On Liapunov functions for single-variable reacting systems displaced from equilibrium

Abstract

We prove that Liapunov functions for a single reactive intermediate evolving toward a nonequilibrium steady state can be obtained from a global potential φ. We consider reactions occurring in a chamber containing a reactant, the intermediate, and a product. Reservoirs connected to the chamber serve to hold the reactant and product concentrations constant, in nonequilibrium proportions. The Liapunov property of φ is significant because of the role it plays in the thermodynamic and stochastic analysis of nonequilibrium systems: φ is defined in terms of the reactive flux to produce the intermediate and the flux to remove the intermediate. The derivative of φ with respect to the concentration of the intermediate yields an effective chemical driving force that is specific to the intermediate, and its time derivative yields a species-specific component of the dissipation that is minimized at steady states. These results hold both near to equilibrium and far from equilibrium for systems with one intermediate, independent of the number of steady states. Local Liapunov functions are also provided by the ‘‘excess dissipation,’’ the second variation in the entropy or in the Helmholtz free energy for the reaction chamber, and quadratic functions introduced in Keizer’s fluctuation–dissipation theory. Linearization of the force and flux expansions for nonequilibrium systems yields an idealized model in which the dissipation decreases monotonically in time and thus provides a Liapunov function for evolution to steady states. This result does not hold for a chemical system approaching a steady state with an arbitrarily small, but macroscopic displacement from equilibrium, even though the series expansions of the reactive fluxes and conjugate thermodynamic forces are closely approximated by truncation at the linear terms. There are always small regions in the immediate vicinity of nonequilibrium steady states where the dissipation increases in time while the system relaxes.