Fluctuations and rare events in stochastic aggregation

Fluctuations and rare events in stochastic
aggregation

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 in
ﬂuids
University College Dublin July 2-6, 2012

./ﬁgures/warwickL

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

Introduction to cluster-cluster aggregation (CCA)

Many particles of one
material dispersed in
another.
Transport: diffusive,
advective, ballistic...
Particles stick together on
contact.
Applications: surface and colloid physics, atmospheric
science, biology, cloud physics, astrophysics...

./ﬁgures/warwickL


Mean-field model: Smoluchowski’s kinetic equation

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

m
∂Nm (t) 1
= dm1 dm2 K (m1 , m2 )Nm1 Nm2 δ(m − m1 − m2 )
∂t 2 0
∞
− dm1 dm2 K (m, m1 )Nm Nm1 δ(m2 − m − m1 )
0
J
+ δ(m − m0 )
m0

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


Mass cascades in CCA with a source and sink

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.
K (m1 , m2 ) = 1.

Stationary state for t → ∞, m m0 (White 1967):
√ β+3
Nm = c J m− 2 .

Describes a cascade of mass from source at m0 to sink at large
m. Analogous to the Richardson cascade in turbulence. ./ﬁgures/warwickL


Stochastic Particle System Model of Aggregation
Consider a lattice in d dimensions with particles of integer
masses. Nt (x, m)=number of particles of mass m on site x at
time t.

Diffusion: Nt (x, m) → Nt (x, m) − 1
Nt (x + n, m) → Nt (x + n, m) + 1
Rate: DNt (x, m)/2d
Aggregation: Nt (x, m1 ) → Nt (x, m1 ) − 1
Nt (x, m2 ) → Nt (x, m2 ) − 1
Nt (x, m1 + m2 ) → Nt (x, m1 + m2 ) + 1
Rate: λNt (x, m1 )Nt (x, m2 )
Injection: Nt (x, m) → Nt (x, m) + 1
Rate: J
./ﬁgures/warwickL


Path integral representation of correlation functions

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

¯
nm = D[zm , zm ] zm e−Seff [zm ,zm ]
¯ (1)

where
T
Seff = dτ dx ¯ ¯
zm (∂t zm − D∆zm − Jδm,1 ) + H[zm + 1, zm ]
0 m

and
1
¯
H[z , z] = − ¯ ¯ ¯
λm1 ,m2 (zm1 +m2 − zm1 zm2 ) zm1 zm2 .
2 m1 ,m2

./ﬁgures/warwickL


Diffusive fluctuations in d ≤ 2

Critical dimension for the theory is 2. Below 2-d diffusive
fluctuations cause mean field theory to fail.
Exact solution (Takayasu 1985)for size distribution in 1-d
(constant kernel):
1 4 3
Nm = c J 3 m − 3 cf MF: Nm ∼ m− 2 .

Renormalisation of reaction rate within perturbative RG
framework ( = 2 − d):

d 2d+2
Nm ∼ J d+2 m− d+2
Leading term in RG expansion is exact in d = 1 ( = 1)! ./figures/warwickL


Multiscaling and intermittency

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

Cn = Nm1 . . . Nmn ∼ m−γn 3
MF: γn = n 2

γ0 = 0.
4
γ1 = 3 in 1-d (exact solution).
γ2 = 3 is exact (analogue of
4
5 -law!).
These are not on a straight line.
RG calculation gives:
γn = 2d+2 n + d+2 n(n−1) + O( 2 )
d+2 2
where = 2 − d.
./ﬁgures/warwickL


Origin of intermittency: anti-correlation between
clusters

“Large particles are large because they have merged with all of
their neighbours".
Due to recurrence of random
walks in d < 2 particles
encounter their neighbours
inﬁnitely often.
Large particles surrounded by
voids (correlation hole) and are
thus spatially correlated.
Spatial intermittency of the mass distribution in low dimensions
is due to sites having atypically few or atypically many particles.
./ﬁgures/warwickL


