Revisiting the Glansdorff-Prigogine criterion for stability within irreversible thermodynamics

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.