Fokker-Planck-Kramers equation for a Brownian gas in a magnetic field

In this work we give an alternative method to calculate the transition probability densities (TPD) for the velocity space, phase space, and Smoluchowsky configuration space of a Brownian gas of charged particles in the presence of a constant magnetic field. Our proposal consists in transforming, by means of a rotation matrix, the Langevin equation of a charged particle in the velocity space into another velocity space where the behavior is quite similar to that of ordinary Brownian motion. A similar strategy is also applied to the phase-space. In consequence, in the transformed space both the Fokker-Planck and Fokker-Planck-Kramers equations are solved following Chandrasekhar's methodology. Our results are compared with those obtained by Czopnik and Garbaczewski [Phys. Rev. E 63, 021105 (2001)].