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We develop a hierarchy of approximations to the master equation for systemsthat exhibit translational invariance and finite-range spatial correlation.Each approximation within the hierarchy is a set of ordinary differentialequations that considers spatial correlations of varying lattice distance; theassumption is that the full system will have finite spatial correlations andthus the behavior of the models within the hierarchy will approach that of thefull system. We provide evidence of this convergence in the context of one- andtwo-dimensional numerical examples. Lower levels within the hierarchy thatconsider shorter spatial correlations, are shown to be up to three orders ofmagnitude faster than traditional kinetic Monte Carlo methods (KMC) forone-dimensional systems, while predicting similar system dynamics and steadystates as KMC methods. We then test the hierarchy on a two-dimensional modelfor the oxidation of CO on RuO2(110), showing that low-order truncations of thehierarchy efficiently capture the essential system dynamics. By consideringsequences of models in the hierarchy that account for longer spatialcorrelations, successive model predictions may be used to establish empiricalapproximation of error estimates. The hierarchy may be thought of as a class ofgeneralized phenomenological kinetic models since each element of the hierarchyapproximates the master equation and the lowest level in the hierarchy isidentical to a simple existing phenomenological kinetic models.

A consistent hierarchy of generalized kinetic equation approximations to the chemical master equation applied to surface catalysis