The Ising ModelPPT课件下载推荐.pptx

The Ising ModelPPT课件下载推荐.pptx

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,TheIsingModel,MathematicalBiologyLecture5JamesA.Glazier,（PartiallyBasedonKooninandMeredith,ComputationalPhysics,Chapter8）,IsingModelBasics,ASimple,ClassicalModelofaMagneticMaterial.ALattice（UsuallyRegular）withaMagnetorClassicalSpinatEachSite,AlignedEitherUp,orDown:

@#@Wouldbe,（inQuantumMechanics）.,TheSpinsInteractwithEachOtherViaaCouplingofStrengthJandtoanExternalAppliedMagneticFieldB.TheTwoSpinInteractionsare:

@#@=J=-J=-J,=J,IsingModelBasicsContinued,TheTotalEnergyoftheSpinsistheHamiltonian:

@#@,IfJ0haveaFerromagnet.EnergyisLowestifall,saretheSame.Favored.,NS,NSIfJ0haveanAntiferromagnet.EnergyisLowestifneighboringsareOpposite.Favored.,IsingModelBasicsConclusion,-1+1+1,TheOne-DimensionalIsingModelisExactlySolubleandIsaHomeworkProbleminGraduate-LevelStatisticalMechanics.TheTwo-DimensionalIsingModelisAlsoExactlySoluble（Onsager）butisImpressivelyMessy.TheThree-DimensionalIsingModelisUnsolved.CanHaveLonger-RangeInteractions,WhichcanhaveDifferentJforDifferentRanges.CanResultinComplexBehaviors,E.g.NeuralNetworks.Similarly,TriangularLatticesandJ0canProduceComplexBehaviors,E.g.FrustrationandSpin-Glasses.+1-1CantSatisfyAllBonds.,Examples,Note:

@#@MaximumEnergyis+24JandMinimum-24Jfor3x3Lattice（AbsorbingBoundaries）.,Thermodynamics,WhatareTheStatisticalPropertiesoftheLatticeataGivenTemperatureT?

@#@DefinetheStateoftheLatticeasaVector:

@#@,Example:

@#@,ForbTcritical）SpinsareEssentiallyRandom.I.e.theProbabilityofAllConfigurationsisEssentiallyEqual.Forbbcritical（I.e.T0,ConfigurationswithAlmostAllSpinsAlignedareMuchMoreProbable.TcriticalistheNelorCurieTemperature.ForJ=1,bcritical0.44orTcritical1.6AsMagnetsareHeated,theirMagnetizationDisappears.,StatisticalMechanics,InThermodynamicsAllStatisticalPropertiesAreDeterminedbythePartitionFunctionZ:

@#@TheProbabilityofaParticularConfigurationis:

@#@,Degeneracy,IntheLowTemperatureLimit,CanhaveMultipleEquivalentofDegenerateStateswiththeLowestEnergy.ThesewillbeEquallyProbable.TheChangefromaLargeNumberofEquiprobableRandomStatestoPicking（Randomly）OneofSeveralDegenerateStatesisaSpontaneousSymmetryBreaking.Example:

@#@,ForVeryLowTemperatures,theProbabilityofFlippingBetweentheTwoStatesisNear0.ForHigherTemperatures,FlippingOccurs（CausesProblemsforSmallMagnets,E.g.inDiskDrives）,IsingMetropolis-BoltzmannDynamics,PickaLatticeSiteatRandomandTrytoSwaptheSpinBetween+1-Example,IfHtHthenAccepttheSwapIfHtHthenAccepttheSwapwithProbabilityaBoltzmannFactorMakingasManySpin-FlipAttemptsasLatticeSitesDefinesOneMonteCarloStepMCS,AlternativeDynamics,GeneratingtheTrialStatesOptimallyisComplexBothDeterministicandRandomAlgorithmsAlternativeDynamicsIncludeKawasakiPickTwoSitesatRandomandSwapTheirSpinsFundamentallyDifferentFromMetropolisSinceTotalNumberofsand-sisConservedThusSamplesaDifferentConfigurationSpaceFromSingle-SpinDynamics,