On the convergence to statistical equilibrium for

arXiv:math-ph/0210039v228Oct2002SubmittedtoJournalofMathematicalPhysics,2002OntheConvergencetoStatisticalEquilibriumforHarmonicCrystalsT.V.Dudnikova1M.V.KeldyshInstituteofAppliedMathematicsRASMoscow125047,Russiae-mail:[e-mail],[e-mail]1,2InstituteofMathematicsViennaUniversityViennaA-1090,Austriae-mail:[e-mail]chnischeUniversit¨atM¨unchenD-80290,Germanye-mail:[e-mail]ynamicsofaharmoniccrystalinddimensionswithncompo-nents,d,narbitrary,d,n≥1,andstudythedistributionμtofthesolutionattimet∈R.Theinitialmeasureμ0hasatranslation-invariantcorrelationmatrix,zeromean,andfinitemeanenergydensity.ItalsosatisfiesaRosenblatt-resp.Ibragimov-Linniktypemixingcondition.ThemainresultistheconvergenceofμttoaGaussianmeasureast→∞.TheproofisbasedonthelongtimeasymptoticsoftheGreen’sfunctionandonBernstein’s“room-corridors”method.1SupportedpartlybytheSTARTproject”NonlinearSchr¨odingerandQuantumBoltzmannEquations”(FWFY137-TEC)ofN.J.MauserandresearchgrantsofDFG(436RUS113/615/0-1)andRFBR(01-01-04002).2OnleaveDepartmentofMechanicsandMathematics,MoscowStateUniversity,Moscow119899,Russia.SupportedpartlybyMax-PlanckInstitutefortheMathematicsinSciences(Leipzig).1IntroductionDespiteconsiderableefforts,theconvergencetoequilibriumforamechanicalsystemhasre-mainedasanextremelydifficultproblem.Ithasbeenrecognizedearlyonthatforaninfinitelyextendedsystem,possiblyontopoflocalhyperbolicity,theflowofstatisticalinformationtoinfinityservesasamechanismforrelaxation.Thetwoprimeexamplesaretheidealgasandtheharmoniccrystal.Weconsiderherethelattercase.Intheharmonicapproximationthecrystalischaracterizedbythedisplacementfieldu(x),wherex∈Γ,ΓisaregularlatticeinRd,andu(x)∈Rnwithndependingonthenumberofatomsintheunitcell.Thefieldu(x)isgovernedbyadiscretewaveequation.Wewillconsiderarbitraryd,nandfornotationalsimplicitysetΓ=Zd.Ourmotivationtoreturntoawellstudiedmodelistoamuchwiderclassofinitialmeasuresthanbefore.Thisprojectrequiresnovelmathematicaltechniques.TheyhavebeendevelopedforthewaveandKlein-GordonequationonRdin[6]-[8],butthediscretestructureposesextradifficulties.Letusbrieflycommentonpreviouswork.In[14]ageneralcriterionisgivenwhichensuresmixingandBernoullinessofthecorrespondingmechanicalflow.TherebytheconvergencetoequilibriumisestablishedforinitialmeasureswhichareabsolutelycontinuouswithrespecttothecanonicalGaussianmeasure.In[14]momentsofthedisplacementfieldarestudied.ThisallowstoreducethespectralanalysisoftheLiouvilleanflowtothespectralpropertiesofthedynamicalgroupdefinedonsolutionsoffiniteenergy.Sincethecrystalisassumedtobehomogeneous,thesespectralpropertiesaredeterminedbythedispersionrelationsωk(θ),k=1,...,n.TheLiouvilleanflowismixingandevenBernoulli,if,exceptforcrossingpoints,eachωk(θ)isareal-analyticfunctionwhichisnotidenticallyconstant.Inparticular,theLebesguemeasureoftheset{θ∈Td:∇ωk(θ)=0}isequaltozero.In[20],forthecased=n=1,initialmeasuresareconsideredwhichhavedistincttemperaturestotheleftandtotheright.In[2],againd=n=1,theconvergencetoequilibriumisprovedforamoregeneralclassofinitialmeasurescharacterizedbyamixingconditionofRosenblatt-resp.Ibragimov-Linniktypeandwhichareasymptoticallytranslation-invarianttotheleftandtotheright.Thedetailedstationaryphaseanalysisof[2]doesnotdirectlygeneralizetod≥2.Ratherwehavetodevelopanovel‘cutoffstrategy’whichmorecarefullyexploitsthemixingconditioninFourierspace.Thisapproachallowsustoalldwithinessencethesameconditionsforthedispersionrelationsasin[14].Ourextensionrequiresthetechniqueofholomorphicfunctionsofseveralcomplexvariables.Inparenthesisweremarkthat,fortheidealgas,R.DobrushinandYu.Suhov[3]firstrealizedtheimportanceofamixingconditionontheinitialmeasure.In[9]itisreplacedbytheconditionoffiniteentropyperunitvolumethusestablishingconvergencewheneverthespecificparticlenumber,energy,andentropyarefinite.Nosuchgeneralresultseemstobeavailablefortheharmoniccrystal.Weoutlineourmainresultandstrategyofproof.Thedisplacementfieldu(x)isthedeviationoftheconfigurationofcrystalatomsfromtheirequilibriumpositions.Assumingthemtobesmallandexpandingtheforcestolinearorderyieldsthediscretelinearwave1equation,¨u(x,t)=−Xy∈ZdV(x−y)u(y,t);u|t=0=u0(x),˙u|t=0=v0(x),x∈Zd.(1.1)Hereu(x,t)=(u1(x,t),...,un(x,t)),u0=(u01,...,u0n)∈Rnandcorrespondinglyforv0.V(x)istheinteraction(orforce)matrix,Vkl(x),k,l=1,...,n.Thedynamics(1.1)isinvariantunderlatticetranslations.LetusdenotebyY(t)=(Y0(t),Y1(t))≡(u(·,t),˙u(·,t)),Y0=(Y00,Y10)≡(u0(·),v0(·)).Then(1.1)takestheformofanevolutionequation,˙Y(t)=AY(t),t∈R;Y(0)=Y0.(1.2)Formally,thisistheHamiltoniansystemsinceAY=JV001!Y=J∇H(Y),J=01−10!.(1.3)HereVisaconvolutionoperatorwiththematrixkernelVandHistheHamiltonianfunctionalH(Y)=12hv,vi+12hVu,ui,Y=(u,v),(1.4)wherethekineticenergyisgivenby12hv,vi=12Xx∈Zd|v(x)|2andthepotentialenergyby12hVu,ui=12Xx,y∈ZdV(x−y)u(y),u(x),·,·beingtherealscalarproductintheEuclideanspaceRn.WeassumethattheinitialdatumY0isarandomelementoftheHilbertspaceHαofrealsequences,seeDefinition2.1below.Y0isdistributedaccordi