Cluster-cluster aggregation with evaporation and deposition, University of Manchester, 02 June 2010

Cluster–Cluster Aggregation (CCA)Stationary State of CCA without Evaporation: p = 0

Stationary State of CCA with Evaporation: p 6= 0Summary and Conclusions

Cluster-cluster aggregation with evaporationand deposition

Colm Connaughton

Mathematics Institute and Centre for Complexity Science,University of Warwick

Collaborators: R. Rajesh (Chennai), Oleg Zaboronski (Warwick)

UoM Theoretical Physics Seminars02 June 2010

Colm Connaughton CCA



Outline

1 Cluster–Cluster Aggregation (CCA)Cluster Aggregation: ApplicationsThe Takayasu Model: A Mathematical Model of CCATakayasu Model with Evaporation

2 Stationary State of CCA without Evaporation: p = 0Kolmogorov Theory of Turbulence: An analogyCorrelations and the Breakdown of Self-Similarity in CCA

3 Stationary State of CCA with Evaporation: p 6= 0Growing PhaseExponential PhaseCritical Phase

4 Summary and Conclusions




Cluster Aggregation: ApplicationsThe Takayasu Model: A Mathematical Model of CCATakayasu Model with Evaporation

Outline









Cluster–Cluster Aggregation: Physical Examples

Particles of one material dis-persed in another. Transport isdiffusive or advective. Interac-tions between particles.

clustering / sedimentation

flocculation

gelation

phase separation

Not to be confused withDiffusion–Limited Aggregation.





Geomorphology: A model of river networks

Scheidegger (1967)

Rivulets flow downhillsoutheast or southwestrandomly (diffusion).

New rivulets appearrandomly (injection).

When rivulets intersectthey combine to producestreams (aggregation).

Interested in distributionof river sizes.





Self-Organised Criticality: Directed Sandpiles

(Dhar and Ramaswamy1989)

Grains added at top. IfO(xi ) = 2 then it topplesand its grains are given toit’s two neighbours 1 leveldown producing an"avalaunche" .

Simplest model of SOC.Avalaunche sizedistribution:

P(s) ∼ s−4/3

Can be mapped to rivernetwork flowing "uphill".





Cluster–Cluster Aggregation: Takayasu Model

Lattice Rd with particles of

integer mass.Nt(x,m)=number of massm on site x at time t .

red-(1-10), green-(10-50),blue-(50-500)

Racz (1985), Takayasu et al.(1988)Diffusion rate: DNt(x,m)/2d

Nt(x,m) → Nt(x,m) − 1

Nt(x + n,m) → Nt(x + n,m) + 1

Aggregation rate:gK (m1,m2)Nt(x,m1)Nt(x,m2)

Nt(x,m1) → Nt(x,m1) − 1

Nt(x,m2) → Nt(x,m2) − 1

Nt(x,m1 + m2) → Nt(x,m1 + m2) + 1

Injection rate: qNt(x,m) → Nt(x,m) + 1





Cluster–Cluster Aggregation: Takayasu Model

Model parameters:

D - diffusion constant

q - mass injection rate

g - reaction rate

mmj k mi

j ∆M ∆

m

M + ∆Mj k∆M k

Physical details are in the kernel: K (m1,m2) ∼ mλ.

Definition

Cn(m1, . . . ,mn)(∆V )n ∏

i dmi = probability of having particlesof masses, mi , in the intervals [mi ,mi + dmi ] in a volume ∆V .

