Environmental Engineering Reference
In-Depth Information
essential steps. There is also a Kubo formula for electrons [53],
which relates electrical conductivity to the electrical current auto-
correlation function. For the electrical case, there is a mechanical
disturbance term [60] in the Hamiltonian, which describes the
energy perturbation due to the applied external electrical field.
The derivation of electrical Kubo formula follows the standard
perturbationtheory.Forthethermalcase,however,thederivationof
GKFislessstraightforwardbutreliesonthestatisticalhypothesisof
localthermalequilibrium,whichcanbedescribedbythelocalspace-
dependent temperature
T
(
r
)
(
r
)]
−
1
.
Consider a system with Hamiltonian
H
and volume
V
at
temperature
T
(
r
) subjected to a small thermal disturbance
=
[
k
B
β
δ
T
(
r
).
The local thermal equilibrium holds and can be described by the
local equilibriumdensity matrix:
ρ
=
exp
d
3
r
β
(
r
)
h
(
r
)
−
/
Z
,
(1.37)
where
Z
is the partition function, and
h
(
r
) is the Hamiltonian
density operator related to the Hamiltonian as
H
=
d
3
r
h
(
r
).
Similarly, the heat current density operator
s
(
r
)isdefinedas
d
3
rs
(
r
),
S
=
(1.38)
where
S
is the total heat current operator. Due to the energy
conservation, these two density operators are related through
∂
h
(
r
)
∂
t
+∇·
s
(
r
)
=
0.
(1.39)
Without
δ
T
(
r
), the system is in thermal equilibrium so that
T
(
r
)
=
T
0
=
[
k
B
β
0
]
−
1
is a constant, and there is no net heat current
(
J
0). Afterapplying thesmallthermal disturbance, the
density matrix becomes
ρ
=
exp
−
β
(
H
+
H
)
/
Z
, (1.40)
where
H
isthesmallperturbationtotheHamiltoniancausedbythe
thermal disturbance. There is a fundamental equality in quantum
mechanics for any two operators
a
and
b
[60] so that
=
Tr[
ρ
0
S
]
=
=
e
β
a
1
+
d
λ
e
−
λ
a
be
λ
(
a
+
b
)
,
β
e
β
(
a
+
b
)
(1.41)
0