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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 |

### Citations

3007 | On the evolution of random graphs,”
- Erdos, Renyi
- 1960
(Show Context)
Citation Context ...del G(N; p), there are N vertices, and each of the \Gamma N 2 \Delta possible edges is present with probability p, independently for different edges. Study of this model goes back to Erdos and R'enyi =-=[28, 29]-=-, and the monograph Bollob'as [18] surveys results up to 1984. Sizes of the connected components have been a classical topic of study. A moment's thought shows that for the kernel K(x; y) = xy the Mar... |

2348 | Random Graphs.
- Bollobás
- 2001
(Show Context)
Citation Context ... a sample of genes in the current generation, and the specific kernel K(x; y) = 1 arises (section 4.2). An area of TP mathematics which turns out to be related to our topic is ffl Random graph theory =-=[18]-=- where the specific kernel K(x; y) = xy has implicitly been studied in great detail: see section 4.4. In addition to the SM study of particular physical phenomena, there is a body of literature (e.g. ... |

450 |
A Course in Combinatorics.
- Lint, Wilson
- 1992
(Show Context)
Citation Context ...uous Smoluchowski coagulation equation. See (49) for the corresponding conjecture for general non-gelling kernels. 4.3 The additive coalescent and the continuum random tree Cayley's formula (see e.g. =-=[97]-=- Chapter 2) says there are N N \Gamma2 trees on N labeled vertices. Pick such a tree T1 at random. To the edges e of T1 attach independent exponential(1) r.v.'sse . Write F (t) for the forest obtained... |

240 |
The continuum random tree
- Aldous
- 1991
(Show Context)
Citation Context ... within distance 1 of a specified point is bounded away from 0 and 1, and we can take a n ! 1 limit. The limit is called the self-similar continuum random tree (SSCRT), and is described rigorously in =-=[4], [5] sect-=-ion 2.5. Figure 4 illustrates the SSCRT: of course, by (statistical) self-similarity there are 20 smaller and smaller branches not shown. The lines in figure 4 are part of the "skeleton" of ... |

209 |
Atmospheric chemistry and physics of air pollution
- Seinfeld
- 1986
(Show Context)
Citation Context ...of application Where possible I cite papers giving an accessible overview of a field, rather than the seminal paper in a field. By far the largest application area is physical chemistry. ffl Aerosols =-=[26, 69]-=-. That is, solid or liquid particles suspended in a gas: such as smoke, smog, dust; water droplet or snowflake formation in clouds. ffl Phase separation in liquid mixtures [1]. ffl Polymerization [30]... |

154 |
The coalescent. Stochastic Process
- Kingman
- 1982
(Show Context)
Citation Context ...for the lengths of these subintervals, the process X is a version of the stochastic coalescent with K(x; y) = 1, and this version is called Kingman's coalescent. The construction goes back to Kingman =-=[47]. Kingman'-=-s coalescent has been used extensively in mathematical population genetics [83], where the emphasis is of the number of "lines of descent" (clusters, in our terminology) and mutations along ... |

152 |
The continuum random tree III
- Aldous
- 1993
(Show Context)
Citation Context ...e construction above and rescale by taking each vertex to have mass 1=N and each edge to have length 1=N 1=2 . The N !1 limit is called the continuum random tree (CRT), and is described rigorously in =-=[5, 7]-=-. (In the SSCRT of Construction 7 the root is attached to an infinite baseline; the CRT here is compact, with total mass 1). Now similarly to Construction 7, at each time \Gamma1 ! t ! 1 construct a P... |

106 |
Line-of-descent and genealogical processes, and their applications in population genetics models.
- Tavaré
- 1984
(Show Context)
Citation Context ...ical entities in physical space, and have been studied as SM mathematics. A different area of application, which has caught the attention of TP mathematicians, is ffl Mathematical population genetics =-=[83]. 5 H-=-ere the entities are `lines of descent", i.e. number of ancestors in past generations of a sample of genes in the current generation, and the specific kernel K(x; y) = 1 arises (section 4.2). An ... |

103 | The continuum random tree. II. An overview.
- Aldous
- 1990
(Show Context)
Citation Context ...in distance 1 of a specified point is bounded away from 0 and 1, and we can take a n ! 1 limit. The limit is called the self-similar continuum random tree (SSCRT), and is described rigorously in [4], =-=[5] section 2-=-.5. Figure 4 illustrates the SSCRT: of course, by (statistical) self-similarity there are 20 smaller and smaller branches not shown. The lines in figure 4 are part of the "skeleton" of the S... |

103 | Brownian excursions, critical random graphs and the multiplicative coalescent.
- Aldous
- 1997
(Show Context)
Citation Context ...esponding to the gelation time T gel = 1 in the deterministic model. Rather detailed rigorous results are known: see [43] for recent exhaustive analysis. We give a probabilistic discussion, following =-=[9]-=-. Recall ML (N) 1 (t) is the mass of the largest cluster in the Marcus-Lushnikov process, that is the size of the largest component of G(N; 1 \Gamma e \Gammat=N ). It is classical [28, 18] that ML (N)... |

87 | The standard additive coalescent
- Aldous, Pitman
- 1998
(Show Context)
Citation Context ...alogy with the discrete case above) the process (X(t); \Gamma1 ! t ! 1) evolves as the stochastic coalescent for K(x; y) = x+y. This process, the standard additive coalescent, is studied in detail in =-=[12]-=-. 33 * * * * * * * * * * * * * * * * * * * * * * * * * Figure 10. The CRT and the additive coalescent 34 4.4 Random graphs and the multiplicative coalescent In the random graph model G(N; p), there ar... |

80 |
A General Mathematics Survey of the Co- agulation Equation,” In
- Drake
- 1972
(Show Context)
Citation Context ...Table 1 gives examples of kernels used in the physical chemistry literature. The table is taken from [74], who cite references to each case. Most of these examples, and others, are explained in Drake =-=[26] section 4-=-.3. Note that we parametrize the "size" of clusters by mass x rather than length l; these kernels are often written in terms of l / x 1=3 instead of x. 3 Table 1 [74] Some specific kernels K... |

70 |
Generalized Poisson Distributions
- Consul
- 1989
(Show Context)
Citation Context ...nfigurationsn(x; 0) = ffi 1 (x); in the continuous case it is a solution arising from infinitesimally small initial clusters (where uniqueness is hardly obvious). Some involve the Borel distribution (=-=[23]-=- sec. 2.7), which for our purposes is best regarded as the total population size Zsin a Galton-Watson branching process with one progenitor and Poisson() offspring distribution. Explicitly, B(; x) j P... |

