Stochastic, analytic and numerical aspects of coagulation processes

Mathematics and Computers in Simulation 62 (2003) 265–275

Stochastic, analytic and numerical aspects ofcoagulation processes

Wolfgang WagnerWeierstrass Institute for Applied Analysis and Stochastics, Mohrenstrasse 39, D-10117 Berlin, Germany

Abstract

In this paper, we review recent results concerning stochastic models for coagulation processes and their relation-ship to deterministic equations. Open problems related to the gelation effect are discussed. Finally, we present somenew conjectures based on numerical experiments performed with stochastic algorithms.© 2003 IMACS. Published by Elsevier Science B.V. All rights reserved.

Keywords:Coagulation; Gelation effect; Stochastic algorithms

1. Introduction

The phenomenon of coagulation occurs in a wide range of applications, e.g. in physics (aggregationof colloidal particles, growth of gas bubbles), meteorology (merging of drops in atmospheric clouds,aerosol transport), chemistry (reacting polymers, soot formation) and astrophysics (formation of starsand planets). We refer to the survey papers[2,8] for more details. The time evolution of the averageconcentration of particles of a given size in some spatially homogeneous physical system is described bySmoluchowski’s coagulation equation[36]

∂

∂tc(t, x) = 1

2

x−1∑y=1

K(x − y, y)c(t, x − y)c(t, y)−∞∑y=1

K(x, y)c(t, x)c(t, y), (1)

wheret ≥ 0 andx = 1,2, . . . . Concentration of particles of sizex increases as a result of coagulationof particles of sizesx − y and y. It decreases if particles of sizex merge with any other particles.The intensity of the process is governed by the (non-negative and symmetric) coagulation kernelK

representing properties of the physical medium.The common stochastic model related to the coagulation equation (1) is a Markov jump process where

two different clusters of sizex and y merge into a single cluster of sizex + y with rateK(x, y)

E-mail address:[email protected] (W. Wagner).URL: http://www.wias-berlin.de/∼wagner

0378-4754/03/$ – see front matter © 2003 IMACS. Published by Elsevier Science B.V. All rights reserved.PII: S0378-4754(02)00236-7

266 W. Wagner / Mathematics and Computers in Simulation 62 (2003) 265–275

(cf. [16,26,28]). The basic relationship between the stochastic model and the deterministic equationis given by the law of large numbers. Qualitative properties of the coagulation equation and its general-izations have been successfully studied using the stochastic approach. We refer to[9,11,17,20,21,30,31]concerning recent results, and especially to[2] as an excellent review. Stochastic particle systems playalso an important role in the numerical treatment ofEq. (1). Many stochastic algorithms are based on theclassical direct simulation process (cf.[7,10,14,15,18,19,34]). Various numerical methods for the coag-ulation equation are reviewed in[33], where also an extended list of references is given. Some stochasticalgorithms (cf.[3,23–25,32]) contain an additional approximation parameter (time step), thus providingsolutions to time discretized versions ofEq. (1).

The purpose of this paper is to review recent results concerning stochastic models for coagulationprocesses and their relationship to deterministic equations. We also discuss open problems related tothe gelation phenomenon. Finally, we present some new conjectures based on numerical experimentsperformed with stochastic algorithms.

2. Stochastic models and approximation

We start from considering the continuous coagulation–fragmentation equation

∂

∂tc(t, x) = 1

2

∫ x

0K(x − y, y)c(t, x − y)c(t, y)dy −

∫ ∞

0K(x, y)c(t, x)c(t, y)dy

+∫ ∞

0f (x, y)c(t, x + y)dy − 1

2

∫ x

0f (x − y, y)c(t, x)dy, (2)

with initial condition c(0, x) = c0(x) ≥ 0, which describes the time evolution of the average concen-tration of particles of sizex > 0. Here, the non-negative and symmetric functionf (x, y) denotes thefragmentation rate of an(x + y)-cluster into clusters of sizex andy. In addition to the effect of coagu-lation, the concentrationc(x, t) increases by fragmentation of an(x + y)-cluster into clusters of sizexandy (third term) and decreases by fragmentation of anx-cluster into clusters of sizey < x andx − y(fourth term). The combined coagulation–fragmentation equation appeared in[29].