∂〈Nm(t)〉∂t

=J

m0δ(m − m0) +

∫ ∞

0dm1dm2 C2(m1,m2) δ(m−m1−m2)

− 2∫ ∞

0dm1dm2 C2(m,m1) δ(m2−m−m1)





Mean-field Theory: Smoluchowski Dynamics

Mean Field Approximation:

C2(m1,m2, t) ≈ Nm1(t)Nm2(t)

Well-mixed. No spatial correlations. Then Nm(t) satisfies theSmoluchowski (1917) kinetic equation :

∂Nm(t)∂t

=

∫ ∞

0dm1dm2K (m1,m2)Nm1Nm2δ(m − m1 − m2)

−

∫ ∞

0dm1dm2K (m,m1)NmNm1δ(m2 − m − m1)

−

∫ ∞

0dm1dm2K (m2,m)NmNm2δ(m1 − m2 − m)

+ (q/m0) δ(m − m0) − DM [Nm]





Takayasu Model with Evaporation

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3

Dep

ositi

on r

ate,

q

Evaporation rate, p

Growing

pha

se, J

>0

Exponential phase, J=0

Upper boundMean fieldNumerics (MF)Numerics (1D)

Evaporation rate: p Nt(x,m)Nt(x,m) → Nt(x,m) − 1

Mass balance is non-trivial ina “closed” system : Krapivsky& Redner (1995)

Similar behaviour in opensystem with injection:Majumdar et al (2000)




Kolmogorov Theory of Turbulence: An analogyCorrelations and the Breakdown of Self-Similarity in CCA

Outline









Kolmogorov 1941 Theory of Turbulence

Reynolds number R = LUν .

Energy injected into largeeddies.

Energy removed from smalleddies at viscous scale.

Transfer by interactionbetween eddies.

Concept of inertial rangeK41 : In the limit of ∞ R, all small scale statistical propertiesdepend only on the local scale, k , and the energy dissipationrate, ǫ. Dimensional analysis :

E(k) = cǫ23 k− 5

3 Kolmogorov spectrum





Structure Functions and the 45-Law

Structure functions : Sn(r) =⟨

(u(x + r) − u(x))n⟩

.Scaling form in stationary state:

limr→0

limν→0

limt→∞

Sn(r) = Cn (ǫr)ζn .

K41 theory gives ζn = n3 .

45 Law : S3(r) = −4

5ǫr . Thus ζ3 = 1 (exact).Colm Connaughton CCA




Stationary State of CCA

1e-30

1e-25

1e-20

1e-15

1e-10

1e-05

1

1 10 100 1000 10000 100000 1e+06

N(o

meg

a)

omega

Spectrum profiles : lambda=1.5 nu=0.5 P=0 Nd=10 IC=N

t=1.079349e-02t=4.844532e-01t=1.384435e+00t=1.999137e+00t=2.331577e+00t=2.474496e+00

Suppose particles havingm > M are removed.

Stationary state is obtained forlarge t when J 6= 0.

Stationary state is a balancebetween injection anddissipation. Constant mass fluxin range [m0,M]

Essentially non-equilibrium: nodetailed balance.





Kolmogorov Theory of CCA (Constant Kernel)

Dimensional analysis λ = 0:

〈Nm〉 = c1 Jx−1D(3−2x)d

d−2 g(d+2)x−2d−2

d−2 m−x

Two possible values for Kolmogorov exponent:

xg =32

xD =2d + 2d + 2

.

Self-similarity of higher order correlation functions:

Cn(m1, . . . ,mn) = cn Jγn−nD(3n−2γn)d

d−2 g(d+2)γn+(2d+2)n

d−2 (m1 . . .mn)−

γnn ,

γgn =

32

n γDn =

2d + 2d + 2

n.





Kolmogorov Solution of Smoluchowski Equation

m2

m1

m 2

m 1δ(

−)

m−m

1 m2

δ(m−

−)

m 1

m 2δ(

−)

m−

m

m

Zakharov Transformation: Nm = Cm−x

(m1,m2) → (mm′

1

m′2,m2

m′2)

(m1,m2) → (m2

m′1,mm′

2

m′1

).

0 =C2

2

∫ ∞

0dm1dm2 K (m1,m2) (m1m2)

−xm2−λ−2x

(

m2x−λ−2 − m2x−λ−21 − m2x−ζ−2

2

)

δ(m − m1 − m2)

x = (λ+ 3)/2. C depends on K . If K (m1,m2)=(m1m2)λ/2:

Nm =

√

J2πg

m−λ+32 .





