{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,5]],"date-time":"2026-04-05T06:44:31Z","timestamp":1775371471673,"version":"3.50.1"},"reference-count":62,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,1,17]],"date-time":"2024-01-17T00:00:00Z","timestamp":1705449600000},"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>It is known that Kaniadakis entropy, a generalization of the Shannon\u2013Boltzmann\u2013Gibbs entropic form, is always super-additive for any bipartite statistically independent distributions. In this paper, we show that when imposing a suitable constraint, there exist classes of maximal entropy distributions labeled by a positive real number \u2135&gt;0 that makes Kaniadakis entropy multi-additive, i.e., S\u03ba[pA\u222aB]=(1+\u2135)S\u03ba[pA]+S\u03ba[pB], under the composition of two statistically independent and identically distributed distributions pA\u222aB(x,y)=pA(x)pB(y), with reduced distributions pA(x) and pB(y) belonging to the same class.<\/jats:p>","DOI":"10.3390\/e26010077","type":"journal-article","created":{"date-parts":[[2024,1,17]],"date-time":"2024-01-17T07:41:28Z","timestamp":1705477288000},"page":"77","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Multi-Additivity in Kaniadakis Entropy"],"prefix":"10.3390","volume":"26","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6699-5889","authenticated-orcid":false,"given":"Antonio M.","family":"Scarfone","sequence":"first","affiliation":[{"name":"Istituto dei Sistemi Complessi\u2014Consiglio Nazionale delle Ricerche (ISC-CNR), c\/o Dipartimento di Scienza Applicata e Tecnologia del Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7954-0549","authenticated-orcid":false,"given":"Tatsuaki","family":"Wada","sequence":"additional","affiliation":[{"name":"Region of Electrical and Electronic Systems Engineering, Ibaraki University, 4-12-1 Nakanarusawa-cho, Hitachi 316-8511, Ibaraki, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2024,1,17]]},"reference":[{"key":"ref_1","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":"Physical A"},{"key":"ref_2","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_3","doi-asserted-by":"crossref","first-page":"036108","DOI":"10.1103\/PhysRevE.72.036108","article-title":"Statistical mechanics in the context of special relativity. II","volume":"72","author":"Kaniadakis","year":"2005","journal-title":"Phys. Rev. E"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/j.physleta.2005.11.056","article-title":"The H-theorem in \u03ba-statistics: Influence on the molecular chaos hypothesis","volume":"352","author":"Silva","year":"2006","journal-title":"Phys. Lett. A"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1016\/j.physa.2006.01.016","article-title":"Towards a relativistic statistical theory","volume":"365","author":"Kaniadakis","year":"2006","journal-title":"Physical A"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"137","DOI":"10.1016\/S0034-4877(10)80028-X","article-title":"A completeness criterion for Kaniadakis, Abe and two-parameter generalized statistical theories","volume":"66","author":"Oikonomou","year":"2010","journal-title":"Rep. Math. Phys."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"356","DOI":"10.1016\/j.physleta.2010.11.057","article-title":"Power-law tailed statistical distributions and Lorentz transformations","volume":"375","author":"Kaniadakis","year":"2011","journal-title":"Phys. Lett. A"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1250064","DOI":"10.1142\/S0217984912500649","article-title":"Physical meaning of the parameters in the two-parameter (\u03ba,\u03b6) generalized statistics","volume":"26","author":"Guo","year":"2012","journal-title":"Mod. Phys. Lett. B"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"052112","DOI":"10.1103\/PhysRevE.95.052112","article-title":"Composition law of \u03ba-entropy for statistically independent systems","volume":"95","author":"Kaniadakis","year":"2017","journal-title":"Phys. Rev. E"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Scarfone, A.M. (2022). Boltzmann configurational entropy revisited in the framework of generalized statistical mechanics. Entropy, 24.","DOI":"10.3390\/e24020140"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"138005","DOI":"10.1016\/j.physletb.2023.138005","article-title":"Is Kaniadakis \u03ba-generalized statistical mechanics general?","volume":"843","author":"Alves","year":"2023","journal-title":"Phys. Lett. B"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1016\/S0378-4371(02)01018-X","article-title":"Deformed