Cellular Automaton Simulation of Polymers - New England Complex ...

backporcupineΤεχνίτη Νοημοσύνη και Ρομποτική

1 Δεκ 2013 (πριν από 3 χρόνια και 10 μήνες)

114 εμφανίσεις

C
E
L
L
U
L
A
R

A
U
T
O
M
A
T
O
N

S
I
M
U
L
A
T
I
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N

O
F

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L
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M
.

A
.

S
M
I
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H
,
a

Y
.

B
A
R
-
Y
A
M
,
b

Y
.

R
A
B
I
N
,
c

N
.

M
A
R
G
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L
U
S
,
a

T
.

T
O
F
F
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L
I
,
a

A
N
D

C
.

H
.

B
E
N
N
E
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T
d

a

M
I
T

L
a
bor
a
t
or
y
f
or

C
om
put
e
r

S
c
i
e
nc
e
,

C
a
m
br
i
d
ge
,

M
A


02139

b

E
C
S
,

44

C
um
m
i
ngt
on
S
t
.
,

B
os
t
on
U
ni
ve
r
s
i
t
y,

B
os
t
on
M
A

02215

c

D
e
pa
r
t
m
e
nt

of

P
hys
i
c
s
,

B
a
r
-
I
l
a
n

U
ni
ve
r
s
i
t
y,

R
a
m
a
t
-
G
a
n
52900,

I
s
r
a
e
l

d

I
B
M

T
.

J
.

W
a
t
s
on
R
e
s
.

C
t
r
.
,

Y
or
k
t
ow
n
H
e
i
ght
s
,

N
Y

11973


A
B
S
T
R
A
C
T


I
n
or
de
r

t
o
i
m
p
r
ove

our

a
bi
l
i
t
y
t
o
s
i
m
ul
a
t
e

t
he

c
om
pl
e
x
be
ha
vi
or

of

pol
ym
e
r
s
,

w
e

i
nt
r
oduc
e

dyna
m
i
c
a
l

m
ode
l
s

i
n
t
he

c
l
a
s
s

of

C
e
l
l
ul
a
r

A
ut
om
a
t
a

(
C
A
)
.


S
pa
c
e

pa
r
t
i
t
i
oni
n
g
m
e
t
hods

e
na
bl
e

us

t
o
ove
r
c
om
e

f
unda
m
e
nt
a
l

obs
t
a
c
l
e
s

t
o
l
a
r
ge

s
c
a
l
e

s
i
m
ul
a
t
i
on
of

c
onne
c
t
e
d
c
ha
i
ns

w
i
t
h
e
xc
l
ude
d
vol
um
e

by
pa
r
a
l
l
e
l

pr
oc
e
s
s
i
ng
c
om
put
e
r
s
.


A

hi
ghl
y
e
f
f
i
c
i
e
nt
,

t
w
o
-
s
pa
c
e

a
l
gor
i
t
hm

i
s

de
vi
s
e
d
a
nd
t
e
s
t
e
d
on
bot
h
C
e
l
l
ul
a
r

A
ut
om
a
t
a

M
a
c
hi
ne
s

(
C
A
M
s
)

a
nd
s
e
r
i
a
l

c
om
put
e
r
s
.

P
r
e
l
i
m
i
na
r
y
r
e
s
ul
t
s

on
t
he

s
t
a
t
i
c

a
nd

dyna
m
i
c

pr
ope
r
t
i
e
s

of

pol
ym
e
r
s

i
n
t
w
o
di
m
e
ns
i
on
s

a
r
e

r
e
por
t
e
d.


I
N
T
R
O
D
U
C
T
I
O
N


P
ol
ym
e
r
s

va
r
y
f
r
om

"
s
i
m
pl
e
"
,

us
ua
l
l
y
s
ynt
he
t
i
c
,

l
i
ne
a
r

c
ha
i
ns

of

i
de
nt
i
c
a
l

m
onom
e
r
s

t
o
hi
ghl
y
c
om
pl
e
x
c
ha
i
ns

c
on
s
i
s
t
i
ng
of

s
e
que
nc
e
s

of

a
m
i
no
-
a
c
i
ds

t
ha
t

f
or
m

t
he

bui
l
di
ng
bl
o
c
ks

o
f

l
i
vi
ng
or
ga
ni
s
m
s
.


R
e
a
l
i
s
t
i
c

dyna
m
i
c
a
l

s
i
m
ul
a
t
i
on
s

of

pol
ym
e
r
s

f
or

s
t
udy
of

s
uc
h
pr
ob
l
e
m
s

a
s

pr
ot
e
i
n
f
ol
di
ng

a
r
e

e
xpe
c
t
e
d
t
o
be

one

of

t
he

m
a
j
or

s
c
i
e
nt
i
f
i
c

unde
r
t
a
ki
ngs

i
n
upc
om
i
ng
ye
a
r
s
.


A
n
a
bi
l
i
t
y
t
o
t
a
ke

a
dva
nt
a
ge

of

m
a
s
s
i
ve
l
y
pa
r
a
l
l
e
l

c
om
put
e
r

a
r
c
hi
t
e
c
t
ur
e
s

w
i
t
h
up
t
o
10
5

pr
oc
e
s
s
or
s

w
oul
d
dr
a
m
a
t
i
c
a
l
y
i
m
pr
ove

t
he

e
f
f
e
c
t
i
ve
ne
s
s

of

s
i
m
ul
a
t
i
ons
.


I
n
or
de
r

t
o
i
l
l
us
t
r
a
t
e

t
he

di
f
f
i
c
ul
t
i
e
s

i
n
s
i
m
ul
a
t
i
ng
pol
ym
e
r

dyna
m
i
c
s

a
n
d
t
he
i
r

r
e
s
ol
ut
i
on,

i
t

pr
ove
s

qui
t
e

f
r
ui
t
f
u
l

a
nd
e
nl
i
ght
e
ni
ng
t
o
c
ons
i
de
r

a
bs
t
r
a
c
t

po
l
ym
e
r

m
ode
l
s
.

O
ne

of

t
he

b
a
s
i
c

pa
r
a
di
gm
s

of

pol
ym
e
r

s
c
i
e
nc
e

i
s

t
ha
t

m
a
ny
of

t
he

dyna
m
i
c
a
l

a
nd
s
t
r
uc
t
ur
a
l

pr
ope
r
t
i
e
s

of

s
uf
f
i
c
i
e
nt
l
y
l
ong
m
a
c
r
om
ol
e
c
ul
e
s

c
a
n
be

unde
r
s
t
ood
w
i
t
hi
n

t
he

f
r
a
m
e
w
or
k
of

a
n
a
bs
t
r
a
c
t

pol
ym
e
r

m
ode
l
.
1
,
2


I
n

t
hi
s

m
ode
l

t
he

pol
ym
e
r

c
ons
i
s
t
s

of

e
l
e
m
e
nt
s

w
hi
c
h
a
r
e

c
onne
c
t
e
d

i
n
s
e
que
nc
e

a
nd

a
voi
d
i
nt
r
udi
ng
on
e
a
c
h
ot
h
e
r
's

s
pa
c
e
.


T
he
r
e

a
r
e

onl
y
t
w
o
e
s
s
e
nt
i
a
l

pa
r
a
m
e
t
e
r
s
,

t
he

l
e
ngt
h
a
nd

t
he

e
xc
l
ude
d
vol
um
e
.


E
ve
n
f
or

t
hi
s

s
i
m
pl
e

m
ode
l

w
he
r
e

a
na
l
yt
i
c
a
l

t
ool
s

s
uc
h

a
s

m
e
a
n
-
f
i
e
l
d,

s
c
a
l
i
ng
a
nd

r
e
nor
m
a
l
i
z
a
t
i
on
gr
oup
m
e
t
hods

pr
ovi
de

a

ba
s
i
c

unde
r
s
t
a
ndi
ng
of

phys
i
c
a
l

pr
ope
r
t
i
e
s
,

c
om
put
e
r

s
i
m
ul
a
t
i
ons

a
r
e

ne
e
de
d

t
o
c
he
c
k
a
na
l
yt
i
c
a
l

r
e
s
ul
t
s

i
n
s
i
m
pl
e

s
y
s
t
e
m
s

a
nd
s
t
udy
c
om
pl
e
x
s
y
s
t
e
m
s

of

de
n
s
e
,

gr
a
f
t
e
d,

br
a
nc
hi
ng,

m
a
t
r
i
x
-
e
m
be
de
d
pol
ym
e
r
s
,

e
t
c
.


F
or

m
or
e

r
e
a
l
i
s
t
i
c

m
ode
l
s
,

c
om
put
e
r

s
i
m
ul
a
t
i
ons

p
r
ovi
de

t
he

onl
y

hope

of

obt
a
i
ni
ng
de
t
a
i
l
e
d
i
nf
or
m
a
t
i
on
.


T
he

c
e
nt
r
a
l

di
f
f
i
c
ul
t
y
i
n
s
i
m
ul
a
t
i
ng
t
he

be
ha
vi
or

of

pol
ym
e
r
s

i
s

t
he

l
a
r
ge

num
b
e
r

of

i
ndi
vi
dua
l

c
om
pone
nt
s

ne
c
e
s
s
a
r
y

t
o
e
f
f
e
c
t

t
he

c
onf
or
m
a
t
i
on
of

a

l
ong

m
a
c
r
om
ol
e
c
ul
e
.



