Free Energy of a Dilute Bose Gas: Lower Bound

TL;DR

This paper establishes a lower bound on the free energy of a dilute homogeneous Bose gas, accounting for interactions and extending previous correlation estimation methods to temperatures near the critical point.

Contribution

It provides a new lower bound on the free energy that incorporates interaction effects and applies uniformly up to near-critical temperatures.

Findings

Lower bound differs from non-interacting case by a specific interaction term

Bound is valid for temperatures up to the order of the critical temperature

Uses coherent states to extend correlation estimation methods

Abstract

A lower bound is derived on the free energy (per unit volume) of a homogeneous Bose gas at density $\rho$ and temperature $T$. In the dilute regime, i.e., when $a^3\rho \ll 1$, 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 $4\pi a (2\rho^2 - [\rho-\rho_c]_+^2)$. Here, $\rho_c(T)$ 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., $T \sim \rho^{2/3}$ 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.