空间齐次 Boltzmann 方程的信息几何形式体系
Information Geometry Formalism for the Spatially Homogeneous Boltzmann Equation
AI总结：
本文用指数 Orlicz 空间构建无限维信息几何框架，将空间齐次 Boltzmann 算子表示为张量积、条件化和边缘化，并几何证明 H 定理。
摘要：
信息几何通过使用指数 Orlicz 空间对相关概率密度流形的切空间进行建模，推广到了无限维情形。我们在此回顾指数流形在一个由相互绝对连续密度构成的适当集合 $\mathcal E$ 上的若干性质。我们特别研究了此背景下 Kullback–Liebler 散度的精细性质。我们还表明，当 $\mathcal E$ 是一组相对于 Maxwell 密度具有有限相对熵的正密度集合时，这一框架非常适合研究空间齐次 Boltzmann 方程。更具体地说，我们从 Boltzmann 算子的 Maxwell 弱形式出发，在几何框架中对其进行分析，将其表示为指数流形中基本运算的复合，即张量积、条件化和边缘化，并以几何方式证明了基本事实，即 H 定理。除 Kullback–Leibler 散度外，我们还通过讨论 Hyvärinen 散度的性质来说明方法的稳健性。这需要将我们的方法推广到包含导数的 Orlicz–Sobolev 空间。
英文摘要：
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 Hyvärinen divergence. This requires to generalise our approach to Orlicz-Sobolev spaces to include derivatives.%
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