Nonlinear Hydrodynamics of a Hard Sphere Fluid Near the Glas(21)

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

alldatauptoeitherthemaximumtimeforwhichwehavedataattheparticularvaluesofn andqunderconsiderationoruptothetimewhereC(q,t)issosmallthatitfadesintothenoise.ThisoccurswhenC(q,t)<0.025exceptinsomeveryfewcaseswherethedatabecomesnoisyatthe0.05level.AsinRef.(39)thestatisticalnoiseseemstohaveanadditivecomponentwhichcausestherelativeerrorstoincreasewhenC(q,t)becomessmaller.WerecallthattherelationbetweenourtimeunitsandEnskogcollisiontimeisgivenby(2.16).Thefactorrelatingthetwoinversetimesvariesfrom1.52atn =0.75to

1.90atn =0.93.Therelationbetweenourunitoflengthandσisthetrivialfactorofσ/h=4.6.

Webeginbyattemptinga ttoastretchedexponentialform:

C(q,t)=e (t/τ0)β(4.1)

wheretheparametersτ0andβarefunctionsofn andq.Thisstepismotivatedinpartbytheexpectationfrompreliminaryinspectionofthedatathatthereisawideregionofqandn valuesforwhichthisformisadequate.Indeedwe ndthatforaconsiderablepartofthedataparticularlyinthelowerdensityregionthisturnsouttobeasatisfactory t.Eveninthecaseswherethedataisnotwell ttedbytheformofEq.(4.1),thenumberτ0(q,n )stillgivesauseful gureofmeritoroverallestimateofthedecaytime.Theresultsofthis tareinTablesIandII.WehaveindicatedintheseTablesthecaseswherethe ttotheaboveformisagood ttothedataandthoseinwhichitisactuallynotthebest tbyenclosingthelattercasesinparentheses.Thequalityofthe tscanbejudgedvisuallyandbytheχ2valuesthatweobtain.

ThesalientpointsoftheresultsinTableIareapparent:thecharacteristictimeisatconstantdensityaverystrongfunctionofq,andithasamaximumatthewavevectorshellsclosetowherethestaticstructurefactorS(q)hasitsmaximum.Alesswellde ned