is 64. We used a model checkpoint with
highest EM on $\mathcal{D}_{\text{dev}}$ for downstream evaluations.
Table 3: Size of datasets used for training and evaluation. Natural Questions | TriviaQA
---|---
$\mathcal{D}_{\text{train}}$ | $\mathcal{D}_{\text{dev}}$ | $\mathcal{D}_{\text{eval}}$ | $\mathcal{D}_{\text{train}}$ | $\mathcal{D}_{\text{dev}}$ | $\mathcal{D}_{\text{eval}}$
20728 | 3048 | 2589 | 11414 | 1695 | 1434
Table 4: Hyperparameters for Seq2SeqTrainer
parameter | value
---|---
learning_rate | 0.00005
lr_scheduler_type | constant_with_warmup
warmup_steps | 1000
weight_decay | 0.01
max_grad_norm | 1.0
max_steps | 15000
per_device_train_batch_size | 1
gradient_accumulation | 2
eval_steps | 500
save_steps | 500
save_strategy | steps
Table 5: Performance of FiD models trained in each mixture of environments on
TriviaQA. The color intensity indicates the relative performance within the
same evaluation mixture of environments (column). Checkmarks indicate which
environment included in each mixture.
${n^{+}}_{\text{eval}}(\rightarrow)$ ${n^{+}}_{\text{train}}(\downarrow)$ | 2 | ✓ | | | | ✓ | ✓ | ✓
---|---|---|---|---|---|---|---|---
5 | | ✓ | | ✓ | | ✓ | ✓
10 | | | ✓ | ✓ | ✓ | | ✓
2 | 5 | 10 | | | | | | | |
✓ | | | | 79.8 | 93.9 | 98.3 | 96.1 | 89.0 | 86.8 | 90.6
| ✓ | | | 75.0 | 93.8 | 98.3 | 96.0 | 86.6 | 84.4 | 89.0
| | ✓ | | 61.7 | 90.0 | 97.7 | 93.8 | 79.7 | 75.8 | 83.1
| ✓ | ✓ | | 73.4 | 92.4 | 97.7 | 95.1 | 85.6 | 82.9 | 87.9
✓ | | ✓ | | 79.1 | 93.3 | 97.5 | 95.4 | 88.3 | 86.2 | 90.0
✓ | ✓ | | | 79.0 | 94.2 | 98.4 | 96.3 | 88.7 | 86.6 | 90.5
✓ | ✓ | ✓ | | 77.8 | 93.4 | 98.0 | 95.7 | 87.9 | 85.6 | 89.7
Since most questions in Natural Questions are annotated with only one ground-
truth answer, we used the first ground-truth answer for each question as a
target output for model training. On the other hand, since questions in
TriviaQA are more exhaustively annotated with paraphrases of ground-truth
answers, we randomly sampled one ground-truth answer that appeared in any of
the input passages as a target output at every training step.
We tokenized each input passage $\tilde{p}_{i}$ described in §2.2 and target
outputs by the tokenizer of t5-base. For both training and inference, to fix
the sequence length of each tokenized passage to 256, we conducted truncation
for longer passages or padding for shorter passages. We did not truncate
target outputs during training and set a maximum length of a predicted answer
to 50 during inference.
In experiments where we subsampled passages to control context quality and
quantity, to reduce the effect of bias in sampled passages, we sampled
different passages at every training step instead of repeatedly using the
fixed set of passages sampled before training.
## Appendix B Details of Experimental Design
We trained three FiD models with different random seeds for each training
environment and conducted evaluations for these models in each evaluation
environment. We sampled five different sets of passages for each evaluation
environment and computed average model performance on these sets of passages.
We independently sampled relevant and irrelevant passages, and thus, the same
set of relevant (or irrelevant) passages was sampled regardless of the number
of irrelevant (or relevant) passages as long as the number of relevant (or
irrelevant) passages was the same.
Table 6: Size of datasets used to train FiD models for the relevant passage annotation $\mathcal{D}_{0,\text{train}}$ | $\mathcal{D}_{0,\text{dev}}$ | $\mathcal{D}_{1,\text{train}}$ | $\mathcal{D}_{1,\text{dev}}$
---|---|---|---
30612 | 4411 | 30589 | 4373
(Natural Questions)
28696 | 4112 | 28489 | 4115
(TriviaQA)
## Appendix C Details of Passage Relevance Annotation by Question Answering
Model
We used FiD models as pre-trained question answering models used in the
passage relevance annotation described in §2.3, and we trained those models as
described in Appendix A except training and development data which we describe
below. For each dataset with original train set $\mathcal{D}_{\text{train}}$
and development set $\mathcal{D}_{\text{dev}}$, we split
$\mathcal{D}_{\text{train}}$ into four sets: $\mathcal{D}_{0,\text{train}}$,
$\mathcal{D}_{0,\text{dev}}$, $\mathcal{D}_{1,\text{train}}$, and
$\mathcal{D}_{1,\text{dev}}$. Then, we respectively trained a FiD model
$\mathcal{M}_{0}$ or $\mathcal{M}_{1}$ with a pair of train and development
data $(\mathcal{D}_{0,\text{train}},\mathcal{D}_{0,\text{dev}})$ or
$(\mathcal{D}_{1,\text{train}},\mathcal{D}_{1,\text{dev}})$. Finally, we
annotated $\mathcal{D}_{0,\text{train}}$ and $\mathcal{D}_{0,\text{dev}}$ with
$\mathcal{M}_{1}$, and $\mathcal{D}_{1,\text{train}}$,
$\mathcal{D}_{1,\text{dev}}$, and $\mathcal{D}_{\text{dev}}$ with
$\mathcal{M}_{0}$. See Table 6 for statistics of the datasets used to train
FiD models for the relevant passage annotation.
Our preliminary experiments showed that the behavior of FiD models differs
when trained with all passages in the original dataset (All), compared to when
trained with only those passages containing a ground-truth answer (Pos). Thus,
we chose a stricter criterion to extract relevant passages. Specifically, we
trained $\mathcal{M}_{0}$ and $\mathcal{M}_{1}$ and annotated passages in each
of All and Pos setting, and we extracted only those passages annotated as a
relevant passage in the both settings.
## Appendix D Full Experimental Results
Full results of the experiments in §3.1 are shown in Figure 6 and Table 7 for
TriviaQA and Figure 7 and Table 8 for Natural Questions.
Full results of the experiments in §3.2 are shown in Figure 8 and Table 9 for
TriviaQA and Figure 9 and Table 10 for Natural Questions.
Results of the experiments in §3.3 are shown in Table 5 for TriviaQA.
Results of the experiments in §4.2 are shown in Figure 10.
Figure 6: Performance of FiD models on TriviaQA with varying training context
quality. Panels represent different evaluation environments with different
$(n_{\text{eval}}^{+},n_{\text{eval}})$ pairs, and a red dashed line shows
corresponding context quality. Red stars represent the best performed models
in the corresponding evaluation environments. Dotted lines show models trained
on the same context quantity $n_{\text{train}}$. Table 7: Exact match (EM) [%]
of FiD models trained in environment
$({n^{+}}_{\text{train}},n_{\text{train}})$ in various evaluation environment
$({n^{+}}_{\text{eval}},n_{\text{eval}})$ in TriviaQA. The standard deviations
for each reported value is denoted with lower subscripts.
| $n_{\text{eval}}$ | 10 | 25 | 60
---|---|---|---|---
${n^{+}}_{\text{eval}}$ | ${n^{+}}_{\text{train}}$ $n_{\text{train}}$ | 10 | 25 | 60 | 10 | 25 | 60 | 10 | 25 | 60
| 1 | $70.9_{3.0}$ | $70.4_{1.5}$ | $70.1_{1.9}$ | $58.5_{3.1}$ | $60.1_{1.6}$ | $61.9_{3.1}$ | $45.7_{3.5}$ | $49.8_{1.8}$ | $53.9_{3.4}$
| 2 | $69.0_{1.2}$ | $72.1_{2.7}$ | $72.4_{1.1}$ | $52.3_{1.9}$ | $60.8_{3.4}$ | $64.1_{2.0}$ | $34.4_{1.9}$ | $49.2_{4.0}$ | $56.1_{2.6}$
1 | 3 | $64.3_{0.5}$ | $70.7_{0.8}$ | $71.4_{1.1}$ | $43.6_{0.5}$ | $57.4_{1.6}$ | $61.8_{1.7}$ | $24.5_{0.2}$ | $43.1_{2.3}$ | $52.2_{2.2}$
| 5 | $52.2_{0.8}$ | $66.9_{0.9}$ | $70.3_{2.0}$ | $29.1_{1.3}$ | $51.0_{1.7}$ | $59.2_{2.9}$ | $15.2_{0.5}$ | $34.1_{2.4}$ | $47.2_{3.3}$
| 10 | $18.0_{0.6}$ | $55.3_{1.2}$ | $63.7_{1.8}$ | $9.5_{0.2}$ | $36.4_{0.5}$ | $50.3_{1.4}$ | $6.4_{0.2}$ | $21.2_{0.4}$ | $36.7_{0.6}$
| 1 | $87.9_{1.4}$ | $86.7_{0.6}$ | $84.9_{0.2}$ | $77.9_{2.2}$ | $77.9_{1.3}$ | $77.9_{1.7}$ | $63.9_{3.3}$ | $66.6_{1.6}$ | $69.9_{3.1}$
| 2 | $89.1_{0.5}$ | $88.5_{1.0}$ | $87.1_{0.4}$ | $76.6_{1.0}$ | $79.8_{2.0}$ | $80.4_{1.1}$ | $56.2_{2.0}$ | $68.1_{3.5}$ | $72.7_{2.1}$
2 | 3 | $86.7_{0.3}$ | $88.7_{0.4}$ | $87.8_{0.4}$ | $71.3_{0.2}$ | $78.6_{1.0}$ | $79.8_{1.1}$ | $45.9_{0.6}$ | $63.8_{2.3}$ | $70.2_{1.5}$
| 5 | $81.2_{0.5}$ | $87.0_{0.2}$ | $87.3_{0.7}$ | $56.9_{1.2}$ | $75.0_{0.5}$ | $78.6_{1.7}$ | $29.6_{1.4}$ | $56.1_{2.2}$ | $66.7_{2.8}$
| 10 | $39.6_{0.6}$ | $80.1_{0.7}$ | $83.0_{1.1}$ | $17.8_{0.6}$ | $61.7_{0.7}$ | $71.7_{1.4}$ | $9.5_{0.3}$ | $39.0_{0.9}$ | $57.0_{1.3}$
| 1 | $93.7_{0.7}$ | $92.1_{0.1}$ | $90.4_{0.4}$ | $86.3_{1.6}$ | $85.4_{0.9}$ | $84.5_{0.8}$ | $73.9_{2.9}$ | $75.5_{1.4}$ | $77.2_{2.2}$
| 2 | $95.4_{0.1}$ | $93.6_{0.2}$ | $92.0_{0.1}$ | $86.9_{0.8}$ | $87.5_{1.3}$ | $87.0_{0.6}$ | $69.7_{1.9}$ | $77.5_{3.0}$ | $80.2_{1.4}$
3 | 3 | $94.3_{0.4}$ | $94.7_{0.2}$ | $92.6_{0.4}$ | $83.6_{0.5}$ | $87.6_{0.4}$ | $87.0_{0.5}$ | $61.9_{0.9}$ | $75.0_{1.7}$ | $79.2_{1.3}$
| 5 | $92.3_{0.4}$ | $93.7_{0.5}$ | $93.1_{0.1}$ | $74.6_{0.7}$ | $85.4_{0.1}$ | $86.4_{0.9}$ | $44.7_{1.1}$ | $69.6_{1.5}$ | $76.5_{2.1}$
| 10 | $59.6_{0.4}$ | $90.8_{0.2}$ | $90.8_{0.7}$ | $27.3_{0.4}$ | $76.1_{0.3}$ | $81.7_{0.8}$ | $12.8_{0.1}$ | $53.6_{0.7}$ | $68.5_{0.8}$
| 1 | $97.7_{0.3}$ | $96.6_{0.1}$ | $95.1_{1.0}$ | $93.3_{0.8}$ | $92.3_{0.4}$ | $91.1_{0.1}$ | $85.4_{1.8}$ | $85.6_{0.7}$ | $85.9_{1.1}$
| 2 | $98.5_{0.1}$ | $97.5_{0.4}$ | $96.1_{0.4}$ | $94.8_{0.4}$ | $93.9_{0.3}$ | $92.6_{0.0}$ | $84.2_{0.9}$ | $87.5_{1.4}$ | $88.5_{0.4}$
5 | 3 | $97.8_{0.1}$ | $98.0_{0.1}$ | $96.6_{0.6}$ | $93.1_{0.1}$ | $94.6_{0.4}$ | $93.2_{0.3}$ | $79.5_{0.6}$ | $86.8_{0.9}$ | $88.3_{0.5}$
| 5 | $97.7_{0.2}$ | $97.6_{0.5}$ | $97.1_{0.2}$ | $90.1_{0.6}$ | $93.8_{0.6}$ | $93.8_{0.1}$ | $67.2_{0.8}$ | $83.5_{0.3}$ | $87.2_{0.8}$
| 10 | $83.8_{0.2}$ | $97.0_{0.3}$ | $96.3_{0.2}$ | $47.4_{0.6}$ | $90.0_{0.2}$ | $91.7_{0.2}$ | $21.9_{0.3}$ | $72.4_{0.2}$ | $82.0_{0.9}$
| 1 | $99.2_{0.2}$ | $98.9_{0.1}$ | $97.8_{0.9}$ | $98.3_{0.1}$ | $97.5_{0.1}$ | $96.3_{0.5}$ | $94.6_{0.6}$ | $93.6_{0.1}$ | $92.9_{0.3}$
| 2 | $99.3_{0.1}$ | $99.0_{0.1}$ | $98.2_{0.1}$ | $98.9_{0.0}$ | $98.3_{0.4}$ | $97.0_{0.3}$ | $95.2_{0.3}$ | $95.2_{0.0}$ | $94.4_{0.2}$
10 | 3 | $99.0_{0.2}$ | $99.2_{0.1}$ | $98.6_{0.4}$ | $98.0_{0.2}$ | $98.5_{0.1}$ | $97.5_{0.4}$ | $93.0_{0.2}$ | $95.6_{0.3}$ | $94.9_{0.3}$
| 5 | $99.2_{0.2}$ | $99.0_{0.3}$ | $98.9_{0.3}$ | $98.1_{0.2}$ | $98.3_{0.6}$ | $97.9_{0.3}$ | $89.6_{0.6}$ | $94.6_{0.7}$ | $95.2_{0.6}$
| 10 | $98.8_{0.1}$ | $98.9_{0.2}$ | $98.4_{0.4}$ | $81.4_{0.4}$ | $97.7_{0.4}$ | $97.5_{0.2}$ | $44.3_{0.3}$ | $90.9_{0.4}$ | $93.5_{0.5}$
Table 8: Exact match (EM) [%] of FiD models trained in environment
$({n^{+}}_{\text{train}},n_{\text{train}})$ in various evaluation environment
$({n^{+}}_{\text{eval}},n_{\text{eval}})$ in Natural Questions. The standard
deviations for each reported value is denoted with lower subscripts.
| $n_{\text{eval}}$ | 10 | 25 | 60
---|---|---|---|---
${n^{+}}_{\text{eval}}$ | ${n^{+}}_{\text{train}}$ $n_{\text{train}}$ | 10 | 25 | 60 | 10 | 25 | 60 | 10 | 25 | 60
| 1 | $69.8_{0.7}$ | $70.3_{0.3}$ | $65.8_{0.3}$ | $58.1_{0.6}$ | $62.4_{0.7}$ | $61.0_{0.1}$ | $45.5_{0.8}$ | $54.1_{1.0}$ | $55.9_{0.4}$
1 | 2 | $65.0_{0.5}$ | $69.6_{0.2}$ | $68.1_{0.5}$ | $45.4_{1.2}$ | $58.6_{0.5}$ | $60.7_{0.8}$ | $24.1_{1.5}$ | $45.7_{0.8}$ | $53.2_{1.2}$
| 3 | $53.9_{1.2}$ | $65.8_{0.2}$ | $67.8_{0.4}$ | $29.0_{0.4}$ | $50.7_{0.8}$ | $58.7_{0.2}$ | $10.6_{0.2}$ | $33.3_{2.1}$ | $49.2_{0.3}$
| 1 | $82.9_{0.4}$ | $80.3_{0.3}$ | $74.0_{1.4}$ | $73.7_{0.2}$ | $74.1_{0.2}$ | $70.8_{0.8}$ | $61.9_{0.5}$ | $67.0_{0.6}$ | $66.5_{0.3}$
2 | 2 | $85.5_{0.1}$ | $83.9_{0.3}$ | $79.9_{0.1}$ | $71.4_{0.5}$ | $75.7_{0.1}$ | $74.5_{0.4}$ | $48.2_{1.4}$ | $64.3_{0.6}$ | $67.8_{0.7}$
| 3 | $83.4_{0.1}$ | $84.2_{0.3}$ | $81.9_{0.7}$ | $60.9_{0.6}$ | $73.9_{0.3}$ | $75.5_{0.8}$ | $29.8_{0.1}$ | $57.6_{1.3}$ | $66.6_{0.7}$
| 1 | $88.6_{0.8}$ | $84.4_{0.5}$ | $77.4_{1.6}$ | $80.8_{0.5}$ | $79.2_{0.2}$ | $74.4_{1.0}$ | $70.3_{0.1}$ | $72.9_{0.4}$ | $70.8_{0.9}$
3 | 2 | $92.2_{0.2}$ | $89.5_{0.6}$ | $84.6_{0.8}$ | $82.4_{0.5}$ | $83.2_{0.5}$ | $79.6_{0.1}$ | $63.5_{1.0}$ | $73.6_{0.5}$ | $74.4_{0.5}$
| 3 | $93.0_{0.1}$ | $91.2_{0.5}$ | $87.6_{0.6}$ | $78.8_{0.3}$ | $83.7_{0.4}$ | $82.1_{0.8}$ | $48.0_{0.2}$ | $70.5_{0.5}$ | $74.8_{0.6}$
Table 9: Exact match (EM) [%] of FiD models trained in environment
$({n^{+}}_{\text{train}},k_{\text{train}})$ in various evaluation environment
$({n^{+}}_{\text{eval}},k_{\text{eval}})$ in TriviaQA. The standard deviations
for each reported value is denoted with lower subscripts.
| $k_{\text{eval}}$ | 1 | 5 | 20
---|---|---|---|---
${n^{+}}_{\text{eval}}$ | ${n^{+}}_{\text{train}}$ $k_{\text{train}}$ | 1 | 5 | 20 | 1 | 5 | 20 | 1 | 5 | 20
1 | 1 | $90.8_{0.2}$ | $90.1_{1.4}$ | $88.6_{0.9}$ | $77.4_{0.3}$ | $76.9_{2.3}$ | $77.5_{2.5}$ | $53.5_{0.3}$ | $57.6_{3.6}$ | $63.4_{3.5}$
2 | $89.0_{0.2}$ | $89.4_{0.6}$ | $88.3_{0.2}$ | $69.0_{0.1}$ | $77.6_{0.4}$ | $77.1_{0.4}$ | $32.2_{1.1}$ | $58.7_{1.0}$ | $63.4_{1.1}$
3 | $88.0_{0.4}$ | $89.4_{0.3}$ | $87.8_{0.8}$ | $68.8_{1.8}$ | $77.1_{0.8}$ | $77.1_{1.5}$ | $34.8_{3.5}$ | $58.3_{1.1}$ | $63.7_{2.9}$
5 | $85.8_{0.3}$ | $87.6_{0.3}$ | | $65.4_{0.8}$ | $74.0_{0.5}$ | | $33.3_{1.3}$ | $54.7_{1.8}$ |
8 | $84.0_{1.1}$ | $86.1_{0.5}$ | | $64.0_{1.6}$ | $72.8_{0.7}$ | | $35.3_{1.9}$ | $53.9_{2.0}$ |
10 | $83.6_{0.6}$ | $84.3_{1.2}$ | | $63.9_{0.6}$ | $70.7_{1.8}$ | | $35.3_{0.7}$ | $52.7_{1.2}$ |
2 | 1 | $95.4_{0.2}$ | $95.1_{0.4}$ | $93.7_{0.6}$ | $85.1_{0.5}$ | $85.8_{1.2}$ | $85.7_{1.4}$ | $61.0_{0.3}$ | $66.4_{3.4}$ | $73.0_{3.2}$
2 | $95.6_{0.1}$ | $95.2_{0.5}$ | $93.9_{0.2}$ | $81.2_{0.2}$ | $87.0_{0.6}$ | $85.9_{0.4}$ | $38.0_{1.2}$ | $68.4_{1.0}$ | $73.9_{1.4}$
3 | $95.0_{0.1}$ | $94.6_{0.5}$ | $93.5_{0.1}$ | $81.1_{1.1}$ | $86.6_{0.7}$ | $86.3_{0.7}$ | $41.5_{3.4}$ | $68.4_{0.7}$ | $74.8_{2.5}$
5 | $93.6_{0.1}$ | $93.9_{0.3}$ | | $77.9_{0.4}$ | $84.9_{0.6}$ | | $40.2_{1.4}$ | $65.6_{2.1}$ |
8 | $92.0_{0.7}$ | $92.8_{0.3}$ | | $75.4_{1.5}$ | $83.1_{0.3}$ | | $42.1_{1.9}$ | $64.6_{1.9}$ |
10 | $91.5_{0.2}$ | $91.6_{0.8}$ | | $74.6_{0.2}$ | $81.3_{1.3}$ | | $41.8_{0.4}$ | $63.3_{1.5}$ |
3 | 1 | $96.9_{0.1}$ | $96.7_{0.3}$ | $95.0_{0.3}$ | $88.2_{0.2}$ | $89.4_{0.9}$ | $88.9_{0.9}$ | $63.5_{0.6}$ | $69.8_{3.2}$ | $76.5_{3.5}$
2 | $97.3_{0.3}$ | $96.6_{0.5}$ | $95.3_{0.2}$ | $84.9_{0.2}$ | $90.6_{0.7}$ | $89.5_{0.3}$ | $39.9_{1.2}$ | $71.3_{0.9}$ | $77.7_{1.2}$
3 | $97.2_{0.1}$ | $96.2_{0.3}$ | $95.2_{0.1}$ | $85.7_{1.3}$ | $90.0_{0.5}$ | $89.6_{0.6}$ | $43.9_{3.5}$ | $71.3_{0.4}$ | $78.5_{2.1}$
5 | $96.2_{0.2}$ | $95.9_{0.1}$ | | $82.9_{0.4}$ | $89.2_{0.2}$ | | $43.0_{1.0}$ | $69.3_{1.6}$ |
8 | $95.1_{0.1}$ | $95.0_{0.4}$ | | $80.9_{1.0}$ | $87.1_{0.3}$ | | $45.4_{2.4}$ | $68.9_{2.4}$ |
10 | $94.4_{0.1}$ | $93.7_{0.5}$ | | $80.2_{0.3}$ | $86.0_{0.7}$ | | $45.1_{0.3}$ | $67.6_{1.0}$ |
5 | 1 | $97.7_{0.2}$ | $98.0_{0.0}$ | $96.8_{0.4}$ | $90.0_{0.3}$ | $92.0_{0.4}$ | $91.6_{0.5}$ | | |
2 | $98.0_{0.1}$ | $97.9_{0.3}$ | $96.9_{0.5}$ | $87.8_{0.3}$ | $93.2_{0.4}$ | $92.3_{0.1}$ | | |
3 | $98.1_{0.2}$ | $97.6_{0.3}$ | $96.9_{0.4}$ | $88.4_{1.1}$ | $92.3_{0.2}$ | $92.4_{0.5}$ | | |
5 | $97.7_{0.2}$ | $97.7_{0.1}$ | | $86.5_{0.3}$ | $92.5_{0.3}$ | | | |
8 | $97.1_{0.2}$ | $97.1_{0.2}$ | | $84.9_{1.1}$ | $90.8_{0.3}$ | | | |
10 | $96.8_{0.1}$ | $96.3_{0.2}$ | | $84.1_{0.4}$ | $90.3_{0.3}$ | | | |
8 | 1 | $98.1_{0.0}$ | $98.7_{0.1}$ | $97.4_{0.6}$ | $91.3_{0.2}$ | $93.7_{0.2}$ | $93.3_{0.2}$ | | |
2 | $98.4_{0.1}$ | $98.5_{0.3}$ | $97.8_{0.3}$ | $89.1_{0.4}$ | $94.5_{0.3}$ | $94.2_{0.1}$ | | |
3 | $98.6_{0.3}$ | $98.2_{0.2}$ | $97.6_{0.3}$ | $90.2_{1.0}$ | $93.9_{0.4}$ | $94.5_{0.2}$ | | |
5 | $98.4_{0.2}$ | $98.4_{0.3}$ | | $88.8_{0.2}$ | $94.2_{0.4}$ | | | |
8 | $97.9_{0.3}$ | $98.0_{0.3}$ | | $87.7_{0.7}$ | $93.3_{0.4}$ | | | |
10 | $97.9_{0.0}$ | $97.4_{0.3}$ | | $87.0_{0.2}$ | $92.6_{0.4}$ | | | |
10 | 1 | $98.5_{0.1}$ | $99.0_{0.1}$ | $98.0_{0.6}$ | $91.9_{0.4}$ | $94.4_{0.1}$ | $94.1_{0.2}$ | | |
2 | $98.6_{0.2}$ | $98.7_{0.2}$ | $98.1_{0.2}$ | $90.2_{0.2}$ | $94.9_{0.3}$ | $95.0_{0.5}$ | | |
3 | $98.7_{0.1}$ | $98.5_{0.3}$ | $98.0_{0.3}$ | $90.9_{1.1}$ | $94.6_{0.4}$ | $95.1_{0.3}$ | | |
5 | $98.7_{0.1}$ | $98.7_{0.1}$ | | $89.6_{0.6}$ | $95.0_{0.5}$ | | | |
8 | $98.3_{0.2}$ | $98.3_{0.3}$ | | $88.7_{0.7}$ | $94.0_{0.3}$ | | | |
10 | $98.1_{0.1}$ | $97.9_{0.3}$ | | $88.4_{0.2}$ | $93.5_{0.5}$ | | | |
Table 10: Exact match (EM) [%] of FiD models trained in environment
$({n^{+}}_{\text{train}},k_{\text{train}})$ in various evaluation environment
$({n^{+}}_{\text{eval}},k_{\text{eval}})$ in Natural Questions. The standard
deviations for each reported value is denoted with lower subscripts.
| $k_{\text{eval}}$ | 1 | 5 | 20
---|---|---|---|---
${n^{+}}_{\text{eval}}$ | ${n^{+}}_{\text{train}}$ $k_{\text{train}}$ | 1 | 5 | 20 | 1 | 5 | 20 | 1 | 5 | 20
1 | 1 | $87.9_{0.5}$ | $88.2_{0.3}$ | $84.3_{0.5}$ | $72.4_{0.7}$ | $76.0_{0.3}$ | $74.7_{0.6}$ | $48.2_{0.6}$ | $58.7_{0.9}$ | $62.9_{1.4}$
2 | $85.1_{0.5}$ | $87.6_{0.1}$ | $83.2_{0.8}$ | $59.5_{1.2}$ | $73.9_{0.4}$ | $73.9_{0.7}$ | $20.7_{1.3}$ | $53.2_{0.6}$ | $62.4_{1.0}$
3 | $82.4_{0.5}$ | $85.8_{0.3}$ | $82.3_{0.7}$ | $56.0_{1.1}$ | $70.9_{0.9}$ | $72.8_{0.6}$ | $20.4_{1.2}$ | $47.5_{1.0}$ | $60.9_{0.6}$
2 | 1 | $91.8_{0.5}$ | $91.1_{0.4}$ | $87.6_{0.6}$ | $80.0_{0.8}$ | $82.3_{0.4}$ | $80.2_{0.5}$ | $54.9_{1.1}$ | $65.7_{0.3}$ | $69.3_{0.9}$
2 | $93.8_{0.0}$ | $92.4_{0.2}$ | $87.7_{1.0}$ | $75.2_{1.0}$ | $83.5_{0.1}$ | $80.6_{0.2}$ | $25.6_{1.5}$ | $62.4_{0.4}$ | $70.7_{0.6}$
3 | $93.1_{0.2}$ | $92.1_{0.2}$ | $87.4_{0.9}$ | $73.9_{0.6}$ | $82.7_{0.6}$ | $80.5_{0.6}$ | $26.3_{1.4}$ | $58.9_{0.8}$ | $70.7_{0.4}$
3 | 1 | $93.3_{0.3}$ | $92.6_{0.3}$ | $89.2_{0.3}$ | $82.6_{0.8}$ | $84.3_{0.4}$ | $81.7_{0.4}$ | $57.3_{0.9}$ | $68.2_{0.0}$ | $71.5_{0.8}$
2 | $96.4_{0.2}$ | $94.6_{0.1}$ | $89.3_{0.8}$ | $80.0_{0.5}$ | $86.7_{0.2}$ | $83.4_{0.4}$ | $27.5_{1.4}$ | $65.7_{0.4}$ | $73.8_{0.4}$
3 | $96.2_{0.3}$ | $94.8_{0.2}$ | $89.5_{0.7}$ | $79.8_{0.1}$ | $87.1_{0.3}$ | $83.3_{0.4}$ | $28.4_{0.9}$ | $63.5_{1.0}$ | $74.4_{0.4}$
Figure 7: Performance of FiD models on Natural Questions with varying training
context quality. Panels represent different evaluation environments with
different $(n^{+}_{\text{eval}},n_{\text{eval}})$ pairs, and a red dashed line
shows corresponding context quality. Red stars represent the best performed
models in the corresponding evaluation environments. Dotted lines show models
trained on the same context quantity $n_{\text{train}}$. Figure 8: Performance
of FiD models on TriviaQA with varying training context quantity. Panels
represent different evaluation environments with different
$(n^{+}_{\text{eval}},k_{\text{eval}})$ pairs, and a red dashed line shows
corresponding context quantity. Red stars represent the best performed models
in the corresponding evaluation environments. Dotted lines show models trained
on the same context quality $\frac{1}{1+k_{\text{train}}}$. Figure 9:
Performance of FiD models on Natural Questions with varying training context
quantity. Panels represent different evaluation environments with different
$(n^{+}_{\text{eval}},k_{\text{eval}})$ pairs, and a red dashed line shows
corresponding context quantity. Red stars represent the best performed models
in the corresponding evaluation environments. Dotted lines show models trained
on the same context quality $\frac{1}{1+k_{\text{train}}}$. Figure 10: Top
panels: Performance of FiD models on TriviaQA with adaptation by the proposed
method (solid lines) and without adaptation (dotted lines). Bottom panels:
Optimal temperature parameter $T^{*}$ selected for each model. Multiple
$T^{*}$ were selected for some context qualities, i.e., training environments,
because we selected $T^{*}$ for each of the three models trained with
different random seeds for each training environment.
Panels represent different evaluation environments with different
$(n^{+}_{\text{eval}},n_{\text{eval}})$ pairs, and a red dashed line shows
corresponding context quality.
|
We can now notice that $\varepsilon\kappa^{2}\big{(}\langle q_{\rm
i}\rangle\llbracket\underline{\mathtt{c}}_{1}\zeta\rrbracket-2\ell\dot{\delta}\langle\underline{\mathtt{c}}_{1}\zeta\rangle\big{)}$
is a lower order term in the sense that
$\varepsilon\kappa^{2}|\langle q_{\rm i}\rangle\llbracket
c_{1}\zeta\rrbracket-2\ell\dot{\delta}\langle
c_{1}\zeta\rangle|\leq\varepsilon^{1/2}\kappa^{1/2}\underline{C}\big{(}\underline{{\mathfrak{E}}}_{\rm
int}+\underline{{\mathfrak{E}}}_{\rm trace}\big{)},$
so that it can be absorbed by the sum of the three energies when
$\varepsilon\kappa$ is small enough to have
$\varepsilon^{1/2}\kappa^{1/2}\underline{C}<1$. For instance, if
$\varepsilon^{1/2}\kappa^{1/2}\underline{C}<1/2$, and denoting
$\widetilde{\underline{{\mathfrak{E}}}}:=\underline{\mathfrak{E}}_{\rm
ext}+\underline{\mathfrak{E}}_{\rm int}+\underline{\mathfrak{E}}_{\rm trace},$
one obtains after a Gronwall estimate
$\widetilde{\underline{{\mathfrak{E}}}}(t)\leq
3\big{[}\widetilde{\underline{{\mathfrak{E}}}}(0)+\varepsilon\underline{C}\int_{0}^{t}\big{(}|f|_{2}^{2}+|(g_{1},g_{2})|^{2}\big{)}\big{]}\exp(\sqrt{\varepsilon}\underline{C}t).$
Since moreover there exists a constant $C_{0}=C_{0}(\frac{1}{h_{\rm
min}},\frac{1}{c_{\rm min}})$ such that
$\widetilde{\underline{\mathfrak{E}}}\leq
C_{0}\big{(}|{\mathtt{Z}}|^{2}+|(\zeta,q,\kappa\partial_{x}q)|_{2}^{2}+\varepsilon\kappa^{3}|(\zeta_{-},\zeta_{+})|^{2}+\varepsilon\kappa^{5}|(\dot{\zeta}_{-},\dot{\zeta}_{+})|^{2}$
and
$|{\mathtt{Z}}|^{2}+|(\zeta,q,\kappa\partial_{x}q)|_{2}^{2}+\varepsilon\kappa^{3}|(\zeta_{-},\zeta_{+})|^{2}+\varepsilon\kappa^{5}|(\dot{\zeta}_{-},\dot{\zeta}_{+})|^{2}\leq
C_{0}\widetilde{\underline{\mathfrak{E}}},$
one deduces the estimate stated in the theorem.
Step 5. Well-posedness. By a straightforward adaptation of the proof of
Theorem 3.3, one can observe that (3.33)-(3.36) can be reformulated as an ODE
for $(\zeta,q,{\mathtt{Z}})\in H^{1}\times
H^{2}({\mathcal{E}})\times{\mathbb{R}}^{3}$ and prove existence and uniqueness
of a solution in this space by Cauchy-Lipschitz’s theorem. For data in
$(\zeta,q,{\mathtt{Z}})\in L^{2}\times
H^{1}({\mathcal{E}})\times{\mathbb{R}}^{3}$, this strategy does not work
directly because the traces $\zeta_{\pm}$ that appear in the component of the
ODE (3.53) for $\langle q_{\rm i}\rangle$ and $\dot{\delta}$ cannot be
controlled by the $L^{2}$ norm of $\zeta$. However, the energy estimate just
proved provides such a control and one can obtain the result by a classical
density argument (as used for instance in the proof of Theorem 3.1.1 in [36],
for hyperbolic initial boundary value problems where the control on the trace
is furnished by using a Kreiss symmetrizer). ∎
## 4\. Return to equilibrium
We now deal with a specific kind of wave-structure interaction that was called
the return to equilibrium problem in [26] and is commonly referred to as ”free
decay test” in engineering. This a situation where the solid is released at
zero speed from an out of equilibrium position ($\delta(t=0)\neq 0$), in a
fluid that is at rest. The solid then oscillates vertically and its motion
sends waves outwards; by this process, the solid loses energy and its
oscillations are damped so that the solid asymptotically stabilizes to its
equilibrium position. Engineers use this free decay test because by measuring
the oscillations of the object, they deduce some buoyancy properties of the
object. More precisely, assuming that the motion of the object satisfies the
phenomenological Cummins equation [11, 30]
(4.1) $M\ddot{\delta}+k\ast\dot{\delta}+a\delta=0$
with $M,a\in\mathbb{R}^{+}$ and $k\in L^{1}_{\text{loc}}(\mathbb{R}^{+})$,
they calibrate these coefficients with experimental measurements. These
measurements are also used to propose nonlinear extensions to (4.1) (by
fitting coefficients with ad hoc nonlinear terms) [39].
Our goal in this section is to study this problem from a mathematical
viewpoint, by proposing a qualitative analysis of the solutions to the
transmission problem (3.2)-(3.5) in the particular configuration corresponding
to the return to equilibrium problem. This approach is expected to lead in
some cases to an equation of the form (4.1), which would provide an analytic
description of the coefficients involved, and also to nonlinear extensions
that could be of interest to engineers.
This program was initiated and achieved in [26] for the (non dispersive)
nonlinear shallow water equations, where it was found that $\delta$ solves a
nonlinear second order ODE without integro-differential term. Still working
with the shallow water equations but in horizontal dimension $d=2$, assuming
radial symmetry and neglecting the nonlinear effects in the exterior region,
it was shown in [6] that the equation on $\delta$ should contain an integro-
differential term. Such a term is also necessary for the nonlinear shallow
water equations in dimension $d=1$ if viscosity is taken into account [33].
The goal of this section is to investigate the contribution of the dispersive
terms of the Boussinesq system to the equation satisfied by $\delta$ in this
specific configuration of the return to equilibrium problem.
From now, we assume that the initial data correspond to the configuration of
the return to equilibrium problem, namely,
(4.2)
$q(t=0)=\zeta(t=0)=0\quad\text{and}\quad\delta(t=0)=\delta_{0},\quad\dot{\delta}(t=0)=0.$
###### Notation 2.
We use throughout this section the same notations as in Section 3, namely, we
write $\kappa=\sqrt{\mu/3}$ and denote by ${\mathfrak{f}}_{\rm sw}$ the
momentum flux of the nonlinear shallow water equations,
${\mathfrak{f}}_{\rm
sw}=\frac{h^{2}-1}{2\varepsilon}+\varepsilon\frac{q^{2}}{h}=\zeta+\varepsilon\big{(}\frac{1}{2}\zeta^{2}+\frac{q^{2}}{h}\big{)}.$
We also recall that the buoyancy frequency $\tau_{\rm buoy}$ is defined in
Appendix A.
We introduce in §4.1 two Cummins operators that allow us to derive an abstract
evolution equation for the solid. We then investigate two specific cases where
it is possible to derive an explicit expression of these operators. The non
dispersive case ($\varepsilon\neq 0$, $\mu=0$) is considered in §4.2 where it
is shown that the motion of the object can be found by solving a simple
nonlinear second order scalar ODE. Waves can then be described by solving an
initial boundary value problem for a scalar Burgers equation. The opposite
case, namely, the linear dispersive case ($\varepsilon=0$, $\mu\neq 0$) is
addressed in §4.3; here again, it is possible to derive an explicit expression
for the Cummins operators leading us to an integro-differential Cummins-type
equation for the motion of the solid; qualitative properties of the solutions,
such as their decay rate are then investigated. Finally, it is shown that the
motion of the waves can be found by solving a nonlocal (in space) perturbation
of the transport equation.
### 4.1. The general Cummins equation
Quite obviously, any smooth solution of the transmission problem (3.2)-(3.5)
with initial condition (4.2) is such that $\zeta$ is an even function while
$q$ is odd – such solutions will be called “symmetric”. This implies that
$\langle q_{\rm i}\rangle=0$ and that the transmission problem can be reduced
into a simpler boundary value problem stated in the following direct corollary
of Theorem 3.1.
###### Corollary 4.1.
Any smooth symmetric solution to the transmission problem (3.2)-(3.5) solves
the following boundary value problem on the half-line $(\ell,\infty)$,
(4.3) $\begin{cases}\partial_{t}\zeta+\partial_{x}q=0\\\
(1-\kappa^{2}\partial_{x}^{2})\partial_{t}q+\partial_{x}{\mathfrak{f}}_{\rm
sw}=0,\end{cases}\quad\mbox{ for }\quad t\geq 0,\quad x\in{\mathcal{E}}^{+},$
with boundary condition
(4.4) $\displaystyle q_{|_{x=\ell}}$ $\displaystyle=-\ell\dot{\delta},$
where $\delta$ solves the ODE
(4.5)
${(\tau_{\mu}(\varepsilon\delta)^{2}+\ell\kappa\frac{1}{h_{+}})}\ddot{\delta}+\delta=\varepsilon\beta(\varepsilon\delta)\dot{\delta}^{2}+{\mathfrak{H}}_{+},$
where $h_{+}=h_{|_{x=\ell}}$, ${\mathfrak{H}}_{+}={\mathfrak{H}}_{|_{x=\ell}}$
and we recall that ${\mathfrak{H}}={\mathfrak{H}}(\zeta,q)$ with
${{\mathfrak{H}}(\zeta,q)=\frac{1}{2}\varepsilon\big{(}\frac{1}{h}\zeta^{2}-\frac{q^{2}}{h^{2}}\big{)}+\frac{1}{h}R_{1}{\mathfrak{f}}_{\rm
sw}},$
and that $\tau_{\mu}(\varepsilon\delta)$ and $\beta(\varepsilon\delta)$ are
defined in Proposition 2.4, namely,
$\displaystyle{{{\tau}_{\mu}(\varepsilon\delta)^{2}}}$
$\displaystyle=\tau_{\rm
buoy}^{2}+\frac{1}{\ell}\int_{0}^{\ell}\frac{x^{2}}{h_{\rm
eq}(x)+\varepsilon\delta}{\rm d}x+{\frac{\kappa^{2}}{h_{\rm
eq}(\ell)+\varepsilon\delta}},$ $\displaystyle\beta(\varepsilon\delta)$
$\displaystyle=\frac{1}{2}\frac{1}{\ell}\int_{0}^{\ell}\frac{x^{2}}{(h_{\rm
eq}(x)+\varepsilon\delta)^{2}}{\rm d}x.$
We know by Proposition 3.2 that if $f$ is a given $C^{1}$ function of time
then there is a unique solution $(\zeta,q)$ to (4.3) with boundary condition
$q_{|_{x=\ell}}=-\ell f$ with initial condition corresponding to the return to
equilibrium problem, namely, $(\zeta,q)(t=0)=(0,0)$. It is in particular
possible to compute the trace of $\zeta$ at $x=\ell$, so that the following
definition makes sense.
###### Definition 4.1 (Cummins operators).
Let $\varepsilon\in\mathbb{R}^{+}$, $\mu=\kappa^{2}/3>0$. Let also $f\in
C^{1}({\mathbb{R}}^{+})$ and $T>0$, and $(\zeta,q)\in
C^{1}\big{(}[0,T;H^{1}({\mathcal{E}}^{+})\times
H^{2}({\mathcal{E}}^{+})\big{)}$ be a solution to (4.3) with boundary
condition $q_{|_{x=\ell}}=-\ell f$ and initial condition
$(\zeta,q)(t=0)=(0,0)$. We define the Cummins operators
${\mathfrak{c}}_{\varepsilon,\mu}$ and ${\mathfrak{C}}_{\varepsilon,\mu}$ as
${\mathfrak{c}}_{\varepsilon,\mu}[f]:=\zeta_{|_{x=\ell}}\quad\mbox{ and
}\quad{\mathfrak{C}}_{\varepsilon,\mu}[f]:={{-}}{{\mathfrak{H}}(\zeta,q)_{|_{x=\ell}}}.$
###### Remark 4.1.
The Cummins operators can be defined for more general cases, for instance, the
solution $(\zeta,q)$ to the initial boundary value problem needs only to be
regular near the boundary $x=\ell$ (regular enough for the trace to make
sense). This allows one to extend the definition of the Cummins operators in
the case $\mu=0$, as done in §4.2 below.
###### Corollary 4.2.
The ODE (4.5) can be reformulated in a compact form as what we shall refer to
as the Cummins equation
(4.6)
${{\big{(}\tau_{\mu}(\varepsilon\delta)^{2}+\ell\kappa\frac{1}{1+\varepsilon{\mathfrak{c}}_{\varepsilon,\mu}[\dot{\delta}]}\big{)}}}\ddot{\delta}+\delta{{{+\mathfrak{C}}_{\varepsilon,\mu}[\dot{\delta}]}}=\varepsilon\beta(\varepsilon\delta)\dot{\delta}^{2},$
with initial conditions $\delta(0)=\delta_{0}$ and $\dot{\delta}(0)=0$.
The equation (4.6) is compact but not simple since the Cummins operators are
nonlinear nonlocal operators which require the resolution of the equations for
the fluid in the exterior domain. In order to get some qualitative insight on
the Cummins equation, we describe it in two limiting cases: in the nonlinear
non dispersive case ($\varepsilon>0$, not necessarily small, and $\mu=0$), and
in the linear, dispersive case ($\varepsilon=0$ and $\mu>0$, not necessarily
small). Note that in both cases, it is not necessary to compute the first
Cummins operator ${\mathfrak{c}}_{\varepsilon,\mu}[\dot{\delta}]$ and that it
is possible to provide an explicit expression of the second one
${\mathfrak{C}}_{\varepsilon,\mu}[\dot{\delta}]$.
### 4.2. The nonlinear non dispersive case
Neglecting the dispersive effects is equivalent to setting
$\mu=\kappa^{2}/3=0$ in the equations (4.3)-(4.6) ; in particular, the model
considered for the propagation of the waves is now the shallow water equations
(4.7) $\begin{cases}\partial_{t}\zeta+\partial_{x}q=0\\\
\partial_{t}q+\varepsilon\partial_{x}\left(\frac{1}{h}q^{2}\right)+h\partial_{x}\zeta=0,\end{cases}\quad\mbox{
for }\quad t\geq 0,\quad x\in{\mathcal{E}}^{+},$
the boundary condition is unchanged
(4.8) $\displaystyle q_{|_{x=\ell}}$ $\displaystyle=-\ell\dot{\delta},$
and the ODE solved by $\delta$ is simplified into
(4.9)
${{\tau_{0}(\varepsilon\delta)^{2}}}\ddot{\delta}+\delta{{{+\mathfrak{C}}_{\varepsilon,0}[\dot{\delta}]}}=-\varepsilon\tau_{0}(\varepsilon\delta)\tau_{0}^{\prime}(\varepsilon\delta)\dot{\delta}^{2},$
where we used the fact that
$\beta(\varepsilon\delta)=-2\tau_{0}(\varepsilon\delta)\tau_{0}^{\prime}(\varepsilon\delta)$
when $\mu=0$ (see (2.28) and (2.29)), and where the definition of the second
Cummins operator has been extended to the case $\mu=0$ as
${\mathfrak{C}}_{\varepsilon,0}[\dot{\delta}]:={{-}}\Big{(}\zeta+\varepsilon\frac{1}{2}\frac{q^{2}}{h^{2}}\Big{)}_{|_{x=\ell}};$
the fact that this definition makes sense follows from the decomposition of
the shallow water invariants into Riemann invariants, as shown in the proof of
the following theorem where an explicit expression of the Cummins operator is
provided. This theorem is a reformulation of Corollary 1 in [26], but with a
slight difference in the function $\gamma$, so that we reproduce a sketch of
the proof111The difference comes from the fact that in [26], the choice of the
boundary condition for the interior pressure was made by assuming that the
jump of pressure at the contact point was purely hydrostatic; as in [33, 5],
we rather use here a choice of the boundary condition on the pressure which is
consistent with the approach used throughout this paper and motivated by the
conservation of total energy, as explained in Corollary 2.1. With the choice
of [26], one would have
${\mathfrak{C}}_{\varepsilon,0}[\dot{\delta}]=-\zeta_{|_{x=\ell}}$ and
consequently
$-\ell\dot{\delta}-\varepsilon\dot{\delta}^{2}\gamma(\varepsilon\dot{\delta})=\frac{1}{\varepsilon}(\sigma_{0}(\varepsilon\frac{\ell}{2}\dot{\delta})^{2}-1)$..
###### Theorem 4.1.
Let $T>0$, $\delta\in C^{2}([0,T])$ and $(\zeta,q)$ be a continuous, piecewise
$C^{1}$ solution of (4.7)-(4.9) on $[0,T]\times(\ell,\infty)$ satisfying the
non vanishing depth condition
$\inf_{[0,T]\times{\mathcal{E}}}h>0\quad\mbox{ and
}\quad\inf_{[0,T]\times{\mathcal{I}}}h_{\rm eq}+\varepsilon\delta>0.$
If moreover $\ell\,\varepsilon\dot{\delta}<2r_{0}$, with
$r_{0}:=\frac{4}{27}$, we have
$\sqrt{h}=\sigma_{0}(\varepsilon\frac{\ell}{2}\dot{\delta})$ with the real
function
$\sigma_{0}(r)=\frac{1}{3}\Big{(}1+C_{-}(r)+C_{+}(r)\Big{)},\quad
C_{\pm}(r)=\frac{3}{2}\Big{(}-4r+2r_{0}\pm 4\sqrt{r(r-r_{0})}\Big{)}^{1/3},$
and the Cummins operator ${\mathfrak{C}}_{\varepsilon,0}$ is given explicitly
by
(4.10)
${\mathfrak{C}}_{\varepsilon,0}[\dot{\delta}]=-\varepsilon^{-1}\Big{(}\sigma_{0}(\varepsilon\frac{\ell}{2}\dot{\delta})-1\Big{)}\Big{(}3\sigma_{0}(\varepsilon\frac{\ell}{2}\dot{\delta})-1\Big{)}=:\ell\dot{\delta}{{+}}\varepsilon\dot{\delta}^{2}\gamma(\varepsilon\dot{\delta}),$
where $\gamma:(-\infty,2r_{0})\to{\mathbb{R}}$ is a smooth function such that
$\gamma(0)=\frac{1}{4}\ell^{2}$ and whose exact expression is given in (4.11)
below.
###### Proof.
The proof of Corollary 1 of [26] is based on the fact that the shallow water
equations can be put in diagonal form,
$\partial_{t}R+(\sqrt{h}+\varepsilon\frac{q}{h})\partial_{x}R=0\quad\mbox{ and
}\quad\partial_{t}L-(\sqrt{h}-\varepsilon\frac{q}{h})\partial_{x}L=0,$
where $R$ and $L$ are respectively the right and left Riemann invariants
$R=\frac{q}{h}+\frac{2}{\varepsilon}(\sqrt{h}-1)\quad\mbox{ and }\quad
L=\frac{q}{h}-\frac{2}{\varepsilon}(\sqrt{h}-1).$
One then notices that with the initial and boundary conditions considered
here, $L$ vanishes identically on $(\ell,\infty)$, which allows one to find
$\sqrt{h}$ in terms of $q$ as a root of the third order polynomial equation in
$\sigma$,
$\sigma^{3}-\sigma^{2}-\varepsilon\frac{1}{2}q=0.$
If $-\varepsilon\frac{1}{2}q<r_{0}$, then, as discussed in [26], the relevant
root is $\sigma_{0}(-\varepsilon\frac{1}{2}q)$.
Recalling that $q_{|_{x=\ell}}=-\ell\dot{\delta}$, we have
$\sqrt{h}=\sigma_{0}(\varepsilon\frac{\ell}{2}\dot{\delta})$. Moreover, $L=0$
implies
$\varepsilon\frac{1}{2}\frac{q^{2}}{h^{2}}_{|_{x=\ell}}=2\,\varepsilon^{-1}\Big{(}\sigma_{0}(\varepsilon\frac{\ell}{2}\dot{\delta})-1\Big{)}^{2}.$
Remarking that
$\zeta_{|_{x=\ell}}=\frac{1}{\varepsilon}(\sigma_{0}(\varepsilon\frac{\ell}{2}\dot{\delta})^{2}-1)$,
one gets
$\displaystyle\Big{(}\zeta+\varepsilon\frac{1}{2}\frac{q^{2}}{h^{2}}\Big{)}_{|_{x=\ell}}$
$\displaystyle=\varepsilon^{-1}(\sigma_{0}-1)(3\sigma_{0}-1)$ (4.11)
$\displaystyle=:-\ell\dot{\delta}{{-}}\varepsilon\dot{\delta}^{2}\gamma(\varepsilon\dot{\delta}),$
where we used the fact that $\sigma_{0}(0)=-\sigma_{0}^{\prime}(0)=1$. The
fact that $\gamma(0)=\frac{1}{4}\ell^{2}$ follows from the observation that
$\sigma_{0}^{\prime\prime}(0)=-4$. ∎
A first corollary is that the motion of the solid can be reduced to a simple
nonlinear ODE, provided that the initial displacement satisfies an upper bound
ensuring that the velocity of the object does not become too big.
