Large deviations for a stochastic model of heat flow

TL;DR

This paper derives the large deviation function for energy profiles in a stochastic harmonic oscillator chain, revealing nonlocal, non-convex features and enhanced fluctuations compared to local equilibrium, extending understanding of nonequilibrium heat flow models.

Contribution

It provides the first derivation of the large deviation function for a stochastic heat conduction model, highlighting its nonlocal and non-convex nature and comparing it to the symmetric exclusion process.

Findings

Large deviation function $S( heta)$ is nonlocal and non-convex.

Normal fluctuations are enhanced compared to local equilibrium.

Features are common in similar nonequilibrium models.

Abstract

We investigate a one dimensional chain of $2N$ harmonic oscillators in which neighboring sites have their energies redistributed randomly. The sites $-N$ and $N$ are in contact with thermal reservoirs at different temperature $\tau_-$ and $\tau_+$. Kipnis, Marchioro, and Presutti \cite{KMP} proved that this model satisfies {}Fourier's law and that in the hydrodynamical scaling limit, when $N \to \infty$, the stationary state has a linear energy density profile $\bar \theta(u)$, $u \in [-1,1]$. We derive the large deviation function $S(\theta(u))$ for the probability of finding, in the stationary state, a profile $\theta(u)$ different from $\bar \theta(u)$. The function $S(\theta)$ has striking similarities to, but also large differences from, the corresponding one of the symmetric exclusion process. Like the latter it is nonlocal and satisfies a variational equation. Unlike the latter it is not convex and the Gaussian normal fluctuations are enhanced rather than suppressed compared to the local equilibrium state. We also briefly discuss more general model and find the features common in these two and other models whose $S(\theta)$ is known.