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Approach to Equilibrium for the Phonon Boltzmann Equation

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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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Authors and Affiliations

  1. UCL, FYMA, chemin du Cyclotron 2, B-1348, Louvain-la-Neuve, Belgium

    Jean Bricmont

  2. Department of Mathematics, Helsinki University, P.O. Box 4, 00014, Helsinki, Finland

    Antti Kupiainen

Authors
  1. Jean Bricmont
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  2. Antti Kupiainen
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Corresponding author

Correspondence to Antti Kupiainen.

Additional information

Communicated by H. Spohn

Partially supported by the Belgian IAP program P6/02.

Partially supported by the Academy of Finland.

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Bricmont, J., Kupiainen, A. Approach to Equilibrium for the Phonon Boltzmann Equation. Commun. Math. Phys. 281, 179–202 (2008). https://doi.org/10.1007/s00220-008-0480-y

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  • DOI: https://doi.org/10.1007/s00220-008-0480-y

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