Frontiers of Physics

, 12:120501 | Cite as

On the ground state energy of the inhomogeneous Bose gas

  • V. B. Bobrov
  • S. A. Trigger
Research Article


Within the self-consistent Hartree–Fock approximation, an explicit in this approximation expression for the ground state energy of inhomogeneous Bose gas is derived as a functional of the inhomogeneous density of the Bose–Einstein condensate. The results obtained are based on existence of the off-diagonal long-range order in the single-particle density matrix for systems with a Bose–Einstein condensate. This makes it possible to avoid the use of anomalous averages. The explicit form of the kinetic energy, which differs from one in the Gross–Pitaevski approach, is found. The obtained form of kinetic energy is valid beyond the Hartree–Fock approximation and can be applied for arbitrary strong interparticle interaction.


Bose condensation elementary excitations single-particle Green function density-density Green function thermodynamic energy 



The authors are grateful to A. G. Khrapak, A. A. Rukhadze, P. P. J. M. Schram and A. G. Zagorodny for helpful discussions. This study was supported by the Russian Science Foundation (Project No. 14-19-01492).


  1. 1.
    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(5221), 198 (1995)ADSCrossRefGoogle Scholar
  2. 2.
    E. A. Cornell and C. E. Wieman, Nobel Lecture: Bose–Einstein condensation in a dilute gas, the first 70 years and some recent experiments, Rev. Mod. Phys. 74(3), 875 (2002)ADSCrossRefGoogle Scholar
  3. 3.
    L. P. Pitaevskii, Bose–Einstein condensation in magnetic traps: Introduction to the theory, Phys. Usp. 41(6), 569 (1998)CrossRefGoogle Scholar
  4. 4.
    E. P. Gross, Structure of a quantized vortex in boson systems, Nuovo Cim. 20(3), 454 (1961)ADSMathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    L. P. Pitaevskii, Zh. Éksp. Teor. Fiz. 40, 646 (1961) [Sov. Phys. JETP 13, 451 (1961).Google Scholar
  6. 6.
    L. P. Pitaevskii, Bose–Einstein condensates in a laser radiation field, Phys. Usp. 49(4), 333 (2006)ADSCrossRefGoogle Scholar
  7. 7.
    E. H. Lieb, R. Seiringer, and J. Yngvason, Bosons in a trap: A rigorous derivation of the Gross–Pitaevskii energy functional, Phys. Rev. A 61(4), 043602 (2000)ADSCrossRefGoogle Scholar
  8. 8.
    E. M. Lifshitz and L. P. Pitaevskii, Statistical Physics, Part 2: Theory of the Condensed State, Oxford: Butterworth-Heinemann, 1980Google Scholar
  9. 9.
    W. H. Bassichis and L. L. Foldy, Analysis of the Bogoliubov method applied to a simple Boson model, Phys. Rev. 133(4A), 935 (1964)ADSCrossRefGoogle Scholar
  10. 10.
    H. Stolz, Theory of interacting bosons without anomalous propagators, Physica A 86(1), 111 (1977)ADSCrossRefGoogle Scholar
  11. 11.
    C. H. Zhang and H. A. Fertig, Superfluidity without symmetry breaking: The time-dependent Hartree–Fock approximation for Bose-condensed condensates, Phys. Rev. A 74(2), 023613 (2006)ADSCrossRefGoogle Scholar
  12. 12.
    P. Navez and K. Bongs, Gap and screening in Raman scattering of a Bose condensed gas, Europhys. Lett. 88(6), 60008 (2009)ADSCrossRefGoogle Scholar
  13. 13.
    V. B. Bobrov, S. A. Trigger, and I. M. Yurin, Coexistence of “bogolons” and the single-particle excitation spectrum with a gap in the degenerate Bose gas, Phys. Lett. A 374(19–20), 1938 (2010)ADSCrossRefMATHGoogle Scholar
  14. 14.