###### Corollary 4.3.
Under the assumptions of the theorem, and with the same notations, let us
assume moreover that
$\varepsilon^{2}\delta_{0}^{2}<{\tau_{0}(\varepsilon|\delta_{0}|)^{2}}\big{(}\frac{2r_{0}}{\ell}\big{)}^{2}.$
Then, using the notations of the theorem, the motion of the solid is found by
solving the nonlinear second order ODE
(4.12)
${{\tau_{0}(\varepsilon\delta)^{2}}}\ddot{\delta}+\ell\dot{\delta}+\delta+\varepsilon{{\left(\tau_{0}(\varepsilon\delta)\tau^{\prime}_{0}(\varepsilon\delta)+\gamma(\varepsilon\dot{\delta})\right)}}\dot{\delta}^{2}=0,$
with initial condition $\delta(0)=\delta_{0}$, $\dot{\delta}(0)=0$.
###### Remark 4.2.
In the linear case ($\varepsilon=0$), this equation is almost the same as
(3.2.12) in [20], the only difference being that the author neglected the
buoyancy frequency $\tau_{\rm buoy}$ in the expression for $\tau_{0}(0)$.
###### Proof.
One just needs to check that the condition
$\ell\varepsilon\dot{\delta}<2r_{0}$, which ensures by Theorem 4.1 that the
Cummins operator takes the form (4.10), is satisfied for all times. Since at
$t=0$, one has $\dot{\delta}=0$, we now that this condition is satisfied for
small times. Since moreover one can deduce from Proposition 3.1 (by setting
$\mu=0$) that
(4.13)
$\tau_{0}(\varepsilon\delta)^{2}\dot{\delta}^{2}+\delta^{2}\leq\delta_{0}^{2},$
one deduces that $|\delta|\leq|\delta_{0}|$ and therefore that
$\dot{\delta}^{2}\leq\tau_{0}(\varepsilon|\delta_{0}|)^{-2}\delta_{0}^{2}$.
The assumption made in the statement of the corollary therefore grants the
result. ∎
The interest of reducing the motion of the solid to an ODE on the surface
displacement is that it is possible to solve it even in situations when
singularity arise in the exterior domain (typically, when shock happen). It is
in particular possible to obtain a global existence result for the ODE (4.12),
while such a result cannot be expected for strong solutions to the full
transmission problem (4.7)-(4.9) due to shock formation. Note that the first
condition on $\delta_{0}$ means that at $t=0$, the solid neither touches the
bottom nor is lifted from a height greater than the height of the water column
under the object when it is at equilibrium.
###### Proposition 4.1.
Let $\delta_{0}\in{\mathbb{R}}$ be such that
$\inf_{{\mathcal{I}}}h_{\rm eq}-\varepsilon|\delta_{0}|>0\quad\mbox{ and
}\quad\varepsilon|\delta_{0}|<\tau_{0}(\varepsilon|\delta_{0}|)\frac{2r_{0}}{\ell}.$
Then there exists a unique global solution $\delta\in
C^{\infty}({\mathbb{R}}^{+})$ to the ODE (4.12) with initial condition
$(\delta,\dot{\delta})_{|_{t=0}}=(\delta_{0},0)$.
###### Remark 4.3.
A byproduct of the proof is that
${\mathfrak{C}}_{\varepsilon,0}[\dot{\delta}]\,\dot{\delta}\geq 0$ and that
this quantity corresponds to the energy transferred at each instant to the
exterior fluid domain, that is, with the notations of Proposition 3.1, one has
$\frac{d}{dt}{\mathfrak{E}}_{\rm
ext}={\mathfrak{C}}_{\varepsilon,0}[\dot{\delta}]\dot{\delta}\geq 0.$
###### Remark 4.4.
The second condition of the proposition is a smallness condition on
$\delta_{0}$, but this condition is not restrictive as it allows $\delta_{0}$
to be of size $O(\varepsilon^{-1})$. As communicated to us by the author it is
possible, under stricter smallness conditions, to prove exponential decay of
the solution of ODEs related to (4.12) using techniques developed in [24].
###### Proof.
There exists a positive time $T>0$ such that on $[0,T)$, there is a solution
$\delta$ such that $\inf_{\mathcal{I}}h_{\rm eq}+\varepsilon\delta>0$ and
$\ell\varepsilon\dot{\delta}<2r_{0}$. We want to show that one can take
$T=+\infty$. As in the proof of Corollary 4.3, this follows from (4.13). We
therefore need to prove that (4.13) holds, without appealing to Proposition
3.1 as in the proof of Corollary 4.3, but by direct manipulations on the
solution to the ODE (4.12). We need the following two lemmas.
###### Lemma 4.1.
The function $\sigma_{0}$ is decreasing on $(-\infty,r_{0})$.
###### Proof of the lemma.
By construction, one has for all $r<r_{0}$,
$\sigma_{0}(r)^{3}-\sigma_{0}(r)^{2}+r=0;$
differentiating this identity yields
$\sigma^{\prime}_{0}(r)(3\sigma_{0}(r)^{2}-2\sigma_{0}(r))=-1,$
so that $\sigma_{0}(r)$ and $3\sigma_{0}(r)^{2}-2\sigma_{0}(r)$ have opposite
sign. It is therefore enough to prove that
$3\sigma_{0}(r)^{2}-2\sigma_{0}(r)>0$ for all $r<r_{0}$. Since
$\sigma_{0}(0)=1$, this quantity is positive at $r=0$ and must therefore
vanish if it changes sign. This means that for some $r_{1}<r_{0}$, one must
have $\sigma_{0}(r_{1})=0$ or $\sigma_{0}(r_{1})=\frac{2}{3}$. Using the cubic
equation solved by $\sigma_{0}(r_{1})$, this implies that $r_{1}=0$ or
$r_{1}=r_{0}$. Both cases have to be excluded because $\sigma_{0}(0)=1\neq 0$
and $r_{1}<r_{0}$ by assumption. The result follows. ∎
###### Lemma 4.2.
If $\dot{\delta}\neq 0$ and $\varepsilon\ell\dot{\delta}<2r_{0}$, the Cummins
operator satisfies
${\mathfrak{C}}_{\varepsilon,0}[\dot{\delta}]\,\dot{\delta}>0.$
###### Proof of the lemma.
Recalling that from (4.10), one has
${\mathfrak{C}}_{\varepsilon,0}[\dot{\delta}]\,\dot{\delta}=-\varepsilon^{-1}(\sigma_{0}(\varepsilon\frac{\ell}{2}\dot{\delta})-1)(3\sigma_{0}(\varepsilon\frac{\ell}{2}\dot{\delta})-1)\dot{\delta},$
the conclusion follows from the previous lemma and the observation that
$\sigma_{0}(0)=1$ and $\sigma_{0}(r_{0})=2/3$. ∎
We can now use the second lemma to conclude: multiplying (4.9) by
$\dot{\delta}$ and integrating in time yields
$\displaystyle\tau_{0}(\varepsilon\delta)^{2}\dot{\delta}^{2}+\delta^{2}$
$\displaystyle=\delta_{0}^{2}-\int_{0}^{t}{\mathfrak{C}}_{\varepsilon,0}[\dot{\delta}]\dot{\delta}<\delta_{0}^{2},$
which implies (4.13); the proposition is therefore proved. ∎
The following Corollary then shows that, once the ODE (4.12) has been solved,
the solution in the exterior domain reduces to a simple initial boundary value
problem for a scalar Burgers-type equation. This is a simple byproduct of the
proof of Theorem 4.1 where it was shown that the nonlinear shallow water
equations were reduced to the scalar equation on the right-going Riemann
invariant.
###### Corollary 4.4.
Under the assumptions of Corollary 4.3, $q$ is found in the exterior domain by
solving the initial boundary value problem
$\begin{cases}\partial_{t}q+\big{(}-\sigma_{0}^{\prime}(-\frac{\varepsilon}{2}q)\sigma_{0}(-\frac{\varepsilon}{2}q)\big{)}^{-1}\partial_{x}q&=0\qquad(t>0,\quad
x>\ell),\\\ q_{|_{t=0}}&=0,\\\ q_{|_{x=\ell}}&=-\ell\dot{\delta},\end{cases}$
with $\delta$ furnished by Proposition 4.1, while $\zeta$ is given in terms of
$q$ by the algebraic expression
(4.14) $\zeta=\frac{1}{\varepsilon}(\sigma_{0}(-\varepsilon q/2)^{2}-1).$
###### Remark 4.5.
More generally, if one wants to compute the waves created by an object in
forced motion, one must solve the same equations as in the corollary, but with
$\delta$ corresponding to this forced motion rather than given by Proposition
4.1.
### 4.3. The linear dispersive case
We have studied in the previous section the situation where dispersive effects
could be neglected ($\mu=\kappa^{2}/3=0$) in front of the nonlinear effects.
We consider here the opposite situation where nonlinear effects are negligible
($\varepsilon=0$) but the dispersive effects are taken into account. That is,
we consider the linear approximation to (4.3)-(4.5). The model considered for
the propagation of the waves is therefore
(4.15) $\begin{cases}\partial_{t}\zeta+\partial_{x}q=0\\\
(1-\kappa^{2}\partial_{x}^{2})\partial_{t}q+\partial_{x}\zeta=0,\end{cases}\quad\mbox{
for }\quad t\geq 0,\quad x\in{\mathcal{E}}^{+},$
the boundary condition is unchanged
(4.16) $\displaystyle q_{|_{x=\ell}}$ $\displaystyle=-\ell\dot{\delta},$
and the ODE solved by $\delta$ is simplified into
(4.17)
${(\tau^{2}_{\mu}+\ell\kappa)}\ddot{\delta}+\delta+{\mathfrak{C}}_{0,\mu}[\dot{\delta}]=0,$
where we recall that according to the definition of the Cummins operator (see
Definition 4.1 and (3.18)),
${\mathfrak{C}}_{0,\mu}[\dot{\delta}]:=-(R_{1}\zeta)_{|_{x=\ell}},$
and where, for the sake of clarity, we simply write throughout this section
$\tau_{\mu}^{2}=\tau_{\mu}(0)^{2}=\tau_{\rm
buoy}^{2}+\frac{1}{\ell}\int_{0}^{\ell}\frac{x^{2}}{h_{\rm
eq}}+\kappa^{2}\frac{1}{h_{\rm eq}}.$
We know by Theorem 3.3 that for all $n\in{\mathbb{N}}$ and $T>0$, there exists
a unique solution $(\zeta,q,\delta)\in
C^{\infty}([0,T];{\mathbb{H}}^{n}\times{\mathbb{R}})$ of (4.15)-(4.17) with
initial conditions (4.2); we want here to analyze the behavior of this
solution. As for the nonlinear non dispersive case in the previous section, we
first provide an explicit expression for the Cummins operator, from which we
are able to derive an uncoupled scalar equation for the evolution of $\delta$,
whose solution can be used to find $\zeta$ and $q$ in the exterior domain
through the resolution of a simpler scalar initial boundary value problem. All
the equations involved in this section are linear, the difficulty coming from
their nonlocal nature.
#### 4.3.1. Preliminary material
In order to give an explicit representation of the Cummins operator
${\mathfrak{C}}_{0,\mu}$, we first need to recall the definition of the Bessel
functions $J_{n}$ (§8.41 in [17])
$J_{n}(t)=\frac{1}{\pi}\int_{0}^{\pi}\cos\big{(}n\theta-t\sin\theta\big{)}{\rm
d}\theta;$
we also define the causal convolution kernels ${\mathcal{K}}^{0}_{\mu}$ and
${\mathcal{K}}^{1}_{\mu}$ as
(4.18)
${\mathcal{K}}^{0}_{\mu}(t)=\frac{1}{\kappa}J_{0}(\frac{t}{\kappa})\quad\mbox{
and
}\quad{\mathcal{K}}^{1}_{\mu}(t)=\frac{1}{t}J_{1}(\frac{t}{\kappa}),\quad\mbox{
for all }t\geq 0,$
and use the following standard notation for the convolution of time causal
functions,
$\forall t\geq 0,\qquad f*g(t)=\int_{0}^{t}f(t-s)g(s){\rm d}s.$
We also need to use the Laplace transform with respect to the time variable,
which we define as
$\mathcal{L}:q\mapsto\hat{q},$
where
$\mathcal{L}[q](s)=\displaystyle\int_{0}^{\infty}q(t)e^{-st}{\rm
d}t\quad\text{ with }\quad
s\in\mathbb{C}_{0}:=\\{s\in\mathbb{C}\,|\,\mathfrak{Re}(s)>0\\}.$
We shall in particular use the following properties on Bessel functions [17]
(4.19)
${\mathcal{L}}^{-1}\big{(}\frac{1}{\sqrt{1+\kappa^{2}s^{2}}}\big{)}={\mathcal{K}}^{0}_{\mu}(t)\quad\mbox{
and }\quad{\mathcal{L}}^{-1}\big{(}\frac{1}{\sqrt{1+\kappa^{2}s^{2}}+\kappa
s}\big{)}={\mathcal{K}}^{1}_{\mu}(t),$
with ${\mathcal{K}}^{0}_{\mu}$ and ${\mathcal{K}}^{1}_{\mu}$ as defined in
(4.18).
#### 4.3.2. Analysis of the equations
Using the linear structure of the equations, one can obtain an explicit
expression for the Cummins operator ${\mathfrak{C}}_{0,\mu}$.
###### Theorem 4.2.
The Cummins operator ${\mathfrak{C}}_{0,\mu}$ is given explicitly by
${\mathfrak{C}}_{0,\mu}[\dot{\delta}]:=\ell\,{\mathcal{K}}^{1}_{\mu}*\dot{\delta},$
where ${\mathcal{K}}^{1}_{\mu}$ is defined in (4.18).
###### Proof.
Applying the Laplace transform to the equations (4.15) and (4.16), which is
possible since all the functions are continuous and bounded in time (as a
consequence of Proposition 3.1), and taking into account that
$\zeta_{|_{t=0}}=q_{|_{t=0}}=0$, this yields
(4.20) $\begin{cases}s\hat{\zeta}+\partial_{x}\hat{q}=0\\\
(1-\kappa^{2}\partial_{x}^{2})s\hat{q}+\partial_{x}\hat{\zeta}=0,\end{cases}\quad\mbox{
and }\quad\widehat{q}_{|_{x=\ell}}=-\ell\widehat{\dot{\delta}}.$
This is an ODE for $(\widehat{\zeta},\widehat{q})$ on the half-line
$(\ell,\infty)$ that can be explicitly solved in terms
$\widehat{\dot{\delta}}$ (note that a representation of the solution in terms
of the Laplace transform in space is also possible [21] but not adapted to our
purpose here; see also [3] for other types of linear dispersive equations);
the formula of the lemma below provides ”right-going” solutions to the linear
Boussinesq equations and it is therefore no surprise that the relationship
between $\widehat{\zeta}$ and $\widehat{q}$ is the same as the one that arises
when imposing transparent boundary conditions as in [23].
###### Lemma 4.3.
There is one and only one solution $(\widehat{\zeta},\widehat{q})$ to (4.20)
that does not grow exponentially at infinity; it is given by
$\begin{cases}\hat{q}(s,x)=\displaystyle-\ell\,\widehat{\dot{\delta}}(s)e^{-\frac{s}{\sqrt{1+\kappa^{2}s^{2}}}(x-\ell)},\\\
\displaystyle\widehat{\zeta}(s,x)=\frac{1}{\sqrt{1+\kappa^{2}s^{2}}}\widehat{q}(s,x),\end{cases}$
where the square root is taken in order to have positive real part.
###### Proof of the lemma.
From (4.20), one deduces
(4.21)
$\partial_{x}^{2}\hat{q}(s)-\frac{s^{2}}{1+\kappa^{2}s^{2}}\hat{q}(s)=0,$
and there are therefore two constants $A(s)$ and $B(s)$ such that
$\displaystyle\hat{q}(s,x)=A(s)e^{-\frac{s}{\sqrt{1+\kappa^{2}s^{2}}}x}+B(s)e^{\frac{s}{\sqrt{1+\kappa^{2}s^{2}}}x}.$
Since exponentially increasing functions are not allowed, we have $B(s)=0$ and
thus
$\hat{q}(s,x)=A(s)e^{-\frac{s}{\sqrt{1+\kappa^{2}s^{2}}}x}.$
Then using the boundary condition on $\widehat{q}$ at $x=\ell$, we find the
expected formula for $\hat{q}$. By using the first equation of (4.20), we get
the formula for $\widehat{\zeta}$. ∎
Let us now remark that for all $f\in L^{2}({\mathcal{E}}^{+})$, one has
$(R_{1}f)_{|_{x=\ell}}=\kappa^{-1}\int_{{\mathcal{E}}^{+}}e^{-\kappa^{-1}(x-\ell)}f(x){\rm
d}x$
so that, using the lemma,
$\displaystyle\widehat{{\mathfrak{C}}_{0,\mu}[\dot{\delta}]}(s)$
$\displaystyle=-(R_{1}\widehat{\zeta})_{|_{x=\ell}}$
$\displaystyle=\displaystyle\frac{\ell\widehat{\dot{\delta}}(s)}{\kappa\sqrt{1+\kappa^{2}s^{2}}}\,\int_{{\mathcal{E}}^{+}}e^{-\big{(}\frac{1}{\kappa}+\frac{s}{\sqrt{1+\kappa^{2}s^{2}}}\big{)}(x-\ell)}{\rm
d}x.$
It follows that
$\widehat{{\mathfrak{C}}_{0,\mu}[\dot{\delta}]}(s)=\frac{\ell}{\sqrt{1+\kappa^{2}s^{2}}+\kappa
s}\,\widehat{\dot{\delta}}(s).$
Using (4.19), this yields
${\mathfrak{C}}_{0,\mu}[\dot{\delta}]=\ell{\mathcal{K}}^{1}_{\mu}*\dot{\delta};$
note also for future use that we also get from the lemma that
$\displaystyle\zeta(t,x)$ $\displaystyle={\mathcal{K}}^{0}_{\mu}*q.$
∎
As in Corollary 4.3 in the non dispersive case, it is possible to determine
the motion of the solid by the resolution of a single scalar equation on
$\delta$; due to the presence of the dispersive terms however, this equation
is no longer an ordinary differential equation but an integro-differential
equation.
###### Corollary 4.5.
The motion of the floating object for the problem (4.15)-(4.17) can be found
directly by solving the linear second order integro-differential equation,
(4.22)
$\big{(}\tau^{2}_{\mu}+\ell\kappa\big{)}\ddot{\delta}+\ell{\mathcal{K}}^{1}_{\mu}*\dot{\delta}+\delta=0,$
with initial conditions $\delta(0)=\delta_{0}$ and $\dot{\delta}(0)=0$.
###### Remark 4.6.
In [33], the authors consider the linearized shallow-water equations with some
viscosity $\nu$. More precisely, they consider (4.15) with
$-\nu\partial_{x}^{2}q$ instead of $-\kappa^{2}\partial_{x}^{2}\partial_{t}q$
in the second equation and find the following Cummins equation
(4.23)
$\tau_{\mu}^{2}\ddot{\delta}+\sqrt{\nu}\ell\delta^{\left(\frac{3}{2}\right)}+\nu\dot{\delta}+\ell{\mathcal{F}}_{\nu}\ast\dot{\delta}+\delta=0\quad\text{with}\quad{\mathcal{F}}_{\nu}:=\mathcal{L}^{-1}\left[\frac{1}{\sqrt{1+\nu
s}+\sqrt{\nu s}}\right],$
and where $\delta^{(\frac{3}{2})}$ stands for the fractional derivative of
order $3/2$ of $\delta$. This equation shares some similarities with (4.22),
in particular the convolution term, although with a different kernel (note
that one gets $\widehat{\mathcal{K}}^{1}_{\mu}(s)$ by replacing $\nu s$ by
$\kappa^{2}s^{2}$ in $\widehat{{\mathcal{F}}}_{\mu}(s)$). One the contrary,
there is in (4.23) a viscous damping term $\nu\dot{\delta}$ that has no
equivalent in (4.22). Note finally that the fractional derivative term
$\sqrt{\nu}\ell\delta^{\left(\frac{3}{2}\right)}$ in (4.23) can be related to
the added mass term ${\ell\kappa\ddot{\delta}}$ in (4.22). Indeed, in the
analysis of [33], this fractional derivative is the leading order term of a
convolution term $\ell F*\dot{\delta}$ with $\widehat{F}(s)=\sqrt{1+\nu s}$.
In the dispersive case, the same analysis would give a symbol
$\sqrt{1+\kappa^{2}s^{2}}$ and the leading order term of the same convolution
would be the dispersive added mass term $\ell\kappa\ddot{\delta}$.
In the linear non dispersive case ($\varepsilon=\mu=0$), Corollary 4.3 shows
that the motion of the object is governed by the same equation as a damped
harmonic oscillator; the return to equilibrium occurs therefore at an
exponential rate. In the presence of dispersion, Corollary 4.5 states that the
motion of the solid is now governed by the integro-differential equation
(4.22) and numerical simulations (see Figure 2) suggest that the decay gets
slower as the dispersion parameter $\kappa=\sqrt{\mu/3}$ increases.
.
Figure 2. Return to equilibrium: evolution of $\delta(t)$ with increasing
value of $\kappa$ and $\delta_{0}=1,\tau_{\rm buoy}=1/6,h_{\rm eq}=1$.
This issue is addressed in the following proposition. In particular, the fact
that $\delta$ belongs to $H^{2}({\mathbb{R}}^{+})$ implies that $\delta$ and
$\dot{\delta}$ tend to zero at infinity, but the third point of the
proposition shows that the decay cannot be stronger than $O(t^{-3/2})$ (as
opposed to the exponential convergence rate in the linear non dispersive
case), bringing a theoretical confirmation to the above numerical
observations.
###### Proposition 4.2.
i. There is a unique solution $\delta\in C^{2}({\mathbb{R}}^{+})\cap
W^{1,\infty}({\mathbb{R}}^{+})$ to (4.22) with initial data
$\delta(0)=\delta_{0}$ and $\dot{\delta}(0)=0$.
ii. Moreover, $\delta\in H^{2}({\mathbb{R}}^{+})$, but for $k\in\\{0,1,2\\}$,
$t\delta^{(k)}\not\in L^{2}({\mathbb{R}}^{+})$.
iii. For all $\alpha>0$ and $k\in\\{0,1,2\\}$ and for all $c>0$ and $T_{0}>0$,
there exists $t>T_{0}$ such that
$|\delta^{(k)}(t)|>ct^{-\frac{3}{2}-\alpha}.$
###### Remark 4.7.
The dispersive delay (convolution) term in (4.22) is responsible for the slow
decay of the solution. Indeed, in the non dispersive limit case $\kappa=0$,
the branching points $s=\pm i\kappa^{-1}$ disappear from the transfer function
$\hat{H}_{\mu}$ derived in (4.24) below, which then becomes
${\hat{H}}_{0}(s):=\frac{\tau^{2}_{0}s+\ell}{{\tau^{2}_{0}\,s^{2}+s\ell+1}},$
whose poles have a strictly negative real part, hence an exponential decay for
$\delta$.
###### Proof.
Since the kernel ${\mathcal{K}}_{1}$ belongs to $L^{1}({\mathbb{R}}^{+})$
(recall that the Bessel function $J_{1}(t)$ decays like $O(t^{-1/2})$), the
proof of the first part of the proposition does not raise any particular
problem. To prove the second part of the proposition, we need a careful
analysis of the transfer function $\widehat{H}_{\mu}$ defined by the relation
$\widehat{\delta}=\widehat{H}_{\mu}\delta_{0}$. Since
$(\delta,\dot{\delta})\in C\cap L^{\infty}(\mathbb{R}^{+})$, the Laplace
transform of $\delta$ and $\dot{\delta}$ are well defined on $\mathbb{C}_{0}$
an after remarking that
$\kappa s+\frac{1}{\sqrt{1+\kappa^{2}s^{2}}+\kappa
s}=\sqrt{1+\kappa^{2}s^{2}},$
the Laplace transform of $\delta$ is, owing to (4.22),
(4.24) $\hat{\delta}={\hat{H}}_{\mu}(s)\delta_{0}\quad\text{ where
}\quad{\hat{H}}_{\mu}(s):=\frac{\tau^{2}_{\mu}s+\ell\sqrt{1+\kappa^{2}s^{2}}}{\tau^{2}_{\mu}\,s^{2}+s\ell\sqrt{1+\kappa^{2}s^{2}}+1}.$
###### Lemma 4.4.
The transfer function ${\hat{H}}_{\mu}$ defined in (4.24) is holomorphic on
$\mathbb{C}_{0}$ and admits only two branching points at $\pm i\kappa^{-1}$.
Moreover all the zeros of the denominator in (4.24) have strictly negative
part.
###### Proof of the lemma.
We denote by $P$ the holomorphic function on $\mathbb{C}_{0}$
$P(s):=\tau^{2}_{\mu}\,s^{2}+s\ell\sqrt{1+\kappa^{2}s^{2}}+1,$
where the square root stands for the square root with positive real part.
Since $P$ has only two singularities which are the branching points $\pm
i\kappa^{-1}$, we can extend it analytically on all the complex plane except
on the cuts $i(-\infty,-\kappa^{-1})$ and $i(\kappa^{-1},\infty)$, and extend
it continuously by a function $P^{*}$ on the imaginary axis by
$P^{*}(i\omega):=\begin{cases}1-\tau^{2}_{\mu}\,\omega^{2}+i\omega\ell\sqrt{1-\kappa^{2}\omega^{2}}&|\omega|\leq\kappa^{-1},\\\
1-\tau^{2}_{\mu}\,\omega^{2}-|\omega|\ell\sqrt{\kappa^{2}\omega^{2}-1}&|\omega|>\kappa^{-1}.\end{cases}$
We first show that $s$ cannot be purely imaginary, then that if a zero is
real, it must be strictly negative, and finally that if a zero is not a real
number, it must satisfy $\Re(s)<0$.
Step 1. The zeros of $P$ cannot be purely imaginary. Indeed, if $\omega$ were
we solution of $P^{*}(i\omega)=0$, then, from the expression of $P(i\omega)$
given above, $\omega^{2}$ would be a real root of the second order polynomial
$(\tau_{\mu}^{4}-\ell^{2}\kappa^{2})X^{2}+(\ell^{2}-2\tau^{2}_{\mu})X+1.$
But the discriminant of this polynomial is
$\Delta=\ell^{4}+4\ell^{2}(\kappa^{2}-\tau^{2}_{\mu})<-\ell^{2}/3$ (since
$\tau^{2}_{\mu}>\kappa^{2}+\ell^{2}/3$), which is negative, implying that the
polynomial cannot have any real root.
Step 2. The zeros of $P$ cannot belong to ${\mathbb{R}}^{+}$ from the simple
observation that $P(\eta)>0$ for all $\eta\in{\mathbb{R}}^{+}$.
Step 3. The zeros of $P$ cannot have positive real part. In order to prove
this, we show here that $\Im(P(s))\neq 0$ for all $s=\eta+i\omega$ with
$s\notin i\mathbb{R}\cup\mathbb{R}$ (the case $s\in{\mathbb{R}}^{+}$ having
been dealt with in Step 2). The imaginary part of $P(s)$ is given by
$\Im[P(s)]=2\tau^{2}_{\mu}\,\eta\,\omega+\eta\,\ell\Im[\sqrt{1+\kappa^{2}s^{2}}]+\omega\,\ell\Re[\sqrt{1+\kappa^{2}s^{2}}].$
By definition of the square root, $\Re[\sqrt{1+\kappa^{2}s^{2}}]\geq 0$ and
the sign of $\Im[\sqrt{1+\kappa^{2}s^{2}}]$ is the same as the sign of the
product $\eta\,\omega$. The following table summarizes the sign of some
quantities in different cases (where $+$ stands for strictly positive, $-$
stands for strictly negative and ind. signifies that the sign is
indeterminate).
$\eta$ | $\omega$ | $2\tau^{2}\,\eta\,\omega$ | $\eta\,\Im[\sqrt{1+\kappa^{2}s^{2}}]$ | $\omega\Re[\sqrt{1+\kappa^{2}s^{2}}$] | $\Im[P(s)]$
---|---|---|---|---|---
- | - | + | - | - | ind.
- | + | - | + | + | ind.
+ | - | - | - | - | -
+ | + | + | + | + | +
Therefore, if $\eta>0$ then $\Im[P(s)]$ is either strictly positive or
strictly negative, so that it does not vanish.
∎
Using Lemma 4.4, we can extend continuously the transfer function
${\hat{H}}_{\mu}$ on the imaginary axis by
${\hat{H}}^{*}_{\mu}(i\omega)=\begin{cases}\displaystyle\frac{\tau_{\mu}^{2}i\omega+\ell\sqrt{1-\kappa^{2}\omega^{2}}}{-\tau^{2}_{\mu}\,\omega^{2}+i\omega\ell\sqrt{1-\kappa^{2}\omega^{2}}+1}\delta_{0},&|\omega|\leq\kappa^{-1},\\\
\displaystyle\frac{\tau^{2}_{\mu}i\omega+i\ell{\rm
sign}(\omega)\sqrt{\kappa^{2}\omega^{2}-1}}{-\tau^{2}_{\mu}\,\omega^{2}-|\omega|\ell\sqrt{\kappa^{2}\omega^{2}-1}+1}\delta_{0},&|\omega|>\kappa^{-1}.\end{cases}$
Integrating $|{\hat{H}}^{*}_{\mu}(i\omega)|^{2}$ over $\mathbb{R}$ we get
$\displaystyle\int_{\mathbb{R}}|{\hat{H}}^{*}_{\mu}(i\omega)|^{2}{\rm
d}\omega=\displaystyle\int_{-\kappa^{-1}}^{\kappa^{-1}}|{\hat{H}}^{*}_{\mu}(i\omega)|^{2}{\rm
d}\omega+\displaystyle\int_{|\omega|>\kappa^{-1}}|{\hat{H}}^{*}_{\mu}(i\omega)|^{2}{\rm
d}\omega.$
The first integral is obviously finite (the denominator in (4.24) does not
vanish on $i{\mathbb{R}}$). The second integral is also finite since
$|{\hat{H}}^{*}_{\mu}(i\omega)|^{2}{\underset{|\omega|\to\infty}{\sim}}\omega^{-2}$.
Then ${\hat{H}}_{\mu}$ belongs to the standard Hardy space
$\mathcal{H}^{2}(\mathbb{C}_{0})$ and by the Paley-Wiener theorem (see Theorem
5.1 below), one has $\delta\in L^{2}({\mathbb{R}}^{+})$.
The same reasoning can be applied to
$\widehat{\dot{\delta}}=\left(\frac{-1}{\tau^{2}_{\mu}\,s^{2}+s\ell\sqrt{1+\kappa^{2}s^{2}}+1}\right)\delta_{0}$
and
$\widehat{\ddot{\delta}}=\left(\frac{-s}{\tau^{2}_{\mu}\,s^{2}+s\ell\sqrt{1+\kappa^{2}s^{2}}+1}\right)\delta_{0}$
so that $\dot{\delta}$ and $\ddot{\delta}$ also belong to
$L^{2}({\mathbb{R}}^{+})$.
Let us now prove that $u(t):=t\delta(t)$ does not belong to
$L^{2}({\mathbb{R}}^{+})$. Denoting by $U$ the Laplace transform of $u$, one
has
$U(s):=(-1)\left(\frac{d}{ds}\right)H_{\mu}(s)\delta_{0}\quad\mbox{ on
}\quad{\mathbb{C}}_{0},$
and the following extension to the imaginary axis holds,
$U^{*}(i\omega)=\begin{cases}\displaystyle(-1)\left(\frac{d}{d\omega}\right)\frac{\tau^{2}_{\mu}i\omega+\ell\sqrt{1-\kappa^{2}\omega^{2}}}{-\tau^{2}_{\mu}\,\omega^{2}+i\omega\ell\sqrt{1-\kappa^{2}\omega^{2}}+1}\delta_{0},&|\omega|\leq\kappa^{-1},\\\
\displaystyle(-1)\left(\frac{d}{d\omega}\right)\frac{\tau^{2}_{\mu}i\omega+i\ell{\rm
sign}(\omega)\sqrt{\kappa^{2}\omega^{2}-1}}{-\tau^{2}_{\mu}\,\omega^{2}-|\omega|\ell\sqrt{\kappa^{2}\omega^{2}-1}+1}\delta_{0},&|\omega|>\kappa^{-1}.\end{cases}$
But $U$ is no longer bounded on the imaginary axis as it contains two non
isolated singularities (of order $-1/2$ is the Puiseux series expansion) at
$\pm i\kappa^{-1}$; the integral
$\displaystyle\int_{-\kappa^{-1}}^{\kappa^{-1}}|U^{*}(i\omega)|^{2}d\omega$
is not finite and thus $U\notin\mathcal{H}^{2}(\mathbb{C}_{0})$. By the Paley-
Wiener theorem, this implies that $u\notin L^{2}(\mathbb{R}^{+})$. The same
reasoning can be applied to
$V(s)=(-1)\left(\frac{d}{ds}\right)\left(\frac{-1}{\tau^{2}_{\mu}\,s^{2}+s\ell\sqrt{1+\kappa^{2}s^{2}}+1}\right)\delta_{0}$
and
$W(s)=(-1)\left(\frac{d}{ds}\right)\left(\frac{-s}{\tau^{2}_{\mu}\,s^{2}+s\ell\sqrt{1+\kappa^{2}s^{2}}+1}\right)\delta_{0},$
which are respectively the Laplace transforms of $t\dot{\delta}$ and
$t\ddot{\delta}$. This completes the proof of the second point of the
proposition.
Let us now prove the third point by contradiction. Assuming that there exists
$C>0$ such that for $t$ large enough
$|\delta^{(k)}(t)|\leq Ct^{-\frac{3}{2}-\alpha},$
one gets
$|t\delta^{(k)}(t)|^{2}\leq Ct^{-1-2\alpha}$
which implies $t\delta^{(k)}\in L^{2}(\mathbb{R}^{+})$, which contradicts the
second point. ∎
In the non dispersive case, we showed in Corollary 4.4 that once the motion of
the object is known, it is possible to find $q$ in the exterior domain by
solving an initial boundary value problem for a Burgers-type scalar equation.
This remains true in the present dispersive linear case, but the initial
boundary value problem one has to solve is now nonlocal in time. Note that as
in Remark 4.5, the corollary can easily be generalized to describe the waves
created by an object in forced motion.
###### Corollary 4.6.
The return to equilibrium problem for the linear Boussinesq equations
(4.15)-(4.17) with initial condition (4.2) can be equivalently formulated as a
scalar nonlocal initial boundary value problem on $q$
(4.25)
$\begin{cases}\partial_{x}q+{\mathcal{K}}^{0}_{\mu}\ast\partial_{t}q&=0\qquad(t>0,\quad
x>\ell),\\\ q_{|_{t=0}}&=0,\\\ q_{|_{x=\ell}}&=-\ell\dot{\delta},\end{cases}$
where ${\mathcal{K}}^{0}_{\mu}$ is defined in (4.18) while $\zeta$ is given in
terms of $q$ by a convolution in time
(4.26) $\zeta={\mathcal{K}}^{0}_{\mu}*q,$
with $\delta$ furnished by Proposition 4.2.
###### Remark 4.8.
The nonlocal initial boundary value problem (4.25) is not standard. The most
convenient way to handle it is to see it as an evolution equation with respect
to $x$ rather than $t$; it then becomes a particular case of the nonlocal
initial boundary value problems considered in Section 5. It is in particular a
consequence of Theorem 5.2 below that (4.25) admits a unique solution $q\in
C({\mathbb{R}}^{+}_{x};H^{1}({\mathbb{R}}^{+}_{x}))\cap
C^{1}({\mathbb{R}}^{+}_{x};L^{2}({\mathbb{R}}^{+}_{x}))$. Moreover,
Proposition 5.2 and Corollary 5.1 imply that the solution if actually of class
$C^{2}({\mathbb{R}}^{+}\times{\mathbb{R}}^{+})$ and infinitely regular with
respect to time, showing that the dispersive terms induce a smoothing effect.
Indeed, when $\kappa=0$, the first equation in (4.25) becomes
(4.27) $\partial_{t}q+\partial_{x}q=0$
and the solution to the initial boundary value problem, explicitly given by
$q(x,t)=\begin{cases}-\ell\dot{\delta}(t-(x-\ell))&\text{ for }t-(x-\ell)\geq
0\\\ 0&\text{ for }t-(x-\ell)<0,\end{cases}$
does not belong to $C^{1}(\mathcal{E}_{+}\times\mathbb{R}^{+})$ because
$\ddot{\delta}(0)=-\frac{1}{\tau^{2}_{\mu}}\neq 0$.
## 5\. The initial boundary value problem for a class of nonlocal transport
equations
As shown in the previous section, the analysis of the return to equilibrium
problem in the linear dispersive case leads to a nonlocal generalization of
the transport equation. The analysis of the initial value problem for such
equations is not standard and we address it in this section. Since this
subject is of interest in its own, we work here with more standard notations.
More precisely, we consider an evolution with respect to the time variable and
a nonlocal term with respect to the space variable (this is the reverse in
§4.3). The domain of consideration is the quadrant $\\{x\geq 0,t\geq 0\\}$.
The typical initial boundary value we shall consider is therefore of the form,
$\begin{cases}\partial_{t}u+{\mathcal{K}}*_{x}\partial_{x}u&=f,\\\
u_{|_{x=0}}&={\underline{u}},\\\ u_{|_{t=0}}&=u^{\rm in}.\end{cases}$
for some convolution kernel ${\mathcal{K}}$ to be made precise later.
After presenting some technical material in §5.1.1 for the functional setting
and the Laplace transform, we remind in §5.2 some very classical facts on the
initial and/or boundary value problems for the standard transport equation,
making a distinction between the case of a positive and a negative velocity.
The nonlocal generalizations of these transport problems, in which
$\partial_{x}u$ is replaced by a nonlocal term
${\mathcal{K}}*_{x}\partial_{x}u$, are addressed in §5.3; in particular
similarities and differences (such as the presence of an additional
compatibility condition and a smoothing effect) with their local counterparts
are commented.
NB. To avoid confusions with the computations performed in §4.3 where the
Laplace transform $\widehat{u}$ was taken with respect to time (with dual
variable $s$) we denote throughout this section by $\widetilde{u}$ the Laplace
transform with respect to $x$ (with dual variable $p=\alpha+i\xi$).
### 5.1. Functional setting and a brief reminder on the Laplace transform
We gather here some definition of functional spaces that play an important
role in the analysis of initial boundary value problems, as well as some
classical facts on the Laplace transform.
#### 5.1.1. Functional setting
In the study of initial boundary value problems for hyperbolic systems of
equations, the space ${\mathbb{X}}^{n}$ plays a central role; it is defined
for all $n\in{\mathbb{N}}$ as
${\mathbb{X}}^{n}=\bigcap_{j=0}^{n}C^{j}({\mathbb{R}}^{+}_{t};H^{n-j}({\mathbb{R}}^{+}_{x}));$
in particular, for all $u\in{\mathbb{X}}^{n}$, one can define for all $t\geq
0$ the quantity
$|||u(t,\cdot)|||_{n}:=\sup_{j+k\leq
n}|\partial_{t}^{j}\partial_{x}^{k}u(t,\cdot)|_{L^{2}({\mathbb{R}}^{+})}.$
Let us also define ${\mathbb{Y}}^{n}$ as
${\mathbb{Y}}^{n}=\bigcap_{j=0}^{n}W^{j,1}_{\rm
loc}({\mathbb{R}}^{+}_{t};H^{n-j}({\mathbb{R}}^{+}_{x})).$
When working with nonlocal transport equations, it is convenient to introduce
weighted versions of these spaces. For any $a\in{\mathbb{R}}$, and
$k\in{\mathbb{N}}$, we introduce therefore
$\displaystyle L^{2}_{a}({\mathbb{R}}^{+})$ $\displaystyle:=\\{u\in L^{2}_{\rm
loc}({\mathbb{R}}^{+}),\,|u|_{L^{2}_{a}}:=\Big{(}\int_{{\mathbb{R}}^{+}}e^{-2ax}|u(x)|^{2}{\rm
d}x\Big{)}^{1/2}<\infty\\},$ $\displaystyle H^{k}_{a}({\mathbb{R}}^{+})$
$\displaystyle:=\\{u\in
L^{2}_{a}({\mathbb{R}}^{+}),\,|u|_{H^{k}_{a}}:=\sum_{l=0}^{k}|\partial_{x}^{l}u|_{L^{2}_{a}}<\infty\\},$
and denote by ${\mathbb{X}}^{n}_{a}$ and ${\mathbb{Y}}^{n}_{a}$ the weighted
version of the spaces ${\mathbb{X}}^{n}$ and ${\mathbb{Y}}^{n}$ obtained by
replacing all $L^{2}({\mathbb{R}}^{+}_{x})$ based spaces by their
$L^{2}_{a}({\mathbb{R}}^{+}_{x})$ analogue: we also write
$|||u(t,\cdot)|||_{a,n}:=\sup_{j+k\leq
n}|\partial_{t}^{j}\partial_{x}^{k}u(t,\cdot)|_{L_{a}^{2}({\mathbb{R}}^{+})}.$
#### 5.1.2. Some results on the Laplace transform
For all $u\in L^{1}_{\rm loc}({\mathbb{R}}^{+})$, the Laplace transform is
defined by
$\widetilde{u}(p)=\int_{0}^{\infty}e^{-px}u(x){\rm d}x,$
for all $p=\alpha+i\xi\in{\mathbb{C}}$ such that this integral converges
absolutely. Using for all $a\in{\mathbb{R}}$ the notation
${\mathbb{C}}_{a}=\\{p\in{\mathbb{C}},\Re p>a\\},$
we can define the Hardy space
$\mathcal{H}^{2}(\mathbb{C}_{a}):=\Big{\\{}U\mbox{\rm{ holomorphic on
}}\mathbb{C}_{a}\,;\,||U||^{2}_{\mathcal{H}^{2}(\mathbb{C}_{a})}:=\sup_{\alpha>a}\int_{\mathbb{R}}|U(\alpha+i\xi)|^{2}d\xi<\infty\Big{\\}}.$
Every function $U\in\mathcal{H}^{2}(\mathbb{C}_{a})$ admits a boundary trace
denoted $U^{*}$ on $a+i{\mathbb{R}}$, that belongs to
$L^{2}(a+i{\mathbb{R}})$, and $\mathcal{H}^{2}(\mathbb{C}_{a})$ is a Hilbert
space for the scalar product
$\langle
F,G\rangle_{\mathcal{H}^{2}(\mathbb{C}_{a})}=\frac{1}{2\pi}\int_{-\infty}^{\infty}F^{*}(a+i\xi)\overline{G^{*}(a+i\xi)}{\rm
d}\xi.$
Recalling that the weighted space $L^{2}_{a}({\mathbb{R}}^{+})$ is defined in
the previous section, we can state the well-known Paley-Wiener theorem.
###### Theorem 5.1.
Let $a\in{\mathbb{R}}$. The Laplace-transform
$\mathcal{L}:\begin{array}[]{lcl}L_{a}^{2}({\mathbb{R}}^{+})\to\mathcal{H}^{2}(\mathbb{C}_{a})\\\
u\mapsto\widetilde{u}\end{array}$
is an isometry between Hilbert spaces.
Recalling that
$\widetilde{\frac{du}{dx}}(p)=p\widetilde{u}(p)-u(0)$
(whenever these quantities make sense), we also have the following
characterization of the weighted Sobolev spaces $H^{k}_{a}({\mathbb{R}}^{+})$.
###### Proposition 5.1.
Let $k\in{\mathbb{N}}$ and
$({\underline{u}}_{0},\dots,{\underline{u}}_{k-1})\in{\mathbb{R}}^{k}$. The
following assertions are equivalent,
* i.
One has $u\in H^{k}_{a}({\mathbb{R}}^{+})$ and for all $0\leq j\leq k-1$,
$\lim_{x\to 0^{+}}\partial_{x}^{j}u(x)={\underline{u}}_{j}$.
* ii.
For all $0\leq j\leq k$, the mapping $p\mapsto
p^{j}\widetilde{u}(p)-\sum_{i=0}^{j-1}p^{j-1-i}{\underline{u}}_{i}$ belongs to
${\mathcal{H}}^{2}({\mathbb{C}}_{a})$ (with the sum taken to be zero if
$j=0$).
Moreover, for all $0\leq j\leq k$, one has
$\widetilde{\partial_{x}^{j}u}=p^{j}\widetilde{u}(p)-\sum_{i=0}^{j-1}p^{j-1-i}{\underline{u}}_{i}$.
### 5.2. Reminder on the standard transport equation
Let us start with some considerations on the standard initial boundary value
problem for the transport equations $\partial_{t}u+\partial_{x}u=f$ (referred
to as right-going case) and $\partial_{t}u-\partial_{x}u=f$ (left-going case).
#### 5.2.1. The right-going case
We consider here the following initial boundary value problem
(5.1) $\begin{cases}\partial_{t}u+\partial_{x}u&=f,\\\
u_{|_{x=0}}&={\underline{u}},\\\ u_{|_{t=0}}&=u^{\rm in},\end{cases}$
with $f\in{\mathbb{Y}}^{1}$, $u^{\rm in}\in H^{1}({\mathbb{R}}^{+})$ and
${\underline{u}}\in H^{1}_{\rm loc}({\mathbb{R}}^{+})$. In order for (5.1) to
admit a solution
$u\in{\mathbb{X}}^{1}=C({\mathbb{R}}^{+}_{t};H^{1}({\mathbb{R}}^{+}_{x}))\cap
C^{1}({\mathbb{R}}^{+}_{t};L^{2}({\mathbb{R}}^{+}_{x}))$, and therefore
continuous on $[0,\infty)\times[0,\infty)$, it is necessary that
${\underline{u}}(t=0)=u^{\rm in}(x=0).$
This compatibility condition is actually sufficient to ensure the existence
and uniqueness of such a solution. Even if the data are more regular, i.e. if
$f\in{\mathbb{Y}}^{n}$, $u^{\rm in}\in H^{n}({\mathbb{R}}^{+})$ and
${\underline{u}}\in H^{n}_{\rm loc}({\mathbb{R}}^{+})$ for some $n>1$, one
cannot expect in general the solution to be in ${\mathbb{X}}^{n}$. It is a
general feature of first order hyperbolic systems that such a regularity is
achieved if and only if $n$ algebraic compatibility conditions are satisfied
(see for instance [4, 36, 37, 18]). Of course, the situation is the same if we
choose to work in the weighted space ${\mathbb{X}}^{n}_{a}$ since the presence
of the weight changes the integrability properties at infinity, but not local
regularity. In the present case, this can easily be checked on the following
explicit representation of the solution
(5.2) $u(t,x)=u^{\rm
in}(x-t)+{\underline{u}}(t-x)+\int_{0}^{t}f(t^{\prime},x-t+t^{\prime}){\rm
d}t^{\prime},$
where $u^{\rm in}$, ${\underline{u}}$ and $f(t,\cdot)$ are extended by zero in
order to be considered as functions defined on the full line ${\mathbb{R}}$
instead of ${\mathbb{R}}^{+}$.
#### 5.2.2. The left-going case
It is well-known that an initial boundary value problem similar to (5.1) is
ill-posed for the left-going transport equation
$\partial_{t}u-\partial_{x}u=f$. Indeed, the initial value problem (without
boundary condition)
(5.3) $\begin{cases}\partial_{t}u-\partial_{x}u&=f,\\\ u_{|_{t=0}}&=u^{\rm
in},\end{cases}$
is well posed, and the solution can be explicitly written as
(5.4) $u(t,x)=u^{\rm in}(x+t)+\int_{0}^{t}f(t^{\prime},x+t-t^{\prime}){\rm
d}t^{\prime};$
in particular, the boundary value ${\underline{u}}$ is given in terms of
$u^{\rm in}$ and $f$ through the relation
${\underline{u}}(t)=u^{\rm in}(t)+\int_{0}^{t}f(t^{\prime},t-t^{\prime}){\rm
d}t^{\prime}$
and therefore cannot be freely prescribed. Note that using this relation in
(5.4), one can express the solution in terms of the boundary data instead of
the initial data, namely,
(5.5)
$u(t,x)={\underline{u}}(x+t)-\int_{0}^{x}f(x+t-x^{\prime},x^{\prime}){\rm
d}x^{\prime}.$
This proves in particular that the following boundary value problem (without
initial condition)
(5.6) $\begin{cases}\partial_{t}u-\partial_{x}u&=f,\\\
u_{|_{x=0}}&={\underline{u}},\end{cases}$
is also well-posed for the left-going transport equation.
We note finally that for the initial value problem (5.3) as well as for the
boundary value problem (5.6) (which are essentially the same by switching the
variables $t$ and $x$) the solution $u$ belongs to ${\mathbb{X}}^{1}$ if the
data are smooth enough without having to impose any compatibility condition,
contrary to what we saw for the right-going case.
### 5.3. The nonlocal transport equation
The aim of this section is to investigate the behavior of nonlocal
perturbations of the right-going and left-going transport equations
respectively given by
(5.7) $\partial_{t}u+{\mathcal{K}}^{0}_{\mu}*_{x}\partial_{x}u=f\quad\mbox{
and }\quad\partial_{t}u-{\mathcal{K}}^{0}_{\mu}*_{x}\partial_{x}u=f,$
where $*_{x}$ stands for the causal convolution with respect to the space
variable,
$\forall x\in{\mathbb{R}}^{+},\qquad
f*_{x}g(x)=\int_{0}^{x}f(x-x^{\prime})g(x^{\prime}){\rm d}x^{\prime}$
and with the Bessel kernel ${\mathcal{K}}^{0}_{\mu}$ as in (4.18); in
particular, we recall that
$\widetilde{{\mathcal{K}}^{0}_{\mu}}(p)=\frac{1}{\sqrt{1+\kappa^{2}p^{2}}}\qquad(\kappa^{2}=\mu/3).$
###### Remark 5.1.
Though we consider here the Bessel kernel ${\mathcal{K}}^{0}_{\mu}$, the
results of this section can easily be adapted to other kernels.
An important feature of the family $({\mathcal{K}}^{0}_{\mu})_{\mu>0}$ is that
it formally converges to the Dirac mass at $x=0$ as $\mu\to 0$, so that the
nonlocal transport equations (5.7) formally converge to the standard right-
going and left-going transport equations respectively, namely,
$\partial_{t}u+\partial_{x}u=f\quad\mbox{ and
}\quad\partial_{t}u-\partial_{x}u=f.$
A natural question is therefore to ask whether the nonlocal initial and/or
boundary value problems have a similar behavior to the behavior of their local
counterpart described in §5.2.
