Power-laws as statistical mixtures

Abstract

Many complex systems are characterized by power-law distributions. In this article, we show that for various examples of power-law distributions, including the two probably most popular ones, the Pareto law for the wealth distribution and Zipf’s law for the occurrence frequency of words in a written text, the power-law tails of the probability distributions can be decomposed into a statistical mixture of canonical equilibrium probability densities of the subsystems. While the interacting units or subsystems have canonical distributions at equilibrium, as predicted by canonical statistical mechanics, the heterogeneity of the shapes of their distributions leads to the appearance of a power-law.