Free Energy of a Dilute Bose Gas: Lower Bound
Tóm tắt
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.
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