Chemistry Reference
In-Depth Information
particular location, x, on the x-axis depends on the number of ways that one
can arrange the forward and backward steps to have the same net difference n.
Since the order of the steps is not important, the number of arrangements is
given by:
!
!
N
N
n
forward
Þ!
5
(6-16)
n
forward
!ð
N
2
n
forward
!
n
backward
!
Since there are N steps and each step has two possible outcomes, the total
number of all possible paths is 2
N
. The probability of the Brownian particle land-
ing at a specific location is
N
!
1
2
N
!
!
P
ð
n
Þ 5
(6-17)
1
1
2
ð
N
1
n
Þ
2
ð
N
2
n
Þ
Equations (6-14) and (6-15)
have been used to replace the number of forward
and backward steps appearing in
Eq. (6-16)
. When N approaches infinity, the
Stirling approximation applies and this means
nN
1
ln
ð
N
!Þ 5
N ln
ð
N
Þ 2
2
ln
ð
2
π
N
Þ
(6-18)
Applying
Eq. (6-18)
to P(n), one obtains, after some algebraic manipula-
tions (see Problem 6-1), the following normalized probability distribution func-
tion over the entire possible range of values of n, which is from minus infinity
to infinity:
n
2
2N
P
ð
n
Þ 5
e
2
p
2
(6-19)
π
N
z
x
y
FIGURE 6.3
The trajectory of a three-dimensional random walker after 1000 steps.
