Statistical-Mechanical Foundation of the Ubiquity of Lévy Distributions in Nature

Abstract

We show that the use of the recently proposed thermostatistics based on the generalized entropic form ${S}_{q}\ensuremath{\equiv}\frac{k(1\ensuremath{-}\ensuremath{\Sigma}{i}^{}{p}_{i}^{q})}{(q\ensuremath{-}1)}$ (where $q\ensuremath{\in}R$, with $q=1$ corresponding to the Boltzmann-Gibbs-Shannon entropy $\ensuremath{-}k\ensuremath{\Sigma}{i}^{}{p}_{i} \mathrm{ln} {p}_{i}$), together with the L\'evy-Gnedenko generalization of the central limit theorem, provide a basic step towards the understanding of why L\'evy distributions are ubiquitous in nature. A consistent experimental verification is proposed.