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Deterministic and Stochastic Models for Coalescence (Aggregation, Coagulation): a Review of the MeanField Theory for Probabilists
 Bernoulli
, 1997
"... 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 ..."
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Cited by 142 (13 self)
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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 wideranging 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 DMS9622859 1 Introduction Models, implicitly or explicitly stochastic, of coalescence (= coagulati...
Coalescent Random Forests
 J. COMBINATORIAL THEORY A
, 1998
"... Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ..."
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Cited by 38 (18 self)
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Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ) such that R k has uniform distribution over the set of all forests of k rooted trees labeled by [n]. Let R n be the trivial forest with n root vertices and no edges. For n k 2, given that R n ; : : : ; R k have been defined so that R k is a rooted forest of k trees, define R k\Gamma1 by addition to R k of a single edge picked uniformly at random from the set of n(k \Gamma 1) edges which when added to R k yield a rooted forest of k \Gamma 1 trees. This coalescent construction is related to a model for a physical process of clustering or coagulation, the additive coalescent in which a system of masses is subject to binary coalescent collisions, with each pair of masses of magnitude...
Approach to SelfSimilarity in Smoluchowski’s Coagulation Equations, preprint
, 2003
"... We consider the approach to selfsimilarity (or dynamical scaling) in Smoluchowski’s equations of coagulation for the solvable kernels K(x, y) = 2, x + y and xy. In addition to the known selfsimilar solutions with exponential tails, there are oneparameter families of solutions with algebraic deca ..."
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Cited by 34 (8 self)
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We consider the approach to selfsimilarity (or dynamical scaling) in Smoluchowski’s equations of coagulation for the solvable kernels K(x, y) = 2, x + y and xy. In addition to the known selfsimilar solutions with exponential tails, there are oneparameter families of solutions with algebraic decay, whose form is related to heavytailed distributions wellknown in probability theory. For K = 2 the size distribution is MittagLeffler, and for K = x + y and K = xy it is a powerlaw rescaling of a maximally skewed αstable Lévy distribution. We characterize completely the domains of attraction of all selfsimilar solutions under weak convergence of measures. Our results are analogous to the classical characterization of stable distributions in probability theory. The proofs are simple, relying on the Laplace transform and a fundamental rigidity lemma for scaling limits.
Massconserving solutions to the discrete coagulationfragmentation model with diffusion
 Nonlinear Anal., 49(3, Ser. A: Theory Methods):297–314
, 2002
"... We consider the following infinite system of reactiondiffusion equations: (1.1) ∂u1 ..."
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Cited by 14 (0 self)
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We consider the following infinite system of reactiondiffusion equations: (1.1) ∂u1
Emergence of the Giant Component in Special MarcusLushnikov Processes
 Random Structures and Algorithms
, 1997
"... Component sizes in the usual random graph process are a special case of the MarcusLushnikov process discussed in the scientific literature, so it is natural to ask how theory surrounding emergence of the giant component generalizes to the MarcusLushnikov process. Essentially no rigorous results ar ..."
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Cited by 13 (4 self)
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Component sizes in the usual random graph process are a special case of the MarcusLushnikov process discussed in the scientific literature, so it is natural to ask how theory surrounding emergence of the giant component generalizes to the MarcusLushnikov process. Essentially no rigorous results are known; we make a start by proving a weak result, but our main purpose is to draw this topic to the attention of random graph theorists. 1 Introduction 1.1 Background At time zero there are n separate "atoms"; as time increases, these atoms coalesce into clusters according to the rule for each pair of clusters, of sizes fx; yg say, they coalesce into a single cluster of size x + y at rate K(x; y)=n where K(x; y) = K(y; x) 0 is some specified rate kernel. This rule specifies a continuoustime finitestate Markov process which we shall call the Research supported by N.S.F. Grant DMS9622859 MarcusLushnikov process. The model was introduced by Marcus [16], and further studied by Lush...
Dynamical scaling in Smoluchowski’s coagulation equation: uniform convergence, preprint 2003
"... Abstract. Smoluchowski’s coagulation equation is a fundamental meanfield model of clustering dynamics. We consider the approach to selfsimilarity (or dynamical scaling) of the cluster size distribution for the “solvable ” rate kernels K(x, y) =2,x+ y, and xy. In the case of continuous cluster size ..."
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Cited by 11 (1 self)
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Abstract. Smoluchowski’s coagulation equation is a fundamental meanfield model of clustering dynamics. We consider the approach to selfsimilarity (or dynamical scaling) of the cluster size distribution for the “solvable ” rate kernels K(x, y) =2,x+ y, and xy. In the case of continuous cluster size distributions, we prove uniform convergence of densities to a selfsimilar solution with exponential tail, under the regularity hypothesis that a suitable moment have an integrable Fourier transform. For discrete size distributions, we prove uniform convergence under optimal moment hypotheses. Our results are completely analogous to classical local convergence theorems for the normal law in probability theory. The proofs rely on the Fourier inversion formula and the solution for the Laplace transform by the method of characteristics in the complex plane.
Selfsimilarity theory of stationary coagulation, Phys. Fluids 14
, 2002
"... A theory of stationary particle size distributions in coagulating systems with particle injection at small sizes is constructed. The size distributions have the form of power laws. Under rather general assumptions, the exponent in the power law is shown to depend only on the degree of homogeneity of ..."
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Cited by 4 (0 self)
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A theory of stationary particle size distributions in coagulating systems with particle injection at small sizes is constructed. The size distributions have the form of power laws. Under rather general assumptions, the exponent in the power law is shown to depend only on the degree of homogeneity of the coagulation kernel. The results obtained depend on detailed and quite sensitive estimates of various integral quantities governing the overall kinetics. The theory provides a unifying framework for a number of isolated results reported previously in the literature. In particular, it provides a more rigorous foundation for the scaling arguments of Hunt, which were based purely on dimensional analysis. 1.
The scaling attractor and ultimate dynamics for Smoluchowski’s coagulation equations
 J. Nonlinear Sci
"... coagulation equations ..."
Gelation and Cluster Growth with ClusterWall Interactions
"... Metallic cluster growth within a reactive polymer matrix is modeled by augmenting coagulation equations to include the influence of side reactions of metal atoms with the polymer matrix: 8 ! : c j = 1 2 P j \Gamma1 k=1 R j \Gammak;k c k c j \Gammak \Gamma c j P 1 k=1 R jk c k \Gamma ffi 1j c j p j = ..."
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Metallic cluster growth within a reactive polymer matrix is modeled by augmenting coagulation equations to include the influence of side reactions of metal atoms with the polymer matrix: 8 ! : c j = 1 2 P j \Gamma1 k=1 R j \Gammak;k c k c j \Gammak \Gamma c j P 1 k=1 R jk c k \Gamma ffi 1j c j p j = 1; 2; : : : p = \Gamma c 1 p; where ? 0 and where c k denotes the concentration of the kth cluster and p denotes the concentration of reactive sites available within the polymer matrix for reaction with metallic atoms. The initial conditions are required to be nonnegative and satisfy P 1 j=1 jc j (0) = 1 and p(0) = p 0 . We assume that R jk = d j ff k ff +(j+k) (j ff +k ff ) d+j+k for 0 ff 1, which encompasses both bond linking kernels ( R jk = j ff k ff ) and surface reaction kernels ( R jk = j ff + k ff ). Our analytical and numerical results indicate that the sidereactions delay gelation in some cases and inhibit gelation in others. We provide numeric...