W
ha
t

m
e
t
hods

s
houl
d
one

us
e

t
o
s
i
m
ul
a
t
e

c
om
pl
e
x
hi
gh
m
ol
e
c
ul
a
r

w
e
i
ght

pol
ym
e
r
s
?


C
onve
nt
i
ona
l

s
i
m
ul
a
t
i
ons

a
r
e

of

t
w
o
t
ype
s
:

m
ol
e
c
u
l
a
r

dyna
m
i
c
s
,
3

a
nd
M
ont
e
-
C
a
r
l
o.
4


M
ol
e
c
ul
a
r

dyna
m
i
c
s

s
i
m
ul
a
t
i
ons

a
r
e

s
ugge
s
t
i
ve

of

r
e
a
l
i
s
t
i
c

N
e
w
t
oni
a
n
dyna
m
i
c
s

of

pol
ym
e
r
s

a
nd
a
r
e

i
m
pl
e
m
e
nt
e
d
by
m
ovi
ng
a
l
l

a
t
om
s

w
i
t
h
s
m
a
l
l

s
t
e
ps

a
c
c
or
di
ng
t
o
f
or
c
e
s

c
a
l
c
ul
a
t
e
d
f
r
om

m
ode
l
e
d
i
nt
e
r
a
t
om
i
c

f
or
c
e
s
.


M
ont
e

C
a
r
l
o
dyna
m
i
c
s

r
e
pr
e
s
e
nt

t
he

dyna
m
i
c
s

of

a
n

e
ns
e
m
bl
e

of

pol
ym
e
r
s

by
s
t
e
ps

w
hi
c
h
t
a
ke

i
nt
o
a
c
c
ount

t
he
r
m
odyna
m
i
c

t
r
a
ns
i
t
i
on
pr
oba
bi
l
i
t
i
e
s
.


B
ot
h
t
e
c
hni
que
s

gi
ve

t
he

s
a
m
e

r
e
s
ul
t
s

f
or

s
t
r
uc
t
ur
e
,

c
on
f
or
m
a
t
i
ona
l

c
ha
nge

a
nd
di
f
f
us
i
on.


A
l
l

a
t
om
s

c
a
n
be

m
ove
d
i
n
pa
r
a
l
l
e
l

(
a
t

t
he

s
a
m
e

t
i
m
e
)

i
n
m
o
l
e
c
ul
a
r

dyna
m
i
c
s
,

w
hi
c
h
t
he
r
e
f
or
e

a
ppe
a
r
s

t
o
be

i
de
a
l
l
y
s
ui
t
e
d
f
or

pa
r
a
l
l
e
l

pr
oc
e
s
s
i
ng
c
om
put
e
r
s
.


H
ow
e
ve
r
,

w
i
t
h
a

pr
oc
e
s
s
or

a
t
t
a
c
he
d
t
o
e
a
c
h

a
t
om
,

c
a
l
c
ul
a
t
i
on
of

t
he

f
o
r
c
e
s

r
e
qui
r
e
s

a

l
a
r
ge

nu
m
be
r

of

c
om
m
un
i
c
a
t
i
ons

be
t
w
e
e
n
pr
oc
e
s
s
or
s
.
5


C
onne
c
t
i
ons

be
t
w
e
e
n
pr
oc
e
s
s
or
s

a
r
e

t
he

l
i
m
i
t
i
ng

f
e
a
t
ur
e

of

pa
r
a
l
l
e
l

c
om
put
e
r
s

a
nd
r
e
s
ol
ut
i
ons

of

c
om
m
uni
c
a
t
i
on
p
r
obl
e
m
s

onl
y
a
ppl
y

t
o
pa
r
t
i
c
ul
a
r

c
om
put
e
r

a
r
c
hi
t
e
c
t
u
r
e
s
.


T
he

ge
ne
r
a
l

a
ppr
oa
c
h
t
o
pa
r
a
l
l
e
l

pr
oc
e
s
s
i
ng
dyna
m
i
c
s

w
e

ha
ve

de
ve
l
ope
d
c
a
n
be

us
e
d
f
or

bot
h
m
ol
e
c
ul
a
r

dyna
m
i
c
s

a
nd
M
ont
e

C
a
r
l
o
s
i
m
ul
a
t
i
ons
.


S
i
nc
e

M
ont
e
-
C
a
r
l
o
doe
s

not

r
e
qui
r
e

t
he

s
pe
c
i
f
i
c
a
t
i
on
of

a
r
t
i
f
i
c
i
a
l

f
o
r
c
e
s

i
n
t
he

a
bs
t
r
a
c
t

m
ode
l
,

w
e

de
s
c
r
i
be

t
he

a
ppr
oa
c
h
i
n
t
he

l
a
ngua
ge

of

M
ont
e
-
C
a
r
l
o
s
t
e
p
dyna
m
i
c
s

f
or

t
hi
s

m
ode
l
.

I
n
M
ont
e
-
C
a
r
l
o
s
i
m
ul
a
t
i
ons

of

a
bs
t
r
a
c
t

pol
ym
e
r

s
t
r
uc
t
ur
e

a
nd
dyna
m
i
c
s
,
4

a

l
ong
c
ha
i
n
c
ons
i
s
t
i
ng
of

a
t
t
a
c
he
d
m
onom
e
r
s

i
s

r
e
pr
e
s
e
nt
e
d
by
t
he

c
oor
di
na
t
e
s

of

e
a
c
h
m
onom
e
r
.


A

s
i
m
ul
a
t
i
on
s
t
e
p
c
ons
i
s
t
s

of

s
e
l
e
c
t
i
ng
a

m
onom
e
r

<
i
>

f
r
om

t
he

pol
ym
e
r

c
ha
i
n
a
nd
p
e
r
f
or
m
i
ng
a

m
ove
.


T
he
r
e

a
r
e

m
a
ny
di
f
f
e
r
e
nt

m
e
t
hods

f
or

de
s
c
r
i
bi
ng
t
he

l
oc
a
l

s
t
r
uc
t
ur
e

of

t
he

pol
ym
e
r

a
nd
t
he

pr
oc
e
s
s

of

e
a
c
h
m
ove
.


H
ow
e
ve
r
,

qui
t
e

ge
ne
r
a
l
l
y,

e
a
c
h
m
ove

i
s

r
e
qui
r
e
d
t
o
pr
e
s
e
r
ve

t
he

c
ons
t
r
a
i
nt
s

t
ha
t

a
r
i
s
e

f
r
om

t
he

t
w
o

pa
r
a
m
e
t
e
r
s

of

a
bs
t
r
a
c
t

pol
ym
e
r
s
,

t
he
i
r

l
e
ngt
h

a
nd
e
xc
l
ude
d
vol
um
e
.

T
he
s
e

c
ons
t
r
a
i
nt
s

a
r
e
:



(
1)


T
he

m
ove

doe
s

not

"
br
e
a
k"

t
he

pol
ym
e
r

c
onne
c
t
i
vi
t
y
-

m
onom
e
r

<
i
>

doe
s

not

di
s
s
oc
i
a
t
e

i
t
s
e
l
f

f
r
om

i
t
s

ne
a
r
e
s
t

ne
i
ghbo
r
s

a
l
ong
t
h
e

c
ha
i
n
(
N
N
C
s
)
.

(
2)


T
he

m
ove

doe
s

not

vi
ol
a
t
e

e
xc
l
ude
d
vol
u
m
e

-

m
onom
e
r

<
i
>

do
e
s

not

ove
r
l
a
p
t
he

vol
um
e

of

a
ny

ot
he
r

m
onom
e
r

<
j
>
.

I
n
na
i
ve

pa
r
a
l
l
e
l

pr
oc
e
s
s
i
ng,

a

s
e
t

of

pr
oc
e
s
s
or
s

i
s

a
s
s
i
gne
d
on
e
-
t
o
-
one

t
o
pe
r
f
or
m

t
he

m
ove
m
e
nt

of

a

s
e
t

of

m
onom
e
r
s
.

E
a
c
h
pr
oc
e
s
s
or

doe
s

not

know

t
he

out
c
om
e

of

t
he

m
ove
m
e
nt

of

t
he

ot
he
r

m
onom
e
r
s
,

i
t

c
a
n
onl
y

know

t
he
i
r

po
s
i
t
i
on
be
f
or
e

t
he

c
ur
r
e
nt

s
t
e
p.


W
i
t
h
t
he

t
w
o
c
ons
t
r
a
i
nt
s

(
1)

a
nd
(
2
)

i
t

ha
s

ge
ne
r
a
l
l
y
be
e
n
c
ons
i
de
r
e
d
i
m
pos
s
i
bl
e

t
o
pe
r
f
or
m

s
ys
t
e
m
a
t
i
c

pa
r
a
l
l
e
l

pr
oc
e
s
s
i
ng

on
pol
ym
e
r

dyna
m
i
c
s

s
i
nc
e

m
ovi
ng
di
f
f
e
r
e
nt

m
onom
e
r
s

a
t

t
he

s
a
m
e

t
i
m
e

i
s

l
i
ke
l
y
t
o
l
e
a
d
t
o
di
s
s
oc
i
a
t
i
on
or

ove
r
l
a
p.


