Free Energy of a Dilute Bose Gas: Lower Bound

A lower bound is derived on the free energy (per unit volume) of a homogeneous Bose gas at density $ρ$ and temperature $T$. In the dilute regime, i.e., when $a^3ρ\ll 1$, where $a$ denotes the scattering length of the pair-interaction potential, our bound differs to leading order from the expression for non-interacting particles by the term $4πa (2ρ^2 - [ρ-ρ_c]_+^2)$. Here, $ρ_c(T)$ denotes the critical density for Bose-Einstein condensation (for the non-interacting gas), and $[ ]_+$ denotes the positive part. Our bound is uniform in the temperature up to temperatures of the order of the critical temperature, i.e., $T \sim ρ^{2/3}$ or smaller. One of the key ingredients in the proof is the use of coherent states to extend the method introduced in [arXiv:math-ph/0601051] for estimating correlations to temperatures below the critical one.

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