Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-field theory for probabilists

Abstract

Consider N particles, which merge into clusters according to the following rule: a cluster of size x and a cluster of size y merge at stochastic rate K(x,y)/N, where K is a specified rate kernel. This Marcus-Lushnikov model of stochastic coalescence and the underlying deterministic approximation given by the Smoluchowski coagulation equations have an extensive scientific literature. Some mathematical literature (Kingman's coalescent in population genetics; component sizes in random graphs) implicitly studies the special cases K(x,y)=1 and K(x,y)=xy. We attempt a wide-ranging survey. General kernels are only now starting to be studied rigorously; so many interesting open problems appear.