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Stochastic oscillators out of equilibrium: scaling limits and correlation estimates

Patrícia Gonçalves, Kohei Hayashi, João Pedro Mangi

TL;DR

This work analyzes a stochastic Bernardin-Stoltz harmonic chain on the 1D torus with exchange noise, establishing diffusive hydrodynamic limits for the conserved volume and energy under diffusive scaling and a weak-asymmetry parameter. The HDL results distinguish between a regime with autonomous heat equations for $\kappa>1$ and a coupled system at the critical value $\kappa=1$, with precise weak-solution formulations and quantitative energy convergence in the latter. The paper also develops a non-equilibrium fluctuation theory for the volume field, proving tightness and convergence to an Ornstein–Uhlenbeck-type limit in equilibrium and providing a detailed martingale framework and quadratic-variation analysis. Central technical contributions include a fourth-moment bound via an $\mathcal{H}^{-1}$-norm and sharp two-point volume-correlation estimates, enabling control of fluctuations and laying groundwork for higher-order correlation studies. The results enhance understanding of how stochastic perturbations interact with Hamiltonian dynamics to shape diffusive macroscopic behavior and fluctuation structures in unbounded, multi-conserved-quantity systems.

Abstract

We consider a purely harmonic chain of oscillators which is perturbed by a stochastic noise. Under this perturbation, the system exhibits two conserved quantities: the volume and the energy. At the level of the hydrodynamic limit, under diffusive scaling, we show that depending on the strength of the Hamiltonian dynamics, energy and volume evolve according to either a system of autonomous heat equations or a non-linear system of coupled parabolic equations. Moreover, for general initial measures, under diffusive scaling, we can characterize the non-equilibrium volume fluctuations. The proofs are based on precise bounds on the two-point volume correlation function and a uniform fourth-moment estimate.

Stochastic oscillators out of equilibrium: scaling limits and correlation estimates

TL;DR

This work analyzes a stochastic Bernardin-Stoltz harmonic chain on the 1D torus with exchange noise, establishing diffusive hydrodynamic limits for the conserved volume and energy under diffusive scaling and a weak-asymmetry parameter. The HDL results distinguish between a regime with autonomous heat equations for and a coupled system at the critical value , with precise weak-solution formulations and quantitative energy convergence in the latter. The paper also develops a non-equilibrium fluctuation theory for the volume field, proving tightness and convergence to an Ornstein–Uhlenbeck-type limit in equilibrium and providing a detailed martingale framework and quadratic-variation analysis. Central technical contributions include a fourth-moment bound via an -norm and sharp two-point volume-correlation estimates, enabling control of fluctuations and laying groundwork for higher-order correlation studies. The results enhance understanding of how stochastic perturbations interact with Hamiltonian dynamics to shape diffusive macroscopic behavior and fluctuation structures in unbounded, multi-conserved-quantity systems.

Abstract

We consider a purely harmonic chain of oscillators which is perturbed by a stochastic noise. Under this perturbation, the system exhibits two conserved quantities: the volume and the energy. At the level of the hydrodynamic limit, under diffusive scaling, we show that depending on the strength of the Hamiltonian dynamics, energy and volume evolve according to either a system of autonomous heat equations or a non-linear system of coupled parabolic equations. Moreover, for general initial measures, under diffusive scaling, we can characterize the non-equilibrium volume fluctuations. The proofs are based on precise bounds on the two-point volume correlation function and a uniform fourth-moment estimate.
Paper Structure (27 sections, 26 theorems, 235 equations)

This paper contains 27 sections, 26 theorems, 235 equations.

Table of Contents

  1. Introduction
  2. Model description and main results
  3. The microscopic model
  4. The dynamics
  5. Invariant measures
  6. Main results
  7. Assumptions on the initial measure
  8. Hydrodynamic limit
  9. Non-equilibrium fluctuations
  10. Hydrodynamic limit
  11. Proof outline
  12. Tightness
  13. Volume empirical measures
  14. Energy empirical measures
  15. Characterization of the limit points
  16. ...and 12 more sections

Key Result

Theorem 2.5

Let $v_0:\mathbb T\to \mathbb R$ and $e_0:\mathbb T\to (0,\infty)$ be bounded measurable functions. Let $\{\mu_N\}_{N\in\mathbb N}$ be a sequence of probability measures on $\Omega_N$ associated to the pair $(v_0,e_0)$, and assume that $\{\mu_N\}_{N\in\mathbb N}$ satisfies Assumption assump:initial_ for each $\sigma\in \{e,v\}$ provided $m>5/2$, where recall that the measure $Q^{v,N}_m$ and $Q^{e,

Theorems & Definitions (47)

  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5
  • Remark 2.6
  • Theorem 2.7
  • Theorem 2.8
  • Remark 2.9
  • Remark 2.10
  • Theorem 2.11
  • ...and 37 more