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 )+ 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
; χ BA ( ω )=
.
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 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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