On the velocity relaxation of a Rayleigh gas

A procedure to test reliability and accuracy of the usual (approximate) differential collision operator (and therefore of the usual kinetic equation) for a Rayleigh gas (or Brownian particles) is presented. The procedure is applied, in the hard sphere model, to the particular case in which the initial heavy-particle velocity distribution is Maxwellian at a temperature T0 different from the equilibrium temperature T. It is found that the most severe limitations to the reliability and accuracy of the usual collision operator originate, in this case, from the truncated Taylor expansion of the ratio between the initial velocity distribution and the equilibrium one. In particular, for light-particle-heavy-particle mass ratios equal to 5 X 10-2, 10-2, 5 X 10-3 and 10-3, the values of T0/T can respectively be only on the intervals [0.89, 1.14], [0.67, 1.48], [0.57, 1.74] and [0.31, 3.25], if a reasonably good accuracy of the usual approximate collision operator is desired. Moreover, since these limitations are found independently of the heavy-particle-light-particle interaction law, their validity is not restricted to the hard sphere model.