#### 5.3.1. The right-going case
We want to address in this section the same kind of initial boundary value
problem as (5.1), but where the space derivative is now replaced by a nonlocal
term, namely, we consider
(5.8)
$\begin{cases}\partial_{t}u+{\mathcal{K}}^{0}_{\mu}*_{x}\partial_{x}u&=f,\\\
u_{|_{x=0}}&={\underline{u}},\\\ u_{|_{t=0}}&=u^{\rm in}.\end{cases}$
As for (5.1), if there exists a solution $u\in{\mathbb{X}}^{1}$ (or more
generally in the weighted version ${\mathbb{X}}^{1}_{a}$ with $a\geq 0$) to
(5.8), then it is continuous at $x=t=0$ and the data must therefore satisfy
the same compatibility condition
(5.9) ${\underline{u}}(t=0)=u^{\rm in}(x=0)$
as for the standard transport equation.
There is however a new compatibility condition that arises here. Indeed, since
${\mathcal{K}}^{0}_{\mu}\in L^{1}_{\rm loc}({\mathbb{R}}^{+})$, the trace of
${\mathcal{K}}^{0}_{\mu}*_{x}\partial_{x}u$ at $x=0$ is well defined if
$\partial_{x}u\in C({\mathbb{R}}^{+}_{t};L^{2}_{\rm
loc}({\mathbb{R}}^{+}_{x}))$, and it must be equal to zero by definition of
the convolution. Taking the trace of the first equation in (5.8), one
therefore finds the following additional compatibility condition for the
existence of solutions with the aforementioned regularity,
(5.10) $\forall
t\in{\mathbb{R}}^{+},\qquad\partial_{t}{\underline{u}}(t)=f(t,0).$
###### Remark 5.2.
The similar procedure applied to the standard transport problem (5.8) yields
the relation
$\partial_{x}u(t,0)=-\partial_{t}{\underline{u}}(t)+f(t,0),$
which is not a compatibility condition but an information on the behavior of
the trace of $\partial_{x}u$ at the boundary.
If these two compatibility conditions are satisfied, the theorem below shows
the well-posedness of the nonlocal initial boundary value problem (5.8). We
recall that the functional spaces have been defined in §5.1.1; note also that
we have to work in weighted spaces here in order to compensate the slow decay
of ${\mathcal{K}}^{0}_{\mu}$ at infinity (which is of order $O(|x|^{-1/2}$)
and that more information on the regularity of the solution is given in
Corollary 5.1 below.
###### Theorem 5.2.
Let $a>0$ and $f\in{\mathbb{Y}}^{1}_{a}$, $u^{\rm in}\in
H_{a}^{1}({\mathbb{R}}_{x}^{+})$ and ${\underline{u}}\in W^{1,1}_{\rm
loc}({\mathbb{R}}^{+}_{t})$. Assume moreover that the compatibility conditions
(5.9) and (5.10) hold. Then there exists a unique solution
$u\in{\mathbb{X}}^{1}_{a}$ to the nonlocal initial boundary value problem
(5.8), and there exists $c_{a}>0$ such that, for all $t\in{\mathbb{R}}^{+}$,
$|u(t,\cdot)|_{H_{a}^{1}({\mathbb{R}}^{+}_{x})}\leq e^{-c_{a}t}|u^{\rm
in}|_{H_{a}^{1}({\mathbb{R}}^{+}_{x})}+\int_{0}^{t}e^{-c_{a}(t-t^{\prime})}\big{[}|f(t^{\prime},\cdot)|_{H_{a}^{1}({\mathbb{R}}^{+}_{x})}+|{\mathcal{K}^{0}_{\mu}}|_{L_{a}^{2}}|{\underline{u}}(t^{\prime})|\big{]}{\rm
d}t^{\prime}.$
If moreover ${\underline{u}}=0$ then the result still holds with $a=c_{a}=0$.
###### Proof.
For the sake of clarity, we simply write ${\mathcal{K}}$ instead of
${\mathcal{K}}^{0}_{\mu}$. Taking the Laplace transform of (5.8) with respect
to space, one gets that
$\partial_{t}\widetilde{u}+\widetilde{{\mathcal{K}}}(p)(p\widetilde{u}-{\underline{u}})=\widetilde{f}\quad\mbox{
on }\quad{\mathbb{R}}^{+}.$
Solving this ODE with initial condition
$\widetilde{u}_{|_{t=0}}=\widetilde{u^{\rm in}}$, one gets the following
expression for $\widetilde{u}$, for all $p\in{\mathbb{C}}_{a}$ and
$t\in{\mathbb{R}}^{+}$,
$\displaystyle\widetilde{u}(t,p)$
$\displaystyle=e^{-p\widetilde{{\mathcal{K}}}(p)t}\widetilde{u^{\rm
in}}(p)+\int_{0}^{t}e^{-p\widetilde{{\mathcal{K}}}(p)(t-t^{\prime})}\widetilde{f}{\rm
d}t^{\prime}+\int_{0}^{t}e^{-p\widetilde{{\mathcal{K}}}(p)(t-t^{\prime})}\widetilde{{\mathcal{K}}}(p){\underline{u}}(t^{\prime}){\rm
d}t^{\prime}$ (5.11)
$\displaystyle=:\widetilde{u_{1}}+\widetilde{u_{2}}+\widetilde{u_{3}}.$
Since the Paley-Wiener Theorem 5.1 states that the Laplace transform is an
isometry between $L_{a}^{2}({\mathbb{R}}^{+})$ and
${\mathcal{H}}^{2}({\mathbb{C}}_{a})$, the following lemma shows that both
$u_{1}$ and $u_{2}$ belong to
$C({\mathbb{R}}^{+}_{t};L^{2}_{a}({\mathbb{R}}^{+}_{x}))$ if $u^{\rm in}\in
L^{2}_{a}({\mathbb{R}}^{x})$ and $f\in L^{1}_{\rm
loc}({\mathbb{R}}^{+}_{t};L^{2}_{a}({\mathbb{R}}^{+}_{x}))$.
###### Lemma 5.1.
Let $a\geq 0$. For all $U\in{\mathcal{H}}^{2}({\mathbb{C}}_{a})$, the mapping
$\begin{array}[]{lcl}{\mathbb{R}}^{+}&\to&{\mathcal{H}}^{2}({\mathbb{C}}_{a})\\\
t&\mapsto&\big{(}p\mapsto
e^{-p\widetilde{{\mathcal{K}}}(p)t}U(p)\big{)}\end{array}$
is well defined and continuous, and for all $t\in{\mathbb{R}}^{+}$,
$\|e^{-p\widetilde{{\mathcal{K}}}(p)t}U\|_{{\mathcal{H}}^{2}({\mathbb{C}}_{a})}\leq\|U\|_{{\mathcal{H}}^{2}({\mathbb{C}}_{a})}$.
If moreover $a>0$ then there exists $c_{a}>0$ such that for all
$t\in{\mathbb{R}}^{+}$,
$\|e^{-p\widetilde{{\mathcal{K}}}(p)t}U\|_{{\mathcal{H}}^{2}({\mathbb{C}}_{a})}\leq
e^{-c_{a}t}\|U\|_{{\mathcal{H}}^{2}({\mathbb{C}}_{a})}.$
###### Proof of the lemma.
Except for the last assertion, we consider only the case $a=0$ since the case
$a>0$ can easily be deduced from it. From the definition of
${\mathcal{H}}^{2}({\mathbb{C}}_{0})$ and Lebesgue’s dominated convergence
theorem, it is sufficient to prove that $e^{-p\widetilde{{\mathcal{K}}}(p)t}$
is holomorphic and bounded on ${\mathbb{C}}_{0}$. The fact that it is
holomorphic directly stems from the explicit expression
$\widetilde{{\mathcal{K}}}(p)=(1+\kappa^{2}p^{2})^{-1/2}$. For the
boundedness, this is a consequence of the fact that
$\Re(p\widetilde{{\mathcal{K}}}(p))\geq 0$ on ${\mathbb{C}}_{0}$, as we now
prove. For all $p=\alpha+i\xi\in\mathbb{C}_{0}$, one computes
(5.12)
$\Re(p\hat{{\mathcal{K}}}(p))=\frac{\alpha\,\Re({\sqrt{1+\kappa^{2}p^{2}}})+\xi\,\Im({\sqrt{1+\kappa^{2}p^{2}}})}{|\sqrt{1+\kappa^{2}p^{2}}|^{2}}.$
Since $\Re(\sqrt{1+\kappa^{2}p^{2}})$ is positive (by definition of the square
root) and the sign of $\Im({\sqrt{1+\kappa^{2}p^{2}}})$ is the same as the
sign of the product $\alpha\,\xi$, one gets the result.
Since we have proved that $\Re(p\widetilde{{\mathcal{K}}}(p))\geq 0$ on
${\mathbb{C}}_{0}$, the last assertion follows if we can prove that
$\Re(p\widetilde{{\mathcal{K}}}(p))$ does not vanish on ${\mathbb{C}}_{a}$ if
$a>0$. Since both terms in the numerator in (5.12) are positive, both must
vanish if $\Re(p\widetilde{{\mathcal{K}}}(s))$ vanishes. Since $\alpha>0$ on
${\mathbb{C}}_{0}$, this implies that there should be
$p=\alpha+i\xi\in{\mathbb{C}}_{a}$ such that $\Re(\sqrt{1+\kappa^{2}p^{2}})=0$
and $\xi\,\Im({\sqrt{1+\kappa^{2}p^{2}}})=0$, which is obviously not possible.
∎
Remarking that for any $a>0$, one has
$\widehat{\mathcal{K}}\in{\mathcal{H}}^{2}({\mathbb{C}}_{a})$, it is also a
direct consequence of the lemma that there is $c_{a}>0$ such that
$\displaystyle\|{\widetilde{u}_{3}}\|_{{\mathcal{H}}^{2}({\mathbb{C}}_{a})}$
$\displaystyle\leq\|{\widetilde{{\mathcal{K}}}}\|_{{\mathcal{H}}^{2}({\mathbb{C}}_{a})}\int_{0}^{t}e^{-c_{a}(t-t^{\prime})}|{\underline{u}}(t^{\prime})|{\rm
d}t^{\prime}.$
Together with the results already proved on $\widetilde{u}_{1}$ and
$\widetilde{u_{2}}$, we deduce (see the Paley-Wiener Theorem 5.1 below) that
$|u(t,\cdot)|_{L_{a}^{2}({\mathbb{R}}^{+}_{x})}\leq e^{-c_{a}t}|u^{\rm
in}|_{L_{a}^{2}({\mathbb{R}}^{+}_{x})}+\int_{0}^{t}e^{-c_{a}(t-t^{\prime})}\big{[}|f(t^{\prime},\cdot)|_{L_{a}^{2}({\mathbb{R}}^{+}_{x})}+|{\mathcal{K}}|_{L_{a}^{2}}|{\underline{u}}(t^{\prime})|\big{]}{\rm
d}t^{\prime}.$
In order to conclude the proof of the theorem, we still need to control
$\partial_{x}u$ and $\partial_{t}u$.
* •
Control of $\partial_{x}u$. We want to show that $\partial_{x}u\in
C({\mathbb{R}}^{+}_{t};L^{2}_{a}({\mathbb{R}}^{+}_{x}))$, or equivalently that
$\widetilde{\partial_{x}u}\in
C({\mathbb{R}}^{+}_{t};{\mathcal{H}}^{2}({\mathbb{C}}_{a}))$. Since
$\widetilde{\partial_{x}u}=p\widetilde{u}-{\underline{u}}$, we consider
$p\widetilde{u}(t,p)=e^{-p\widetilde{{\mathcal{K}}}(p)t}p\widetilde{u^{\rm
in}}(p)+\int_{0}^{t}e^{-p\widetilde{{\mathcal{K}}}(p)(t-t^{\prime})}p\widetilde{f}{\rm
d}t^{\prime}+\int_{0}^{t}e^{-p\widetilde{{\mathcal{K}}}(p)(t-t^{\prime})}p\widetilde{{\mathcal{K}}}(p){\underline{u}}(t^{\prime}){\rm
d}t^{\prime}.$
Writing $p\widetilde{u^{\rm in}}=\widetilde{\partial_{x}u^{\rm in}}+u^{\rm
in}(0)$, $p\widetilde{f}(t,p)=\widetilde{\partial_{x}f}(t,p)+f(t,0)$, we can
remark that
$\displaystyle\int_{0}^{t}e^{-p\widetilde{{\mathcal{K}}}(p)(t-t^{\prime})}p\widetilde{{\mathcal{K}}}(p){\underline{u}}(t^{\prime}){\rm
d}t^{\prime}$
$\displaystyle=\int_{0}^{t}\partial_{t^{\prime}}\big{(}e^{-p\widetilde{{\mathcal{K}}}(p)(t-t^{\prime})}\big{)}{\underline{u}}(t^{\prime}){\rm
d}t^{\prime}$
$\displaystyle={\underline{u}}(t)-e^{-p\widetilde{{\mathcal{K}}}(p)t}{\underline{u}}(0)-\int_{0}^{t}e^{-p\widetilde{{\mathcal{K}}}(p)(t-t^{\prime})}\partial_{t}{\underline{u}}(t^{\prime}){\rm
d}t^{\prime},$
from which we deduce that
$\displaystyle\widetilde{\partial_{x}u}(t,p)=$ $\displaystyle
e^{-p\widetilde{{\mathcal{K}}}(p)t}\widetilde{\partial_{x}u^{\rm
in}}(p)+\int_{0}^{t}e^{-p\widetilde{{\mathcal{K}}}(p)(t-t^{\prime})}\widetilde{\partial_{x}f}{\rm
d}t^{\prime}$ $\displaystyle+e^{-p\widetilde{{\mathcal{K}}}(p)t}\big{(}u^{\rm
in}(0)-{\underline{u}}(0)\big{)}+\int_{0}^{t}e^{-p\widetilde{{\mathcal{K}}}(p)(t-t^{\prime})}\big{(}f(t^{\prime},0)-\partial_{t}{\underline{u}}(t^{\prime})\big{)}{\rm
d}t^{\prime}.$
While the first two components of the right-hand side belong to
$C({\mathbb{R}}^{+}_{t};{\mathcal{H}}^{2}({\mathbb{C}}_{a}))$ by Lemma 5.1,
the last two ones do not, unless the compatibilty conditions given in the
statement of the Theorem are satisfied, in which case these two components
cancel and the result follows together with the upper bound
$|\partial_{x}u(t,\cdot)|_{L^{2}_{a}}\leq|\partial_{x}u^{\rm
in}|_{L^{2}_{a}}+\int_{0}^{t}e^{-c_{a}(t-t^{\prime})}|\partial_{x}f(t^{\prime},\cdot)|_{L^{2}_{a}}{\rm
d}t^{\prime}.$
* •
Control of $\partial_{t}u$. Using the equations, one has
$\displaystyle|\partial_{t}u|_{L^{2}_{a}}$
$\displaystyle\leq|{\mathcal{K}}*_{x}\partial_{x}u|_{L^{2}_{a}}+|f|_{L^{2}_{a}}$
$\displaystyle\leq|{\mathcal{K}}|_{L^{1}_{a}}|\partial_{x}u|_{L^{2}_{a}}+|f|_{L^{2}_{a}},$
with $L^{1}_{a}=L^{1}({\mathbb{R}}^{+},e^{-ax}{\rm d}x)$, showing as needed
that $\partial_{t}u\in
C({\mathbb{R}}^{+}_{t};L^{2}_{a}({\mathbb{R}}^{+}_{x}))$.
The theorem follows easily. ∎
###### Remark 5.3.
As explained above in Remark 5.1, the initial boundary value problem (5.8) can
be seen as a nonlocal perturbation of the standard transport problem (5.1)
toward which it formally converges when $\mu\to 0$. There seems however to be
some discrepancy because two compatibility conditions, namely, (5.9) and
(5.10), are needed to ensure the existence of solutions
$u\in{\mathbb{X}}^{1}_{a}$ to (5.8), while the sole compatibility condition
(5.9) is sufficient to get a similar result for the standard transport problem
(5.1). One should explain why the second compatibility condition (5.10)
disappears in the formal limit $\mu=0$.
The reason is that (5.10) is here to ensure continuity of the solution at the
boundary $x=0$. Indeed, by the initial value theorem, we know that $\lim_{x\to
0^{+}}u(t,x)=\lim_{p\in{\mathbb{C}}_{a},|p|\to\infty}p\widetilde{u}(t,p)$, and
we therefore get from the Laplace representation formula (5.11) that
$\lim_{x\to 0^{+}}u(t,x)=e^{-\frac{t}{\kappa}}u^{\rm
in}(0)+\int_{0}^{t}e^{-\frac{t-t^{\prime}}{\kappa}}f(t^{\prime},0){\rm
d}t^{\prime}+\int_{0}^{t}e^{-\frac{t-t^{\prime}}{\kappa}}\frac{1}{\kappa}{\underline{u}}(t^{\prime}){\rm
d}t^{\prime}$
where we used the fact that
$\lim_{p\in{\mathbb{C}}_{a},|p|\to\infty}p\widetilde{{\mathcal{K}}}(p)=\kappa^{-1}$;
after an integration by parts, the right-hand side can be written
${\underline{u}}(t)+e^{-\frac{t}{\kappa}}\big{(}u^{\rm
in}(0)-{\underline{u}}(0)\big{)}+\int_{0}^{t}e^{-\frac{t-t^{\prime}}{\kappa}}\big{(}f(t^{\prime},0)-\partial_{t}{\underline{u}}(t^{\prime})\big{)}{\rm
d}t^{\prime},$
so that, if the first compatibility condition (5.9) is satisfied, one has
$\lim_{x\to
0^{+}}u(t,x)-{\underline{u}}(t)=\int_{0}^{t}e^{-\frac{t-t^{\prime}}{\kappa}}\big{(}f(t^{\prime},0)-\partial_{t}{\underline{u}}(t^{\prime})\big{)}{\rm
d}t^{\prime},$
which is nonzero if the second compatibility condition is not satisfied, hence
a lack of continuity at $x=0$ (there would therefore be a Dirac mass at $x=0$
is the expression for $\partial_{x}u(t,\cdot)$ that would therefore not be in
$L^{2}_{a}({\mathbb{R}}^{+}_{x})$ as seen in the proof). However, one readily
observes that
$\lim_{\mu\to
0}\int_{0}^{t}e^{-\frac{t-t^{\prime}}{\kappa}}\big{(}f(t^{\prime},0)-\partial_{t}{\underline{u}}(t^{\prime})\big{)}{\rm
d}t^{\prime}=0\qquad(\kappa^{2}=\mu/3),$
so that this discontinuity shrinks to zero in the limit $\mu\to 0$, explaining
why the second compatibility condition is no longer necessary in the endpoint
case $\mu=0$.
Before going further, we recall that there are two possibilities to define
fractional derivatives of order $\alpha\in(0,1)$ on ${\mathbb{R}}^{+}$ using
the convolution kernel
${\mathfrak{K}}_{\alpha}(x)=x^{-\alpha}/{\Gamma(1-\alpha)}$ with
$\alpha\in(0,1)$ and $\Gamma$ the Euler Gamma function, namely, the Riemann-
Liouville and Caputo derivatives, defined respectively as
$D^{\alpha}_{\rm
RL}u=\partial_{x}\big{(}{\mathfrak{K}}_{\alpha}*_{x}u)\quad\mbox{ and }\quad
D^{\alpha}_{\rm C}u={\mathfrak{K}}_{\alpha}*_{x}\partial_{x}u.$
In the nonlocal initial boundary value problem (5.8), the space derivative
$\partial_{x}u$ in the standard transport equation has been replaced by the
nonlocal term ${\mathcal{K}}^{0}_{\mu}*\partial_{x}u$ which can be considered
as a generalized derivative of Caputo type, with the kernel
${\mathfrak{K}}_{\alpha}$ replaced by the Bessel kernel
${\mathcal{K}}^{0}_{\mu}$. It is noteworthy that working with the Riemann-
Liouville version of this operator, namely
$\partial_{x}\big{(}{\mathcal{K}}^{0}_{\mu}*_{x}u)$, the situation is
drastically different. Indeed, as shown in the following proposition, it is
not possible to impose a boundary data anymore since the knowledge of the
initial data suffices to fully determine the solution; in other words, the
initial value problem
(5.13)
$\begin{cases}\partial_{t}u+\partial_{x}\big{(}{\mathcal{K}}^{0}_{\mu}*_{x}u)&=f,\\\
u_{|_{t=0}}&=u^{\rm in},\end{cases}$
is well posed on ${\mathbb{R}}^{+}_{t}\times{\mathbb{R}}^{+}_{x}$. In
particular, the trace of the solution at the boundary $x=0$ is determined by
$f$ and $u^{\rm in}$ and therefore cannot be imposed. We also show that if the
data $u^{\rm in}$ and $f$ are smoother, then the solution is in
${\mathbb{X}}_{a}^{2}$, but generally not in ${\mathbb{X}}_{a}^{3}$ or higher
in the absence of additional compatibility condition (but we show however that
the regularity in time can be higher).
###### Proposition 5.2.
Let $a>0$, $n=1$ or $2$, and $f\in{\mathbb{Y}}^{n}_{a}$ and $u^{\rm in}\in
H_{a}^{n}({\mathbb{R}}_{x}^{+})$. Then there exists a unique solution
$u\in{\mathbb{X}}_{a}^{n}$ to the nonlocal initial boundary value problem
(5.13). Moreover, one has $u(t,\cdot)_{|_{x=0}}={\underline{u}}(t)$ for all
$t\in{\mathbb{R}}^{+}$, with ${\underline{u}}(t)$ given by
${\underline{u}}(t)=e^{-\frac{t}{\kappa}}u^{\rm
in}(0)+\int_{0}^{t}e^{-\frac{t-t^{\prime}}{\kappa}}f(t^{\prime},0){\rm
d}t^{\prime}.$
If in addition $f\in
C^{q}({\mathbb{R}}^{+}_{t};H^{n}_{a}({\mathbb{R}}^{+}_{x}))$ for some
$q\in{\mathbb{N}}$ then one also has $u\in
C^{q+1}({\mathbb{R}}^{+}_{t};H^{n}_{a}({\mathbb{R}}^{+}_{x}))$.
###### Remark 5.4.
Comparing the representation of the solution given in (5.14) below to the
representation of the solution to the initial boundary value problem (5.8)
given in (5.11), one can check that they are both the same if
${\underline{u}}=0$, which is not surprising since one can compute
$\partial_{x}({\mathcal{K}}^{0}_{\mu}*_{x}u)(t,x)=({\mathcal{K}}^{0}_{\mu}*_{x}\partial_{x}u)(t,x)+{\mathcal{K}}^{0}_{\mu}(x){\underline{u}}(t),$
so that the Caputo and Riemann-Liouville nonlocal initial boundary value
problem coincide when ${\underline{u}}=0$.
###### Proof.
As previously done, we simply write ${\mathcal{K}}={\mathcal{K}}_{\mu}^{0}$.
Taking the Laplace transform of (5.13) one readily gets
(5.14)
$\widetilde{u}(t,p)=e^{-p\widetilde{{\mathcal{K}}}(p)t}\widetilde{u^{\rm
in}}(p)+\int_{0}^{t}e^{-p\widetilde{{\mathcal{K}}}(p)(t-t^{\prime})}\widetilde{f}(t^{\prime},p){\rm
d}t^{\prime};$
by the initial value theorem, one gets that $\lim_{x\to
0^{+}}u(t,x)={\underline{u}}(t)$, with ${\underline{u}}$ as in the statement
of the theorem.
For all $j$ and $l$, one deduces from the above formula for $\widetilde{u}$
that
$p^{j}\widetilde{\partial_{t}^{l}u}=(-p\widehat{{\mathcal{K}}}(p))^{l}\big{[}e^{-p\widehat{{\mathcal{K}}}(p)t}p^{j}\widetilde{u^{\rm
in}}+\int_{0}^{t}e^{-p\widetilde{\mathcal{K}}(p)(t-t^{\prime})}p^{j}\widetilde{f}\big{]}+\sum_{m=1}^{l}(-p\widehat{{\mathcal{K}}}(p))^{l-m}p^{j}\widetilde{\partial_{t}^{m-1}f}.$
Replacing in this expression
$p^{j}\widetilde{v}=\widetilde{\partial_{x}^{j}v}+\sum_{i=0}^{j-1}p^{j-1-i}(\partial_{x}^{i}v)_{|_{x=0}}$
for $v=u^{\rm in}$, $\widetilde{f}$, $\widetilde{\partial_{t}^{m-1}f}$, we
obtain
$p^{j}\widetilde{\partial_{t}^{l}u}=\sum_{i=0}^{j-1}p^{j-1-i}U_{li}(p)+F_{lj}(t,p)$
(using the convention that the summation is zero if $j-1<0$) with
$\displaystyle U_{li}(p):=$
$\displaystyle(-p\widetilde{{\mathcal{K}}}(p))^{l}\big{(}\partial_{x}^{i}u^{\rm
in}(0)+\int_{0}^{t}e^{-p\widetilde{\mathcal{K}}(p)(t-t^{\prime})}(\partial_{x}^{i}f)(t^{\prime},0){\rm
d}t^{\prime}\big{)}$
$\displaystyle+\sum_{m=1}^{l}(-p\widetilde{{\mathcal{K}}}(p))^{l-m}(\partial_{t}^{m-1}\partial_{x}^{i}f)_{|_{x=0}}$
and
$F_{lj}(t,p):=(-p\widehat{{\mathcal{K}}}(p))^{l}\big{[}\widetilde{\partial_{x}^{j}u^{\rm
in}}+\int_{0}^{t}e^{-p\widetilde{\mathcal{K}}(p)(t-t^{\prime})}\widetilde{\partial_{x}^{j}f}\big{]}+\sum_{m=1}^{l}(-p\widehat{{\mathcal{K}}}(p))^{l-m}\widetilde{\partial_{t}^{m-1}\partial_{x}^{j}f}.$
Remarking that $\lim_{|p|\to\infty}p\widehat{{\mathcal{K}}}(p)=\kappa^{-1}$,
and introducing ${\underline{u}}_{li}=\lim_{|p|\to\infty}U_{li}(p)$, namely,
${\underline{u}}_{li}=(-\kappa)^{-l}\big{(}\partial_{x}^{i}u^{\rm
in}(0)+\int_{0}^{t}e^{-\frac{t-t^{\prime}}{\kappa}}(\partial_{x}^{i}f)(t^{\prime},0){\rm
d}t^{\prime}\big{)}\\\
+\sum_{m=1}^{l}(-\kappa)^{-l+m}(\partial_{t}^{m-1}\partial_{x}^{i}f)_{|_{x=0}}$
(of course, ${\underline{u}}_{00}={\underline{u}}$), we can write
$p^{j}\widetilde{\partial_{t}^{l}u}-\sum_{i=0}^{j-1}p^{j-1-i}{\underline{u}}_{li}(p)=\sum_{i=0}^{j-1}p^{j-1-i}\big{(}U_{li}(p)-{\underline{u}}_{li}\big{)}+F_{lj}(t,p).$
From Proposition 5.1, we can deduce that $\partial_{x}^{j}\partial_{t}^{l}u$
belongs to $C({\mathbb{R}}^{+}_{t};L^{2}_{a}({\mathbb{R}}^{+}_{x}))$ if the
right-hand side of the above equality is in
$C({\mathbb{R}}^{+}_{t};{\mathcal{H}}^{2}({\mathbb{C}}_{a}))$. This is obvious
for $F_{lj}$ under the assumptions made in the statement of the proposition
(see Lemma 5.1); for the summation, the problem reduces to determine whether
the mapping $p\mapsto
p^{j-1}\big{(}p\widehat{{\mathcal{K}}}(p)-\kappa^{-1}\big{)}$ belongs to
${\mathcal{H}}^{2}({\mathbb{C}}_{a})$ or not. This mapping being holomorphic
on ${\mathbb{C}}_{a}$, we just need to check that it is square integrable on
$a+i{\mathbb{R}}$. Recalling that
$\widehat{{\mathcal{K}}}(p)=\frac{1}{\sqrt{1+\kappa^{2}p^{2}}}$, and using the
fact that for all $p\in{\mathbb{C}}_{a}$ one has $\sqrt{p^{2}}=p$, one has
$p^{j-1}\big{(}p\widehat{{\mathcal{K}}}(p)-\kappa^{-1}\big{)}\sim-\frac{1}{2\kappa^{3}}p^{j-3}$
at infinity; the mapping is therefore square integrable on $a+i{\mathbb{R}}$
if and only if $j\leq 2$, hence the results. ∎
As a corollary, we can exhibit a smoothing effect for the nonlocal transport
problem (5.8) that does not exist for the standard transport problem (5.1).
Indeed, as one can easily check on the explicit expression (5.2), even if the
data $u^{\rm in}$, ${\underline{u}}$ and $f$ are very smooth, the solution is
not $C^{1}({\mathbb{R}}^{+}\times{\mathbb{R}}^{+})$ if the additional
compatibility condition $\partial_{t}{\underline{u}}(0)=-\partial_{x}u^{\rm
in}(0)+f(0,0)$ is not imposed. There is a smoothing effect for the nonlocal
problem in the sense that the solution constructed in Theorem 5.2 actually
belongs to ${\mathbb{X}}_{a}^{3}\subset
C^{2}({\mathbb{R}}^{+}\times{\mathbb{R}}^{+})$ without any additional
compatibility condition if the data are smooth enough. Note that using the
last statement of Proposition 5.2, the proof shows that additional regularity
in time on $\partial_{x}f$ would yield additional regularity in time on
$\partial_{x}u$.
###### Corollary 5.1.
Under the assumptions of Theorem 5.2, if moreover $f\in{\mathbb{Y}}^{n}_{a}$,
$u^{\rm in}\in H_{a}^{n}({\mathbb{R}}_{x}^{+})$ and ${\underline{u}}\in
W^{n,1}_{\rm loc}({\mathbb{R}}^{+}_{t})$ for $n=2$ or $3$, then the solution
$u$ provided by the theorem belongs to ${\mathbb{X}}^{n}_{a}$.
###### Proof.
Taking the space derivative of the nonlocal transport equation in (5.8), it is
easy to see that $v=\partial_{x}u$ solves the initial boundary value problem
$\begin{cases}\partial_{t}v+\partial_{x}\big{(}{\mathcal{K}}*_{x}v)=\partial_{x}f,\\\
v_{|_{t=0}}=\partial_{x}u^{\rm in}.\end{cases}$
It follows therefore from Proposition 5.2 that
$\partial_{x}u\in{\mathbb{X}}^{n-1}$. We are therefore left to prove that
$\partial_{t}^{j}u\in C({\mathbb{R}}_{t}^{+};L^{2}_{a}({\mathbb{R}}^{+}_{x}))$
for $1\leq j\leq n$; this easily follows from the observation that
$\partial_{t}^{j}u=-{\mathcal{K}}*_{x}\partial_{t}^{j-1}\partial_{x}u+\partial_{t}^{j-1}f$
and from the fact that ${\mathcal{K}}\in L^{1}_{a}({\mathbb{R}}^{+})$. ∎
#### 5.3.2. The left-going case
As for the right-going case in the previous section, we want to consider a
nonlocal perturbation of the standard transport problem in which the space
derivative $\partial_{x}$ is replaced by a nonlocal term
${\mathcal{K}}^{0}_{\mu}*\partial_{x}$. As recalled in §5.2, for the standard
left-going transport equation, one has to consider either the initial value
problem or the boundary value problem. While both cases are symmetric in the
case of the standard transport equation, this is no longer the case and, as we
shall see, the boundary value problem leads simpler expressions. We therefore
consider here its nonlocal analogue (see Remark 5.6 below for the nonlocal
analogue of the initial value problem),
(5.15)
$\begin{cases}\partial_{t}u-{\mathcal{K}}^{0}_{\mu}*_{x}\partial_{x}u&=f,\\\
u_{|_{x=0}}&={\underline{u}}.\end{cases}$
As for the boundary value problem (5.6) for the standard left-going transport
equation, there is no compatibility condition like (5.9) since $u^{\rm in}$ is
not prescribed. On the other hand, the analysis leading to the second
compatibility condition (5.10) remains valid, and it is still necessary to
have
(5.16) $\forall
t\in{\mathbb{R}}^{+},\qquad\partial_{t}{\underline{u}}(t)=f(t,0)$
in order to expect a solution $u$ that belongs to ${\mathbb{X}}^{1}$. In the
statement below, we use the notation
$H^{1}_{a}({\mathbb{R}}^{+}_{t}\times{\mathbb{R}}^{+}_{x}):=H^{1}({\mathbb{R}}^{+}_{t};L^{2}_{a}({\mathbb{R}}^{+}_{x}))\cap
L^{2}({\mathbb{R}}^{+}_{t};H^{1}_{a}({\mathbb{R}}^{+}_{x}))$
(note that the assumptions on the time dependence of $f$ and ${\underline{u}}$
are chosen in order to ensure the convergence of the integral term over the
range $(t,+\infty)$ and that they could easily be weakened).
###### Theorem 5.3.
Let $a>0$, $f\in H^{1}_{a}({\mathbb{R}}^{+}_{t}\times{\mathbb{R}}^{+}_{x})$,
and ${\underline{u}}\in H^{1}({\mathbb{R}}^{+}_{t})$. Assume moreover that the
compatibility condition (5.16) holds. Then there exists a unique solution
$u\in{\mathbb{X}}^{1}_{a}$ to the nonlocal boundary value problem (5.15), and
there exists $c_{a}>0$ such that, for all $t\in{\mathbb{R}}^{+}$,
$|u(t,\cdot)|_{H_{a}^{1}({\mathbb{R}}^{+}_{x})}\leq\int_{t}^{\infty}e^{c_{a}(t-t^{\prime})}\big{[}|f(t^{\prime},\cdot)|_{H_{a}^{1}({\mathbb{R}}^{+}_{x})}+|{\mathcal{K}^{0}_{\mu}}|_{L_{a}^{2}}|{\underline{u}}(t^{\prime})|\big{]}{\rm
d}t^{\prime}.$
###### Proof.
Still denoting ${\mathcal{K}}={\mathcal{K}}_{\mu}^{0}$ and following the same
procedure as for the proof of Theorem 5.2, one readily finds that
$\widetilde{u}(t,p)=-\int_{t}^{\infty}e^{p\widetilde{{\mathcal{K}}}(p)(t-t^{\prime})}\big{(}\widetilde{f}(t^{\prime},p)-\widetilde{{\mathcal{K}}}(p){\underline{u}}(t^{\prime})\big{)}{\rm
d}t^{\prime};$
as for the right-going case, one can check that the compatibility condition
(5.16) is necessary for the continuity of the solution at $x=0$. We omit the
proof which is an easy adaptation of the proof of Theorem 5.2. ∎
###### Remark 5.5.
As for the standard boundary transport problem (5.6), the initial data is
determined in terms of the source term $f$ and the boundary data
${\underline{u}}$ by evaluating the Laplace representation formula given in
the proof at $t=0$, namely,
(5.17) $\widetilde{u^{\rm
in}}(p)=-\int_{0}^{\infty}e^{-p\widetilde{{\mathcal{K}}_{\mu}^{0}}(p)t^{\prime}}\big{(}\widetilde{f}(t^{\prime},p)-\widetilde{{\mathcal{K}}_{\mu}^{0}}(p){\underline{u}}(t^{\prime})\big{)}{\rm
d}t^{\prime}.$
In the limit case $\mu=0$ (and therefore
$\widehat{{\mathcal{K}}}_{\mu}(p)=1$), one can check that the representation
formula of the proof is equivalent to (5.5); the additional compatibility
condition (5.16) that is not necessary for (5.6) also disappears at the limit
along a mechanism similar to the one described in Remark 5.3.
###### Remark 5.6.
For the standard left-going transport equation, the initial value problem
(5.4) and the boundary value problem (5.5) can be treated in a totally
symmetric case by switching the variables $t$ and $x$. The presence of the
nonlocal term breaks this symmetry, and the nonlocal initial value problem
would be more delicate to deal with than the boundary value problem addressed
above. In particular, one would need to find ${\underline{u}}$ in terms of $f$
and $u^{\rm in}$ by solving the nonlocal equation (5.17).
## Appendix A Non dimensionalization of the equations
We show here how to derive the dimensionless equations of motion used
throughout this paper. To begin with, the Boussinesq-Abbott system describing
the propagation of weakly nonlinear waves in a fluid of mean depth $h_{0}$ and
with a pressure $P_{\rm atm}+\underline{P}$ exerted at the surface ($P_{\rm
atm}$ is a constant reference value for the atmospheric pressure) is given by
(A.1) $\begin{cases}\partial_{t}\zeta+\partial_{x}q=0,\\\
(1-\frac{h_{0}^{2}}{3}\partial_{x}^{2})\partial_{t}q+\partial_{x}\big{(}\frac{1}{h}q^{2}\big{)}+gh\partial_{x}\zeta=-\frac{h}{\rho}\partial_{x}\underline{P}\qquad(h=h_{0}+\zeta).\end{cases}$
###### Remark A.1.
Introducing the hydrodynamic pressure $\Pi$ as
(A.2) $\Pi=\underline{P}+\rho g\zeta,$
and alternative formulation of (A.1) is
(A.3) $\begin{cases}\partial_{t}\zeta+\partial_{x}q=0,\\\
(1-\frac{h_{0}^{2}}{3}\partial_{x}^{2})\partial_{t}q+\partial_{x}\big{(}\frac{1}{h}q^{2}\big{)}=-\frac{h}{\rho}\partial_{x}\Pi;\end{cases}$
we shall sometimes use this alternative formulation under the floating object.
Let us now consider the equations for the solid. We recall that we consider
here a floating object with vertical lateral walls located at $x=\pm\ell$
($\ell>0$) and allowed to move only vertically (heave motion). There is
therefore only one degree of freedom for the motion of the solid which can be
fully deduced from the signed distance $\delta(t)$ between the center of mass
$G=\big{(}x_{G},z_{G}(t)\big{)}$ and its equilibrium position $G_{\rm
eq}=(x_{G},z_{G,{\rm eq}})$, namely, $\delta=z_{G}(t)-z_{G,{\rm eq}}$.
Let us also assume that the water depth below the object is given at
equilibrium by a nonnegative single valued function $x\mapsto h_{\rm eq}(x)$;
the part of the bottom of the object in contact with the water (the wetted
surface) is therefore given at all time $t$ by the graph of the function
${\zeta}_{\rm w}$ defined as
(A.4) $\zeta_{\rm w}(t,x)=\delta(t)+h_{\rm eq}(x)-h_{0}.$
Newton’s equation for a body of mass $m$ that only moves vertically and
subject to gravity and hydrodynamic forces is given by
(A.5) $m\ddot{\delta}+mg=\displaystyle\int_{-\ell}^{\ell}\underline{P}_{\rm
i}(t,x){\rm d}x,$
where $\underline{P}_{\rm i}(t,x)$ is the pressure exerted by the fluid on the
object at the point $(x,\zeta_{\rm w}(t,x))$. Note that at equilibrium, the
pressure is hydrostatic, $\underline{P}_{\rm i}=-\rho g(h_{\rm eq}-h_{0})$, so
that
$m=\rho\int_{-\ell}^{\ell}(h_{0}-h_{\rm eq}(x)){\rm
d}x\qquad\mbox{(Archimedes' principle)},$
and we can rewrite Newton’s equation under the form
$m\ddot{\delta}=\displaystyle\int_{-\ell}^{\ell}\big{(}\underline{P}_{\rm
i}(t,x)+\rho g(h_{\rm eq}-h_{0})\big{)}{\rm d}x.$
By definition of the hydrodynamic pressure, its value $\Pi_{\rm i}$ in the
interior domain $(-\ell,\ell)$ is given by
$\Pi_{\rm i}=\underline{P}_{\rm i}+\rho g\zeta_{\rm w},$
from which we infer, using (A.4),
(A.6) $\tau_{\rm buoy}^{2}\ddot{\delta}+\delta=\displaystyle\frac{1}{2\rho
g\ell}\int_{-\ell}^{\ell}\Pi_{\rm i}(t,x){\rm d}x,$
where $2\pi\tau_{\rm buoy}$ is the buoyancy period defined through
$\tau_{\rm buoy}^{2}=\frac{m}{2\ell\rho g}.$
We now proceed to derive dimensionless versions of (A.1), (A.4), (A.5). We
recall that $h_{0}$ denotes the water depth at rest, and also denote by $a$
and $L$ the typical amplitude of the waves and a typical horizontal scale
respectively. For the Boussinesq-Abbott equations (A.1), we use the following
scalings
$\widetilde{x}=\frac{x}{L},\quad\widetilde{z}=\frac{z}{h_{0}},\quad\widetilde{t}=\frac{t}{L/\sqrt{gh_{0}}},\quad\widetilde{\zeta}=\frac{\zeta}{a},\quad\widetilde{q}=\frac{q}{a\sqrt{gh_{0}}},\quad\widetilde{\underline{P}}=\frac{\underline{P}}{\rho
gh_{0}}$
and consequently $\widetilde{h}=1+\varepsilon\widetilde{\zeta}$. We also
introduce the nonlinearity and shallowness parameters $\varepsilon$ and $\mu$
as
$\varepsilon=\frac{a}{h_{0}},\qquad\mu=\frac{h_{0}^{2}}{L^{2}}.$
For the sake of clarity the tildes used to denote dimensionless quantities are
omitted throughout this paper. The system (A.1) thus becomes
(A.7) $\begin{cases}\partial_{t}\zeta+\partial_{x}q=0,\\\
(1-\frac{1}{3}\mu\partial_{x}^{2})\partial_{t}q+\varepsilon\partial_{x}\big{(}\frac{1}{h}q^{2}\big{)}+h\partial_{x}\zeta=-\frac{1}{\varepsilon}h\partial_{x}\underline{P}\qquad(h=1+\varepsilon\zeta).\end{cases}$
###### Remark A.2.
The dimensionless form of the hydrodynamic pressure is naturally
$\widetilde{\Pi}=\frac{\Pi}{\rho
gh_{0}}=\widetilde{\underline{P}}+{\varepsilon}\widetilde{\zeta},$
so that the dimensionless version of the alternative formulation (A.3) is
(omitting the tildes)
(A.8) $\begin{cases}\partial_{t}\zeta+\partial_{x}q=0,\\\
(1-\frac{\mu}{3}\partial_{x}^{2})\partial_{t}q+\varepsilon\partial_{x}\big{(}\frac{1}{h}q^{2}\big{)}=-\frac{h}{\varepsilon}\partial_{x}\Pi;\end{cases}$
In order to derive the dimensionless versions of (A.4), (A.5) and (A.5), we
also need the following scalings
$\widetilde{\zeta}_{\rm w}=\frac{\zeta_{\rm
w}}{a},\qquad\widetilde{\delta}=\frac{\delta}{a},\qquad\widetilde{h}_{\rm
eq}=\frac{h_{\rm eq}}{h_{0}},\qquad\widetilde{m}=\frac{m}{2\ell\rho
h_{0}},\qquad\widetilde{\tau}_{\rm buoy}=\frac{\tau_{\rm
buoy}}{L/\sqrt{gh_{0}}},\qquad\widetilde{\ell}=\frac{\ell}{L}$
so that, omitting again the tildes for the sake of readability, we can rewrite
(A.4) and (A.5) as
(A.9) $\zeta_{\rm w}(t,x)=\delta(t)+\frac{1}{\varepsilon}\big{(}h_{\rm
eq}(x)-1\big{)}.$
and
(A.10) $\tau_{\rm
buoy}^{2}\ddot{\delta}+\frac{1}{\varepsilon}m=\displaystyle\frac{1}{\varepsilon}\frac{1}{2\ell}\int_{-\ell}^{\ell}\underline{P}_{\rm
i}(t,x){\rm d}x;$
note that in these dimensionless coordinates, the coordinates of the vertical
sides of the object are $x=\pm\ell$ and that Archimedes’ principle reads in
dimensionless form as
$m=\frac{1}{2\ell}\int_{-\ell}^{\ell}(1-h_{\rm eq}).$
Finally, the dimensionless version of A.6 is
$\tau_{\rm
buoy}^{2}\ddot{\delta}+\delta=\displaystyle\frac{1}{\varepsilon}\frac{1}{2\ell}\int_{-\ell}^{\ell}\Pi_{\rm
i}(t,x){\rm d}x.$
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|
# A Bilevel Optimization Framework For fuel-constrained UAV-UGV Cooperative
Routing: Planning and Experimental Validation
Md Safwan Mondal1, Subramanian Ramasamy1, Pranav Bhounsule1 *This work was
supported by ARO grant W911NF-14-S-003.1Md Safwan Mondal, Subramanian Ramasamy
and Pranav A. Bhounsule are with the Department of Mechanical and Industrial
Engineering, University of Illinois Chicago, IL, 60607 USA<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract
Fast moving unmanned aerial vehicles (UAVs) are well suited for aerial
surveillance, but are limited by their battery capacity. To increase their
endurance UAVs can be refueled on slow moving unmanned ground vehicles (UGVs).
The cooperative routing of UAV-UGV to survey vast regions within their speed
and fuel constraints is a computationally challenging problem, but can be
simplified with heuristics. Here we present multiple heuristics to enable
feasible and sufficiently optimal solutions to the problem. Using the UAV fuel
limits and the minimum set cover algorithm, the UGV refueling stops are
determined. These refueling stops enable the allocation of mission points to
the UAV and UGV. A standard traveling salesman formulation and a vehicle
routing formulation with time windows, dropped visits, and capacity
constraints is used to solve for the UGV and UAV route, respectively.
Experimental validation of the approach on a small-scale testbed shows the
efficacy of the approach.
## 1 INTRODUCTION
The integration of Unmanned Aerial Vehicles (UAVs) and Unmanned Ground
Vehicles (UGVs) has been increasingly utilized in various applications, such
as search and rescue, surveillance, and reconnaissance missions, and
transportation [1, 2, 3, 4] etc. One of the most significant challenges in
such cooperative routing is the limited fuel capacity of UAVs, which restricts
their operational time and range. However, effective cooperation between UAVs
and UGVs can enhance the mission efficiency and extend the coverage range of
UAVs, thereby enabling them to achieve longer range and persistent operations.
The complexity of such cooperative routing for UAV-UGV systems lies in its
combinatorial nature, making it computationally challenging to solve the
formulation with exact methods. Therefore, suitable heuristics should be used
to achieve high-quality solutions quickly.
In this paper, we propose a bi-level optimization framework for solving the
fuel-constrained UAV-UGV cooperative routing problem that optimizes the
operational time and fuel consumption of both vehicles. To validate our
proposed algorithm, we conducted hardware testing that provides practical
insights into the real-world application and feasibility of our proposed
approach. By maximizing the efficiency of the cooperative system, our
algorithm has the potential to overcome the limitations posed by fuel capacity
and speed constraints, enabling successful implementation of UAV-UGV
cooperative routing in a variety of applications.
### 1-A Related works
(a) given mission scenario with MSC
(b) first subproblem
(c) second subproblem
Figure 1: Minimum set cover algorithm and task allocation technique a) Given
mission scenario with minimum set cover algorithm. Blue circle indicates
radial coverage of UAV. Here, three refuel stops (including starting depot)
can cover entire mission b) First subproblem where UGV moves from starting
depot to first refuel stop and UAV missions points within radial coverage are
assigned. c) Second subproblem where UGV moves from first refuel stop to
second refuel stop and UAV missions points within radial coverage are
assigned.
Fuel-constrained vehicle routing problem of UAVs has been an area of
considerable research. Several studies have investigated the routing of
multiple fuel-constrained UAVs with recharging on fixed depots. Levy et al.
[5] investigated this routing of multiple fuel-constrained UAVs with fixed
recharging depots using variable neighborhood descent (VND) and variable
neighborhood search (VNS) heuristics to find feasible solutions for large
instances. Similarly, Sundar et al. [6] developed a mixed-integer linear
programming model (MILP) that can be solved using an off-the-shelf MILP
solver. Instead of fixed depot, Maini et al. [7] addressed cooperative routing
of a single UAV-UGV system, with the UGV serving as a mobile charging point
for the UAV on a road. Unlike previous studies, the authors developed a greedy
heuristic method to find the rendezvous locations of recharging along the UGV
route. Manyam et al. [8] investigated the cooperative routing problem of a
team comprising a UAV and UGV subject to communication constraints. They
formulated the problem as a mixed-integer linear program (MILP) and developed
a branch-and-cut algorithm to solve the problem optimally.
Researchers extended the UAV-UGV cooperative vehicle routing problem by
solving it in a tiered two-echelon approach. To solve the two-echelon
cooperative routing problem, Luo et al. [9] proposed a binary integer
programming model with two heuristics. Liu et al. [10] developed a two-stage
route based framework to optimize both the truck’s main route and the drone’s
adjoint flying routes for a truck and drone based parcel delivery system. They
created a hybrid heuristic that combined nearest neighbor and cost-cutting
strategies to quickly build a viable solution. In our previous research [11,
12], we explored a hierarchical bi-level optimization framework for
cooperative routing of multiple fuel-constrained UAVs and a single UGV. The
framework employs K-means clustering at the outer level to generate UGV visit
points, which can be connected using the traveling salesman problem (TSP) to
obtain the UGV route. At the inner level, a vehicle routing problem with
capacity constraints, time windows, and dropped visits for the UAV is
formulated and solved using the UGV path. In an extension to our work [13], we
demonstrated that optimizing the heuristics parameters with Genetic Algorithm
(GA) and Bayesian Optimization (BO) methods can lead to a significant
improvement in the quality of the solutions. But the past framework was
scenario specific making it hard to generalize for any unknown scenario, which
was driving force for this study to come up with a robust optimization
framework that can be generalized and implemented experimentally on hardware.
On the experimental front, few significant works have been done to demonstrate
localization and mapping of UAVs in indoor environments. Nigam et al. [14, 15,
16] investigated for high-level scalable control techniques for unmanned
aerial vehicles (UAVs) for performing persistent surveillance in an uncertain
stochastic environment in a hardware testbed. Two UAVs were used by Frew et
al. [17] to demonstrate road following, obstacle avoidance, and convoy
protection in a fligt testing, while Jodeh et al. [18] provided an overview of
cooperative control algorithms of heterogeneous UAVs by the Air Force Research
Laboratories (AFRL). Leahy et al. [19, 20] experimentally validated their
proposed method for automating persistent surveillance missions involving
multiple vehicles. Automata-based techniques were used to generate collision-
free motion plans and vector fields were created for use with a differential
flatness-based controller, allowing vehicle flight and deployment to be fully
automated according to the motion plans. They used charging platforms for the
UAVs for truly persistent missions. Boeing’s Vehicle Swarm Technology
Laboratory (VSTL) [16, 21, 22, 23] and MIT’s RAVEN laboratory [24] testbed
have conducted significant UAV flight testing demonstrations in indoor lab
scale setups.
The novelty of our work lies in the development of a robust bi-level
optimization framework that uses minimum set cover algorithm and a task
allocation technique to solve UAV-UGV cooperative routing in two echelon
fashion. We have validated our proposed methodology with a hardware flight
testing in lab scale experimental setup. To this end, we present following
novel contributions: 1) The overall framework uses bilevel optimization with
task allocation technique for mission allocation and constrained programming
based routing solvers. 2) The task allocation technique based on minimum set
cover algorithm divides the entire problem into decoupled subproblems which
radically simplifies the overall problem. 3) A constraint programming-based
formulation for vehicle routing problem with time windows, dropped visits, and
fuel constraints enables quick solutions of each subproblem. 4) Hardware
validation of our work demonstrates the practical feasibility and real-world
applicability of our proposed algorithms and methods.
