The crossover to the KPZ equation

Abstract

In this paper we consider the one-dimensional weakly asymmetric simple exclusion process under the invariant state $\nu_{\rho}$: the Bernoulli product measure of parameter $\rho\in{(0,1)}$. We show that the limit density field is governed by an Ornstein-Uhlenbeck process for strength asymmetry $n^{2-\gamma}$ if $\gamma\in(1/2,1)$, while for $\gamma=1/2$ it is an energy solution of the KPZ equation. From this result we obtain that the fluctuations of the current of particles are Gaussian distributed for $\gamma\in(1/2,1)$, while for $\gamma=1/2$ the limit distribution is written in terms of the KPZ equation.