Fractional kinetics for relaxation and superdiffusion in a magnetic field

Abstract

We propose fractional Fokker-Planck equation for the kinetic description of relaxation and superdiffusion processes in constant magnetic and random electric fields. We assume that the random electric field acting on a test charged particle is isotropic and possesses non-Gaussian Levy stable statistics. These assumptions provide us with a straightforward possibility to consider formation of anomalous stationary states and superdiffusion processes, both properties are inherent to strongly non-equilibrium plasmas of solar systems and thermonuclear devices. We solve fractional kinetic equations, study the properties of the solution, and compare analytical results with those of numerical simulation based on the solution of the Langevin equations with the noise source having Levy stable probability density. We 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.

Lévy Flights in a Steep Potential Well

“…An impressive experimental evidence of Lévy processes was reported by the group of Walther in the study of the position of a single ion in a one-dimensional optical lattice, in which diverging fluctuations could be observed in the kinetic energy [9]. From a phenomenological point of view, LFs have been used to describe the dynamics observed in plasmas [10] or in molecular collisions [11]. They have also been successfully applied to describe the statistics encountered in the spatial gazing patterns of bacteria [12], albatross birds [13], or even spidermonkeys [14].…”

Lévy stable noise-induced transitions: stochastic resonance, resonant activation and dynamic hysteresis

“…with stationary , independent increments), the assumption about its Gaussianity can be easily violated. The examples range from the description of the dynamics in plasmas [5], diffusion in energy space, self-diffusion in micelle systems, exciton and charge transport in polymers under conformational motion and incoherent atomic radiation trapping -to the spectral analysis of paleoclimatic [6] or economic data [7], motion in optimal search strategies among randomly distributed target sites [8], fluorophore diffusion as studied in photo-bleaching experiments, interstellar scintillations [9], ratcheting devices [10,11] and many others [12].…”

Escape from hypercube driven by multi-variate α-stable noises: role of independence

“…Nevertheless, the description of 2D escape kinetics significantly differs from 1D. Bi-variate α-stable noises are defined by the spectral measure Λ(·) which determines the escape kinetics along with the associated fractional diffusion equation [37][38][39]. Here, we use two most natural escape scenarios, i.e.…”