Ising model
The Ising model (German pronunciation: [ˈiːzɪŋ]) (or Lenz-Ising model or Ising-Lenz model), named after the physicists Ernst Ising and Wilhelm Lenz, is a mathematical model of ferromagnetism in statistical mechanics. The model consists of discrete variables that represent magnetic dipole moments of atomic "spins" that can be in one of two states (+1 or −1). The spins are arranged in a graph, usually a lattice (where the local structure repeats periodically in all directions), allowing each spin to interact with its neighbors. Neighboring spins that agree have a lower energy than those that disagree; the system tends to the lowest energy but heat disturbs this tendency, thus creating the possibility of different structural phases. The model allows the identification of phase transitions as a simplified model of reality. The two-dimensional square-lattice Ising model is one of the simplest statistical models to show a phase transition. The Ising model was invented by the physicist Wilhelm Lenz, who gave it as a problem to his student Ernst Ising. The one-dimensional Ising model was solved by alone in his 1924 thesis; it has no phase transition. The two-dimensional square-lattice Ising model is much harder and was only given an analytic description much later, by Lars Onsager. It is usually solved by a transfer-matrix method, although there exist different approaches, more related to quantum field theory. In dimensions greater than four, the phase transition of the Ising model is described by mean-field theory. The Ising problem without an external field can be equivalently formulated as a graph maximum cut (Max-Cut) problem that can be solved via combinatorial optimization. (Wikipedia).