Multiplication with some test functionϕ and integration with respect to the size variablex transformthe right-hand side ofEq. (2)into∫ ∞

0

∫ ∞

0

[1

2ϕ(x + y)− ϕ(x)

]K(x, y)c(t, x)c(t, y)dy dx

+∫ ∞

0

∫ x

0

[ϕ(y)− 1

2ϕ(x)

]f (x − y, y)c(t, x)dy dx. (3)

Here, the identity∫ ∞

0

∫ ∞0 ψ(x, y)dy dx = ∫ ∞

0

∫ x0 ψ(x− y, y)dy dx has been used. Having in mind (3),

one obtains a weak version of the coagulation–fragmentationEq. (2),

d

dt

∫Z

ϕ(x)P (t,dx) =∫Z

∫Z

[1

2ϕ(x + y)− ϕ(x)

]K(x, y)P (t,dy)P (t,dx)

+∫Z

∫Z

[ϕ(y)− 1

2ϕ(x)

]F(x,dy)P (t,dx). (4)

W. Wagner / Mathematics and Computers in Simulation 62 (2003) 265–275 267

The size space is eitherZ = (0,∞), corresponding toEq. (2), or Z = {1,2, . . . }, corresponding toEq. (1). An elementP of the space of continuous measure-valued functionsC([0,∞),M(Z)) is calleda solution if it satisfiesEq. (4), for all continuous function with compact supportϕ ∈ Cc(Z), and∫ t

0

[∫Z

∫Z

K(x, y)P (s,dx)P (s,dy)+∫Z

F(x,Z)P (s,dx)

]ds <∞ (5)

for all t ≥ 0.The fragmentation measureF is concentrated on{y ∈ Z : y < x} and satisfies the symmetry assump-

tion ∫Z

ϕ(y)F (x,dy) =∫Z

ϕ(x − y)F (x,dy), ∀x ∈ Z, (6)

for measurable functionsϕ that are bounded on each interval(0, x). Note thatEq. (6), with ϕ(y) = y,implies∫

Z

yF(x,dy) = x2F(x,Z), ∀x ∈ Z. (7)

In particular, the choice

F(x,dy) = χ(0,x)(y)f (x − y, y)dy,whereχA denotes the indicator function of a setA, corresponds to expression(3).

The solutionP(t,dx) of Eq. (4)represents the flow of concentration in the size spaceZ. The total massof the system is determined as

∫Z

xP(t,dx). We call the function

P (t,dx) = xP(t,dx), t ≥ 0, (8)

the mass flow and consider themass flow equation

d

dt

∫Z

ϕ(x)P (t,dx) =∫Z

∫Z

[ϕ(x + y)− ϕ(x)]K(x, y)y

P (t,dy)P (t,dx)

+∫Z

∫Z

[ϕ(y)− ϕ(x)] yxF(x,dy)P (t,dx). (9)

A function P ∈ C([0,∞),M(Z)) is called a solution if it satisfiesEq. (9)for all ϕ ∈ Cc(Z) and∫ t

0

[∫Z

∫Z

K(x, y)

xyP (s,dx)P (s,dy)+

∫Z

F(x,Z)

xP (s,dx)

]ds <∞, (10)

for all t ≥ 0. Eq. (9)was studied in[12] for the pure coagulation case. The discrete version ofEq. (9)(with F = 0) has been used in[3] for constructing a discretized in time stochastic algorithm forEq. (1).

