Shock fluctuations in the asymmetric simple exclusion process

Summary We consider the one dimensional nearest neighbors asymmetric simple exclusion process with ratesq andp for left and right jumps respectively;q<p. Ferrari et al. (1991) have shown that if the initial measure isv _, , a product measure with densities and to the left and right of the origin respectively, <, then there exists a (microscopic) shock for the system. A shock is a random positionX _t such that the system as seen from this position at timet has asymptotic product distributions with densities and to the left and right of the origin respectively, uniformly int. We compute the diffusion coefficient of the shockD=lim _t t ^–1(E(X _t )²–(EX _t )²) and findD=(p–q)(–)^–1((1–)+(1–)) as conjectured by Spohn (1991). We show that in the scale the position ofX _t is determined by the initial distribution of particles in a region of length proportional tot. We prove that the distribution of the process at the average position of the shock converges to a fair mixture of the product measures with densities and . This is the so called dynamical phase transition. Under shock initial conditions we show how the density fluctuation fields depend on the initial configuration.     

Multiparticle space-time transitions in the totally asymmetric simple exclusion process

We study correlation functions of the totally asymmetric simple exclusion process (TASEP) in discrete time with backward sequential update. We prove a determinant formula for the generalized Green’s function describing transitions between particle positions at given instants. As an example, we calculate the current correlation function, i.e., the joint probability distribution of times required by each particle to travel a given distance. An asymptotic analysis shows that current fluctuations converge to the Airy ₂ process.     

Upper bound on the occupation time in the simple exclusion process

We consider the occupation time variance in the asymmetric exclusion process. In the case where the mean m ≠ 0 (m is the mean hopping rate) and ρ = 1/2 (ρ is the filling probability for a state), we find that the variance is bounded above by O(t ^3/2 ). __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 156, No. 1, pp. 147–158, July, 2008.     

Nonequilibrium scaling limit for a tagged particle in the simple exclusion process with long jumps

We prove an invariance principle for a tagged particle in a simple exclusion process with long jumps out of equilibrium. © 2008 Wiley Periodicals, Inc.     

Shocks in the asymmetric exclusion process

Summary In this paper, we consider limit theorems for the asymmetric nearest neighbor exclusion process on the integers. The initial distribution is a product measure with asymptotic density at - and at +. Earlier results described the limiting behavior in all cases except for 0<<1/2, +=1. Here we treat the exceptional case, which is more delicate. It corresponds to the one in which a shock wave occurs in an associated partial differential equation. In the cases treated earlier, the limit was an extremal invariant measure. By contrast, in the present case the limit is a mixture of two invariant measures. Our theorem resolves a conjecture made by the third author in 1975 [7]. The convergence proof is based on coupling and symmetry considerations.Research supported in part by NSF Grant DMS 83-1080Research supported in part by NSF Grant MCS 83-00836     

Shock fluctuations in asymmetric simple exclusion

Summary The one dimensional nearest neighbors asymmetric simple exclusion process in used as a microscopic approximation to the Burgers equation. We study the process with rates of jumpsp>q to the right and left, respectively, and with initial product measure with densities < to the left and right of the origin, respectively (with shock initial conditions). We prove that a second class particle added to the system at the origin at time zero identifies microscopically the shock for all later times. If this particle is added at another site, then it describes the behavior of a characteristic of the Burgers equation. For vanishing left density (=0) we prove, in the scale t^1/2, that the position of the shock at timet depends only on the initial configuration in a region depending ont. The proofs are based on laws of large numbers for the second class particle.     

Invariant measures for the strong-facilitated exclusion process

Consider a generalized model of the facilitated exclusion process, which is a onedimensional exclusion process with a dynamical constraint that prevents the particle at site x from jumping to x+1 (or x-1) if the sites x-1, x-2 (or x+1, x+2) are empty. It is nongradient and lacks invariant measures of product form. The purpose of this paper is to identify the invariant measures and to show that they satisfy both exponential decay of correlations and equivalence of ensembles. These properties will...     

Distributional limits for the symmetric exclusion process

Strong negative dependence properties have recently been proved for the symmetric exclusion process. In this paper, we apply these results to prove convergence to the Poisson and Gaussian distributions for various functionals of the process.     

Microscopic structure of the k-step¶exclusion process

We review the hydrodynamics and discuss the shock, rarefaction fan and contact discontinuity at a microscopic level for a one-dimensional totally asymmetric k-step exclusion process. In particular we define a microscopical object that identifies the shock in the decreasing case. Received: 9 April 2002     

Asymptotic behavior of a tagged particle in simple exclusion processes

We review in this article central limit theorems for a tagged particle in the simple exclusion process. In the first two sections we present a general method to prove central limit theorems for additive functional of Markov processes. These results are then applied to the case of a tagged particle in the exclusion process. Related questions, such as smoothness of the diffusion coefficient and finite dimensional approximations, are considered in the last section.     

Pointwise ergodic theorems for the symmetric exclusion process

Summary Almost sure convergence theorems are proved for Cesaro averages of continous functions in the case of the symmetric exclsion processes in dimension d≧3. For the occupation time of a single site the same result is proved in all dimensions. Partially supported by CNPq     

Hydrodynamics and large deviation for simple exclusion processes

We prove the hydrodynamical limit for weakly asymmetric simple exclusion processes. A large deviation property with respect to this limit is established for the symmetric case. We treat also the situation where a slow reaction (creation and annihilation of particles) is present.     

Hydrodynamical limit for spatially heterogeneous simple exclusion processes

Summary.   We prove hydrodynamical limit for spatially heterogeneous, asymmetric simple exclusion processes on Z ^d . The jump rate of particles depends on the macroscopic position x through some nonnegative, smooth velocity profile α(x). Hydrodynamics are described by the entropy solution to a spatially heterogeneous conservation law of the form

Fluctuation–dissipation equation of asymmetric simple exclusion processes

Summary. We consider asymmetric simple exclusion processes on the lattice Zopf; ^d in dimension d≥3. We denote by L the generator of the process, ∇ the lattice gradient, η the configuration, and w the current of the dynamics associated to the conserved quantity. We prove that the fluctuation–dissipation equation w=Lu+D∇η has a solution for some function u and some constant D identified to be the diffusion coefficient. Intuitively, Lu represents rapid fluctuation and this equation describes a decomposition of the current into fluctuation and gradient of the density field, representing the dissipation. Using this result, we proved rigorously that the Green-Kubo formula converges and it can be identified as the diffusion coefficient. Received: 14 May 1996 / In revised form: 20 February 1997     

Diffusive limit of a tagged particle in asymmetric simple exclusion processes

Invariance principles are proved under diffusive scaling for the centered position of a tagged particle in the simple exclusion process with asymmetric nonzero drift jump probabilities in dimensions d ≥ 3. The method of proof is by martin‐gale techniques which rely on the fact that symmetric random walks are transient in high dimensions. © 2000 John Wiley & Sons, Inc.