Environmental Engineering Reference
In-Depth Information
and the reciprocal relation
∞
1
2
π
f
(
ω
)
e
−iωt
dω.
f
(
t
)=
(3
.
1
.
6
b
)
−∞
In order to take advantage of the causality condition (3.1.5), we shall
consider the Laplace transform of
φ
BA
(
t
) (the usual
s
is replaced by
−
iz
):
χ
BA
(
z
)=
∞
0
φ
BA
(
t
)
e
izt
dt.
(3
.
1
.
7
a
)
z
=
z
1
+
iz
2
is a complex variable and, if
0
e
−t
dt
is assumed
|
φ
BA
(
t
)
|
0
+
, the converse relation is
to be finite in the limit
→
∞
+
i
1
2
π
χ
BA
(
z
)
e
−izt
dz
φ
BA
(
t
)=
;
>
0
.
(3
.
1
.
7
b
)
−∞
+
i
When
φ
BA
(
t
) satisfies the above condition and eqn (3.1.5), it can readily
be shown that
χ
BA
(
z
) is an analytic function in the upper part of the
complex
z
-plane (
z
2
>
0).
In order to ensure that the evolution of the system is uniquely de-
termined by
ρ
0
=
ρ
(
−∞
)and
f
(
t
), it is necessary that the external
perturbation be turned on in a smooth, adiabatic way. This may be
accomplished by replacing
f
(
t
)in(4)by
f
(
t
)
e
t
,>
0. This force
vanishes in the limit
t
→−∞
, and any unwanted secondary effects may
be removed by taking the limit
0
+
. Then, with the definition of the
→
'generalized' Fourier transform
∞
e
iωt
e
−t
dt,
B
(
ω
)
B
(
t
)
B
= lim
→
0
+
−
(3
.
1
.
8)
−∞
eqn (3.1.4) is transformed into
B
(
ω
)
=
χ
BA
(
ω
)
f
(
ω
)
,
(3
.
1
.
9
a
)
where
χ
BA
(
ω
) is the boundary value of the analytic function
χ
BA
(
z
)on
the real axis:
χ
BA
(
ω
) = lim
→
0
+
χ
BA
(
z
=
ω
+
i
)
.
(3
.
1
.
9
b
)
χ
BA
(
ω
) is called the frequency-dependent or
generalized susceptibility
and is the Fourier transform, as defined by (3.1.8), of the response func-
tion
φ
BA
(
t
).
The mathematical restrictions (3.1.5) and (3.1.7) on
φ
BA
(
t
)have
the direct physical significance that the system is respectively causal
and stable against a small perturbation. The two conditions ensure that
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