If some measure-valued functionsP and P satisfy Eq. (8), then P is a solution of the mass flowequation (9) if and only ifP is a solution of the coagulationEq. (4). Indeed,P ∈ C([0,∞),M(Z)) iffP ∈ C([0,∞),M(Z)) (vague topology onM(Z)), andEq. (5)is satisfied iff (10) is satisfied. Moreover,


using symmetry, one obtains∫Z

∫Z

[1

2ϕ(x + y)(x + y)− ϕ(x)x

]K(x, y)P (t,dx)P (t,dy)

=∫Z

∫Z

[ϕ(x + y)x − ϕ(x)x]K(x, y)xy

P (t,dx)P (t,dy)

=∫Z

∫Z

[ϕ(x + y)− ϕ(x)]K(x, y)y

P (t,dx)P (t,dy),

for anyϕ ∈ Cc(Z). Finally, usingEq. (7), one obtains∫Z

∫Z

[yϕ(y)− 1

2xϕ(x)

]F(x,dy)P (t,dx)

=∫Z

∫Z

[y

xϕ(y)− 1

2ϕ(x)

]F(x,dy)P (t,dx) =

∫Z

∫Z

[ϕ(y)− ϕ(x)] yxF(x,dy)P (t,dx),

and the equivalence follows.First we approximate the solution ofEq. (4)by a sequence of particle systems. For everyn = 1,2, . . . ,

define

Sn ={p = 1

n

N∑i=1

δxi , xi ∈ Z, N = 1,2, . . .

},

and consider anSn-valued jump processUn(t) with generatorK + F, where

KΦ(p) = 1

2n

∑1≤i,j≤N

[Φ(JK(p, i, j))−Φ(p)]K(xi, xj ), (11)

with

JK(p, i, j) = p + 1

n(δxi+xj − δxi − δxj ), (12)

is the coagulation operator, and

FΦ(p) = 1

2

N∑i=1

∫Z

[Φ(JF (p, i, y))−Φ(p)]F(xi,dy),

with

JF (p, i, y) = p + 1

n(δy + δxi−y − δxi ),

is the fragmentation operator. For

Φ(p) =∫Z

ϕ(z)p(dz), p ∈ Sn, ϕ ∈ Cc(Z), (13)

one obtains∫Z

ϕ(z)Un(t,dz) =∫Z

ϕ(z)Un(0,dz)+∫ t

0(KΦ + FΦ)(Un(s))ds +Mnϕ(t), (14)


whereMnϕ is some martingale term. The representations

KΦ(p) =∫Z

∫Z

K(x, y)

[1

2ϕ(x + y)− ϕ(x)

]p(dx)p(dy)

and (cf.Eq. (6))

FΦ(p) =∫Z

∫Z

[ϕ(y)− 1

2ϕ(x)

]F(x,dy)p(dx)

indicate the connection betweenEqs. (4) and (14), suggesting the property

limn→∞U

(n)(t) = P(t).

Rigorous results concerning this transition can be found, e.g. in[11,20,30]. Note that the proofs of theseresults provide existence theorems for the deterministic limiting equation as a by-product.

Next, we construct a particle approximation to the solution of the mass flow equation (9), which, dueto the equivalence of the equations, provides an alternative approximation to the solution ofEq. (4). Weintroduce the generatorK + F, where

KΦ(p) = 1

n

∑1≤i,j≤N

[Φ(JK(p, i, j))−Φ(p)]K(xi, xj )xj

, (15)

with

JK(p, i, j) = p + 1

n(δxi+xj − δxi ), (16)

is the modified coagulation operator, and

FΦ(p) =N∑i=1

∫Z

[Φ(JF (p, i, y))−Φ(p)] yxiF (xi,dy),

with

JF (p, i, y) = p + 1

n(δy − δxi ),

is the modified fragmentation operator. Considering test function(13), one obtains the expressions

KΦ(p) =∫Z

∫Z

[ϕ(x + y)− ϕ(x)]K(x, y)y

p(dx)p(dy)

and

FΦ(p) =∫Z

∫Z

[ϕ(y)− ϕ(x)] yxF(x,dy)p(dx)

so that a martingale representation analogous toEq. (14)indicates the connection withEq. (9). A rigoroustransition has been carried out in[12] for the pure coagulation case.


