# The Ising Model

### Statistical Physics

Statistical Physics attempts to predict the properties of complex systems containing many interacting components undergoing random thermal motions. These might be molecules in a gas, atoms in a magnet, polymers in solution or a host of other interesting physical systems. One way in which physicists can gain insight into these difficult systems is by constructing idealised models which, it is hoped, will exhibit some of the interesting features of real systems but are simpler to study.

### A Simple Model

Perhaps the most famous of these simplified models is the two-dimensional Ising model which may be used to model the behavior of simple magnets. The model consists of a set of magnetic spins arranged on a regular square lattice. Each spin can in one of two states - which you can think of as `up' and `down'. The energy of the system is determined by the sum of elementary interactions between a spin and its neighbours on the lattice.

At very low temperatures the system will be found in its lowest energy state in which all the spins are, for example, up. As the system is heated the spins start to jiggle around and complicated motions result. The goal of statistical physics is to not to predict all of these detailed motions but only to calculate certain average properties of these motions, for example, how many spins on average are pointing up, what is the mean energy etc.

### Computer Simulation

One way to predict these averages is to simulate the Ising model on a computer. Our applet illustrates this for a small (8 by 8) lattice. The circles correspond to spins whose state up or down is shown by their color (black or red). Once the applet starts you should see that the spins are suffering random fluctuations from red to black and vice versa. The size of these fluctuations is determined by the temperature - more accurately by the inverse temperature 1/T. The slider allows you to change the temperature freely from zero (infinitely hot!) to 1 (a fairly low temperature). The Pause and Resume buttons allow you to stop the motion and look at the spins for a while. You should notice that for small values of this parameter black and red circles are equally probable - there is no net magnetization at sufficiently high temperature. But at for large values of the parameter the system prefers one color over another - it exhibits a net magnetization. Notice that for values about 0.4 the system cannot decide what to do - the net magnetization undergoes large fluctuations. Close to this temperature the system is said to undergo a phase transition .