Towards a Derivation of Fourier’s Law for Coupled Anharmonic Oscillators

Abstract

We consider a Hamiltonian system made of weakly coupled anharmonic oscillators arranged on a three dimensional lattice Z(2n) x Z(2), and subjected to stochastic forcing mimicking heat baths of temperatures T-1 and T-2 on the hyperplanes at O 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.