Velocity fluctuations in cooling granular gases

arXiv:cond-mat/0302418v1[cond-mat.stat-mech]20Feb2003Velocityfluctuationsincoolinggranulargases.AndreaBaldassarri1,UmbertoMariniBettoloMarconi2,andAndreaPuglisi31INFMUdrRoma1,UniversityofRome“LaSapienza”,piazzaleAldoMoro2,I-00185,Roma,Italy2DipartimentodiMatematicaeFisicaandINFMUdrCamerino,UniversityofCamerino,ViaMadonnadelleCarceriI-62032Camerino,Italy3INFMCenterforStatisticalMechanicsandComplexity,University“LaSapienza”,piazzaleAldoMoro2,I-00185Roma,ItalySummary.Westudytheformationandthedynamicsofcorrelationsintheve-locityfieldfor1Dand2Dcoolinggranulargaseswiththeassumptionofnegligibledensityfluctuations(“HomogeneousVelocity-correlatedCoolingState”,HVCS).Itisshownthatthepredictionsofmeanfieldmodelsfailwhenvelocityfluctuationsbe-comeimportant.ThestudyofcorrelationsisdonebymeansofmoleculardynamicsandintroducinganInelasticLatticeMaxwellModels.ThislatticemodelisabletoreproduceallthepropertiesoftheHomogeneousCoolingStateandseveralfeaturesoftheHVCS.Moreoveritallowsveryprecisemeasurementsofstructurefunctionsandothercrucialstatisticalindicators.Thestudysuggeststhatboththe1Dandthe2Ddynamicsofthevelocityfieldarecompatiblewithadiffusivedynamicsatlargescalewithamorecomplexbehavioratsmallscale.In2Dtheissueofscaleseparation,whichisofinterestinthecontextofkinetictheories,isaddressed.1IntroductionTheinelastichardspheresmodel[1]withoutenergyinput,initiallypreparedinahomogeneousstate,exhibits,afterarapidtransient,aregimecharac-terizedbyhomogeneousdensityandaprobabilitydistributionofvelocitiesthatdependsontimeonlythroughthetotalkineticenergy(globalgranulartemperatureTg(t)),i.e.ascalingvelocitydistribution.ThisisthesocalledHomogeneousCoolingState(HCS)orHaffregime[2].Ithasbeenshownbyseveralauthors[3–6]thatthisstateisunstablewithrespecttoshearandclusteringinstabilities:structurescanformthatseemtominimizedissipation,mainlyintheformofvelocityvorticesandhighdensityclusters.Theseinsta-bilitiesgrowondifferentspaceandtimescales,sothatonecaninvestigatethemseparately.Severaltheorieshavebeenproposedtotakeintoaccounttheemergenceofstructuresingranulargases.Someofthesearemorefun-damental,becausederivethecorrelationfunctionsdirectlyfromthekineticequations[7];others,thatassumethevalidityofthehydrodynamicdescrip-tion,deservethenameofmesoscopictheories[8];othersarephenomenolog-icaltheoriesthatsuggestanalogieswithBurgersequation[9,10]orspinodaldecompositionmodels[11]ormodecouplingtheories[12].Someofthesethe-oriescandescribethebehaviorofthecoolinggranulargasfardeeplyintothe2AndreaBaldassarrietal.correlatedregime,givingpredictionsfortheasymptoticdecayofenergy.Inthispaper,afterabriefsummaryoftheresultsofthesetheories,weshowhowthedeparturefromtheHCScanbewellmodeledbyaclassofmod-elsobtainedplacingonalatticetheoriginalhomogeneousInelasticMaxwellModel.Thesemodelshavethedisadvantagesofbeingconceivedundertheas-sumptionofnegligibledensityfluctuationsandthereforecanbeusefulonlyinthedescriptionofthefirstinstability,i.e.thegrowthofvelocityfluctuations.Insection2wereviewtheHCS,itsinstabilitiesandtheexistingtheories.Insection3webrieflydiscusstheInelasticMaxwellModel[13–15],whichisastartingpointfortheintroductionoftheInelasticLatticeMaxwellModels(ILMM).Insection4andinsection5theanalysisandresultsoftheILMMinoneandtwodimensionsarereviewed[14,16].Finallyinsection6conclusionsaredrawn.2InstabilitiesoftheHomogeneousCoolingStateAcoolinggranulargasinddimensionsisdefinedasanensembleofNgrains,i.e.hardobjects(rodsifd=1,disksifd=2,spheresifd=3)oflinearsize(diameter)σ,placedinavolumeVwithperiodicboundaryconditions.Thegrainsevolvefreelyandinteractwitheachotherthroughinstantaneousbinaryinelasticcollisions.Therulethatgivesthevelocitiesafterthecollisionasfunctionsofthevelocitiesbeforethecollisionisthedefinitionoftheparticulargranulargasmodel.Inthiscaseweusethemodelwithconstantrestitutioncoefficientwithoutrotationaldegreesoffreedomandsetthemassofthegrainsm=1.Othermodelshavebeenconsideredintheliterature[1].Thecollisionrulebetweenaparticlewithvelocityvandonewithvelocityv∗forthismodelis:v′=v−1+r2[(v−v∗)·ˆn]ˆn(1)wheretheprimedvelocityisthepost-collisionalone,ˆn=(r−r∗)/|r−r∗|istheunitvectorinthedirectionjoiningthecentersofcollidingparticles,andristherestitutioncoefficientandtakesvaluesbetween0and1.Whenr=1thecollisioniselastic.Usually(innumericalorrealexperiments)granulargasesarepreparedinahomogeneoussituation:uniform-randompositionsofgrains,Gaussianoruniform-randominitialvelocitywithnopreferreddirection.Ithappensthat,ifthesystemislargeenoughortheinelasticitylargeenough,theimposedhomogeneityisbrokenafteracertaintime.Themoreacceptedscenarioisatwotimehomogeneitybreaking:atatimetsthevelocityfieldbecomesunstabletotheformationofshearbands,thenatatimetctsthedensityfieldbecomesunstabletowardtheformationofhighdensityclusters.Coolinggranulargases32.1ThehomogeneouscoolingstateAgranulargasevolvingfromahomogeneouslyrandomstatelosesmemoryofitsinitialconditionafteratimeoftheorderofonecollisionperparticleandrapidlyenterstheHomogeneousCoolingRegime.ThisregimeisexpectedtobewelldescribedbythegranularBoltzmannEquation(se