Stochastic Models for Second-Order Chemical Reaction Kinetics. The Equilibrium State

Stochastic models for a number of reversible chemical reactions involving one or two second-order steps are examined and exact solutions which involve standard orthogonal polynomials are found for the equilibrium states. The expected values and variances of the numbers of molecules present in the systems at equilibrium are expressed in terms of these polynomials. Comparison of the stochastic mean value and the classical deterministic expression, shows that they are equivalent for most systems of physical interest, that is when large numbers of molecules are present. The relative fluctuation is found to be O (N−½), where N is of the order of the number of molecules present in the system, in agreement with the results of equilibrium statistical mechanics.