Anomalous Transport in Velocity Space, from Fokker-Planck to General Equation

TL;DR

This paper develops a general equation for anomalous diffusion in velocity space based on the master equation and probability transition functions, addressing both long-tailed and rapidly decreasing cases, with applications to non-equilibrium systems.

Contribution

It introduces a new general equation for velocity space anomalous transport, extending previous coordinate space approaches to handle various PTF behaviors and non-equilibrium conditions.

Findings

Derived a new equation for momentum distribution evolution.

Analyzed stationary and non-stationary solutions.

Showed violation of Einstein relation in non-equilibrium cases.

Abstract

The problem of anomalous diffusion in momentum (velocity) space is considered based on the master equation and the appropriate probability transition function (PTF). The approach recently developed by the author for coordinate space, is applied with necessary modifications to velocity space. A new general equation for the time evolution of the momentum distribution function in momentum space is derived. This allows the solution of various problems of anomalous transport when the probability transition function (PTF) has a long tail in momentum space. For the opposite cases of the PTF rapidly decreasing as a function of transfer momenta (when large transfer momenta are strongly suppressed), the developed approach allows us to consider strongly non-equilibrium cases of the system evolution. The stationary and non-stationary solutions are studied. As an example, the particular case of the Boltzmann-type PT-function for collisions of heavy and light particles with the determined (prescribed) distribution function, which can be strongly non-equilibrium, is considered within the proposed general approach. The appropriate diffusion and friction coefficients are found. The Einstein relation between the friction and diffusion coefficients is shown to be violated in these cases.