Isothermal molecular-dynamics calculations

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).