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Induced gelation in a two-site spatial coagulation

model

Rainer Siegmund-Schultze∗ and Wolfgang Wagner†

April 9, 2008

Abstract

A two-site spatial coagulation model is considered. Particles ofmasses m and n at the same site form a new particle of mass m +n at rate mn . Independently, particles jump to the other site at aconstant rate. The limit (for increasing particle numbers) of this modelis expected to be non-deterministic after the gelation time, namely, oneor two giant particles randomly jump between the two sites. Moreover,a new effect of induced gelation is observed - the gelation happeningat the site with the larger initial number of monomers immediatelyinduces gelation at the other site. Induced gelation is shown to beof logarithmic order. The limiting behaviour of the model is derivedrigorously up to the gelation time, while the expected post-gelationbehaviour is illustrated by a numerical simulation.

1 Introduction

We consider a minimal spatial model of coagulation with the gelling kernelmn describing the coagulation rate and an independent motion of particlesbetween two sites 0 and 1 which happens at a constant rate for each particleirrespective of its mass. So the basic model is a continuous time Markovjump process

{ξt,m,j} , t ≥ 0 , m = 1, 2, . . . , j = 0, 1 .

∗Technische Universitat Berlin, Institut fur Mathematik, Straße des 17. Juni 136,

D-10623 Berlin, Germany†Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, D-10117

Berlin, Germany

1

A state of the process is given by two sequences of non-negative integers,both with only a finite number of non-zero entries, describing the numbers ofparticles of different masses at the two sites. As for the initial distribution,we consider the deterministic state given by the two length-one sequences{N0}, {N1} meaning a start with N0, N1 monomers at the two sites. Denotethe total number of monomers by

N = N0 + N1 .

Given a state (ξm,j) , the jump rate (depending on the scaling parameter N)has the form

1

2N(N − 1) + κN

∑

m,j

ξm,j . (1)

The transition is defined as follows: Either, with probability

κ∑

m,j

ξm,j

1

2(N − 1) + κ

∑

m,j

ξm,j

−1

,

a randomly selected particle (an aggregation of monomers) jumps to theopposite site, or with the remaining probability two monomers are randomlychosen and the corresponding particles coalesce provided they are not thesame particle and they are at the same site. This means that an unorderedpair of different particles of masses m and n at the same site coalesces at ratemn . Hence our model is a (minimal) spatial generalization of the Marcus-Lushnikov stochastic coalescent ML(N)(t) with monomeric initial condition([13], [12]).

It is clear that we can describe the evolution as well as a colored graphevolution (with two colors): Start with N0 isolated red nodes and N1 isolatedgreen nodes. Add at constant rate 1 an edge for each unordered pair ofnodes, where edges between differently colored nodes are instantly removedas well as multiple edges, and let (maximal) connected components changetheir color at constant rate κN .

We investigate the limiting behaviour of this model for N tending toinfinity while

N1/N0 → λ , (2)

where λ is a given parameter. The expected behaviour is notably morecomplex compared to the well-known case, where all particles live at thesame site (mean-field model or homogeneous model). The mean-field model

2

has a deterministic limit, described by a modified Smoluchowski system ofdifferential equations (with the modification being urged by the effect ofgelation, meaning that, after rescaling time by t → t/N , at gelation timetgel = 1 a giant particle of mass comparable to the total mass suddenlyappears in a phase transition and modifies the evolution). The more generalmodel considered here is expected to have a stochastic limit after gelationtime, caused by the fact that the motion of (at most two) giant particles isno longer governed by the law of large numbers. Moreover, a new effect ofinduced gelation is observed: The gelation happening at the site with thelarger initial concentration of monomers immediately induces a gelation atthe other site. We give a rigorous derivation of the limiting behaviour up tothe gelation time which is based mainly on the cutting techniques developedin [14], [15].

We emphasize that a mathematically rigorous derivation of the limitingbehaviour after gelation in our model can be expected to be quite difficultsince the existing techniques either are based on a martingale approach andrely heavily upon the finiteness of the expected particle size per monomer([14]), which breaks down at the gelation time, or they make use of randomgraph techniques ([2]) which admit no (or at least no immediate) generaliza-tion to the spatial model. Also, as will be seen later, the speed of convergenceto the limit process is very slow, namely of logarithmic order in the totalparticle number, because induced gelation is an effect of logarithmic effec-tiveness. Hence, as far as post-gelation behaviour is concerned (Section 4),this account is of mainly phenomenological nature and will deliver evidencefor the expected phenomena by numerical simulation. In the rigorous partwe use the notions ’gelation time’ and ’gelation’ in a weaker sense, meaningthe explosion of the second moment.

2 Limiting behaviour of the spatial model before

gelation

Here we study the two-site model at pre-gelation times. The parameter λindicates the dependence on the initial condition (2).

Theorem 1 Let κ > 0. Then there is a non-random time tλgel > 0,

such that the rescaled quantities N−1ξt/N,m,j of our model with asymptoticparameter λ do uniformly converge in probability to deterministic quantities

3

cλt,m,j fulfilling a two-site analogue of Smoluchowski’s coagulation equation:

·cλ

t,m,j =1

2

∑

0<n<m

n(m − n)cλt,n,jc

λt,m−n,j −

∑

n>0

mncλt,m,jc

λt,n,j (3)

−κcλt,m,j + κcλ

t,m,1−j

with initial condition

cλ0,1,0 =

1

1 + λ, cλ

0,1,1 =λ

1 + λ, cλ

0,m,j = 0 , m > 1 . (4)

Moreover, tλgel is characterized by the fact that the two quantities

σλt,j =

∑

m>0

m2cλt,m,j , j = 0, 1 , (5)

reach infinity for the first time.

By a strong maximal positive solution of the system of ODE (3) weunderstand a solution defined in an interval [0, t) which obeys the properties:

(a) 0 ≤ cλt,m,j for all m ∈ N, j = 0, 1, t ∈ [0, t)

(b) the second moments (5) exist for t ∈ [0, t) and fulfil

∫ T

0σλ

s,jds < ∞ , T < t , j = 0, 1

(c) 0 < t ≤ +∞ is the largest value with properties (a), (b).

Theorem 2 The two-site Smoluchowski equation (3) has a unique strongmaximal positive solution with t = tλgel.

Next, let us consider the differential equation describing (as will be seenlater, at the end of the proof of Theorem 2) the behaviour of the secondmoments at both sites at pre-gelation times

·σ

λ

t,j = (σλt,j)

2 + κ(σλ

t,1−j − σλt,j

), (6)

with initial condition

σλ0,0 = (1 + λ)−1 , σλ

0,1 = λ(1 + λ)−1. (7)

4

Equation (6) is autonomous and extends in an obvious way the well-known

equation·σ = σ2 valid in the pre-gelation regime for the mean-field model.

Writing x(t) for σλt,0 and y(t) for σλ

t,1 we get the following system of twoODE’s

·x = x2 + κ (y − x) ,

·y = y2 + κ(x − y) (8)

with

x0 = (1 + λ)−1 ≥ 0 , y0 = λ(1 + λ)−1 ≥ 0 . (9)

Theorem 3 If κ > 0, then there is a positive tgel ≤ 2 such that x, y aresmooth in [0, tgel) and we have x, y → +∞ as t → tgel simultaneously.

Theorem 4 If κ > 0 and x0 > y0 then x > y for all t ∈ [0, tgel) andx ∼ (tgel − t)−1 as t → tgel while y ∼ −κ log(tgel − t).

Hence (except for κ = 0) both second moments (expected particle massesat site j per monomer) σλ

t,j , j = 0, 1 become infinite simultaneously (this istrivial for λ = 1), an effect we call induced gelation in view of the mechanismlying behind this: If λ 6= 1, the immigration shortly before gelation time oflarge particles from the site with the larger initial number of particles (i.e.site 0 in the case 0 ≤ λ < 1) forces a gelation on the other site. Thenumerical studies with particle numbers up to 109 show that this inducedgelation exhibits some special features which distinguish it clearly from the’genuine’ gelation taking place at the first site. In fact, for λ < 1 the growthof σλ

t,0 is like (t − tgel)−1 while the growth of σλ

t,1 is only of logarithmic

order κ log(t − tgel)−1. Moreover, the proof of Theorem 4 indicates that,

approaching tgel, the growth of σλt,1 is finally completely caused by particles

with large masses immigrating from site 0, while the ’genuine’ growth bycoagulation at the site 1 can be asymptotically neglected.

On the other hand, it is clear from the logarithmic character of the in-duced gelation that it can be expected to be practically perceptible onlyfor particle numbers even the logarithm of which is large enough. Take forinstance 109 particles at site 0 and 0 particles at site 1 with κ = 0.3. As isknown from the one-site model, at gelation time the size of the giant particlecan be expected to be about 10

2

39 = 106. So in fact the mass of immigrat-

ing particles up to gelation time is bounded by this quantity. Therefore themaximum increase of the (empirical) value for σλ

t,1 is most probably less than6κ log 10 ≈ 4. That is an estimate of the maximal increase induced by site

5

0 shortly before gelation happens there. The gelation at site 1 has to be ac-complished now by coagulation of particles at site 1 , since the immigrationof massive particles from site 0 rather rapidly stops after gelation, becausethese are rapidly consumed by the giant particle. The estimated remainingtime it takes for site 1 to reach gelation is now about 1/4, meaning thatin fact even for 109 particles a perceptible delay occurs until the inducedgelation actually takes place. Making κ larger does not help too much, be-cause then the tendency to reach equilibrium between both sites is stronger,such that it becomes harder to distinguish the induced from the ’genuine’gelation, which would have taken place anyway at site 1, if migration wasstopped shortly before tgel. Nonetheless, as we will see (interpreting thenumerical studies), it is for the given choice of parameters not difficult torecognize the induced gelation by some characteristic properties appearingafter tgel.