exponentials and logarithms in generalized thermostatistics","volume":"316","author":"Naudts","year":"2002","journal-title":"Physical A"},{"key":"ref_13","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_14","doi-asserted-by":"crossref","first-page":"624","DOI":"10.3390\/e15020624","article-title":"Entropic forms and related algebras","volume":"15","author":"Scarfone","year":"2013","journal-title":"Entropy"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"1691","DOI":"10.1016\/j.physleta.2014.04.030","article-title":"Analysis of fractal groups of the type d-(m,r)-Cantor within the framework of Kaniadakis statistics","volume":"378","author":"Souza","year":"2014","journal-title":"Phys. Lett. A"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"2812","DOI":"10.3390\/e17052812","article-title":"On the \u03ba-deformed cyclic functions and the generalized Fourier series in the framework of the \u03ba-algebra","volume":"17","author":"Scarfone","year":"2015","journal-title":"Entropy"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"2776","DOI":"10.3390\/math10152776","article-title":"Lie Symmetries of the nonlinear Fokker-Planck equation based on weighted Kaniadakis entropy","volume":"10","author":"Hirica","year":"2022","journal-title":"Mathematics"},{"key":"ref_18","first-page":"35","article-title":"Bose-Einstein condensation in the framework of \u03ba-statistics","volume":"325","author":"Aliano","year":"2003","journal-title":"Physical A"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"126","DOI":"10.1016\/j.physa.2004.03.084","article-title":"Thermodynamic stabilities of the generalized Boltzmann entropies","volume":"340","author":"Wada","year":"2004","journal-title":"Physical A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"452","DOI":"10.1016\/j.physleta.2016.12.019","article-title":"Thermodynamic properties of the blackbody radiation: A Kaniadakis approach","volume":"381","author":"Lourek","year":"2017","journal-title":"Phys. Lett. A"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1140\/epjd\/s10053-023-00676-8","article-title":"Dust ion acoustic bi-soliton, soliton, and shock waves in unmagnetized plasma with Kaniadakis-distributed electrons in planar and nonplanar geometry","volume":"77","author":"Raut","year":"2023","journal-title":"Eur. Phys. J. D"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"1108","DOI":"10.1016\/j.asr.2022.08.062","article-title":"Gas-dynamic approach to the theory of non-linear ion-acoustic waves in plasma with Kaniadakis\u2019 distributed species","volume":"71","author":"Dubinov","year":"2023","journal-title":"Adv. Space Res."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"891","DOI":"10.1515\/zna-2021-0164","article-title":"Periodic and localized structures in dusty plasma with Kaniadakis distribution","volume":"76","author":"Khalid","year":"2021","journal-title":"Z. Naturforsch. A"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"A85","DOI":"10.1051\/0004-6361\/201323344","article-title":"A method to deconvolve stellar rotational velocities","volume":"565","author":"Rial","year":"2014","journal-title":"Astron. Astrophys."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"666","DOI":"10.1016\/j.physa.2012.10.022","article-title":"Non-Gaussian statistics, Maxwellian derivation and stellar polytropes","volume":"392","author":"Bento","year":"2013","journal-title":"Physical A"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"L48","DOI":"10.1088\/0004-637X\/696\/1\/L48","article-title":"Non-gaussian statistics and stellar rotational velocities of main-sequence field stars","volume":"696","author":"Carvalho","year":"2009","journal-title":"Astrophys. J."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1471","DOI":"10.1016\/j.net.2021.10.003","article-title":"A novel analytical solution of the deformed Doppler broadening function using the Kaniadakis distribution and the comparison of computational efficiencies with the numerical solution","volume":"54","author":"Martinez","year":"2022","journal-title":"Nucl. Eng. Technol."