Derivation of Hartree’s theory for mean-field Bose gases
[Dérivation de la théorie de Hartree pour des gaz de bosons dans le régime de champ moyen]
Journées équations aux dérivées partielles (2013), article no. 7, 21 p.

Dans cet article, nous présentons des résultats obtenus avec Phan Thành Nam, Nicolas Rougerie, Sylvia Serfaty et Jan Philip Solovej. Nous considérons un système de N bosons qui interagissent avec un potentiel d’intensité 1/N (on parle de régime de champ moyen). Dans la limite où N, nous montrons que le premier ordre du développement des valeurs propres du Hamiltonien à N corps est donné par la théorie non linéaire de Hartree, alors que l’ordre suivant est donné par l’opérateur de Bogoliubov. Nous discutons également en détails du phénomène de condensation de Bose-Einstein dans de tels systèmes.

This article is a review of recent results with Phan Thành Nam, Nicolas Rougerie, Sylvia Serfaty and Jan Philip Solovej. We consider a system of N bosons with an interaction of intensity 1/N (mean-field regime). In the limit N, we prove that the first order in the expansion of the eigenvalues of the many-particle Hamiltonian is given by the nonlinear Hartree theory, whereas the next order is predicted by the Bogoliubov Hamiltonian. We also discuss the occurrence of Bose-Einstein condensation in these systems.

DOI : 10.5802/jedp.103
Classification : 35Q40, 81Q99
Mots clés : Hartree theory, mean-field limit, Bose-Einstein condensation, quantum de Finetti theorem
Lewin, Mathieu 1

1 CNRS & Université de Cergy-Pontoise (UMR 8088) 95000 Cergy-Pontoise, France.
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Lewin, Mathieu. Derivation of Hartree’s theory for mean-field Bose gases. Journées équations aux dérivées partielles (2013), article  no. 7, 21 p. doi : 10.5802/jedp.103. http://www.numdam.org/articles/10.5802/jedp.103/

[1] Aftalion, Amandine Vortices in Bose–Einstein Condensates, Progress in nonlinear differential equations and their applications, 67, Springer, 2006 | MR | Zbl

[2] Aftalion, Amandine; Blanc, Xavier; Dalibard, Jean Vortex patterns in a fast rotating Bose-Einstein condensate, Phys. Rev. A, Volume 71 (2005) no. 2, pp. 023611 http://link.aps.org/abstract/PRA/v71/e023611 | DOI

[3] Aftalion, Amandine; Blanc, Xavier; Nier, Francis Lowest Landau level functional and Bargmann spaces for Bose-Einstein condensates, J. Funct. Anal., Volume 241 (2006) no. 2, pp. 661-702 | MR | Zbl

[4] Ammari, Zied; Nier, Francis Mean Field Limit for Bosons and Infinite Dimensional Phase-Space Analysis, Annales Henri Poincaré, Volume 9 (2008), pp. 1503-1574 (10.1007/s00023-008-0393-5) | MR | Zbl

[5] Ammari, Zied; Nier, Francis Mean field propagation of Wigner measures and BBGKY hierarchies for general bosonic states, J. Math. Pures Appl., Volume 95 (2011) no. 6, pp. 585-626 | MR | Zbl

[6] Bach, Volker Ionization energies of bosonic Coulomb systems, Lett. Math. Phys., Volume 21 (1991) no. 2, pp. 139-149 | DOI | MR | Zbl

[7] Bach, Volker; Lewis, Roger; Lieb, Elliott H.; Siedentop, Heinz On the number of bound states of a bosonic N-particle Coulomb system, Math. Z., Volume 214 (1993) no. 3, pp. 441-459 | DOI | MR | Zbl

[8] Bardos, Claude; Golse, François; Mauser, Norbert J. Weak coupling limit of the N-particle Schrödinger equation, Methods Appl. Anal., Volume 7 (2000) no. 2, pp. 275-293 (Cathleen Morawetz: a great mathematician) | MR | Zbl

[9] Benguria, R.; Lieb, E. H. Proof of the Stability of Highly Negative Ions in the Absence of the Pauli Principle, Physical Review Letters, Volume 50 (1983), pp. 1771-1774 | DOI

[10] Bogoliubov, N. N. On the Theory of Superfluidity, J. Phys. (USSR), Volume 11 (1947), pp. 23

[11] Calogero, F. Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials, J. Mathematical Phys., Volume 12 (1971), pp. 419-436 | MR | Zbl

