Chemistry Reference
In-Depth Information
of the random walker at any other instant (the random walker has no memory);
that is, the motion at time
t
τ
is
statistically independent
of the motion at time
t
.
The integral Eq.(4) can, however, equally well be written by Taylor's theorem as:
±
∞
−∞
∂
n
f
∂x
n
∞
∞
τ
n
n
!
∂
n
f
∂t
n
n
φ
(
, τ
)
d
n
!
=
n
=
0
n
=
0
∞
2
n
∂
2
n
f
∂x
2
n
(2
n
)!
=
(5)
n
=
0
by the definition of an average. Moreover, on account of Eq.(3)
2
n
+
1
=
0
(6)
Equation (5), which is the simplest form of the Kramers-Moyal expansion [15],
is entirely equivalent to the Smoluchowski integral equation. Now let us suppose
that
and
τ
approach zero (extremely small displacements in infinitesimally short
times) in such a way that:
2
lim
=
D
(7)
2
τ
→
0
τ
→
0
and let us further suppose that all terms (
τ
2
) and (
τ
4
) and higher vanish in Eq. (5).
Then this equation formally becomes the diffusion equation [13, 17, 18]:
D
∂
2
f
∂x
2
∂f
∂x
=
(8)
where
fdx
=
f
(
x, t
|
x
0
,t
0
)
dx
(9)
is the conditional probability that the random walker is in
x
dx
at time
t
given that it was at
x
0
at time
t
0
. The neglect of the higher-order terms in
→
x
+
2
n
as
τ
0 may be justified as follows. We recall that the probability distribution
φ
(
, τ
) of the elementary displacements in time
τ
arises from the continual buf-
feting of the random walker by the very large number of impacts by the molecules
of the surrounding medium. Thus the resulting displacement
of the walker is the
sum
of the elementary displacements arising from the molecular collisions (sup-
posed statistically independent), which take place in time
τ
so that the central limit
theorem of probability theory applies. The central limit theorem may be stated
as follows [15], let
→
{
ξ
i
}
be a sequence of
n
independent random variables, each