T
he

f
or
m
e
r

c
ons
t
r
a
i
nt

onl
y
r
e
s
t
r
i
c
t
s

t
he

pa
r
a
l
l
e
l

m
ot
i
on

of

ne
a
r
e
s
t

ne
i
ghbor
s

a
nd

t
hus

c
a
n
be

ove
r
c
om
e

us
i
ng
s
i
m
pl
e

a
l
gor
i
t
hm
s
.


I
n
c
ont
r
a
s
t
,

t
he

"
non
-
l
oc
a
l
"

e
xc
l
ude
d
vol
um
e

c
on
s
t
r
a
i
nt

r
e
s
t
r
i
c
t
s

t
he

pa
r
a
l
l
e
l

m
ot
i
on
of

any

t
w
o
m
onom
e
r
s
,

pr
e
s
e
nt
i
ng
a

f
unda
m
e
nt
a
l

di
f
f
i
c
ul
t
y

f
or

pa
r
a
l
l
e
l

p
r
o
c
e
s
s
i
ng.


C
E
L
L
U
L
A
R

A
U
T
O
M
A
T
O
N


D
Y
N
A
M
I
C
S


W
e

pr
opos
e

a

ge
ne
r
a
l

w
a
y
t
o
ove
r
c
om
e

t
hi
s

di
f
f
i
c
ul
t
y
by

r
e
c
ogni
z
i
ng
t
ha
t

pol
ym
e
r

i
nt
e
r
a
c
t
i
ons

a
r
e

l
oc
al


i
n
s
pac
e
.


T
he

pol
ym
e
r

c
a
n
c
oi
l

s
o
a
s

t
o
br
i
ng
a
ny
t
w
o
m
onom
e
r
s

i
nt
o

c
ont
a
c
t
,

ye
t
,

a
t

a
ny
pa
r
t
i
c
ul
a
r

t
i
m
e
,

t
he

onl
y
pos
s
i
bl
e

i
nt
e
r
a
c
t
i
ons

a
r
e

be
t
w
e
e
n
m
onom
e
r
s

w
hi
c
h

a
r
e

ne
a
r
by
i
n
s
pa
c
e
.


A
s

a

ge
ne
r
a
l

r
ul
e
,

m
onom
e
r
s

s
uf
f
i
c
i
e
nt
l
y
di
s
t
a
nt

i
n

s
pa
c
e

f
r
om

e
a
c
h
ot
he
r

c
a
n
be

m
ove
d
i
n
p
a
r
a
l
l
e
l

w
i
t
hout

a
ny
pos
s
i
bi
l
i
t
y
of

i
nt
e
r
f
e
r
i
ng
i
n
t
hat

s
t
e
p
.


T
w
o
m
onom
e
r
s

c
a
n

m
ove

i
nde
pe
nde
nt
l
y
i
f

t
he

di
s
t
a
nc
e

be
t
w
e
e
n
t
he
m

i
s

gr
e
a
t
e
r

t
ha
n
2(
l

+

r
)
,

w
he
r
e

l

i
s

t
he

s
t
e
p

l
e
ngt
h
a
nd
r

i
s

t
he

e
xc
l
ude
d
vol
um
e

r
a
di
us
.


T
hi
s

c
a
n
be

r
e
a
l
i
z
e
d
by
pa
r
t
i
t
i
oni
ng
s
pa
c
e

a
s

i
n
F
i
g.

1.


I
n
e
a
c
h
s
ha
de
d

r
e
gi
on,

of

l
i
ne
a
r

di
m
e
ns
i
on
(
l
+
r
)
,

w
e

s
e
l
e
c
t

a

m
onom
e
r

(
w
hos
e

c
e
nt
e
r

l
i
e
s

i
n

t
ha
t

s
ha
de
d
r
e
gi
on)
.


T
he
n
w
e

m
ove

a
l
l

t
he

s
e
l
e
c
t
e
d
m
onom
e
r
s

i
nde
pe
nde
nt
l
y.


F
i
na
l
l
y,

w
e

s
hi
f
t

t
he

s
ha
de
d

r
e
gi
ons

be
f
or
e

t
he

ne
xt

s
e
l
e
c
t
i
on
pr
oc
e
s
s
.


T
hi
s

br
i
e
f

de
s
c
r
i
pt
i
on
s
pe
c
i
f
i
e
s

e
nt
i
r
e
l
y

t
he

m
odi
f
i
c
a
t
i
on
of

a
n
e
xi
s
t
i
ng
non
-
pa
r
a
l
l
e
l

a
l
gor
i
t
hm

i
nt
o
a

pa
r
a
l
l
e
l

one
.


H
ow
e
ve
r
,

not
e

t
ha
t

t
he
r
e

i
s

a
n
a
ddi
t
i
ona
l

c
om
put
a
t
i
ona
l

t
a
s
k:


t
he

t
a
bul
a
t
i
on
of

m
onom
e
r
s

i
n
e
a
c
h
r
e
gi
on,

w
hi
c
h
m
us
t

be

upda
t
e
d
w
i
t
h
e
a
c
h
m
ove
.


T
hi
s

up
da
t
e

c
a
n
be

pe
r
f
or
m
e
d

e
f
f
i
c
i
e
nt
l
y

a
nd
i
n

pa
r
a
l
l
e
l
.


T
he

i
de
a

of

s
pa
c
e

or
i
e
nt
e
d
dyna
m
i
c
s

i
s

m
a
ni
f
e
s
t

i
n
t
he

ve
r
y
ge
ne
r
a
l

c
a
t
e
gor
y
of

dyna
m
i
c
a
l

m
ode
l
s

know
n
a
s

C
e
l
l
ul
a
r

A
ut
om
a
t
a

(
C
A
)
6


of

w
hi
c
h
t
he

be
s
t

kno
w
n
e
xa
m
pl
e

i
s

C
onw
a
y's

"
G
a
m
e

of

L
i
f
e
"
.
7



A
n
a
ut
om
a
t
on
c
ons
i
s
t
s

of

a

r
ul
e

f
or

s
pe
c
i
f
yi
ng
t
he

s
t
a
t
e

of

a

s
ys
t
e
m

a
f
t
e
r

a

t
i
m
e

i
nt
e
r
va
l

(
de
f
i
ne
d
t
o

be

one

uni
t

o
f

t
i
m
e
)

i
n

t
e
r
m
s

of

i
t
s

p
r
e
vi
ous

s
t
a
t
e
.


I
n
a

s
i
m
pl
i
f
i
e
d
f
or
m
,

di
vi
di
ng
s
pa
c
e

i
nt
o
c
e
l
l
s
,

a
nd
c
on
s
i
de
r
i
ng
e
a
c
h
c
e
l
l

t
o
ha
ve

onl
y
t
w
o
pos
s
i
bl
e

va
l
u
e
s

O
N

a
n
d

O
F
F
,

t
he

s
t
a
t
e

of

a

c
e
l
l

i
s

de
t
e
r
m
i
ne
d

by
t
he

c
ondi
t
i
on
of

i
t
s

ne
i
ghbor
hood
a
t

t
he

pr
e
vi
ous

t
i
m
e
.


D
i
f
f
e
r
e
nt

r
ul
e
s

f
or

de
t
e
r
m
i
ni
ng

t
he

s
t
a
t
e

of

t
he

c
e
l
l

de
s
c
r
i
be

di
f
f
e
r
e
nt

dyna
m
i
c
s
.


T
he

C
e
l
l
ul
a
r

A
ut
om
a
t
on
i
s

a

s
uf
f
i
c
i
e
nt
l
y
ge
ne
r
a
l

c
on
s
t
r
uc
t
i
on
t
ha
t

i
t

i
s

be
l
i
e
v
e
d
t
o
c
a
pt
ur
e

t
he

e
s
s
e
n
c
e

of

dyna
m
i
c
a
l

s
ys
t
e
m
s

f
or

f
or
m
a
l

i
nve
s
t
i
ga
t
i
ons
.
8
,
6


S
pe
c
i
f
i
c

a
ut
om
a
t
a

ha
ve

be
e
n
us
e
d
t
o
m
ode
l

s
ys
t
e
m
s

of

i
nt
e
r
e
s
t

i
n
B
i
ol
ogy,

C
he
m
i
s
t
r
y
a
nd
P
hy
s
i
c
s
.


C
A

a
r
e

i
de
a
l

f
o
r

s
i
m
ul
a
t
i
on

on
pa
r
a
l
l
e
l

c
om
put
e
r
s
.


M
or
e
ove
r
,

t
he
r
e

e
xi
s
t

C
e
l
l
ul
a
r

A
ut
om
a
t
a

M
a
c
hi
ne
s
9

w
hi
c
h
a
r
e

c
om
put
e
r
s

s
pe
c
i
f
i
c
a
l
l
y
de
s
i
gne
d
t
o
s
i
m
ul
a
t
e

a

c
e
l
l
ul
a
r

a
ut
om
a
t
on
f
or

s
t
udy
a
nd
a
na
l
ys
i
s
.

