arXiv:0908.0109 · Journal of Mathematical Physics

A lower bound on the ground state energy of dilute Bose gas

Ji Oon Lee, Jun Yin

Published May 1, 2010• 7 citations

Abstract

Consider an N-boson system interacting via a two-body repulsive short-range potential V in a three dimensional box Λ of side length L. We take the limit N,L→∞ while keeping the density ρ=N/L3 fixed and small. We prove a new lower bound for its ground state energy per particle, E(N,Λ)/N≥4πaρ[1−O(ρ1/3|log ρ|3)], as ρ→0, where a is the scattering length of V.

DOI: 10.1063/1.3376639

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arXiv:0908.0109 · Journal of Mathematical Physics

A lower bound on the ground state energy of dilute Bose gas

Ji Oon Lee, Jun Yin

Published May 1, 2010• 7 citations

arXiv HTML PDF

Abstract

DOI: 10.1063/1.3376639

Related papers

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 $\ensuremath{\lambda}$.

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arXiv:1401.5960Jan 23, 20141 citation

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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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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 $$e(\rho ) \ge 4\pi a \rho ^2 (1- C \sqrt{\rho a^3} \,)$$ .

arXiv:1901.00539Communications in Mathematical PhysicsMar 12, 202018 citations

The Ground State of the Bose Gas

Élliott H. Lieb, Robert Seiringer, Jan Philip Solovej +1 more

Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density, $ρ$ , can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous -- and this fact is used in all modeling. In such cases, the leading term in the energy/particle is $2π\hbar^2 a ρ/m$ where $a$ is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange $N^{7/5}$ law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has recently been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two-dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, two other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the `traps' actually used in the experiments. For this system it is also possible to prove complete Bose condensation, as we have shown. Another topic is a proof that Foldy's 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy.Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density, $ρ$, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous -- and this fact is used in all modeling. In such cases, the leading term in the energy/particle is $2π\hbar^2 a ρ/m$ where $a$ is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange $N^{7/5}$ law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has recently been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two-dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, two other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the `traps' actually used in the experiments. For this system it is also possible to prove complete Bose condensation, as we have shown. Another topic is a proof that Foldy's 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy.

arXiv:math-ph/0204027Apr 11, 20025 citations

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 $\ensuremath{\lambda}$.

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.

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.

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.

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} \,)

Now that the low temperature properties of quantum-mechanical many-body systems (bosons) at low density,

ρ

, can be examined experimentally it is appropriate to revisit some of the formulas deduced by many authors 4-5 decades ago. For systems with repulsive (i.e. positive) interaction potentials the experimental low temperature state and the ground state are effectively synonymous -- and this fact is used in all modeling. In such cases, the leading term in the energy/particle is

2π\hbar^2 a ρ/m

where

a

is the scattering length of the two-body potential. Owing to the delicate and peculiar nature of bosonic correlations (such as the strange

N^{7/5}

law for charged bosons), four decades of research failed to establish this plausible formula rigorously. The only previous lower bound for the energy was found by Dyson in 1957, but it was 14 times too small. The correct asymptotic formula has recently been obtained by us and this work will be presented. The reason behind the mathematical difficulties will be emphasized. A different formula, postulated as late as 1971 by Schick, holds in two-dimensions and this, too, will be shown to be correct. With the aid of the methodology developed to prove the lower bound for the homogeneous gas, two other problems have been successfully addressed. One is the proof by us that the Gross-Pitaevskii equation correctly describes the ground state in the `traps' actually used in the experiments. For this system it is also possible to prove complete Bose condensation, as we have shown. Another topic is a proof that Foldy's 1961 theory of a high density Bose gas of charged particles correctly describes its ground state energy.