Generating Function Formula of Heat Transfer in Harmonic Networks

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We consider heat transfer across an arbitrary harmonic network connected to two heat baths at different temperatures. The network has N positional degrees of freedom, of which Nₗ are connected to a bath at temperature Tₗ and Nᵣ are connected to a bath at temperature Tᵣ. We derive an exact formula for the cumulant generating function for heat transfer between the two baths. The formula is valid even for Nₗ ≠ Nᵣ and satisfies the Gallavotti-Cohen fluctuation symmetry. Since harmonic crystals in three dimensions are known to exhibit different regimes of transport such as ballistic, anomalous and diffusive, our result implies validity of the fluctuation theorem in all regimes. Our exact formula provides a powerful tool to study other properties of nonequilibrium current fluctuations.