arXiv:2603.13084

The Lee-Huang-Yang energy for a dilute gas of hard spheres: an upper bound

Giulia Basti, Morris Brooks, Serena Cenatiempo, Alessandro Olgiati, Benjamin Schlein

Published Mar 13, 2026• 0 citations

Abstract

We consider a quantum gas consisting of $N$ hard spheres with radius $\frak{a} > 0$, obeying bosonic statistics and moving in the box $Λ= [0;L]^3$ with periodic boundary conditions. We are interested in the ground state energy per unit volume in the thermodynamic limit, with $N, L \to \infty$ at fixed density $ρ= N / L^3$. We derive an upper bound for the ground state energy density, matching the famous Lee-Huang-Yang formula, up to lower order terms, in the dilute limit $ρ\frak{a}^3 \ll 1$.

DOI: 10.48550/arxiv.2603.13084

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arXiv:2603.13084

The Lee-Huang-Yang energy for a dilute gas of hard spheres: an upper bound

Giulia Basti, Morris Brooks, Serena Cenatiempo, Alessandro Olgiati, Benjamin Schlein

Published Mar 13, 2026• 0 citations

arXiv HTML PDF

Abstract

We consider a quantum gas consisting of

N

hard spheres with radius

\frak{a} > 0

, obeying bosonic statistics and moving in the box

Λ= [0;L]^3

with periodic boundary conditions. We are interested in the ground state energy per unit volume in the thermodynamic limit, with

N, L \to \infty

at fixed density

ρ= N / L^3

. We derive an upper bound for the ground state energy density, matching the famous Lee-Huang-Yang formula, up to lower order terms, in the dilute limit

ρ\frak{a}^3 \ll 1

DOI: 10.48550/arxiv.2603.13084

Related papers

Upper Bound for the Ground State Energy of a Dilute Bose Gas of Hard Spheres

Giulia Basti, Serena Cenatiempo, Alessandro Giuliani +3 more

Abstract We consider a gas of bosons interacting through a hard-sphere potential with radius $\mathfrak {a}$ a in the thermodynamic limit. We derive an upper bound for the ground state energy per particle at low density. Our bound captures the leading term $4\pi \rho \mathfrak {a}$ 4 π ρ a and shows that corrections are smaller than $C \rho \mathfrak {a} (\rho {{\mathfrak {a}}}^3)^{1/2}$ C ρ a ( ρ a 3 ) 1 / 2 , for a sufficiently large constant $C > 0$ C > 0 . In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy of a dilute gas of hard spheres is, in fact, of the order $\rho \mathfrak {a}(\rho {{\mathfrak {a}}}^3)^{1/2}$ ρ a ( ρ a 3 ) 1 / 2 , in agreement with the Lee–Huang–Yang prediction.Abstract We consider a gas of bosons interacting through a hard-sphere potential with radius $$\mathfrak {a}$$ a in the thermodynamic limit. We derive an upper bound for the ground state energy per particle at low density. Our bound captures the leading term $$4\pi \rho \mathfrak {a}$$ 4 π ρ a and shows that corrections are smaller than $$C \rho \mathfrak {a} (\rho {{\mathfrak {a}}}^3)^{1/2}$$ C ρ a ( ρ a 3 ) 1 / 2 , for a sufficiently large constant $$C > 0$$ C > 0 . In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy of a dilute gas of hard spheres is, in fact, of the order $$\rho \mathfrak {a}(\rho {{\mathfrak {a}}}^3)^{1/2}$$ ρ a ( ρ a 3 ) 1 / 2 , in agreement with the Lee–Huang–Yang prediction.

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Consider $N$ bosons in a finite box $\ensuremath{\Lambda}={[0,L]}^{3}\ensuremath{\subset}{\mathbb{R}}^{3}$ interacting via a two-body non-negative soft potential $V=\ensuremath{\lambda}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ with $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ fixed and $\ensuremath{\lambda}>0$ small. We will take the limit $L,N\ensuremath{\rightarrow}\ensuremath{\infty}$ by keeping the density $\ensuremath{\varrho}=N∕{L}^{3}$ fixed and small. We construct a variational state, which gives an upper bound on the ground-state energy per particle $\ensuremath{\epsilon}$, $\ensuremath{\epsilon}\ensuremath{\leqslant}4\ensuremath{\pi}\ensuremath{\varrho}a[1+(128∕15\sqrt{\ensuremath{\pi}}){(\ensuremath{\varrho}{a}^{3})}^{1∕2}{S}_{\ensuremath{\lambda}}]+O({\ensuremath{\varrho}}^{2}\ensuremath{\mid}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varrho}\ensuremath{\mid})$, as $\ensuremath{\varrho}\ensuremath{\rightarrow}0$, with a constant satisfying $1\ensuremath{\leqslant}{S}_{\ensuremath{\lambda}}\ensuremath{\leqslant}1+C\ensuremath{\lambda}$. Here $a$ is the scattering length of $V$ and thus depends on $\ensuremath{\lambda}$. In comparison, the prediction by Lee and Yang [Phys. Rev. 105, 1119 (1957)] and Lee, Huang, and Yang [Phys. Rev. 106, 1135 (1957)] asserts that ${S}_{\ensuremath{\lambda}}=1$ independent of $\ensuremath{\lambda}$.Consider $N$ bosons in a finite box $\ensuremath{\Lambda}={[0,L]}^{3}\ensuremath{\subset}{\mathbb{R}}^{3}$ interacting via a two-body non-negative soft potential $V=\ensuremath{\lambda}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ with $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ fixed and $\ensuremath{\lambda}>0$ small. We will take the limit $L,N\ensuremath{\rightarrow}\ensuremath{\infty}$ by keeping the density $\ensuremath{\varrho}=N∕{L}^{3}$ fixed and small. We construct a variational state, which gives an upper bound on the ground-state energy per particle $\ensuremath{\epsilon}$, $\ensuremath{\epsilon}\ensuremath{\leqslant}4\ensuremath{\pi}\ensuremath{\varrho}a[1+(128∕15\sqrt{\ensuremath{\pi}}){(\ensuremath{\varrho}{a}^{3})}^{1∕2}{S}_{\ensuremath{\lambda}}]+O({\ensuremath{\varrho}}^{2}\ensuremath{\mid}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varrho}\ensuremath{\mid})$, as $\ensuremath{\varrho}\ensuremath{\rightarrow}0$, with a constant satisfying $1\ensuremath{\leqslant}{S}_{\ensuremath{\lambda}}\ensuremath{\leqslant}1+C\ensuremath{\lambda}$. Here $a$ is the scattering length of $V$ and thus depends on $\ensuremath{\lambda}$. In comparison, the prediction by Lee and Yang [Phys. Rev. 105, 1119 (1957)] and Lee, Huang, and Yang [Phys. Rev. 106, 1135 (1957)] asserts that ${S}_{\ensuremath{\lambda}}=1$ independent of $\ensuremath{\lambda}$.

