Convergence to the Stochastic Burgers Equation from a degenerate microscopic dynamics

TL;DR

This paper proves the convergence of certain one-dimensional interacting particle systems with degenerate jump rates to the stochastic Burgers equation, introducing a new proof of the second order Boltzmann-Gibbs principle.

Contribution

It provides a novel proof of the Boltzmann-Gibbs principle for models with blocked configurations and no spectral gap assumptions, advancing understanding of hydrodynamic limits.

Findings

Convergence established for models with degenerate jump rates

Blocked configurations are exponentially rare under equilibrium

Dynamical mechanisms enable particle exchange despite degeneracy

Abstract

In this paper we prove the convergence to the stochastic Burgers equation from one-dimensional interacting particle systems, whose dynamics allow the degeneracy of the jump rates. To this aim, we provide a new proof of the second order Boltzmann-Gibbs principle introduced in [Gon\c{c}alves, Jara 2014]. The main technical difficulty is that our models exhibit configurations that do not evolve under the dynamics - the blocked configurations - and are locally non-ergodic. Our proof does not impose any knowledge on the spectral gap for the microscopic models. Instead, it relies on the fact that, under the equilibrium measure, the probability to find a blocked configuration in a finite box is exponentially small in the size of the box. Then, a dynamical mechanism allows to exchange particles even when the jump rate for the direct exchange is zero.