Particle motion in stochastic force fields

Abstract

Abstract Usually Fokker-Planck or master equations are used as kinetic equations for the problem of particle motion in stochastic force fields. With the Stratonovich calculus also a kinetic equation is derived which formally looks like a Fokker-Planck equation, but differs in some essential aspect. Moreover, we get an operator equation which is more suitable to derive moment equations than the kinetic equations. As a starting point for the derivation of the Fokker-Planck and master equations it is common to use the Chapman-Kolmogorov equation or the Liouville equation with Markov assumptions. Because this is not necessary for our derivation with the Stratonovich calculus we also give derivations for the former kinetic equations without starting from the Chapman-Kolmogorov equation and show how the Markov assumption could be taken into account at the end of the derivation. In particular we study the motion of charged particles in a stochastic magnetic field. This example shows that significant differences between the various kinetic equations always should be expected in the case of a nonvanishing correlation time of the stochastic force field and nonlinear fluctuation equations.