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

Soumyabrata Saha, S. Jangid+3 more|arXiv (Cornell University)|Jan 5, 2026

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TL;DR

The authors develop a path-integral–based coarse-graining framework that derives fluctuating hydrodynamics from microscopic stochastic lattice gases and the Dean-Kawasaki equation, yielding explicit transport coefficients D(ρ) and σ(ρ) and validating with numerics.

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 naïve 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.

Motivation & Objective

• Develop a systematic coarse-graining method to derive fluctuating hydrodynamics from microscopic stochastic dynamics.
• Show the essential role of local-equilibrium averages beyond naive gradient expansions.
• Extend the framework to interacting Brownian particles via Dean-Kawasaki and recover fluctuating hydrodynamics.
• Provide explicit transport coefficients D(ρ) and σ(ρ) for various lattice models and compare with known results.
• Validate theoretical predictions with numerical simulations.

Proposed method

• Use Martin-Siggia-Rose-Janssen-De Dominicis (MSRJD) path-integral formalism to represent the stochastic diffusion equation for the density.
• Start from microscopic lattice-gas dynamics (gradient models with a single conserved density) and write the exact microscopic path probability.
• Average over local equilibrium measures at smoothly varying density to obtain the coarse-grained hydrodynamic action.
• Perform a second-order gradient expansion while preserving the gradient structure of the current, yielding D(ρ) and σ(ρ).
• Derive the fluctuating hydrodynamic equation ∂tρ = ∂x(D(ρ)∂xρ) + (1/√ℓ)∂x(√σ(ρ) η) and relate coefficients via fluctuation-dissipation relations when appropriate.
• Extend the approach to the Dean-Kawasaki equation for interacting Brownian particles and extract the resulting mobility and diffusivity.

(b) Variance of the current

Experimental results

Research questions

• RQ1 How can one systematically derive fluctuating hydrodynamics from underlying stochastic microscopic dynamics for diffusive systems with a single conserved density?
• RQ2 What is the role of local-equilibrium averages in coarse-graining beyond naive gradient expansions?
• RQ3 How do transport coefficients D(ρ) and σ(ρ) emerge for different microscopic models (lattice exclusion, partial exclusion, inclusion) and for Brownian particles?
• RQ4 Can the method be extended to coarse-grain the Dean-Kawasaki equation and reveal the correct dependence of mobility and diffusivity on density?

Key findings

• A bottom-up coarse-graining procedure recovers fluctuating hydrodynamics for diffusive lattice gases with a single conserved density.
• Local-equilibrium averaging is essential and nontrivial, going beyond simple gradient expansions to obtain correct coefficients.
• Explicit transport coefficients D(ρ) and σ(ρ) are derived for multiple lattice models, including SSEP, SSDEP, SSMEP, SSPEP, and their Brownian-particle counterparts.
• The framework reveals a gradient structure of microscopic currents and yields known results for D(ρ) and σ(ρ) that match existing literature (e.g., for exclusion models, Brownian hard rods).
• Numerical simulations corroborate the theoretical fluctuating-hydrodynamics predictions across the considered models.

(c) Variance of the tracer-position