Ising model

This is an applet shows the Ising model in action. Ising model may be used to describe the behavior of simple magnets.

Our system is composed by NxN square cells (you can alter the size of the lattice at the Input Panel), which represents one type of state i.e. for spins there are two states up and down. Once you press the button Start, the simulation starts from a random configuration and equilibrates. If the default values have not been changed the system will converge by the elimination of one of the species (spins). Why? Let's have a look how the algorithms work.

Metropolis: we choose a random cell and temporarily change its state. Now we measure the energy of the chosen cell; if the energy is reduced then we accept the change; if not then we can still accept the change only with a Boltzmann probability. If we do not accept the change we undo the initial change.

Kawasaki: we choose two random cells with opposite states and we temporarily swap their states. Now we measure the energies of the chosen cells; if the total energy is reduced then we accept the change; if not then we can still accept the change only with a Boltzmann probability. If we do not accept the change we undo the initial changes.

It is obvious that in Metropolis algorithm do not keep the population of the states fixed and one of the populations will dominate! There are two ways to undermine complete domination of one state spins. Either reduce the value J (at the Input Panel), which represents the energy interaction between two neighbors or increase the system's temperature T, which indirectly undermines the energy interactions. For J = 0.42 and T = 1.0, we will observe that the spins start to cluster but neither will be able to dominate. Feel free to play with these values!

In Kawasaki algorithm the populations of the spins remain fixed. The system will try to equilibrate by minimizing the surface contact between the populations. High temperature T and low J will undermine that.