On the Derivation of Fourier’s Law

Abstract

We discuss derivation of Fourier's law of heat conduction from a micro-scopic Hamiltonian dynamics. The model consists of weakly coupled anharmonic oscillators arranged on a three dimensional lattice and subjected to a stochastic forc-ing on the boundary. We introduce a truncation of the system of equations satisfied by correlation functions of the stationary state of the system which leads to a non-linear generalized Boltzman equation for the two-point stationary correlation func-tions. 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.