The Takayasu Model in Low Dimensions

1e-06

1e-05

0.0001

0.001

0.01

0.1

1

1 10 100 1000

P(m

)

m

Spatially extendedMean field

m-4/3

m-3/2

In d ≤ 2. Mean fieldscaling exponents are notcorrect.

In 1-d x = 32 becomes

x = 43 (exact).

Reason is development ofspatial correlationsgenerated by recurrenceproperty of random walks.





Spatial Correlations

0 x 100

1 x 10-3

2 x 10-3

3 x 10-3

4 x 10-3

5 x 10-3

0 20 40 60 80 100

Den

sity

aut

o-co

rrel

atio

n

x

Regular Diffusionβ=2.0 Levy diffusionβ=1.6 Levy diffusionβ=1.0 Levy diffusionRandom hopping

Visualising spatial correlations:

Definition

Pm(x) = Probability of finding aparticle of mass greater thanm at a distance x from aparticle of mass m.

Heavy particles develop zonesof exclusion.Aggregation of heavy particlesis suppressed relative to MFestimates.





A Theoretical Approach

1 A set {ni ,m}m∈Z+

xi∈Rd determines a configuration.

2 Write a Master equation for time evolution of P({ni ,m}).3 Convert master equation into a Schrodinger equation :

ddt

|ψ(t)〉 = −H[ai ,m,a†i ,m] |ψ(t)〉

using Doi’s formalism. Path integral representation gives acontinuous field theory having critical dimension 2:

Nm(t) =

∫

DφDφ∗φ(m, t) e−Seff[φ,φ∗,t,D,g,J]

4 Use standard techniques to compute correlation functions.





Renormalisation of reaction rate

Mean-field answer obtained from summing tree diagrams but ind ≤ 2, loops are divergent as t → ∞.The only loop diagrams which correct the average density arethose which renormalise the reaction rate :

Resumming: g → gR(m)

Nm ∼ 〈φm〉 = (J/D)d

d+2 m− 2d+2d+2

xD is renormalised mean field exponent (Rajesh and Majumdar(2000) by other means).





Renormalisation of correlation functions in d < 2

+

+

+

�1�2�2

�1�2�1

�1 �1�2 �2

First diagram gives MF answer :〈Rµ1Rµ2〉 = 〈Rµ1〉〈Rµ2〉.

Singularities in third and fourthdiagram are removed by λ→ λR .

Singularity in second is not.

Higher correlations also requiremultiplicative renormalisation.

Final result : Cn(m1, . . . ,mn) ∼ m−γ(n)

γ(n) =

(

2d + 2d + 2

)

n +

(

ǫ

d + 2

)

n(n − 1)

2+ O(ǫ2).

where ǫ = 2 − d .





Multi-scaling of higher order correlation functions

0

2

4

6

8

10

0 1 2 3 4 5

γ (n

)

n

γkolm(n)one loop

Montecarlo measure-ments of multiscalingexponents in Takayasumodel.

RG calculation showspresence of multi-scalingin the particle distributionfor high masses.

Compare exponentsobtained from ǫ-expansionwith measurements fromMonte-Carlo simulationsin d = 1.

Why is agreement sogood?

Note special property ofn = 2...





Analogue of the 45-Law

γ(2) = 3 is an exact - a counterpart of the 4/5 law.Confirms multiscaling in this model without usingǫ-expansion.Stationary state:

0 =

∫ ∞

0dm1dm2 C(m1,m2) δ(m−m1−m2)

−

∫ ∞

0dm1dm2 C(m,m1) δ(m2−m−m1)

−

∫ ∞

0dm1dm2 C(m,m2), δ(m1−m2−m)

Scaling form : C(m1,m2) = (m1m2)−hψ(m1/m2).

Zakharov transformation and constant flux give h = 3.




Growing PhaseExponential PhaseCritical Phase

Outline









Nonequilibrium Phase Transition

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3

Dep

ositi

on r

ate,

q

Evaporation rate, p

Growing

pha

se, J

>0

Exponential phase, J=0

Upper boundMean fieldNumerics (MF)Numerics (1D)

Low evaporation: growingphase - M(t) ∼ t .

