Velocity Distributions and Correlations in Homogen

arXiv:cond-mat/0105322v2[cond-mat.soft]11Jun2001VelocityDistributionsandCorrelationsinHomogeneouslyHeatedGranularMediaSungJoonMoon∗,M.D.Shattuck†,andJ.B.SwiftCenterforNonlinearDynamicsandDepartmentofPhysics,TheUniversityofTexasatAustin,Austin,TX78712(February1,2008)Wecomparethesteadystatevelocitydistributionsfromourthree-dimensionalinelastichardspheremoleculardynamicssimulationforhomogeneouslyheatedgranularmedia,withthepredic-tionsofameanfield-typeEnskog-Boltzmannequationforinelastichardspheres[vanNoije&Ernst,Gran.Matt.1,57(1998)].Althoughwefindqualitativeagreementforallvaluesofdensityandinelasticity,thequantitativedisagreementapproaches∼40%athighinelasticityordensity.Bycontrastthepredictionsofthepseudo-Maxwellmoleculemodel[Carrillo,Cercignani&Gamba,Phys.Rev.E,62,7700(2000)]arebothqualitativelyandquantitativelydifferentfromthoseofoursimulation.Wealsomeasureshort-rangeandlong-rangevelocitycorrelationsexhibitingnon-zerocorrelationsatcontactbeforethecollision,andbeingconsistentwithaslowalgebraicdecayoveradecadeintheunitofthediameteroftheparticle,proportionaltor−(1+α),where0.2α0.3.Theexistenceofthesecorrelationsimplythefailureofthemolecularchaosassumptionandthemeanfieldapproximation,whichisresponsibleforthequantitativedisagreementoftheinelastichardspherekinetictheory.I.INTRODUCTIONGranularmaterialsarecollectionsofnoncohesivemacroscopicdissipativeparticlesandareencounteredinnatureandintheindustry[1].Thesematerialsexhibitawidevarietyofphenomenadependingontheexter-nalforcing.Therapidgranularflowregime,wherethecollisionsaremodeledasinstantaneousbinaryinelasticcollisions,isreminiscentofagasofhardspheres.Thus,acommontheoreticalapproachforthisregimeasafirstor-derapproximationistomodelthesystembymeansofthekineticandthecontinuumequationsforsmoothinelas-tichardsphereswithavelocity-independentcoefficientofrestitution[2].Inthekinetictheoryapproach,themeanfield-typeBoltzmannorEnskog-Boltzmannequa-tionforinelastichardspheresisused,andmosttech-niquesaredirectlytransposedfromthekinetictheoryofagasofelastichardspheres[3].Itisknownthatthisformulationisareasonabledescriptionfornearlyelastic(1−e2≪1,whereeisthenormalcoefficientofrestitu-tion)anddilutecases.However,thesetheoreticalmodelsincludeapproximationssuchasthetruncationofseriesexpansionorhierarchy,andtheintroductionofequationclosure.Thedissipativenatureofthecollisionmodifiesthephysicsinanontrivialway,andtheaccuracyandthelimitationofthemeanfield-typekineticdescriptionwiththeseapproximationsisnotyetknown.Theextensionofthetheoryforthemoreinelasticanddensecase,includ-ingsurfacefriction,isoneofmajorgoalsofthecurrentinelastickinetictheory[4].Thereisanattractorinthephasespaceofthegran-ularmedia,becauseinelasticcollisionsdissipatekineticenergy;intheabsenceofanexternalenergysource,agranularmediumlosesitskineticenergythroughcolli-sionsandbecomesastaticpile.Toreachasteadystateoranoscillatorystate,asystemofgranularmediarequiresanexternalenergysource.Inthispaper,weinvestigatethesteadystatevelocitydistributionsandvelocitycorre-lationsofspatiallyhomogeneousgranularmediasubjecttoavolumetricGaussianwhitenoiseforcing.Thissys-temhasbeenstudiedbyseveralauthors[5,6,12,18]asareferencesystemforthekinetictheoryofgranularmedia.Inthissystem,particlescollideinelastically,andexecuteBrownianmotionbetweencollisions;themotionisanal-ogoustoBrowniandynamicsofhardspheresuspension,butherethekineticenergyisdissipatedduetothein-elasticcollisionsratherthanhydrodynamicdrag.Thissystemisfarfromequilibrium,andthesteadystateve-locitydistributiondeviatesfromtheMaxwell-Boltzmann(MB)distribution.Thevelocitydistributionofthissys-temwasfirsttheoreticallystudiedbyvanNoijeetal.[5],whoobtainedtheapproximatesolutionstotheinelastichardsphereEnskog-BoltzmannequationwithaGaus-sianwhitenoiseforcing.TheirresultsweretestedagainsttheDirectSimulationMonteCarlo(DSMC)methodoftheinelasticEnskog-Boltzmannequation[6],andagoodagreementwasfound;itconfirmedtheaccuracyoftheapproximateanalysisofvanNoijeetal.,becausetheva-lidityoftheDSMCmethodreliesonthevalidityoftheinelastic(Enskog-)Boltzmannequation[7].Inthecurrentstudy,weusealargemoleculardynamics(MD)simulation(usingupto477,500particles)ofinelas-tichardspherestoinvestigatethesteadystatevelocitydistributions,andquantitativelyexaminetheaccuracyoftheinelasticEnskog-Boltzmannequationmodelforthissystem.Ourmethodisfreefromtheassumptionsusedintheinelastickinetictheory,andithasanadvantagethatcorrelationscandevelopbuthasadisadvantagethatitmayhaveafinitesimulationsizeeffect.Wealsocompareourresultswiththeoreticalpredic-tionsofCarrilloetal.[8],whoobtainedthesteadystatevelocitydistributionbyusingthepseudo-Maxwell1moleculemodel[9].TheMaxwellmoleculemodelisthespecialcaseoftheinversepowerlawmodelfortheinter-particlepotential,whosepotentialhastheformofV(r)∼r−4,whereristheinter-particledistance.Thismodelhasbeenwidelyusedforanalyticalstudiesasareferencesystemformorerealisticsystems,becausethemodelfacilitatescalculationsinvolvinglinearizedBoltz-mannoperator.Thepseudo-MaxwellmoleculemodelisaninelasticanalogoftheMaxwellmoleculem