Crossover to the KPZ equation

TL;DR

This paper analyzes the transition from weakly asymmetric exclusion processes to the KPZ equation in one dimension, identifying the critical crossover point and describing the limiting behaviors of the density field.

Contribution

It characterizes the crossover regime to the KPZ equation depending on asymmetry strength, establishing the limiting equations for the density field at the critical point.

Findings

Density field solves Ornstein-Uhlenbeck equation for b3a0b3a0a0(1/2,1]

At b3=1/2, density field is an energy solution of KPZ

Crossover for particle current derived from density field behavior

Abstract

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 $an^{2-\gamma}$ ($a,\gamma>0$) and it occurs at $\gamma=1/2$. We show that the density field is a solution of an Ornstein-Uhlenbeck equation if $\gamma\in(1/2,1]$, while for $\gamma=1/2$ it is an energy solution of the KPZ equation. The corresponding crossover for the current of particles is readily obtained.