Environmental Engineering Reference
In-Depth Information
where
λ
is a phase space variable. With this equality, the numerator
of density matrix in Eq. 1.40 can be approximated to the first order
as
=
e
−
β
H
1
−
β
d
λ
e
λ
H
H
e
−
λ
H
.
e
−
β
(
H
+
H
)
(1.42)
0
The small perturbation to the Hamiltonian can be expressed as
d
3
r
δ
T
(
r
)
h
(
r
)
1
T
H
=
dt
d
3
r
δ
T
(
r
)
∇·
s
(
r
)
1
T
=−
dt
d
3
r
1
T
=
∇
·
T
(
r
)
s
(
r
)
dt
∇
T
(
r
)
·
S
,
1
T
=
(1.43)
where we have used the integrated form of Eq. 1.39, performed an
integration by part, and assumed the temperature gradient to be a
spatial constant.
With the small thermal disturbance, the net heat current
becomes
∞
dt
β
0
S
(0))
e
−
λ
H
S
(
t
)
,
(1.44)
wheretheangularbracketdenotestheensembleaverage.According
to the Fourier'slaw
λ
e
λ
H
(
1
TV
=
ρ
=−
∇
·
J
Tr[
S
]
d
T
0
J
μ
∇
υ
T
,
κ
μυ
=−
(1.45)
μ
υ
where
are two Cartesian indices, thermal conductivity can
bewritten in the tensorform as
and
τ
dt
β
0
λ
e
λ
H
S
υ
(
t
)
e
−
λ
H
S
μ
(
t
)
1
TV
κ
μυ
=
lim
τ
→∞
lim
V
d
→∞
0
τ
dt
β
0
d
λ
e
λ
H
S
υ
(0)
e
−
λ
H
S
μ
(
t
)
1
TV
=
lim
τ
→∞
lim
V
→∞
0
τ
dt
β
0
d
λ
S
υ
(
−
i
λ
)
S
μ
(
t
)
,
1
TV
=
lim
τ
→∞
lim
V
(1.46)
→∞
0
where we have assumed that in the equilibrium steady state HCACF
in Eq. 1.46 only depends on the time difference (
t
t
t
),
=
−
