Microscopic Structure of Travelling Waves in the Asymmetric Simple Exclusion Process

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.