Equilibrium states for interactions with acoustic phonons

Abstract

The motivation of this paper is a result of A. Majorana ([6], [12]) concerning the interactions with optical phonons. He proved that there were equilibrium states which were not Maxwellian distributions (or Fermi-Dirac distributions in the nonlinear case). Then the diffusion approximation of the nonlinear electron-optical phonon collision mechanisms, studied by P.A. Markowich, F, Poupaud and C. Schmeiser in [10], gave a new energy dependent drift-diffusion model. Using this work, we are interested in the interactions with acoustic phonons : we prove here that the collision operator thermalizes towards Maxwellian (or Fermi-Dirac) distributions. Hence we can predict fluid limits similar to the standard drift-diffusion models.