3 Properties of the mean field model

Consider the case κ = 0, N = N0, i.e. the mean field model. Then the graphis completely coloured red for all times and we are in the situation consideredin great detail in the monograph [2]. There the random graph model G(N, p)is studied which describes a graph with N nodes, where for each pair ofnodes an edge is present with probability p, completely independent fromthe presence or non-presence of the other edges. Since the process describedabove connects an unordered pair of different monomers at constant rate 1,the probability of the corresponding edge to be present at time t is 1− e−t.Hence the distribution of {ξt,m,0}m>0 is given by the joint distribution ofmaximally connected components of sizes m = 1, 2, 3... in the random graphG(N, 1 − e−t).

As is well-known in both the coagulation interpretation (cf. the compre-hensive paper [1]) and the random graph set-up ([5], [2]) the time-rescaled(t → t/N) model undergoes at t = 1 (that is at time 1/N in the old scale) aphase transition called gelation, which is characterized by the occurrence ofa single ’giant particle’ of mass ξmax

t = Θ(N) (comparable to N in the sensethat (ξmax

t )−1 = O(N−1)) for t > 1. Up to this gelation time tgel = 1 therescaled particle numbers N−1ξt,m,0,m = 1, 2, ... tend in probability to de-terministic quantities ct,m fulfilling the Smoluchowski system of differentialequations

·ct,m =

1

2

∑

0<n<m

n(m − n)ct,nct,m−n − mct,m

∑

n>0

nct,n (10)

6

which conserves the total (rescaled) mass 1, i.e.∑

m>0 mct,m ≡ 1, t ≤ 1.At later times the limiting behaviour is no longer described by this set ofequations. As expected, the Smoluchowski system does no longer conservethe mass for t > tgel , but it does not take into account the strong effect ofthe giant particle on the rest of the system. The actual limiting quantitiesof the stochastic coalescent differ from the solutions of (10), though sharingthe property that total mass gets lost.

The proper system of equations is

·ct,m =

1

2

∑

0<n<m

n(m − n)ct,nct,m−n − mct,m (11)

Observe that in contrast to the infinite system of equations (10) the system(11) has the nice property that the finite subsystems obtained for m ≤ M areautonomous. It is obtained from (10) by replacing the quantity

∑n>0 nct,n

by 1, which simply means that particles are aggregated to others (includingthe giant one) at constant rate 1, representing the invariant rescaled totalmass.

The solution to (11) can be given in explicit terms for the case c0,1 = 1,i.e. the monodisperse initial condition:

ct,m = tm−1mm−2e−tm/m! (12)

The convergence of the rescaled stochastic coalescent to (12), i.e. the solutionof (11), was shown rigorously ([2]) for all t ≥ 0.

The quantities {mct,m}m>0, together with a mass of ζt,∞ := 1−∑

m>0 mct,m

assigned to the value +∞, form the Borel distribution with parameter t, i.e.the distribution of the total progeny of a Galton-Watson process with Pois-son offspring of parameter t. Observe that for t > 1 the branching processis super-critical and hence the total offspring is infinite with positive prob-ability, leading to a mass defect:

ζt,∗ :=∑

m>0

mct,m < 1 , (13)

reflecting gelation.While for the system (10) the second moment σt :=

∑m>0 m2ct,m reaches

infinity exactly at tgel and remains infinite, for the system (11) this secondmoment is infinite only at tgel (see Fig. 1). In fact, σt is just a reducedsecond moment, with the gel mass not being included.

The second moment has a more immediate interpretation as a firstmoment : It is the average mass of the particle where a randomly chosen

7

Figure 1: Plot of ζt,∗ and σt

monomer is aggregated into. So at gelation time this average mass becomesinfinite, and it returns to finite values only because monomers ’get lost’ inthe giant particle.

Let us look at the behaviour of the system (11) in some more detail.For this sake it is convenient to change from the consideration of particlenumbers to the mass flow equation which is obtained if we consider thequantities ζt,m := mct,m representing the mass concentrated in particles ofsize m at time t. Now the evolution equation becomes

·ζt,m =

1

2m

∑

0<n<m

ζt,nζt,m−n − mζt,m. (14)

We introduce the (probability) generating function

u(x, t) :=∑

m>0

ζt,mxm , 0 ≤ x ≤ 1 , 0 ≤ t < +∞ . (15)

Observe that due to the gelation effect at times t > 1 the total mass ζt,∗ =u(1, t) is less than 1, hence u is for t > 1 no longer a probability generatingfunction in the strict sense. The evolution of u comprises all necessaryinformation about the limit system including the mass of the gel-particle

8

Figure 2: Plot of u(x, t), 0 ≤ t ≤ 3, 0 ≤ x ≤ 1

ζt,∞ = 1−u(1, t) and the expected particle size per monomer σt = ∂∂xu(1, t).

From a well-known formula for the probability generating function of thetotal progeny of a Galton-Watson process (cf. [8]) we derive the followingsemi-explicit formula for u:

x = uet(1−u). (16)

Now observe some interesting consequences of this solution. First observethat for each t (16) has the trivial solution u = 1 for x = 1, and it is notdifficult to check that for t ≤ 1 this is the only solution of 1 = uet(1−u)

fulfilling 0 ≤ u ≤ 1, while for t > 1 there is a second solution with 0 < u < 1representing the true value of u(1, t) = ζt,∗ = 1 − ζt,∞, see Figs. 2, 3.

Observe that the singularity at t = 1 appears as a cusp catastrophe inthe full solution of (16), and the existence of the gel-particle seems to findsome reflection in the solution u(x, t) ≡ 1.

So for t ≥ 1 the total mass is given by the inverse of the function

u → t =− log u

1 − u. (17)

Also, it is easy to derive from this that the initial slope of ζt,∗ at t = 1 is−2, resp. the initial growth speed of the gel-particle is 2, a fact mentionedin [2, p.123] (where it has to be taken into account that the time scalingthere differs from ours by a factor of two, such that his value is 4). Then,in course of time ζt,∗ tends to zero exponentially fast.

9

Figure 3: Extended plot of u(x, t), 0 ≤ t ≤ 3, 0 ≤ x ≤ 2.5

Another implication of (16) is a relation for the expected particle size

per monomer σt which reads as σt =ζt,∗

1−tζt,∗. From this it is easy to derive

the asymptotics of σ in a right neighbourhood of 1 as σt ∼ (t − 1)−1 whichshows that |t − 1|−1 is the asymptotics for σ near t = 1 (for t < 1 thisformula is well-known to be the exact expression for σ).

4 Expected behaviour of the spatial model after

induced gelation

The following remarks are only heuristical, but they are supported by theresults of numerical simulations.

It is clear, that after gelation the spatial model will behave in a morecomplex way, since the limiting model cannot be expected to be non-random.In fact, there will be two gel particles at the two sites immediately after gela-tion, and these are not subject to a law of large numbers in their migrationbehaviour. Consequently, the limiting behaviour after tgel must be expectedto be characterized by jumps of the giant particle(s) from one site to the

10

other. They immediately coalesce if they happen to be at the same site.The further time evolution of the quantities cλ

t,m,j strongly depends on thepresence or absence of a giant particle at j.

Hence, in contrast to the (mean field) stochastic coalescent the (asymp-totic) gelation effect does not only imply a loss of total mass (caught by thegiant particle) but additionally the spatial model shows up a stochastic lim-iting behaviour. So the limit model cannot be described by a (deterministic)kinetic equation similar to (11).

In order to describe the limiting object as a Markov process it is conve-nient to include the two gel-particles with their masses. This is not necessaryin the mean-field model since the gel-particle simply represents the lost mass,whereas in our situation the distribution of the lost mass between the giantparticles and their random locations are important.

In order to unify the set-up, we consider from this moment rescaledmasses N−1mξt,m,j instead of rescaled particle numbers N−1ξt,m,j . Withthis change of the point of view the Smoluchowski system valid for t ≤ tgel

gets modified to a mass flow equation (writing ζλt,m,j for mcλ

t,m,j)

·ζ

λ

t,m,j =1

2m

∑

0<n<m

ζλt,n,jζ

λt,m−n,j−mζλ

t,m,j

∑

n>0

ζλt,n,j−κζλ

t,m,j+κζλt,m,1−j , (18)

for 0 ≤ t ≤ tgel . Let us denote the masses of the gel-particles by ζλt,∞,j, i ∈

{0, 1}. They are zero for t ≤ tgel. Finally, let πj(t), j ∈ {0, 1} , t ≥ 0 betwo completely independent renewal processes with exponential waiting time(parameter κ−1), with πj(t) denoting the time since the last renewal eventoccurred. These two processes are the only random source of the limitingmodel. We get the stochastic evolution equations

·ζ

λ

t,m,j =1

2m

∑

0<n<m

ζλt,n,jζ

λt,m−n,j (19)

−mζλt,m,j

(ζλt,∞,j +

∑

n>0

ζλt,n,j

)− κζλ

t,m,j + κζλt,m,1−j

11

and

ζλt,∞,j = 1{π1−j(t)≥πj (t)}

∫ t

t−πj(t)ζλs,∞,j

∑

n>0

nζλs,n,jds + (20)

1{π1−j(t)<πj (t)}

[ζλt−π1−j(t),∞,j +

ζλt−π1−j(t),∞,1−j +

∫ t

t−π1−j(t)ζλs,∞,j

∑

n>0

nζλs,n,jds

],

for t ≥ 0 . This means in more rigorous terms that the limiting processis a Markov jump process with piecewise continuous trajectories, where thejumps are determined by the (external) renewal processes πj(t), and betweenjumps the trajectory is described by an ODE (i.e. deterministic), while atthe discrete jump times t1, t2, ... given by πj(t) a gel-particle of mass ζλ

ti,∞,j

changes from j to 1− j and forms at site 1− j together with the gel-particlethere (if there is one) a larger gel-particle of mass ζλ

ti,∞,j +ζλti,∞,1−j. Observe

that we need to write the deterministic evolution equation between jumpsas an integral equation since another gelation may have time to occur at thegiven site as long as ζλ

t,∞,j is zero with the consequence that discontinuities

in the derivative occur. The trajectories of ζλt,m,j are piecewise differentiable

between the jumps.So, from gelation time up to the first jump time happening after gelation

(the pre-gelation jumps given by πj(t) are irrelevant) the whole evolution iscompletely deterministic and may be described by (19) plus the equation forthe gel-particle’s masses (20) which can be rewritten as ζλ

t,∞,j = ζλt,∞,jσ

λt,j .

Just as in the mean field model after gelation the equation (6)

·σ

λ

t,j = (σλt,j)

2 + κ(σλ

t,1−j − σλt,j

)(21)

has to be modified, taking into account the strong effect of the gel-particle,and ceases to be autonomous:

·σ

λ

t,j = (σλt,j)

2 + κ(σλ

t,1−j − σλt,j

)− ζλ

t,∞,j

∑

n>0

n2ζλs,n,j (22)

So it involves the second moment of the mass distribution, resp. the ’thirdmoment’ in the traditional notation (referring to c instead of ζ). So, whileσζ∞ gives the speed of growth of ζ∞ (of the gel-particle), the speed ofdescent of σ now includes ρζ∞, where ρ is the second moment of the mass

12

distribution, this being valid in the mean-field model as well as in our two-site model. The above differential equation for σ may be expected to be validexcept for the jump times (where right and left derivative are different) andexcept for the gelation time(s) (in our case there is more than one of themwith positive probability as explained above). In the latter case it becomesmeaningless since σ exhibits a singularity (and the indefinite expression 0·∞is involved).

In view of the fact that the consideration of the (probability) generatingfunction of the mass distribution in the one-site (homogeneous) situationgives a detailed picture of the behaviour of the limit model also after gelation,it is tempting to try this method also in our case to retrieve informationabout the deterministic behaviour until the first jump happens. So onemight hope for some more information about the asymptotics of ζ∞ and σafter tgel. Of course, it should clearly be expected that at the inducing sitethe asymptotics of σ remains unchanged as in the one-site case |t − tgel|

−1

due to the fact that the influence of the induced site is negligible near tgel.It is less clear what will happen at the induced site in a right neighbourhoodof tgel, since here the effects of feeding σ (at a logarithmic order, by the stilllarge value of its counterpart at the inducing site) and of reducing σ by the(comparatively small) gel-particle are competitive.

Let us introduce the generating functions

u(x, t, j) =∑

m

ζλt,m,jx

m. (23)

Then the evolution equations for ζλt,m,j imply the following equation for u,

which should be valid until the first jump happens (after that u itself willbe random)

∂

∂tu(x, t, j) = x(u(x, t, j) − ζλ

t,∞,j − u(1, t, j))∂

∂xu(x, t, j)

+κ(u(x, t, 1 − j) − u(x, t, j)).

Observe that the first, non-linear term differs from the one-site situation bythe appearance of the total mass ζλ

t,∞,j + u(1, t, j) at the given site instead

of simply 1 in the one-site expression x(u − 1) ∂∂xu. The new linear term is

due to the motion of particles. Of course this equation is not autonomousand has to be supplemented by

d

dtζλt,∞,j = ζλ

t,∞,j ·∂

∂xu(1, t, j), (24)

13

which is just the rewritten evolution equation of ζλt,∞,j from above. Obvi-

ously, this system of PDE looks much less promising for having a (semi-)explicit solution, like it was the case in the one-site model.

We emphasize again, that all these considerations are heuristical thoughlikely to be valid, until a rigorous proof of the convergence to the limitingprocess is available.

We defined gelation for our model in a weak sense meaning that σ tendsto infinity, which is strong evidence (but of course no rigorous proof) forthe simultaneous appearance of gel particles at the first gelation instance ifκ > 0. In fact, the possible later gelation events cannot be induced gelations,since the presence of a gel-particle at the other site keeps the second momentthere at finite values. So the first gelation is particularly interesting (and theproof of the rigorous limit theorem can be expected to be difficult mainlybecause of this special circumstance).

Bear in mind, that, for N being large but finite, only a tiny portion oflogarithmic order of the quantity σ feeds the induced gelation at the othersite, or, to say it in different words, there is only a small time window aroundgelation time where many very large particles aggregate at the site where the’genuine’ gelation just started, so only a small part of them can be expectedto migrate to the other site to induce gelation there. In fact, it is quite likelythat just as in the mean field model the largest particles (except for the gel-particle) are of size ≤ N2/3 and these ’mesoscopic’ particles are present onlyfor a short time around before and after gelation. They carry the growthof the σ. The respective part of them which succeed in jumping to theother site during their short time of existence is likely to be of logarithmicorder in view of the calculations made in the proof of Theorem 4. Thisleads to the situation, that, in contrast to the inducing site’s growth of theexpected particle mass per monomer to ’infinity’ (in fact, some power Nα,α ≤ 2

3), the growth of the expected particle mass at the induced site is carriedonly by the few very large particles which immigrated and will probably beof logarithmic order in N . Therefore three effects should be likely to beobserved even for very large N of the order of Loschmidt’s constant or evenof the order 10100:

1) the induced gelation happens with a significant delay of the order(log N)−1,

2) since it is carried by only few large ’foreign’ particles , in contrastto the inducing gelation the growth of σ in the (delay) pre-gelation intervalcan be expected to show visible erratic deviations (caused by jumps from/tothe other site and coalescences between the large particles) from the law oflarge numbers and

14

3) since the initial growth speed of the gel-particle is governed by thevalue of σ at gelation time, which is maximally of logarithmic order in N atthe induced site, the second gel-particle’s growth will be slow (we referredin section 3 to the fact that in the mean-field case and hence probably alsoat the inducing site the (unrescaled) primary slope of the gel-particle’s massis of order 2N).

Though at the time being no quantum computer was available to usto study simulations with 10100 particles (or only 1023), the simulationswith N = 109 rather clearly show the predicted peculiarities of the inducedgelation, see Fig. 4. In this simulation, the behaviour at site 0 is clearlygoverned by the law of large numbers, except for the random jump of thegiant particle from 1 to 0 at time 5.86. At site 1 the boost of the secondmoment induced by site 0 around the (inducing) gelation time is visible, butnot too strong. The delay between inducing and induced gelation is aboutone time unit, and during this delay time the second moment at site 1 ismainly carried by just a few particles, which fact causes strong fluctuationsvisible as some large jumps respectively a ’Brownian-like’ behaviour aroundthe (induced) gelation. As criterion for gelation a threshold of size 2 · 106

for the largest particle was chosen, i.e. 2N2/3.

5 Remarks concerning other gelling kernels

The situation for general gelling kernels (instead of mn) is more compli-cated, even in the mean field case. Existence of gelling solutions has beenconjectured for a long time (see the discussion in [1]). Rigorous results weregiven in [10], [6]. The asymptotic behaviour of the stochastic model wasstudied, e.g., in [10], [14], [15], [3], [7]. Convergence has been proved in thepre-gelation interval. After gelation, the Smoluchowski equation might haveto be modified (dependent on the kernel). Convergence is established underthe assumption that uniqueness holds. The asymptotic size of the largestparticle is still an open problem. It has been studied in [16], [1] (see also thecorresponding sections in [3], [19]).

Here we add some heuristic comments concerning the particle behaviouraround gelation, explaining the peculiarity of the mn-kernel in comparisonto several other gelling kernels.

15

Figure 4: Simulation with one billion particles, λ = 0, κ = 0.29

16

Kernel (mn)α for 1/2 < α < 1

If there was a giant particle of size γN , it would absorb the particles of fixedsize m at a rate

N−1(γmN)α =γαmα

N1−α

i.e. at an asymptotically vanishing rate. So it looks a bit surprising that itoccurs at all. The explanation should be some kind of food chain. Just as inthe α = 1 case, in the pre-gelation phase the coagulation process shifts thetotal mass progressively to particles of larger size (individual mass), lettingat the gelation point the mass flow ’into infinity’ reach an asymptoticallypositive speed. That means that the speed at which mass is lost from theset of all particles of size m ≤ B does no longer go to zero as B tends toinfinity (at a suitable slower speed than N). Hence a positive part of themass accumulates at particles of a size which grows to infinity as N goesto infinity. An indicator of that situation is that the mean particle size permonomer (’second moment’ in the usual terminology) σ grows to infinity.It is not a priori obvious, what should be the typical size of the particlescarrying that lost mass.

Anyway, it might be conjectured that just as in the α = 1 case the ’lostmass’ changes the regime abruptly by bringing the second moment back tofinite values and stopping the ’active’ transfer (from the point of view ofbounded-size particles) of mass to infinity. If this was true, the lost masswould actively absorb mass from the finite size particles at positive speed,preferring large particles (that is the reason, why the second moment dropsto finite values for α = 1; it should be noted that ’finite’ always means here:remaining bounded for N tending to infinity and does not refer to a limitmodel). As was mentioned above, the absorbing power of the giant particleis not enough to do this. For α < 1 it must be taken into account that,while the rate at which a single particle of size γN absorbs m−particles isgiven by γαmα

N1−α , for the same mass γN being distributed on γN1−β particlesof size Nβ the absorbing force is given by

N−1γN1−β(mNβ)α = γmαNβ(α−1).

The exponent is still negative, indicating that, though splitting the mass intomany smaller particles increases the absorbing force, the totality of ’infinite-size’ particles is unable to actively withdraw mass from the bounded region.

Consequently, gelation will go on in a way very different from the criticalcase α = 1: The bounded size particles do actively transfer mass to infinityeven after the gelation point, and this mass is transported through a kind

17

of food chain at positive speed to infinity, reaching the giant particle onlyat the end of this chain. Observe, that the mass transfer from particles ofsize Nβ directly to the giant size N particle is given by a rate

N−1(NβN)α = Nα(1+β)−1

which tends to zero unless β ≥ 1α − 1. So the food chain should be located

between exponents 0 and 1α − 1.

It should be expected that, in contrast to the critical case α = 1, thesecond moment σ keeps being infinite after the gelation point.

As for a corresponding spatial model (with mass-independent jump rate),the behaviour should be significantly different compared to the critical mncase: The portion of mass concentrated on finite size particles might behavedeterministically after gelation, too, because of a law of large numbers. Thisshould be a kind of deterministic spatial Smoluchowski equation. Since thegiant particle(s) at the spatial sites is/are unable to directly consume massfrom below the threshold 1

α − 1, the random jumps performed by them (orit) should be without effect on the rest of the mass.

Kernel mαn + mnα for α < 1

The behaviour will be similar to the mn-case, since the giant particle is ableto withdraw mass from the bounded region at positive speed in this case.

Kernel mα + nα for α > 1

For any α > 1 the kernel mα + nα can be expected to be instantaneouslyand completely gelling ([1], [11]). In fact: For large N after an arbitraryshort time an arbitrary large particle of size N ′ is present with arbitrarylarge probability (of course at the beginning 1 << N ′ << N). This particleby the law of large numbers grows like the solution of the ODE x = xα,starting at initial value N ′. So it takes an arbitrary short time to consumethe rest of the population. Hence no interesting spatial model should beexpected.

Kernel mαnβ + mβnα for α + β > 1

If one of the parameters is greater than one, the argument of the case mα +nα, α > 1 applies. Otherwise, if α, β < 1, it seems, that anything said aboutthe (mn)α case could be valid here, too.

18

6 Proofs

We start with quoting a result on differential inequalities in Rd which will

be used in the following considerations at several places. Let for x, y ∈ Rd

the notation x ≤ y mean that xi ≤ yi for i = 1, 2, ..., d. A function g : Rd →

Rd is called quasi-monotone increasing (with respect to this natural semi-

ordering) if for each z ∈ Rd, 0 ≤ z from x ≤ y and 〈z, x〉 = 〈z, y〉 it follows

that 〈z, g(x)〉 ≤ 〈z, g(y)〉. In fact, we shall employ this definition only for thecases d = 1 (where any function is quasi-monotone increasing) and d = 2,where it simply means that gi(x) is a monotone increasing function of x3−i

when xi is fixed, for i = 1, 2. Then we have (cf. [17] or [9])

Theorem 5 Let f : [a, b]×Rd → R

d be a function such that f(t, ·) is quasi-monotone increasing and locally Lipschitz continuous for each t ∈ [a, b], andlet u, v : [a, b] → R

d be differentiable. Then

·u(t) − f(t, u(t)) ≤

·v(t) − f(t, v(t)) , t ∈ [a, b] (25)

andu(a) ≤ v(a) (26)

implyu(t) ≤ v(t) , t ∈ [a, b] . (27)

6.1 Proof of Theorem 3

First observe that the theorem about existence and uniqueness of ODE en-sures that there is a unique maximal local smooth solution to this evolutionequation up to some 0 < Tmax ≤ +∞, and if Tmax < +∞ then necessarilythe vector (x, y) becomes unbounded near Tmax. Consider S := x + y. Itfulfils the equation S = x2 +y2 ≥ 1

2(x+y)2 = 12S2. The maximal solution of

the corresponding ODE is (1− 12t)−1, 0 ≤ t < 2. Hence we see (using Theo-

rem 5 in the case d = 1) that Tmax ≤ 2 is finite. Let tgel := Tmax. We know,that at least one of the x, y will be unbounded near tgel. Assume it is thefunction x. Since x ≥ x2 − κx ≥ −κx and y ≥ y2 − κy ≥ −κy, the functionsare is non-negative for t > 0 (again by Theorem 5). Furthermore, since x isunbounded near tgel, this estimate combined with Theorem 5 shows that itis in fact monotonically increasing near tgel and hence tends to +∞. Assumey is bounded near tgel. Then there is some δ such that y2 + κ (x − y) > 1

2κxfor t > tgel − δ implying y > 1

2κx for t > tgel − δ.

19

Now represent x as g(tgel − t)−1 in a left neighbourhood of tgel with gbeing smooth in a right neighbourhood (0, δ) of 0 where we may use thesame δ as above. We get

·x = x2 + κ (y − x) = g(tgel − t)−2 ·

g(tgel − t) = x2 ·g(tgel − t) ,

so that

·g(t) = [(x(tgel − t))2 + κ(y(tgel − t) − x(tgel − t))]/(x(tgel − t))2 → 1

as t → 0 by assumption. Hence g(t) = c(t)t with c being bounded fromabove and below by a positive constant C (resp. its inverse C−1) in a rightneighbourhood of 0. We derive

y(tgel − t) = y(tgel − δ) +

∫ tgel−t

tgel−δ

·y(u)du > y(tgel − δ) +

κ

2

∫ tgel−t

tgel−δx(u)du

= y(tgel − δ) +1

2κ

∫ tgel−t

tgel−δc(tgel − u)−1(tgel − u)−1du

≥1

2κC

∫ tgel−t

tgel−δ(tgel − u)−1du =

1

2κC(ln(δ) − ln t).

The last expression is unbounded as t → 0 in contradiction to the assump-tion. Hence we know that y is unbounded near tgel, too. As above we mayconclude that y → +∞ as t → tgel monotonically. We have shown theassertion.

6.2 Proof of Theorem 4

Consider D = x − y, S = x + y and assume x0 > y0. These functions fulfilthe quadratic ODE

·S =

1

2S2 +

1

2D2 ,

·D = SD − 2κD

with initial condition S0 = 1, 0 < D0 ≤ S0. Since S is obviously monoton-ically increasing we get D = SD − 2κD ≥ (1 − 2κ)D from which we inferby Theorem 5 that D > 0 is always fulfilled. Hence the site with originallymore monomers has always a larger expected particle size. And, since Sgoes to infinity monotonically near tgel, we conclude that D is positive neartgel, so D increases monotonically near tgel, too. Assume D is bounded near

20

tgel. As above, represent S as f(tgel − t)−1 in a left neighbourhood of tgel

which yields

·S =

1

2S2 +

1

2D2 = f(tgel − t)−2

·f(tgel − t) = S2

·f(tgel − t) (28)

and we may infer again that·f would tend to 1

2 near 0 and as above wewould obtain a contradiction, since in this case the integral

∫ tgel−t

tgel−δS(u)D(u)du ≥ D(tgel − δ)

∫ tgel−t

tgel−δS(u)du (29)

diverges near tgel. So in fact D increases monotonically to infinity near tgel.Let Q := D

S ≤ 1. Then it is easy to see that Q fulfils

·Q =

1

2D(1 − Q2) − 2κQ . (30)

If Q was bounded away from 1 near tgel we might conclude from this differ-ential equation that it is strictly increasing near tgel, hence having a limitQ0 < 1. With

·S =

1

2S2 +

1

2D2 = f(tgel − t)−2

·f(tgel − t) = S2

·f(tgel − t) (31)

we would conclude that·f would tend to 1

2(1 + Q20) near 0 and hence∫ tgel−t

tgel−δ S(u)du as well as∫ tgel−ttgel−δ D(u)du would diverge near tgel. This would

imply the divergence of Q near tgel in view of (30), a contradiction. We haveshown that 1 is an accumulation point of Q near tgel. If it was not the limit,Q would necessarily assume arbitrary large negative values in contradictionto the boundedness from below by −2κQ. We have shown that Q → 1 ast → tgel. This means that y/x → 0 as t → tgel, provided x0 > y0. Repet-ing the calculations some lines above, this time with the confirmed value

Q0 = 1, we get that S = f(tgel − t)−1, where·f → 1 as t → 0. So there are

smooth (in a right neighbourhood of -and excluding- 0) functions c, c1, c2 alltending to 1 as t → 0 such that

S = c1(tgel − t)(tgel − t)−1 , D = c2(tgel − t)(tgel − t)−1 ,

x = c(tgel − t)(tgel − t)−1

if x0 > y0. It remains to calculate the asymptotics of y.

21

We consider the quantity q := y/x < 1 which fulfils·q = q2(x−κ)−qx+κ.

We know that q is positive but tends to zero near tgel, and qx = y tends to

infinity. Hence for t sufficiently close to tgel we have·q < −(1 − δ)qx with

arbitrary small δ. So

·

[log q] < −(1−δ)x < −(1−δ)c(tgel−t)(tgel−t)−1 < −(1−2δ)(tgel−t)−1 (32)

for t sufficiently close to tgel. Consequently we get log q < c0 + (1 −2δ) log(tgel − t) (for some c0) in a neighbourhood of tgel. So q < ec0(tgel −t)1−2δ and hence y < ec0(tgel − t)−2δ. This proves that y is not only o(x)but even o(xε) for each ε > 0.

We return to the original equation·y = y2 + κ(x − y). Now it is obvious

that for y0 < x0 the increase of y near tgel is (asymptotically) completelycaused by the migration term κx, while the coagulation term y2 can beneglected. For any positive δ we find for t sufficiently close to tgel: (1 −

δ)κx <·y < (1 + δ)κx and the x-asymptotics yields

−(1 − 2δ)κ log(tgel − t) < k0 + y < −(1 + 2δ)κ log(tgel − t) (33)

in a sufficiently small neighbourhood of tgel, where k0 is a constant dependingon δ. Therefore we finally deduced the logarithmic asymptotics of y =−c3(tgel − t)κ log(tgel − t) with c3 tending to 1 as t → 0.

6.3 Proof of Theorem 2

Existence

Fix some T, 0 < T < tλgel. Consider the following system of ODE

·cλ

t,m,j =1

2

∑

0<n<m

n(m − n)cλt,n,jc

λt,m−n,j − mcλ

t,m,jζλt,∗,j

−κcλt,m,j + κcλ

t,m,1−j (34)

·ζ

λ

t,∗,j = −κζλt,∗,j + κζλ

t,∗,1−j.

with initial condition

cλ0,1,0 = ζλ

0,∗,0 =1

1 + λ, cλ

0,1,1 = ζλ0,∗,1 =

λ

1 + λ, cλ

0,m,j = 0 , m > 1 .

(35)It has obviously a unique solution for all t since, given B ∈ N, the equationsfor m < B, together with the equations for ζλ

t,∗,j form a finite autonomous

22

system. First observe that all quantities cλt,m,j , ζ

λt,∗,j are non-negative. In

fact, this is easily seen for ζλt,∗,j which can be given in explicit terms:

ζλt,∗,0 =

1

2

(1 +

1 − λ

1 + λe−2κt

), ζλ

t,∗,1 =1

2

(1 −

1 − λ

1 + λe−2κt

). (36)

To see it for the cλt,m,j , let us first modify the initial condition to cλ

0,m,j =c,m > 1, where c > 0 is an arbitrary fixed value. It is enough to prove thenon-negativity of the corresponding solutions for all times t > 0 in thesecases in view of the theorem about the continuous dependence of solutionsof ODE on the initial data. Assume the opposite. Then there would be somec such that at least one of the cλ

t,m,j reaches 0 for the first time at some time

0 < t0. We get the differential inequalities cλt,m,j ≥ (−m − κ)cλ

t,m,j beingvalid for 0 ≤ t ≤ t0 taking into account (36). But now we immediately inferfrom Theorem 5 that cλ

t0,m,j > 0 in contradiction to our assumption. Hence

indeed all cλt,m,j are non-negative for all times.

Next, consider the quantities

ζB,λt,∗,j :=

∑

m<B

mcλt,m,j and δB,λ

t,j := ζB,λt,∗,j − ζλ

t,∗,j.

We have

·ζ

B,λ

t,∗,j =1

2

∑

k+l<B

(k + l)klcλt,k,jc

λt,l,j − ζλ

t,∗,j

∑

m<B

m2cλt,m,j +

κ(ζB,λt,∗,1−j − ζB,λ

t,∗,j

)

≤∑

k,l<B

k2lcλt,k,jc

λt,l,j − ζλ

t,∗,j

∑

m<B

m2cλt,m,j + κ

(ζB,λt,∗,1−j − ζB,λ

t,∗,j

)

=(ζB,λt,∗,j − ζλ

t,∗,j

) ∑

m<B

m2cλt,m,j + κ

(ζB,λt,∗,1−j − ζB,λ

t,∗,j

)(37)

which implies

·δB,λ

t,j ≤ δB,λt,j

∑

m<B

m2cλt,m,j + κ

(δB,λt,1−j − δB,λ

t,j

). (38)

The initial values δB,λ0,j are zero. So we may conclude, again by Theorem 5,

that δB,λt,j is non-positive yielding

ζB,λt,∗,j ≤ ζλ

t,∗,j ≤ 1. (39)

23

Next, consider the quantity

σB,λt,j :=

∑

m<B

m2cλt,m,j . (40)

We have

·σ

B,λ

t,j =1

2

∑

k+l<B

(k + l)2klcλt,k,jc

λt,l,j − ζλ

t,∗,j

∑

m<B

m3cλt,m,j +

κ(σB,λ

t,1−j − σB,λt,j

)

≤∑

k,l<B

k3lcλt,k,jc

λt,l,j +

(σB,λ

t,j

)2− ζλ

t,∗,j

∑

m<B

m3cλt,m,j +

κ(σB,λ

t,1−j − σB,λt,j

)(41)

=(σB,λ

t,j

)2+(ζB,λt,∗,j − ζλ

t,∗,j

) ∑

m<B

m3cλt,m,j + κ

(σB,λ

t,1−j − σB,λt,j

).

Hence, by the above estimate ζB,λt,∗,j ≤ ζλ

t,∗,j we arrive at

·σ

B,λ

t,j ≤(σB,λ

t,j

)2+ κ

(σB,λ

t,1−j − σB,λt,j

). (42)

The function defined by fj(x) := x2j + κ(x3−j − xj) is quasi-monotone in-

creasing. Consequently by Theorem 5, (σB,λt,j )j=0,1 is dominated in [0, tλgel)

by the solution of the ODE (8)-(9), for all B. So (σλt,j)j=0,1 is bounded

from above by this solution, too. Hence σλt,j is integrable in any compact

sub-interval of [0, tλgel).Next, consider again the relation (37) which implies

ζB,λt,∗,j = ζB,λ

0,∗,j +

∫ t

0

{1

2

∑

k+l<B

(k + l)klcλs,k,jc

λs,l,j −

ζλs,∗,j

∑

m<B

m2cλs,m,j + κ

(ζB,λs,∗,1−j − ζB,λ

s,∗,j

)}ds.

The quantity 12

∑k+l<B(k + l)klcλ

s,k,jcλs,l,j converges monotonically (as B →

∞) towards1

2

∑

k,l

(k + l)klcλs,k,jc

λs,l,j = σλ

t,j

∑

m

mcλs,m,j (43)

24

for each s ∈ [0, tλgel), in the same way∑

m<B m2cλs,m,j converges towards

σλt,j , and ζB,λ

s,∗,j towards

ζ∞,λs,∗,j :=

∑

m

mcλs,m,j ≤ ζλ

t,∗,j ≤ 1 (44)

(by (39)). So from Lebesgue’s theorem about monotone convergence weinfer, taking into account the uniform boundedness of all these quantities in[0, T ], T < tλgel

ζ∞,λt,∗,j = ζ∞,λ

0,∗,j +

∫ t

0

{σλ

s,j(ζ∞,λs,∗,j − ζλ

s,∗,j) + κ(ζ∞,λs,∗,1−j − ζ∞,λ

s,∗,j

)}ds. (45)

Let δ∞,λs,j := ζ∞,λ

s,∗,j−ζλs,∗,j ≤ 0 (by (39)) and dλ

s = |δ∞,λs,0 +δ∞,λ

s,1 | = −δ∞,λs,0 −δ∞,λ

s,1

and we get (bearing in mind that for each s we have ζλs,∗,0 + ζλ

s,∗,1 = 1 inview of (34))

dλt =

∫ t

0

{−σλ

s,0δ∞,λs,0 − σλ

s,1δ∞,λs,1

}ds ≤ C

∫ t

0dλ

s ds (46)

for some constant C depending on T , since σλs,j is uniformly bounded for 0 ≤

s ≤ T < tλgel. Since dλs is bounded by 2, we get from (46) for s ≤ 1

2C that dλs is

bounded by 1 and by induction that it must be zero in [0, 12C ], which, arguing

by repetition, leads to the conclusion that dλs is zero everywhere in [0, tλgel).

Hence we have ζ∞,λs,∗,j ≡ ζλ

s,∗,j everywhere in [0, tλgel). Now, substituting ζ∞,λs,∗,j =∑

m mcλs,m,j for ζλ

s,∗,j in (34) we see that the unique solution of (34) also

satisfies (3) in [0, tλgel). We proved that this solution is positive and strong

in the sense of integrability of σλt,j in any compact interval contained in

[0, tλgel).

Uniqueness

Let a strong positive solution (cλt,m,j)m∈N,j=0,1,t∈[0,t) , t < tgel , of (3) be given.

Using again the notations ζB,λt,∗,j :=

∑m<B mcλ

t,m,j and ζ∞,λt,∗,j :=

∑m mcλ

t,m,j ,from (3) we infer in the same way as in (37)

·ζ

B,λ

t,∗,j ≤(ζB,λt,∗,j − ζ∞,λ

t,∗,j

) ∑

m<B

m2cλt,m,j + κ

(ζB,λt,∗,1−j − ζB,λ

t,∗,j

)

which in view of ζB,λt,∗,j ≤ ζ∞,λ

t,∗,j yields·ζ

B,λ

t,∗,j ≤ κ(ζB,λt,∗,1−j − ζB,λ

t,∗,j

)and con-

sequently ζB,λt,∗,0 + ζB,λ

t,∗,1 ≤ 0. Thus the sum of the two positive quantities

25

ζB,λt,∗,j , j = 0, 1 is always less than or equal to its initial value 1 (for B > 1).

Letting B tend to infinity we arrive at the conclusion that both ζ∞,λt,∗,0, ζ

∞,λt,∗,1

are not greater than one for 0 ≤ t < t. Now we have

ζB,λt,∗,j = ζB,λ

0,∗,j +

∫ t

0

1

2

∑

k+l<B

(k + l)klcλs,k,jc

λs,l,jds

−

∫ t

0ζ∞,λs,∗,j

∑

m<B

m2cλs,m,jds +

∫ t

0κ(ζB,λs,∗,1−j − ζB,λ

s,∗,j

)ds.

With B → ∞ the first integral on the right-hand side tends towards

∫ t

0

1

2

∑

k,l

(k + l)klcλs,k,jc

λs,l,jds =

∫ t

0σλ

s,jζ∞,λs,∗,jds ≤

∫ t

0σλ

s,jds < +∞ (47)

for each 0 ≤ t < t by the assumption that the solution is strong. The sameargument yields the finiteness of the second integral, which tends to thesame limit

∫ t0 σλ

s,jζ∞,λs,∗,jds. So we arrive at the system of two equations

ζ∞,λt,∗,j = ζ∞,λ

0,∗,j +

∫ t

0κ(ζ∞,λs,∗,1−j − ζ∞,λ

s,∗,j

)ds , j = 0, 1

from which we may easily conclude that the bounded quantities ζ∞,λt,∗,j are

continuous and even continuously differentiable and fulfil the ODE

·ζ∞,λ

t,∗,j = −κζ∞,λt,∗,j + κζ∞,λ

t,∗,1−j (48)

with initial condition

ζ∞,λ0,∗,0 =

1

1 + λ, ζ∞,λ

0,∗,1 =λ

1 + λ. (49)

Therefore our solution also fulfils (34) which has a unique solution for all t.So the maximal strong positive solution must be unique.

Maximality

We still have to show that the maximality implies t = tλgel. Knowing al-

ready that the solution to (34) in [0, tλgel) is positive and strong (in view

of the fact derived above that σλt,j is bounded from above by the contin-

uous solution of (6)), we just need to show that t cannot be larger than

26

tλgel. For this sake we first prove that the second moments (σλt,j)j=0,1, as an-

nounced above, fulfil (6) for 0 ≤ t < tλgel. Denote by ρB,λt,j , ρλ

t,j the quantities∑m<B m3cλ

t,m,j ,∑

m m3cλt,m,j , respectively, given the solution to (34). We

get, in an analogous way as in (41) for 0 ≤ t ≤ T < tλgel

·ρ

B,λ

t,j =1

2

∑

k+l<B

(k + l)3klcλt,k,jc

λt,l,j − ζλ

t,∗,j

∑

m<B

m4cλt,m,j +

κ(ρB,λ

t,1−j − ρB,λt,j

)

≤∑

k,l<B

k4lcλt,k,jc

λt,l,j + 3

∑

k,l<B

k3l2cλt,k,jc

λt,l,j − ζλ

t,∗,j

∑

m<B

m4cλt,m,j +

κ(ρB,λ

t,1−j − ρB,λt,j

)

= 3ρB,λt,j σB,λ

t,j +(ζB,λt,∗,j − ζλ

t,∗,j

) ∑

m<B

m4cλt,m,j + κ

(ρB,λ

t,1−j − ρB,λt,j

)

≤ 3ρB,λt,j C + κ

(ρB,λ

t,1−j − ρB,λt,j

),

where C is a constant depending on T , where we made use of the bounded-ness of σλ

t,j in compact sub-intervals of [0, tλgel). From this, in the same way

as for the second moment, we derive that all ρB,λt,j and hence also ρλ

t,j arebounded by the solution of the coupled system of ODE

·x = 3Cx + κ (y − x) ,

·y = 3Cy + κ(x − y)

with x0 = 11+λ , y0 = λ

1+λ in [0, T ]. Now we may rewrite the first equationin (41):

σB,λt,j = σB,λ

0,j +

∫ t

0

1

2

∑

k+l<B

(k + l)2klcλs,k,jc

λs,l,jds

−

∫ t

0ζλs,∗,jρ

B,λs,j ds + κ

∫ t

0

(σB,λ

s,1−j − σB,λs,j

)ds.

With B → ∞ the first integral on the right hand side tends monotonicallytowards

∫ t

0

1

2

∑

k,l

(k + l)2klcλs,k,jc

λs,l,jds =

∫ t

0

(ζλs,∗,jρ

λs,j + (σB,λ

s,j )2)

ds

being finite for t < tλgel, and the second integral converges monotonicallytowards ∫ t

0ζλs,∗,jρ

λs,jds

27

which is finite for these t, too. So we may pass to the limit on both sidesand get

σλt,j = σλ

0,j +

∫ t

0(σλ

s,j)2ds + κ

∫ t

0

(σλ

s,1−j − σλs,j

)ds.

Thus the σλt,j are continuous and even continuously differentiable in [0, tλgel)

and fulfil (6). Now from Proposition 4 we easily infer that the condition thatσλ

t,j be integrable for each compact sub-interval of [0, t) (strong solution)

implies t ≤ tλgel.

6.4 Proof of Theorem 1

We are now prepared to prove Theorem 1. At this point we take the pointof view that the configuration at time t of our particle system is given by afinite counting measure on N × {0, 1},

Xt =∑

m,j

ξt,m,jδ(m,j) (50)

Let us, just for the moment, renounce the assumption that the initial con-figuration be monomeric, i.e. ξ0,m,j may be positive for m > 1.

In order to prove our limit theorem we consider a coupled system oftruncated versions of the process {Xt} indexed by B ∈ N = N ∪ {∞} wherethe dynamics of {XB

t } are given as follows:1. First define

XB0 :=

∫

N×{0,1}1m<Bδ(m,i)X0(d(m, i)) +

∫

N×{0,1}1m≥Bmδ(B,0)X0(d(m, i)),

(51)i.e. the particles of sizes less than B remain unchanged, while at each site themasses of all particles larger than B are combined and this mass is assignedto (B, 0). X0 can be represented as a finite sum

∑l∈J δ(ml,jl), where J is a

finite index set, and each XB0 can be written as

∑

l∈J(B)

δ(ml,jl) + XB0 ({(B, 0)})δ(B,0) , (52)

where J(B) is the subset of J corresponding to particles of size smaller thanB. Obviously X∞

0 = X0.2. It is easily seen that, by simply cutting each Xt in an analogous

way, the corresponding quantities will not form a Markov process except for

28

κ = 0 (which corresponds to the one-site case). We intend to let XB0 evolve

Markovian, thereby keeping it in close distance to Xt for B large, i.e. tointroduce a coupled evolution of (Xt,X

Bt ) where both marginal processes

are Markovian and their distance can be controlled to be small for largeB. To perform this, first introduce a new discrete parameter assigned tothe particles taking two possible values v and h (’visible’, ’hidden’). Thatmeans, instead of N × {0, 1} we choose N×{0, 1}×{v,h} as phase space. Allparticles present in X0 with masses below B are marked visible, the rest(with masses ≥ B) is marked hidden. This gives XB

0 . Given a path (Xt)t≥0

we construct (XBt )t≥0 as follows, not changing the projection to N×{0, 1}

which is (Xt)t≥0:a) the mark ’hidden’ is dominant, i.e. the combination of two particles

occurring in (Xt)t≥0 produces a hidden particles iff at least one of them washidden, and jumping to the other site does not change the status hidden orvisible

b) if a particle of mass at least B is generated by the combination oftwo particles, it is marked hidden, and

c) additionally, particles become ’hidden’ by a newly introduced inter-action between particles of different sites: given XB

t , a ’visible’ particle ofmass m < B at site j becomes ’hidden’ at rate

m

∫

N×{0,1}×{v,h}k1i=1−j,q=hX

Bt (d(k, i, q)). (53)

So the interaction concerns only the newly introduced mark, and has theeffect that any pair of particles with masses m,k at different sites assigns themark ’hidden’ to the first particle at rate mk, supposed the second particleis already hidden. This represents a Markovian refinement of the originalprocess.

Now we define XBt by

XBt =

∫

N×{0,1}×{v,h}

(δ(k,i)1k<B,q=v + δ(B,0)k1q=h

)XB

t (d(k, i, q)). (54)

This means, that only visible particles are present in XBt at locations < B,

whereas all hidden particles (in particular those with masses at least B) arecollected to form the mass at B at site 0. Observe that the total rate atwhich a visible particle of mass m becomes ’hidden’ is given by the productof m with the total mass of all other hidden particles.

3. So, if we do not intend to describe the joint dynamics of (Xt,XBt ) but

only that of XBt we may do it as follows, referring to the representation (52):

29

Consider a family (TBl,l′)l,l′∈J(B),l<l′,jl=jl′

of independent exponentially dis-tributed random variables with parameter mlml′ , a second (independent ofthe first) family (SB

l )l∈J(B) of independent exponentially distributed random

variables with parameter κN , and a third independent family (UBl )l∈J(B)

of independent exponentially distributed random variables with parametermlX

B0 ({(B, 0)}). These families are finite in view of the finiteness of J . Let

T = minl,l′

(TBl,l′ , S

Bl , UB

l ) . (55)

Define XBt = XB

0 for t < T and

XBT =

XB0 − δ(ml ,jl) − δ(ml′ ,jl′)

+ δ(ml+ml′ ,jl) if T = TBl,l′ ,

ml + ml′ < BXB

0 − δ(ml ,jl) − δ(ml′ ,jl′)+ (ml + ml′)δ(B,0) if T = TB

l,l′ ,

ml,ml′ < B,ml + ml′ ≥ B

XB0 − δ(ml ,jl) + δ(ml,1−jl) if T = SB

l ,ml < B

XB0 − δ(ml ,jl) + mlδ(B,0) if T = UB

l

XB0 otherwise

(56)Starting this construction anew with XB

T instead of XB0 we finally define XB

t

for arbitrary t ≥ 0, thus arriving at a Markov jump processes {XBt }. Observe

that {X∞t } = {X

ν(X0)+1t } = {Xt}, where ν(X0) :=

∫N×{0,1} mX0(d(m, i)) is

the total mass.The evolution of a single XB

t may be described as follows:

• Particles of sizes m,n at the same site coalesce at rate mn, supposedm,n < B. If m + n < B, a new particle of size m + n is generated,whereas for m + n ≥ B the mass m + n is added to the mass alreadyaggregated at (B, 0).

• Particles of size less than B change their location at rate κN .

• Finally, particles of size m < B are deleted at a rate given by theproduct of their size and the mass already aggregated at (B, 0) andtheir mass is added to the aggregated mass.

The evolution conserves the total mass

ν(X0) ≡ νB(XB0 ) :=

∫

N×{0,1},m<BmXB

0 (d(m, i)) + XB0 ({(B, 0)}) , (57)

30

being independent of B.Let, for B ∈ N, MB denote the space of finite signed measures on

(N ∩ [1, B]) × {0, 1}

and define LB : MB → MB by⟨f, LB(µ)

⟩=

1

2

∫

(N×{0,1})2

[f(x + y, j)1x+y<B + f(B, 0)(x + y)1x+y≥B −

f(x, j) − f(y, j)]1x,y<B,j=k · xy(µ × µ)(d[x, j], [y, k])

+

∫

N×{0,1}µ({(B, 0)}) [f(B, 0)x − f(x, j)] 1x<Bxµ(d[x, j])

+ κ

∫

N×{0,1}[f(x, 1 − j) − f(x, j)] 1x<Bµ(d[x, j]),

for any bounded function f on N×{0, 1}. Fix a finite measure µ0 on N×{0, 1}with

ν(µ0) =

∫

N×{0,1}xµ0(d[x, j]) < +∞ (58)

and define

µB0 :=

∫

N×{0,1}1m<Bδ(m,i)µ0(d(m, i)) +

∫

N×{0,1}1m≥Bmδ(B,0)µ0(d(m, i)) .

(59)It is easy to see that for each B ∈ N the equation

µBt = µB

0 +

∫ t

0LB(µB

s )ds (60)

has a unique solution which is a continuous map µB(·) : t → MB . In fact,

writing

µBt =

∑

i≤B,j∈{0,1}

cBt,i,jδ(i,j)

results in a finite approximation of the two-site Smoluchowski system:

·cB

t,i,j =1

2

∑

k+m=i

kmcBt,k,jc

Bt,m,j − icB

t,i,j

∑

k<B

kcBt,k,j − icB

t,B,0cBt,i,j +

κ(cBt,i,1−j − cB

t,i,j) for i < B and (61)

·cB

t,B,0 =1

2

∑

k+m≥Bk,m<B

(k + m)kmcBt,k,jc

Bt,m,j + cB

t,B,0

∑

k<B

k2cBt,k,j.

31

This system of B first order ODE’s obviously has a unique global solution.We may apply Theorem 5 in the same way as it was done in the first part ofthe proof of Theorem 2 to see that the solution is non-negative. Moreover,it is not difficult to check that (61) conserves the total mass

ν(µ0) ≡ νB(µB0 ) :=

∫

N×{0,1},m<BmµB

0 (d(m, i)) + µB0 ({(B, 0)}) (62)

being independent of B.

Proposition 6 The solution of the approximating system (61) with initialcondition

cB0,1,0 =

1

1 + λ, cB

0,1,1 =λ

1 + λ, cB

0,m,j = 0 , m > 1 (63)

fulfils

B−1∑

i=1

i|cλt,i,j − cB

t,i,j | ≤ cBt,B,0 ≤ B−1C0 , 0 ≤ t ≤ T < tλgel, (64)

where C0 is a constant depending on T and λ.

P r o o f. Let us recall the two-site modified Smoluchowski equation(omitting the upper index λ for simplicity)

·ct,i,j =

1

2

∑

k+m=i

kmct,k,jct,m,j − iζt,∗,ict,i,j − κct,i,j + κct,i,1−j (65)

·ζt,∗,j = −κζt,∗,j + κζt,∗,1−j

which we considered for the initial condition

c0,1,0 = ζ0,∗,0 =1

1 + λ, c0,1,1 = ζ0,∗,1 =

λ

1 + λ, c0,m,j = 0 , m > 1 .

(66)The solution obeys ζt,∗,j =

∑k kct,k,j up to tgel as it was shown in the proof

of Theorem 2. Moreover, using the quantity

ζ(B)t,j =

∑

k≥B

kct,k,j = ζt,∗,j −∑

k<B

kct,k,j , (67)

32

we may infer for t < tgel from (65)

·ct,i,j =

1

2

∑

k+m=i

kmct,k,jct,m,j − ict,i,j

∑

k<B

kct,k,j − ict,i,jζ(B)t,j −

κct,i,j + κct,i,1−j for i < B , t < tgel and (68)

·ζ(B)

t,j =1

2

∑

k+m≥Bk,m<B

(k + m)kmct,k,jct,m,j + ζ(B)t,j

∑

k<B

k2ct,k,j −

κζ(B)t,j + κζ

(B)t,1−j .

DefineβB

t,i,j := ct,i,j − cBt,i,j , i < B . (69)

Subtracting (61) from (68) we get

·β

B

t,i,j =1

2

∑

k+m=i

kmct,k,jct,m,j −1

2

∑

k+m=i

kmcBt,k,jc

Bt,m,j

−ict,i,j

∑

k<B

kct,k,j + icBt,i,j

∑

k<B

kcBt,k,j

−ict,i,jζ(B)t,j + icB

t,i,jcBt,B,0 + κ(βB

t,i,1−j − βBt,i,j)

=1

2

∑

k+m=i

km(βBt,k,jβ

Bt,m,j + 2cB

t,k,jβBt,m,j)

−i∑

k<B

k(βBt,i,jct,k,j + cB

t,i,jβBt,k,j)

−i(βBt,i,jζ

(B)t,j + cB

t,i,jζ(B)t,j − cB

t,B,0cBt,i,j) + κ(βB

t,i,1−j − βBt,i,j) .

The mass conservation law for (61) and (68) implies

cBt,B,0 = 1 −

1∑

l=0

∑

k<B

kcBt,k,l = ζ

(B)t,j + ζ

(B)t,1−j +

1∑

l=0

∑

k<B

k(ct,k,l − cBt,k,l) .

33

Thus we obtain

·β

B

t,i,j =1

2

∑

k+m=i

km(βBt,k,jβ

Bt,m,j + 2cB

t,k,jβBt,m,j) + κ(βB

t,i,1−j − βBt,i,j)

−i∑

k<B

k(βBt,i,jct,k,j + cB

t,i,jβBt,k,j)

−i

(βB

t,i,jζ(B)t,j −

(ζ(B)t,1−j +

∑

k<B

k(βBt,k,j + βB

t,k,1−j)

)cBt,i,j

)

=1

2

∑

k+m=i

km(βBt,k,jβ

Bt,m,j + 2cB

t,k,jβBt,m,j) + κ(βB

t,i,1−j − βBt,i,j)

−i∑

k<B

kβBt,i,jct,k,j

−i

(βB

t,i,jζ(B)t,j −

(ζ(B)t,1−j +

∑

k<B

kβBt,k,1−j

)cBt,i,j

).

Finally we arrive at the following finite system of ODE:

·

βBt,i,j =

1

2

∑

k+m=i

km(βBt,k,jβ

Bt,m,j + 2cB

t,k,jβBt,m,j) + κ(βB

t,i,1−j − βBt,i,j)

−iβBt,i,j

(ζ(B)t,j +

∑

k<B

kct,k,j

)+ icB

t,i,j

(ζ(B)t,1−j +

∑

k<B

kβBt,k,1−j

).

with initial condition βB0,i,j = 0 , which has a unique solution. Again we may

apply Theorem 5 in the same way as it was done in the first part of the proofof Theorem 2 to see that the solution is non-negative, taking into accountthe boundedness of the total mass of the solution to (68) by 1 and the factthat this solution was already proved to be non-negative. Hence we havethe relation

cBt,i,j ≤ ct,i,j , i < B , t ≥ 0. (70)

ThusB−1∑

i=1

|ct,i,j − cBt,i,j| =

B−1∑

i=1

(ct,i,j − cBt,i,j) (71)

and since both systems conserve the total mass we get

B−1∑

i=1

i|ct,i,j − cBt,i,j| = cB

t,B,0 − ζ(B)t,0 − ζ

(B)t,1 ≤ cB

t,B,0. (72)

34

So, in order to estimate the total deviation of the solution of (61) withrespect to that of (65) for masses less than B it is enough to estimate cB

t,B,0.We get from (61) and (70) for any t ≤ T < tgel

·cB

t,B,0 =1

2

∑

k+m≥Bk,m<B

(k + m)kmcBt,k,jc

Bt,m,j + cB

t,B,0

∑

k<B

k2cBt,k,j

≤1

2

∑

k+m≥Bk,m<B

(k + m)kmct,k,jct,m,j + cBt,B,0

∑

k<B

k2ct,k,j

≤1

2

∑

k+m≥Bk,m<B

(k + m)kmct,k,jct,m,j + cBt,B,0(σt,0 + σt,1)

≤1

2

∑

k+m≥Bk,m<B

(k + m)kmct,k,jct,m,j + cBt,B,0C

where C is a constant depending on T as it was shown in the proof ofTheorem 2 that the second moments σt,j = σλ

t,j are continuous functionsin each compact sub-interval of [0, tgel). Hence in [0, T ] we have (using thenotation [x] for the integer part of a real number x , and ζt,∗,0 + ζt,∗,1 ≡ 1

35

for the fourth line)

·cB

t,B,0 ≤∑

k+m≥Bk,m<B

k2mct,k,jct,m,j + cBt,B,0C

≤∑

k+m≥B

k2mct,k,jct,m,j + cBt,B,0C

≤∑

k≥[B/2]m

k2mct,k,jct,m,j +∑

m≥[B/2]k

k2mct,k,jct,m,j + cBt,B,0C

≤∑

k≥[B/2]

k2ct,k,j + (σt,0 + σt,1)∑

m≥[B/2]

mct,m,j + cBt,B,0C

≤∑

k≥[B/2]

k2ct,k,j + C∑

m≥[B/2]

mct,m,j + cBt,B,0C

≤ [B/2]−1

∑

k≥[B/2]

k3ct,k,j + C∑

m≥[B/2]

m2ct,m,j

+ cB

t,B,0C

≤ [B/2]−1

(∑

k

k3ct,k,j + C(σt,0 + σt,1)

)+ cB

t,B,0C

≤ [B/2]−1

(∑

k

k3ct,k,j + C2

)+ cB

t,B,0C

≤ [B/2]−1(C ′ + C2

)+ cB

t,B,0C

as, again in the proof of Theorem 2, it was shown that also the third momentis bounded uniformly in compact sub-intervals of [0, tgel). Hence we see thatthe positive quantity cB

t,B,0 in [0, T ] is bounded (in view of Theorem 5) bythe solution of the differential equation

·x = [B/2]−1

(C ′ + C2

)+ Cx (73)

with initial condition x(0) = 0, which is given by

x(t) = [B/2]−1C−1(C ′ + C2

) (eCt − 1

). (74)

Therefore, there is a constant C0 depending on T such that cBt,B,0 ≤ B−1C0

in [0, T ]. This proves the proposition. �

Consider the process (50) and the corresponding coupled family of ap-proximations constructed above. Set

Xt = N−1Xt/N , XBt = N−1XB

t/N . (75)

36

Let d be a metric on the set of all finite measures on N × {0, 1} whichgenerates the weak topology, i.e. 〈f, µn〉 → 〈f, µ〉 if and only if d(µn, µ) → 0for each bounded function on N × {0, 1}. We may choose d in a way thatd(µ, µ′) ≤ ‖µ − µ′‖ for each pair of measures, where ‖.‖ denotes the totalvariation norm.

Proposition 7 Assume

N0/N →1

1 + λ, N1/N →

λ

1 + λas N → ∞ . (76)

Then, for all t ≥ 0,sups≤t

d(XBs , µB

s ) → 0 (77)

in probability as N → ∞.

P r o o f. The sequence of laws corresponding to(XB

t

)t≥0

is tight,

and any weak limit point (with respect to N) is concentrated on the set ofsolutions of the equation (60) (cf., e.g., [4, Th. 2.3]). Since the solution (µB

t )of this equation is unique, we get convergence in distribution, which impliesthe statement of the proposition (cf., e.g., [18, Sect. 3]). �

The following theorem and its proof adapt Theorem 4.4 of [14] to oursituation. It is a stronger version of Theorem 1.

Theorem 8 Assume (76) and consider the solution (µt)0≤t<tλgel

of the Smolu-

chowski equation up to tλgel . Then for all t < tλgel

sups≤t

d(xXs(d[x, j]), xµs(d[x, j])) → 0 (78)

in probability, as N → ∞ .

P r o o f. For simplicity, we omit the upper index λ and write dx insteadof d[x, j] . Fix δ and t < tgel. By Proposition 6 we find some B such thatµB

t ({(B, 0)}) < δ/4. Since the measures are supported on a compact set,Proposition 7 implies

sups≤t

d(xXBs (dx ∩ [1, B)), xµB

s (dx ∩ [1, B))) → 0 (79)

37

in probability as N → ∞. Using that the derivative of µBs ({(B, 0)}) is

non-negative, we have by Proposition 6 for s ≤ t

∥∥xµs(dx) − xµBs (dx ∩ [1, B))

∥∥ ≤ 2µBs ({(B, 0)}) ≤ 2µB

t ({(B, 0)}) < δ/2.(80)

On the other hand,∥∥∥xXs(dx) − xXB

s (dx ∩ [1, B))∥∥∥ ≤ 2XB

s ({(B, 0)}) (81)

by the fact that the evolutions of particles smaller than B coincide for {Xt}and {XB

t } except for those which are hidden and aggregated at (B, 0) in{XB

t }, and by conservation of the total mass. We may proceed as follows,bearing in mind that the aggregated mass is increasing in time

∥∥∥xXs(dx) − xXBs (dx ∩ [1, B))

∥∥∥ ≤ 2XBt ({(B, 0)})

≤ 2µBt ({(B, 0)}) + 2|XB

t ({(B, 0)}) − µBt ({(B, 0)})|

≤ δ/2 + 2∥∥∥XB

t − µBt

∥∥∥ .

So, using the property of d to be dominated by the variation distance, weobtain

d(xXs(dx), xµs(dx)) ≤∥∥∥xXs(dx) − xXB

s (dx ∩ [1, B))∥∥∥+

d(xXBs (dx ∩ [1, B)), xµB

s (dx ∩ [1, B))) +∥∥xµs(dx) − xµBs (dx ∩ [1, B))

∥∥

≤ δ + 2∥∥∥XB

t − µBt

∥∥∥+ d(xXBs (dx ∩ [1, B)), xµB

s (dx ∩ [1, B)))

which gives

P

(sups≤t

d(xXs(dx), xµs(dx)) > δ

)→ 0 (82)

by Proposition 7, bearing in mind that all metrics considered here are equiv-alent on the finite-dimensional vector space corresponding to measures on{1, 2, ..., B} × {0, 1}. �

Acknowledgements

The authors would like to thank Hans Babovsky (Ilmenau) for interestingdiscussions. Support by Deutsche Forschungsgemeinschaft (grant BA 922/7-1) is gratefully acknowledged.

38

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