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: (in Quantum Mechanics Would be ). 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=J

Ising Model Basics—Continued The Total Energy of the ‘Spins’ is the Hamiltonian: If J>0 have a Ferromagnet. Energy is Lowest if all  are the Same. Favored. If J<0 have an Antiferromagnet. Energy is Lowest if neighboring  are Opposite. Favored. N S N S N S S N

The One-Dimensional Ising Model is Exactly Soluble and Is a Homework Problem in Graduate- Level Statistical Mechanics. The Two-Dimensional 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 J<0 can Produce Complex Behaviors, E.g. Frustration and Spin- Glasses. Can’t Satisfy All Bonds. Ising Model Basics—Conclusion +1 +1

Examples Note: Maximum Energy is +24J and Minimum -24J for 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 Vector: (i,j) (1,j)(2,j)(3,j) (i,1) (i,2) +1 (i,3) Example:

For  T critical ) Spins are Essentially Random. I.e. the Probability of All Configurations is Essentially Equal. For  >  critical (I.e. T 0, Configurations with Almost All Spins Aligned are Much More Probable. T critical is the Neél or Curie Temperature. For J=1,  critical ~0.44 or T critical ~1.6 As Magnets are Heated, their Magnetization Disappears. Statistical Mechanics In Thermodynamics All Statistical Properties Are Determined by the Partition Function Z : The Probability of a Particular Configuration is:

Degeneracy In the Low Temperature Limit, Can have Multiple Equivalent of Degenerate States 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 Between the Two States is Near 0. For HigherTemperatures, Flipping Occurs (Causes Problems for Small Magnets, E.g. in Disk Drives) +1

Ising Metropolis-Boltzmann Dynamics Pick a Lattice Site at Random and Try to Swap the Spin Between +1  -1. Example:  If H t <H 0 then Accept the Swap. If H t >H 0 then Accept the Swap with Probability: a Boltzmann Factor. Making as Many Spin-Flip Attempts as Lattice Sites Defines One Monte Carlo Step (MCS)

Alternative Dynamics Generating the Trial States Optimally is Complex. Both Deterministic and Random Algorithms. Alternative Dynamics Include Kawasaki (Pick Two Sites at Random and Swap Their Spins). Fundamentally Different From Metropolis Since Total Number of +1s and -1s is Conserved. Thus Samples a Different Configuration Space From Single-Spin Dynamics.