Page 351 - Numerical Methods for Chemical Engineering

340 7 Probability theory and stochastic simulation

Using the fact that the statistical properties of the velocity are independent of absolute
time,

V x (t 1 )V x (t 2 ) = V x (t 1 − t 2 )V x (0) (7.128)
→ 0, we have
and taking the limit τ V x
V x (t 1 )V x (t 2 ) = 2Dδ(t 1 − t 2 ) (7.129)

Therefore,
t
t
' ' t ' ' t
2
[ x(t)] = [2Dδ(t 1 − t 2 )]dt 1 dt 2 = 2D δ(t 1 − t 2 )dt 1 dt 2
0 0 0 0
' t
= 2D (1)dt 2 = 2Dt (7.130)
0
The mean-squared displacement varies linearly with time, and D is indeed the diffusivity,
as commonly defined.

The Langevin equation
We again consider the 1-D motion of a spherical particle, and now include a conservative
force arising from an external potential energy field U(x). The equation of motion is then
dU
dV x
m =−ζ V x − + F R (t) (7.131)
dt dx
= m/ζ, we achieve this limit by letting m → 0,
We want to take the limit τ V x → 0. As τ V x
but retain the constant value of the drag constant ζ (i.e., we neglect inertial effects). The
equation of motion then becomes
dx 1 dU 1
V x = =− + F R (t) (7.132)
dt ζ dx ζ
This is known as the Langevin equation. Assuming that the statistical properties of the
random force are independent of U(x), we analyze F R (t) in the absence of an external
potential, where
dx 1
V x (t) = = F R (t) (7.133)
dt ζ
The autocorrelation of this equation yields the statistical properties of F R (t),
2
2
F R (t)F R (0) = ζ V x (t)V x (0) = 2Dζ δ(t)
(7.134)
F R (t) = 0
We next reintroduce the potential U(x) and integrate the Langevin equation over an interval
from time t to time t + t,
1 ' t+ t dU 1 ' t+ t

x(t + t) − x(t) =− dt + F R (t )dt (7.135)
ζ t dx x(t ) ζ t