The Ising Model
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.
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.
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 .
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