Annals of Mathematics

The energy of dilute Bose gases

Søren Fournais, Jan Philip Solovej

Published Nov 1, 2020• 82 citations

Abstract

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.

DOI: 10.4007/annals.2020.192.3.5

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Annals of Mathematics

The energy of dilute Bose gases

Søren Fournais, Jan Philip Solovej

Published Nov 1, 2020• 82 citations

Publisher

Abstract

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.

DOI: 10.4007/annals.2020.192.3.5

Related papers

The energy of dilute Bose gases

Søren Fournais, Jan Philip Solovej

For a dilute system of non-relativistic bosons interacting through a positive $L^1$ potential $v$ with scattering length $a$ we prove that the ground state energy density satisfies the bound $e(ρ) \geq 4πa ρ^2 (1+ \frac{128}{15\sqrtπ} \sqrt{ρa^3} +o(\sqrt{ρa^3}\,))$ , thereby proving the Lee-Huang-Yang formula for the energy density.For a dilute system of non-relativistic bosons interacting through a positive $L^1$ potential $v$ with scattering length $a$ we prove that the ground state energy density satisfies the bound $e(ρ) \geq 4πa ρ^2 (1+ \frac{128}{15\sqrtπ} \sqrt{ρa^3} +o(\sqrt{ρa^3}\,))$, thereby proving the Lee-Huang-Yang formula for the energy density.

arXiv:1904.06164Apr 12, 20190 citations

A Simple 2nd Order Lower Bound to the Energy of Dilute Bose Gases

Birger Brietzke, Søren Fournais, Jan Philip Solovej

For a dilute system of non-relativistic bosons interacting through a positive, radial potential v with scattering length a we prove that the ground state energy density satisfies the bound $e(\rho ) \ge 4\pi a \rho ^2 (1- C \sqrt{\rho a^3} \,)$ .For a dilute system of non-relativistic bosons interacting through a positive, radial potential v with scattering length a we prove that the ground state energy density satisfies the bound $$e(\rho ) \ge 4\pi a \rho ^2 (1- C \sqrt{\rho a^3} \,)$$ .

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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

The Ground State Energy of a Two-Dimensional Bose Gas

Søren Fournais, Théotime Girardot, Lukas Junge +2 more

Abstract We prove the following formula for the ground state energy density of a dilute Bose gas with density $\rho$ ρ in 2 dimensions in the thermodynamic limit $\begin{aligned} e^{\text {2D}}(\rho ) = 4\pi \rho ^2 Y\Big (1 - Y \vert \log Y \vert + \Big ( 2\Gamma + \frac{1}{2} + \log (\pi ) \Big ) Y \Big ) + o(\rho ^2 Y^{2}), \end{aligned}$ e 2D ( ρ ) = 4 π ρ 2 Y ( 1 - Y | log Y | + ( 2 Γ + 1 2 + log ( π ) ) Y ) + o ( ρ 2 Y 2 ) , as $\rho a^2 \rightarrow 0$ ρ a 2 → 0 . Here $Y= |\log (\rho a^2)|^{-1}$ Y = | log ( ρ a 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.Abstract We prove the following formula for the ground state energy density of a dilute Bose gas with density $$\rho $$ ρ in 2 dimensions in the thermodynamic limit $$\begin{aligned} e^{\text {2D}}(\rho ) = 4\pi \rho ^2 Y\Big (1 - Y \vert \log Y \vert + \Big ( 2\Gamma + \frac{1}{2} + \log (\pi ) \Big ) Y \Big ) + o(\rho ^2 Y^{2}), \end{aligned}$$ e 2D ( ρ ) = 4 π ρ 2 Y ( 1 - Y | log Y | + ( 2 Γ + 1 2 + log ( π ) ) Y ) + o ( ρ 2 Y 2 ) , as $$\rho a^2 \rightarrow 0$$ ρ a 2 → 0 . Here $$Y= |\log (\rho a^2)|^{-1}$$ Y = | log ( ρ a 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.

Communications in Mathematical PhysicsFeb 23, 202410 citations

Lee-Huang-Yang Energy for Dilute Bose Gases: an Upper Bound for General Potentials

Giulia Basti, Morris Brooks, Serena Cenatiempo +2 more

arXiv:2609.25314Sep 21, 20260 citations

Ground-state energy of a low-density Bose gas: A second-order upper bound

László Erdős, Benjamin Schlein, Horng‐Tzer Yau

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 $\ensuremath{\lambda}$.

arXiv:0806.4873Physical Review ANov 19, 200870 citations

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.

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

L^1

potential

v

with scattering length

a

we prove that the ground state energy density satisfies the bound

e(ρ) \geq 4πa ρ^2 (1+ \frac{128}{15\sqrtπ} \sqrt{ρa^3} +o(\sqrt{ρa^3}\,))

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

For a dilute system of non-relativistic bosons interacting through a positive, radial 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- C \sqrt{\rho a^3} \,)

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.

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

\rho

ρ in 2 dimensions in the thermodynamic limit

\begin{aligned} e^{\text {2D}}(\rho ) = 4\pi \rho ^2 Y\Big (1 - Y \vert \log Y \vert + \Big ( 2\Gamma + \frac{1}{2} + \log (\pi ) \Big ) Y \Big ) + o(\rho ^2 Y^{2}), \end{aligned}

e 2D ( ρ ) = 4 π ρ 2 Y ( 1 - Y | log Y | + ( 2 Γ + 1 2 + log ( π ) ) Y ) + o ( ρ 2 Y 2 ) , as

\rho a^2 \rightarrow 0

ρ a 2 → 0 . Here

Y= |\log (\rho a^2)|^{-1}

Y = | log ( ρ a 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.

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 $\ensuremath{\lambda}$.