70 |
The birth of the giant component. Random Structures and Algorithms
- Janson, Knuth, et al.
- 1993
(Show Context)
Citation Context ...re deeply into the behavior of the stochastic model around the "critical time" corresponding to the gelation time T gel = 1 in the deterministic model. Rather detailed rigorous results are k=-=nown: see [43]-=- for recent exhaustive analysis. We give a probabilistic discussion, following [9]. Recall ML (N) 1 (t) is the mass of the largest cluster in the Marcus-Lushnikov process, that is the size of the larg... |

62 |
Systems in Stochastic Equilibrium,
- Whittle
- 1986
(Show Context)
Citation Context ...ous addition of new particles. [74] In fact, much of the literature we cite relating to existence of solutions of the Smoluchowski coagulation equations deals with such more general settings. Whittle =-=[104]-=- provides a mathematical introduction to reversible mean-field models of coalescing and fragmentation, and Ernst [30] relates this topic to broader topics in statistical physics. (c) There has been mu... |

59 |
Fractal Growth Phenomena, World Scientific Pub.
- Vicsek
- 1992
(Show Context)
Citation Context ...dy, mostly using Monte Carlo simulation, of fractal structure of cluster-cluster aggregation models, in the spirit of the well known DLA model of cluster growth by adding single particles. See Vicsek =-=[99]-=- Chapter 8 for a survey. In the setting of the Smoluchowski coagulation equations this possibility has classically been ignored (in specifying rate kernels, it is often assumed that clusters are spher... |

56 | Tree-valued Markov chains derived from Galton-Watson processes
- Aldous, Pitman
- 1998
(Show Context)
Citation Context ... rigorous formalization of the notion of "the history of coalescences containing a typical particle in the Smoluchowski coagulation equation". Detailed study of the cases K(x; y) = x + y and=-= xy is in [13]-=-. The construction extends without essential change to the continuous setting. Here property (*) becomes at each time t there is a stationary ergodic point process on the line, in which the inter-poin... |

53 | Coalescent random forests,
- Pitman
- 1999
(Show Context)
Citation Context ...; 0st ! 1) is the Marcus-Lushnikov process associated with the additive kernel K(x; y) = x + y, with monodisperse initial conditions. This construction was apparently first explicitly given by Pitman =-=[63]-=-, although various formulas associated with it had previously been developed in the 32 combinatorial literature [62, 105] and the SM literature [40]. Here are two examples of simple formulas. The numb... |

50 |
Asymptotic fringe distributions for general families of random trees
- Aldous
- 1991
(Show Context)
Citation Context ...howed that it arises as a n ! 1 limit of uniform random n-vertex trees. The finite-n property that "the tree has the same distribution relative to each vertex" extends to the limit infinite =-=tree: see [3]-=- for one statement of this invariance property. Now regard each edge as appearing at a random time T with exponential(1) distribution, independently for different edges. At times 0 ! t ! 1 we see a co... |

48 | Construction of Markovian coalescents
- Evans, Pitman
- 1998
(Show Context)
Citation Context ...the decreasing ordering x 1sx 2s: : : of cluster-masses, so the state space becomes the infinite-dimensional simplex; (ii) identifying fx i g with the measure P x i ffi x i (\Delta). Evans and Pitman =-=[33] give a ca-=-reful account of the "technical bookkeeping" issues involved in formalization (e.g. one wishes to track the previous history of mergers of a particular cluster at time t). The details are no... |

46 |
On tree census and the giant component in sparse random graphs, Random Structures and Algorithms
- Pittel
- 1990
(Show Context)
Citation Context ...Watson branching process). Barbour [15] proved the central limit theorem: for fixed x, N \Gamma1=2 (ML (N) (x; t)(t) \Gamma Nn(x; t)) d ! Normal(0; n(x; t)(1+(t \Gamma 1)x 2 n(x; t))) (38) and Pittel =-=[64]-=- established joint convergence to the mean-zero Gaussian process with covariances (t \Gamma 1)xyn(x; t)n(y; t); y 6= x: (39) (Note that these results are stated in the literature for tree-components, ... |

40 |
Asymptotics in the random assignment problem
- Aldous
- 1992
(Show Context)
Citation Context ...\Gammat x \Gamma3=2 e \Gammae \Gamma2t x=2 the formula we recorded in table 2. 21 Construction 8 K(x; y) = xy; discrete. This construction, illustrated in figure 5, was used for different purposes in =-=[6]-=-. OE OE @ @ @ @ @ \Gamma \Gamma \Gamma \Gamma \Gamma @ @ @ @ H H H H H H H H \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Phi \Gamma \Gamma \Gamma \Gamma 1 2 3 11 12 13 5.3 1.7 0.4 3.6 0.3 0.7 2.4 21 22 23 1.8... |

40 | Theoretical Mathematics": Towards a Cultural Synthesis of
- Jaffe, Quinn
- 1993
(Show Context)
Citation Context ... In TP mathematics one is supposed to have explicit assumptions and conclusions, as well as a rigorous argument linking them. A lively recent debate on these matters can be found in the discussion of =-=[42]. Much of -=-this survey deals with issues which have not been studied systematically as TP mathematics, and our "open problems" concern proofs of matters which mostly appear implicitly or explicitly in ... |

39 |
Theory of molecular size distribution and gel formation in branched-chain polymers,
- Stockmayer
- 1943
(Show Context)
Citation Context ... represents the Stockmayer model in which sol and gel do not interact, and in this case the post-gelation solution has the surprisingly simple form n(x; t) = n(x; 1)t \Gamma1 ; ts1 which goes back to =-=[80]-=-. This hasn't been studied in the random graphs literature, but a probabilistic elaboration will be given in [10] section 3.7. 4.5 Combinatorial approaches A different approach to the monodisperse Mar... |

34 |
Molecular Size Distribution in Three Dimensional Polymers
- Flory
- 1941
(Show Context)
Citation Context ...asy way to take limits to obtain a construction for the continuous case K(x; y) = 23 xy. Rather sophisticated ideas are needed -- see section 4.4. Remark. There is a long history, going back to Flory =-=[34]-=-, of using branching process models in the theory of polymerization. The idea that certain such models were equivalent to the Smoluchowski coagulation equation with certain kernels seems to have emerg... |