T
w
o
r
e
c
e
nt

e
f
f
or
t
s

ha
ve

be
e
n
di
r
e
c
t
e
d
a
t

us
i
ng
a
u
t
om
a
t
a

i
n
t
he

de
s
c
r
i
pt
i
on
o
f

pol
ym
e
r
s

or

s
t
r
i
ngs
.


C
hopa
r
d
1
0

s
how
e
d
f
or

t
he

f
i
r
s
t

t
i
m
e

how

e
xt
e
nde
d
obj
e
c
t
s

s
uc
h
a
s

s
t
r
i
ngs

c
oul
d
b
e

s
i
m
ul
a
t
e
d.


K
oe
l
m
a
n
1
1

a
da
pt
e
d
a

l
a
t
t
i
c
e

ga
s

C
A

dyna
m
i
c
s

by
i
nc
or
por
a
t
i
ng
a

non
-
a
ut
om
a
t
on
dyna
m
i
c
a
l

s
t
e
p
w
hi
c
h
a
t
t
a
c
he
d
m
onom
e
r
s

t
oge
t
he
r

t
o
f
o
r
m

a

pol
ym
e
r
.

S
t
a
nda
r
d
C
A

a
r
e

not

w
e
l
l

s
ui
t
e
d
t
o

t
he

de
s
c
r
i
pt
i
on
of

s
ys
t
e
m
s

w
i
t
h
c
ons
t
r
a
i
nt
s

or

c
ons
e
r
va
t
i
on
l
a
w
s
.


F
or

e
xa
m
pl
e
,

i
f

w
e

w
a
nt

t
o

c
ons
e
r
ve

t
he

num
be
r

of

O
N

s
i
t
e
s

w
e

m
us
t

e
s
t
a
bl
i
s
h
a

r
ul
e

w
h
e
r
e

t
ur
ni
ng
of
f

one

s
i
t
e

i
s

t
i
e
d
t
o
t
ur
ni
ng
on
a
not
he
r

s
i
t
e
.


A

m
odi
f
i
c
a
t
i
on
of

C
A

de
ve
l
ope
d
t
o
de
s
c
r
i
be

s
uc
h
s
ys
t
e
m
s

i
s

M
a
r
gol
us

dyna
m
i
c
s
,
9

w
he
r
e

t
he

r
ul
e

upda
t
e
s

a

w
hol
e

ne
i
ghbor
hood
r
a
t
he
r

t
ha
n
a

s
i
ngl
e

c
e
l
l
.


T
he
n,

a

c
ons
e
r
va
t
i
on
l
a
w

w
hi
c
h
hol
ds

i
n
t
he

ne
i
ghbor
hood
upda
t
e

(
t
he

num
be
r

of

O
N

s
i
t
e
s
)

a
l
s
o
hol
ds

gl
oba
l
l
y.

W
e

ha
ve

de
ve
l
ope
d
a

C
A

M
a
r
gol
us

dyna
m
i
c
s

f
o
r

t
he

s
i
m
ul
a
t
i
on
of

pol
ym
e
r
s
.


F
or

e
xa
m
pl
e
,

i
n
t
w
o
di
m
e
ns
i
ons
:

O
N

c
e
l
l
s

r
e
pr
e
s
e
nt

m
onom
e
r
s
,

a
nd
a

pol
ym
e
r

i
s

d
e
s
c
r
i
be
d
by

a

s
e
t

of

m
onom
e
r
s

w
hi
c
h
t
ouc
h
e
i
t
he
r

a
t

c
or
ne
r
s

or

on
e
d
ge
s

of

t
he

c
e
l
l
s
.


I
n
a

c
ha
i
n
pol
ym
e
r

(
F
i
g.

2)

e
a
c
h
m
onom
e
r

ha
s

t
w
o
s
uc
h
ne
i
ghbor
s

e
xc
e
pt

f
or

t
he

e
nds

w
hi
c
h
ha
ve

onl
y
one
.


W
e

s
e
l
e
c
t

a

s
ub
-
l
a
t
t
i
c
e

of

c
e
l
l
s

(
i
l
l
us
t
r
a
t
e
d
by
c
i
r
c
l
e
s

i
n
F
i
g.

2)

s
e
pa
r
a
t
e
d
by
3
c
e
l
l
s

i
n
e
a
c
h
di
r
e
c
t
i
on.


I
f

a

m
onom
e
r

i
s

l
oc
a
t
e
d
i
n

one

of

t
he
s
e

c
e
l
l
s
,

i
t

c
a
n

b
e

m
ove
d
up,

dow
n,

r
i
ght
,

or

l
e
f
t

a
s

pe
r
m
i
t
t
e
d

by
t
he

c
ons
t
r
a
i
nt
s

of

m
a
i
nt
a
i
ni
ng
c
onne
c
t
i
vi
t
y

a
nd
e
xc
l
ude
d
vol
um
e
.


P
e
r
m
i
t
t
e
d
m
ove
m
e
nt
s

a
r
e

i
l
l
us
t
r
a
t
e
d
by
a
r
r
ow
s
.


M
ove
m
e
nt
s

not

a
l
l
ow
e
d

by
e
xc
l
ude
d
vol
um
e

a
r
e

i
ndi
c
a
t
e
d
by
X
s

i
n
F
i
g
.

2.


A

m
ove
m
e
nt

c
or
r
e
s
ponds

t
o
a
n
upda
t
e

of

t
he

3x3
pl
a
que
t
t
e

a
r
ound
a

s
e
l
e
c
t
e
d
s
i
t
e
;

t
he

i
nf
or
m
a
t
i
on
f
or

t
he

upda
t
e

i
s

c
ont
a
i
ne
d
i
n
a

l
a
r
ge
r

5x5
r
e
gi
on.


A
l
l

3x3
pl
a
que
t
t
e
s

c
a
n
be

upda
t
e
d
s
i
m
ul
t
a
ne
ous
l
y.


F
i
na
l
l
y
,

t
he

s
e
l
e
c
t
e
d
s
ubl
a
t
t
i
c
e

(
c
i
r
c
l
e
s
)

i
s

s
hi
f
t
e
d
be
f
or
e

t
he

ne
xt

s
e
t

of

upda
t
e

s
t
e
ps

(
pr
e
f
e
r
a
bl
y
r
a
ndom
l
y
t
o
a
voi
d
pos
s
i
bl
e

c
or
r
e
l
a
t
i
ons

i
n
t
he

m
ot
i
on)
.


I
n
t
he

s
i
m
ul
a
t
i
on
of

l
ong

pol
ym
e
r
s
,

s
i
nc
e

w
e

a
r
e

i
nt
e
r
e
s
t
e
d
i
n

t
he

a
s
ym
pt
ot
i
c

be
h
a
vi
or

w
he
r
e

l
oc
a
l

pr
ope
r
t
i
e
s

a
r
e

uni
m
por
t
a
nt
,

a
nd

s
i
nc
e

t
he

r
e
l
a
xa
t
i
on
t
i
m
e

i
nc
r
e
a
s
e
s

dr
a
m
a
t
i
c
a
l
l
y
w
i
t
h
t
he

num
be
r

of

m
onom
e
r
s
,

i
t

i
s

i
m
por
t
a
nt

t
o
h
a
ve

a

m
ode
l

w
he
r
e

t
he

hi
gh
m
ol
e
c
ul
a
r

w
e
i
ght

l
i
m
i
t

i
s

r
e
a
l
i
z
e
d
f
or

m
ode
r
a
t
e

num
be
r
s

o
f

m
ono
m
e
r
s
.


A
s

f
o
r

r
e
a
l

pol
ym
e
r
s
,

t
he

l
ong
-
l
e
ngt
h

be
ha
vi
or

i
s

r
e
a
c
h
e
d
w
he
n

t
he

d
e
t
a
i
l
s

of

t
h
e

l
oc
a
l

pr
ope
r
t
i
e
s

be
c
om
e

uni
m
por
t
a
nt
.


T
hus

w
e

c
hoos
e

t
he

l
oc
a
l

dyna
m
i
c
s

t
o
m
i
ni
m
i
z
e

t
he

i
nf
l
ue
nc
e

of

l
oc
a
l

c
ons
t
r
a
i
nt
s

on
t
he

dyna
m
i
c
s
.


T
he
r
e

a
r
e

t
w
o

c
ha
r
a
c
t
e
r
i
s
t
i
c

t
ype
s

of

l
oc
a
l

dy
na
m
i
c
a
l

be
ha
vi
or

of

a

po
l
ym
e
r

-

m
ot
i
on

pe
r
pe
ndi
c
ul
a
r

t
o
t
he

pol
ym
e
r

a
nd
m
ot
i
on
a
l
ong
t
he

pol
ym
e
r

c
ont
our

(
w
hi
c
h
i
nvol
ve
s

l
o
c
a
l

l
e
ngt
h
c
ha
nge
s

i
n
t
he

l
ong
pol
ym
e
r
)
.


