Levy Flight Superdiffusion: An Introduction

TL;DR

This paper introduces Levy flight superdiffusion as a self-similar Levy process, generalizing Brownian motion through stable distributions and fractional derivatives, with applications to stationary distributions and barrier crossing in complex systems.

Contribution

It presents a comprehensive introduction to Levy flight superdiffusion, deriving the fractional Fokker-Planck equation and analyzing stationary states and transition times.

Findings

Derived the fractional Fokker-Planck equation for Levy flights.

Analyzed stationary probability distributions in monostable potentials.

Provided expressions for nonlinear relaxation times in barrier crossing.

Abstract

After a short excursion from discovery of Brownian motion to the Richardson "law of four thirds" in turbulent diffusion, the article introduces the L\'{e}vy flight superdiffusion as a self-similar L\'{e}vy process. The condition of self-similarity converts the infinitely divisible characteristic function of the L\'{e}vy process into a stable characteristic function of the L\'{e}vy motion. The L\'{e}vy motion generalizes the Brownian motion on the base of the $\alpha$-stable distributions theory and fractional order derivatives. The further development of the idea lies on the generalization of the Langevin equation with a non-Gaussian white noise source and the use of functional approach. This leads to the Kolmogorov's equation for arbitrary Markovian processes. As particular case we obtain the fractional Fokker-Planck equation for L\'{e}vy flights. Some results concerning stationary probability distributions of L\'{e}vy motion in symmetric smooth monostable potentials, and a general expression to calculate the nonlinear relaxation time in barrier crossing problems are derived. Finally we discuss results on the same characteristics and barrier crossing problems with L\'{e}vy flights, recently obtained with different approaches.