Cluster expansions for diffusion-controlled reactions

Abstract

The customary treatments of diffusion-controlled reactions involve some approximate assumptions to reduce the problem to two molecules. It is possible, however, to set up the full hierarchy of equations for the distribution functions for all molecules and to investigate the approximations in a systematic way by expanding in appropriate parameters. We carry this out for the simple reaction X + X A and find the standard theory by omitting third and higher powers of the density. Another expansion, valid for fast diffusion but not confined to low density, is based on the hierarchy for correlation functions. Finally, an expansion for high densities is given, in which case the hierarchy can be solved although it does not terminate.