Approach to equilibrium for the phonon Boltzmann equation

TL;DR

This paper analyzes how solutions to the phonon Boltzmann equation approach equilibrium, demonstrating diffusive behavior and deriving the Fourier law from microscopic dynamics of anharmonic oscillators.

Contribution

It provides a rigorous proof of diffusive relaxation to equilibrium and the derivation of Fourier's law from the kinetic limit of anharmonic oscillator lattices.

Findings

Solutions tend to equilibrium diffusively

The Fourier law is derived from microscopic dynamics

Fast variables are slaved to slow conserved quantities

Abstract

We study the asymptotics of solutions of the Boltzmann equation describing the kinetic limit of a lattice of classical interacting anharmonic oscillators. We prove that, if the initial condition is a small perturbation of an equilibrium state, and vanishes at infinity, the dynamics tends diffusively to equilibrium. The solution is the sum of a local equilibrium state, associated to conserved quantities that diffuse to zero, and fast variables that are slaved to the slow ones. This slaving implies the Fourier law, which relates the induced currents to the gradients of the conserved quantities.