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Information-Thermodynamic Bound on Information Flow in Turbulent Cascade

We investigate the nature of information flow in turbulence from an information thermodynamic viewpoint. For the shell model with thermal noise, we show that information of large-scale eddies is transferred to small scales along with the energy cascade. The information transfer rate is characterized by the large-eddy turnover time. Furthermore, we numerically show that the information-thermodynamic efficiency is extremely low compared to other information processing systems such as Maxwell's demon. This result implies that transferring information from large to small scales involves enormous thermodynamic costs, indicating the poor performance of turbulence as an information processing system.

Introduction.-Various phenomena observed in complicated systems such as the earth system [1][2][3] and chemical reaction network [4] can be regarded as nonequilibrium cooperative phenomena that emerge from many-body interactions. To elucidate and control the dynamics behind such phenomena, it is pertinent to focus on information transfer between components constituting the system. Particularly in mesoscopic systems affected by thermal fluctuations, the nature of such information transfer can be described by information thermodynamics [5]. Information thermodynamics is essentially stochastic thermodynamics for subsystems [6,7] and provides constraints that are consistent with thermodynamics on the exchange of heat and information between subsystems. Recently, it has been applied to information processing at the cellular level in biological systems [8][9][10][11][12] and even to deterministic chemical reaction networks [13].
Turbulent cascade is also a nonequilibrium cooperative phenomenon that emerges from extremely complicated interactions. In fully developed three-dimensional fluid turbulence, kinetic energy is transferred conservatively from large to small scales [14]. This energy cascade phenomenon can be described intuitively as the successive generation of smaller vortices by the stretching of larger vortices. Along with the energy cascade, fluctuations of small-scale quantities (e.g., the energy dissipation rate) follow those of large-scale quantities (e.g., the energy injection rate) with a time delay that corresponds to the large-eddy turnover time [15][16][17]. Moreover, the energy cascade induces chaos synchronization of smallscale motions, where small-scale velocity field is slaved to the chaotic dynamics of large-scale velocity field [18][19][20][21][22]. These phenomena suggest that information flow manifests the intrinsic characteristics of turbulence dynamics.
These observations motivate us to explore the nature of the information transfer across scales associated with turbulent cascade. Revealing the details of the information transfer process in turbulence may not only provide insights into the mechanism underlying various phenomena that characterize turbulence dynamics, but also allow comparative studies with other information processing systems. While turbulence has been studied in various contexts from such an information-theoretic viewpoint over recent decades [23][24][25][26][27][28][29][30][31][32][33][34], the nature of the information flow associated with turbulent cascade has not yet been elucidated. In particular, no previous studies have considered the effects of the thermal fluctuations that are inherent in fluids on information transfer, even though the thermal fluctuations affect the turbulence dynamics significantly [35][36][37][38][39][40]. In this regard, information thermodynamics may enable us to obtain fundamental constraints on information flow while taking the effects of thermal fluctuations into account.
Here, we aim to reveal the laws that govern information flow in fully developed three-dimensional fluid turbulence using information thermodynamics. To this end, we employ fluctuating hydrodynamics to explicitly take into account thermal fluctuations that are inherent in fluids [41,42]. Specifically, we use the stochastic Sabra shell model, which is a simplified caricature of the fluctuating Navier-Stokes equations in wave number space. This model has recently been used to investigate the effects of thermal noise on turbulence [36,37].
In this Letter, we show that large-scale eddies destroy information about small-scale eddies while transferring energy to small scales; in contrast, the small-scale eddies gain information about large-scale eddies while receiving energy from large scales. We numerically illustrate our findings and show that the rate of information transfer is characterized by the large-eddy turnover time. Furthermore, our numerical simulations suggest that the corresponding information-thermodynamic efficiency is quite low compared to other information processing systems such as Maxwell's demon. This implies that transferring information from large to small scales involves enormous thermodynamic costs, indicating the poor performance of turbulence as an information processing system.
Setup.-We consider the Sabra shell model with thermal noise [36,37,43]. Let u n (t) ∈ C be the "velocity" at time t with the wave number k n = k 0 2 n (n = 0, 1, · · · , N ). The time evolution of the complex shell variables u := {u n } is given by the following Langevin equation: with the scale-local nonlinear interactions given by Here, ν > 0 represents the kinematic viscosity, f n ∈ C denotes the external body force that acts only at large scales n ≤ n f < N , and ξ n ∈ C is the zero-mean white Gaussian noise that satisfies ξ n (t)ξ * n (t ) = 2δ nn δ(t − t ). The specific form of the thermal noise term satisfies the fluctuation-dissipation relation of the second kind, where T denotes the absolute temperature, k B the Boltzmann constant, and ρ the mass "density".