33 |
Stochastic coalescence
- Marcus
- 1968
(Show Context)
Citation Context ...a1 ; n i \Gamma1; n i+1 ; : : : ; n j \Gamma1 ; n j \Gamma1; n j+1 ; : : : ; n i+j \Gamma1 ; n i+j +1; n i+j+1 ; : : :) with rate K(i; j)n i n j =N . This model was perhaps first introduced by Marcus =-=[56]-=-, and re-introduced by several authors [36, 81], in particular by Lushnikov [54] as a model of gelation. We call it the Marcus-Lushnikov processsML (N) (t). The state of this process at time t may be ... |

33 |
Scaling solutions of Smoluchowski’s coagulation equation
- Dongen, Ernst
- 1988
(Show Context)
Citation Context ... where the specific kernel K(x; y) = xy has implicitly been studied in great detail: see section 4.4. In addition to the SM study of particular physical phenomena, there is a body of literature (e.g. =-=[93, 87, 95]-=- in J. Statistical Physics) devoted to mathematical study, with varying levels of rigor, of the kind of models we discuss here. When we get to discussing detailed mathematical results, we shall most o... |

32 |
On an infinite set of non-linear differential equations
- McLeod
- 1962
(Show Context)
Citation Context ... : : : ; 0s1: (5) table 2: formulas for n(x; t) K(x; y) 1 x + y xy discrete (1 + t 2 ) \Gamma2 ( t 2+t ) x\Gamma1 e \Gammat B(1 \Gamma e \Gammat ; x) x \Gamma1 B(t; x) 0st ! 1 [76] 0st ! 1 [37] 0sts1 =-=[57]-=- continuous 4t \Gamma2 exp(\Gamma2x=t) (2) \Gamma1=2 e \Gammat x \Gamma3=2 e \Gammae \Gamma2t x=2 (2) \Gamma1=2 x \Gamma5=2 e \Gammat 2 x=2 0 ! t ! 1 [67] \Gamma1 ! t ! 1 \Gamma1 ! t ! 0 We included t... |

28 |
Drei Vortrage uber Diffusion, Brownsche Bewegung und Koagulation von
- Smoluchowski
- 1916
(Show Context)
Citation Context ...Gammax =x!; x = 1; 2; 3; : : : ; 0s1: (5) table 2: formulas for n(x; t) K(x; y) 1 x + y xy discrete (1 + t 2 ) \Gamma2 ( t 2+t ) x\Gamma1 e \Gammat B(1 \Gamma e \Gammat ; x) x \Gamma1 B(t; x) 0st ! 1 =-=[76]-=- 0st ! 1 [37] 0sts1 [57] continuous 4t \Gamma2 exp(\Gamma2x=t) (2) \Gamma1=2 e \Gammat x \Gamma3=2 e \Gammae \Gamma2t x=2 (2) \Gamma1=2 x \Gamma5=2 e \Gammat 2 x=2 0 ! t ! 1 [67] \Gamma1 ! t ! 1 \Gamm... |

26 |
The Self-Preserving Particle Size Distribution for Coagulation by Brownian Motion
- Friedlander, Wang
- 1966
(Show Context)
Citation Context ...number rigorous results. 1.3 Digression: TP and SM mathematics To ask "what is known" in this subject leads inexorably to philosophical issues concerning pure and applied mathematics. A typi=-=cal paper [30, 35, 101] we cite from the sc-=-ientific literature would be described by a layman as "mathematics" rather than "science": it is devoted to analysis of a mathematical model rather than description of experimental... |

26 |
Some new aspects of coagulation theory
- Lushnikov
- 1978
(Show Context)
Citation Context ...+j \Gamma1 ; n i+j +1; n i+j+1 ; : : :) with rate K(i; j)n i n j =N . This model was perhaps first introduced by Marcus [56], and re-introduced by several authors [36, 81], in particular by Lushnikov =-=[54]-=- as a model of gelation. We call it the Marcus-Lushnikov processsML (N) (t). The state of this process at time t may be written in two equivalent ways. We may write ML (N) (x; t) for the (random) numb... |

25 |
Proof of dynamical scaling in Smoluchowski’s coagulation equation with constant kernel
- Kreer, Penrose
- 1994
(Show Context)
Citation Context ...(see e.g. [24] Prop. 9.3.1) imply that G(t)L t converges in distribution to the exponential(1=m 1 (0)) distribution, establishing (25). Without this easy probability argument, rather tedious analysis =-=[50] seems nee-=-ded to prove (25). Remark. The key feature of these two constructions is that there is an "invariance" property (stationarity under shifts of the line) which enables us to define the determi... |

24 |
Coagulation in finite systems
- Lushnikov
- 1978
(Show Context)
Citation Context ...arly, there is a discrete reformulation of the construction of Kingman 's coalescent which enables one to write down a partition function [16]. In the additive case, such results go back to Lushnikov =-=[55]-=-, and have two somewhat different extensions. Hendricks et al. [40] give an expression for the partition function for the kernel K(x; y) = A +B(x + y). And implicit in [55] (see [40, 20] for clearer e... |

22 |
Smoluchowski’s theory of coagulation in colloids holds rigorously in the Boltzmann–Grad-limit
- Lang, Xanh
- 1980
(Show Context)
Citation Context ...art of a more detailed rigorous stochastic model. To establish these rigorously as a limit of the type of model in section 1.1 is a topic called hydrodynamicssor propagation of chaos. Lang and Nguyen =-=[52]-=- study a model of discrete particles performing Brownian motion in 3 dimensions, coalescing when they approach within a fixed distance, the diffusion rate of clusters being unaffected by cluster size.... |

20 |
Random labelled trees and their branching networks
- Grimmett
- 1980
(Show Context)
Citation Context ...individuals as unit masses and the parent-child relationships as edges. This construction gives a random infinite tree, with root ? at position 0, illustrated in the bottom part of figure 3. Grimmett =-=[38] first des-=-cribed this tree, and showed that it arises as a n ! 1 limit of uniform random n-vertex trees. The finite-n property that "the tree has the same distribution relative to each vertex" extends... |

20 |
Comparison of analytical and physical modeling of planetesimal accumulation,
- Wetherill
- 1990
(Show Context)
Citation Context ...number rigorous results. 1.3 Digression: TP and SM mathematics To ask "what is known" in this subject leads inexorably to philosophical issues concerning pure and applied mathematics. A typi=-=cal paper [30, 35, 101] we cite from the sc-=-ientific literature would be described by a layman as "mathematics" rather than "science": it is devoted to analysis of a mathematical model rather than description of experimental... |

