{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T01:31:07Z","timestamp":1760059867586,"version":"build-2065373602"},"reference-count":13,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2025,7,17]],"date-time":"2025-07-17T00:00:00Z","timestamp":1752710400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>The thermal Doppler broadening of spectral profiles for particle populations in the absence or presence of potential fields can be described by kappa distributions. The kappa distribution provides a replacement for the Maxwell\u2013Boltzmann distribution, which can be considered as a generalization for describing systems characterized by local correlations among their particles, as found in space and astrophysical plasmas. This paper presents all special cases of kappa distributions as members of a general pathway family of densities introduced by Mathai. The aim of the present paper is to bring to attention the application of various forms of the kappa distribution, its various special cases and its generalizations, which, in scalar-variable and multivariate situations, belong to a general family of distributions known as Mathai\u2019s pathway models, comprising three different families of functions, namely the generalized type-1 beta, type-2 beta and gamma families. Through one parameter, known as the pathway parameter, one will be able to reach all the three families of functions and the stages of transitioning from one family to another. After pointing out the connection of multivariate (vector-variate) kappa distributions to the multivariate pathway model, the multivariate kappa distribution is extended to the real matrix-variate case by working out the various forms and by evaluating the normalizing constants of the various forms of the matrix-variate case explicitly. It is also pointed out that the pathway models are available for the scalar, vector and rectangular matrix-variate cases in the real domain as well as in the complex domain.<\/jats:p>","DOI":"10.3390\/axioms14070539","type":"journal-article","created":{"date-parts":[[2025,7,17]],"date-time":"2025-07-17T11:25:17Z","timestamp":1752751517000},"page":"539","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Extended d-D Kappa Distribution"],"prefix":"10.3390","volume":"14","author":[{"given":"Arak M.","family":"Mathai","sequence":"first","affiliation":[{"name":"Department of Mathematics and Statistics, McGill University, Montreal, QC H3A 2K6, Canada"}]},{"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 Centre, A-1400 Vienna, Austria"}]}],"member":"1968","published-online":{"date-parts":[[2025,7,17]]},"reference":[{"key":"ref_1","unstructured":"Livadiotis, G. (2017). Kappa Distribution: Theory and Applications in Plasmas, Elsevier. [1st ed.]."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"153","DOI":"10.1007\/s11207-010-9640-2","article-title":"Kappa distributions: Theory and applications in space plasmas","volume":"267","author":"Pierrard","year":"2010","journal-title":"Sol. Phys."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1607","DOI":"10.1002\/2014JA020825","article-title":"Statistical background and properties of kappa distributions in space plasmas","volume":"120","author":"Livadiotis","year":"2015","journal-title":"J. Geophys. Res."},{"key":"ref_4","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_5","unstructured":"Binsack, J.H. (1966). Plasma Studies with the IMP-2 Satellite. 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Rep., 13.","DOI":"10.1038\/s41598-023-36080-w"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"317","DOI":"10.1016\/j.laa.2004.09.022","article-title":"A pathway to matrix-variate gamma and normal densities","volume":"396","author":"Mathai","year":"2005","journal-title":"Linear Algebra Its Appl."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/S0378-4371(03)00019-0","article-title":"Superstatistics","volume":"322","author":"Beck","year":"2003","journal-title":"Physica A"},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Mathai, A.M. (1997). Jacobians of Matrix Transformations and Functions of Matrix Argument, World Scientific Publishing.","DOI":"10.1142\/3438"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Mathai, A.M., Provost, S.B., and Haubold, H.J. (2022). Multivariate Statistical Analysis in the Real and Complex Domains, Springer Nature.","DOI":"10.1007\/978-3-030-95864-0"}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/539\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,9]],"date-time":"2025-10-09T18:11:18Z","timestamp":1760033478000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/7\/539"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,7,17]]},"references-count":13,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2025,7]]}},"alternative-id":["axioms14070539"],"URL":"https:\/\/doi.org\/10.3390\/axioms14070539","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,7,17]]}}}