An analog of the Boltzmann H-theorem (a Liapunov function) for systems of coupled chemical reactions

Abstract

A proof is given that the function Σc ́(k)ln c(k) c(k), where c(k) is the volume concentration of the kth species and c(k) its value at the stationary state, is always negative (and zero at the stationary state) for homogeneous chemical reaction systems, regardless of whether or not detailed balance holds at the stationary state. The proof depends on the assumption that all reactions may be decomposed into elementary, single-step reactions which obey "mass action" rate equations. The function is the time derivative of a (positive definite) Liapunov function, Σ{ c(k)lnc(k) c(k)-c(k)+c(k)}, which establishes that such systems will always approach their respective stationary states, and that these stationary states are stable. It follows that such systems can never exhibit limit cycle behavior. The results apply to systems open to matter and energy fluxes, as well as to closed systems, so long as the concentrations (or intensities) of the material (or energetic) species to which the system is permeable do not change with time. For systems in which the stationary states are true equilibria, the Liapunov function reduces approximately to (1/RT times) the excess free energy per unit volume. © 1967.