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Theoretical investigation of synchronous totally asymmetric exclusion processes on lattices with multiple-input-single-output junctions

In this paper, we investigate the dynamics of synchronous totally asymmetric exclusion processes on lattices with a multiple-input single-output (MISO) junction, which consists of m subchains for the input and one main chain for the output. A MISO junction is a type of complex geometry that is relevant to many biological processes as well as vehicular and pedestrian traffic flow. A mean-field approach is developed to deal with the junction that connects the subchains and the main chain. Theoretical results for stationary particle currents, density profiles, and a phase diagram have been obtained. It is found that the phase boundary moves toward the left in the phase diagram with an increase of the number of subchains. The nonequilibrium stationary states, stationary-state phases, and phase boundaries are determined by the boundary conditions of the system as well as by the number of subchains. The density profiles obtained from computer simulations show very good agreement with our theoretical analysis.


I. INTRODUCTION
Non-equilibrium transport phenomena have attracted much attention of physicists since physical principles underlying these phenomena could be revealed in terms of phases and phase transitions [1,2]. Totally asymmetric simple exclusion processes (TASEPs) serve as a basic model for non-equilibrium systems and have been widely applied in the study of transport phenomena in chemistry, physics and biology, for example, particle diffusion through membrane channels [3], the kinetics of synthesis of proteins [4], polymer dynamics in dense media [5], gel electrophoresis [6], intracellular transport of motor proteins moving along cytoskeletal filaments [7], vehicular traffic [1,8] and ant traffic [9].
TASEPs are non-equilibrium one-dimensional lattice models in which particles move along one direction with hardcore interactions. The exact solution of TASEPs has been obtained by using the matrix product ansatz (MPA) [10] and the Bethe ansatz [11], respectively. Recently, some extensions of TASEPs focus on coupling with Langmuir kinetics [12,13,14,15,16,17,18,19,20,21] and/or lattice geometries [22,23,24,25,26,27,28], as well as bottleneck-induced transport phenomena such as in [29,30]. Ref. [29] investigated diffusive compartments as bottlenecks in a driven transport. They found that when a diffusive bottleneck is at the boundary of a system, the system cannot reach a maximal current phase; when a diffusive bottleneck is in the interior of the system, the system is dictated by the diffusive bottleneck and has a maximal current defined by the bottleneck. More recently, Ref. [30] introduced a bottleneck phase to describe the phenomenon that the current is independent of boundary conditions.
In this paper, we focus on one special lattice geometry -junctions, which are widely observed in many real physical systems. Those junctions are formed for various reasons. For example, (i) variation of the number of protofilaments on a microtubule in vitro [31]; (ii) transport of vesicles in a branching axon or dendrite [32]; (iii) merging of two or more roads; (iv) data through hubs (e.g., switches, routers) on the Internet. This kind of lattice geometry can cause congestion, e.g., in the traffic of molecular motors, vehicles or data packets. Traffic congestion of molecular motors could lead to some human diseases such as Alzheimers [33] and some neurodegenerative diseases [34]. Vehicular traffic congestion can pollute environment, increase fuel consumption.
Inspired by this wide range of possible applications, we investigate the dynamics of synchronous (i.e., in parallel update) totally asymmetric exclusion processes on lattices with a multi-input-single-output (MISO) junction (see Fig.  1). The parallel updating procedure has been typically adopted in modeling vehicular and pedestrian traffic [1,8]. Multiple inputs exhibit more complex interactions between particles at junction points than two inputs. In reality, it can be observed that several traffic lanes merge into one lane and multiple protofilaments come together to form one protofilament [31]. However, they have not been understood well from the viewpoint of theoretical analysis.
Theoretical calculations, along with a mean-field approximation, are developed. The phase diagram is presented and density profiles are investigated. A phenomenological domain wall theory, based on Refs. [23,36], is used to predict phase boundaries. Computer simulations are also conducted. Note that TASEPs on lattices with Y-junctions (e.g., two-input-single-out junctions) in a random updating procedure has been investigated in Ref. [23]. Y-junctions can be seen as a special case of multi-input-single-output junctions.
The paper is organized as follows. In Section II, we give a description of a synchronous TASEP model with an MISO junction, theoretical calculations as well as the mean-field approximation are developed. In Section III, we analyze the phase boundaries using a phenomenological domain wall theory. In Section IV, the results of our theoretical calculations and computer simulations are presented. Finally, we give our conclusions in Section V.