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Søren Fournais, Théotime Girardot, Lukas Junge +2 more

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.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.

arXiv:2206.11100Jun 22, 20220 citations

We consider a quantum gas consisting of

N

hard spheres with radius

\frak{a} > 0

, obeying bosonic statistics and moving in the box

Λ= [0;L]^3

with periodic boundary conditions. We are interested in the ground state energy per unit volume in the thermodynamic limit, with

N, L \to \infty

at fixed density

ρ= N / L^3

. We derive an upper bound for the ground state energy density, matching the famous Lee-Huang-Yang formula, up to lower order terms, in the dilute limit

ρ\frak{a}^3 \ll 1

Abstract We consider a gas of bosons interacting through a hard-sphere potential with radius

\mathfrak {a}

a in the thermodynamic limit. We derive an upper bound for the ground state energy per particle at low density. Our bound captures the leading term

4\pi \rho \mathfrak {a}

4 π ρ a and shows that corrections are smaller than

C \rho \mathfrak {a} (\rho {{\mathfrak {a}}}^3)^{1/2}

C ρ a ( ρ a 3 ) 1 / 2 , for a sufficiently large constant

C > 0

C > 0 . In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy of a dilute gas of hard spheres is, in fact, of the order

\rho \mathfrak {a}(\rho {{\mathfrak {a}}}^3)^{1/2}

ρ a ( ρ a 3 ) 1 / 2 , in agreement with the Lee–Huang–Yang prediction.

We consider a gas of bosons interacting through a hard-sphere potential with radius

\frak{a}

in the thermodynamic limit. We derive a simple upper bound for the ground state energy per particle at low density. Our bound captures the leading term

4πρ\frak{a}

and shows that corrections are smaller than

C ρ\frak{a} (ρ\frak{a}^3)^{1/2}

, for a sufficiently large constant

C > 0

. In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy is, in fact, of the order

ρ\frak{a} (ρ\frak{a}^3)^{1/2}

, in agreement with the Lee-Huang-Yang prediction.

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

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

Consider

N

bosons in a finite box $\ensuremath{\Lambda}={[0,L]}^{3}\ensuremath{\subset}{\mathbb{R}}^{3}$ interacting via a two-body non-negative soft potential $V=\ensuremath{\lambda}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ with $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}$ fixed and $\ensuremath{\lambda}>0$ small. We will take the limit $L,N\ensuremath{\rightarrow}\ensuremath{\infty}$ by keeping the density $\ensuremath{\varrho}=N∕{L}^{3}$ fixed and small. We construct a variational state, which gives an upper bound on the ground-state energy per particle $\ensuremath{\epsilon}$, $\ensuremath{\epsilon}\ensuremath{\leqslant}4\ensuremath{\pi}\ensuremath{\varrho}a[1+(128∕15\sqrt{\ensuremath{\pi}}){(\ensuremath{\varrho}{a}^{3})}^{1∕2}{S}_{\ensuremath{\lambda}}]+O({\ensuremath{\varrho}}^{2}\ensuremath{\mid}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}\ensuremath{\varrho}\ensuremath{\mid})$, as $\ensuremath{\varrho}\ensuremath{\rightarrow}0$, with a constant satisfying $1\ensuremath{\leqslant}{S}_{\ensuremath{\lambda}}\ensuremath{\leqslant}1+C\ensuremath{\lambda}$. Here

a

is the scattering length of

V

and thus depends on $\ensuremath{\lambda}$. In comparison, the prediction by Lee and Yang [Phys. Rev. 105, 1119 (1957)] and Lee, Huang, and Yang [Phys. Rev. 106, 1135 (1957)] asserts that ${S}_{\ensuremath{\lambda}}=1$ independent of $\ensuremath{\lambda}$.

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

ρ

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.