Heavy ions in light gases in an electric field: II. Time-dependent solutions of the Fokker-Planck equation both in the absence and in the presence of a magnetic field

Abstract

Kihara's time-dependent solution of the approximate Boltzmann equation (Fokker-Planck equation) for heavy ions in light gases in static electric and magnetic fields is extended in order to treat also the cases in which the electric field varies in time. However, since Kihara's solution refers only to the particular situation in which the initial ion velocity distribution is a (shifted) Maxwellian distribution at the equilibrium temperature, a new, quite general method to solve the Fokker-Planck equation with arbitrary initial conditions is also presented. This method reduces the problem to the field-free case, and yields, obviously, the extended Kihara solution as particular case. In the framework of the theory based on the Fokker-Planck equation, our present results show that, both in the absence and in the presence of electric and or magnetic fields (and when diffusion phenomena are negligible), an ion velocity distribution which is initially Maxwellian (at any temperature) around any ion mean velocity, maintains its Maxwellian form during the relaxation process. Moreover, stationary and non-stationary ion velocity distributions in static or alternating electric fields and static magnetic fields are also explicitly obtained.