20 |
A Global Existence Theorem for Smolu- chowski’s Coagulation Equations
- White
- 1980
(Show Context)
Citation Context ... m 0 (t) is decreasing and m 1 (t) is constant) d dt m 2 (t)sk 0 (2m 2 (t)m 1 (0) +m 2 1 (0)) implying m 2 (t) ! 1 for all t ! 1. It is not difficult to make this type of argument rigorous. See White =-=[102]-=- and Heilmann [39] for the discrete case, and chapters 3 and 4 of Dubovskii [27] for the continuous case (in the more general setting of coagulation and fragmentation), and extensive references to the... |

20 |
On the average behavior of set merging algorithms (Extended abstract),
- Yao
- 1976
(Show Context)
Citation Context ...itial conditions. This construction was apparently first explicitly given by Pitman [63], although various formulas associated with it had previously been developed in the 32 combinatorial literature =-=[62, 105]-=- and the SM literature [40]. Here are two examples of simple formulas. The number D (N) (t) of clusters satisfies D (N) (t) \Gamma 1 d = Binomial(N \Gamma 1; e \Gammat ) The cluster-sizes (in random o... |

19 |
Kinetics of polymerization
- Ziff
- 1980
(Show Context)
Citation Context ... interpretation of gelation is that after the critical time, a strictly positive proportion of mass lies in infinitemass clusters, the gel. One can model post-gelation behavior by explicitly modeling =-=[106]-=- interaction between the gel and the sol (finite-mass clusters), but in this survey we shall only consider pre-gelation behavior, except for brief comments on the K(x; y) = xy case in section 4.4. 10 ... |

18 |
The solution of the coagulating equation for cloud droplets in a rising air
- Golovin
- 1963
(Show Context)
Citation Context ... = 1; 2; 3; : : : ; 0s1: (5) table 2: formulas for n(x; t) K(x; y) 1 x + y xy discrete (1 + t 2 ) \Gamma2 ( t 2+t ) x\Gamma1 e \Gammat B(1 \Gamma e \Gammat ; x) x \Gamma1 B(t; x) 0st ! 1 [76] 0st ! 1 =-=[37]-=- 0sts1 [57] continuous 4t \Gamma2 exp(\Gamma2x=t) (2) \Gamma1=2 e \Gammat x \Gamma3=2 e \Gammae \Gamma2t x=2 (2) \Gamma1=2 x \Gamma5=2 e \Gammat 2 x=2 0 ! t ! 1 [67] \Gamma1 ! t ! 1 \Gamma1 ! t ! 0 We... |

17 |
The stochastic coalescence model for cloud droplet growth,
- Gillespie
- 1972
(Show Context)
Citation Context ...a1 ; n j \Gamma1; n j+1 ; : : : ; n i+j \Gamma1 ; n i+j +1; n i+j+1 ; : : :) with rate K(i; j)n i n j =N . This model was perhaps first introduced by Marcus [56], and re-introduced by several authors =-=[36, 81]-=-, in particular by Lushnikov [54] as a model of gelation. We call it the Marcus-Lushnikov processsML (N) (t). The state of this process at time t may be written in two equivalent ways. We may write ML... |

16 |
Poisson convergence and random graphs
- Barbour
- 1982
(Show Context)
Citation Context ...1 EML (N) (x; t)(t) ! n(x; t) as N !1 for fixed xs1; t ! 1 (37) is classical [28] and easy (relative to a given vertex, the random graph looks locally like a Galton-Watson branching process). Barbour =-=[15]-=- proved the central limit theorem: for fixed x, N \Gamma1=2 (ML (N) (x; t)(t) \Gamma Nn(x; t)) d ! Normal(0; n(x; t)(1+(t \Gamma 1)x 2 n(x; t))) (38) and Pittel [64] established joint convergence to t... |

16 |
Analytic studies of cloud droplet coalescence,
- Scott
- 1968
(Show Context)
Citation Context ...me explicit solutions for the three special kernels in table 2 and the general bilinear kernel K(x; y) = A + B(x + y) + Cxy, under polydisperse initial conditions, are discussed in [26] section 6.3., =-=[17, 68, 71, 77, 78, 84]-=-. Some discussion of physically reasonable or mathematically tractable initial configurations (e.g. in the continuous setting, the Gamma, log-Normal or n(x; 0) = x \Gammafi ; xsx 0 densities) in the c... |

15 |
Coagulation processes with a phase transition
- Ernst, Ziff, et al.
- 1984
(Show Context)
Citation Context ...rge clusters at the critical time, rather than in absolute terms. The Smoluchowski coagulation equations with K(x; y) = xy have been studied many times in the SM literature. In particular, Ziff et al =-=[107, 31] observe (41) a-=-rising as a "scaling limit" as t " 1 for the monodisperse initial conditions, and discuss the extent to which this remains true for more general 37 initial conditions, and also discuss ... |

14 | Emergence of the giant component in special Marcus-Lushnikov processes, Random Structures and Algorithms 12
- Aldous
- 1998
(Show Context)
Citation Context ..., and consider s(t) = X x f(x)n(x; t): It is easy to check d dt s(t) = s 2 (t) and hence s(t) = (1 \Gamma t) \Gamma1 ; 0st ! 1. This suggests (but does not quite prove) that T gel = 1. It is shown in =-=[8]-=- that for kernels of this special form (46) one can use stochastic calculus to analyze the Marcus-Lusnikov process and prove the conclusion of Open Problem 12. 5.3 Dynamical scaling and entrance bound... |

14 | The entrance boundary of the multiplicative coalescent
- Aldous, Limic
- 1998
(Show Context)
Citation Context ...uess that the standard multiplicative coalescent is in some sense the "essentially unique" version of the multiplicative coalescent on \Gamma1 ! t ! 1 satisfying (43): the precise result is =-=proved in [11]-=-. The convergence (40) extends to polydisperse Marcus-Lushnikov processes as follows. For r = 2; 3 write oe r (N) = P i (ML (N) i (0)) r , and write c(N) = (oe 3 (N)) 2=3 =(oe 2 (N)) 2 . The appropria... |

