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 I 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 p_ to right density p +, p_ < p +, which travel with velocity ( 2 p 1 ) ( 1 p + p_). 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 p + at an exponential rate as n ~ + oo, with a characteristic length which becomes independent of p when p/( 1 p) > x / P + ( 1 p _ )/p _ ( 1 p + ). For a special value of the asymmetry, given by p / ( l p ) = p +(1 p _ ) / p_ ( 1 p § ), the measure is Bernoulli, with density p_ on the left and p + on the right. In the weakly asymmetric limit, 2 p 1 --, 0, the microscopic width of the shock diverges as ( 2 p 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.