DynamicalReplicaTheoryforDiluteSpinSystems：稀自旋系统的动力学复制理论

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1、DynamicalReplicaTheoryforDiluteSpinSystemsJonHatchettRIKENBSIIncollaborationwithI.PérezCastillo,A.C.C.Coolen&N.S.SkantzosMotivationOrderparameterflowinphysicalsystemsNon-equilibriumphenomenaTemperaturecyclingSlowrelaxationsNon-detailedbalancesystemsBiologicalmodelsComplexnetworksAnalysisof

2、algorithmsModelDetailsLocallytree-likestructureStochasticdynamicsaccordingtolocalfieldParallelSequential(Glauber,Metropolis)LangevinIsingorsoftspinsEvolutionofObservablesObservablesofintereste.g.WantevolutiononfinitetimesIngeneral,e.g.dependsonEvolutionofProbabilityStatevectorLinearequatio

3、nforE.g.MasterequationGeneralsetofobservableswithdistributionObtainKramers-MoyalexpansionforAssumptionsofDRTRetainLiouvilleterminKMexpansionExactifobservablesarewell-chosenDeterministicevolutionoforderparametersSelf-averagingobservablesatalltimesExpectthistobeexact(forcertainobservables)Eq

4、uipartitioninginthesubshellaverageIngeneral,thisisanapproximationProbabilityManifoldsManifoldofdistributionsondimS=2^N-1IdentifyobservableswithcoordinatesMisanexponentialfamilydimM=#observablesGeometryofSandMTwoimportantcoordinatesystemsforMExponentialfamilycoordinatesMixturefamilycoordina

5、tesConnectedviaLegendretransformationGeodesicforexponentialfamilyGeodesicformixturefamilyWhatdoAssumptionsdo?SMminimizesm-projectione-flatsubmanifoldWhatdoAssumptionsdo?IfthentheoryisexactE.g.atequilibriumForsomesetsofinitialconditionsForsomesimplersystems,observablesetsThisconditionisnotn

6、ecessaryforexactnessApproximationis“bestpossible”Sinceisunknown,theKLdivergenceisnotaccessibleGlauberDynamicsMasterequationHamiltonianEvolutionofobservablesGlauberDynamicsSimplechoiceofobservables(m,e)FindsimpleODEs:isthedistributionoflocalfieldsinasystemwithmagnetisationandenergyAlgorithm

7、GivenfindconjugatefieldsGenerallyachallenginginverseproblemWorkonagivendisorderinstanceRequiresbeliefpropagationtoconvergeGivenfindfielddistributionStandardequilibriumproblemCalculateandSolveodesviastandardmethodsComparisonwithSimulations3-regularspinglass(rsp