{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,15]],"date-time":"2025-10-15T10:33:53Z","timestamp":1760524433724,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,11,25]],"date-time":"2022-11-25T00:00:00Z","timestamp":1669334400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Conselho Nacional de Desenvolvimento Cient\u00edfico e Tecnol\u00f3gico (CNPq)"},{"name":"Funda\u00e7\u00e3o Carlos Chagas Filho de Amparo \u00e0 Pesquisa do Estado do Rio de Janeiro (FAPERJ)"},{"name":"Coordena\u00e7\u00e3o de Aperfei\u00e7oamento de Pessoal de N\u00edvel Superior-Brasil (CAPES)"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Several generalizations or extensions of the Boltzmann\u2013Gibbs thermostatistics, based on non-standard entropies, have been the focus of considerable research activity in recent years. Among these, the power-law, non-additive entropies Sq\u2261k1\u2212\u2211ipiqq\u22121(q\u2208R;S1=SBG\u2261\u2212k\u2211ipilnpi) have harvested the largest number of successful applications. The specific structural features of the Sq thermostatistics, therefore, are worthy of close scrutiny. In the present work, we analyze one of these features, according to which the q-logarithm function lnqx\u2261x1\u2212q\u221211\u2212q(ln1x=lnx) associated with the Sq entropy is linked, via a duality relation, to the q-exponential function characterizing the maximum-entropy probability distributions. We enquire into which entropic functionals lead to this or similar structures, and investigate the corresponding duality relations.<\/jats:p>","DOI":"10.3390\/e24121723","type":"journal-article","created":{"date-parts":[[2022,11,25]],"date-time":"2022-11-25T02:29:23Z","timestamp":1669343363000},"page":"1723","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Entropy Optimization, Generalized Logarithms, and Duality Relations"],"prefix":"10.3390","volume":"24","author":[{"given":"Angel R.","family":"Plastino","sequence":"first","affiliation":[{"name":"CeBio y Departamento de Ciencias B\u00e1sicas, Universidad Nacional del Noroeste de la Prov\u00edncia de Buenos Aires, UNNOBA, CONICET, Roque Saenz Pe\u00f1a 456, Junin B6000, Argentina"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9387-9194","authenticated-orcid":false,"given":"Constantino","family":"Tsallis","sequence":"additional","affiliation":[{"name":"Centro Brasileiro de Pesquisas F\u00edsicas and National Institute of Science and Technology for Complex Systems, Rua Xavier Sigaud 150, Rio de Janeiro 22290-180, RJ, Brazil"},{"name":"Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA"},{"name":"Complexity Science Hub Vienna, Josefst\u00e4dter Stra\u00dfe 39, 1080 Vienna, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7532-3881","authenticated-orcid":false,"given":"Roseli S.","family":"Wedemann","sequence":"additional","affiliation":[{"name":"Instituto de Matem\u00e1tica e Estat\u00edstica, Universidade do Estado do Rio de Janeiro, Rua S\u00e3o Francisco Xavier 524, Rio de Janeiro 20550-900, RJ, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-0104-2355","authenticated-orcid":false,"given":"Hans J.","family":"Haubold","sequence":"additional","affiliation":[{"name":"Office for Outer Space Affairs, United Nations, Vienna International Center, 1400 Vienna, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,11,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"264","DOI":"10.3390\/encyclopedia2010018","article-title":"Entropy","volume":"2","author":"Tsallis","year":"2022","journal-title":"Encyclopedia"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"120601","DOI":"10.1103\/PhysRevLett.122.120601","article-title":"Maximum entropy principle in statistical inference: Case for non-Shannonian entropies","volume":"122","author":"Jizba","year":"2019","journal-title":"Phys. Rev. Lett."},{"key":"ref_3","doi-asserted-by":"crossref","unstructured":"Naudts, J. (2011). Generalised Thermostatistics, Springer.","DOI":"10.1007\/978-0-85729-355-8"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"495","DOI":"10.1080\/00107510902823517","article-title":"Generalised information and entropy measures in physics","volume":"50","author":"Beck","year":"2009","journal-title":"Contemp. Phys."},{"key":"ref_5","unstructured":"Tsallis, C. (2009). Introduction to Nonextensive Statistical Mechanics\u2014Approaching a Complex World, Springer."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"20006","DOI":"10.1209\/0295-5075\/93\/20006","article-title":"A comprehensive classification of complex statistical systems and an axiomatic derivation of their entropy and distribution functions","volume":"93","author":"Hanel","year":"2011","journal-title":"EPL"},{"key":"ref_7","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. Stat. Phys."