Note the following intuitive derivation of the approximating particle system for the mass flow equation.Consider a system of independent particles determined by the generator

QΦ(p) =N∑i=1

∫Z

[Φ(JQ(p, i, y))−Φ(p)]Q(xi,dy), (17)

with

JQ(p, i, y) = p + 1

n(δxi+y − δxi ),

whereQ denotes some jump measure. Obviously, the corresponding limiting equation is

d

dt

∫Z

ϕ(x)P (t,dx) =∫Z

∫Z

[ϕ(x + y)− ϕ(x)]Q(x,dy)P (t,dx). (18)

Note that the mass flowEq. (9)has the form (18) with appropriately chosenQ. In particular, the coagulationterm corresponds to

Q(x,dy) = K(x, y)y

P (t,dy).

After replacing formallyP by the empirical measure of the particle system, the generator(17) takes theform (15). This connection suggests how to construct a non-linear Markov process related to the massflow equation (9). Such a process is determined by a stochastic equation with coefficients depending onthe law of the solution. This approach has been carried out in[5,6] for the pure coagulation case, and in[22] for the case including fragmentation.

3. Numerical algorithms

Note that a basic element of the convergence proofs mentioned in the previous section is uniquenessof the solution to the limiting equation. However this uniqueness has been established[30] only up to thegelation point

tgel = inf {t ≥ 0 :m1(t) < m1(0)}, where m1(t) =∞∑x=1

xc(t, x), (19)

which is finite for sufficiently fast increasing coagulation kernels. Thus, the problem of convergenceafter that point is open for general kernels, and numerical observations concerning the behaviour of thestochastic processes are of special interest.

In this section, we discuss the problem how to determine the time evolution of the massm1 for gellingkernels, using stochastic algorithms. Here, we restrict our considerations to the case of pure coagulation.

Kernels satisfying

K(Cx,Cy) = CεK(x, y), ∀C, x, y > 0, (20)

are called homogeneous, with exponent of homogeneityε. The following properties have been proved inparticular cases and are widely expected to be true in general (cf.[20,21], and the recent paper[13]). If


ε ≤ 1 then there is no gelation, whileε > 1 implies gelation, i.e.tgel <∞. Moreover, ifε > 2 then thereis instantaneous gelation, i.e.tgel = 0. The special case of the multiplicative kernel

K(x, y) = xy (21)

has been intensively studied in the framework of random graph theory (see[2], Section 4.4). In particular,it has been proved in[4] that the standard stochastic model (cf.Eqs. (11) and (12)) converges to adeterministic limit, which coincides with the solution of the Smoluchowski equation (1) before thegelation point, but is different from the solution after that point. Moreover, it has been established thatthe asymptotic behaviour of the maximal single clusterM

(n)1 (t) in the system changes qualitatively at the

gelation point, namely it becomes of ordern after tgel while it is of lower order before that point. Thus,for the kernel(21), one obtains

limn→∞

M(n)1 (t)

n> 0, ∀ t > tgel, (22)

but the property (cf.Eq. (19))

m1(t) = m1(0)− limn→∞

M(n)1 (t)

n, ∀ t ≥ 0, (23)

does not hold.The common stochastic algorithm for the coagulation equation (4) is based on the process with the

generator (11), (12). Due to its physical interpretation, we call this simulation proceduredirect simulationalgorithm. Note that, if the current state of the process is represented by particlesx1(t), ... , xN(t)(t), thenfunctionals are approximated by∫

Z

ϕ(x)P (t,dx) ∼ 1

n

N(t)∑i=1

ϕ(xi(t)), t ≥ 0, (24)

for some test functionϕ. Obviously, this algorithm is mass-preserving. However, property(22)suggeststo study the gelation problem for general coagulation kernels using the maximal component in the system,namely measuring the quantity

M(n)1 (t)

n, t ≥ 0, (25)

for different values ofn.The simulation procedure based on the process with the generator (15), (16) is calledmass flow algo-

rithm. If the current state of the process is represented by particlesx1(t), ... , xN(t)(t), then functionals ofthe solution are approximated by (contrast this with (24))

∫Z

ϕ(x)P (t,dx) ∼ 1

n

N(t)∑i=1

ϕ(xi(t))

xi(t), t ≥ 0. (26)

An interesting aspect of the mass flow model is the emergence of infinite clusters in finite time for gellingkernels. It has been conjectured in[12] that the (random) explosion timesτn∞ of the stochastic systemconverge (asn→ ∞) to the gelation timetgel. This would connect the gelation effect (a property of the


limiting equation) with the explosion phenomenon of a stochastic process, thus representing the physicalinterpretation of gelation. In this respect, we refer to recently announced results[27] concerning explosionbehaviour of some appropriately scaled tagged particle in the direct simulation process.

