Nonlinear Hydrodynamics of a Hard Sphere Fluid Near the Glas(11)

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

whereF[ρ]isoftheRYform

F[ρ]=Fl[ρ0]+kBT 35: drdr′C(r r′)δρδρ′ (2.2)dr(ρlog(ρ/ρ0) δρ) (1/2)

In(2.1)and(2.2)ρisthenumberdensity eldandδρ≡ρ ρ0thedeviationofthat eldfromitsaveragevalueρ0.Flisthefreeenergyoftheuniformliquid,TisthetemperatureandkBtheBoltzmannconstant.Inthe rstequationm0denotesthemassofahardsphere.Finally,C(r r′)isthedirectcorrelationfunction.Theinclusionofthisfunctioninthefreeenergyensuresthatuponlinearizationofthelogarithminthe rsttermontherighthandsideof(2.2)oneobtainstheusualexpressionforthestaticstructurefactorofasimple uidintermsofC.ForhardspheresasimpleexpressionforC(r r′)canbeobtainedinthePercus-Yevickapproximation37:

ξ<1(2.3a)

(2.3b)C(ξ)= λ1 6ηfλ2ξ (1/2)ηfλ1ξ2;C(ξ)=0;ξ>1

whereξ≡|r r′|/σ,ηfisthepackingfraction:

ηf=(π/6)ρ0σ3≡(π/6)n

and:

λ1=(1+2ηf)2/(1 ηf)4

λ2= (1+ηf/2)2/(1 ηf)4(2.4)(2.5)(2.6)

Wehavewrittenρ0inthedenominatoroftheg2terminEq.(2.1),ratherthanρasinRef.(39),sothatthefulldensitydependenceofthefreeenergyisgivenbytheRYform.AspointedoutinRef.(41),ifweusedρwewouldobtainuponfunctionalintegrationwithrespecttothegaussianvariableg2alog(ρ)contributioninvolvingdensity uctuationsassociatedwiththekineticenergy.ThesearealreadyincludedinEq.(2.2).