同步涌現(xiàn)系綜序參量方法及其應(yīng)用

同步涌現(xiàn)：系綜序參量方法及其應(yīng)用鄭志剛，華僑大學(xué)系統(tǒng)科學(xué)研究所Email:IntroductionOrder-parameterdynamicsEnsembleorderparameter(EOP)ApproachKuramoto-SakaguchistarnetworksEOPapproach:ExtensionsConcludingRemarksOutlineKeywords:系綜序參量Ensembleorderparameter統(tǒng)計(jì)獨(dú)立性條件StatisticalIndependenceCondition主項(xiàng)近似Dominating-termapproximationReductionism:gloryanddilemma物理學(xué)發(fā)展300年：還原論----系統(tǒng)的拆裝:一切都?xì)w結(jié)為最基本的組成部分和決定它們行為的最基本的規(guī)律。最輝煌成就：原子結(jié)構(gòu)的探索原子是由原子核和電子構(gòu)成的，原子核是由中子和質(zhì)子構(gòu)成的，中子和質(zhì)子是由夸克構(gòu)成的，這些粒子的物理性質(zhì)是由它們之間的相互作用決定的。I.IntroductionsuperconductivitysuperfluidicityQuantumHalleffectJosephsoneffectBrainProtein多了就是不一樣涌現(xiàn)Emergence:Moreisdifferent平衡相變：對(duì)稱性自發(fā)破缺氣-液-固相變，鐵磁相變VanderWaals非平衡體系：非平衡相變與自組織IlyaPrigogine:DissipativeStructuresHermannHaken:Synergetics遠(yuǎn)平衡態(tài)：原有熱力學(xué)分支失穩(wěn)，出現(xiàn)新的結(jié)構(gòu)非平衡相變--自組織理論耗散結(jié)構(gòu)理論，協(xié)同學(xué)非平衡為有序之源，非平衡的世界豐富多彩StatisticalMechanics+NonlinearDynamics

→SystemsScience（系統(tǒng)科學(xué)）集體行為：大自由度系統(tǒng)出現(xiàn)集體行為時(shí)，系統(tǒng)的整體行為收縮到一個(gè)低維子空間（序參量空間）。協(xié)同學(xué)（H.Haken)：slavingprinciple----消去快變量，慢變量（序參量）主宰整體行為自組織理論（I.Prigogine）：利用宏觀變量方程來(lái)分析熱力學(xué)系統(tǒng)的非平衡行為與轉(zhuǎn)變同步：有序的涌現(xiàn)1673,ChristiaanHuygens相互作用擺鐘的同步擺動(dòng)螢火蟲的同步閃光1680,EngelbertKaempfer1935,HughSmith[HughSmith(1935),“Synchronousflashingoffireflies,”Science82:51.TheoreticalRoutetoSynchronizationWinfree的開創(chuàng)性工作（1967）A.T.Winfree,TheGeometryofBiologicalTime,Springer,NewYork,1980.

Kuramoto’sWorkfrom1975Y.Kuramoto,ChemicalOscillations,Waves,andTurbulence,Springer,Berlin,1984.KuramotoApproach同步轉(zhuǎn)變：非平衡相變序參量?jī)煞N解:Incoherentstate:homogeneousdistributionofphasesCoherentstate:lockingofphases兩種解在不同耦合強(qiáng)度下的穩(wěn)定性不同WhenN→∞,positionofthephasedistributionfunctionintotwoparts:自洽方程SuccessfulUnsuccessful不同集體態(tài)的穩(wěn)定性非定態(tài)的集體態(tài)研究同步理論研究的三個(gè)不同層次提出了同步分岔樹、非局域同步、部分同步的概念提出了同步與相空間維數(shù)轉(zhuǎn)變的關(guān)系----------------------------------------------------著作：《混沌控制》非線性科學(xué)叢書,上?？萍冀逃霭嫔?2000).《耦合非線性系統(tǒng)的合作行為與時(shí)空動(dòng)力學(xué)》（高教出版社，2004）BookChapter：Synchronizationofcoupledphaseoscillators,Chapter9,p293-327,in“AdvancesinElectricalEngineeringResearch.Volume1”(NovaSciencePublishingHouse,2011).微觀動(dòng)力學(xué)層面：我們已有的研究工作論文：Phys.Rev.Lett.81,5318-5321(1998);Inter.J.Bif.&Chaos10,10,2399-2414(2000);Phys.Rev.E62,402-408(2000);62,7501–7504(2000);62,7882-7885(2000);62,3552-3557(2000);65,056211(2002);66,036208(2002);67,026223(2003);68,037202(2003);79,056210(2009);Europhys.Lett.74,2,229-235(2006);87,50006(2009);Chaos18,043109(2008);Sci.Rep.3,3522(2013);