14 | Instantaneous gelation in coagulation dynamics
- Carr, Costa
- 1992
(Show Context)
Citation Context ...2 ! 1 such that c 1 (xy) fl=2sK(x; y)sc 2 xy: Moreover it is believed that for fl ? 2 the kernel is instantaneously gelling, that is T gel = 0. Precisely, an argument in [88] (rewritten rigorously in =-=[22]-=- in the discrete case) shows that T gel = 0 provided 9fl ? 2ff ? 2 such that x ff + y ffsK(x; y)s(xy) fl=2 8x; y: 2.4 Self-similarity Consider the continuous setting, and as above suppose the kernel K... |

13 |
Exact solutions for random coagulation processes
- Hendriks, Spouge, et al.
- 1985
(Show Context)
Citation Context ...ion was apparently first explicitly given by Pitman [63], although various formulas associated with it had previously been developed in the 32 combinatorial literature [62, 105] and the SM literature =-=[40]-=-. Here are two examples of simple formulas. The number D (N) (t) of clusters satisfies D (N) (t) \Gamma 1 d = Binomial(N \Gamma 1; e \Gammat ) The cluster-sizes (in random order) of Y (N) (t), given D... |

12 |
Exact solution of the non-linear Boltzmann equation for Maxwell models, Phys
- Ernst
- 1979
(Show Context)
Citation Context ...number rigorous results. 1.3 Digression: TP and SM mathematics To ask "what is known" in this subject leads inexorably to philosophical issues concerning pure and applied mathematics. A typi=-=cal paper [30, 35, 101] we cite from the sc-=-ientific literature would be described by a layman as "mathematics" rather than "science": it is devoted to analysis of a mathematical model rather than description of experimental... |

12 |
in coagulating systems
- Dongen, Ernst
- 1987
(Show Context)
Citation Context ... where the specific kernel K(x; y) = xy has implicitly been studied in great detail: see section 4.4. In addition to the SM study of particular physical phenomena, there is a body of literature (e.g. =-=[93, 87, 95]-=- in J. Statistical Physics) devoted to mathematical study, with varying levels of rigor, of the kind of models we discuss here. When we get to discussing detailed mathematical results, we shall most o... |

11 | Tree-valued Markov chains and Poisson-Galton-Watson distributions - Aldous - 1998 |

10 |
On the scalar transport equation
- McLeod
- 1964
(Show Context)
Citation Context ... for the monodisperse initial conditions, and discuss the extent to which this remains true for more general 37 initial conditions, and also discuss different models of post-gelation behavior. McLeod =-=[58]-=- showed that with the initial configuration n(x; 0) = ffe \Gammaffx the continuous Smoluchowski coagulation equation has explicit solution n(x; t) = ffe \Gamma(t+ff)x I 1 (2xff 1=2 x 1=2 ) x 2 t 1=2 o... |

10 | Coagulation and branching process models of gravitational clustering
- Sheth, Pitman
- 1997
(Show Context)
Citation Context ...ph by Dubovskii [27] focuses narrowly on mathematical issues of existence and uniqueness of 7 solutions. Snapshots of recent research results and interests are provided by the introductions to papers =-=[70, 74, 85]-=-. We now give a quick overview of three aspects of the Smoluchowski coagulation equations; exact solutions, gelation and self-similar solutions. 2.2 Exact solutions It has long been recognized that th... |

10 | Cluster size distribution in irreversible aggregation at large times - Ernst, Dongen - 1985 |

9 |
On the possible occurrence of instantaneous gelation in Smoluchowski's coagulation equation
- Dongen
- 1987
(Show Context)
Citation Context ...provided 9fl ? 1; 0 ! c 1 ; c 2 ! 1 such that c 1 (xy) fl=2sK(x; y)sc 2 xy: Moreover it is believed that for fl ? 2 the kernel is instantaneously gelling, that is T gel = 0. Precisely, an argument in =-=[88]-=- (rewritten rigorously in [22] in the discrete case) shows that T gel = 0 provided 9fl ? 2ff ? 2 such that x ff + y ffsK(x; y)s(xy) fl=2 8x; y: 2.4 Self-similarity Consider the continuous setting, and... |

8 |
The extent of correlations in a stochastic coalescence process,
- Bayewitz, Yerushalmi, et al.
- 1974
(Show Context)
Citation Context ...by an elementary recurrence formula ([18] exercise 7.1). Similarly, there is a discrete reformulation of the construction of Kingman 's coalescent which enables one to write down a partition function =-=[16]-=-. In the additive case, such results go back to Lushnikov [55], and have two somewhat different extensions. Hendricks et al. [40] give an expression for the partition function for the kernel K(x; y) =... |

8 |
The stable doubly infinite pedigree process of supercritical branching populations. Zeitschrift fur Wahrscheinlichkeitstheorie und verwandte Gebiete
- Nerman, Jagers
- 1984
(Show Context)
Citation Context ...es exactly as (27). (We mention this because the analogous assertion for pure coalescence is false -- cf. section 4.1). The issue of self-similar solutions (clearly analogous to stable type structure =-=[60]-=- for supercritical branching processes) for pure fragmentation was studied by Brennan and Durrett [19]. We summarize their results in Appendix A.3. An initially promising idea is to use branching proc... |

8 |
Solutions and critical times for the monodisperse coagulation equation when aij = A+B(i
- Spouge
- 1983
(Show Context)
Citation Context ...4 for its interpretation. There has also been considerable attention paid to the general bilinear kernel K(x; y) = A + B(x + y) + Cxy, for which some more complicated explicit solutions are available =-=[86, 90, 77, 78]-=-. 2.3 Gelation Consider the second moment of cluster mass m 2 (t) = R 1 0 x 2 n(x; t)dx. Because a coalescence fx; yg ! x + y increases the sum-of-squares-of-masses by 2xy, the continuous Smoluchowski... |

8 |
On the occurrence of a gelation transition in Smoluchowski’s coagulation equation
- Dongen, Ernst
- 1986
(Show Context)
Citation Context ...in table 1 is homogeneous, with exponents \Gamma 1 6 ; 0; 1 6 ; 7 9 ; 1; 4 3 . Principle 1 says that for fls1 the kernel is non-gelling. For some time it has been widely accepted in the SM literature =-=[92, 95]-=- that for fl ? 1 the kernel is gelling. The first general rigorous result was given only recently by Jeon [44, 45], who showed that T gel ! 1 provided 9fl ? 1; 0 ! c 1 ; c 2 ! 1 such that c 1 (xy) fl=... |

