Ising model

 转自:http://en.wikipedia.org/wiki/Ising_model

The Ising model (English pronunciation: /ˈaɪsɪŋ/,German: [iːzɪŋ]) is amathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables called spins that can be in one of two states. The spins are arranged in a lattice or graph, and each spin interacts at most with its nearest neighbors. The goal is to findphase changes in the Ising model, as a simplified model of phase changes in real substances. The two-dimensionalsquare lattice Ising model is one of the simplest statistical models to show a phase change.

The Ising model was invented by the physicist Wilhelm Lenz (1920) who gave it as a problem to his studentErnst Ising after whom it is named. The one-dimensional Ising model has no phase change and was in 1924 already solved by Ernst Ising himself. The two-dimensional square lattice Ising model is much harder and, in the case of zero magnetic field, was given a complete analytic description much later, byLars Onsager (1944). It is usually solved by atransfer-matrix method, although there exist different approaches, more related toquantum field theory.

In dimensions greater than four, the phase-transition of the Ising model is described bymean field theory.

Istrail (2000) showed that computing thefree energy of an arbitrary subgraph of an Ising model on a lattice of dimension three or more is computationally intractable so that no method of approximation will allow computation of the thermodynamic properties of arbitrary subgraphs in dimensions higher than two.