Environmental Engineering Reference
In-Depth Information
We introduce the Fourier transforms
Z
Z
1
1
G
(
p
)
D
f
(
z
)exp(
ipz
)
dz
,
g
(
p
)
D
'
(
z
)exp(
ipz
)
dz
,
(1.75)
1
1
which can be inverted to give
Z
1
Z
1
1
2
1
2
f
(
z
)
D
G
(
p
)exp(
ipz
)
dp
,
'
(
z
)
D
g
(
p
)exp(
ipz
)
dp
.
π
π
1
1
Equation (1.75) yields
Z
Z
1
1
1,
g
0
(0)
g
00
(0)
z
k
,
g
(0)
D
'
(
z
)
dz
D
D
i
z
'
(
z
)
dz
D
i z
k
,
D
1
1
(1.76)
where
z
k
and
z
k
are the mean shift and the mean square shift of the variable after
one step. From (1.74) and (1.76) it follows that
exp
z
k
!
n
Z
1
X
n
Y
g
n
(
p
),
G
(
p
)
D
ip
1
'
(
z
k
)
dz
k
D
k
D
1
k
D
1
and hence
Z
Z
1
1
1
2
1
2
g
n
(
p
)exp(
ipz
)
dp
f
(
z
)
D
D
exp(
n
ln
g
C
ipz
)
dp
.
π
π
1
1
Since
n
1, the integral converges at small
p
.Expandingln
g
in a power series
of
p
,wehave
ln
1
z
k
p
2
/2
(
z
k
)
2
p
2
/2 ,
z
k
p
2
/2
ln
g
D
i z
k
p
D
i z
k
p
C
which gives
Z
1
exp
h
ip
(
nz
k
nz
k
p
2
/2
i
dp
1
2
f
(
z
)
D
z
)
π
1
2
exp
.
z
)
2
1
(
z
D
p
2
(1.77)
2
Δ
2
π
Δ
In this exp
res
sion,
z
D
nz
k
is the mean shift of the variable for
n
steps,
z
2
D
nz
k
,
(
z
)
2
is the root-mean-square deviation of this value. Formula (1.77)
is called the normal distribution, or the Gaussian distribution. It is valid if the
principal contributio
n t
o the integral in (1.76) comes from small values of
p
,that
is
, i
f
z
k
p
2
D
z
2
and
Δ
1and
z
k
p
2
1. Because this integral is determined by a range
nz
k
p
2
1, the Gaussian distribution is valid for a large number of steps
n
1.
