Shock profiles for the asymmetric simple exclusion process in one dimension

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_± at an exponential rate as n → ± ∞, with a characteristic length which becomes independent of p when p/(1 - p) > √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 → 0, 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.