### 1-B Problem Description
The aim of the problem is to perform an cooperative mission involves visiting
a set of designated mission points ( see figure 1(a)) $x_{n_{i}}\in
X\equiv\left(x_{i},t_{i}\right)$, using either the UGV road-based visit or the
UAV aerial flyover, where $x_{i}\in R^{2}$ is a position on the ground plane;
$t_{i}$ is the timestamp of the last visit to the node. The cost of travel
between any two mission points is defined as the time required to travel from
one point to another, $c_{ij}=t_{j}-t_{i}$. The UGV ${g_{i}\in
G\equiv(\tau_{i},f_{i},x_{i})}$, and UAV ${a_{i}\in
A\equiv(\tau_{i},f_{i},x_{i})}$, are heterogeneous in nature, with the UAV
having a higher velocity but a lower fuel capacity compared to the UGV. During
the mission, the vehicles are represented by current task or state $\tau_{i}$,
fuel level $f_{i}$ and position $x_{i}$. Unlike UAV, the UGV is restricted to
traveling along the road network, and the fuel consumption rate of both
vehicles is a function of the velocity of the vehicle. The UAV can be
recharged by the UGV at any refueling stop or at the starting depot, with the
recharging time dependent on the fuel level of the UAV. The UGV is assumed to
have an infinite fuel capacity due to its larger fuel capacity compared to the
UAV. With the described assumptions, the objective is to find the quickest
route for the UAV and UGV to visit all mission points together, with the
starting depot being both the starting and ending point, while ensuring that
the UAV never runs out of fuel.
To find the time-optimized route, the following goals must be achieved:
1. 1.
Identification of suitable stop locations where UAV will synchronize with UGV
to get recharged to cover all the mission points, i.e, $x_{r}\equiv
x_{i}=x_{j}$, here, $x_{i}\in a_{i},x_{j}\in g_{i}$.
2. 2.
Determination of optimal times during the mission, when UAV, UGV will meet at
those refuel stops i.e, $t_{r}\equiv t_{i}=t_{j}$, here, $t_{i}\in A,t_{j}\in
G$.
3. 3.
Determination of the optimal routes $\tau_{i}\in A,\ \ \tau_{j}\in G$ for both
the UGV and UAV based on the refueling locations $x_{r}$ and times $t_{r}$.
## 2 OPTIMIZATION FRAMEWORK
For the UAV-UGV cooperative routing, we proposed a bilevel optimization
framework, which is shown in Fig. 2. The framework is designed as a two-level
hierarchical structure, where at the higher level, we determine the UGV route
using the “UGV First, UAV Second” heuristic method, which involves
prioritizing the UGV route and then constructing the UAV route based on it. To
ensure the feasibility of the cooperative route, it is critical to locate
suitable refueling sites along the UGV route. We employed a minimum set cover
algorithm to identify the best locations for refueling. The inner-level UAV
route was built based on the UGV route by dividing the entire scenario into
subproblems what could be solved by modeling them as an energy constrained
vehicle routing problem (E-VRP).
Figure 2: optimization framework
### 2-A Outer level: UGV route
Maini et al. [7] demonstrated that in order to establish a viable cooperative
route, it is necessary to ensure that at least one refueling stop is located
within the UAV fuel coverage radius for each data point. Thus to determine the
minimum number of refueling stops required to cover the entire mission
scenario, we can adopt the minimum set cover algorithm (MSC). This is a well-
established problem that can be solved using a variety of methods, including
greedy heuristics [7]. However, in this study, we proposed a constraint
programming formulation for minimum set cover algorithm. For a same scenario,
we employed both greedy method and constraint programming approach
individually to solve the minimum set cover problem where constraint
programming method outperformed greedy heuristics.
#### 2-A1 Greedy heuristics method
Using a greedy heuristic approach can help to reduce the complexity of the
minimum set cover problem. This method starts with the mission scenario points
that require coverage $x_{n}$ and the UAV’s fuel capacity $F$ as the inputs,
as an output we aim of determining the minimum number of mission points that
can serve as refueling stops $x_{r}$ to cover the entire mission points
$x_{n}$. In other words, given a set of mission points
$X=\\{x_{1},x_{2},...,x_{n}\\}$ the goal is to identify a subset
$X_{r}=\\{x_{1},x_{2},...,x_{r}\\}$ that has the least number of elements to
act as refueling stops to cover the entire scenario. The greedy algorithm
selects the initial point as the first refueling stop and then sequentially
adds the mission points that cover the greatest number of other mission points
to the refueling stop set until all points are covered.
Although the greedy heuristic can produce an optimal result for a minimum set
cover problem quickly, there is possibility of multiple optimal solutions for
a given scenario. Since we are implementing a bilevel optimization framework,
it is essential to consider all the other optimal solutions of the outer level
algorithm. As it is not possible to acquire all optimal solutions using the
greedy heuristic, we employed the constraint programming method. This approach
can rapidly generate multiple optimal results, if any are present. Also, the
greedy method may result into locally optimal solution which can be overcome
through an alternate constraint programming formulation of the minimum set
cover problem.
#### 2-A2 Constraint programming method
The minimum set cover problem can be transformed into a linear integer
programming model using the constraint programming method (CP method). The
objective function in Eq. 2.1 determines the minimum number of refueling
stops, and the constraint function Eq. 2.2 ensures that each mission point
$x_{i}$ has at least one refueling stop $x_{r}$ assigned to it. The decision
variable $y_{i}$ determines whether a mission point will be selected as a
refueling stop or not. We utilized Google’s OR-Tools™ CP-SAT solver [25] to
solve this linear integer formulation, and it can record multiple optimal
solutions if they exist. After identifying the refueling stop locations, a UGV
route can be created by connecting these stops on the road network. We can
solve a simple travelling salesman problem (TSP), considering the refueling
stops, to determine an optimal UGV route. Once the optimal UGV route is
established, we can proceed to the inner loop UAV routing. Objective,
$\operatorname{min}\sum_{i=1}^{r}y_{i}$ (2.1)
s.t.,
$\sum_{j=1}^{r}y_{j}\geq 1,\forall x_{i},\ i=1,\ldots,n$ (2.2)
$y_{i}\in\\{0,1\\},\forall i=1,\ldots,n$ (2.3)
### 2-B Inner level: UAV route
At the inner level of our framework, we employed a task allocation technique
to divide the entire mission scenario into independent subproblems which were
solved individually as an energy constrained vehicle routing problems (E-VRP).
#### 2-B1 Task allocation technique
Given the scenario and the obtained UGV route from outer loop MSC algorithm,
we can divide the entire problem into $n-1$ number of subproblems ($n$ =
number of refuel stops with starting depot) with an assumption that UGV
travels only between two refuel stops in each subproblem. Before the
subproblem division, each mission point is assigned to its nearest refuel stop
(including starting depot) that covers it. In the subproblems, the first
refuel stop is the origin node and the second refuel stop is the destination
node of UGV route. The sub-problems are decoupled from each other by
allocating separate mission points in them. The UAV mission points assigned to
the destination refuel stop under each subproblem are allocated to that
subproblem. Only, for the first subproblem the mission points assigned to both
origin and destination node should be allocated to it.
Figure 1 demonstrate the process of subproblem division and mission
allocation. Figure 1(a) shows refuel stop locations along the UGV route
obtained from outer level MSC algorithm for a given scenario, the first
subproblem (figure 1(b)) is created by taking the starting depot as the origin
node and the refuel stop 1 as the destination node. The UAV mission points
covered by origin node (starting depot) and destination node (refuel stop 1)
are assigned for subproblem 1. Similarly, the second subproblem (figure 1(c))
is created by taking the refuel stop 1 as origin node and refuel stop 2 as
destination node and the mission points covered by the destination node
(refuel stop 2) are assigned for this subproblem. Once we get an independent
set of subproblems through task allocation, we tried to solve each subproblem
by modeling it as energy constrained vehicle routing problem (E-VRP).
#### 2-B2 E-VRP model
The formulation of the E-VRP can be described with a graph theory. Consider an
undirected graph $G=(V,E)$ where $V$ is the set of vertices
$V=\\{0,1,2,...k\\}$ and $E$ is the set of edges between the vertices $i$ and
$j$ as $E=\\{(i,j)\,\|\ i,j\,\in\,V,i\neq j\\}$. The non-negative arc cost
between the vertices $i$ and $j$ is expressed as $c_{ij}$ and $x_{ij}$ is a
binary decision variable whose value will be 1 if a vehicle travels from $i$
to $j$, and 0 otherwise. The objective function of the E-VRP problem is
indicated by Eq. 2.4 which minimizes the total travel time, the other
formulation of the constraints like fuel constraint, time window constraint
and generic constraints of the E-VRP can be found here [13].
$\min\sum_{i}\sum_{j}c_{ij}x_{ij}\quad\forall i,j\in V\\\ $ (2.4)
Again, we used Google OR-Tools ™ as our heuristic implementation for solving
the E-VRP model with constrained programming (CP). OR-Tools ™ uses a search
tree, local search, and meta-heuristics to identify feasible and optimal
solutions. The heart of OR-Tools ™ is a CP-SAT solver that employs
DecisionBuilder to find an initial feasible solution using the Path Cheapest
Arc Strategy [26]. OR-Tools ™ then performs a local search to find the best
solution in the neighborhood of the current solution. This local search uses
move operators to rewire nodes and check for feasibility and cost, and the
moves are repeated until a termination criterion is met [12].
## 3 RESULTS
(a)
(b)
Figure 3: UAV & UGV trajectory obtained from bilevel optimization with Greedy
and CP method at the outer loop. Numerical and alphabetical order shows the
UAV and UGV motion respectively. a) CP method based trajectory b) greedy
method based trajectory.
Figure 3 provides an illustrative example of the input and output of the
problem at hand. The input, depicted in Figure 3, consists of mission points
denoted by black crosses. The UAV and UGV must both initiate and terminate at
the starting depot, while ensuring that all mission points are covered either
by the UAV or UGV. The UAV can recharge at the depot or at designated
refueling sites from the UGV. The UAV and UGV have fixed velocities of 10 m/s
and 4 m/s, respectively, and their fuel capacities are 287.7 kJ and 25.01 MJ,
respectively.
To carry out the optimizations, we employed Python 3 and OR Tools ™ library,
which ran on a 3.7 GHz Intel Core i9 processor with 48 GB RAM on a 64-bit
operating system. For the scenario, two types of cooperative routes (if
different) were generated by implementing the Greedy method and the CP method
at the outer loop of the suggested framework. The UGV-only route, where only a
UGV completes the whole mission was also determined for the scenario. Based on
the metrics of total mission completion time and total energy consumption, the
impact of collaboration between the UAV and UGV on mission execution was
assessed by comparing the cooperative route with the UGV only route which
served as the upper limit.
Table I shows the improvement that was achieved through cooperative routing of
UAV-UGV on the mission scenario. Cooperative routing is extremely energy
efficient. Both CP method and greedy method at the outer loop showed positive
improvement reducing total energy consumption in the mission by 36-39%.
However, for total mission time greedy method at the outer loop had a negative
impact. This is due to position of refuel stops ( see trajectory in figure
3(b) ) what made the UAV to take frequent detours (6 times) for recharging at
the refuel sites elongating the total mission time. However, appropriate
refuel stop locations obtained CP method ( see trajectory in figure 3(a) )
helped UAV to complete its route with less recharging detour (4 times), which
effectively reduced total mission time.
Further insights about the trajectory of the cooperative route can be drawn
from Table II. As discussed earlier, greedy results in longer UAV travel time
which ultimately costs higher mission time. Energy consumption of UGV is low
in greedy method as UAV is visiting majority of mission points compare to CP
method. In sum, both CP method and greedy heuristics are capable of providing
feasible cooperative route for constrained complex mission scenario; however
CP method outperforms greedy method at the cost of some computational
efficiency.
TABLE I: Impact of the optimal solution of the cooperative routing. Metrics | | Cooperative
---
routing
| UGV
---
only route
Improvement (%)
| | CP
---
method
| Greedy
---
method
| | CP
---
method
| Greedy
---
method
| Time
---
consumption
(min.)
200 | 272 | 233 | 14.16 | -16.74
| Energy
---
consumption
(MJ)
21.98 | 21.14 | 34.69 | 36.62 | 39.06
TABLE II: Comparison between trajectories of CP method and greedy heuristics Metrics | CP method | Greedy method
---|---|---
Total time (min) | 200 | 272
Computational time (min) | 9 | 4
UGV results | |
Travel time (minutes) | 200 | 272
Energy consumed (MJ) | 20.79 | 19.52
Mission visited | 22 | 18
UAV results | |
Travel time (minutes) | 100 | 136.203
Energy consumed (kJ) | 1186.464 | 1618.092
Recharging stops on UGV | 3 | 6
Recharging stops on Depot | 1 | 0
Missions visited | 22 | 26
## 4 EXPERIMENT DESIGN
The most stringent way of validating a framework is by hardware demonstration,
but it often has limitations in terms of scope and variety. Therefore,
simulation results are often utilized to assess method performance, while
experiments are used to verify some of those outcomes. The validation of a
surveillance planning framework through experiments is especially significant
as it is a highly challenging problem to solve. We devised a small laboratory
scale scenario to test the proposed framework methodology with just one UAV
and UGV. In order to fully achieve autonomy during the experiment, each robot
must be capable of independently locating, planning, and executing its desired
route to each location without relying on external input. Our localization,
planning, and control strategies are built on the hardware’s sensing,
processing, and communication capabilities. However, developing an appropriate
experimental system poses various challenges as we need to consider the
integration of software, hardware, and communication to accomplish the task.
We have separately described the individual blocks of the hardware
architecture (see figure 4) as follows:
Figure 4: Hardware architecture
(a)
(b)
Figure 5: Comparison between simulation and experimental results. In the
trajectory, the alphabetical order represents the direction of motion of the
UAV
(a)
(b)
Figure 6: Experiment instances a) UAV, UGV are moving towards their
designated locations b) UAV landed on UGV for recharging.
1. 1.
Hardware: For our UAV, we chose the DJI Tello quadcopter due to its compact
size and affordability. It is a small quadrotor measuring 9.8 x 9.2 x 4.1cm
and weighing 80 g, including the battery (Li-Po, 1100mAh, 3.8V) and 3 inch
propellers. The drone comes with a built-in flight controller that offers
basic features such as flight stabilization and attitude regulation,
translational velocity control, and simple trajectory execution. This flight
controller is closed hardware and operated using a dedicated command protocol.
However, a higher level of autonomy can be achieved by using an external
ground-based computer that uses received telemetry and video feed to control
the drone with the same communication protocol.
For the UGV, we built a small raspberry-based omnidirectional car with a
landing pad for charging the UAV. The UGV can be controlled by controlling its
four wheels through the raspberry pi.
2. 2.
Control & communication: A wireless 2.4 GHz 802.11n WiFi connection was used
to communicate with the drone. The approach makes use of the official Tello
SDK 2.0. The UDP port is used to send text messages to the drone programming
interface. To create the application, we used the SDK and the low-level Python
library DJItelloPy. Wireless 5 GHz Wifi communication was also established
with the Raspberry Pi for controlling the UGV.
3. 3.
Central manager: The final element of the system design is the centralized
system manager. Based on the input scenario, this manager solves a fuel
constrained vehicle routing problem using the proposed bi-level optimization
framework to generate separate routes for the UAV and UGV. Then the manager
communicates with the UAV and the UGV, assigns respective tasks to them to
begin the surveillance. The manager continuously monitors their progress by
collecting the positional data of the vehicles from motion capture system. The
UGV runs on an open loop control and stops on its assigned stop locations on
the route. While the central system enforce an feedback control on the UAV to
navigate it correctly and make a successful landing on the UGV during
recharge.
4. 4.
Experiment scenario: The experiments were conducted in the Robotics and Motion
Lab at the University of Illinois Chicago. The lab has a designated flight
area equipped with a motion capture system that serves as a reference for the
position of reflective markers placed on the quadrotor and the ground vehicle.
This enables real-time localization of the robots during the experiment. The
positional data of the vehicles can be obtained at a rate of 100 Hz, with a
latency of less than 9 ms. A mission scenario was created by selecting 12
different points over an area of $4m\times 4m$ for the UAV, a road network was
designed for UGV and a fuel constraint was introduced by limiting the UAV’s
flight time in a single recharge (endurance limit). For this experimental
setup the endurance limit was setup to be 50 seconds, the UAV and UGV speed
was 0.20 m/s and 0.15 m/s respectively This required the UAV to visit the UGV
at regular intervals for recharging in order to complete the mission. However
no real recharging took place, it was only hypothesized that UAV gets
recharged instantly when it lands on the UGV. The UGV road network was also
designed to be challenging, with the farthest points on the mission requiring
the UAV to operate near its maximum endurance limit, thus testing the
robustness of the proposed framework.
### 4-A Flight test results
Multiple trials of the experiment were carried out on the scenario. The
algorithm was fed with the locations of the mission points of the UAV and the
road network points as input. The outer-loop of the algorithm determined the
UGV traversal path with refueling spots in space and time, while the inner-
loop of the framework generated the UAV route. Both the routes were provided
to the individual agents from the central manager, and the agents started
performing their missions. The purpose of the experiment was to verify the
feasibility of the algorithm’s output and to determine if the multi-agent
experiment could be successfully carried out with our experimental
architecture. The motion capture system was used to track the positional data
of the UAV and UGV, which was processed to produce the experimental route. The
figure 5(a) shows a comparison between the simulation route and the
experimental route. During the experiment, due to dynamics, the UAV drifted
away in some places but successfully managed to visit the mission points and
get recharged from the UGV by landing on it, because of the feedback control.
The endurance limit constraint was also tested, and it was observed that the
maximum flight time in a single recharge was always below the maximum limit in
figure 5(b). Dynamics of the UAV played an important role in the experiment
which was compensated by considering buffer time period in the modeling of
take off and landing of the UAV in the simulation counterpart. Instances of
the flight test can be seen in figure 6.
## 5 CONCLUSIONS
We conclude that a bilevel optimization framework with suitably designed
heuristics is an effective method for solving cooperative routing problems.
Our heuristics involved solving for UGV route first using the minimum set
cover algorithm and traveling salesman followed by the UAV route based on
mission allocation and a vehicle routing formulation. We found constraint
programming outperform greedy heuristics to solve the minimum set cover
algorithm although the former takes more computational time than the latter.
Experimental validation of the framework on a small testbed shows a close
match between simulation and hardware and confirming the proposed approach.
## ACKNOWLEDGMENT
The authors would like to thank Jean-Paul F. Reddinger, James M. Dotterweich,
and Marshal A. Childers from DEVCOM Army Research Laboratory, Aberdeen Proving
Grounds, Aberdeen, MD 21005 USA and James D. Humann is with DEVCOM Army
Research Laboratory, Los Angeles, CA, 90094 USA for providing solutions that
helped improve the formulation and solutions.
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|
that in turn acts as vector subtraction (tail-to-tail). When coupled to a
system of coordinates, we saw that we must be able to describe vectors in
terms of their lengths along each basis vector — we call these lengths the
components of the vector. Later on, to be consistent between algebra and
geometry, the we found rules for scalar multiplication and the dot product.
Finally, using areas of parallelograms, we built the component form of the
cross product AND inserted the idea of handedness into our coordinate systems
by defining cross product relationships between our basis vectors. Table 1.1
lists the equations that are the most important (and general) throughout this
chapter. I included the equation references to bring you back to the
discussion where we derived them, just in case you need some context to
refresh your memory about what each equation means.
Sprinkled throughout the chapter are several proofs that I recommended that
you try to do. All of them are practically identical to what I have already
done in the chapter, although they may have more components or a couple more
steps. I sincerely suggest that you try to do a few of them just to get a feel
for how these proofs are done on your own — in physics and math, one of the
best ways to deeply understand derivations or proofs is by putting in the time
to do it out for yourself (although in my experience, a physics and math
educations leaves very little time for anything but perpetual confusion…).
I know that for many of you reading this chapter, my lack of numerical values
will be unsettling — I will only get more algebraic as the chapters progress.
I do this on purpose though. My first reason is that it is frankly easier once
you get used to it. I do remember that the phase transition between needing
numbers in math and exclusively using letters is not a smooth one. It will
take time to master — probably as much time as it did for you when you first
started using the symbol $\pi$ instead of $3.14159$ back in the day. The
advantage to only using letters or symbols is that you eventually need not
worry about how the intermediate numerical values affect the outcome of your
mathematics. By extension, the necessary variables will be left in your
physical models of the natural world giving a deeper insight into how the
universe works. Unfortunately, unless you already have a handle on the
physics, injecting numerical values at intermediate steps will obfuscate this
insight. But again, it takes time (a.k.a. practice) to get used to doing
everything algebraically. Hopefully this chapter and the following will serve
as an external perturbation to make your phase transition that much easier.
TABLE 1.1: A summary of the important and general equations derived for vector
operations.
Equation Description | Equation Formula | Text Reference
---|---|---
3-Dimensional Cartesian Vector | $\vec{v}=v_{x}\hat{x}+v_{y}\hat{y}+v_{z}\hat{z}$ | Eq. 1.3
$n$-Dimensional Cartesian Vector | $\vec{v}=v_{1}\hat{x}_{1}+v_{2}\hat{x}_{2}+\dots=\sum\limits_{j=1}^{n}v_{j}\hat{x}_{j}$ | Eq. 1.4
Angle between $xy$ Components | $\tan\theta_{xy}=\dfrac{v_{y}}{v_{x}}$ | Eq. 1.5
Length/Magnitude of Vector | $|\vec{v}|=v=\sqrt{\sum\limits_{j=1}^{n}v_{j}^{2}}$ | Eq. 1.8
Sum of Two Vectors | $\vec{u}+\vec{v}=\sum\limits_{j=1}^{n}(u_{j}+v_{j})\hat{x}_{j}$ | Eq. 1.13
Difference of Two Vectors | $\vec{u}-\vec{v}=\sum\limits_{j=1}^{n}(u_{j}-v_{j})\hat{x}_{j}$ | Eq. 1.14
Scalar Multiplication | $a\vec{v}=\sum\limits_{j=1}^{n}(av_{j})\hat{x}_{j}$ | Eq. 1.15
Unit vector in Direction of $\vec{v}$ | $\hat{v}=\dfrac{\vec{v}}{|\vec{v}|}=\dfrac{\vec{v}}{v}$ | Eq. 1.16
Dot Product | $\vec{u}\cdot\vec{v}=\sum\limits_{j=1}^{n}u_{j}v_{j}=uv\cos{\theta}$ | Eq. 1.21
Projection of $\vec{v}$ onto $\vec{u}$ | $\textrm{proj}_{\hat{u}}(\vec{v})=\dfrac{\vec{u}\cdot\vec{v}}{|\vec{u}|}=\dfrac{\vec{u}\cdot\vec{v}}{u}$ | Eq. 1.23
Length/Magnitude and Dot Product | $|\vec{v}|=v=\sqrt{\vec{v}\cdot\vec{v}}$ | Eq. 1.25
Magnitude of Cross Product | $|\vec{v}\times\vec{u}|=uv\sin\theta$ | Eq. 1.27
Right-Handed Cartesian Basis | $\begin{aligned} \hat{x}\times\hat{y}&=\hat{z}\\\ \hat{z}\times\hat{x}&=\hat{y}\\\ \hat{y}\times\hat{z}&=\hat{x}\end{aligned}$ | Eq. 1.36
3-Dimensional Cross Product | $\begin{aligned} \vec{u}\times\vec{v}&=\;\;\;\;\left(v_{y}u_{z}-v_{z}u_{y}\right)\hat{x}\\\ &\;\;\;\;+\left(v_{z}u_{x}-v_{x}u_{z}\right)\hat{y}\\\ &\;\;\;\;+\left(v_{x}u_{y}-v_{y}u_{x}\right)\hat{z}\end{aligned}$ | Eq. 1.37
Definition of Linear Operator | $\mathcal{O}(a\vec{u}+b\vec{v})=a\mathcal{O}(\vec{u})+b\mathcal{O}(\vec{v})$ | Eq. 1.39
## Chapter 2 Complex Algebra
In this chapter, we will start a perhaps unfamiliar form of mathematics for
many of you, but it is crucial to not only our understanding of quantum
phenomena, but also our understanding of differential equations, Fourier
Analysis of signals, all kinds of waves, and various forms of data analysis.
By the end of this chapter, we want to be able to add complex numbers to our
set of “game pieces” that we have been developing, and then eventually get to
the point that we can create functions of complex numbers. This chapter should
leave you in a pretty good position to understand most, if not all, the
complex mathematics you will cover in your undergraduate physics curriculum.
This chapter should also set you up to begin to learn the calculus of complex-
valued functions later on in your mathematics career.
### 2.1 The Lie of Imaginary Numbers
Before we get going, there is a common misconception that I want to clear up.
There is no such distinction between real and imaginary numbers in the
colloquial sense; that is, there is no set of objects that are somehow
tangible that we call the real numbers, $\mathds{R}$, versus the somehow
intangible objects called the imaginary numbers, $\mathds{I}$. These sets of
objects are certainly distinct mathematically, but that is due to a rotation
rather than some metaphysical and mystical separation that seems to exist by
calling two things real and imaginary.
The reason why I want to address this is because the term “imaginary” has a
totally different connotation in normal life than it does in mathematics. To
be clear, at some point in the development of our algebra system, some
mathematicians like Rene Decartes did truly believe that imaginary numbers
really were not a thing, but a rather convenient way out of an otherwise
harder problem [ComplexNumberHistory]. However, mathematicians today
effectively only use the word as a label whose name bears no deep meaning. We
credit people like Gauss, Cauchy, Euler, and Riemann, among others, for
changing the way we think about these objects mathematically. Gauss showed
that imaginary numbers are truly just an extension of the more conventional
real numbers, and had even tried to get them rebranded as lateral numbers
instead. His quote on this subject is below [Gauss_Quote].
> That this subject [imaginary numbers] has hitherto been surrounded by
> mysterious obscurity, is to be attributed largely to an ill adapted
> notation. If, for example, $+1$, $-1$, and the square root of $-1$ had been
> called direct, inverse and lateral units, instead of positive, negative and
> imaginary (or even impossible), such an obscurity would have been out of the
> question.
The problem with taking the word “imaginary” too literally in physics makes
the interpretation of certain natural phenomena appear as fake or something
akin to pseudoscience. For example, when describing an electron’s spin, the
imaginary (lateral!) unit $i=\sqrt{-1}$ appears when talking about the
projection of the spin vector along the $y$-axis. If we were to naively see
the presence of $i$, we might be tempted to conclude that there is something
mystical about this part of our physical world. Or even worse, we might
conclude that we could never measure the $y$ component of the spin because it
is imaginary! But this interpretation is not true. Gauss’ idea of lateral
numbers can be used to more appropriately explain the appearance of $i$ in
electron spins. In this case, as you will learn later in your physics career,
we can only ever know precisely the projection of an electron’s spin along one
axis in space; we denote the forward direction with a $+$ sign and the
backward direction with a $-$ sign. However, we can measure the statistical
effects of the spin’s vector components in the other two dimensions in 3D
space. What this means is we have a total of 3 sets of distinct pairs of basis
vectors in this spin space, which is supposed to have physical meaning in all
of 3D space. Without going into too much linear algebra, we essentially need 4
components to represent the final two perpendicular axes in 3D space — but
this is impossible with only the real numbers! Real components can only ever
give you the magnitude and direction that a vector points along a particular
line, as we discussed in the chapter on Vectors. Hence, our very real
measurements of the natural world force us to extend the real numbers to
include their lateral counterparts in order to accurately describe electron
spin.
Without further ado, I will quit my (legitimate) grumbling, and proceed with
our introduction to the world of complex algebra.
### 2.2 Some Important Definitions
Before we move on with more algebra, we need to lay down some ground rules for
these things that I’m calling complex numbers. The first time they are
typically introduced (although not the first time they were ever contrived
[ComplexNumberHistory]) is with the standard defining equation
$\displaystyle x^{2}+1=0,$ (2.1)
where one solves for $x$ to find $x=\pm\sqrt{-1}$. Since we showed in Section
1.2.1 that for any real number111Remember that we use the symbol $\mathds{R}$
to represent the set of all reals and the symbol $\in$ to mean “is an element
of.” $a\in\mathds{R}$, $a^{2}>0$, then we find that there can be no real
number $x$ that satisfies $x^{2}+1=0$. Thus we define the imaginary unit as
$\displaystyle i=\sqrt{-1}.$ (2.2)
Likewise, we could easily define the real unit with the following equation
$x^{2}-1=0$, and so we obtain $x=\pm 1$. These ideas fit in with our
understanding that $1$ represents “unit” length along a number line. We will
define any imaginary number $\alpha$ as $\alpha=ai$, where $a\in\mathds{R}$.
Such an object would solve the quadratic $x^{2}+a^{2}=0$ for $x$. Likewise, a
real number $a\in\mathds{R}$ would solve the quadratic $x^{2}-a^{2}=0$ for
$x$.
When we move to more complicated quadratics, we have an equation that looks
something like $ax^{2}+bx+c=0$, whose solutions are given by the quadratic
formula
$\displaystyle x=-\frac{b}{2a}\pm\frac{1}{2a}\sqrt{b^{2}-4ac}.$ (2.3)
If we pay attention to the quantity called the discriminant, $D=b^{2}-4ac$, we
should take note there are exactly three cases for $D$ given by the ordering
property of the reals. They are given explicitly as
$\displaystyle D>0$ $\displaystyle\Rightarrow b^{2}>4ac,$ $\displaystyle D=0$
$\displaystyle\Rightarrow b^{2}=4ac,$ (2.4) $\displaystyle D<0$
$\displaystyle\Rightarrow b^{2}<4ac.$
The third case, $D<0$, implies that we will again have a negative number
inside of a square-root, and so this implies that there will be an imaginary
unit involved in some way. Written explicitly, when $D<0$, then it must be
true that $-D>0$. Then
$\displaystyle
x=-\frac{b}{2a}\pm\frac{1}{2a}\sqrt{D}=-\frac{b}{2a}\pm\frac{1}{2a}\sqrt{-(-D)}=-\frac{b}{2a}\pm\frac{i}{2a}\sqrt{-D}=-\frac{b}{2a}\pm\frac{i}{2a}\sqrt{4ac-b^{2}}$
(2.5)
What we have now is something that hopefully is a little jarring to you,
especially if you have never seen complex algebra before. We have an
expression that is somehow telling us to add a real number $-b/2a$ with the
imaginary number $i\sqrt{4ac-b^{2}}/2a$. But can we? There is mathematically a
pretty large distinction between real numbers and imaginary numbers; the
square (and any other nonzero even power) of a real number is always positive.
Meanwhile, the square (and any other nonzero even power) of an imaginary
number is always negative. We proved the former, whereas we had to define the
latter. So what gives?
The way we deal with this conundrum is actually by defining something new. It
turns out that if you were jarred before, you were right, because there is no
way to add a nonzero real number with a nonzero imaginary number and get
either a purely real or imaginary number out. The sets of objects are just too
different. All of their differences can pretty much be reduced to the fact
that
$\displaystyle 1^{2}=1\textrm{ and }i^{2}=-1,$ (2.6)
and so we get around this issue by saying each of these numbers is totally
distinct from one another, but they can be combined to form a greater set of
numbers; just as the basis vectors $\hat{x}$ and $\hat{y}$ are totally
distinct, but can be combined to form a plane. Actually, if we take the real
unit and the imaginary unit as basis vectors, we can create the complex plane,
denoted by $\mathds{C}$. Furthermore, we define a complex number $z$ as the
vector sum of a real part $z=\mathrm{Re}(z)$ and an imaginary part
$z=\mathrm{Im}(z)$ like the following
$\displaystyle z=\mathrm{Re}(z)+i\,\mathrm{Im}(z).$ (2.7)
It is important to see then that this definition constrains both the real part
and the imaginary part to be real numbers! Please read that sentence again —
it confuses a lot of people. Even though the imaginary part of a complex
number is called the imaginary part, it itself is real. It represents the
projection (see Eq. 1.23) of the complex number in the direction of $i$,
whereas the real part is the projection of the complex number in the direction
of $1$.
Using this vector-like interpretation of complex numbers, then it must be true
the so-called “real axis” and “imaginary axis” must together span a complex
plane, just like the $x$-axis and $y$-axis span the $xy$-plane. By convention,
we denote $x=\mathrm{Re}(z)$ and $y=\mathrm{Im}(z)$, so we can write any
complex number $z\in\mathds{C}$ as
$\displaystyle z\in\mathds{C}\textrm{ if and only if }z=x+iy,\textrm{ where
}i=\sqrt{-1}\textrm{ and }x,y\in\mathds{R}.$ (2.8)
Since the complex numbers form a plane, then we can more easily see what Gauss
was talking about when he claimed that the imaginary unit should be instead
named the lateral unit. The existence of this plane just means that the real
numbers are accompanied by another orthogonal axis that has had the misfortune
of having us silly humans call them “imaginary”. The lateral numbers are
present with or without us claiming they are figments of our imaginations —
they are an extension of the real numbers into the generalized complex plane.
With that said, I will continue to refer to them as imaginary numbers just so
you get accustomed to the vernacular. But please do not think any of this is
some kind of fantasy. If you are willing to accept the existence of the real
number line, it is clear there must exist an accompanying imaginary (lateral!)
number line.
### 2.3 Conjugates and Magnitudes
A question that we can now ask since we have defined complex numbers is
whether there is a way to construct the real and imaginary parts of any
complex number computationally. In other words, if we know know any real
number $x$ and any real number $y$, we can compute a complex number $z=x+iy$.
That’s old news. The new question is whether we can find another complex
number $z^{\ast}$ if we know already know a complex number $z$ such that we
could calculate $x=\mathrm{Re}(z)$ and $y=\mathrm{Im}(z)$. If we were to
assume that this were possible, then we would have to start by clarifying that
if $z^{\ast}$ were complex, then $z^{\ast}=u+iv$ for some combination of real
numbers $u$ and $v$. Thus,
$\displaystyle z$ $\displaystyle=x+iy$ $\displaystyle z^{\ast}$
$\displaystyle=u+iv$
As of right now, we have two separate equations with two unknowns. By
“separate”, I mean that as of right now, we don’t have an equation to relate
any of the variables. Hence, we are free to choose that
$x=\mathrm{Re}(z)=(z+z^{\ast})/2$. Thus,
$\displaystyle
x=\mathrm{Re}(z)=\frac{1}{2}\left(z+z^{\ast}\right)=\frac{1}{2}\left[(x+u)+i(y+v)\right]=\frac{x+u}{2}+i\,\frac{y+v}{2}$
Since $x$ is purely real, then that means that it cannot be imaginary. Thus
the coefficient attached to $i$ must vanish; in other words, $y+v=0\Rightarrow
v=-y$. This means that
$\displaystyle
x=\frac{x+u}{2}+i\,\frac{y+v}{2}=\frac{x+u}{2}+i0=\frac{x+u}{2}$
By multiplying both sides by 2 and subtracting over the remaining x we then
find $x=u$. This means we have an expression for $z^{\ast}$, given $z$:
$\displaystyle\textrm{if }z=x+iy\textrm{ then }z^{\ast}=x+i(-y)=x-iy.$ (2.9)
The complex number $z^{\ast}$ is a very helpful quantity — so helpful, in
fact, that it is given the name of the complex conjugate to $\mathbf{z}$.
Additionally, this complex conjugate is unique, as based on our rules for the
real numbers in Section 1.2.1, there exists only one $v=-y\in\mathds{R}$ if
$y\in\mathds{R}$222This uniqueness property justifies our phrasing of the
complex conjugate instead of a complex conjugate.. Notice that
$z^{\ast}=x-iy=x+(-i)y$. This means that whenever we want a complex conjugate
of $z=x+iy$, then all we have to do is replace every $i$ with $-i$. If it is
not immediately clear why this is true, it’s because we have defined complex
numbers in such a way that any number may be written as a real part plus $i$
times an imaginary part, and then we defined the complex conjugate in terms of
those arbitrary real and imaginary parts. This rule is particularly helpful
for when you have a rather nasty function of complex variables, but you need
the conjugate to compute something useful.
Now what would be useful to compute with a complex conjugate? For starters, we
know that we can find the real part of $z$ (or $z^{\ast}$) with the following
formula.
$\displaystyle\mathrm{Re}(z)=\frac{z+z^{\ast}}{2}.$ (2.10)
We can actually calculate the imaginary part of $z$ as well. I leave it to you
to verify that
$\displaystyle\mathrm{Im}(z)=\frac{z-z^{\ast}}{2i}.$ (2.11)
To show this, set $y=\mathrm{Im}(z)$ and use Eq. 2.9 to solve for $y$. If the
division by $i$ is weird for you, then instead use
$1/i=(1/i)(i/i)=i/(i^{2})=i/(-1)=-i$. The third part of Example 2.3 shows
another helpful tool that the complex conjugate provides.
Consider the complex numbers $z=x+iy$ and $w=u+iv$, where
$x,y,u,v\in\mathds{R}$. We seek the real and imaginary parts of $z\pm w$,
$zw$, and $z/w$. 1. We start with $z\pm w$. $\displaystyle z\pm w$
$\displaystyle=(x+iy)\pm(u+iv)$ $\displaystyle=(x\pm u)+i(y\pm v).$ (2.12)
Hence, $\mathrm{Re}(z\pm w)=x\pm u$ and $\mathrm{Im}(z\pm w)=y\pm v$. 2. Next,
we compute $zw$. $\displaystyle zw$ $\displaystyle=(x+iy)(u+iv)$
$\displaystyle=xu+ixv+iyu+i^{2}yv$ $\displaystyle=(xu-yv)+i(xv+yu).$ (2.13)
Thus, $\mathrm{Re}(zw)=xu-yv$ and $\mathrm{Im}(zw)=xv+vu$. 3. Lastly, we seek
the quantity $z/w$ (this one is a little tricky). $\displaystyle\frac{z}{w}$
$\displaystyle=\frac{x+iy}{u+iv}$
$\displaystyle=\frac{x+iy}{u+iv}\,\frac{w^{\ast}}{w^{\ast}}$
$\displaystyle=\left(\frac{x+iy}{u+iv}\right)\left(\frac{u-iv}{u-iv}\right),\textrm{
Using Eq. \ref{eq: Complex Conjugate} for }w$
$\displaystyle=\frac{(x+iy)(u-iv)}{(u+iv)(u-iv)}$
$\displaystyle=\frac{(xu+yv)+i(xv-yu)}{u^{2}+v^{2}}$
$\displaystyle=\frac{xu+yv}{u^{2}+v^{2}}+i\,\frac{xv-yu}{u^{2}+v^{2}}.$ (2.14)
Therefore we have
$\displaystyle\mathrm{Re}\left(\frac{z}{w}\right)=\frac{xu+yv}{u^{2}+v^{2}}\textrm{
and }\mathrm{Im}\left(\frac{z}{w}\right)=\frac{xv-yu}{u^{2}+v^{2}}$ It is
important to note that since we can establish clear real and imaginary parts
for each of the arithmetic operations above — addition, subtraction,
multiplication, and division — these quantities are also complex numbers, by
definition. Thus, when we add, subtract, multiply, or divide complex numbers,
we will always get complex numbers back!
For a next calculation, let’s find the product of $z$ with its conjugate
$z^{\ast}$ which is often denoted as $z^{\ast}z$; we do so by direct
substitution.
$\displaystyle z^{\ast}z$ $\displaystyle=(x-iy)(x+iy)$
$\displaystyle=x^{2}-iyx+ixy-i^{2}y^{2}$
$\displaystyle=x^{2}-(-1)^{2}y^{2}+i(xy-yx)$ $\displaystyle=x^{2}+y^{2}+i0$
$\displaystyle=x^{2}+y^{2}.$
Outright, this formula may not look very impressive, so let’s try
$\sqrt{z^{\ast}z}$:
$\displaystyle\sqrt{z^{\ast}z}=\sqrt{x^{2}+y^{2}}=\sqrt{[\textrm{Re}(z)]^{2}+[\textrm{Im}(z)]^{2}}.$
(2.15)
Hopefully this catches your eye as the Pythagorean Theorem, or even more
importantly, the equation for the length of a two-dimensional vector Eq. 1.6,
where the $x$-component of the vector is just $x$ and the $y$-component is
just $y$! Using Eq. 2.15, called the complex modulus of $z$, we can calculate
a real modulus and and imaginary modulus if we consider only a purely real
number $x$ or purely imaginary number $\Upsilon=iy$.
$\displaystyle\sqrt{x^{\ast}x}$ $\displaystyle=\sqrt{x^{2}+0}=|x|,\textrm{ a
real number has no imaginary part}$
$\displaystyle\sqrt{\Upsilon^{\ast}\Upsilon}$
$\displaystyle=\sqrt{0+y^{2}}=|y|,\textrm{ an imaginary number has no real
part}$
where the absolute value bars are necessary because we implicitly took the
positive square root, and so our moduli should be positive for arbitrary
$x,y\in\mathds{R}$. But these expressions are just the magnitudes of the real
numbers $x$ and $y$. In other words, the complex modulus is the generalization
of magnitude — a.k.a. absolute value — in the complex plane! In other words,
we may write
$\displaystyle|z|=\sqrt{z^{\ast}z}=\sqrt{[\textrm{Re}(z)]^{2}+[\textrm{Im}(z)]^{2}},$
(2.16)
when talking about the magnitude of any complex number.
Before moving on, there are a couple of more things I want to talk about.
Firstly, what happens if we need to find the complex conjugate of a sum (or
difference)? Suppose we have two complex numbers $z$ and $w$, as we do in the
first part of Example 2.3, and we want to find $(z\pm w)^{\ast}$. We could do
so directly as shown below using Eq. 2.12,
$\displaystyle(z\pm w)^{\ast}=(x\pm u)+(-i)(y\pm
v)=(x-iy)\pm(u-iv)=z^{\ast}\pm w^{\ast}.$ (2.17)
Notice that we replaced the $+i$ in Eq. 2.12 with the $-i$ to find the
conjugate initially. Hence, when we have a sum (or difference) of two complex
numbers, then the conjugate of the sum (or difference) is just the sum (or
difference) of the conjugates. How about a product of two complex numbers?
Here we use Eq. 2.13,
$\displaystyle(zw)^{\ast}$ $\displaystyle=(xu-yv)+(-i)(xv+yu),$
$\displaystyle=(xu+i^{2}yv)-i(xv+yu),$ $\displaystyle=xu+(iy)(iv)-ixv-
iyu,\textrm{ combine like-terms in $x$ and $-iy$}$
$\displaystyle=x(u-iv)-iy(u-iv),$ $\displaystyle=(x-iy)(u-iv),$
$\displaystyle=z^{\ast}\,w^{\ast}.$ (2.18)
(Note derivation is a little tricky because I needed to remember that
$-1=i^{2}$ in the second step and then group one $i$ with $y$ and the other
with $v$ in the third step.) This relation shows that the conjugate of the
product is simply the product of the conjugates! At this point, however, we
have enough information to be certain that complex conjugation is NOT a linear
operation for complex numbers (see Eq. 1.39 for the definition of a linear
operator). To see this more clearly, we first extend our definition of a
linear operator to the complex plane
$\displaystyle\mathcal{O}(a\vec{z}+b\vec{w})=a\mathcal{O}(\vec{z})+b\mathcal{O}(\vec{w}),\;\;\textrm{for
all }a,b\in\mathds{C}.$ (2.19)
For right now, consider the complex vectors $\vec{z}$ and $\vec{w}$ as being
normal vectors whose components are complex-valued. The specific details on
these complex vectors are not totally necessary (we’ll save that for linear
algebra…) for right now. What is important here is that if we have to compute
$(a\vec{z}+b\vec{w})^{\ast}$, then we would have
$\displaystyle(a\vec{z}+b\vec{w})^{\ast}$
$\displaystyle=(a\vec{z})^{\ast}\pm(b\vec{w})^{\ast},\textrm{ by Eq. \ref{eq:
conjugate z pm w}}$ (2.20) $\displaystyle=a^{\ast}\,\vec{z}^{\ast}\pm
b^{\ast}\,\vec{w}^{\ast}.\textrm{ by Eq. \ref{eq: conjugate zw}}$ (2.21)
The only way this is equal to $a(\vec{z}^{\ast})+b(\vec{w}^{\ast})$ is if
$a=a^{\ast}$ and $b=b^{\ast}$, implying that $a,b\in\mathds{R}$. But since our
complex vectors have complex-valued components, and we have already shown that
multiplication between complex numbers produced complex numbers, it follows
that in general the condition that both $a$ and $b$ are real is NOT general.
Thus, in general, complex conjugation is not linear333This fact is
specifically exploited in quantum mechanics and quantum field theory when
dealing with time-reversal symmetry in real physical systems.!
We have shown that the conjugate of the sum is the sum of the conjugates (Eq.
2.17) and the conjugate of the product is the product of the conjugates (Eq.
2.18). Show now that the conjugate of the quotient is the quotient of the
conjugates. In other words, show that $(z/w)^{\ast}=z^{\ast}/w^{\ast}$. (Hint:
either start with $z^{\ast}/w^{\ast}$, substitute in $-i$ for the $+i$, and
then show that it is the conjugate of Eq. 2.14, OR start with the conjugate of
Eq. 2.14 and manipulate it algebraically into $z^{\ast}/w^{\ast}$. I think
both methods take similar amounts of algebra…)
With purely real numbers, it is true that $|a|=|-a|$ for all $a\in\mathds{R}$.
Thus, magnitudes are not unique; however, we can use a sign difference to
order our real numbers from least to greatest. This problem focuses on whether
we can do a similar thing in the complex plane — if we could, then it would be
true that only $z$ and $-z$ share the same magnitude, just like the reals. We
used $\sqrt{z^{\ast}z}$ as the definition of the magnitude of a complex
number. Show that the complex numbers cannot be ordered like the reals by
finding at least one other complex number that shares the same magnitude of
$z$ (and $-z$). For example, show that $|z|=|z^{\ast}|$ for all
$z=x+iy\in\mathds{C}$. (Hint: to avoid confusing yourself with all the $z$s
and asterisks, define $w=z^{\ast}$ and then find $|w|$ using Eq. 2.16.
Finally, compare your result with Eq. 2.15.)
### 2.4 Cartesian versus Polar Representations
The fact that the real and imaginary axes form a complex plane implies that we
should be able to draw any complex number in a geometric plane. Further, since
we have a Pythagorean Theorem-type relationship for the magnitude of the
complex number (and $1$ is totally distinct from $i$), we are able to infer
that the real and imaginary axes are orthogonal to one another — just like the
$x$-axis and the $y$-axis. Figure 2.1 shows a possible complex number $z=x+iy$
in the complex plane. Additionally, if we know the components $x$ and $y$,
then we could easily draw the conjugate of $z$, namely $z^{\ast}=x-iy$. It is
important to note that the points $z$ and $z^{\ast}$ are the complex numbers;
meanwhile, based on our knowledge of vectors, we could easily draw an arrow
from the origin to each complex number, where the length of each vector would
the the magnitude (or length) of the complex number. Now for some
trigonometry.
FIG. 2.1: If we consider $z=x+iy$, then we can draw it and its conjugate,
$z^{\ast}=x-iy$, in the complex plane.
From the definition of sines, cosines, and tangents (see Section 1.3.3), we
can use the geometrical right angle between the real and imaginary axes to
define the angle of the the complex number, or the so-called argument of a
complex number, denoted by $\mathrm{arg}(z)=\theta$. As is shown in Fig. 2.1,
$\theta$ is the angular elevation of the complex number above the $+x$-axis.
Since we know that $\mathrm{Im}(z^{\ast})=-\mathrm{Im}(z)$ while the real
parts are identical, we know that $\mathrm{arg}(z^{\ast})=-\mathrm{arg}(z)$.
Using these angles we can then determine
$\displaystyle\cos\theta$
$\displaystyle=\frac{\mathrm{adjacent}}{\mathrm{hypotenuse}}=\frac{x}{|z|}=\frac{x}{\sqrt{x^{2}+y^{2}}},$
(2.22) $\displaystyle\sin\theta$
$\displaystyle=\frac{\mathrm{opposite}}{\mathrm{hypotenuse}}=\frac{y}{|z|}=\frac{y}{\sqrt{x^{2}+y^{2}}}.$
(2.23)
By solving the second equalities for $x$ and $y$ and then substituting these
quantities in $z=x+iy$, we find
$\displaystyle z=|z|\cos\theta+i|z|\sin\theta=|z|(\cos\theta+i\sin\theta).$
(2.24)
Note that this picture is consistent with just substituting in $-i$ for $+i$
to obtain a conjugate, because
$\displaystyle
z^{\ast}=|z^{\ast}|[\cos\theta+(-i)\sin\theta]=|z|[\cos(-\theta)+i\sin(-\theta)].$
(2.25)
where we have used the result of Problem 2.3 to set $|z^{\ast}|=|z|$, and then
we used the even and odd symmetry of the sinusoids:
$\cos(-\theta)=\cos(\theta)$ and $\sin(-\theta)=-\sin\theta$.
Equation 2.24 shows that any complex number $z=x+iy$ can be represented as the
product of a radial part, and an angular part, as shown below
$\displaystyle z=\underbrace{|z|}_{\textrm{Radial
Part}}(\underbrace{\cos\theta+i\sin\theta}_{\textrm{Angular Part}}),$ (2.26)
as a complement to the real-and-imaginary-part representations. Again, for the
sake of completeness, the radial part is the modulus of the complex number
while the angular part is the argument of the complex number. Take note that
while the radial part is a real number like $\mathrm{Re}(z)$ and
$\mathrm{Im}(z)$, the angular part is still complex!
Thus, we have written any complex number in either a real-and-imaginary
representation and in a radial-and-angular representation. These terms,
although straight-to-the-point, are fairly clunky, so instead we name them the
Cartesian and Polar representations of a complex number, respectively. The
“pole” in this case is the distance (magnitude/length) the complex number is
from the origin. A much more useful (and elegant) way to write a complex
number $z=x+iy\in\mathds{C}$ and its conjugate in the polar representation is
actually with an imaginary exponential $\textrm{e}^{i\theta}$, given as
$\displaystyle z$ $\displaystyle=r\textrm{e}^{i\theta},$ (2.27) $\displaystyle
z^{\ast}$ $\displaystyle=r\textrm{e}^{-i\theta},$ (2.28)
where
$\displaystyle r$ $\displaystyle=|z|=\sqrt{x^{2}+y^{2}},$ (2.29)
$\displaystyle\theta$
$\displaystyle=\mathrm{arg}(z)=\arctan\left(\frac{y}{x}\right).$ (2.30)
Of course, to write such a thing, it would have to be true that
$\displaystyle\textrm{e}^{i\theta}=\cos\theta+i\sin\theta,$ (2.31)
where $\textrm{e}\approx 2.71828182846$ is the base of the natural logarithm.
Surprisingly, this result is true for all values of $\theta$! Remarkably, this
result is not too difficult to prove even though it took a supergenius like
Leonhard Euler444Pronounced oiler, unlike how they said it in The Imitation
Game, much to my chagrin… to first do it555Hence, it is usually called the
Euler identity. — it can be done with an introductory understanding of Taylor
Series in a Calculus II course — although it is a bit beyond the scope of this
chapter666Don’t worry! We will come back to it later on and YOU will prove it
in Problem 3.4.2 (I help you along though)..
