Asymmetric simple exclusion process (ASEP) has become a very prominent model in studies of non-equilibrium systems, because it plays the same role as Ising model in equilibrium systems.^[1] The ASEP was first introduced in the description of kinetics of biopolymerization.^[2] Despite its simplicity, the model has been extensively applied in descriptions of protein synthesis,^[3] dense media polymer dynamics,^[4] gel electrophoresis,^[5] membrane channels diffusion,^[6] traffic flow,^[7–11] information flow,^[12,13] and so on.

The totally asymmetric simple exclusion process (TASEP) is the simplest limit of ASEP. In one-dimensional TASEP, particles move in one direction and obey the hard-core exclusion principle. Each site of the model has two states, occupied by one particle or empty. However, many realistic situations are not in one dimension, for example, transport along parallel or intersected lanes is very common in vehicle traffic or intracellular transport. Therefore, studies of TASEP have been extended from one-lane to multi-lane.^[14–18]

Spontaneous symmetry breaking (SSB) is one phenomenon observed in TASEP, and it has attracted many scientific researchers’ interests in recent decades. This phenomenon was first observed in the ‘bridge model’.^[19] In this model, there are two species of particles hopping in the single lane and the hopping directions are opposite. It has been found that the asymmetric stationary states could exist under the symmetrical conditions for the two species of particles. However, it is unclear how the system can be transformed to the symmetry-breaking state.^[20] Later, Pronina et al. investigated two-parallel-lane TASEPs.^[21] There are also two species of particles in the model. The particles hop in opposite directions without changing lanes and obey the narrow entrance principle. Monte Carlo (MC) simulations showed that one asymmetric phase does not exist. However, the theoretical analysis indicated that the asymmetric phase exists. In the theoretical analysis, the simple mean-field (SMF) method was adopted, while the correlation of the two lanes was ignored. Tian et al. studied the system with the cluster mean-field (CMF) method. In the analysis, the correlation of two sites was firstly considered, then more sites of the two lanes were considered. Those theoretical analyses all indicated that the asymmetric phase does not exist.^[22,23] However, the mechanism of the occurrence of SSB is still not well understood.^[15]

Yuan et al.^[18] investigated another two-lane TASEP. In Ref. [18], two lanes interact at a crossing site, and there are three different update rules of particles, which correspond to three models: A, B, and C. Model A can be used in simulation of motion of molecular motor, and models B and C can be used in simulation of vehicle traffic. In the models B and C, the asymmetric HD/HL phase exists, which has been verified by MC simulations and CMF analysis.^[24,25] It means that the SSB does exist in both models. While in the model A, simulations show that there are only symmetric phases. It is still not known exactly whether the SSB (i.e., the asymmetric HD/HL phase) exists in the model. To clarify its existence, more careful investigations are needed.

In Ref. [18], the models were analyzed by SMF. The asymmetric phases can not be investigated due to the ignorance of correlation. In the present paper, we carry out CMF analysis to study the asymmetric phase. In the CMF method, the correlation of sites is considered. When the correlation exists, the CMF analysis works better than the SMF analysis. We have considered the correlation of three and five sites in Refs. [24,25], respectively. It has been found that one boundary of asymmetric phase is determined by the difference of the upstream segment densities of the two lanes. In this paper, five sites are considered in the CMF analysis due to the update rules of particles. It is found that the densities of the two lanes are equal, which indicates that SSB does not exist in the system. The rest of this paper is organized as follows. In Section 2, the model is described in detail. In Section 3, the results of CMF analysis and MC simulations are discussed. In Section 4, conclusions are presented.

Figure 1 shows a schematic view of the model. In the system, there are two intersected lanes. Lane 1 and lane 2 are in horizontal and vertical directions, respectively. Both lanes have L sites. Lane 1 is numbered from 1 to L, and lane 2 from L + 1 to 2L. Each site can be vacant or occupied by one particle. The updates of particles are random. The particles can enter lane 1 or lane 2 with a probability α, and they can leave lane 1 or lane 2 with a probability β. In the bulk (except site C), the particles can hop forward if the next site is empty.

Fig. 1.

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	Fig. 1. (a) Schematic view of the model. The allowed moves are indicated by arrows and the prohibited hoppings are shown by crossed arrows. α and β correspond to the entrance and exit probabilities. (b) The illustration of cluster mean-field analysis. α1 and α2 (β1 and β2) denote effective injection (removal) probabilities.

In the model, if a particle of site C is chosen, the particle hops to site C₂ (C₄) if the site C₂ (C₄) is vacant and site C₄ (C₂) is occupied. When the sites C₂ and C₄ are both vacant, the particle hops to site C₂ or C₄ with an equal probability 0.5.

In Ref. [18], it has been found that there are three symmetric phases in the system, see Fig. 2. In these symmetric phases, the two lanes are in the same states and their densities are the same. When α < β and α < λ₁, the two lanes are in low densities (LD) and the system is in a symmetric LL phase. The densities of the two lanes are equal to α. When β < α and β < λ₁, the two lanes are in high densities (HD) and the system is in a symmetric HH phase. The densities of the two lanes are equal to 1 − β. When α > λ₁ and β > λ₁, the system is in symmetric HL phase. The downstream segments of the intersection of the lanes are in low densities and the upstream segments are in high densities. It is found that λ₁ ≈ 0.428. Figure 3 shows the density profiles of the three phases.

