Nonlinear Hydrodynamics of a Hard Sphere Fluid Near the Glas(17)

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

IV)byperformingaportionofthecalculations(withreducedstatistics)atN=25.Ourchoiceoftheratioσ/hwhich xesthelengthscalefortheproblem,isdictatedbytwoconcerns.The rstisthatwewishtobeabletostudythedependenceofthedynamicsonwavevectorinthemainregionofinterestfromthepointofviewofthestaticstructurefactorS(q)≡S(q,t=0).Thus,wewishtochooseourunitoflengthsothatthemainpeakinS(q)fallsinthemiddlepartoftherangeofwavevectorswithinthe rstBrillouinzoneofthecomputationallattice.Secondly,toavoidcrystallizationatthehigherdensitiesstudied,wehavefoundthatweneedNandσtobeincommensurate.Selectingσ/h=4.6leavesqmaxwellawayfromthezoneedgeforalldensities,neartheQ=8shell,andisclearlyincommensuratewithN.Thedensityrangewehaveinvestigatedincludesn =0.5,0.75≤n ≤0.90at0.05intervals,andn =0.93.The rstoftheseiswellwithinthediluteliquidregion,andwasusedonlytocheckthatlimit.AsexplainedinRef.(39),itisnecessarytoincludetheparameterλinordertorepresenttheactual uctuationsthroughgaussiannoise.Itsprecisevalueisnotcrucial,sinceitessentiallyamountstoachoiceofthenormalizationofthestatic uctuationsS(q),butitclearlymustbesmallsincethedensitymustalwaysbepositive.Wehavetakenhereλ=0.001.TheremainingparametersKandηarefunctionsofn andmaybecalculatedasdiscussedinthepreviousSection.

Weturnnowtotheveryimportantquestionofdatacollection.Weareparticularlyconcernedwithensuringthatwithinstatisticalerrortheaveragescollectedbeequilibrated,thatis,stationaryinthetimescalesstudied.We ndthat,withthedatacollectionprocedureasoutlinedbelow,theinitialconditionsthatweusetobegintheintegrationoftheequationsofthemotionareunimportant(exceptinthattheydeterminetosomeextentthedurationoftransientbehavior)andweusuallytakethemtobea atdistributionofnequaltoitsaveragevalue,n0,andvanishingcurrents.Wethenmonitorthecurrent-current