Communications in Mathematical Physics

, Volume 368, Issue 2, pp 723–776 | Cite as

Bose–Einstein Condensation in a Dilute, Trapped Gas at Positive Temperature

  • Andreas Deuchert
  • Robert SeiringerEmail author
  • Jakob Yngvason
Open Access


We consider an interacting, dilute Bose gas trapped in a harmonic potential at a positive temperature. The system is analyzed in a combination of a thermodynamic and a Gross–Pitaevskii (GP) limit where the trap frequency \({\omega}\), the temperature T, and the particle number N are related by \({N \sim (T / \omega)^{3} \to\infty}\) while the scattering length is so small that the interaction energy per particle around the center of the trap is of the same order of magnitude as the spectral gap in the trap. We prove that the difference between the canonical free energy of the interacting gas and the one of the noninteracting system can be obtained by minimizing the GP energy functional. We also prove Bose–Einstein condensation in the following sense: The one-particle density matrix of any approximate minimizer of the canonical free energy functional is to leading order given by that of the noninteracting gas but with the free condensate wavefunction replaced by the GP minimizer.



Partial financial support by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 694227), and by the Austrian Science Fund (FWF), project Nr. P 27533-N27, is gratefully acknowledged.


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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • Andreas Deuchert
    • 1
  • Robert Seiringer
    • 1
    Email author
  • Jakob Yngvason
    • 2
  1. 1.Institute of Science and Technology Austria (IST Austria)KlosterneuburgAustria
  2. 2.Faculty of PhysicsUniversity of ViennaViennaAustria

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