PHYSICAL REVIEW BVOLUME 32, NUMBER 91NOVEMBER 1985
Iso he mal molecula -dynamics calcula ions
Luis F. Rull
Depa amen o F sica Teo ica, Facul ad de Fi'sica, Uni e sidad de Se illa,
41012Se illa, Spain
Juan J. Mo ales and F ancisco Cuad os
Seccion de FI'sicas, Facul ad de Ciencias, Uni e sidad de Ex emadu a,
06071 Badajoz, Spain
(Recei ed 30 Janua y 1985; e ised manusc ip ecei ed 28 May 1985)
%e ha e pe o med long- ime uns o molecula -dynamics compu e simula ions o a wo-dimensional
Lenna d-Jones sys em, wi hou any scaling p ocedu e. The he modynamic p ope ies show spon aneous
luc ua ions excep when he sys em is a om he mel ing zone.
In a ecen pape , 'Tox ae d s udied c i ically aspecial
compu a ional echnique: "iso he mal-isoba ic" molecula
dynamics (T-p MD) (used by Ab aham and Koch' o in es-
iga e mel ing in a wo-dimensional Lenna d-Jones sys em).
By using he calcula ion o he iso he mal comp essibili y
om he luc ua ions o olume, Tox ae d es ablished ha
he scaling p ocedu e was inco ec and canno be used o
ob ain de ini i e s a emen s abou he na u e o he phase
ansi ion. Tox ae d no ed ha he T-p MD calcula ions,
inspi ed by he N-p-T-ensemble Mon e Ca lo me hod, a e
ob ained om Laplace ans o ms o he canonical pa i ion
unc ion, and his ans o ma ion has o be pe o med o
cons an ex ensi e s a e a iables, excluding he olume.
In an analysis o he ensembles used in molecula -
dynamics simula ions, Lado ound ha , in he classical lim-
i , o in e ac ing sys ems wi h ha d-co e in e ac ions,
Mon e Ca lo and molecula -dynamics calcula ions should
yield he same equa ion o s a e wi hou co ec ions, i.e.,
BlnQ~ I
(PP )N T+--
V()V (IS/ )N- -Ei
whe e he con igu a ional in eg al Q~ is de ined by
Q~ =Vd " 8'( "),
whe e
~( ~) 0, U( )=oo,
1, U( )=0,
and, as usual, P=(ksT) ', wi h he only assump ions be-
ing he basic pos ula es o equilib ium s a is ical mechanics.
The le -hand side (N-V-T) co esponds o he Mon e
Ca lo calcula ion, ha is, wi h he numbe o pa icles,
olume, and empe a u e cons an , and he igh -hand side
(N VE) co espo-nd-s o he molecula dynamics calcula-
ion, wi h he numbe o pa icles, olume, and ene gy con-
s an .
Nose4 has ecen ly made an exhaus i e s udy o he en-
sembles in molecula dynamics o ep oduce bo h he
canonical and he iso he mal-isoba ic p obabili y densi ies in
phase space. The physical sys em o in e es consis s o N
pa icles, o which an ex e nal mac oscopic a iable and i s
LLI
1
LLI
l—
~OQ ~ l p++M+ 01~+0 oe++I ~
~~~~~~
0.7,.
20 40 60 '80
10 TIME STEPS
FIG. 1. Reduced empe a u e s ime o a wo-dimensional liquid a p ~=1.00.
32 6050 1985 The Ame ican Physical Socie y
BRIEF REPORTS 6051
1,3'
eee4ee ~eeeeeeeee ~e eeeee4eee eeee eee ee eeeeeeeeeeeeeee eel $
eeeee ee.ee ~ee
ee eeee ~
T1
0.7
20 60 80
10 TINE STEPS
FIG. 2. Same as Fig. 1, bu o a wo-dimensional solid e y close o he mel ing zone. The densi y is p m=1. 14.
conjuga e momen um a e added, pe mi ing he ene gy o
he sys em o luc ua e in away simila o Haile and
Gup a's esul s wi h a he mal ba h. '
In his B ie Repo -, we p esen some esul s o wo long
uns in wo dimensions close o and a om he mel ing
zone, which we e pe o med in o de o in es iga e he luc-
ua ions o he he modynamic a iables du ing he ime o .
e olu ion o he sys em. Ou simula ions ha e been ca ied
ou wi hou any scaling p ocedu es ( o p essu e and em-
pe a u e). Jus as i is inco ec o pe o m asimula ion us-
ing he cons an -p essu e me hod in he mel ing zone, i is
also inco ec o use he cons an - empe a u e me hod, be-
cause he e a e p oblems ela ed o luc ua ions in space and
ime independen o he na u e o he phase ansi ion ( i s
o second o de ).
In ou calcula ions we ha e chosen asys em o %=256
pa icles in e ac ing wi h a unca ed Lenna d-Jones po en-
ial a , =2.5 ~ ( is he posi ion o he po en ial
minimum) .
