Energy Loss to a Cold Background Gas. I. Higher Order Corrections to the Fokker-Planck Operator for a Lorentz Gas

The relaxation of an isotropic distribution of test particles in a homogeneous background gas is considered when the mass ratio is not necessarily very small. For the most part, the temperature of the background gas molecules is assumed to be zero, although a method is presented for including nonzero-temperature effects. The Boltzmann collision integral is represented by an infinite-series differential operator which, for all force laws, reduces to the usual Fokker-Planck equation when terms of second order in the mass ratio are discarded. For the case of Coulomb interactions, the usual Fokker-Planck equation is obtained if either the second-order mass-ratio terms or the terms of order $\frac{1}{\ln \Lambda }$ are discarded. A random-walk analysis is used to obtain a differential operator which agrees with the infinite-series differential operator when third-order terms in the mass ratio are discarded. When the background-gas temperature approaches zero, the usual Fokker-Planck equation predicts that an initial delta-function distribution will always remain a delta function during the relaxation process; whereas it is shown that both the random-walk analysis and the infinite-series differential operator give exact values for the dispersion of the initial delta function.