Structural transitions in systems with a triple-well potential

Objectives. Recently studied phenomena in condensed matter physics have prompted new insights into the dynamic theory of crystals. The results of numerous experimental data demonstrate the impossibility of their explanation within the framework of linear models of the dynamics of many-particle systems, resulting in the necessity to account for nonlinear effects. Analyzing the dynamics of systems in condensed matter physics containing a sufficiently large number of particles shows that modes of motion can undergo changes depending on the potential of interparticle interaction. This is also reflected in the presence of domains with essentially chaotic phase space having a number of degrees of freedom N ≥ 1.5 and a certain set of interparticle interaction parameters. However, it is not only the dynamic model that appears to be strongly nonlinear. A similar nature of motion can be also observed in a static nonlinear many-particle system. The paper aims to study the influence of the external field specified by the interatomic triple-well potential on the equilibrium structure of a chain of interacting atoms.

Methods. Methods of Hamiltonian mechanics are used.

Results. Analytical expressions are obtained and analyzed for determining the phase portrait of the equilibrium structure of a chain of interacting atoms for various values of the parameter characterizing the local potential of the field in which each atom of the chain moves. Phase portraits of the equilibrium structure of the system are constructed in continuous and discrete representations of the equilibrium equations for various values of the parameter characterizing the local potential of the field in which each atom of the chain moves.

Conclusions. It is shown that both periodic and random chaotic arrangements of atoms are implemented depending on the magnitude of the external field.

1. Gleick J. Khaos. Sozdanie novoi nauki (Chaos. Making a New Science). Transl. from Engl. Moscow: AST: Corpus; 2022. 432 p. (in Russ.). ISBN 978-5-17-138645-0 [Gleick J. Chaos: Making a New Science. Open Road Medea; 2011. 360 p.]

2. Grinchenko V.T., Matsypura V.T., Snarskii A.A. Vvedenie v nelineinuyu dinamiku: Khaos i fraktaly (Introduction to Nonlinear Dynamics: Chaos and Fractals). M.: LENAND; 2021. 280 p. (in Russ.). ISBN 978-5-9710-6410-7

3. Zaslavskii G.M. Fizika khaosa v gamil’tonovykh sistemakh (Physics of Chaos in Hamiltonian Systems). Moscow, Izhevsk: Institute of Computer Research; 2004. 286 p. (in Russ.). ISBN 5-93972-342-X

4. Akhmadulina E.N., Savin E.S. Proton oscillation in a three-minimum potential. Fine Chem. Technol. (Vestnik MITHT). 2012;7(6):88–91 (in Russ.). Available from URL: https://www.elibrary.ru/pnmfxv

5. Ragushina M.D., Savin E.S. Dynamics of systems allowing structural phase transitions. Fine Chem. Technol. 2016;11(2): 81–85 (in Russ.). https://doi.org/10.32362/2410-6593-2016-11-2-81-85, available from URL: https://www.elibrary.ru/tugqvj

6. Ivanchenko M.V., Kozinov E.A., Volokitin V.D., Liniov A.V., Meyerov I.B., Denisov S.V. Classical bifurcation diagrams by quantum means. Ann. Der Phys. 2017;529(8):1600402. https://doi.org/10.1002/andp.201600402

7. Ivanchenko M.V. Dissipative quantum chaos. In: Nanoelectronics, Nanophotonics and Nonlinear Physics: Collection of proceedings of the 14th All-Russian Conference of Young Scientists. Saratov: Techno-Décor; 2019. P. 96 (in Russ.). Available from URL: https://www.elibrary.ru/pdgagn

8. Arnold L., Boxier P. Stochastic bifurcation: instructive examples in dimension one. In: M. Pinsky and V. Wihstutz (Eds.). Diffusion Processes and Related Problems in Analysis. Volume II: Stochastic Flows. Volume 27: Progress in Probability. Birkhäuser, Boston Basel Stuttgart; 1992. P. 241–255. https://doi.org/10.1007/978-1-4612-0389-6_10

9. Yusipov I.I., Denisov S.V., Ivanchenko M.V. Bifurcations and chaos in open quantum systems. Radiophys. Quantum El. 2023;66(1):63–76. https://doi.org/10.1007/s11141-023-10276-6, available from URL: https://www.elibrary.ru/skhwmx [Original Russian Text: Yusipov I.I., Denisov S.V., Ivanchenko M.V. Bifurcations and chaos in open quantum systems. Izvestiya vysshikh uchebnykh zavedenii. Radiofizika. 2023;66(1):71–85 (in Russ.). Available from URL: https://www.elibrary.ru/zmnges]

10. Poot M., van der Zant H.S.J. Mechanical systems in the quantum regime. Phys. Rep. 2012;511(5):273–335. http://doi.org/10.1016/j.physrep.2011.12.004

11. Savin E.S. Model of the transition of polyethylene to a mesophase state. Polym. Sci. Ser. A. 2000;42(6):637–641. Available from URL: https://www.elibrary.ru/lfybdp [Original Russian Text: Savin E.S. Model of the transition of polyethylene to a mesophase state. Vysokomole-kulyarnye Soedineniya. Seriya A. 2000; 42(6):974–979 (in Russ.). Available from URL: https://www.elibrary.ru/mpggtv]

12. Haken H. Sinergetika. Ierarkhiya neustoichivostei v samoorganizuyushchikhsya sistemakh i ustroistvakh (Synergetics. Hierarchy of Instabilities in Self-Organizing Systems and Devices). Transl. from Engl. Moscow: Mir; 1985. 424 p. (in Russ.). [Haken H. Synergetics: An Introduction. Berlin: Springer Verlag; 1978. 394 p.]

13. Beloshapkin V.V., Tretyakov A.G., Zaslavsky G.M. Disorder of Particle Chains as a Dynamical Problem of Transition to Chaos: Analogy to Simulation Induced Chaos. Communications on Pure and Applied Mathematics. 1994;47(1):39–46. https://doi.org/10.1002/cpa.3160470104, available from URL: https://www.elibrary.ru/zztesn

14. Chernikov A.A., Natenzon M.Ya., Petrovichev B.A., Sagdeev R.Z., Zaslavsky G.M. Strong changing of adiabatic invariants, KAM-tori and Web-tori. Phys. Lett. A. 1988;129(7):377–380. https://doi.org/10.1016/0375-9601(88)90006-0

15. Gradshtein M.S., Ryzhik I.M. Tablitsy integralov, summ, ryadov i proizvedenii ( Tables of Integrals, Sums, Series and Products). Moscow: Fizmatgiz; 1971. 1108 p. (in Russ.).

16. Baterman H., Erdelyi A. Vysshie transtsendentnye funktsii. Ellipticheskie i avtomorfnye funktsii. Funktsii Lame i Mat’e (Higher transcendental functions. Elliptic and automorphic functions. Lamé and Mathieu functions). Transl. from Engl. Moscow: Nauka; 1967. 300 p. (in Russ.). [Baterman H., Erdelyi A. Higher Transcendental Functions. V. 3. McGraw Hill; 1955. 312 p.]