Revisiting the Glansdorff-Prigogine criterion for stability within irreversible thermodynamics

Glansdorff and Prigogine proposed a decomposition of the entropy production rate, which has more recently been rediscovered in various constructions of nonequilibrium statistical mechanics. 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 δ2EP and they suggested as sufficient stability criterion for a nonequilibrium macroscopic condition that δ2EP be positive. In joint work with Karel Netočný we find that their excess entropy production rate is itself the time-derivative of a deformed free energy functional [18]. The positivity of the excess δ2EP, 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 δ2EP immediately implies a Clausius (in)equality for the excess heat. We have proposed a nonperturbative version [16] using a modified excess entropy production that we also review. A final and related question concerns the operational meaning of fluctuation functionals, nonequilibrium free energies, and how they make their entrée in irreversible thermodynamics.