A
n

e
f
f
e
c
t
i
v
e

a
ppoa
c
h
t
o
m
i
ni
m
i
z
e

t
he

i
nf
l
ue
nc
e

of

l
oc
a
l

s
t
r
uc
t
ur
e

i
s

t
o
a
l
l
ow

l
oc
a
l

c
ha
nge
s

i
n
pol
ym
e
r

l
e
ng
t
h.


I
n
or
de
r

t
o
e
na
bl
e

m
onom
e
r
s

t
o
be

m
or
e

f
l
e
xi
bl
e

i
n
t
he
i
r

m
o
t
i
on
w
e

ha
ve

ge
ne
r
a
l
i
z
e
d
t
he

C
A

dyna
m
i
c
s

j
us
t

de
s
c
r
i
be
d.

I
n
a

ge
ne
r
a
l
i
z
e
d
f
or
m
ul
a
t
i
on
of

pol
ym
e
r

dyna
m
i
c
s
,

w
e

de
f
i
ne

a

r
e
gi
on
V

of

c
e
l
l
s

a
r
ound

e
a
c
h
m
onom
e
r

a
s

i
t
s

"
bondi
ng
ne
i
ghbor
hood"
.


A

pol
ym
e
r

i
s

c
ons
t
r
uc
t
e
d
w
he
r
e

e
a
c
h
m
onom
e
r

ha
s

i
t
s

N
N
C
s

i
n
t
he

r
e
gi
on
V

a
bout

i
t
.


T
he

dyna
m
i
c
s

i
s

de
f
i
ne
d
s
i
m
pl
y
by
r
e
qui
r
i
ng
t
ha
t

t
he

m
ot
i
on
of

a

m
onom
e
r

be

a
l
l
ow
e
d
onl
y
i
f

i
t
s

m
ov
e
m
e
nt

t
o
a

ne
w

pos
i
t
i
on
(
s
e
l
e
c
t
e
d
a
t

r
a
ndom

f
r
om

a

m
ove
m
e
nt

r
e
gi
on
M

a
r
ound
i
t
)

doe
s

not

c
ha
nge

i
t
s

N
N
C
s
.


T
hi
s

pr
e
s
e
r
ve
s

bot
h
c
onne
c
t
i
vi
t
y
(
pr
e
ve
nt
i
ng
l
os
s

or

c
ha
nge

of

a

n
e
i
ghbor
)

a
nd
e
xc
l
u
de
d
vol
um
e

(
pr
e
ve
nt
i
ng
t
he

a
ddi
t
i
on
of

a

ne
i
ghbor
)
.


I
n
t
he

or
i
gi
na
l

C
A

dyna
m
i
c
s

(
F
i
g.

2)

V

w
a
s

t
he

3x3
r
e
gi
on
a
r
ound
t
he

m
onom
e
r
.


U
s
i
ng
t
he

ge
ne
r
a
l

f
or
m
ul
a
t
i
on

V

c
a
n
b
e

l
a
r
ge
r
,

f
or

e
xa
m
pl
e

a

5x5
r
e
gi
on.


N
N
C
s

no
l
onge
r

ne
e
d
t
o
be

a
dj
a
c
e
nt

i
n
s
pa
c
e
:


t
he

bonde
d
n
e
i
ghbor
s

of

a

m
onom
e
r

a
r
e

de
f
i
ne
d
s
ol
e
l
y
b
y

be
i
ng
w
i
t
hi
n
V
.


T
he
n
m
onom
e
r
s

c
a
n
s
e
pa
r
a
t
e

by
one

s
pa
c
e

a
nd
r
e
m
a
i
n
N
N
C
s
,

a
s

i
n
t
he


F
i
gur
e

1:


I
l
l
us
t
r
a
t
i
on

of

a
n
a
bs
t
r
a
c
t

pol
ym
e
r

c
om
pos
e
d
of

m
onom
e
r
s

w
hi
c
h

a
r
e

c
onne
c
t
e
d
t
o
ne
i
ghbor
s

a
nd
do
not

ove
r
l
a
p
"
e
xc
l
ude
d"

vol
um
e
s
.


T
he

m
ot
i
on
of

m
onom
e
r
s

i
s

r
e
s
t
r
i
c
t
e
d
s
o
t
ha
t

t
he
y
do

not

de
t
a
c
h
f
r
om

ne
i
ghbo
r
s

or

ove
r
l
a
p
w
i
t
h

a
ny
ot
he
r

m
onom
e
r
.


M
ovi
ng
a
ny
t
w
o
m
onom
e
r
s

a
t

t
he

s
a
m
e

t
i
m
e

c
a
n
l
e
a
d
t
o
i
na
dve
r
t
e
nt

ove
r
l
a
ppi
ng.


H
ow
e
ve
r
,

i
f

onl
y
one

m
onom
e
r

i
s

s
e
l
e
c
t
e
d
f
r
om

e
a
c
h

s
ha
de
d
r
e
gi
on
t
he

m
onom
e
r
s

s
e
l
e
c
t
e
d
c
a
n

be

m
ove
d
a
t

t
he

s
a
m
e

t
i
m
e

w
i
t
hout

c
ha
nc
e

of

a
c
c
i
de
nt
a
l

ove
r
l
a
p.


T
he

s
ha
de
d
r
e
gi
ons

c
a
n
t
he
n
be

s
hi
f
t
e
d
s
o
a
l
l

m
onom
e
r
s

c
a
n
be

m
ove
d.
x
x

F
i
gur
e

2:


I
l
l
us
t
r
a
t
i
on
of

a

c
e
l
l
ul
a
r

pol
ym
e
r

m
ode
l

w
hi
c
h
c
a
n
be

m
ove
d
us
i
ng
M
a
r
gol
us

C
e
l
l
ul
a
r

A
ut
om
a
t
on

dyna
m
i
c
s
.


M
onom
e
r
s

a
r
e

c
ons
i
de
r
e
d

a
t
t
a
c
he
d
i
f

t
he
y
a
r
e

t
ouc
hi
ng
by
e
i
t
he
r

f
a
c
e
s

or

c
or
ne
r
s
.

M
ove
s

a
r
e

onl
y

or
i
gi
na
t
i
ng
i
n
t
he

s
qua
r
e
s

m
a
r
ke
d
by

c
e
nt
r
a
l

c
i
r
c
l
e
s

i
n
t
he

m
i
ddl
e

of

3x3

ne
i
ghbor
hoods

s
e
pa
r
a
t
e
d
by
buf
f
e
r
s

a
nd

pos
s
i
bl
e

m
ove
s

a
r
e

gi
ve
n
by
t
he

a
r
r
ow
s

.


M
ove
s

w
hi
c
h

a
r
e

not

a
l
l
ow
e
d
du
e

t
o

e
xc
l
ude
d
vol
um
e

a
r
e

m
a
r
ke
d
by
a
n

"
x"

i
n
t
he

t
a
r
ge
t

s
qua
r
e
.




f
l
uc
t
ua
t
i
ng
bond
m
e
t
hod.
1
2
,
1
3


T
hi
s

e
na
bl
e
s

l
oc
a
l

l
e
n
gt
h
c
ha
nge
s
,

i
m
pr
ovi
ng
t
he

s
i
m
ul
a
t
i
on
of

hi
gh
m
ol
e
c
ul
a
r

w
e
i
ght

pol
ym
e
r
s
.


F
or

pa
r
a
l
l
e
l

pr
oc
e
s
s
i
ng,

m
onom
e
r
s

c
a
n
be

m
ove
d

i
nde
pe
nde
nt
l
y
a
t

a

di
s
t
a
nc
e

w
he
r
e

t
w
o
m
onom
e
r
s

c
a
n
be

m
ove
d
t
ow
a
r
ds

e
a
c
h
ot
he
r

w
i
t
hout

one

e
nt
e
r
i
ng
t
he

ot
he
r
's

bond
i
ng
ne
i
ghbor
hood

V
.


T
W
O
-
S
P
A
C
E

A
L
G
O
R
I
T
H
M


T
he

c
ons
i
de
r
a
t
i
on
of

m
or
e

f
l
e
xi
bl
e

pol
ym
e
r

dyna
m
i
c
s

l
e
a
ds

u
s

t
o
a

s
e
c
ond
ge
ne
r
a
l

c
l
a
s
s

of

dyna
m
i
c
s

w
hi
c
h

e
na
bl
e
s

a

di
f
f
e
r
e
nt

pa
r
t
i
t
i
oni
ng
of

t
he

pol
ym
e
r

f
or

s
i
m
ul
a
t
i
ons
.


T
he

s
i
m
pl
e
s
t

w
a
y
t
o
de
s
c
r
i
be

t
hi
s

dyna
m
i
c
s

i
n
t
w
o
di
m
e
n
s
i
o
ns

i
s

t
o
c
on
s
i
de
r

a

pol
ym
e
r

on

t
w
o
pa
r
a
l
l
e
l

pl
a
ne
s

(
s
e
e

F
i
g.

3)
.


T
he

m
on
om
e
r
s

of

t
he

pol
ym
e
r

a
l
t
e
r
na
t
e

be
t
w
e
e
n
t
he

pl
a
ne
s

s
o
t
ha
t

odd
-
num
be
r
e
d
m
onom
e
r
s

a
r
e

on

one

pl
a
ne

a
nd
e
ve
n
-
num
be
r
e
d
m
onom
e
r
s

a
r
e

on

t
he

ot
h
e
r
.


