Tsallis, Constantino and Gell-Mann, Murray and Sato, Yuzuru
(2005)
*Asymptotically scale-invariant occupancy of phase space makes the entropy S_q extensive.*
Proceedings of the National Academy of Sciences of the United States of America, 102
(43).
pp. 15377-15382.
ISSN 0027-8424.
PMCID PMC1266086.
doi:10.1073/pnas.0503807102.
https://resolver.caltech.edu/CaltechAUTHORS:20150824-151627203

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## Abstract

Phase space can be constructed for N equal and distinguishable subsystems that could be probabilistically either weakly correlated or strongly correlated. If they are locally correlated, we expect the Boltzmann-Gibbs entropy S_(BG) ≡ -k Σ_i p_i ln p_i to be extensive, i.e., S_(BG)(N) ∝ N for N → ∞. In particular, if they are independent, S_(BG) is strictly additive, i.e., S_(BG)(N) = NS_(BG)(1), ∀N. However, if the subsystems are globally correlated, we expect, for a vast class of systems, the entropy S_q ≡ k[1 - Σi p^q_i]/(q - 1) (with S_1 = S_(BG)) for some special value of q ≠ 1 to be the one which is extensive [i.e., S_q(N) ∝ N for N → ∞]. Another concept which is relevant is strict or asymptotic scale-freedom (or scale-invariance), defined as the situation for which all marginal probabilities of the N-system coincide or asymptotically approach (for N → ∞) the joint probabilities of the (N - 1)-system. If each subsystem is a binary one, scale-freedom is guaranteed by what we hereafter refer to as the Leibnitz rule, i.e., the sum of two successive joint probabilities of the N-system coincides or asymptotically approaches the corresponding joint probability of the (N - 1)-system. The kinds of interplay of these various concepts are illustrated in several examples. One of them justifies the title of this paper. We conjecture that these mechanisms are deeply related to the very frequent emergence, in natural and artificial complex systems, of scale-free structures and to their connections with nonextensive statistical mechanics. Summarizing, we have shown that, for asymptotically scale-invariant systems, it is S_q with q ≠ 1, and not S_(BG), the entropy which matches standard, clausius-like, prescriptions of classical thermodynamics.

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Additional Information: | © 2005 National Academy of Sciences. Contributed by Murray Gell-Mann, July 25, 2005. We are grateful to R. Hersh for pointing out to us that the joint-probability structure of one of our discrete models is analogous to that of the Leibnitz triangle. We have also benefited from very fruitful remarks by J. Marsh and L. G. Moyano. Y.S. was supported by the Postdoctoral Fellowship at Santa Fe Institute. Support from SI International and AFRL is acknowledged as well. Finally, the work of one of us (M.G.M.) was supported by the C.O.U.Q. Foundation and by Insight Venture Management. The generous help provided by these organizations is gratefully acknowledged. Author contributions: C.T. and M.G.-M. designed research; C.T. and Y.S. performed research; C.T., M.G.-M., and Y.S. analyzed data; and C.T. and M.G.-M. wrote the paper. | |||||||||||||||

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Issue or Number: | 43 | |||||||||||||||

PubMed Central ID: | PMC1266086 | |||||||||||||||

DOI: | 10.1073/pnas.0503807102 | |||||||||||||||

Record Number: | CaltechAUTHORS:20150824-151627203 | |||||||||||||||

Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20150824-151627203 | |||||||||||||||

Official Citation: | Constantino Tsallis, Murray Gell-Mann, and Yuzuru Sato Asymptotically scale-invariant occupancy of phase space makes the entropy Sq extensive PNAS 2005 102 (43) 15377-15382; published ahead of print October 17, 2005, doi:10.1073/pnas.0503807102 | |||||||||||||||

Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | |||||||||||||||

ID Code: | 59863 | |||||||||||||||

Collection: | CaltechAUTHORS | |||||||||||||||

Deposited By: | George Porter | |||||||||||||||

Deposited On: | 25 Aug 2015 18:11 | |||||||||||||||

Last Modified: | 10 Nov 2021 22:26 |

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