In order to be able to perform calculations with the mass flow algorithm beyond the gelation point, weintroduce some truncation parameterbn such that particles with size bigger thanbn are removed from thesystem. This corresponds to a modification of the coagulation jump(16)by

p − 1

nδxi + χ(0,bn)(xi + xj )

1

nδxi+xj .

Since the truncated mass flow algorithm is not mass-preserving, (26) suggests a natural approximationof the mass, namely

N (n)(t)

n, t ≥ 0. (27)

It has been conjectured in[12] that

m1(t) = limn→∞

N (n)(t)

n, ∀ t ≥ 0. (28)

The numerical algorithms consist in generating trajectories of the corresponding process and calculatingestimators(25) and (27), respectively. The results in the figures below are averaged over ten independentruns. Both algorithms are initialized according to the monodisperse initial condition

c0(1) = 1, c0(x) = 0, x = 2,3, . . . .

For an efficient generation of trajectories the coagulation kernel is replaced by some majorant kernel, andfictitious jumps are introduced. This leads to an easy calculation of the time step and to an independentgeneration of the collision partners. For more details of this common numerical approach we refer to[10].

As test examples, we consider the two classes of kernels

K(1)α,β(x, y) = 1

2[xαyβ + xβyα], 0 ≤ β ≤ α ≤ 1, (29)

and[37]

K(2)γ (x, y) = 2(xy)γ

(x + y)γ − xγ − yγ , 1< γ ≤ 2. (30)

The exponents of homogeneity (cf.Eq. (20)) areε = α + β for the kernel(29)andε = γ for the kernel(30). Note that the multiplicative kernel(21) is contained in both classes(29) (α = β = 1) and(30)(γ = 2). For kernel(30), it has been proved in[1] that the moment of orderγ explodes att = 1, “stronglysuggesting”tgel = 1 for all γ .

An additional parameter characterizing growth properties of the coagulation kernel was used in[35],namely a coefficientκ such that

limy→∞

1

yε−κlimx→∞

K(x, y)

xκ∈ (0,∞), ε

2≤ κ ≤ ε.


Note thatκ = α for the kernel(29). An easy computation implies

limx→∞

K(2)γ (x, y)

x= 2

γyγ−1

so thatκ = 1 for the kernel(30).Spouge’s conjecture[34] says that for kernels(29) with α = β ∈ (0.5,1) the mass of the gel is

approximated by the maximal cluster, i.e.Eq. (23)and, in particular,(22) hold. On the contrary, there isAldous’ conjecture ([2], Section 5.2) thatM(n)1 (t) = o(n) after the gelation point, i.e.(22)does not hold,for homogeneous kernels(20)with exponent 1< ε < 2.

Numerical experiments in[10,12] have indicated that for the kernel(30) (noteε = γ andκ = 1)property(22) holds, butEq. (23)does not. On the other hand, for the kernel(29) with α = β ∈ (0.5,1)(noteε = 2α andκ < 1) numerical observations did not provide any suggestion concerning the asymptoticbehaviour. Similar slow convergence has been observed in[19] for the caseα = β > 1, related toinstantaneous gelation[21]. Thus, the casesκ = 1 andκ < 1 seem to be qualitatively different, whenε ∈ (1,2). We conjecture that in the caseκ = 1 property(22) holds, butEq. (23)does not, in analogywith the case of the multiplicative kernel(21). Numerical results for the kernel(29) with α = 1 andβ = 0.5 (noteε = 1.5 andκ = 1) are displayed inFig. 1. On the left-hand side curves for the estimator(25) are given forn = 105, 106, 107. On the right-hand side curves for the estimator(27) are given forn = 103, 104, 105 andbn = 100n. In both graphs numerical convergence is observed and, taking intoaccount(28), the quantitative results provide a further illustration of the conjecture.

