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

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′, u′)-coordinates, and makes it possible to re-orientate the magnetic field along an appropriate direction (say along the z′-axis). The second one is a time-dependent rotation which transforms the (r′, u′)-coordinates into other (r″, u′)-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″, u″) phase-space, following our methodology developed in Ref.