Ising model: Difference between revisions
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The '''Ising model''' is also known as the '''Lenz-Ising''' model. For a history of the Lenz-Ising model see | The '''Ising model''' is also known as the '''Lenz-Ising''' model. 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> | |||
The Ising model is commonly defined over an ordered lattice. | 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. | ||
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* [[1-dimensional Ising model]] (exact solution) | * [[1-dimensional Ising model]] (exact solution) | ||
==2-dimensional Ising model== | ==2-dimensional Ising model== | ||
Solved by [[Lars Onsager]] in 1944 | Solved by [[Lars Onsager]] in 1944 | ||
<ref>[http://dx.doi.org/10.1103/PhysRev.65.117 Lars Onsager "Crystal Statistics. I. A Two-Dimensional Model with an Order-Disorder Transition", Physical Review '''65''' pp. 117 - 149 (1944)]</ref> | |||
<ref>[http://dx.doi.org/10.1103/PhysRev.88.1332 M. Kac and J. C. Ward "A Combinatorial Solution of the Two-Dimensional Ising Model", Physical Review '''88''' pp. 1332-1337 (1952)]</ref> | |||
<ref>Rodney J. Baxter "Exactly Solved Models in Statistical Mechanics", Academic Press (1982) ISBN 0120831821 Chapter 7 (freely available [http://tpsrv.anu.edu.au/Members/baxter/book/Exactly.pdf pdf])</ref>. | |||
[[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== | ==3-dimensional Ising model== | ||
Sorin Istrail has shown that the solution of Ising's model cannot be extended into three dimensions for any lattice | Sorin Istrail has shown that the solution of Ising's model cannot be extended into three dimensions for any lattice | ||
<ref>[http://www.sandia.gov/LabNews/LN04-21-00/sorin_story.html Three-dimensional proof for Ising model impossible, Sandia researcher claims to have shown]</ref> | |||
<ref>[http://dx.doi.org/10.1145/335305.335316 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)]</ref> | |||
==ANNNI model== | ==ANNNI model== | ||
The '''axial next-nearest neighbour Ising''' (ANNNI) model | The '''axial next-nearest neighbour Ising''' (ANNNI) model <ref>[http://dx.doi.org/10.1016/0370-1573(88)90140-8 Walter Selke "The ANNNI model — Theoretical analysis and experimental application", Physics Reports '''170''' pp. 213-264 (1988)]</ref> is used to study alloys, adsorbates, ferroelectrics, magnetic systems, and polytypes. | ||
==See also== | ==See also== | ||
*[[Critical exponents]] | *[[Critical exponents]] | ||
*[[Potts model]] | *[[Potts model]] | ||
==References== | ==References== | ||
<references/> | |||
[[Category: Models]] | [[Category: Models]] | ||
Revision as of 17:56, 30 November 2009
The Ising model is also known as the Lenz-Ising model. For a history of the Lenz-Ising model see [1] [2] The Ising model is commonly defined over an ordered lattice. Each site of the lattice can adopt two states, . 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.
where is the Boltzmann constant, is the temperature, indicates that the sum is performed over nearest neighbors, and indicates the state of the i-th site, and 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] [4] [5]. 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 [6] [7]
ANNNI model
The axial next-nearest neighbour Ising (ANNNI) model [8] is used to study alloys, adsorbates, ferroelectrics, magnetic systems, and polytypes.
See also
References
- ↑ 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)
- ↑ 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)