Next, we compare the kernel

1

2β − 1K(1)1,β(x, y) = 1

2(2β − 1)[xyβ + xβy] (31)

and the kernelK(2)1+β (cf. Eq. (30)). Note that for these kernels bothε andκ, as well as the value atx = y = 1, are the same. Numerical results are displayed inFig. 2for β = 0.5 (left) andβ = 0.8 (right).They were obtained using the estimator(27) with n = 105 andbn = 10n. The dashed lines correspondto the kernel(31), and the solid lines to the kernel(30) with γ = 1 + β. The curves for both kernelsare rather similar to each other. In particular, using a scaling argument andtgel = 1 for kernel(30), thequantity 1/(2β − 1) provides a surprisingly good approximation for the gelation time corresponding tokernel(29)with α = 1 and different values ofβ.

Fig. 1. EstimatorsM(n)

1 (t)/n (left) andN (n)(t)/n (right) for kernel(1/2)[xy0.5 + x0.5y].


Fig. 2. EstimatorN (n)(t)/n for kernels [1/(2β − 1)]K(1)1,β (dashed line) andK(2)1+β (solid line) withβ = 0.5 (left) andβ = 0.8(right).

Acknowledgements

The author is grateful to Andreas Eibeck (Berlin) for many interesting discussions on the subject ofthis paper, and for assistance in the numerical experiments.

References

[1] D. Aldous, Emergence of the giant component in special Marcus–Lushnikov processes, Random Struct. Algorithms 12 (2)(1998) 179–196.

[2] D.J. Aldous, Deterministic and stochastic models for coalescence (aggregation and coagulation): a review of the mean-fieldtheory for probabilists, Bernoulli 5 (1) (1999) 3–48.

[3] H. Babovsky, On a Monte Carlo scheme for Smoluchowski’s coagulation equation, Monte Carlo Methods Appl. 5 (1)(1999) 1–18.

[4] E. Buffet, J.V. Pulé, Polymers and random graphs, J. Statist. Phys. 64 (1–2) (1991) 87–110.[5] M. Deaconu, N. Fournier, E. Tanré, A pure jump Markov process associated with Smoluchowski’s coagulation equation,

Technical Report 6, Institut Elie Cartan, Nancy, 2001.[6] M. Deaconu, N. Fournier, E. Tanré, Study of a stochastic particle system associated with the Smoluchowski coagulation

equation, Technical Report 15, Institut Elie Cartan, Nancy, 2001.[7] E.R. Domilovskii, A.A. Lushnikov, V.N. Piskunov, Monte Carlo simulation of coagulation processes, Dokl. Akad. Nauk

SSSR, Ser. Phys. Chem. 240 (1) (1978) 108–110 (in Russian).[8] R.L. Drake, A general mathematical survey of the coagulation equation, in: G. Hidy, J. Brock (Eds.), Topics in Current

Aerosol Research, Part 2, Pergamon Press, Oxford, 1972, pp. 201–376.[9] R. Durrett, B.L. Granovsky, S. Gueron, The equilibrium behavior of reversible coagulation–fragmentation processes, J.

Theoret. Probab. 12 (2) (1999) 447–474.[10] A. Eibeck, W. Wagner, An efficient stochastic algorithm for studying coagulation dynamics and gelation phenomena, SIAM

J. Sci. Comput. 22 (3) (2000) 802–821.[11] A. Eibeck, W. Wagner, Approximative solution of the coagulation–fragmentation equation by stochastic particle systems,

Stochastic Anal. Appl. 18 (6) (2000) 921–948.[12] A. Eibeck, W. Wagner, Stochastic particle approximations for Smoluchowski’s coagulation equation, Ann. Appl. Probab.

11 (4) (2001) 1137–1165.[13] M. Escobedo, S. Mischler, B. Perthame, Gelation in coagulation and fragmentation models, Technical Report DMA-01-19,

Ecole Normale Superieure, Paris, 2001.[14] A.L. Garcia, C. Van den Broeck, M. Aertsens, R. Serneels, A Monte Carlo simulation of coagulation, Phys. A 143 (1987)

535–546.[15] D.N. Gillespie, An exact method for numerically simulating the stochastic coalescence process in a cloud, J. Atm. Sci. 32

(1975) 1977–1989.


