The Ising ModelPPT课件下载推荐.pptx

1、,The Ising Model,Mathematical Biology Lecture 5 James A.Glazier,（Partially Based on Koonin and Meredith,Computational Physics,Chapter 8）,Ising Model Basics,A Simple,Classical Model of a Magnetic Material.A Lattice（Usually Regular）with a Magnet or Classical Spin at Each Site,Aligned Either Up,or Down

2、:#Would be,（in Quantum Mechanics）.,The Spins Interact with Each Other Via a Coupling of Strength J and to an External Applied Magnetic Field B.The Two Spin Interactions are:#=J=-J=-J,=J,Ising Model BasicsContinued,The Total Energy of the Spins is theHamiltonian:#,If J0 have a Ferromagnet.Energy is L

3、owest if all,s are the Same.Favored.,NS,NSIf J0 have an Antiferromagnet.Energy is Lowest if neighboring s are Opposite.Favored.,Ising Model BasicsConclusion,-1+1+1,The One-Dimensional Ising Model is Exactly Soluble and Is a Homework Problem in Graduate-Level Statistical Mechanics.The Two-Dimensional

4、 Ising Model is Also Exactly Soluble（Onsager）but is Impressively Messy.The Three-Dimensional Ising Model is Unsolved.Can Have Longer-Range Interactions,Which can have Different J for Different Ranges.Can Result in Complex Behaviors,E.g.Neural Networks.Similarly,Triangular Lattices and J0 can Produce

5、 Complex Behaviors,E.g.Frustration and Spin-Glasses.+1-1Cant Satisfy AllBonds.,Examples,Note:#Maximum Energy is+24J and Minimum-24Jfor 3 x 3 Lattice（Absorbing Boundaries）.,Thermodynamics,What are The Statistical Properties of the Lattice at a Given Temperature T?#Define the State of the Lattice as a

6、 Vector:#,Example:#,For bTcritical）Spins are Essentially Random.I.e.the Probability of All Configurations is Essentially Equal.For bbcritical（I.e.T0,Configurations with Almost All Spins Aligned are Much More Probable.Tcritical is the Nel or Curie Temperature.For J=1,bcritical0.44 or Tcritical1.6As M

7、agnets are Heated,their Magnetization Disappears.,Statistical Mechanics,In Thermodynamics All Statistical Properties Are Determined by the Partition Function Z:#The Probability of a Particular Configurationis:#,Degeneracy,In the Low Temperature Limit,Can have Multiple Equivalent of Degenerate States

8、 with the Lowest Energy.These will be Equally Probable.The Change from a Large Number of Equiprobable Random States to Picking（Randomly）One of Several Degenerate States is a Spontaneous Symmetry Breaking.Example:#,For Very Low Temperatures,the Probability of Flipping Betweenthe Two States is Near 0.

9、For Higher Temperatures,Flipping Occurs（Causes Problems for Small Magnets,E.g.in Disk Drives）,Ising Metropolis-Boltzmann Dynamics,Pick a Lattice Site at Random and Try to Swap the Spin Between+1-Example,If HtH then Accept the SwapIf HtH then Accept the Swap with Probabilitya Boltzmann FactorMaking a

10、s Many Spin-Flip Attempts as Lattice Sites Defines One Monte Carlo Step MCS,Alternative Dynamics,Generating the Trial States Optimally is ComplexBoth Deterministic and Random AlgorithmsAlternative Dynamics Include Kawasaki Pick Two Sites at Random and Swap Their Spins Fundamentally Different From Metropolis Since Total Number of s and-s is Conserved Thus Samples a Different Configuration Space From Single-Spin Dynamics,