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

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FREETRANSPORTLIMITFORN-PARTICLESDYNAMICSWITHSINGULARANDSHORTRANGEPOTENTIAL.J.BARRE´ANDP.E.JABINAbstract.Westudythelimitofsystemsofinteractingparticles,whenthenumberofparticlebecomesverylarge.Thesupportoftheinteractionvanishesasthenumberofparticlesgoestoinﬁnity,sothatthenaturallimitisjustfreetransport,butnolimitationisassumedaboutthestrengthoftheinteraction.Weobtainexplicitestimatesforthenumberofparticleseﬀectivelyinteractinganddescribethewaytheydoit.1.IntroductionWestudythedynamicsofmanyinteractingparticlesinthelimitofaninﬁnitenumberofparticles.Theforceactingoneachparticleisasumofpair-wiseinteractionwiththeotherparticles.ThekindofdynamicsthatisexpectedatthelimitdependsonthescalingoftheforcetermwithrespecttothenumberofparticlesN.HereweconsiderveryshortrangeinteractioninthesensethattheforcebetweentwoparticlesvanishesiftheirdistanceislargerthanRwith,indimension3NR2<<1.Inthisscalingtheformallimitissimplyfreetransport:eachparticlemoveswithitsinitialvelocity.Indeedaformalcomputationeasilyshowsthatonaverageaparticleshouldneverundergoacollision:i.e.thenumberofparticlescomingatadistancelessthanRtoanotherisnegligibleinfrontofN,inatimeintervaloforder1.ThisisasimpliﬁedproblemforthemoreinterestingcaseNR2=constwheretheobtentionofcollisionalmodels(ofBoltzmanntype)isconjectured.TheﬁrstrigourousstepinthatdirectionwasobtainedbyLanfordinacelebratedwork[12](seealso[13]).Thisresultnever-thelesssuﬀersfromtwoimportantrestrictions:Firsttheparticlesarehardspheres(theyinteractwiththepotentialΦ(x)=+∞if|x|≤RandΦ=0if|x|>R).Andsecondthelimitisonlyvalidforatime2000MathematicsSubjectClassiﬁcation.Kinetictheoryofgases;Interactingparticlessystems.1

2J.BARRE´ANDP.E.JABINsmallwithrespecttotheaveragetimeittakesforoneparticletohaveacollision(orthemeanfreepathafterrescaling).ThissecondrestrictionwasimprovedbyCercignani,IllnerandPul-virentiin[5](seealso[10]);Thetimeintervalofvaliditywasstillﬁniteandoforderthemeanfreepaththough.Bothresultsdealwithhardspheresalthoughextensionstorepulsivepotentialsarementioned(see[13])butunpublished(asfarasweknow).Thoseresultsareeasilyextendedtoourscaling.Thelimitationonthetimeintervalisnomorearealissueasthemeanfreepathtendstoinﬁnity.Howevertheproofswouldstillrequirehardspheresinteraction(oratleastarepulsivepotential).Notethatinthecaseofanarbitrarypotential,thevelocityboundsneededtosubstantiatetheintuitionbehindtheformalscalingargumentarenoteasytoprove;inaddition,theveryconceptofcollisionneedsacarefuldeﬁnition.WestartwithProp.2.4whichjustiﬁesthefreetransportlimit,stillonlyforrepulsivepotentials.Theinterestismainlyinthesimplicityoftheproofastheresultisnotreallynew(althoughstrictlyspeakingneverstatedbeforeforapotentialwithsolittleregularity).ThemainresultisTheorem2.5.ItunfortunatelydealswithanevenshorterrangeR<<N−3/5butitassumesvirtuallynothingonthepotential(itdoesnotneedtoberepulsiveanditsscalecouldbeaslargeasonewants)andthereforeitisnotatallincludedinpreviouscontributions.Moreoverwenotonlyprovethefreetransportlimitbutalsodescribeallpossiblecollisionsequencesbetweenparticlesandtheresultisconsequentlymuchmoreprecise.Thecoreoftheproofistoensurethatthebuildupofcorrelationsbetweenparticlesdoesnotdestroythevalidityofsimplescalingarguments.Theorem2.5leavesopenthequestionofnontriviallimitintherangeN−1/2<<R<N−3/5(forwhichwehavenoparticularcluetooﬀer).Theparticlesvelocitiescouldthenbecomeunboundedandthelimitwouldnotnecessarilybefreetransport(astheprobabilityofinteractingwithanotherparticlewouldincreaseaccordingly).Finallyandforthesakeofcompleteness,letusmentionthatthelimitdynamicisalsoanimportantandfertileﬁeldofstudyinthecaseoflongrangepotentials(andobviouslywithaweakerscaling).Vlasov-likekineticequationsarethenexpected.Theﬁrstrigorousre-sultswereobtainedbyBraunandHepp[4],NeunzertandWick[14]andDobrushin[6],andconcernveryregularinteractionpotentials.WealsorefertoSpohn[17]wheretheWassersteindistanceisusedtoob-taintheconvergence.Tworeferencesrelatedtonumericalsimulationswithparticles’methodsareforinstanceVictoryandAllen[18]andWollman[19].Alltheseworksrequirearegularinteractionkernel(at

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