T
he

pol
ym
e
r

a
nd
i
t
s

dyna
m
i
c
s

a
r
e

de
s
c
r
i
be
d
a
s

i
n

t
he

pr
e
c
e
di
ng
pa
r
a
gr
a
ph
e
xc
e
pt

t
ha
t

t
he

N
N
C
s

of

a

m
onom
e
r

a
r
e

l
oc
a
t
e
d
i
n
t
he

ot
he
r

pl
a
ne

a
nd
e
xc
l
ude
d
vol
um
e

w
i
t
h
r
e
s
pe
c
t

t
o
non
-
N
N
C

m
onom
e
r
s

i
s

onl
y
e
nf
or
c
e
d
w
i
t
h
r
e
s
pe
c
t

t
o
m
ono
m
e
r
s

i
n
t
he

ot
he
r

pl
a
ne
.


S
i
m
pl
y
s
t
a
t
e
d,

us
i
ng

t
he

l
a
ngua
ge

of

t
he

pr
e
c
e
e
di
ng
p
a
r
a
gr
a
ph,

t
he

bondi
ng
ne
i
ghbor
hood
of

a

m
onom
e
r

i
n
one

pl
a
ne

i
s

l
oc
a
t
e
d
e
nt
i
r
e
l
y
i
n
t
he

ot
he
r

pl
a
ne
.


U
s
i
ng
t
hi
s

dyna
m
i
c
s

a
n
a
ddi
t
i
ona
l

f
l
e
xi
bi
l
i
t
y
i
s

a
c
hi
e
ve
d
be
c
a
u
s
e

ne
i
ghbor
i
ng
m
onom
e
r
s

c
a
n
be

"
on
t
op
of

e
a
c
h
ot
he
r
"

s
o
t
ha
t

e
ve
n
a

3x3

ne
i
ghbor
hood
dyna
m
i
c
s

a
l
l
ow
s

l
oc
a
l

e
xpa
ns
i
on

a
nd
c
ont
r
a
c
t
i
on.


M
or
e

i
nt
e
r
e
s
t
i
ngl
y,

i
t

i
s

pos
s
i
bl
e

i
n

t
hi
s

dyna
m
i
c
s

t
o

m
ove

a
l
l

of

t
he

m
on
om
e
r
s

i
n
one

pl
a
ne

a
t

t
he

s
a
m
e

t
i
m
e

w
i
t
hout

c
onc
e
r
n
f
or

t
he
i
r

i
nt
e
r
f
e
r
e
nc
e

be
c
a
us
e

bot
h
c
onne
c
t
i
vi
t
y
a
nd
e
xc
l
ude
d
vol
um
e

a
r
e

i
m
pl
e
m
e
nt
e
d

t
hr
ough
i
nt
e
r
a
c
t
i
ons

w
i
t
h

t
he

ot
he
r

pl
a
ne
.


T
hi
s

i
s

a
n
a
l
t
e
r
na
t
i
ve

t
o

t
he

c
onc
e
pt

o
f

pa
r
t
i
t
i
oni
ng
s
pa
c
e
,

t
hr
ough
a

s
pe
c
i
a
l

pa
r
t
i
t
i
oni
ng
of

t
he

pol
ym
e
r
.


T
he

a
bi
l
i
t
y
t
o
upda
t
e

1/
2
of

t
he

m
onom
e
r
s

s
i
m
ul
t
a
ne
ous
l
y
i
m
pl
i
e
s

t
ha
t

t
hi
s

dyna
m
i
c
s

i
s

e
f
f
i
c
i
e
nt

f
or

i
m
pl
e
m
e
nt
a
t
i
on
on

bot
h
C
A

m
a
c
hi
ne
s

a
nd
ot
he
r

pa
r
a
l
l
e
l

pr
oc
e
s
s
i
ng
c
om
put
e
r
s

s
uc
h
a
s

H
ype
r
c
ube
s

a
nd
C
onne
c
t
i
on
M
a
c
hi
ne
s
.


I
t

m
a
y

e
ve
n
be

pos
s
i
bl
e

t
o
t
a
ke

f
ul
l

a
dva
nt
a
ge

of

c
onve
nt
i
ona
l

ve
c
t
or

pr
oc
e
s
s
or
s
.

T
o
s
how

t
ha
t

a
l
l

t
he

m
onom
e
r
s

i
n
one

s
pa
c
e

c
a
n
be

m
ove
d
i
nde
pe
nde
nt
l
y,

w
e

m
us
t

s
ho
w

t
ha
t

t
he
i
r

m
ot
i
on
c
a
nnot

r
e
s
ul
t

i
n

e
i
t
he
r

br
e
a
ki
ng

t
he

pol
ym
e
r

or

vi
ol
a
t
i
ng
e
xc
l
ude
d
vol
um
e
.


S
i
nc
e

e
a
c
h
m
onom
e
r

i
s

m
ove
d

pr
e
s
e
r
vi
ng
i
t
s

N
N
C
s
,

t
he

pol
ym
e
r

c
a
n
not

be

br
oke
n.


E
xc
l
ude
d

vol
um
e

i
s

di
f
f
e
r
e
nt

f
or

t
w
o
m
onom
e
r
s

w
i
t
hi
n
a

pl
a
ne

a
nd
f
or

t
w
o
m
onom
e
r
s

i
n
oppos
i
t
e

pl
a
ne
s
.


F
or

t
w
o
m
onom
e
r
s

i
n

oppos
i
t
e

pl
a
ne
s
,

t
he

e
xc
l
ude
d
vol
um
e

i
s

t
he

ne
i
ghbor
hood
vol
um
e

a
nd

pr
e
ve
nt
i
ng
a

c
ha
nge

i
n
num
be
r

of

N
N
C
s

a
l
s
o
pr
e
ve
nt
s

vi
ol
a
t
i
on
of

e
xc
l
ude
d
vol
um
e
.


F
or

t
w
o

m
onom
e
r
s

i
n
t
he

s
a
m
e

pl
a
ne

e
xc
l
ude
d
vol
um
e

i
s

j
us
t

t
he

r
e
qui
r
m
e
nt

t
ha
t

t
w
o
m
onom
e
r
s

do
no
t

m
ove

ont
o
t
he

s
a
m
e

s
i
t
e

(
t
he
y
c
a
n
be

a
dj
a
c
e
nt

s
i
nc
e

t
he
y
a
r
e

ne
ve
r

N
N
C
s
)
.


I
n
a

pr
oof

by

c
ont
r
a
di
c
t
i
on
t
ha
t

t
w
o
m
onom
e
r
s

c
a
nnot

m
ove

ont
o
t
he

s
a
m
e

s
i
t
e
,

a
s
s
um
e

t
w
o
m
onom
e
r
s

w
e
r
e


t
o
m
ove

t
o
t
he

s
a
m
e

s
i
t
e
.


I
n
t
hi
s

s
t
a
t
e

t
he
y
w
i
l
l

h
a
ve

t
he

s
a
m
e

N
N
C
s

a
nd,

s
i
nc
e

our

a
l
gor
i
t
hm

e
xpl
i
c
i
t
l
y
pr
e
ve
nt
s

a
ny
t
w
o
m
onom
e
r
s

f
r
om

ha
vi
ng
t
he

s
a
m
e

N
N
C
s
,

t
he

a
bove

s
i
t
ua
t
i
on
c
a
n
no
t

a
r
i
s
e
.


F
i
g.


3:


S
c
he
m
a
t
i
c

i
l
l
us
t
r
a
t
i
on
of

a

t
w
o
-
pl
a
ne

pol
ym
e
r

us
e
d
i
n
t
w
o
-
pl
a
ne

pol
ym
e
r

dyna
m
i
c
s

de
s
c
r
i
be
d
i
n
t
he

t
e
xt
.



M
onom
e
r
s

on
t
he

upp
e
r

pl
a
ne

a
r
e

s
how
n
a
s

f
i
l
l
e
d
c
i
r
c
l
e
s
,

m
onom
e
r
s

on

t
he

l
ow
e
r

p
l
a
ne

a
r
e

s
how
n
a
s

ope
n
c
i
r
c
l
e
s
.


M
onom
e
r
s

a
r
e

a
t
t
a
c
he
d
on
l
y
t
o
m
onom
e
r
s

i
n
t
he

ot
he
r

pl
a
ne
.


T
he

a
t
t
a
c
he
m
e
nt

ne
i
gbor
hood
of

e
a
c
h
m
onom
e
r

i
s

a

3x3

r
e
gi
on
of

c
e
l
l
s

l
oc
a
t
e
d
i
n
t
he

oppos
i
t
e

pl
a
ne
.


A

l
i
ght
l
y
s
ha
de
d
r
e
gi
on
i
ndi
c
a
t
e
s

t
he

a
t
t
a
c
he
m
e
nt

ne
i
ghbor
hood
of

t
he

bl
a
c
k
m
onom
e
r

m
a
r
ke
d
w
i
t
h
a

w
hi
t
e

dot
.


