Perturbation theory for the breakdown of mean-field kinetics in oscillatory reaction-diffusion systems

Spatially distributed, nonequilibrium chemical systems described by a Markov chain model are considered. The evolution of such systems arises from a combination of local birth-death reactive events and random walks executed by the particles on a lattice. The parameter gamma, the ratio of characteristic time scales of reaction and diffusion, is used to gauge the relative contributions of these two processes to the overall dynamics. For the case of relatively fast diffusion, i.e., gamma much less than 1, an approximate solution to the Markov chain in the form of a perturbation expansion in powers of gamma is derived. Kinetic equations for the average concentrations that follow from the solution differ from the mass-action law and contain memory terms. For a reaction-diffusion system with Willamowski-Rossler reaction mechanism, we further derive the following two results: (a) in the limit of gamma-->0, these memory terms vanish and the mass-action law is recovered; (b) the memory kernel is found to assume a simple exponential form. A comparison with numerical results from lattice gas automaton simulations is also carried out. (C) 1998 American Institute of Physics. [S0021-9606(98)51225-1].