FREE TRANSPORT LIMIT FOR N PARTICLES DYNAMICS WITH SINGULAR AND SHORT RANGE

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FREE TRANSPORT LIMIT FOR N-PARTICLES DYNAMICS WITH SINGULAR AND SHORT RANGE POTENTIAL. J. BARRE AND P. E. JABIN Abstract. We study the limit of systems of interacting particles, when the number of particle becomes very large. The support of the interaction vanishes as the number of particles goes to infinity, so that the natural limit is just free transport, but no limitation is assumed about the strength of the interaction. We obtain explicit estimates for the number of particles effectively interacting and describe the way they do it. 1. Introduction We study the dynamics of many interacting particles in the limit of an infinite number of particles. The force acting on each particle is a sum of pair-wise interaction with the other particles. The kind of dynamics that is expected at the limit depends on the scaling of the force term with respect to the number of particles N . Here we consider very short range interaction in the sense that the force between two particles vanishes if their distance is larger than R with, in dimension 3 N R2 << 1. In this scaling the formal limit is simply free transport : each particle moves with its initial velocity. Indeed a formal computation easily shows that on average a particle should never undergo a collision : i.e. the number of particles coming at a distance less than R to another is negligible in front of N , in a time interval of order 1.

• collision


• mal computation


• transport limit


• interaction


• all particles


• limit dynamic


• vlasov equation


• particles


• between particles


• vlasov-like kinetic equations

Sujets

• Collision
• Interaction
• Vlasov equation
• Particle

Informations

• Publié par: pefav
• Nombre de lectures: 15
• Langue: English

Extrait

FREETRANSPORTLIMITFORN-PARTICLES DYNAMICSWITHSINGULARANDSHORTRANGE POTENTIAL. J.BARRE ´ ANDP.E.JABIN Abstract. Westudythelimitofsystemsofinteractingparticles, whenthenumberofparticlebecomesverylarge.Thesupportof theinteractionvanishesasthenumberofparticlesgoestoinfinity, sothatthenaturallimitisjustfreetransport,butnolimitationis assumedaboutthestrengthoftheinteraction.Weobtainexplicit estimatesforthenumberofparticleseffectivelyinteractingand describethewaytheydoit. 1. Introduction Westudythedynamicsofmanyinteractingparticlesinthelimitof aninfinitenumberofparticles.Theforceactingoneachparticleis asumofpair-wiseinteractionwiththeotherparticles.Thekindof dynamicsthatisexpectedatthelimitdependsonthescalingofthe forcetermwithrespecttothenumberofparticlesN.Hereweconsider veryshortrangeinteractioninthesensethattheforcebetweentwo particlesvanishesiftheirdistanceislargerthanRwith,indimension3 NR 2 <<1. Inthisscalingtheformallimitissimplyfreetransport:eachparticle moveswithitsinitialvelocity.Indeedaformalcomputationeasily showsthatonaverageaparticleshouldneverundergoacollision:i.e. thenumberofparticlescomingatadistancelessthanRtoanotheris negligibleinfrontofN,inatimeintervaloforder1. ThisisasimplifiedproblemforthemoreinterestingcaseNR 2 = constwheretheobtentionofcollisionalmodels(ofBoltzmanntype) isconjectured.Thefirstrigourousstepinthatdirectionwasobtained byLanfordinacelebratedwork[12](seealso[13]).Thisresultnever- thelesssuffersfromtwoimportantrestrictions:Firsttheparticlesare hardspheres(theyinteractwiththepotentialΦ(x)=+ ∞ if | x |≤ R andΦ=0if | x | >R).Andsecondthelimitisonlyvalidforatime 2000MathematicsSubjectClassification.Kinetictheoryofgases;Interacting particlessystems. 1

2J.BARRE ´ ANDP.E.JABIN smallwithrespecttotheaveragetimeittakesforoneparticletohave acollision(orthemeanfreepathafterrescaling). ThissecondrestrictionwasimprovedbyCercignani,IllnerandPul- virentiin[5](seealso[10]);Thetimeintervalofvaliditywasstillfinite andoforderthemeanfreepaththough.Bothresultsdealwithhard spheresalthoughextensionstorepulsivepotentialsarementioned(see [13])butunpublished(asfarasweknow). Thoseresultsareeasilyextendedtoourscaling.Thelimitation onthetimeintervalisnomorearealissueasthemeanfreepath tendstoinfinity.Howevertheproofswouldstillrequirehardspheres interaction(oratleastarepulsivepotential).Notethatinthecaseof anarbitrarypotential,thevelocityboundsneededtosubstantiatethe intuitionbehindtheformalscalingargumentarenoteasytoprove;in addition,theveryconceptofcollisionneedsacarefuldefinition. WestartwithProp.2.4whichjustifiesthefreetransportlimit,still onlyforrepulsivepotentials.Theinterestismainlyinthesimplicity oftheproofastheresultisnotreallynew(althoughstrictlyspeaking neverstatedbeforeforapotentialwithsolittleregularity). ThemainresultisTheorem2.5.Itunfortunatelydealswithaneven shorterrangeR<<N − 3/5 butitassumesvirtuallynothingonthe potential(itdoesnotneedtoberepulsiveanditsscalecouldbeas largeasonewants)andthereforeitisnotatallincludedinprevious contributions.Moreoverwenotonlyprovethefreetransportlimitbut alsodescribeallpossiblecollisionsequencesbetweenparticlesandthe resultisconsequentlymuchmoreprecise.Thecoreoftheproofis toensurethatthebuildupofcorrelationsbetweenparticlesdoesnot destroythevalidityofsimplescalingarguments.Theorem2.5leaves openthequestionofnontriviallimitintherangeN −1/2 <<R<N −3/5 (forwhichwehavenoparticularcluetooffer).Theparticlesvelocities couldthenbecomeunboundedandthelimitwouldnotnecessarilybe freetransport(astheprobabilityofinteractingwithanotherparticle wouldincreaseaccordingly). Finallyandforthesakeofcompleteness,letusmentionthatthe limitdynamicisalsoanimportantandfertilefieldofstudyinthe caseoflongrangepotentials(andobviouslywithaweakerscaling). Vlasov-likekineticequationsarethenexpected.Thefirstrigorousre- sultswereobtainedbyBraunandHepp[4],NeunzertandWick[14] andDobrushin[6],andconcernveryregularinteractionpotentials.We alsorefertoSpohn[17]wheretheWassersteindistanceisusedtoob- taintheconvergence.Tworeferencesrelatedtonumericalsimulations withparticles’methodsareforinstanceVictoryandAllen[18]and Wollman[19].Alltheseworksrequirearegularinteractionkernel(at