{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,25]],"date-time":"2026-02-25T04:21:53Z","timestamp":1771993313695,"version":"3.50.1"},"reference-count":15,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2022,5,16]],"date-time":"2022-05-16T00:00:00Z","timestamp":1652659200000},"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 R\u00e9nyi entropy is a generalization of the usual concept of entropy which depends on a parameter q. In fact, R\u00e9nyi entropy is closely related to free energy. Suppose we start with a system in thermal equilibrium and then suddenly divide the temperature by q. Then the maximum amount of work the system can perform as it moves to equilibrium at the new temperature divided by the change in temperature equals the system\u2019s R\u00e9nyi entropy in its original state. This result applies to both classical and quantum systems. Mathematically, we can express this result as follows: the R\u00e9nyi entropy of a system in thermal equilibrium is without the \u2018q\u22121-derivative\u2019 of its free energy with respect to the temperature. This shows that R\u00e9nyi entropy is a q-deformation of the usual concept of entropy.<\/jats:p>","DOI":"10.3390\/e24050706","type":"journal-article","created":{"date-parts":[[2022,5,16]],"date-time":"2022-05-16T13:06:23Z","timestamp":1652706383000},"page":"706","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["R\u00e9nyi Entropy and Free Energy"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0609-9836","authenticated-orcid":false,"given":"John C.","family":"Baez","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of California, Riverside, CA 92507, USA"}]}],"member":"1968","published-online":{"date-parts":[[2022,5,16]]},"reference":[{"key":"ref_1","first-page":"547","article-title":"On measures of information and entropy","volume":"1","author":"Neyman","year":"1906","journal-title":"Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability 1960, Berkeley, CA, USA, 20 June\u201330 July 1960"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"053015","DOI":"10.1088\/1367-2630\/13\/5\/053015","article-title":"On the work value of information","volume":"13","author":"Dahlsten","year":"2011","journal-title":"New J. 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Sci."},{"key":"ref_7","unstructured":"Baez, J.C. (2011, February 10). R\u00e9nyi Entropy and Free Energy. Azimuth. Available online: http:\/\/johncarlosbaez.wordpress.com\/2011\/02\/10\/rnyi-entropy-and-free-energy\/."},{"key":"ref_8","unstructured":"Beck, C., and Schl\u00f6gl, F. (1995). Thermodynamics of Chaotic Systems, Cambridge U. Press."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Principe, J. (2010). R\u00e9nyi\u2019s entropy, divergence and their nonparametric estimators. Information Theoretic Learning: R\u00e9nyi\u2019s Entropy and Kernel Perspectives, Springer.","DOI":"10.1007\/978-1-4419-1570-2"},{"key":"ref_10","unstructured":"Baez, J.C. (2022, March 15). R\u00e9nyi Entropy and Free Energy. Available online: https:\/\/arxiv.org\/abs\/1102.2098v3."},{"key":"ref_11","unstructured":"Polettini, M. (2022, March 15). R\u00e9nyi Entropy and Free Energy. Matteoeo. 10 February 2011. Available online: https:\/\/web.archive.org\/web\/20120124091413\/http:\/\/tomate.blogsome.com\/2011\/02\/10\/renyi-entropy-and-free-energy\/."},{"key":"ref_12","unstructured":"Downes, E. (2022, May 15). Comment on R\u00e9nyi Entropy and Free Energy. Azimuth. Available online: http:\/\/johncarlosbaez.wordpress.com\/2011\/02\/10\/rnyi-entropy-and-free-energy\/#comment-4065."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"326","DOI":"10.1016\/S0375-9601(96)00832-8","article-title":"A Note on the q-deformation-theoretic Aspect of the Generalized Entropies in Nonextensive Physics","volume":"224","author":"Abe","year":"1997","journal-title":"Phys. Lett. A"},{"key":"ref_14","unstructured":"van Dam, W., and Hayden, P. (2022, May 15). R\u00e9nyi-Entropic Bounds on Quantum Communication, Sec. 4.1: R\u00e9nyi Entropy. Available online: http:\/\/arxiv.org\/abs\/quant-ph\/0204093."},{"key":"ref_15","unstructured":"Cheung, P., and Kac, V. (2002). Quantum Calculus, Springer."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/5\/706\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T23:11:08Z","timestamp":1760137868000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/5\/706"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,5,16]]},"references-count":15,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2022,5]]}},"alternative-id":["e24050706"],"URL":"https:\/\/doi.org\/10.3390\/e24050706","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,5,16]]}}}