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arXiv:2406.19886 · Forum of Mathematics Sigma

Second-order energy expansion of Bose gases with three-body interactions

Morris Brooks

Published Jan 1, 2025• 0 citations
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Abstract

Abstract We provide a second-order energy expansion for a gas of N bosonic particles with three-body interactions in the Gross-Pitaevskii regime. We especially confirm a conjecture by Nam, Ricaud, and Triay in [25], where they predict the subleading term in the asymptotic expansion of the ground state energy to be of the order $\sqrt {N}$ . In addition, we show that low-energy states satisfy Bose-Einstein condensation with a rate of the order $N^{-\frac {3}{4}}$ .

DOI
10.1017/fms.2025.10113

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Abstract We provide a second-order energy expansion for a gas of N bosonic particles with three-body interactions in the Gross-Pitaevskii regime. We especially confirm a conjecture by Nam, Ricaud, and Triay in [25], where they predict the subleading term in the asymptotic expansion of the ground state energy to be of the order N\sqrt {N}N​ . In addition, we show that low-energy states satisfy Bose-Einstein condensation with a rate of the order N−34N^{-\frac {3}{4}}N−43​ .
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).
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 For a translation invariant system of N bosons in the Gross–Pitaevskii regime, we establish a precise bound for the ground state energy ENE_NEN​ E N . While the leading, order N , contribution to ENE_NEN​ E N has been known since (Lieb et al. in Phys Rev A 61:043602, 2000; Lieb and Yngvason in Phys Rev Lett 80:2504–2507, 1998) and the second order corrections (of order one) have been first determined in Boccato et al. (Acta Math 222(2):219–335, 2019), our estimate also resolves the next term in the asymptotic expansion of ENE_NEN​ E N , which is of the order (log⁡N)/N(\log N) / N(logN)/N ( log N ) / N .
We consider a gas of bosons interacting through a three-body hard-core potential in the thermodynamic limit. We derive an upper bound on the ground state energy of the system at the leading order using a Jastrow factor. Our result matches the lower bound proven by Nam-Ricaud-Triay (J Math Phys 63:071903, 2022) and therefore resolves the leading order. Moreover, a straightforward adaptation of our proof can be used for systems interacting via combined two-body and three-body interactions to generalise Theorem 1.2 from (Ann. Henri Poincaré, 2026) to hard-core potentials.
Abstract We consider a gas of N weakly interacting bosons in the ground state. Such gases exhibit Bose–Einstein condensation. The binding energy is defined as the energy it takes to remove one particle from the gas. In this article, we prove an asymptotic expansion for the binding energy, and compute the first orders explicitly for the homogeneous gas. Our result addresses in particular a conjecture by Nam (Lett Math Phys 108(1):141–159, 2018), and provides an asymptotic expansion of the ionization energy of bosonic atoms.
Various ground state properties such as chemical potential, differential and total energy per particle etc. of Bose-Einstein condensate of 10000 85 Rb atoms with varying repulsive self-interaction energy have been reported considering not only two-body interaction but also including the higher order terms of the low-density energy expansion of homogeneous Bose gas in the Ginzburg, Pitaevskii and Gross (GPG) equation. These include hard-core approximation of the bosons neglected earlier and the three-body interaction terms. The study is more general as it includes the terms beyond logarithm in energy density expansion. It is also shown that such a consideration does not violate the lower bound predicted earlier in which the 'constant' beyond logarithm term in the three-body interaction was neglected.