umu.sePublikationer
Ändra sökning
RefereraExporteraLänk till posten
Permanent länk

Direktlänk
Referera
Referensformat
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Annat format
Fler format
Språk
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Annat språk
Fler språk
Utmatningsformat
  • html
  • text
  • asciidoc
  • rtf
Series expansion for the density of states of the Ising and Potts models
Umeå universitet, Teknisk-naturvetenskaplig fakultet, Matematik och matematisk statistik.
Manuskript (Övrigt vetenskapligt)
Identifikatorer
URN: urn:nbn:se:umu:diva-2344OAI: oai:DiVA.org:umu-2344DiVA, id: diva2:140299
Tillgänglig från: 2007-05-10 Skapad: 2007-05-10 Senast uppdaterad: 2010-01-13Bibliografiskt granskad
Ingår i avhandling
1. On the Ising problem and some matrix operations
Öppna denna publikation i ny flik eller fönster >>On the Ising problem and some matrix operations
2007 (Engelska)Doktorsavhandling, sammanläggning (Övrigt vetenskapligt)
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.

Ort, förlag, år, upplaga, sidor
Umeå: Matematik och matematisk statistik, 2007. s. 7
Serie
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 37
Nyckelord
Ising problem, phase tansition, matrix multiplicatoin, matrix inversion
Nationell ämneskategori
Diskret matematik
Identifikatorer
urn:nbn:se:umu:diva-1129 (URN)987-91-7264-323-9 (ISBN)
Disputation
2007-05-31, MA 121, MIT, Umeå, 13:15 (Engelska)
Opponent
Handledare
Tillgänglig från: 2007-05-10 Skapad: 2007-05-10 Senast uppdaterad: 2011-04-21Bibliografiskt granskad

Open Access i DiVA

Fulltext saknas i DiVA

Av organisationen
Matematik och matematisk statistik

Sök vidare utanför DiVA

GoogleGoogle Scholar

urn-nbn

Altmetricpoäng

urn-nbn
Totalt: 237 träffar
RefereraExporteraLänk till posten
Permanent länk

Direktlänk
Referera
Referensformat
  • apa
  • ieee
  • modern-language-association-8th-edition
  • vancouver
  • Annat format
Fler format
Språk
  • de-DE
  • en-GB
  • en-US
  • fi-FI
  • nn-NO
  • nn-NB
  • sv-SE
  • Annat språk
Fler språk
Utmatningsformat
  • html
  • text
  • asciidoc
  • rtf