An exact solution of a one-dimensional asymmetric exclusion model with open boundaries

A simple asymmetric exclusion model with open boundaries is solved exactly in one dimension. The exact solution is obtained by deriving a recursion relation for the steady state: if the steady state is known for all system sizes less thanN, then our equation (8) gives the steady state for sizeN. Using this recursion, we obtain closed expressions (48) for the average occupations of all sites. The results are compared to the predictions of a mean field theory. In particular, for infinitely large systems, the effect of the boundary decays as the distance to the power –1/2 instead of the inverse of the distance, as predicted by the mean field theory.     

Bethe ansatz solution of the asymmetric exclusion process with open boundaries

We derive the Bethe ansatz equations describing the complete spectrum of the transition matrix of the partially asymmetric exclusion process with the most general open boundary conditions. For totally asymmetric diffusion we calculate the spectral gap, which characterizes the approach to stationarity at large times. We observe boundary induced crossovers in and between massive, diffusive, and Kardar-Parisi-Zhang scaling regimes.     

Exact diffusion constant of a one-dimensional asymmetric exclusion model with open boundaries

For the 1D fully asymmetric exclusion model with open boundary conditions, we calculate exactly the fluctuations of the current of particles. The method used is an extension of a matrix technique developed recently to describe the equatime steady-state properties for open boundary conditions and the diffusion constant for particles on a ring. We show how the fluctuations of the current are related to non-equal-time correlations. In the thermodynamic limit, our results agree with recent results of Ferrari and Fontes obtained by working directly in the infinite system. We also show that the fluctuations of the current become singular when the system undergoes a phase transition with discontinuities along the first-order transition line.     

Totally asymmetric exclusion processes at constrained m-input n-output junction points

Totally asymmetric exclusion processes at constrained m-input n-output junction points under random update are studied by theoretical calculation and computer simulation in this paper. At the junction points, the hopping rate of particles from m-input parallel lattices to n-output parallel lattices is assumed to be equal to r/n （0 〈 r 〈 1 ）. The mean-field approach and Monte Carlo simulations show that the phase diagram can be classified into three regions at any value of r. More interestingly, there is a threshold rc = n（ 1 - √1 - m/n）/m. In the cases of r 〉 re and r 〈 rc, qualitatively different phases exist in the system. With the increase of the value of m/n, the regions of （LD, LD） and （MC, LD） or （HD, LD） decrease, and the （HD, HD） is the only phase that increases in the region （LD stands for low density, HD stands for high density, and MC for maximal current）. Stationary current and density profiles are calculated, showing that they are in good agreement with Monte Carlo simulations.     

A simulation study of an asymmetric exclusion model with open and periodic boundaries for parallel dynamics

The effect of jumping rate probability on the phase diagram of an asymmetric exclusion model is studied by numerical simulations. Density, current and velocity of particles are calculated for parallel dynamics. In the open boundaries case for one species of particles (particles 1), a passage from first to second order transition occurs by decreasing the jumping rate. In the periodic boundaries case, by introducing another species of particle (particle 2) which plays the role of obstacle for particles 1, the average velocity of particles 1 increases with increasing the jumping rate for small density. While the average velocity of particle 2 decreases for small and intermediate densities. Received: 21 October 1997     

Two-lane totally asymmetric exclusion processes with particle creation and annihilation

This paper studies two-lane totally asymmetric simple exclusion processes (TASEP) coupled with particle creation and annihilation in one of the two lanes (lane 2). The dependence of the density profiles of both lanes on the lane-change rate Ω$\Omega$ is investigated. It is shown that with the increase of Ω$\Omega$, a complex behavior on both lanes occurs. Synchronization of the shocks in both lanes occurs when Ω$\Omega$ exceeds a threshold _Ωc≈10${\Omega }_c\approx 10$. A boundary layer is also observed at the left boundary as a finite-size effect. The situations arising from very large creation/annihilation rate and very small creation/annihilation rate are also investigated. The mean-field analysis is presented and it agrees well with Monte Carlo simulations.     

Phase transitions in a driven lattice gas at two temperatures

By suitably combining the uniformly driven lattice gas and the two-temperature kinetic Ising model, we obtain a generalized model that allows us to probe a variety of nonequilibrium phase transitions, including a type not previously observed. This new type of transition involves longitudinally ordered steady states, which are phase-segragated states with interface normalsparallel to the drive. Using computer simulations on a two-dimensional lattice gas, we map out the structure of the phase diagram, and the nature of the transitions, in the three-dimensional space of the drive and the two temperatures. While recovering anticipated results in most cases, we find one surprise, namely, that the transition from disorder to longitudinal order is continuous. Unless it turns out to be very weakly first order, this result is inconsistent with the expectation of field-theoretic renormalization group calculations.     

