Environmental Engineering Reference
In-Depth Information
magnetic susceptibility, and thereby indirectly to the corresponding first-
derivatives, like the specific heat and the magnetization. Consequently,
most physical properties of a macroscopic system near equilibrium may
be described in terms of the correlation functions.
As a supplement to the response function
φ
BA
(
t − t
), we now in-
troduce the
Green function
, defined as
t
)
B
(
t
);
A
(
t
)
G
BA
(
t
−
≡
(3
.
3
.
12)
i
h
θ
(
t
[
B
(
t
)
, A
(
t
)]
t
)
t
)
.
≡−
−
=
−
φ
BA
(
t
−
This Green function is often referred to as the
double-time
or the
retarded
Green function (Zubarev 1960), and it is simply our previous response
function, but with the opposite sign. Introducing the Laplace transform
G
BA
(
z
) according to (3.1.7), we find, as before, that the corresponding
Fourier transform is
B
;
A
G
BA
(
ω
)
≡
ω
= lim
→
0
+
G
BA
(
z
=
ω
+
i
)
∞
(3
.
3
.
13)
G
BA
(
t
)
e
i
(
ω
+
i
)
t
dt
=
= lim
→
0
+
−
χ
BA
(
ω
)
.
−∞
(0)
We note that, if
A
and
B
are dimensionless operators, then
G
BA
(
ω
)or
χ
BA
(
ω
) have the dimensions of inverse energy.
If
t
= 0, the derivative of the Green function with respect to
t
is
h
δ
(
t
)
d
dt
G
BA
(
t
)=
i
[
B
(
t
)
, A
]
[
d B
(
t
)
/dt , A
]
−
+
θ
(
t
)
h
δ
(
t
)
.
i
i
h
θ
(
t
)
[
B, A
]
[[
B
(
t
)
,
]
, A
]
=
−
−
H
A Fourier transformation of this expression then leads to the
equation
of motion
for the Green function:
B
;
A
ω
−
[
B,H
];
A
ω
=
[
B, A
]
.
hω
(3
.
3
.
14
a
)
The sux
ω
indicates the Fourier transforms (3.3.13), and
hω
is short-
hand for
h
(
ω
+
i
)with
0
+
. In many applications,
A
and
B
are
the same (Hermitian) operator, in which case the r.h.s. of (3
.
3
.
14
a
)van-
ishes and one may proceed to the second derivative. With the condition
that
→
[[[
A
(
t
)
,
]
, A
]
[[
A
(
t
)
,
]
,
[
A,
H
]
,
H
−
H
H
is
]]
, the equation
[
A,
];
A
H
ω
leads to
of motion for the Green function
(
hω
)
2
A
;
A
[
A,
]; [
A,
[[
A,
]
, A
]
ω
+
H
H
]
ω
=
H
.
(3
.
3
.
14
b
)
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