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arXiv:2505.15976

A Lee-Huang-Yang type expansion for the thermodynamic energy density of a dilute mixture of Bose gases

Marco Olivieri

Published May 21, 2025• 0 citations

Abstract

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.

DOI: 10.48550/arxiv.2505.15976

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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 prove the following formula for the ground state energy density of a dilute Bose gas with density

ρ

in

2

dimensions in the thermodynamic limit \begin{align*} e^{\rm{2D}}(ρ) = 4πρ^2 Y\left(1 - Y \vert \log Y \vert + \left( 2Γ+ \frac{1}{2} + \log(π) \right) Y \right) + o(ρ^2 Y^{2}). \end{align*} Here

Y= |\log(ρa^2)|^{-1}

and

a

is the scattering length of the two-body potential. This result in

2

dimensions corresponds to the famous Lee-Huang-Yang formula in

3

dimensions. The proof is valid for essentially all positive potentials with finite scattering length, in particular it covers the crucial case of the hard core potential.

For a dilute system of non-relativistic bosons interacting through a positive, compactly supported,

L^1

-potential

v

with scattering length

a

we prove that the ground state energy density satisfies the bound

e(\rho) \ge 4\pi a \rho^2(1 + \frac{128}{15\sqrt{\pi}} \sqrt{\rho a^3} + o(\sqrt{\rho a^3}))

, thereby proving the Lee-Huang-Yang formula for the energy density.

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 \in L^3 (\mathbb {R}^3)

, gives better rates and, in our opinion, is substantially simpler.

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.

For a system of

N

Fermions of spin

1/2

, with its interaction potential of scattering length

a

, the classical Huang-Yang formula states that the energy density

e(ρ)

is of the form \begin{equation*} e(ρ)=\frac{3}{5}(3π^2)^{\frac{2}{3}}ρ^{\frac{5}{3}}+2πaρ^2 +\frac{12}{35}(11-2\ln2)3^{\frac{1}{3}}π^{\frac{2}{3}}a^2ρ^{\frac{7}{3}} +o(ρ^{\frac{7}{3}}). \end{equation*} We consider a general system of

N

Fermions of spin

1/\mathbf{q}

with the scattering length of the interaction potential at the scale

a\propto N^{-1}

, that is, in the Gross-Pitaevskii regime. We prove the 2nd order ground state energy approximation corresponding to the Huang-Yang formula. The thermodynamic limit case shares a similar logic, and could be dealt with in a separate paper.