arXiv:2402.05646

Ground state energy of the low density Bose gas with two-body and three-body interactions

François Louis Antoine Visconti

Published Feb 8, 2024• 0 citations

Abstract

In the present paper we study the low density Bose gas in the thermodynamic limit interacting via two-body and three-body interaction potentials. We prove that the leading order of the ground state energy is entirely characterised by both the scattering length of the two-body potential and the scattering energy of the three-body potential. The corresponding result for two-body interactions was proven in seminal papers of Dyson (1957) and of Lieb-Yngvason (1998), and the result for three-body interactions was proven by Nam-Ricaud-Triay (2022). The present result resolves a conjecture of Nam-Ricaud-Triay (2022).

DOI: 10.48550/arxiv.2402.05646

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arXiv:2402.05646

Ground state energy of the low density Bose gas with two-body and three-body interactions

François Louis Antoine Visconti

Published Feb 8, 2024• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.2402.05646

Related papers

Ground State Energy of Dilute Bose Gases with Two-Body and Three-Body Interactions

François Louis Antoine Visconti

Abstract We study dilute Bose gases in the thermodynamic limit interacting via two-body and three-body interaction potentials. We prove that the leading order of the thermodynamic ground state energy is entirely characterized by both the scattering length of the two-body potential and the scattering energy of the three-body potential. The corresponding result for two-body interactions was proven in seminal papers of Dyson (1957) [1] and Lieb–Yngvason (1998) [2], and the result for three-body interactions was proven much more recently by Nam–Ricaud–Triay (2022) [3]. The present result resolves a conjecture of Nam–Ricaud–Triay (2022) [4].Abstract We study dilute Bose gases in the thermodynamic limit interacting via two-body and three-body interaction potentials. We prove that the leading order of the thermodynamic ground state energy is entirely characterized by both the scattering length of the two-body potential and the scattering energy of the three-body potential. The corresponding result for two-body interactions was proven in seminal papers of Dyson (1957) [1] and Lieb–Yngvason (1998) [2], and the result for three-body interactions was proven much more recently by Nam–Ricaud–Triay (2022) [3]. The present result resolves a conjecture of Nam–Ricaud–Triay (2022) [4].

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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

\sqrt {N}

. In addition, we show that low-energy states satisfy Bose-Einstein condensation with a rate of the order

N^{-\frac {3}{4}}

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.

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.