Basic properties.-The nonlinear term B n (u, u * ) satisfies the following relation: Hence, the energy balance equation reads where := n f n=0 u n f * n + u * n f n /2 denotes the energy injection rate. In the steady state, the energy injection rate balances the dissipation rate as follows: Here, we have ignored the energy injection due to the thermal noise by noting that the kinetic energy is much larger than the thermal energy over a wide range of scales in turbulence [36,37]. Shell models are known to exhibit rich temporal and multiscale statistics that are similar to those observed in real turbulent flow [44]. Most importantly, the model exhibits energy cascade. To see this, we consider the time evolution of the large-scale energy at the scales k n ≤ "learning" destruction Energy cascade where Π K denotes the scale-to-scale energy flux from the large scales k n ≤ K to the small scales k n > K: Because the viscous dissipation is negligible at scales much larger than the Kolmogorov dissipation scale η ≡ k −1 ν := ν 3/4 −1/4 , the last two terms on the righthand side of (6) can be ignored within the inertial range k f K k ν , where k f := k n f . Hence, in the steady state, we obtain The energy is thus transferred conservatively from large to small scales within the inertial range. Note that while the condition k f < K instead of k f K is sufficient, we use the conventional definition for the inertial range.
Information-theoretic quantities.-Here, we introduce important quantities that characterize the information transfer among the shell variables. To quantify the information transfer across scales, we define the large-scale and small-scale shell variables as U < K := {u n , u * n |n ≤ n K } and U > K := {u, u * } \ U < K , respectively (see Fig. 1). The strength of the correlation between U < K and U > K is quantified by the mutual information (MI) [45]: These inequalities are the main result of this Letter, which will be derived later. The inequality (12) implies that the large-scale shell variables U < K are destroying information about the small-scale variables U > K while transferring kinetic energy to small scales. In contrast, (13) states that the small-scale shell variables U > K are "learning" about U < K while receiving the kinetic energy from large scales (see Fig. 1). In particular, the maximum LR is given by ρ /k B T .
Numerical simulation.-We here numerically illustrate the main result by estimating the LR. We set N = 22 and n f = 1 to ensure that the external force acts only on 0th and 1st shells of the total 23 shells. The values of the external force and the other parameters are chosen following [36,37] so that the achieved Reynolds number (Re) and the ratio of the thermal energy to the kinetic energy at the Kolmogorov dissipation scale are both comparable to the typical values in the atmospheric boundary layer, i.e., Re ∼ 10 6 with k B T /ρu 2 η ∼ 10 −8 , where u η := ( ν) 1/4 denotes the characteristic velocity at the Kolmogorov dissipation scale. To investigate the Reynolds number dependence of the LR, we have also performed a simulation for a low Reynolds number, Re ∼ 10 5 , by setting N = 19. For both N = 19 and 22 cases, we have used N samp = 3 × 10 5 samples in the following averaging and estimation. Further details of the numerical simulation are given in [48]. Figure 2(a) shows the energy spectrum E n := |u n | 2 ss /2 in the steady state. The achieved Reynolds numbers are Re 9.25 × 10 4 and 1.46 × 10 6 for the two cases N = 19 and 22, respectively. From this figure, we can see that the spectrum is consistent with the Kolmogorov spectrum in the inertial range, E n ∝ k −2/3 n , while it exhibits the equipartition of energy in the dissipation range, E n = k B T /ρ.
We now estimate the LR defined by (10) and (11). To this end, we first estimate the MI. Note that the naive binning approach is not feasible in this case because it requires estimation of the 2(N + 1)-dimensional probability density p t (U < K , U > K ). Instead, we use the so-called Kraskov-Stögbauer-Grassberger (KSG) estimator [49][50][51][52], which has the advantage that it does not require estimation of the underlying probability density. The KSG estimator uses the distances to the k-th nearest neighbors of the sample points in the data to detect the structures of the underlying probability distribution. While we set k = 4 here, following [49], essentially the same result can be obtained for other values of k. Because the KSG estimator is based on the local uniformity assumption of the probability density, the estimated value approaches the true value as N samp → ∞ when this assumption is satisfied. Figure 2(b) shows the estimated MIÎ . Its standard deviation is also estimated to be ∼ 10 −3 by subsampling [51,52], which lies within the marker size. Notably, the MI is almost independent of K in the inertial range, while it takes relatively large and small values in the injection and dissipation scales, respectively. This result reflects the dynamics that is scale-invariant in the inertial range while affected by external forces and thermal fluctuations in the injection and dissipation scales.
The LR can be estimated by substitutingÎ KSG into (10) or (11). Note that this procedure requires high accuracy in the estimation of the MI because the LR is defined through increments in the MI. Because it is not feasible to increase the number of samples, we instead take the approach of using the largest possible time increment ∆t. Correspondingly, we focus only on l < K , because l > K is defined through increments in the small-scale shell variables U > K , which are fast variables relative to U < K . We therefore estimate l < K , as defined by (10), usinĝ Because we are interested in K within the inertial range, we choose ∆t such that it is smaller than the smallest time scale in the inertial range. Therefore, we set ∆t = 0.1τ η , where τ η := η/u η denotes the typical time scale at the Kolmogorov dissipation scale. Note that ∆t is different from the time step δt := 10 −5 τ η used in solving (1) numerically.
In Fig. 2(c), we show the estimated LRl < K in units of the inverse of the large-eddy turnover time τ L := 1/k 0 u rms , where u 2 rms := N n=0 |u n | 2 ss . For Re 10 5 , we find that τ L 181τ η , while for Re 10 6 , we find that τ L 734τ η . From this figure, we can see that the LR takes negative values for K within the inertial range. Because (ρ /k B T )τ L 7.79 × 10 9 for Re 10 5 and 3.15 × 10 10 for Re 10 6 , (12) is indeed satisfied. Interestingly, the lower bound of (12) is a loose bound. By noting that l < K = −l > K in the steady state, this result states that the information-thermodynamic efficiency de-fined as l > K k B T /ρ is quite low [53]. In other words, the small-scale eddies acquire information about the largescale eddies at a relatively high thermodynamic cost. This property is in contrast to other information processing systems such as Maxwell's demon [11,47,53] and thus characterizes turbulence dynamics. Furthermore, Fig. 2(c) suggests that the LR can be scaled as l < K ∼ C/τ L in the inertial range, where C denotes a Reindependent dimensionless constant. By noting that τ L can be interpreted as the characteristic time scale for the largest eddies to be stretched into smaller eddies, this result implies that the information of large-scale eddies is transferred to small scales by this stretching process.
Derivation of the main result.-The derivation of the main result is based on the second law of information thermodynamics for bipartite systems [53]. Specifically, we first formulate the second law of information thermodynamics for (1) by dividing the shell variables {u, u * } into the two groups U < K and U > K , and then we take the inviscid limit ν → 0.
Let S[U < K ] be the Shannon entropy of the largescale shell variables, which is defined by S The entropy increase in the heat bath associated with U < K is given by the sum of the contributions of each shell [54][55][56]: where • denotes the multiplication in the sense of Stratonovich [57]. Similarly, let S[U > K ] andṠ > env be the Shannon entropy and entropy increase in the heat bath associated with the small-scale shell variables U > K , respectively. The second law of information thermodynamics is then given by [56] If U < K and U > K are independent, then l < K = l > K = 0 and the standard second law of thermodynamics follows. In contrast, if they are correlated, then l < K and l > K can be either positive or negative.
We now assume that the system is in the steady state and set K to be within the inertial range k f K k ν . Then,Ṡ < env andṠ > env can be expressed in terms of the energy flux (7) aṡ where we have used Π k N = 0, which follows from (3). By substituting these expressions into (16) and (17) and by noting that Π K ss → as K/k ν → 0 and l < K + l > K = 0 in the steady state, we arrive at the main result (12) and (13).
Concluding remarks.-We have investigated the nature of information flow in turbulence from an information thermodynamic viewpoint. Thermal fluctuations are crucial in deriving the universal relation (12) and (13) for information flow. On the other hand, it should be noted that the information flow itself is mainly governed by the large-scale dynamics rather than by the thermal fluctuations. In fact, the LR in the deterministic case (T = 0) is found to be almost the same as in the noisy case (T > 0) in the inertial range [58]. This result is consistent with the fact that the LR can be scaled as l < K ∼ C/τ L in the inertial range.
