Local Equation of State and Velocity Distributions

arXiv:cond-mat/0402104v2[cond-mat.stat-mech]7Jul2004LocalequationofstateandvelocitydistributionsofadrivengranulargasOlafHerbst,PeterM¨uller,MatthiasOtto,andAnnetteZippeliusInstitutf¨urTheoretischePhysik,Georg-August-Universit¨at,D–37077G¨ottingen,Germany(Dated:April9,2004)Wepresentevent-drivensimulationsofagranulargasofinelasticharddiskswithincompletenormalrestitu-tionintwodimensionsbetweenvibratingwalls(withoutgravity).Wemeasurehydrodynamicquantitiessuchasthestresstensor,densityandtemperatureprofiles,aswellasvelocitydistributions.Relatingthelocalpressuretothelocaltemperatureandlocaldensity,weconstructalocalconstitutiveequation.Forstronginelasticitiesthelocalconstitutiverelationdependsonglobalsystemparameters,likethevolumefractionandtheaspectratio.Formoderateinelasticitiestheconstitutiverelationisapproximatelyindependentofthesystemparametersandcanhenceberegardedasalocalequationofstate,eventhoughthesystemishighlyinhomogeneouswithhet-erogeneoustemperatureanddensityprofilesarisingasaconsequenceoftheenergyinjection.Concerningthelocalvelocitydistributionswefindthattheydonotscalewiththesquarerootofthelocalgranulartemperature.Moreoverthehigh-velocitytailsaredifferentforthedistributionofthex-andthey-componentofthevelocity,andevendependonthepositioninthesample,theglobalvolumefraction,andthecoefficientofrestitution.PACSnumbers:45.70.-n,51.30.+i,51.10.+yI.INTRODUCTIONThephysicsofvibro-fluidizedgranularmaterialsisfarfrombeingfullyunderstood.Inparticular,theapplicabilityofhy-drodynamics[1,2,3,4,5]isstillanobjectofdebate[6,7,8].Despitethesimilaritiestothehydrodynamicsofelastichard-spheresystems,concerninge.g.theappearanceofinstabili-ties[9,10,11,12],amaindifferencetoordinaryfluidsisthefactthatcontinuousenergyinjectionisvitaltomaintainastationarystate.Otherwiseagasofinelasticsphereswouldcollapse,evenintheabsenceofgravity.Thenon-trivialna-tureofthisstationarystatehasbeenelucidatedinexperimentsofvibratinggrains.Amongthemoststrikingfeaturesfoundarenon-Gaussianvelocitydistributions[13,14,15,16]andclusterformation[17,18,19].Furthermore,severalattemptshavebeenmadetoextracteitheranequationofstateormoregenerallyascalingrelationforthethermodynamicvariables[20,21].Thespecificationofthedrivingmechanismisacrucialin-gredientforanymodeldescribingvibratedgranularfluids.Arathersimpleapproach,thoughlessappealingfromanexper-imentalpointofview,utilizesstochasticbulkheatingbyun-correlatedrandomforces,whichactoneveryparticleateveryinstantoftime[22,23,24,25].In[26]randomrestitutioncoefficientswereconsideredwithaprobabilitydistributional-lowingforvaluesbothsmallerandgreaterthanone.However,thisyieldsnon-universalpropertiesdependingonthespecificformofthisdistribution.Themultiplicativebulkdrivingde-finedintermsofstochasticcollisionrulesin[27]issimilarinspirit.Ascomparedtobulkdriving,adrivingmechanismwhichactsonlyataboundaryofthesystemismuchclosertoexperiments.In[2]theenergyinfluxattheboundaryhasbeenmodeledbyaheuristicallymotivatedansatzfortheheatcur-rent,while[28,29]assumeheatingattheboundarythroughthermalwalls.Inthepresentpaperthedrivingmechanismconsistsinincrementingaparticle’svelocitybyaconstantamountaftereverycollisionwithawall.Thiswayofdrivingwasusedbeforeine.g.[30,31],andwithslightmodificationsin[32,33,34].Onetheoreticalapproachtostudygranularsystemsatfluiddensitiesrestsonkinetictheory,seee.g.[5,35,36,37]andreferencestherein.Particularapplicationstodrivengranulargasescanbefoundin[23,26,27,28,38].MuchofthisworkhasbeenbasedontheBoltzmannequation,modifiedforinelasticcollisions.Onemethodtosolvethisnonlinearequationisbasedonthelocalequilibriumdistribution[39],whichisonlyknownfortheelasticcase.Forsystemswithstronglyinelasticcollisions,thestationarystateisunknownsothatasystematicdiscussionoftransportpropertieswithinkinetictheoryisseverelyhampered.Hydrodynamicstudies[2,11,12,30]havebeenmoti-vatedpartlybythesearchforanunderstandingoftemper-atureanddensityprofiles[2],butalsobyexperimentsonhydrodynamic-likeinstabilities[9].Inthehydrodynamicap-proachthequestionarises,howtorelatethepressuretothedensityandtemperature.Severalequationsofstatefortheglobalquantitieshavebeenproposed,eitherinterpolatingbe-tweenthehighandlowdensitylimit[2]orinvokingtheBoltz-mannequation[31]forinelasticallycollidingparticles.Itisnotclear,whetheranequationofstatealsoholdsforthelo-calhydrodynamicfieldsinastronglydriven,non-equilibriumsystem,wherethefieldsarestronglyinhomogeneous.Mostsimulationsofvibratedgranulargaseshavebeenbasedeitheronevent-drivenmoleculardynamics[31,40,41]orthedirect-simulationMonteCarlomethod[28,30,42,43].Ifthesystemisdriventhroughtheboundaries,inhomoge-neousdensityandtemperatureprofilesaremeasured.Forlowdensitiesthecomputedtemperatureprofilesagreewellwithhydrodynamictheory[2,30],whereasformoderateorhighdensitiestheprofilesarenotwellunderstoodwiththeexcep-tionofalmostelasticallycollidingparticles.Thefullstresstensor,includingpotentialcontributions,hasonlybeencom-putedforfreelycoolingsystems[44].Insimulationsofdrivengranulargases[28]thecollisionalpartofthestresstensorhasnotbeenmeasu