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Moradpour, H., Javaherian, M., Namvar, E., and Ziaie, A.H. (2022). Gamow temperature in Tsallis and Kaniadakis statistics. Entropy, 24.","DOI":"10.3390\/e24060797"},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"107914","DOI":"10.1016\/j.anucene.2020.107914","article-title":"Quasi-Maxwellian interference term functions","volume":"151","author":"Guedes","year":"2021","journal-title":"Ann. Nucl. Energy"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"106960","DOI":"10.1016\/j.anucene.2019.106960","article-title":"New analytical formulations for the Doppler broadening function and interference term based on Kaniadakis distributions","volume":"135","author":"Martinez","year":"2020","journal-title":"Ann. Nucl. Energy"},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"314","DOI":"10.1140\/epjc\/s10052-022-10285-1","article-title":"Modified Friedmann equations from Kaniadakis entropy and cosmological implications on baryogenesis and 7Li-abundance","volume":"82","author":"Luciano","year":"2022","journal-title":"Eur. Phys. J. C"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"137553","DOI":"10.1016\/j.physletb.2022.137553","article-title":"Modified cosmology from the thermodynamics of apparent horizon","volume":"835","author":"Nojiri","year":"2022","journal-title":"Phys. Lett. B"},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1712","DOI":"10.3390\/e24121712","article-title":"Gravity and cosmology in Kaniadakis statistics: Current status and future challenges","volume":"24","author":"Luciano","year":"2022","journal-title":"Entropy"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"102085","DOI":"10.1016\/j.newast.2023.102085","article-title":"Kaniadakis agegraphic dark energy","volume":"105","author":"Kumar","year":"2024","journal-title":"New Astron."},{"key":"ref_35","doi-asserted-by":"crossref","unstructured":"Azhar, N., Rani, S., and Jawad, A. (2023). Cosmic and thermodynamic consequences of Kaniadakis holographic dark energy in Brans\u2013Dicke gravity. Entropy, 25.","DOI":"10.3390\/e25040576"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"2350074","DOI":"10.1142\/S0219887823500743","article-title":"Statefinder hierarchy of Kaniadakis holographic dark energy with composite null diagnostic","volume":"20","author":"Singh","year":"2023","journal-title":"Int. J. Geom. Methods Mod. Phys."},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"137189","DOI":"10.1016\/j.physletb.2022.137189","article-title":"Early and late universe holographic cosmology from a new generalized entropy","volume":"831","author":"Nojiri","year":"2022","journal-title":"Phys. Lett. B"},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2250152","DOI":"10.1142\/S0217732322501528","article-title":"Kaniadakis holographic dark energy in Brans-Dicke cosmology","volume":"37","author":"Ghaffari","year":"2022","journal-title":"Mod. Phys. Lett. A"},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"2250078","DOI":"10.1142\/S021827182250078X","article-title":"Cosmographic and thermodynamic analysis of Kaniadakis holographic dark energy","volume":"31","author":"Rani","year":"2022","journal-title":"Int. J. Mod. Phys. D"},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"7213","DOI":"10.3390\/e17107213","article-title":"Dualistic Hessian structures among the thermodynamic potentials in the \u03ba-hermostatistics","volume":"17","author":"Wada","year":"2015","journal-title":"Entropy"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"375009","DOI":"10.1088\/1751-8121\/aba98a","article-title":"Thermodynamic geometry of Kaniadakis statistics","volume":"53","author":"Mohammadzadeh","year":"2020","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_42","doi-asserted-by":"crossref","unstructured":"Scarfone, A.M., Matsuzoe, H., and Wada, T. (2018). Information geometry of \u03ba-exponential families: Dually-flat, Hessian and Legendre structures. Entropy, 20.","DOI":"10.3390\/e20060436"},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"1204","DOI":"10.3390\/e17031204","article-title":"Information geometry on the \u03ba-thermostatistics","volume":"17","author":"Wada","year":"2015","journal-title":"Entropy"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"1080","DOI":"10.1016\/j.jmaa.2015.05.065","article-title":"A statistical manifold modeled on Orlicz spaces using Kaniadakis \u03ba-exponential models","volume":"431","year":"2015","journal-title":"J. Math. Anal."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"275002","DOI":"10.1088\/1751-8113\/47\/27\/275002","article-title":"Legendre structure of kappa-thermostatistics revisited in the framework of information geometry","volume":"47","author":"Scarfone","year":"2014","journal-title":"J. Phys. A Math. Theor."