Exact solution of a one-dimensional Boltzmann equa

arXiv:cond-mat/0602558v2[cond-mat.stat-mech]21Apr2006Exactsolutionofaone-dimensionalBoltzmannequationforagranulartracerparticleJ.PiaseckiaJ.TalbotbP.ViotcaInstituteofTheoreticalPhysics,UniversityofWarsaw,Ho˙za69,00-681Warsaw,PolandbDepartmentofChemistryandBiochemistry,DuquesneUniversity,Pittsburgh,PA15282-1530,USAcLaboratoiredePhysiqueTh´eoriquedelaMati`ereCondens´ee,Universit´ePierreetMarieCurie,4,placeJussieu,75252ParisCedex,05FranceAbstractWeconsideraone-dimensionalsystemconsistingofagranulartracerparticleofmassMinabathofthermalizedparticleseachofmassm.Whenthemassratio,M/m,isequaltothecoefficientofrestitution,α,thesystemmapstoaaone-dimensionalelasticgas.Inthiscase,Boltzmannequationcanbesolvedexactly.Wealsoobtainexpressionsforthevelocityautocorrelationfunctionandthediffusioncoefficient.NumericalsimulationsoftheBoltzmannequationareperformedforM/m6=αwherenoanalyticalsolutionisavailable.Itappearsthatthedynamicalfeaturesremainqualitativelysimilartothosefoundintheexactlysolvablecase.Keywords:Boltzmannequation;granulargas;Green-Kuborelation.PACS:05.20.Dd,45.70.-n,1IntroductionGranulargasesconsistofparticlesthatundergodissipativecollisions,makingthemaparadigmofnon-equilibriumstatisticalmechanics[1,2].Theabsenceofmicroscopicreversibility(ineachcollision,kineticenergyislost)andthecontractionofthephasespacevolumeleadtospecificpropertieswhichdifferfromelasticallycollidingthermalsystems.Theseincludenon-gaussianstatis-tics,modifiedhydrodynamics,andabsenceofequipartition[1,2,3,4,5,6].Comparedtoequilibrium,ourknowledgeofnonequilibriumstatisticalmechan-icsisstillincomplete,andexactresultsarerare.Inthispaper,weconsideraPreprintsubmittedtoPhysicaA6February2008one-dimensionalmodelofatracerparticlethatundergoesdissipativecollisionswithparticlesofathermalizedbath.ThekineticdescriptionisprovidedbytheBoltzmannequation.Weshowthatwhenthemassratioofthetracertoabathparticle,M/m,isequaltothecoefficientofrestitutionα,thereexistsamappingwiththeBoltzmannequationofidenticalelastichardrods,thesolutionofwhichwasderivedalmostthirtyyearsago[7].WeproposehereasimplerwayofsolvingtheBoltzmannequationandwediscusstheresultsinthecontextofourmodel,i.e.agranularparticleinathermalizedbath.Inaddition,weobtainthevelocityautocorrelationfunctionaswellasthediffusioncoefficient.NumericalsimulationsoftheBoltzmannequationshowthatthequalitativekineticfeaturesarenotstronglydependentonthevalueofthecoefficientofrestitution,implyingthattheexactsolutionprovidesthecharacteristicsofthesystemalsowhenM/m6=α,whereananalyticsolutionisnotavailable.2ThemodelWeconsideraone-dimensionalgasofidenticalhardrodsofmassmintowhichoneinsertsagranularhardrodofmassM[8].Particleshavehardcoreinteractionsandcollisionsbetweenbathparticlesareelastic.But,collisionsbetweenthetracerparticleandthebathparticlesareassumedinelasticandcharacterizedbyacoefficientofrestitution,α≤1.Inaddition,weassumethatastationary,Gaussianvelocitydistributionisimposedonthebathparti-cles,compensatingforthekineticenergylostduringcollisionswiththetracerparticle.Foreachcollision,momentumisconservedsothatforacollisionbetweenabathparticleandthetracerparticle,onehasMV∗+mv∗=MV+mv(1)wheretheupper-casevelocitycorrespondstothetracerparticleandthelower-casevelocitytothebathparticle,andtheasterisksdenotepost-collisionalquantities.Atthemomentofimpact,therelativevelocitychangessignandshrinks(forα1)accordingtothecollisionruleV∗−v∗=−α(V−v)(2)BycombiningEqs.(1)and(2),thevelocitiesofcollidingparticlesaftercollision2aregivenbyV∗=(μ−α)V+(1+α)v1+μ(3)v∗=(1+α)V+(μ−1−α)v1+μ−1(4)whereμ=M/m.Notethatthetracermasscanbealsotakenasequaltothefluidparticlemassbyintroducinganeffectiverestitutioncoefficient[11].TheprecollisionalvelocitiesV∗∗andv∗∗,correspondingtotheinversecollision(V∗∗,v∗∗)→(V,v),canbeexpressedintermsofVandvbyreplacinginEqs.(3)and(4)thecoefficientαbyα−1.3StructureoftheBoltzmannequationatα=M/mThekineticdescriptionofthismodelisprovidedbytheBoltzmannequationaccordingtowhichtheprobabilitydistributionofthetracerparticleevolvesas∂∂t+V∂∂R!f(R,V,t)==ρZ+∞−∞dc|V−c|hα−2f(R,V∗∗,t)φ(c∗∗)−f(R,V,t)φ(c)i(5)whereRisthepositionofthetracerparticle,φ(c)denotestheequilibriumMaxwelldistributionofthebathparticles,andρtheirnumberdensity.NotethatEq.(5),althoughlinearinthecaseofatracerparticle,hasnotyetbeensolvedexacty.Exactsolutionshave,however,beenfoundintheframeworkoftheinelasticMaxwellmodelwherethecollisionfrequency,whichoccursintheBoltzmanncollisionalterm,isassumedindependentoftherelativevelocity[9,10].ThefirstobjectiveofthispaperistoshowthatanexactsolutionofEq.(5)canbeobtainedwhenthecoefficientofrestitutionequalsthemassratioofthetracertoabathparticle,α=μ=M/m.Inthiscase,theBoltzmannequationforthedistributionofthegranulartracerparticleturnsouttobemathemat-icallyidenticaltotheBoltzmannequationdescribingataggedelasticparticle3inathermalizedbathofmechanicallyidenticalparticles.Thisfact,shownbe-low,opensthewaytoanexactsolution.However,thephysicalsituationsaretotallydifferent:evolutionofadissipativesystemtowardsastationarystateinonecaseandevolutionofaconservativesystemtowardsthermaleq