(110a) Reduction of Stochastic On-Lattice Chemical Kinetics Models to Well-Mixed Descriptions Via Singular Perturbation

Well-mixed and lattice-based descriptions of stochastic chemical kinetics have been extensively used in the literature and realizations of the corresponding stochastic processes are obtained by the Gillespie stochastic simulation algorithm and lattice kinetic Monte Carlo algorithms, respectively. However, the two frameworks remain disconnected. We present a novel approach that allows us to bridge these frameworks in the limiting case of fast diffusion. The core of our approach is a singular perturbation methodology applicable to the general discrete master equation. Starting from such an equation that captures single species adsorption, desorption, diffusion and chemical reaction on a lattice, we apply singular perturbation to derive a reduced stochastic model in the fast Fickian diffusion limit. We thus show that fast diffusion results in effectively well-mixed kinetics, in which the lumped rate of bimolecular reactions depends on the number of neighbors of a site on the lattice. Moreover, we propose a mapping between the stochastic propensities and the deterministic rates of the well-mixed vessel and lattice dynamics that illustrates the hierarchy of models and the key parameters that enable model reduction. In the presence of energetic interactions, both the reduced stochastic model and the deterministic model at the thermodynamic limit capture in detail the short-range energetic interactions. On the other hand, we demonstrate the potentially poor performance of the Bragg-Williams mean-field models in the presence of such interactions.