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1、CommunicationsinCommun.math.Phys.56,125—146(1977)MathematicalPhysics©bySpringer-Verlag1977TheVlasovDynamicsandItsFluctuationsintheIfNLimitofInteractingClassicalParticlesW.BraunandK.HeppPhysicsDepartment,ETH,CH-8093Zurich,SwitzerlandAbstract.ForclassicalTV-particlesystemswithpairinteractionN
2、"1XΦ(^ΐ~3j)trιeVlasovdynamicsisshowntobethew*-limitasl^i^J^NN-+QO.Propagationofmolecularchaosholdsinthislimit,andthefluctuationsofintensiveobservablesconvergetoaGaussianstochasticprocess.§1.IntroductionConsidertheNewtonianequationx(ί,α,μ)=fμ(dfc)f(jc(ί,α,μ)-2c(ί,&,μ))(1.1)foraparticlewithin
3、itialconditionz(0,α,μ)=(x(0,α,μx(0,a,μ))=a=(q,p)(1.2)interactingviaaregular2-bodyforceF(q)=—Fφ(g)=—E(—%)withotherparticleshavinginitialconditionsdistributedoverarealBorelmeasureμonIR6.ThisframeworkcontainsthecanonicaldynamicsofNmasspoints*;,(*,«*)=Σf(*„(*,αtf)-xjί,%)),(1.3)m=1wherel^n^Nand
4、withinitialconditionaN=(al9...,aN).For,letμaN(dd)=£δan(da)and2c(ί,α,μα2V)bethesolutionof(1.1).Thennxn(ί,αjv)=x(t,αn,μaN)(1.4)isthesolutionof(1.3).Ontheotherhandforμf(dά)=f(ά)datheNewtonianEquation(1.1)alsosolvestheVlasovEquation[1]:(ί,α)=-E(ί,α)-(ί,α)Jdcίf(t,a')F(Q-f)(1.5)102W.BraunandK.Hep
5、pwithinitialcondition/(O,a)=f(a)eC1,ifJdaf(d)^1.Forthispurposeweobservethatthemappinga-^z(t,a,μ)iscanonical.Thenaneasycalculation(2.11)showsthat/(ί,z(t9a,μf))=f(a)satisfies(1.5).WeremarkthattheJV-particledynamics(1.3)andtheVlasovdynamics(1.5)representdisjointspecializationsof(1.1)formeasu
6、resμ"Nandμf(da)=f(a)da.WeexpectthattheVlasovdynamicsdescribesasymptoticallythetimeevolutionoftheparticlenumberdensityin1R6oftheΛΓ-bodysystemwithinitialcondition<%,wheneverμ^N-1Σ<5an—/(i6)n=lforJV->oo.TheNewtonianequationsforμ*Nareequivalentto(1.3)ifthe2-bodyforceFisreplacedbyN'1^.Weshallsee
7、inSection2that(1.6)impliesz(ί,α,μ#)->z(ί,α,//).HencetheVlasovdynamicsdoesindeeddescribethecon-tinuumlimitforpointparticleconfigurationswheretheinteractionenergyisscaleddowntoretainafiniteenergyperparticle.TheVlasovdynamicsshouldalsodescribeasymptotically
