# 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)
- Nucleation
- Domain Coarsening

Last modified: June 1, 1997
James P. Sethna, sethna@lassp.cornell.edu,
http://www.lassp.cornell.edu/sethna/sethna.html

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