najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2025-16-3-261-273najo-313Research ArticleMATHEMATICSМАТЕМАТИКАFree energy and entropy for the constructive Gibbs measures of the Ising model on the Cayley tree of order threeСвободная энергия и энтропия конструктивных мер Гиббса для модели Изинга на дереве Кэли порядка триhttps://orcid.org/0000-0003-2987-7714РахматуллаевМ. М.RahmatullaevM. M.

Muzaffar M. Rahmatullaev

4-b, University str, 100174, Tashkent

100000, Tashkent

mrahmatullaev@rambler.ruhttps://orcid.org/0009-0000-7212-1598БурхоноваЗ. А.BurxonovaZ. A.

Zulxumor A. Burxonova

161, Boburshox str, 160107, Namangan

12, Islom Karimov str, 160103, Namangan

 

zulxumorburxonova4@gmail.comV. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of Sciences; New Uzbekistan UniversityUzbekistanNamangan State University; Namangan State Technical UniversityRussian Federation202529062025163261273Copyright © Rahmatullaev M.M., Burxonova Z.A., 20252025Рахматуллаев М.М., Бурхонова З.А.Rahmatullaev M.M., Burxonova Z.A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/313

In this paper, we identify non-translation-invariant constructive Gibbs measures for the Ising model on a third-order Cayley tree, which differ from known ones. We provide the conditions for the existence of at least two distinct Gibbs measures, which implies that a phase transition occurs. The free energies and entropies corresponding to the identified measures are calculated. These free energies and entropies are then compared with the known ones and shown to differ from them.

В этой статье мы изучаем нетрансляционно-инвариантные конструктивные меры Гиббса для модели Изинга на дереве Кэли третьего порядка, которые отличаются от известных мер Гиббса. Приведены условия для существования по крайней мере двух различных мер Гиббса, который определяют существование фазового перехода. Вычислены свободные энергии и энтропии для найденных мер. Полученные свободные энергии и энтропии сравниваются с известными результатами, и показано, что они отличаются от свободных энергии и энтропии для других мер Гиббса.