II. THE MODEL AND MEAN-FIELD APPROXIMATION
An MISO junction is illustrated in Fig. 1. The system consists of m subchains for input and one main chain (chain m + 1) for output connected by junction points-sites N on the subchains and site N + 1 on the main chain. Each subchain and the main chain includes N sites. For simplification, inter-lane transitions between subchains are not permitted in this paper. Particles are assumed to move from the left to the right. An occupation variable, τ ℓ,i , denotes the state of the ith site in the ℓth subchain (ℓ = 1, 2, ..., m) and the main chain (ℓ = m + 1). τ ℓ,i = 1 (or τ ℓ,i = 0) means that site τ ℓ,i is occupied (or empty). The system updates all particles in parallel by the following rules (see Fig. 1): • i = 1. (i) If τ ℓ,1 = 0, a particle enters the system at rate α; or (ii) If τ ℓ,1 = 1 and τ ℓ,2 = 0, then the particle at site (ℓ, 1) moves into site (ℓ, 2); or (iii) If both τ ℓ,1 = 1 and τ ℓ,2 = 1, then the particle at site (ℓ, 1) does not move.
• i = N . (i) If sites N of k subchains (1 < k ≤ m) are occupied by k particles at the same time, particles have the same priority to hop to site N + 1. However, only one particle will enter site N + 1 at any single time step, providing that site N + 1 is empty; or (ii) If only one site N of the subchains is occupied by a particle, the particle can directly hop to site N + 1 providing that site N + 1 is empty.
Exactly solvable results of an one-dimensional synchronous TASEP have been obtained in Ref. [37,38]. We briefly recall these results, as the solution of our synchronous TASEP with a junction can be derived from them. There are three phases (low density (LD), high density (HD) and maximal current(MC)) and a transition line when α = β. The MC J = 0.5 can only be reached at α = β = 1 [38].
• When α < β ≤ 1, a low-density (LD) phase is obtained with where J is the system current; ρ is the bulk density; ρ 1 (ρ N ) is the particle density at the first (last) site.
Based on the above results, we develop exactly solvable results for TASEPs with an MISO junction. For an MISO junction, as the current is conserved through the system, we have: where J ℓ (ℓ = 1, 2, ..., m) is the current on the ℓth subchain; J m+1 is the current of the main chain. Each of the m subchains of the MISO junction can be seen as a synchronous TASEP with injection rate α and ejection rate β ef f , while the main chain can be seen as a synchronous TASEP with injection rate α ef f and ejection rate β. α ef f and β ef f can be written as These m subchains should have the identical phases when particles on the m subchains merge into the main chain with the same priority. Our computer simulations also support this prediction. Thus, the stationary state of the system can be obtained by combining the possible phases that exist in each of these subchains and the main chain. As each single chain may have three possible phases (LD, HD and MC), due to the equivalence of these subchains, the number of possible stationary phases of the system is equal to 3 2 = 9. In other words, a stationary state can be one of the following nine phases: the ( • The (LD, HD) phase. The conditions for this case are as follows: From Eqs. (1) and (2), the stationary properties of this phase are given by: According to Eq. (3), mJ 1 = J m+1 , we get: However, α ef f and β ef f are not solvable from above equations. In other words, we cannot calculate the bulk density through the above equations. The density will be solved through the domain wall theory in Section III. • The (LD, MC) phase. This phase corresponds to the following conditions: According to Eqs. (1) and (3), we obtain: Obviously, α = 1/(2m − 1) < 1/2 when m ≥ 2. That is, Eq. (8) is satisfied. Thus, the (LD, MC) phase can exist in the system when: Again, we cannot calculate the bulk density through above equations. The density will also be solved through domain wall theory in Section III. • The (LD, LD) phase. The following conditions should be satisfied: According to Eq. (1), the stationary current and density are given by: Using Eqs. (3) and (4), α ef f and β ef f are expressed as: Since α ef f ≤ 1 and , the system is in the (LD, LD) phase when: • The (HD, HD) phase. The conditions for this case are given by: The current and density of this phase in a stationary state are: From Eqs. (3) and (16), we obtain β ef f = β/[m + (m − 1)β]. Thus, the system is in the (HD, HD) phase when: • The (HD, MC) phase. The corresponding conditions for this phase are: According to Eqs. (2-3), we obtain Thus, the (HD, MC) phase can exist in the system when: • (HD, LD) phase. The conditions of existence for this phase can be written as The corresponding expressions for stationary current and density are According to Eq. (3), we have From Eqs. (4) and (22), α ef f and β ef f can be rewritten as follows Substituting Eq. (24) into Eq. (23), we obtain α ef f = 1 + m 2 /4 + m/2 − 1 and β ef f = 1 + m 2 /4 − m/2. It can be seen that values of α ef f and β ef f are determined by the number of subchains m, independent of α and β. This indicates that the (HD, LD) phase cannot be represented in the α − β plane. In other words, the (HD, LD) phase does not exist in the system. In fact, when the subchains are in the high density phase, particles at site N will hop to site N + 1 at almost each time step, which leads to α ef f ≈ 1. Thus, it is impossible for the main chain to maintain the low density phase. is also shown in Fig. 2(b). From Fig. 2 HD) phases, the density change on the subchains is discontinuous, while the density change of the main chain is continuous. Also, for the transition from the (LD, MC) phase to the (HD, MC) phase, the density change on the subchains is discontinuous, while the density profile on the main chain is unchanged.
