Chemistry Reference
In-Depth Information
express MD (Newton's law) in the form of Eqs. [33], the authors first define
the distributions
179
X
r
ð
x
0
m
i
d
ð
x
0
x
i
Þ
;
t
Þ¼
i
X
v
ð
x
0
Þ
d
ð
x
0
x
i
Þ
;
t
Þ¼
v
i
ð
t
½
34
i
(
)
2
X
i
X
1
ð
x
0
m
i
v
i
þ
d
ð
x
0
x
i
Þ
~
e
;
t
Þ¼
i
f
½
x
i
ð
t
Þ
x
j
ð
t
Þ
6¼
j
and the fluxes
2
X
i
1
s
a
;
b
ð
x
0
x
i
b
x
j
b
Þ
;
t
Þ¼
f
a
ð
x
i
x
j
Þð
6¼
j
ð
1
0
d
f
x
0
½
x
j
þ
l
ð
x
i
x
j
Þg
d
l
½
35
4
X
i
1
j
ð
x
0
j
ð
v
i
þ
v
j
Þ
f
ð
x
j
x
i
Þð
x
i
x
j
Þ
;
t
Þ¼
6¼
ð
1
0
d
f
x
0
½
x
j
þ
l
ð
x
i
x
j
Þg
d
l
where
f
is the interaction potential and
f
ð
x
j
x
i
Þ
is the force between the
i
th
and the
j
th particles. We then have
q
t
q
r
x
0
s
¼
0
½
36
j
¼
r
0
q
t
~
e
þr
x
0
0
Equations [36] are numerically solved using any appropriate macroscopic solver.
In Ref. 176 the authors used the Nessyahu and Tadmor algorithm, which is
formulated over a staggered grid.
180
With this specific macrosolver, the fluxes
s
and
j
are required as input data at each macroscopic time step and for each
grid point
x
k
. These fluxes are obtained by performing local MD simulations
that are constrained by the local macroscopic variables
A
,
v
,and
e
. After the
MD system equilibrates, the fluxes are evaluated by time/ensemble averaging.
It is important to notice that the microscopic model does not have to be solved
everywhere, but rather only over a small region near where the data estima-
tion is carried out. The first step in setting up the local MD simulations is find-
ing an atomic configuration that is consistent with the local macroscopic
variables. This is accomplished using the Cauchy-Born rule (see discussion
of the QC method above), i.e., generating the basis vectors for the deformed
