Travelling waves and long-time behaviour of the weakly asymmetric exclusion process

Abstract

We consider the one-dimensional weakly asymmetric nearest neighbour exclusion process and study its macroscopic behaviour on a longer time scale than that for which the deterministic approximation by the Burgers equation holds. We show that this long-time behaviour is completely determined by the initial particle configuration, and does not depend on the randomness of the evolution. Moreover, we examine in detail a special case for which a travelling wave is observed with stable profile but randomly fluctuating space location.