J.Phys.A47,125101(2014).建立序參量動(dòng)力學(xué)：耦合振子高維相空間動(dòng)力學(xué)（微觀描述）的低維描述（宏觀描述）研究序參量動(dòng)力學(xué)與分岔：集體行為相變與涌現(xiàn)Why？大量微觀自由度競(jìng)爭(zhēng)的結(jié)果，序參量的涌現(xiàn)過(guò)程很重要在很多非平衡系統(tǒng)中，宏觀序參量并不唯一，并非不隨時(shí)間變化(e.g.,limitcycles,chaos,……)熱力學(xué)極限Thermodynamiclimit-----統(tǒng)計(jì)力學(xué)有限系統(tǒng)---漲落---系綜方法我們的使命：序參量分析的動(dòng)力學(xué)基礎(chǔ)是快慢變量競(jìng)爭(zhēng)，物理學(xué)基礎(chǔ)是系統(tǒng)的微觀對(duì)稱性與對(duì)稱破缺（類似于相變）序參量動(dòng)力學(xué)的穩(wěn)定性與分岔對(duì)應(yīng)于宏觀態(tài)及其相變吸引域及其變化決定了同步轉(zhuǎn)變的連續(xù)性（爆炸性同步）TheOtt-AntonsenApproachE.OttandT.M.Antonsen,Lowdimensionalbehavioroflargesystemsofgloballycoupledoscillators,Chaos18,037113(2008).Mean-fieldKuramotoModelIntroducinganorderparameterII.Order-parameterdynamicsMoregenerally一組廣義序參量HomogeneousparametersinhomogeneousparametersIdenticaloscillatorsThermodynamiclimitContinuityequation所有宏觀量都可以通過(guò)微觀變量的統(tǒng)計(jì)平均得到。Theorderparameters:MicroscopicdynamicsStatisticsMacroscopicdescriptionFourierexpansionAsimplecaseOnehasOtt-AntonsenAnsatz!Wegetasingleequationofsingleorder-parametereq.:InvariantmanifoldTheinfinitelymanydynamicalequationsoforderparametersarereducedtoasingleequationonthismanifoldandthecorrespondingphasedensityistheso-calledPoissonkerneldistribution:Poissonmanifold:Ifonechoosestheinitialstateonthismanifold,thestatewillneverevolveoutofit.Emergence:DimensionReductionInvariantmanifold:Infact,itisnotanansatz.SupposeOnegetsOA!III.Ensembleorderparameter(EOP)ApproachArequirementforstatisticaltreatmentofpopulationsofoscillators:Howtodoforfinitenumberofoscillators?OrderparameterThesameasthethermodynamic-limitcase.ItseemsthatifoneinsertstheOAAnsatz,onegetsthesameequationoforderparameter.Doesitwork?

