Upper Bound for the Free Energy of Dilute Bose Gases at Low Temperature

TL;DR

This paper establishes an upper bound for the free energy of dilute Bose gases at low temperature, matching recent lower bounds and confirming the Lee–Huang–Yang correction order.

Contribution

It provides a new upper bound for the free energy of dilute Bose gases at low temperature, complementing existing lower bounds and confirming the Lee–Huang–Yang order.

Findings

Upper bound matches the Lee–Huang–Yang order

Confirms the free energy estimate at low temperature

Complements recent lower bounds

Abstract

We consider a Bose gas at density $\rho > 0$, interacting through a repulsive potential $V \in L^2 (\mathbb{R}^3)$ with scattering length $\mathfrak{a} > 0$. We prove an upper bound for the free energy of the system, valid at low temperature $T \lesssim \rho \mathfrak{a}$. Combined with the recent lower bound obtained in \cite{HabHaiNamSeiTri-23}, our estimate resolves the free energy per unit volume up to and including the Lee--Huang--Yang order $\mathfrak{a} \rho^2 (\rho \mathfrak{a}^3)^{1/2}$.