Brownian motion in a magnetic field and in the presence of additional external forces

Abstract

Our purpose in this paper is to solve exactly the Fokker-Planck-Kramers equation of a charged particle (heavy-ion) embedded in a fluid and under the influence of mechanical and electromagnetic forces. In this work the magnetic field is assumed to be constant and pointing along any direction of a Cartesian reference frame; the mechanical and electrical forces are both space-independent, but in general time-dependent. Our proposal relies upon two transformations of the Langevin equation associated with the charged particle's phase-space $( r, u)$. The first one is a fixed rotation which transforms the $( r, u)$-coordinates into other $( r^{\prime}, u^{\prime})$-coordinates, and makes it possible to re-orientate the magnetic field along an appropriate direction (say along the $z^{\prime}$-axis). The second one is a time-dependent rotation which transforms the $( r^{\prime}, u^{\prime})$-coordinates into other $( r^{\prime\prime}, u^{\prime\prime})$-coordinates, in which the resulting Langevin equation strongly resembles that of ordinary Brownian motion in the presence of external forces. Under these circumstances, the Fokker-Planck-Kramers equation can immediately be solved in the $( r^{\prime\prime}, u^{\prime\prime})$ phase-space, following our methodology developed in Ref. [Phys. Rev. E 76 (2007) 021106].