Chemistry Reference
In-Depth Information
Boltzmann) a detailed kinetic model, namely, collisions of hard spheres, as de-
scribed by Mazo [21]. Unlike Einstein, whose theory is essentially statistical and
follows from the central limit theorem, Smoluchowski used the dynamics of the
particle motion in a specific dynamic way. The link between the two methods was
provided by Langevin in 1908. His idea was [15, 21] that a suspended particle in
a fluid is acted upon by both systematic and random forces due to the molecules
of the solvent. This force can be expressed as a sum of its average value and a
fluctuation about this average value that causes the unceasing haphazard charac-
ter of the Brownian motion. Thus Langevin's starting point, which he deemed
“infinitely more simple than that of Einstein”, is the Newtonian equation for the
random variables (
x, v
). These variables now become the stochastic differential
equation:
m
dv
dx
dt
=
dt
+
ζv
=
λ
(
t
)
v
(18)
in which the fluctuating force
λ
(
t
) (Gaussian white noise) has the following prop-
erties:
λ
(
t
)
=
0
(19)
λ
(
t
1
)
λ
(
t
2
)
=
2
kTζδ
(
t
1
−
t
2
)
(20)
Here
δ
denots the Dirac-delta function, where the angular braces denote the
statistical average of
λ
over its realizations. Moreover, Isserlis's theorem [15]
concerning averages of products of Gaussian random variables must be satisfied.
Equation(18), which is the equation of motion of the random variables
x
and
v
, when averaged over the realizations of the phase path (
x
,
v
), then yields (for
convenience setting
t
0
=
0)
[
x
(
t
)
x
(0)]
2
ζ
e
−
ζ
|
t
|
/m
kTm
ζ
2
m
−
|
t
−
=
1
+
(21)
(This result was actually obtained for the first time by Ornstein in 1918 [21,
22]) and Eq.(21) is the mean-square differentiable, so that the root mean square
(rms). velocity of the Brownian particle now exists. It is equivalent to Einstein's
result only at times well in excess of
m/ζ
, which is the frictional time. The reason
for this is [21] which states that Einstein worked only in configuration space of the
Brownian particle. He did not actually introduce the velocity of the particle, except
in so far as to demonstrate that for the pollen particles he envisaged, the time scales
were so long that the last two terms in Eq.(21) were negligible. This assumption is
equivalent to stating that the particles thermalize exceedingly rapidly. By working
in the complete phase space [21] and introducing the concept of random variables
Langevin was also able to find the velocity relaxation.
