{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:34:47Z","timestamp":1760243687512,"version":"build-2065373602"},"reference-count":23,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2012,7,27]],"date-time":"2012-07-27T00:00:00Z","timestamp":1343347200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We derive a two-parameter family of generalized entropies, Spq, and means mpq. To this end, assume that we want to calculate an entropy and a mean for n non-negative real numbers {x1,\u2026,xn}. For comparison, we consider {m1,\u2026,mk} where mi = m for all i = 1,\u2026,k and where m and k are chosen such that the lp and lq norms of {x1,\u2026,xn} and {m1,\u2026,mk} coincide. We formally allow k to be real. Then, we define k, log k, and m to be a generalized cardinality kpq, a generalized entropy Spq, and a generalized mean mpq respectively. We show that this family of entropies includes the Shannon and R\u00e9nyi entropies and that the family of generalized means includes the power means (such as arithmetic, harmonic, geometric, root-mean-square, maximum, and minimum) as well as novel means of Shannon-like and R\u00e9nyi-like forms. A thermodynamic interpretation arises from the fact that the lp norm is closely related to the partition function at inverse temperature \u03b2 = p. Namely, two systems possess the same generalized entropy and generalized mean energy if and only if their partition functions agree at two temperatures, which is also equivalent to the condition that their Helmholtz free energies agree at these two temperatures.<\/jats:p>","DOI":"10.3390\/e14081317","type":"journal-article","created":{"date-parts":[[2012,7,28]],"date-time":"2012-07-28T03:13:59Z","timestamp":1343445239000},"page":"1317-1342","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Equivalence of Partition Functions Leads to Classification of Entropies and Means"],"prefix":"10.3390","volume":"14","author":[{"given":"Michel S.","family":"Elnaggar","sequence":"first","affiliation":[{"name":"Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario N2L 3G1, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Achim","family":"Kempf","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2012,7,27]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2307\/1401301","article-title":"On the foundations of information theory","volume":"33","year":"1965","journal-title":"Rev. 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