instantaneous gelation in smoluchowski's coagulation equation revisited, american physical...
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American Physical Society March Meeting, Portland Oregon, March 19 2010TRANSCRIPT
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Instantaneous Gelation in Smoluchowski’sCoagulation Equation Revisited
R.C. Ball1 C. Connaughton1 T.H.M. Stein2 O.Zaboronski1
1University of Warwick, UK2University of Reading, UK
APS March Meeting, Portland19 March 2010
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Cluster aggregation: (m1) + (m2)→ (m1 + m2)
Size distribution: Nm(t) satisfies (mean field) :
∂tNm =
∫ m
0dm1K (m1,m −m1)Nm1Nm−m1
− 2Nm
∫ ∞0
dm1K (m,m1)Nm1 + J δ(m − 1)
(Smoluchowski, 1916)
Many applications exhibit scaling:
Micro-physics encoded in kernel:K (am1,am2) = aµ+νK (m1,m2)
K (m1,m2) ∼ mµ1 mν
2 m1�m2
Typical size, s(t):N(m, t) ∼ s(t)aF (m/s(t))
F (x) ∼ x−y x� 1
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Gelation Transition (Lushnikov [1977], Ziff [1980])
M(t) for K (m1,m2) = (m1m2)3/4.
M(t) =∫∞
0 m N(m, t) dm isformally conserved. However forµ+ ν > 1:
M(t) <∫ ∞
0m N(m,0) dm t > t∗
Best studied by introducingcut-off, Mmax, and studying limitMmax →∞. (Laurencot [2004])
Instantaneous GelationIf ν > 1 then t∗ = 0. (Van Dongen & Ernst [1987])May be complete: M(t) = 0 for t > 0. Example :K (m1,m2) = m1+ε
1 + m1+ε2 for ε > 0.
Mathematically pathological?
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Instantaneous Gelation in Regularised System
Yet there are applications with ν > 1: differential sedimentation,gravitationally clustering.
M(t) for K (m1,m2) = (m1m2)3/4.
With cut-off, Mmax, regularizedgelation time, t∗Mmax
, is clearlyidentifiable.t∗Mmax
decreases as Mmaxincreases.Van Dongen & Ernst recovered inlimit Mmax →∞.
Decrease of t∗Mmaxas Mmax is very slow (numerics suggest
logarithmic 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,R∞
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.M(t) ∼ t (due to source) butN(t) ∼ 1/t .
If the exponent ν > 1 then big clusters are so “hungry" that theyeat all smaller clusters at a rate which diverges with the cut-off,Mmax. (cf “addition model" (Brilliantov & Krapivsky [1992],Laurencot [1999])).
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Stationary State in the presence of a source
With cut-off, a stationary state may be reached if a source ofmonomers is present (Horvai et al [2007]) even though no suchstate exists in the unregularized system.
Non-local stationary state (theory vs
numerics) for ν = 3/2.
Introduce model kernelK (m1,m2) = max(mν
1 ,mν2).
Stationary state expressible via arecursion relation.Asymptotics:
N(m) ∼ A m−ν m� 1
Amplitude vanishes slowly asMmax →∞Decay near monomer scaleseems exponential.
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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 Mmaxincreases.Heuristic explanation in terms of“reset” mechanism.
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Conclusions
As has been known since the 1980’s, the Mmax →∞ limitof the regularized Smoluchowski equation is pathologicalfor kernels having ν > 1 but lots of interesting thingshappen along the way.Weak dependence of the regularized system on the valueof Mmax 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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