Reaction-Diffusion Model for A + A Reaction

We formulate an approach to the A + A {yields} products reaction that is based on a reaction-diffusion equation frequently used for the A + B problem but requires an appropriate generalization for the A + A problem. This approach leads naturally to the proportionality of the rate of the reaction to the pair correlation function evaluated at r = a, where a is the diameter of the reacting particles. In other words, the reaction rate is proportional to the probability that two A particles are sufficiently close. We also present numerical simulations in one and two dimensions in order to check our predictions. We confirm the well-known anomalous rate law in one dimension (the anomalies are marginal in two dimensions) and the proportionality of the reaction rate to the two-particle correlation function. Our simulations show that the rate of the reaction is indeed determined entirely by the spatial distribution of a very small shell of particles around a given reactant particle. Anomalous kinetics is a direct reflection of the deviation of the spatial distribution of this small shell from a random configuration. We also present simulation results that confirm the predicted distance and time scaling of the pair correlationmore » function in one dimension. 92 refs., 3 figs.« less