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On Condensations in the Bogoliubov Weakly Imperfect Bose Gas

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Abstract

We show that condensation in the Bogoliubov weakly imperfect Bose gas (WIBG) may appear in two stages. If interaction is such that the pressure of the WIBG does not coincide with the pressure of the perfect Bose gas (PBG), then the WIBG may manifest two kinds of condensations: nonconventional Bose condensation in zero mode, due to the interaction (the first stage), and conventional (generalized) Bose–Einstein condensation in modes next to the zero mode due to the particle density saturation (the second stage). Otherwise the WIBG manifests only the latter kind of condensation.

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REFERENCES

  1. N. N. Bogoliubov, On the theory of superfluidity, J. Phys. (USSR) 11:23 (1947).

    Google Scholar 

  2. N. N. Bogoliubov, Lectures on Quantum Statistics Vol I: Quantum Statistics (Gordon and Breach, Science Publishers, New York/London/Paris, 1970).

    Google Scholar 

  3. N. Angelescu, A. Verbeure, and V. A. Zagrebnov, On Bogoliubov's model of superfluidity, J. Phys. A: Math. Gen. 25:3473 (1992).

    Google Scholar 

  4. J.-B. Bru and V. A. Zagrebnov, Exact phase diagram of the Bogoliubov weakly imperfect Bose gas, Phys. Lett. A 244:371 (1998).

    Google Scholar 

  5. J.-B. Bru and V. A. Zagrebnov, Exact solution of the Bogoliubov Hamiltonian for weakly imperfect Bose gas, J. Phys. A: Math. Gen. A 31:9377 (1998).

    Google Scholar 

  6. J.-B. Bru and V. A. Zagrebnov, Quantum interpretation of thermodynamic behaviour of the Bogoliubov weakly imperfect Bose gas, Phys. Lett. A 247:37 (1998).

    Google Scholar 

  7. J. Ginibre, On the asymptotic exactness of the Bogoliubov approximation for many bsons systems, Commun. Math. Phys. 8:26 (1968).

    Google Scholar 

  8. R. Griffiths, A proof that the free energy of a spin system is extensive, J. Math. Phys. 5:1215 (1964).

    Google Scholar 

  9. K. Hepp and E. H. Lieb, Equilibrium statistical mechanics of matter interacting with the quantized radiation field, Phys. Rev. A 8:2517 (1973).

    Google Scholar 

  10. T. Michoel and A. Verbeure, Nonextensive Bose-Einstein condensation model, J. Math. Phys. 40:1268 (1999).

    Google Scholar 

  11. J.-B. Bru and V. A. Zagrebnov, Exactly soluble model with two kinds of Bose-Einstein condensations, Physica A 268:309 (1999).

    Google Scholar 

  12. M. van den Berg and J. T. Lewis, On generalized condensation in the free boson gas, Physica A 110:550 (1982).

    Google Scholar 

  13. M. van den Berg, On boson condensation into an infinite number of low-lying levels, J. Math. Phys. 23:1159 (1982).

    Google Scholar 

  14. M. van den Berg, J. T. Lewis, and J. V. Pulè, A general theory of Bose-Einstein condensation in the free boson gas, Helv. Phys. Acta 59:1271 (1986).

    Google Scholar 

  15. Vl. V. Papoyan and V. A. Zagrebnov, On generalized Bose-Einstein condensation in the almost-ideal boson gas, Helv. Phys. Acta 63:557 (1990).

    Google Scholar 

  16. J. T. Lewis, Mark Kac seminar on probability and physics: The Large Deviation Principle in Statistical Mechanics, syllabus 17 (Centrum voor Wiskunde en Informatica CWI, Amsterdam, 1985-1987).

    Google Scholar 

  17. J. T. Lewis and C.-E. Pfister, Thermodynamic probability theory: some aspects of large deviations, Russian Math. Surveys 50(2):279 (1995).

