Upper bound for the grand canonical free energy of the Bose gas in the Gross-Pitaevskii limit for general interaction potentials

Abstract

We consider a homogeneous Bose gas in the Gross--Pitaevskii limit at temperatures that are comparable to the critical temperature for Bose--Einstein condensation. Recently, an upper bound for the grand canonical free energy was proved in arXiv:2305.19173 [math-ph] capturing two novel contributions. First, the free energy of the interacting condensate is given in terms of an effective theory describing the probability distribution of the number of condensed particles. Second, the free energy of the thermally excited particles equals that of a temperature-dependent Bogoliubov Hamiltonian. We extend this result to a more general class of interaction potentials, including interactions with a hard core. Our proof follows a different approach than the one in arXiv:2305.19173 [math-ph]: we model microscopic correlations between the particles by a Jastrow factor, and exploit a cancellation in the computation of the energy that emerges due to the different length scales in the system.

arXiv comment51 pages

References in corpus (29)

• 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
• Observation of grand-canonical number statistics in a photon Bose-Einstein condensate
• The Second Order Upper Bound for the Ground Energy of a Bose Gas
• Justification of c-Number Substitutions in Bosonic Hamiltonians
• Optimal Rate for Bose-Einstein Condensation in the Gross-Pitaevskii Regime
• Ground states of large bosonic systems: The gross-pitaevskii limit revisited
• Gross-Pitaevskii Dynamics for Bose-Einstein Condensates
• Ground State Asymptotics of a Dilute, Rotating Gas
• Bose-Einstein Condensation Beyond the Gross-Pitaevskii Regime
• Superfluidity in Dilute Trapped Bose Gases
• Interaction Effects on Number Fluctuations in a Bose-Einstein Condensate of Light
• Bose-Einstein Condensation with Optimal Rate for Trapped Bosons in the Gross-Pitaevskii Regime
• The Thermodynamic Pressure of a Dilute Fermi Gas
• Excitation Spectrum for Bose Gases beyond the Gross-Pitaevskii Regime
• The Bogoliubov free energy functional II. The dilute limit
• Free Energy of a Dilute Bose Gas: Lower Bound
• Fluctuations of the Bose-Einstein condensate
• Bose-Einstein Condensation in a Dilute, Trapped Gas at Positive Temperature
• Free Energies of Dilute Bose gases: upper bound
• Gross-Pitaevskii Limit of a Homogeneous Bose Gas at Positive Temperature
• A second order upper bound for the ground state energy of a hard-sphere gas in the Gross-Pitaevskii regime
• The excitation spectrum of two dimensional Bose gases in the Gross-Pitaevskii regime
• The Bose gas in a box with Neumann boundary conditions
• 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. I. Lower bound
• The free energy of the two-dimensional dilute Bose gas. II. Upper bound

Cited by in corpus (2)

• Third order corrections to the ground state energy of a Bose gas in the Gross-Pitaevskii regime
• The excitation spectrum of a Bose gas with an impurity in the Gross-Pitaevskii regime