Brownian particles in an external field: Asymptotic distribution functions and mean square displacement

Abstract

A simple procedure is proposed by which the long-time form of the distribution of large (or heavy) ions in a fluid, in a time-varying electric field, is obtained as asymptotic solution of the Fokker-Planck (or Klein-Kramers) equation. In this way, it is shown that, when the initial ion distribution is the product of a delta function in position space times a shifted Maxwellian in velocity space, the asymptotic ion distribution, at sufficiently large times, coincides with the asymptotic form of the corresponding fundamental solution of the Fokker-Planck equation. Moreover, it is shown that a simplified (even if incorrect) form of the ion distribution can successfully be used to obtain correct values of a large class of average quantities. In this connection, the proper, asymptotic formula for the ion mean square displacement in time-varying electric fields is widely discussed and compared to the corresponding result following from the appropriate diffusion equation.