Fluctuations and rare events in stochastic aggregation

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Fluctuations and rare events in stochasticaggregation

Colm Connaughton

Mathematics Institute and Centre for Complexity Science,University of Warwick, UK

Collaborators: R. Rajesh (Chennai), R. Tribe (Warwick), O. Zaboronski (Warwick).

IUTAM Syposium - Common aspects of extreme events influids

University College Dublin July 2-6, 2012

http://www.slideshare.net/connaughtonc Stochastic aggregation


Introduction to cluster-cluster aggregation (CCA)

Many particles of onematerial dispersed inanother.Transport: diffusive,advective, ballistic...Particles stick together oncontact.

Applications: surface and colloid physics, atmosphericscience, biology, cloud physics, astrophysics...



Mean-field model: Smoluchowski’s kinetic equation

Cluster size distribution, Nm(t), satisfies the kinetic equation :

Smoluchowski equation :

∂Nm(t)∂t

=12

∫ m

0dm1dm2K (m1,m2)Nm1Nm2δ(m −m1 −m2)

−∫ ∞

0dm1dm2K (m,m1)NmNm1δ(m2 −m −m1)

+J

m0δ(m −m0)

Source of monomers: input mass flux J.Assumes no correlations.Very extensive literature on this equation.



Mass cascades in CCA with a source and sink

K (m1,m2) = 1.

Kernel is often homogeneous:

K (am1,am2) = aβ K (m1,m2)

Today we will mostly consider the con-stant kernel (β = 0):

K (m1,m2) = λ.

Stationary state for t →∞, m� m0 (White 1967):

Nm = c√

J m−β+3

2 .

Describes a cascade of mass from source at m0 to sink at largem. Analogous to the Richardson cascade in turbulence.



Stochastic Particle System Model of Aggregation

Consider a lattice in d dimensions with particles of integermasses. Nt (x,m)=number of particles of mass m on site x attime t .

Diffusion: Nt (x,m)→ Nt (x,m)− 1Nt (x + n,m)→ Nt (x + n,m) + 1

Rate: DNt (x,m)/2d

Aggregation: Nt (x,m1)→ Nt (x,m1)− 1Nt (x,m2)→ Nt (x,m2)− 1Nt (x,m1 + m2)→ Nt (x,m1 + m2) + 1

Rate: λNt (x,m1)Nt (x,m2)

Injection: Nt (x,m)→ Nt (x,m) + 1Rate: J



Path integral representation of correlation functions

Stochastic particle system can be reformulated as a statisticalfield theory (Doi, Peliti, Zeldovich, Ovchinnikov, Cardy...)

〈nm〉 =

∫D[zm, zm] zm e−Seff[zm,zm] (1)

where

Seff =

∫ T

0dτdx

[∑m

zm(∂tzm − D∆zm − Jδm,1) +H[zm + 1, zm]

]

and

H[z, z] = −12

∑m1,m2

λm1,m2(zm1+m2 − zm1 zm2) zm1zm2 .



Diffusive fluctuations in d ≤ 2

Critical dimension for the theory is 2. Below 2-d diffusivefluctuations cause mean field theory to fail.Exact solution (Takayasu 1985)for size distribution in 1-d(constant kernel):

〈Nm〉 = c J13 m−

43 cf MF: Nm ∼ m−

32 .

Renormalisation of reaction rate within perturbative RGframework (ε = 2− d):

〈Nm〉 ∼ Jd

d+2 m−2d+2d+2

Leading term in RG expansion is exact in d = 1 (ε = 1)!



Multiscaling and intermittency

For d ≤ 2 higher order mass correlation functions exhibitmulti-scaling with m:

Cn = Nm1 . . .Nmn ∼ m−γn MF: γn = n 32

γ0 = 0.γ1 = 4

3 in 1-d (exact solution).γ2 = 3 is exact (analogue of45 -law!).These are not on a straight line.RG calculation gives:γn = 2d+2

d+2 n + εd+2

n(n−1)2 + O(ε2)

where ε = 2− d .



Origin of intermittency: anti-correlation betweenclusters

“Large particles are large because they have merged with all oftheir neighbours".

Due to recurrence of randomwalks in d < 2 particlesencounter their neighboursinfinitely often.Large particles surrounded byvoids (correlation hole) and arethus spatially correlated.

Spatial intermittency of the mass distribution in low dimensionsis due to sites having atypically few or atypically many particles.



Rare events in stochastic aggregation

Can we calculate probabilities of atypical configurations?Spatial problem is hard. Start with a simpler 0-d problem:

Initial condition: M particles of size 1. No source.Particles aggregate at rate λ.What is annihilation time, τ , when 1 particle remains?

