Free Energies of Dilute Bose gases: upper bound

TL;DR

This paper establishes an upper bound on the free energy of dilute Bose gases, complementing previous lower bounds, and precisely characterizes the free energy difference in the low-density limit, especially near the Bose-Einstein condensation threshold.

Contribution

It provides a rigorous upper bound on the free energy of dilute Bose gases, confirming the leading order behavior in the low-density regime and near the critical density for Bose-Einstein condensation.

Findings

Matching upper and lower bounds confirm the free energy difference formula

The leading term of free energy difference is proportional to the scattering length and density

Results apply in the low-density limit where $a^3 ho o 0$

Abstract

We derive a upper bound on the free energy of a Bose gas system at density $\rho$ and temperature $T$. In combination with the lower bound derived previously by Seiringer \cite{RS1}, our result proves that in the low density limit, i.e., when $a^3\rho\ll 1$, where $a$ denotes the scattering length of the pair-interaction potential, the leading term of $\Delta f$ the free energy difference per volume between interacting and ideal Bose gases is equal to $4\pi a (2\rho^2-[\rho-\rhoc]^2_+)$. Here, $\rhoc(T)$ denotes the critical density for Bose-Einstein condensation (for the ideal gas), and $[\cdot ]_+$ $=$ $\max\{\cdot, 0\}$ denotes the positive part.