Representations and divergences in the space of probability measures and stochastic thermodynamics

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 Gibbsian canonical distribution yields many results in classical thermomechanics. An affine structure based on the canonical representation and two divergences are introduced in the space of probability measures. It is shown that thermodynamic work, as a conditional expectation, is indicative of the RN derivative between two energy representations being singular. The entropy divergence and the heat divergence yield respectively a Massieu–Planck potential based and a generalized Carnot inequalities.