Page 370 - Numerical Methods for Chemical Engineering

Markov chains and processes; Monte Carlo methods 359

fluctuations in the order parameter, a mean-field approximation, we identify the thermody-
namic prediction of the uniform order parameter ϕ as that minimizing the mean-field free
energy
1 2 4
f MF ( ϕ ) =−w ϕ + r ϕ + u ϕ (7.247)
2
3
Taking ∂ f MF /∂ ϕ = 0, we have −w + 2r ϕ MF + 4u ϕ = 0. In the absence of an
MF
external field, w = 0, this becomes

0, r > 0, T > T ∗
2 6

2 ϕ MF r + 2u ϕ = 0 ⇒ ϕ MF = −r (7.248)
MF
± ,r < 0, T < T ∗
2u
*
Thus T is the predicted transition temperature from mean-field theory. This mean-field
picture of the phase transition is only valid when c is so large that spatial modulations in
the local order parameter field are suppressed; however, we see from (7.246) that this is
unlikely, especially when the dimension d is small.
One way to sample the fluctuations in the order parameter, and thus model their effect
upon the phase transition, is to propose a stochastic model for the order parameter field
such as the time dependent Ginzburg–Landau model A (TDGL-A) dynamics:
∂ϕ δH
=− + η(t, x) (7.249)
∂t δϕ
where > 0. The functional derivative of H [ϕ(x)] is

H[ϕ(x) + εδ(x − x )] − H[ϕ(x)]

δH 3 2
= lim = [−w + rϕ + 4uϕ − c∇ ϕ]| x
δϕ ε→0 ε
x
(7.250)
and η(t, x) is a random noise field. Here, we have taken the “naive” approach of dividing
the field differential by dt, as this is common in the physics community, even though the
time derivative is not well defined. Equation (7.249) is a stochastic PDE, and to discretize
it, we place a uniform grid of points x m , where m is a unique label. If ϕ m is the field value
at m, we discretize (7.249) as the set of ordinary SDEs

δH
dϕ m =− dt + η m dt
δϕ
x m
(7.251)
δH 3
2
=−w(x m ) + rϕ m + 4uϕ − c∇ ϕ
m
δϕ x m
x m
2
We use finite differences to relate ∇ ϕ| xm to an algebraic difference of field values at
neighboring grid points. Let us compare (7.251) to the set of SDEs for Brownian motion in
multiple dimensions,
1 ∂U 1
dx m =− dt + F R,m (t)dt (7.252)
ζ ∂x m ζ
where the random force vector has the statistical properties

F R,m (t) = 0 F R,m (t)F R,n (t ) = 2k b T ζδ(t − t )δ mn (7.253)
We see a strong resemblance between (7.251) and (7.252), except that (7.251) has the
functional derivative evaluated at x m rather than the “traditional” derivative of the energy