Ising Model

• Every little box of the spin field represents one of the two possible states S_i=-1, 1 (white/blue).
• The energy is calculated from the formula E=-Sum_<i,j>S_iS_j where <i,j> symbolizes all pairs of nearest neighbours on the lattice.
• At infinite temperature the energy per spin (E/N, where N=L² is the number of spins) is zero. At zero temperature, all the spins are parallel and the energy per spin is -2.
• You can control the temperature either by typing a (positive) real number into the temperature field or by adjusting the thermometer by mouse.
• The critical temperature of the two dimensional Ising model is T_Crit=2/ln(1+sqrt2) = 2.269 . Initially the temperature is set to this value.
• The magnetization is simply the mean of all spins.

Observe the following:

1. At temperatures well above the critical temperatures, the spin arrangement converges to a nearly random arrangement, independent of the starting state: "Init cold", "Init warm" or "Init hot", and fluctuates quickly. We say that, above the critical temperature, there is a single thermodynamic state and this has zero magnetization. The spin arrangement is truly random at infinite temperature.
2. If you start below the critical temperature with "Init cold" (i.e. all the S_i=-1) you will see that just a few small cluster of blue (i.e. S_i=1) spins appear, and there is a non-zero (negative) magnetization. If we had started the simulation with all the S_i 1 (blue) then there would have been a net positive magnetization. We see that, below the critical temperature, there are two thermodynamic states (the "up spin" state with positive magnetization and the "down spin" state negative magnetization) and the system stays in one or the other depending on how the spins are initialized.
3. If you start below the critical temperature with "Init hot" or "Init warm" then you see that the system initially cannot make up its mind whether to go into the "up spin" or "down spin" state. Large clusters of each spin form. Eventually, if you let the simulation run for a long time, one of the states will win. Which one wins depends on the random thermal fluctuations. There is equal probability for it to be the "up spin" or "down spin" state.
4. For temperatures near the transition temperature, there are large clusters of spins with the same orientation, which fluctuate only very slowly. This is because the "correlation length" of an infinitely large system diverges at the critical point.

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Acknowledgement:
This applet is copied with only cosmetic changes from a program by Bernd Nottelmann at http://planck.uni-muenster.de:8080/java/ising.html.
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