Approach to equilibrium for the phonon Boltzmann equation

Jean Bricmont, Antti Kupiainen

    Research output: Contribution to journalArticleScientificpeer-review

    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.
    Original languageEnglish
    JournalCommunications in Mathematical Physics
    Volume281
    Issue number1
    Pages (from-to)179-202
    Number of pages24
    ISSN0010-3616
    DOIs
    Publication statusPublished - 2008
    MoE publication typeA1 Journal article-refereed

    Fields of Science

    • 114 Physical sciences
    • 111 Mathematics

    Cite this

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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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    Approach to equilibrium for the phonon Boltzmann equation. / Bricmont, Jean; Kupiainen, Antti.

    In: Communications in Mathematical Physics, Vol. 281, No. 1, 2008, p. 179-202.

    Research output: Contribution to journalArticleScientificpeer-review

    TY - JOUR

    T1 - Approach to equilibrium for the phonon Boltzmann equation

    AU - Bricmont, Jean

    AU - Kupiainen, Antti

    PY - 2008

    Y1 - 2008

    N2 - 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.

    AB - 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.

    KW - 114 Physical sciences

    KW - 111 Mathematics

    U2 - 10.1007/s00220-008-0480-y

    DO - 10.1007/s00220-008-0480-y

    M3 - Article

    VL - 281

    SP - 179

    EP - 202

    JO - Communications in Mathematical Physics

    JF - Communications in Mathematical Physics

    SN - 0010-3616

    IS - 1

    ER -