Advances in the studies of anomalous diffusion in velocity space

TL;DR

This paper derives a generalized Fokker-Planck equation to model anomalous diffusion in velocity space, accounting for long-tailed probability transition functions and diverse particle interactions, including dusty plasmas and Coulomb collisions.

Contribution

It introduces a new generalized Fokker-Planck equation applicable to various particle interactions with long-tailed transition functions, extending previous models.

Findings

Derived a generalized Fokker-Planck equation for velocity space.

Applied the theory to dusty plasma and Coulomb collisions.

Identified a new term due to velocity-dependent cross-sections.

Abstract

A generalized Fokker-Planck equation is derived to describe particle kinetics in specific situations when the probability transition function (PTF) has a long tail in momentum space. The equation is valid for an arbitrary value of the transferred in a collision act momentum and for the arbitrary mass ratio of the interacting particles. On the basis of the generalized Fokker-Planck equation anomalous diffusion in velocity space is considered for hard sphere model of particle interactions, Coulomb collisions and interactions typical for dusty plasmas. The example of dusty plasma interaction is peculiar in way that it leads to a new term in the obtained Fokker-Planck-iike equation due to the dependence of the differential cross-section on the relative velocity. The theory is also applied to diffusion of heavy particles in the ambience of light particles with a prescribed power-type velocity distribution function. In general, the theory is applicable to consideration of anomalous diffusion in velocity space if the typical velocity of one sort of particles undergoing diffusion is small compared to the typical velocity of the background particles.