×

zbMATH — the first resource for mathematics

Phase transition and chaos: \(p\)-adic Potts model on a Cayley tree. (English) Zbl 1355.37106
Summary: In our previous investigations, we have developed the renormalization group method to \(p\)-adic models on Cayley trees, this method is closely related to the investigation of dynamical system associated with a given model. In this paper, we are interested in the following question: how is the existence of the phase transition related to chaotic behavior of the associated dynamical system (this is one of the important question in physics)? To realize this question, we consider as a toy model the \(p\)-adic \(q\)-state Potts model on a Cayley tree, and show, in the phase transition regime, the associated dynamical system is chaotic, i.e. it is conjugate to the full shift. As an application of this result, we are able to show the existence of periodic (with any period) \(p\)-adic quasi Gibbs measures for the model. This allows us to know that how large is the class of \(p\)-adic quasi Gibbs measures. We point out that a similar kind of result is not known in the case of real numbers.

MSC:
37N20 Dynamical systems in other branches of physics (quantum mechanics, general relativity, laser physics)
46S10 Functional analysis over fields other than \(\mathbb{R}\) or \(\mathbb{C}\) or the quaternions; non-Archimedean functional analysis
39A70 Difference operators
47S10 Operator theory over fields other than \(\mathbb{R}\), \(\mathbb{C}\) or the quaternions; non-Archimedean operator theory
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anashin, V.; Khrennikov, A., Applied algebraic dynamics, (2009), Walter de Gruyter Berlin, New York · Zbl 1184.37002
[2] Areféva, I. Y.; Dragovic, B., Volovich i.v. p-adic summability of the anharmonic oscillator, Phys Lett B, 200, 512-514, (1988)
[3] Areféva, I. Y.; Dragovic, B.; Frampton, P. H.; Volovich, I. V., The wave function of the universe and p-adic gravity, Int J Mod Phys A, 6, 4341-4358, (1991) · Zbl 0733.53039
[4] Avetisov, V. A.; Bikulov, A. H.; Kozyrev, S. V., Application of p-adic analysis to models of spontaneous breaking of the replica symmetry, J Phys A Math Gen, 32, 8785-8791, (1999) · Zbl 0957.82034
[5] Baxter, R. J., Exactly solved models in statistical mechanics, (1982), Academic Press London · Zbl 0538.60093
[6] Besser, A.; Deninger, C., p-adic Mahler measures, J Reine Angew Math, 517, 19-50, (1999) · Zbl 0937.11056
[7] Dragovich, B.; Khrennikov, A.; Kozyrev, S. V.; Volovich, I. V., On p-adic mathematical physics, p-Adic Numbers Ultrametric Anal Appl, 1, 1-17, (2009) · Zbl 1187.81004
[8] Fan, A. H.; Li, M. T.; Yao, J. Y.; Zhou, D., Strict ergodicity of affine p-adic dynamical systems on z_p, Adv Math, 214, 666-700, (2007) · Zbl 1121.37010
[9] Fan, A. H.; Liao, L. M.; Wang, Y. F.; Zhou, D., p-adic repellers in q_p are subshifts of finite type, C R Math Acad Sci Paris, 344, 219-224, (2007) · Zbl 1108.37016
[10] Fisher, M. E., The renormalization group in the theory of critical behavior, Rev Mod Phys, 46, 597-616, (1974)
[11] Freund, P. G.O.; Olson, M., Non-Archimedian strings, Phys Lett B, 199, 186-190, (1987)
[12] Ganikhodjaev, N. N.; Mukhamedov, F. M.; Rozikov, U. A., Phase transitions of the Ising model on \(\mathbb{Z}\) in the p-adic number field, Uzbek Math J, 4, 23-29, (1998)
[13] Gandolfo, D.; Rozikov, U.; Ruiz, J., On p-adic Gibbs measures for hard core model on a Cayley tree, Markov Proc Rel Top, 18, 701-720, (2012) · Zbl 1281.82006
[14] Georgii, H. O., Gibbs measures and phase transitions, (1988), Walter de Gruyter Berlin · Zbl 0657.60122
[15] Gyorgyi, G.; Kondor, I.; Sasvari, L.; Tel, T., Phase transitions to chaos, (1992), World Scientific Singapore · Zbl 0875.82005
[16] Khamraev, M.; Mukhamedov, F. M., On p-adic λ-model on the Cayley tree, J Math Phys, 45, 4025-4034, (2004) · Zbl 1064.82006
[17] Khrennikov, A., p-adic valued probability measures, Indag Mathem NS, 7, 311-330, (1996) · Zbl 0872.60002
[18] Khrennikov, A., p-adic valued distributions in mathematical physics, (1994), Kluwer Academic Publisher Dordrecht · Zbl 0833.46061
[19] Khrennikov, A., Non-Archimedean analysis: Quantum paradoxes, dynamical systems and biological models, (1997), Kluwer Academic Publisher Dordrecht · Zbl 0920.11087
[20] Khrennikov, A., Generalized probabilities taking values in non-Archimedean fields and in topological groups, Russ J Math Phys, 14, 142-159, (2007) · Zbl 1124.60002
[21] Khrennikov, A., Cognitive processes of the brain: an ultrametric model of information dynamics in unconsciousness, p-Adic Numbers Ultrametric Anal Appl, 6, 293-302, (2014) · Zbl 1354.91125
