2023-Sustainable Industrial Processing Summit
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 - GROUND STATE
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: 477
Topic: 38
Abstract:
In 1938, the authors: Yakov Ilyich Frenkel and Tatyana Abramovna Kontorova (LFTI) put forward the Frenkel-Kontorova model (FK model) [1], which served as the basis for the creation of many theories of highly nonlinear processes [2].
It is of interest to develop the FK model in order to expand the scope of its natural science applications.
The FK model is an elastically periodic chain of atoms (CH) in a periodic potential.
CH - is a one-dimensional sequence of N point atoms of masses m with coordinates {xi} and period β, interconnected by elastic springs with the law of elastic dispersion Φ(x). Φ(x) - most often this is Hooke's law, 
.
The periodic potential V(x) has even symmetry and period a=1.
In most previous works [1,2], it was assumed 
that solutions would be simplified by eliminating boundary effects from consideration. As it turned out [3,4], this assumption is wrong: - in the discrete FK model there is no small parameter 1/N, so it is necessary to find exact solutions taking into account the position of its boundary atoms.
The potential energy of the FK system has the form:
where N is the number of atoms in the chain, N=2K+1; xi is the distance of the i-th atom to the center of the chain.
In [1] 
.
If β=0, then in the ground state of the FK model the point CH is at the minimum of the potential V(x=0). When β≠0 the position of the central atom does not change x0=0. The positions of the remaining atoms relative to the center are not even and are determined from a system of nonlinear equilibrium equations.
- system of N equilibrium equations. Taking into account x-i = - xi, we have:
At x0=0, all coordinates xi and β are functions of x1, x1 ∈ [0, 0.5], therefore, solutions for the ground state of the FK model can be obtained by minimizing U(x1) with respect to x1 [3,4]. Comparing the exact solutions [3,4] with their continuum approximations [5], we found that the properties of the ground state of the discrete FK model coincide with the properties of its continuum approximation only in the homogeneity region.
Keywords:
FK model; Basic state; Exact solutionReferences:
[1] Ya.I. Frenkel, T. Kontorova JETP, 8, 1340, (1938)[2] O.M. Braun, Y.S. Kivshar. The Frenkel-Kontorova Model, Springer (2004)
[3] A.K. Abkaryan, A.Yu. Babushkin, B.S. Dobronets, V.S. Krasikov, A.N. Filonov FTT, 2, (2016)
[4] A.Yu. Babushkin, A.K. Abkaryan, B.S. Dobronets, V.S. Krasikov, A.N. Filonov FTT, 9, (2016)
[5] V. L. Pokrovskii and A. L. Talapov, Sov. Phys. JETP 48(3), 579 (1978)