Revisiting the Glansdorff-Prigogine criterion for stability within irreversible thermodynamics

TL;DR

This paper revisits the Glansdorff-Prigogine stability criterion in irreversible thermodynamics, showing that the excess entropy production rate relates to a local free energy functional whose monotonicity indicates stability.

Contribution

It demonstrates that the excess entropy production rate is the time derivative of a local free energy functional, linking stability criteria to the monotonicity of this functional in nonlinear diffusions.

Findings

Excess entropy production rate equals the time derivative of a local free energy.

Positivity of excess entropy production rate implies monotonic decrease of free energy.

Relation established between excess entropy production and Clausius heat inequalities.

Abstract

Glansdorff and Prigogine (1970) proposed a decomposition of the entropy production rate, which today is mostly known for Markov processes as the Hatano-Sasa approach. Their context was irreversible thermodynamics which, while ignoring fluctuations, still allows a somewhat broader treatment than the one based on the Master or Fokker-Planck equation. Glansdorff and Prigogine were the first to introduce a notion of excess entropy production rate $\delta^2$EP and they suggested as sufficient stability criterion for a nonequilibrium macroscopic condition that $\delta^2$EP be positive. We find for nonlinear diffusions that their excess entropy production rate is itself the time-derivative of a local free energy which is the close-to-equilibrium functional governing macroscopic fluctuations. The positivity of the excess $\delta^2$EP, for which we state a simple sufficient condition, is therefore equivalent with the monotonicity in time of that functional in the relaxation to steady nonequilibrium. There also appears a relation with recent extensions of the Clausius heat theorem close-to-equilibrium. The positivity of $\delta^2$EP immediately implies a Clausius (in)equality for the excess heat. A final and related question concerns the operational meaning of fluctuation functionals, nonequilibrium free energies, and how they make their entr\'ee in irreversible thermodynamics.