Scaling theory and exactly solved models in the kinetics of irreversible aggregation

Abstract

Abstract The scaling theory of irreversible aggregation is discussed in some detail. First, we review the general theory in the simplest case of binary reactions. We then extend consideration to ternary reactions, multispecies aggregation, inhomogeneous situations with arbitrary size dependent diffusion constants as well as arbitrary production terms. A precise formulation of the scaling hypothesis is given as well as a general theory of crossover phenomena. The consequences of this definition are described at length. The specific issues arising in the case in which an infinite cluster forms at finite times (the so-called gelling case) are discussed, in order to address discrepancies between theory and recent numerical work. Finally, a large number of exactly solved models are reviewed extensively with a view to pointing out precisely in which sense the scaling hypothesis holds in these various models. It is shown that the specific definition given here will give good results for almost all cases. On the other hand, we show that it is usually possible to find counterexamples to stronger formulations of the scaling hypothesis.