Asymptotic solutions of the nonlinear Boltzmann eq

arXiv:cond-mat/0304608v1[cond-mat.stat-mech]27Apr2003AsymptoticsolutionsofthenonlinearBoltzmannequationfordissipativesystemsM.H.Ernst1andR.Brito21InstituutvoorTheoretischeFysica,UniversiteitUtrecht,Postbus80.195,3508TDUtrecht,TheNetherlands.Email:[e-mail]2DeptodeF´ısicaAplicadaI,UniversidadComplutense,28040Madrid,Spain.Email:[e-mail](v,t)ofthenonlinearBoltzmannequationind-dimensionsarestudiedforanewclassofdissipativemodels,calledinelasticrepul-sivescatterers,interactingthroughpseudo-powerlawrepulsions,characterizedbyastrengthparameterν,andembeddinginelastichardspheres(ν=1)andinelas-ticMaxwellmodels(ν=0).Thesystemsareeitherfreelycoolingwithoutenergyinputordrivenbythermostats,e.g.whitenoise,andapproachstablenonequilib-riumsteadystates,ormarginallystablehomogeneouscoolingstates,wherethedata,vd0(t)F(v,t)plottedversusc=v/v0(t),collapseonascalingorsimilaritysolutionf(c),wherev0(t)isther.m.s.velocity.Thedissipativeinteractionsgener-ateoverpopulatedhighenergytails,describedgenericallybystretchedGaussians,f(c)∼exp[−βcb]with0b2,whereb=νwithν0infreecooling,andb=1+12νwithν≥0whendrivenbywhitenoise.Powerlawtails,f(c)∼1/ca+d,areonlyfoundinmarginalcases,wheretheexponentaistherootofatranscen-dentalequation.Thestabilitythresholddependonthetypeofthermostat,andisforthecaseoffreecoolinglocatedatν=0.MoreoverweanalyzeaninelasticBGK-typekineticequationwithanenergyde-pendentcollisionfrequencycoupledtoathermostat,thatcapturesallqualitativepropertiesofthevelocitydistributionfunctioninMaxwellmodels,aspredictedbythefullnonlinearBoltzmannequation,butfailsforharderinteractionswithν0.1IntroductionClassickinetictheory[1–5]dealswithelasticparticleswithenergyconservingdynamics.Thesystemisdescribedbythesingleparticledistributionfunc-tion,whosetimeevolutionisgovernedbythenonlinearBoltzmannequation.Theasymptoticstatesofsuchsystemsfollowtheuniversallawsofthermody-namics,andthedistributionfunctionistheMaxwellBoltzmanndistribution.Thisscenariodoesnotapplytodissipativesystems,whereenergyislostininelasticinteractions.Inelasticsystemstheapproachtoasymptoticstatesischaracterizedbyakineticstageofrapidrelaxationinvelocityspacetoalocallyhomogeneousequilibriumstate,followedbyahydrodynamicstageofslowapproachtoagloballyhomogeneousequilibriumstate.Thetimescaleinthekineticstageisthemeanfreetimetmfbetweencollisions.Inthekinetictheoryofinelastic2M.H.ErnstandR.Britosystems[6–12]thetypeofdecaydependsontheenergysupplytothedissi-pativesystem.Withoutenergysupplythereisfirstakineticstageofrapidrelaxationonthetimescaletmftoalocallyhomogeneousadiabaticstate,thehomogeneouscoolingstate,describedbyscalingorsimilaritysolutionswithaslowlychangingparameter,atleastforweaklyinelasticsystems.Withenergysupplytheevolutionismoresimilartotheelasticcasewith,however,equilibriumstatesreplacedbynon-equilibriumsteadystates.ThevelocitydistributionsintheseadiabaticorsteadystatesareverydifferentfromaMaxwellBoltzmanndistribution.Thesubsequentstageofevolutioninvolvestransportphenomenaandcomplexhydrodynamicphenomenaofclusteringandpatternformation[12,13].Theinterestingranularmatteringeneralhasstronglystimulatednewdevelopmentsinthekinetictheoryofgranularfluidsandgases,whichshowsurprisingnewphysics.Agranularfluidisacollectionofsmallorlargemacro-scopicparticles,withshortrangerepulsivehardcoreinteractions,inwhichenergyislostininelasticcollisions,andthesystemcoolswhennotdriven.Whenrapidlydriven,gravitycanbeneglected.Thedynamicsisbasedonbinarycollisionsandballisticmotionbetweencollisions,whichconserveto-talmomentum.Sothesesystemscanbeconsideredtobeagranularfluidorgas.Theprototypicalmodelfortheseso-calledrapidgranularflowsisafluidorgasofperfectlysmoothmono-disperseinelastichardspheres,anditsnon-equilibriumbehaviorcanbedescribedbythenonlinearBoltzmannequa-tion[6–12].Theinelasticcollisionsaremodeledbyacoefficientofrestitutionα(0α1),where1−α2
measuresthedegreeofinelasticity.Thisreviewfocusesonthefirststageofevolution,andstudiesthevelocitydistributionF(v,t)inspatiallyhomogeneousstatesofinelasticsystems.Forthatreasonmostofthecitations,giveninthisarticle,onlyrefertokinetictheorystudiesofF(v,t),andnottostudiesoftransportproperties.Therevival[14–26]inkinetictheoryofinelasticsystemshasbeenstronglystim-ulatedbytheincreasingsophisticationofexperimentaltechniques[27,28],whichmakedirectmeasurementsofvelocitydistributionsfeasibleinnon-equilibriumsteadystates.Inthisreviewwealsoincludeinelasticgeneral-izations[29]oftheclassicalrepulsivepowerlawinteractions[2–4],whichembedboththeinelastichardspheres(ν=1),aswellastherecentlymuchstudied[30–40]inelasticMaxwellmodels(ν=0)inasingleclassofmodels,parametrizedbyanexponentν.Thisexponentcharacterizesthedependenceofthecollisionfrequenciesontheenergyofimpactatcollision.Infact,thekinetictheoryforsuchmodelsisofinterestinitsownright,asthemajorityofinter-particleinteractionsinmacroscopicsystemsinvolvesomeeffectsofinelasticity.Ourgoalistoexposethegenericanduniversalfeaturesofthevelocitydistributionsindissipativefluids,andtocomparethemwithc