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Towards a derivation of Fourier’s law for coupled anharmonic oscillators. (English) Zbl 1136.82026

Summary: We consider a Hamiltonian system made of weakly coupled anharmonic oscillators arranged on a three dimensional lattice \({\mathbb{Z}}_{2N} \times \mathbb{Z}^2\), and subjected to stochastic forcing mimicking heat baths of temperatures \(T_{1}\) and \(T_{2}\) on the hyperplanes at 0 and N. We introduce a truncation of the Hopf equations describing the stationary state of the system which leads to a nonlinear equation for the two-point stationary correlation functions. We prove that these equations have a unique solution which, for \(N\) large, is approximately a local equilibrium state satisfying Fourier law that relates the heat current to a local temperature gradient. The temperature exhibits a nonlinear profile.

MSC:

82C20 Dynamic lattice systems (kinetic Ising, etc.) and systems on graphs in time-dependent statistical mechanics
82C70 Transport processes in time-dependent statistical mechanics
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