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

TL;DR

This paper proves that in a chain of harmonic oscillators with a point Langevin thermostat and rarefied noise, the energy distribution converges to a kinetic equation after rescaling, extending previous work without scattering.

Contribution

It extends previous models by including energy, momentum, and volume conserving noise, deriving a kinetic limit for the phonon energy distribution in a harmonic chain.

Findings

Energy density converges to a solution of a linear kinetic equation.

Interface conditions at the thermostat involve reflection, transmission, and absorption.

Results generalize previous models without inter-particle scattering.

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 [3], where a thermostatted harmonic chain (with no inter-particle scattering) has been considered.