On the «determinization» of stochastic processes with increasing degrees of freedom in the system

DOI: 10.34759/trd-2023-128-07

Аuthors

Khatuntseva O. N.

e-mail: olga.khatuntseva@rsce.ru

Abstract

The issue of the origin and existence of non-deterministic, i.e. stochastic processes in various physical, biological, social and other systems is an extremely complex and interesting task. Despite the long-standing interest in them on the part of scientists, there are still discussions on whether «stochasticity» is the true antipode of «determinism», or it reflects only a certain degree of our ignorance about the system or process under study. These problems are being conjugated, in particular, with the issues on predetermination of the various systems dynamics and the Universe as a whole.

After the new branches of science emergence in the last century and, quantum mechanics, in particular, many issues related to the «stochasticity» have lost their urgency, since the concept of «uncertainty» has become fixed at the micro-level as one of the fundamental scientific concepts. However, the issue of the uncertainty «loss» while transition from micro to macro systems remains an open scientific problem.

The article raises questions on the fundamental possibility of the stochastic process emergence in the systems described by the deterministic autonomous differential equations obeying Cauchy’s theorem on existence and uniqueness, as well as questions related to the possibility of «determinization» of stochastic systems with the number of degrees of freedom increase.

The author shows that not only deterministic chaos with a structure difficult to analyze and interpret in the systems of autonomous differential equations, but true non-determinism, i.e.«stochasticity» may occur stipulated by the incompatibility of differential equations due to the finiteness of the time step. This phenomenon may occur herewith at any arbitrarily small, but finite step in time.

The author managed to demonstrate the work that that an increase in the degrees of freedom in the system under consideration leads to its «determinization». This phenomenon allows, in particular, answering the question on why stochastic systems with a large number of degrees of freedom, such as, developed turbulent fluid flows, large star clusters (such as galaxies), etc., are quite deterministic on average and demonstrate a stable dynamic state.

Keywords:

chaos, autonomous differential equations, system of Lorentz equations, stochastic processes, turbulence

References

1. Khatuntseva O.N. Estestvennye i tekhnicheskie nauki, 2017, no. 11, pp. 255-257.
2. Khatuntseva O.N. Trudy MAI, 2020, no. 112. URL: http://trudymai.ru/eng/published.php?ID=116313. DOI: 10.34759/trd-2020-112-1
3. Pierre Bergé, Yves Pomeau, Christian Vidal. L’ordre dans le chaos: Vers une approche déterministe de la turbulence, 1997, 352 p.
4. Magnitskii N.A., Sidorov S.V. Novye metody khaoticheskoi dinamiki (New methods for chaotic dynamics), Moscow, Editorial URSS, 2004, 318 p.
5. Pchelintsev A.N. Sibirskii zhurnal vychislitel’noi matematiki, 2014, vol. 17, no. 2, pp. 191–201.
6. Khatuntseva O.N. Trudy MAI, 2022, no. 123. URL: https://trudymai.ru/eng/published.php?ID=165492. DOI: 10.34759/trd-2022-123-08
7. Khatuntseva O.N. Trudy MAI, 2022, no. 122. URL: https://trudymai.ru/eng/published.php?ID=164194. DOI: 10.34759/trd-2022-122-07
8. Khatuntseva O.N. Trudy MAI, 2021, no. 118. URL: http://trudymai.ru/eng/published.php?ID=158211. DOI: 10.34759/trd-202-118-02
9. Khatuntseva O.N. Generalized Analytical Solution of the Problem of Determining the Universal Profile of the Turbulent Flow of an Incompressible Fluid, A. Ladyzhenskaya centennial conference on PDE’s. Saint Petersburg, 2022, pp. 61.
10. Schlichting H. Boundary layer theory, London, 1955, 535 p.
11. Spalart P.R. Strategies for turbulence modeling and simulation, International Journal of Heat and Fluid Flow, 2000, vol. 21, no. 3, pp. 252-263.
12. Dauchot O., Daviaud F. Finite-amplitude perturbation and spots growth mechanism in plane Couette flow, Physics of Fluids, 1995, no. 7, pp. 335-343.
13. Bottin S., Daviaud F., Manneville P., Dauchot O. Discontinuous transition to spatiotemporal intermittency in plane Couette flow, Europhysics Letters, 2007, no. 43 (2), pp. 171-176.
14. Tuckerman Laurette S., Kreilos T, Schrobsdorff H., Schneider Tobias M., Gibson John F. Turbulent-laminar patterns in plane Poiseuille flow, Physics of Fluids, 2015, DOI:10.1063/1.4900874
15. Orszag Steven A., Kells Lawrence C. Transition to turbulence in plane Poiseuille and plane Couette flow, Journal of Fluid Mechanics, 1980, no. 96, pp. 59-205.
16. Larina E.V., Kryukov I.A., Ivanov I.E. Trudy MAI, 2016, no. 91. URL: http://trudymai.ru/eng/published.php?ID=75565
17. Dehaeze F., Barakos G.N., Batrakov A.S., Kusyumov A.N., Mikhailov S.A. Trudy MAI, 2012, no. 59. URL: http://trudymai.ru/eng/published.php?ID=34840
18. Kravchuk M.O., Kudimov N.F., Safronov A.V. Trudy MAI, 2015, no. 82. URL: http://trudymai.ru/eng/published.php?ID=58536
19. Do S.Z. Trudy MAI, 2014, no. 75. URL: http://trudymai.ru/eng/published.php?ID=49670
20. Vu M.Kh., Popov S.A., Ryzhov Yu.A. Trudy MAI, 2012, no. 53. URL: http://trudymai.ru/eng/published.php?ID=29361
21. Menter F.R. Zonal two equation k-w turbulence models for aerodynamic flows, AIAA Paper, 1993, N93-2906, pp. 21.
22. Shih T.-H., Liou W.W., Shabbir A., Yang Z., and Zhu J. A New k-e Eddy-Viscosity Model for High Reynolds Number Turbulent Flows — Model Developmentand Validation, Computers Fluids, 1995, vol. 24, no. 3, pp. 227-238.
23. Launder B.E., Reece G.J., Rodi W. Progress in the Development of a Reynolds-Stress Turbulence Closure, Journal of Fluid Mechanics, April 1975, vol. 68, no. 3, pp. 537-566.
24. Wilcox David C. Turbulence Modeling for CFD. Second edition, Anaheim: DCW Industries, 1998, 174 p.

Download