GrossPitaevskii Limit of a Homogeneous Bose Gas at Positive Temperature
Abstract
We consider a dilute, homogeneous Bose gas at positive temperature. The system is investigated in the GrossPitaevskii limit, where the scattering length a is so small that the interaction energy is of the same order of magnitude as the spectral gap of the Laplacian, and for temperatures that are comparable to the critical temperature of the ideal gas. We show that the difference between the specific free energy of the interacting system and the one of the ideal gas is to leading order given by 4 π a (2 ϱ^{2}ϱ_{0}^{2}) . Here ϱ denotes the density of the system and ϱ_{0} is the expected condensate density of the ideal gas. Additionally, we show that the oneparticle density matrix of any approximate minimizer of the Gibbs free energy functional is to leading order given by the one of the ideal gas. This in particular proves BoseEinstein condensation with critical temperature given by the one of the ideal gas to leading order. One key ingredient of our proof is a novel use of the Gibbs variational principle that goes hand in hand with the cnumber substitution.
 Publication:

Archive for Rational Mechanics and Analysis
 Pub Date:
 March 2020
 DOI:
 10.1007/s00205020014894
 arXiv:
 arXiv:1901.11363
 Bibcode:
 2020ArRMA.236.1217D
 Keywords:

 Mathematical Physics;
 Condensed Matter  Quantum Gases
 EPrint:
 Archive for Rational Mechanics and Analysis, 236(3), 12171271 (2020)