It is initially a little strange to go from the Cartesian representation to
the polar representation, but ultimately using the polar representation is
more convenient than the Cartesian representation (otherwise physicists
wouldn’t bother with it!). So this problem is designed to have you practice
the conversion for a few important numbers in physics. (a) Show that
$i=\textrm{e}^{i\pi/2}$ by arguing $r(i)=1$ and $\mathrm{arg}(i)=\pi/2$. (b)
Show that, in the Cartesian representation,
$\sqrt{2}\,\textrm{e}^{-i\pi/4}=1-i$ (c) Consider the complex number
$z=\textrm{e}^{i\pi}$. What is $z+1$ in both Polar and Cartesian
representation?
Using the polar representation, it is possible for us to construct two of the
most widely used expressions in all of physics — we are going to rewrite sine
and cosine in terms of exponentials. To start consider the complex number of
unit magnitude given by Eq. 2.31. Then it must be true
$\displaystyle\mathrm{Re}\left(\textrm{e}^{i\theta}\right)=\cos\theta\in\mathds{R},$
(2.32)
$\displaystyle\mathrm{Im}\left(\textrm{e}^{i\theta}\right)=\sin\theta\in\mathds{R}.$
(2.33)
But by Eq. 2.10 and Eq. 2.11, we can write the real and imaginary parts of any
complex number as a superposition of it and its complex conjugate. Then,
$\displaystyle\cos\theta$
$\displaystyle=\frac{\textrm{e}^{i\theta}+\textrm{e}^{-i\theta}}{2},$ (2.34)
$\displaystyle\sin\theta$
$\displaystyle=\frac{\textrm{e}^{i\theta}-\textrm{e}^{-i\theta}}{2i}.$ (2.35)
Part of the reason why these relationships are so useful, is it allows us to
treat trigonometric functions — objects that are defined geometrically — as
exponential functions instead, which then allow us to employ a slew of useful
(and quick) multiplication, differentiation, and integration rules to
otherwise algebraically cumbersome functions. In physics, this polar
representation of trigonometric functions helps us describe certain geometries
and spaces in terms of essentially successive multiplications. As you can see
in the chapter on Fourier Analysis, being able to convert sines and cosines
into combinations of exponential functions makes otherwise impossible algebra
much simpler. As we continue through this book, I will highlight more
locations where these can be immediately implemented to make your life easier
because the sooner you begin to feel comfortable with Eqs. 2.34 and 2.35, the
faster you will begin to see through the mathematics of difficult subjects
like signals, optics, and quantum mechanics to understand the underlying
phenomena in a much more precise way. There is something potentially
unsettling by writing geometric formulas as exponential formulas; it seems to
imply there is a clear rotational aspect of multiplying two numbers, rather
than the simpler stretching-and-shrinking interpretation that was valid with
the real numbers. We study this more in the next section.
### 2.5 Multiplication = Dilation + Rotation
Let’s consider two complex numbers in polar representation, given by
$z=r\textrm{e}^{i\theta}$ and $w=\rho\textrm{e}^{i\phi}$. We will calculate
their product and difference in the polar representation.
$\displaystyle zw$
$\displaystyle=\left(r\textrm{e}^{i\theta}\right)\left(\rho\textrm{e}^{i\phi}\right)=r\rho\textrm{e}^{i\theta}\textrm{e}^{i\phi}=(r\rho)\,\textrm{e}^{i(\theta+\phi)},$
(2.36) $\displaystyle\frac{z}{w}$
$\displaystyle=\frac{r\textrm{e}^{i\theta}}{\rho\textrm{e}^{i\phi}}=\frac{r}{\rho}\,\textrm{e}^{i\theta}\textrm{e}^{-i\phi}=\left(\frac{r}{\rho}\right)\,\textrm{e}^{i(\theta-\phi)},$
(2.37)
Thus, by Eq. 2.36, multiplying a complex number by another is equivalent to
dilating the magnitude of the first by the second and rotating the first
complex number counter-clockwise by the second’s argument. Figure 2.2 shows
the geometry behind multiplication. Likewise, since division is the
multiplicative inverse, we should not be surprised by Eq. 2.37 which says that
dividing a complex number by another is equivalent to constricting777My word
choice for “anti-dilating.” the magnitude of the first by the second and
rotating the first complex number clockwise by the second’s argument.
FIG. 2.2: When we multiply the complex number $z=r\textrm{e}^{i\theta}$ by
$w=\rho\textrm{e}^{i\phi}$, the magnitude of $z$ is dilated by a factor of
$\rho$, while the angle of $z$ is rotated by $\mathrm{arg}(w)=\phi$. Thus
$|zw|=r\rho$ and $\mathrm{arg}(zw)=\theta+\phi$.
What is often the case, at least in physics, when we deal with complex
numbers, we are usually only taking about ones with unit magnitude — that is,
the complex modulus is equal to one. Any complex number of the form in Eq.
2.31 fits this description. The reason why these numbers are so important is
that they do not dilate the modulus of any complex number they are
multiplying. Instead, they exclusively rotate the complex number they are
multiplying. Figure 2.3 serves to help you get some intuition to how the
multiplication = rotation bit works with more concrete numerical examples like
$\pm 1$ and $\pm i$. Very often, physicists will talk of the phase of some
quantity (such as in electromagnetic theory, optics, or quantum mechanics),
and refer to the entire complex number $\textrm{e}^{i\theta}$ as this phase.
To be perfectly precise, the actual angle (argument) is the phase, NOT the
entire complex number. However, as we have seen by Eq. 2.36 where $r=1$,
multiplying by the entire number $z=\textrm{e}^{i\theta}$ only changes the
total phase of the product, since it rotates the original number by
$\mathrm{arg}(z)=\theta$. Thus, in terms of things we can experimentally
detect (and therefore know are truly there), we can only find the resultant
change of phase due to the multiplication of a complex number by another of
unit magnitude.
FIG. 2.3: In this figure, we multiply each of the following points along the
complex unit circle by the imaginary (lateral!) unit $i=\textrm{e}^{\pi i/2}$:
$(1,0)$, $(0,i)$, $(-1,0)$, and $(0,-i)$. Starting with $(1,0)$, we have
$1\cdot i=i=\textrm{e}^{\pi i/2}$ over the solid red arc. Hence, multiplying
$1$ by $i$, rotates $1$ by $\pi/2$ radians ($90^{\mathrm{o}}$). Next, we
multiply $i$ by $i$ in the dashed blue arc. But $i^{2}=-1=\textrm{e}^{\pi
i/2+\pi i/2}=\textrm{e}^{\pi i}$. Thus, again, multiplying by $i$ rotates our
starting complex number $i$ by $\pi/2$ radians. I leave you to confirm the
other two rotations for yourself.
Our ability to detect these so-called phase differences is actually rather
remarkable. For example, in both classical and quantum mechanics, magnetic
fields exert a torque on any charged object with angular momentum. This torque
causes the charged object to precess in a circle; in other words, the charged
object will behave similarly to how a toy top begins to wobble in circles
around its central axis before it falls over due to gravity. In this case, the
field that generates the wobble is the magnetic field instead of the
gravitational field. Anyway, in quantum mechanics, a typical experiment to
measure the phase difference goes something like this: generate a beam of
identical particles, split the beam into two parts, do something to one beam
and leave the other alone, then recombine the beams and see if anything
happens. So one such experiment deals with measuring the intensity of
particles with inherent angular momentum (spin) after splitting up the beam
and having one part travel through a magnetic field over some distance. The
amount those particles wobble due to the magnetic field then acts as the phase
difference between the particles in the beam. As you will eventually learn,
the intensity of the recombined beam is a function of the phase difference,
meaning we can experimentally vary either the strength of the magnetic field
or increase the distance the beam travels through it, and then change the
intensity of the resulting beam of particles! For an example of such an
experiment, check out [spin_interferometry].
Before proceeding, let’s take note of one special case of multiplication:
exponentiation. In other words, we can raise any real number $a\in\mathds{R}$
to the $n\in\mathds{R}$ power by successively multiplying $a$ by itself $n$
times. For example, if $a=2$ and $n=4$, then $a^{n}=2^{4}=2\cdot 2\cdot 2\cdot
2=16$. Likewise, if we have a negative exponent, then we divide successively,
while if $n$ has a noninteger fractional part, we take the appropriate root
$(2^{1/3}=\sqrt[3]{2})$. Using the polar representation of a complex number
$z=r\textrm{e}^{i\theta}$, we can generalize exponentiation rather
straightforwardly as
$\displaystyle
z^{n}=r^{n}\,(\textrm{e}^{i\theta})^{n}=r^{n}\,\textrm{e}^{in\theta}.$ (2.38)
By setting a new angle $\psi=n\theta$, and a new radial part as
$s=r^{n}\in\mathds{R}$, we have
$\displaystyle z^{n}=s\,\textrm{e}^{i\psi}.$ (2.39)
Making use of Euler’s Identity (Eq. 2.31), we see that
$\displaystyle z^{n}=s(\cos\psi+i\sin\psi)=r^{n}(\cos n\theta+i\sin n\theta).$
(2.40)
Since the quantity $z^{n}$ can be written in terms of a radial and angular
part, it must be complex-valued, as those parts yield real and imaginary
parts. Thus $z^{n}\in\mathds{C}$ which means our rules for complex numbers
lead to closure under exponentiation! In other words, we cannot ever possibly
end up with a non-complex-valued quantity purely by exponentiation888This is a
very good thing, mind you, for it essentially gives us motivation for starting
to see if more complicated algebraic and transcendental functions always
return complex numbers..
There is, meanwhile, a much more subtle identity that we may not have
initially noticed while showing $z^{n}\in\mathds{C}$. We made use of a
statement known as De Moivre’s Theorem that says for any integer $n$
$\displaystyle\left(\cos\theta+i\sin\theta\right)^{n}=\cos(n\theta)+i\sin(n\theta).$
(2.41)
Looking closely at Eq. 2.41, we see that this statement is actually incredibly
complicated. It manages to relate something as easy to compute as raising a
number to an integer power $n$ to the much more difficult trigonometric
functions, cosine and sine. But more importantly De Moivre’s Theorem puts the
exponent $n$ inside of the argument of the trig function! Usually, there is no
clear way to translate the argument of a trig function to anything outside of
the function — for example $\cos(2x)\neq 2\cos x\neq(\cos x)^{2}$ for every
value of $x$. Trigonometric functions just don’t behave nicely like this.
However, by making use of the complex plane, De Moivre’s Theorem gives us a
way of applying algebraically simply operations to compute otherwise very
difficult, if not outright impossible, quantities.
I do want to emphasize that De Moivre’s Theorem only applies for integer
exponents, whereas Eq. 2.40 applies for all possible values of $n$ (even
complex ones!). What gives? Where is there a difference? We look into this
question next.
#### 2.5.1 Roots of Unity
Let’s consider the case in De Moivre’s Theorem where $n=1/2$. In this
particular case, then if De Moivre’s Theorem were to hold, then it would be
true (but it is not)
$\displaystyle\left(\cos\theta+i\sin\theta\right)^{1/2}=\cos\left(\frac{\theta}{2}\right)+i\sin\left(\frac{\theta}{2}\right).$
Let’s choose the easier case of $\theta=0$ and $\theta=2\pi$ since on the
left-hand side, we will have
$\displaystyle(\cos 0+i\sin 0)^{1/2}=(\cos 2\pi+i\sin 2\pi)^{1/2}=1^{1/2}.$
This follows since $\sin 0=\sin 2\pi=0$ and $\cos 0=\cos 2\pi=1$. There is no
problem yet. The issue arises when we look at the right-hand side to find
$\displaystyle\cos\left(\frac{0}{2}\right)+i\sin\left(\frac{0}{2}\right)=\cos
0$ $\displaystyle=1$
$\displaystyle\cos\left(\frac{2\pi}{2}\right)+i\sin\left(\frac{2\pi}{2}\right)=\cos\pi$
$\displaystyle=-1.$
(Remember that $\sin\pi=0$.) So, if De Moivre’s Theorem were to be trusted, we
would find that $1^{1/2}=1=-1$. In other words, we would find that our answer
implies that $1=-1$ and this contradicts our rules for real numbers.
This result is not too surprising from an Algebra II point-of-view since we
already know that both $(1)^{2}$ and $(-1)^{2}=1$, which we normally write
instead as $1^{1/2}=\pm 1$. But this means that the square-root function is
mutli-valued, and in general the $n^{\mathrm{th}}$-root is also multi-valued.
The problem with De Moivre’s Theorem is that it does not explicitly account
for this phenomenon. We could easily generalize it to account for the multi-
valuedness in exponentiation by rational exponents.
We start with the so-called Roots of Unity as they are both fundamental and
something that we have already developed the motivation for. Let’s start off
easy, and find the cubic roots of unity. We will proceed in the same way as we
did before with the square-roots of unity. Here we have
$\displaystyle\left(\cos\theta+i\sin\theta\right)^{1/3}=\cos\left(\frac{\theta}{3}\right)+i\sin\left(\frac{\theta}{3}\right).$
Like before, we choose $\theta=0$ and $\theta=2\pi$ because $\cos 0=\cos
2\pi=1$. There is actually another case we can consider, too:
$\theta=4\pi\Rightarrow\cos 4\pi=1$. Calculating each of these cases, we have
$\displaystyle\cos\left(\frac{0}{3}\right)+i\sin\left(\frac{0}{3}\right)=1$
$\displaystyle\cos\left(\frac{2\pi}{3}\right)+i\sin\left(\frac{2\pi}{3}\right)=-\frac{1}{2}+i\frac{\sqrt{3}}{2}$
$\displaystyle\cos\left(\frac{4\pi}{3}\right)+i\sin\left(\frac{4\pi}{3}\right)=-\frac{1}{2}-i\frac{\sqrt{3}}{2}$
And so the cubic-roots of unity we found are
$\displaystyle
1^{1/3}\in\left\\{1,-\frac{1}{2}+i\frac{\sqrt{3}}{2},-\frac{1}{2}-i\frac{\sqrt{3}}{2}\right\\}$
Hopefully, you aren’t satisfied with this derivation of the cubic-roots of
unity so far because I just arbitrarily decided to include the $4\pi$ part
when I didn’t include it for the square-roots. The reason why I was able to
include $4\pi$ for the cubic-roots and not for the square-roots is because of
the oscillatory behavior of sines and cosines — they repeat themselves every
$2\pi$ radians. So essentially, when looking for $\theta$ values that would
show the roots of unity are indeed multi-valued, I needed to watch out for two
criteria. First, I needed to make sure $\textrm{e}^{i\theta}=1$ since we are
taking about roots of unity, AND I needed to make sure that the right-hand
side did not repeat itself. Let’s analyze each criterion independently, and by
doing so, we will generalize the square and cubic cases to the
$n^{\mathrm{th}}$-root.
If we want $\textrm{e}^{i\theta}=1$, then we need $\cos\theta+i\sin\theta=1$.
But the value $1$ is totally real, hence the sine coefficient attached to the
$i$ must vanish for our $\theta$ values. This condition holds for
$\theta\in\\{0,\pm\pi,\pm 2\pi,\pm 3\pi,\dots\\}$. Next, we are dealing with
roots of unity, not roots of negative-unity. Thus, we can only allow for even
multiplies of $\pi$ so that $\cos(2\pi m)=1$, where $m\in\\{0,\pm 1,\pm 2,\pm
3,\dots\\}$. So this explains why I kept choosing $\theta=0,2\pi$ and then
$4\pi$. But why not the negatives, too? The answer to that comes again from
the even symmetry of the cosine function:
$\displaystyle\cos(-\theta)=\cos(\theta).$
In other words, the cosine functions ignore the overall negative sign inside
of their argument and produce the same result either way. Hence, we could
include the negative values, $-2\pi,-4\pi,$ et cetera, but we would ultimately
always recover the same set of possible $\theta$ values.
Now we move onto the second criterion which says that $\theta$ cannot make the
right-hand side repeat itself. We use the first criterion that we had
established where all of the roots of unity will have $\theta$ values that are
an even multiple of $2\pi$. Let’s call this multiple $m$ and write $1^{1/n}$
as
$\displaystyle 1^{1/n}=\cos\left(\frac{2\pi m}{n}\right)+i\sin\left(\frac{2\pi
m}{n}\right),\;\;m\in\\{0,1,2,\dots\\}.$
If we start with $m=0$, then we will always find $1^{1/n}=1$, as we expect,
since $1^{n}=1$. Then, as we continue to check through all of the possible
multiples we have, we find the first repeating value at $m=n$:
$\displaystyle 1^{1/n}\underbrace{=}_{m=n}\cos\left(\frac{2\pi
n}{n}\right)+i\sin\left(\frac{2\pi n}{n}\right)=\cos 2\pi=1$
But this returns the exact same value as $m=0$. Furthermore, any multiple of
$n$ will ALWAYS be the same as the $m=0$ case. Meanwhile, all of the possible
multiples up to $n$ are totally allowed because, in general,
$\displaystyle\frac{2\pi m}{n}\neq 2\pi,$
the only case where equality holds is when $m=n$. Thus, our
$n^{\mathrm{th}}$-roots of unity are given as
$\displaystyle 1^{1/n}=\cos\left(\frac{2\pi m}{n}\right)+i\sin\left(\frac{2\pi
m}{n}\right),\;\;m\in\\{0,1,\dots,n-1\\},$ (2.42)
in the Cartesian representation. In the polar representation, we pack the
sinusoids into the exponential function to find
$\displaystyle 1^{1/n}=\mathrm{exp}\left(\frac{2\pi
mi}{n}\right),\;\;m\in\\{0,1,\dots,n-1\\}.$ (2.43)
(Note that $\mathrm{exp}(x)=\textrm{e}^{x}$ for all $x$. I used the exp
representation because the equation would have looked gross if I used e
instead.)
FIG. 2.4: A few plots of the symmetry in the roots of unity. The orange points
are the complex numbers centered around their common origin. The vectors
attached to each point is to help with our geometrical interpretation of the
complex plane. The regular polygons within each unit circle represent the
internal symmetry within the set of the $n^{\mathrm{th}}$-roots of unity.
The $n^{\mathrm{th}}$-roots of unity are quite nice geometrically because they
chop up the unit circle ($2\pi$ radians) into regular $n$-gons by dividing the
full $2\pi$ radians into equally-spaced $2\pi/n$ intervals. A few of these are
shown in Fig. 2.4. What is more beautiful from a physical point-of-view is
that since the roots of unity form regular $n$-gons inscribed within a unit
circle, their sum must vanish. This can be directly applied to study
cylindrically (and polygonally) symmetrical systems in physics; for example,
point like-charges (masses) arranged in some polygonal shape whose net
electric (gravitational) fields must vanish at their geometrical center. In
more computational applications in physics, this property allows us to compute
discrete Fourier Transforms of signals which is a mathematical operation that
allows us to understand the signal in terms of its frequency-dependence
instead of its time-dependence999The Fourier Analysis chapter in this book
does not deal with the discrete transform, but the tools developed in the
chapter in the continuum can help you understand the intuition behind the
discrete version..
To show how the roots sum to zero explicitly, recall the square-roots of
unity: $1$ and $-1$. Together they form a line (lame), but their sum is
$1+(-1)=0$. We can also do the cubic-roots of unity that form an equilateral
triangle (less lame):
$\displaystyle
1-\frac{1}{2}+i\frac{\sqrt{3}}{2}-\frac{1}{2}-i\frac{\sqrt{3}}{2}=1-1+i0=0.$
Now let’s generalize (disclaimer: this will be one of the more abstract things
so far).
Using Eq. 2.42, find the Cartesian representation of the $4^{\mathrm{th}}$ and
$5^{\mathrm{th}}$ roots of unity and show that the sum of their roots is zero.
Consider the sum of the $n^{\mathrm{th}}$-roots of unity in polar
representation, written as $S_{n}$, and given as
$\displaystyle S_{n}=1+\textrm{e}^{2\pi i/n}+\dots+\textrm{e}^{2\pi
i(n-1)/n}=\sum_{m=0}^{n-1}\textrm{e}^{2\pi im/n}$
If we look closely at the summation, we will see that we actually have a
finite geometric series, defined generally as
$\displaystyle S_{n}=1+a+a^{2}+\dots+a^{n-1}=\sum_{m=0}^{n-1}a^{m},$ (2.44)
for some value $a$. In this case, $a=\textrm{e}^{2\pi i/n}$ and
$\displaystyle S_{n}=\sum_{m=0}^{n-1}\left(\textrm{e}^{2\pi i/n}\right)^{m}$
To determine what this finite sum is, we can employ a few
mathemagical101010Bad pun? tricks that I highly recommend you work out with
me. Knowing how they work WILL come in handy later on in your physics career,
at least when it comes to handling finite sums. If you work these tricks out
with me, then you are prone to remembering them in the future. Here we go.
We will calculate the quantity $1-S_{n}$ using the definition of $S_{n}$ given
in Eq. 2.44, and at the end we will substitute in $a=\textrm{e}^{2\pi i/n}$.
$\displaystyle 1-S_{n}$ $\displaystyle=1-\sum_{m=0}^{n-1}a^{m}$
$\displaystyle=1-(1+a+a^{2}+\dots a^{n-1})$
$\displaystyle=-(a+a^{2}+\dots+a^{n-1})$ $\displaystyle=-a(1+a+\dots+a^{n-2})$
$\displaystyle=-aS_{n-1}.$
In the last equality, I used the definition of the finite geometric series
again, except since the sum only goes to $n-2$, then that means the proper
subscript on $S$ is $n-1$ since $(n-1)-1=n-2$. In this form, we cannot proceed
because we needed $S_{n}$, NOT $S_{n-1}$. Thus we need to figure out a way to
relate the two different finite sums.
Let’s consider now $S_{n-1}=1+a+\dots+a^{n-2}$. If we look again at Eq. 2.44,
we will see that if we were to add $a^{n-1}$ to $S_{n-1}$, we would have
$\displaystyle S_{n-1}+a^{n-1}=1+a+\dots+a^{n-2}+a^{n-1}=1+a+\dots
a^{n-1}=S_{n}.$
Hence, $S_{n}-a^{n-1}=S_{n-1}$. Now we are going to substitute this expression
in for $S_{n-1}$ inside the last equality for $1-S_{n}$ and isolate $S_{n}$.
$\displaystyle 1-S_{n}=-aS_{n-1}=-a\left(S_{n}-a^{n-1}\right)=-aS_{n}+a^{n}.$
By moving the $S_{n}$ on the left-hand side to the right-hand side we have
$\displaystyle 1=S_{n}-aS_{n}+a^{n}\Rightarrow 1-a^{n}=(1-a)S_{n},$
and finally we have
$\displaystyle S_{n}=\frac{1-a^{n}}{1-a},$ (2.45)
which is the exact way to calculate a finite geometric series for any value of
$a\neq 1$ (the formula blows up at $a=1$ because if $a$ were 1 then we would
have a $S_{n}-S_{n}=0$ step in our derivation, thus we had to implicitly
assume that $a\neq 1$). Finally, we substitute in $a=\textrm{e}^{2\pi i/n}\neq
1$:
$\displaystyle S_{n}(\textrm{e}^{2\pi i/n})=\frac{1-(\textrm{e}^{2\pi
i/n})^{n}}{1-\textrm{e}^{2\pi i/n}}=\frac{1-\textrm{e}^{2\pi
in/n}}{1-\textrm{e}^{2\pi i/n}}=\frac{1-\textrm{e}^{2\pi
i}}{1-\textrm{e}^{2\pi i/n}}=\frac{1-1}{1-\textrm{e}^{2\pi i/n}}=0.$ (2.46)
And so it is true that the sum of all of the $n^{\mathrm{th}}$-roots of unity
will always be identically zero! Furthermore, since a geometric series is
built from multiplying the same object $a=\textrm{e}{2\pi i/n}$ by itself a
bunch of time, and we know that multiplication by a complex number of unit
magnitude is a pure rotation, then the complex number $\textrm{e}^{2\pi i/n}$
represents a symmetry present in the set of the $n^{\mathrm{th}}$-roots of
unity. The symmetry here is that our set of complex numbers are invariant
under a rotation of $2\pi/n$ radians — i.e. the number of points looks
identical if we rotate all of them by the same $2\pi/n$ angle! For example, if
we have the square-roots $\\{1,-1\\}$ and we rotate each number through the
complex plane by $2\pi/2=\pi$ radians, we have
$\displaystyle\\{1,-1\\}\xrightarrow{\textrm{rotate by }\pi\textrm{
radians}}\\{-1,1\\},$
as shown in Since the rotation $\textrm{e}^{\pi i}$ maps the set of square-
roots onto themselves, this rotation is the symmetry I was talking about that
is directly embedded within the set of square-roots of unity.
FIG. 2.5: The symmetry in the square-roots of unity is a rotation by $\pi$
radians since the set of roots is identical under this rotation.
Since the ambiguity in De Moivre’s Theorem is taken care of, we can return to
Eq. 2.40 to account for all of the different possible roots of $z$.
Specifically, we can talk about any root $1/n$, we have
$\displaystyle z^{1/n}$
$\displaystyle=\left(r\textrm{e}^{i\theta}\right)^{1/n}$
$\displaystyle=r^{1/n}\cdot 1^{1/n}\cdot\textrm{e}^{i\theta/n}$
$\displaystyle=r^{1/n}\,\textrm{e}^{2\pi
im/n}\,\textrm{e}^{i\theta/n},\;\;m\in\\{0,1,\dots,n-1\\}$
$\displaystyle=r^{1/n}\,\mathrm{exp}\left(i\frac{\theta+2\pi
m}{n}\right),\;\;m\in\\{0,1,\dots,n-1\\}$
$\displaystyle=r^{1/n}\left[\cos\left(\frac{\theta+2\pi
m}{n}\right)+i\sin\left(\frac{\theta+2\pi
m}{n}\right)\right],\;\;m\in\\{0,1,\dots,n-1\\}$ (2.47)
Then, by extension, a complex number $z$ to any rational power $p/q$ (for
example 4/3) can be written as
$\displaystyle z^{p/q}$
$\displaystyle=\left[r^{1/q}\,\mathrm{exp}\left(i\frac{\theta+2\pi
m}{q}\right)\right]^{p},\;\;m\in\\{0,1,\dots,q-1\\}$
$\displaystyle=r^{p/q}\,\mathrm{exp}\left(ip\frac{\theta+2\pi
m}{q}\right)^{p},\;\;m\in\\{0,1,\dots,q-1\\}$
$\displaystyle=r^{p/q}\left\\{\cos\left[\frac{p}{q}(\theta+2\pi
m)\right]+i\sin\left[\frac{p}{q}(\theta+2\pi
m)\right]\right\\},\;\;m\in\\{0,1,\dots,q-1\\},$ (2.48)
The power with raising a complex number to a rational exponent is that we can
then use this as a basis for raising a complex number to an irrational
exponent since we can always successively approximate an irrational number
with a rational number. For example, $\sqrt{2}\approx 1.414=1414/1000$. But
how about raising a complex number to a complex exponent? We will study this
and more in the next section.
### 2.6 Functions of a Complex Variable
In this section, we will study a few algebraic and transcendental functions of
a complex variable. Unfortunately, we will not have time to study these
functions beyond just domain and range, but rest assured that understanding
the inputs and outputs of complex functions is totally sufficient for a B.S.
in physics. As one studies either more math or more physics, ideas from
Complex Analysis become relevant if not actually crucial, whether they be in
the form of circuit analysis, fluid mechanics, or quantum field theory. I have
some references for anyone interested at this point in their studies
[complex_analysis_wikipedia_2018, beck_marchesi_pixton_sabalka_2017,
arnold_complexanalysis] (feel free to come back at a later date for them
though!).
I have used the term “map” before when talking about functions, and perhaps
you’ve heard others use that word in the same context. In more colloquial
settings, people use maps to get from one place to another, and in this
section the connection between a function and an everyday map will become
clearer. We will see that functions, particularly complex-valued functions,
serve to get from one place in the complex plane to another. To illustrate
this idea, let’s consider the following example where we see how regions of
the complex plane are connected through the function, or mapping,
$w(z)=z^{2}$.
Consider the complex-valued function $w(z)=z^{2}$. We want to study $w$ over a
unit square in the first quadrant of the complex plane. To start, we consider
$z=x+iy$ in the Cartesian representation. Then, $\displaystyle
w(z)=z^{2}=(x+iy)^{2}=x^{2}-y^{2}+2ixy.$ (2.49) If we define $w=u+iv$, then we
have $u=x^{2}-y^{2}$ and $v=2xy$ for the real and imaginary parts of $w$,
respectively. Now we define the boundary of the unit square for our input, or
domain, of interest in the complex plane $\displaystyle
z=\begin{cases}x+i0,&x\in[0,1]\\\ 1+iy,&y\in[0,1]\\\ x+1i,&x\in[0,1]\\\
0+iy,&y\in[0,1]\end{cases}$ If we substitute these coordinates for $z$ into
$w$, we have $\displaystyle w(z)=\begin{cases}x^{2},&x\in[0,1],\;y=0\\\
1-y^{2}+2iy,&x=1,\;y\in[0,1]\\\ x^{2}-1+2ix,&x\in[0,1],\;y=1\\\
-y^{2},&x=0,\;y\in[0,1]\end{cases}$ The “mapping”, in this case, comes in when
we rewrite $w(z)$ into its own set of real and imaginary parts, given by
$w=u+iv$. By looking at the four same regions, we have $\displaystyle
w(\textrm{unit cube})\in\begin{Bmatrix}u=x^{2},\;&v=0\\\ u=1-y^{2},\;&v=2y\\\
u=x^{2}-1,\;&v=2x\\\
u=-y^{2},\;&v=0\end{Bmatrix}=\begin{Bmatrix}u,&u\in[0,1]\\\
u=1-\frac{1}{4}v^{2},&v\in\left[0,2\right]\\\
u=\frac{1}{4}v^{2}-1,&v\in\left[0,2\right]\\\ u,&u\in[-1,0]\end{Bmatrix}$ It
is definitely difficult to visualize how the equation above defines a map, at
least not in this representation. Figure 2.6 shows how the equation above can
serve as a set of instructions of how to turn one plot, or regions of the
complex plane, is connected to another through the $w(z)=z^{2}$ function.
FIG. 2.6: In the left plot, we define the unit square in the $z$-plane. It is
through the $w(z)=z^{2}$ mapping that we connect the unit square to the
square-squared region in the $w$-plane in the right plot. The colored curves
represent sets of the complex plane that are connected through the
$w(z)=z^{2}$ “map.”
From Example 2.6, we can conclude a couple of things. First, complex-valued
functions unite different regions of the complex plane. And second,
specifically for this quadratic function, we chose to only consider the unit
square. We could have chosen a square of size 2, or $\pi$, or a gazillion.
Then $w$ would also increase in size by Eq. 2.49. I will leave it to you to
pick particular points on the square of side length $s$ and plug it into Eq.
2.49 to see how large the region in the $w$-plane becomes (hint: try the
points $(s,0),\;(s,s),$ and $(0,s)$). Since we can choose a square of any
size, we can artificially stretch the region in the $w$-plane to infinity as
we increase the domain in the $z$-plane. Notice though, that as long as our
domain is only in the first quadrant in the $z$-plane, we will only ever be in
the first or second quadrants in the $w$-plane. This region is sometimes
referred to as the upper-half plane. As you may see after doing Problem 2.6,
other regions will map to fill the lower-half plane, and by extension, we can
stretch regions out as well to fill the entire lower-half plane. Hence, the
function $w(z)=z^{2}$ maps the entire complex plane into the entire complex
plane! Actually, functions such as these are specifically named entire
functions since they are defined111111Technically, functions are entire only
if they converge everywhere in the plane. However, convergence is beyond the
scope of this chapter, so understanding entire functions as simply being
defined everywhere is good enough for right now. everywhere in the plane.
Since we have begun to understand complex mappings in terms of complex domains
and ranges, we seek to study a few more fairly commonly-used complex functions
in physics.
Use Eq. 2.49 to show that the square defined by $\displaystyle
z=\begin{cases}x+i0,&x\in[-1,0],\;y=0\\\ -1+iy,&x=-1,\;y\in[0,1]\\\
x+1y,&x\in[-1,0],\;y=1\\\ 0+iy,&x=0,\;y\in[0,1]\end{cases}$ maps to the same
region in Fig. 2.6, except that it is reflected over the $u$-axis into the
lower-half plane.
#### 2.6.1 Polynomials
For the first type of function, we consider functions of the form
$\displaystyle
w(z)=P_{n}(z)=\sum_{j=0}^{n}a_{j}z^{j}=a_{0}+a_{1}z+a_{2}z^{2}+\dots+a_{n}z^{n},$
(2.50)
where all of the constant coefficients $\\{a_{j}\\}$ are arbitrary complex
numbers, but each exponent is a nonnegative integer. This class of functions,
often denoted by the symbol $P_{n}$ for “polynomial of
$n^{\mathrm{th}}$-order,” is the complex generalization of the real
polynomials that you already know. This means that instead of only allowing
ourselves to input real numbers into a polynomial, we now are allowing
ourselves to insert a complex number $z=r\textrm{e}^{i\theta}$ as the argument
of the function (in other words, an element from the domain). With this value
of $z$ and by Eq. 2.40, we have
$\displaystyle
P_{n}\left(z=r\textrm{e}^{i\theta}\right)=\sum_{j=0}^{n}a_{j}r^{j}\textrm{e}^{ij\theta}=a_{0}+a_{1}r\textrm{e}^{i\theta}+a_{2}r^{2}\textrm{e}^{2i\theta}+\dots+a_{n}r^{n}\textrm{e}^{in\theta}.$
(2.51)
It is important to note that we are entirely allowed to compute every single
one of these terms using the polar representations of the coefficients, if we
had them given. Since each term is defined and computable, then we are
definitely allowed to add all of their corresponding real and imaginary parts.
Thus, for any $z\in\mathds{C}$ that we throw at this thing, we will always be
able to find its output as a complex number. Thus a general polynomial maps
the complex plane onto121212To be clear, there is a difference between the
words into and onto. A function from set $A$ to set $B$, $f:A\rightarrow B$,
is said to be onto if every element of $B$ is in the range of $f$. Pictorally,
this means that there are no parts of $B$ that are not connected by $f$ to
$A$. Otherwise, mathematicians use the word into. the complex plane. Our
conclusion is important because, just like with real numbers, complex
polynomials are actually relatively easy functions to handle, whether it be in
calculus, computational physics, mathematical modeling, et cetera, because we
can handle them term-wise. Other functions, (like the transcendental
functions) are not so nice.
Consider the complex quadratic, $\displaystyle
P_{2}(z)=i+\textrm{e}^{i\pi/4}z-2z^{2}.$ Use either the Cartesian or the polar
representation to evalutate $P_{2}(z)$ at the following points: $z_{1}=1$,
$z_{2}=-i\pi$, $z_{3}=1-i$. Also, write down the set of coefficients
$\\{a_{j}\\}$ (this part is NOT a trick. I just want to make sure you know
what the set of cofficients is).
#### 2.6.2 Complex Exponentiation Revisted
The last time we dealt with complex exponentials was with deriving the
rational generalization to De Moivre’s Theorem in Eq. 2.48. We were explicit,
however, that the exponent itself had to be a rational number
$p/q\in\mathds{R}$, where $p$ and $q\neq 0$ are integers along the real axis.
Now we have to ask the question of whether it is possible to raise a complex
number $z$ to a constant complex exponent $\xi$. In other words, we want to
study the function $w(z)=z^{\xi}$.
Since $\xi\in\mathds{C}$, we choose to work with it in its Cartesian
representation. Let $\alpha=\mathrm{Re}(\xi)$ and $\beta=\mathrm{Im}(\xi)$
such that $\xi=\alpha+i\beta$. Thus
$\displaystyle w(z)=z^{\xi}=z^{\alpha+i\beta}=z^{\alpha}\,z^{i\beta}.$
Now, we will make the substitution of $z=r\textrm{e}^{i\theta}$ into the
equation above.
$\displaystyle w(z)$
$\displaystyle=\left(r\textrm{e}^{i\theta}\right)^{\alpha}\left(r\textrm{e}^{i\theta}\right)^{i\beta},$
$\displaystyle=\left(r^{\alpha}\textrm{e}^{i\theta\cdot\alpha}\right)\,\left(r^{i\beta}\,\textrm{e}^{i\theta\cdot
i\beta}\right),$
$\displaystyle=r^{\alpha}\textrm{e}^{i\alpha\theta}\,r^{i\beta}\textrm{e}^{-\beta\theta},$
$\displaystyle=r^{\alpha+i\beta}\textrm{e}^{-\beta\theta}\,\textrm{e}^{i\alpha\theta}.$
Now, before proceeding further, since $r$ is a real number, that means it lies
entirely along the real axis, and we can use our regular logarithms on it,
since those are defined over the positive real axis. Thus, we choose to write
$r=\textrm{e}^{\ln r}$, as the (real) functions $\textrm{e}^{x}$ and $\ln x$
are inverse functions such that $\textrm{e}^{\ln x}=x=\ln(\textrm{e}^{x})$.
When we make this substitution into $w$, we can derive our final result.
$\displaystyle w(z)$
$\displaystyle=\textrm{e}^{\ln(r)\cdot(\alpha+i\beta)}\textrm{e}^{-\beta\theta}\,\textrm{e}^{i\alpha\theta},$
$\displaystyle=\textrm{e}^{\alpha\ln(r)+i\beta\ln(r)}\textrm{e}^{-\beta\theta}\,\textrm{e}^{i\alpha\theta},$
$\displaystyle=\textrm{e}^{\alpha\ln(r)-\beta\theta}\,\textrm{e}^{i[\beta\ln(r)+\alpha\theta]},$
$\displaystyle=r^{\alpha}\,\textrm{e}^{-\beta\theta}\,\textrm{e}^{i[\beta\ln(r)+\alpha\theta]}.$
(2.52)
It is important to note that the long string of symbols given in Eq. 2.52 is
something that is entirely computable. What I mean by that is this expression
is written in the polar representation of a complex number, where
$\displaystyle|w(z)|$
$\displaystyle=r^{\alpha}\textrm{e}^{-\beta\theta}\in\mathds{R}^{+},$
$\displaystyle\mathrm{arg}[w(z)]$
$\displaystyle=\beta\ln(r)+\alpha\theta\in\mathds{R}.$
Hence Eq. 2.52 is already in the form of a complex number, regardless of which
values of $\alpha$ and $\beta$ we choose! To be clear, if we were to convert
back to a Cartesian representation then we would need to be careful with any
ambiguities in the rational exponents, as is given in Eq. 2.48. But otherwise,
we are free to map the complex plane $z$ onto131313Since there are many
possible roots for complex exponents, the $z$-plane actually gets mapped onto
the $w$-plane in multi-valued ways, which we won’t go into. These
peculiarities (translated to “headaches”in math-speak) are known as branch
cuts, and special note really should be paid to them from a mathematical
point-of-view. However, in physics, they are only noted if necessary. the
complex plane $w$ via complex exponentiation.
Use the polar representation and Eq. 2.52 to show that
$i^{i}=\textrm{e}^{-\pi/2}$. This example shows that, while complex
exponentiation maps the complex plane onto the complex plane, exponentiation
does NOT necessarily map imaginary numbers into other imaginary numbers. This
is interesting because exponentiation will always map real numbers into other
real numbers.
#### 2.6.3 Complex Logarithms
Since we have covered the case of exponentiation, it follows that we should
cover inverse exponentiation, that is, we need to discuss how to find the
logarithm of a complex number. To study this function, consider $w(z)=\ln(z)$,
where $\ln$ denotes “natural logarithm141414In many advanced mathematics, the
“common log”, denoted by $\log$, is used instead of $\ln$. However, the
“common log” of a number $x$ in physics is almost always given by
$\ln(x)/\ln(10)=\log_{10}(x)$. ”. Just as before, this problem is most easily
tackled in the polar representation.
$\displaystyle
w(z)=w(r\textrm{e}^{i\theta})=\ln(r\textrm{e}^{i\theta})=\ln(r)+\ln(\textrm{e}^{i\theta})=\ln(r)+i\theta.$
(2.53)
To be clear, in this derivation, we implicitly borrowed the logarithm of a
product rule from the real numbers, $\ln(ab)=\ln(a)+\ln(b)$, and we implicitly
enforce that the natural logarithm is the function-inverse of
$\textrm{e}^{x}$. It is very important to note, however, that ANY complex
number $\xi=r_{\xi}\textrm{e}^{i\theta_{\xi}}$ is invariant under a full
rotation about the origin — therefore $\xi$ invariant under a rotation of
$2\pi n$, where $n$ is an integer. This is equivalent to saying
$\xi\,\textrm{e}^{2\pi ni}=\xi\cdot 1=\xi$. However, the exponents are
additive, therefore
$\displaystyle\xi\textrm{e}^{2\pi
ni}=r_{\xi}\textrm{e}^{i\theta_{\xi}}\textrm{e}^{2\pi
ni}=r_{\xi}\textrm{e}^{i(\theta_{\xi}+2\pi n)}$
But if we apply this idea to $z=r\textrm{e}^{i\theta}$, then we have
$z=r\textrm{e}^{i(\theta+2\pi n)}$. Thus, when we take a logarithm, we have
$\displaystyle w(z)=\ln(r)+i(\theta+2\pi n)=\ln(r)+i\theta+2\pi ni.$ (2.54)
But wait. From Eq. 2.53, have
$\displaystyle w(z)=\ln(r)+i\theta=\ln(r)+i\theta+2\pi ni,$ (2.55)
which seems to imply that $2\pi ni=0$ if this equation is to be true. Since
$2\pi$ and $i$ are both nonzero quantities, this means that $n$ would have to
be zero! However, we said $n$ can be any integer, not just zero. What gives?
Did we accidentally stumble on a contradiction within complex algebra?
So the truth is that, with our current system of complex numbers, we actually
did find a contradiction, and it has to do with our interpretation of
multiplication as rotation and $\textrm{e}^{2\pi ni}=1$ in the polar
representation. But this contraction is not a totally new idea — it is similar
to the idea that rational exponents are multivalued. It turns out that in the
complex plane, the logarithm also accrues multiple values for the same input.
In this sense, the full logarithm is not a function since for any input there
are many outputs (so it does not pass the complex-version of the vertical line
test). How do we deal with this problem? We use the idea of branch cuts to
save the day, where a “branch cut” is a fancy name for specifying a particular
range of the logarithm over a particular domain. In fact, this approach is
identical to the one you learned in trigonometry, where we would only specify
the inverse cosine ($\arccos(x)=\cos^{-1}(x)$) or inverse sine
($\arcsin{}(x)=\sin^{-1}(x)$) on the interval $x\in[-1,1)$, since there are
(infinitely) many values of $x$ such that $\cos(x)=\pm 1$ or $\sin(x)=\pm 1$.
Although this is (sort of) passing-the-buck on the issue of having the
logarithm be mutlivalued, it DOES allow us to retain our interpretation of
multiplication as rotations in the complex plane — we just need to be careful
about which branch we’re talking about when using the logarithm.
So the problem with the logarithm is due to the multiplicity, or rotational
symmetry, surrounding the $2\pi n$ angle. Thus, it helps to define a principal
branch of the logarithm, so there really isn’t a lot of ambiguity surrounding
our equations. All we need to do is define the imaginary part of the logarithm
over a $2\pi$-interval to keep things consistent, since it is the imaginary
part in Eq. 2.53 that has the multiple values. The interval we choose may
initially seem odd, but actually is more intuitive in physics, and is given by
$\theta_{principal}\in[-\pi,\pi)$. Therefore, the principal logarithm is
defined as
$\displaystyle\mathrm{Ln}(z)=\mathrm{Ln}(r\textrm{e}^{i\theta})=\ln(r)+i\theta,\;\theta\in[-\pi,\pi).$
(2.56)
Here, the capital $\mathrm{L}$ signifies the special branch of the multivalued
logarithm given in Eq. 2.53. Notice though that there is a lowercase
$\mathrm{l}$ in the real part of Eq. 2.56. This is because $r$ is a positive
real number, and we can always take a normal, non-multivalued logarithm of any
positive real number. All of the multivaluedness, again, is only due to the
imaginary part.
You have probably been told for your whole life that it is impossible to take
a logarithm of a negative number. Unfortunately, that is just not true. Use
Eq. 2.53 defined for $\mathrm{arg}(z)\in[\pi,3\pi)$ to calculate $\ln(-1)$. To
check whether your answer makes sense, suppose you compute $\ln(-1)$, and let
us denote it by $w=\ln(-1)$. This would imply that $\textrm{e}^{w}=-1$, or
$\textrm{e}^{w}+1=0$. Do you recognize this equation? Next, compute $\ln(-a)$,
where $a$ is a positive real number. Hence, $-a$ represents any point along
the negative real axis with an absolute value of $a$. For example, if $a=9$,
then I would be asking you to evaluate $\ln(-9)$. Since the number $-a$
represents any negative real number, what can you conclude about $\ln(-a)$? Is
it ever a purely real number (Hint: No it isn’t.)? Is this why you’ve been
taught that you cannot take the logarithm of a negative number (Hint: Yes it
is.)?
#### 2.6.4 Hyperbolic Trigonometry
The final class of functions that I want to discuss is within the set of
functions that most people refer to as hyperbolic trigonometry. The geometry
is actually remarkably simple to jump to from normal, or circular,
trigonometry. The latter is governed by the unit circle, defined by
$x^{2}+y^{2}=1$. We are instead going to be interested in the so-called unit
hyperbola, defined by $x^{2}-y^{2}=1$. Although the geometry may be
interesting, I am going to be honest with you: the only time I’ve ever
actually seen it be relevant is in orbital mechanics where trajectories
matter, or general relativity, where the geometry of spacetime is the
dynamical variable of interest (in other fields of physics, like Physics 1,
the dynamical variables are position and velocity as a functions of time).
With that said, the actual hyperbolic functions themselves are often
incredibly useful tools to use in physics. They, for example, are used to
describe how the electrostatic potential changes in space from a source. I
have included Fig. 2.7 to help you see what unit hyperbola looks like, and how
the hyperbolic trigonometric functions relate triangles to it.
FIG. 2.7: The unit hyperbola $x^{2}-y^{2}=1$ parameterized by the functions
$x=\cosh t$ and $y=\sinh t$. Think of the parameter $t$ as the time it takes
for the black point to move from the point $(x,y)=(1,0)$ to the point
$(x,y)=(x(t),y(t))$ along the unit hyperbola. FIG. 2.8: Plots of the
hyperbolic cosine (Eq. 2.57) and hyperbolic sine (Eq. 2.58) functions over a
few real numbers $x$.
The most commonly used hyperbolic functions are the hyperbolic cosine and the
hyperbolic sine functions, defined by
$\displaystyle\cosh{x}$
$\displaystyle=\frac{\textrm{e}^{x}+\textrm{e}^{-x}}{2},$ (2.57)
$\displaystyle\sinh{x}$
$\displaystyle=\frac{\textrm{e}^{x}-\textrm{e}^{-x}}{2},$ (2.58)
for any $x\in\mathds{R}$. Here the extra $\mathrm{h}$ that’s attached is the
hyperbolic part, and the $\cosh$ notation is pronounced cah-sh while the
$\sinh$ part is often pronounced sin-ch. I’m not 100% sure why there is an
added ch in the $\sinh$ pronounciation, but my best guess is that sin-h is
hard to say. Since I know that these real-valued functions will be brand new
to most of you reading, I included a plot of them in Fig. 2.8 so you can
visualize them. Notice that as $x\rightarrow+\infty$, the hyperbolic sine and
hyperbolic cosine become equal, whereas on the other side of the real axis,
the $\sinh$ and $\cosh$ approach values equal magnitudes but opposite signs.
Note that if we set $x=\cosh(t)$ and $y=\sinh(t)$, then
$\displaystyle x^{2}-y^{2}$ $\displaystyle=(\cosh t)^{2}-(\sinh t)^{2},$
$\displaystyle=\left(\frac{\textrm{e}^{t}+\textrm{e}^{-t}}{2}\right)^{2}-\left(\frac{\textrm{e}^{t}-\textrm{e}^{-t}}{2}\right)^{2},$
$\displaystyle=\frac{1}{4}(\textrm{e}^{2t}+\textrm{e}^{-2t}+2)-\frac{1}{4}(\textrm{e}^{2t}+\textrm{e}^{-2t}-2),$
$\displaystyle=\frac{1}{4}(\textrm{e}^{2t}+\textrm{e}^{-2t}-\textrm{e}^{2t}+\textrm{e}^{-2t}+2+2),$
$\displaystyle=1$
which shows that these definitions of the $\cosh$ and $\sinh$ appropriately
parameterize the unit hyperbola with the parameter $t$. To better understand
the parameterization, imagine $t$ as a unit of time, and $x(t)$ and $y(t)$ are
the $x,y$ positions of a point moving along the unit hyperbola.
It is also really important to notice that the $\cosh$ is an even function,
whereas the $\sinh$ is an odd function, in the sense that an even function
$f(x)$ has the property $f(-x)=f(x)$ while odd functions have the property
$f(-x)=-f(x)$. I will prove these properties for the $\cosh$ and $\sinh$ to
you below.
$\displaystyle\cosh(-x)$
$\displaystyle=\frac{\textrm{e}^{(-x)}+\textrm{e}^{-(-x)}}{2}=\frac{\textrm{e}^{-x}+\textrm{e}^{x}}{2}=\frac{\textrm{e}^{x}+\textrm{e}^{-x}}{2}=\cosh(x).$
$\displaystyle\sinh(-x)$
$\displaystyle=\frac{\textrm{e}^{(-x)}-\textrm{e}^{-(-x)}}{2}=\frac{\textrm{e}^{-x}-\textrm{e}^{x}}{2}=\frac{-(\textrm{e}^{x}-\textrm{e}^{-x})}{2}=-\sinh(x).$
I wanted to point these properties out to you because they actually
identically mimic those of the normal trigonometric functions, where the
cosine is even and the sine is odd! There are actually a slew of other
analogous properties between circular trigonometry and hyperbolic
trigonometry, but I will not address them here. From here I am going to shift
our attention to uniting circular trig with hyperbolic trig through the
complex plane.
We defined a hyperbolic cosine and a hyperbolic sine. Recall that the normal
(circular) tangent is the quotient of the sine and cosine. Use this quotient
to derive the hyperbolic tangent function over the real numbers $x$, denoted
by $\tanh{x}$ and pronounced tan-ch, and then sketch it so you see what it
looks like. Based on your expression for the $\tanh$, prove that it is an odd
function over the reals.
