Environmental Engineering Reference
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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> e −iE α t/h .
A
α >e iE α t/h
|
B
|
|
|
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> ( 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>
= lim
0 +
( n α
n α ) ,
E α
E α
ih
αα
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