Shock Profiles for the Partially Asymmetric Simple Exclusion Process

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 $\rho_-$ to right density $\rho_+$, $\rho_-\sqrt{\rho_+(1-\rho_-)/\rho_-(1-\rho_+)}$. When $p/(1-p)=\rho_+(1-\rho_-)/\rho_-(1-\rho_+)$ the measure is Bernoulli, with density $\rho_-$ on the left and $\rho_+$ 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 space-dependent density profile described by the viscous Burgers equation with a well-defined distribution for the location of the second class particle.