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Incidence of q statistics in rank distributions

Yalcin, G. Cigdem and Robledo, Alberto and Gell-Mann, Murray (2014) Incidence of q statistics in rank distributions. Proceedings of the National Academy of Sciences of the United States of America, 111 (39). pp. 14082-14087. ISSN 0027-8424. PMCID PMC4191817.

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We show that size-rank distributions with power-law decay (often only over a limited extent) observed in a vast number of instances in a widespread family of systems obey Tsallis statistics. The theoretical framework for these distributions is analogous to that of a nonlinear iterated map near a tangent bifurcation for which the Lyapunov exponent is negligible or vanishes. The relevant statistical–mechanical expressions associated with these distributions are derived from a maximum entropy principle with the use of two different constraints, and the resulting duality of entropy indexes is seen to portray physically relevant information. Whereas the value of the index α fixes the distribution’s power-law exponent, that for the dual index 2 − α ensures the extensivity of the deformed entropy.

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Additional Information:© 2015 National Academy of Sciences. Contributed by Murray Gell-Mann, June 27, 2014 (sent for review May 29, 2014) G.C.Y. and A.R. gratefully acknowledge the hospitality of the Santa Fe Institute. Support by Dirección General de Asuntos del Personal Académico-Universidad Nacional Autónoma de México-IN100311 and Consejo Nacional de Ciencia y Technología CB-2011-167978 is acknowledged. G.C.Y. was supported by the Scientific Research Projects Coordination Unit of Istanbul University, Project 36529. M.G.-M. acknowledges the generous support of Insight Venture Partners and the Bryan J. and June B. Zwan Foundation. Author contributions: G.C.Y., A.R., and M.G.-M. designed research; performed research; and wrote the paper. The authors declare no conflict of interest.
Funding AgencyGrant Number
Universidad Nacional Autónoma de México (UNAM)IN100311
Consejo Nacional de Ciencia y Technología (CONACYT)CB-2011-167978
Istanbul Univesity36529
Insight Venture PartnersUNSPECIFIED
Bryan J. and June B. Zwan FoundationUNSPECIFIED
Subject Keywords:rank-ordered data; generalized entropies
PubMed Central ID:PMC4191817
Record Number:CaltechAUTHORS:20150824-092812844
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Official Citation:G. Cigdem Yalcin, Alberto Robledo, and Murray Gell-Mann Incidence of q statistics in rank distributions PNAS 2014 111 (39) 14082-14087; published ahead of print September 4, 2014, doi:10.1073/pnas.1412093111
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:59833
Deposited By: George Porter
Deposited On:24 Aug 2015 17:44
Last Modified:24 Aug 2015 17:44

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