https://doi.org/10.1140/epjb/s10051-024-00764-7
Regular Article - Statistical and Nonlinear Physics
Typicality, entropy and the generalization of statistical mechanics
1
Institute of Biology, University Graz, Holteigasse 6, 8010, Graz, Austria
2
Complexity Science Hub Vienna, Josefstädter Strasse 39, 1080, Vienna, Austria
3
CS, Medical University of Vienna, Spitalgasse 23, 1090, Vienna, Austria
4
FNSPE, Czech Technical University in Prague, Břehová 7, 115 19, Prague, Czech Republic
Received:
9
June
2024
Accepted:
2
August
2024
Published online:
29
August
2024
When at equilibrium, large-scale systems obey conventional thermodynamics because they belong to microscopic configurations (or states) that are typical. Crucially, the typical states usually represent only a small fraction of the total number of possible states, and yet the characterization of the set of typical states—the typical set—alone is sufficient to describe the macroscopic behavior of a given system. Consequently, the concept of typicality, and the associated Asymptotic Equipartition Property allow for a drastic reduction of the degrees of freedom needed for system’s statistical description. The mathematical rationale for such a simplification in the description is due to the phenomenon of concentration of measure. The later emerges for equilibrium configurations thanks to very strict constraints on the underlying dynamics, such as weekly interacting and (almost) independent system constituents. The question naturally arises as to whether the concentration of measure and related typicality considerations can be extended and applied to more general complex systems, and if so, what mathematical structure can be expected in the ensuing generalized thermodynamics. In this paper, we illustrate the relevance of the concept of typicality in the toy model context of the “thermalized” coin and show how this leads naturally to Shannon entropy. We also show an intriguing connection: The characterization of typical sets in terms of Rényi and Tsallis entropies naturally leads to the free energy and partition function, respectively, and makes their relationship explicit. Finally, we propose potential ways to generalize the concept of typicality to systems where the standard microscopic assumptions do not hold.
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© The Author(s), under exclusive licence to EDP Sciences, SIF and Springer-Verlag GmbH Germany, part of Springer Nature 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