High evaporation:exponential phase -M(t) ∼ constant.

Critical line q = qc(p)separates the tworegimes.

Mean field: qMFc (p) = p + 2 − 2

√

p + 1

Upper bound: qc(p) ≤12

(

p − 2 +

√

p2 + 4)





Growing phase in mean field

Most aspects of system are amenable to analytic analysis atmean field level.

10-5

10-4

10-3

10-2

10-1

100

101

0 200 400 600 800 1000

Agg

rega

tion

Flu

x

m

lattice: 1000, λ=10.0, q=1.0

p=0.0p=0.815p=0.415p=0.315p=0.215

Mass flux.

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

100 101 102 103

Mas

s di

strib

utio

n

m

lattice: 1000, λ=10.0, q=1.0

p=0.0p=0.815p=0.415p=0.315p=0.215m-3/2

m-5/2

Mass distribution.





Growing Phase in 1D

Aside from C2(m) (known from constant flux) we have noanalytic results for the 1-D case yet. Numerically observe thesame multiscaling exponents.

0

0.1

0.2

0.3

0.4

0.5

100 101 102 103 104 105 106

J agg

m

q=1.00

q=0.75

q=0.50

q≈qc

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

0 2 4 6 8 10 12 14

ln[P

k(m

)]

ln(m)

k=1

k=2

k=3

-1.33-3.00-5.04

0

1

2

3

4

5

0 0.5 1 1.5 2 2.5 3

γ nn

k=1

SimulationTheory

Conjecture that growing phase is in same universality class asthe original Takayasu model (mass flux is modified).





Exponential Phase

Mass 0 1 m

I m

J agg (m) J agg (1)

J ev (m)

10-12

10-10

10-8

10-6

10-4

10-2

100

0 5 10 15 20 25 30 35 40 45

m

JaggP(m)

10-10

10-8

10-6

10-4

10-2

100

0 10 20 30 40 50m

Jagg

pmP(m+1)

If p < pc the massdistribution decaysexponentially andJagg→ 0 as m → ∞.

Theory gives the meanfield result:

P(m + 1) ∼1

p mJ(m)

agg

which numerics suggest istrue for d < 2.

Looks more like detailedbalance.





Critical Phase

10-10

10-8

10-6

10-4

10-2

100

100 101 102 103

Nm

m

lattice: 100000, λ=10.0, q=0.22 p=1.0

NmC2(m)

m-5/2

m-4

N(m) and C2(m) at thecritical point (mean field).

If p = pc the stationarymass flux Jagg decays as apower law as m → ∞.

At mean field level:

Nm ∼ m− 52

Krapivsky & Redner(1995).

Exponent is modified ind = 1. Numerics gives1.83.




Outline








Conclusions

CCA is a broadly interesting and useful model in physicsand elsewhere.

There are useful analogies with turbulent systems.

In d ≤ 2 diffusive fluctuations dominate the dynamicsleading to a breakdown of mean-field theory andemergence of spatially correlated structures.

Introduction of weak evaporation doesn’t change much.

Stronger evaporation triggers transition from growing toexponential phase.




References

Colm Connaughton, R. Rajesh and Oleg Zaboronski1 "Phases of Evaporation–Deposition Models", To appear,

(2010)2 "Constant Flux Relation for Driven Dissipative Systems",

Phys. Rev. Lett. 98, 080601 (2007)3 "Cluster-Cluster Aggregation as an Analogue of a

Turbulent Cascade", Physica D, Volume 222, 1-2 (2006)4 "Breakdown of Kolmogorov Scaling in Models of Cluster

Aggregation", Phys. Rev. Lett. 94, 194503 (2005)5 "Stationary Kolmogorov solutions of the Smoluchowski

aggregation equation with a source term", Phys. Rev. E69, 061114 (2004)


Cluster-cluster aggregation with evaporation and deposition, University of Manchester, 02 June 2010

Documents

Transcript of Cluster-cluster aggregation with evaporation and deposition, University of Manchester, 02 June 2010