Page 362 - Numerical Methods for Chemical Engineering

Brownian dynamics and SDEs 351

Some care must be taken if the diffusivity is itself a function of position, as the correct
microscopic balance for the concentration field is
∂c ∂ ∂ ∂c
=− [J c c] + D(x) (7.204)
∂t ∂x ∂x ∂x
Applying the chain rule,
∂ ∂ ∂ ∂c ∂ dD
[Dc] = D + c (7.205)
∂x ∂x ∂x ∂x ∂x dx
we rewrite (7.204) as the correct form of the Fokker–Planck equation for a position-
dependent diffusivity D(x):
∂c ∂ dD ∂ 2
=− J c + c + [D(x)c(t, x)] (7.206)
∂t ∂x dx ∂x 2
The corresponding Langevin equation is
dD 1/2
1
dx = J c (x) + dt + [2D(x)] dW t (7.207)
dx
The additional deterministic contribution from the position-dependent diffusivity is known
as spurious drift.

The Einstein relation

We are now ready to derive the proper relationship between the drag constant ζ and the
diffusivity D. If the diffusion is constant, the Fokker–Planck equation (7.202) takes the
form
1
∂p ∂ dU ∂p
= p(t, x) + D (7.208)
∂t ∂x ζ dx ∂x
As t →∞, the system approaches a stable steady state for D > 0 at which

d 1 dU dp
p(x) + D = 0 (7.209)
dx ζ dx dx
Integrating yields
1 dU dp
p(x) + D = constant = 0 (7.210)
ζ dx dx
In the latter equality we use the fact that since the probability field is conserved (there is no
source term), the net flux (convective and diffusive) is zero everywhere at the steady state.
We want this condition to be satisfied by the equilibrium Boltzmann distribution (7.180).
Taking the derivative of p eq (x) yields

dp eq d −1 U(x) 1 −1 1 dU U(x) 1 dU
= Z exp − = Z − exp − =− p eq
dx dx k b T k b T dx k b T k b T dx
(7.211)
Substituting this into the no-flux condition (7.210) we obtain
1 dU 1 dU 1 dU 1 D
0 = p eq + D − p eq = − p eq (7.212)
ζ dx k b T dx dx ζ k b T