Differential kinetic equations for a Rayleigh gas with inelastic collisions

Starting from the collision integral of the appropriate generalized Boltzmann equation (Waldmann–Trübenbacher equation), a differential collision operator for a Rayleigh gas with inelastic collisions, i.e. for heavy (atomic) particles dilutely dispersed in a light molecular background gas, is obtained. The procedure is based on the assumption that the heavy particles are not too far from the thermal equilibrium with the background gas, and leads to an approximate operator which is correct up to (and including) the first-order terms in the ratio between the light-particle mass and the sum of the masses of a light particle and of a heavy particle. The obtained operator reduces to the usual Fokker–Planck collision operator when only elastic collisions are considered. All the steps of the procedure are briefly discussed and the use of the new operator in approximate (differential) kinetic equations appropriate to some possible physical situations is examined. Finally, the rather abstract kinetic equation (of the Fokker–Planck type) previously obtained by Mazo is led to its explicit final form and criticized.