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arXiv · 1409.7428

Incidence of $q$-statistics in rank distributions

G. Cigdem Yalcin·Alberto Robledo·Murray Gell-Mann
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Abstract

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. While the value of the index $\alpha $ fixes the distribution's power-law exponent, that for the dual index $2-\alpha $ ensures the extensivity of the deformed entropy.

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G. Cigdem Yalcin, Alberto Robledo, Murray Gell-Mann. 2014-09-25. Incidence of $q$-statistics in rank distributions. https://doi.org/10.1073/pnas.1412093111

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