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## Revista mexicana de física E

*Print version* ISSN 1870-3542

#### Abstract

JIMENEZ-AQUINO, J.I; ROMERO-BASTIDA, M and PEREZ-GUERRERO NOYOLA, A.C. **Brownian motion in a magnetic field and in the presence of additional external forces**.* Rev. mex. fís. E* [online]. 2008, vol.54, n.1, pp.81-86.
ISSN 1870-3542.

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. [*Phys. Rev. E ***76** (2007) 021106]

**Keywords
:
**(FP) Fokker-Planck; (FPK) Fokker-Planck-Kramers.