Large deviations of heat flow in harmonic chains

Abstract

We consider heat transport across a harmonic chain connected at its two ends to white-noise Langevin reservoirs at different temperatures. In the steady state of this system the heat $Q$ flowing from one reservoir into the system in a finite time $τ$ has a distribution $P(Q,τ)$. We study the large time form of the corresponding moment generating function $ \sim g(λ) e^{τμ(λ)}$. Exact formal expressions, in terms of phonon Green's functions, are obtained for both $μ(λ)$ and also the lowest order correction $g(λ)$. We point out that, in general a knowledge of both $μ(λ)$ and $g(λ)$ is required for finding the large deviation function associated with $P(Q,τ)$. The function $μ(λ)$ is known to be the largest eigenvector of an appropriate Fokker-Planck type operator and our method also gives the corresponding eigenvector exactly.