Environmental Engineering Reference
In-Depth Information
and these symmetry relations allow us to write
A
(
t
)
A
χ
AA
(
ω
0
)cos(
ω
0
t
)+
χ
AA
(
ω
0
)sin(
ω
0
t
)
−
=
f
0
{
}
.
The part of the response which is in phase with the external force is pro-
portional to
χ
AA
(
ω
0
), which is therefore called the reactive component.
The rate of energy absorption due to the field is
d
dt
H
A
(
t
)
Q
=
=
∂
H
/∂t
=
−
∂f/∂t,
which shows that the
mean
dissipation rate is determined by the out-of-
phase response proportional to
χ
AA
(
ω
):
Q
=
2
f
0
ω
0
χ
AA
(
ω
0
)
(3
.
3
.
2)
and
χ
AA
(
ω
) is therefore called the absorptive part of the susceptibility.
If the eigenvalues
E
α
and the corresponding eigenstates
|
α>
for
the Hamiltonian
H
0
) are known, it is possible to derive an explicit
expression for
χ
BA
(
ω
). According to the definition (3.2.10),
H
(=
Z
Tr
e
−βH
[
e
iHt/h
Be
−iHt/h
, A
]
=
K
BA
(
t
)=
i
h
1
Z
αα
e
−βE
α
e
iE
α
t/h
i
h
1
B
A
α
>e
−iE
α
t/h
<α
|
<α
|
|
|
α>
α> e
−iE
α
t/h
.
A
α
>e
iE
α
t/h
<α
|
B
−
<α
|
|
|
Interchanging
α
and
α
in the last term, and introducing the population
factor
Z
=
α
1
Z
e
−βE
α
e
−βE
α
,
n
α
=
;
(3
.
3
.
3
a
)
we get
h
αα
K
BA
(
t
)=
i
B
A
α
>< α
|
n
α
)
e
i
(
E
α
−E
α
)
t/h
,
<α
|
|
|
α>
(
n
α
−
(3
.
3
.
3
b
)
and hence
∞
K
BA
(
t
)
e
i
(
w
+
i
)
t
dt
χ
BA
(
ω
) = lim
→
0
+
0
(3
.
3
.
4
a
)
B
A
α
>< α
|
<α
|
|
|
α>
= lim
→
0
+
(
n
α
−
n
α
)
,
E
α
−
E
α
−
hω
−
ih
αα
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