Anomalous velocity distributions in inelastic Maxw

arXiv:cond-mat/0310406v2[cond-mat.stat-mech]10Feb2004AnomalousvelocitydistributionsininelasticMaxwellgasesR.BritoDeptodeFisicaAplicadaIandGISC,UniversidadComplutense28040Madrid,SpainEmail:[e-mail]tischeFysica,UniversiteitUtrechtPostbus80.195,3508TDUtrecht,TheNetherlandsEmail:[e-mail]eorystudyinvestigatingtheeffectsofinelasticityonthestructureofthenon-equilibriumstates,inparticularonthebehaviorofthevelocitydistributioninthehighenergytails.StartingpointisthenonlinearBoltzmannequationforspatiallyhomogeneoussys-tems,whichsupposedlydescribesthebehaviorofthevelocitydistributionfunctionindissipativesystemsaslongasthesystemremainsinthehomogeneouscoolingstate,i.e.onrelativelyshorttimescalesbeforetheclusteringandsimilarinstabilitiesstarttocreatespatialinhomogeneities.Thisisdoneforthetwomostcommonmodelsfordissipativesystems,i.e.inelastichardspheresandinelasticMaxwellparticles.Thereisastrongemphasisonthelattermodelsbecausethatistheareawheremostoftheinterestingnewdevelopmentsoccurred.InsystemsofMaxwellparti-clesthecollisionfrequencyisindependentoftherelativevelocityofthecollidingparticles,andinhardspheresystemsitislinear.Wethendemonstratetheexistenceofscalingsolutionsforthevelocitydistributionfunction,F(v,t)∼v0(t)−df((v/v0(t)),wherev0isther.m.s.velocity.Thescalingformf(c)showsoverpopulationinthehighenergytails.Inthecaseoffreelycoolingsystemsthetailsareofalgebraicform,f(c)∼c−d−a,wheretheexponentamayormaynotdependonthedegreeofinelasticity,andinthecaseofforcedsystemsthetailsareofstretchedGaussiantypef(v)∼exp[−β(v/v0)b]withb2.1IntroductionTheinterestingranularfluids[1]andgaseshasledtoagreatrevivalinkinetictheoryofdis-sipativesystems[2,3,4,5],inparticularinthenon-equilibriumsteadystatesofsuchsystems.Agranularfluid[6]isacollectionofsmallorlargemacroscopicparticleswithshortrangehardcorerepulsions,whichloseenergyininelasticcollisions,andthesystemcoolswithoutconstantenergyinput.AsenergyisnotconservedininelasticcollisionsGibbs’equilibriumstatisticalmechanicsisnotapplicable,andnon-equilibriumstatisticalmechanicsandkinetictheoryforsuchsystemshavetobedevelopedtodescribeandunderstandthewealthofinterestingphenomenadiscoveredinsuchsystems.Theinelasticityisresponsibleforalotofnewphysics,suchasclusteringandspatialheterogeneities[7],inelasticcollapseandthedevelopmentofsingularitieswithinafinitetime[8,9],spontaneousformationsofpatternsandphasetransitions[10],overpopulatednon-Gaussianhighenergytailsindistributionfunctions[11,12,13],breakdownofmolecularchaos[7,14],1singlepeakinitialdistributionsdevelopingintostabletwo-peakdistributionsastheinelasticitydecreases,atleastinone-dimensionalsystems[15,16].Thesephenomenahavebeenstudiedinlaboratoryexperiments[13],byMolecularDynamics[7,14,15]orMonteCarlosimulations[17,18,15,19],andbykinetictheorymethods(seerecentreview[20]andreferencestherein).Theprototypicalmodelforgranularfluidsorgasesisasystemofmono-disperse,smoothinelastichardspheres,whichloseafractionoftheirrelativekineticenergyineverycollision,proportionaltothedegreeofinelasticity(1−α2),whereαwith0≤α≤1isthecoefficientofrestitution.Themodelisawell-definedmicroscopicNparticlemodel,whichcanbestudiedbymoleculardynamics,andbykinetictheory.ThesingleparticledistributionfunctioncanbedescribedbythenonlinearBoltzmannequationforinelastichardspheres[2,3].Thepresentarticlepresentsareviewofkinetictheorystudies,dealingwiththeearlystagesoflocalrelaxationofthevelocitydistributionF(v,t),andweavoidthelongtimehydrodynamicregimewheregradientsindensityandflowfieldsareimportant.So,werestrictourstudytospatiallyhomogeneousstates.Withoutenergysupplythesesystemsarefreelycooling[21,11,12].WhenenergyissuppliedtothesystemasourceorforcingtermisaddedtotheBoltzmannequation[22,14,12,18],andthekineticequationallowssteadystatesolutions,whichdependonthemodeofenergysupply[22,14,12,18].Afreelyevolvinginelasticgasorfluidrelaxeswithinameanfreetimetoahomogeneouscool-ingstate,whereitcanbedescribedbyascalingorsimilaritysolutionoftheBoltzmannequation,F(v,t)∼(1/v0(t))df(v/v0(t)).Suchsolutionsdependonasinglescalingvariablec=v/v0(t)wherev0(t)isther.m.s.velocityorinstantaneouswidthofthedistribution.ThisearlyevolutioniscomparabletotherapiddecayofthedistributionfunctiontoaMaxwell-Boltzmanndistributioninanspatiallyhomogenouselasticsystem.However,insystemsofelasticparticlessimilaritysolutionsofthenonlinearhomogeneousBoltzmannequationdonotcontrolthelongtimebehav-iorofF(v,t)[23].Theearlierstudies[21,11,12]oftheseproblemsweremainlyfocussingoninelastichardspheres,whichistheproto-typicalmodelforinelasticgasesandfluids,andontheextremelysimplifiedinelasticBGK-modelswithasinglerelaxationtime[24].Morerecentlysimplifiedstochasticmodelshavebeenintroduced[25,26,27]totacklethenonlinearBoltzmannequation,whilekeepingtheessentialphysicsoftheinelasticcollisions.Unfortunatelythemicroscopicdynamicsofthesestochasticmodelsisonlydefinedforvelocityvariables,andthemodelscannotbestudiedinphasespaceusingtheN−particlemethodsofstatisticalmechanicsandmoleculard