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

Abstract

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)].

Charged Brownian particles: Kramers and Smoluchowski equations and the hydrothermodynamical picture

“…In his celebrated 1943 Brownian motion paper [11], Chandrasekhar outlined the method for solving a Brownian particle in a general field of force. It took approximately sixty years to report exact solutions for the Brownian motion of a charged particle in uniform and static electric and/or magnetic fields [71][72][73][74][75][76][77][78][79][80][81][82][83][84][85] (see also some previous related works [86,87]). …”

Classical and quantum Brownian motion in an electromagnetic field

“…From Eqs. (14) and (2), separating Z(t) into real and imaginary parts, one obtains the average coordinates,…”

“…the origin of cosmic rays). The present paper focuses on Brownian motion in a constant magnetic field and a spatially homogeneous, possibly time‐dependent, electromagnetic field, a problem studied so far in various papers both in the classical and in the quantum regime.…”

Propagator integral method for plasma physics kinetic equations with practical applications

“…For this equation, also known as the Klein-Kramers equation [62][63][64], many works have applied integral methods in an open space [35,41,44,65] providing good result and capturing dynamics that classical schemes would miss, e.g., the parasite effective diffusion in the q 1 q 1 and q 1 q 2 transversal directions. In here, the PIM is used to solve this type of equations in an open domain.…”

“…The Dirac delta is necessary to compute the propagator, meanwhile Heaviside function appears in its time-integral, which is necessary to include non-homogeneous terms. For these problems, the following numerical implementations of these functions are assumed: for the Dirac delta 63) and for the Heaviside function…”