Quantitative homogenization of interacting particle systems

TL;DR

This paper proves that finite-volume approximations of the bulk diffusion matrix for certain reversible, non-gradient interacting particle systems in continuous space converge algebraically, using methods inspired by elliptic homogenization.

Contribution

It introduces a novel approach to quantitative homogenization for non-gradient particle systems, including new inequalities of independent interest.

Findings

Finite-volume approximations converge at an algebraic rate.

Develops modifications of Caccioppoli and multiscale Poincaré inequalities.

Applicable to reversible, non-gradient particle systems in continuous space.

Abstract

For a class of interacting particle systems in continuous space, we show that finite-volume approximations of the bulk diffusion matrix converge at an algebraic rate. The models we consider are reversible with respect to the Poisson measures with constant density, and are of non-gradient type. Our approach is inspired by recent progress in the quantitative homogenization of elliptic equations. Along the way, we develop suitable modifications of the Caccioppoli and multiscale Poincar\'e inequalities, which are of independent interest.