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

coarsediscretelatticeweusea nermeshthande nedontheoriginallatticetoimprovetheaccuracyoftheintegration.Theproceduresemployedtointegratetheremainingsetofdi erentialequationsovertimeandgeneratethegaussiannoiseareidenticaltothoseusedinRef.(39)andreferencescitedthere.

WewillfocusouranalysisonthetimedependenceofthedynamicstructurefactorS(q,t):

S(q,t)=dxe3iq·(x x′)<δn(x,0)δn(x′,t)>(3.1)

Speci cally,wewillconsidertheangularaverageofS(q,t).Onacubiclattice,itisappropriatetode nethee ectivelengthofq,q,as:

q2=2(3 cosqx cosqy cosqz),(3.2)

andweperformangularaveragesofq-dependentquantitiesbyaveragingovervaluesofqinthe rstBrillouinzone,inasphericalshellofmeanradius(asgivenbyq)correspondingtothatofthevector(πQ/N,0,0)andthicknessπ/N.ThevalueofQrangesfrom1toapproximately31/2N,althoughonlyasmallerrangeisfreeof nitesizee ects.Wewill,forsimplicityofnotation,denotequantitiesaveragedinthiswaybysimplydroppingthevectorsymbolfromthewavevectorargument:S(q,t),andoftenwewillindicatethevaluesofqbythe‘shellnumber’Q.

Nextweturntoourchoicesforparametervalues.Thecorrelationfunctionsthatweareinterestedinarespatiallyshortranged.Itisthereforenotnecessarytouseextremelylargelatticesizes.TheresultspresentedwereobtainedusingN=15.AsinRef.(39),thisprovedtobeadequate.Wecheckedthat nitesizee ectsdonota ectthedynamicsinthewavevectorregionforwhichresultsarepresentedhere,(5<Q<15,seeSection