[12] Calogero, F.; Marchioro, C. Lower bounds to the ground-state energy of systems containing identical particles, J. Mathematical Phys., Volume 10 (1969), pp. 562-569 | MR

[13] Choquet, Gustave Lectures on analysis. Vol 2. Representation theory, Mathematics lecture note series, W.A. Benjamin, Inc, New York, 1969 | Zbl

[14] Christandl, Matthias; König, Robert; Mitchison, Graeme; Renner, Renato One-and-a-half quantum de Finetti theorems, Comm. Math. Phys., Volume 273 (2007) no. 2, pp. 473-498 | DOI | MR | Zbl

[15] Cornean, H. D.; Derezinski, J.; Zin, P. On the infimum of the energy-momentum spectrum of a homogeneous Bose gas, J. Math. Phys., Volume 50 (2009) no. 6, pp. 062103 http://link.aip.org/link/?JMP/50/062103/1 | DOI | MR | Zbl

[16] De Finetti, Bruno Funzione caratteristica di un fenomeno aleatorio, Atti della R. Accademia Nazionale dei Lincei, 1931 (Ser. 6, Memorie, Classe di Scienze Fisiche, Matematiche e Naturali)

[17] de Finetti, Bruno La prévision : ses lois logiques, ses sources subjectives, Ann. Inst. H. Poincaré, Volume 7 (1937) no. 1, pp. 1-68 | Numdam | MR | Zbl

[18] Dereziński, J.; Napiórkowski, M. Excitation spectrum of interacting bosons in the mean-field infinite-volume limit, Annales Henri Poincaré (2014), pp. 1-31 | DOI

[19] Diaconis, P.; Freedman, D. Finite exchangeable sequences, Ann. Probab., Volume 8 (1980) no. 4, pp. 745-764 http://www.jstor.org/stable/2242823 | MR | Zbl

[20] Dynkin, E. B. Classes of equivalent random quantities, Uspehi Matem. Nauk (N.S.), Volume 8 (1953) no. 2(54), pp. 125-130 | MR | Zbl

[21] Elgart, Alexander; Erdős, László; Schlein, Benjamin; Yau, Horng-Tzer Gross-Pitaevskii equation as the mean field limit of weakly coupled bosons, Arch. Ration. Mech. Anal., Volume 179 (2006) no. 2, pp. 265-283 | DOI | MR | Zbl

[22] Elgart, Alexander; Schlein, Benjamin Mean field dynamics of boson stars, Comm. Pure Appl. Math., Volume 60 (2007) no. 4, pp. 500-545 | MR | Zbl

[23] Erdös, L.; Schlein, B.; Yau, H.-T. Ground-state energy of a low-density Bose gas: A second-order upper bound, Phys. Rev. A, Volume 78 (2008) no. 5, pp. 053627 | DOI

[24] Erdős, László; Schlein, Benjamin; Yau, Horng-Tzer Rigorous derivation of the Gross-Pitaevskii equation with a large interaction potential, J. Amer. Math. Soc., Volume 22 (2009) no. 4, pp. 1099-1156 | DOI | MR | Zbl

[25] Fannes, M.; Spohn, H.; Verbeure, A. Equilibrium states for mean field models, J. Math. Phys., Volume 21 (1980) no. 2, pp. 355-358 | DOI | MR | Zbl

[26] Fröhlich, Jürg; Knowles, Antti; Schwarz, Simon On the mean-field limit of bosons with Coulomb two-body interaction, Commun. Math. Phys., Volume 288 (2009) no. 3, pp. 1023-1059 | DOI | MR | Zbl

[27] Ginibre, J.; Velo, G. The classical field limit of scattering theory for nonrelativistic many-boson systems. I, Commun. Math. Phys., Volume 66 (1979) no. 1, pp. 37-76 http://projecteuclid.org/getRecord?id=euclid.cmp/1103904940 | MR | Zbl

[28] Girardeau, M. Relationship between systems of impenetrable bosons and fermions in one dimension, J. Mathematical Phys., Volume 1 (1960), pp. 516-523 | MR | Zbl

[29] Giuliani, Alessandro; Seiringer, Robert The ground state energy of the weakly interacting Bose gas at high density, J. Stat. Phys., Volume 135 (2009) no. 5-6, pp. 915-934 | DOI | MR | Zbl