We now provide some technical remarks on estimation of the LR. Although the KSG estimator used here is asymptotically unbiased for N samp → ∞, there are a sample-size-dependent bias and a k-dependent bias for a finite N samp in general [51,52]. In our case, we have found that the magnitude ofÎ . This may be because the probability distribution p t (U < K , U > K ) is skewed and has heavy tails, thus violating the local uniformity condition [51]. Nevertheless, we have confirmed that the sign ofl < K does not depend on the choice of k [52]. It should also be noted that the number of samples N samp used here is not sufficient for high accurate estimation of the LR because the standard deviation of the estimated MI is comparable to its increment. In other words, if we naively estimate the error bar of the LRl < K by using the estimated standard deviation of the MI, it is of the same order asl < K itself. It is therefore desirable to perform the numerical calculations with higher accuracy while taking the bias into account.
Although we have focused on the Sabra shell model, similar results to those presented here would hold for other turbulence models, including the fluctuating Navier-Stokes equations [41,42]. Because turbulent cascade is a ubiquitous phenomenon found in quantum fluids [59][60][61][62], supercritical fluids near a critical point [63], elastic bodies [64,65], and even spin systems [66][67][68], it would be interesting to investigate the nature of the information flow in these various systems.
We thank Masanobu Inubushi, Wouter J. T. Bos, Susumu Goto, and Shin-ichi Sasa for fruitful discussions. We also thank Dmytro Bandak and Gregory L.

S1. DERIVATION OF THE SECOND LAW OF INFORMATION THERMODYNAMICS
In this section, we explain the derivation of the second law of information thermodynamics for the stochastic Sabra shell model.

A. Formulation of the second law of thermodynamics
First, we formulate the standard second law of thermodynamics. Let S[u, u * ] := − dudu * p t (u, u * ) ln p t (u, u * ) be the system entropy, where dudu * := n dRe[u n ]dIm[u n ], and p t (u, u * ) denotes the probability distribution function. Here, we use the notation S[u, u * ] to indicate the relevant random variables (u, u * ) although S[u, u * ] is not a function of (u, u * ). The time evolution of p t (u, u * ) is governed by the following Fokker-Planck equation [1]: where A n (u, u * ) := B n (u, u * ) − νk 2 n u n + f n , and J n (u, u * ) denotes the probability current, Therefore, the average rate of change of the system entropy is given by In the last line, we have introducedṠ n [u, u * ], which is given bẏ where the over-dot denotes the rates of change of observables that are not a time derivative of a state function. Let ∆s env be the stochastic medium entropy production in an infinitesimal time interval [t, t + dt]. To identify ∆s env , we impose the local detailed balance condition [2,3]: Here, the transition probability density p(u , u * , t + dt|u, u * , t) is given by, in the Ito scheme, [1] p(u , u * , t + dt|u, u where du := u − u with u(t) = u and u(t + dt) = u . Similarly, p(−u, −u * , t + dt| − u , −u * , t) is given by where A ir n (u, u * ) and A rev n (u, u * ) denote the irreversible and reversible parts of A n (u, u * ), respectively: By substituting (S7) and (S8) into (S6), we obtain where Note that, by using the Stratonovich product [4], this expression can be simply rewritten as (S13) The average medium entropy production can be calculated as where the conditional average ∆s env |u, u * can be evaluated by replacing du n (resp. du * n ) with A n (u, u * )dt (resp. A * n (u, u * )dt) in ∆s env . Then, the medium entropy production rate reads whereṠ env n denotes the contribution from the n-th shell: By combining (S4) and (S15), the second law of thermodynamics can be expressed as where J ir n (u, u * ) denotes the irreversible current given by Furthermore, since the total system {u, u * } can be considered as a N + 1 multipartite systems, i.e., each shell variable (u n , u * n ) experiences independent noise, it follows that the second law holds for each shell variable individually [5]: (S19)

B. Derivation of the second law of information thermodynamics
We now derive the second law of information thermodynamics for the bipartite systems U < K and U > K [5,6]. First, we rewriteṠ n [u, u * ] for n ≤ n K aṡ where we have used p t (u, u * ) = p t (U < K , U > K ) and ∂p t (U > K )/∂u n = ∂p t (U > K )/∂u * n = 0 for n ≤ n K . Note thaṫ S n [U < K |U > K ] is related to the learning rate through the following relation: From (S19)-(S21), we obtain the second law of information thermodynamics for Similarly, we can derive the second law of information thermodynamics for U > K : In this section, we explain the details of the numerical simulation. After describing the setup, the details of the KSG estimator are explained. In particular, we provide a detailed explanation of the method used to estimate the variance and bias of the KSG estimator. We also present the result in the deterministic case (T = 0).

