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Entropy 2015, 17(6), 4323-4363; doi:10.3390/e17064323

Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation

1
Department of Economics and Statistics, Università di Torino & Collegio Carlo Alberto, Corso Unione Sovietica, 218/bis, 10134 Torino, Italy
2
Castro Statistics, Collegio Carlo Alberto, Via Real Collegio 30, 10024 Moncalieri, Italy
*
Author to whom correspondence should be addressed.
Academic Editor: Giorgio Kaniadakis
Received: 15 February 2015 / Revised: 12 June 2015 / Accepted: 16 June 2015 / Published: 19 June 2015
(This article belongs to the Special Issue Entropic Aspects in Statistical Physics of Complex Systems)
View Full-Text   |   Download PDF [424 KB, uploaded 19 June 2015]   |  

Abstract

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 Ɛ 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 Ɛ 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’s 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 Hyvärinen divergence. This requires us to generalize our approach to Orlicz–Sobolev spaces to include derivatives. View Full-Text
Keywords: information geometry; Orlicz space; spatially homogeneous Boltzmann equation; Kullback-Leibler divergence; Hyvärinen divergence information geometry; Orlicz space; spatially homogeneous Boltzmann equation; Kullback-Leibler divergence; Hyvärinen divergence
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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Lods, B.; Pistone, G. Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation. Entropy 2015, 17, 4323-4363.

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