дерево Кэлимодель Изингамера Гиббсасвободная энергияэнтропияCayley treeIsing modelGibbs measurefree energyentropyThe authors thank Obid Sh. Karshiboev for valuable advice and an anonymous referee for helpful suggestions that improved the paper.ReferencesBleher P.M. and Ganikhodjaev N.N. On pure phases of the Ising model on the Bethe lattice. Theory Probab. Appl., 1990, 35, P. 216–227.Bleher P.M. and Ganikhodjaev N.N. On pure phases of the Ising model on the Bethe lattice. Theory Probab. Appl., 1990, 35, P. 216–227.Rozikov U.A. Gibbs measures on Cayley trees, (Singapore: World Scientific), 2013.Rozikov U.A. Gibbs measures on Cayley trees, (Singapore: World Scientific), 2013.Zachary S. Countable state space Markov random fields and Markov chains on trees, Ann.Prob., 1983, 11, P. 894–903.Zachary S. Countable state space Markov random fields and Markov chains on trees, Ann.Prob., 1983, 11, P. 894–903.Ganikhodjaev N.N., Rozikov U.A. A description of periodic extremal Gibbs measures of some lattice models on the Cayley tree. Theor. Math. Phys., 1997, 111, P. 480–486.Ganikhodjaev N.N., Rozikov U.A. A description of periodic extremal Gibbs measures of some lattice models on the Cayley tree. Theor. Math. Phys., 1997, 111, P. 480–486.Rozikov U.A. Partition structures of the group representation of the Cayley tree into cosets by finite-index normal subgroups and their applications to the description of periodic Gibbs distributions. Theor. Math. Phys., 1997, 112, P. 929–933.Rozikov U.A. Partition structures of the group representation of the Cayley tree into cosets by finite-index normal subgroups and their applications to the description of periodic Gibbs distributions. Theor. Math. Phys., 1997, 112, P. 929–933.Rozikov U.A. Describtion uncountable number of Gibbs measures for inhomogeneous Ising model. Theor. Math. Phys., 1999, 118, P. 95–104.Rozikov U.A. Describtion uncountable number of Gibbs measures for inhomogeneous Ising model. Theor. Math. Phys., 1999, 118, P. 95–104.Bleher P.M., Ganikhodjaev N.N. On pure phases of the Ising model on the Bethe lattice. Theory Probab.Appl., 1990, 35, P. 216–227.Bleher P.M., Ganikhodjaev N.N. On pure phases of the Ising model on the Bethe lattice. Theory Probab.Appl., 1990, 35, P. 216–227.Rozikov U.A., Rahmatullaev M.M. Description of weakly periodic Gibbs measures of the Ising model on the Cayley tree. Theor. Math. Phys., 2008, 156, P. 1218–1227.Rozikov U.A., Rahmatullaev M.M. Description of weakly periodic Gibbs measures of the Ising model on the Cayley tree. Theor. Math. Phys., 2008, 156, P. 1218–1227.Rozikov U.A., Rahmatullaev M.M. On weak periodic Gibbs measures of the Ising model on a Cayley tree, Dokl. AN Uzbekistan, 2008, 4, P. 12–15.Rozikov U.A., Rahmatullaev M.M. On weak periodic Gibbs measures of the Ising model on a Cayley tree, Dokl. AN Uzbekistan, 2008, 4, P. 12–15.Rahmatullaev M.M. Ising model on trees: (k0)- non translation-invariant Gibbs measures. Journal of Physics: Conference Series, 2017, 819, P. 012019.Rahmatullaev M.M. Ising model on trees: (k0)- non translation-invariant Gibbs measures. Journal of Physics: Conference Series, 2017, 819, P. 012019.Rakhmatullaev M.M., Samijonova N.D. Translation-invariant p-adic quasi Gibbs Measures for the Potts Model with an external field on the Cayley Tree. Nanosystems: Physics, Chemistry, Mathematics, 2025, 16(2), P. 164–175.Rakhmatullaev M.M., Samijonova N.D. Translation-invariant p-adic quasi Gibbs Measures for the Potts Model with an external field on the Cayley Tree. Nanosystems: Physics, Chemistry, Mathematics, 2025, 16(2), P. 164–175.Rahmatullaev M.M., Khakimov O.N., Tukhtabaev A.M. A p-adic generalized Gibbs measure for the Ising model on a Cayley tree. Theoretical and Mathematical Physics, 2019, 201(1), P. 1521–1530.Rahmatullaev M.M., Khakimov O.N., Tukhtabaev A.M. A p-adic generalized Gibbs measure for the Ising model on a Cayley tree. Theoretical and Mathematical Physics, 2019, 201(1), P. 1521–1530.Rahmatullaev M.M., Tukhtabaev A.M. Non periodic