{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:39:50Z","timestamp":1760240390890,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"5","license":[{"start":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T00:00:00Z","timestamp":1558915200000},"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>An entropic functional S is said additive if it satisfies, for any two probabilistically independent systems A and B, that     S ( A + B ) = S ( A ) + S ( B )     [...]<\/jats:p>","DOI":"10.3390\/e21050538","type":"journal-article","created":{"date-parts":[[2019,5,27]],"date-time":"2019-05-27T11:19:27Z","timestamp":1558955967000},"page":"538","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Nonadditive Entropies and Complex Systems"],"prefix":"10.3390","volume":"21","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8290-8183","authenticated-orcid":false,"given":"Andrea","family":"Rapisarda","sequence":"first","affiliation":[{"name":"Dipartimento di Fisica e Astronomia and INFN Sezione di Catania, Via S. Sofia 64, 95123 Catania, Italy"},{"name":"Complexity Science Hub Vienna, Josefst\u00e4dterstrasse 39, A-1090 Vienna, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stefan","family":"Thurner","sequence":"additional","affiliation":[{"name":"Complexity Science Hub Vienna, Josefst\u00e4dterstrasse 39, A-1090 Vienna, Austria"},{"name":"Section for the Science of Complex Systems, CeMSIIS, Medical University of Vienna, Spitalgasse 23, A-1090 Vienna, Austria"},{"name":"Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Constantino","family":"Tsallis","sequence":"additional","affiliation":[{"name":"Complexity Science Hub Vienna, Josefst\u00e4dterstrasse 39, A-1090 Vienna, Austria"},{"name":"Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA"},{"name":"Centro Brasileiro de Pesquisas F\u00edsicas and National Institute of Science and Technology of Complex Systems, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, Brazil"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2019,5,27]]},"reference":[{"key":"ref_1","unstructured":"Boltzmann, L. (2019, May 27). Weitere Studien uber das Warmegleichgewicht unter Gas molekulen (Further Studies on Thermal Equilibrium Between Gas Molecules). Available online: http:\/\/www.eoht.info\/page\/Further+Studies+on+the+Thermal+Equilibrium+of+Gas+Molecules."},{"key":"ref_2","first-page":"188","article-title":"On the Relation of a General Mechanical Theorem to the Second Law of Thermodynamics","volume":"Volume 2","author":"Brush","year":"1966","journal-title":"Kinetic Theory: Irreversible Processes"},{"key":"ref_3","unstructured":"Gibbs, J.W. (1981). Elementary Principles in Statistical Mechanics\u2014Developed with Especial Reference to the Rational Foundation of Thermodynamics (C. Scribner\u2019s Sons, New York, NY, USA, 1902; Yale University Press: New Haven, CT, USA, 1948), OX Bow Press."},{"key":"ref_4","unstructured":"Gibbs, J.W. (1948). The Collected Works of J.Willard Gibbs (Vol. 1: Thermodynamics), Yale University Press."},{"key":"ref_5","first-page":"273","article-title":"Thermodynamik Quantenmechanischer Gesamtheiten","volume":"1927","year":"1927","journal-title":"Nachrichten von der Gesellschaft der Wissenschaften zu Gottingen"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1002\/j.1538-7305.1948.tb01338.x","article-title":"A Mathematical Theory of Communication","volume":"27","author":"Shannon","year":"1948","journal-title":"Bell Syst. Tech. J."},{"key":"ref_7","unstructured":"Shannon, C.E., and Weaver, W. (1949). The Mathematical Theory of Communication, University of Illinois Press."},{"key":"ref_8","first-page":"547","article-title":"On measures of information and entropy","volume":"Volume 1","year":"1961","journal-title":"Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability"},{"key":"ref_9","unstructured":"R\u00e9nyi, A. (1970). Probability Theory, North-Holland Co."},{"key":"ref_10","first-page":"28","article-title":"New non-additive measures of entropy for discrete probability distributions","volume":"10","author":"Sharma","year":"1975","journal-title":"J. Math. Sci."},{"key":"ref_11","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_12","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_13","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1016\/S0375-9601(98)00572-6","article-title":"A family of non-extensive entropies","volume":"246","author":"Borges","year":"1998","journal-title":"Phys. Lett. A"},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/S0375-9601(98)00500-3","article-title":"Distributions and channel capacities in generalized statistical mechanics","volume":"247","author":"Landsberg","year":"1998","journal-title":"Phys. Lett. A"},{"key":"ref_15","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. J. Phys."},{"key":"ref_16","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":"Phys. A"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"405","DOI":"10.1016\/S0378-4371(01)00184-4","article-title":"Non linear kinetics underlying generalized statistics","volume":"296","author":"Kaniadakis","year":"2001","journal-title":"Phys. A"},{"key":"ref_18","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_19","doi-asserted-by":"crossref","first-page":"113301","DOI":"10.1063\/1.2801996","article-title":"Two-parameter generalization of the logarithm and exponential functions and Boltzmann-Gibbs-Shannon entropy","volume":"48","author":"Schwammle","year":"2007","journal-title":"J. Math. Phys."