[16] D.T. Gillespie, The stochastic coalescence model for cloud droplet growth, J. Atm. Sci. 29 (1972) 1496–1510.[17] S. Gueron, The steady-state distributions of coagulation–fragmentation processes, J. Math. Biol. 37 (1) (1998) 1–27.[18] F. Guias, A Monte Carlo approach to the Smoluchowski equations, Monte Carlo Methods Appl. 3 (4) (1997) 313–326.[19] F. Guias, A direct simulation method for the coagulation–fragmentation equations with multiplicative coagulation kernels,

Monte Carlo Methods Appl. 5 (4) (1999) 287–309.[20] I. Jeon, Existence of gelling solutions for coagulation–fragmentation equations, Commun. Math. Phys. 194 (3) (1998)

541–567.[21] I. Jeon, Spouge’s conjecture on complete and instantaneous gelation, J. Statist. Phys. 96 (5–6) (1999) 1049–1070.[22] B. Jourdain, Non-linear processes associated with the discrete Smoluchowski coagulation fragmentation equation, Technical

Report 207, ENPC-CERMICS, Marne-la-Vallé, 2001.[23] A. Kolodko, K. Sabelfeld, W. Wagner, A stochastic method for solving Smoluchowski’s coagulation equation, Math.

Comput. Simul. 49 (1–2) (1999) 57–79.[24] A.A. Kolodko, W. Wagner, Convergence of a Nanbu type method for the Smoluchowski equation, Monte Carlo Methods

Appl. 3 (4) (1997) 255–273.[25] K. Liffman, A direct simulation Monte Carlo method for cluster coagulation, J. Comput. Phys. 100 (1992) 116–127.[26] A.A. Lushnikov, Certain new aspects of the coagulation theory, Izv. Akad. Nauk SSSR, Ser. Fiz. Atm. Okeana 14 (10)

(1978) 738–743.[27] P. March, A tagged particle in coagulation processes, in: Stochastic Models for Coagulation Processes, Mathematisches

Forschungsinstitut Oberwolfach, Report 39, 2001, p. 6 (seehttp://www.mfo.de/Meetings/#T0134c).[28] A.H. Marcus, Stochastic coalescence, Technometrics 10 (1) (1968) 133–148.[29] Z.A. Melzak, A scalar transport equation, Trans. Am. Math. Soc. 85 (1957) 547–560.[30] J.R. Norris, Smoluchowski’s coagulation equation: uniqueness, non-uniqueness and a hydrodynamic limit for the stochastic

coalescent, Ann. Appl. Probab. 9 (1) (1999) 78–109.[31] J.R. Norris, Cluster coagulation, Commun. Math. Phys. 209 (2) (2000) 407–435.[32] K.K. Sabelfeld, Stochastic models for coagulation of aerosol particles in intermittent turbulent flows, Math. Comput. Simul.

47 (2–5) (1998) 85–101.[33] K.K. Sabelfeld, S.V. Rogazinskii, A.A. Kolodko, A.I. Levykin, Stochastic algorithms for solving Smoluchowski coagulation

equation and applications to aerosol growth simulation, Monte Carlo Methods Appl. 2 (1) (1996) 41–87.[34] J.L. Spouge, Monte Carlo results for random coagulation, J. Colloid Interf. Sci. 107 (1) (1985) 38–43.[35] P.G.J. van Dongen, M.H. Ernst, Cluster size distribution in irreversible aggregation at large times, J. Phys. A 18 (14) (1985)

2779–2793.[36] M. von Smoluchowski, Drei Vorträge über Diffusion, Brownsche Molekularbewegung und Koagulation von Kolloidteilchen.

Phys. Z. 17 (1916) 557–571, 585–599[37] R.M. Ziff, Kinetics of polymerization, J. Statist. Phys. 23 (2) (1980) 241–263.

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Stochastic, analytic and numerical aspects of coagulation processes