On the velocity relaxation of a Rayleigh gas

Two methods of solution of the usual (Fokker-Planck) kinetic equation for a Rayleigh gas are discussed. One is that based on the so-called, and well-known, “fundamental solution” of the same equation, while the other profits by the expansion of the heavy-particle velocity distribution in eigenfunctions of the Fokker-Planck collision operator. The particular case is considered in which the initial heavy-particle velocity distribution is Maxwellian (around a given flow velocity) at a temperature generally different from that of equilibrium. Moreover, the inadequacies of the usual Fokker-Planck equation in describing the relaxation processes are individuated and discussed devoting particular attention to situations in which the initial heavy-particle velocity distribution is anisotropic. In this regard, exploiting the circumstance that the eigenfunctions of the Fokker-Planck collision operator are also eigenfunctions of the Boltzmann collision operator in the Maxwell model, an enlightening comparison between our results and those of the exact Boltzmann theory is presented.