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"785","DOI":"10.1093\/imamat\/hxm039","article-title":"Lambert function and a new non-extensive form of entropy","volume":"72","author":"Shafee","year":"2007","journal-title":"IMA J. Appl. Math."},{"key":"ref_21","unstructured":"Tsallis, C. (2009). Introduction to Nonextensive Statistical Mechanics\u2014Approaching a Complex World, Springer."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"2516","DOI":"10.1016\/j.physleta.2009.05.026","article-title":"Entropies based on fractional calculus","volume":"373","author":"Ubriaco","year":"2009","journal-title":"Phys. Lett. A"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"021121","DOI":"10.1103\/PhysRevE.84.021121","article-title":"Group entropies, correlation laws, and zeta functions","volume":"84","author":"Tempesta","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_24","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":"Europhys. Lett."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"2487","DOI":"10.1140\/epjc\/s10052-013-2487-6","article-title":"Black hole thermodynamical entropy","volume":"73","author":"Tsallis","year":"2013","journal-title":"Eur. Phys. J. C"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"22","DOI":"10.1016\/j.aop.2015.12.008","article-title":"A new entropy based on a group-theoretical structure","volume":"366","author":"Curado","year":"2016","journal-title":"Ann. Phys."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"180","DOI":"10.1016\/j.aop.2015.08.013","article-title":"Beyond the Shannon-Khinchin formulation: The composability axiom and the universal-group entropy","volume":"365","author":"Tempesta","year":"2016","journal-title":"Ann. Phys."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"375002","DOI":"10.1088\/1751-8121\/aad57b","article-title":"Statistical mechanics of exploding phase spaces: Ontic open systems","volume":"51","author":"Jensen","year":"2018","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Rodriguez, A., Nobre, F.D., and Tsallis, C. (2019). d-dimensional classical Heisenberg model with arbitrarily-ranged interactions: Lyapunov exponents and distributions of momenta and energies. Entropy, 21.","DOI":"10.3390\/e21010031"},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Curado, E.M.F., Nobre, F.D., and Plastino, A. (2018). Associating an entropy with power-law frequency of events. Entropy, 20.","DOI":"10.3390\/e20120940"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Viallon-Galinier, L., Combe, G., Richefeu, V., and Atman, A.P.F. (2018). Emergence of shear bands in confined granular systems: Singularity of the q-statistics. Entropy, 20.","DOI":"10.3390\/e20110862"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Hanel, R., and Thurner, S. (2018). Maximum configuration principle for driven systems with arbitrary driving. Entropy, 20.","DOI":"10.3390\/e20110838"},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Obregon, O., Lopez, J.L., and Ortega-Cruz, M. (2018). On quantum superstatistics and the critical behavior of nonextensive ideal Bose gases. Entropy, 20.","DOI":"10.3390\/e20100773"},{"key":"ref_34","doi-asserted-by":"crossref","unstructured":"Ibrahim, R.W., and Darus, M. (2018). Analytic study of complex fractional Tsallis\u2019 entropy with applications in CNNs. Entropy, 20.","DOI":"10.3390\/e20100722"},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Zhao, P., Pan, J., Zhou, B., Wang, J., and Song, Y. (2018). Hedging for the Regime-Switching Price Model Based on Non-Extensive Statistical Mechanics. Entropy, 20.","DOI":"10.3390\/e20040248"},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Cetin, K., Afsar, O., and Tirnakli, U. (2018). Generalized Pesin-like identity and scaling relations at the chaos threshold of the R\u00f6ssler system. Entropy, 20.","DOI":"10.3390\/e20040216"},{"key":"ref_37","doi-asserted-by":"crossref","unstructured":"Zhao, P., Zhou, B., and Wang, J. (2018). Non-Gaussian closed form solutions for geometric average asian options in the framework of non-extensive statistical mechanics. Entropy, 20.","DOI":"10.3390\/e20010071"},{"key":"ref_38","unstructured":"Jensen, H.J., and Tempesta, P. (2018). Group entropies: From phase space geometry to entropy functionals via group theory. Entropy, 20."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/5\/538\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T12:55:14Z","timestamp":1760187314000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/21\/5\/538"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,5,27]]},"references-count":38,"journal-issue":{"issue":"5","published-online":{"date-parts":[[2019,5]]}},"alternative-id":["e21050538"],"URL":"https:\/\/doi.org\/10.3390\/e21050538","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2019,5,27]]}}}