The existence of straight-line paths, invariant vectors, and invariant tensors characterizing nonequilibrium state distributions during chemical reactions

We have deduced the existence of straight-line reaction paths for various reaction systems. The general conditions for their existence in isomerization systems, dissociation-recombination reactions, and bimolecular reactions are discussed. We have also derived the invariant quantity that characterizes the internal-state distributions of either reactants or products when the phenomenological rate constant is steady. An invariant vector is obtained for first-order reactions, and an invariant tensor is obtained for second-order reactions. Analytic expressions for these invariant quantities are given for several cases. The application and usefulness of an invariant vector is illustrated for an example reaction by using it to calculate the internal-state population distributions at any time during the forward or reverse reactions from the internal-state distributions at any one time during either. These calculations check satisfactorily against the distributions obtained by numerical solution of the master equation.