DIFFUSION LIMIT FOR A KINETIC EQUATION WITH A THERMOSTATTED INTERFACE

International audienceWe 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 ρ(t, y) of a heat equation with the boundary condition ρ(t, 0) ≡ T

Diffusion limit for a kinetic equation with a thermostatted interface

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 $\rho(t,0)\equiv T$

The free path in a high velocity random flight process associated to a Lorentz gas in an external field

We investigate the asymptotic behavior of the free path of a variable density random flight model in an external field as the initial velocity of the particle goes to infinity. The random flight models we study arise naturally as the Boltzmann-Grad limit of a random Lorentz gas in the presence of an external field. By analyzing the time duration of the free path, we obtain exact forms for the asymptotic mean and variance of the free path in terms of the external field and the density of scatterers. As a consequence, we obtain a diffusion approximation for the joint process of the particle observed at reflection times and the amount of time spent in free flight.Comment: 30 page

Diffusion approximation of radiative transfer problems with interfaces

Abstract We derive the diffusion approximation of transport equations with discontinuities at interfaces. The transport equations model the energy density of acoustic waves. The waves are reflected and transmitted at the interface between different media, which leads to discontinuities of the energy density across the interface. The diffusion approximation, which is valid inside each region is not correct at the vicinity of the interface. However using interface layer analysis, we prove that the transport solution can be approximated by a diffusion term plus an interface layer which decays exponentially fast. We derive systematically the correct form of the interface conditions for this diffusion term