Environmental Engineering Reference
In-Depth Information
is also called a response function.
A
is a shorthand notation for
A
(
t
=0).
The inverse response function
K
AB
(
t
), which determines
A
(
t
)
caused
f
(
t
)
B
,is
by the perturbation
H
1
=
−
K
AB
(
t
)=
i
[
A
(
t
)
, B
]
h
=
−
K
BA
(
−
t
)
,
and
K
BA
(
t
) can be expressed in terms of the corresponding causal re-
sponse functions as
K
BA
(
t
)=
φ
BA
(
t
)
t>
0
for
−
φ
AB
(
−
t
)
f r
t<
0
.
The susceptibility is divided into two terms, the reactive part
2
χ
BA
(
z
)+
χ
AB
(
z
∗
)
,
1
χ
BA
(
z
)=
χ
AB
(
z
∗
)
−
≡
−
(3
.
2
.
11
a
)
and the absorptive part
2
i
χ
BA
(
z
)
z
∗
)
,
1
χ
BA
(
z
)=
χ
AB
(
z
∗
)
−
−
≡
−
χ
AB
(
−
(3
.
2
.
11
b
)
so that
χ
BA
(
z
)=
χ
BA
(
z
)+
iχ
BA
(
z
)
(3
.
2
.
11
c
)
and, according to the Kramers-Kronig relation (3.1.10),
∞
∞
χ
BA
(
ω
)
ω
−
χ
BA
(
ω
)
ω
−
χ
BA
(
ω
)=
1
1
π
P
dω
;
χ
BA
(
ω
)=
dω
.
π
P
−
ω
ω
−∞
−∞
(3
.
2
.
11
d
)
In these equations,
χ
AB
(
ω
) is the boundary value obtained by taking
z
=
ω
+
i
, i.e. as lim
→
0
+
χ
AB
(
−
z
∗
=
−
−
ω
+
i
), corresponding to the
z
∗
), like
χ
AB
(
z
), is analytic in the upper half-
plane. The appropriate Laplace transform of
K
BA
(
t
) with this property
is
condition that
χ
AB
(
−
K
BA
(
z
)=
∞
−∞
K
BA
(
t
)
e
i
(
z
1
t
+
iz
2
|t|
)
dt
=
∞
0
∞
φ
AB
(
t
)
e
−iz
∗
t
dt.
φ
BA
(
t
)
e
izt
dt
−
0
Hence
K
BA
(
z
)=2
iχ
BA
(
z
)
.
(3
.
2
.
12)
Next we introduce the dynamic
correlation function
, sometimes re-
ferred to as the
scattering function
. It is defined as follows:
B
(
t
)
A
B
A
B A
(
B
A
S
BA
(
t
)
≡
−
=
−
t
)
−
,
(3
.
2
.
13)
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