I
t
s

t
w
o
ne
i
ghbo
r
s

a
r
e

l
oc
a
t
e
d
i
n
t
he

a
t
t
a
c
he
m
e
nt

ne
i
gbor
hood.


A
t
t
a
c
hm
e
nt
s

a
r
e

i
ndi
c
a
t
e
d
by
l
i
ne

s
e
gm
e
nt
s

be
t
w
e
e
n

m
onom
e
r
s
.


T
he

t
w
o
-
s
pa
c
e

a
l
gor
i
t
h
m

m
a
y
be

us
e
d
i
n
t
hr
e
e

di
m
e
ns
i
ons

by
c
on
s
i
de
r
i
ng
t
he

pol
ym
e
r

t
o
be

i
n
a

doubl
e

t
hr
e
e
-
di
m
e
ns
i
ona
l

s
pa
c
e
.


T
he

doubl
e
-
s
pa
c
e

pol
ym
e
r

dyna
m
i
c
s

c
a
n

be

m
a
pp
e
d
ont
o

a

s
i
ngl
e
-
s
pa
c
e

dyna
m
i
c
s

w
i
t
h
a
n
unus
ua
l

i
m
pl
e
m
e
nt
a
t
i
on
of

e
xc
l
ude
d
vol
um
e
.


T
hi
s

di
f
f
e
r
e
nc
e

of

t
he

l
oc
a
l

i
nt
e
r
a
c
t
i
on
doe
s

not

a
f
f
e
c
t

t
he

a
s
ym
pt
o
t
i
c

s
t
r
uc
t
ur
a
l

a
nd
dyna
m
i
c
a
l

be
ha
vi
or
,

w
hi
c
h
t
he
r
e
f
or
e

i
s

i
n

t
he

s
a
m
e

uni
ve
r
s
a
l
i
t
y
c
l
a
s
s

a
s

ot
he
r

a
bs
t
r
a
c
t

pol
ym
e
r

m
ode
l
s
.


T
hus

t
he

s
e
c
ond

ge
ne
r
a
l

a
l
gor
i
t
hm

w
e

ha
ve

de
s
c
r
i
be
d
i
s

a

hi
ghl
y
e
f
f
i
c
i
e
nt

t
e
c
hni
que

f
o
r

s
i
m
ul
a
t
i
ons

of

a
bs
t
r
a
c
t

pol
ym
e
r

pr
obl
e
m
s
.


H
ow
e
ve
r
,

t
he

pol
y
m
e
r

pa
r
t
i
t
i
oni
ng
us
e
d
i
s

not

obvi
ous
l
y
a
ppl
i
c
a
bl
e

t
o
t
he

dyna
m
i
c
s

of

r
e
a
l
i
s
t
i
c

pol
ym
e
r

m
ode
l
s
,

w
hi
c
h
m
a
y
s
t
i
l
l

be

pa
r
a
l
l
e
l
i
z
e
d
by
t
he

us
e

of

t
he

pr
e
vi
ous
l
y
de
s
c
r
i
be
d
s
pa
c
e

pa
r
t
i
t
i
oni
ng.



R
E
S
U
L
T
S


W
e

ha
ve

pe
r
f
or
m
e
d
t
e
s
t

s
i
m
ul
a
t
i
ons

of

t
he

t
w
o
-
s
p
a
c
e

a
l
gor
i
t
hm

on
t
he

pr
e
s
e
nt

s
t
a
t
e
-
of
-
t
he
-
a
r
t

C
e
l
l
ul
a
r

A
ut
om
a
t
a

M
a
c
hi
ne
,

C
A
M

6,

a
nd
on
S
P
A
R
C

s
t
a
t
i
ons
.


T
he

r
e
s
ul
t
s

of

bot
h

s
i
m
ul
a
t
i
ons

a
r
e

pr
e
s
e
nt
e
d
i
n
F
i
g.

4
w
he
r
e

w
e

pl
ot

t
he

r
a
di
us

of

gyr
a
t
i
on

of

a

po
l
ym
e
r

a
s

a

f
unc
t
i
on
of

i
t
s

c
ont
our

l
e
ngt
h
a
nd
c
om
pa
r
e

w
i
t
h
t
he

e
xa
c
t

r
e
s
ul
t

i
n
t
w
o

di
m
e
ns
i
ons
,

R
~
L




.


W
e

a
l
s
o
us
e
d
t
he

a
l
gor
i
t
hm

t
o
c
a
l
c
ul
a
t
e

t
he

l
onge
s
t

r
e
l
a
xa
t
i
on
t
i
m
e

o
f

t
he

c
ha
i
n
.

F
or

s
i
m
pl
i
c
i
t
y
of

a
na
l
ys
i
s

onl
y
t
he

S
P
A
R
C

d
a
t
a

i
s

pr
e
s
e
nt
e
d
i
n

F
i
g.

5,


a
nd
c
om
pa
r
e
d
w
i
t
h
t
he

R
ou
s
e

m
ode
l

pr
e
di
c
t
i
on,

!

~

L
2
.
5

.



F
i
na
l
l
y,

w
e

w
oul
d
l
i
ke

t
o
c
om
m
e
nt

on
t
h
e

e
f
f
i
c
i
e
n
c
y
of

t
he

pr
opos
e
d
i
m
pl
e
m
e
nt
a
t
i
on

of

C
A

a
l
gor
i
t
hm
s

on
pa
r
a
l
l
e
l

a
nd
s
e
r
i
a
l

m
a
c
hi
ne
s
.



A
l
l

o
f

t
he

a
l
gor
i
t
hm
s


di
s
c
us
s
e
d

a
bove

s
c
a
l
e

w
i
t
h

t
he

num
be
r

of

m
ove
s

t
o
r
e
l
a
x
a

po
l
ym
e
r
,

M
!

~

L

!

~

"
L
2
#
+
2
,


w
he
r
e

2
#
+
2

e
qua
l
s

3.
5
i
n
t
w
o
di
m
e
ns
i
ons
,

a
nd
3.
2
i
n
t
hr
e
e

di
m
e
ns
i
ons
;

!

i
s

t
he

r
e
l
a
xa
t
i
on
t
i
m
e
,

L

i
s

t
he

l
e
ngt
h
o
f

t
he

pol
ym
e
r
,

a
nd
"

de
pe
nds

on
pol
ym
e
r

l
oc
a
l

s
t
r
uc
t
ur
e
.


I
n
a

c
onve
nt
i
ona
l

a
l
gor
i
t
hm

t
he

c
a
l
c
ul
a
t
i
on
t
i
m
e

s
c
a
l
e
s

a
s

M
!
(
t
m

+

L
t
x
)
;

t
m

i
s

t
he

t
i
m
e

t
o
e
xe
c
ut
e

a

m
ove

a
nd
t
x

i
s

t
he

e
xc
l
ude
d
vol
u
m
e

t
e
s
t

t
i
m
e
.


I
n
s
pa
c
e

pa
r
t
i
t
i
oni
ng
t
hi
s

be
c
om
e
s

M
!
(
t
m
,
x
,
s

+

t
c
V
/
V
p
)
/
P
;


P

i
s

t
he

num
be
r

o
f

pr
oc
e
s
s
or
s
,

t
m
,
x
,
s

i
s

t
he

t
i
m
e

t
o
pe
r
f
or
m

a

m
ove

a
nd
e
xc
l
ude
d
vo
l
u
m
e

t
e
s
t

a
nd
m
a
i
nt
a
i
n
t
he

s
pa
c
e

pa
r
t
i
t
i
oni
ng
,

t
c

i
s

t
he

t
i
m
e

t
o
c
he
c
k
w
he
t
he
r

a

pa
r
t
i
t
i
on
i
s

oc
c
upi
e
d,

a
nd
V
/
V
p

i
s

t
he

r
a
t
i
o
of

s
i
m
ul
a
t
i
on
vol
um
e

t
o
f
i
l
l
e
d
vol
um
e

(
t
he

s
e
c
ond
t
e
r
m

m
a
y
of
t
e
n
be

e
l
i
m
i
na
t
e
d
by
c
oa
r
s
e

gr
a
i
ni
ng)
.


I
n
t
he

C
A

t
w
o
-
s
pa
c
e

pol
ym
e
r

m
ode
l

on
a

C
A
M

t
he

t
i
m
e

s
c
a
l
e
s

a
s

M
!

t
m
/
(
P
V
p
/
V
)

a
nd
on

c
onne
c
t
i
on
m
a
c
hi
ne

or

hype
r
c
ube

a
s

M
!

t
m
/
P
.


I
n
a

C
A
M
,

t
m

i
s

of

o
r
de
r

1
(
one

t
i
m
e

s
t
e
p
pe
r

m
ove
)
,

w
hi
l
e

f
or

ot
he
r

pa
r
a
l
l
e
l

m
a
c
hi
ne
s

t
m

i
s

s
ubs
t
a
nt
i
a
l
l
y
l
a
r
ge
r
,

t
h
ough
e
m
pt
y
s
pa
c
e

ne
e
d
not

be

pr
oc
e
s
s
e
d.


