Nonlinear Hydrodynamics of a Hard Sphere Fluid Near the Glas(18)

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

correlationsasafunctionofrunningtimet0.AfterarelativelyshorttimetKoforder10thecurrentcorrelationsreachtheirequilibriumvalueasgivenbytheequipartitiontheorem.ItmightbetemptingtoassumethatthedensitycorrelationshavealsoequilibratedbythattimeandsuchanassumptionissometimesmadeinMDwork,butwe ndthatforoursystematleastthisassumptiondoesnothold.

Tostudythedensitycorrelations,westore,forrunningtimest0≥tK,wheret0isthetimemeasuredfromtheinitiationofthecomputation,atalargenumberofperiodictimebins,theproductsoftheformδn(x,t0)δn(x′,t0+t)forallxx′.WethenmonitorthesphericallyaveragedspatialFouriertransform,S(q,t,t0)ofthequantity:

S(x,x′,t,t0)=<δn(x,t0)δn(x′,t0+t)>(3.3)

wheretheaverageisunderstoodtobeoveranumbernboftimebinsseparatedbyaninterval t.ThetimerangecoveredbytheaveragingprocessistR=nb t.InorderforS(q,t,t0)tobeanadequateapproximationtothethermodynamicaverageS(q,t)itisrequiredthatitbenotonlyindependentoft0,butalsoindependentoftRwithinstatisticalerror.Dependenceonthet0indicatesthepresenceofatransient.DependenceontRindicatesthattheaveragingtimeistooshortforergodicitytohold.Averyimportantpointisthatwe ndthattheminimumvalueofthetransienttimefordensity uctuationsisnottK,butitisoftheorderoftheslowestcharacteristicdecaytimet inthesystem.AsdiscussedinthenextSection,t isastronglyincreasingfunctionofdensity,andismuchlongerthantheequilibrationtimeforthekineticenergy.Similarly,itisnecessaryfortheaveragetoincludearangetRoforderofseveraltimest .We ndthatS(q,t,t0)athigherdensitieshasconsiderableoscillationsoverttimerangessmallerthant .Asoneincreases