Description of Extreme Gibbs Measures for the Ising Model with Three Interactions

• Authors: H. Akin , N. Ganikhodjaev , S. Temir , S. Uguz
• Languages of publication: EN

Abstracts

EN

In this paper, we consider an Ising model with three competing interactions (nearest neighbor, next-nearest neighbor, and ternary prolonged neighbor) on the Cayley tree of order two, investigated by Ganikhodjaev et al. We study translation-invariant Gibbs measures of the Ising model with these competing interactions. Also, we investigate the set of the extreme Gibbs measures called Markov random fields with memory 2 of the model.

Keywords

EN

05.50.+q; 64.60.-i; 64.60.De; 75.10.Hk

Discipline

• 75.10.Hk: Classical spin models
• 64.60.De: Statistical mechanics of model systems (Ising model, Potts model, field-theory models, Monte Carlo techniques, etc.)
• 64.60.-i: General studies of phase transitions(see also 63.70.+h Statistical mechanics of lattice vibrations and displacive phase transitions; for critical phenomena in solid surfaces and interfaces, and in magnetism, see 68.35.Rh, and 75.40.-s, respectively)
• 05.50.+q: Lattice theory and statistics (Ising, Potts, etc.)(see also 64.60.Cn Order-disorder transformations, and 75.10.Hk Classical spin models)

• Publisher: Institute of Physics, Polish Academy of Sciences
• Journal: Acta Physica Polonica A
• Year: 2013
• Volume: 123
• Issue: 2
• Pages: 484-487

Dates

published

2013-02

Contributors

author

H. Akin

• Zirve University, Faculty of Education, Department of Mathematics, Gaziantep, 27260, Turkey

author

N. Ganikhodjaev

• Department of Computational and Theoretical Sciences, Faculty of Science, IIUM, 25200 Kuantan, Malaysia

author

S. Temir

• Harran University, Arts and Science Faculty, Department of Mathematics, Sanliurfa, 63120, Turkey

author

S. Uguz

• Harran University, Arts and Science Faculty, Department of Mathematics, Sanliurfa, 63120, Turkey

References

• [1] J.W. Gibbs, Elementary Principles in Statistical Mechanics, Yale University Press, New Haven 1902
• [2] L. Boltzmann, Lectures on the Theory of Gases, Gauthier-Villars, Paris, tome I 1902, tome II 1905, reedition Jean Gabay, Paris 1987 (in French)
• [3] N. Ganikhodjaev, H. Akın, S. Uğuz, S. Temir, J. Concrete Appl. Math. 9, 26 (2011)
• [4] N. Ganikhodjaev, H. Akın, S. Uğuz, S. Temir, Phase Transit. 84, 1045 (2011)
• [5] N. Ganikhodjaev, H. Akın, S. Uğuz, S. Temir, J. Stat. Mech. 2011, P03025 (2011)
• [6] J. Vannimenus, Z. Phys. B 43, 141 (1981)
• [7] M. Fannes, A. Verbeure, Commun. Math. Phys. 96, 115 (1984)
• [8] H.O. Georgii, Gibbs Measures and Phase Transitions, de Gruyter Stud. Math., Vol. 9, Walter de Gruyter, Berlin 1988
• [9] Ch.J. Preston, Gibbs States on Countable Sets, Cambridge Univ. Press, Cambridge 1974
• [10] F. Spitzer, Ann. Probab. 3, 387 (1975)

Identifiers

DOI

10.12693/APhysPolA.123.484