Ising model
The Ising model [1] (also known as the Lenz-Ising model) is commonly defined over an ordered lattice. Each site of the lattice can adopt two states, 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 \in \{-1, +1 \}} . 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 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, 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.
For a detailed and very readable history of the Lenz-Ising model see the following references:[2] [3] [4].
1-dimensional Ising model
- 1-dimensional Ising model (exact solution)
2-dimensional Ising model
Solved by Lars Onsager in 1944 [5] [6] [7]. Rudolf Peierls had previously shown (1935) that, contrary to the one-dimensional case, the two-dimensional model must have a phase transition.
3-dimensional Ising model
Sorin Istrail has shown that the solution of Ising's model cannot be extended into three dimensions for any lattice [8] [9]
ANNNI model
The axial next-nearest neighbour Ising (ANNNI) model [10] is used to study alloys, adsorbates, ferroelectrics, magnetic systems, and polytypes.
See also
References
- ↑ Ernst Ising "Beitrag zur Theorie des Ferromagnetismus", Zeitschrift für Physik A Hadrons and Nuclei 31 pp. 253-258 (1925)
- ↑ S. G. Brush "History of the Lenz-Ising Model", Reviews of Modern Physics 39 pp. 883-893 (1967)
- ↑ 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)
- ↑ 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)
- ↑ Lars Onsager "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition", Physical Review 65 pp. 117 - 149 (1944)
- ↑ M. Kac and J. C. Ward "A Combinatorial Solution of the Two-Dimensional Ising Model", Physical Review 88 pp. 1332-1337 (1952)
- ↑ Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 7 (freely available pdf)
- ↑ 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)
- ↑ Walter Selke "The ANNNI model — Theoretical analysis and experimental application", Physics Reports 170 pp. 213-264 (1988)