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Quantitative equilibrium fluctuations for interacting particle systems

Chenlin Gu, Jean-Christophe Mourrat, Maximilian Nitzschner

Abstract

We consider a class of interacting particle systems in continuous space of non-gradient type, which are reversible with respect to Poisson point processes with constant density. For these models, a rate of convergence was recently obtained in 10.1214/22-AOP1573 for certain finite-volume approximations of the bulk diffusion matrix. Here, we show how to leverage this to obtain quantitative versions of a number of results capturing the large-scale fluctuations of these systems, such as the convergence of two-point correlation functions and the Green-Kubo formula.

Quantitative equilibrium fluctuations for interacting particle systems

Abstract

We consider a class of interacting particle systems in continuous space of non-gradient type, which are reversible with respect to Poisson point processes with constant density. For these models, a rate of convergence was recently obtained in 10.1214/22-AOP1573 for certain finite-volume approximations of the bulk diffusion matrix. Here, we show how to leverage this to obtain quantitative versions of a number of results capturing the large-scale fluctuations of these systems, such as the convergence of two-point correlation functions and the Green-Kubo formula.
Paper Structure (11 sections, 11 theorems, 141 equations)

This paper contains 11 sections, 11 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Euclidean space
  4. Configuration space
  5. Homogenization of the bulk diffusion matrix
  6. Semigroup and diffusion on configuration space
  7. Two-scale expansion for elliptic equation
  8. Quantitative estimate of two-point function and semigroup
  9. Two-scale expansion of parabolic equation
  10. Regularization effect
  11. Convergence rate of Green--Kubo formulas

Key Result

Theorem 1.1

There exists an exponent $\beta(d,\Lambda,\rho) > 0$ and a constant $C(d,\Lambda,\rho) < + \infty$ such that for every $f, g \in L^1({\mathbb{R}^d}) \cap L^2({\mathbb{R}^d})$ and $t > s > 0$, we have

Theorems & Definitions (25)

  • Theorem 1.1: Quantitative asymptotics for two-point functions
  • Theorem 1.2: Quantitative Green--Kubo formula
  • Lemma 2.1
  • proof
  • Proposition 2.2: Elementary properties of the semigroup
  • proof
  • Lemma 3.1
  • proof
  • Proposition 3.2: Two-scale expansion of elliptic equation
  • proof
  • ...and 15 more