Lévy flights on the half line

Area of Science:

• Statistical Physics
• Stochastic Processes
• Non-Brownian Motion

Background:

• Lévy flights exhibit anomalous diffusion, deviating from standard Brownian motion.
• Understanding particle behavior near boundaries is crucial in many physical systems.
• The fractional Fokker-Planck equation governs Lévy flight dynamics.

Purpose of the Study:

• To determine the probability distribution function (PDF) of Lévy flight positions.
• To investigate the effect of an absorbing wall at the origin on this PDF.
• To develop an analytical solution for the PDF using a perturbation approach.

Main Methods:

• Perturbation scheme expanding in ε = 2-α around the Brownian motion solution (α = 2).
• Solving the fractional Fokker-Planck equation associated with Lévy flights.
• Numerical simulations and exact computation for α = 1 to validate results.

Main Results:

• An explicit analytical solution for the PDF is derived, exact to the first order in ε.
• The study conjectures the precise asymptotic behavior of the PDF for any α.
• First subleading corrections to the asymptotic behavior are included.

Conclusions:

• The perturbation method provides an accurate analytical approximation for Lévy flight PDFs near an absorbing wall.
• The derived solution and conjectured asymptotic behavior are well-supported by numerical and exact computations.
• This work offers insights into anomalous diffusion processes in confined systems.