A Caltech Library Service

Generalized entropies and logarithms and their duality relations

Hanel, Rudolf and Thurner, Stefan and Gell-Mann, Murray (2012) Generalized entropies and logarithms and their duality relations. Proceedings of the National Academy of Sciences of the United States of America, 109 (47). pp. 19151-19154. ISSN 0027-8424. PMCID PMC3511158. doi:10.1073/pnas.1216885109.

[img] PDF - Published Version
See Usage Policy.

[img] PDF - Supplemental Material
See Usage Policy.


Use this Persistent URL to link to this item:


For statistical systems that violate one of the four Shannon–Khinchin axioms, entropy takes a more general form than the Boltzmann–Gibbs entropy. The framework of superstatistics allows one to formulate a maximum entropy principle with these generalized entropies, making them useful for understanding distribution functions of non-Markovian or nonergodic complex systems. For such systems where the composability axiom is violated there exist only two ways to implement the maximum entropy principle, one using escort probabilities, the other not. The two ways are connected through a duality. Here we show that this duality fixes a unique escort probability, which allows us to derive a complete theory of the generalized logarithms that naturally arise from the violation of this axiom. We then show how the functional forms of these generalized logarithms are related to the asymptotic scaling behavior of the entropy.

Item Type:Article
Related URLs:
URLURL TypeDescription CentralArticle Information
Additional Information:© 2012 National Academy of Sciences. Freely available online through the PNAS open access option. Contributed by Murray Gell-Mann, September 28, 2012 (sent for review August 6, 2012) R.H. and S.T. thank the Santa Fe Institute and M.G.-M. thanks the Aspen Center for Physics for their hospitality. Support for this work was provided in part by National Science Foundation Grant 1066293, Insight Venture Partners, and the Bryan J. and June B. Zwan Foundation (M.G.-M.). The authors declare no conflict of interest. This article contains supporting information online at
Funding AgencyGrant Number
Insight Venture PartnersUNSPECIFIED
Bryan J. and June B. Zwan FoundationUNSPECIFIED
Subject Keywords:classical statistical mechanics; correlated systems
Issue or Number:47
PubMed Central ID:PMC3511158
Record Number:CaltechAUTHORS:20150824-101319749
Persistent URL:
Official Citation:Rudolf Hanel, Stefan Thurner, and Murray Gell-Mann Generalized entropies and logarithms and their duality relations PNAS 2012 109 (47) 19151-19154; published ahead of print November 5, 2012, doi:10.1073/pnas.1216885109
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:59838
Deposited By: George Porter
Deposited On:24 Aug 2015 17:56
Last Modified:10 Nov 2021 22:26

Repository Staff Only: item control page