Ising Model -- from Eric Weisstein's World of Physics

A simple model used in statistical mechanics. The Ising model tries to imitate behaviour in which individual elements (e.g., atoms, animals, protein folds, biological membrane, social behavior, etc.) modify their behavior so as to conform to the behavior of other individuals in their vicinity. The Ising model has more recently been used to model phase separation in binary alloys and spin glasses. In biology, it can model neural networks, flocking birds, or beating heart cells. It can also be applied in sociology. More than 12,000 papers have been published between 1969 and 1997 using the Ising model.

This Ising model was proposed in the 1924 doctoral thesis of Ernst Ising, a student of W. Lenz. Ising tried to explain certain empirically observed facts about ferromagnetic materials using a model of proposed by Lenz (1920). It was referred to in Heisenberg's (1928) paper which used the exchange mechanism to describe ferromagnetism. The name became well-established with the publication of a paper by Peierls (1936), which gave a non-rigorous proof that spontaneous magnetization must exist. A breakthrough occurred when it was shown that a matrix $Eric Weisstein's World of Math$ formulation of the model allows the partition function to be related to the largest eigenvalue $Eric Weisstein's World of Math$ of the matrix $Eric Weisstein's World of Math$ (Kramers and Wannier 1941, Montroll 1941, 1942, Kubo 1943). Kramers and Wannier (1941) calculated the Curie temperature using a two-dimensional Ising model, and a complete analytic solution was subsequently given by Onsager (1944).

Adams, C. C. §7.4 and 8.3 in The Knot Book. New York: W. H. Freeman, 1994.

Cipra, B. A. "An Introduction to the Ising Model." Amer. Math. Monthly 94, 937-959, 1987.

De La Harpe, P. and Jones, V. F. R. "Graph Invariants Related to Statistical Mechanical Models: Examples and Problems." J. Combin. Th., 207-227, 1995.

Ge, M.-L.; Hu, L.; and Wang, Y. "Knot Theory, Partition Function and Fractals." J. Knot Th. Ramifications 5, 37-54, 1996.

Heisenberg, W. "Zur Theorie des Ferromagnetismus." Zeitschr. f. Physik 49, 619-636, 1928. [German].

Ising, E. "Beitrag zur Theorie des Ferromagnetismus." Zeitschr.. f. Physik 31, 253-258, 1925. [German].

Jones, V. F. R. "On Knot Invariants Related to Some Statistical Mechanical Models." Pacific J. Math. 137, 311-334, 1989.

Kindermann, R. and Snell, J. L. "Markov Random Fields and their Applications." J. Amer. Math. Soc. 1, 1-22.

Kobe, S. "Ernst Ising--Physicist and Teacher." http://www.physik.tu-dresden.de/itp/members/kobe/isingcor.ps.

Kramers, H. A. and Wannier, G. H. "Statistics of the Two-Dimensional Ferromagnet. Part I." Phys. Rev. 60, 252-262, 1941.

Kubo, E. "An Analytic Method in Statistical Mechanics." Busseiron Kenkyu 1, 1-13, 1943. [Japanese].

Lenz, W. "Beitrag zum Verständnis der magnetischen Eigenschaften in festen Körpern." Phys. Zeitschr. 21, 613-615, 1920.

Montroll, E. "Statistical Mechanics of Nearest Neighbor Systems." J. Chem. Phys. 9, 706, 1941.

Montroll, E. J. Chem. Phys. 10, 61, 1942.

Onsager, L. "Crystal Statistics. I. A Two-Dimensional Model with a Order-Disorder Transition." Phys. Rev. 65, 117-149, 1944.

Peierls, R. "On Ising's Model of Ferromagnetism." Proc. Cambridge Phil. Soc. 32, 477-481, 1936.