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:
˙ λ
N = − N2 N(0) = M.
2
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 . ./figures/warwickL


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

¯
P(N(T ) = 1) = D[zm , zm ] e−Seff [zm ,zm ]
¯ (2)

where
T
Seff = dt ¯
zm ∂t zm ¯
+ H[zm , zm ]
0 m

+ ¯ ¯
(zm zm − M ln zm ) δm,1 δ(t) − ln zm δm,M δ(t − T )
m

and
1
¯
H[z , z] = − ¯ ¯ ¯
λm1 ,m2 (zm1 +m2 − zm1 zm2 ) zm1 zm2 .
2 m1 ,m2 ./ﬁgures/warwickL


Instanton equations for early annihilation
If probability of the rare event of interest is concentrated on a
single trajectory (“instanton") then Seff can be estimated using
the Laplace method. Trajectory satisﬁes:
δH 1
˙
zm = − = ¯ ¯
λm1 ,m2 (δm,m1 +m2 − zm1 δm,m2 − zm2 δm,m1 ) zm1 zm2
¯m
δz 2m
1 ,m2

˙ δH 1
¯
zm = =− ¯ ¯ ¯
λm1 ,m2 (zm1 +m2 − zm1 zm2 )(zm1 δm,m2 + zm2 δm,m1 )
δzm 2m m2
1,

with boundary conditions:
¯
zm (0) zm (0) = M δm,1 ¯
zm (T ) zm (T ) = 1 δm,M .
Integrals of motion:
E = H(z c , z c ) (’Instanton energy’)
¯
c ¯c
M = m mzm zm (Mass)
Find that:
./ﬁgures/warwickL
Seff = −E · t + boundary terms

Solution of the instanton equations for constant kernel

Task: find the value of E for which the boundary conditions are
¯
satisfied. For constant kernel, can solve for N = m zm (t)zm (t):

λ
∂t N = − N 2 + E, N(0) = M N(T ) = 1.
2
Solution:
Know E < 0 so let E = − λ p2 .
2

λ 2 M
N(t) = p tan p (t − t0 ) t0 = tan−1 .
2 λp p
π
For M → ∞ and t → 0, we find p ∼ λt .
Obtain
π2
P(N(T ) = 1) ∼ e− 2 λ T for T τMF and M 1.
./figures/warwickL


Late annihilation
The opposite regime of late annihilation is not described by
previous instanton equations. For T τMF then
P(N(T ) > 1) ≈ P(N(T ) = 2). Exact equation for ∂t N(t) :
λ
∂t N = − N (N − 1)
2
For late times N(t) = 1 + n(t) with n(t) ∈ {0, 1} and
λ
∂t n = − ( n 2 + n )
2
= −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 ./ﬁgures/warwickL


Statistics of mass flux fluctuations
Analogue of the mass flux for toy model: J = M (τ is the
τ
annihilation time). JMF = τM = Mλ . Consider relative size of
MF 2
fluctuations of J above and below JMF .

M π 2 J+
P(J > J+ ) = P(τ < ) ∼ e− 2λM as M → ∞
J+
M − λM
P(J < J− ) = P(τ > ) ∼ e J− as M → ∞
J−

Take J+ = LJMF , J− = JMF . Fluctuations are not symmetric with
L
respect to L:
P(J > JMF L) π2
∼ e−( 4 −1)L
P(J < JMF /L)
Large fluxes are exponentially less probable than small ones.
Reminiscent of Gallavotti-Cohen Fluctuation Relation for
entropy production in dynamical systems. ./figures/warwickL


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

Fluctuations and rare events in stochastic aggregation

Empfohlen

Empfohlen

Weitere ähnliche Inhalte

Was ist angesagt?

Was ist angesagt? (20)

Ähnlich wie Fluctuations and rare events in stochastic aggregation

Ähnlich wie Fluctuations and rare events in stochastic aggregation (20)

Kürzlich hochgeladen

Kürzlich hochgeladen (20)

Fluctuations and rare events in stochastic aggregation