Equilibrium perturbations for stochastic interacting systems

TL;DR

This paper studies how small perturbations in equilibrium states of stochastic systems like exclusion processes and anharmonic chains evolve, showing they follow Burgers equations under certain conditions.

Contribution

It demonstrates the evolution of equilibrium perturbations in stochastic systems follows Burgers equations, extending understanding of their macroscopic behavior.

Findings

Perturbed quantities follow Burgers equations in the exclusion process.

In anharmonic chains, perturbations evolve according to two decoupled Burgers equations.

Results hold in the smooth regime under specific parameter constraints.

Abstract

We consider the equilibrium perturbations for two stochastic systems: the $d$-dimensional generalized exclusion process and the one-dimensional chain of anharmonic oscillators. We add a perturbation of order $N^{-\alpha}$ to the equilibrium profile and speed up the process by $N^{1+\kappa}$ for parameters $0<\kappa\le\alpha$. Under some additional constraints on $\kappa$ and $\alpha$, we show the perturbed quantities evolve according to the Burgers equation in the exclusion process, and to two decoupled Burgers equations in the anharmonic chain, both in the smooth regime.