On the hydrodynamics of active matter models on a lattice

TL;DR

This paper explores the mathematical derivation of hydrodynamic limits for active matter models on lattices, highlighting differences between gradient and non-gradient models and providing insights for both physics and mathematics communities.

Contribution

It links microscopic lattice gas dynamics to macroscopic hydrodynamic equations for active matter, emphasizing the mathematical challenges and differences in model types.

Findings

Derived hydrodynamic limits for various active lattice gases.

Identified key differences between gradient and non-gradient models.

Provided guidance on designing microscopic models with correct macroscopic behavior.

Abstract

Active matter has been widely studied in recent years because of its rich phenomenology, whose mathematical understanding is still partial. We present some results, based on [8, 17] linking microscopic lattice gases to their macroscopic limit, and explore how the mathematical state of the art allows to derive from various types of microscopic dynamics their hydrodynamic limit. We present some of the crucial aspects of this theory when applied to weakly asymmetric active models. We comment on the specific challenges one should consider when designing an active lattice gas, and in particular underline mathematical and phenomenological differences between gradient and non-gradient models. Our purpose is to provide the physics community, as well as member of the mathematical community not specialized in the mathematical derivation of scaling limits of lattice gases, some key elements in defining microscopic models and deriving their hydrodynamic limit.