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 = 0
Summary and Conclusions

Cluster-cluster aggregation with evaporation
and deposition

Colm Connaughton

Mathematics Institute and Centre for Complexity Science,
University of Warwick
Collaborators: R. Rajesh (Chennai), Oleg Zaboronski (Warwick)

UoM Theoretical Physics Seminars
02 June 2010

Colm Connaughton CCA


Outline
1 Cluster–Cluster Aggregation (CCA)
Cluster Aggregation: Applications
The Takayasu Model: A Mathematical Model of CCA
Takayasu Model with Evaporation
2 Stationary State of CCA without Evaporation: p = 0
Kolmogorov Theory of Turbulence: An analogy
Correlations and the Breakdown of Self-Similarity in CCA
3 Stationary State of CCA with Evaporation: p = 0
Growing Phase
Exponential Phase
Critical Phase
4 Summary and Conclusions



Outline
Growing Phase
Exponential Phase
Critical Phase



Cluster–Cluster Aggregation: Physical Examples

Particles of one material dis-
persed in another. Transport is
diffusive or advective. Interac-
tions between particles.
clustering / sedimentation
ﬂocculation
gelation
phase separation
Not to be confused with
Diffusion–Limited Aggregation.



Geomorphology: A model of river networks

Scheidegger (1967)
Rivulets ﬂow downhill
southeast or southwest
randomly (diffusion).
New rivulets appear
randomly (injection).
When rivulets intersect
they combine to produce
streams (aggregation).
Interested in distribution
of river sizes.



Self-Organised Criticality: Directed Sandpiles
Grains added at top. If
O(xi ) = 2 then it topples
and its grains are given to
it’s two neighbours 1 level
down producing an
"avalaunche" .
Simplest model of SOC.
Avalaunche size
distribution:

P(s) ∼ s−4/3

(Dhar and Ramaswamy Can be mapped to river
1989) network ﬂowing "uphill".



Cluster–Cluster Aggregation: Takayasu Model
Lattice Rd with particles of Racz (1985), Takayasu et al.
integer mass. (1988)
Nt (x, m)=number of mass Diffusion rate: DNt (x, m)/2d
m on site x at time t. 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

red-(1-10), green-(10-50), Injection rate: q
blue-(50-500) Nt (x, m) → Nt (x, m) + 1


Cluster–Cluster Aggregation: Takayasu Model
Model parameters:
D - diffusion constant ∆M j ∆M k ∆M j+ ∆M k
q - mass injection rate m
mj mk mi
g - reaction rate
Physical details are in the kernel: K (m1 , m2 ) ∼ mλ .
Deﬁnition
Cn (m1 , . . . , mn )(∆V )n i dmi = probability of having particles
of masses, mi , in the intervals [mi , mi + dmi ] in a volume ∆V .
∞
∂ Nm (t) J
= δ(m − m0 ) + dm1 dm2 C2 (m1 , m2 ) δ(m−m1 −m2 )
∂t m0 0
∞
− 2 dm1 dm2 C2 (m, m1 ) δ(m2 −m−m1 )
0


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 the
Smoluchowski (1917) kinetic equation :
∞
∂Nm (t)
= dm1 dm2 K (m1 , m2 )Nm1 Nm2 δ(m − m1 − m2 )
∂t 0
∞
− dm1 dm2 K (m, m1 )Nm Nm1 δ(m2 − m − m1 )
0
∞
− dm1 dm2 K (m2 , m)Nm Nm2 δ(m1 − m2 − m)
0
+ (q/m0 ) δ(m − m0 ) − DM [Nm ]




2.5
Upper bound
Mean field
Numerics (MF)
Evaporation rate: p Nt (x, m)
2
Numerics (1D) Nt (x, m) → Nt (x, m) − 1
Deposition rate, q

1.5
0
J>

Mass balance is non-trivial in
e,
as
ph

a “closed” system : Krapivsky
g

1
in
w
ro
G

0.5 se,
J=0 & Redner (1995)
pha
ntial
Exp
one Similar behaviour in open
0
0 0.5 1 1.5 2 2.5 3 system with injection:
Evaporation rate, p
Majumdar et al (2000)


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

Outline
Growing Phase
Exponential Phase
Critical Phase



Kolmogorov 1941 Theory of Turbulence
LU
Reynolds number R = ν .
Energy injected into large
eddies.
Energy removed from small
eddies at viscous scale.
Transfer by interaction
between eddies.
Concept of inertial range
K41 : In the limit of ∞ R, all small scale statistical properties
depend only on the local scale, k, and the energy dissipation
rate, ǫ. Dimensional analysis :
2 5
E (k) = cǫ 3 k − 3 Kolmogorov spectrum



4
Structure Functions and the 5 -Law
Structure functions : Sn (r ) = (u(x + r ) − u(x))n .
Scaling form in stationary state:
lim lim lim Sn (r ) = Cn (ǫr )ζn .
r →0 ν→0 t→∞

K41 theory gives ζn = n .
3

4
5 Law : S3 (r ) = − 4 ǫr . Thus ζ3 = 1 (exact).
5


Stationary State of CCA

Suppose particles having
1
Spectrum profiles : lambda=1.5 nu=0.5 P=0 Nd=10 IC=N

t=1.079349e-02
m > M are removed.
t=4.844532e-01

Stationary state is obtained for
t=1.384435e+00
t=1.999137e+00
t=2.331577e+00
1e-05 t=2.474496e+00

1e-10
large t when J = 0.
Stationary state is a balance
N(omega)

1e-15

between injection and
1e-20

dissipation. Constant mass ﬂux
1e-25

in range [m0 , M]
1e-30
1 10 100 1000
omega
10000 100000 1e+06
Essentially non-equilibrium: no
detailed balance.



