Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation

TL;DR

This paper applies information geometry to the spatially homogeneous Boltzmann equation, using exponential Orlicz spaces to analyze the operator and prove fundamental properties like the H-theorem in a geometric framework.

Contribution

It introduces a geometric approach to the Boltzmann equation using infinite-dimensional information geometry and extends the analysis to include divergences like Hyv"arinen divergence.

Findings

Geometric formulation of the Boltzmann operator as elementary operations

Proof of the H-theorem within the geometric framework

Extension to Orlicz-Sobolev spaces for divergence analysis

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 $\mathcal E$ 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 $\mathcal E$ is a set of positive densities with finite relative entropy with respect to the Maxwell density. More precisely, we analyse 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 Hyv\"arinen divergence. This requires to generalise our approach to Orlicz-Sobolev spaces to include derivatives.%