Nonlinear Hydrodynamics of a Hard Sphere Fluid Near the Glas(15)

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

rsttermofFn.Thesedi erencesthenarisebecauseofthedi erentformofδF/δjwhichresultsinmorecomplicatedexpressionsforthestreamingvelocities.Ourexpressionscor-rectlyreducetothecontinuityandNavier-Stokesequationsinthelinearlimit.Ingeneral,onecaninterpret(2.20)asthecontinuityequationifthe eldgisthoughtofasρ0timesthevelocity.Thecurrentdensityisthennotgivensimplybyg,however.Thesequestionsdonota ect,obviously,thevalidityoftheconclusionsobtainedfromthismodel.

AsalientfeatureofEq.(2.21)ofconsiderableimportanceisthattheterminδFn/δnwhichinvolvesthedirectcorrelationfunctionCisanintegraloverspacewithrangeσ.Thus,wehavetosolveinthiscasenotmerelyasetofpartialdi erentialequationswithstochastictermsbutasfarasthespatialdependenceisconcernedanintegrodi erentialequation.Thiscomplicationmakesthisadi cultproblemtosolvenumerically.Themethodsusedwillbeexplainedbelow.

III.METHODS

InthisSectionwediscussthemethodsweusetosolveourequationsofmotion,thephysicalquantitieswefocuson,thedatacollectionmethodsandstatistics,therangeofparametervalueswehavestudied,andrelatedmatters.

Ourobjectiveistostudythedynamiccorrelationsofoursystemasde nedbelow.InordertodosowesolvenumericallyEqns.(2.20)and(2.21)onathreedimensionalcubiclatticeofsizeN3.ThemaincomplicationwemustconsideristhespatialintegralinvolvingthedirectcorrelationfunctionC(r).ForeachintegrationstepthisintegralmustbedoneN3timessinceitiscomputedoveraspherewithitsoriginateachoneofthelatticesites.Toavoidrepeatedcalculationswecreate,intheinitializationofourprogram,atablelistingforeachlatticesitethelocationofallneighboringsitestobeintegratedoverandtheircorrespondingvaluesofC(r).Additionally,sincethesphereisimbeddedona