Exact and explicit probability densities for one-sided Lévy stable distributions

Area of Science:

• Probability Theory
• Statistical Physics
• Anomalous Diffusion

Background:

• Lévy stable distributions are fundamental in modeling random systems.
• Understanding anomalous diffusion and fractional kinetics requires precise probability distributions.
• Existing solutions for these distributions are limited for certain parameter ranges.

Purpose of the Study:

• To derive exact and explicit expressions for one-sided, heavy-tailed Lévy stable probability distributions (gα(x)).
• To cover all rational indices α = l/k < 1, where k and l are positive integers.
• To provide a tool for fine-tuning α for specific experimental applications.

Main Methods:

• Analytical derivation of probability distribution functions.
• Expressing solutions in terms of known mathematical functions.
• Verification against previously established results for k ≤ 4.

Main Results:

• Exact and explicit expressions for gα(x) are provided for all α = l/k < 1.
• All known results for k ≤ 4 are reproduced.
• Numerous new exact solutions are presented for k > 4.

Conclusions:

• The derived expressions offer a comprehensive set of solutions for Lévy stable distributions.
• These results facilitate the adaptation of gα(x) to various experimental scenarios.
• The work advances the study of anomalous diffusion and fractional kinetics.