Geometric calculations on probability manifolds from reciprocal relations in Master equations

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.