T
he
r
e
f
or
e
,

w
he
n
t
he

ne
w

ge
ne
r
a
t
i
on
m
odul
a
r

3
-
di
m
e
ns
i
ona
l

C
A
M
-
8
be
c
om
e
s

ope
r
a
t
i
ona
l

w
i
t
h

4
m
i
l
l
i
on
c
e
l
l
s

a
nd
25
m
i
l
l
i
on
c
e
l
l

upda
t
e
s

pe
r

s
e
c
ond
pe
r

m
odul
e

i
t

w
i
l
l

be

i
de
a
l

f
or

s
i
m
ul
a
t
i
on
of

hi
gh
de
ns
i
t
y

pol
ym
e
r

m
e
l
t
s

w
he
r
e

c
onve
nt
i
ona
l

c
om
put
e
r
s

a
r
e

a
ppr
oa
c
hi
ng
t
he
i
r

l
i
m
i
t
.
1
4

0.1
1
10
10
1
10
100
1000
!
L
2
10
3
10
4
10
5

F
i
gur
e

5:

S
i
m
ul
a
t
i
ons

of

t
he

r
e
l
a
xa
t
i
on
t
i
m
e

!

of

a

pol
ym
e
r

a
s

a

f
unc
t
i
on
of

pol
ym
e
r

l
e
ngt
h

L
,

t
e
s
t
i
ng
t
he

t
w
o
-
s
pa
c
e

a
l
gor
i
t
hm
.


S
i
m
u
l
a
t
i
ons

w
e
r
e

pe
r
f
or
m
e
d
on

a

S
P
A
R
C

s
t
a
t
i
on.


T
he

a
s
ym
pt
ot
i
c

f
i
t

i
ndi
c
a
t
e
d
by
t
he

da
s
he
d
l
i
ne

w
a
s

obt
a
i
ne
d
a
s

!
~
0.
12
L
2
.
5
5

c
ons
i
s
t
e
nt

w
i
t
h
t
h
e

R
ous
e

m
ode
l

e
xpone
nt

of

2.
5
.


T
he

s
m
a
l
l

p
r
e
f
a
c
t
or

i
ndi
c
a
t
e
s

t
he

e
f
f
i
c
i
e
nc
y
o
f

t
he

t
w
o
-
s
pa
c
e

0.1
1
10
100
1
10
100
1000
R
g
L

F
i
gur
e

4:


S
i
m
ul
a
t
i
ons

of

t
he

r
a
di
us

of

gyr
a
t
i
on
o
f

a

pol
ym
e
r

R
g
,

a
s

a

f
unc
t
i
on
of

t
he

pol
ym
e
r

l
e
ngt
h
L

(
one

l
e
s
s

t
ha
n
t
he

num
be
r

of

m
onom
e
r
s
)
,

t
e
s
t
i
ng
t
he

t
w
o
-
s
pa
c
e

a
l
gor
i
t
hm

on
a

w
or
ks
t
a
t
i
on
(
o
)

a
nd
on
C
A
M
-
6
(
*
)

t
he

c
ur
r
e
nt

ge
ne
r
a
t
i
on
C
e
l
l
ul
a
r

A
ut
om
a
t
a

M
a
c
hi
ne

(
s
oon
t
o
be

r
e
pl
a
c
e
d
by
C
A
M
-
8)
.


A
n
a
s
ym
pt
ot
i
c

f
i
t

i
ndi
c
a
t
e
d
by
t
he

da
s
he
d
l
i
ne

i
s

R
g
~
0.
8

L
0
.
7
5
1

c
o
ns
i
s
t
e
nt

w
i
t
h
t
he

e
xa
c
t

e
xpone
nt

0.
75.


A
gr
e
e
m
e
nt

w
i
t
h
t
he

a
s
ym
pt
ot
i
c

va
l
ue
s

a
r
e

r
e
a
c
he
d
f
or

r
e
m
a
r
ka
bl
y
s
m
a
l
l

pol
ym
e
r
s

of

l
e
ngt
h

L
=
2
.

a
l
gor
i
t
hm
.


P
ol
y
m
e
r
s

of

l
e
ngt
h
30
-
50

a
l
r
e
a
dy
a
pp
r
oa
c
h
t
he

a
s
ym
pt
ot
i
c

be
ha
vi
or
.


T
hi
s

r
e
s
e
a
r
c
h
i
s

s
uppor
t
e
d
i
n

pa
r
t

b
y
t
he

U
S
-
I
s
r
a
e
l

B
i
na
t
i
ona
l

S
c
i
e
nc
e

F
ounda
t
i
on.



R
E
F
E
R
E
N
C
E
S
:


1



P
.

G
.

de
G
e
nne
s

"
S
c
a
l
i
ng
C
onc
e
pt
s

i
n
P
ol
ym
e
r

P
hys
i
c
s
"

(
C
or
ne
l
l

U
ni
v.

P
r
e
s
s
,

I
t
ha
c
a

1979)

2



M
.

D
oi

a
nd
S
.

F
.

E
dw
a
r
ds

"
T
he
or
y
of

P
ol
ym
e
r

D
yna
m
i
c
s
"

(
O
xf
or
d
S
c
i
e
nc
e

P
ubl
i
c
a
t
i
ons
,

O
xf
or
d
1986)

3


J
.

A
.

M
c
C
a
m
m
on
a
nd
S
.

C
.

H
a
r
ve
y
"
D
yna
m
i
c
s

of

P
r
ot
e
i
ns

a
nd
N
uc
l
e
i
c

A
c
i
ds
"

(
C
a
m
br
i
dge

U
ni
v.

P
r
e
s
s
,

C
a
m
br
i
dge

1987)

4

"
M
ont
e
-
C
a
r
l
o
M
e
t
hods

i
n
S
t
a
t
i
s
t
i
c
a
l

P
hys
i
c
s
"
,

2
nd
e
d.
,

K
.

B
i
nde
r

e
d.

(
S
pr
i
nge
r
-
V
e
r
l
a
g,

B
e
r
l
i
n
1986)
,
"
A
ppl
i
c
a
t
i
ons

of

M
ont
e
-
C
a
r
l
o
M
e
t
ho
ds

i
n
S
t
a
t
i
s
t
i
c
a
l

P
hys
i
c
s
"
,

2nd
e
d.
,

K
.

B
i
nde
r

e
d.

(
S
p
r
i
nge
r
-
V
e
r
l
a
g
,

B
e
r
l
i
n

1987)

5



B
.

M
.

B
oghos
i
a
n,

C
om
put
e
r
s

i
n

P
hys
i
c
s

4,

14

(
1
990)

6

"
T
he
or
y
a
nd
A
ppl
i
c
a
t
i
ons

o
f

C
e
l
l
ul
a
r

A
ut
om
a
t
a
"
,

S
.

W
ol
f
r
a
m

E
d.
,
(
W
or
l
d
S
c
i
e
nt
i
f
i
c
,

S
i
nga
por
e
,

1986
)

7


M
.

G
a
r
dne
r
,

M
a
t
he
m
a
t
i
c
a
l

G
a
m
e
s
,

S
c
i
.

A
m
e
r
.

(
F
e
b.

M
a
r
.
,

A
pr
.

1971;

J
a
n.

1972
)

8


"
C
e
l
l
ul
a
r

A
ut
om
a
t
a
"
,

D
.

F
a
r
m
e
r
,

T
.

T
o
f
f
ol
i

a
nd
S
.

W
ol
f
r
a
m

(
e
ds
)
,

(
N
o
r
t
h
-
H
ol
l
a
nd,

1984
)
,

or

T
.

T
o
f
f
ol
i
,

C
e
l
l
ul
a
r

A
ut
om
a
t
on
m
e
c
ha
ni
c
s
"
,

P
hD

T
he
s
i
s
,

L
ogi
c

of

C
om
put
e
r
s

G
r
oup,

U
.

of

M
i
c
hi
ga
n
(
1977)

9


T
.

T
of
f
ol
i

a
nd
N
.

M
a
r
gol
us
,

"
C
e
l
l
ul
a
r

A
ut
om
a
t
a

M
a
c
hi
ne
s
:

A

N
e
w

E
nvi
r
onm
e
nt

f
or

M
ode
l
i
ng"
,

(
M
I
T

P
r
e
s
s
,

1987)

1
0


B
.

C
hopa
r
d,

J
.

of

P
hys
.

A


23,

1671
(
1990
)

1
1


J
.

M
.

V
.

A
.

K
oe
l
m
a
n,

P
hys
.

R
e
v.

L
e
t
t
.

64
,

1915

(
1990)

1
2


I
.

C
a
r
m
e
s
i
n
a
nd

K
.

K
r
e
m
e
r
,

M
a
c
r
om
ol
e
c
ul
e
s

21,

2819
(
1988)

1
3


H
.

P
.

W
i
t
t
m
a
nn

a
nd
K
.

K
r
e
m
e
r
,

C
om
put
e
r

P
hys
i
c
s

C
om
m
uni
c
a
t
i
ons

61,

309
(
1990)

1
4


K
.

K
r
e
m
e
r

a
nd
G
.

S
.

G
r
e
s
t
,

J
.

C
he
m
.

P
hys
.

92,

50
57
(
1990)