We conduct a numerical study of the dynamic behavior of a dense hard sphere fluid by deriving and integrating a set of Langevin equations. The statics of the system is described by a free energy functional of the Ramakrishnan-Yussouff form. We find that th

thedensityanestimatefort canbefoundbyextrapolationfromthelowerdensities,sincethebehavioroft withdensityturnsouttoobeytheVogel-Fulcherlaw22.

Workingwithintheseconstraints,obtainingstatisticallyreliableresultsstillrequiresaveragingoveralargenumberoftimebinswhichmeansverylargetotalrunningtimes.Inaddition,inordertoeliminateanypossibilityofspuriouscorrelationsduetoapeculiartransient,wehaverepeatedthewholeprocedurethreeto vetimesateachdensity.Theresultspresentedherecorrespondtoacombinedtotalofbetween1000and3900bins,dependingonthedensity,atalldensitieswepresentresultsforexceptn =0.75whereatotalof600wastaken.Theseverylargenumbers,muchlargerthanthecorrespondingnumbersinRef.(39),shouldbeconsideredascomparabletothe‘numberofruns’inastandardsimulationforanonequilibriumproblemsuchasspinodaldecompositionandleadtoverygoodqualitydata.Thecostofobtainingthedatarisesaccordingly,ofcourse:atotalof200hoursofCray2andCrayX/MPtimewererequired.

Wehavealsoveri edthatthequantityS(q,t=0)calculatedfollowingtheaboveprocedureandoverthetimerangesjustdescribedisconsistentwiththepurelystaticresult.Todothis, rstwecalculatethestaticresult:weevaluatenumerically,usingfastFouriertransforms,thediscreteFouriertransformC(q)of(2.3)forasystemofthesizeconsidered,andweobtainfromthatresultthestaticresultSs(q)∝1/(1 n C(q)).Toobtainthisequationonemustexpandthe rsttermin(2.12)tosecondorderinδn.ThecomparisonbetweenS(q)andSs(q)isshowninFig.1.Wecanseethatthetworesultsareinverygoodagreementatthedensitiesplotted.ThePYstaticvaluesareknowntobe(seee.g.Fig2inRef.(42))ingoodagreementwithMDresultsinthisdensityrange.OurresultsarethenalsoinagreementwithMDinthislimit.Athigherdensities,thecomputedresultrepresentsahigherdegreeoforderthanthatcalculatedfromthestatics.