Successive defects asymmetric simple exclusion processes with particles of arbitrary size

This paper uses various mean-field approaches and the Monte Carlo simulation to calculate asymmetric simple exclusion processes with particles of arbitrary size in the successive defects system. In this system, the hopping probability p (p<1) and the size d of particles are not constant. Through theoretical calculation and computer simulation, it obtains the exact theoretical results and finds that the theoretical results are in agreement with the computer simulation. These results are helpful in analysing the effect of traffic with different hopping probabilities p and sizes d of particle.     

Effect of unequal injection rates on asymmetric exclusion processes with junction

In this paper, we investigate the effect of unequal injection rates on totally asymmetric simple exclusion processes (TASEPs) with a 2-input 1-output junction and parallel update. A mean-field approach is developed to deal with the junction that connects two sub-chains and the single main chain. We obtain the stationary particle currents, density profiles and phase diagrams. Interestingly, we find that the number of stationary-state phases is changeable depending on the value of α₁ (α₁ is the injection rate on the first sub-chain). When α₁ > 1/3, there are seven stationary-state phases in the system, however when α₁< 1/3, only six stationary-state phases exist in the system. The theoretical calculations are shown to be in agreement with Monte Carlo simulations.     

Effect of unequal injection rates and different hopping rates on asymmetric exclusion processes with junction

In this paper, the effects of unequal injection rates and different hopping rates on the asymmetric simple exclusion process (ASEP) with a 2-input 1-output junction are studied by using a simple mean-field approach and extensive computer simulations. The steady-state particle currents, the density profiles, and the phase diagrams are obtained. It is shown that with unequal injection rates and different hopping rates, the phase diagram structure is qualitatively changed. The theoretical calculations are in good agreement with Monte Carlo simulations.     

Existence of spontaneous symmetry breaking in two-lane totally asymmetric simple exclusion processes with an intersection

We study two-lane totally asymmetric simple exclusion processes(TASEPs) with an intersection. Monte Carlo simulations show that only symmetric phases exist in the system. To verify the existence of asymmetric phases, we carry out a cluster mean-field analysis. Analytical results show that the densities of the two upstream segments of the intersection site are always equal, which indicates that the system is not in asymmetric phases. It demonstrates that the spontaneous symmetry breaking does not exist in the system. The density profiles and the boundaries of the symmetric phases are also investigated. We find that the cluster mean-field analysis shows better agreement with simulations than the simple mean-field analysis where the correlation of sites is ignored.     

Effects of detachment and size of particles in totally asymmetric simple exclusion processes

In this article, the effects of irreversible detachments of particles in totally asymmetric simple exclusion processes (TASEPs) with extended particles which occupy more than one lattice site, are investigated. First, an approximate mean-field theory is used to calculate phase diagrams and density profiles. The results show that the detachment and the size of the particles have distinct effects on the stationary phases in the two sublattices divided by the detachment, especially in the mc/hd and the hd/hd phase. Here, symbols “mc”, “hd”, and “ld” are initials of maximal current, high density, and low density, respectively, and “mc/hd” represents the stationary state that the left sublattice is in the maximum-current (mc) phase while the right sublattice is in the high density (hd) phase. When the detachment rate is very large, there are four stationary phases, including ld/ld, ld/hd, mc/ld, and mc/hd phases. When the value of the detachment rate is in the middle range, the hd/hd phase occurs, and hence there exist five stationary phases. When the rate is very small, the mc/hd phase disappears, and there are only four phases again. These theoretical results qualitatively and quantitatively agree with computer Monte Carlo simulations especially in the case of large value of the detachment rate.     

Finite-size effects for phase segregation in a two-dimensional asymmetric exclusion model with two species

We investigate the stationary states of a two-dimensional lattice gas model with exclusion, in the presence of an external field. The lattice is populated by equal numbers of positively and negatively charged particles. An analytical mean-field approach and Monte Carlo simulations give strong evidence of the fact that at any finite density the only relevant stationary state of the system in the thermodynamic limit is inhomogeneous, consisting of a strip of particles transverse to the field. In the inhomogeneous phase, the density profiles and the current measured by Monte Carlo simulations are closely related to those found in mean field. The same is true for the finite-size behavior of the system.     

