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

Abstract

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.

arXiv comment52 pages; additional references added

References in corpus (22)

• Theory of Bose-Einstein condensation in trapped gases
• Bose-Einstein condensation of atoms in a uniform potential
• Proof of Bose-Einstein Condensation for Dilute Trapped Gases
• Bosons in a Trap: A Rigorous Derivation of the Gross-Pitaevskii Energy Functional
• Derivation of the Gross-Pitaevskii Equation for Rotating Bose Gases
• Ground State Energy of the Low Density Bose Gas
• Quantum depletion of a homogeneous Bose-Einstein condensate
• Near Resonant Spatial Images of Confined Bose-Einstein Condensates in the '4D' Magnetic Bottle
• Rigorous Derivation of the Gross-Pitaevskii Equation with a Large Interaction Potential
• Quantitative Derivation of the Gross-Pitaevskii Equation
• Ground states of large bosonic systems: The gross-pitaevskii limit revisited
• Ground State Energy of the Low Density Fermi Gas
• Can a Bose gas be saturated?
• The Thermodynamic Pressure of a Dilute Fermi Gas
• Free Energy of a Dilute Bose Gas: Lower Bound
• The Canonical Perfect Bose Gas in Casimir Boxes
• Gibbs measures of nonlinear Schrödinger equations as limits of many-body quantum states in dimensions $d\le 3$
• Fluctuations of the Bose-Einstein condensate
• Free Energies of Dilute Bose gases: upper bound
• A family of monotone quantum relative entropies
• Note on a Family of Monotone Quantum Relative Entropies
• Correlation inequalities for noninteracting Bose gases

Cited by in corpus (12)

• Optimal rate of condensation for trapped bosons in the Gross-Pitaevskii regime
• Gross-Pitaevskii Limit of a Homogeneous Bose Gas at Positive Temperature
• Ground state energy of a dilute two-dimensional Bose gas from the Bogoliubov free energy functional
• A path-integral analysis of interacting Bose gases and loop gases
• Upper bound for the grand canonical free energy of the Bose gas in the Gross-Pitaevskii limit
• Finite-Size Effects with Boundary Conditions on Bose-Einstein Condensation
• 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
• On the effect of repulsive interactions on Bose-Einstein condensation in the Luttinger-Sy model
• Second Order Expansion of Gibbs State Reduced Density Matrices in the Gross-Pitaevskii Regime
• Thermodynamics of inhomogenous imperfect quantum gases in harmonic traps