Mathematical model of ferromagnetism in statistical mechanics
Statistical mechanics
Thermodynamics
Kinetic theory
Particle statistics
Spin–statistics theorem
Indistinguishable particles
Maxwell–Boltzmann
Bose–Einstein
Fermi–Dirac
Parastatistics
Anyonic statistics
Braid statistics
Thermodynamic ensembles
NVE Microcanonical
NVT Canonical
µVT Grand canonical
NPH Isoenthalpic–isobaric
NPT Isothermal–isobaric
Models
Debye
Einstein
Ising
Potts
Potentials
Internal energy
Enthalpy
Helmholtz free energy
Gibbs free energy
Grand potential / Landau free energy
Scientists
Maxwell
Boltzmann
Bose
Gibbs
Einstein
Ehrenfest
von Neumann
Tolman
Debye
Fermi
v
t
e
The Ising model (or Lenz–Ising 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.[1]
The Ising model was invented by the physicist Wilhelm Lenz (1920), who gave it as a problem to his student Ernst Ising. The one-dimensional Ising model was solved by Ising (1925) alone in his 1924 thesis;[2] 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 (1944). 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 model for greater dimensions was also explored with respect to various tree topologies in the late 1970s, culminating in an exact solution of the zero-field, time-independent Barth (1981) model for closed Cayley trees of arbitrary branching ratio, and thereby, arbitrarily large dimensionality within tree branches. The solution to this model exhibited a new, unusual phase transition behavior, along with non-vanishing long-range and nearest-neighbor spin-spin correlations, deemed relevant to large neural networks as one of its possible applications.
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.
^See Gallavotti (1999), Chapters VI-VII.
^Ernst Ising, Contribution to the Theory of Ferromagnetism
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