Fokker-Planck Equation for Fractional Systems

TL;DR

This paper extends the Fokker-Planck equation to fractional systems by generalizing phase space concepts using fractional integrals, leading to a new framework for non-Hamiltonian systems with fractional powers.

Contribution

It introduces a fractional phase space interpretation and derives generalized Bogoliubov and Fokker-Planck equations for fractional power systems.

Findings

Fractional phase space can be interpreted as a space with noninteger dimension.

Derived generalized Bogoliubov equations for fractional systems.

Obtained a fractional Fokker-Planck equation from the Liouville equation.

Abstract

The normalization condition, average values and reduced distribution functions can be generalized by fractional integrals. The interpretation of the fractional analog of phase space as a space with noninteger dimension is discussed. A fractional (power) system is described by the fractional powers of coordinates and momenta. These systems can be considered as non-Hamiltonian systems in the usual phase space. The generalizations of the Bogoliubov equations are derived from the Liouville equation for fractional (power) systems. Using these equations, the corresponding Fokker-Planck equation is obtained.