Environmental Engineering Reference
In-Depth Information
using (3.2.5) and defining
A
0
(
t
)=
e
iH
0
t/h
Ae
−iH
0
t/h
.
According to (3.2.6), taking into account the boundary condition, the
time-dependent density operator is
ρ
(
t
)=
e
−iH
0
t/h
t
−∞
ρ
I
(
t
)
dt
+
ρ
0
e
iH
0
t/h
d
dt
(3
.
2
.
7)
t
i
h
[
A
0
(
t
−
t
)
,ρ
0
]
f
(
t
)
dt
,
=
ρ
0
+
−∞
to first order in the external perturbations. This determines the time
dependence of, for example,
B
as
=Tr
(
ρ
(
t
)
ρ
0
)
B
B
(
t
)
B
−
−
h
Tr
t
−∞
t
)
,ρ
0
]
Bf
(
t
)
dt
i
[
A
0
(
t
−
=
and, utilizing the invariance of the trace under cyclic permutations, we
obtain, to leading order,
t
Tr
ρ
0
[
B, A
0
(
t
−
t
)]
f
(
t
)
dt
i
h
B
(
t
)
B
−
=
−∞
(3
.
2
.
8)
t
−∞
i
h
[
B
0
(
t
)
, A
0
(
t
)]
0
f
(
t
)
dt
.
=
A comparison of this result with the definition (3.1.4) of the response
function then gives
t
)=
i
[
B
(
t
)
, A
(
t
)]
t
)
φ
BA
(
t
−
h
θ
(
t
−
,
(3
.
2
.
9)
where the unit step function,
θ
(
t
)=0or1when
t<
0or
t>
0 respec-
tively, is introduced in order to ensure that
φ
BA
satisfies the causality
principle (3.1.5). In this final result, and below, we suppress the index 0,
but we stress that both the variations with time and the ensemble aver-
age are thermal-equilibrium values determined by
H
0
, and are unaffected
by the external disturbances. This expression in terms of microscopic
quantities, is called
the Kubo formula
for the response function (Kubo
1957, 1966).
The expression (3.2.9) is the starting point for introducing a number
of useful functions:
K
BA
(
t
)=
i
i
h
[
B
(
t
)
, A
]
[
B, A
(
h
=
−
t
)]
(3
.
2
.
10)
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