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Stochastic Burgers Equation from Non-Product Stationary Measures via a Generalised Second-Order Boltzmann-Gibbs Principle

Patrícia Gonçalves, Maria Chiara Ricciuti, Gunter Schütz

TL;DR

The paper extends the Boltzmann-Gibbs framework to conservative lattice systems with non-product stationary measures, removing the need for a spectral gap or equivalence-of-ensembles by leveraging correlation-decay bounds. It introduces a generalised second-order Boltzmann-Gibbs principle proven via a multi-scale induction that controls non-invariant corrections, and applies it to the Katz-Lebowitz-Spohn model to show convergence of equilibrium density fluctuations to the stationary energy solution of the stochastic Burgers equation. The work also clarifies the role of degenerate dynamics by restricting to configurations with mobile clusters and demonstrates consistency with mode-coupling predictions, including a KPZ-to-diffusive crossover in driven systems. Together, these results provide a rigorous route from non-product invariant measures to macroscopic SPDE dynamics and open avenues for non-gradient and non-translation-invariant extensions.

Abstract

We prove a generalised second-order Boltzmann-Gibbs principle for conservative interacting particle systems on a lattice whose stationary measures are not of product type and not invariant under particle jumps. The result, which requires neither a spectral gap bound nor an equivalence of ensembles, extends the classical framework to settings with correlated invariant measures and is based on quantitative bounds for the correlation decay. As an application, we show that the equilibrium density fluctuations of the Katz-Lebowitz-Spohn model with a given choice of parameters converge, under diffusive scaling, to the stationary energy solution of the stochastic Burgers equation.

Stochastic Burgers Equation from Non-Product Stationary Measures via a Generalised Second-Order Boltzmann-Gibbs Principle

TL;DR

The paper extends the Boltzmann-Gibbs framework to conservative lattice systems with non-product stationary measures, removing the need for a spectral gap or equivalence-of-ensembles by leveraging correlation-decay bounds. It introduces a generalised second-order Boltzmann-Gibbs principle proven via a multi-scale induction that controls non-invariant corrections, and applies it to the Katz-Lebowitz-Spohn model to show convergence of equilibrium density fluctuations to the stationary energy solution of the stochastic Burgers equation. The work also clarifies the role of degenerate dynamics by restricting to configurations with mobile clusters and demonstrates consistency with mode-coupling predictions, including a KPZ-to-diffusive crossover in driven systems. Together, these results provide a rigorous route from non-product invariant measures to macroscopic SPDE dynamics and open avenues for non-gradient and non-translation-invariant extensions.

Abstract

We prove a generalised second-order Boltzmann-Gibbs principle for conservative interacting particle systems on a lattice whose stationary measures are not of product type and not invariant under particle jumps. The result, which requires neither a spectral gap bound nor an equivalence of ensembles, extends the classical framework to settings with correlated invariant measures and is based on quantitative bounds for the correlation decay. As an application, we show that the equilibrium density fluctuations of the Katz-Lebowitz-Spohn model with a given choice of parameters converge, under diffusive scaling, to the stationary energy solution of the stochastic Burgers equation.
Paper Structure (23 sections, 18 theorems, 111 equations, 2 figures)

This paper contains 23 sections, 18 theorems, 111 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Statement of Results
  3. The Generalised Second-Order Boltzmann-Gibbs Principle
  4. Assumptions
  5. Main Result
  6. Application to the KLS Model
  7. Dynamics and Invariant Measure
  8. Stationary Energy Solutions of the Stochastic Burgers Equation
  9. Convergence of Equilibrium Fluctuations
  10. Proof of the Generalised Second-Order Boltzmann-Gibbs Principle
  11. One-Block Estimate
  12. Doubling the Box
  13. Renormalisation Step
  14. Proof of Theorem \ref{['thm:gen_bg']}
  15. Equilibrium Fluctuations of the KLS Model
  16. ...and 8 more sections

Key Result

Theorem 2.1

Let Assumptions 1, 2 and 3 hold.

Figures (2)

  • Figure 1: Microscopic dynamics of the KLS model with $\kappa_N=-1$.
  • Figure 2: Example of a path from a configuration $\eta$ in $\mathscr{G}_{\ell_0}(x+\ell)$ to the configuration $\eta^{x+y, x+z}$ that swaps the occupation state of sites $x+y$ and $x+z$.

Theorems & Definitions (40)

  • Theorem 2.1: Generalised Second-Order Boltzmann-Gibbs Principle
  • Remark 2.2
  • Corollary 2.3
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6: Stationarity
  • Definition 2.7: Energy Estimate
  • Proposition 2.8
  • Definition 2.9: Stationary Energy Solution of the Stochastic Burgers Equation
  • Proposition 2.10
  • ...and 30 more