------No.Inthecaseofafinitenumberofoscillators,thePoissonmanifoldsatisfyingtheOAansatzisnotattainable.Exceptthesynchronousstate,allthestatesofthesystemoffiniteoscillatorslieoutofthisPoissonmanifold.Wecanbuildanothersystemwiththesameorder-parameterequationsbutnotlimitedbythefinitesize.-----Theensemblemethod.LetusconsideranensemblewithMidenticalsystemsofNoscillatorsobeyingthesamedynamicalequationsandsameparameters.Theinitialphasesofoscillatorsarechosenfromthesamedistribution,whichmakesthemeanvaluesoforderparameters{αn}overtheensemblehavethelimitwhenthenumberofsamplingM→∞.Oneneedstoperformensembleaveragetotheaboveequation:Unclosed,notself-consistentTypicalcaseI:

CouplingfunctionwithonecomponentEOPequations:StatisticalIndependenceGeneralizedOAansatzThevalidityoftheaboveEOPequationdependslargelyonthedegreeofstatisticalindependence.ErrorfunctionIV.Kuramoto-SakaguchistarnetworksIngredientsofheterogeneousnetworks,e.g.,thescale-freenetworksByintroducingOrderparameterThesystemfallsintotheframeworkdiscussedabove-----thefirstthreenonzeroFouriercoeffcientsofthecouplingfunction.Finitesize:EnsembleorderparameterschemeIntroducingIfResultforstatisticalindependenceapprox.TheinitialconditionsarechosenfromthePoissonkernel.TheensembleorderparametersevolvesonaninvariantmanifoldForsystemsoffiniteoscillators,theensembleorderparametermethodplaysthedominantroleinrevealingthekeycollectivestructureofcoupledoscillatorsystems.ForadetailedstudyofsynchronizationdynamicsonstarnetworksintheframeworkofEOP,pleasereferto:CanXu,JianGao,YutingSun,XiaHuang,andZhigangZheng*,Sci.Rep.5,12039;YutingSun,JianGao,CanXu,XiaHuang,andZhigangZheng*,Ensembleorderparameteranalysisofsynchronizationonstarnetworks,PRE(2015).WhenmorethanthreeFouriercoefficientsofthecouplingfunctionarenonzero,thetraditionalOAschememayfail.BUTtheEOPapproachisstillapplicable!Thepriceoftheensembleorderparameterdescriptionistheerrorinducedbythestatisticalindependenceassumption,whosevaliditycanbecheckedbynumericalresults.V.EOPapproach:Extensions1.ThecaseofHigherordercouplingonlyEOPeqs.OAsolutioninvariantmanifoldsub-manifoldofthePoissonmanifold2.Mixtureofhigherordercouplings:theDominating-termassumptionCouplingfunctionwiththefirstfivenonzeroFouriercoefficientsEOPequations:OAsolutionTrivialDoesOA-likesolutionexist?thisconditioncanbesatisfiedaroundthevicinityofsynchronizationsatesandincoherentstates.Ingeneral,ifwechoosetheinitialconditionsas,andalongtheevolution,thedifferencesofhighertermsαn

withα_1^ncankeepsmallenough,theconsistentconditioncanbesatisfiednaturally.Dominating-termAssumptionThefirstequationisgotfromthedynamicsofthefirstorderparameter.Thesecondequationisgotfromthehigher-orderorderparameters.FortherelationofapproximatedOAansatz,thehighertermsaremuchsmallerthanthefirstone,thefirstequationcanbeconsideredasthedominatingterm.ExampleVI.ConcludingremarksWefocusedonthetheoreticaldescriptionoflow-dimensionalorder-parameterdynamicsofthecollectivebehaviorsincoupledphaseoscillators.Wederivedtheclosedformofdynamicalequationsfororderparameters,wherethePoissonkernelbeinganinvariantmanifoldandtheimprovedOAansatz.EOPapproachissuitableforsystemsofbothfiniteandinfinite,identicalandnonidenticaloscillators.Inourapproach,thescopeoftheOAansatzisdeterminedastwoparts,i.e.,thelimitofinfinitelymanyoscillatorsandtheconditionthatonlythefirstthreeFouriercoeffcientsofthecouplingstrengtharenonzero.合作者博士生：徐燦碩士生：高健

孫玉庭

向海蓉J.Gao,C.Xu,Y.Sun,andZ.Zheng*,Orderparameteranalysisforlow-dimensionalbehaviorsofcoupledphase-oscillators,ScientificReports(2016,submitted);