Thermodynamic and stochastic theory for nonequilibrium systems with multiple reactive intermediates: The concept and role of excess work

Abstract

We continue our development of a global thermodynamic and stochastic theory of open chemical systems far from equilibrium with an analysis of a broad class of isothermal, multicomponent reaction mechanisms with multiple steady states, studied under the assumption of local equilibrium. We generalize species-specific affinities of reaction intermediates, obtained in prior work for nonautocatalytic reaction mechanisms, to autocatalytic kinetics and define with these affinities an excess free energy differential Fφ. The quantity Fφ is the difference between the work required to reverse a spontaneous concentration change and the work available when the same concentration change is imposed on a system in a reference steady state. The integral of Fφ is in general not a state function; in contrast, the function φdet obtained by integrating Fφ along deterministic kinetic trajectories is a state function, as well as an identifiable term in the time-integrated dissipation. Unlike the total integrated dissipation, φdet remains finite during the infinite duration of the system’s relaxation to a steady state and hence φdet can be used to characterize that process. The variational relation δφ≥0 is shown to be a necessary and sufficient thermodynamic criterion for a stable steady state in terms of the excess work of displacement of the intermediates and φdet is a Liapunov function in the domain of attraction of such steady states. Based on these results and earlier work with nonautocatalytic and equilibrating systems, we hypothesize that the stationary distribution of the master equation may be obtained in the form Ps=𝒩 exp(−φdet/kT) and provide an analytical argument for this form for macroscopic systems. This generalizes the Einstein fluctuation formula to multivariable systems with multiple steady states, far from equilibrium. In the following article, the utility of the approximation to Ps for systems with single or multiple intermediates, and single or multiple steady states is shown by comparison with numerical solutions of the master equation.