p-adic generalized Gibbs measure for Ising model, p-Adic Numbers, Ultrametric Analysis and Applications, 2019, 11(4), P. 319–327.Rahmatullaev M.M., Tukhtabaev A.M. Non periodic p-adic generalized Gibbs measure for Ising model, p-Adic Numbers, Ultrametric Analysis and Applications, 2019, 11(4), P. 319–327.Rahmatullaev M.M., Tukhtabaev A.M. Some non-periodic p-adic generalized Gibbs measures for the Ising model on a Cayley tree of order k. Mathematical Physics, Analysis and Geometry, 2023, 26(22).Rahmatullaev M.M., Tukhtabaev A.M. Some non-periodic p-adic generalized Gibbs measures for the Ising model on a Cayley tree of order k. Mathematical Physics, Analysis and Geometry, 2023, 26(22).Rozikov U.A., Rakhmatullaev M.M. Ising model on Cayley tree: a new class of Gibbs measures and their comparison with known ones. J. Stat. Mech., 2017, P. 093205.Rozikov U.A., Rakhmatullaev M.M. Ising model on Cayley tree: a new class of Gibbs measures and their comparison with known ones. J. Stat. Mech., 2017, P. 093205.Mavlonov I.M., Sattarov A.M., Karimova S.A., Haydarov F.H. On solutions to nonlinear integral equation of the Hammerstein type and its applications to Gibbs measures for continuous spin systems. Nanosystems: Physics, Chemistry, Mathematics, 2024, 15(1), P. 23–30.Mavlonov I.M., Sattarov A.M., Karimova S.A., Haydarov F.H. On solutions to nonlinear integral equation of the Hammerstein type and its applications to Gibbs measures for continuous spin systems. Nanosystems: Physics, Chemistry, Mathematics, 2024, 15(1), P. 23–30.Rahmatullaev M.M., Egamov D.O. Translation-invariant Gibbs measures for the mixed spin-1/2 and spin-1 Ising model with an external field on a Cayley tree. Nanosystems: Physics, Chemistry, Mathematics, 2024, 15(5), P. 576–585.Rahmatullaev M.M., Egamov D.O. Translation-invariant Gibbs measures for the mixed spin-1/2 and spin-1 Ising model with an external field on a Cayley tree. Nanosystems: Physics, Chemistry, Mathematics, 2024, 15(5), P. 576–585.Friedli S., Velenik Y. A Concrete Mathematical Introduction, Statistical Mechanics of Lattice Systems, Cambridge University Press, 2017.Friedli S., Velenik Y. A Concrete Mathematical Introduction, Statistical Mechanics of Lattice Systems, Cambridge University Press, 2017.Georgii H.O. Gibbs Measures and Phase Transitions.de Gryuter,Berlin (1998)Georgii H.O. Gibbs Measures and Phase Transitions.de Gryuter,Berlin (1998)Feynman R.P. There’s plenty of room at the bottom, Engineering and Science, 1960, 5(23), P. 22–36.Feynman R.P. There’s plenty of room at the bottom, Engineering and Science, 1960, 5(23), P. 22–36.Gandolfo D., Rahmatullaev M.M., Rozikov U.A. Ruiz J. On Free Energies of the Ising Model on the Cayley Tree. J. Stat Phys, 2013, 150, P. 1201–1217.Gandolfo D., Rahmatullaev M.M., Rozikov U.A. Ruiz J. On Free Energies of the Ising Model on the Cayley Tree. J. Stat Phys, 2013, 150, P. 1201–1217.Semkin S.V., Smagin V.P., Tarasov V.S. Frustrations in a Diluted Ising Magnet on the Bethe Lattice. J. Exp. Theor. Phys., 2022, 134, P. 713–719.Semkin S.V., Smagin V.P., Tarasov V.S. Frustrations in a Diluted Ising Magnet on the Bethe Lattice. J. Exp. Theor. Phys., 2022, 134, P. 713–719.Semkin S.V., Smagin V.P. Free energy, entropy, and magnetization of a onedimensional Ising model of a diluted magnet. Theor. Math. Phys., 2023, 217, P. 17881794.Semkin S.V., Smagin V.P. Free energy, entropy, and magnetization of a onedimensional Ising model of a diluted magnet. Theor. Math. Phys., 2023, 217, P. 17881794.Mukhamedov F., Akin H., and Khakimov O. Gibbs measures and free energies of IsingVannimenus model on the Cayley tree. Journal of Statistical Mechanics: Theory and Experiment, 2017.Mukhamedov F., Akin H., and Khakimov O. Gibbs measures and free energies of IsingVannimenus model on the Cayley tree. Journal of Statistical Mechanics: Theory and Experiment, 2017.Akin H. A novel computational method of the free energy for the Ising model on Cayley tree of order three. Chin. J. Phys., 2022, 77, P. 2276–2287.Akin