Okay, back to complex numbers. Consider the function $w(z)=\cosh{z}$. This
time, we will choose to work with $z$ in its Cartesian representation so that
$z=x+iy$. Then, using Eq. 2.57, we have
$\displaystyle w(z)$ $\displaystyle=\cosh(x+iy),$
$\displaystyle=\frac{\textrm{e}^{x+iy}+\textrm{e}^{-(x+iy)}}{2},$
$\displaystyle=\frac{\textrm{e}^{x}\textrm{e}^{iy}+\textrm{e}^{-x}\textrm{e}^{-iy}}{2},$
$\displaystyle=\frac{\textrm{e}^{x}\left(\cos y+i\sin
y\right)+\textrm{e}^{-x}\left(\cos y-i\sin y\right)}{2},$
$\displaystyle=\frac{\left(\textrm{e}^{x}+\textrm{e}^{-x}\right)\cos
y+i\left(\textrm{e}^{x}-\textrm{e}^{-x}\right)\sin y}{2},$
$\displaystyle=\frac{\textrm{e}^{x}+\textrm{e}^{-x}}{2}\,\cos
y+i\frac{\textrm{e}^{x}-\textrm{e}^{-x}}{2}\,\sin y,$
$\displaystyle=\cosh(x)\cos(y)+i\sinh(x)\sin(y).$ (2.59)
Thus, we have shown that the $\cosh$ of a complex argument mixes circular and
hyperbolic trigonometry. Furthermore, let’s assume that $z=iy$, meaning it is
a purely imaginary (lateral) number. Then
$\displaystyle\cosh(iy)=\cos(y).$ (2.60)
This relationship goes the other way as well. If we consider $\cos(iy)$ then
we have from Eq. 2.34
$\displaystyle\cos(iy)=\frac{\textrm{e}^{i(iy)}+\textrm{e}^{-i(iy)}}{2}=\frac{\textrm{e}^{-y}+\textrm{e}^{y}}{2}=\frac{\textrm{e}^{y}+\textrm{e}^{-y}}{2}=\cosh(y).$
(2.61)
Since the imaginary arguments can be used to convert between circular and
hyperbolic trig using these equations, we conclude that hyperbolic triangles
are really just a lateral/adjacent form of straight-line versions. This
conclusion could have actually been reached with the unit circle itself,
$x^{2}+y^{2}=1$, if we changed the $y$ part to an imaginary number
$y\rightarrow iy$, thus converting the unit circle to
$x^{2}+(iy)^{2}=x^{2}-y^{2}=1$, the unit hyperbola! Perhaps if we had not
given $i=\sqrt{-1}$ the name of “imaginary”, then we could have known more
about geometry, in general, immediately from the start.
I derived the relationships between the hyperbolic cosine and the circular
cosine. Using an almost identical treatment, derive the following equations:
$\displaystyle\sinh(x+iy)$ $\displaystyle=\sinh(x)\cos(y)+i\cosh(x)\sin(y),$
(2.62) $\displaystyle\sinh(iy)$ $\displaystyle=i\sin(y),$ (2.63)
$\displaystyle\sin(iy)$ $\displaystyle=i\sinh(y).$ (2.64)
### 2.7 Concluding Remarks
In this chapter, the fundamentals of complex algebra were introduced and used
to generalize a few commonly used real functions into the complex plane.
Knowing how to use complex numbers to study physical systems is particularly
helpful. For example, it is pretty common to treat all electromagnetic waves
as traveling complex exponentials given by
$\textrm{e}^{i(\vec{k}\cdot\vec{r}-\omega t)}$, where $\vec{k}$ is the
wavenumber of the wave, $\vec{r}$ is the position of the wavefront, $\omega$
is the frequency of the wave, and $t$ is the time. To be clear, the
electromagnetic waves are purely real objects meaning that all of their
physical properties would be quantities that we would associate with the real
line (they are not lateral quantities, whatever those may be…). So why are
these imaginary/lateral parts relevant in physics? It turns out that
exponentials have a plethora of useful properties in calculus and they are
much more manageable than the trigonometric waves defined by sines and
cosines. Thus, we usually only need to worry about the real part of the
electric and magnetic fields. Meanwhile, whenever the waves move through a
material, it turns out that the $\vec{k}$ can be identified as having both a
real part and a lateral part, where the latter actually plays a role in the
heat-loss of the wave in that medium (this property is called attenuation). In
other words, knowing how to split up complex numbers into real and imaginary
(lateral) parts and more can tell us very important things about how physical
objects interact with one another. And this case is one where we use complex
numbers to make our mathematical modeling simpler — it does not include
situations in physics where the whole complex plane is absolutely necessary,
such as anything where the word “quantum” is involved.
There is still a lot of topics within the topic of complex numbers at large,
but this chapter hopefully gives you a solid foundation in something that,
once you get the hang of it, will truly make that math you use in physics much
simpler, and by extension, your life much easier. Not to mention, complex
algebra at least makes things like numbers a little prettier to look at.
I have included a table of helpful equations (Table 2.1) for you to reference
for whenever you need them later on in your career. They also have a reference
to the text where they were described.
TABLE 2.1: A summary of the important and general equations in Complex
Algebra.
Equation Description | Equation Formula | Text Reference
---|---|---
Definition of Complex Numbers | $z=x+iy,\;\;x,y\in\mathds{R}$ | Eq. 2.8
Definition of Complex Conjugate | $z=x-iy,\;\;x,y\in\mathds{R}$ | Eq. 2.9
Real Part of a Complex Number | $\mathrm{Re}(z)=\dfrac{z+z^{\ast}}{2}\in\mathds{R}$ | Eq. 2.10
Imaginary Part of a Complex Number | $\mathrm{Im}(z)=\dfrac{z-z^{\ast}}{2i}\in\mathds{R}$ | Eq. 2.11
Addition of Complex Numbers | $z\pm w=(x\pm u)+i(y\pm v)$ | Eq. 2.12
Product of Complex Numbers | $zw=(xu-yv)+i(xv+yu)$ | Eq. 2.13
Quotient of Complex Numbers | $\dfrac{z}{w}=\dfrac{xu+yv}{u^{2}+v^{2}}+i\,\dfrac{xv-yu}{u^{2}+v^{2}}$ | Eq. 2.14
Modulus of a Complex Number | $|z|=\sqrt{z^{\ast}z}=\sqrt{[\textrm{Re}(z)]^{2}+[\textrm{Im}(z)]^{2}}$ | Eq. 2.16
Polar Representation | $z=|z|\textrm{e}^{i\,\mathrm{arg}(z)}$ | Eq. 2.27
Argument of a Complex Number | $\mathrm{arg}(z)=\arctan\left[\dfrac{\mathrm{Im}(z)}{\mathrm{Re}(z)}\right]$ | Eq. 2.30
Euler’s Identity | $\textrm{e}^{i\theta}=\cos\theta+i\,\sin\theta$ | Eq. 2.31
Cosine Formula | $\cos\theta=\dfrac{\textrm{e}^{i\theta}+\textrm{e}^{-i\theta}}{2}$ | Eq. 2.34
Sine Formula | $\sin\theta=\dfrac{\textrm{e}^{i\theta}-\textrm{e}^{-i\theta}}{2i}$ | Eq. 2.35
Multiplication as Dilation and Rotation | $zw=(r\rho)\,\textrm{e}^{i(\theta+\phi)}$ | Eq. 2.36
De Moivre’ Theorem (for Integer $n$) | $(\cos\theta+i\sin\theta)^{n}=\cos(n\theta)+i\sin(n\theta)$ | Eq. 2.41
$n^{\mathrm{th}}$ Roots of Unity | $1^{1/n}=\exp\left(\dfrac{2\pi mi}{n}\right),\;\,m\in\\{0,1,\dots,n-1\\}$ | Eq. 2.43
$n$-Term Geometric Series | $S_{n}=\sum\limits_{m=0}^{n-1}a^{m}=\dfrac{1-a^{n}}{1-a},\;a\neq 1$ | Eq. 2.45
Sum of $n^{\mathrm{th}}$ Roots of Unity | $\sum\limits_{m=0}^{n-1}\left(\textrm{e}^{2\pi i/n}\right)^{m}=0$ | Eq. 2.46
General Complex Exponentiation | $z^{\alpha+i\beta}=\left(r\textrm{e}^{i\theta}\right)^{\alpha+i\beta}=r^{n}\textrm{e}^{-\beta\theta}\,\textrm{e}^{i[\beta\ln(r)+\alpha\theta]}$ | Eq. 2.52
Principal Logarithm | $\mathrm{Ln}(z)=\mathrm{Ln}(r\textrm{e}^{i\theta})=\ln(r)+i\theta,\;\;\theta\in[-\pi,\pi)$ | Eq. 2.56
## Chapter 3 Calculus of a Single Variable
Up until now, I have not assumed that you have any prior knowledge of
calculus, mostly because it is frankly unnecessary to understand vectors and
complex algebra. Thus, everything that we have covered is totally static. It,
in no way, describes anything that can possibly change at all. For example, if
the position vector of a particular object is known at a particular time, we
would thus far only have the tools to describe the object’s vector at that
time. Sure, we could resolve the vector into its components, we could talk
about the direction of the vector, we could talk about it being orthogonal to
other vectors, but we would have NO WAY of talking about how that vector is
changing at that time. Hence could not speak of how the object will move in
time.
You might be wondering why we could not just reference the object’s velocity
vector at that time, since surely the velocity will tell us about how the
position vector changes. As we will see, this is absolutely true. However,
none of the mathematical tools we’ve discussed so far allow us to describe
this change. Meanwhile, you can definitely intuit how position will change
because you have experienced changes in your everyday life. Our goal in this
chapter is to develop your natural intuition into something mathematically
precise and immutable — our goal is to develop calculus.
As a heads up, chapter is going to be dense. Like, degenerate matter dense111A
classical model of the degeneracy will probably suffice though.. We are going
to cover essentially the same amount of material that one would see as they
progressed from a Calculus I and II sequence of courses. Although we will
cover a lot of material, it will be condensed to a form that is sufficient for
implementation in physics. Truth be told, once you understand derivatives and
integrals from a geometric/changey perspective, then the number of dimensions
rarely makes a difference to a physicist. Sure, it might add a few extra terms
here and there, but the actual modeling of the natural world using rates of
change is what matters to us, and it is almost always easy, if not trivial, to
discuss the laws of physics using calculus. That is not to say that actually
solving problems in physics is trivial, it isn’t, but the nontriviality of the
problem solving in physics is partly due to the inclusion of our clunky form
of algebra with calculus. The other part is usually just because nature is
smarter than us.
I plan to introduce many calculus topics at are relevant for physics in ways
that are easy to visualize. I am not going to get too bogged down in the
details about which functions have which derivatives or antiderivatives,
although I will talk about a few important ones. Calculus really should not be
about the memorization of a bunch of special-case formulas; when it is, it is
usually very hard for students to see its value or applicability. Thus, I will
reference outside sources for specific formulas if they are not quick and easy
to derive (tables of derivatives and integrals are ubiquitous and freely
available on the Internet). Although the transition can be hard, once you are
able to see beyond the mess of special cases, using calculus to describe the
universe will be effortless, and you will probably wonder how you didn’t
already describe it is such a fluent way222Okay, so maybe this is a little
anecdotal and waxing poetic, but, hey, if using calculus were worse than what
we had before, why would we still be using it?.
### 3.1 Position, Velocity, and Acceleration
People usually have an almost instinctive understanding of how things move
(with the exception of feathers and bowling balls in free fall).
Unfortunately, it took an incredible amount of time for us to develop a means
of communicating our understanding with others in a precise way. We can thank
people like Galileo and Newton for figuring that last bit out for us. In
historical hindsight though, developing the crucial ideas of calculus are
pretty straightforward.
We are going to start by exploiting the position, velocity, and acceleration
of some object (maybe a car or volleyball or bug) and we will be careful to
remember that each is a vector. Furthermore, let us assume that we know the
position vectors at a few different times, and let us denote these vectors by
$\vec{r}(t)$. Using vector addition (recall tip-to-tail), we can then define a
vector that changes the position at one time $t$ to one at a later time
$t+\Delta t$. Here this $\Delta t$ represents an increment in time, for
example, it could represent a year, a century, a second, a femtosecond, a unit
of Planck time, et cetera. Physically, this increment in time would be
whatever time difference it takes to get the object from one position to the
next. This change vector at time $t$ is going to be denoted as
$\Delta\vec{r}(t)$ and connects the position at $t$, $\vec{r}(t)$, to the
position at $t+\Delta t$, $\Delta\vec{r}(t+\Delta t)$, as is shown in Fig.
3.1.
FIG. 3.1: If the position of an object is known at two times, perhaps at $t$
and $t+\Delta t$, it is possible to describe the change in position from
$\vec{r}(t)$ to $\vec{r}(t+\Delta t)$ as $\Delta\vec{r}(t)=\vec{r}(t+\Delta
t)-\vec{r}(t)$.
A question that follows from this set up would be something like, “Is it
possible to describe how quickly the object moved from $\vec{r}(t)$ to
$\vec{r}(t+\Delta t)$?” Intuitively, we might propose that the change in
position, or the position increment, $\Delta\vec{r}(t)$, is proportional to
the increment in time, $\Delta t$. For example, if we were running at a
specific pace for a longer period of time, we will travel farther. However,
time is scalar, whereas the change in position is definitely a vector by Fig.
3.1. Thus, we are left to define the velocity333For anyone who already knows
calculus, this is the average velocity. of the object at time $t$ as the
vector responsible for the change in the object’s position. This is denoted as
$\vec{v}(t)$ and can be written mathematically as
$\displaystyle\Delta\vec{r}(t)=\vec{r}(t+\Delta t)-\vec{r}(t)=\vec{v}(t)\Delta
t,$ (3.1)
or if we divide both sides by the scalar time increment, we have
$\displaystyle\vec{v}(t)=\frac{\Delta\vec{r}(t)}{\Delta
t}=\frac{\vec{r}(t+\Delta t)-\vec{r}(t)}{\Delta t},$ (3.2)
It is important to remember that at this point the $(t)$ things everywhere are
meant to symbolize that the letter immediately before is a function of $t$; it
does not mean multiply everything by $t$. For example, $\vec{v}(t)$ means that
the velocity is a function of time $t$ rather than multiply $\vec{v}$ and the
scalar $t$.
By looking at Eq. 3.2, we see that the equation has the form of a slope, in
that it is a function that has a change divided by a change, or more
colloquially, it has a rise-over-run type form. Furthermore, we can see where
the units of miles-per-hour or meters-per-second come from, since we multiply
the position vector measured in units of length by the scalar $1/\Delta t$
which has units of per-time. Also, using this definition, since the time
increment takes on the value of whatever time-recording device we have allows,
if there is no change in position, we conclude that there is zero velocity.
Likewise, if there is ever a time where the velocity is zero, then there will
be no resulting change in position. Alright, now that the relationship between
changes in position and velocity is established between two points in time,
what happens when we introduce more points? Or from a more scientific
perspective, what happens when we measure the position of an object at a
greater number of points in time?
FIG. 3.2: If the position of an object is known at three times, perhaps at
$t$, $t+\Delta t$, and $t+2\Delta t$, it is possible to describe the change in
the change in the position from $\vec{r}(t)$ to $\vec{r}(t+2\Delta t)$ as
$\Delta\Delta\vec{r}(t+\Delta t)=\Delta\vec{r}(t+\Delta t)-\Delta\vec{r}(t)$.
Since the change in position is a measure of velocity by Eq. 3.2, then this
figure shows the existence in the change in velocity, also known as the
acceleration.
For simplicity, let’s assume we know the position of a particle at three
points in time: $t$, $t+\Delta t$, and $t+2\Delta t$. Then, just as we did
before in Fig. 3.1, it is possible for us to connect the position at $t$ with
that at $t+\Delta t$, and we can now connect the position at $t+\Delta t$ with
that at $t+2\Delta t$. A picture of this scenario is shown in Fig. 3.2, and in
this figure, it is evident that there are really two changes in position:
$\displaystyle\Delta\vec{r}(t)$ $\displaystyle=\vec{r}(t+\Delta
t)-\vec{r}(t),$ $\displaystyle\Delta\vec{r}(t+\Delta t)$
$\displaystyle=\vec{r}(t+2\Delta t)-\vec{r}(t+\Delta t).$
Thus, we may ask “what is the change in the change in position?” Let us denote
this change of change in position as $\Delta\Delta\vec{r}(t+\Delta t)$, since
the change of change position occurs at $t+\Delta t$. Let’s denote the change
of change as $\Delta\Delta$, and evaluate it directly
$\displaystyle\Delta\Delta\vec{r}(t+\Delta t)=\Delta\vec{r}(t+\Delta
t)-\Delta\vec{r}(t)$
If we now assume that this change is proportional to some vector rate, just
like we did when we came up with the notion of velocity, and denote this rate
as $\vec{a}(t+\Delta t)$, then we can posit that
$\displaystyle\Delta\Delta\vec{r}(t+\Delta t)=\vec{a}(t+\Delta t)\Delta
t^{2},$
Here $\Delta t^{2}=(\Delta t)^{2}$. Solving for $\vec{a}(t+\Delta t)$
directly, we have
$\displaystyle\vec{a}(t+\Delta t)=\frac{\Delta\vec{r}(t+\Delta
t)-\Delta\vec{r}(t)}{\Delta t^{2}}=\frac{\frac{\Delta\vec{r}(t+\Delta
t)}{\Delta t}-\frac{\Delta\vec{r}(t)}{\Delta t}}{\Delta t}.$
In the second equality, I moved one of the $\Delta t$s into the numerator to
make two fractions because these fractions are defined explicitly in Eq. 3.2.
Using this definition, we have
$\displaystyle\vec{a}(t+\Delta t)=\frac{\vec{v}(t+\Delta t)-\vec{v}(t)}{\Delta
t},$ (3.3)
which we recognize as the definition of acceleration444Average acceleration to
be precise. Again, this equation is one that has the form of a slope. Further,
we would say that if there is a change in velocity, then there must have been
an acceleration to cause it. However, since the acceleration is related to the
change of change in position, we could also say that a “second-order” change
in position is due to the existence of an acceleration.
So what have we learned by measuring more points? Well, it appears that each
time we measure a new point, we can define a new rate associated with the
addition of a new position measurement. This is generally true for position
measurements. Actually, the names for a few of the even higher order changes
($3^{rd}-6^{th}$) are the jerk, snap, crackle, and pop
[thompson_systems_technology]. We also know that the little time increment
$\Delta t$ is some measurement, but its actual value depends on the time-
measuring device we have (for example, atomic clocks have smaller $\Delta t$s
than grandfather clocks). In principle, we could have exactly one position
measurement for every single time measurement we have, and let’s assume we
have $N$ total of these position measurements. Then we could write them down
in a table of sorts as
$\displaystyle\begin{array}[]{c|c|ccccc}\textbf{Measure:}&\textrm{Time
}t&t&t+\Delta t&t+2\Delta t&\dots&t+(N-1)\Delta t\\\
\hline\cr\textbf{Measure:}&\textrm{Position
}\vec{r}&\vec{r}(t)&\vec{r}(t+\Delta t)&\vec{r}(t+2\Delta
t)&\dots&\vec{r}(t+(N-1)\Delta t)\end{array}$
And so each time there is a change in position, we would presumably have a
corresponding velocity measurement, and additionally, each time we have a
change in velocity we would have an acceleration. So we would have two more
rows in that table
$\displaystyle\begin{array}[]{c|c|ccccc}\textbf{Measure:}&\textrm{Time
}t&t&t+\Delta t&t+2\Delta t&\dots&t+(N-1)\Delta t\\\
\hline\cr\textbf{Measure:}&\textrm{Position
}\vec{r}&\vec{r}(t)&\vec{r}(t+\Delta t)&\vec{r}(t+2\Delta
t)&\dots&\vec{r}(t+(N-1)\Delta t)\\\
\hline\cr\textbf{Calculate:}&\textrm{Velocity }\vec{v}&-&\vec{v}(t+\Delta
t)&\vec{v}(t+2\Delta t)&\dots&\vec{v}(t+(N-1)\Delta t)\\\
\hline\cr\textbf{Calculate:}&\textrm{Acceleration }\vec{a}&-&\vec{a}(t+\Delta
t)&\vec{a}(t+2\Delta t)&\dots&-\end{array}$
Here, the dashed marks appear since there are no changes initially for either
the position or velocity; thus there are no velocities or accelerations at
this time. To better help visualize this table, imagine that all of these
measurements came from an object’s trajectory as that shown in Fig. 3.3, where
the object starts at an initial position of $\vec{r}_{0}$ and moves to a final
position of $\vec{r}_{f}$. In the figure, the blue points represent the
positions as functions of time (remember that these are all vectors, hence the
ghostly vectors to $\vec{r}_{0}$ and $\vec{r}_{f}$), the red dashed vectors
represent the velocities as functions of time, and the green dotted vectors
represent the accelerations as functions of time.
FIG. 3.3: The position, velocity, and acceleration of particle as a function
of time $t$ and it meanders from an initial position $\vec{r}_{0}$ to a final
position $\vec{r}_{f}$. During the trip, it can have many different velocities
and accelerations, whose directions are represented by the red dashed arrows
and the green dotted arrows, respectively.
An important highlight from drawing out all of these position measurements is
that the velocity vectors can really be seen to connect all of the position
measurements together. More precisely, if we wanted to know the position of
the object at any time measurement, let’s say at the $j^{\mathrm{th}}$ time
increment, then we could find it by starting at $\vec{r}_{0}$ and then adding
all of the changes in position up until the $j^{\mathrm{th}}$ increment. In
other terms,
$\displaystyle\vec{r}(t+j\Delta t)$
$\displaystyle=\vec{r}_{0}+\Delta\vec{r}(t)+\Delta\vec{r}(t+\Delta
t)+\Delta\vec{r}(t+2\Delta t)+\dots\Delta\vec{r}(t+(j-1)\Delta t),$
$\displaystyle=\vec{r}_{0}+\vec{v}(t)\Delta t+\vec{v}(t+\Delta t)\Delta
t+\vec{v}(t+2\Delta t)\Delta t+\dots+\vec{v}(t+(j-1)\Delta t)\Delta t.$
All of the $\Delta t$s here show up from the implicit definition of the
velocity at a point. The equation above can be written much more succinctly
with summation notation, and so we have
$\displaystyle\vec{r}(t+j\Delta
t)=\vec{r}_{0}+\sum_{k=0}^{j-1}\vec{v}(t+k\Delta t)\Delta t.$ (3.4)
Although it is harder to visualize from Fig. 3.3 (but not impossible!), there
is actually an analogous summation relationship that exists to relate the
velocity at $t+j\Delta t$ to all of the intermediate accelerations:
$\displaystyle\vec{v}(t+j\Delta
t)=\vec{v}_{0}+\sum_{k=0}^{j-1}\vec{a}(t+k\Delta t)\Delta t.$ (3.5)
These equations have the form of the area of a rectangle, where $\Delta t$ is
the width of the rectangle, and the velocity or acceleration would be the
height of the rectangle. Furthermore, we actually add up all of the
intermediate areas to find something new.
In short, this is all that calculus is: either evaluating slopes or adding up
a bunch of boxes. If we want to describe how quickly something changes in
terms of another variable — for example, if we want to find how fast position
changes as a function of time — we would use a slope. If, on the other hand,
we wanted to describe the aggregate effects of many successive actions — for
example, if we want to find out where our final position is after moving with
a bunch of different velocities — we would add boxes. Even further, we can
undo the effect of taking a slope by adding a bunch of areas together; an
operational inversion analogous to how multiplying something by $4$ and then
dividing the result by $4$ returns the original something. And that is truly
all of calculus. There really is nothing else to conceptualize.
However, like most things, this conceptualization is easier said than done.
And that’s okay. As we progress through this chapter, we will study many more
applications of these ideas and refine them into some theorems. Fortunately
for us, the context of this chapter will largely be the natural world, and so
if the mathematics ever gets too messy, we can always find real-world
analogies to bolster our intuition and understand which message the math is
trying to convey.
### 3.2 Continuity
We have discussed how slopes and box-addition represent function-transitions
between position, velocity, and acceleration. But everything we did was built
on the assumption that we were taking measurements of time with some device
that only measured in increments of $\Delta t$. While this picture is accurate
from a numerical perspective, and definitely from an experimental point-of-
view, it is not necessarily physical and certainly not mathematical.
Physically, objects have positions at intermediate values between the points
we measure. We unfortunately are limited by how quickly we can measure
something’s position; however, this is no way implies that objects only move
in discrete ways. In principle, if we were able to “take more measurements”
then we could describe an object’s position, velocity, or acceleration at any
time $t$ that we wanted.
To better illustrate this idea, we assume that something only moves over a
fixed time interval, perhaps $t_{f}-t_{0}$ and that we have made a total of
$N$ measurements. Then the time increment would be
$\displaystyle\Delta t=\frac{t_{f}-t_{0}}{N}.$
Notice that this time increment gets smaller when the number of samples $N$
gets larger. If this is not clear, choose $t_{f}-t_{0}$ to be your favorite
number (physicists really like $1$) and then divide by successively larger
numbers — for example, $1/1=1$, $1/2=0.5$, $1/3=0.33\dots$, …, $1/100=0.01$, …
So as long as the $t_{f}-t_{0}$ is a fixed number, $\Delta t$ will get very
small as $N$ gets very large.
The small time increment is helpful, because it allows us to probe more time-
values. For example, if $t_{0}=1$, $t_{f}=2$, and $N=10$, then we can talk
about times like
$\displaystyle t_{0}+\Delta t$ $\displaystyle=1+1/10=1.1,$ $\displaystyle
t_{0}+2\Delta t$ $\displaystyle=1+2/10=1.2,$ $\displaystyle\vdots$
$\displaystyle t_{0}+(N-1)\Delta t$ $\displaystyle=1+9/10=1.9$
But we could never reach a time value like $1.11$, since our time increment is
larger than that extra $0.01$. In other words, our resolution is not good
enough to see a time increment of $0.01$. So what can we do? Instead of only
taking $10$ measurements, how about we take $100$? Then we could have
$\displaystyle t_{0}+\Delta t$ $\displaystyle=1+1/100=1.01,$ $\displaystyle
t_{0}+2\Delta t$ $\displaystyle=1+2/100=1.02,$ $\displaystyle\vdots$
$\displaystyle t_{0}+(N-1)\Delta t$ $\displaystyle=1+99/100=1.99$
By the same argument, we could jump down to a resolution of 0.001 by taking
1000 measurements to have
$\displaystyle t_{0}+\Delta t$ $\displaystyle=1+1/1000=1.001,$ $\displaystyle
t_{0}+2\Delta t$ $\displaystyle=1+2/1000=1.002,$ $\displaystyle\vdots$
$\displaystyle t_{0}+(N-1)\Delta t$ $\displaystyle=1+999/1000=1.999$
Hence, it is clear that by increasing the number of samples, we can know the
position over smaller time increments and therefore observe what happens at
those intermittent steps that we would have been incapable of observing
before. So now the question is: “how many samples do we need to get a time
increment to probe time up to any arbitrary level of precision $\epsilon$?”
To solve this problem, we just need the time increment to be less than this
arbitrary precision $\epsilon$ (before $\epsilon\in\\{0.1,0.01,0.001\\}$).
Thus, what we do is posit that $\Delta t<\epsilon$, or
$\displaystyle\frac{t_{f}-t_{0}}{N}<\epsilon\Rightarrow\frac{t_{f}-t_{0}}{\epsilon}<N,$
(3.6)
The second inequality555This inequality rule can be tricky for a lot of people
the first time they see it. Most people know that if you multiply an
inequality by a negative sign, then you must flip the inequality. The same is
true for division. For example, it is true that $1/2<1/1$, but it is not true
that $2<1$, when we invert the inequality. is our answer. I recommend you test
it for yourself.
The beautiful thing about this idea is that $\epsilon$ can literally be
anything, as long as it’s positive. That means we could increase it or
decrease it at will — this is the math equivalent of increasing or decreasing
the volume of your phone666Or whatever the cool sound-emitting technological
marvel you future people use., except there is no upper or lower bound! Thus,
we could, at least in principle, probe time measurements to any level of
precision we want. All we would need to do is take enough measurements.
In mathematics, we give this idea the name, limit of a sequence. In this case,
the sequence would be $\Delta t$, and the elements of the sequence would be
each $\Delta t$ with a incrementally higher $N$ value substituted in. In this
case, we would say that the limit of $\Delta t$ would be zero since we can
definitively get within a precision of $\epsilon$ of zero using the
inequalities above. More formally, we would say that the limit of some
sequence $a_{N}$, denoted by $L$, exists by the following criterion:
> The sequence $a_{N}$ is said to converge to the limit $L$ if there exists
> positive integers $n$ and $M$ such that $|a_{n}-L|<\epsilon$ for every
> $\epsilon>0$, whenever $n>M$.
When the limit does in fact exist, we say that
$\displaystyle\lim_{N\rightarrow\infty}a_{N}=L,$ (3.7)
where the calculation of how many measurements are required to reach a
precision of $\epsilon$ is given by the positive integer $M$ and the “taking
enough measurements” is taken into account when $N\rightarrow\infty$ in Eq.
3.7.
Knowing more and more precise time measurements is great, but what does it get
us? How can we be sure that there will truly be a position measurement for us
to take if we zoom in on the temporal axis with smaller and smaller time
increments? To the best of my knowledge, in physics, we assume that this is
always true for moving objects. But this would imply that as we more and more
precise time measurements, we could get more and more precise position
measurements. Or at least, if the position at one point in time is known, then
the position as $t+\Delta t$ should be nearby that at $t$. Otherwise, the
object would somehow jump/teleport to another location randomly. And that
would be weird/not anything we have ever observed, so in physics we assume
that we can always zoom in infinitely far for all spatial and temporal
variables ($N\rightarrow\infty$).
But how would we represent this mathematically? For this case, we will assume
that we have taken infinitely many temporal measurements so that we can zoom
in arbitrarily around some special point in time, let’s call it $t^{\prime}$.
If we have zoomed in to some arbitrary level of precision, now denoted by
$\delta t$777The lowercase delta is because our precision is tiny, so I
figured a little variable would be appropriate., then we can probe any time
values within what’s called an open ball of radius $\delta t$. The ball itself
is defined as the set of all $t$-values that are within the radius $\delta t$
of the central point $t^{\prime}$, and for those who are interested it is
written as
$\displaystyle\mathcal{B}(t^{\prime},\delta
t)=\\{t\in\mathds{R}:|t-t^{\prime}|<\delta t\\}.$
In a single dimension though, this ball is just the open interval
$t\in(t^{\prime}-\delta t,t^{\prime}+\delta t)$ along the real line. It is a
circular disk in two dimensions, and actually is a ball (sphere) in three
dimensions. In four and higher dimensions, it is harder to visualize888Yes,
this is a dare to prove me wrong..
Anyway, we want to be sure that an object doesn’t suddenly teleport elsewhere,
so we need the position of an object at time $t$ to be nearby the position at
time $t^{\prime}$. Therefore, if we can find a position ball whose radius
$|\delta\vec{r}|$ is somehow the result of the $\delta t$ precision we chose,
then we can guarantee that the position of the object at time $t$ will always
be within the vicinity of the position at time $t^{\prime}$. And most
importantly, means
$\displaystyle|\vec{r}(t)-\vec{r}(t^{\prime})|<|\delta\vec{r}|,$ (3.8)
where $|\delta\vec{r}|$ is a finite number, of again, arbitrary precision.
This whole idea that zooming in far on the independent variable consequently
leads to us freely zooming on the dependent variable is what is known as
continuity. To be more mathematically precise:
> A function $f:\mathds{R}\rightarrow\mathds{R}$ is said to be continuous at
> $x^{\prime}$ if there exists a $\delta>0$ for every $\epsilon>0$ such that
> for every $x$ where $|x-x^{\prime}|<\delta$ then implies
> $|f(x)-f(x^{\prime})|<\epsilon$.
When the above statement holds, we call the output $f(x^{\prime})$ the limit
of $f(x)$ as $x\rightarrow x^{\prime}$ which is written as
$\displaystyle\lim_{x\rightarrow x^{\prime}}f(x)=f(x^{\prime}).$ (3.9)
Sometimes, the limit is instead defined as $F$ so we can study functions where
$f(x^{\prime})$ need not be defined.
The definition above is what you may have seen/heard of before — it is the
$\epsilon$-$\delta$ definition of continuity — but it is strictly in one-
dimension, whereas actually everything else up until this point was in an
arbitrarily high number of spatial dimensions; so maybe we were taking about
an object only moving along a single axis (one-dimensional), or on a curve on
a surface (two-dimensional), or even in a trajectory through space (three-
dimensional). Here we condensed back to one-dimension with the
$\epsilon$-$\delta$ definition of continuity, because the mathematical
precision in higher dimensions is actually a little too nuanced to jump in
with immediately. Actually, to be completely honest, I don’t think the
definition, as it is written above, has ever come up in a physics class that
I’ve taken. However, the idea of being able to zoom in as far as we like does
come up constantly; that is why I spent so much time developing the intuition
behind the $\epsilon$-$\delta$ definition of continuity. Nevertheless, I do
want to unpack that definition a little, because it will be extremely relevant
for things like derivatives later on.
First things first, the statement $f:\mathds{R}\rightarrow\mathds{R}$
translates to “function $f$ that has real number inputs (first $\mathds{R}$)
and then outputs real numbers (second $\mathds{R}$).” This means we are only
dealing with the good-ol’ one-dimensional functions that you learned in high
school. The next thing that is listed is that this definition of continuity
only applies for functions at a single point $x^{\prime}$ in the domain. It
says nothing about the continuity of $f$ at nearby points. The actual
requirement for continuity is that there has to be an $\delta>0$, a.k.a. an
arbitrarily small number, to zoom in on $x$ nearby $x^{\prime}$ when we zoom
in on $f(x^{\prime})$ itself to a chosen precision of $\epsilon$. Therefore,
this definition posits that the existence of the $\epsilon$ precision
guarantees the existence of the $\delta$ precision in the independent
variable. This means that continuity allows us to continue taking smaller and
smaller independent-variable increments to measure smaller and smaller
dependent-variable increments, just like our intuition with an object’s
position informed us. So this definition, although I agree that it is pretty
intangible the first time you see it, incorporates all of what we’ve discussed
in the section so far, but it does it in a much more concise manner.
Example 3.2 shows how we might use the definition of continuity to show simple
functions are continuous. The truth is that it usually takes a different
algebra tricks to establish the continuity of individual functions, and
oftentimes these tricks are NOT obvious. If you ever take a course in
mathematical analysis, you will see it. Therefore, it usually helps to
establish the continuity of general classes of functions, like that of sums,
products, or compositions, because we can then argue that whatever messy
function we have is truly just a combination of continuous functions. Thus, I
am going to prove a few theorems to you regarding the continuity of sums,
products, and compositions.
Disclaimer: the proofs will be pretty abstract, and so if it is too hard to
follow at first, that’s okay. The fundamentally important continuity rules in
physics will be the numbered limit equations that follow each proof, but I
wouldn’t be able to live with myself if I just told you what they were without
justification. I mean, this is supposed to be science after all, right?
Definitions are great, but their utility really comes in applying them. In
this example, I will prove to you that constant and linear functions are
continuous. We first start with a constant function. Let’s assume our constant
function looks like $f(x)=A$ for all $x\in\mathds{R}$. If you don’t like the
$A$ here, replace it with the number $\pi$ and then, at the end, replace all
the $\pi$s with $A$s. Okay, so the proof goes something like this: Consider
the point $x^{\prime}\in\mathds{R}$ and the open ball
$\mathcal{B}(f(x^{\prime}),\epsilon)$ centered at $f(x^{\prime})$ with radius
$\epsilon>0$. More visually, this means we have zoomed in on the dependent
variable $f$ to a precision of $\epsilon$. We want to show that there exists a
$\delta>0$ such that $|x-x^{\prime}|<\delta$, no matter what $\epsilon$
precision we choose. Now consider the absolute difference
$|f(x)-f(x^{\prime})|$, where I will substitute in the values of $x$ and
$x^{\prime}$ into $f(x)=A$. $\displaystyle|f(x)-f(x^{\prime})|=|A-A|=0$ But by
definition, $\epsilon>0$. If we let $\delta=\epsilon>0$, then for every $x$
such that $|x-x^{\prime}|<\delta$, we have
$\displaystyle|f(x)-f(x^{\prime})|=|A-A|=0<\delta=\epsilon$ This shows that
$f$ is continuous at $x^{\prime}$. However, since this argument holds for all
$x^{\prime}\in\mathds{R}$, we conclude that constants functions are continuous
over all of the real numbers. For the next proof, we look at the linear
function $f(x)=bx$ for all $x\in\mathds{R}$, where $b\neq 0$ is a constant
coefficient. The proof is similar in set up to the one above, but its
execution will probably look very different to you the first time you see it.
Remember that the key is to show the existence of any $\delta>0$ for every
conceivable $\epsilon>0$. Consider the point $x^{\prime}\in\mathds{R}$ and the
open ball $\mathcal{B}(f(x^{\prime}),\epsilon)$ centered at $f(x^{\prime})$
with radius $\epsilon>0$. We will argue that for every $\epsilon>0$, we can
find a $\delta=\epsilon/|b|>0$ such that when $|x-x^{\prime}|<\delta$, we have
$|f(x)-f(x^{\prime})|<\epsilon$. $\displaystyle|f(x)-f(x^{\prime})|$
$\displaystyle=|bx-bx^{\prime}|$ $\displaystyle=|b(x-x^{\prime})|$
$\displaystyle=|b|\,\underbrace{|x-x^{\prime}|}_{<\delta}$
$\displaystyle<|b|\,\frac{\epsilon}{|b|}$ $\displaystyle=\epsilon$ Thus,
$|f(x)-f(x^{\prime})|<\epsilon$ when $|x-x^{\prime}|<\delta$, for all
$\delta>0$; thereby completing the proof that $f(x)=bx$ is continuous at
$x^{\prime}$. Further, since $x^{\prime}$ is arbitrary, again, this argument
holds for all of the real numbers. We conclude that linear functions are
continuous over all of the real numbers.
Before we proceed to this point, we need one very important theorem concerning
the absolute function: the so-called triangle inequality. It states simply
that for any $x,y\in\mathds{R}$, we have
$\displaystyle|x+y|\leq|x|+|y|.$ (3.10)
A proof of this statement is straightforward, and follows from the fact that
$x^{2}=(|x|)^{2}$ for all real numbers. It goes as something like this,
$\displaystyle(|x|+|y|)^{2}$ $\displaystyle=(|x|)^{2}+(|y|)^{2}+2|x|\,|y|$
$\displaystyle\geq(|x|)^{2}+(|y|)^{2}+2xy,\textrm{ since either $x$ or $y$
could possibly be negative}$ $\displaystyle=x^{2}+y^{2}+2xy$
$\displaystyle=(x+y)^{2}$ $\displaystyle=(|x+y|)^{2}.$
Thus, $(|x+y|)^{2}\leq(|x|+|y|)^{2}$. By taking the square root of both sides,
we have the triangle inequality; a theorem that gets its name because it
essentially says that if a triangle is made up of three sides, one side cannot
have a magnitude greater than the sum of the other two. This inequality will
help us establish that sums of continuous functions are continuous.
Now for continuity in summation. To prove this, consider the function
$h(x)=f(x)+g(x)$, where $f(x)$ and $g(x)$ are both continuous at $x^{\prime}$.
Since both $f$ and $g$ are continuous at $x^{\prime}$, then there must exist
$\delta$-precisions for both functions for every $\epsilon$-precision, namely
$\delta_{f}$ and $\delta_{g}$, respectively, such that
$\displaystyle|x-x^{\prime}|<\delta_{f}\textrm{ implies }|f(x)-f(x^{\prime})|$
$\displaystyle<\frac{\epsilon}{2},$
$\displaystyle|x-x^{\prime}|<\delta_{g}\textrm{ implies }|g(x)-g(x^{\prime})|$
$\displaystyle<\frac{\epsilon}{2},$
Then, by substitution into $h$, we have
$\displaystyle|h(x)-h(x^{\prime})|$
$\displaystyle=|f(x)+g(x)-f(x^{\prime})-g(x^{\prime})|$
$\displaystyle=|f(x)-f(x^{\prime})+g(x)-g(x^{\prime})|$
$\displaystyle\leq|f(x)-f(x^{\prime})|+|g(x)-g(x^{\prime})|,\textrm{ thanks to
the triangle inequality}$
$\displaystyle<\frac{\epsilon}{2}+\frac{\epsilon}{2},\textrm{ whenever
}|x-x^{\prime}|<\min\\{\delta_{f},\;\delta_{g}\\},$ $\displaystyle=\epsilon$
Thus, when we define $\delta=\min\\{\delta_{f},\;\delta_{g}\\}$, where the
$\min$ function outputs the smaller value of $\delta_{f}$ and $\delta_{g}$,
then we have that $|x-x^{\prime}|<\delta$ implies
$|h(x)-h(x^{\prime})|<\epsilon$ for every $\epsilon>0$. This means that when
$f$ and $g$ are continuous at $x^{\prime}$, so is the sum of the two. I want
to emphasize that this conclusion implies the following limit rule:
$\displaystyle\lim_{x\rightarrow
x^{\prime}}\left[f(x)+g(x)\right]=\left[\lim_{x\rightarrow
x^{\prime}}f(x)\right]+\left[\lim_{x\rightarrow x^{\prime}}g(x)\right].$
(3.11)
Hence, the limit of the sum is the sum of the limits. Furthermore, if (and
only if) both $f$ and $g$ are continuous over all of the real numbers, then so
is their sum $h$.
We next want to prove that the product of two functions that are continuous at
$x^{\prime}$ is also continuous at that point when two functions are
defined999The defined condition guarantees that I can write something like
$f(x^{\prime})$ and not have is accidentally blow up to infinity. at
$x^{\prime}$. Now let $h(x)=f(x)g(x)$, and let the definitions of $\delta_{f}$
and $\delta_{g}$ be modified a little from what is given above as
$\displaystyle|x-x^{\prime}|<\delta_{f}\textrm{ implies }|f(x)-f(x^{\prime})|$
$\displaystyle<\frac{\epsilon}{2(1+|g(x^{\prime})|)},$
$\displaystyle|x-x^{\prime}|<\delta_{g}\textrm{ implies }|g(x)-g(x^{\prime})|$
$\displaystyle<\frac{\epsilon}{2(1+|f(x^{\prime})|)}.$
These odd looking definitions will come in handy later on so that we will find
$|h(x)-h(x^{\prime})|<\epsilon$ and not some function of $\epsilon$. Then we
have by substitution,
$\displaystyle|h(x)-h(x^{\prime})|$
$\displaystyle=|f(x)g(x)-f(x^{\prime})g(x^{\prime})|,$
$\displaystyle=|f(x)g(x)+0-f(x^{\prime})g(x^{\prime})|,$
$\displaystyle=|f(x)g(x)-f(x^{\prime})g(x)+f(x^{\prime})g(x)-f(x^{\prime})g(x^{\prime})|,$
$\displaystyle=\left|\left[f(x)g(x)-f(x^{\prime})g(x)\right]+\left[f(x^{\prime})g(x)-f(x^{\prime})g(x^{\prime})\right]\right|,$
$\displaystyle\leq|f(x)-f(x^{\prime})|\,|g(x)|+|f(x^{\prime})|\,|g(x)-g(x^{\prime})|,\textrm{
via the triangle inequality}$
$\displaystyle<\frac{\epsilon}{2(1+|g(x^{\prime})|)}\,|g(x)|+|f(x^{\prime})|\,\frac{\epsilon}{2(1+|f(x^{\prime})|)},\textrm{
when }|x-x^{\prime}|<\min\\{\delta_{f},\;\delta_{g}\\}.$
Before continuing, we must figure out a way to deal with the $|g(x)|$ because
we do not know the specific value of $g$ at $x$ — we only know that
$g(x^{\prime})$ is defined and that $g$ is continuous at $x^{\prime}$. But
since $g$ is continuous at $x^{\prime}$, we are free to zoom in as far as we
like on $g(x^{\prime})$ and we are guaranteed to be able to find an
$x$-interval that corresponds to our $g$-precision. Thus, what we will do is
choose to limit our $g$-precision to be less than a set number, for example,
$1$. Then, by the continuity of $g$, there exists a $\delta_{1}$ such that
$\displaystyle|x-x^{\prime}|<\delta_{1}\textrm{ implies }|g(x)-g(x^{\prime})|$
$\displaystyle<1.$
Then we can say using the triangle inequality that
$\displaystyle|g(x)|=|g(x)+0|=|g(x)-g(x^{\prime})+g(x^{\prime})|\leq|g(x)-g(x^{\prime})|+|g(x^{\prime})|<1+|g(x^{\prime})|,$
where the last inequality holds for every $x$ such that
$|x-x^{\prime}|<\delta_{1}$. Okay, so now that we have these conditions in
place, and if we remember that $|f(x^{\prime})|\leq 1+|f(x^{\prime})|$, we can
define our $\delta$-precision such that
$\delta=\min\\{\delta_{f},\;\delta_{g},\;\delta_{1}\\}$. Then for every $x$
such that $|x-x^{\prime}|<\delta$, we have
$\displaystyle|h(x)-h(x^{\prime})|$
$\displaystyle<\frac{\epsilon}{2(1+|g(x^{\prime})|)}\,|g(x)|+|f(x^{\prime})|\,\frac{\epsilon}{2(1+|f(x^{\prime})|)},$
$\displaystyle\leq\frac{\epsilon\,(1+|g(x^{\prime})|)}{2(1+|g(x^{\prime})|)}+\frac{\epsilon\,(1+|f(x^{\prime})|)}{2(1+|f(x^{\prime})|)},$
$\displaystyle=\frac{\epsilon}{2}+\frac{\epsilon}{2}$
$\displaystyle=\epsilon.$
Since we were able to find an $\delta>0$ for every $\epsilon>0$ such
that$|x-x^{\prime}|<\delta$ implies/leads to $|h(x)-h(x^{\prime})|<\epsilon$,
then we can conclude that $h(x)=f(x)g(x)$ is continuous at $x^{\prime}$. This
time, the corresponding limit equation would look like
$\displaystyle\lim_{x\rightarrow
x^{\prime}}\left[f(x)g(x)\right]=\left[\lim_{x\rightarrow
x^{\prime}}f(x)\right]\,\left[\lim_{x\rightarrow x^{\prime}}g(x)\right],$
(3.12)
or the limit of a product is the product of limits. Just as was the case for
the the sum of continuous functions, if both $f$ and $g$ are continuous over
all of the reals, so is their product. However, if only one function is
continuous everywhere, then the product is NOT continuous everywhere — it,
too, is limited101010Nice pun, right? to only be continuous for the values
$x^{\prime}$ where both of its factors are.
The last continuity rule deals with compositions of functions. As a brief
review compositions are sometimes written as $g(f(x))$ or $g\circ f(x)$. They
are functions that look something like this: if $g(x)=x^{2}$ and $f(x)=4\sin
x$, then $g(f(x))=(4\sin x)^{2}$. Thus, the notation of $g(f(x))$ really just
means put whatever the output of $f$ is into $g$ as the input. Although this
class of functions may seem like more of a special case compared to sums or
products (it did to me the first time I learned of it), compositions of
functions appear everywhere in both calculus and physics. Thus, knowing how
they behave is imperative.
Our goal in particular is to show that compositions of continuous functions
are also continuous. To do this, we then need to show that a function
$h(x)=g(f(x))$ is continuous at the point $x^{\prime}$, given that $f$ is
continuous at $x^{\prime}$ and $g$ is continuous at $f(x^{\prime})$. To make
this proof a little clearer, define
$\displaystyle u=f(x),\;u^{\prime}=f(x^{\prime}).$
By the continuity of $g$ at $u^{\prime}$, we know that for every $\epsilon>0$,
there exists a $\delta_{u}>0$ such that
$\displaystyle|u-u^{\prime}|=|f(x)-f(x^{\prime})|<\delta_{u}\textrm{ implies
}|g(u)-g(u^{\prime})|<\epsilon.$
Then by the continuity of $f$ at $x^{\prime}$, we know that for every
$\epsilon_{f}$-precision, there exists a $\delta>0$ such that
$\displaystyle|x-x^{\prime}|<\delta\textrm{ implies
}|f(x)-f(x^{\prime})|=|u-u^{\prime}|<\epsilon_{f}.$
Therefore, since we want something that looks like
$\displaystyle|x-x^{\prime}|<\delta\textrm{ implies
}|h(x)-h(x^{\prime})|=|g(f(x))-g(f(x^{\prime}))|<\epsilon,$
then we choose $\epsilon_{f}=\delta_{u}$ such that our final statement reads
as
$\displaystyle|x-x^{\prime}|<\delta\textrm{ implies
}|f(x)-f(x^{\prime})|=|u-u^{\prime}|<\delta_{u}\textrm{ implies
}|g(u)-g(u^{\prime})|=|h(x)-h(x^{\prime})|<\epsilon.$
for every $\epsilon>0$. As long as both $f$ and $g$ are continuous everywhere,
then the composite function will be continuous everywhere. If there are any
points where $g$ is not continuous, then the composite function will not be
continuous at those points111111There is a huge caveat here: sometimes
functions that are not specifically defined at a point can still look
continuous at that point. These types of function discontinuities are called
removable discontinuities or holes.. The corresponding limit equation for this
composition property can be written as
$\displaystyle\lim_{x\rightarrow
x^{\prime}}\left[g(f(x))\right]=g\left(\lim_{x\rightarrow
x^{\prime}}f(x)\right),$ (3.13)
which can be summarized as: the limit of the composition is the composition of
the limits.
Again, these continuity rules (Eqs. 3.11, 3.12, and 3.13) are incredibly
useful tools to determine if any function is continuous, or at least
continuous at a specified point.
This problem is designed to show you how to use the continuity rules defined
by Eqs. 3.11, 3.12, and 3.13, given that any constant functions or linear
functions are continuous from Example 3.2. As a quick example of how to use
these continuity rules, by 3.11, it follows that the line $f(x)=bx+A$, where
$b$ and $A$ are constants, is continuous for every single real number input,
$x$. This is because both $bx$ and $A$ are continuous everywhere from Example
3.2, therefore their sum is also continuous everywhere. (a) For the first part
of the problem, use Eq. 3.12 to argue that $f(x)=bx^{2}$ is continuous
everywhere. (b) Then, using Eq. 3.11, argue that any parabola
$f(x)=ax^{2}+bx+c$ is continuous everywhere. (c) Now argue that any cubic
polynomial $P_{3}(x)=a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}$ is continuous
everywhere, where every element of the set of coefficients
$\\{a_{n}\\}=\\{a_{0},a_{1},a_{2},a_{3}\\}$ is constant. (d) Finally, use the
same argument to show that any $n^{\mathrm{th}}$ order polynomial, denoted by
$P_{n}$, and written as
$P_{n}(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\dots+a_{2}x^{2}+a_{1}x+a_{0}$ and
accompanied by the set of constant coefficients
$\\{a_{n}\\}=\\{a_{0},a_{1},a_{2}\dots,a_{n-1},a_{n}\\}$ is continuous
everywhere.
Although the ability to potentially take an infinite amount of measurements is
great, it is honestly more helpful to know, for example, how an object is
moving because we can then predict where it will be in the next instant in
time. In physics, equations like these are called equations of motion, and the
|
# Stick-slip and convergence of feedback-controlled systems with Coulomb
friction
Michael Ruderman<EMAIL_ADDRESS>University of Agder, 4604-Norway
###### Abstract
An analysis of stick-slip behavior and trajectory convergence in feedback-
controlled motion systems with discontinuous Coulomb friction is provided. A
closed-form parameter-dependent stiction region, around an invariant
equilibrium set, is proved to be always reachable and globally attractive. It
is shown that only asymptotic convergence can be achieved, with at least one
but mostly an infinite number of consecutive stick-slip cycles, independent of
the initial conditions. Theoretical developments are supported by a number of
numerical results with dedicated convergence examples.
