Environmental Engineering Reference
In-Depth Information
The pair of equations (3.3.14) will be the starting point for our applica-
tion of linear response theory.
According to the definition (3.2.10) of
K
BA
(
t
), and eqn (3.2.12),
K
BA
(
ω
)=2
iχ
BA
(
ω
)=
2
iG
BA
(
ω
)
.
−
We may write
∞
i
π
i
h
[
B
(
t
)
, A
]
χ
BA
(
ω
)
e
−iωt
dω
=
(3
.
3
.
15)
−∞
and, setting
t
= 0, we obtain the following
sum rule
:
∞
h
π
[
B, A
]
χ
BA
(
ω
)
dω
=
,
(3
.
3
.
16)
−∞
which may be compared with the value obtained for the equal-time corre-
lation function
B A
B
A
, (3.3.7). The Green function in (3
.
3
.
14
a
)
must satisfy this sum rule, and we note that the thermal averages in
(3
.
3
.
14
a
) and (3.3.16) are the same. Equation (3.3.16) is only the first
of a whole series of sum rules.
The
n
th time-derivative of
B
(
t
) may be written
−
dt
n
B
(
t
)=
i
n
d
n
n
B
(
t
)
B
(
t
)
, B
(
t
)]
.
L
L
≡
H
ith
[
h
Taking the
n
th derivative on both sides of eqn (3.3.15), we get
∞
iω
)
n
χ
BA
(
ω
)
e
−iωt
dω
=
i
n
+1
i
π
n
B
(
t
)
, A
]
(
−
[
L
.
h
−∞
Next we introduce the normalized
spectral weight function
∞
χ
BA
(
ω
)
ω
1
χ
BA
(0)
1
π
F
BA
(
ω
)
dω
=1
.
(3
.
3
.
17
a
)
The normalization of
F
BA
(
ω
) is a simple consequence of the Kramers-
Kronig relation (3
.
2
.
11
d
). The
n
th order moment of
ω
, with respect to
the spectral weight function
F
BA
(
ω
), is then defined as
F
BA
(
ω
)=
,
where
−∞
BA
=
∞
−∞
ω
n
ω
n
F
BA
(
ω
)
dω,
(3
.
3
.
17
b
)
which allows the relation between the
n
th derivatives at
t
=0tobe
written
χ
BA
(0)
(
hω
)
n
+1
1)
n
n
B, A
]
BA
=(
−
[
L
.
(3
.
3
.
18
a
)
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