Free Energy of a Dilute Bose Gas: Lower Bound

Abstract

A lower bound is derived on the free energy (per unit volume) of a homogeneous Bose gas at density $\varrho$ and temperature T. In the dilute regime, i.e., when $a^3\varrho \ll 1$ , 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 $4{\pi}a ( 2{\varrho^2}-[\varrho-\varrho_c]_+^2 )$ . Here, $\varrho_c(T)$ denotes the critical density for Bose-Einstein condensation (for the non-interacting gas), and $[\, \cdot \, ]_+ = \max\{ \, \cdot\, , 0\}$ denotes the positive part. Our bound is uniform in the temperature up to temperatures of the order of the critical temperature, i.e., T ~ $\varrho$ ^2/3 or smaller. One of the key ingredients in the proof is the use of coherent states to extend the method introduced in [17] for estimating correlations to temperatures below the critical one.