###### keywords:
Coulomb friction , limit cycles , PID control , convergence analysis , sliding
mode , discontinuities
## 1 Introduction
Feedback-controlled motion systems are mostly subject to nonlinear friction,
and the direction-dependent Coulomb friction force plays a crucial role owing
to a (theoretical) discontinuity at velocity zero-crossings. Although the more
complex dynamic friction laws (see, e.g., [1, 2, 3] and references therein)
allow frictional discontinuity to be bypassed during analysis, the basic
Coulomb friction phenomenon continues to present the same challenges in terms
of controller convergence, especially in the presence of an integral control
action. Associated stick-slip behavior and so-called frictional limit cycles
were formerly addressed in [4]. An algebraic prediction of stick-slip, with a
large set of parametric equalities, was compared to the describing function
method, while the Coulomb plus static friction law was assumed for avoiding
discontinuity at the velocity zero-crossing. An explicit solution for
friction-generated limit cycles has also been proposed in [5], necessitating
static friction approximation (to avoid discontinuity) and also requiring the
stiction friction which is larger than the Coulomb friction level. Despite
including explicit analysis of state trajectories for both sticking and
slipping phases, no straightforward conclusions about the appearance and
convergence of stick-slip behavior have been reported. The appearance of
friction-induced (so-called hunting) limit cycles has been briefly addressed
in [6], for the assumed LuGre [7] and so-called switch [8] friction models.
Note that a former analysis of stick-slip behavior and associated friction-
induced limit cycles can be found in [9]. An explanation of how a
proportional-feedback-controlled motion with Coulomb friction comes to
sticking was subsequently shown in [10] by using the invariance principle.
Stick-slip behavior, as an observable phenomenon known in control engineering
practice, was highlighted in [1], and several other control studies have since
attempted to analyze and compensate such processes. For instance, a related
analysis of under-compensation and over-compensation of static friction was
reported in [11]. Issues associated with a slow (creeping-like) convergence of
the feedback-controlled motion in the presence of Coulomb friction have been
addressed and experimentally demonstrated in [12]. More recently, the
convergence problems (of PID feedback control) have been well demonstrated,
with accurate experiments reported, in [13] while attempting to reduce
settling errors by the reset integral control [14]. The related analysis has
been reported before in [15]. Despite a number of experimental observations
and elaborated studies reported in the literature, it appears that no general
consensus has been established in relation to friction-induced stick-slip
cycles in feedback-controlled systems with Coulomb friction. In particular,
questions arise over when and under which conditions the stick-slip cycles
occur, and how a PID-controlled motion will converge to zero equilibrium in
the presence of Coulomb friction (with discontinuity). Note that the problem
of slow convergence in vicinity to a set reference position is of particular
relevance for advanced motion control systems, see, e.g., [16], but in the PID
design and tuning, see, e.g., [17], the associated issues are not widely
accepted and have yet to be formalized, that despite huge demand from
precision engineering control tasks. This gap should not come as a complete
surprise, however, given the nontrivial friction microdynamics, visible in
experimental studies [18, 19, 20], and the uncertain and time-varying friction
behavior, see, e.g., [21].
This paper is intended to contribute to the analysis of feedback-controlled
system convergence with Coulomb friction and, thus, to understanding of stick-
slip cycles occurring in servo mechanisms. In order to keep the analysis as
general as possible and to clarify the principal phenomenon, we have assumed
the classical Coulomb friction law with discontinuity. This led us
(unavoidably) to variable structure system dynamics, distinguishing between
the modes of motion sticking and slipping. At the same time, we show that all
state trajectories always remain continuous and almost always differentiable
(except finite switching between both modes). We provide theorems and identify
the conditions to demonstrate the sticking region around zero equilibrium
reachable and globally attractive. The main contribution of analysis is
further reinforced by several illustrative numerical examples.
### 1.1 Problem statement
Throughout the paper, we will deal with the feedback-controlled systems
described by
$\ddot{x}(t)+K_{d}\dot{x}(t)+K_{p}x(t)+K_{i}\int x(t)dt+F(t)=0,$ (1)
where the derivative, proportional and integral feedback gains are $K_{d}$,
$K_{p}$ and $K_{i}$, respectively. The nonlinear friction (that with
discontinuity) is denoted by $F$, and the set-value control problem is reduced
to the convergence problem for a non-zero initial condition, i.e., $x(0)\neq
0$. Furthermore, we use the following simplifications of the system plant
without loss of generality: the relative motion of an inertial body with unity
mass is considered in the generalized $(x,\dot{x})$ coordinates. The inherent
system damping (including linear viscous friction) and stiffness (of restoring
spring elements) are incorporated (if applicable) into $K_{d}>0$ and
$K_{p}>0$, respectively. There are no actuator constraints, so feedback for
the integral output error is directly applicable via the gain factor
$K_{i}>0$.
The control problem (1) has long been associated with issues of slow and/or
cyclic convergence of $x(t)$ in the vicinity of steady-state for the set
reference value. This (sometimes called hunting behavior or even hunting limit
cycles) has been addressed in analysis and also observed in several controlled
positioning experiments, e.g., [1, 4, 5, 6, 12, 13]. These phenomena seem to
be associated with integral control action and nonlinear (Coulomb-type)
friction within a vanishing region around equilibria, where the potential
field of proportional feedback weakens and cannot provide $x(t)\rightarrow 0$
at certain (application-required) times $t<\mathrm{const}$. The hunting
behavior is directly cognate with the stick-slip, where a smooth (continuous)
motion alternates with a sticking phase of zero or slowly creeping
displacement. Stick-slip appearance, parametric conditions and convergence in
semi-stable limit cycles are the focus of our study, while we assume the
Coulomb friction force with discontinuity.
## 2 Stiction due to discontinuous Coulomb friction
In this Section, we will analyze the stick-slip behavior of the (1) system,
for which the classical Coulomb friction with discontinuity is represented by
$F(\dot{x})=F_{c}\,\mathrm{sign}(\dot{x})$. Here, the Coulomb friction
coefficient is $F_{c}>0$, and the sign operator is defined by
$\mathrm{sign}(z)=\left\\{\begin{array}[]{ll}1,&\;z>0,\\\
\left[1,1\right],&\;z=0,\\\ -1,&\;z<0.\\\ \end{array}\right.$ (2)
Note that (2) constitutes an ideal relay with instantaneous switching upon
change of the input sign. We also note that for a zero-displacement rate, the
friction equation becomes an inclusion $F(0)\in[-F_{c},F_{c}]$ in the Filippov
sense [22] when seeking the corresponding solution.
We will consider the feedback-controlled system in a minimal state-space
representation, as follows:
$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle Ax+Bu,$ (3)
$\displaystyle y$ $\displaystyle=$ $\displaystyle Cx,$ (4) $\displaystyle u$
$\displaystyle=$ $\displaystyle-\mathrm{sign}(y).$ (5)
Note that in this way, we also approach the system notation provided in [23]
for analysis of relay feedback systems (RFSs). Introducing the state vector
$x=(x_{1},x_{2},x_{3})^{T}\in\mathbb{R}^{3}$ of the integral, output and
derivative errors, (1) can be rewritten as (3)-(5), with the system matrix
$A=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\ -K_{i}&-K_{p}&-K_{d}\\\
\end{array}\right),$ (6)
and input and output distribution vectors
$B=\left(\begin{array}[]{c}0\\\ 0\\\ F_{c}\\\ \end{array}\right),\quad
C^{T}=\left(\begin{array}[]{c}0\\\ 0\\\ 1\\\ \end{array}\right)$ (7)
correspondingly.
### 2.1 Without integral feedback
Firstly, we consider the (3)-(7) system without an integral feedback action,
meaning $K_{i}=0$. In this case, the phase-plane
$(x_{2},x_{3})\in\mathbb{R}^{2}$ is divided into two regions
$P^{+}=\\{x\in\mathbb{R}^{2}:x_{3}>0\\},\quad
P^{-}=\\{x\in\mathbb{R}^{2}:x_{3}<0\\}$ (8)
by the discontinuity manifold $S=\\{x\in\mathbb{R}^{2}:x_{3}=0\\}$. It can be
seen that in the discontinuity manifold $S$, the vector fields of the state
value $x_{s}$ are given by
$\displaystyle f^{+}(x_{s})$ $\displaystyle=\overset{x\in
P^{+}}{\underset{x\rightarrow
x_{s}}{\lim}}(Ax+Bu)=\left(\begin{array}[]{c}0\\\
-K_{p}x_{2}-F_{c}\end{array}\right),$ (11) $\displaystyle f^{-}(x_{s})$
$\displaystyle=\overset{x\in P^{-}}{\underset{x\rightarrow
x_{s}}{\lim}}(Ax+Bu)=\left(\begin{array}[]{c}0\\\
-K_{p}x_{2}+F_{c}\end{array}\right)$ (14)
and are pointing in opposite directions within $|x_{2}|\leq F_{c}K_{p}^{-1}$.
In contrast, outside of this region (denoted by $S_{0}$ in Figure 1), both
vector fields are pointing in the same direction, towards $P^{+}$ for
$x_{2}<-F_{c}K_{p}^{-1}$ and towards $P^{-}$ for $x_{2}>F_{c}K_{p}^{-1}$.
Figure 1: Phase portrait of $(x_{2},x_{3})$-trajectories of the (3)-(7) system
without integral feedback, attracted to $S_{0}$ from various initial values
Since both vector fields are normal to the manifold $S$, neither smooth motion
nor sliding mode can occur for the $(x_{2},x_{3})\in S_{0}$ trajectories,
which means that any trajectory reaching $S_{0}$ will remain there
$\forall\>t\rightarrow\infty$. Thus, $S_{0}$ constitutes the largest invariant
set of equilibrium points for (3)-(7) without integral control action. Note
that this has also been shown in [10] and is well known when a relative motion
with Coulomb friction is controlled by the proportional-derivative (PD)
feedback only. In this case, the set value error can be reduced by increasing
$K_{p}$ but cannot be driven to zero as long as $F_{c}\neq 0$. The phase
portraits of the trajectories converging to $S_{0}$ are exemplary shown in
Figure 1.
### 2.2 With integral feedback
When allowing for $K_{i}\neq 0$, it is intuitively apparent that having
reached point $x(t_{s})\in S_{0}$ at $t=t_{s}$, the trajectory cannot remain
there for all times $t_{s}<t<\infty$. While for the motion states
$x_{3}(t_{s})=0$ and $x_{2}(t_{s})=\mathrm{const}\neq 0$, the integral control
effort $K_{i}x_{2}(t_{s})\int^{t}_{t_{s}}dt$ grows continuously and, at some
finite time ($t_{c}>t_{s}$), should lead to the breakaway [24] and a new onset
of continuous motion. This alternating phase, upon system sticking, is often
referred to as _slipping_ , cf., e.g., [5], so that a stick-slip motion [1, 3]
appears, also in the form of limit cycles. In order to analyze the friction-
induced limit cycles, sometimes denoted as hunting-limit cycles, cf. [6], we
firstly need to look into the system dynamics during stiction, i.e., for
$t_{s}<t<t_{c}$. Recall that in this phase, the system (3)-(7) yields as
continuously switching, owing to $x_{3}=0$ and to the discontinuous relay
nonlinearity in the feedback loop. The developments given below are motivated
by analysis of the existence of fast switches, provided in [23] for RFS, while
the obtained results rely on the sliding-mode principles, see, e.g., [25, 26].
Consider the switching variable (or, more generally, surface) $S=Cx(t)=0$, for
which the sliding mode should occur on the manifold $S$. This requires that
the existence and reachability condition, cf. [25],
$\dot{S}S\leq-\eta|S|$ (15)
is fulfilled, where $\eta$ is a small positive constant.
###### Theorem 1.
Given is the control system (3)-(7) with Coulomb friction. The system is
sticking at $x_{3}=0$ iff
$|K_{i}x_{1}|+|K_{p}x_{2}|\leq F_{c}.$ (16)
###### Proof.
The system remains sticking as long as it is in the sliding mode for which
(15) is fulfilled. The sliding-mode condition (15) can be rewritten as
$\dot{S}\,\mathrm{sign}(S)\leq-\eta,$ (17)
while the time derivative of the sliding surface is
$\dot{S}=(CAx\pm CB)=\bigl{(}CAx-\mathrm{sign}(S)CB\bigr{)},$ (18)
depending on the sign of $Cx$. Substituting (18) into (17) results in
$\displaystyle CAx$ $\displaystyle\leq$ $\displaystyle
CB-\eta\qquad\hbox{for}\quad\mathrm{sign}(S)>0,$ (19) $\displaystyle-CAx$
$\displaystyle\leq$ $\displaystyle
CB-\eta\qquad\hbox{for}\quad\mathrm{sign}(S)<0.$ (20)
Since $CB,\eta>0$, the inequalities (19) and (20) can be summarized in
$|CAx|\leq CB-\eta.$ (21)
Evaluating (21) with $x_{3}=0$ and $0\neq\eta\rightarrow 0^{+}$ results in
(16) and completes the proof. ∎
###### Remark 1.
The condition obtained using the Theorem 1 is equivalent to the set of
attraction $\\{x\in S:|CAx|<|CB|\\}$ for $CB>0$ demonstrated in [23, Section
4].
Now, we are interested in the state dynamics during stiction, which means
within the sliding mode. Since staying in the sliding mode (corresponding to
$S\equiv 0$ on the switching surface) requires
$\dot{S}=C\dot{x}=CA\dot{x}+CBu=0\qquad\hbox{for}\quad t_{s}<t<t_{c},$ (22)
one obtains the so-called equivalent control as
$u_{e}=-(CB)^{-1}CAx.$ (23)
Recall that an equivalent control, [26], is the linear one (i.e., with no
relay action) which is required to maintain the system in an ideal sliding
mode without fast-switching. Consequently, substituting (23) into (3) results
in the equivalent system dynamics
$\dot{x}_{e}=\bigl{[}I-B(CB)^{-1}C\bigr{]}Ax=OAx,$ (24)
which governs the state trajectories as long as the system remains in the
sliding mode. $O$ is the so-called projection operator of the original system
dynamics, satisfying the properties $CO=0$ and $OB=0$. Evaluating (24) with
(6) and (7) yields the system dynamics during stiction as
$\left(\begin{array}[]{c}\dot{x}_{1}\\\ \dot{x}_{2}\\\ \dot{x}_{3}\\\
\end{array}\right)=\left(\begin{array}[]{ccc}0&1&0\\\ 0&0&1\\\ 0&0&0\\\
\end{array}\right)\left(\begin{array}[]{c}x_{1}(t_{s})\\\ x_{2}(t_{s})\\\ 0\\\
\end{array}\right).$ (25)
It can be seen that neither relative displacement nor its rate will change
when the system is sticking, although the integral error grows according to
$x_{1}(t)=x_{1}(t_{s})+x_{2}(t_{s})\int\limits^{t_{c}}_{t_{s}}dt.$ (26)
It can be noted that if $K_{i}=0$ then the condition (16), correspondingly the
inequality $|CAx|\leq|CB|$, reduces to $|x_{2}|\leq F_{c}K_{p}^{-1}$, while
the sliding mode (25) reduces to the zero dynamics of the system in stiction
(the results we obtained in Section 2.1).
### 2.3 Region of attraction
The given Theorem 1 provides the necessary (and sufficient) condition as the
(3)-(7) system remains sticking. Yet it is also necessary to demonstrate
global attraction of state trajectories to the stiction region. Recall that
the latter corresponds to the subset
$S_{0}=\\{x\in\mathbb{R}^{3}\,:\,x_{3}=0,\,|K_{i}x_{1}|+|K_{p}x_{2}|\leq
F_{c}\\}$ (27)
where the sliding mode occurs (cf. (16) and (25)).
Firstly, we will explore the persistence of the sliding mode, meaning we will
prove whether the system can stay incessantly inside of
$S_{0}^{s}=\\{S_{0}\,:\,x_{2}\neq 0\\}$, i.e., in all times
$t_{s}<t\rightarrow\infty$. By making $(x_{1},x_{2})$-projection of
$x\in\mathbb{R}^{3}$, one can show that (16) results in a rhombus, as
schematically illustrated in Figure 2.
Figure 2: Rhombus-shape, in $(x_{1},x_{2})$-projection, of the $S_{0}$-region
of attraction, with vector-field during the stiction mode and example of an
entering and leaving trajectory at $t_{s}$ and $t_{c}$, respectively
The indicated vector field is unambiguous due to the integral control action
(cf. the sliding-mode dynamics (25)). This means that after reaching
$S_{0}^{s}$ at $t_{s}$, any trajectory will leave it at $t_{c}$ having hit the
boundary of $S_{0}$. Denoting the point of reaching $S_{0}$ by
$x(t_{s})\equiv(x^{s}_{1},x^{s}_{2},0)$, one can calculate the new point of
leaving $S_{0}^{s}$ as
$x_{1}(t_{c})=:x^{c}_{1}=x^{s}_{2}\bigl{(}F_{c}|x^{s}_{2}|^{-1}-K_{p}\bigr{)}K^{-1}_{i}.$
(28)
Correspondingly, from (26) and (28) one obtains the time of leaving
$S_{0}^{s}$ as
$t_{c}=\Bigl{[}x^{s}_{2}\bigl{(}F_{c}|x^{s}_{2}|^{-1}-K_{p}\bigr{)}K^{-1}_{i}-x^{s}_{1}+x^{s}_{2}t_{s}\Bigr{]}(x^{s}_{2})^{-1}.$
(29)
From the above, it can be recognized that if $K_{i}\rightarrow 0$, the
stiction region $S_{0}$ blows to the entire $(x_{1},x_{2})$-subspace and,
consequently, $t_{c}\rightarrow\infty$. This means that a system trajectory
will never leave $S_{0}$ having reached it (a result fully in line with what
was demonstrated in Section 2.1). On other hand, if allowing for
$K_{i}\rightarrow\infty$ the time instant $t_{c}\rightarrow t^{+}_{s}$,
according to (29) with $x^{s}_{1}\rightarrow 0$, and this is due to $S_{0}$
collapsing to $\mathrm{proj}_{x_{2}}S_{0}$.
Let us now demonstrate that $S_{0}$ is globally attractive for all initial
values outside of $S_{0}$, meaning
$\forall\>x(t_{0})\in\mathbb{R}^{3}\backslash S_{0}$. As with the eigen-
dynamics in (3) and (6), which is linear, one can ensure the global
exponential stability by analyzing the characteristic polynomial
$s^{3}+K_{d}s^{2}+K_{p}s+K_{i}=0,$ (30)
and applying the standard Routh-Hurwitz stability criterion. The control
parameters condition
$K_{d}K_{p}>K_{i}$ (31)
should then be satisfied, guaranteeing that all eigenvalues $\lambda_{i}$ of
the (6) system matrix have $\mathrm{Re}\\{\lambda_{i}\\}<0$ with
$i=1,\ldots,3$. Then, the resulting (switched) $\dot{x}=Ax\mp B$ subsystems
behave as asymptotically stable in both subspaces
$\\{x\in\mathbb{R}^{3}\backslash S_{0}:x_{3}\gtrless 0\\}$ correspondingly. It
should be noted that the condition of the above parameters is conservative,
since the Coulomb friction itself is always dissipative, independently of
whether $x_{3}>0$ or $x_{3}<0$. This can be shown by considering the
dissipated energy
$\bar{V}(t)=-F(t)\int\dot{x}(t)dt=-F(t)\bar{x},$ (32)
which is equivalent to a mechanical work provided by the constant friction
force $F$ along a unidirectional displacement $\bar{x}$. Taking the time
derivative of (32) and substituting the Coulomb friction law results in
$\displaystyle\frac{d}{dt}\bar{V}(t)$ $\displaystyle=$
$\displaystyle-\frac{d}{dt}F(t)\bar{x}-F(t)\frac{d}{dt}\bar{x}$ (33)
$\displaystyle=$ $\displaystyle
0-F_{c}\mathrm{sign}\bigl{(}\dot{x}(t)\bigr{)}\dot{x}(t)=-F_{c}\bigl{|}\dot{x}(t)\bigr{|}.$
Therefore, $\dot{\bar{V}}(t)<0$ for all $x_{3}(t)\neq 0$. This quite intuitive
yet relevant condition reveals the relay feedback (5) as an additional (rate-
independent) damping, which contributes to stabilization of the closed-loop
dynamics (3)-(7). This result will be further used for proof of the Corollary
1. Notwithstanding this, we will keep the conservative stability condition
(31) as the sufficient (but not necessary) one. This appears reasonable due to
the usually uncertain Coulomb friction parameter and, hence, in order for
increasing overall robustness of the feedback control system. The following
example should, however, exemplify the additionally stabilizing behavior of
the Coulomb friction, even when (31) is violated.
###### Example 1.
Consider the (3)-(7) system with $K_{d}=0$, $K_{p}=100$ and $K_{i}=1$. The
eigenvalues of the $A$ system matrix are $\lambda_{1}=-0.01$,
$\lambda_{2,3}=0.0005\pm 10j$, which implies the linear subsystem is
asymptotically unstable. It should also be noted that (31) is not fulfilled.
To evaluate the trajectories of the (3)-(7) system, one can use the particular
solution
$x(t)=e^{At}x(\tau)+A^{-1}\bigl{(}e^{At}-I\bigr{)}Bu,$ (34)
for the constant control $u=\mp 1$, which corresponds to the (5) relay,
switched in the $x_{3}>0$ and $x_{3}<0$ subspaces. The initial values
$x(\tau)=[x_{1}(t),x_{2}(t),0]^{T}$ at $t=\tau$ should be reassigned each time
the trajectory crosses $(x_{1},x_{2})$-plane, meaning the relay switches at
$x_{3}=0$ outside of $S_{0}$. The $x_{3}$-state trajectory, with an initial
value $x_{3}(0)=10$, is shown in Figure 3, once for $F_{c}=0$ (solid red line)
and once for $F_{c}=1$ (blue dash-dot line).
Figure 3: Velocity trajectories (from $x_{3}(0)=10$) of the Example 1 system
with $F_{c}=0$ (solid red line) and $F_{c}=1$ (blue dash-dot line)
It is easy to recognize, here, that even low-amplitude Coulomb friction
($F_{c}=1$ compared to the proportional feedback gain $K_{p}=100$) leads to
stabilization of the, otherwise, unstable closed-loop control system.
###### Corollary 1.
Consider the (3)-(7) system with the control parameters satisfying (31). The
stiction region (27), given by Theorem 1, is globally attractive for all
initial values outside of this region: $x(t_{0})\in\mathbb{R}^{3}\backslash
S_{0}$.
###### Proof.
By virtue of the passivity theorem, e.g., [27, 28], the feedback
interconnection of energy dissipating systems is also energy dissipating.
Since (3), (4) is dissipative when (31) is fulfilled, and (5) is also
dissipative for $x_{3}\neq 0$, their feedback interconnection yields
dissipative almost everywhere (except $x_{3}=0$) outside of $S_{0}$. This
implies that any $x(t)$ system trajectory, starting from the outside of
$S_{0}$, converges continuously, and some ball $\mathcal{B}\equiv\|x\|$ around
the origin shrinks over time:
$\|x(t_{2})\|<\|x(t_{1})\|\quad\forall\quad
t_{2}>t_{1},\>x\in\mathbb{R}^{3}\backslash S_{0}.$ (35)
For some $t_{3}>t_{2}$, the shrinking circle
$\mathrm{proj}_{(x_{1},x_{2})}\mathcal{B}\subseteq S_{0}$, and for $t_{4}\geq
t_{3}$, zero velocity $x_{3}(t_{4})=0$ will be reached. This implies
$x(t_{4})\in S_{0}$, which completes the proof.
∎
###### Remark 2.
The sliding-mode condition (17), which results in $|CAx|\leq CB$ and,
correspondingly, proves Theorem 1, constitutes the existence and reachability
condition for $S_{0}$, which is necessary but not sufficient. This is because
(16) does not contain any requirements imposed on the $K_{d}$-parameter value.
Theorem 1 and Corollary 1 constitute the necessary and sufficient conditions
for $S_{0}$ to be both globally reachable and attractive from outside of
$S_{0}$.
## 3 Analysis of stick-slip convergence
In this Section, we will analyze the convergence behavior of stick-slip
trajectories in the (3)-(7) system. Recall that having reached $S_{0}^{s}$ at
$t_{s}$, the $x(t)$ trajectory will leave it at $t_{c}$, given by (29), which
is due to the progressing $|x_{1}(t)|$ value, which will (unavoidably) violate
the stiction condition (16). To show (qualitatively) how state trajectories
evolve during a stick-slip cycle, consider the triple-integrator chain (see
Figure 4(a)), which arises out of the closed-loop dynamics (1).
(a)
(b) (c)
(d)
Figure 4: Phase portrait at stick-slip; (a) triple-integrator chain, (b)
$(x_{1},x_{2})$-projection during sticking, (c) $(x_{2},x_{3})$-projection
during slipping, (d) typical trajectory during one stick-slip cycle
Eliminating the time argument, which is a standard procedure for a phase-plane
construction, one can write in general terms
$x_{n}dx_{n}=\dot{x}_{n}dx_{n-1},\quad n=\\{3,2\\},$ (36)
for the first and second (from the left) integrator. For a unidirectional
motion (here, $\mathrm{sign}(x_{3})=1$, for instance, is assumed) and
piecewise constant approximation $\dot{x}_{n}=\mathrm{const}$, one obtains
$\displaystyle x_{1}$ $\displaystyle=$ $\displaystyle
x^{2}_{2}(2x_{3})^{-1}-x_{1}^{min},$ (37) $\displaystyle x_{2}$
$\displaystyle=$ $\displaystyle x^{2}_{3}(2\dot{x}_{3})^{-1}-x_{2}^{c}$ (38)
after integrating the left- and right-hand sides of (36). Obviously, for
$t\geq t_{c}$, the $x_{1}(t)$-trajectory evolves parabolically with a
dependency of $x_{2}(t)$, while $x_{3}(t)$ is square-root-dependent on
$x_{2}(t)$ (see Figures 4(b) and (c) correspondingly). Note that the
increasing and decreasing segments of the corresponding trajectories are both
asymmetric, effectively due to non-constant $\dot{x}_{n}$-value as the motion
evolves. At the same time, one can stress that the extremum $x_{1}^{min}$
(here, minimal due to the assumed positive sign of velocity) always lies on
the $x_{1}$-axis (cf. Figure 4(b)) owing to $x_{1}=\int x_{2}dt$ and
$\mathrm{sign}(x_{3})=\mathrm{const}$. Differently, the
$(x_{2},x_{3})$-projection of the $x(t)$-trajectory can be shifted along the
$x_{2}$-axis, while it always ends in $x_{3}(t_{s})=0$ for $x(t)\in S_{0}$
(cf. Figure 4(c)). The resulting alternation of the stick-slip phases is
schematically shown in Figure 4(d).
###### Proposition 1.
Having reached $S_{0}$, the (3)-(7) system does not leave
$\Omega\in\mathbb{R}^{3}$ with $\mathrm{proj}_{(x_{1},x_{2})}\Omega\subseteq
S_{0}$ and converges _asymptotically_ to
$x(t)\underset{t\rightarrow\infty}{=}\bigl{\\{}(x_{1},0,0):\,|x_{1}|\leq
F_{c}K_{i}^{-1}\bigr{\\}}$ with multiple (or at least one) stick-slip cycles.
The stick-slip cycles can occur with either zero-crossing of $x_{2}$ or
keeping the same $\mathrm{sign}(x_{2}(t_{s,1}))$, where $t_{s,1}$ is the time
instant when $x(t)$ reaches $S_{0}$ for the first time.
When first disregarding the friction side effect, i.e., $F_{c}=0$, it is well
understood that non-overshoot of the set reference value cannot be reached
independently of the assigned control parameters, provided $K_{i}\neq 0$ and
the initial conditions are such that either $x_{1}(0)=0$ or
$\mathrm{sign}(x_{1}(0))=\mathrm{sign}(x_{2}(0))$. This becomes evident since
the integral error state $x_{1}(t)$ accumulates the output error over time.
That is, in order for $|x_{1}(t)|$ to start decreasing, at least one change of
the $x_{2}(t)$-sign is required. An exception is when
$\mathrm{sign}(x_{1}(0))\neq\mathrm{sign}(x_{2}(0))$, which allows both
$x_{1}(t)$ and $x_{2}(t)$ to converge to zero from the opposite directions.
Thus, at least one overshoot should appear, even if all control gains are
assigned to have the real poles only; cf. later Example 4.
When the Coulomb friction becomes effective, i.e., $F_{c}\neq 0$, the system
can change to stiction again, and also without overshoot of $x_{2}=0$ after
starting to slip at $x(t_{c})$. This means that a motion trajectory lands
again onto $S_{0}$ at time $t_{s,2}>t_{c}>t_{s,1}$ and with
$\mathrm{sign}(x_{2}(t_{s,2}))=\mathrm{sign}(x_{2}(t_{s,1}))$. Since the
system with $K_{d},F_{c}>0$ is dissipative, the energy level is
$V(t_{s,2})<V(t_{c})$, meaning the motion trajectory $x(t)$ always lands onto
$S_{0}$ closer to the origin than it was leaving $S_{0}$ in $x(t_{c})$. Note
that the system energy within $S_{0}$ can be expressed by the potential field
of the proportional and integral control errors, yielding
$V(t)=\frac{1}{2}K_{i}x_{1}^{2}+\frac{1}{2}K_{p}x_{2}^{2}\quad\hbox{for}\quad
t\in[t_{s},\ldots,t_{c}].$ (39)
One can recognize that the energy level (39), of the system in stiction, is an
ellipse
$\frac{x_{2}^{2}}{a^{2}}+\frac{x_{1}^{2}}{b^{2}}=1\quad\hbox{with}\quad
a^{2}=2VK_{p}^{-1},\;b^{2}=2VK_{i}^{-1}.$ (40)
Since the energy $V(t_{c})$ is bounded by $S_{0}$, cf. Figure 2, one can show
that the semi-major axis $a\leq F_{c}K_{p}^{-1}$ and semi-minor axis $b\leq
F_{c}K_{i}^{-1}$. From the system dissipativity and global attractiveness of
$S_{0}$, cf. Corollary 1, it follows that the trajectory becomes sticking
again, meaning $x(t)\in S_{0}$ for $t\geq t_{s,2}>t_{c}$. Since
$V(t_{s,2})<V(t_{c})$, the ellipse (40) shrinks as $a$ and $b$ become smaller
(both being proportional to $V(t)$). It is important to notice that during
system sticking, on the contrary, the energy level increases as
$V(t_{c})>V(t_{s,1})$. This is logical since the integral control action
pushes additional energy into the control loop when the system is at a
standstill, thus leading to a breakaway and allowing for the motion to restart
again once the sticking trajectory reaches the $S_{0}$-boundary.
###### Remark 3.
When the state trajectory reattains the stiction region $x(t_{s,2})\in S_{0}$
without overshoot, meaning
$\mathrm{sign}(x_{2}(t_{s,2}))=\mathrm{sign}(x_{2}(t_{c}))$, the system is
over-damped by the Coulomb friction. Otherwise,
$\mathrm{sign}(x_{2}(t_{s,2}))\neq\mathrm{sign}(x_{2}(t_{c}))$ means the
system is said to be under-damped by the Coulomb friction. A special (but not
feasible, as will be shown) case of $x_{2}(t_{s,2})=0$, meaning the system
reaches equilibrium $S_{0}\backslash S_{0}^{s}$ and remains there
$\forall\>t\geq t_{s,2}$, is analyzed below by proving Theorem 2.
###### Theorem 2.
The (3)-(7) system, with control parameters satisfying (31) and $F_{c}>0$,
converges asymptotically to invariant set $\Lambda=\\{(x_{1},0,0):|x_{1}|\leq
F_{c}K_{i}^{-1}\\}$ by the number of stick-slip cycles $1\leq N<\infty$. There
are no system parameter values and stick-slip initial conditions
$(x_{1}(t_{s,n}),x_{2}(t_{s,n}))$ with $n\in N$ which allow $\Lambda$ to be
reached by the end of the next stick-slip cycle at time
$t_{s,n+1}>t_{c,n}>t_{s,n}$.
###### Proof.
Convergence to $\Lambda$ follows from system dissipativity during the slipping
and shrinking ellipse (40), which implies an ever-decreasing energy level by
the end of one stick-slip cycle, meaning $V(t_{s,n+1})<V(t_{s,n})$. This
implies $|x_{2}(t_{s,n+1})|<|x_{2}(t_{s,n})|$ for $n\in N$ and ensures such
$x(t)$-trajectory, which starts slipping at $t_{c,n}$ and lands closer to the
origin at $t_{s,n+1}$ than before (at $t_{s,n}$).
The proof of the second part of Theorem 2, which says it is impossible to
reach the invariant equilibrium set $\Lambda$ after one particular stick-slip
cycle, follows through contradiction. For this purpose, we should first assume
that there is a particular setting
$\bigl{(}K_{p},K_{i},K_{d},F_{c},x_{1}(t_{s,n}),x_{2}(t_{s,n})\bigr{)}$ for
which the state trajectory $x(t_{s,n+1})\in\Lambda$, i.e. in the next stiction
phase at the finite time $t_{s,n+1}>t_{c,n}>t_{s,n}$. The initial conditions
of a slipping phase are always given, cf. with Section 2.3, by
$\displaystyle x_{2}(t_{c,n})$ $\displaystyle=$ $\displaystyle
x_{2}(t_{s,n}),$ (41) $\displaystyle x_{1}(t_{c,n})$ $\displaystyle=$
$\displaystyle\frac{F_{c}}{K_{i}}-\frac{K_{p}}{K_{i}}x_{2}(t_{c,n})\>\hbox{ in
1st quad.,}$ (42) $\displaystyle x_{1}(t_{c,n})$ $\displaystyle=$
$\displaystyle-\frac{F_{c}}{K_{i}}-\frac{K_{p}}{K_{i}}x_{2}(t_{c,n})\>\hbox{
in 3rd quad.}$ (43)
This becomes apparent when inspecting the stiction phase dynamics (25), (26)
and $S_{0}$-boundary, cf. Figure 2. For reaching $\Lambda$ at a final time
instant $\psi=t_{s,n+1}$, while starting at $\tau=t_{c,n}$, an explicit
particular solution of
$0=C\Bigl{[}e^{A\psi}\left(\begin{array}[]{c}x_{1}(\tau)\\\\[-1.42271pt]
x_{2}(\tau)\\\\[-1.42271pt]
0\end{array}\right)+A^{-1}(e^{A\psi}-I)Bu-\left(\begin{array}[]{c}x_{1}(\psi)\\\\[-1.42271pt]
0\\\\[-1.42271pt] 0\end{array}\right)\Bigr{]}$ (44)
with $u=\pm 1$, should exist, cf. with (34). Due to the symmetry of solutions,
we will consider the 1st quadrant of $S_{0}$ only, i.e., with the above
initial condition (42) and $u=+1$ correspondingly, without loss of generality
when solving (44). Recall that the matrix exponential
$e^{A\psi}=\sum^{\infty}_{k=0}\frac{(A\psi)^{k}}{k!}$ (45)
has to be evaluated to find an explicit solution for (44). Substituting the
initial condition equalities (41) and (42) into (44), we solve (44) with
respect to $x_{2}(\tau)$, and that for an increasing $k=[1,\ldots,40]$. For
all the solutions evaluated with the help of the Symbolic Math ToolboxTM, it
was found that (44) has no initial-value solution other than zero, meaning
$x_{2}(\tau)=x_{2}(t_{s,n})=0$. This means that there are no other initial
conditions than zero for which a stick-slip cycle could lead to
$x(t)\in\Lambda$ at $t=t_{s,n+1}$. This contradicts our initial assumption
that such initial conditions exist and, hence, completes the proof.
∎
###### Remark 4.
Since no relative motion occurs during a stiction phase, cf. Section 2.2, the
trajectories solution (44) represents the single descriptor of the system
dynamics, determining convergence of trajectories during the slipping phases.
One can recognize that discontinuous Coulomb friction contributes as a
constant piecewise-continuous input term $u$, into the solution of
trajectories $x(t)$ at $t_{c,n}<t<t_{s,n+1}$. Thus, it comes as no surprise
that the stick-slip convergence appears only asymptotically, meaning either
within one or an (theoretically) infinite number of stick-slip cycles. We also
note that this is independent of whether $x(t)$ reattains $S_{0}$ with or
without overshooting.
## 4 Numerical examples
The following numerical examples serve to illustrate and evaluate the above
analysis. A dedicated numerical simulation of the stick-slip dynamics is
developed by implementing (25), (26) and (34), while the conditions of Theorem
1 provide switching between the piecewise smooth trajectories of the
alternating slipping and sticking phases of relative motion of the (3)-(7)
system.
###### Example 2.
Consider the (3)-(7) system with $K_{d}=20$, $K_{p}=100$, $K_{i}=1000$ and
varying $F_{c}=\\{50,75,100\\}$. The initial values are assigned as
$x(0)=\\{0,-1.1,0\\}$, corresponding to a classical positioning task for the
feedback- controlled system (1). Note that $|x_{2}(0)|>F_{c}K_{p}^{-1}$ so
that the trajectories start outside of $S_{0}$ and are, therefore, inherently
in the slipping phase. The transient and convergence responses for all three
Coulomb friction values are shown opposite to one another in Figure 5, cf.
qualitatively with an experimental convergence pattern reported in [13, Fig
4].
(a) (b)
Figure 5: Output response of Example 2: transient phase $t=[0,\ldots,12]$ sec
(a), convergence phase $t=[12,\ldots,120]$ sec (b)
###### Example 3.
Consider the (3)-(7) system with $K_{d}=10$, $K_{p}=1040$, $K_{i}=8000$ and
$F_{c}=100$. The initial values $x(0)=(0,-0.15,0)$ are assigned to be close to
but still outside of the $S_{0}$-region. The linear damping $K_{d}$ is
selected, with respect to $F_{c}$, so that the system exhibits only one
initial overshoot; but the stick and slip phases alternate without changing
the sign of $x_{2}$. The output displacement response is shown in Figure 6(a).
The stick-slip convergence without zero-crossing is particularly visible on
the logarithmic scale in Figure 6(b). Note that after the series of stick-slip
cycles, a further evaluation of the alternating dynamics (about $10^{-13}$ in
order of magnitude) is no longer feasible, due to a finite time step and
corresponding numerical accuracy, cf. 1st quadrant of the $S_{0}$-rhombus in
Figure 2.
(a) (b)
Figure 6: Output (a) and logarithmic (b) response of Example 3
###### Example 4.
Consider the (3)-(7) system with $K_{d}=56$, $K_{p}=1040$, $K_{i}=6400$ and
$F_{c}=100$. Note that the control gains are assigned in such a way that the
linear dynamics (3) and (6) reveal a double real pole at $\lambda_{1,2}=-20$
and one in its vicinity at $\lambda_{3}=-16$. This ensures that all states
converge fairly simultaneously towards zero, once the $\mathrm{sign}(x_{3})$
remains unchanged. For the initial conditions, $x_{1}(0),x_{3}(0)=0$ and the
varying initial displacement is $x_{2}(0)=\\{-0.2,-0.25,-0.3,-0.35\\}$. Note
that all $x(0)$ are outside of $S_{0}$, while the transient overshoot lands
(in all cases) within $S_{0}$, thus leading to the first stiction after
overshoot; see Figure 7. One can recognize that the integral state requires
quite different times before the system then passes again into slipping.
During the slipping phase, all states converge asymptotically towards zero,
provided $F_{c}$ remains constant. Here, it is important to notice that in the
real physical systems, a varying $F_{c}$-value and so-called frictional
adhesion, see, e.g., [29], at extremely low velocities, will both lead to the
system passing into a sticking phase again, therefore, further provoking
multiple stick-slip cycles. Even though it is not the case here, with our
ideal Coulomb friction assumption, the Theorem 2 still holds, since there is
only an asymptotic convergence after at least one stick-slip cycle has
occurred.
Figure 7: Output response of Example 4 for different initial values
###### Example 5.
Consider the (3)-(7) system with $K_{d}=20$, $K_{p}=100$, $K_{i}=1000$ and
$F_{c}=50$. The initial condition $x(0)=(0,-0.5,0)$ is assigned to be on the
boundary of $S_{0}$, thus leading to short initial slipping and, then,
realizing a large number of stick-slip cycles by long-term simulation with
$t=[0,\ldots,100000]$ sec. The output is shown as logarithmic absolute value
(due to the alternating sign) over the logarithmic time argument in Figure 8.
One can recognize that each consequent sticking phase gets closer to the
origin, while the stick-slip period grows exponentially, cf. logarithmic
timescale. This confirms an asymptotic convergence with stick-slip cycles, cf.
Theorem 2.
Figure 8: Output response of Example 5, the logarithmic absolute value over
the logarithmic time argument
## 5 Conclusions
An analysis of stick-slip behavior during convergence of feedback-controlled
motion with Coulomb friction has been developed. The most general case of
frictional discontinuity at velocity zero-crossing has been assumed, and the
parametric conditions for appearance of the stiction region around the
equilibrium (set independently of the initial values) are derived. Theorem 1
and Corollary 1 proved the stiction region to be globally reachable and
attractive. Theorem 2 stated that convergence is only asymptotically possible
and occurs with at least one but mostly an infinite number of stick-slip
cycles in a sequence. In particular, an ’ideal’ convergence of the control
configuration with all real poles in a neighborhood appears with one initial
stick-slip cycle, followed by an asymptotic convergence without new stiction
phases. The number of illustrative numerical examples, with different initial
conditions and parameter settings, argue in favor of the developed analysis
and provide additional insight into stick-slip mechanisms of controlled motion
with Coulomb friction.
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|
[1]
inst1]organization=UK Atomic Energy Authority,addressline=Culham Science
Centre, city=Abingdon, postcode=OX14 3DB, country=UK
inst2]organization=Department of Materials, University of
Manchester,city=Manchester, postcode=M13 9PL, country=UK
inst5]organization=Department of Materials Physics, Eötvös
University,postcode=PO Box 32, H-1518, city=Budapest, country=Hungary
inst6]organization=Deutsches Elektronen-Synchrotron DESY,addressline =
Notkestr. 85, postcode=22607, city=Hamburg, country=Germany
inst7]organization=Monash University, city=Clayton, postcode =VIC 3800,
country = Australia
[1]Corresponding author
# Dislocation density transients and saturation in irradiated zirconium
Andrew R. Warwick<EMAIL_ADDRESS>Rhys Thomas M. Boleininger Ö. Koç
G. Zilahi G. Ribárik Z. Hegedues U. Lienert T. Ungar C. Race M. Preuss
P. Frankel S. L. Dudarev [ [ [ [ [
###### Abstract
Zirconium alloys are widely used as the fuel cladding material in pressurised
water reactors, accumulating a significant population of defects and
dislocations from exposure to neutrons. We present and interpret synchrotron
microbeam X-ray diffraction measurements of proton-irradiated Zircaloy-4,
where we identify a transient peak and the subsequent saturation of
dislocation density as a function of exposure. This is explained by direct
atomistic simulations showing that the observed variation of dislocation
density as a function of dose is a natural result of the evolution of the
dense defect and dislocation microstructure driven by the concurrent
generation of defects and their subsequent stress-driven relaxation. In the
dynamic equilibrium state of the material developing in the high dose limit,
the defect content distribution of the population of dislocation loops,
coexisting with the dislocation network, follows a power law with exponent
$\alpha\approx 2.2$. This corresponds to the power law exponent of
$\beta\approx 3.4$ for the distribution of loops as a function of their
diameter that compares favourably with the experimentally measured values of
$\beta$ in the range $3\leq\beta\leq 4$.
###### keywords:
zirconium irradiation dislocations defects
Dislocation density evolution in irradiated Zr has been predicted and
measured.
A transient peak and subsequent saturation in dislocation density has been
observed.
Dislocation loop diameters in heavily irradiated Zr are power law distributed.
## 1 Introduction
In the core of modern boiling (BWR) or pressurized (PWR) water reactors, the
uranium dioxide fuel assemblies are immersed in circulating pressurised water
and thus it is critical that only the heat produced by the fission reactions
is transported by the coolant and there is no contamination of the coolant
from the radioactive fuel itself. Hence, the fuel is cladded to protect the
reactor environment from contamination, be that during reactor operation or in
transit. From the design choices that date back over fifty years (Rickover et
al., 1975), zirconium alloys are currently employed as the uranium dioxide
fuel cladding in water-cooled reactors. Containing more than 95 wt% Zr, these
alloys are mostly pure zirconium, chosen for its low neutron absorption cross
section (Pomerance, 1951). Small amounts of Sn, Nb, Fe and/or Cr in the alloys
help protect against corrosion and improve structural integrity (Lemaignan,
2012; Onimus and Bechade, 2012).
Figure 1: Dislocation density as a function of dose observed experimentally
using microbeam synchrotron X-ray diffraction (XRD) measurements of proton-
irradiated Zircaloy-4, compared to predictions derived from simulations of
pure $\alpha$-Zr performed using the creation relaxation algorithm technique.
Four experimental samples with nominal doses of 0.1, 0.5, 1.0 and 2.0 dpa were
scanned across the variable dose regime. The simulation results are scaled by
a factor of 0.1 and averaged over three different interatomic potentials (see
text).
The elastic neutron scattering cross-section for a Zr nucleus is, however,
similar to that of other elements in the Periodic table (Sears, 2006), and a
prolonged exposure to neutron irradiation results in the accumulation of a
considerable amount of microscopic radiation defects generated from the atomic
recoils initiated by collisions with neutrons. This gives rise to the
deterioration of mechanical and physical properties and stimulates dimensional
changes (Holt, 1988; Onimus and Bechade, 2012). The high energy
$>1\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ neutrons produced by the fissile
uranium oxide fuel (Nicodemus and Staub, 1953) collide with Zr nuclei,
initiating collision cascades that displace atoms from their lattice sites,
rearrange the crystal structure, and generate crystal lattice defects (Domain
and Legris, 2005). Defects accumulate with increasing exposure to neutron
irradiation, in the form of pairs of self-interstitial and vacancy defects
(Frenkel pairs) as well as in the form of clusters of defects that eventually
coalesce into large-scale defects, such as dislocation loops and dislocations
(Warwick et al., 2021). At temperatures above
300-$350\text{\,}\mathrm{\SIUnitSymbolCelsius}$ where point defect diffusion
occurs at an appreciable rate comparable or faster rates than the dose rate,
it is important to consider random Brownian motion of defects to interfaces,
grain boundaries, dislocations or other point defect clusters, whereas at
lower temperatures the microstructure of accumulating defects is dominated by
other factors. Given the significance of the life-limiting effect of
structural changes on the properties of zirconium cladding, there is
significant research effort aimed at improving the performance of structural
zirconium alloys in the operating environment of a fission reactor (Adamson et
al., 2019; Zinkle and Was, 2013).
The exposure of a material to energetic particles is often quantified by the
notion of dose, typically expressed in the units of ‘displacement per atom’
(dpa). Dpa is a simple measure of exposure of a material to radiation and
represents the average number of times each atom in the material has been a
part of a Frenkel self-interstitial-vacancy defect pair. Typically, a
zirconium alloy cladding is exposed to $\sim
15\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$ over the five years of service
(Zinkle and Was, 2013). Producing an accurate safety case involves the
identification of the microstructure formed at a given dose, temperature, and
externally applied load.
In particular, the formation of dislocation loops and dislocations is known to
play a critical role in the resulting deterioration of the cladding’s
structural and mechanical properties. For instance, in the large dose limit,
high densities of defects and dislocations accumulate in the cladding, causing
embrittlement. Another important degradation mode is the so-called
‘irradiation-induced growth’ (IIG) that arises from the anisotropy of the
hexagonal close packed (hcp) crystal structure of zirconium ($\alpha$-Zr) and
zirconium alloys (Griffiths, 2020; Onimus et al., 2022). This crystal
structure is stable up to and beyond the reactor core’s operating temperature
range of $280\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to
$350\text{\,}\mathrm{\SIUnitSymbolCelsius}$, exhibiting an hcp-bcc instability
only at significantly higher temperatures above
$860\text{\,}\mathrm{\SIUnitSymbolCelsius}$ (Willaime and Massobrio, 1989).
With increasing dose, the interstitial and vacancy-type dislocation loops with
the $\frac{1}{3}\langle 2\bar{1}\bar{1}0\rangle$ Burgers vectors, inhabiting
the prismatic crystallographic planes, form a population of the so-called ‘a
loops’. At relatively low temperatures, this strongly correlates with observed
elongation along the ‘a’ $\langle 2\bar{1}\bar{1}0\rangle$ and contraction
along the ‘c’ $\langle 0001\rangle$ crystallographic directions, saturating at
doses larger than $1\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$ (Holt, 1988). At
temperatures above $\sim 300\text{\,}\mathrm{\SIUnitSymbolCelsius}$ and doses
exceeding $\sim 5\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$, large vacancy-type
‘c loops’ with Burgers vectors $\frac{1}{2}\langle 0001\rangle$ or
$\frac{1}{6}\langle 20\bar{2}3\rangle$ appear in the basal crystallographic
planes. Accompanying this onset of formation of the c-type vacancy loops, the
magnitudes of a and c strains increase linearly with dose in a phenomenon
called ‘the breakaway growth’ (Choi and Kim, 2013). Whilst it is known that
dislocations have substantial elastic relaxation volume (Dudarev and Ma, 2018;
Boleininger et al., 2022) giving rise to significant dimensional changes
(Christensen et al., 2020), the present understanding of this IIG is mostly
phenomenological and existing models are unable to predict, from first
principles, the variation of the dislocation content consistent with, or at
least verified by, the experimental data.
Below we present experimental observations and predictions derived from
simulations, showing that the density of dislocations saturates in the
temperature and dose range where dimensional changes exhibit saturation. This
is summarised in Figure 1 illustrating the dislocation densities
experimentally measured in proton-irradiated Zircaloy-4
(Zr-1.5%Sn-0.2%Fe-0.1%Cr) together with the simulation data for irradiated
$\alpha$-Zr plotted as a function of dose. In agreement with observations
performed using ion irradiation (Yu et al., 2017), we find that the density of
dislocations evolves through a transient at doses
$<1\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$ before saturating at larger doses.
The microstructures produced by our simulations indicate that at very low
doses $\ll 1\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$, small dislocation loops
form, which subsequently grow and coalesce into a complex interconnected
dislocation network developing at moderate doses
$<1\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$. The dislocation network
eventually forms complete crystallographic planes (Mason et al., 2020;
Boleininger et al., 2023) and partly dissociates into large dislocation loops
at high doses $\gg 1\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$. The formation of
dislocation loops and dislocations is principally driven by the stress-
mediated clustering of self-interstitial atoms, where in the high dose limit
the self-interstitial cluster population size distribution follows a power law
$p(N)\propto 1/N^{2.2}$ developing in the saturation regime corresponding to
high neutron exposure. Below, we show that this fact, also confirmed by
experimental observations (Ungár et al., 2021), has important implications for
the interpretation of experimental observations of dislocation loops and for
our understanding of the dynamics of microstructural evolution in the
irradiated cladding.
Our manuscript is structured by detailing our methods in §2 and presenting our
results in §3 before summarising key conclusions in §4. The concept and
relevant definitions of dislocation density are discussed in §2.1. Details of
our experimental set-up and and an overview of the cmwp line profile analysis
software are given in §2.2 followed by an explanation of our simulation
method, its range of validity and our choice of settings in §2.3. We show how
the simulated microstructure correlates to the dislocation density profile
shown in Figure 1 in §3.1 in addition to characterising the evolution of the
power law distribution of dislocation sizes. Finally, the stored energy
associated with dislocation content that drives the changing microstructure as
a function of dose is investigated in §3.2.
