Download Rate equations for coagulation beyond the mean field approximation

Rate equations for coagulation (and other
processes) beyond the mean field
approximation
Ian Ford
Department of Physics and Astronomy and
London Centre for Nanotechnology
UCL
Featuring the work of....
• Christiane Losert-Valiente Kroon
• James Burnett
• Johnathan Lau
• Danial Khan
on complex population dynamics
(funded by EPSRC, Leverhulme Trust and UCL)
A simple problem in coagulation: 1+11
N (t )
N(t)
t
N particles at time t, coagulation rate  N ( N  1) / 2
so
d N
dt
  N ( N  1)
d N
dt
  N 2  N
A mean field approximation can avoid a hierarchy problem
Instead solve:
d N
dt
even though

  N
2

 N   N
2
for small mean populations
Traditional (Gillespie) Monte Carlo
• Start with N0 particles.
• Combine two into one after a time selected from
a suitable probability distribution
• Continue until a chosen total time has elapsed
• Repeat from the beginning and gather statistics
on
Example: Monte Carlo
simulation of 100000
particles coagulating
according to a product
coagulation kernel
Why can’t we just add some noise to the
traditional rate equation, and then take an
average?
dN
  N ( N  1)   (t )
dt
Yes, but it has to be complex noise!
And hence N has to be complex!!
A nano-LED
Box model
p
n
Hole
transport
annihilation
p
n
Electron
transport
Transfer according to diffusion and drift
in a self-consistent electric field
Traditional drift-diffusion-reaction equations:
Extension to traditional approach:
+ complex noise
+ complex noise
and then average over noise,
in order to treat a small system
Where do these complex population rate
equations come from?
1. Fock space representation of probability
distribution P(N,t),
–
–
evolving under raising and lowering operators;
formulation of mean population as a path integral:
Doi-Peliti-Cardy.
2. Poisson representation of probability distribution:
–
Gardiner-Drummond.
Route 1: Fock space representation
• Probability P(N,t) that there are N particles in the
system at time t. Represent as state:
• This evolves with time according to the stochastic
dynamics: express this sequence of transitions as
a path integral; manipulate; gives SDE.
Route 2: Poisson representation
• Express probability that there are N particles in the
system as time t
• Choose
such that P( N ,0)   N N 0
• a superposition of Poisson distributions, over a
contour of complex means, with evolving
weighting function for f (,t)
Master equations for P(N,t) lead to a Fokker-Planck
equation for f (,t), and then to a stochastic
differential equation for :
d N
Rather like
dt
  N
2
but with a complex noise
And then the mean population is
• Contrast with mean field solution
if
d N
dt
  N
2
‘All’ we have to do is
1. Solve SDE for the complex stochastic
population (t) with initial condition (t) = 0
2. Add up contributions from a complex contour
of initial conditions
3. Average over the noise! Yikes!
Re 0 
Im(0 )
Re( (t ))
Im( (t ))
t
Analytical treatment of 1+11 coagulation
• Solve the stochastic differential equation for (t):
• And average over the noise W(t):
c.f.
Average over contour of 0
• Saddle point at
0  N 0
So we get the traditional result
N (t )   (t )
0  N 0
for early time
N0

1  N 0t
but there are deviations when the mean population
approaches unity.
Trivial problem of
coagulation of two initial
particles!
Exact solutions available for arbitrary N0:
An initial 12 particles....
A numerical example: A+AC reaction on a
nanosurface
detachment
• Standard kinetics:
dN A
2
 j   N A  2 N A
dt
reaction
attachment
• complex population kinetics:
d A
2
1/ 2
 j   A  2  A  i (2 )  A (t )
dt
Low attachment rate j: one path
Re  A (t ) 
time t
With averaging over many paths, and for
initial condition N0=0
Re A (t )
time t
Standard kinetics
N A (t )
(small j)
0.7
0.6
0.5
0.4
0.3
0.2
0.1
-0.00001
0.00001
0.00002
0.00003
0.00004
0.00005
time t
Small mean population: mean field approximation
predicts too small a mean population
Average population disturbed by strange
instabilities
We have let in a monster!
• When the complex population is negative, the drift
term points towards minus infinity.....
Luckily, we can tame the monster...
• Invent a new SDE for ’(t) without pathological
drift term, e.g.
• Evolve a stochastic variable (t) such that
• Cameron-Martin-Girsanov Theorem tells us that
i.e.  is the RadonNikodym derivative
Nano-LED model: zero dimensions
1-d nano-LED model
Coupled to a similar chain of boxes for electrons
When mean populations in a box are of order
unity:
• Mean populations raised with respect to mean
field approximation
• Mean annihilation rate reduced
• An array of nano-LEDs is less efficient than a
single large LED of same cross section.
Extended coagulation kinetics
dg
g 1

1
   j , g  j jg  j    j , g jg  iC g g (t )
dt
j 1 2
j 1
Noise term
Nucleation kinetics
• Extended Becker-Döring equations
.
.
dg
dt
  g 11g 1   gg   g1g   g 1g 1  iC g g (t )
.
.
.
.
Noise terms
Conclusions
• Small system stochastic problems tackled by a
minimal extension to the traditional, mean field
rate equations.
– Hence a common approach for large and small
mean populations
• Makes the population complex, and requires
careful averaging
• Analytic results emerge
• Numerical results possible, as long as
instabilities are controlled
• Is this better than Gillespie Monte Carlo?
Or is it completely perverse?