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 (e ^{− λQ}~ 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.