7 |
On Lushnikov’s model of gelation
- Buffet, Pulé
- 1990
(Show Context)
Citation Context ...back to Lushnikov [55], and have two somewhat different extensions. Hendricks et al. [40] give an expression for the partition function for the kernel K(x; y) = A +B(x + y). And implicit in [55] (see =-=[40, 20]-=- for clearer expositions) is the following result. Lemma 9 Consider the Marcus-Lushnikov process with K(x; y) = xf(y) + yf(x) for some f , with monodisperse initial configuration. Then p(n; t) = N ! Y... |

7 |
Polymers and random graphs
- Buffet, Pulé
- 1991
(Show Context)
Citation Context ...\Gamma1=3 ) transition, from the viewpoint of the breakdown in the Gaussian approximation. It is not clear exactly when the explicit connection with random graph theory was made in the SM literature: =-=[21]-=- is one account from 1991. Conversely, even present-day accounts of random graphs [43] make no mention of the Smoluchowski coagulation equation connection. Let us briefly discuss post-gelation behavio... |

7 | Evolution of coagulating systems - Lushnikov - 1973 |

7 |
Roessel, The mass-conserving solutions of Smoluchowski’s coagulation equation: The general bilinear kernel
- Shirvani, van
- 1992
(Show Context)
Citation Context ...me explicit solutions for the three special kernels in table 2 and the general bilinear kernel K(x; y) = A + B(x + y) + Cxy, under polydisperse initial conditions, are discussed in [26] section 6.3., =-=[17, 68, 71, 77, 78, 84]-=-. Some discussion of physically reasonable or mathematically tractable initial configurations (e.g. in the continuous setting, the Gamma, log-Normal or n(x; 0) = x \Gammafi ; xsx 0 densities) in the c... |

6 | Splitting intervals. II. Limit laws for lengths - Brennan, Durrett - 1987 |

6 |
Post-gelation solutions to Smoluchowski’s coagulation equation
- Bak, Heilmann
- 1994
(Show Context)
Citation Context ...ing and m 1 (t) is constant) d dt m 2 (t)sk 0 (2m 2 (t)m 1 (0) +m 2 1 (0)) implying m 2 (t) ! 1 for all t ! 1. It is not difficult to make this type of argument rigorous. See White [102] and Heilmann =-=[39]-=- for the discrete case, and chapters 3 and 4 of Dubovskii [27] for the continuous case (in the more general setting of coagulation and fragmentation), and extensive references to the literature. The e... |

6 | Numerical solution of the Smoluchowski kinetic equation and asymptotics of the distribution function - Krivitsky - 1995 |

6 |
Limit theorems for the number of trees of a given size in a random forest
- Pavlov
- 1977
(Show Context)
Citation Context ...itial conditions. This construction was apparently first explicitly given by Pitman [63], although various formulas associated with it had previously been developed in the 32 combinatorial literature =-=[62, 105]-=- and the SM literature [40]. Here are two examples of simple formulas. The number D (N) (t) of clusters satisfies D (N) (t) \Gamma 1 d = Binomial(N \Gamma 1; e \Gammat ) The cluster-sizes (in random o... |

6 |
The development of structure in the expanding universe
- Silk, White
- 1978
(Show Context)
Citation Context ...ase separation in liquid mixtures [1]. ffl Polymerization [30]. The kernels shown in table 1 arise in such areas. Other areas include ffl Astronomy: formation of large-scale structure in the universe =-=[73]-=-; formation of protostellar clusters within galaxies [72, 14]; formation of planets within solar systems [101] ffl Biological entities, e.g. algae [2]. ffl Bubble swarms [79]. In all these settings, t... |

6 |
Solution of the coagulation equations in the case of a bilinear coefficient of adhesion of particles. Soviet Physics: Doklady
- Trubnikov
- 1971
(Show Context)
Citation Context ...4 for its interpretation. There has also been considerable attention paid to the general bilinear kernel K(x; y) = A + B(x + y) + Cxy, for which some more complicated explicit solutions are available =-=[86, 90, 77, 78]-=-. 2.3 Gelation Consider the second moment of cluster mass m 2 (t) = R 1 0 x 2 n(x; t)dx. Because a coalescence fx; yg ! x + y increases the sum-of-squares-of-masses by 2xy, the continuous Smoluchowski... |

5 |
Aggregation and gelation I analytical solution for CST and batch
- Smit, Hounslow, et al.
- 1994
(Show Context)
Citation Context ...he paper, time t is always a continuous variable; discrete and continuous refer to cluster masses. Table 1 gives examples of kernels used in the physical chemistry literature. The table is taken from =-=[74], who cite-=- references to each case. Most of these examples, and others, are explained in Drake [26] section 4.3. Note that we parametrize the "size" of clusters by mass x rather than length l; these k... |

5 | Solutions of Smoluchowski’s Coagulation Equation at Large Cluster Sizes”, Physica A - Dongen, Ernst - 1987 |

4 |
Gelation phenomena
- Jeon
- 1997
(Show Context)
Citation Context ...kernel is non-gelling. For some time it has been widely accepted in the SM literature [92, 95] that for fl ? 1 the kernel is gelling. The first general rigorous result was given only recently by Jeon =-=[44, 45]-=-, who showed that T gel ! 1 provided 9fl ? 1; 0 ! c 1 ; c 2 ! 1 such that c 1 (xy) fl=2sK(x; y)sc 2 xy: Moreover it is believed that for fl ? 2 the kernel is instantaneously gelling, that is T gel = 0... |

4 |
An inverse problem in agglomeration kinetics
- Muralidar, Ramkrishna
- 1986
(Show Context)
Citation Context ...arity [98]. ffl Numerical methods. See [49] for references, and [66] for Monte Carlo procedures. 14 ffl The inverse problem. How to estimate the rate kernel K from experimental or observational data. =-=[2, 59, 75]. (b) This-=- paper deals only with "pure coalescing" models. But much of the scientific literature considers coalescence together with other effects, in particular ffl fragmentation (splitting) [27] ffl... |

4 |
Stochastic algorithms for solving the smolouchovsky coagulation equation and application to aerosol growth simulation, Monte Carlo Methods and Applications 2
- Sablefeld, Rogasinsky, et al.
- 1996
(Show Context)
Citation Context ... ; xsx 0 densities) in the context of the "applied" kernels in table 1 can be found in [26] section 4.5. ffl Time to approach self-similarity [98]. ffl Numerical methods. See [49] for refere=-=nces, and [66] for Monte-=- Carlo procedures. 14 ffl The inverse problem. How to estimate the rate kernel K from experimental or observational data. [2, 59, 75]. (b) This paper deals only with "pure coalescing" models... |

