oscillatory kinetics in cluster-cluster aggregation
DESCRIPTION
Workshop "Interacting particle models in the physical, biological and social sciences", University of Warwick, May 02-03 2013TRANSCRIPT
Oscillatory kinetics in cluster-clusteraggregation
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 andsocial sciences
University of Warwick02-03 May 2013
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)
Aggregation phenomena : motivation
Many particles of onematerial dispersed inanother.Transport is diffusive oradvective.Particles stick together oncontact.
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 ofsuch systems in the presence of a source and sink of particles.
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)
Mean-field theory: the Smoluchowski equation
Assume the system is statistically homogeneous andwell-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)Nm1Nm2δ(m −m1 −m2)
− 2∫ ∞
0dm1dm2K (m,m1)NmNm1δ(m2 −m −m1)
+ J δ(m −m0)
Notation: In many applications kernel is homogeneous:K (am1,am2) = aλ K (m1,m2)
K (m1,m2) ∼ mµ1 mν
2 m1�m2.
Clearly λ = µ+ ν.http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)
Some example kernels
Brownian coagulation of ideal polymer in a good solvent (ν ≈ 35 ,
≈= −35 ):
K (m1,m2) =
(m1
m2
) 35
+
(m2
m1
) 35
+ 2
Gravitational settling of spherical droplets in laminar flow(ν = 4
3 , µ = 0) :
K (m1,m2) =
(m
131 + m
132
)2 ∣∣∣∣m 231 −m
232
∣∣∣∣Differential rotation driven coalescence (Saturn’s rings) (ν = 2
3 ,µ = −1
2 ) :
K (m1,m2) =
(m
131 + m
132
)2√m−1
1 + m−12
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)
Stationary solutions of the Smoluchowski equationwith a source and sink
Add monomers at rate, J.Remove those with m > M.Stationary state is obtained forlarge t which balancesinjection and removal.Constant mass flux in range[m0,M]
Model kernel:
K (m1,m2) =12
(mµ1 mν
2+mν1mµ
2 )
Stationary state for t →∞, m0 � m� M (Hayakawa 1987):
Nm =
√J (1− (ν − µ)2) cos((ν − µ)π/2)
2πm−
λ+32 .
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)
(Stockmayer) Regularisation and universality
Removing particles with m > M provides an explicit(non-conservative) regularisation. All integrals are finite.
∂tNm =12
∫ m
1dm1K (m1,m −m1)Nm1Nm−m1
− Nm
∫ M−m
1dm1K (m,m1)Nm1 + J δ(m − 1)
− DM [Nm(t)]
where
DM [Nm(t)] = Nm
∫ M
M−mdm1K (m,m1)Nm1 (1)
If |ν − µ| < 1 stationary state is universal - independent of M asM →∞ (Hayakawa solution). If |ν − µ| > 1 stationary state isnon-universal.
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)
Nonlocal kinetic equation (Horvai et al [2007])
Smoluchowski equation can be written (ignore source and sinkterms for now):
∂Nm
∂t=
∫ m2
1[F (m −m1,m1)− F (m,m1)] d m1
−∫ M
m2
F (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 −m1,m1)− F (m,m1)] = −m1N(m1, t)∂
∂m[K (m,m1) N(m, t)]
+O(m21).
We obtain an (almost) PDE.http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)
Nonlocal kinetic equation (Horvai et al [2007])
∂Nm
∂t= −Dµ+1
∂
∂m[mνN(m, t)]− Dν N(m, t)
where
Dµ+1 =∫ m
21 mµ+1
1 N(m1, t)dm1 →∫ M
1mµ+1
1 N(m1, t)dm1 = Mµ+1
Dν =∫ M
m mν1N(m1, t)dm1 →
∫ M
1mν
1N(m1, t)dm1 = Mν
Solving for stationary state:
Nm = C exp[β
γm−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)
Asymptotic solution of the nonlocal kinetic equation
β can be determined self-consistently since the momentsMµ+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
1dm m DM [Nm] ,
giving
C =
√J log Mγ
Mfor M � 1
Various assumptions are now verifiable a-posteriori.
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)
Asymptotic solution of the nonlocal kinetic equation
Nonlocal stationary state is not like the Hayakawa solution:
Stationary state (theory vs numerics).
Stationary state has theasymptotic form for M � 1:
Nm =
√2Jγ log M
MMm−γ
m−ν .
Stretched exponential for smallm, power law for large m.Stationary particle density:
N =
√J(
M −MM−γ)
M√γ log M
∼
√J
γ log Mas M →∞.
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)
Dynamical approach to the nonlocal stationary state
Total density, N(t), vs timefor ν = 3
2 , µ = − 32 .
Stochastic simulation
Numerics indicate that dynamicsare non-trivial when the cascadeis nonlocal (|ν − µ| > 1).Observe collective oscillations ofthe total density of clusters forlarge times.Are these oscillations slowlydecaying transients or is thestationary state unstable?Phenomenon is robust to noiseand visible in stochasticsimulations (in 0-d at least).
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)
Linear instability of the stationary state
Total density, N(t), vs timefor ν = 3
2 , µ = − 32 .
Linear stability analysis
We developed an iterativeprocedure which allowed us tocalculate the stationaryspectrum, N∗m, exactly.Using this we did a(semi-analytic) linear stabilityanalysis of the exactstationary state.Concluded that the nonlocalstationary state is linearlyunstable.Movie of density contrast,Nm(t)/N∗m.
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)
Instability has a nontrivial dependence on parameters
Instability growth rate vs M
Instability growth rate vs ν.
Linear stability analysis of thestationary state for |ν − µ| > 1reveals presence of a Hopfbifurcation as M is increased.Contrary to intuition, dependence ofthe growth rate on the exponent ν isnon-monotonic.Oscillatory behaviour seemingly dueto an attracting limit cycle embeddedin this very high-dimensionaldynamical system.
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)
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 = ms(t) ,
where
a = −ν + µ+ 32
s(t) ∼ t2
1−ν−µ .
Period estimated as the time, τM ,required for s(τM) ≈ M. Amplitude,AM , estimated as the mass suppliedin time τM :
τM ∼ M1−ν−µ
2 AM ∼ J M1−ν−µ
2 .
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)
Conclusions and open questions
Summary:Stationary solution of Smoluchowski equation with sourceinvestigated in regime |ν − µ| > 1.Size distribution can be calculated asymptotically and hasa novel functional form.Amplitude of state vanishes as the dissipation scale grows.Stationary state can become unstable so the long-timebehaviour 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 waveturbulence?
http://www.slideshare.net/connaughtonc Phys. Rev. Lett. 109, 168304 (2012)