---
abstract: '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 [@kors], where a thermostatted harmonic chain (with no inter-particle scattering) has been considered.'
author:
- 'Tomasz Komorowski[^1]'
- 'Stefano Olla[^2]'
title: Kinetic limit for a chain of harmonic oscillators with a point Langevin thermostat
---

Introduction {#intro}
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The mathematical analysis of macroscopic energy transport in anharmonic chain of oscillators [constitutes a very hard mathematical problem]{}, see [@spohn2006]. One approach to it is to replace the non-linearity by a stochastic exchange of momentum between nearest neighbor particles [in such a way 