A. Setup
To evaluate the inertial range straightforwardly, we first nondimensionalize the equation (1) with the Kolmogorov dissipation scale η and the velocity scale u η := ( ν) 1/4 by settinĝ u n := u n /u η ,k n := ηk n ,t := t/τ η ,ξ n := (u η /η) −1/2 ξ n ,f n := f n /F, where τ η := η/u η denotes the typical time scale at the Kolmogorov dissipation scale, and F denotes the typical magnitude of the force per mass. The nondimensionalized equation (1) readṡ whereB Here, θ η := k B T /ρu 2 η denotes the ratio of the thermal energy to the kinetic energy at the Kolmogorov dissipation scale, and F η := F η/u 2 η denotes the nondimensionalized magnitude of the force. Correspondingly, we set the shell index to be n = M, · · · , R with M = −[(3/4) log 2 (Re)] and R = N − M , so that k 0 = 1 corresponds to the Kolmogorov dissipation scale.
We use a slaved 3/2-strong-order Ito-Taylor scheme [7] with the time-step δt := 10 −5 , which is smaller than the viscous time scale at the highest wavenumberτ vis := 1/k 2 R ∼ 10 −4 . The parameter values are set to the same values used in [8,9], which are consistent with the typical values in the atmospheric boundary layer (ABL). Specifically, the range of shell numbers is chosen as n = −15, · · · , 7 so that the achieved Reynolds number is comparable to the typical value in the ABL of Re ∼ 10 6 . To investigate the Re dependence of the LR, we also perform the numerical simulation for Re ∼ 10 5 by setting n = −12, · · · , 7. In both cases, the dimensionless temperature is chosen as θ η := 2.328 × 10 −8 , and the external force acts only on the first two shells, i.e., n f = −14 for Re ∼ 10 6 and −11 for Re ∼ 10 5 . The values of the external forces are adjusted such thatû rms := R n=M |û n | 2 ∼ 10 2 andˆ := R n=Mk 2 n |û n | 2 1 [8,9]: for Re ∼ 10 6 , while for Re ∼ 10 5 , In the averaging of the energy spectrum and the estimation of the MI, we use N samp = 3 × 10 5 samples. For the case of Re ∼ 10 6 , these samples are obtained by sampling 100 snapshots at timet = 1000i (i = 1, 2, · · · , 100) for each of the 3000 noise realizations. That is, for each of the 3000 independent runs, we sample 100 snapshots. Here, the time interval of the sampling, 1000, is chosen to be larger than one large-eddy turnover time τ L /τ η 734 < 1000. Similarly, for the case of Re ∼ 10 5 , N samp = 3 × 10 5 samples are obtained by sampling 100 snapshots at timet = 500i (i = 1, 2, · · · , 100) for each of the 3000 noise realizations, where the time interval, 500, is chosen to be larger than one large-eddy turnover time τ L /τ η 181 < 500.

B. The KSG estimator
The KSG estimator for the mutual information I[X : Y ] (either or both of the random variables X and Y can be multidimensional) is defined as follows [10]: where k ∈ N denotes the parameter of the KSG estimator, ψ is the digamma function, N samp denotes the total number of samples, and n (k) α (i) denotes the α extent of the smallest hyper-rectangle in the (x, y) space centered at the i-th sample (x i , y i ) that contains k of its neighboring samples. While any norms can be used for α j − α i , we use the standard Euclidean norm here.
Note that k is the only free parameter of the KSG estimator. By varying k, we can detect the structure of the underlying probability distribution in different spatial resolutions. To choose the optimal k (if it exists), we must estimate both the standard deviation and the bias of the KSG estimator [11].

C. Estimation of the variance of the KSG estimator
In this section, we explain the method used to estimate of the variance of the KSG estimator based on the subsampling approach proposed by Holmes and Nemenman [11]. This method is based on the fact that the variance of any function that is an average of N i.i.d. random variables scales as 1/N . Therefore, we write the variance of the KSG estimator as We estimate B (k) via a subsampling approach. Specifically, we first partition the N samp = N samples into n nonoverlapping subsets of equal size as much as possible. LetÎ (S33) This is our estimate of Var N/n [Î (k) KSG ] = nB (k) /N . Finally, we estimate B (k) by using maximum likelihood estimation. In doing so, we first calculate σ 2 k,N/n for various n ( = 1, 2, · · · , L). Then, from Cochran's theorem, (n − 1)σ 2 k,N/n /Var N/n [Î (k) KSG ] follows the χ 2 -distribution with n − 1 degrees of freedom: By assuming independence of We then obtain the maximum likelihood estimator: . (S36) By combining (S32) and (S36), we can estimate the variance of the KSG estimator to beB (k) /N samp .

