Title: Geometric calculations on probability manifolds from reciprocal relations in Master equations Show Chinese title

Title: 概率流形上的几何计算来自主方程中的倒易关系

Abstract: Onsager reciprocal relations are widely used to model irreversible processes in complex systems in physics. Recently, it has been studied that Onsager principles for master equations on finite states introduce a class of Riemannian metrics in a probability simplex, named probability manifolds. We refer to these manifolds as finite-state generalized Wasserstein-$2$ spaces. In this paper, we study geometric calculations in probability manifolds, deriving the Levi-Civita connection, gradient, Hessian, and parallel transport, as well as Riemannian and sectional curvatures. We present two examples of geometric quantities in probability manifolds. These include Levi-Civita connections from the chemical monomolecular triangle reaction and sectional, Ricci and scalar curvatures in Wasserstein space on a simplex set with a three-point lattice. Show Chinese abstract

Abstract: 昂萨格互易关系被广泛用于模拟物理中复杂系统中的不可逆过程。 最近的研究表明，有限状态主方程的昂萨格原理引入了一类概率单纯形上的黎曼度量，称为概率流形。 我们把这些流形称为有限状态广义昂萨格-$2$空间。 在本文中，我们研究了概率流形中的几何计算，推导了黎曼联络、梯度、Hessian 和平行移动，以及黎曼曲率和截面曲率。 我们给出了概率流形中两个几何量的例子。 这些包括从化学单分子三角反应得到的黎曼联络，以及三点格点上的单纯形集上Wasserstein空间中的截面、Ricci和数量曲率。