Page 353 - Numerical Methods for Chemical Engineering

342 7 Probability theory and stochastic simulation

Let us take a random walk in one dimension, stepping a distance l every δt time interval.
After n such steps, the elapsed time is t = n(δt). The mean-squared displacement during
the random walk is
t

2
2
2
[ x(t)] = l n = l (7.145)
δt
Now, to be a model of diffusive motion, we must have the mean-squared displacement
growing linearly with elapsed time; therefore, we set
√
2
l = δt l = δt (7.146)
so that
2
[ x(t)] = t (7.147)
In the limit δt → 0, this random walk becomes a Wiener process. A Wiener process has
√
the same long-time behavior as a random walk with steps of δt each δt time period, but
the steps are taken infinitely close together. This is unphysical; however, when modeling
Brownian diffusion, we are really only interested in behavior on time scales longer than the
velocity autocorrelation time.
√
Note that for δt «1, δt » δt. The “derivative” of this process for small, but finite δt,
is approximately
√
x l δt 1
∼ = = √ (7.148)
t δt δt δt
Thus, as δt → 0, the “derivative” of x(t) diverges. In fact, it diverges so badly that the
integral
' t+δt
dx
x(t + δt) − x(t) = dt (7.149)
t dt t
is not defined according to the rules of deterministic calculus.

Stochastic Differential Equations (SDEs)
The lack of a proper definition for (7.149) means that we cannot apply the traditional rules
of calculus to Brownian motion; rather, we must use the special rules of stochastic calculus.
Thus, integrals of the form of (7.137) and ODEs of the form of (7.132) are not to be defined
using deterministic calculus as we have done above. Let us now write (7.132) in a form that
is well defined by multiplying it by dt,

1 dU 1
dx =− dt + F R (t)dt (7.150)
ζ dx ζ
For a small time interval, (7.144) yields
1 1/2
F R dt = dX R = (2D) dW t (7.151)
ζ