{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,3]],"date-time":"2026-06-03T05:13:25Z","timestamp":1780463605466,"version":"3.54.1"},"reference-count":75,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T00:00:00Z","timestamp":1774310400000},"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>In this paper, we clarify several issues concerning the abstract geometrical formulation of thermodynamics on non-compact symmetric spaces U\/H that are the mathematical model of hidden layers in the new paradigm of Cartan Neural Networks. We introduce a clear-cut distinction between the generalized thermodynamics associated with Integrable Dynamical Systems and the challenging proposal of Gibbs probability distributions on U\/H provided by generalized thermodynamics \u00e0 la Souriau. Our main result is the proof that U\/H.s supporting such Gibbs distributions are only the K\u00e4hler ones. Furthermore, for the latter, we solve the problem of determining the space of temperatures, namely, of Lie algebra elements for which the partition function converges. The space of generalized temperatures is the orbit under the adjoint action of U of a positivity domain in the Cartan subalgebra Cc\u2282H of the maximal compact subalgebra H\u2282U. We illustrate how our explicit constructions for the Poincar\u00e9 and Siegel planes might be extended to the whole class of Calabi\u2013Vesentini manifolds utilizing Paint Group symmetry. Furthermore, we claim that Rao\u2019s, Chentsov\u2019s, and Amari\u2019s Information Geometry and the thermodynamical geometry of Ruppeiner and Lychagin are the very same thing. In particular, we provide an explicit study of thermodynamical geometry for the Poincar\u00e9 plane. The key feature of the Gibbs probability distributions in this setup is their covariance under the entire group of symmetries U. The partition function is invariant against U transformations, and the set of its arguments, namely the generalized temperatures, can always be reduced to a minimal set whose cardinality is equal to the rank of the compact denominator group H\u2282U.<\/jats:p>","DOI":"10.3390\/e28040365","type":"journal-article","created":{"date-parts":[[2026,3,25]],"date-time":"2026-03-25T10:07:13Z","timestamp":1774433233000},"page":"365","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Thermodynamics \u00e0 la Souriau on K\u00e4hler Non-Compact Symmetric Spaces for Cartan Neural Networks"],"prefix":"10.3390","volume":"28","author":[{"given":"Pietro G.","family":"Fr\u00e9","sequence":"first","affiliation":[{"name":"Dipartimento di Fisica, Universit\u00e0 di Torino, Via P. Giuria 1, I-10125 Torino, Italy"},{"name":"Additati & Partners Consulting s.r.l.,Via Filippo Pacini 36, I-51100 Pistoia, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9775-2115","authenticated-orcid":false,"given":"Alexander S.","family":"Sorin","sequence":"additional","affiliation":[{"name":"Center for Quantum Science and Technology, Tel-Aviv University, Tel Aviv 69978, Israel"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2510-9507","authenticated-orcid":false,"given":"Mario","family":"Trigiante","sequence":"additional","affiliation":[{"name":"Dipartimento DISAT, Politecnico di Torino, Corso Duca degli Abruzzi 24, I-10129 Torino, Italy"},{"name":"Istituto Nazionale di Fisica Nucleare (INFN), Sezione di Torino, I-10125 Torino, Italy"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2026,3,24]]},"reference":[{"key":"ref_1","unstructured":"Bruzzo, U., Fr\u00e9, P.G., and Trigiante, M. 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