Nonlinear Hydrodynamics of a Hard Sphere Fluid Near the Glas(14)

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

Vji= n(x) iδFn

n0 jjj(x) ijj(x)

t

and:

ji

δn (1/n0) j+(1/n0) (nj)=0(2.20) j(jijj) (1/n0) jjj ijj+(1/n0)η 2ji+Θi(2.21)

whereηisthebareshearviscosityinourunits45.Thenoise eldsΘi(x,t)satisfythesecond uctuation-dissipationtheoremintheform:

<Θi(x,t)Θj(x′,t′)>= 2Kληn0δi,j 2δ(r r′)δ(t t′).(2.22)

Werecallthatintheseequations,andintheremainderofthepaper,thetimeismeasuredinunitsasgivenin(2.16).In(2.22)theangularbracketsdenotethethermody-namicaverage.Thequantityλisadimensionlessmeasureoftheequilibrium uctuations.Theaveragevaluen0isrelatedton =nσ3throughn0=n (h/σ)3.Forhardspheres,onecanwriteηintermsofKandthedensityintheform:

η=(h/σ)246√π)][g(σ) 1+0.8(2πn /3)+0.761(2πn /3)2g(σ)](2.23)Theequationsofmotion(2.20)and(2.21)areslightlymorecomplicatedthanthecor-respondingequationsforthemodelinRef.(39).Thereisanadditionalconvectionterminthesecondequationandsomeadditionalfactorsofn/n0.Thesecomplicationscanbedirectlytraceddowntothedi erentdensitydependenceofthe rsttermin(2.11)asdis-cussedabove,thatis,tothefactthatakineticenergycontributionisnowincludedinthe