Page 361 - Numerical Methods for Chemical Engineering

350 7 Probability theory and stochastic simulation

†
If we define the adjoint operator L such that
t
+∞ +∞
' '
†
[L t F(x)]p(t, x|t , x )dx = F(x)[L p(t, x|t , x )]dx (7.194)
t
−∞ −∞
then (7.193) becomes
+∞ +∞ ∂
' '
†
F(x)[L p(t, x|t , x )]dx = F(x) p(t, x|t , x ) dx (7.195)
t
−∞ −∞ ∂t
and hence the transition probability distribution is governed by the forward Kolmogorov
equation
∂
†

p(t, x|t , x ) = L p(t, x|t , x ) (7.196)
t
∂t
For the operator (7.186), one can show by integration by parts that
∂ 1 ∂ 2
† 2
L =− a(t, x) + [b(t, x)] (7.197)
t
∂x 2 ∂x 2
and thus
2
∂ ∂ 1 ∂ 2

p(t, x|t , x ) = − a(t, x) + [b(t, x)] p(t, x|t , x ) (7.198)
∂t ∂x 2 ∂x 2
Using the relation
'
+∞

p(t, x) = p(t, x|t , x )p(t , x )dx (7.199)
−∞
wemultiply(7.198)by p(t , x )andintegrateoverall x toobtaintheFokker–Planckequation

for p(t, x),
∂p ∂ 1 ∂ 2 2
=− [a(t, x)p(t, x)] + {[b(t, x)] p(t, x)} (7.200)
∂t ∂x 2 ∂x 2
For 1-D Brownian motion with the SDE (7.152),
1 dU 2
a(t, x) = J c (x) =− [b(t, x)] = 2D (7.201)
ζ dx
the Fokker–Planck equation is
∂p ∂ ∂ 2
=− [J c (x)p(t, x)] + {Dp(t, x)} (7.202)
∂t ∂x ∂x 2
This looks somewhat like a convection/diffusion equation, where the first term is the con-
vective flux due to the presence of the external potential and the second term is the diffusion
caused by the random Brownian motion. If we have a system of N noninteracting particles,
each governed by an independent SDE (7.152), the concentration field of the particles is
c(t, x) = Np(t, x) and is governed by
∂c ∂ ∂ 2
=− [J c (x)c(t, x)] + {Dc(t, x)} (7.203)
∂t ∂x ∂x 2
Thus, we see that there is a strong relationship between macroscopic diffusion and micro-
scopic Brownian motion.