  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.