[30] Gottlieb, Alex D. Examples of bosonic de Finetti states over finite dimensional Hilbert spaces, J. Stat. Phys., Volume 121 (2005) no. 3-4, pp. 497-509 | DOI | MR | Zbl

[31] Grech, Philip; Seiringer, Robert The Excitation Spectrum for Weakly Interacting Bosons in a Trap, Comm. Math. Phys., Volume 322 (2013) no. 2, pp. 559-591 | DOI | MR

[32] Hartree, D. R. The wave-mechanics of an atom with a non-Coulomb central field. Part I. Theory and methods., Proc. Camb. Phil. Soc., Volume 24 (1928), pp. 89-312

[33] Hepp, K. The classical limit for quantum mechanical correlation functions, Comm. Math. Phys., Volume 35 (1974) no. 4, pp. 265-277 | MR

[34] Hewitt, Edwin; Savage, Leonard J. Symmetric measures on Cartesian products, Trans. Amer. Math. Soc., Volume 80 (1955), pp. 470-501 | MR | Zbl

[35] Hoffmann-Ostenhof, Maria; Hoffmann-Ostenhof, Thomas Schrödinger inequalities and asymptotic behavior of the electron density of atoms and molecules, Phys. Rev. A, Volume 16 (1977) no. 5, pp. 1782-1785 | MR

[36] Hudson, R. L.; Moody, G. R. Locally normal symmetric states and an analogue of de Finetti’s theorem, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete, Volume 33 (1975/76) no. 4, pp. 343-351 | MR | Zbl

[37] Kiessling, Michael K.-H. The Hartree limit of Born’s ensemble for the ground state of a bosonic atom or ion, J. Math. Phys., Volume 53 (2012) no. 9, pp. 095223 http://link.aip.org/link/?JMP/53/095223/1 | DOI | MR

[38] Knowles, Antti; Pickl, Peter Mean-field dynamics: singular potentials and rate of convergence, Commun. Math. Phys., Volume 298 (2010) no. 1, pp. 101-138 | DOI | MR | Zbl

[39] Lewin, Mathieu Geometric methods for nonlinear many-body quantum systems, J. Funct. Anal., Volume 260 (2011), pp. 3535-3595 | DOI | MR | Zbl

[40] Lewin, Mathieu; Nam, Phan Thành; Rougerie, Nicolas Derivation of Hartree’s theory for generic mean-field Bose gases, Adv. Math., Volume 254 (2014), pp. 570-621 | DOI

[41] Lewin, Mathieu; Nam, Phan Thành; Schlein, Benjamin Fluctuations around Hartree states in the mean-field regime (2013) (arXiv eprint) | arXiv

[42] Lewin, Mathieu; Nam, Phan Thanh; Serfaty, Sylvia; Solovej, Jan Philip Bogoliubov spectrum of interacting Bose gases, Comm. Pure Appl. Math., Volume in press (2013)

[43] Lieb, Elliott H. Exact analysis of an interacting Bose gas. II. The excitation spectrum, Phys. Rev. (2), Volume 130 (1963), pp. 1616-1624 | MR | Zbl

[44] Lieb, Elliott H.; Liniger, Werner Exact analysis of an interacting Bose gas. I. The general solution and the ground state, Phys. Rev. (2), Volume 130 (1963), pp. 1605-1616 | MR | Zbl

[45] Lieb, Elliott H.; Seiringer, Robert Derivation of the Gross-Pitaevskii equation for rotating Bose gases, Commun. Math. Phys., Volume 264 (2006) no. 2, pp. 505-537 | MR | Zbl

[46] Lieb, Elliott H.; Seiringer, Robert; Solovej, Jan Philip; Yngvason, Jakob The mathematics of the Bose gas and its condensation, Oberwolfach Seminars, Birkhäuser, 2005 | MR | Zbl

[47] Lieb, Elliott H.; Solovej, Jan Philip Ground state energy of the one-component charged Bose gas, Commun. Math. Phys., Volume 217 (2001) no. 1, pp. 127-163 | DOI | MR | Zbl

[48] Lieb, Elliott H.; Solovej, Jan Philip Ground state energy of the two-component charged Bose gas., Commun. Math. Phys., Volume 252 (2004) no. 1-3, pp. 485-534 | MR | Zbl

[49] Lieb, Elliott H.; Thirring, Walter E. Gravitational collapse in quantum mechanics with relativistic kinetic energy, Ann. Physics, Volume 155 (1984) no. 2, pp. 494-512 | MR

