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

K A Penson¹, K Górska

¹Laboratoire de Physique Théorique de la Matière Condensée (LPTMC), Université Pierre et Marie Curie, CNRS UMR 7600, Tour 13-5ième étage, Boîte Courrier 121, 4 place Jussieu, F 75252 Paris Cedex 05, France. penson@lptl.jussieu.fr

Physical Review Letters

|January 15, 2011

Summary

Researchers present exact expressions for one-sided, heavy-tailed Lévy stable probability distributions (Lévy stable distributions) with index α. These findings are crucial for understanding anomalous diffusion and fractional kinetics in random systems.

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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.