D. Estimation bias of the KSG estimator
Although the KSG estimator is asymptotically unbiased for sufficiently regular probability distributions as N samp → ∞, both sample-size-dependent bias and k-dependent bias generally exist for a finite N samp [11]. The sample-sizedependent (resp. k-dependent) bias can be detected by comparing the sample-size-dependence (resp. k-dependence) of the estimated MI with its standard deviation. If the sample-size-dependence (resp. k-dependence) of the estimated MI is much larger than its standard deviation, then a sample-size-dependent (resp. k-dependent) bias may be present.
Note that k is related to the spatial resolution in detecting the structure of the underlying probability distribution. For large k, because the fine structure of the probability distribution cannot be detected, we would expect the MI to be underestimated. At the same time, because n x (i) and n y (i) both increase with increasing k, the standard deviation of the estimated MI will be smaller for large k. If there is no k-dependent bias, we can choose the optimal k such that there is no sample-size-dependence compared to the standard deviation and the standard deviation is the smallest. Figure S1 shows bias of the KSG estimatorÎ as a function of 1/N samp = n/N with N = 10 5 in the case of Re ∼ 10 6 . Here, we use n = 2, 3, · · · , 10, following [11]. The wave number K is within the inertial range, K = k 10 (left), and at the Kolmogorov dissipation scale, K = k 15 (right). The error bars are estimated by using the unbiased sample variance (S33). From (S32) and (S36), the standard deviation ofÎ is estimated to be ∼ 10 −3 for N samp ∼ 10 5 . It can be seen from Fig. S1 that, while there is no significant sample-size-dependent bias, a k-dependent bias does exist. In particular, the estimated MI is underestimated as k is increased. Figure S2 shows the scale dependences of the estimated MI and LR for k = 4, 10, 20, 50 with N samp = 2 × 10 5 in the case of Re ∼ 10 6 . Here, k = 4 is chosen because k = 2, 3, 4 are recommended in [10]. This results clearly show that the estimated MI and LR are underestimated as k is increased. Therefore, it is difficult to choose the optimal k in this case. The important point here is that the estimated LR is negative for K within the inertial range for all k. As mentioned in the main text, we remark that the error bar of the LRl < K is of the same order asl < K itself if we naively estimate it by using the estimated standard deviation of the MI. (c) Scale dependence of the estimated LR l < K . Note that it is plotted in units of the inverse of τL. In all panels, the dotted and dashed lines represent the Kolmogorov dissipation scale kν = 1/η and injection scale k f , respectively. The noisy case is the same as the one presented in the main text (cyan line in Fig. 2).

E. Deterministic case
Here, we present the result in the deterministic case (T = 0). We calculate the deterministic case by setting θ η = 0 in the Ito-Taylor scheme (Sec. S2 A). Other parameters are the same as in the noisy case with N = 22. As independent initial-data, we use the snapshots in the noisy case at timet = 10 5 . We have used N samp = 2 × 10 5 samples in the following averaging and estimation. These samples are obtained by sampling 100 snapshots at timê t = 1000i (i = 1, 2, · · · , 100) for each of the 2000 independent runs. Figure S3(a) shows the energy spectrum E n := |u n | 2 ss /2 in the steady state. The achieved Reynolds number in the deterministic case is Re 1.46 × 10 6 . In the inertial range, both the deterministic and noisy cases exhibit the Kolmogorov spectrum. In the dissipation range, in contrast, the deterministic case shows a rapid exponential decay. Figure S3(b) shows the estimated MIÎ (k) KSG [U < K (t) : U > K (t)] with k = 4. Its standard deviation is also estimated to be ∼ 10 −3 by subsampling, which lies within the marker size. While the deterministic case is almost the same as the noisy case in the inertial range, it takes a finite value even in the dissipation range. In other words, the correlation between large and small scales is not destroyed because of the absence of thermal fluctuations.
In Fig. S3(c), we show the estimated LRl < K in units of the inverse of the large-eddy turnover time τ L . In the deterministic case, we find that τ L 732τ η . From this figure, we can see that the LR in the deterministic case takes almost the same value as in the noisy case in the inertial range. This result implies that the information flow itself is mainly governed by the large-scale dynamics rather than by the thermal fluctuations. In contrast, it takes finite negative value in the dissipation range. In other words, in the absence of thermal fluctuations, the information flow