Introduction to the Ising Model

The Ising Model is one of the pillars of statistical mechanics. Each site can have two values (red/white, 1/0, +/-, ...), and neighboring sites have an energetic preference to be the same value. As a system of +/- spins, it is a model for magnetism: like iron, there is a temperature (the Neel point) above which the magnetization "melts" away. Run at high temperatures (say above 3) to see the melted state: run at low temperatures (below 2) to see the magnetized state.

Thought of as sites either occupied or vacant (1/0) on a lattice, it is a model for the liquid-gas transition: dense regions of occupied "liquid" are surrounded by dilute regions of mostly "gas". Our simulation isn't ideal for visualizing this, because we allow atoms to be created and destroyed: one of the possible projects involves writing a version which only allows sites to flip in pairs, moving atoms from occupied to empty sites. Try running at 2.4 for a while (to generate a "clumped up" state), and then drop to 2.0 or so. For a limited time, you will likely observe fairly well defined red regions and white regions: think of a red fluid with a few red vapor atoms in the white regions. As you run for longer, you'll see either the vapor or the fluid win: the last drop will evaporate, or the last bubble will collapse. Of course, real liquid/vapor transitions are in three dimensions: another project will be to write a version of the program which runs a three-dimensional Ising model.

Other Ising Model Presets

Description of the Model.
Phase Diagram
Magnetization M(T)
Domain Coarsening

Last modified: June 1, 1997

James P. Sethna,,

Statistical Mechanics: Entropy, Order Parameters, and Complexity, now available at Oxford University Press (USA, Europe).