Approach to equilibrium for the phonon Boltzmann equation

Bricmont, Jean;Kupiainen, Antti
(2008) Communications in Mathematical Physics — Vol. 281, n° 1, p. 179-202 (2008)

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Authors
  • Bricmont, JeanUCLouvain
    Author
  • Kupiainen, AnttiUCLouvain
    Author
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.
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Bricmont, J., & Kupiainen, A. (2008). Approach to equilibrium for the phonon Boltzmann equation. Communications in Mathematical Physics, 281(1), 179-202. https://doi.org/10.1007/s00220-008-0480-y (Original work published 2008)