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Generalized entropies and the transformation group of superstatistics

Hanel, Rudolf and Thurner, Stefan and Gell-Mann, Murray (2011) Generalized entropies and the transformation group of superstatistics. Proceedings of the National Academy of Sciences of the United States of America, 108 (16). pp. 6390-6394. ISSN 0027-8424. PMCID PMC3080995.

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Superstatistics describes statistical systems that behave like superpositions of different inverse temperatures β, so that the probability distribution is p(ε_i) α ∫_0^∞ f(β)e^(-β(ε_i))dβ, where the “kernel” f(β) is nonnegative and normalized [∫f(β)dβ = 1]. We discuss the relation between this distribution and the generalized entropic form S = Σ_i s(p_i). The first three Shannon–Khinchin axioms are assumed to hold. It then turns out that for a given distribution there are two different ways to construct the entropy. One approach uses escort probabilities and the other does not; the question of which to use must be decided empirically. The two approaches are related by a duality. The thermodynamic properties of the system can be quite different for the two approaches. In that connection, we present the transformation laws for the superstatistical distributions under macroscopic state changes. The transformation group is the Euclidean group in one dimension.

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Additional Information:© 2011 National Academy of Sciences. Contributed by Murray Gell-Mann, March 3, 2011 (sent for review February 17, 2011) R.H. and S.T. thank the Santa Fe Institute for hospitality. M.G.-M. acknowledges the generous support of Mr. Jerry Murdock and The Bryan J. and June B. Zwan Foundation. In addition, he is greatly indebted to Professor E.G.D. Cohen for valuable conversations. Author contributions: R.H., S.T., and M.G.-M. performed research and wrote the paper. The authors declare no conflict of interest.
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Bryan J. and June B. Zwan FoundationUNSPECIFIED
Subject Keywords:classical statistical mechanics; correlated systems; thermodynamics
Issue or Number:16
PubMed Central ID:PMC3080995
Record Number:CaltechAUTHORS:20150824-103421520
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Official Citation:Rudolf Hanel, Stefan Thurner, and Murray Gell-Mann Generalized entropies and the transformation group of superstatistics PNAS 2011 108 (16) 6390-6394; published ahead of print April 4, 2011, doi:10.1073/pnas.1103539108
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:59841
Deposited By: George Porter
Deposited On:24 Aug 2015 17:58
Last Modified:03 Oct 2019 08:50

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