Large deviations of heat flow in harmonic chains

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 $\tau$ has a distribution $P(Q,\tau)$. We study the large time form of the corresponding moment generating function $ \sim g(\lambda) e^{\tau\mu (\lambda)}$. Exact formal expressions, in terms of phonon Green's functions, are obtained for both $\mu(\lambda)$ and also the lowest order correction $g(\lambda)$. We point out that, in general a knowledge of both $\mu(\lambda)$ and $g(\lambda)$ is required for finding the large deviation function associated with $P(Q,\tau)$. The function $\mu(\lambda)$ is known to be the largest eigenvector of an appropriate Fokker-Planck type operator and our method also gives the corresponding eigenvector exactly.