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Entropy 2015, 17(5), 2853-2861; doi:10.3390/e17052853

Conceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems

1
Centro Brasileiro de Pesquisas Físicas and National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, 22290-180, Rio de Janeiro - RJ, Brazil
2
Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, NM, USA 
Academic Editor: Antonio M. Scarfone
Received: 9 April 2015 / Accepted: 4 May 2015 / Published: 5 May 2015
(This article belongs to the Collection Advances in Applied Statistical Mechanics)
View Full-Text   |   Download PDF [248 KB, uploaded 6 May 2015]

Abstract

It is by now well known that the Boltzmann-Gibbs-von Neumann-Shannon logarithmic entropic functional (\(S_{BG}\)) is inadequate for wide classes of strongly correlated systems: see for instance the 2001 Brukner and Zeilinger's {\it Conceptual inadequacy of the Shannon information in quantum measurements}, among many other systems exhibiting various forms of complexity. On the other hand, the Shannon and Khinchin axioms uniquely mandate the BG form \(S_{BG}=-k\sum_i p_i \ln p_i\); the Shore and Johnson axioms follow the same path. Many natural, artificial and social systems have been satisfactorily approached with nonadditive entropies such as the \(S_q=k \frac{1-\sum_i p_i^q}{q-1}\) one (\(q \in {\cal R}; \,S_1=S_{BG}\)), basis of nonextensive statistical mechanics. Consistently, the Shannon 1948 and Khinchine 1953 uniqueness theorems have already been generalized in the literature, by Santos 1997 and Abe 2000 respectively, in order to uniquely mandate \(S_q\). We argue here that the same remains to be done with the Shore and Johnson 1980 axioms. We arrive to this conclusion by analyzing specific classes of strongly correlated complex systems that await such generalization. View Full-Text
Keywords: nonadditive entropies; nonextensive statistical mechanics; strongly correlated random variables; Shore and Johnson axioms nonadditive entropies; nonextensive statistical mechanics; strongly correlated random variables; Shore and Johnson axioms
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Tsallis, C. Conceptual Inadequacy of the Shore and Johnson Axioms for Wide Classes of Complex Systems. Entropy 2015, 17, 2853-2861.

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