Inelastic scattering models in transport theory and their small mean free path analysis

In this paper we perform an asymptotic analysis of a singularly perturbed linear Boltzmann equation with inelastic scattering operator in the Lorentz gas limit, when the parameter corresponding to the mean free path of particles is small. The physical model allows for two-level held particles (ground state and excited state), so that scattering test particles trigger either excitation or de-excitation processes, with corresponding loss or gain of kinetic energy. After examining the main properties of the collision mechanism, the compressed Chapman-Enskog expansion procedure is applied to find the asymptotic equation when the collisions are dominant. A peculiarity of this inelastic process is that the collision operator has an infinite dimensional null-space. On the hydrodynamic level this is reflected in the small mean free path approximation being rather a family of diffusion equations than a single equation, asis the case in classical transport theory. Also the appropriate hydrodynamic quantity turns out to be different from the standard macroscopic density.