Kolmogorov Theory of CCA (Constant Kernel)
Dimensional analysis λ = 0:
(3−2x )d (d +2)x −2d −2
Nm = c1 J x−1 D d −2 g d −2 m−x

Two possible values for Kolmogorov exponent:

3 2d + 2
xg = xD = .
2 d +2
Self-similarity of higher order correlation functions:
(3n−2γn )d (d +2)γn +(2d +2)n γn
Cn (m1 , . . . , mn ) = cn J γn −n D d −2 g d −2 (m1 . . . mn )− n ,

g 3 D 2d + 2
γn = n γn = n.
2 d +2


Kolmogorov Solution of Smoluchowski Equation
Zakharov Transformation: Nm = Cm−x
) 1
m2 −m
−m

′
mm1 m2
δ(m
2

m (m1 , m2 ) → ( ′ , ′ )

)
m2 m2
δ(

−m
2
m
−m

−m
′
m2 mm2
1
−m

1
δ(m

(m1 , m2 ) → ( , ′ ).
2
)

m m1 ′
m1 m1
∞
C2
0= dm1 dm2 K (m1 , m2 ) (m1 m2 )−x m2−λ−2x
2 0
2x−λ−2 2x−λ−2 2x−ζ−2
m − m1 − m2 δ(m − m1 − m2 )

x = (λ + 3)/2. C depends on K . If K (m1 , m2 )=(m1 m2 )λ/2 :
J λ+3
Nm = m− 2 .
2πg


The Takayasu Model in Low Dimensions

1
In d ≤ 2. Mean ﬁeld
Spatially extended
Mean field
-4/3
m-3/2
scaling exponents are not
0.1 m
correct.
0.01
3
In 1-d x = 2 becomes
P(m)

x = 4 (exact).
0.001

3
0.0001
Reason is development of
1e-05
spatial correlations
1e-06
1 10 100 1000 generated by recurrence
m
property of random walks.



Spatial Correlations

Visualising spatial correlations:

5 x 10-3
Regular Diffusion
Deﬁnition
β=2.0 Levy diffusion

4 x 10-3
β=1.6 Levy diffusion Pm (x) = Probability of ﬁnding a
β=1.0 Levy diffusion
particle of mass greater than
Density auto-correlation

Random hopping

3 x 10-3
m at a distance x from a
2 x 10-3
particle of mass m.

1 x 10-3 Heavy particles develop zones
0 x 100
of exclusion.
0 20 40
x
60 80 100
Aggregation of heavy particles
is suppressed relative to MF
estimates.



A Theoretical Approach
m∈Z +
1 A set {ni,m }x ∈Rd determines a conﬁguration.
i
2 Write a Master equation for time evolution of P({ni,m }).
3 Convert master equation into a Schrodinger equation :

d
|ψ(t) = −H[ai,m , a† ] |ψ(t)
i,m
dt
using Doi’s formalism. Path integral representation gives a
continuous ﬁeld theory having critical dimension 2:

∗ ,t,D,g,J]
Nm (t) = DφDφ∗ φ(m, t) e−Seff [φ,φ

4 Use standard techniques to compute correlation functions.



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

Resumming: g → gR (m)
d 2d +2
Nm ∼ φm = (J/D) d +2 m− d +2

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


Renormalisation of correlation functions in d < 2
½ ½
First diagram gives MF answer :
+ Rµ1 Rµ2 = Rµ1 Rµ2 .
¾ ¾
Singularities in third and fourth
½ ½ diagram are removed by λ → λR .
½

+ +
Singularity in second is not.
¾
Higher correlations also require
¾ ¾
multiplicative renormalisation.
Final result : Cn (m1 , . . . , mn ) ∼ m−γ(n)

2d + 2 ǫ n(n − 1)
γ(n) = n+ + O(ǫ2 ).
d +2 d +2 2

where ǫ = 2 − d .



Multi-scaling of higher order correlation functions
RG calculation shows
10
presence of multi-scaling
γkolm(n)

8
one loop
in the particle distribution
6
for high masses.
γ (n)

4
Compare exponents
2
obtained from ǫ-expansion
0
with measurements from
0 1 2 3 4 5
n Monte-Carlo simulations
Montecarlo measure- in d = 1.
ments of multiscaling Why is agreement so
exponents in Takayasu good?
model. Note special property of
n = 2...



