A Diffusion-Limited Reaction

Fluctuations in diffusion-controlled reactions lack the necessary features for a mesoscopic description. We show how the correlations dominate the dynamics by juxtaposing the macroscopic dynamics of a cellular automaton model for the diffusion-controlled limit to the deterministic diffusion-reaction rate equation for the same reaction. A more detailed N-body master equation is then presented in which explicit diffusion-controlled limits are explained. 1 What Is Special About Diffusion-Limited Reactions? Common diffusion-reaction-rate equations are similar in structure and concept to the hydrodynamic equations for simple fluids and appear to be a simpler paradigm for general purposes. To the statistical physicist, however, systems with ongoing chemical reactions pose a more fundamental challenge than simple fluids in that two familiar properties of fluctuations cannot be taken for granted there: a) slow variation in comparison to the fast microscopic (i.e. molecular) dynamics due to conservation laws on the microscopic level, b) continuous variation, as it is found if the fluctuating quantities are sums of many small contributions which may vary more or less independently (e.g. all sorts of densities). Fluctuations with these two properties allow an extension of the deterministic dynamics of continuously varying macroscopic observables into mesoscopic stochastic dynamic laws without touching on the microscopic discrete details. Fluctuations in the mass-, momentum-, and energydensities in simple fluids are prime examples amenable to such treatment. Diffusion-reaction rate equations, too, are kinetic laws for densities, named concentrations, just like the hydrodynamic kinetic equations. This formal analogy to does not carry very far, however, because the concentrations are not conserved under the microscopic dynamics (property (a)) and, in general, do not vary slowly. So these equations apply only in regimes where the effect of the reactions is small compared to transport over macroscopic distances. In particular, this excludes the diffusion-controlled regime [1-9] far from equilibrium, where the reaction rate is limited by transport of the single reactant particles towards each other. There the microscopic correlations 1 There may exist other quantities which are conserved under the microscopic dynamics, which do not necessarily vary continuously. A Diffusion-Limited Reaction 89 may dominate the dynamics so that working assumptions which are common for fluctuations in fluids may turn out as unphysical. In the following, we start with two extreme models of a simple diffusionreaction system with a reversible autocatalytic reaction, namely the meanfield diffusion-reaction-rate equation and a stochastic cellular automaton model for the diffusion-controlled limit case to show how the macroscopic dynamics deviate in both cases. We then explain a slightly more general model, in which we discuss explicitly ways to take diffusion-controlled limits. The reduced limit-dynamics in our example depends on a special conservation law, which determines the convenient choice of variables. It corresponds conceptually to the hydrodynamic level of description in fluids although there is no formal similarity to the BBGKY-hierarchy of kinetic equations for the multiple point densities (mixed moments). We conclude with open questions for future research and a “message”. 2 Two Descriptions of a Diffusion-Reaction System We consider a common autocatalytic process, like it is found in the famous pattern-forming Belouzov-Zhabotinsky-reaction: