instantaneous gelation in smoluchwski's coagulation equation revisited, conference in memory of...

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Instantaneous Gelation in Smoluchowski’sCoagulation Equation Revisited

Colm Connaughton

Mathematics Institute and Centre for Complexity Science,University of Warwick, UK

Joint work with Robin Ball, Thorwald Stein and Oleg Zaboronski.

Conference in Memory of Carlo CercignaniIHP Paris

11 February 2011

http://www.slideshare.net/connaughtonc arxiv:1012.4431 cond-mat.stat-mech

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Cluster aggregation: (m1) + (m2)→ (m1 + m2)

Physical picture:Large "cloud" of particles moving around (eg by diffusion).Particles merge irreversibly on contact.Rate of merging of particles with masses, m1 and m2 isK (m1,m2). Kernel K (m1,m2) encodes microphysics.

Size distribution, Nm(t), is the average density of clusters ofmass m at time t .

Smoluchowski equation(mean field) :

∂tNm(t) =

∫ m

0dm1K (m1,m −m1)Nm1(t)Nm−m1(t)

− 2Nm(t)∫ ∞

0dm1K (m,m1)Nm1(t) + J δ(m − 1)

(Smoluchowski, 1916)


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A regularised Smoluchowski equation

In this talk we will make frequent use of a regularisedSmoluchowski equation:

∂tNm =

∫ m

1dm1K (m1,m −m1)Nm1Nm−m1

− 2 Nm

∫ M−m

1dm1K (m,m1)Nm1 + J δ(m − 1)

− DM [Nm(t)]

where

DM [Nm(t)] = 2 Nm

∫ M

M−mdm1K (m,m1)Nm1 (1)

Regularisation corresponds to removing clusters having m > Mand enforcing a minimum mass which we can take to be 1.Explicitly breaks mass conservation.


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Scaling Solutions of the Smoluchowski equation

In many applications kernelis a homogeneous function.Denote degree by λ:K (am1,am2) = aλK (m1,m2)

Asymptotic behaviour:

K (m1,m2) ∼ mµ1 mν

2 m1�m2

Scaling solutions have the form

N(m, t) ∼ s(t)aF (m/s(t))

where s(t) is the typical cluster size and a is a dynamicalscaling exponent. Often the scaling function, F (x), is a powerlaw at small x :

F (x) ∼ x−y x� 1


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Gelation Transition (Lushnikov [1977], Ziff [1980])

M1(t) for K (m1,m2) = (m1m2)3/4.

M1(t) =∫∞

0 m N(m, t) dm isformally conserved. However forλ > 1:

M1(t) <∫ ∞

0m N(m,0) dm t > t∗

Best studied by introducingcut-off, M, and studying limitM →∞. (Laurencot [2004])

Instantaneous GelationIf ν > 1 then t∗ = 0. (Van Dongen & Ernst [1987])May be complete: M1(t) = 0 for t > 0. Example :K (m1,m2) = m1+ε

1 + m1+ε2 for ε > 0.

Mathematically pathological?What about gravitational clustering, differentialsedimentation?


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Instantaneous Gelation in Regularised System

M(t) for K (m1,m2) = m321 + m3/2

2 .

With cut-off, M, regularizedgelation time, t∗M , is clearlyidentifiable.t∗M decreases as M increases.Van Dongen & Ernst recovered inlimit M →∞.

Decrease of t∗M as M is very slow (numerics suggestlogarithmic decrease). This suggests such models arephysically reasonable.Consistent with related results of Krapivsky and Ben-Naimand Krapivsky [2003] on exchange-driven growth.


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Nonlocal Interactions Drive Instantaneous Gelation

Total density,∫∞

0 N(m, t) dm for

K (m1,m2) = m1 + m2 and source.

Instantaneous gelation is drivenby the runaway absorbtion ofsmall clusters by large ones.This is clear from the analyticallytractable (but non-gelling)marginal kernel, m1 + m2, withsource of monomers for which

dNdt

= −2M1(t)N(t) + J.

M1(t) ∼ t (due to source) but

N(t) =12

√Jπ Erfi(

√J t) e−Jt2 ∼ 1/t .

Big clusters are so “hungry" that they eat all smaller clustersfaster than they are added. If ν > 1, this process runs away.


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Cluster aggregation with a source of monomers

Add monomers at rate, J.Suppose particles havingm > M are removed.Stationary state is obtainedfor large t .Stationary state is abalance between injectionand removal. Constantmass flux in range [m0,M]

Essentially non-equilibrium:no detailed balance.


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Finding the stationary state

Write the Smoluchoswki equation in the following way:

∂Nm(t)∂t

=

∫ ∞0

dm1dm2K (m1,m2)Nm1Nm2δ(m −m1 −m2)

−∫ ∞

0dm1dm2K (m,m1)NmNm1δ(m2 −m −m1)

−∫ ∞

0dm1dm2K (m,m2)NmNm2δ(m1 −m2 −m)

Assume stationary power law : Nm = c m−x

0 = c2∫ ∞

0dm1dm2K (m1,m2)(m1 m2)−xδ(m −m1 −m2)

− c2∫ ∞

0dm1dm2K (m,m1)(m m1)−xδ(m2 −m −m1)

− c2∫ ∞

0dm1dm2K (m,m2)(m m2)−xδ(m1 −m2 −m)


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Finding the stationary state: the exponent

Zakharov Transformation:

(m1,m2) → (mm′1m′2

,m2

m′2)

(m1,m2) → (m2

m′1,mm′2m′1

).

0 =c2

2

∫ ∞0

dm1dm2 K (m1,m2) (m1m2)−xmλ+2−2x(m2x−λ−2 −m2x−λ−2

1 −m2x−λ−22

)δ(m −m1 −m2)

Clearlyx = (λ+ 3)/2,

makes the collision integral vanish.


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Finding the stationary state: the amplitude

Amplitude, c, depends on K but is explicitly computable:

c =

√√√√ 2 JdI(x)

dx

∣∣∣x=λ+3

2

where

I(x) =

∫ 1

0dµK (µ,1−µ) (µ(1−µ))−x

(1− µ2x−λ−2 − (1− µ)2x−λ−2

).

Convergent for |µ− ν| < 1. For example, ifK (m1,m2)=(m1m2)λ/2:

c =

√J

2π.

independent of λ (maybe surprising?).


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Instantaneously gelling case (regularised) with source

A stationary state may be reached if a source of monomers ispresent (Horvai et al [2007]).

Stationary state (theory vs numerics)

for ν = 3/2.

Model kernelK (m1,m2) = max(m1,m2)ν .Stationary state has theasymptotic form for M � 1:

Nm =

√J log Mν−1

MMm1−ν

m−ν .

Stretched exponential for smallm, power law for large m.Stationary particle density:

N =

√J(

M −MM1−ν)

M√

log Mν−1∼

√J

log Mν−1 as M →∞.


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Where does this strange formula come from?

Smoluchowski equation can be written (ignore source and sinkterms for now):

∂Nm

∂t=

∫ m2

1[F (m,m −m1)− F (m,m1)] d m1

−∫ M

mF (m,m1) dm1

whereF (m1,m2) = K (m1,m2) N(m1, t) N(m2, t)

Taylor expand the first term to first order in m1:

[F (m,m −m1)− F (m,m1)] = −m1N(m1, t)∂

∂m[mνN(m, t)]+O(m2

1).

We obtain an (almost) PDE.


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∂Nm

∂t= −D1

∂

∂m[mνN(m, t)]− Dν N(m, t)

where

D1 =∫ m

21 m1N(m1, t)dm1 →

∫ M

1m1N(m1, t)dm1 = M1

Dν =∫ M

m mν1N(m1, t)dm1 →

∫ M

1mν

1N(m1, t)dm1 = Mν

If M1 and Mν did not depend on Nm, we would have thestationary solution:

Nm = C exp[β

m1−ν

ν − 1

]m−ν

where C is an arbitrary constant and

β =Mν

M1.


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The ratio β can be determined self-consistently since themoments M1 and Mν can be calculated explicitly in termsof Γ-functions.Resulting consistency condition gives:

β ∼ log Mν−1 for M � 1.

Constant, C, is fixed by requiring global mass balance:

J =

∫ M

1dm DM [Nm] ,

giving

C =

√J log Mν−1

Mfor M � 1

Various assumptions are now verifiable a-posteriori.


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Approach to Stationary State is non-trivial

Total density vs time forK (m1,m2) = m1+ε

1 + m1+ε2 .

“Q-factor" for ν = 0.2.

Numerics indicate that theapproach to stationary state isnon-trivial.Collective oscillations of the totaldensity of clusters.Numerical measurements of theQ-factor of these oscillationssuggests that they are long-livedtransients. Last longer as Mincreases.Heuristic explanation in terms of“reset” mechanism.


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Conclusions

As has been known since the 1980’s, the M →∞ limit ofthe regularized Smoluchowski equation is pathological forkernels having ν > 1 but lots of interesting things happenalong the way.Weak dependence of the regularized system on the valueof M mean that such kernels are reasonable physicalmodels (of intermediate asymptotics).Regularized system reaches a stationary state in thepresence of a source of monomers which is not of theusual “cascade" type.Long-lived oscillatory transient seems to result frominteraction between the cut-off and the source (“reset"phenomenon).


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