Variational principle for fractional kinetics and the L\'evy Ansatz

TL;DR

This paper develops a variational principle for fractional kinetics using an auxiliary-field formalism, providing solutions that encompass subdiffusion to superdiffusion phenomena via the Lévý Ansatz.

Contribution

It introduces a novel variational approach for fractional kinetic equations, applying it to spatio-temporal fractional Fokker-Planck equations with the Lévý Ansatz.

Findings

Unified description of subdiffusion and superdiffusion

Variational solutions for fractional Fokker-Planck equations

Analysis of probability distribution motion under periodic drift

Abstract

A variational principle is developed for fractional kinetics based on the auxiliary-field formalism. It is applied to the Fokker-Planck equation with spatio-temporal fractionality, and a variational solution is obtained with the help of the L\'evy Ansatz. It is shown how the whole range from subdiffusion to superdiffusion is realized by the variational solution, as a competing effect between the long waiting time and the long jump. The motion of the center of the probability distribution is also analyzed in the case of a periodic drift.