Steady-State L\'evy Flights in a Confined Domain

TL;DR

This paper derives a generalized fractional Fokker-Planck equation for Levy flights in confined domains, analytically solves it for steady states, and reveals boundary concentration phenomena explained by the beta distribution.

Contribution

It introduces a generalized framework for Levy flights in bounded systems and provides analytical solutions for their steady-state distributions.

Findings

Levy flights in a potential well follow a beta distribution.

The probability density becomes singular at the boundaries.

Boundary concentration of particles is explained by the derived distribution.

Abstract

We derive the generalized Fokker-Planck equation associated with a Langevin equation driven by arbitrary additive white noise. We apply our result to study the distribution of symmetric and asymmetric L\'{e}vy flights in an infinitely deep potential well. The fractional Fokker-Planck equation for L\'{e}vy flights is derived and solved analytically in the steady state. It is shown that L\'{e}vy flights are distributed according to the beta distribution, whose probability density becomes singular at the boundaries of the well. The origin of the preferred concentration of flying objects near the boundaries in nonequilibrium systems is clarified.