From the above analysis, we can see the non-equilibrium stationary states, stationary-state phases and the phase boundary are determined by the boundary conditions of the system as well as by the number of subchains. In other words, the dynamics of the system is determined by its environment and its own structure.

III. DOMAIN WALL DYNAMICS
A phenomenological domain wall (i.e., shock) theory to explain phase behavior of a TASEP in a random update procedure with open boundaries is introduced in Ref. [36]. A domain wall is a phase boundary connected by two possible stationary states. The wall can have a random walk through the system with a drift speed defined as follows [36]: where J and ρ are the currents and densities in the two phases; '+'('-') denotes the phase to the right (left) of the domain wall. When V > 0, the domain wall moves to the right, while the domain wall travels to the left when V < 0. For instance, if α < β in a TASEP, the domain wall first appears at the left end and will drift to the right later. The wall will pass through the system, which eventually leads to the system being in a low-density stationary state. When V = 0, it implies that the domain wall has no net drift between two possible phases. In this case, stationary density profiles are linearly increased and the domain wall can exist anywhere with equal probability in the system. In this paper, the line specified by α = β/[m + (m − 1)β] corresponds to the coexistence of the (LD, LD) and (HD, HD) phases. However, the simple approximation theory may not fully reflect the correlations near the junctions as indicated in Ref. [23]. The domain wall theory is adopted in order to derive the phase boundary and the density profile in the bulk. This theory has also been used in Ref. [23].
To determine the position of the domain wall in the system, we define x as x = i/(2N ), where i is the site index and 2N is the length of the system. In the range of 0 < x ≤ 1, the domain wall moves at rate v L in the left subsystem (the sub-chains). In the range of 1 < x ≤ 2, the domain wall moves at rate v R in the right subsystem (the main chain), see Fig. (3).
v L and v R can be given by utilizing Eq. (25): where: As a result, v L and v R are rewritten as: Similarly to Ref. [23], q L (q R ) is denoted as a probability to find the domain wall at any position in the left (right) subsystem. For a special site, i, in the left (right) subsystem, the probability is obviously equal to q L /N (q R /N ). Then, at the junction point, we have: In addition, normalized q L and q R are satisfied with: Combining Eqs. (29) and (30), we obtain: According to Eq. (28), Eq. (31) implies that: Accordingly, the probabilities of the domain walls falling in certain zones in the left and right subsystems are also given by: and Thus, the density at any position in the system becomes: Finally, from Eqs. (32)- (35), one can obtain: and Densities in the boundary conditions can be calculated as ρ(x = 0) L = α/(1 + α) and ρ(x = 2) R = 1/(1 + β). These results are completely identical with theoretical analysis in Refs. [37,38]. At the junction point N , the densities are equal to ρ( Note that in the transition line between the (LD, LD) and (HD, HD) phases, we obtain the relationship α = β/[m+(m−1)β].

IV. SIMULATION RESULTS AND DISCUSSIONS
To validate our theoretical analysis, computer simulations are conducted. Here, we only present a synchronous TASEP with a Y-type junction, that is m = 2. The numbers of sites of the subchains and the main chain are all equal to 1,000. In simulations, stationary density profiles are obtained by averaging 10 8 sampling at each site. The first 10 5 N time steps are discarded to let the transient time out.
The density profiles for the (LD, LD), (HD, HD) and (HD, MC) phases are shown in Fig. 4. We only illustrate the density properties of subchain 1 and the main chain since the density properties of the other subchain is essentially the same as subchain 1. It is found that there is a good agreement between Monte Carlo simulations (MCS) and mean field (MF) analysis (see Figs. 4(a)-(e)), which verifies our theoretical investigations.
