The kinetics of a number of prototypical diffusion-controlled reactions are studied, primarily through the application of scaling approaches. Our goal is to investigate the effects of spatial inhomogeneities in the particle densities on the reaction kinetics. In general, we find that there exists an upper critical dimension below which the kinetics cannot be described by a rate-equation analysis, an approach which gives the kinetic laws of a mean-field approximation. Below the upper critical dimension, spatial inhomogeneities in the particle densities give rise to new fluctuation-dominated kinetics. For irreversible reactions, with bimolecular decay A+B→inert being a simple generic example, universal kinetic laws are obtained which are a function only of the spatial dimension, the number of particles needed to initiate a reaction, and the nature of the particle-conservation laws for the system. We also consider the generalization of bimolecular decay to a multistate process where particles of variable ‘‘heights’’ and traps of variable ‘‘depths’’ react. This phenomenologically rich example serves as an illustrative testing ground for scaling ideas. The introduction of a small reaction probability on bimolecular decay provides an example of a dramatic crossover from an initial mean-field decay to the fluctuation-dominated decay law at large times. When the possibility of a reverse reaction is allowed in bimolecular decay, the available evidence suggests that there is a power-law approach to the equilibrium state, rather than an exponential approach predicted by the rate equation. Many of our new theoretical predictions are tested by extensive numerical simulations.

• Received 25 January 1985

DOI:https://doi.org/10.1103/PhysRevA.32.435