Exact solution of the master equation for the asymmetric exclusion process

Using the Bethe ansatz, we obtain the exact solution of the master equation for the totally asymmetric exclusion process on an infinite one-dimensional lattice. We derive explicit expressions for the conditional probabilitiesP(x₁,...,x_N;t/y ₁,...,y_N; 0) of findingN particles on lattices sitesx ₁,...,x_N at timet with initial occupationy ₁,...,y_N at timet=0.     

Exact results for the asymmetric simple exclusion process with a blockage

We present new results for the current as a function of transmission rate in the one-dimensional totally asymmetric simple exclusion process (TASEP) with a blockage that lowers the jump rate at one site from one tor<1. Exact finitevolume results serve to bound the allowed values for the current in the infinite system. This proves the existence of a nonequilibrium phase transition, corresponding to an immiscibility gap in the allowed values of the asymptotic densities which the infinite system can have in a stationary state. A series expansion inr, derived from the finite systems, is proven to be asymptotic for all sufficiently large systems. Padé approximants based on this series, which make specific assumptions about the nature of the singularity atr=1, match numerical data for the infinite system to 1 part in 10⁴.     

Relaxation Times in the ASEP Model Using a DMRG Method

We compute the largest relaxation times for the totally asymmetric exclusion process (TASEP) with open boundary conditions with a DMRG method. This allows us to reach much larger system sizes than in previous numerical studies. We are then able to show that the phenomenological theory of the domain wall indeed predicts correctly the largest relaxation time for large systems. Besides, we can obtain results even when the domain wall approach breaks down, and show that the KPZ dynamical exponent z=3/2 is recovered in the whole maximal current phase.     

A Fredholm Determinant Representation in ASEP

In previous work (Tracy and Widom in Commun. Math. Phys. 279:815–844, 2008) the authors found integral formulas for probabilities in the asymmetric simple exclusion process (ASEP) on the integer lattice ℤ. The dynamics are uniquely determined once the initial state is specified. In this note we restrict our attention to the case of step initial condition with particles at the positive integers ℤ⁺ and consider the distribution function for the mth particle from the left. In Tracy and Widom (Commun. Math. Phys. 279:815–844, 2008) an infinite series of multiple integrals was derived for the distribution. In this note we show that the series can be summed to give a single integral whose integrand involves a Fredholm determinant. We use this determinant representation to derive (non-rigorously, at this writing) a scaling limit.     

Entropy of Open Lattice Systems

We investigate the behavior of the Gibbs-Shannon entropy of the stationary nonequilibrium measure describing a one-dimensional lattice gas, of L sites, with symmetric exclusion dynamics and in contact with particle reservoirs at different densities. In the hydrodynamic scaling limit, L → ∞, the leading order (O(L)) behavior of this entropy has been shown by Bahadoran to be that of a product measure corresponding to strict local equilibrium; we compute the first correction, which is O(1). The computation uses a formal expansion of the entropy in terms of truncated correlation functions; for this system the k ^th such correlation is shown to be O(L ^−k+1). This entropy correction depends only on the scaled truncated pair correlation, which describes the covariance of the density field. It coincides, in the large L limit, with the corresponding correction obtained from a Gaussian measure with the same covariance.     

Bethe Ansatz Solution of Discrete Time Stochastic Processes with Fully Parallel Update

We present the Bethe ansatz solution for the discrete time zero range and asymmetric exclusion processes with fully parallel dynamics. The model depends on two parameters: p, the probability of single particle hopping, and q, the deformation parameter, which in the general case, |q| < 1, is responsible for long range interaction between particles. The particular case q = 0 corresponds to the Nagel-Schreckenberg traffic model with v _max = 1. As a result, we obtain the largest eigenvalue of the equation for the generating function of the distance travelled by particles. For the case q = 0 the result is obtained for arbitrary size of the lattice and number of particles. In the general case we study the model in the scaling limit and obtain the universal form specific for the Kardar-Parisi-Zhang universality class. We describe the phase transition occurring in the limit p→ 1 when q < 0.     

Self-diffusion in one-dimensional lattice gases in the presence of an external field

We study the motion of a tagged particle in a one-dimensional lattice gas with nearest-neighbor asymmetric jumps, withp (respectively,q),p > q, the probability to jump to the right (left). It was shown in Ref. 6 that the fluctuations in the position of the tagged particle behave normally; (X)²Dt. Here we compute explicitly the diffusion coefficient. We findD=(1-)(p-q). where is the gas density. The result confirms some recent conjectures based on theoretical arguments and computer experiments.Partially supported by NSF grant No. DMR81-14726.Partially supported by CNR.Partially supported by CNPq, grant No. 201682-83.     

From Vicious Walkers to TASEP

We propose a model of semi-vicious walkers, which interpolates between the totally asymmetric simple exclusion process and the vicious walkers model, having the two as limiting cases. For this model we calculate the asymptotics of the survival probability for m particles and obtain a scaling function, which describes the transition from one limiting case to another. Then, we use a fluctuation-dissipation relation allowing us to reinterpret the result as the particle current generating function in the totally asymmetric simple exclusion process. Thus we obtain the particle current distribution asymptotically in the large time limit as the number of particles is fixed. The results apply to the large deviation scale as well as to the diffusive scale. In the latter we obtain a new universal distribution, which has a skew non-Gaussian form. For m particles its asymptotic behavior is shown to be as y→−∞ and as y→∞.     

Matrix Representation of the Stationary Measure for the Multispecies TASEP

In this work we construct the stationary measure of the N species totally asymmetric simple exclusion process in a matrix product formulation. We make the connection between the matrix product formulation and the queueing theory picture of Ferrari and Martin. In particular, in the standard representation, the matrices act on the space of queue lengths. For N>2 the matrices in fact become tensor products of elements of quadratic algebras. This enables us to give a purely algebraic proof of the stationary measure which we present for N=3.     

