A Detailed Fluctuation Theorem for Heat Fluxes in Harmonic Networks Out of Thermal Equilibrium

Abstract

We continue the investigation, started in [JPS], of a network of harmonic oscillators driven out of thermal equilibrium by heat reservoirs. We study the statistics of the fluctuations of the heat fluxes flowing between the network and the reservoirs in the nonequilibrium steady state and in the large time limit. We prove a large deviation principle for these fluctuations and derive the fluctuation relation satisfied by the associated rate function.performed by the Langevin force f i during the time interval [0, t ] as the amount of heat injected in the network by the i th -reservoir during this period. We denote by Ξ = R ∂I the vector space where the heat currents Φ(t ) = (Φ i (t )) i ∈∂I take their values and write the associated Euclidean inner product as 〈ξ,The main results of the present work concern the statistics of the Ξ-valued process {Φ(t )} t ≥0 induced by the stationary Markov process generated by the system of stochastic differential equationṡ q = ∇ p H (q, p),ṗ = −∇ q H (q, p) + f , with appropriate initial conditions. More precisely, and under a controllability condition which ensures the existence and uniqueness of an invariant measure for this Markov process:• We identify a subspace L ⊂ Ξ characterized by the fact that for ξ ∈ L one has 〈ξ, Φ(t )〉 = Q ξ (q(t ), p(t )) − Q ξ (q(0), p (0)) (1.3) Parts of this work were performed during the visits of M.D. and M.H. at the University of Toulon and of C.-A.P. at the University of Sfax. We thank the CPT, the University of Toulon and the Mathematics Department of Sfax for their hospitality and support. M.H. and C.-A.P. are also grateful to the Centre de Recherches Mathématiques de l'Université de Montréal for its hospitality and the Simons foundation and CNRS for their support during their stay in Montréal in the fall 2018.