Workshop "Interacting particle models in the physical, biological and social sciences", University of Warwick, May 02-03 2013

1. Oscillatory kinetics in cluster-cluster
aggregation
Colm Connaughton
Mathematics Institute and Centre for Complexity Science,
University of Warwick, UK
Collaborators: R. Ball (Warwick), P.P. Jones (Warwick), R. Rajesh (Chennai), O.
Zaboronski (Warwick).
Interacting particle models in the physical, biological and
social sciences
University of Warwick
02-03 May 2013
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)

2. Aggregation phenomena : motivation
Many particles of one
material dispersed in
another.
Transport is diffusive or
advective.
Particles stick together on
contact.
Applications: surface physics, colloids, atmospheric science,
earth sciences, polymers, bio-physics, cloud physics.
This talk:
Today we will focus on the mean field statistical dynamics of
such systems in the presence of a source and sink of particles.
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)

3. Mean-field theory: the Smoluchowski equation
Assume the system is statistically homogeneous and
well-mixed so that there are no spatial correlations.
Cluster size distribution, Nm(t), satisfies the kinetic equation :
Smoluchowski equation :
∂Nm(t)
∂t
=
∞
0
dm1dm2K(m1, m2)Nm1
Nm2
δ(m − m1 − m2)
− 2
∞
0
dm1dm2K(m, m1)NmNm1
δ(m2 − m − m1)
+ J δ(m − m0)
Notation: In many applications kernel is homogeneous:
K(am1, am2) = aλ
K(m1, m2)
K(m1, m2) ∼ mµ
1 mν
2 m1 m2.
Clearly λ = µ + ν.
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)

4. Some example kernels
Brownian coagulation of ideal polymer in a good solvent (ν ≈ 3
5 ,
≈= −3
5 ):
K(m1, m2) =
m1
m2
3
5
+
m2
m1
3
5
+ 2
Gravitational settling of spherical droplets in laminar flow
(ν = 4
3 , µ = 0) :
K(m1, m2) = m
1
3
1 + m
1
3
2
2
m
2
3
1 − m
2
3
2
Differential rotation driven coalescence (Saturn’s rings) (ν = 2
3 ,
µ = −1
2 ) :
K(m1, m2) = m
1
3
1 + m
1
3
2
2
m−1
1 + m−1
2
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)

5. Stationary solutions of the Smoluchowski equation
with a source and sink
Add monomers at rate, J.
Remove those with m > M.
Stationary state is obtained for
large t which balances
injection and removal.
Constant mass flux in range
[m0, M]
Model kernel:
K(m1, m2) =
1
2
(mµ
1 mν
2+mν
1mµ
2 )
Stationary state for t → ∞, m0 m M (Hayakawa 1987):
Nm =
J (1 − (ν − µ)2) cos((ν − µ) π/2)
2π
m−λ+3
2 .
Require mass flux to be local: |µ − ν| < 1. But what if it isn’t?
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)

6. (Stockmayer) Regularisation and universality
Removing particles with m > M provides an explicit
(non-conservative) regularisation. All integrals are finite.
∂t Nm =
1
2
m
1
dm1K(m1, m − m1)Nm1
Nm−m1
− Nm
M−m
1
dm1K(m, m1)Nm1
+ J δ(m − 1)
− DM [Nm(t)]
where
DM [Nm(t)] = Nm
M
M−m
dm1K(m, m1)Nm1
(1)
If |ν − µ| < 1 stationary state is universal - independent of M as
M → ∞ (Hayakawa solution). If |ν − µ| > 1 stationary state is
non-universal.
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)

7. Nonlocal kinetic equation (Horvai et al [2007])
Smoluchowski equation can be written (ignore source and sink
terms for now):
∂Nm
∂t
=
m
2
1
[F(m − m1, m1) − F(m, m1)] d m1
−
M
m
2
F(m, m1) dm1
where
F(m1, m2) = K(m1, m2) N(m1, t) N(m2, t)
Taylor expand the first term to first order in m1:
[F(m − m1, m1) − F(m, m1)] = −m1N(m1, t)
∂
∂m
[K(m, m1) N(m, t)]
+O(m2
1).
We obtain an (almost) PDE.
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)

