B. Derrida, J. L. Lebowitz and E. R. Speer
Shock Profiles for the Asymmetric Simple Exclusion Process
in One Dimension
(94K, LaTeX)
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 $\rho_-$ to right density $\rho_+$, $\rho_-<\rho_+$,
which travel with velocity $(2p-1)(1-\rho_+-\rho_-)$. 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
$\rho_{\pm}$ at an exponential rate as $n\to\pm\infty$, with a
characteristic length which becomes independent of $p$ when
$p/(1-p)>\sqrt{\rho_+(1-\rho_-)/\rho_-(1-\rho_+)}$. For a special
value of the asymmetry, given by
$p/(1-p)=\rho_+(1-\rho_-)/\rho_-(1-\rho_+)$, 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 density profile described by the viscous Burgers
equation with a well-defined distribution for the location of the
second class particle.