The sum over histories representation for chemical kinetics: a quantitative theory based on chemical pathways

A new representation for chemical kinetics based on a sum over histories formulation is discussed. The description of the time-dependent chemistry of a reaction network is provided by chemical pathways defined at a molecular level. Using this methodology, the quantitative time evolution of the kinetics is described by enumerating the most important pathways followed by a chemical moiety such as a tagged atom. An explicit formula for the pathway probabilities is derived which takes the form of an integral over a time-ordered product. This expression has a simple and computationally efficient Monte Carlo representation which permits the method to be applied to a wide range of problems. For small reaction networks, the chemical pathways can be enumerated using graph theoretic methods. More complicated networks can be explored using random walks computed from a stochastic algorithm. The workings of the method are illustrated using a simple network of 20 chemical species which react via first-order kinetics. The application of the sum over histories representation to problems in surface catalysis and hydrogen combustion provide more realistic applications.