# American Institute of Mathematical Sciences

April  2018, 11(2): 239-278. doi: 10.3934/krm.2018013

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

 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

Received  May 2016 Revised  December 2016 Published  January 2018

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

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.

Citation: Tomasz Komorowski, Łukasz Stȩpień. Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation. Kinetic & Related Models, 2018, 11 (2) : 239-278. doi: 10.3934/krm.2018013
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