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Deterministic and Stochastic Models for Coalescence (Aggregation, Coagulation): a Review of the Mean-Field Theory for Probabilists (1997)
| Venue: | Bernoulli |
| Citations: | 222 - 13 self |
BibTeX
@ARTICLE{Aldous97deterministicand,
author = {David J. Aldous},
title = {Deterministic and Stochastic Models for Coalescence (Aggregation, Coagulation): a Review of the Mean-Field Theory for Probabilists},
journal = {Bernoulli},
year = {1997},
volume = {5},
pages = {3--48}
}
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Abstract
Consider N particles, which merge into clusters according to the 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 MarcusLushnikov 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. Keywords. branching process, coalescence, continuum tree, densitydependent Markov process, gelation, random graph, random tree, Smoluchowski coagulation equation Research supported by N.S.F. Grant DMS96-22859 1 Introduction Models, implicitly or explicitly stochastic, of coalescence (= coagulati...
Keyphrases
mean-field theory stochastic model random graph specified rate kernel continuum tree extensive scientific literature wide-ranging survey size merge introduction model mathematical literature special case marcuslushnikov model general kernel densitydependent markov process many interesting open problem underlying deterministic approximation random tree n.s.f. grant dms96-22859 population genetics stochastic coalescence smoluchowski coagulation equation smoluchowski coagulation equation research
