Nonlinear Hydrodynamics of a Hard Sphere Fluid Near the Glas(12)

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

Itisconvenienttochooseunitsatthispoint,whichwillsimplifysubsequentexpres-sions.Wetakem0astheunitofmass.Wechoosealsoaunitoflength,h,whichweshalllateridentifywiththelatticeconstantofthecomputationallattice.Fortheunitoftimewechooset0suchthat:

t0=h/c(2.7)

wherecisthespeedofsound.Thischoiceoftheunitoftimeismotivatedasfollows:thecharacteristicphonontimeforthesystemistp=1/(cq)whereqisawavevector.Hencetp/t0=1/(qh).Weshallchoosebelowtheratioσ/hsothatqhisoforderunityinthewavevectorrangeofinterest.Hencethechoice(2.7)correspondstoacharacteristictimeofordertp.ThesamechoicewastakeninRef.(39).

Wethende nethedimensionlessquantities:

x=r/h(2.8)

n=ρh3

j=gh3/c(2.9)(2.10)

andintroducealsothedimensionlessfreeenergyF[n,j]intermsoftheabovequantities:

F[n,j]=(1/2) dxj2

m0c2

(2.13)