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The Ising model (; ), named after the physicist Ernst Ising, 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, allowing each spin to interact with its neighbors. 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.〔See , Chapters VI-VII.〕 The Ising model was invented by the physicist , who gave it as a problem to his student Ernst Ising. The one-dimensional Ising model has no phase transition and was solved by himself in his 1924 thesis.〔(Ernst Ising, ''Contribution to the Theory of Ferromagnetism'' )〕 The two-dimensional square lattice Ising model is much harder, and was given an analytic description much later, by . 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. ==Definition== Consider a set of lattice sites Λ, each with a set of adjacent sites (e.g. a graph) forming a ''d''-dimensional lattice. For each lattice site ''k'' ∈ Λ there is a discrete variable σ''k'' such that σ''k'' ∈ , representing the site's spin. A ''spin configuration'', σ = (σ''k'')''k''∈Λ is an assignment of spin value to each lattice site. For any two adjacent sites ''i'', ''j'' ∈ Λ one has an ''interaction'' ''J''''ij''. Also a site ''j'' ∈ Λ has an ''external magnetic field'' ''h''''j'' interacting with it. The ''energy'' of a configuration σ is given by the Hamiltonian function : where the first sum is over pairs of adjacent spins (every pair is counted once). The notation '' : where β = (''kBT'')−1 and the normalization constant : is the partition function. For a function ''f'' of the spins ("observable"), one denotes by : the expectation (mean value) of ''f''. The configuration probabilities ''P''β(σ) represent the probability of being in a state with configuration σ in equilibrium. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Ising model」の詳細全文を読む スポンサード リンク
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