Phase Diagram of a Two-Species Lattice Model with a Linear Instability

We discuss the properties of a one-dimensional lattice model of a driven system with two species of particles in which the mobility of one species depends on the density of the other. This model was introduced by Lahiri and Ramaswamy ( Phys . Rev. Lett ., 79 , 1150 (1997)) in the context of sedimenting colloidal crystals, and its continuum version was shown to exhibit an instability arising from linear gradient couplings. In this paper we review recent progress in understanding the full phase diagram of the model. There are three phases. In the first, the steady state can be determined exactly along a representative locus using the condition of detailed balance. The system shows phase separation of an exceptionally robust sort, termed strong phase separation, which survives at all temperatures. The second phase arises in the threshold case where the first species evolves independently of the second, but the fluctuations of the first influence the evolution of the second, as in the passive scalar problem. The second species then shows phase separation of a delicate sort, in which long-range order coexists with fluctuations which do not damp down in the large-size limit. This fluctuation-dominated phase ordering is associated with power law decays in cluster size distributions and a breakdown of the Porod law. The third phase is one with a uniform overall density, and along a representative locus the steady state is shown to have product measure form. Density fluctuations are transported by two kinematic waves, each involving both species and coupled at the nonlinear level. Their dissipation properties are governed by the symmetries of these couplings, which depend on the overall densities. In the most interesting case, the dissipation of the two modes is characterized by different critical exponents, despite the nonlinear coupling.     

Zone inhomogeneity with the random asymmetric simple exclusion process in a one-lane system

In this paper we use theoretical analysis and extensive simulations to study zone inhomogeneity with the random asymmetric simple exclusion process (ASEP). In the inhomogeneous zone, the hopping probability is less than 1. Two typical lattice geometries are investigated here. In case A, the lattice includes two equal segments. The hopping probability in the left segment is equal to 1, and in the right segment it is equal to p, which is less than 1. In case B, there are three equal segments in the system; the hopping probabilities in the left and right segments are equal to 1, and in the middle segment it is equal to p, which is less than 1. Through theoretical analysis, we can discover the effect on these systems when p is changed.     

Thermodynamics and dynamics of systems with long-range interactions

We review simple aspects of the thermodynamic and dynamical properties of systems with long-range pairwise interactions (LRI), which decay as 1/r^d+σ at large distances r in d dimensions. Two broad classes of such systems are discussed. (i) Systems with a slow decay of the interactions, termed “strong” LRI, where the energy is super-extensive. These systems are characterized by unusual properties such as inequivalence of ensembles, negative specific heat, slow decay of correlations, anomalous diffusion and ergodicity breaking. (ii) Systems with faster decay of the interaction potential, where the energy is additive, thus resulting in less dramatic effects. These interactions affect the thermodynamic behavior of systems near phase transitions, where long-range correlations are naturally present. Long-range correlations are often present in systems driven out of equilibrium when the dynamics involves conserved quantities. Steady state properties of driven systems with local dynamics are considered within the framework outlined above.     

The dynamics of cargo driven by molecular motors in the context of asymmetric simple exclusion processes

We consider the dynamics of cargo driven by a collection of interacting molecular motors in the context of an asymmetric simple exclusion process (ASEP). The model is formulated to account for (i) excluded-volume interactions, (ii) the observed asymmetry of the stochastic movement of individual motors and (iii) interactions between motors and cargo. Items (i) and (ii) form the basis of ASEP models and have already been considered to study the behavior of motor density profile [A. Parmeggiani, T. Franosch, E. Frey, Phase Coexistence in driven one-dimensional transport, Phys. Rev. Lett. 90 (2003) 086601-1-086601-4]. Item (iii) is new. It is introduced here as an attempt to describe explicitly the dependence of cargo movement on the dynamics of motors in this context. The steady-state solutions of the model indicate that the system undergoes a phase transition of condensation type as the motor density varies. We study the consequences of this transition to the behavior of the average cargo velocity.     

Computer simulation of shock waves in the completely asymmetric simple exclusion process

We study the evolution of the completely asymmetric simple exclusion process in one dimension, with particles moving only to the right, for initial configurations corresponding to average density _– ( ₊) left (right) of the origin, _{– +}. The microscopic shock position is identified by introducing a second-class particle. Results indicate that the shock profile is stable, and that the distribution as seen from the shock positionN(t) tends, as time increases, to a limiting distribution, which is locally close to an equilibrium distribution far from the shock. Moreover , withV=1– _–– ₊, as predicted, and the dispersion ofN(t), ²(t), behaves linearly, for not too small values of ₊– _–, i.e., , whereS is equal, up to a scaling factor, to the valueS _WA predicted in the weakly asymmetric case. For ₊= _– we find agreement with the conjecture .Dedicated to the memory of Paola Calderoni.