4 |
Time-lag for attainment of the self-preserving particle size distribution by coagulation
- Vemury, Kusters, et al.
- 1994
(Show Context)
Citation Context ...rical methods that certain of the kernels arising in table 1 (e.g. K(x; y) = (x 1=3 + y 1=3 )(x \Gamma1=3 + y \Gamma1=3 ) [35]) appear to have self-similar solutions which are roughly log-Normal. See =-=[61, 98, 49]-=- for recent work. It is sometimes stated in the SM literature that this is to be expected when large-small coalescences (rather than large-large) predominate. I haven't found any convincing mathematic... |

4 |
Real space measurement of structure in phase separating binary fluid mixtures
- White, Wiltzius
- 1995
(Show Context)
Citation Context ...e regarded as old-fashioned: current research focuses on more physically realistic models. But to show this topic is not completely moribund, we mention recent experimental work of White and Wiltzius =-=[103]-=-, recounted for laymen in The Economist [1]. Certain liquids (e.g. olive oil and alcohol) mix at high temperature but separate at low temperature; how in detail does separation occur as temperature is... |

3 |
On the stochastic approach to cluster size distribution during particle coagulation I: Asymptotic expansion in the deterministic limit
- Donnelly, Simons
- 1993
(Show Context)
Citation Context ... \Gamma e \Gamma2x )+ q 2=3(1 \Gamma e \Gamma2x )Z (36) where Z is standard Normal, independent of B 0 . Variance calculations of this type for the constant-rate stochastic coalescent can be found in =-=[87, 25]-=-, only the latter making the explicit connection with Kingman's coalescent. There is a simple connection between Kingman's coalescent and Construction 4 (which exhibited the solution of the continuous... |

3 |
Scaling Theory for Ballistic Aggregation
- Jiang, Leyvraz
- 1993
(Show Context)
Citation Context ... often presupposes that physical parameters are within certain ranges, so taking mathematical limits may not make much sense if we cannot preserve such constraints. Conversely, there is SM discussion =-=[46, 85]-=- of models in the spirit of section 1.1 where the Smoluchowski coagulation equations do not provide satisfactory solutions over time-intervals of interest. A more substantial body of recent mathematic... |

3 |
Kinetics of aerosol formation in the free molecule regime in presence of condensable vapor
- Koutzenogii, Levykin, et al.
- 1996
(Show Context)
Citation Context ...rical methods that certain of the kernels arising in table 1 (e.g. K(x; y) = (x 1=3 + y 1=3 )(x \Gamma1=3 + y \Gamma1=3 ) [35]) appear to have self-similar solutions which are roughly log-Normal. See =-=[61, 98, 49]-=- for recent work. It is sometimes stated in the SM literature that this is to be expected when large-small coalescences (rather than large-large) predominate. I haven't found any convincing mathematic... |

3 |
Solutions and critical times for the polydisperse coagulation equation when a(x, y
- Spouge
- 1983
(Show Context)
Citation Context ...4 for its interpretation. There has also been considerable attention paid to the general bilinear kernel K(x; y) = A + B(x + y) + Cxy, for which some more complicated explicit solutions are available =-=[86, 90, 77, 78]-=-. 2.3 Gelation Consider the second moment of cluster mass m 2 (t) = R 1 0 x 2 n(x; t)dx. Because a coalescence fx; yg ! x + y increases the sum-of-squares-of-masses by 2xy, the continuous Smoluchowski... |

3 |
Stochastic coagulation equation and validity of the statistical coagulation equation
- Tanaka, Nakazawa
- 1993
(Show Context)
Citation Context ...a1 ; n j \Gamma1; n j+1 ; : : : ; n i+j \Gamma1 ; n i+j +1; n i+j+1 ; : : :) with rate K(i; j)n i n j =N . This model was perhaps first introduced by Marcus [56], and re-introduced by several authors =-=[36, 81]-=-, in particular by Lushnikov [54] as a model of gelation. We call it the Marcus-Lushnikov processsML (N) (t). The state of this process at time t may be written in two equivalent ways. We may write ML... |

3 |
Validity of the stochastic coagulation equation and runaway growth of protoplanets
- Tanaka, Nakazawa
- 1994
(Show Context)
Citation Context ...amma (N \Gamma k)f(k)b k (t) with b k (0) = 1 (k=1) . van Dongen and Ernst [93, 87] give the most detailed SM treatment of the special cases of the stochastic coalescent: see also Tanaka and Nagazawa =-=[81, 82]-=-. 5 Stochastic coalescence with general kernels In sections 4.2 - 4.4 we saw that the behavior of the Marcus-Lushnikov process and the stochastic coalescent for the three special kernels is mostly wel... |

3 | A mathematical study of the particle size distribution of coagulating disperse systems - Wang - 1966 |

2 |
Approximation and parameter estimation problems for algal aggregation models
- Ackleh, Fitzpatrick, et al.
- 1984
(Show Context)
Citation Context ...rmation of large-scale structure in the universe [73]; formation of protostellar clusters within galaxies [72, 14]; formation of planets within solar systems [101] ffl Biological entities, e.g. algae =-=[2]-=-. ffl Bubble swarms [79]. In all these settings, the clusters were physical entities in physical space, and have been studied as SM mathematics. A different area of application, which has caught the a... |

2 |
An Introduction to the Theory of Point Proceses
- Daley, Vere-Jones
- 2002
(Show Context)
Citation Context ...e density of L t as n(x; t)=G(t), the analysis above shows that n(x; t) satisfies 18 the Smoluchowski coagulation equation. But classical (and easy) results on thinning of renewal processes (see e.g. =-=[24]-=- Prop. 9.3.1) imply that G(t)L t converges in distribution to the exponential(1=m 1 (0)) distribution, establishing (25). Without this easy probability argument, rather tedious analysis [50] seems nee... |

2 |
Scaling dynamics of aerosol coagulation
- Olivier, Sorensen, et al.
- 1992
(Show Context)
Citation Context ...rical methods that certain of the kernels arising in table 1 (e.g. K(x; y) = (x 1=3 + y 1=3 )(x \Gamma1=3 + y \Gamma1=3 ) [35]) appear to have self-similar solutions which are roughly log-Normal. See =-=[61, 98, 49]-=- for recent work. It is sometimes stated in the SM literature that this is to be expected when large-small coalescences (rather than large-large) predominate. I haven't found any convincing mathematic... |