Fig. 2.

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	Fig. 2. Phase diagram of two-lane TASEPs with an intersection. Solid lines are obtained from MC simulations. Black dashed lines represent the results from SMF analysis. Red dashed lines correspond to the results from CMF analysis.

Fig. 3.

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	Fig. 3. Density profiles of the three symmetric phases. The results from MC simulations are shown by solid lines and those from CMF analysis are shown by dashed lines. (a) Symmetric HL phase with α = 0.8, β = 0.7, (b) Symmetric HH phase with α = 0.7, β = 0.2. (c) Symmetric LL phase with α = 0.2, β = 0.7.

The boundaries of these phases have been obtained through analysis of the symmetric HL phase. In the analysis, the SMF method was adopted. The system was divided into four segments (i.e., I, II, III, and IV), and each segment was regarded as single-lane TASEP, see Fig. 1(a). It has been found that when

The SMF analytical results are shown in Fig. 2. We can see that the deviation between the analytical and simulation results exists. This is because of the ignorance of the correlation of sites. In other two models of Ref. [18], both MC simulations and theoretical analysis indicated that the SSB phenomenon does exist. When SSB occurs, the two lanes are in HD and HL phases, respectively, and their densities are not equal. However, the existence of the asymmetric phase of the present model can not be verified by the SMF analysis. Motivated by this, the CMF analysis is adopted to investigate the existence.

In the analysis, five sites (i.e., sites C, C₁, C₂, C₃, and C₄) are considered due to the update rule of the model, see Fig. 1(b). Q_{τ1τ2τ3τ4τ5} is defined as the probability of site C₃ in state τ₁, site C in state τ₂, site C₄ in state τ₃, site C₁ in state τ₄, and site C₂ in state τ₅. Sites C₁, C₂, C₃, and C₄ have two states: empty or occupied, and τ₁, τ₃, τ₄, τ₅ can be 1 and 0 (1 represents occupation, 0 represents empty). Site C has three states, and τ₂ can be 1, 2, and 0 (1 represents the state occupied by a particle moving in lane 1, 2 represents the state occupied by a particle moving in lane 2, and 0 represents the state of empty). The effective entrance rates into sites C₁ and C₃ are denoted by α₁ and α₂, and the effective exiting rates of particles from sites C₂ and C₄ are denoted by β₁ and β₂.

We first assume that the asymmetric phase exists. Then equations of Q_{τ1τ2τ3τ4τ5} can be obtained. Take Q₀₀₀₀₀ for example,

Fig. 4.

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	Fig. 4. The relationship between ρ1 (ρ2) and β. The scattered points are from the CMF analysis, βc represents the value of the top boundary of the symmetric HH phase.

It can be seen that ρ₁ is always equal to ρ₂ with the change of β, which violates the assumed asymmetric phase. In addition, the results of MC simulations do not change with the system size, which means that the phases of the present model are also symmetric. From both CMF analysis and MC simulations, we demonstrate that the asymmetric phase does not exist in the system, which indicates that the SSB does not exist in the system.

The boundaries between these symmetric phases can also be analyzed. In the symmetric phases, due to the symmetry, α₁ = α₂, β₁ = β₂, J₁ = J₂, J₃ = J₄, ρ₁ = ρ₂. We have 53 unknown quantities including α₁ (α₂), β₁ (β₂), J₁ (J₂), J₃ (J₄), ρ₁ (ρ₂) and the 48 probabilities Q_{τ1τ2τ3τ4τ5}. We also have 53 equations including Eqs. (3)–(5), (7), (9), (12), (14), and (A1)–(A46). These equations can be solved, and the boundary between the symmetric HL and HH pahses can be determined by

In this work, we have investigated two-lane TASEPs with an intersection under open boundaries. The random update is adopted. In the model, particles can change moving lanes with a probability 0.5 at the intersection site. The MC simulations show that the phases of the system are symmetric. To verify the existence of asymmetric phase, theoretical analysis is needed.

The system has been investigated by the SMF analysis. However, due to the ignorance of the correlation of sites, the asymmetric phase cannot be analyzed. Motivated by this, the CMF analysis is adopted to investigate the existence of the asymmetric phase, and five sites including the intersection site and four sites next nearest to it are considered. It is firstly assumed that the asymmetric phase exists. Through theoretical analysis, it is found that the densities of the two upstream segments of the intersection are always equal. It means that the two segments are symmetric, which violates the assumption that the two lanes are asymmetric. The MC simulations also indicate that the phases of the present model are symmetric. Therefore, it demonstrates that the asymmetric phase (spontaneous symmetry breaking) does not exist in the system.

The boundaries of the symmetric phases can also be analyzed. It has been found that the CMF analytical results are in excellent agreement with the simulations. By considering the correlation, the CMF analysis shows better agreement with the MC simulations than the SMF analysis where the correlation is ignored. We also investigate the density profiles when the system is in a symmetric HL phase, and it has been shown that both the CMF analytical and simulation results are also in excellent agreement.