Fractional kinetics for relaxation and superdiffusion in a magnetic field

Abstract

Fractional Fokker–Planck equation is proposed for the kinetic description of relaxation and superdiffusion processes in constant magnetic and random electric fields. It is assumed that the random electric field acting on a test charged particle is isotropic and possesses non-Gaussian Levy stable statistics. These assumptions provide one with a straightforward possibility to consider formation of anomalous stationary states and superdiffusion processes, both properties are inherent to strongly nonequilibrium plasmas of solar systems and thermonuclear devices. The fractional kinetic equation is solved, the properties of the solution are studied, and analytical results are compared with those of numerical simulation based on the solution of the Langevin equations with a noise source having Levy stable probability density. It is found, in particular, that the stationary states are essentially non-Maxwellian ones and, at the diffusion stage of relaxation, the characteristic displacement of a particle grows superdiffusively with time and is inversely proportional to the magnetic field.