8. Nonlocal kinetic equation (Horvai et al [2007])
∂Nm
∂t
= −Dµ+1
∂
∂m
[mν
N(m, t)] − Dν N(m, t)
where
Dµ+1 =
m
2
1 mµ+1
1 N(m1, t)dm1 →
M
1
mµ+1
1 N(m1, t)dm1 = Mµ+1
Dν =
M
m mν
1N(m1, t)dm1 →
M
1
mν
1N(m1, t)dm1 = Mν
Solving for stationary state:
Nm = C exp
β
γ
m−γ
m−ν
where γ = ν − µ − 1, C is an arbitrary constant and
β =
Mν
Mµ+1
.
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)

9. Asymptotic solution of the nonlocal kinetic equation
β can be determined self-consistently since the moments
Mµ+1 and Mν become Γ-functions.
Resulting consistency condition gives:
β ∼ γ log M for M 1.
Constant, C, is fixed by requiring global mass balance:
J =
M
1
dm m DM [Nm] ,
giving
C =
J log Mγ
M
for M 1
Various assumptions are now verifiable a-posteriori.
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)

10. Asymptotic solution of the nonlocal kinetic equation
Nonlocal stationary state is not like the Hayakawa solution:
Stationary state (theory vs numerics).
Stationary state has the
asymptotic form for M 1:
Nm =
2Jγ log M
M
Mm−γ
m−ν
.
Stretched exponential for small
m, power law for large m.
Stationary particle density:
N =
√
J M − MM−γ
M γ log M
∼
J
γ log M
as M → ∞.
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)

11. Dynamical approach to the nonlocal stationary state
Total density, N(t), vs time
for ν = 3
2
, µ = − 3
2
.
Stochastic simulation
Numerics indicate that dynamics
are non-trivial when the cascade
is nonlocal (|ν − µ| > 1).
Observe collective oscillations of
the total density of clusters for
large times.
Are these oscillations slowly
decaying transients or is the
stationary state unstable?
Phenomenon is robust to noise
and visible in stochastic
simulations (in 0-d at least).
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)

12. Linear instability of the stationary state
Total density, N(t), vs time
for ν = 3
2
, µ = − 3
2
.
Linear stability analysis
We developed an iterative
procedure which allowed us to
calculate the stationary
spectrum, N∗
m, exactly.
Using this we did a
(semi-analytic) linear stability
analysis of the exact
stationary state.
Concluded that the nonlocal
stationary state is linearly
unstable.
Movie of density contrast,
Nm(t)/N∗
m.
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)

13. Instability has a nontrivial dependence on parameters
Instability growth rate vs M
Instability growth rate vs ν.
Linear stability analysis of the
stationary state for |ν − µ| > 1
reveals presence of a Hopf
bifurcation as M is increased.
Contrary to intuition, dependence of
the growth rate on the exponent ν is
non-monotonic.
Oscillatory behaviour seemingly due
to an attracting limit cycle embedded
in this very high-dimensional
dynamical system.
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)

14. Scaling of period and amplitude of oscillation with M
Self-simliarity of mass pulse
Oscillations for different M
Oscillations are a sequence of
”resets” of self-similar pulses:
Nm(t) = s(t)a
F(ξ) with ξ = m
s(t),
where
a = −
ν + µ + 3
2
s(t) ∼ t
2
1−ν−µ .
Period estimated as the time, τM,
required for s(τM) ≈ M. Amplitude,
AM, estimated as the mass supplied
in time τM:
τM ∼ M
1−ν−µ
2 AM ∼ J M
1−ν−µ
2 .
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)

15. Conclusions and open questions
Summary:
Stationary solution of Smoluchowski equation with source
investigated in regime |ν − µ| > 1.
Size distribution can be calculated asymptotically and has
a novel functional form.
Amplitude of state vanishes as the dissipation scale grows.
Stationary state can become unstable so the long-time
behaviour of the cascade dynamics is oscillatory.
Questions:
Do any physical systems really behave like this?
What happens in spatially extended systems?
Other examples of oscillatory kinetics? Eg in wave
turbulence?
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)