Automodel solutions for L\'evy flight-based transport on a uniform background

TL;DR

This paper introduces an automodel solution for non-stationary superdiffusive transport characterized by Lévy flights, validated through numerical comparisons in 1D and 3D cases with various spectral line shapes.

Contribution

The paper develops a novel automodel solution for superdiffusive transport with power-law step-length distributions, validated against numerical solutions in multiple physical scenarios.

Findings

Automodel solution accurately describes superdiffusive transport with Lévy flights.

Validation through numerical solutions in 1D and 3D cases.

Effective for various spectral line shapes in resonance radiation transfer.

Abstract

A wide class of non-stationary superdiffusive transport on a uniform background with a power-law decay, at large distances, of the step-length probability distribution function (PDF) is shown to possess an automodel solution. The solution for Green function is constructed using the scaling laws for the propagation front (relevant-to-superdiffusion average displacement) and asymptotic solutions far beyond and far in advance of the propagation front. These scaling laws are determined essentially by the long-free-path carriers (L\'evy flights). The validity of the suggested automodel solution is proved by its comparison with numerical solutions in the one-dimensional (1D) case of the transport equation with a simple long-tailed PDF with various power-law exponents and in the 3D case of the Biberman-Holstein equation of the resonance radiation transfer for various (Doppler, Lorentz, Voight and Holtsmark) spectral line shapes.