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1140\/epjb\/e2009-00154-y","article-title":"\u03ba-Exponential models from the geometrical viewpoint","volume":"70","author":"Pistone","year":"2009","journal-title":"Eur. Phys. J. B"},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"022112","DOI":"10.1103\/PhysRevE.99.022112","article-title":"Analysis of human DNA through power-law statistics","volume":"99","author":"Costa","year":"2019","journal-title":"Phys. Rev. E"},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"38004","DOI":"10.1209\/0295-5075\/108\/38004","article-title":"A \u03ba-statistical analysis of the Y-chromosome","volume":"108","author":"Souza","year":"2014","journal-title":"EPL"},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"842","DOI":"10.1016\/j.physleta.2013.01.032","article-title":"Maximum entropy principle for Kaniadakis statistics and networks","volume":"377","author":"Moreira","year":"2013","journal-title":"Phys. Lett. A"},{"key":"ref_50","doi-asserted-by":"crossref","first-page":"360","DOI":"10.1016\/j.physa.2014.04.009","article-title":"A \u03ba-deformed model of growing complex networks with fitness","volume":"407","author":"Stella","year":"2014","journal-title":"Physical A"},{"key":"ref_51","doi-asserted-by":"crossref","unstructured":"Clementi, F., Gallegati, M., and Kaniadakis, G. (2009). A \u03ba-generalized statistical mechanics approach to income analysis. J. Stat. Mech., P02037.","DOI":"10.1088\/1742-5468\/2009\/02\/P02037"},{"key":"ref_52","doi-asserted-by":"crossref","first-page":"261","DOI":"10.1140\/epjb\/e2012-30239-3","article-title":"Exploiting the flexibility of a family of models for taxation and redistribution","volume":"85","author":"Bertotti","year":"2012","journal-title":"Eur. Phys. J. B"},{"key":"ref_53","doi-asserted-by":"crossref","first-page":"29","DOI":"10.1016\/j.physleta.2015.09.004","article-title":"Common origin of power-law tails in income distributions and relativistic gases","volume":"380","author":"Modanese","year":"2016","journal-title":"Phys. Lett. A"},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1250038","DOI":"10.1142\/S0219024912500380","article-title":"The minimal \u03ba-entropy martingale measure","volume":"15","author":"Trivellato","year":"2012","journal-title":"Int. J. Theor. Appl. Financ."},{"key":"ref_55","doi-asserted-by":"crossref","first-page":"3471","DOI":"10.3390\/e15093471","article-title":"Deformed exponentials and applications to finance","volume":"15","author":"Trivellato","year":"2013","journal-title":"Entropy"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"3051","DOI":"10.1016\/j.physa.2013.03.015","article-title":"A maximum (non-extensive) entropy approach to equity options bid-ask spread","volume":"392","author":"Tapiero","year":"2013","journal-title":"Physical A"},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"45","DOI":"10.1143\/PTPS.162.45","article-title":"Canonical partition function for anomalous systems described by the \u03ba-entropy","volume":"162","author":"Scarfone","year":"2006","journal-title":"Prog. Theor. Phys. Suppl."},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"404","DOI":"10.1016\/j.physleta.2006.02.057","article-title":"Thermal and mechanical equilibrium among weakly interacting systems in generalized thermostatistics framework","volume":"355","author":"Scarfone","year":"2006","journal-title":"Phys. Lett. A"},{"key":"ref_59","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":"Landsberge","year":"1998","journal-title":"Phys. Lett. A"},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"21001","DOI":"10.1209\/0295-5075\/ad0764","article-title":"Entropy defect: Algebra and thermodynamics","volume":"144","author":"Livadiotis","year":"2023","journal-title":"Europhys. Lett."},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"9033","DOI":"10.1038\/s41598-023-36080-w","article-title":"Entropy defect in thermodynamics","volume":"13","author":"Livadiotis","year":"2023","journal-title":"Sci. Rep."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"026123","DOI":"10.1103\/PhysRevE.72.026123","article-title":"Thermodynamic equilibrium and its stability for microcanonical systems described by the Sharma-Taneja-Mittal entropy","volume":"72","author":"Scarfone","year":"2005","journal-title":"Phys. Rev. E"}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/1\/77\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T13:48:32Z","timestamp":1760104112000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/26\/1\/77"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,17]]},"references-count":62,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2024,1]]}},"alternative-id":["e26010077"],"URL":"https:\/\/doi.org\/10.3390\/e26010077","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,1,17]]}}}