Gross-Pitaevskii 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 Gross-Pitaevskii (GP) 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 $4\pi a\left(2{\varrho }^2-{\varrho }_0^2\right)$. Here $\varrho$ denotes the density of the system and ${\varrho }_0$ is the expected condensate density of the ideal gas. Additionally, we show that the one-particle 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 Bose-Einstein 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 c-number substitution.

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• Upper bound for the grand canonical free energy of the Bose gas in the Gross-Pitaevskii limit
• The free energy of the two-dimensional dilute Bose gas. II. Upper bound
• Semiclassical approximation and critical temperature shift for weakly interacting trapped bosons
• Upper bound for the grand canonical free energy of the Bose gas in the Gross-Pitaevskii limit for general interaction potentials
• Third order corrections to the ground state energy of a Bose gas in the Gross-Pitaevskii regime
• On the effect of repulsive interactions on Bose-Einstein condensation in the Luttinger-Sy model
• Bose-Einstein condensation on hyperbolic spaces
• Second Order Expansion of Gibbs State Reduced Density Matrices in the Gross-Pitaevskii Regime
• Differential Equations of Quantum Mechanics