A Gallavotti–Cohen-Type Symmetry in the Large Deviation Functional for Stochastic Dynamics

We extend the work of Kurchan on the Gallavotti–Cohen fluctuation theorem, which yields a symmetry property of the large deviation function, to general Markov processes. These include jump processes describing the evolution of stochastic lattice gases driven in the bulk or through particle reservoirs, general diffusive processes in physical and/or velocity space, as well as Hamiltonian systems with stochastic boundary conditions. For dynamics satisfying local detailed balance we establish a link between the average of the action functional in the fluctuation theorem and the macroscopic entropy production. This gives, in the linear regime, an alternative derivation of the Green–Kubo formula and the Onsager reciprocity relations. In the nonlinear regime consequences of the new symmetry are harder to come by and the large deviation functional difficult to compute. For the asymmetric simple exclusion process the latter is determined explicitly using the Bethe ansatz in the limit of large N.     

Phase Separation in a Bidirectional Two-Lane Asymmetric Exclusion Process

This paper studies a bidirectional two-lane asymmetric exclusion process, in which particles move in opposite direction on the two lanes. Interaction between the two lanes is implemented as follows: particle hops with rate p when there is a particle at the same site in the other lane, otherwise it hops with rate 1. It is shown that under periodic boundary conditions, a plateau will form on the fundamental diagram if p<1. This plateau corresponds to a phase separation phenomenon. We have compared the phase separation with those reported in previous works, and it is shown that the mechanism of phase separation in our model is different from previous ones. A possible phase separation mechanism is proposed, i.e., the system always tries to maximize the probability that particles could hop with rate 1. A simple mean field approximation and a 2-cluster mean field approach have been applied to calculate the steady current. It is shown that the results of the 2-cluster mean field approach are much closer to the simulations.     

Branching exclusion process on a strip

We consider a model of stochastically interacting particles on an infinite strip of ²; in this model, known as a branching exclusion process, particles jump to each empty neighboring site with rate /4 and also can create a new particle with rate 1/4 at each one of these sites. The initial configuration is assumed to have a rightmost particle and we study the process as seen from the rightmost vertical line occupied. We prove that this process has exactly one invariant measure with the property thatH, the number of empty sites to the left of the rightmost particle, has an exponential moment. This refines a result presented by Bramson {eaet al.}, who proved that ford=1,H is finite with probability 1.     

A Two-Parametric Family of Asymmetric Exclusion Processes and Its Exact Solution

A two-parameter family of asymmetric exclusion processes for particles on a one-dimensional lattice is defined. The two parameters of the model control the driving force and effect which we call pushing, due to the fact that particles can push each other in this model. We show that this model is exactly solvable via the coordinate Bethe Ansatz and show that its N-particle S-matrix is factorizable. We also study the interplay of the above effects in determining various steady state and dynamical characteristics of the system.     

The Asymmetric Avalanche Process

An asymmetric stochastic process describing the avalanche dynamics on a ring is proposed. A general kinetic equation which incorporates the exclusion and avalanche processes is considered. The Bethe ansatz method is used to calculate the generating function for the total distance covered by all particles. It gives the average velocity of particles which exhibits a phase transition from an intermittent to continuous flow. We calculated also higher cumulants and the large deviation function for the particle flow. The latter has the universal form obtained earlier for the asymmetric exclusion process and conjectured to be common for all models of the Kardar–Parisi–Zhang universality class.     

Superdiffusivity of Occupation-Time Variance in 2-dimensional Asymmetric Exclusion Processes with Density ρ = 1/2

We compute that the growth of the occupation-time variance at the origin up to time t in dimension d = 2 with respect to asymmetric simple exclusion in equilibrium with density ρ = 1/2 is in a certain sense at least tlog (log t) for general rates, and at least t(log t)^1/2 for rates which are asymmetric only in the direction of one of the axes. These estimates give a complement to bounds in the literature when d = 1, and are consistent with an important conjecture with respect to the transition function and variance of “second-class” particles.Research supported in part by NSA H982300510041 and NSF-DMS 0504193     

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.     

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.     

Exact Solution of a Cellular Automaton for Traffic

We present an exact solution of a probabilistic cellular automaton for traffic with open boundary conditions, e.g., cars can enter and leave a part of a highway with certain probabilities. The model studied is the asymmetric exclusion process (ASEP) with simultaneous updating of all sites. It is equivalent to a special case (v _max=1) of the Nagel–Schreckenberg model for highway traffic, which has found many applications in real-time traffic simulations. The simultaneous updating induces additional strong short-range correlations compared to other updating schemes. The stationary state is written in terms of a matrix product solution. The corresponding algebra, which expresses a system-size recursion relation for the weights of the configurations, is quartic, in contrast to previous cases, in which the algebra is quadratic. We derive the phase diagram and compute various properties such as density profiles, two-point functions, and the fluctuations in the number of particles (cars) in the system. The current and the density profiles can be mapped onto the ASEP with other time-discrete updating procedures. Through use of this mapping, our results also give new results for these models.     

Phase transitions in an exactly soluble one-dimensional exclusion process

We consider an exclusion process with particles injected with rate at the origin and removed with rate at the right boundary of a one-dimensional chain of sites. The particles are allowed to hop onto unoccupied sites, to the right only. For the special case of = = 1 the model was solved previously by Derridaet al. Here we extend the solution to general , . The phase diagram obtained from our exact solution differs from the one predicted by the mean-field approximation.