Mean-field equation for total number of particles:

N = −λ2

N2 N(0) = M.

Solution: N(t) = 2M/(2 + Mλ t)⇒ τMF = 2/λ for large M.

Two types of rare events in limit M � 1:Fast annihilation: P(N(τ) = 1) with τ � τMF.Slow annihilation: P(N(τ) > 1) with τ � τMF.



Path integral formula for probabilities

Path integral formula for probabilities is similar to before butincludes boundary terms:

P(N(T ) = 1) =

∫D[zm, zm] e−Seff[zm,zm] (2)

where

Seff =

∫ T

0dt

[(∑m

zm ∂tzm

)+H[zm, zm]

+∑

m

((zmzm −M ln zm) δm,1δ(t)− ln zm δm,M δ(t − T )

)]and

H[z, z] = −12

∑m1,m2

λm1,m2(zm1+m2 − zm1 zm2) zm1zm2 .



Instanton equations for early annihilation

If probability of the rare event of interest is concentrated on asingle trajectory (“instanton") then Seff can be estimated usingthe Laplace method. Trajectory satisfies:

zm = − δHδzm

=12

∑m1,m2

λm1,m2 (δm,m1+m2 − zm1δm,m2 − zm2δm,m1) zm1zm2

˙zm =δHδzm

= −12

∑m1, m2

λm1,m2(zm1+m2 − zm1 zm2)(zm1δm,m2 + zm2δm,m1)

with boundary conditions:

zm(0) zm(0) = M δm,1 zm(T ) zm(T ) = 1 δm,M .

Integrals of motion:E = H(zc , zc) (’Instanton energy’)M =

∑m mzc

mzcm (Mass)

Find that:Seff = −E · t + boundary terms



Solution of the instanton equations for constant kernel

Task: find the value of E for which the boundary conditions aresatisfied. For constant kernel, can solve for N =

∑m zm(t)zm(t):

∂tN = −λ2

N2 + E , N(0) = M N(T ) = 1.

Solution:Know E < 0 so let E = −λ

2 p2.

N(t) = p tan[pλ

2(t − t0)

]t0 =

2λp

tan−1 Mp.

For M →∞ and t → 0, we find p ∼ πλt .

Obtain

P(N(T ) = 1) ∼ e−π2

2λ T for T � τMF and M � 1.



Late annihilation

The opposite regime of late annihilation is not described byprevious instanton equations. For T � τMF thenP(N(T ) > 1) ≈ P(N(T ) = 2). Exact equation for ∂t〈N(t)〉:

∂t〈N〉 = −λ2〈N (N − 1)〉

For late times N(t) = 1 + n(t) with n(t) ∈ {0,1} and

∂t〈n〉 = −λ2

(〈n2〉+ 〈n〉)

= −λ 〈n〉 since 〈n2〉 = 〈n〉⇒ 〈n〉 ∼ e−λt

⇒ P(n(t) = 1) ∼ e−λt

Probability of late annihilation

P(N(T ) > 1) ∼ e−λt for T � τMF



Statistics of mass flux fluctuations

Analogue of the mass flux for toy model: J = Mτ (τ is the

annihilation time). JMF = MτMF

= Mλ2 . Consider relative size of

fluctuations of J above and below JMF.

P(J > J+) = P(τ <MJ+

) ∼ e−π2J+2λM as M →∞

P(J < J−) = P(τ >MJ−

) ∼ e−λM

J− as M →∞

Take J+ = LJMF, J− = JMFL . Fluctuations are not symmetric with

respect to L:P(J > JMFL)

P(J < JMF/L)∼ e−(π

24 −1)L

Large fluxes are exponentially less probable than small ones.Reminiscent of Gallavotti-Cohen Fluctuation Relation forentropy production in dynamical systems.



Conclusions

Cluster-cluster aggregation exhibits non-equilibriumstatistical dynamics which are closely analogous toturbulence with a mass cascade playing the role of theenergy cascade.CCA with diffusive transport has a critical dimension of 2.Unlike turbulence, the RG flow for this system has a stableperturbative fixed point in d < 2 of order ε = 2− d .Mass distribution exhibits spatial intermittency due toanti-correlation between particles. Multi-scaling exponentscan be calculated by RG.Probabilities of rare configurations can be calculated using“instanton" method (at least in 0-d). Instanton equationsare very closely related to mean-field equations.Fluctuations of the mass flux exhibit a (geometric)asymmetry which is very reminiscent of various “fluctuationrelations" in other non-equilibrium systems.


Fluctuations and rare events in stochastic aggregation

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