Environmental Engineering Reference
In-Depth Information
Figure 1.17
ThermalconductivityofthecrystallineSiliconat1000Kversus
super cell size (
N
×
N
×
N
unit cells) for different methods. The solid line
draws the experimental result at the same temperature from Ref. [66].
The frequency-domain approach is an indirect way to handle
GKF in frequency-domain by using Fourier transform and Wiener-
Khinchin theorem. From mathematical point of view, one can
considerGKFinEq.1.48astheFouriertransformofHCACF,andthus
can further define spectral thermal conductivity as
∞
1
3
k
B
T
2
V
e
i
ω
t
,
κ
(
ω
)
=
dt
S
(0)
·
S
(
t
)
(1.60)
0
in which the conventional thermal conductivity can be regarded as
thespectralthermalconductivityinthestaticlimit
ω
→
0.Similarly,
spectral heat current can bedefined as
∞
dt
S
(
t
)
e
i
ω
t
.
S
(
ω
)
=
(1.61)
0
Wiener-Khinchin theorem [67] asserts that in a stationary
randomprocess,thepowerspectraldensityistheFouriertransform
of the corresponding autocorrelation function. Thus, according to
Wiener-Khinchin theorem, the heat current power spectral density
can bewritten as
∞
2
e
i
ω
t
.
|
|
S
(
ω
)
=
dt
S
(0)
·
S
(
t
)
(1.62)
0
