[Physique mathématique des gaz de bosons dilués]
Nous discutons des progrès récents dans l’analyse mathématique des gaz de bosons dilués. Nous passons en revue les résultats obtenus en une, deux et trois dimensions, mais l’accent sera mis sur le cas tridimensionnel. En chaque dimension, nous disposons d’un développement asymptotique à deux termes de la densité d’énergie de l’état fondamental, sous la forme d’expressions ne dépendant que de la longueur de diffusion du potentiel. En dimension trois, il s’agit de la célèbre formule de Lee–Huang–Yang. En dimension deux et en dimension trois, la limite diluée est dans le régime d’interaction faible, alors qu’en dimension un, elle est au contraire en interaction forte. Nous esquissons brièvement les difficultés mathématiques rencontrées et passons en revue quelques problèmes encore ouverts dans ce domaine.
We discuss recent progress in the mathematical analysis of dilute Bose gases. We review results in one to three dimensions, but the focus will be on three dimensions. In all dimensions we have a two term asymptotic expansion of the ground state energy density by an expression that depends only on the scattering length of the potential. In dimension three this is the celebrated Lee–Huang–Yang formula. In dimensions two and three the dilute limit is a weakly interacting regime whereas in dimension one it is rather strongly interacting. We sketch briefly the mathematical difficulties and review some remaining open problems in the field.
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Mots-clés : États fondamentaux des gaz de bosons, Gaz atomiques froids, Condensation de Bose–Einstein, Approximation de Bogolioubov, Formule de Lee–Huang–Yang
Jan Philip Solovej 1
CC-BY 4.0
@article{CRPHYS_2025__26_G1_339_0,
author = {Jan Philip Solovej},
title = {Mathematical physics of dilute {Bose} gases},
journal = {Comptes Rendus. Physique},
pages = {339--348},
publisher = {Acad\'emie des sciences, Paris},
volume = {26},
year = {2025},
doi = {10.5802/crphys.247},
language = {en},
}
Jan Philip Solovej. Mathematical physics of dilute Bose gases. Comptes Rendus. Physique, Volume 26 (2025), pp. 339-348. doi : 10.5802/crphys.247. https://comptes-rendus.academie-sciences.fr/physique/articles/10.5802/crphys.247/
[1] The ground state energy of dilute Bose gas in potentials with positive scattering length, Commun. Math. Phys., Volume 295 (2010) no. 1, pp. 1-27 (English) | DOI
[2] The Mathematics of the Bose Gas and its Condensation, Oberwolfach Seminars, 34, Birkhäuser, Basel, 2005 (English)
[3] Die Wellenfunktion und Geschwindigkeitsverteilung des entarteten gases, Z. Phys., Volume 56 (1929) no. 11–12, pp. 778-789 | DOI
[4] Eigenvalues and eigenfunctions of a Bose system of hard spheres and its low-temperature properties, Phys. Rev., Volume 106 (1957) no. 6, pp. 1135-1145 | DOI
[5] Ground-state energy of a hard-sphere gas, Phys. Rev., II. Ser., Volume 106 (1957), pp. 20-26 (English) | DOI
[6] Ground state energy of the low density Bose gas, Phys. Rev. Lett., Volume 80 (1998) no. 12, pp. 2504-2507 | DOI
[7] The second order upper bound for the ground energy of a Bose gas, J. Stat. Phys., Volume 136 (2009) no. 3, pp. 453-503 (English) | DOI
[8] A new second-order upper bound for the ground state energy of dilute Bose gases, Forum Math. Sigma, Volume 9 (2021), e74 | DOI
[9] The energy of dilute Bose gases, Ann. Math., Volume 192 (2020) no. 3, pp. 893-976 | DOI
[10] The energy of dilute Bose gases. II: the general case, Invent. Math., Volume 232 (2023) no. 2, pp. 863-994 (English) | DOI
[11] Upper bound for the ground state energy of a dilute Bose gas of hard spheres, Arch. Ration. Mech. Anal., Volume 248 (2024) no. 6, 100 (English) | DOI
[12] A second order upper bound for the ground state energy of a hard-sphere gas in the Gross–Pitaevskii regime, Commun. Math. Phys., Volume 399 (2023) no. 1, pp. 1-55 (English) | DOI
