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

Abstract

We consider an infinite chain of coupled harmonic oscillators whose Hamiltonian dynamics is perturbed by a random exchange of momentum between particles such that total energy and momentum are conserved, modeling collision between atoms. This random exchange is rarefied in the limit, that corresponds to the hypothesis that in the macroscopic unit time only a finite number of collisions takes place (the Boltzmann-Grad limit). Furthermore, the system is in contact with a Langevin thermostat at temperature T through a single particle. 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 caused by the presence of the thermostat. The paper extends the results of [15], where a harmonic chain (with no inter-particle scattering) in contact with a Langevin thermostat has been considered.