Fokker-Planck-Kramers equations of a heavy ion in presence of external fields.
In this work, we use the same strategy studied in our previous work [J. I. Jiménez-Aquino and M. Romero-Bastida, Phys. Rev. E 74, 041117 (2006)] to solve exactly the Fokker-Planck (FP) and Fokker-Planck-Kramers (FPK) equations of a charged Brownian particle in a fluid (a heavy ion in a light gas) under the influence of external fields: a constant magnetic field and, in general, time-varying mechanical and electric fields. In our proposal, a time-dependent rotation matrix is introduced to transform the Langevin equation in the phase-space (r,u) to a new space (r',u') . As a result, the transformed Langevin equations are very similar to those of ordinary Brownian motion in the presence of those time-varying external forces only, without the magnetic field; therefore, the associated FP and FPK equations can easily be solved in those transformed spaces. To solve these equations, we use the methods of solution developed by Chandrasekhar in the field-free case of ordinary Brownian motion. We also calculate a more general transition probability density in the velocity space by assuming an initial heavy-ion Maxwellian distribution at a temperature generally different from that corresponding to equilibrium, the same as that used by Ferrari [Physica A 163, 596 (1990)].
Jiménez-Aquino
J I
JI
Departamento de Física, Universidad Autónoma Metropolitana-Iztapalapa, Apartado Postal 55-534, Codigo Postal 09340, México, Distrito Federal, Mexico. ines@xanum.uam.mx
Romero-Bastida
M
M