Environmental Engineering Reference
In-Depth Information
current can be simplified as[58]
2
i
,
j
i
=
j
6
i
,
j
,
k
i
=
j
,
j
=
k
1
1
=
r
ij
(
F
ij
·
+
(
r
ij
+
r
ik
)(
F
ijk
·
S
v
i
)
v
i
),
(1.51)
which is computationally more e
cient as the calculation of local
energy isnotrequired.
1.6.2
Different Implementations
The main di
culty in implementing GKF in practical calculations
arises from how to carry out the time integral up to infinity. In this
section, we review the existing different implementations of GKF in
literature. Most of them can be categorized into two types: one is
the time-domain approach, and the other is the frequency-domain
approach.
The time-domain approach is to handle GKF in time-domain.
The most straightforward way is the direct integration method
[63], which replaces the integral with summation and numerically
records HCACF in time-domain as
Cor
(
t
)
=
S
υ
(0)
S
μ
(
t
)
N
−
m
S
υ
(
nt
0
)
S
μ
((
n
+
m
)
t
0
)
,
1
N
−
m
=
(1.52)
=
n
0
where
t
0
isthetimestep,
N
isthetotalnumberoftimestepsinEMD
simulations, and
m
=
/
t
0
is the integer number for time
t
. In this
method, the infinite integral in GKF is replaced by a summation up
to afinite cut-off time
t
τ
c
τ
c
t
0
k
B
T
2
V
κ
μυ
=
Cor
(
t
),
(1.53)
t
=
0
In addition to direct integration, several alternatives have also
been proposed by making use of certain statistical properties of
HCACF.Forinstance,Li
etal.
suggestedtofitHCACFaccordingtothe
singleexponentialfunction in the time interval[
τ
1
,
τ
2
] [56]
ge
−
t
/τ
0
,
Cor
(
t
)
=
(1.54)
