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The '''Ising model''' <ref>[http://dx.doi.org/10.1007/BF02980577 Ernst Ising "Beitrag zur Theorie des Ferromagnetismus", Zeitschrift für Physik A Hadrons and Nuclei '''31''' pp. 253-258 (1925)]</ref> (also known as the '''Lenz-Ising''' model) is commonly defined over an ordered lattice.  
The '''Ising model''' is also known as the '''Lenz-Ising''' model <ref>[http://dx.doi.org/10.1007/BF02980577 Ernst Ising "Beitrag zur Theorie des Ferromagnetismus", Zeitschrift für Physik A Hadrons and Nuclei '''31''' pp. 253-258 (1925)]</ref>. For a history of the Lenz-Ising model see <ref>[http://dx.doi.org/10.1103/RevModPhys.39.883  S. G. Brush "History of the Lenz-Ising Model", Reviews of Modern Physics '''39''' pp. 883-893 (1967)]</ref>
<ref>[http://dx.doi.org/10.1007/s00407-004-0088-3 Martin Niss "History of the Lenz-Ising Model 1920-1950: From Ferromagnetic to Cooperative Phenomena", Archive for History of Exact Sciences '''59''' pp. 267-318 (2005)]</ref>
<ref>[http://dx.doi.org/10.1007/s00407-008-0039-5 Martin Niss "History of the Lenz–Ising Model 1950–1965: from irrelevance to relevance", Archive for History of Exact Sciences '''63''' pp. 243-287 (2009)]</ref>.
The Ising model is commonly defined over an ordered lattice.  
Each site of the lattice can adopt two states, <math>S \in \{-1, +1 \}</math>. Note that sometimes these states are referred to as ''spins'' and the values are referred to as ''down'' and ''up'' respectively.  
Each site of the lattice can adopt two states, <math>S \in \{-1, +1 \}</math>. Note that sometimes these states are referred to as ''spins'' and the values are referred to as ''down'' and ''up'' respectively.  
The energy of the system is the sum of pair interactions
The energy of the system is the sum of pair interactions
between nearest neighbors.
between nearest neighbors.
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where <math>k_B</math> is the [[Boltzmann constant]], <math>T</math> is the [[temperature]],  <math> \langle ij \rangle </math> indicates that the sum is performed over nearest neighbors, and
where <math>k_B</math> is the [[Boltzmann constant]], <math>T</math> is the [[temperature]],  <math> \langle ij \rangle </math> indicates that the sum is performed over nearest neighbors, and
<math> S_i </math> indicates the state of the i-th site, and <math> K </math> is the coupling constant.
<math> S_i </math> indicates the state of the i-th site, and <math> K </math> is the coupling constant.
 
For a detailed and very readable history of the Lenz-Ising model see the following references:<ref>[http://dx.doi.org/10.1103/RevModPhys.39.883  S. G. Brush "History of the Lenz-Ising Model", Reviews of Modern Physics '''39''' pp. 883-893 (1967)]</ref>
<ref>[http://dx.doi.org/10.1007/s00407-004-0088-3 Martin Niss "History of the Lenz-Ising Model 1920-1950: From Ferromagnetic to Cooperative Phenomena", Archive for History of Exact Sciences '''59''' pp. 267-318 (2005)]</ref>
<ref>[http://dx.doi.org/10.1007/s00407-008-0039-5 Martin Niss "History of the Lenz–Ising Model 1950–1965: from irrelevance to relevance", Archive for History of Exact Sciences '''63''' pp. 243-287 (2009)]</ref>.
==1-dimensional Ising model==
==1-dimensional Ising model==
:''Main article: [[1-dimensional Ising model]]''
:''Main article: [[1-dimensional Ising model]]''
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