A phenomenological domain wall (DW) theory developed in Section III is used to calculate the density profiles of phase boundaries such as the (LD, HD) and (LD, MC) phases (see Fig. 5). The results obtained from the domain wall theory show an agreement with computer simulations. When α and β both increase and also maintain α = β/[m + (m − 1)β], the system keeps in the (LD, HD) phase until β = 1; the slope of the density profiles for x < 1 decreases until the slope reduces to 0.5, while the slope of the density profiles for 1 < x < 2 also decrease until the slope decreases to 0. For instance, the slope decreases from 0.588 to 0.542 (also see Eq. (32)) when α increases from 0.1 to 0.2 (see Figs. 5(a) and (b)). Finally, the slopes of density profiles of the subchains become 0.5 and the slope of density profile of the main chain becomes 0 when α = 1/3 and β = 1 (see Fig. 5(c)). Additionally, Monte Carlo simulations, theoretical calculations and domain wall theory all show that, when α = 1/3 and β = 1 (i.e., the transition phase between the other four phases), the main chain is in the maximal current phase.
Density profiles of the systems for m = 2 and m = 3 with the synchronous update scheme are simulated and compared (see Fig. 6). According to Eq. (7), the phase boundary between the (LD, LD) and (HD, HD) phases can be described as α = β/(2 + β) for m = 2 and α = β/(3 + 2β) for m = 3. Fig. 6 shows that both systems are in the (LD, LD) phase when α = 0.1 and β = 0.8. However, when α increases (e.g., α = 0.2), the system for m = 2 is still in the (LD, LD) phase, while the system for m = 3 is in the (HD, HD) phase (see Fig. 6(b)). This is due to the phase boundary between the (LD, LD) and (HD, HD) phases moving towards the left when m increases (see Fig. 2(a)). Density profiles in the (HD, HD) phase for both m = 2 and m = 3 are shown in Fig. 6(c). Compared with Fig. 6(a) and (c), it can be seen that the density profiles of the subchains of both systems are the same when both systems in the (LD, LD) phase, while the density profiles of the main chains of both systems are the same when both systems are in the (HD, HD) phase. Fig. 6(d) illustrates that the system is in the (LD, LD) phase for m = 2, while it is in the (HD, MC) phase for m = 3.
We can also see the similarities and differences between the phase diagram of the system with the synchronous/parallel update scheme (see Fig. 2(b)) and that of the system with the random update scheme (see Fig.  3 in [23]). One can see that the structures of the phase diagrams are similar. All have five phases in their phase diagrams. Also, increasing the number of subchains (i.e., inputs) only shifts the transition line between the (LD, LD) phase and the (HD, HD) phase that does not fall on the boundaries of the phase diagrams. However, the differences in the phase diagrams include: (i) the (HD, MC) phase region in the phase diagram of the system with the random update scheme reduces to a line in that of the system with the synchronous update scheme; and (ii) the line of the (LD, MC) phase in the phase diagram of the system with the random update scheme reduces to a point in the phase diagram of the system with the synchronous update scheme. Fig. 7 shows the differences in the density profiles of the systems with the synchronous update scheme and the system with the random update scheme when m = 2. In Fig. 7(a), these two systems are in the (LD, LD) phase when α = 0.1 and β = 0.8. When α is increased to 0.2 and β is unchanged, the system with the synchronous update scheme is still in the (LD, LD) phase, while the phase of the system with the random update scheme becomes the (HD, MC) phase (see Fig. 7(b)). Fig. 7(c) shows the system in the (HD, HD) phase in both systems when α = 0.8 and β = 0.32. With the increase of β (e.g., β = 0.8),the phase of the system with the random update scheme changes to the (HD, MC) phase, while it still keeps in the (HD, HD) phase in the other system. Note that the system also exhibits a particle-hole symmetry. Since particles moving forward at junction points with the same priority is equivalent to holes moving backward at the same priority. Also, our method can be used to analyze synchronous TASEPs with a single-input-multi-output (SIMO) junction. Other inhomogeneous synchronous TASEPs can be investigated in the similar way. For instance, it would be interesting to study an MISO junction where these parallel subchains are dynamically different.

V. SUMMARY AND CONCLUSIONS
Multi-input-single-output (MISO) junctions are relevant to many biological processes as well as vehicular and pedestrian traffic flow. Synchronous totally asymmetric exclusion processes (TASEPs) with an MISO junction are investigated in this paper. The theoretical solutions, mean-field approximation, domain wall theory are developed. We also compare the phase diagrams and density profiles between the system with the synchronous update scheme and the system with the random update scheme. The main differences in the phase diagrams include: (i) the (HD, MC) phase region in the phase diagram of the system with the random update scheme reduces to a line in that of the system with the synchronous update scheme; and (ii) the line of the (LD, MC) phase in the phase diagram of the system with the random update scheme reduces to a point in the phase diagram of the system with the synchronous update scheme.
The approach used in this paper can be used directly to analyze TASEPs with a single-input-multi-output (MIMO)