{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,29]],"date-time":"2026-07-29T03:29:07Z","timestamp":1785295747512,"version":"3.55.0"},"reference-count":56,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2014,7,30]],"date-time":"2014-07-30T00:00:00Z","timestamp":1406678400000},"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>The paper studies the \u201cLagrangian temperature\u201d defined through the entropy maximization in the canonical ensemble, which is the negative inverse Lagrangian multiplier corresponding to the constraint of internal energy. The Lagrangian temperature is derived for systems out of thermal equilibrium described by kappa distributions such as space plasmas. The physical meaning of temperature is manifested by the equivalency of two different definitions, that is, through Maxwell\u2019s kinetic theory and Clausius\u2019 thermodynamics. The equivalency of the two definitions is true either for systems at thermal equilibrium described by Maxwell distributions or for systems out of thermal equilibrium described by kappa distributions, and gives the meaning of the actual temperature, that is, the real or measured temperature. However, the third definition, that of the Lagrangian temperature, coincides with the primary two definitions only at thermal equilibrium, and thus, in the general case of systems out of thermal equilibrium, it does not represent the actual temperature, but it is rather a function of this. The paper derives and examines the exact expression and physical meaning of the Lagrangian temperature, showing that it has essentially different content to what is commonly thought. This is achieved by: (i) maximizing the entropy in the continuous description of energy within the general framework of non-extensive statistical mechanics, (ii) using the concept of the  \u201cN-particle\u201d kappa distribution, which is governed by a special kappa index that is invariant of the degrees of freedom and the number of particles, and (iii) determining the appropriate scales of length and speed involved in the phase-space microstates. Finally, the paper demonstrates the behavior of the Lagrangian against the actual temperature in various datasets of space plasmas.<\/jats:p>","DOI":"10.3390\/e16084290","type":"journal-article","created":{"date-parts":[[2014,7,30]],"date-time":"2014-07-30T09:19:07Z","timestamp":1406711947000},"page":"4290-4308","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":53,"title":["\u201cLagrangian Temperature\u201d: Derivation and Physical Meaning for Systems Described by Kappa Distributions"],"prefix":"10.3390","volume":"16","author":[{"given":"George","family":"Livadiotis","sequence":"first","affiliation":[{"name":"Southwest Research Institute, 6220 Culebra Rd, San Antonio, TX-78238, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2014,7,30]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"183","DOI":"10.1007\/s11214-013-9982-9","article-title":"Understanding Kappa Distributions: A Toolbox for Space Science and Astrophysics","volume":"175","author":"Livadiotis","year":"2013","journal-title":"Space Sci. 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