Stochastic oscillators out of equilibrium: scaling limits and correlation estimates

TL;DR

This paper studies a stochastic harmonic oscillator chain, revealing how energy and volume evolve under different conditions and characterizing volume fluctuations out of equilibrium using correlation bounds.

Contribution

It provides a detailed analysis of the hydrodynamic limits and fluctuation behavior of a stochastic oscillator system with conserved quantities.

Findings

Energy and volume follow heat equations or coupled parabolic equations depending on dynamics strength.

Characterization of non-equilibrium volume fluctuations under diffusive scaling.

Establishment of bounds on correlation functions and moments for the system.

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.