Ising model: Difference between revisions
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between nearest neighbors. | between nearest neighbors. | ||
<math> \frac{U}{k_B T} = - K \sum_{\langle ij \rangle} S_i S_j </math> | :<math> \frac{U}{k_B T} = - K \sum_{\langle ij \rangle} S_i S_j </math> | ||
where <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 | <math> S_i </math> indicates the state of the i-th site, and <math> K </math> is the coupling constant. | ||
==1-dimensional Ising model== | ==1-dimensional Ising model== | ||
* [[1-dimensional Ising model]] (exact solution) | * [[1-dimensional Ising model]] (exact solution) | ||
Revision as of 16:28, 22 January 2008
The Ising model is also known as the Lenz-Ising model. For a history of the Lenz-Ising model see Refs. 1 and 2. The Ising model is commonly defined over an ordered lattice. Each site of the lattice can adopt two states: either UP (S=+1) or DOWN (S=-1).
The energy of the system is the sum of pair interactions between nearest neighbors.
- Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{U}{k_B T} = - K \sum_{\langle ij \rangle} S_i S_j }
where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k_B} is the Boltzmann constant, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle T} is the temperature, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle ij \rangle } indicates that the sum is performed over nearest neighbors, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_i } indicates the state of the i-th site, and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K } is the coupling constant.
1-dimensional Ising model
- 1-dimensional Ising model (exact solution)
2-dimensional Ising model
Solved by Lars Onsager in 1944.
3-dimensional Ising model
Sorin Istrail has shown that the solution of Ising's model cannot be extended into three dimensions for any lattice:
- Three-dimensional proof for Ising model impossible, Sandia researcher claims to have shown
- Sorin Istrail "Statistical mechanics, three-dimensionality and NP-completeness: I. Universality of intracatability for the partition function of the Ising model across non-planar surfaces", Proceedings of the thirty-second annual ACM symposium on Theory of computing pp. 87 - 96 (2000)
ANNNI model
The axial next-nearest neighbour Ising (ANNNI) model is used to study alloys, adsorbates, ferroelectrics, magnetic systems, and polytypes.