Stochastic Free Energies, Conditional Probability and Legendre Transform for Ensemble Change

TL;DR

This paper develops a mathematical framework connecting stochastic free energies, conditional probabilities, and Legendre transforms to describe ensemble changes in thermodynamics, applicable to Markov processes with or without detailed balance.

Contribution

It derives the Legendre transform from stochastic thermodynamics and relates free energies of fluctuating and fixed ensembles through relative entropy and thermodynamic limits.

Findings

Derived the Legendre transform from stochastic free energies.

Established the relation between fluctuating and fixed ensemble free energies.

Applied the theory to ideal gases and microcanonical systems.

Abstract

This work extends a recently developed mathematical theory of thermodynamics for Markov processes with, and more importantly, without detailed balance. We show that the Legendre transform in connection to ensemble changes in Gibbs' statistical mechanics can be derived from the stochastic theory. We consider the joint probability $p_{XY}$ of two random variables X and Y and the conditional probability $p_{X|Y=y^*}$, with $y^*=<Y>$ according to $p_{XY}$. The stochastic free energies of the XY system (fluctuating Y ensemble) and the $X|Y$ system (fixed Y ensemble) are related by the chain rule for relative entropy. In the thermodynamic limit, defined as $V,Y\to\infty$ (where one assumes Y as an extensive quantity, while V denotes the system size parameter), the marginal probability obeys $p_Y(y)\to \exp(-V\phi(z))$ with $z=y/V$. A conjugate variable $\mu=-\phi(z)/z$ naturally emerges from this result. The stochastic free energies of the fluctuating and fixed ensembles are then related by $F_{XY}=F_{X|Y=y^*}-\mu y^*$, with $\mu=\partial F_{X|Y=y}/\partial y$. The time evolutions for the two free energies are the same: $d[F_{XY}(t)]/dt=\partial [F_{X|Y=y^*(t)}(t)]/ \partial t$. This mathematical theory is applied to systems with fixed and fluctuating number of identical independent random variables (idea gas), as well as to microcanonical systems with uniform stationary probability.