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Gell-Mann, M., and Tsallis, C. (2004). Nonextensive Entropy: Interdisciplinary Applications, Oxford University Press.","DOI":"10.1093\/oso\/9780195159769.001.0001"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1765","DOI":"10.3390\/e13101765","article-title":"The nonadditive entropy Sq and its applications in physics and elsewhere: Some remarks","volume":"13","author":"Tsallis","year":"2011","journal-title":"Entropy"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Tsallis, C. (2019). Beyond Boltzmann-Gibbs-Shannon in physics and elsewhere. Entropy, 21.","DOI":"10.3390\/e21070696"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"S\u00e1nchez Almeida, J. (2022). The principle of maximum entropy and the distribution of mass in galaxies. Universe, 8.","DOI":"10.3390\/universe8040214"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1016\/S0375-9601(98)00572-6","article-title":"A family of nonextensive entropies","volume":"246","author":"Borges","year":"1998","journal-title":"Phys. Lett. A"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1089","DOI":"10.1088\/0305-4470\/32\/7\/002","article-title":"Maximum entropy approach to stretched exponential probability distributions","volume":"32","author":"Anteneodo","year":"1999","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"056125","DOI":"10.1103\/PhysRevE.66.056125","article-title":"Statistical mechanics in the context of special relativity","volume":"66","author":"Kaniadakis","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"2067","DOI":"10.3390\/e12092067","article-title":"Superstatistics and gravitation","volume":"12","year":"2010","journal-title":"Entropy"},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Amig\u00f3, J.M., Balogh, S.G., and Hern\u00e1ndez, S. (2018). A brief review of generalized entropies. Entropy, 20.","DOI":"10.3390\/e20110813"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"50005","DOI":"10.1209\/0295-5075\/133\/50005","article-title":"An overview of generalized entropic forms","volume":"133","author":"Ilic","year":"2021","journal-title":"EPL"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"257","DOI":"10.1016\/S0375-9601(96)00942-5","article-title":"On the universality of thermodynamics\u2019 Legendre transform structure","volume":"226","author":"Plastino","year":"1997","journal-title":"Phys. Lett. A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"299","DOI":"10.1016\/S0378-4371(97)00175-1","article-title":"Some general relations in arbitrary thermostatistics","volume":"242","author":"Mendes","year":"1997","journal-title":"Physical A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1590\/S0103-97331999000100003","article-title":"General aspects of the thermodynamical formalism","volume":"29","author":"Curado","year":"1999","journal-title":"Braz. Journ. Phys."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"6675","DOI":"10.1063\/1.532184","article-title":"The role of information measures in the determination of the maximum entropy-minimum norm solution of the generalized inverse problem","volume":"38","author":"Plastino","year":"1997","journal-title":"J. Math. Phys"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1140\/epjb\/e2005-00372-3","article-title":"Dynamical thermostatting and statistical ensembles","volume":"48","author":"Roston","year":"2005","journal-title":"Eur. Phys. J. B"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"250","DOI":"10.1006\/aphy.1996.5659","article-title":"A dynamical thermostatting approach to nonextensive canonical ensembles","volume":"255","author":"Plastino","year":"1997","journal-title":"Ann. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"026106","DOI":"10.1103\/PhysRevE.67.026106","article-title":"Constructing a statistical mechanics for Beck-Cohen superstatistics","volume":"67","author":"Tsallis","year":"2003","journal-title":"Phys. Rev. E"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"060101","DOI":"10.1103\/PhysRevE.101.060101","article-title":"Thermodynamics from first principles: Correlations and nonextensivity","volume":"101","author":"Saadatmand","year":"2020","journal-title":"Phys. Rev. E"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"L69","DOI":"10.1088\/0305-4470\/24\/2\/004","article-title":"Generalized statistical mechanics: Connection with thermodynamics","volume":"24","author":"Curado","year":"1991","journal-title":"J. Phys. A"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"534","DOI":"10.1016\/S0378-4371(98)00437-3","article-title":"The role of constraints within generalized nonextensive statistics","volume":"261","author":"Tsallis","year":"1998","journal-title":"Phys. A"},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Gazeau, J.P., and Tsallis, C. (2019). Moebius transforms, cycles and q-triplets in statistical mechanics. Entropy, 21.","DOI":"10.3390\/e21121155"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"298628","DOI":"10.1155\/2011\/298628","article-title":"Mittag-Leffler Functions and Their Applications","volume":"2011","author":"Haubold","year":"2011","journal-title":"J. Appl. Math."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"123101","DOI":"10.1088\/1742-5468\/aa967f","article-title":"Uniqueness and characterization theorems for generalized entropies","volume":"2017","author":"Enciso","year":"2017","journal-title":"J. Stat. Mech."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1038\/s41598-020-60188-y","article-title":"Universality classes and information-theoretic measures of complexity via group entropies","volume":"10","author":"Tempesta","year":"2020","journal-title":"Sci. Rep."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"013303","DOI":"10.1063\/1.3063640","article-title":"On the maximum entropy principle and the minimization of the Fisher information in Tsallis statistics","volume":"50","author":"Furuichi","year":"2009","journal-title":"J. Math. Phys."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/12\/1723\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:26:29Z","timestamp":1760145989000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/24\/12\/1723"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,11,25]]},"references-count":32,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["e24121723"],"URL":"https:\/\/doi.org\/10.3390\/e24121723","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2022,11,25]]}}}