Environmental Engineering Reference
In-Depth Information
get the following important relation between the correlation function
and the susceptibility:
1
e
−βhω
χ
BA
(
ω
)
,
S
BA
(
ω
)=2
h
(3
.
2
.
18)
1
−
which is called the
fluctuation-dissipation theorem
.Th srela ionex-
presses explicitly the close connection between the spontaneous fluctu-
ations in the system, as described by the correlation function, and the
response of the system to external perturbations, as determined by the
susceptibility.
The calculations above do not depend on the starting assumption
that
B
(or
A
) is a physical observable, i.e. that
B
should be equal to
B
†
. This has the advantage that, if the Kubo formula (3.2.9) is taken to
be the starting point instead of eqn (3.1.4), the formalism applies more
generally.
3.3 Energy absorption and the Green function
In this section, we first present a calculation of the energy transferred
to the system by the external perturbation
Af
(
t
) in (3.1.2),
incidentally justifying the names of the two susceptibility components
in (3.2.11). The energy absorption can be expressed in terms of
χ
AA
(
ω
)
and, without loss of generality,
A
may here be assumed to be a Hermitian
operator, so that
A
=
A
†
.Inthiscase,
f
(
t
) is real, and considering a
harmonic variation
H
1
=
−
f
0
e
iω
0
t
+
e
−iω
0
t
with
f
(
t
)=
f
0
cos (
ω
0
t
)=
2
f
0
=
f
0
,
then
as
∞
−∞
e
i
(
ω−ω
0
)
t
dt
=2
πδ
(
ω
f
(
ω
)=
πf
0
{
δ
(
ω
−
ω
0
)+
δ
(
ω
+
ω
0
)
}
,
−
ω
0
)
,
and we have
f
0
χ
AA
(
ω
0
)
e
iω
0
t
+
χ
AA
(
ω
0
)
e
−iω
0
t
.
A
(
t
)
A
=
2
−
−
The introduction of
A
=
B
=
A
†
in (3.2.15), and in the definition
(3.2.11), yields
χ
AA
(
ω
)
∗
=
χ
AA
(
ω
)=
χ
AA
(
−
ω
)
(3
.
3
.
1)
χ
AA
(
ω
)
∗
=
χ
AA
(
ω
)=
χ
AA
(
−
−
ω
)
,
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