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Kinetic limit for a chain of harmonic oscillators with a point Langevin thermostat

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ko-kin-boundary-submit.pdf (511.3Kb)
Date
2019-10
Publisher city
Paris
Publisher
Cahier de recherche CEREMADE, Université Paris-Dauphine
Publishing date
2019
Collection title
Cahier de recherche CEREMADE, Université Paris-Dauphine
Link to item file
https://hal.archives-ouvertes.fr/hal-02305797
Dewey
Sciences connexes (physique, astrophysique)
Sujet
Langevin thermostat; oscillator
URI
https://basepub.dauphine.fr/handle/123456789/20129
Collections
  • CEREMADE : Publications
Metadata
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Author
Komorowski, Tomasz
89537 Institut of Mathematics - Polish Academy of Sciences [PAN]
Olla, Stefano
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Type
Document de travail / Working paper
Item number of pages
52
Abstract (EN)
We consider an infinite chain of coupled harmonic oscillators with a Langevin thermostat attached at the origin and energy, momentum and volume conserving noise that models the collisions between atoms. The noise is rarefied in the limit, that corresponds to the hypothesis that in the macroscopic unit time only a finite number of collisions takes place (Boltzmann-Grad limit). We prove that, after the hyperbolic space-time rescaling, the Wigner distribution, describing the energy density of phonons in space-frequency domain, converges to a positive energy density function W (t, y, k) that evolves according to a linear kinetic equation, with the interface condition at y = 0 that corresponds to reflection, transmission and absorption of phonons. The paper extends the results of [3], where a thermostatted harmonic chain (with no inter-particle scattering) has been considered.

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