[13] Dynamics and thermodynamics of the low-temperature strongly interacting Bose gas, Phys. Rev. Lett., Volume 107 (2011), 135301 | DOI
[14] The equation of state of a low-temperature Fermi gas with tunable interactions, Science, Volume 328 (2010) no. 5979, pp. 729-732 | DOI
[15] Two-dimensional system of hard-core bosons, Phys. Rev. A, Volume 3 (1971) no. 3, pp. 1067-1073 | DOI
[16] The ground state energy of a dilute two-dimensional Bose gas, J. Stat. Phys., Volume 103 (2001) no. 3–4, pp. 509-526 | DOI
[17] Hard-disc Bose gas, Phys. Lett. A, Volume 68 (1978) no. 1, pp. 12-14 | DOI
[18] Dilute Bose gas in two dimensions: density expansions and the Gross–Pitaevskii equation, Phys. Rev. E, Volume 64 (2001), 027105 | DOI
[19] Pseudopotential method and dilute hard “sphere” Bose gas in dimensions 2, 4 and 5, Europhys. Lett., Volume 84 (2008) no. 4, 40001 | DOI
[20] Ground state energy of the two-dimensional weakly interacting Bose gas: first correction beyond Bogoliubov theory, Phys. Rev. Lett., Volume 102 (2009), 180404 | DOI
[21] The ground state energy of a two-dimensional Bose gas, Commun. Math. Phys., Volume 405 (2024) no. 3, 59 | DOI
[22] Ground state energy of dilute Bose gases in 1D, Commun. Math. Phys., Volume 406 (2025) no. 2, 27 | DOI
[23] Ground state of a Bose system of hard spheres, Phys. Rev., Volume 115 (1959) no. 6, pp. 1390-1404 | DOI
[24] Three-boson problem at low energy and implications for dilute Bose–Einstein condensates, Phys. Rev. A, Volume 78 (2008), 013636 | DOI
[25] Third order corrections to the ground state energy of a Bose gas in the Gross–Pitaevskii regime (2024) | arXiv
[26] On the theory of superfluidity, Proc. Inst. Math. Kiev (1947) no. 9, pp. 89-103 Rus. Trans Izv. Akad. Nauk Ser. Fiz. 11, 77 (1947), Eng. Trans. J. Phys. (USSR), 11, 23 (1947)
[27] The Bogoliubov free energy functional. I: existence of minimizers and phase diagram, Arch. Ration. Mech. Anal., Volume 229 (2018) no. 3, pp. 1037-1090 (English) | DOI
[28] The Bogoliubov free energy functional. II: the dilute limit, Commun. Math. Phys., Volume 360 (2018) no. 1, pp. 347-403 (English) | DOI
[29] Ground-state energy of a low-density Bose gas: a second-order upper bound, Phys. Rev. A, Volume 78 (2008), 053627 | DOI
[30] Proof of Bose–Einstein condensation for dilute trapped gases, Phys. Rev. Lett., Volume 88 (2002), 170409 | DOI
[31] Complete Bose–Einstein condensation in the Gross–Pitaevskii regime, Commun. Math. Phys., Volume 359 (2018) no. 3, pp. 975-1026 (English) | DOI
[32] Bogoliubov theory in the Gross–Pitaevskii limit, Acta Math., Volume 222 (2019) no. 2, pp. 219-335 (English) | DOI
[33] Optimal rate for Bose–Einstein condensation in the Gross–Pitaevskii regime, Commun. Math. Phys., Volume 376 (2020) no. 2, pp. 1311-1395 | DOI
[34] Bose–Einstein condensation beyond the Gross–Pitaevskii regime, Ann. Henri Poincaré, Volume 22 (2021) no. 4, pp. 1163-1233 (English) | DOI
[35] Length scales for BEC in the dilute Bose gas, EMS Press, Berlin (2021), pp. 115-133 | DOI
[36] The XY has long-range order for all spins and all dimensions greater than one, Phys. Rev. Lett., Volume 61 (1988) no. 22, pp. 2582-2584 | DOI
[37] Bose–Einstein quantum phase transition in an optical lattice model, Phys. Rev. A, Volume 70 (2004), 023612 | DOI
[38] Observation of Bose–Einstein condensation in a dilute atomic vapor, Science, Volume 269 (1995) no. 5221, pp. 198-201 | DOI
[39] Bose–Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., Volume 75 (1995) no. 22, pp. 3969-3973 | DOI
[40] Bose–Einstein condensation of atoms in a uniform potential, Phys. Rev. Lett., Volume 110 (2013), 200406 | DOI
[41] Ultracold scattering properties of the short-lived Rb isotopes, Phys. Rev. A, Volume 59 (1999) no. 2, pp. 1303-1308 | DOI
[42] Exact analysis of an interacting bose gas. I. The general solution and the ground state, Phys. Rev., Volume 130 (1963) no. 4, pp. 1605-1616 | DOI
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