Ising model

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The Ising model, named after the physicist Ernst Ising, is a mathematical model in statistical mechanics. It can be represented on a graph where its configuration space is the set of all possible assignments of +1 or -1 to each vertex of the graph. The graph can exhibit periodic boundary conditions or free space boundary conditions depending on the system being modelled. To complete the model, a function, E(e) must be defined, giving the difference between the energy of the "bond" associated with the edge when the spins on both ends of the bond are opposite and the energy when they are aligned. It is also possible to include an external magnetic field.

At a finite temperature, T, the probability of a configuration is proportional to

<math>e^{-\sum_e E(e)}</math>.

See partition function (statistical mechanics). A full mathematical development of the Ising model and its solution in 1D is given in the article on the Potts model.

In his 1925 PhD thesis, Ising solved the model for the 1D case. In one dimension, the solution admits no phase transition. On the basis of this result, he incorrectly concluded that his model does not exhibit phase behaviour in any dimension.

Most solutions involve using the Metropolis-Hastings algorithm running inside a Monte Carlo loop. Depending on the complexity only adjacent vertices can be taken into account or for long-range models other vertices can be included.

The Ising model undergoes a phase transition between an ordered and a disordered phase in 2 dimensions or more. In 2 dimensions, the Ising model has a strong/weak duality (between high temperatures and low ones) called the Kramers-Wannier duality. The fixed point of this duality is at the second-order phase transition temperature.

While the Ising model is an extremely simplified description of ferromagnetism, its importance is underscored by the fact that other systems can be mapped exactly or approximately to the Ising system. The grand canonical ensemble formulation of the lattice gas model, for example, can be mapped exactly to the canonical ensemble formulation of the Ising model. The mapping allows one to exploit simulation and analytical results of the Ising model to answer questions about the related models.

The Ising model in two dimensions, and in the absence of an external magnetic field, was analytically solved at the critical point in 1944 by Lars Onsager but the 3D Ising model has not been analytically solved and is thought to be computationally intractable.

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References

Barry M. McCoy and Tai Tsun Wu, The Two-Dimensional Ising Model, (1973) Harvard University Press, Cambridge Massachusetts, ISBN 674-91440-6.
non-primitive mean spherical approximation model, fluid phase equilibria, 209(2003)1,13-27, Morteza Lotfikian

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