Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation

Tomasz Komorowski; Łukasz Stȩpień

doi:10.3934/krm.2018013

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2018, Volume 11, Issue 2: 239-278. Doi: 10.3934/krm.2018013

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Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation

Tomasz Komorowski^1, , and
Łukasz Stȩpień^2,

1.
Institute of Mathematics, Polish Academy Of Sciences, ul. Śniadeckich 8, 00-656, Warsaw, Poland
2.
Lublin University of Technology, Nadbystrzycka 38A, 20-618 Lublin, Poland

* Corresponding author: Tomasz Komorowski
* Corresponding author: Tomasz Komorowski

Received: April 30, 2016

Revised: November 30, 2016

Published: March 2018

Both authors acknowledge the support of the Polish National Science Center grant UMO-2012/07/B/ST1/03320

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Abstract

We consider a one dimensional infinite chain of harmonic oscillators whose dynamics is weakly perturbed by a stochastic term conserving energy and momentum and whose evolution is governed by an Ornstein-Uhlenbeck process. We prove the kinetic limit for the Wigner functions corresponding to the chain. This result generalizes the results of [7] obtained for a random momentum exchange that is of a white noise type. In contrast with [7] the scattering term in the limiting Boltzmann equation obtained in the present situation depends also on the dispersion relation.

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Mathematics Subject Classification: Primary: 82C44; Secondary: 82C20, 82C70.

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