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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DEVELOPMENT OF THE FRENKEL-KONTOROVA MODEL AIM MODEL
Alexander Filonov1; 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: 485
Topic: 38
Abstract:
A dynamic theory of ordering of the DFK model hole dislocations is put forward.
The results of the developed FK and DFK models are incorporated into the AIM theory to answer the question: how does the process of ordering dislocations of the DFK model (two interacting elastically periodic chains of N atoms) proceed into a commensurate crystal?
It is known that after a strong tug at the ends of one of the chains of the DFK model, half of the atoms of the first chain leave the region of interaction with the second. As a result: the remaining half of the atoms form with the atoms of the second chain a commensurate crystal with a double period. Each element of this crystal is a Frenkel-Kontorova (FK) hole dislocation [1].
A FK is a formation that does not decay in space and has a number of properties that coincide with the characteristics of a point particle, namely: – mass M; incompressible size (equal to 2); kinetic and potential energy, etc.
A commensurate crystal after a jerk is not formed immediately, but as atoms fly out of the region of interaction of chains, during the movement of dislocations to the center of the system.
On a chain (length N), dislocations appear at its edges and are arranged in pairs symmetrically relative to the center.
It is of interest to write down the spatiotemporal equation of the FK ordering process, starting from the departure of the first two edge atoms of the stretched chain to the departure of its last L= N/2 atoms.
Emitted atoms are, in the AIM theory, countable characteristics of FK dislocations, analogues of moments in time.
Let us associate the emitted J-atom of the DFK chain with a dislocation with number J at its end, J≤J0, J0 = L/2. The remaining FK dislocations numbered i, 1 ≤ i < J, are located between the center of the chain and its edge. FK move towards the center of the chain.
We will describe the dynamics of dislocation ordering depending on the number of the ejected J-atom using the AIM* model. * (AIM- ab initio mundu (lat.) – from the beginning of the world)
The theory states:
1. DFK dislocation with number i is an analogue of the i-th moment of time, located at the moment of time J at a distance 
from the center of the system.
2. R(J) - discrete Lagrangian of the AIM model.
R(J) has the form:
3. 
determined from the equation:
4. Our goal is to find all values of ; 1≤ i 0, 
, with the final solution:
where L is the main parameter of the model.
5. Parameters MJ, V(J) are found from the system of inequalities:
where MJ is the mass of atoms in a chain of 2J dislocations, atoms are called DFK dislocations; 
- period of an elastic-periodic chain, with an elasticity coefficient equal to 1;
V(J) is the periodic potential in which the dislocation chain is located.
Thus, the AIM theory is an open FK model with an increasing number of particles and with “running” ones, i.e. J-dependent periods, masses and potential amplitudes.
As shown earlier: - in systems with periodic potentials, inhomogeneous dynamic solutions inevitably arise that are not destroyed. Consequently, in a system with potential (1) it is impossible to obtain a homogeneous solution (3) as the final result.
In this regard, it is necessary to introduce additional terms into Lagrangian (1), ensuring the destruction of nonlinear excitations.
We believe that the initial terms of the “destruction mechanism” should be the first and second harmonics V(J) with the main and doubled periods alternating on (off) depending on the parity of J.
We expect that in a system with the first and second harmonics alternately turning on (off) the previous one-dimensional solitons stop, but over time two-dimensional non-decaying dynamic excitations are formed.
After some time JK ~ J0 in (1), the following harmonics VK(J) are turned on (off). At points J = JK on the temporary dislocation chain of the AIM system, phase transitions occur with a change in the spatial dimension of dynamic excited states.
To summarize, for the AIM model we write:
AIM - model
where 
is the “destruction mechanism”, with each moment of time divided into K-instants, with the corresponding harmonics of the external potential.
From general considerations it follows that VK(J) ~ V0(J), V0(J0) = 0. The phase transition points JK are determined by inequalities (4).
The AIM model is a cosmological application.
From the Big Jerk through the Big Bang and Beyond
From a cosmological point of view, the number of moments in time is equal to the optimally round number, i.e. 
.
Let’s compare the generally accepted concepts with the concepts of the AIM model:
1. “Matter” – energy excitations on the time chain;
2. “Dark energy” - one-dimensional phase of matter (stationary);
3. “Dark matter” - two-dimensional phase of matter (stationary);
4. “Visible matter” - three-dimensional phase of matter (dynamic);
5. The next phase is four-dimensional, etc.
Let us estimate the phase composition of matter at different stages of the development of the Universe.
Let X be the dynamic weight part of the Universe, then for a state of K phases we write:
X+KX+K2X+K3X+…+KK-1X=1; those. X= (K-1)/(KK-1).
When K=3, X+3X+9X=1; X3≈8%. Based on estimates of modern cosmology, for time intervals Tk we have:
T3=30 billion years; T2=90 billion years; T1=180 billion years; T4=7.5 billion years; T5=1.5 billion years...; T = ∑Tk ≈ 310 billion years.
The AIM+ model assumes the return of the emitted atoms of the DFK –chain to the point of their departure, with the formation of two interacting subsystems in the AIM model.