[50] Lieb, Elliott H.; Yau, Horng-Tzer The Chandrasekhar theory of stellar collapse as the limit of quantum mechanics, Commun. Math. Phys., Volume 112 (1987) no. 1, pp. 147-174 | MR | Zbl

[51] Lions, Pierre-Louis The concentration-compactness principle in the calculus of variations. The locally compact case, Part I, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 1 (1984) no. 2, pp. 109-149 | Numdam | Zbl

[52] Lions, Pierre-Louis The concentration-compactness principle in the calculus of variations. The locally compact case, Part II, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 1 (1984) no. 4, pp. 223-283 | Numdam | Zbl

[53] Lions, Pierre-Louis Mean-Field games and applications (2007) (Lectures at the Collège de France, unpublished) | Zbl

[54] Petz, D.; Raggio, G. A.; Verbeure, A. Asymptotics of Varadhan-type and the Gibbs variational principle, Comm. Math. Phys., Volume 121 (1989) no. 2, pp. 271-282 http://projecteuclid.org/getRecord?id=euclid.cmp/1104178067 | MR | Zbl

[55] Pickl, P. A simple derivation of mean-field limits for quantum systems, Lett. Math. Phys., Volume 97 (2011) no. 2, pp. 151-164 | MR | Zbl

[56] Raggio, G. A.; Werner, R. F. Quantum statistical mechanics of general mean field systems, Helv. Phys. Acta, Volume 62 (1989) no. 8, pp. 980-1003 | MR | Zbl

[57] Rodnianski, Igor; Schlein, Benjamin Quantum fluctuations and rate of convergence towards mean field dynamics, Commun. Math. Phys., Volume 291 (2009) no. 1, pp. 31-61 | DOI | MR | Zbl

[58] Seiringer, Robert The excitation spectrum for weakly interacting bosons, Commun. Math. Phys., Volume 306 (2011) no. 2, pp. 565-578 | DOI | MR | Zbl

[59] Seiringer, Robert; Yngvason, Jakob; Zagrebnov, Valentin A Disordered Bose-Einstein condensates with interaction in one dimension, J. Stat. Mech., Volume 2012 (2012) no. 11, pp. P11007 http://stacks.iop.org/1742-5468/2012/i=11/a=P11007

[60] Solovej, Jan Philip Asymptotics for bosonic atoms, Lett. Math. Phys., Volume 20 (1990) no. 2, pp. 165-172 | DOI | MR | Zbl

[61] Solovej, Jan Philip Upper bounds to the ground state energies of the one- and two-component charged Bose gases, Commun. Math. Phys., Volume 266 (2006) no. 3, pp. 797-818 | MR | Zbl

[62] Spohn, Herbert Kinetic equations from Hamiltonian dynamics: Markovian limits, Rev. Modern Phys., Volume 52 (1980) no. 3, pp. 569-615 | MR | Zbl

[63] Størmer, Erling Symmetric states of infinite tensor products of C * -algebras, J. Functional Analysis, Volume 3 (1969), pp. 48-68 | MR | Zbl

[64] Sutherland, B. Quantum Many-Body Problem in One Dimension: Ground State, J. Mathematical Phys., Volume 12 (1971), pp. 246-250 | DOI

[65] Sutherland, B. Quantum Many-Body Problem in One Dimension: Thermodynamics, J. Mathematical Phys., Volume 12 (1971), pp. 251-256 | DOI

[66] van den Berg, M.; Lewis, J. T.; Pulé, J. V. The large deviation principle and some models of an interacting boson gas, Comm. Math. Phys., Volume 118 (1988) no. 1, pp. 61-85 http://projecteuclid.org/getRecord?id=euclid.cmp/1104161908 | MR | Zbl

[67] Werner, R. F. Large deviations and mean-field quantum systems, Quantum probability & related topics (QP-PQ, VII), World Sci. Publ., River Edge, NJ, 1992, pp. 349-381 | MR | Zbl

[68] Yau, Horng-Tzer; Yin, Jun The second order upper bound for the ground energy of a Bose gas, J. Stat. Phys., Volume 136 (2009) no. 3, pp. 453-503 | DOI | MR | Zbl

[69] Yngvason, J. The interacting Bose gas: A continuing challenge, Phys. Particles Nuclei, Volume 41 (2010), pp. 880-884 | DOI

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