On Dynamical Behavior of the p-adic λ-Ising Model on Cayley Tree
DOI:
https://doi.org/10.15407/mag15.03.321Ключові слова:
p-адичнi числа, p-адична квазiмiра Гiббса, динамiчнi системи, дерево КейлiАнотація
У даній статті ми продовжуємо вивчення деяких властивостей $p$-aдичної $\lambda$-ізінгової моделі змішаного типу, що вивчалася в [3]. У тій роботі існування $p$-адичних мір Гібса і фазових переходів досліджувалося в моделі на дереві Кейлі другого порядка. У даній статті ми вивчаємо динамічну поведінку нерухомих точок, які було знайдено в [3]. Основним результатом є те, що ми довели, що нерухома точка $u_0$ є атрактором, а інші нерухомі точки $u_{1,2}$ репелентними нерухомими точками для $p$-aдичної $\lambda$-ізінгової моделі змішаного типу. На додаток описано розмір басейну атрактора для нерухомої точки $u_0$.Mathematics Subject Classification: 58F12, 46S10, 12J12, 11S99, 54H20, 30D05.
Посилання
R.J. Baxter, Exactly Solved Models in Statistical Mechanics, Academic Press, London, 1982.
S. De Smedt and A. Khrennikov, A p-adic behavior of dynamical systems, Rev. Mat. Complut. 12 (1999), 301–323. https://doi.org/10.5209/rev_REMA.1999.v12.n2.17103
M. Dogan, Phase transition of the mixed type p-adic λ-Ising model on the Cayley tree, p-Adic Numbers Ultrametric Anal. Appl. 10 (2018), No. 4, 276–286. https://doi.org/10.1134/S2070046618040040
N.N. Ganikhodjaev, F.M. Mukhamedov, and U.A. Rozikov, Phase transitions of the Ising model on Z in the p-adic number field, Uzbek. Mat. Zh. 4 (1998), 23–29 (Russian).
H.O. Georgii, Gibbs Measures and Phase Transitions, De Gruyter Studies in Mathematics, 9, Walter de Gruyter & Co., Berlin, 1988.
M. Khamraev, F.M. Mukhamedov, and U.A. Rozikov, On uniqueness of Gibbs measure for p-adic λ-model on the Cayley tree, Lett. Math. Phys. 70 (2004), 17–28. https://doi.org/10.1007/s11005-004-3500-7
A. Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Mathematics and its Applications, 427, Kluwer Academic Publishers, Dordrecht, 1997. https://doi.org/10.1007/978-94-009-1483-4
A.Yu. Khrennikov and S. Ludkovsky, Stochastic processes on non-Archimedean spaces with values in non-Archimedean fields, Markov Process. Related Fields 9 (2003), 131–162.
N. Koblitz, p-Adic Numbers, p-Adic Analysis and Zeta-Function, Graduate Texts in Mathematics, 58, Springer-Verlag, New York-Heidelberg, 1977. https://doi.org/10.1007/978-1-4684-0047-2
J. Lubin, Non-Archimedean dynamical systems, Compositio Math Math. 94 (1994), 321–346.
F. Mukhamedov, On factor associated with the unordered phase of λ-model on a Cayley tree, Rep. Math. Phys. 53 (2004), 1–18. https://doi.org/10.1016/S0034-4877(04)90001-8
F. Mukhamedov, A dynamical system appoach to phase transitions p-adic Potts model on the Cayley tree of order two, Rep. Math. Phys. 70 (2012), 385–406. https://doi.org/10.1016/S0034-4877(12)60053-6
F. Mukhamedov, On dynamical systems and phase transitions for q + 1-state p-adic Potts model on the Cayley tree, Math. Phys. Anal. Geom. 16 (2013), 49–87. https://doi.org/10.1007/s11040-012-9120-z
F. Mukhamedov, Recurrence equations over trees in a non-Archimedean context, p-Adic Numbers Ultrametric Anal. Appl. 6 (2014), 310–317. https://doi.org/10.1134/S2070046614040062
F. Mukhamedov and H. Akin, On non-Archimedean recurrence equations and their applications, J. Math. Anal. Appl. 423 (2015), 1203–1218. https://doi.org/10.1016/j.jmaa.2014.10.046
F. Mukhamedov, M. Dogan, and H. Akin, On chaotic behaviour of the p-adic generalized Ising mapping and its application, J. Difference Equ. Appl. 23 (2017), 1542–1561. https://doi.org/10.1080/10236198.2017.1340468
F. Mukhamedov and U.A. Rozikov, On rational p-adic dynamical systems, Methods of Funct. Anal. and Topology 10 (2004), 21–31.
F. Mukhamedov, M. Saburov, and O. Khakimov, On p-adic Ising–Vannimenus model on an arbitraray order Cayley tree, J. Stat. Mech. Theory Exp. (2015), No. 5, P05032. https://doi.org/10.1088/1742-5468/2015/05/P05032
M. Ostilli, Cayley trees and Bethe lattices: A concise analysis for mathematicians and physicists, Phys. A 391 (2012), 3417–3423. https://doi.org/10.1016/j.physa.2012.01.038
U.A. Rozikov, Description of limit Gibbs measures for λ-models on the Bethe lattice, Siberian Math. J. 39 (1998), 373–380. https://doi.org/10.1007/BF02677521
U.A. Rozikov, Gibbs Measures on Cayley Trees, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2013. https://doi.org/10.1142/8841
M. Saburov and M.A. Khameini, Quadratic equations over p-adic fields and their applications in statistical mechanics, ScienceAsia 41 (2015), 209–215. https://doi.org/10.2306/scienceasia1513-1874.2015.41.209
A.N. Shiryaev, Probability, Nauka, Moscow, 1980 (Russian).
