On the derivation of Fourier's law for coupled anharmonic oscillators

TL;DR

This paper derives Fourier's law for a 3D lattice of weakly coupled anharmonic oscillators under stochastic forcing, showing the emergence of a local equilibrium state with a nonlinear temperature profile.

Contribution

It introduces a novel truncation of Hopf equations leading to a nonlinear equation for correlations, proving the existence of a local equilibrium state satisfying Fourier's law.

Findings

Unique solution to the correlation equations for large N

Emergence of Fourier's law in the model

Nonlinear temperature profile observed

Abstract

We study the Hamiltonian system made of weakly coupled anharmonic oscillators arranged on a three dimensional lattice and subjected to a stochastic forcing mimicking heat baths of temperatures T_1 and T_2 on the hyperplanes at x_1=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 equlibrium state satisfying Fourier law that relates the heat current to a local temperature gradient. The temperature exhibits a nonlinear profile.