## 2 Theory and Methodology
### 2.1 Dislocation density
There are multiple ways to define the density of dislocation lines $\rho$ in a
deformed or irradiated material. For all the measurements and computations in
this study, $\rho$ is defined as a scalar ratio of the total dislocation line
length to volume $V$ containing the dislocations, namely (Hull and Bacon,
2011)
$\displaystyle\rho:=\frac{1}{V}\int_{\perp\in
V}{\left|\mathrm{d}\mathbf{l}\right|}.$ (1)
Here $\mathbf{l}$ is a position vector on a dislocation line such that
$\mathrm{d}\mathbf{l}=\bm{\xi}\mathrm{d}s$ for unit tangent vector $\bm{\xi}$
and arc length $s$, and the integration is performed with respect to the arc
length over all the dislocation lines $\perp$ in $V$. The choice of volume is
somewhat arbitrary but, for a given resolution, is expected to reflect the
average amount of dislocations present. For example, in a TEM micrograph this
can be chosen to be a region contained in the image and in a molecular
dynamics simulation one may use the entire simulation cell.
Another possible definition is the areal density $\rho_{A}$ that measures the
number of dislocation lines crossing an open surface as (Hull and Bacon, 2011)
$\displaystyle\rho_{A}:=\frac{1}{A}\int_{\perp\in
A}{\left|\mathrm{d}\mathbf{S}\cdot\bm{\xi}\right|},$ (2)
where $\mathrm{d}\mathbf{S}$ is a vector area element of $A$ with direction
normal to the surface and, similar to Equation 1, there is a dependence on the
choice of the surface. If all the dislocations are co-linear and perpendicular
to the chosen surface, Equations 1 and 2 produce the same value. Whilst for an
arbitrary distribution of dislocations this is often not the case, generally
both measures do not differ by much more than a factor of two (Schoeck, 1962).
Needless to say, a dislocation is not solely defined by its tangent vector and
it is noteworthy that neither Equation 1 nor Equation 2 contain any
information about the Burgers vector $\mathbf{b}$ of the dislocation.
Significant physical quantities such as dislocation energy and the Peach-
Koehler force both depend on $\mathbf{b}$. The Nye tensor $\bm{\alpha}$ is a
tensorial measure of dislocation density that is a linear function of lattice
curvature (Nye, 1953) and is also a function of position $\mathbf{x}$ such
that (Jones et al., 2016)
$\displaystyle\bm{\alpha}(\mathbf{x}):=\int_{\perp}{\delta\left(\mathbf{x}-\mathbf{l}\right)\mathbf{b}(\mathbf{l})\otimes\mathrm{d}\mathbf{l}},$
(3)
where $\otimes$ denotes the tensor product, integration is performed over all
of the dislocation lines in the system, and $\delta\left(\mathbf{x}\right)$ is
the Dirac delta distribution defined by the property
$\displaystyle\int\mathrm{d}^{3}\mathbf{x}^{\prime}\,\delta(\mathbf{x}^{\prime}-\mathbf{x})f(\mathbf{x}^{\prime})=f(\mathbf{x}),$
(4)
for an arbitrary well-behaved function $f(\mathbf{x})$. Whilst full
information about the dislocation content is contained in Equation 3,
attempting to average $\bm{\alpha}$ over a volume $V$ can be problematic.
Essentially, this stems from the fact that the integral of
$\mathrm{d}\mathbf{l}$ along a dislocation segment contained in $V$ that
starts and ends at $\mathbf{x}_{a}$ and $\mathbf{x}_{b}$ respectively is
$\mathbf{x}_{b}-\mathbf{x}_{a}$. Thus, any information pertaining to curvature
of a dislocation line is lost and closed paths in particular, i.e. dislocation
loops, provide no contribution to the volume average (Arsenlis and Parks,
1999; Mandadapu et al., 2014). The dislocations sections that integrate to
zero are referred to as Statistically Stored Dislocations (SSD) and the
surviving contributions are the Geometrically Necessary Dislocations (GND).
Experimental techniques that infer GND content, such as micro-beam Laue
measurements and high resolution electron back-scattered diffraction,
implicitly make use of Equation 3 (Das et al., 2018). As it is difficult to
characterise the entire population of dislocations using Equation 3,
throughout this manuscript we have chosen to use the scalar measure given by
$\rho$ in Equation 1 which is the same definition as that employed in our
X-ray line profile analysis.
### 2.2 Experiment
Dislocation densities were measured in four $3\text{\,}\mathrm{mm}$ $\times$
$1\text{\,}\mathrm{mm}$ $\times$ $0.5\text{\,}\mathrm{mm}$ Zircaloy-4 samples
(composition Zr-0.17Fe-1.24Sn-0.10Cr) proton-irradiated to different doses.
The samples possessed a recrystallised equiaxed microstructure with a low
dislocation density of $<1\text{\times}{10}^{14}\text{\,}\mathrm{m}^{-2}$ and
a characteristic ‘split-basal’ texture due to processing, where the basal
poles are aligned along the normal direction (ND), with a $\pm$30 degree tilt
towards the transverse direction. Irradiation induced growth strains in
similarly textured Zircaloys are known to saturate at doses below $\sim
10\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$ at
$320\text{\,}\mathrm{\SIUnitSymbolCelsius}$ (Adamson et al., 2019) and thus we
may expect dislocation densities to display a similar pattern of evolution in
these samples.
The ND face of each sample was proton irradiated with
$2\text{\,}\mathrm{M}\mathrm{e}\mathrm{V}$ protons at
$350\text{\,}\mathrm{\SIUnitSymbolCelsius}$ at the University of Manchester’s
Dalton Cumbrian Facility, UK. The temperature of the samples during
irradiation was monitored via a thermal imaging camera in order to hold it
within $\pm 10\text{\,}\mathrm{\SIUnitSymbolCelsius}$ of the target
temperature. Unlike neutrons, the Coulomb interaction between the protons and
the target material results in the shallow penetration of protons into the
material. The resulting radiation exposure, quantified by the dose and dose
rate, varies significantly as a function of depth, with the dose rate being of
the order of $10^{-5}$ dpa/s. The dose profile was calculated using the quick
Kinchin-Pease setting in srim (Ziegler et al., 2010) with the lattice binding
energy and threshold displacement energy set to
$0\text{\,}\mathrm{e}\mathrm{V}$ and $40\text{\,}\mathrm{e}\mathrm{V}$
respectively (Stoller et al., 2013). A typical dose vs. depth profile in one
of our Zircaloy-4 samples consists of a plateau region extending $\sim
10\text{\,}\mathrm{\SIUnitSymbolMicro m}$ from the surface where the dose and
dose rate are approximately constant before sharply rising and falling to zero
at a region corresponding to protons coming to rest in the material, called
the Bragg peak and located at $\sim
30\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$. The samples were irradiated
such that the doses at 60% of the Bragg peak depth from the surface, termed
‘nominal doses’, were 0.1, 0.5, 1 and
$2\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$ respectively. Within the first
$30\text{\,}\mathrm{\SIUnitSymbolMicro m}$ of each sample from the irradiated
surface, the calculated dose and dose rate vary from their nominal values by
factors of 0.6 to 8.5 thus allowing us to measure data spanning over a wide
range of irradiation exposures.
Using a small X-ray beam of $2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ $\times$
$100\text{\,}\mathrm{\SIUnitSymbolMicro m}$ cross section, the samples were
scanned in cross-section from the surface to a depth of
$50\text{\,}\mathrm{\SIUnitSymbolMicro m}$ within the sample at
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}$ increments at the P21.2 beamline at
the PETRA III synchrotron facility at DESY in Hamburg, Germany. The samples
were translated perpendicular to the scanning direction by
$200\text{\,}\mathrm{\SIUnitSymbolMicro m}$ during each scan to improve grain
statistics and reduce the spottiness of the pattern. The set-up of the
diffraction experiment and sample geometry is shown in Figure 2.
(a)
(b)
Figure 2: (a) : Experimental geometry for measuring line profiles from
Zircaloy-4 samples on the P21.2 beamline at the DESY synchrotron, Hamburg,
Germany. (b): Variation of dislocation density and dose as a function of depth
for the nominal $0.1\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$ sample.
Dislocation densities were extracted from the line profiles using the
Convolutional Multiple Whole Profile (cmwp) software (Ribárik et al., 2020).
The cmwp software models the line profile intensity $I(q)$, where $q$ is the
wavevector magnitude, as a convolution of intensities arising from
instrumental effects, size broadening, and, in particular, strain broadening
due to dislocations. Instrumental effects were determined using a LaB6
standard specimen, and the size broadening was determined assuming a log-
normal size distribution of coherently scattering grains (Ribárik et al.,
2020). Here, we outline the model underpinning the cmwp software, whereas for
a broader context we refer an interested reader to reviews of the method
(Wilkens, 1970; Ungár et al., 1999; Ribárik et al., 2020).
In the theory of X-ray line profile analysis, the Fourier components of the
broadened intensity peak profile corresponding to a reciprocal lattice vector
$G$, denoted $I^{D}_{G}$, are related to the strain distribution by
$\displaystyle\mathcal{F}\left\\{I^{D}_{G}(q)\right\\}(L)=\exp\left[-2\pi^{2}G^{2}L^{2}\langle\epsilon^{2}_{G,L}\rangle\right],$
(5)
where $L$ is the Fourier variable and $\langle\epsilon^{2}_{G,L}\rangle$ is
the mean-square strain. Wilkens (1970) derived an expression for
$\langle\epsilon^{2}_{G,L}\rangle$ by numerical methods arising from the
‘restrictedly random distribution’ concept of dislocations. Dislocation lines
are assumed to be parallel in the sub-areas of equal size $A$ perpendicular to
the line direction. A number $N_{\perp}$ of dislocation lines with equal
numbers of positive and negative Burgers vectors occupy random positions in a
plane normal to the dislocation lines. The characteristic linear size of the
sub-areas is chosen to be proportional to a parameter termed the effective
outer cut-off radius $R_{e}$. The dislocation density $\rho$ is then defined
by Wilkens (1970) as the areal density of dislocations given by Equation 2
that, as discussed in §2.1, is equivalent to the volume density of
dislocations defined by Equation 1 for this specific dislocation
configuration. The dipole character of the distribution is determined by the
arrangement parameter $M=R_{e}\sqrt{\rho}$ . Whilst $R_{e}$ is one of the
fitting parameters in the cmwp software, thus affecting the value of $M$, this
article is principally concerned with the determination of $\rho$ and thus we
do not discuss the arrangement parameter further.
An expression for the mean-square strain for a restricted random distribution
of dislocations was derived by Wilkens (1970)
$\displaystyle\langle\epsilon^{2}_{G,L}\rangle=\frac{\rho
Cb^{2}}{4\pi}f(\eta),$ (6)
where $C$ is a parameter that is refined by the profile fitting cmwp
algorithm, termed the dislocation contrast factor. The value of $C$ was
evaluated for dislocations in $\alpha$-Zr by Balogh et al. (2016). The Wilkens
function $f(\eta)$, where $\eta=L/R_{e}$, has the following asymptotic forms
in the limit of small and large $\eta$, respectively
$\displaystyle f(\eta)\sim\begin{cases}\ln{\eta}&,\,\eta\rightarrow 0\\\
\frac{1}{\eta}&,\,\eta\rightarrow\infty.\end{cases}$ (7)
The numerically obtained formula for $f(\eta)$ may be found in Wilkens (1970).
Although the Wilkens model is formally derived assuming that a dislocation
configuration is composed of straight lines, the model is in fact able to
mimic the statistical properties of distributions of curved dislocations
(Kamminga and Delhez, 2000; Groma and Borbély, 2004). When compared to
transmission electron microscope (TEM) measurements of irradiated Zircaloy-2,
the cmwp software was able accurately follow the dislocation density evolution
as a function of dose (Seymour et al., 2017), and the cmwp approach has now
become an accepted tool for determining dislocation densities (Ungár et al.,
2021; Topping et al., 2018) as well as other microstructural features (Ungár
et al., 2021) in irradiated Zircaloys. The cmwp software evaluates parameters
describing effects of both the specimen size and dislocation broadening of
diffraction intensity peaks by first employing a statistical Monte Carlo
optimisation followed by the Marquardt-Levenberg non-linear least squares
algorithm, see Ribárik et al. (2020). All the peaks in the interval from
$0\text{\,}\mathrm{n}\mathrm{m}^{-1}$ $<q<$
$13\text{\,}\mathrm{n}\mathrm{m}^{-1}$ were included in the fitting procedure
and the uncertainty in $\rho$ was quantified according to the quality of fit
as described by Ribárik et al. (2020). The variation of dislocation density
with depth calculated by cmwp is shown in Figure 2(b) that, when mapped to the
dose calculated by srim at a given depth, enables plotting the dislocation
density as a function of dose as illustrated in Figure 1.
### 2.3 Simulation
The accumulation of defects and microstructural evolution as a function of
dose was simulated using the Creation Relaxation Algorithm (CRA) (Derlet and
Dudarev, 2020). The CRA exploits the separation of timescales associated with
relatively fast stress driven and comparatively slow thermally activated
evolution of defect microstructure. This results in a simple algorithm where,
starting with a perfect crystal structure of $\alpha$-Zr, the Frenkel pairs of
defects are created at random and the microstructure is subsequently relaxed
via direct energy minimisation such that the system evolves purely through the
action of internal stresses arising from the generation of defects. The dose
is measured in units of ‘canonical displacement per atom’ (cdpa) computed as
the ratio of the total number of Frenkel pairs generated by the algorithm to
the number of atoms in the system. CRA simulations assume that vacancies
remain effectively immobile, in turn also immobilising the dislocation part of
the microstructure (Arakawa et al., 2020), leaving the internal fluctuating
stresses as the only remaining factor driving the migration and clustering of
self-interstitial atom defects. In (Warwick et al., 2021) we identified the
approximate temperature and dose rate range where the simulation method
retains its validity when applied to $\alpha$-Zr.
The IIG strains observed in neutron irradiated zirconium alloys with initially
low dislocation densities tend to saturate with increasing dose at
temperatures less than $\sim 300\text{\,}\mathrm{\SIUnitSymbolCelsius}$, see
Adamson et al. (2019). At temperatures above $\sim
300\text{\,}\mathrm{\SIUnitSymbolCelsius}$, saturation persists over shorter
intervals of dose before the strain magnitudes start increasing linearly as a
part of the breakaway growth phenomenon (Holt, 1988). A significant change of
pattern of thermal evolution has also been found above $\sim
300\text{\,}\mathrm{\SIUnitSymbolCelsius}$ in proton irradiated Zircaloy-2
when samples irradiated to $2\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$ were
annealed for $1\text{\,}\mathrm{h}$ at various temperatures (Topping et al.,
2018). The X-ray line profile measurements performed by Topping et al. (2018)
showed that the a-loop density significantly decreased only at temperatures
above $\sim 300\text{\,}\mathrm{\SIUnitSymbolCelsius}$.
These data offer a valuable insight into the timescales on which thermally
activated processes, including vacancy migration, drive the evolution of
heavily irradiated zirconium. Earlier (Warwick et al., 2021), noting that the
rates of thermally activated processes follow the Arrhenius law (Vineyard,
1957; Landauer and Swanson, 1961; Allnatt and Lidiard, 1993), we showed that
the annealing experiment data imply that the characteristic activation energy
$E_{a}$ for the processes primarily responsible for the observed thermally
activated behaviour must be close to $\sim 2\text{\,}\mathrm{e}\mathrm{V}$.
Also, as described in §2.2, the proton-irradiation defect production dose rate
$\dot{\phi}$ at all depths in our experiments is high and close to $10^{-5}$
dpa s-1. Given this high dose rate, we can estimate an upper bound $\tilde{T}$
on the range of temperatures where the rate of migration of defects stimulated
directly by irradiation is higher than the rate of thermally activated
migration of defects. For a given activation energy $E_{a}$, using the dose
rate model by Nordlund et al. (2018), we find that the two rates are
comparable if
$\displaystyle\dot{\phi}\left({2E_{d}\over
E_{a}}\right)\approx\nu\exp{\left(-\frac{E_{a}}{k_{B}\tilde{T}}\right)},$ (8)
where $E_{d}$ is the threshold displacement energy required for forming a
defect, the attempt frequency is ${\nu\approx\omega_{D}/2\pi}=5.84\times
10^{12}$ s-1, given the Debye frequency ${\omega_{D}=3.67\times
10^{13}\,\mathrm{s}^{-1}}$ (Zarestky, 1979) and $k_{B}=0.861\times 10^{-4}$
eV/K is the Boltzmann constant. Taking $E_{a}=2$ eV and $E_{d}=40$ eV, and
solving equation (8) for $\tilde{T}$, we find $\tilde{T}=625$ K$\approx
350\text{\,}\mathrm{\SIUnitSymbolCelsius}$. Below this temperature, the
eigenrate of thermal relaxation of microstructure is lower than the rate at
which defects are driven by irradiation. Hence, at temperatures below
$\tilde{T}$, the defect structures generated by irradiation evolve
predominantly through fast athermal stress relaxation (Derlet and Dudarev,
2020).
The above esitmate for $\tilde{T}$$\sim
350\text{\,}\mathrm{\SIUnitSymbolCelsius}$ is close to the temperature at
which Topping et al. (2018) observed the occurrence of a significant change in
the thermal response of microstructure during annealing. Notably, the change
of pattern of breakaway growth at $\sim
300\text{\,}\mathrm{\SIUnitSymbolCelsius}$ noted by Holt (1988) was observed
at significantly lower dose rates than those characterising our proton
irradiation experiments. Hence, the above temperature must reflect the
fundamental scale of activation energies associated with microstructural
evolution of Zircaloys under irradiation.
The migration energy of individual vacancies in pure elemental $\alpha$-Zr
(Varvenne et al., 2014) of $\sim 0.5\text{\,}\mathrm{eV}$ is too low to
account for the observed behaviour, and while the thermal diffusion of
vacancies and other point defects affect microstructural evolution, reducing
the overall concentration of defects noted in our analysis, experimental
observations indicate the presence of a rate-limiting process with a higher
activation energy that stabilises the observed dense defect microstructures at
temperatures as high as $300\text{\,}\mathrm{\SIUnitSymbolCelsius}$.
High activation energies are known to be associated with the formation of
immobile vacancy-impurity clusters involving carbon or nitrogen (Fu et al.,
2008; Terentyev et al., 2014; Theodorou et al., 2022). In bcc iron, the
effective migration energy of vacancies is defined by the energy of
dissociation of a cluster involving a vacancy and a carbon dimer (Paxton,
2014), and this dissociation energy can be as high as 2.22 eV (Kabir et al.,
2010), far higher than the activation energy of 0.55 eV characterising vacancy
migration in pure elemental Fe (Fu et al., 2005). Given that the
characteristic formation and migration energies of defects in Zr and Fe are
nearly the same (Dudarev, 2013), the high effective activation energy of the
order of 2 eV seen in experiments on Zr likely result from the impurity effect
similar to that found in Fe. As noted by Kabir et al. (2010), at relatively
low temperatures the vacancy-carbon dimer complexes are immobile, making the
dissociation temperature of these complexes one of the key parameters
determining the response of a material to radiation exposure.
CRA simulations (Derlet and Dudarev, 2020) or the simulations involving the
production of defects by successive collision cascade events (Mason et al.,
2021; Granberg et al., 2023; Boleininger et al., 2023) do not imply the
absence of mobility of defects. Self-interstitial atom defects exhibit the
non-Arrhenius mobility (Dudarev, 2008) and their motion is strongly affected
by elastic strain fields (Dudarev et al., 2010), resulting in the rapid
clustering of these defects into interstitial dislocation loops and,
subsequently, into a dense entangled network of dislocations (Derlet and
Dudarev, 2020; Boleininger et al., 2022). The latter forms spontaneously at
doses above approximately $0.3\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$ (Mason
et al., 2020). The clustering of self-interstitial defects into dislocation
loops and dislocations stems from the fact that this is a highly energetically
favourable process, releasing up to $E^{f}_{SIA}\approx 3$ eV per self-
interstitial coalescence event (Domain and Legris, 2005; Dudarev, 2013).
The fact that it is the SIA formation energy, fundamentally related to the
strong elastic interaction between the self-interstitial defects, that drives
the evolution of microstructure at relatively low tempeatures rather than the
diffusion of self-interstitial per se, is confirmed by finite-temperature
simulations by Chartier and Marinica (2019). The simulations were performed at
300K and hence included the thermal diffusion of self-interstitial atom
defects, but still exhibited the same pattern of evolution as that predicted
by the CRA simulations (Derlet and Dudarev, 2020). This is confirmed by
experimental observations by Wang et al. (2023) showing the trends similar to
those found in simulations, even though in tungsten the diffusion of self-
interstitial defects occurs at temperatures as low as 27 K (Ehrhart et al.,
1991; Ma and Dudarev, 2019).
The above analysis of ab initio data and experimental information shows that
the temperature interval over which the dynamics of microstructural evolution
changes from the low-temperature mode dominated by microscopic stress
fluctuations (Derlet and Dudarev, 2020) to the high-temperature mode dominated
by the Arrhenius thermally activated diffusion (Allnatt and Lidiard, 1993), in
zirconium alloys spans approximately from
$300\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to
$400\text{\,}\mathrm{\SIUnitSymbolCelsius}$, as illustrated particularly well
by Fig. 8 from Topping et al. (2018). The qualitative picture of
microstructural evolution of zirconium irradiated by high-energy protons can
now be summarised as follows. Proton irradiation produces relatively low
energy recoils, generating defects in the form of Frenkel pairs or small
defect clusters (Boleininger et al., 2023). Self-interstitial atom defects
coalesce into dislocation loops and a dislocation network, whereas vacancies
either diffuse and recombine with the interstitial dislocation loops or
extended dislocation structures, or form immobile vacancy-impurity clusters
(Kabir et al., 2010). These vacancy-impurity clusters immobilise and stabilise
the dislocation microstructure (Arakawa et al., 2020), but dissociate in the
temperature interval from $300\text{\,}\mathrm{\SIUnitSymbolCelsius}$ to
$400\text{\,}\mathrm{\SIUnitSymbolCelsius}$. Over this temperature interval,
the mode of microstructural evolution changes from that dominated by stress
fluctuations and coalescence of self-interstitial defects to the mode
dominated by vacancy diffusion. Over the same interval of temperatures, the
IIG changes from a mode exhibiting saturation to that of runaway growth. Our
observations exhibit the formation of a dense dislocation network, suggesting
that the experimental conditions correspond to the low-temperature rather than
the high-temperature mode of microstructural evolution. The selection of a
simulation approach below reflects and recognises this fact. An algorithm for
modelling microstructural evolution at higher temperatures has to include the
treatment of microscopic stress fluctuations as well as diffusion and
interaction of vacancies and impurities. The development of such an algorithm
remains a challenge for future studies.
The CRA was implemented in the molecular dynamics program lammps (Plimpton,
1995) 111https://lammps.sandia.gov, 3 Mar and 29 Oct 2020 stable builds. For
this study, unless stated otherwise, we present results averaged across all
three Embedded Atom Method (EAM) potentials developed in Ref. (Mendelev and
Ackland, 2007), as is the case in Figure 1. Whilst there are variations
between the potentials with respect to their predicted formation energies and
elastic properties of self-interstitials and vacancies (Mendelev and Ackland,
2007; Varvenne et al., 2014; Varvenne and Clouet, 2017), we have found that
all three potentials qualitatively produce the same macroscopic dimensional
changes and microstructural evolution under the CRA (Warwick et al., 2021).
Furthermore, a similar study also employed the CRA on Zr and predicted the
same trends with a different potential (Tian et al., 2021).
Our simulations employ periodic boundary conditions and supercells containing
$\sim$ 2M and $\sim$ 10M atoms, with the cell edges parallel to the
$[2\bar{1}\bar{1}0]$, $[\bar{1}2\bar{1}0]$ and $[0001]$ directions. Energy
minimisation was performed using a combination of the conjugate gradient and
FIRE algorithms (Bitzek et al., 2006) such that the relaxed force on any atom
was smaller than $1\text{\,}\mathrm{m}\mathrm{e}\mathrm{V}\text{\AA}^{-1}$. In
the interest of computational efficiency, the simulation cell shape and size
was kept fixed during relaxation. Whilst these boundary conditions result in a
macroscopic internal stress of $\sim
1\text{\,}\mathrm{G}\mathrm{P}\mathrm{a}$, the dimensional changes that would
occur if the cell shape relaxed may nevertheless be accurately computed using
linear elasticity theory; furthermore, the microstructure resulting from
relaxing under zero pressure is similar to that under zero strain (Warwick et
al., 2021; Tian et al., 2021). Dislocations were identified directly from
atomic positions using the Dislocation eXtraction Algorithm (DXA) (Stukowski
et al., 2012). This is achieved by assigning crystal structure types to each
atom using common neighbour analysis (Faken and Jónsson, 1994). Given the
crystal structure of the hcp reference crystal, Burgers circuits are drawn
around regions containing atoms assigned to non-hcp crystal structure in order
to compute Burgers vectors and dislocation lines. Dislocation densities are
computed according to Equation 1.
In order to enable a closer comparison between our simulations and line
profile analysis experiments, we simulated the intensity profile of powder
diffraction patterns for all of the CRA microstructures using the Debye
equation (Debye, 1915) where the intensity of scattered X-rays is proportional
to
$\displaystyle I(q)=\sum_{i,j}{\mathrm{sinc}\left(2\pi qr_{ij}\right)},$ (9)
for wavevector magnitude $q$, and the sum runs over all the pairs of atoms
positioned at $\mathbf{r}_{i}$ and $\mathbf{r}_{j}$ separated by distance
${r_{ij}=|\mathbf{r}_{i}-\mathbf{r}_{j}|}$. Furthermore, in Equation 9 it is
assumed that the atomic form factor is the same for all atoms in the system.
The time required for computing all the pairwise distances $r_{ij}$ for a
system of $N$ atoms scales unfavourably as $N^{2}$ and thus we parallelised
the task. The line profile was calculated without using periodic boundary
conditions and thus the powder was treated as if it were composed of randomly
oriented nano-grains as large as the simulation box. $I(q)$ was computed over
the domain $3\text{\,}\mathrm{n}\mathrm{m}^{-1}$ $\leq q\leq$
$13\text{\,}\mathrm{n}\mathrm{m}^{-1}$ spanning all peaks up to the
$\\{22\bar{4}0\\}$ reflections and the wavenumbers were sampled every
$2\text{\times}{10}^{-3}\text{\,}\mathrm{n}\mathrm{m}^{-1}$.
Data were visualised and processed with the ovito (Stukowski, 2010) and
paraview (Ahrens et al., 2005) software packages. For more details, please
refer to our recent publication (Warwick et al., 2021).
## 3 Results and Discussion
### 3.1 Dislocation structure and distribution
(a)
(b)
(c)
Figure 3: Dislocation microstructures calculated by the CRA simulations using
the MA1 potential with 10M atoms. Green, orange and red dislocations indicate
$\langle 2\bar{1}\bar{1}0\rangle,\langle 1\bar{1}00\rangle$ and unusual
Burgers vectors, respectively. Dots are self-interstitial atoms identified
with the Wigner-Seitz analysis and boundaries of interstitial clusters are
rendered as transparent surfaces. 3(a): Self interstitial point defects and
small dislocation loops at very low dose 0.02 cdpa. 3(b) : Dense dislocation
network at low dose 0.13 cdpa. 3(c) : Extended dislocation network at high
dose 2.00 cdpa with percolating interstitial cluster.
The peak and saturation of dislocation density shown in Figure 1 can be
readily understood by inspecting the spatial configuration of dislocations
generated by our simulations. The nature of defects evolving through internal
stresses in the CRA results in dislocations being almost exclusively formed by
the agglomeration of self-interstitial atoms whilst vacancies remain immobile
and generally form small clusters containing $\mathcal{O}(10)$ vacancies that
are not large enough to relax into dislocation loops. We do find a small
number of much larger clusters containing $\mathcal{O}(100)$ vacancies.
However, these are fundamentally interstitial in origin as the large vacancy
clusters are nothing but vacant spaces in the crystallographic planes formed
by the self-interstitial defects, see Mason et al. (2020); Boleininger et al.
(2023). The renders shown in Figure 3 suggest that the dislocation structure
evolves in three stages. At low dose (Figure 3(a)), small loops form before
coalescing into a dislocation network (Figure 3(b)) at which point the
dislocation density saturates. At high doses (Figure 3(c)), full interstitial-
type atomic planes form and the dislocation network fragments into loops,
resulting in a drop in the dislocation density. The visualised microstructures
were rendered from simulations employing the MA1 potential. When comparing the
interatomic potentials, we discovered that the MA2 and MA3 potentials produce
large populations of twinned regions (Warwick et al., 2021). It seems likely
that these are artefacts of the potentials since such defects are not commonly
observed in experiment. As was noted by Warwick et al. (2021), for the MA3
potential in particular, a large proportion of dislocations coalesce into
these twinned regions whose volume fraction also features a transient peak
followed by saturation. The twinned regions are composed of dense arrays of
dislocations and thus the pattern of evolution of dislocation structures and
saturation in density is common to all three potentials. Thus, the
microstructures derived from MA1 simulations are presented in order to avoid
needlessly complicating our discussion.
Figure 4: Simulated $\\{1\bar{1}02\\}$ X-ray diffraction peak profile as a
function of dose computed from the MA1 microstructures. The variation in the
intensity of the peak and its centre shift is highlighted by the red path.
Application of the cmwp software to our simulated line profiles also shows
that the dislocation density saturates as a function of dose, approaching a
limiting value of $\sim 10^{15}$ m-2.
CRA simulations tend to overestimate the defect content in the high dose limit
(Boleininger et al., 2023), however the qualitative trends observed under a
variety of conditions are predicted accurately (Mason et al., 2020, 2021;
Warwick et al., 2021). This overestimation arises from the lack of re-
crystallisation induced by collision cascades. Producing the primary knock on
atoms with kinetic energies close to the threshold displacement energy results
in defect densities very close to those predicted by the CRA whilst much
larger recoil energies result in defect densities lower by approximately a
factor of approximately 10, see Boleininger et al. (2023). Thus, in agreement
with the analysis by Mason et al. (2020, 2021); Boleininger et al. (2023)
where the defect content was independently assessed using the experimentally
measured deuterium retention, we have scaled down the dislocation density in
Figure 1 by a factor of 10 to enable a direct comparison of our experimental
and simulation data. After applying this scaling it can be seen that the CRA
indeed predicts a qualitatively accurate variation dislocation density
profile. The occurrence of a transient peak of dislocation density at a
moderate dose was also noted earlier in simulations of iron and tungsten
(Chartier and Marinica, 2019; Derlet and Dudarev, 2020).
In order to enable a closer comparison with experiment we simulated a powder
diffraction line profile for all of our CRA microstructures as described in
§2.3. Figure 4 illustrates the evolution of the $\\{1\bar{1}02\\}$ diffraction
peak profile as a function of dose, reflecting the eventual saturation of the
microstructure in the peak profile seen in the limit of high dose. We observe
that in the transient regime, the peak intensity drops before rising and
settling at a steady value. Furthermore, the peak broadens at high dose and
shifts to higher scattering angles, indicating lattice compression. The
formation of extra atomic planes due to the coalescence of dislocation loops
does not result in the volumetric expansion but instead causes lattice
compression because the simulations are performed at constant cell shape and
size. Under zero pressure boundary conditions, we expect the peak centre to
shift to lower wave-numbers instead.
Saturation of the peak profile has been quantified by extracting the
dislocation density from the data using the cmwp software. cmwp is known to
infer dislocation densities that are larger than those determined from TEM
images and this has been attributed to the ability of X-ray line profile
analysis to resolve small loops in power law distributed dislocation loops
(Ungár et al., 2021, 2021). Interestingly, we find that the dislocation
densities computed by cmwp and DXA nevertheless differ by approximately an
order of magnitude although the character of variation of the observed and
simulated dislocation densities as functions of dose are the same. We note
that the difference between the cmwp software and DXA results brings attention
to an important question: at what size is a dislocation loop too small to be
counted as a dislocation object?
Usually, dislocations are considered to be the sources of long-range strain
fluctuations responsible for the X-ray peak line profile broadening detected
in irradiated materials (Wilkens, 1970). On the other hand, small dislocation
loops produce shorter-range strains and in the far-field limit they are
equivalent to point defects, with the associated strain fields resulting in
the Huang diffuse scattering, producing a relatively uniform increase in the
scattered intensity in X-ray diffraction patterns (Simmons and Baluffi, 1958).
Furthermore, when a dislocation loop is so small that the loop diameter is
comparable to the core width of a dislocation then the loop is mostly
comprised of core atoms and its structure can no longer be described using
conventional elasticity (Boleininger et al., 2018; Boleininger and Dudarev,
2019). Thus, one would expect the resulting strains to be unlike those
associated with linear elastic fields of dislocation loops and the core
effects to be significant. Modelling the strain broadening effects associated
with small dislocation loops requires further analysis and we defer it to a
future study. A large proportion of the dislocation objects found in the
simulated microstructures are so small that they should be treated as point
defects by CMWP and would also not be easily detected in TEM images (Zhou et
al., 2006). Determining the nature of such defects could be highly relevant to
determining mechanisms that cause the complex high dose phenomena such as
breakaway growth.
(a)
(b)
Figure 5: 5(a) : Histogram of self-interstitial cluster population sizes
detected in MA1 microstructures containing 10M atoms corresponding to Figures
3. The population sizes of clusters, $N$, follow a power law distribution.
$N_{\mathrm{SIA}}$ denotes the total number of self-interstitial atoms
detected by the Wigner-Seitz algorithm. The data are presented on a log-log
scale. Lines of best fit were calculated using the maximum likelihood
estimates (MLE) for the exponent in a power law distribution stated in Eq. 10.
5(b): Evolution of the MLEs for the power law exponent $\alpha$ as a function
of dose averaged over all three interatomic MA potentials. Shaded regions
represent confidence intervals of the MLE. Data for the doses lower than
$1\text{\,}\mathrm{c}\mathrm{d}\mathrm{p}\mathrm{a}$ are presented on a log-
linear scale whereas in the saturation regime
$>1\text{\,}\mathrm{c}\mathrm{d}\mathrm{p}\mathrm{a}$ a linear-linear scale is
employed. $\alpha$ was averaged over doses larger than
$1\text{\,}\mathrm{c}\mathrm{d}\mathrm{p}\mathrm{a}$ to arrive at the value
$\langle\alpha\rangle=2.28\pm 0.01$ indicated by the horizontal dashed blue
line for a 2M cell, and $\langle\alpha\rangle=2.23\pm 0.01$ (dashed green
line) for a 10M cell.
Size distributions of dislocation loops are often investigated in experiment
(Yi et al., 2015) and so for the purposes of comparison and characterising the
spatial arrangement of dislocations we examined the distribution of defect
cluster - dislocation loop sizes. As the dislocations in this simulation are
mostly of interstitial type, we can gain insight into the statistics of
dislocation structures by examining the statistics of interstitial defect
clusters. Figure 5(a) shows the frequency distribution of clusters containing
$N$ interstitials as calculated by ovito for the doses corresponding to the
renders in Figure 3. For clusters with population sizes $N<10$, the bin widths
of the histogram are equal to 1 whereas at larger cluster population sizes the
bin widths increase logarithmically (Milojević, 2010). The raw data for the
cluster population size frequencies were averaged over the bin widths to
produce the step chart shown. Visually, the histograms appear to follow a
straight line on a log-log scale, suggesting a power law distribution.
Furthermore, simulations employing the CRA have shown evidence for self-
organised critical behaviour (Derlet and Dudarev, 2020) and thus it is
reasonable to test the cluster population size power law distribution
hypothesis such that the probability mass function for clusters containing $N$
interstitials is given by
$p(N)=\frac{1}{\zeta(\alpha)N^{\alpha}}$ (10)
where the Riemann zeta function (Heynsworth and Goldberg, 1965) is defined as
$\zeta(\alpha)=\sum_{k=1}^{\infty}1/k^{\alpha}$ for exponent $\alpha>1$.
We may define an exponent that best fits the data shown in Figure 5(a) via a
maximum likelihood estimation. The likelihood function
$\displaystyle\mathcal{L}\left(\\{N_{i}|i\in[1..N_{\mathrm{tot}}]\\}|p(N|\alpha)\right)=\prod_{i=1}^{N_{\mathrm{tot}}}p(N_{i}|\alpha)$
(11)
returns the probability of observing the $N_{\mathrm{tot}}$ measured data
points $\\{N_{i}|i\in[1..N_{\mathrm{tot}}]\\}$ if they were produced from a
distribution $p(N|\alpha)$ of given parameter(s) $\alpha$. The MLE is produced
by determining the value of $\alpha_{\mathrm{MLE}}$ maximising
$\ln\mathcal{L}$ such that:
$\displaystyle\left.\frac{\partial\ln\mathcal{L}}{\partial\alpha}\right\rvert_{\alpha=\alpha_{\mathrm{MLE}}}=0.$
(12)
Equation 12 was solved numerically using the python package powerlaw (Alstott
et al., 2014) and as shown in Figure 5(b), the MLEs and associated standard
error for $\alpha$ exhibit transients over a range of doses corresponding to
the formation of a dislocation network by the coalescence of loops. The
minimum of this transient appears to be correlated with the peak in
dislocation density before saturating to a constant value at high doses. The
twinned regions that emerge when employing the MA2 and MA3 potentials, as
described at the beginning of §3.1, are also interstitial defect clusters.
Therefore they are included in this statistical analysis and we find the same
trend and remarkably similar values of $\alpha$ across all three MA
potentials. Averaging across the potentials for doses larger then
$1\text{\,}\mathrm{c}\mathrm{d}\mathrm{p}\mathrm{a}$ in the saturation regime,
we find the MLE for exponent $\alpha=2.28\pm 0.01$ for a 2M simulation cell
and $\alpha=2.23\pm 0.01$ for a 10M simulation cell. These values are close
but slightly higher than the exponents found in simulations and observations
of collision cascades in tungsten in the limit of low dose (Sand et al., 2013;
Yi et al., 2015).
Figure 6: Estimation of dislocation density from interstitial cluster
perimeter density. $N$ corresponds to the number of atoms in the cluster and
the perimeter of a given cluster is estimated according to Equation 13.
An immediate observation and the consequence of the power law statistics is
that the overwhelming majority of clusters are small. Thus, even the order of
magnitude of the calculated dislocation density is sensitive to the choice one
makes for a minimum detectable size of a dislocation loop. To show this,
assume that clusters with cluster population sizes $N$ lying in a range
${N_{\text{min}}<N<N_{\text{max}}}$ correspond to a circular platelet of
interstitial atoms i.e. a dislocation loop (Gilbert et al., 2008). The
$N_{\text{min}}$ and $N_{\text{max}}$ thresholds correspond to cluster
population sizes that are either sufficiently small to be treated as point
defects, or are large enough to be close to the threshold for forming a
percolating atomic plane consisting of interconnected dislocation loops.
Assume that each platelet contains one extra atom per atomic string in the
direction of the Burgers vector (Boleininger et al., 2018; Boleininger and
Dudarev, 2019), and involves $N$ atoms with volume $\Omega$ equal to the
atomic volume of $\alpha$-Zr. Using the formula $({\bf b}\cdot{\bf A})$ for
the volume of a dislocation loop, its perimeter can then be estimated as
$P_{S}=2\pi\sqrt{{N\Omega\over\pi b}},$ (13)
where $b$ is the length of the Burgers vector along the normal to the loop
habit plane. Here, we assume that the platelets form $a$-type dislocation
loops such that $b$ is equal to the $a$ lattice parameter.
Summing the perimeters corresponding to cluster population size distributions
between various choices of $N_{\text{min}}$ and $N_{\text{max}}$ provides a
measure of the difference in dislocation density due to different choices of
thresholds. Typically, a dislocation core extends over about five interatomic
distances (Boleininger et al., 2018; Boleininger and Dudarev, 2019) and thus
one would reasonably argue that $N_{\text{min}}$ should at least be greater
than five. Furthermore, we should not include the fully formed planes
corresponding to a cluster percolating through the periodic boundaries, thus
setting $N_{\text{max}}=10^{3}$. In Figure 6 we observe an order of magnitude
difference between the results involving the counting of all the viable
clusters smaller than the population size of the percolating cluster
($5<N<10^{3}$), and the values found by counting all the clusters but
increasing the lower threshold to exclude clusters containing less than 100
self-interstitial atoms. This example illustrates how the power law
distribution of defect sizes in highly irradiated microstructures can
profoundly affect how dislocations are counted and their density quantified.
We may also employ Equation 13 to estimate the power law exponent for the
distribution of loops with respect to loop diameter $D$ where it is apparent
that, assuming that all the loops have the same Burgers vector $b$, $N\propto
D^{2}$. Treating $N$ as a continuous variable, we derive the probability
density function as a function of loop diameter
$\displaystyle p_{D}(D)=\left\lvert\frac{\mathrm{d}N}{\mathrm{d}D}\right\rvert
p(N(D))\propto\frac{1}{D^{2\alpha-1}},$ (14)
where $\alpha$ is the exponent entering Equation 10. Hence the diameter of
circular loops is expected to be power law distributed with exponent
$\beta=2\alpha-1$. Using the data derived from the CRA simulations in the high
dose limit illustrated in Figure 5(b) for a 10M atom cell, we find that
$\beta=3.46$. In experiments performed by Ungár et al. (2021) dislocation
diameter data measured by TEM were combined with an XRD line profile
measurement of the total dislocation density. When fitting the dislocation
size distribution to a power law the exponent was found to lie in the range
$3\leq\beta\leq 4$, which agrees well with the values derived from our
simulations.
Concluding this section, we note that power law exponent values $\alpha=2$ and
$\beta=3$ appear to represent natural low limits characterising the power law
statistics of populations of dislocation loops in a heavily irradiated
material. Indeed, any power law distribution with $\alpha<2$ would imply a
divergent total count of interstitial defects contained in the loops, a
paradox that can only be resolved by recognising that loops of very large size
are nothing but extra crystallographic atomic planes, which through elastic
interactions would tend to modify the finite-size part of the dislocation loop
distribution so as to make it normalisable, in this way steering the value of
exponent $\alpha$ towards and above the limiting value of 2.
### 3.2 Stored energy
Whilst analysing the configuration of atoms helps to describe how the
microstructure evolves, on its own this approach provides limited insight into
why the observed processes take place. Evidently, the repeated creation and
relaxation of defects forces the microstructure into a state that has the
energy higher than that of a single crystal. By examining the excess energy
$E_{exc}$ that the system is driven to, we may determine how the accumulation
of point defects produces the observed dislocation density. For an irradiated
microstructure at dose $\phi$ containing $N$ atoms with total energy
$E_{total}(\phi)$ we may measure the excess (stored) energy as
$\displaystyle E_{exc}(\phi):=E_{total}(\phi)-NE_{coh},$ (15)
where $E_{coh}$ is the cohesive energy of $\alpha$-Zr.
Our simulations are carried out under zero global strain boundary conditions.
Whilst this is computationally convenient, in reality specimens are often
irradiated under zero applied stress, allowing the body as a whole to undergo
strain. Assuming linear elasticity, we may correct for this and remove the
stored elastic energy $E_{el}$ from our simulation results. The defects
produced by radiation damage induce eigenstrains, also known as residual
strains, that act as sources of elastic strain. Let $\epsilon_{ij}^{0}$ denote
the elastic strain that would come about under zero applied stress boundary
conditions. In this case, the potential energy of the body is lowered by doing
work equal to $\frac{1}{2}\sigma_{ij}\epsilon_{ij}^{0}$ where Hooke’s law
determines the stress $\sigma_{ij}=C_{ijkl}\epsilon_{kl}^{0}$ and the elastic
constants tensor is $C_{ijkl}$. Enforcing zero global strain requires that the
integral of the strain $\epsilon_{ij}$ over the body with volume $V$ is zero
or, equivalently, that one has zero volume averaged strain
$\displaystyle\langle\epsilon_{ij}\rangle:=\frac{1}{V}\int_{V}\mathrm{d}^{3}\mathbf{x}^{\prime}\,\epsilon_{ij}(\mathbf{x}^{\prime})=0.$
(16)
This boundary condition is satisfied by the strain
$\displaystyle\epsilon_{ij}(\mathbf{x})=\epsilon_{ij}^{0}(\mathbf{x})-\langle\epsilon^{0}_{ij}\rangle.$
(17)
Applying Hooke’s law to Equation 17, we observe that the system is under a
state of global stress
$\langle\sigma_{ij}\rangle=C_{ijkl}\langle\epsilon^{0}_{kl}\rangle$ and we may
treat this as the stress that develops in our simulations. Therefore, we can
compute the stored elastic energy
$\displaystyle E_{el}=\frac{1}{2}\langle\sigma_{ij}\rangle
S_{ijkl}\langle\sigma_{kl}\rangle,$ (18)
where $S_{ijkl}$ is the elastic compliance tensor (Warwick et al., 2021)
related to $C_{ijkl}$ by
$S_{ijpq}C_{pqkl}=\frac{1}{2}\left(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right)$.
We find across all the employed potentials that $E_{el}$ accounts only for a
small fraction $<10\%$ of the stored energy, meaning that the vast majority of
stored energy is contained in the point defect centres, dislocation cores and
fluctuations arising from the elastic fields of these defects. Indeed, when a
high dose snapshot was explicitly relaxed under zero pressure we found that
the difference in excess energy to that corrected for by our estimate of
$E_{el}$ is of the same order of magnitude. Furthermore, the microstructure
did not change significantly providing more indication of the minor role of
the global elastic energy.
(a)
(b)
(c)
Figure 7: Contributions of microstructural defects to the excess energy.
Results shown are for the MA1 potential using 10 million atoms. 7(a):
Comparison of vacancy energy $E_{vac}$ estimated by assuming all $N_{V}$
vacancies are isolated with formation energy $E^{f}_{1vac}$ (blue curve) with
$E_{vac}$ calculated via explicitly relaxing pristine $\alpha$-Zr containing
the same number and arrangement of vacancies. 7(b) : Breakdown of total excess
energy $E_{exc}$ into elastic $E_{el}$, interstitial $E_{int}$ and vacancy
$E_{vac}$ contributions. 7(c): Comparison of interstitial excess energy,
concentration and dislocation density that exhibits the similarity in profile
between all three quantities. A peak occurs in all three profiles near
$0.1\text{\,}\mathrm{c}\mathrm{d}\mathrm{p}\mathrm{a}$.
The remaining stored energy is confined in the two classes of defects
associated with vacancies and interstitials. Given that vacancies do not
cluster significantly, we may estimate their energetic contribution as
$\displaystyle E_{vac}(\phi)=N_{vac}(\phi)E^{f}_{1vac},$ (19)
where, for dose $\phi$, the number of vacancies is denoted $N_{vac}$ and
$E^{f}_{1vac}$ is the formation energy of a single vacancy. Values of
$E^{f}_{1vac}$ for each of the potentials used in this study may be found in
Mendelev and Ackland (2007). To check the validity of Equation 19 we isolated
the $N_{vac}$ vacancies identified by Wigner-Seitz analysis and subsequently
relaxed an $N$-atom supercell of pristine $\alpha$-Zr containing the same
number of vacancies in the same positions. The simulation cell was relaxed and
the formation energy was computed as
$\displaystyle E_{vac}=E^{total}_{vac}-(N-N_{vac})E_{coh},$ (20)
where $E^{total}_{vac}$ is the resulting total energy. Figure 7(a) summarises
this process and shows the resulting formation energy for these arrangements
of vacancies. Thus we observe that Equation 19 is a good approximation,
providing further evidence that the majority of $E_{vac}$ is due to isolated
vacancies.
The contribution to $E_{exc}$ from the formation energy of small interstitial
clusters and dislocation content may now be calculated as
$\displaystyle E_{int}=E_{exc}-E_{el}-E_{vac}.$ (21)
In Figure 7(b) we show the relative proportion of elastic, interstitial and
vacancy contributions to the total excess energy where it is evident that the
elastic contribution is in the minority, vacancy contributions dominate and
the interstitial contribution follows a similar trend to the dislocation
density profile shown in Figure 1. The profile of $E_{int}$ is shown in Figure
7(c) together with the interstitial concentration and dislocation density in
order to highlight their correlation with each other.
Recently, Differential Scanning Calorimetry (DSC) experiments were performed
to measure the stored energy of irradiated titanium as a means of inferring
the number of defects present (Hirst et al., 2022). The measurements indicated
stored energies associated with irradiation induced defects to be on the order
of $0.1\text{\,}\mathrm{J}\mathrm{g}^{-1}$. Their analysis also allowed the
authors to infer the presence of defects that are invisible to TEM imaging. At
$0.07\text{\,}\mathrm{c}\mathrm{d}\mathrm{p}\mathrm{a}$ in Figure 7(c), we
find that the peak in specific energy associated with $E_{int}$ is
$25\text{\,}\mathrm{J}\mathrm{g}^{-1}$ that subsequently drops and plateaus at
high dose to $12\text{\,}\mathrm{J}\mathrm{g}^{-1}$. At elevated temperatures,
the defect content is likely to be $\sim 10\%$ of that calculated in our
simulations cf. (Mason et al., 2020, 2021) and thus we expect the associated
stored energy in Zr to be comparable to $1\text{\,}\mathrm{J}\mathrm{g}^{-1}$.
## 4 Conclusions
In summary, we have performed experiments and simulations showing that the
dislocation density in irradiated zirconium and zircaloys exhibits a peak at a
moderate dose and then saturates at doses greater than
$1\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$. Simulations indicate that this
occurs in a regime of dose rate and temperature where microstructural
evolution is predominantly driven by stress relaxation. The material enters a
critical state at $\sim 1\text{\,}\mathrm{d}\mathrm{p}\mathrm{a}$, where
interstitial clusters grow to a sufficient size to percolate the volume of the
material. At high dose, the population of smaller clusters and dislocation
loops is distributed as a function of cluster defect content according to a
power law statistics with the exponent close to $\alpha\approx 2.2$. As a
function of defect diameter, this results in a power law distribution of
defect clusters with exponent of $\beta\approx 3.5$, which compares favourably
with the range of values $3\leq\beta\leq 4$ derived from experimental
observations (Ungár et al., 2021).The analysis highlights the significance of
precise definition of defect sizes included in the measured dislocation
densities. Irrespectively of the statistics of dislocation structures, the
trend in the dislocation density evolution in zirconium irradiated at
temperatures below $\sim 350^{\circ}\>\mathrm{C}$ is clear, the dislocation
density saturates as a function of dose.
## 5 Acknowledgements
This work received funding from the RCUK Energy Programme Grant No.
EP/W006839/1 and MIDAS EPSRC Grant No. EP/S01702X/1, and was partially carried
out within the framework of the EUROfusion Consortium,funded by the European
Union via the Euratom Research and Training Programme (Grant Agreement No
101052200 — EUROfusion). Views and opinions expressed are however those of the
authors only and do not necessarily reflect those of the European Union or the
European Commission. Neither the European Union nor the European Commission
can be held responsible for them. We acknowledge DESY (Hamburg, Germany), a
member of the Helmholtz Association HGF, for the provision of experimental
facilities. We gratefully acknowledge the use of the high-performance
computing facility MARCONI (Bologna, Italy) provided by EUROfusion, and
computing resources supplied by the IRIS (STFC) Consortium. This work also
received support from the EPSRC Access to HPC Programme on the ARCHER2 UK
National Supercomputing Service (http://www.archer2.ac.uk).
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Subsets and Splits
Filtered Text Samples
Retrieves 100 samples of text containing the specific phrase "You are a helpful assistant", providing limited insight into the dataset.
Helpful Assistant Text Samples
Returns a limited set of rows containing the phrase 'helpful assistant' in the text, providing basic filtering of relevant entries.