H. A novel computational method of the free energy for the Ising model on Cayley tree of order three. Chin. J. Phys., 2022, 77, P. 2276–2287.Akin H. Calculation of some thermodynamic quantities for the Ising model on a kth order Cayley tree. Physica B, 2023, 662.Akin H. Calculation of some thermodynamic quantities for the Ising model on a kth order Cayley tree. Physica B, 2023, 662.Akin H. Investigation of thermodynamic properties of mixed-spin (2,1/2) Ising and Blum-Capel models on a Cayley tree. Chaos, Solitons, Fractals, 2024, 184, P. 114980.Akin H. Investigation of thermodynamic properties of mixed-spin (2,1/2) Ising and Blum-Capel models on a Cayley tree. Chaos, Solitons, Fractals, 2024, 184, P. 114980.Akin H. Calculation of the free energy of the Ising model on a Cayley tree via the self-similarity method. Axioms, 2022, 11, P. 703.Akin H. Calculation of the free energy of the Ising model on a Cayley tree via the self-similarity method. Axioms, 2022, 11, P. 703.Akin H. Quantitative behavior of (1,1/2)-MSIM on a Cayley tree. Chin. J. Phys., 2023, 83, P. 501–514.Akin H. Quantitative behavior of (1,1/2)-MSIM on a Cayley tree. Chin. J. Phys., 2023, 83, P. 501–514.Akin H. The classification of disordered phases of mixed spin (2,1/2) Ising model and the chaoticity of the corresponding dynamical system, Chaos, Solitons, Fractals, 2023, 167.Akin H. The classification of disordered phases of mixed spin (2,1/2) Ising model and the chaoticity of the corresponding dynamical system, Chaos, Solitons, Fractals, 2023, 167.Rahmatullaev M.M., Burxonova Z.A. Constructive Gibbs measures for the Ising model on the Cayley tree. Reports of Mathematical Physics, 2024, 93(3), P. 361–370.Rahmatullaev M.M., Burxonova Z.A. Constructive Gibbs measures for the Ising model on the Cayley tree. Reports of Mathematical Physics, 2024, 93(3), P. 361–370.Rozikov U.A., Rahmatullaev M.M. Weakly periodic ground states and Gibbs measures for the Ising model with competing interactions on the Cayley tree. Theor. Math. Phys., 2009, 160, P. 1291–1299.Rozikov U.A., Rahmatullaev M.M. Weakly periodic ground states and Gibbs measures for the Ising model with competing interactions on the Cayley tree. Theor. Math. Phys., 2009, 160, P. 1291–1299.Rahmatullaev M.M., Dekhkanov J. Existence weakly periodic Gibbs measures for the Ising model on the Cayley tree of order two. Vladikavkaz Mathematical Journal, 2021, 23(4), P. 77–88.Rahmatullaev M.M., Dekhkanov J. Existence weakly periodic Gibbs measures for the Ising model on the Cayley tree of order two. Vladikavkaz Mathematical Journal, 2021, 23(4), P. 77–88.Rahmatullaev M.M. On weakly periodic states and Gibbs measures of the Ising model on the Cayley tree, Fan, Tashkent, 2024. (in Russian)Rahmatullaev M.M. On weakly periodic states and Gibbs measures of the Ising model on the Cayley tree, Fan, Tashkent, 2024. (in Russian)Preston C. Gibbs States on Countable Sets. Cambridge University Press, London, 1974.Preston C. Gibbs States on Countable Sets. Cambridge University Press, London, 1974.Esfarjani K., Mansoori G.A. Statistical mechanical modeling and its application to nanosystems. Handbook of Theoretical and Computational Nanotechnology, 2006, II(14), P. 1-45.Esfarjani K., Mansoori G.A. Statistical mechanical modeling and its application to nanosystems. Handbook of Theoretical and Computational Nanotechnology, 2006, II(14), P. 1-45.Hill L. Thermodynamics of Small Systems, W. A. Benjamin, New York, 1963.Hill L. Thermodynamics of Small Systems, W. A. Benjamin, New York, 1963.Hill T.L. A Different Approach to Nanothermodynamics. Nano Letters, 1, 2001, P. 273–275.Hill T.L. A Different Approach to Nanothermodynamics. Nano Letters, 1, 2001, P. 273–275.Mansoori G.A. Proceedings of the first Conference on Nanotechnology-The Next Industrial Revolution, 2002, 2, P. 345.Mansoori G.A. Proceedings of the first Conference on Nanotechnology-The Next Industrial Revolution, 2002, 2, P. 345.

The authors declare that there are no conflicts of interest present.