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TRANSPORTCOEFFICIENTSINSOMESTOCHASTICMODELSOFTHEREVISEDENSKOGEQUATIONJACEKPOLEWCZAKANDGEORGESTELLAbstract.AstochasticmodeloftherevisedEnskogequationisconsidered.AchoiceofthesmearingfunctionsuggestedbytheworkofLeegwater([1])isusedtoapplythemodeltotherepulsivepartoftheLennard-Jonespotentialandtheinverse-powersoft-spherepotential.Thevirialcoecientsobtainedfromtheequilibriumpropertiesofthemodelsareinexcellentagreementwiththeknownexactcoecientsforthesemodels.ThetransportcoecientsfortherepulsiveLennard-Jones(RLP)modelarealsocomputedandappeartobeofcomparableaccuracytotheEnskog-theorycoecientsapplieddirectlytoahard-spheresystem,althoughexactresultsfortheRLPwithwhichtomakeanextensivecomparisonarenotyetavailable.Thepressureandthetransportcoecientsobtainedfromthemodel(shearviscosity,thermalconductivity,andself-diusion)arecomparedwiththepressureandthecorrespondingtransportcoecientspredictedbytheEnskogandsquare-wellkinetictheories.1.IntroductionOneoftheimportantgoalsofanykinetictheoryisitscapabilitytocomputetransportcoecientsofauid.ThekinetictheorybasedontheBoltzmannequation,validintheregimeofclose{to{zerodensities(dilute{gasregime),producesthetransportcoecientsofanidealuid.Formoderatelydense,hard{spheregases,thekinetictheorybasedonthestandardorrevisedEnskogequations([2],[3],[4])describes,onthehydrodynamicallevel,anon{idealuidwithtransportcoecientshavinganonvanishingbulkviscosity.Inordertoaccountformorerealisticpotentials(otherthanhard{corepotentials)severalextensionsoftherevisedEnskogequationhavebeenproposed.Sincesmoothpotentialscanbeapproximatedbyasequenceofstepfunctions,thesquare-wellpotentialSWcanbeconsideredasthelowest-orderapproximationofthistype.For0,0aR,andSW(r)=8:1;forra,;forarR,0forrR.(1.1)theauthorsin[5],[6],[7],[8],[9],and[10]havedevelopednewmodelsofkinetictheories.Recentonesarebasedonthemaximizationofthene{grainedentropy,subjecttoappropriateconstraints,whichyieldswhathascometobecalledkineticvariationaltheory(KVT).Useofasetofincreas-inglystringentconstraintswhichareasubsetoftheexactconstraintsintheproblemseemtoprovideKeywordsandphrases.Kinetictheoryofgases,Enskogtheory,hardspheressystems,Lennard-Jonespotential,transportcoecients.WefondlydedicatethisworktoBobDorfman,whohaslentG.S.asympatheticeartovariousnotionsofdisequi-libriumforthelastfourdecades.TheauthorsgratefullyacknowledgethesupportofthisworkbytheDivisionofChemicalSciences,OceofBasicEnergySciences,OceofEnergyResearch,U.S.DepartmentofEnergy.12JACEKPOLEWCZAKANDGEORGESTELLincreasinglyaccuratequantitativeassessmentsoftransportcoecientsandrelatedmacroscopicquan-tities.Inparticular,theversionthatisbaseduponalocalenergy{densityconstraint(calledbytheauthorsKVTIII,becauseitisgeneratedbythethirdinthesequenceofconstraintsmentionedabove)hasbeeninvestigatedindetailandthetransportcoecientsobtainedfromtheKVTIIIareingoodagreementwithexperimentaldataandcomputersimulationresultsformoderatelydenseuids.Althoughthisprogresshasbeenencouraging,therehavebeennofullysatisfactoryextensionsofthesetheoriestomorerealisticpairpotentialsthathavesoftratherthanhardcoresandsmoothlyvaryingattractivetailswithnodiscontinuities.Karkheck,StellandXu[11]appliedKVTIIItoatruncatedLennard-Jones(LJ)potentialtoobtainformalKVTIIIexpressionsforthefullLJlimit,inwhichthepointoftruncationgoestoinnity,butthetemporalapproachtoequilibriuminKVTIIIhascertainpathologicalfeaturesinthislimit.AlthoughtheKVTIIItransportcoecientsthemselvesarerobustandgenerallyusefulforamoderatelydenseLJuid,theKVTIIIdoesarelativelypoorjobincapturingthequantitativebehaviorofthediusioncoecientasafunctionoftemperature.Subsequently,inhisthesis,J.A.Leegwater[1]consideredanewapproachtoobtainingtheself-diusioncoecientofanLJsystemfromEnskog-theoryexpressionsintowhichaweighingfactorW(r)isintroducedtoaccommodatethesoftLJpotential.TheresultingexpressionhastheformD=38n1R10W(r)y(r)dr
kBTm1=2;(1.2)wherey(r)=expLJ(r)gLJ2(r)isthecavityfunction,LJistheLennard-Jonespotential,LJ(r)=4dr12dr6#;(1.3)gLJ2(r)istheequilibriumradialcorrelationfunctionoftheLennard-Jonessystem,andW(r)isacertainweightingfunctionthatwasobtainedbysolvingapproximatelythetwoparticleequationsofmotion.BecausetheLJy(r)isverywellapproximatedbyahard-spherey(r)forasuitablychosenhard-spherediameter,onecansimplyuseahard-spherey(r)in(1.2).Leegwater'sideawastogeneralizetheexpressionfortheEnskogvalueoftheself-diusioncoecient,DEnskog=38n1a2gHS2(a+)kBTm1=2;(1.4)(heregHS2(a+)isthecontactvalueofhard-spheres'radialcorrelationfunction)tocontinuouspoten-tials,byanadhocinsertioninto(1.4)thefactorgLJ2,weightedbythefunctionW(r).Furthermore,sinceLeegwater'sW(r)satisesthenormalizationcondition1Z0W(r)r2dr=1;(1.5)theapproximation(1.2)reducesto(1.4)inthehard-spherelimit.Inthisworkweprovidenewargumentsforvalidityoftheapproximateformula(1.2)fortheself{diusioncoecient.Inaddition,wederivethecorrespondingformulasforothertransportcoe-cients,notconsideredbyLeegwater.Westartbyintroducingaclassofthekineticequationsthathavebuiltintotheirstructuretheweig