Chemistry Reference
In-Depth Information
path of a molecule) the Brownian motion trajectories appear on average to be the
same. That is, the trajectory of a given Brownian particle is a random fractal [20]
(dilation invariant object). The scaling of the steps or segments of the trajectory
with magnification defines the fractal dimension, which in this case according to
the right-hand side of Eq.(14) is 2.
To continue, Einstein, essentially by considering the Brownian motion of a
particle in a potential well
V
(
x
) [cf. Eq.(17)] and requiring that ultimately the
Boltzmann distribution of positions must prevail in the well so that
f
=
0 in Eq.(17)
(implying in the present context that the diffusion current
j
is zero) determined
the diffusion coefficient
D
. Hence, he obtained the famous formula for the mean-
squared displacement:
[
x
(
t
)
x
(
t
0
)]
2
=
2
kT
ζ
−
|
t
−
t
0
|
(16)
In writing Eq. (16), it is assumed that the viscous drag on the particle is given
by Stokes' law for the viscous drag coefficient of a sphere of radius
a
moving
in a fluid of viscosity
η
, namely,
ζ
6
πηa
. Equation (16), which connects the
mean-square fluctuations in the displacement of the Brownian particle (and so
the mean-thermal energy
kT
with the dissipative coupling
ζ
) to the heat bath
is essentially the fluctuation-dissipation theorem [15]. In 1908, Perrin [14, 15,
21] successfully calculated the Avogadro number from observations of the mean-
square displacement of a Brownian particle thus confirming Eq. (16). Equation (8)
in the presence of a potential
V
(
x
), which reads in continuity equation form:
=
D
∂f
∂f
∂t
∂
∂x
f
ζ
∂V
∂x
∂j
∂x
=
∂x
+
=−
(17)
is called the Smoluchowski differential equation, usually abbreviated to just the
Smoluchowski equation. The diffusion current vanishes in thermal equilibrium,
that is, when the Boltzmann distribution has been established in the potential well.
The equilibrium situation discussed by Einstein, where the drift current due to the
force of potential
V
is exactly balanced by the diffusion current due to the Brownian
motion, thus maintains the Boltzmann distribution in the well which is in contrast
to the nonequilibrium situation. This situation is encountered in Kramers theory
of the escape of particles over potential barriers due to the shuttling action of
Brownian motion, which is used to calculate reaction rates [15]. Here
j
, which
is the overbarrier current, equals a constant reflecting the fact that the escaping
particles disturb the Boltzmann distribution in the well.
We remark that Einstein's approach to the Brownian motion is based on sta-
tistical assumptions of a general nature, not fixed to a specific model [21]. How-
ever, Smoluchowski's investigations, which yielded essentially the same results,
were published some months later than Einstein's. He considered (in the spirit of
