Diffusion limit for a kinetic equation with a thermostatted interface

Abstract

We consider a linear phonon Boltzmann equation with a reflecting/transmitting/absorbing interface. This equation appears as the Boltzmann-Grad limit for the energy density function of a harmonic chain of oscillators with inter-particle stochastic scattering in the presence of a heat bath at temperature $ T $ in contact with one oscillator at the origin. We prove that under the diffusive scaling the solutions of the phonon equation tend to the solution $ \rho(t, y) $ of a heat equation with the boundary condition $ \rho(t, 0)\equiv T $.