{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,19]],"date-time":"2025-11-19T14:51:44Z","timestamp":1763563904316,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2015,6,19]],"date-time":"2015-06-19T00:00:00Z","timestamp":1434672000000},"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>Information Geometry generalizes to infinite dimension by modeling the tangent  space of the relevant manifold of probability densities with exponential Orlicz spaces.  We review here several properties of the exponential manifold on a suitable set \u0190 of  mutually absolutely continuous densities. We study in particular the fine properties of the  Kullback-Liebler divergence in this context. We also show that this setting is well-suited  for the study of the spatially homogeneous Boltzmann equation if \u0190 is a set of positive  densities with finite relative entropy with respect to the Maxwell density. More precisely,  we analyze the Boltzmann operator in the geometric setting from the point of its Maxwell\u2019s  weak form as a composition of elementary operations in the exponential manifold, namely  tensor product, conditioning, marginalization and we prove in a geometric way the basic  facts, i.e., the H-theorem. We also illustrate the robustness of our method by discussing,  besides the Kullback-Leibler divergence, also the property of Hyv\u00e4rinen divergence. This  requires us to generalize our approach to Orlicz\u2013Sobolev spaces to include derivatives.<\/jats:p>","DOI":"10.3390\/e17064323","type":"journal-article","created":{"date-parts":[[2015,6,19]],"date-time":"2015-06-19T10:23:33Z","timestamp":1434709413000},"page":"4323-4363","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation"],"prefix":"10.3390","volume":"17","author":[{"given":"Bertrand","family":"Lods","sequence":"first","affiliation":[{"name":"Department of Economics and Statistics, Universit\u00e0 di Torino & Collegio Carlo Alberto, Corso Unione Sovietica, 218\/bis, 10134 Torino, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Giovanni","family":"Pistone","sequence":"additional","affiliation":[{"name":"Castro Statistics, Collegio Carlo Alberto, Via Real Collegio 30, 10024 Moncalieri, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,6,19]]},"reference":[{"key":"ref_1","unstructured":"Amari, S.i., and Nagaoka, H. 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