Page 352 - Numerical Methods for Chemical Engineering

Brownian dynamics and SDEs 341

Assuming the gradient of the potential to be constant over the interval t,
1 dU
x(t + t) − x(t) =− ( t) + X R (t, t + t) (7.136)
ζ dx
where we have defined the random displacement due to the random force,
1 ' t+ t

X R (t, t + t) = F R (t )dt (7.137)
ζ t
The Wiener process

Let us consider the statistical properties of the random displacement (7.137). First, we see
that the average displacement is zero:
1 ' t+ t

X R (t, t + t) = F R (t ) dt = 0 (7.138)
ζ t
Next, we compute the autocorrelation function of X R :
7 t+ t
t + t 8

1 1 '
'

X R (t, t + t)X R (t , t + t) = F R (t 1 )dt 1 F R (t 2 )dt 2
ζ t ζ t
' t + t (

t+ t
1
'
= F R (t 1 )F R (t 2 ) dt 2 dt 1
ζ 2 t t
(7.139)
2
Substituting F R (t 1 )F R (t 2 ) = 2Dζ δ(t 1 − t 2 ) yields
(

' t+ t ' t + t

X R (t, t + t)X R (t , t + t) = 2D δ(t 1 − t 2 )dt 2 dt 1 (7.140)
t t

If t = t,as t → 0, the right-hand side of (7.140) goes to zero as there is no t 2 ∈ [t , t +

t] that equals any t 1 ∈ [t, t + t]. But, if t = t, then
' t+ t ' t+ t 1
X R (t, t + t)X R (t, t + t) = 2D δ(t 1 − t 2 )dt 2 dt 1
t t
' t+ t
= 2D {1} dt 1 = 2D( t) (7.141)
t
Therefore, the correlation function of the displacement due to the random force during a
time interval t is
X R (t, t + t)X R (t , t + t) = 2D( t)δ(t − t ) (7.142)

We see that the statistical properties of this random displacement depend upon the value of
the diffusivity and upon the time step t. We separate these two dependences by defining
the random variable W t with the statistical properties,

W t = 0 W t W t = ( t)δ(t − t ) (7.143)

such that
X R (t, t + t) = (2D) 1/2 W t (7.144)
W t is said to be the finite increment of a Wiener process.