{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,10]],"date-time":"2026-02-10T17:55:39Z","timestamp":1770746139246,"version":"3.49.0"},"reference-count":131,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2020,5,26]],"date-time":"2020-05-26T00:00:00Z","timestamp":1590451200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The concept of duality of probability distributions constitutes a fundamental \u201cbrick\u201d in the solid framework of nonextensive statistical mechanics\u2014the generalization of Boltzmann\u2013Gibbs statistical mechanics under the consideration of the q-entropy. The probability duality is solving old-standing issues of the theory, e.g., it ascertains the additivity for the internal energy given the additivity in the energy of microstates. However, it is a rather complex part of the theory, and certainly, it cannot be trivially explained along the Gibb\u2019s path of entropy maximization. Recently, it was shown that an alternative picture exists, considering a dual entropy, instead of a dual probability. In particular, the framework of nonextensive statistical mechanics can be equivalently developed using q- and 1\/q- entropies. The canonical probability distribution coincides again with the known q-exponential distribution, but without the necessity of the duality of ordinary-escort probabilities. Furthermore, it is shown that the dual entropies, q-entropy and 1\/q-entropy, as well as, the 1-entropy, are involved in an identity, useful in theoretical development and applications.<\/jats:p>","DOI":"10.3390\/e22060594","type":"journal-article","created":{"date-parts":[[2020,5,27]],"date-time":"2020-05-27T05:35:52Z","timestamp":1590557752000},"page":"594","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["Nonextensive Statistical Mechanics: Equivalence Between Dual Entropy and Dual Probabilities"],"prefix":"10.3390","volume":"22","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7655-6019","authenticated-orcid":false,"given":"George","family":"Livadiotis","sequence":"first","affiliation":[{"name":"Division of Space Science and Engineering, Southwest Research Institute, San Antonio, TX 78238, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,5,26]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"479","DOI":"10.1007\/BF01016429","article-title":"Possible generalization of Boltzmann-Gibbs statistics","volume":"52","author":"Tsallis","year":"1988","journal-title":"J. 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