On the stability of mass action reactions in open systems

Abstract

We consider the stability of chemical systems whose evolution is governed by mass action reactions. We show, with no reference to the second law of thermodynamics, that the stability of equilibrium in a closed system follows from the fact that one can express the kinetic matrix for the linearized equations near equilibrium in the form of a matrix times its transpose. This is a necessary and sufficient condition that the matrix be positive indefinite (some of the eigenvalues are necessarily zero due to the existence of conservation relations in a closed system). We then extend this approach to open systems where one finds that a component of the kinetic matrix cannot be expressed as the product of a matrix times its transpose. It is this matrix, having a simple expression in terms of stoichiometric matrices, thus expressing the basic mechanisms of the chemistry, that is potentially responsible for instabilities, oscillations, and chaos far from equilibrium. Examination of this matrix leads naturally to autocatalytic reactions as candidates for interesting behavior far from equilibrium. We conclude that all of the interesting features of mass action kinetics can be determined directly from the kinetic equations and the principle of detailed balance with no additional constraints coming uniquely from thermodynamics.