Crossover to the KPZ equation

We characterize the crossover regime to the KPZ equation for a class of one-dimensional weakly asymmetric exclusion processes. The crossover depends on the strength asymmetry $a{n}^{2-\gamma }$ ($a,\gamma >0$) and it occurs at $\gamma =1\mathrm{/}2$. We show that the density field is a solution of an Ornstein-Uhlenbeck equation if $\gamma \in (1\mathrm{/}2,1]$, while for $\gamma =1\mathrm{/}2$ it is an energy solution of the KPZ equation. The corresponding crossover for the current of particles is readily obtained.