Page 350 - Numerical Methods for Chemical Engineering

Brownian dynamics and SDEs 339

force and allow the drag force to dissipate the kinetic energy. The particle motion then
follows

dV x ζt
m =−ζ V x ⇒ V x (t) = V x (0) exp − (7.120)
dt m
Therefore, the velocity correlation time is
m 4 3 1 2ρR 2
= = πρR = (7.121)
τ V x
ζ 3 6πµR 9µ
3
−3
3
Fortheexamplecaseofaneutrallybuoyantparticleinwater, ρ = 10 kg/m ,µ = 10 Pa s.
For a very small particle with a diameter of 10 nm, on the size of macromolecules, the
velocity autocorrelation time is
3
3
(10 kg/m )(10 −8 m) 2 3−16+3 −10
∼ = 10 s = 10 s (7.122)
τ V x −3
(10 Pa s)
Even for a much larger particle of 100 µm = 10 −4 m, the correlation time is still quite
short,
3
3
(10 kg/m )(10 −4 m) 2 3−8+3 −2
∼ = 10 s = 10 s (7.123)
τ V x
(10 −3 Pa s)
, the results will
Now, if we take velocity measurements at times much further apart than τ V x
be independent and uncorrelated from each other. From the above analysis, we see that the
correlation times of a particle in water are quite short. Thus, in many applications where
,we
we are only concerned with the dynamics of the particle on time scales larger than τ V x
can neglect the effect of velocity correlation and derive our governing equation in the limit
→ 0. Even in this limit, however, we must have a nonzero value of V x (0)V x (0) =
τ V x
2
[V x (0)] ; therefore, we write our approximate velocity autocorrelation as
V x (t)V x (0) = 2Dδ(t) (7.124)
where δ(t)isthe Dirac delta function:
1 x 2 ' +∞
δ(t) = lim √ exp − f (t)δ(t)dt = f (0) (7.125)
σ→0 σ 2π 2σ 2
−∞
Strictly speaking, the Dirac delta function is not a function at all, but is rather defined solely
through the integral relation in (7.125). See the discussion of the theory of distributions in
Stakgold (1979).
To see that D is consistent with the common definition of the diffusivity, we compute the
average displacement over the time period [0, t],
t
'
x(t) = V x (t 1 )dt 1 (7.126)
0
with
t
= ' t ' t > ' ' t
2
[ x(t)] = V x (t 1 )dt 1 V x (t 2 )dt 2 = V x (t 1 )V x (t 2 ) dt 1 dt 2 (7.127)
0 0 0 0