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arXiv:2304.02405

The free energy of dilute Bose gases at low temperatures

Florian Haberberger, Christian Hainzl, Phan Thành Nam, Robert Seiringer, Arnaud Triay

Published Apr 5, 2023• 3 citations
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Abstract

We consider a low density Bose gas interacting through a repulsive potential in the thermodynamic limit. We justify, as a rigorous lower bound, a Lee--Huang--Yang type formula for the free energy at suitably low temperatures, where the modified excitation spectrum leads to a second order correction of the same order as the Lee--Huang--Yang correction to the ground state energy.

DOI
10.48550/arxiv.2304.02405

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We consider a low density Bose gas interacting through a repulsive potential in the thermodynamic limit. We justify, as a rigorous lower bound, a Lee--Huang--Yang type formula for the free energy at suitably low temperatures, where the modified excitation spectrum leads to a second order correction of the same order as the Lee--Huang--Yang correction to the ground state energy.
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.
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.
We consider a dilute gas in 3D composed of two species of bosons interacting through positive inter-species and intra-species pairwise potentials. We prove a second order expansion for the energy density in the thermodynamic limit. For the case of compactly supported, integrable potentials, we derive the correct second order of the expansion. If we make the further assumption of having soft potentials, we also derive the correct coefficient of the second order and the resulting formula is coherent with the physics literature. If we let the density and scattering length of one of the species go to zero, we obtain the Lee-Huang-Yang formula for one species of bosons. The paper also contains a proof of BEC for a mixture of bosons in a box with length scale larger than the Gross-Pitaevskii one.
We present an overview of the approach to establish a lower bound to the ground state energy for the dilute, interacting Bose gas in a periodic box. In this paper, the size of the box is larger than the Gross–Pitaevskii length scale. The presentation includes both the two- and three-dimensional cases, and catches the second-order correction, i.e. the Lee–Huang–Yang term. The calculation on a box of this length scale is the main step to calculate the energy in the thermodynamic limit. However, the periodic boundary condition simplifies many steps of the argument considerably compared to the localized problem coming from the thermodynamic case.