arXiv:2206.11100

The Ground State Energy of a Two-Dimensional Bose Gas

Søren Fournais, Théotime Girardot, Lukas Junge, Morin, L., Marco Olivieri

Published Jun 22, 2022• 0 citations

Abstract

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.

DOI: 10.48550/arxiv.2206.11100

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The Ground State Energy of a Two-Dimensional Bose Gas

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arXiv:2206.11100

The Ground State Energy of a Two-Dimensional Bose Gas

Søren Fournais, Théotime Girardot, Lukas Junge, Morin, L., Marco Olivieri

Published Jun 22, 2022• 0 citations

arXiv HTML PDF

Abstract

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.

DOI: 10.48550/arxiv.2206.11100

Related papers

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.

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Annals of MathematicsNov 1, 202082 citations

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

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arXiv:2505.15976May 21, 20250 citations

The Ground State Energy of a Dilute Bose Gas in Dimension n >3

Anders Aaen

We consider a Bose gas in spatial dimension $n>3$ with a repulsive, radially symmetric two-body potential $V$ . In the limit of low density $ρ$ , the ground state energy per particle in the thermodynamic limit is shown to be $(n-2)|\mathbb S^{n-1}|a^{n-2}ρ$ , where $|\mathbb S^{n-1}|$ denotes the surface measure of the unit sphere in $\mathbb{R}^n$ and $a$ is the scattering length of $V$ . Furthermore, for smooth and compactly supported two-body potentials, we derive upper bounds to the ground state energy with a correction term $(1+Cγ)8π^4a^6ρ^2|\ln(a^4ρ)|$ in dimension $n=4$ , where $γ:=\int V(x)|x|^{-2}\, dx$ , and a correction term which is $\mathcal{O}(ρ^2)$ in higher dimensions.We consider a Bose gas in spatial dimension $n>3$ with a repulsive, radially symmetric two-body potential $V$. In the limit of low density $ρ$, the ground state energy per particle in the thermodynamic limit is shown to be $(n-2)|\mathbb S^{n-1}|a^{n-2}ρ$, where $|\mathbb S^{n-1}|$ denotes the surface measure of the unit sphere in $\mathbb{R}^n$ and $a$ is the scattering length of $V$. Furthermore, for smooth and compactly supported two-body potentials, we derive upper bounds to the ground state energy with a correction term $(1+Cγ)8π^4a^6ρ^2|\ln(a^4ρ)|$ in dimension $n=4$, where $γ:=\int V(x)|x|^{-2}\, dx$, and a correction term which is $\mathcal{O}(ρ^2)$ in higher dimensions.

arXiv:1401.5960Jan 23, 20141 citation

A new second-order upper bound for the ground state energy of dilute Bose gases

Giulia Basti, Serena Cenatiempo, Benjamin Schlein

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.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.

arXiv:2101.06222Forum of Mathematics SigmaJan 1, 202144 citations

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

ρ

2

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.

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, 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.

We consider a Bose gas in spatial dimension

n>3

with a repulsive, radially symmetric two-body potential

V

. In the limit of low density

ρ

, the ground state energy per particle in the thermodynamic limit is shown to be

(n-2)|\mathbb S^{n-1}|a^{n-2}ρ

, where

|\mathbb S^{n-1}|

denotes the surface measure of the unit sphere in

\mathbb{R}^n

and

a

is the scattering length of

V

. Furthermore, for smooth and compactly supported two-body potentials, we derive upper bounds to the ground state energy with a correction term

(1+Cγ)8π^4a^6ρ^2|\ln(a^4ρ)|

in dimension

n=4

, where

γ:=\int V(x)|x|^{-2}\, dx

, and a correction term which is

\mathcal{O}(ρ^2)

in higher dimensions.

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

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