Menu
Publications
2024
2023
2022
2021
2020
2019
2018
2017
2016
2015
2014
2013
2012
2011
2010
2009
2008
2007
2006
2005
2004
2003
2002
2001
Editor-in-Chief
Nikiforov
Vladimir O.
D.Sc., Prof.
Partners
doi: 10.17586/2226-1494-2020-20-3-432-437
DETERMINISTIC SYSTEMS WITH NATURAL QUANTIZATION
Read the full article
';
Article in Russian
For citation:
Abstract
For citation:
Golovina V.V., Groshikov E.S., Rymkevich P.P. Deterministic systems with natural quantization. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2020, vol. 20, no. 3, pp. 432–437 (in Russian). doi: 10.17586/2226-1494-2020-20-3-432-437
Abstract
Subject of Research. The research of deterministic systems is a topical problem of natural science. The paper presents an approach for behavior study of the deterministic systems. The work is aimed at creation of the evolution equation for deterministic systems, which shows that part of the macroscopic processes complies with quantum logic. Method. A new algebraic approach is proposed based on non-commutative algebra. It is shown that for any deterministic system in case of setting its change for a short time period, it is possible to create a system of differential equations describing the evolution of a given system in time. Implemented apparatus of non-commutative multiplication is an alternative to the operator calculus for quantum mechanics. Main Results. An associative non-commutative ring is built describing the evolution of arbitrary deterministic system. The proposed algebra is an isomorphic Heisenberg algebra. It is shown that all elements of algebraic rings are functions of numerical variables unlike the mathematical apparatus of quantum mechanics and, therefore, it is possible to give them different physical meaning. Practical Relevance. An example of differential equation creation is considered describing the motion of a classical particle in the presence of random forces. The obtained equation describes the probability density of a classical particle location at an arbitrary point of time in phase space.
Keywords: system, non-commutative multiplication, transfer matrix, differentiation operator, physical model, Fourier transform
References
References
-
Andronov A.A. Vitt A.A., Khaikin S.E. Oscillation Theory. Moscow, Nauka Publ., 1981, 568 p. (in Russian)
-
Guckenheimer J., Holmes Ph. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, 1990, 459 p.
-
Kapitca S.P., Kurdiumov S.P. Synergetics and Future Forecasts. Moscow, Nauka Publ., 1997, 285 p. (in Russian)
-
Kniazeva E.N., Kurdiumov S.P. Fundamental Principles of Synergetics: Blow-Up Regimes, Self-Organization and Tempo Structures. St. Petersburg, Aletejja Publ., 2002, 414 p. (in Russian)
-
Rymkevich P.P. Introduction to properties propagation theory. Proc. XXVII Summer International School “Analysis and synthesis of nonlinear mechanics of oscillatory systems”, St. Petersburg, 2000, pp. 455–497. (in Russian)
-
Gorshkov A.S., Makarov A.G., Romanova A.A., Rymkevitch P.P. Estimation of the average period of the nonstationary heat transmission through multi layered textile and clothing industry fabrics. The News of higher educational institutions. Technology of Light Industry, 2011, no. 4, pp. 44–45. (in Russian)
-
Rymkevich P., Makarov A., Basenko V., Liashenko V., Shafarenko Yu. The chart method of the solution of the one-dimensional non-stationary tasks in the theory of heat and mass transfer. Vestnik of St. Petersburg State University of Technology and Desig. Series 1. Natural and Technical Sciences, 2015, no. 4, pp. 8–12. (in Russian)
-
Gardiner C.W. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer, 1985.
-
Rymkevich P.P., Gorshkov A.S. Transport Theory. St. Petersburg, SPbPU Publ., 2015, 120 p. (in Russian)
-
Rymkevich P.P., Gorshkov A.S. Transfer Equation of Additive Properties in Quantum Mechanics. LAP LAMBERT Academic Publishing, 2015, 128 p. (in Russian)
-
Maslov V.P. Method of Operators. Moscow, Nauka Publ., 1973, 621 p. (in Russian)
-
Feinman R.P. On the operator calculus applied in quantum electrodynamics. Problemy sovremennoj fiziki, 1955, vol. 3, pp. 37–79. (in Russian)
-
Karasev M.V., Maslov V.P. Nonlinear Poisson Brackets. Geometry and Quantization. Moscow, Nauka Publ., 1991, 365 p. (in Russian)
-
Maslov V.P. Application of the method of ordered operators to obtain exact solutions. Theoretical and Mathematical Physics, 1977, vol. 33, no. 2, pp. 960–976. doi: 10.1007/BF01036594
-
Karasev M.V., Maslov V.P. Algebras with general commutation relations and their applications. II. Unitary-nonlinear operator equations. Journal of Soviet Mathematics, 1981, vol. 15, no. 3, pp. 273–368. doi: 10.1007/BF01083679
-
Maslov V.P., Nazaikinskii V.E. Algebras with general commutation relations and their applications. I. Pseudodifferential equations with increasing coefficients. Journal of Soviet Mathematics, 1981, vol. 15, no. 3, pp. 167–273. doi: 10.1007/BF01083678
-
Karasev M.V. Weyl and ordered calculus of noncommuting operators. Mathematical Notes of the Academy of Sciences of the USSR, 1979, vol. 26, no. 6, pp. 945–958. doi: 10.1007/BF01142081
-
Anderson R.F.V. The Weyl functional calculus. Journal of Functional Analysis, 1969, vol. 4, no. 2, pp. 240–267. doi: 10.1016/0022-1236(69)90013-5
-
Berezin F.A. Quantization. Mathematics of the USSR-Izvestiya, 1974, vol. 8, no. 5, pp. 1109–1165. doi: 10.1070/IM1974v008n05ABEH002140
-
Lifshitc E.M., Pataevskii L.P. Theoretical Physics. Tutorial. V. 9. Statistical Physics. Part 2. Condensed State Theory. Moscow, Nauka Publ., 1978, 497 p. (in Russian)
-
Landau L.D., Lifshitc E.M. Theoretical Physics. Tutorial. V. 4. Statistical Physics. Part. 1. Moscow, NaukaPubl.,1976, 583 p. (in Russian)
-
Rymkevich P.P., Golovina V.V., Altukhov A.I. Motion equation averaging in potential autonomous systems. Scientific and Technical Journal of Information Technologies, Mechanics and Optics, 2020, vol. 20, no. 1, pp 141–146. (in Russian). doi: 10.17586/2226-1494-2020-20-1-141-146
