Power-Laws as Statistical Mixtures
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.
KeywordsOccurrence Frequency Wealth Distribution Statistical Mixture Partial Distribution Boltzmann Entropy
M.P., E.H., and A.C. acknowledge support from the Estonian Science Foundation Grant no. 9462 and the Institutional Research Funding IUT (IUT39-1) of the Estonian Ministry of Education and Research. A.C. acknowledges financial support from grant number BT/BI/03/004/2003(C) of Government of India, Ministry of Science and Technology, Department of Biotechnology, Bioinformatics Division. L.M. acknowledges the Estonian Research Council for supporting his work with the grant PUTJD110.
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