Slow time evolution of two-time-scale reaction-diffusion systems: the physical origin of nondiffusive transport

We study, from a mesoscopic point of view, the slow time-scale dynamics of a mixture of chemicals in which there is a chemical reaction that occurs much faster than all other processes, including diffusion. For a simple paradigmatic model reaction, it is possible to find a reduced set of dynamical equations analytically. This procedure, which yields the same mean field equations as the macroscopic approach described by Strier and Dawson [J. Chem. Phys, 112, 825 (2000)], clarifies the physical origin of some of the terms that appear in the reduced reaction-diffusion equations, such as "negative density dependent cross diffusion terms," whose actual meaning is hard to assess within the macroscopic framework. We also present a two-time-scale reactive lattice gas automaton with which it is possible to check the validity of the analytical results and the conditions under which the reduced description holds. Using this lattice gas we also show how the differential interaction with immobile species can give rise to the formation of stable Turing patterns in a system where all the other chemicals diffuse approximately at the same rate.