A consistence-stability approach to hydrodynamic limit of interacting particle systems on lattices

TL;DR

This paper introduces a new, simple, and quantitative method based on a consistence-stability approach in Wasserstein distance for proving the hydrodynamic limit of certain interacting particle systems on lattices, specifically in one dimension.

Contribution

It presents a novel, simplified, and quantitative method that avoids block estimates for establishing hydrodynamic limits in lattice particle systems.

Findings

Method applies to zero-range and Ginzburg-Landau processes with Kawasaki dynamics.

Convergence rate is quantitative and uniform in time.

Proof relies on a consistence-stability approach in Wasserstein distance.

Abstract

This is a review based on the presentation done at the seminar Laurent Schwartz in December 2021. It is announcing results in the forthcoming [Menegaki-Mouhot-Marahrens'22]. This work presents a new simple quantitative method for proving the hydrodynamic limit of a class of interacting particle systems on lattices. We present here this method in a simplified setting, for the zero-range process and the Ginzburg-Landau process with Kawasaki dynamics, in the parabolic scaling and in dimension $1$. The rate of convergence is quantitative and uniform in time. The proof relies on a consistence-stability approach in Wasserstein distance, and it avoids the use of the ``block estimates''.