期刊名称：Proceedings of the National Academy of Sciences
印刷版ISSN：0027-8424
电子版ISSN：1091-6490
出版年度：2014
卷号：111
期号：39
页码：14082-14087
DOI：10.1073/pnas.1412093111
语种：English
出版社：The National Academy of Sciences of the United States of America
摘要：SignificanceThe contents presented are of prime importance to the field of generalized statistical mechanics. We fulfill a longstanding need of exhibiting the kind of abundant real-world data that match the formal developments in this subject. These are size-rank distributions for which we provide a solid bridge between experimental data and theory. Also, this work delivers a working explanation for the existing duality between the two Tsallis-type entropy expressions that generalize the canonical expression. One relates to the distribution's power-law exponent whereas the other ensures entropy extensivity. The generalized entropies arise from a drastic reduction of configurations available to the system. We argue that this phase-space contraction is farthest for ranked data of the Zipf type. 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 fixes the distribution's power-law exponent, that for the dual index 2 - ensures the extensivity of the deformed entropy.