    A. M. Ettouhami, Re-examining Bogoliubov’s theory of an interacting Bose gas, Prog. Theor. Phys. 127(3), 453 (2012)ADSCrossRefMATHGoogle Scholar
  15. 15.
    V. B. Bobrov and S. A. Trigger, Structure factor and distribution function of degenerate Bose gases without anomalous averages, J. Low Temp. Phys. 170(1–2), 31 (2013)ADSCrossRefGoogle Scholar
  16. 16.
    V. B. Bobrov, S. A. Triger, and P. Schram, Sov. Phys. JETP 80, 853 (1995)ADSGoogle Scholar
  17. 17.
    V. B. Bobrov and S. A. Trigger, On the properties of systems with Bose–Einstein condensate in the Coulomb model of matter, Bull. Lebedev Phys. Inst. 42(1), 13 (2015)ADSCrossRefGoogle Scholar
  18. 18.
    V. B. Bobrov, A. G. Zagorodny, and S. A. Trigger, Coulomb interaction potential and Bose–Einstein condensate, Low Temp. Phys. 41, 901 (2015)ADSCrossRefGoogle Scholar
  19. 19.
    L. P. Kadanoff and G. Baym, Quantum Statistical Mechanics, New York: Benjamin, 1962MATHGoogle Scholar
  20. 20.
    N. N. Bogolubov and N. N. Jr Bogolubov, Introduction to Quantum Statistical Mechanics, New York: Gordon and Breach, 1992MATHGoogle Scholar
  21. 21.
    V. B. Bobrov, S. A. Trigger, and A. Zagorodny, Virial theorem, one-particle density matrix, and equilibrium condition in an external field, Phys. Rev. A 82(4), 044105 (2010)ADSCrossRefGoogle Scholar
  22. 22.
    V. L. Ginzburg and L. D. Landau, Zh. Eksp. Teor. Fiz. 20, 1064 (1950) (English transl.: L. D. Landau, Collected Papers, Oxford: Pergamon, 1965, p. 546)Google Scholar
  23. 23.
    O. Penrose and L. Onsager, Bose–Einstein condensation and liquid helium, Phys. Rev. 104(3), 576 (1956)ADSCrossRefMATHGoogle Scholar
  24. 24.
    C. N. Yang, Concept of off-diagonal long-range order and the quantum phases of liquid He and of superconductors, Rev. Mod. Phys. 34(4), 694 (1962)ADSMathSciNetCrossRefGoogle Scholar
  25. 25.
    V. B. Bobrov, S. A. Trigger, and A. G. Zagorodny, The off-diagonal long-range order and inhomogeneous Bose–Einstein condensate, Dokl. Phys. 60(4), 147 (2015)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, Theory of Bose–Einstein condensation in trapped gases, Rev. Mod. Phys. 71(3), 463 (1999)ADSCrossRefGoogle Scholar
  27. 27.
    P. Hohenberg and W. Kohn, Inhomogeneous electron gas, Phys. Rev. 136(3B), B864 (1964)ADSMathSciNetCrossRefGoogle Scholar
  28. 28.
    R. O. Jones and O. Gunnarsson, The density functional formalism, its applications and prospects, Rev. Mod. Phys. 61(3), 689 (1989)ADSCrossRefGoogle Scholar
  29. 29.
    N. N. Bogoliubov, On the theory of superfluidity, J. Phys. (USSR) 11, 23 (1947)MathSciNetGoogle Scholar
  30. 30.
    V. B. Bobrov, A. Zagorodny, and S. A. Trigger, Coulomb interaction potential and Bose–Einstein condensate, Low Temp. Phys. 41(11), 901 (2015)ADSCrossRefGoogle Scholar
  31. 31.
    N. Navon, S. Piatecki, K. Günter, B. Rem, T. C. Nguyen, F. Chevy, W. Krauth, and C. Salomon, Dynamics and thermodynamics of the low-temperature strongly interacting Bose gas, Phys. Rev. Lett. 107(13), 135301 (2011)ADSCrossRefGoogle Scholar
  32. 32.
    T. L. Ho and Q. Zhou, Chromatin remodelling during development, Nature 463(7280), 1057 (2010)CrossRefGoogle Scholar
  33. 33.
    L. D. Landau and E. M. Lifshitz, Statistical Physics, Part 1, Oxford: Butterworth-Heinemann, 1980MATHGoogle Scholar

Copyright information

© Higher Education Press and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Joint Institute for High TemperaturesRussian Academy of SciencesMoscowRussia

Personalised recommendations