## The Annals of Probability

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

### Microscopic Structure of Travelling Waves in the Asymmetric Simple Exclusion Process

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

#### 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