Fokker-Planck Equation with Fractional Coordinate Derivatives

TL;DR

This paper derives a fractional Fokker-Planck equation from a generalized Kolmogorov-Feller framework, modeling systems with long-range interactions and power-law correlated particle steps.

Contribution

It introduces a novel derivation of the Fokker-Planck equation with fractional derivatives based on long-range interactions and power-law step correlations.

Findings

Derivation of fractional Fokker-Planck equation with order 1<α<2

Method uses successive approximations and averaging over fast variables

Applicable to systems with long-range interactions and power-law correlations

Abstract

Using the generalized Kolmogorov-Feller equation with long-range interaction, we obtain kinetic equations with fractional derivatives with respect to coordinates. The method of successive approximations with the averaging with respect to fast variable is used. The main assumption is that the correlator of probability densities of particles to make a step has a power-law dependence. As a result, we obtain Fokker-Planck equation with fractional coordinate derivative of order $1<\alpha<2$.