4
Analogue of the 5 -Law
γ(2) = 3 is an exact - a counterpart of the 4/5 law.
Conﬁrms multiscaling in this model without using
ǫ-expansion.
Stationary state:
∞
0 = dm1 dm2 C(m1 , m2 ) δ(m−m1 −m2 )
0
∞
− dm1 dm2 C(m, m1 ) δ(m2 −m−m1 )
0
∞
− dm1 dm2 C(m, m2 ), δ(m1 −m2 −m)
0

Scaling form : C(m1 , m2 ) = (m1 m2 )−h ψ(m1 /m2 ).
Zakharov transformation and constant ﬂux give h = 3.

Growing Phase
Exponential Phase
Critical Phase

Outline
Growing Phase
Exponential Phase
Critical Phase


Growing Phase
Exponential Phase
Critical Phase

Nonequilibrium Phase Transition
2.5
Upper bound
Low evaporation: growing
2
Mean field
Numerics (MF) phase - M(t) ∼ t.
Numerics (1D)

High evaporation:
Deposition rate, q

1.5
J >0 exponential phase -
e,
as
ph
g

1
M(t) ∼ constant.
in
w
ro
G

0
, J=
0.5
tia l ph
ase Critical line q = qc (p)
onen
Exp
0 separates the two
0 0.5 1 1.5 2 2.5 3
Evaporation rate, p regimes.

MF
Mean ﬁeld: qc (p) = p + 2 − 2 p+1

1
Upper bound: qc (p) ≤ p−2+ p2 + 4
2


Growing Phase
Exponential Phase
Critical Phase

Growing phase in mean field

Most aspects of system are amenable to analytic analysis at
mean field level.
lattice: 1000, λ=10.0, q=1.0 lattice: 1000, λ=10.0, q=1.0
101 101
p=0.0
100 p=0.815
100 p=0.415
-1 p=0.315
10
10-1 p=0.215
Aggregation Flux

Mass distribution
10-2 m-3/2
m-5/2
10-2 10-3

10-4
10-3 p=0.0
p=0.815 10-5
10-4 p=0.415
p=0.315 10-6
p=0.215
10-5 10-7 0
0 200 400 600 800 1000 10 101 102 103
m m

Mass flux. Mass distribution.


Growing Phase
Exponential Phase
Critical Phase

Growing Phase in 1D

Aside from C2 (m) (known from constant flux) we have no
analytic results for the 1-D case yet. Numerically observe the
same multiscaling exponents.
0.5 0
q=1.00 k=1
-10
0.4 -20 k=2

-30
k=3
0.3 q=0.75 -40

ln[Pk(m)]
Jagg

-50 5 Simulation
0.2 4 Theory
-60
3

γn
-70
q=0.50 2
0.1 -80 1
0
-1.33
q≈qc -90 -3.00
0 0.5 1 1.5 2 2.5 3
0 n -5.04
0 1 2 3 4 5 6 -100
10 10 10 10 10 10 10 0 2 4 6 8 10 12 14
m ln(m)

Conjecture that growing phase is in same universality class as
the original Takayasu model (mass flux is modified). k=1


Growing Phase
Exponential Phase
Critical Phase

Exponential Phase
If p < pc the mass
distribution decays
Im
Jev(m)
exponentially and
Jagg (1)
Jagg (m)
Jagg → 0 as m → ∞.
Theory gives the mean
0 1 m Mass
10
0 ﬁeld result:
100
Jagg
10-2 pmP(m+1)
-2
10 10-4
10-6
1 (m)
10
-4 10-8 P(m + 1) ∼ Jagg
10
-6
10-10
0 10 20 30 40 50
pm
m

-8
10

-10
which numerics suggest is
10

-12
Jagg
P(m)
true for d < 2.
10
0 5 10 15 20 25 30 35 40 45
m Looks more like detailed
balance.


Growing Phase
Exponential Phase
Critical Phase

Critical Phase

If p = pc the stationary
lattice: 100000, λ=10.0, q=0.22 p=1.0
100
mass flux Jagg decays as a
Nm
C2(m)
m-5/2
power law as m → ∞.
10-2
m-4
At mean field level:
10-4
Nm

5
10-6
Nm ∼ m − 2
10-8

Krapivsky & Redner
10-10
100 101 102 103
m
(1995).
Exponent is modified in
N(m) and C2 (m) at the
d = 1. Numerics gives
critical point (mean field).
1.83.



Conclusions

CCA is a broadly interesting and useful model in physics
and elsewhere.
There are useful analogies with turbulent systems.
In d ≤ 2 diffusive ﬂuctuations dominate the dynamics
leading to a breakdown of mean-ﬁeld theory and
emergence of spatially correlated structures.
Introduction of weak evaporation doesn’t change much.
Stronger evaporation triggers transition from growing to
exponential phase.



References

Colm Connaughton, R. Rajesh and Oleg Zaboronski
1 "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. E
69, 061114 (2004)


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

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