On the velocity relaxation of a Rayleigh gas

A careful analysis of the assumptions and approximations underlying the derivation of the usual kinetic equation for a Rayleigh gas (or a Brownian particle) is performed. The passage from the exact Boltzmann collision operator to its approximate differential form is thus investigated. It is shown that the exact operator can be replaced by the approximate differential one only when proper conditions on the initial heavy-particle velocity distribution are satisfied. From this analysis it follows that the usual kinetic equation is unable to describe the initial stages of the relaxation of an initial δ-function or of a maxwellian distribution at a temperature much lower than — or, also, for non-maxwellian interactions, much higher than — the equilibrium temperature. In any case, the ratio of the initial velocity distribution to the equilibrium one cannot present a fine structure, or too appreciable deviations from a polynomial form of first or second degree in the velocity components, in velocity-space regions which have linear dimensions which are not large compared with the heavy-particle root-mean-square velocity change per elastic collision. Moreover, except that in the Maxwell model, the initial mean energy of the heavy particles cannot too largely exceed the equilibrium value.