Magnetization M(T) in the Ising Model

Start the system in a spin up state (by running at low temperatures with H>0). Find the average of the magnetization <M> at a series of temperatures leading up to the critical temperature T_c where M appears to approximately vanish. You'll need to equilibrate at each temperature before starting the averaging: watch M(t) and run for a while at each temperature until you stop seeing a monotonic decrease in M. (Near the critical point, you'll find that M can flip from positive to negative, even at temperatures where it won't stay near zero. You'll need to go to larger system sizes if you want good data near there.) Make a file of your values of temperature and magnetization: while you're at it, you might copy down the mean energy <E> and the fluctuations <(M-<M>)^2> and (<E-<E>)^2> too, for studying the susceptibility, and specific heat). Plot M(T). Does it look like M vanishes with a power law?

Don't get manic about it the first round: do a quick job to get the overall feel for the problem. You'll probably want to come back and add more data points near T_c, or perhaps run longer for better equilibration and/or better statistical averaging. Just remember to start with an aligned state if you're studying T<T_c: it takes a long time to reach ferromagnetic equilibrium if you start from a random state.

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).