2 |
A statistical model for the initial stellar mass function
- Silk, Takahashi
- 1979
(Show Context)
Citation Context ... [30]. The kernels shown in table 1 arise in such areas. Other areas include ffl Astronomy: formation of large-scale structure in the universe [73]; formation of protostellar clusters within galaxies =-=[72, 14]-=-; formation of planets within solar systems [101] ffl Biological entities, e.g. algae [2]. ffl Bubble swarms [79]. In all these settings, the clusters were physical entities in physical space, and hav... |

2 |
Aggregation and gelation III: Numerical classification of kernels and case studies of aggregation and growth
- Smit, Hounslow, et al.
- 1995
(Show Context)
Citation Context ...arity [98]. ffl Numerical methods. See [49] for references, and [66] for Monte Carlo procedures. 14 ffl The inverse problem. How to estimate the rate kernel K from experimental or observational data. =-=[2, 59, 75]. (b) This-=- paper deals only with "pure coalescing" models. But much of the scientific literature considers coalescence together with other effects, in particular ffl fragmentation (splitting) [27] ffl... |

2 |
Dynamics and growth of particles undergoing ballistic coalescence
- Trizac, Hansen
- 1996
(Show Context)
Citation Context ...ph by Dubovskii [27] focuses narrowly on mathematical issues of existence and uniqueness of 7 solutions. Snapshots of recent research results and interests are provided by the introductions to papers =-=[70, 74, 85]-=-. We now give a quick overview of three aspects of the Smoluchowski coagulation equations; exact solutions, gelation and self-similar solutions. 2.2 Exact solutions It has long been recognized that th... |

1 |
On coagulation and the stellar mass function
- Allen, Bastien
- 1995
(Show Context)
Citation Context ... [30]. The kernels shown in table 1 arise in such areas. Other areas include ffl Astronomy: formation of large-scale structure in the universe [73]; formation of protostellar clusters within galaxies =-=[72, 14]-=-; formation of planets within solar systems [101] ffl Biological entities, e.g. algae [2]. ffl Bubble swarms [79]. In all these settings, the clusters were physical entities in physical space, and hav... |

1 |
The exact solution of the coagulation equation with kernel K ij
- Binglin
- 1987
(Show Context)
Citation Context ...me explicit solutions for the three special kernels in table 2 and the general bilinear kernel K(x; y) = A + B(x + y) + Cxy, under polydisperse initial conditions, are discussed in [26] section 6.3., =-=[17, 68, 71, 77, 78, 84]-=-. Some discussion of physically reasonable or mathematically tractable initial configurations (e.g. in the continuous setting, the Gamma, log-Normal or n(x; 0) = x \Gammafi ; xsx 0 densities) in the c... |

1 |
Mathematical Theory of Coagulation. Global Analysis Research
- Dubovskii
- 1994
(Show Context)
Citation Context ... 59, 75]. (b) This paper deals only with "pure coalescing" models. But much of the scientific literature considers coalescence together with other effects, in particular ffl fragmentation (s=-=plitting) [27]-=- ffl removal of clusters (sedimentation, condensation, crystalization) [41] ffl continuous addition of new particles. [74] In fact, much of the literature we cite relating to existence of solutions of... |

1 | Coagulation in a continuously stirred tank reactor
- Hendriks, Ziff
- 1985
(Show Context)
Citation Context ... of the scientific literature considers coalescence together with other effects, in particular ffl fragmentation (splitting) [27] ffl removal of clusters (sedimentation, condensation, crystalization) =-=[41]-=- ffl continuous addition of new particles. [74] In fact, much of the literature we cite relating to existence of solutions of the Smoluchowski coagulation equations deals with such more general settin... |

1 |
Some condensation processes of McKean type
- Knight
- 1971
(Show Context)
Citation Context ... = e 2t . In this connection, it has long been known that the two kernels x + y and max(x; y) have no self-similar solution satisfying (14) and R 1 0 /(x)dx ! 1: see Drake [26] section 6.4 and Knight =-=[48]-=- respectively. If a unique self-similar solution exists, it is natural to expect convergence to self-similarity from rather general initial configurations. See section 3.1 for a simple proof for K(x; ... |

1 |
Theoretical aspects of the size distribution of fog droplets
- Schumann
- 1940
(Show Context)
Citation Context ...amma1 B(t; x) 0st ! 1 [76] 0st ! 1 [37] 0sts1 [57] continuous 4t \Gamma2 exp(\Gamma2x=t) (2) \Gamma1=2 e \Gammat x \Gamma3=2 e \Gammae \Gamma2t x=2 (2) \Gamma1=2 x \Gamma5=2 e \Gammat 2 x=2 0 ! t ! 1 =-=[67]-=- \Gamma1 ! t ! 1 \Gamma1 ! t ! 0 We included the conventional attributions of 4 of these formulas, which have been rediscovered many times. The remaining two continuous solutions arise by rescaling ti... |

1 |
A model for simultaneous coalescence of bubble clusters
- Stewart, Crowe, et al.
- 1993
(Show Context)
Citation Context ...structure in the universe [73]; formation of protostellar clusters within galaxies [72, 14]; formation of planets within solar systems [101] ffl Biological entities, e.g. algae [2]. ffl Bubble swarms =-=[79]-=-. In all these settings, the clusters were physical entities in physical space, and have been studied as SM mathematics. A different area of application, which has caught the attention of TP mathemati... |

1 |
An exact solution of the Smoluchowski equation and its correspondence to the solution of the continuous equation
- Treat
- 1990
(Show Context)
Citation Context |

1 |
Size distribution in the polymerization model A f RB g
- Dongen, Ernst
- 1984
(Show Context)
Citation Context |

1 | Tail distribution of large clusters from the coagulation equation - Dongen, Ernst - 1987 |

1 |
Kinetics of gelation and
- Ziff, Ernst, et al.
- 1983
(Show Context)
Citation Context ...rge clusters at the critical time, rather than in absolute terms. The Smoluchowski coagulation equations with K(x; y) = xy have been studied many times in the SM literature. In particular, Ziff et al =-=[107, 31] observe (41) a-=-rising as a "scaling limit" as t " 1 for the monodisperse initial conditions, and discuss the extent to which this remains true for more general 37 initial conditions, and also discuss ... |