[22] Khrennikov, A.; Ludkovsky, S., Stochastic processes on non-Archimedean spaces with values in non-Archimedean fields, Markov Process Relat Fields, 9, 131-162, (2003) · Zbl 1017.60045
[23] Khrennikov, A.; Mukhamedov, F.; Mendes, J. F.F., On p-adic Gibbs measures of countable state Potts model on the Cayley tree, Nonlinearity, 20, 2923-2937, (2007) · Zbl 1139.46050
[24] Khrennikov, A. Y.; Nilsson, M., p-adic deterministic and random dynamical systems, (2004), Kluwer Dordreht
[25] Khrennikov, A. Y.; Yamada, S.; van Rooij, A., Measure-theoretical approach to p-adic probability theory, Ann Math Blaise Pascal, 6, 21-32, (1999) · Zbl 0941.60010
[26] Khrennikov, A.; Yurova, E., Criteria of ergodicity for p-adic dynamical systems in terms of coordinate functions, Chaos Solitons Fractals, 60, 11-30, (2014) · Zbl 1348.37128
[27] Koblitz, N., p-adic numbers, p-adic analysis and zeta-function, (1977), Springer Berlin · Zbl 0364.12015
[28] Ludkovsky, S. V., Non-Archimedean valued quasi-invariant descending at infinity measures, Int J Math Math Sci, 2005, 23, 3799-3817, (2005) · Zbl 1106.46058
[29] Mukhamedov, F., On existence of generalized Gibbs measures for one dimensional p-adic countable state Potts model, Proc Steklov Inst Math, 265, 165-176, (2009) · Zbl 1187.82027
[30] Mukhamedov, F., On p-adic quasi Gibbs measures for \(q + 1\)-state Potts model on the Cayley tree, P-adic Numbers Ultametric Anal Appl, 2, 241-251, (2010) · Zbl 1268.82007
[31] Mukhamedov, F., A dynamical system approach to phase transitions p-adic Potts model on the Cayley tree of order two, Rep Math Phys, 70, 385-406, (2012) · Zbl 1271.82018
[32] Mukhamedov, F., On dynamical systems and phase transitions for \(q + 1\)-state p-adic Potts model on the Cayley tree, Math Phys Anal Geom, 16, 49-87, (2013) · Zbl 1280.46047
[33] Mukhamedov, F., On strong phase transition for one dimensional countable state p-adic Potts model, J Stat Mech, P01007, (2014)
[34] Mukhamedov, F., Renormalization method in p-adic λ-model on the Cayley tree, Int J Theor Phys, 54, 3577-3595, (2015) · Zbl 1334.82046
[35] Mukhamedov, F.; Akin, H., On p-adic Potts model on the Cayley tree of order three, Theor Math Phys, 176, 1267-1279, (2013) · Zbl 1286.82004
[36] Mukhamedov, F.; Dogan, M., On p-adic λ-model on the Cayley tree II: phase transitions, Rep Math Phys, 75, 25-46, (2015) · Zbl 1321.82020
[37] Mukhamedov, F. M.; Rozikov, U. A., On Gibbs measures of p-adic Potts model on the Cayley tree, Indag Math NS, 15, 85-100, (2004) · Zbl 1161.82311
[38] Mukhamedov, F. M.; Rozikov, U. A., On inhomogeneous p-adic Potts model on a Cayley tree, Infin Dimens Anal Quantum Probab Relat Top, 8, 277-290, (2005) · Zbl 1096.82007
[39] Ostilli, M., Cayley trees and Bethe lattices: A concise analysis for mathematicians and physicists, Physica A, 391, 3417-3423, (2012)
[40] Rahmatullaev, M. M., The existence of weakly periodic Gibbs measures for the Potts model on a Cayley tree, Theor Math Phys, 180, 1019-1029, (2014) · Zbl 1308.82014
[41] Rivera-Letelier, J., Dynamics of rational functions over local fields, Astérisque, 287, 147-230, (2003) · Zbl 1140.37336
[42] Rozikov, U. A., Gibbs measures on Cayley trees, (2013), World Scientific Singapore · Zbl 1278.82002
[43] Rozikov, U. A.; Khakimov, O. N., Description of all translation-invariant p-dic Gibbs measures for the Potts model on a Cayley tree, Markov Process Relat Fields, 21, 177-204, (2015) · Zbl 1332.82018
[44] Rozikov, U. A.; Khakimov, O. N., p-adic Gibbs measures and Markov random fields on countable graphs, Theor Math Phys, 175, 518-525, (2013) · Zbl 1286.82013
[45] Rozikov, U. A.; Khakimov, R., Periodic Gibbs measures for the Potts model on the Cayley tree, Theor Math Phys, 175, 699-709, (2013) · Zbl 1286.82006
[46] Rozikov, U.; Sattorov, I. A., On a nonlinear p-adic dynamical system, P-Adic Numbers Ultram Anal Appl, 6, 54-65, (2014) · Zbl 1347.37148
[47] Saburov, M.; Ahmad, M. A., On descriptions of all translation invariant p-adic Gibbs measures for the Potts model on the Cayley tree of order three, Math Phys Anal Geom, 18, 26, (2015) · Zbl 06552604
[48] Vladimirov, V. S.; Volovich, I. V.; Zelenov, E. I., p-adic analysis and mathematical physics, (1994), World Scientific Singapore · Zbl 0812.46076
[49] Volovich, I. V., Number theory as the ultimate physical theory, p-Adic Numbers Ultrametric Anal Appl, 2, 77-87, (2010) · Zbl 1258.81074
[50] Volovich, I. V., \(p -\)adic string, Class Quantum Gravity, 4, L83-L87, (1987)
[51] Wilson, K. G.; Kogut, J., The renormalization group and the ϵ- expansion, Phys Rep, 12, 75-200, (1974)
[52] Woodcock, C. F.; Smart, N. P., p-adic chaos and random number generation, Exp Math, 7, 333-342, (1998) · Zbl 0920.11052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.