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Generalization of the Gross–Pitaevskii equation for multi-particle cases

Nicolai Bogoliubov, Jr.

Steklov Institute of Mathematics of the Russian Academy of Sciences, Moscow 119991, Russia

E-mail Address: nikolai_bogolubov@hotmail.com

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Mukhayo Rasulova

Institute of Nuclear Physics Academy of Sciences of Uzbekistan, Tashkent 100214, Uzbekistan

E-mail Address: rasulova@live.com

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, and

Tohir Akramov

https://orcid.org/0000-0002-2463-5874

National University of Uzbekistan named after M. Ulugbek, Institute of Nuclear Physics Academy of Sciences of Uzbekistan, Tashkent 100214, Uzbekistan

E-mail Address: tohir.akramov@yahoo.com

Corresponding author.

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https://doi.org/10.1142/S0217979221503112Cited by:0 (Source: Crossref)

Abstract

A new method is proposed to obtain Gross–Pitaevskii equation by the chain of Bogoliubov–Born–Green–Kirkwood–Yvon (BBGKY) quantum kinetic equations. In that sense, we investigate the dynamics of a quantum system including infinite number of identical particles which interact via a (special) pair potential on the form of Dirac delta-function.

Keywords:

PACS: 05.30.−d, 51.10.+y

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References

1. A. Einstein, Berl. Veg. 22, 261 (1924); 23, 3 (1925); 23, 18 (1925). Google Scholar
2. L. P. Pitaevskii , UFN 168(6), 641 (1998). Crossref, Web of Science, Google Scholar
3. E. P. Gross, Nuovo Cimento 20, 454 (1961); Math. Phys. 4, 195 (1963). Google Scholar
4. L. P. Pitaevsky , J. Exp. Theor. Phys. 40, 646 (1961). Google Scholar
5. N. N. Bogoliubov , Izv. AN USSR, Ser. Phys. 11(1), 77 (1947). Google Scholar
6. N. N. Bogoliubov , Vestnik Moskovskogo Univ. 7, 43 (1947). Google Scholar
7. N. N. Bogoliubov , Lectures on Quantum Statistics (Radyanska shkola, 1949). Google Scholar
8. C. J. Pethick and H. Smith , Bose–Einstein Condensation in Dilute Gases (University Press, Cambridge, 2008). Crossref, Google Scholar
9. V. B. Bobrov, S. A. Trigger and I. M. Yurin , Phys. Lett. A374, 1938 (2010). Crossref, Web of Science, ADS, Google Scholar
10. V. B. Bobrov and S. A. Trigger , Bose-Einstein condensate wave function and non-linear Schrodinger equation, Bull. Lebedev Phys. Inst. 43, 266 (2016). Crossref, Web of Science, ADS, Google Scholar
11. V. B. Bobrov, A. G. Zagorodny and S. A. Trigger , Low Temp. Phys. 43, 343 (2017). Crossref, Web of Science, ADS, Google Scholar
12. N. N. Bogoliubov , Problemi dynamicheskoy theorii v statisticheskoy fizike (Gostehizdat, M.-L., 1946). Google Scholar
13. N. N. Bogoliubov and N. N. Bogoliubov (Jr.) , Introduction to Quantum Statistical Mechanics (Nauka, Moscow, 1984). Google Scholar
14. M. Yu. Rasulova , The BBGKY hierarchy of quantum kinetic equations and its application in cryptography, Phys. Part. Nucl. 51(4), 781 (2020). Crossref, Web of Science, Google Scholar
15. L. Erdös, B. Schlein and H.-T. Yau , Derivation of the Gross-Pitaevskii hierarchy for the dynamics of Bose-Einstein condensate, Commun. Pure Appl. Math. 59(12), 1659 (2006). Crossref, Web of Science, Google Scholar
16. L. Erdös, B. Schlein and H.-T. Yau , Derivation of the cubic non-linear Schrodinger equation from quantum dynamics of many-body systems, Invent. Math. 167, 515 (2007). Crossref, Web of Science, ADS, Google Scholar
17. L. Erdös, B. Schlein and H.-T. Yau , Derivation of the Gross-Pitaevskii equation for the dynamics of Bose-Einstein condensate, Annals on Mathematics, Princeton University and Institute for Advanced Study 172(1), 291 (2010). Google Scholar
18. N. Bogoliubov (Jr.) et al., A chain of kinetic equations of Bogoliubov–Born–Green–Kirkwood–Yvon and its application to non-equilibrium complex systems, Multi-Chaos, Fractal, and Multi-Fractional Artificial Intelligence of Different Complex Systems, (Elsevier, 2021). Google Scholar
19. E. H. Lieb and W. Liniger , Exast analysis of interacting Bose gas. The general solution and the ground state, Phys. Rev. 130, 1605 (1963). Crossref, Web of Science, ADS, Google Scholar
20. M. Yu. Rasulova and A. K. Vidibida, Kinetic equations for distribution functions and density matrices, ITP-27P Preprint (1976). Google Scholar
21. N. N. Bogoliubov and M. Yu. Rasulova , The cauchy problem for the hierarchy of quantum kinetic equations for correlation matrices with coulomb potential, Appl. Math. Inform. Sci. 10(11), 2365 (2016). Crossref, Google Scholar
22. D. Ruelle , Statistical Mechanics, Rigorous Results (Mir, Moscow, 1971). Google Scholar
23. D. Ya. Petrina and V. Z. Enolskii , Oscillations of one-dimensional systems, Dopovidi Akad. Nauk Ukrainskoi Rsr Seriya a-Fiziko-Matematichni Ta Technichni Nauki 8, 756 (1976). Google Scholar