The Annals of Probability

Microscopic Structure of Travelling Waves in the Asymmetric Simple Exclusion Process

P. A. Ferrari, C. Kipnis, and E. Saada

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Abstract

The one-dimensional nearest neighbor asymmetric simple exclusion process has been used as a microscopic approximation for the Burgers equation. This equation has travelling wave solutions. In this paper we show that those solutions have a microscopic structure. More precisely, we consider the simple exclusion process with rate $p$ (respectively, $q = 1 - p)$ for jumps to the right (left), $\frac{1}{2} < p \leq 1$, and we prove the following results: There exists a measure $\mu$ on the space of configurations approaching asymptotically the product measure with densities $\rho$ and $\lambda$ to the left and right of the origin, respectively, $\rho < \lambda$, and there exists a random position $X(t) \in \mathbb{Z}$, such that, at time $t$, the system "as seen from $X(t)$," remains distributed according to $\mu$, for all $t \geq 0$. The hydrodynamical limit for the simple exclusion process with initial measure $\mu$ converges to the travelling wave solution of the inviscid Burgers equation. The random position $X(t)/t$ converges strongly to the speed $\nu = (1 - \lambda - \rho)(p - q)$ of the travelling wave. Finally, in the weakly asymmetric hydrodynamical limit, the stationary density profile converges to the travelling wave solution of the Burgers equation.

Article information

Source
Ann. Probab. Volume 19, Number 1 (1991), 226-244.

Dates
First available in Project Euclid: 19 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1176990542

Digital Object Identifier
doi:10.1214/aop/1176990542

Mathematical Reviews number (MathSciNet)
MR1085334

Zentralblatt MATH identifier
0725.60113

JSTOR
links.jstor.org

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Asymmetric simple exclusion process Burgers equation microscopic travelling waves

Citation

Ferrari, P. A.; Kipnis, C.; Saada, E. Microscopic Structure of Travelling Waves in the Asymmetric Simple Exclusion Process. Ann. Probab. 19 (1991), no. 1, 226--244. doi:10.1214/aop/1176990542. https://projecteuclid.org/euclid.aop/1176990542


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