Environmental Engineering Reference
In-Depth Information
and used
t
he time
d
ependence of operator in Heisenberg picture:
a
(
t
)
=
e
i
H
t
a
(0)
e
−
i
H
t
. Notice that the order of the limits in Eq. 1.46
is important: the volume limit
V
→∞
must be taken first, followed
bythetimelimit
τ
→∞
[61].Forthesakeofsimplicity,wewillomit
thelimitsinEq.1.46andputthetimelimittoinfinityinthefollowing
discussions.
In the classical limit
→
0, Eq. 1.46 can befurther simplified to
∞
dt
β
0
d
λ
S
υ
(0)
S
μ
(
t
)
1
TV
κ
μυ
=
0
∞
dt
S
υ
(0)
S
μ
(
t
)
.
1
k
B
T
2
V
=
(1.47)
0
For bulk materials with cubic symmetry, thermal conductivity is
usually expressed in scalar form as
∞
1
3
k
B
T
2
V
κ
=
dt
S
(0)
·
S
(
t
)
,
(1.48)
0
inwhichthermalconductivityisaveragedoverthreediagonalterms.
For low-dimensional materials without cubic symmetry, such as
nanowire and nanotube, thermal conductivity should be calculated
according to Eq. 1.47.
According to Hardy's formulation [62], heat current can be
defined as [63]
dt
i
d
S
=
r
i
(
t
)
ε
i
(
t
),
(1.49)
where
r
i
(
t
)and
ε
i
(
t
) denote the time-dependent coordinate and
totalenergyofatom
i
,respectively.TakeSWpotentialasanexample,
heat current can be further written as
2
i
,
j
6
i
,
j
,
k
1
1
S
=
v
i
ε
i
+
r
ij
(
F
ij
·
v
i
)
+
(
r
ij
+
r
ik
)(
F
ijk
·
v
i
),
i
=
=
=
i
j
i
j
,
j
k
(1.50)
where
v
i
is the velocity of atom
i
,and
F
ij
and
F
ijk
denote the
two-body and three-body force, respectively. The first term on the
right-hand side of Eq. 1.50 describes the heat convection typically
occurringinfluids,whiletheresttermsdescribetheheatconduction
betweentheatoms,whichisdominantpartinsolids.Therefore,heat
