Page 371 - Numerical Methods for Chemical Engineering

360 7 Probability theory and stochastic simulation

with respect to the grid value. For a lattice of cells with a volume v c around each point,
if we use quadrature to approximate the functional H[ϕ(x)] as a function H(ϕ) of the grid
values,
∂ H

δH −1
≈ v (7.254)
c
δϕ ∂ϕ m
x m
so that the analogous form to (7.252) is
∂ H

−1
dϕ m =− v c dt + η m dt (7.255)
∂ϕ m
Thus, we associate ζ −1 ⇔ v −1 and by analogy to (7.253) define the statistical properties
c
of η m to be
−1
η m (t) = 0 η m (t)η n (t ) = 2k b T v δ m,n δ(t − t ) (7.256)

c
−1
As v c → 0,v δ m, n → δ(x m − x n ), so that the statistical properties of the random noise
c
field η(t, x) in (7.249) are

η(t, x) = 0 η(t, x)η(t , x ) = 2k b T δ(x − x )δ(t − t ) (7.257)

The SDE for the field value at each grid point is then

δH −1 1/2 (m)

dϕ m =− dt + 2k b T v c dW t (7.258)
δϕ
x m
(m)
where the dW t are independent Wiener processes. TDGL A 2D.m uses this method to
sample the order-parameter field fluctuations at a specified temperature. For more on field
theory applications in statistical physics, consult Chaikin & Lubensky (2000).
Monte Carlo integration
In Chapter 4, we considered a simple method to estimate by Monte Carlo simulation the
value of a definite integral
'
= f (x)dx (7.259)

We consider here an alternative method employing importance sampling that may be used
when we can compute the volume V of easily. We start by writing (7.259) as the integral
over all space

'
1, if x ∈
= H (x) f (x)dx H (x) = (7.260)
R N 0, if x /∈
Then, we write the integral in terms of the average of f (x)over , which we determine by
Monte Carlo simulation,
N S
1 [ j]
= V f f = f x (7.261)
N S
j=1
[j]
The {x }, uniformly distributed in , are obtained from Metropolis Monte Carlo sampling