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Computation of the Ising partition function for two-dimensional square grids
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Department of Physics, AlbaNova University Center, KTH, SE-106 91 Stockholm, Sweden.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Department of Physics, AlbaNova University Center, KTH, SE-106 91 Stockholm, Sweden.
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2004 (English)In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, ISSN 1063-651X, E-ISSN 1095-3787, Vol. 69, no 4, 19- p., 046104Article in journal (Refereed) Published
Abstract [en]

An improved method for obtaining the Ising partition function for $n \times n$ square grids with periodic boundary is presented. Our method applies results from Galois theory in order to split the computation into smaller parts and at the same time avoid the use of numerics. Using this method we have computed the exact partition function for the $320 \times 320$-grid, the $256 \times 256$-grid, and the $160 \times 160$-grid, as well as for a number of smaller grids. We obtain scaling parameters and compare with what theory prescribes.

Place, publisher, year, edition, pages
New York: American Physical Society through the American Institute of Physics , 2004. Vol. 69, no 4, 19- p., 046104
National Category
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-7684DOI: 10.1103/PhysRevE.69.046104OAI: oai:DiVA.org:umu-7684DiVA: diva2:147355
Available from: 2008-01-11 Created: 2008-01-11 Last updated: 2017-12-14Bibliographically approved
In thesis
1. On the Ising problem and some matrix operations
Open this publication in new window or tab >>On the Ising problem and some matrix operations
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

The first part of the dissertation concerns the Ising problem proposed to Ernst Ising by his supervisor Wilhelm Lenz in the early 20s. The Ising model, or perhaps more correctly the Lenz-Ising model, tries to capture the behaviour of phase transitions, i.e. how local rules of engagement can produce large scale behaviour.

Two decades later Lars Onsager solved the Ising problem for the quadratic lattice without an outer field. Using his ideas solutions for other lattices in two dimensions have been constructed. We describe a method for calculating the Ising partition function for immense square grids, up to linear order 320 (i.e. 102400 vertices).

In three dimensions however only a few results are known. One of the most important unanswered questions is at which temperature the Ising model has its phase transition. In this dissertation it is shown that an upper bound for the critical coupling Kc, the inverse absolute temperature, is 0.29 for the tree dimensional cubic lattice.

To be able to get more information one has to use different statistical methods. We describe one sampling method that can use simple state generation like the Metropolis algorithm for large lattices. We also discuss how to reconstruct the entropy from the model, in order to obtain parameters as the free energy.

The Ising model gives a partition function associated with all finite graphs. In this dissertation we show that a number of interesting graph invariants can be calculated from the coefficients of the Ising partition function. We also give some interesting observations about the partition function in general and show that there are, for any N, N non-isomorphic graphs with the same Ising partition function.

The second part of the dissertation is about matrix operations. We consider the problem of multiplying them when the entries are elements in a finite semiring or in an additively finitely generated semiring. We describe a method that uses O(n3 / log n) arithmetic operations.

We also consider the problem of reducing n x n matrices over a finite field of size q using O(n2 / logq n) row operations in the worst case.

Place, publisher, year, edition, pages
Umeå: Matematik och matematisk statistik, 2007. 7 p.
Series
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 37
Keyword
Ising problem, phase tansition, matrix multiplicatoin, matrix inversion
National Category
Discrete Mathematics
Identifiers
urn:nbn:se:umu:diva-1129 (URN)987-91-7264-323-9 (ISBN)
Public defence
2007-05-31, MA 121, MIT, Umeå, 13:15 (English)
Opponent
Supervisors
Available from: 2007-05-10 Created: 2007-05-10 Last updated: 2011-04-21Bibliographically approved

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Häggkvist, RolandAndrén, DanielLundow, Per-HåkanMarkström, Klas
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Department of Mathematics and Mathematical Statistics
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Physical Review E. Statistical, Nonlinear, and Soft Matter Physics: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics
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