Shock Profiles for the Asymmetric Simple Exclusion Process in One Dimension

Abstract

The asymmetric simple exclusion process (ASEP) on a one-dimensional lattice is a system of particles which jump at rates $p$ and $1-p$ (here $p>1/2$) to adjacent empty sites on their right and left respectively. The system is described on suitable macroscopic spatial and temporal scales by the inviscid Burgers' equation; the latter has shock solutions with a discontinuous jump from left density $ρ_-$ to right density $ρ_+$, $ρ_-<ρ_+$, which travel with velocity $(2p-1)(1-ρ_+-ρ_-)$. In the microscopic system we may track the shock position by introducing a second class particle, which is attracted to and travels with the shock. In this paper we obtain the time invariant measure for this shock solution in the ASEP, as seen from such a particle. The mean density at lattice site $n$, measured from this particle, approaches $ρ_{\pm}$ at an exponential rate as $n\to\pm\infty$, with a characteristic length which becomes independent of $p$ when $p/(1-p)>\sqrt{ρ_+(1-ρ_-)/ρ_-(1-ρ_+)}$. For a special value of the asymmetry, given by $p/(1-p)=ρ_+(1-ρ_-)/ρ_-(1-ρ_+)$, the measure is Bernoulli, with density $ρ_-$ on the left and $ρ_+$ on the right. In the weakly asymmetric limit, $2p-1\to0$, the microscopic width of the shock diverges as $(2p-1)^{-1}$. The stationary measure is then essentially a superposition of Bernoulli measures, corresponding to a convolution of a density profile described by the viscous Burgers equation with a well-defined distribution for the location of the second class particle.