## DIMACS TR: 97-08

## Shock Profiles for the Partially
Asymmetric Simple Exclusion Process

### Authors: B. Derrida, J. L. Lebowitz and E. R. Speer

**
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
$p/(1-p)>\sqrt{\rho_+(1-\rho_-)/\rho_-(1-\rho_+)}$. 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.

Paper Available at:
ftp://dimacs.rutgers.edu/pub/dimacs/TechnicalReports/TechReports/1997/97-08.ps.gz

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