A bottom-up approach to fluctuating hydrodynamics: Coarse-graining of stochastic lattice gases and the Dean-Kawasaki equation

TL;DR

This paper presents a systematic derivation of fluctuating hydrodynamics from microscopic stochastic lattice gases and the Dean-Kawasaki equation, emphasizing local equilibrium and coarse-graining techniques.

Contribution

It introduces a path-integral based coarse-graining method for stochastic lattice gases and extends this approach to interacting Brownian particles via the Dean-Kawasaki equation.

Findings

Coarse-graining recovers fluctuating hydrodynamics with controlled approximations.

Local-equilibrium averages are crucial for accurate derivation.

Revealed a density-dependent mobility and thermodynamic pressure-driven diffusivity.

Abstract

Fluctuating hydrodynamics provides a quantitative, large-scale description of many-body systems in terms of smooth variables, with microscopic details entering only through a small set of transport coefficients. Although this framework has been highly successful in characterizing macroscopic fluctuations and correlations, a systematic derivation of fluctuating hydrodynamics from underlying stochastic microscopic dynamics remains obscure for broad classes of interacting systems. For stochastic lattice gas models with gradient dynamics and a single conserved density, we develop a path-integral based coarse-graining procedure that recovers fluctuating hydrodynamics in a controlled manner. Our analysis highlights the essential role of local-equilibrium averages, which go beyond na\"ive mean-field-type gradient expansions. We further extend this approach to interacting Brownian particles by coarse-graining the Dean-Kawasaki equation, revealing a mobility proportional to the density and a diffusivity determined by the thermodynamic pressure.