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)