Langevin equation versus kinetic equation: Subdiffusive behavior of charged particles in a stochastic magnetic field

The running diffusion coefficient D(t) is evaluated for a system of charged particles undergoing the effect of a fluctuating magnetic field and of their mutual collisions. The latter coefficient can be expressed either in terms of the mean square displacement (MSD) of a test particle, or in terms of a correlation between a fluctuating distribution function and the magnetic field fluctuation. In the first case a stochastic differential equation of Langevin type for the position of a test particle must be solved; the second problem requires the determination of the distribution function from a kinetic equation. Using suitable simplifications, both problems are amenable to exact analytic solution. The conclusion is that the equivalence of the two approaches is by no means automatically guaranteed. A new type of object, the ‘‘hybrid kinetic equation’’ is constructed: it automatically ensures the equivalence with the Langevin results. The same conclusion holds for the generalized Fokker–Planck equation. The (Bhatnagar–Gross–Krook) (BGK) model for the collisions yields a completely wrong result. A linear approximation to the hybrid kinetic equation yields an inexact behavior, but represents an acceptable approximation in the strongly collisional limit.