    Google Scholar 

  18. E. Størmer, Symmetric states of infinite tensor products of C*-algebra, J. Funct. Anal. 3:48 (1969).

    Google Scholar 

  19. M. Fannes, J. T. Lewis, and A. Verbeure, Symmetric states of composite systems, Lett. Math. Phys. 15:255 (1988).

    Google Scholar 

  20. N. G. Duffield, H. Ross, and R. F. Werner, Macroscopic limiting dynamics of a class of inhomogeneous mean field quantum systems, Ann. Inst. Henri Poincaré 56:143 (1992).

    Google Scholar 

  21. J. Manuceau and A. Verbeure, Quasi-free states of the C.C.R.—algebra and Bogoliubov transformations, Commun. Math. Phys. 9:293 (1968).

    Google Scholar 

  22. O. Brattelli and D. W. Robinson, Operator Algebras and Quantum Statistical Mechanics, Vol II, 2nd ed. (Springer-Verlag, New York, 1996).

    Google Scholar 

  23. E. B. Davies, The thermodynamic limit for an imperfect boson gas, Comm. Math. Phys. 28:69 (1972).

    Google Scholar 

  24. J. T. Lewis, J. V. Pulé, and V. A. Zagrebnov, The large deviation principle for the Kac distribution, Helv. Phys. Acta 61:1063 (1988).

    Google Scholar 

  25. N. Angelescu and A. Verbeure, Variational solution of a superfluidity model, Physica A 216:388 (1995).

    Google Scholar 

  26. N. Angelescu, A. Verbeure, and V. A. Zagrebnov, Superfluidity III, J. Phys. A: Math. Gen. 30:4895 (1997).

    Google Scholar 

  27. M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis (Academic Press, New York and London, 1972).

    Google Scholar 

  28. M. van den Berg and J. T. Lewis, On the free boson gas in a weak external potential, Commun. Math. Phys. 81:475 (1981).

    Google Scholar 

  29. J. V. Pulé, The free boson gas in a weak external potential, J. Math. Phys. 24:138 (1983).

    Google Scholar 

  30. M. van den Berg, On condensation in the free-bosons gas and the spectrum of the Laplacian, J. Stat. Phys. 31:623 (1983).

    Google Scholar 

  31. A. Einstein, Quantentheorie des einatomigen idealen Gases, Sitzungsberichte der Preussischen Akademie der Wissenschaften I:3 (1925).

    Google Scholar 

  32. R. M. Ziff, G. E. Uhlenbeck, and M. Kac, The ideal Bose-Einstein gas, revisited, Phys. Rep. 32:169 (1977).

    Google Scholar 

  33. K. B. Davis, M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett. 75:3969 (1995).

    Google Scholar 

  34. M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Observation of Bose-Einstein condensation in a dilute atomic vapor, Science 269:198 (1995).

    Google Scholar 

  35. C. C. Bradley, C. A. Sackett, J. J. Tollet, and R. G. Hulet, Evidence of Bose-Einstein condensation in an atomic gas with attractive interactions, Phys. Rev. Lett. 75:1687 (1995).

    Google Scholar 

  36. D. Ruelle, Statistical Mechanics: Rigorous Results (Benjamin, New York, 1969).

    Google Scholar 

  37. N. N. Bogoliubov (Jr.), J. G. Brankov, V. A. Zagrebnov, A. M. Kurbatov, and N. S. Tonchev, Some classes of exactly soluble models of problems in Quantum Statistical Mechanics the method of the approximating Hamiltonian, Russian Math. Surveys 39(6):1 (1984).

    Google Scholar 

  38. M. Fannes and A. Verbeure, Correlation inequalities and equilibrium states I, Commun. Math. Phys. 55:125 (1977).

    Google Scholar 

  39. M. Fannes and A. Verbeure, Correlation inequalities and equilibrium states II, Commun. Math. Phys. 57:165 (1977).

    Google Scholar 

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Bru, JB., Zagrebnov, V.A. On Condensations in the Bogoliubov Weakly Imperfect Bose Gas. Journal of Statistical Physics 99, 1297–1338 (2000). https://doi.org/10.1023/A:1018692823463

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