<s>
Periodic	B-Algorithm
boundary	I-Algorithm
conditions	I-Algorithm
(	O
PBCs	O
)	O
are	O
a	O
set	O
of	O
boundary	O
conditions	O
which	O
are	O
often	O
chosen	O
for	O
approximating	O
a	O
large	O
(	O
infinite	O
)	O
system	O
by	O
using	O
a	O
small	O
part	O
called	O
a	O
unit	O
cell	O
.	O
</s>
<s>
PBCs	O
are	O
often	O
used	O
in	O
computer	B-Application
simulations	I-Application
and	O
mathematical	O
models	O
.	O
</s>
<s>
The	O
topology	B-Architecture
of	O
two-dimensional	O
PBC	O
is	O
equal	O
to	O
that	O
of	O
a	O
world	O
map	O
of	O
some	O
video	O
games	O
;	O
the	O
geometry	O
of	O
the	O
unit	O
cell	O
satisfies	O
perfect	O
two-dimensional	O
tiling	O
,	O
and	O
when	O
an	O
object	O
passes	O
through	O
one	O
side	O
of	O
the	O
unit	O
cell	O
,	O
it	O
re-appears	O
on	O
the	O
opposite	O
side	O
with	O
the	O
same	O
velocity	O
.	O
</s>
<s>
In	O
computer	B-Application
simulations	I-Application
,	O
one	O
of	O
these	O
is	O
the	O
original	O
simulation	O
box	O
,	O
and	O
others	O
are	O
copies	O
called	O
images	O
.	O
</s>
<s>
The	O
minimum-image	B-Algorithm
convention	I-Algorithm
is	O
a	O
common	O
form	O
of	O
PBC	O
particle	O
bookkeeping	O
in	O
which	O
each	O
individual	O
particle	O
in	O
the	O
simulation	O
interacts	O
with	O
the	O
closest	O
image	O
of	O
the	O
remaining	O
particles	O
in	O
the	O
system	O
.	O
</s>
<s>
Born	O
–	O
von	O
Karman	O
boundary	O
conditions	O
are	O
periodic	B-Algorithm
boundary	I-Algorithm
conditions	I-Algorithm
for	O
a	O
special	O
system	O
.	O
</s>
<s>
Similarly	O
,	O
the	O
wavelength	O
of	O
sound	O
or	O
shock	O
waves	O
and	O
phonons	B-Language
in	O
the	O
system	O
is	O
limited	O
by	O
the	O
box	O
size	O
.	O
</s>
<s>
Maintenance	O
of	O
the	O
minimum-image	B-Algorithm
convention	I-Algorithm
also	O
generally	O
requires	O
that	O
a	O
spherical	O
cutoff	O
radius	O
for	O
nonbonded	O
forces	O
be	O
at	O
most	O
half	O
the	O
length	O
of	O
one	O
side	O
of	O
a	O
cubic	O
box	O
.	O
</s>
<s>
The	O
size	O
of	O
the	O
simulation	O
box	O
must	O
also	O
be	O
large	O
enough	O
to	O
prevent	O
periodic	O
artifacts	O
from	O
occurring	O
due	O
to	O
the	O
unphysical	O
topology	B-Architecture
of	O
the	O
simulation	O
.	O
</s>
<s>
A	O
cube	B-Application
or	O
rectangular	O
prism	O
is	O
the	O
most	O
intuitive	O
and	O
common	O
choice	O
,	O
but	O
can	O
be	O
computationally	O
expensive	O
due	O
to	O
unnecessary	O
amounts	O
of	O
solvent	O
molecules	O
in	O
the	O
corners	O
,	O
distant	O
from	O
the	O
central	O
macromolecules	O
.	O
</s>
<s>
For	O
simulations	O
in	O
2D	O
and	O
3D	O
space	O
,	O
cubic	O
periodic	B-Algorithm
boundary	I-Algorithm
condition	I-Algorithm
is	O
most	O
commonly	O
used	O
since	O
it	O
is	O
simplest	O
in	O
coding	O
.	O
</s>
<s>
In	O
computer	B-Application
simulation	I-Application
of	O
high	O
dimensional	O
systems	O
,	O
however	O
,	O
the	O
hypercubic	B-Operating_System
periodic	B-Algorithm
boundary	I-Algorithm
condition	I-Algorithm
can	O
be	O
less	O
efficient	O
because	O
corners	O
occupy	O
most	O
part	O
of	O
the	O
space	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
hypercubic	B-Operating_System
periodic	B-Algorithm
boundary	I-Algorithm
condition	I-Algorithm
corresponds	O
to	O
the	O
hypercubic	B-Operating_System
lattice	O
packing	O
.	O
</s>
<s>
The	O
implementation	O
of	O
these	O
high	O
dimensional	O
periodic	B-Algorithm
boundary	I-Algorithm
conditions	I-Algorithm
is	O
equivalent	O
to	O
error	B-Error_Name
correction	I-Error_Name
code	I-Error_Name
approaches	O
in	O
information	O
theory	O
.	O
</s>
<s>
Under	O
periodic	B-Algorithm
boundary	I-Algorithm
conditions	I-Algorithm
,	O
the	O
linear	O
momentum	B-Algorithm
of	O
the	O
system	O
is	O
conserved	O
,	O
but	O
Angular	O
momentum	B-Algorithm
is	O
not	O
.	O
</s>
<s>
Conventional	O
explanation	O
of	O
this	O
fact	O
is	O
based	O
on	O
Noether	O
's	O
theorem	O
,	O
which	O
states	O
that	O
conservation	O
of	O
angular	O
momentum	B-Algorithm
follows	O
from	O
rotational	O
invariance	O
of	O
Lagrangian	O
.	O
</s>
<s>
However	O
,	O
this	O
approach	O
was	O
shown	O
to	O
not	O
be	O
consistent	O
:	O
it	O
fails	O
to	O
explain	O
the	O
absence	O
of	O
conservation	O
of	O
angular	O
momentum	B-Algorithm
of	O
a	O
single	O
particle	O
moving	O
in	O
a	O
periodic	O
cell	O
.	O
</s>
<s>
Lagrangian	O
of	O
the	O
particle	O
is	O
constant	O
and	O
therefore	O
rotationally	O
invariant	O
,	O
while	O
angular	O
momentum	B-Algorithm
of	O
the	O
particle	O
is	O
not	O
conserved	O
.	O
</s>
<s>
The	O
periodic	O
cell	O
exchanges	O
mass	O
momentum	B-Algorithm
,	O
angular	O
momentum	B-Algorithm
,	O
and	O
energy	O
with	O
the	O
neighboring	O
cells	O
.	O
</s>
<s>
When	O
applied	O
to	O
the	O
microcanonical	O
ensemble	O
(	O
constant	O
particle	O
number	O
,	O
volume	O
,	O
and	O
energy	O
,	O
abbreviated	O
NVE	O
)	O
,	O
using	O
PBC	O
rather	O
than	O
reflecting	O
walls	O
slightly	O
alters	O
the	O
sampling	O
of	O
the	O
simulation	O
due	O
to	O
the	O
conservation	O
of	O
total	O
linear	O
momentum	B-Algorithm
and	O
the	O
position	O
of	O
the	O
center	O
of	O
mass	O
;	O
this	O
ensemble	O
has	O
been	O
termed	O
the	O
"	O
molecular	O
dynamics	O
ensemble	O
"	O
or	O
the	O
NVEPG	O
ensemble	O
.	O
</s>
