Ising model
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The Ising model, named after the physicist Ernst Ising, is a 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. 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 than when they are aligned. It's also possible to have 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).
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 behavior in all dimensions.
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 in 1944 by Lars Onsager.
See also
Further reading
- Clough, John; Douthett, Jack; and Krantz, Richard (2000). "Maximally Even Sets: A Discovery in Mathematical Music Theory is Found to Apply in Physics", Bridges: Mathematical Connections in Art, Music, and Science, p.193-200, Conference Proceedings 2000, ed. Reza Sarhangi. Winfield, Kansas: Central Plain Book Manufacturing.
External links
- Barry A. Cipra, "The Ising model is NP-complete", SIAM News, Vol. 33, No. 6; online edition (.pdf) (http://www.siam.org/siamnews/07-00/ising.pdf)
- Reports why the Ising model can't be solved exactly in general, since non-planar Ising models are NP-complete.
- Monte Carlo simulations (http://www.cp.tn.tudelft.nl/research/ising/): a java applet implementing various methods.
- Science World article on the Ising Model (http://scienceworld.wolfram.com/physics/IsingModel.html)
- An Ising Applet by Syracuse University (http://www.phy.syr.edu/courses/ijmp_c/Ising.html)
- Phase transitions on lattices (http://ibiblio.org/e-notes/Perc/contents.htm)
- Nature news article: The Ising on the cake (http://sugp.caltech.edu/sorin/The%20Ising%20on%20the%20cake.htm)
- Three-dimensional proof for Ising Model impossible, Sandia researcher claims (http://www.sandia.gov/media/NewsRel/NR2000/ising.htm)de:Ising-Modell