The ini ial posi ions o he pa icles o med a wo-
dimensional iangula la ice o densi ies p 2=1.00 and
p ~=1.14. To in eg a e he equa ions o mo ion we used
he algo i hm p oposed by Tox ae d. This algo i hm is
e y accu a e, bu in oduces a d i in he ene gy because
he equa ions a e no e e sible in ime; his ene gy-d i
p oblem can be sol ed by escaling he eloci ies o he pa -
icles h ough he hea capaci y. In ou simula ions his was
no necessa y, because he maximum d i is less han 10 4,
and consequen ly he inc emen in he empe a u e is
kshT/e —
=10 ', and, in addi ion, he usual pe iodic boun-
da y condi ions we e used.
Bo h sys ems e ol ed in exac ly he same way. A e
e e y en ime s eps (upda ing he able o he nea es
neighbo s), he eloci ies we e eno malized o gi e he e-
~e
ee ~e~eeeee ~.e~ee ee ~
ee ~~ee ~~ee~ee ee e
40
10 TINE STEPS
60 80
FIG. 3. Reduced p essu e s ime o a wo-dimensional liquid a p ~ =1.00.
60S2 BRIEF REPORTS 32
10
I
9"~ ~Oy
~y~~
~0
~y~O ~~0
~~~y
~0~~y
~4~01 ~l40 ~y$0 ~y ~g~~~O QQ ~~O+ eqi
~OO ~ee ~~QO ~I
20
1Q TI&E STEPS
FIG, 4. Same as Fig. 3, bu o a wo-dimensional solid e y close o he mel ing zone. The densi y is p m=1. 14.
duced empe a u e AT/a=1. 00 (e is he minimum o he
I.enna d-Jones po en ial). The escaling p ocedu e was pe -
o med du ing he i s 8000 ime s eps. A e ha he e o-
lu ion o he sys em was ee, wi hou any scaling p o-
cedu e, he he modynamic p ope ies being hen ob ained
as ime a e ages o each 800 ime s eps. The i ial p es-
su e was ob ained o he unca ed Lenna d-Jones po en-
ial, no o a ull Lenna d-Jones po en ial.
The s a e poin s ha we ha e chosen co espond o he
luid s a e (p =1.00) and he solid s a e (p ' =1.14), he
la e e y close o he mel ing zone loca ed he modynami-
cally by Tox ae d. 'Tox ae d calcula ed he poin whe e
luid and solid ha e equal chemical po en ials co esponding
o he empe a u e ks T/e =1.00, he di e ence in ou case
being ha we ha e pe o med long uns (40000 ime s eps
o a luid and 80000 ime s eps o asolid).
Figu es 1-4 show he esul s o ou simula ions. Each
poin in hese igu es co esponds o an a e age o e 800
ime s eps. In he liquid sys em (Figs. 1and 3) he em-
pe a u e and p essu e show he " egula " luc ua ion ypical
o he liquid sys em, and he sys em is in equilib ium once
ee e olu ion commences. Bu o he sys em close o he
mel ing zone, equilib ium is eached a e 20000 ime s eps
(Figs. 2and 4), and he empe a u e spon aneously d ops
wice (a ows), and consequen ly he p essu e ises.
Tox ae d' wonde ed i he equilib ium s a e o he sys em
could be ob ained in a ime in e al o 10000 ime s eps.
-Ou answe is no, because in he mel ing zone (o close o
i ) he sys em spon aneously goes back and o h be ween
wo di e en poin s o s a e.
I seems e iden ha a om a ansi ion in liquid o
solid sys ems, compu e simula ions using he T-p MC
me hod a e eliable because he luc ua ions a e small.
Howe e , luc ua ions play an impo an ole in he mel ing
zone; so o ob ain ade ini i e s a emen abou he na u e o
he phase ansi ion one should use MD simula ions o ee
e olu ion, wi hou any scaling p ocedu e, i.e.,wi hou em-
pe a u e o p essu e cons ain s. These cons ain s imply
ha he sys em is no isola ed, bu is in con ac wi h an en-
e gy ese oi and ha he ene gy is ans e ed by agen-
e alized o ce.'
The coope a ion o he Uni e si y o Se illa Compu e
Cen e is g a e ully acknowledged.
S.Tox ae d, Phys. Re . B29, 2821 (1984).
F. F. Ab aham and S. %.Koch, Phys. Re . B29, 2824 (1984), and
e e ences ci ed he ein.
F. Lado, J. Chem. Phys. 75, 5461 (1981).
4S. Nose, Mol, Phys. 52, 255 (1984); J. Chem. Phys. 81, 511 (1984).
5J. M. Haile and S. Gup a, J. Chem. Phys. 79, 3067 (1983).
S. Tox ae d, J. Compu . Phys. 47, 444 (1982).
7S. Tox ae d, Phys. Re . A24, 2735 (1981).