Incidence of $q$-statistics in rank distributions

TL;DR

This paper demonstrates that size-rank distributions with power-law decay in various systems follow Tsallis statistics, derived from a maximum entropy principle, revealing a duality of entropy indexes linked to physical properties.

Contribution

It introduces a theoretical framework connecting rank distributions to Tsallis statistics via a maximum entropy approach with dual constraints, highlighting the physical significance of entropy index duality.

Findings

Distributions obey Tsallis statistics near tangent bifurcation

Dual entropy indexes relate to distribution properties

Extensivity of deformed entropy is ensured by dual index

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.