Anomalous Kinetics in Velocity Space: equations and models

TL;DR

This paper derives and solves equations describing anomalous diffusion in momentum space, applicable to non-equilibrium systems, and provides formulas for stationary and non-stationary cases considering different probability transition functions.

Contribution

It introduces a general framework for anomalous diffusion in velocity space, including equations for both stationary and non-stationary cases, and accounts for different types of probability transition functions.

Findings

Derived equations for anomalous diffusion in momentum space.

Provided solutions for stationary and non-stationary cases.

Applicable to systems with long-tailed and short-range PT-functions.

Abstract

Equation for anomalous diffusion in momentum space, recently obtained in the recent paper (S.A. Trigger, ArXiv 0907.2793 v1, [cond-matt. stat.-mech.], 16 July 2009) is solved for the stationary and non-stationary cases on basis of the appropriate probability transition function (PTF). Consideration of diffusion for heavy particles in a gas of the light particles can be essentially simplified due to small ratio of the masses of the particles. General equation for the distribution of the light particles, shifted in velocity space, is also derived. For the case of anomalous diffusion in momentum space the closed equation is formulated for the Fourier-component of the momentum distribution function. The effective friction and diffusion coefficients are found also for the shifted distribution. If the appropriate integrals are finite the equations derived in the paper are applicable for both cases: the PT-function with the long tails and the short range PT-functions in momentum space. In the last case the results are equivalent to the Fokker-Planck equation. Practically the new results of this paper are applicable to strongly non-equilibrium physical systems.