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Advances in Mathematics

Upper bound for the free energy of dilute Bose gases at low temperature

Florian Haberberger, Christian Hainzl, Benjamin Schlein, Arnaud Triay

Published Feb 3, 2026• 3 citations
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Abstract

We consider a Bose gas at density ρ > 0 , interacting through a repulsive potential V ∈ L 2 ( R 3 ) with scattering length a > 0 . We prove an upper bound for the free energy of the system, valid at low temperature T ≲ ρ a . Combined with the recent lower bound obtained in [18] , our estimate resolves the free energy per unit volume up to and including the Lee–Huang–Yang order a ρ 2 ( ρ a 3 ) 1 / 2 .

DOI
10.1016/j.aim.2026.110825

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We consider a Bose gas at density ρ > 0 , interacting through a repulsive potential V ∈ L 2 ( R 3 ) with scattering length a > 0 . We prove an upper bound for the free energy of the system, valid at low temperature T ≲ ρ a . Combined with the recent lower bound obtained in [18] , our estimate resolves the free energy per unit volume up to and including the Lee–Huang–Yang order a ρ 2 ( ρ a 3 ) 1 / 2 .
Abstract We consider a dilute Bose gas in the thermodynamic limit and prove a lower bound on the free energy for low temperatures which is in agreement with the conjecture of Lee–Huang–Yang on the excitation spectrum of the system. Combining techniques of Fournais and Solovej (Invent Math 232(2):863–994, 2023) and Haberberger et al. (The free energy of dilute Bose gases at low temperatures, 2023), we give a simpler and shorter proof resolving the case of strong interactions, including the hard-core potential.
We prove an upper bound on the free energy of a two-dimensional homogeneous Bose gas in the thermodynamic limit. We show that for a2ρ ≪ 1 and βρ ≳ 1, the free energy per unit volume differs from the one of the non-interacting system by at most 4πρ2|lna2ρ|−1(2−[1−βc/β]+2) to leading order, where a is the scattering length of the two-body interaction potential, ρ is the density, β is the inverse temperature, and βc is the inverse Berezinskii–Kosterlitz–Thouless critical temperature for superfluidity. In combination with the corresponding matching lower bound proved by Deuchert et al. [Forum Math. Sigma 8, e20 (2020)], this shows equality in the asymptotic expansion.
A lower bound is derived on the free energy (per unit volume) of a homogeneous Bose gas at density and temperature T. In the dilute regime, i.e., when a 3 ≪ 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 4πa(2 2 −[−c] 2 +). Here, c(T) denotes the critical density for Bose-Einstein condensation (for the non-interacting gas), and [ ·]+ = max { · , 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 ∼ 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. 1 Introduction and Main Result The advance of experimental techniques for studying ultra-cold atomic gases
Abstract We establish an upper bound for the ground state energy per unit volume of a dilute Bose gas in the thermodynamic limit, capturing the correct second-order term, as predicted by the Lee–Huang–Yang formula. This result was first established in [20] by H.-T. Yau and J. Yin. Our proof, which applies to repulsive and compactly supported V∈L3(R3)V \in L^3 (\mathbb {R}^3)V∈L3(R3) , gives better rates and, in our opinion, is substantially simpler.