Stochastic Burgers equation from long range exclusion interactions

TL;DR

This paper derives the stochastic Burgers equation as the limit of one-dimensional exclusion processes with long-range jumps and asymmetry, revealing the macroscopic behavior of density fluctuations.

Contribution

It establishes the convergence of long-range exclusion processes with asymmetry to the stochastic Burgers equation under diffusive scaling.

Findings

Density fluctuations converge to the stochastic Burgers equation

The limit process is the unique energy solution of the equation

Asymmetry strength scales as rac{1}{\,\sqrt{n}} with system size

Abstract

We consider one-dimensional exclusion processes with long jumps given by a transition probability of the form $p_n(\cdot)=s(\cdot)+\gamma_na(\cdot)$, such that its symmetric part $s(\cdot)$ is irreducible with finite variance and its antisymmetric part is absolutely bounded by $s(\cdot).$ We prove that under diffusive time scaling and strength of asymmetry $\sqrt n \gamma_n \to_{n\to\infty} b\neq 0$, the equilibrium density fluctuations are given by the unique energy solution of the stochastic Burgers equation.