Representations and Divergences in the Space of Probability Measures and Stochastic Thermodynamics Liu Honga,b, Hong Qiana,∗, Lowell F. Thompsona aDepartment of Applied Mathematics, University of Washington, Seattle, WA 98195-3925, U.S.A. bZhou Pei-Yuan Center for Applied Mathematics, Tsinghua University, Beijing, 100084, P.R.C. Abstract Radon-Nikodym (RN) derivative between two measures arises naturally in the affine structure of the space of probability measures with densities. Entropy, free energy, relative entropy, and entropy production as mathematical concepts associated with RN derivatives are introduced. We identify a simple equation that connects two measures with densities as a possible mathematical basis of the entropy balance equation that is central in nonequilibrium thermodynamics. Application of this formalism to Gibb- sian canonical distribution yields many results in classical thermomechanics. An affine structure based on the canonical represenation and two divergences are introduced in the space of probability measures. It is shown that thermodynamic work, as a condi- tional expectation, is indictive of the RN derivative between two energy represenations being singular. The entropy divergence and the heat divergence yield respectively a Massieu-Planck potential based and a generalized Carnot inequalities. Keywords: Radon-Nikodym derivative, affine structure, space of probability measures, heat divergence 2010 MSC: 60-xx, 80-xx, 82-xx 1. Introduction arXiv:1902.09766v2 [cond-mat.stat-mech] 14 Apr 2020 A subtle distinction exists between the prevalent approach to stochastic processes in traditional applied mathematics and the physicist’s perspective on stochastic dy- ∗Corresponding author Email addresses:
[email protected] (Liu Hong),
[email protected] (Hong Qian),
[email protected] (Lowell F.