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METHODS OF FUNCTIONAL ANALYSIS FOR THE DISCRETE VELOCITY MODEL OF THE BBGKY HIERARCHY OF EQUATIONS OF SYMMETRIC SYSTEMS OF PARTICLES INTERACTING AS HARD SPHERES

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Abstract

The BBGKY hierarchy of equations was derived by Bogoliubov, Born, Green, Kirkwood, Yvon.


References

  1. Illner R., Pulvirenti M. A derivation of the BBGKY hierarchy for hard sphere particle systems // Transp. Theory and Statist. Phys. – 1987. – Vol. 16, № 7. – P. 997-1012.
  2. Krönke, S., & Schmelcher, P. (2018). Born-Bogoliubov-Green-Kirkwood-Yvon hierarchy for ultracold bosonic systems. Physical Review A, 98(1), 013629.
  3. Lenard, A. (1960). On Bogoliubov's kinetic equation for a spatially homogeneous plasma. Annals of Physics, 10(3), 390-400.
  4. Morrison, P. J. (1984). The Hamiltonian structure of the BBGKY hierarchy equations. Contemporary Mathematics, 28.
  5. Hubal, H. M. (2016). The convergence of the series of the solution of the cauchy problem for the bbgky hierarchy of equations in many-kind particle systems. International Journal of Pure and Applied Mathematics, 108(4), 957-965.
  6. Hubal, H.M. (2019) Mathematical description of the non-equilibrium state of symmetric particle systems. International Journal of Applied Mathematics, 32(5), 767–774.
  7. Lanford III O. E. Time evolution of large classical systems // Lecture Notes in Phys. – Berlin: Springer-Verlag. – 1975. – Vol. 38. – P. 1-111.
  8. Cercignani C., Illner R., Pulvirenti M. The mathematical theory of dilute gases // Appl. Math. Sci. – New York: Springer-Verlag, 1994. – Vol. 106. – vii, 347 pp.
  9. Gallavotti G., Lanford III O. E., Lebowitz J. L. Thermodynamic limit of time-dependent correlation functions for one-dimensional systems // J. Math. Phys. – 1970. – Vol. 11, № 9. – P. 2898-2905.
  10. Loeb, L. B. (2004). The kinetic theory of gases. Courier Corporation.
  11. Vereshchagin, G. V., & Aksenov, A. G. (2017). Relativistic kinetic theory: with applications in astrophysics and cosmology. Cambridge University Press.
  12. Baalrud, S. D., & Daligault, J. (2019). Mean force kinetic theory: A convergent kinetic theory for weakly and strongly coupled plasmas. Physics of Plasmas, 26(8).
  13. Borgioli G., Lauro G., Monaco R. On the discrete velocity models of the Enskog equation // Nonlinear kinetic theory and mathematical aspects of hyperbolic systems / Ed. V. Boffi, F. Bampi, G. Toscani. – Singapore: World Scientific. – 1993. – P. 38-47.
  14. Hubal, H. M. (2010). The Fokker-Planck Equation for the System "Brownian Particle in Thermostat" Based on the Presented Probability Approach. J. Math. Phys., Anal., Geom., 6(1), 48–55.
  15. Hubal, H. M. (2014). The Einstein Law for the System ”Brownian Particle in Thermostat” Based on the Presented Probability Approach. International Journal of Pure and Applied Mathematics, 93(6), 885-895.
  16. Marchioro C., Pellegrinotti A., Pulvirenti M. Remarks on the existence of non equilibrium dynamics // Colloquia Math. Soc. Janos Bolyai, 10. Random Fields. Rigorous Results in Statist. Mech. and Quantum Field Theory. – Amsterdam: North-Holland Publ. Co. – 1981. – Vol. II. – P. 733-746.
  17. Spohn H. Large scale dynamics of interacting particles. – Berlin: Springer-Verlag, 1991. – xi, 342 pp.
  18. Uchiyama K. Derivation of the Boltzmann equation from particle dynamics // Hiroshima Math. J. – 1988. – Vol. 18, № 2. – P. 245-297.
  19. Villani, C. (2002). A review of mathematical topics in collisional kinetic theory. Handbook of mathematical fluid dynamics, 1(71-305), 3-8.
  20. Yvon J. La théorie statistique des fluides et l'équation d'état. – Paris: Hermann et Cie , 1935. – 49 p.
  21. Губаль, Г. М. (2013). LATEX як видавнича система для створення математичних текстів і для програмування. Комп'ютерно-інтегровані технології: освіта, наука, виробництво, (12), 23-26.
  22. Губаль, Г. М. (2013). Стратегії для створення математичної статті у видавничій системі LATEX. Комп'ютерно-інтегровані технології: освіта, наука, виробництво, (13), 10-13.
  23. De Marco, A. (2019). Graphics for LATEX users. ArsTEXnica, 65.
  24. Goossens, M., Rahtz, S. P., & Mittelbach, F. (1997). The LATEX graphics companion: illustrating documents with TEX and PostScript (Vol. 3). Addison-Wesley Professional.
  25. Grätzer, G. (2014). Practical LaTeX. Springer.
  26. Griffiths, D. F., & Higham, D. J. (2016). Learning LaTeX. Society for Industrial and Applied Mathematics.
  27. Hubal, H. M. (2018). Mathematical texts and figures in the LATEX system. Computer Integrated Technologies: Education, Science, Production, (32), 90-94.
  28. Kottwitz, S. (2021). LaTeX beginner’s guide. Packt Publishing, USA, (354 p.).
  29. Nakashima, H., & Saito, Y. (2007). How to Typeset Your Papers in LATEX (Version 6). IPSJ Journal.
  30. Губаль, Г. М. (2014). Особливості створення інтерактивних математичних тестів у видавничій системі LATEX. Комп’ютерно-інтегровані технології: освіта, наука, виробництво, (15), 9-13.
  31. Hubal, H.M. (2023) Improvement of references and footnotes in mathematical and other texts by creating macros in the LaTeX programming language. International Journal on Information Technologies & Security, (15(3)), 15–22.
  32. Onyenwe, I. E., Onyedinma, E., Ikechukwu-Onyenwe, O. O., Agbata, O., & Tubo, F. N. (2024). Integration of LaTeX formula in computer-based test application for academic purposes. arXiv preprint arXiv:2402.01660.
  33. Hubal, Н. (2019). Construction and study of the system of differential equations that describes oscillatory chemical reactions based on diffusion. Computer-Integrated Technologies: Education, Science, Production, (34), 32-36.
  34. Hubal, H.M. (2021). Mathematical modeling of biochemical processes rates in biological systems. Computer-Integrated Technologies: Education, Science, Production, (42), 43–49.
  35. Губаль, Г. М. (2022). Математичне дослідження стійкості особливих точок систем диференціальних рівнянь, які описують швидкості біохімічних процесів. Наукові нотатки, (73), 29-39.
  36. Hubal, H. (2020). Construction and study of the system of differential equations that describes the mutual synchronization of coupled self-oscillating chemical systems. Computer-Integrated Technologies: Education, Science, Production, (41), 30-34.
  37. Gerasimenko V. I., Ryabukha T. V., Stashenko M. O. (2004) On the structure of expansions for the BBGKY hierarchy solutions // J. Phys. A: Math. and General, Vol. 37, № 42. – P. 9861-9872.

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