# Representations and Divergences in the Space of Probability Measures and Stochastic Thermodynamics

## TL;DR

This paper explores the mathematical structure of probability measures and divergences, linking them to thermodynamic concepts like entropy and free energy, and applies this to classical thermodynamics and nonequilibrium systems.

## Contribution

It introduces a formalism connecting probability measure divergences with thermodynamic quantities, providing new insights into entropy balance and energy representations.

## Key findings

- Derived a simple equation linking measures with densities to entropy balance.
- Introduced divergences leading to generalized Carnot inequalities.
- Applied formalism to Gibbs distribution for classical thermomechanics.

## 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 represenation and two divergences are introduced in the space of probability measures. It is shown that thermodynamic work, as a conditional 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.