2023-Sustainable Industrial Processing Summit
SIPS2023 Volume 13. Intl. Symp on Physics, Mathematics and Multiscale Mechanics
| Editors: | F. Kongoli, A. B. Bhattacharya, A.C. Pandey, G. Sandhu, F. Quattrocchi, L. Sajo-Bohus, S. Singh, H.S. Virk, R.M. Santilli, M. Mikalajunas, E. Aifantis, T. Vougiouklis, P. Mandell, E. Suhir, D. Bammann, J. Baumgardner, M. Horstemeyer, N. Morgan, R. Prabhu, A. Rajendran |
| Publisher: | Flogen Star OUTREACH |
| Publication Year: | 2023 |
| Pages: | 298 pages |
| ISBN: | 978-1-989820-96-4 (CD) |
| ISSN: | 2291-1227 (Metals and Materials Processing in a Clean Environment Series) |
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FRENKEL-KONTOROVA MODEL - DYNAMICS
Alexander Filonov1; Artur Abkaryan2; Aleksandr Ivanenko2;1INSTITUTE OF NONFERROUS METALS AND MATERIALS SCIENCE SIBERIAN FEDERAL UNIVERSITY KRASNOYARSK PR. IMENI GAZETY KRASNOYARSKII RABOCHII 95 RUSSIAN FEDERATION, Красноярск, Russian Federation; 2INSTITUTE OF ENGINEERING PHYSICS AND RADIOELECTRONICS, SIBERIAN FEDERAL UNIVERSITY, Krasnoyarsk, Russian Federation;
Type of Paper: Regular
Id Paper: 480
Topic: 38
Abstract:
In [1], the FK model (β=1) was written in the continuum approximation, after which an exact analytical solution was found for a soliton moving at speed w (Frenkel-Kontorova dislocation).
In [1] there is no answer to the question about the presence in the system of other non-dislocation solutions that are localized in space and do not decay in time.
Such solutions were found in exact numerical calculations.
Let us write the Hamiltonian as the sum of kinetic K and potential U energies, with the law of elastic dispersion of the general form:
Let us consider not only Hooke’s law, but also a function that does not allow the intersection of atoms in space (x≠o), for example, with two local minima x≈α and x=1+β:
Assuming discrete time - t = j h, j = 1.2 ..., where h is the time step - we have solutions to the system of Newton’s equations for the k-th CH atom of the form:
System of equations (3) is an algorithm for constructing dynamic solutions of the FK model.
In [2], the “Chain” program with algorithm (2)-(3) constructs dynamic solutions of the FK model.
An example of a dynamic solution with Hooke's law of elastic dispersion is considered and in it an energy excitation that does not decay in time, moving with a non-uniform speed and with an energy lower than the rest energy of the dislocation, is found.
When considering the dynamics of the FK model [2]: - α=0.5, β=0, γ=0.044, V0=0.03, Δ=1, a phase transition from β- to α-phase was found, with a decrease in the size of the CH by almost two times.
Conclusions:
1 The dynamics of the FK model and the dynamics of its continuum approximation do not always and everywhere coincide.
2 For the original discrete model, the result of an exact solution of the string limit may turn out to be erroneous. For example, in [3] an exact expression was obtained for the partition function of the FK model in the continuum approximation. Based on the above, it can be argued that this solution is not applicable to the original FK model.
3 If we accept that in local field theories ∇φ - this is a gradient analogue of Hooke’s chain, then we assume that in the center of the black hole matter with a changed metric and with fields collapsing to the size ɑN is grouped.
Keywords:
FC model; Dynamics; Exact solutionsReferences:
[1] Ya.I. Frenkel, T. Kontorova JETP, 8, 1340, (1938)[2] A.Yu. Babushkin, A.K. Abkarian et al. “Program for calculating an elastic-periodic chain” (2014). Patent No. 2014616693
[3] A.N. Filonov, G.M. Zaslavsky Phys. Lett, 85 A, 237, 1981