When environment, such as temperature, changes, the system which consists of many interacting elements may shift to the state where the law of a micro level is broken.
For example, if a hot metal is cooled, the magnetization will appear spontaneously. That is, the small magnetization which each atom has will be gathered in the specific direction. Since the small magnetization of each atom does not necessary like a specific direction with the law of a micro level, the spontaneous magnetization is a specific character for many atoms which interact each other.
In the Ising model, each atom is in a lattice site and the magnetization takes only two opposite states like north or south. And each atom only interacts with the nearest neighbor sites so that the direction of their magnetizations become equal when the temperature falls. However, the direction of either is not necessarily preferred.
Although it is in fact a simple model, the phenomenon that the whole atoms take a same direction occurs at a certain temperature. This corresponds to the appearance of the spontaneous magnetization above-mentioned. Thus, the Ising model is very well researched as a fundamental model in such the phenomenon.
In the present demonstration, we can see the state of the Ising model on the two-dimensional square lattice at a temperature. Please change the temperature by the vertical slide, you can see how the state change. The state of magnetization of each lattice point is expressed with red and white. At low temperature, the big island appears rapidly. It corresponds to the spontaneous magnetization.
In this simulation, we can change algorithms which generates the states. Especially the Swendsen-Wang or Wolff algorithm can sample various states at a small step. These algorithm are often used in the recent Monte Carlo simulations.
$Id: monte-en.html 77 2011-07-29 09:28:41Z kenji $